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Las diagrammas suivants illustrant la mAthoda. 1 2 3 1 a 3 4 5 6 '« ■ * it -JTSTT-Jjr- ADAMS' NEW * ARITHMSTIO,^ 8VITED TO HALIFAX CURRENCY; ••Y-:'(< IN ^IIICH THE .«*--,' PRINCIPLES OF OPERATING BY NUMBERS Vv./. ■^'*^-* ARE " "■■ ■ ^'' AVAXi'B'TZOA.ZiIi'S' nXVIiAZXTBD, ' 1 BTNTBBTZOAI.It'S' APVZt^BD: THUS . . i;;.-;-W ■>: ^* COMBINING THE ADVANTAGES -> TO BE DKRiriD BOTH FROM || THE INDUCTIVE AND SYNTHETIC MODE OF INSTRUCTING: ^ THE WHOLE MAI>X rAMILI4R|BY A. GREAT VARIKTT OF VSEFDt AND IMTERBSTIMG EXAMPLES, CALCULATED AT ONCE TO ENGAGE THE PUPIL IN THE STUDY, AND TO GIVE HIM A FULL KNOWLEDGE OF FIGURES IN THEIR APPLICATION TO ALL THE PRACTICAL PURPOSES OF LIFE. DBSIQNED FOH THIS USB OF SCHOOLS AND ACADEMIES IN THE I f- t-i ■ -^ BY DANIEL. ADAMS, M, D. ■TJLirtTBAD, Xi. 0. PUBLISHED BY WALTON &, GAYIX>BD. /.r. ^« :>l ,*'> X ■ H , _jt . f . (I I ; , t ;■ tt ir";" ' .> ' '!. i1 > r ■ *l ^■^ '* ''■^iy ?7; ?«• u ,f «■■ ■• > c> '-? 1, 'i '. J I 'U i'9 Thkrk arc two inctliodsof teaciiing: ih» agnthetic Mi ihc uHolytic, In tho synthetic method, the pupil is firat preaeoted witli a general view of the science he is studying, and afterwards with the particulars uf which it consists. The analytic method reverses this order : the pupil is first presented with the fartieularg, from which he is led, by certain natural acd easy gradations, to those views which are more general and comprehensive. The Scholar's Arithmetic published in 1801 , is synthetic. If that is a fault of the work, it is u fauJt of the times in which it appeared. The analytic or inductive method of teaching, ns now applied to ele- mentary instruction, is among the improvements of later years. Its introduction is ascribed to Pestalozzi, a distinguished teacher in Switzerland. It has been applied to arithmetic, with great ingenuity, by Mr. Colbukn, in our own country. \ The analytic is unquestionably the beat method of oc^umn^ know- ledge ; the synthetic is the best method of recapitulating or reviewing it. In a treatise designed for school education, both methods are use- ful . Such is the plan of the present undertaking which the author, occupied as ho is with other objects and pursuits, would willingly have foiborne, but that, the demandf for the Scholar's Arithmetic still con- tinuing, an obligation, incurred by long-continued and extended pat- ronage, did not allow him to decline the labor of a revisal, which should adapt it to the present more enlightened views of teaching this I science in our schools. In doing this, however, it has been necessary to make it a new work. In the execution of this design, an anaZ^«i« of each rule is first given, 1 containing a familiar explanation of its various principles ; aflcr which iibllows asvntheais of these principles, with questions in form of a sup- fplement. Nothing is taught dogmatically ; no technical term is used [till it has first been defined, nor any principle inculcated without a pre- vious devolopement of its truth; and the pupil is made to understand Ithe reason of each process as he proceeds. The examples under each rule are mostly of a practical nature, be- ginning with those that are very eimy, und gruduallv advancing to iiose more difficult, till one is introduced cuntairiiiig uirgor numbers, md which is not easily solvod in tUv niiiiil ; tliun in a plain, familiar lanner, the pupil is shown how tlio su!iiti .^^qo I i *) as* Mv'.fj f.'s iU' Ut **■*•=. 1 - ' " ri* "!% ' a* *s^»i '>t«tOBt-K^-T-^|g5"T.tV^' r PUBLISHERS' PREFACE. «'^^' ~ 9M, k4.t 4^ ? : »f . fr.**» ^ .•' *.*' S' *-^ •"!*;••'■ ■^■aj!" *-■*'< 'i?S iJ^ -». •-, ^»f> Tub author ut'the folluwing practical treatise upon Artth- luetic, has made himself favorably known in th« United States and to a considerable extent in the Canadas, for % i^reat number of years, by his works, designed for the use of Academies and primary schools. The "Scholars' Arith- motic," published in the year 1801, continued in almost uni- ' versal use, until within a very short time past. But juster views beginning to prevail, and sounder principles becoming established in the public mind, upon thcsubject of elementa- ry education, a revision of the work seemed necessary. At this time, " Adams' New Arithmetic," was published.— ^ This seems evidently to have been prepared with much care. The author has recognized in it throughout, this important law in relation to the mind, that it must first be made ac- quainted with particular facts, or there will be no ability to arrive at correct general conclusions. Particular examples are therefore given upon each subject, and from them, in a manner obvious to the young mind, all the general . rules are deduced. In other words, the author has carefully and prudently pursued, in his book, what is called the analytic method. The care used in defining necessary terms, which might not be quite clear, the practical character of the ex- amples given under each rule, the methodical disposit "i of the different parts of each subject, and of the different st b- jects, the general perspicuity, simplicity and accuracy of the work, render it invaluable to the pupil. It is due the author to observe, that *• Adams' New Arith- metic," for its adaptation to the capacities of young and or- dinary minds, is justly considered the best practical treatise which has been offered to the public. In the present edition, the main purpose in view was to adapt Adam's work to the currency of the British Provin- ces. No separate article, as in the original, has been al- lotted to Federal Money ; for iliis the pupil has been refer- red to Decimal Fractions, in which also almost all the ex- amples will be found in the money of the United States. — Additional examples in the compound rules have been giv -:..-;.;■ A 2 ,^ ' " ^ - . , -^ - ,,- 6 PVBLI81feR«' PRRFACB. nn, and the old ohes retained, under the tide of Halifax currencjr ; and generally, throughout the book, where de- nominations of moner occur, Halifax currency has been subatitutod for Federal Money. The rules and examples in Reduction of Currencies have been essentially changed; and in Reduction, aAer the Ta- ble of English Money, which is called the Table of Hali- fax Currency, a list of the Gold and Silver Coins current in the Province, has been inserted. This may be depended upon as entirely accurate. The tables of French, and Dry, Long, Square, and Solid Measure, have been given—and ^ what are the weights and measures established by law in this Province is also stated. The most novel feature in the book will be found in the rule of Interest. Certainly an innovation, but it is believed, an improvement, has been made. The pounds in any giv- en sum upon which interest is to be cast, are left to stand as the units, and the shillings and pence are reduced to decimal parts of a pound. The interest is then obtained the same as in Federal Money, and the decimal parts in the result reduced to shillings and pence. It is considered, that this method is more simple, and concise, and will be found in practice to be more convenient than any other, fiut set- ting aside considerations of temporary convenience, if this change and attempted amelioration, shall assist in some very slight degree in turning men's minds toward the Decimal Ratio, and inducing them to look forward to a period when all the denominations of money, weights and measures, throughout the world, shall be expressed in decimals, it cannot be affirmed that no beneiit has been obtained. The importance of the principal and essential alteration in the book, viz. the adaptation of it to the currency of the country, will not /til to be observed by every one. It is indeed singular, that hitherto, no Canadian Arithmetic in the English language, has been published. Mercantile, agri- cultural, and generally the business men of the country, will be aware of a benefit to be realized, and it is consider- ed that somethmg also bearing a relation to political advan- tage, may be in the result. JStmstead, L. C. Aug. 1, 1833. ^pf^ ^ p t'- 4^ > Addition, . . •• Alliialion, Arilhinetical IVogrcMion, Compound Numbora, Addition of, . Subtraction of, . — Multiplication and Diviaion of, Coiumiuion, Coins, Table of, Cube Root, Extraction of, Division, Duodecimals, Multiplication of, Equation of Payments , Evolution, Federal Money, Fractions, ' Proper and Improper, To change an improper fraction to a whole or mixed number, '. . ■ To reduce a whole or mixed number to an improper fraction, - .... To reduce a fraction to ill lowest terms, To divide a fraction by a whole number, ■ To multiply a fraction by a whole number, — — — ^ whole number by a fraction, • one fraction by another, ' ^. . To divide a whole number by a fraction, I one fraction by anothei, Addition and subtraction of fractions, Reduction of . . do. Decimal Fractions, . . AdJition and subtraction of, Multiplication of, Division of, . . -Reduction of. Reduction of vulgar fractioos to decimals, Fellowship, Halifax Currency, Insurance, . • . Interest, Compound, Involution, Multiplication, Simple, Numeration, Proportion, or the Rule of Three, Compound, fY' Pmee. 14 21H 58 80 rt5 8U 152 5f» 211 nf> 197 1U8 171 :202 C4 1»« 100 100 101 109 105 167 W.) 110 112 114 116 121 128 132 136 137 143 140 188 59 152 150 164 201 28 9 174 182 % ..;,-> IVrmulation, iMt ••.... Uutiucliun, . . " - . , . r.' Tahhpi of Money, Weights, MoudurM, Ac, Hodiirtiuii III' ('urrntirifi, . ^, ^*^ »„ Undo, (11 tlin Ki'Intion uf NuiiiImth, • • • HubtrortKiti, Hinipip, Squtiru Hiiut, Extraction of, \' 50—78 .*« 141) > 173 ^^ MlSCKM.ANKOfS KXMAl'I.r.S. Hnrt«i, ex. 21— .Ti. I Position, ex. H!»— lOH 'I'o iinii tlio uiuu ofu Sqiiuri! or I'nrnllRlo|E(rnm, ox. llH — ITi'l u Triaiiglu, iix. If).")— |.V.». Having tlie Dinmotnr of n Circle, to fnui lliu (virciiniluKnicc ; or hav- ing tlio Cirnimforcnco, to find the Dintnittiir, «.\. 171 — 175, To find tho Area ofa Circle, ex. 17(t— 171>. aClobo.ox. 1^0, IHI Tu lind thu iSoliil ContontH of a (ilobo, ux. 1H2, Idl. i ' ('ylindcr, ex. 1^5 — 187. -' I'yrnmid, or Cone, «x. IH8, IflO. any Irregular Hody, cx- ijoy. i. | Miiclianicnl Poworn, ox. 1!>2— ^M. Forn^s oi ISolci, Uuceipls, Orders and Bills of ParrcN. page, 25h J[Jnok Kct'pinf;. ;.:-. or^y ■k fl-«il. ■>i. ' t . .; 1 ■ ' ' .](>¥it.'^''l\'>ii}:.vni--^ i(4-,-».^* I' m i:.'."- ' ,■•■ .'F, . ^«^ik .■=^'^^tmam: ji:aawaiMiBWii(0^ l-<0. :r.4.'f 1 X. Ufa. -20J. f?«, 25M ■•■'(' 1 ,". ^av..:j. ••»■,;>."' ^ Mht 12 HUMERATION. < IT 3 ; > same manner as tens and hundreds of unitt. To thousands succeed milliona, billionH, &c., to each of which, as to units and to thousands, are appropriated three places,* as exhi- bited in the fqUyrin^ examples : JM a o -a ■e CO 3 O *^ H s /^.A,^> rw^-^^ /x.a.^> /x.a.^% /%>s.^»\ /^-^,^^, 09 "5 a 3 00 09 'O 91 ^ 0* OB -a 5E ** cj •O «, 09 "d _ «« a-= S S(»flS«fi3«B35c3gC Example 1st. 3 174592837463512 EXAHPLK 2d. 3, _• o fl •TS O 2 S «- 1 7 4, 5 9 2, 8 3 7, 4 6 3, 5 1 2 .2 "9 oJ S ,2 B o o -^ COM V 'C B C -2 S -C -2 « = 5 W5 o H "^ « c 0.5-^ O.S.2 g.«-.s g,5 3 CUSS oPQ w o n »H To facilitate the reading of large numbers, it is frequently practised to point tht^m off into periods of three figures each, as in the 2d example. The names and the order of the pe- riods being known, this division enables us to read num- bers consisting of many figures as easily as we can read three figures only. Thus, the above examples are read 3 (three) Quadrillions, 174 ( on© hundred seventy-four) Tril- lions, 592 (five hundred ninety-two) Billions, 837 (eight hundred thirty-seven) Millions, 463 (four hundred sixty- three) Thousands, 512 ( five hundred and twelve.) After the same manner are read the numbers contained in the following ^ -«><■' ■ i\di * This Ib according to the French method of countinc. The Engliib, after hundreds of milhons, instead of proceedinc to oillioni, reckon thousands, t«ns and hundreds of thousands of mtlhons, appropriating SIX places, instead of three, to millioni, billions, dtc. ^£3di^ m 3. > thousands ai to units * as exbi c o « C S S S ► 3512 5 3, 5 1 2 5 « ' B S S 'C • J frequently figures each, r of the pe- I read num- re can read ) are read 3 -four) Tril- 837 (eight idred sixty- re.) iontained in The English, lions, r^kon sppropriatioi IT 3. ■.:.♦/#>. NUMEltATIOlf. /f.vVT>\,i; iir § k. C 9 Q PS SI'S NUMERATION TABLE. ; ^ ^ ^ >v^i^. Those words at the head of the table are applicabte to any sum or ',M-ui\, number, and roust be committed ,y^, T perfectly to memory, so as to be , readily applied on any occasion. ' .a 3 « a . . . 7 ♦ ;«.' l! ! 'li);^ ,tr.i 1 8 8 . 9 5 8 3 6,1 ..86 .432 7 5 4 6 2 3 7 1 6 2 7 5^4 9 Of theie characters, I, 2, 3, 4, 5, 6, 7, 8, 9, 0, the nine Jirst are some- times called significant figures, or digits, in distinction from the last, which, of itself, is of no value, yet, placed at the right hand of another figure, it increases the value of that figure in the same ten fold pro- portion as if it had been followed by any one of the significant figures. Note. Should the pupil find any difiiculty in reading the following numbers, let him first transcribe them, and point them off into periods. ^* /^ 7^^ '_ ^^ ^ ...:"„. ,. ../ !; .' 5768 >,,. 52831209 '*f!I'' 2^731401^ 34120 vVjr 175264013 ;„;? 5203845761204,^ 701602 3456720834 1347812067301^ 6539285 25037026531 341246801734526 * The expressing of numbers, (as now shown,) by figures, is called Notation. The reading of any number set down in figures, is called Numeration, isr^m *;"» ^irmmq xumv ^ >ii After being able to read correctly all the numbers in the foregoing table, the pupil may proceed to expitss the fol- lowing numbers by figures : 1. Seventy-six. "* ' r'>*»i'?f>'t^, V^i. 2. Eight hundred and seven. '^ a&i«^:i'^l;:tff^s^-\^ 3. Twelve hundred, (that is, one thousand and twohun dred.) * . •»• 4. Eighteen hundred. ' '^'^^M • :^h. vo-* lni« .ftf*!-- B H ADDITION or 8I1IPLB RUMBERS . If 3, 4. 5. Twenty-seven hundred and nineteen. 6. Forty-nine hundred and sixty. 7. Ninety-two thousand and forty -five. *•» •*= ^ 8. One hundred thousand. 9. Two millions, eighty thousands, and seven hundreds. 10. One hundred millions, one hundred thousand, one hundred and one. ..,' .^:. 11. Finy -two millions, six thousand/ and twenty. 12. Six billions, seven millions, eight thousand, and nine hundred. 13. Ninety-four billions, eighteen thousand, one hundred and seventeen. 14. One hundred thirty -two billions, two hundred mill- ions, and nine. 15. Five trillions, sixty billions, twelve millions, and ten thousand. 16. Seven hundred trillions, eighty-six bilUgns, and seven millions. , ,„^_ ... , t' ' OF SIMPLE NUMBERS. # J IT 4. 1. James had five peaches, his mother gave him 3 peaches more ; how many peaches had he then 1 ,,.{.-» :• 2. John bought one book for 9 pence, and another for 6 pence ; how many pence did he give for both 1 3. Peter bought a waggon for 10 shillings, and sold it so as to gain 4 shillings ; how many shillings did he get for it 1 4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left ; how many walnuts had he at first 1 5. A man bought a carriage for 54 pounds ; he expended 8 pounds in repairs, and then sold it so as to gain 5 pounds; how many pounds did he get for the carriage 1 T' 6. A man bought 3 yoke of oxen ; for the first he gave 16 pounds, for the second he gave 18 pounds, and for the third he gave 20 pounds ; how many pounds did he give , for the three 1 7. Samuel bought an orange for four pence, and some walnuts for three pence ; then he bought a knife for 1 shill- ing, and a book for 4 shillings ; how many shillings did he spend, and !iow many pence 1 CM 7 3,4. IT 4. ADDITlOlf OP SIMPLE IfUMBRRl. 16 hundreds, sand, one and nine hundred red mill. >) and ten nd seven '6 him 3 ^er for 6 lold it so t for it 1 ler, and upended pounds; for the te give ■1 .1 8. A man had 3 calves worth 10 shillings each, 4 calves worth 15 shillings each, and 7 calves worth 2 pounds each; how many calves had he 1 9. A man sold a cow for 4 pounds, some corn for 5 pounds, wheat for 7 pounds, and butter for 2 pounds ; how many pounds must he receive 1 The putting together two or more numbers, (as in the foregoing examples,) so as to make one lohole number, is called jiddition, and the whole number is called the sum, or atnount. 10. One man owes me 5 pounds, another 6 pounds, anoth- er 14 pounds, and another 3 pounds ; what is the amount due to me 1 11. What is the amount of 4, 3, 7, 2, 8, and 9 pounds ! 12. In a certain school 9 study grammar, 15 study arith* roetic, 20 attend to writing, and 12 study geography ; what is the whole number of scholars 1 Signs. A cross, -f-, one line horizontal and the other per- pendicular, is the sign of addition. It shows that humben, with this sign between them, are to be added together. It is sometimes read pltu^ which is a Latin word, signifying more. Two parallel, horizontal lines, es=, are the sign of equality. It signifies that the number befbre it is equal to the number qfier it. Thus, 54-3a8 is read 5 and 3 are 8 ; or, 6 plus (that is, more) 3 is equal to 8. In this manner let the pupil be instrueted to commit the following ADDITION TABLE. ^-|-0< 2 + r «4-2! 2 + 3 = 2-f4. 2 + 5: 2 + 6( «4-7 = 2 + 8. 3 + 9: 2[3 + 0< 3 3-fl: 4 3 + 2« 53 + 3: 6;3-j-4: 73 + 6. 8,3 + 6 = 9 3 + 7 = 10|3 + 8s ll|3 + 9s 3 4H -0= 4 4 4l -1= 5 5 4- -2= 6 6 4- -3= 7 7 4- -4= 8 8 4- -5= 9 9 4 + 6=: 10 10 4 + 7 = 11 11 4 + 8 = 12 12, 4-f -9 = 13 5 + 0i 5 + 1' 5 + 2i 5 + 3: 5 + 4: 5 + 6: 5 + 6: 5 + 7« 6+8 = 6 + 9 = : 5 : 6 : 7 : 8 I 9 10 11 12 13 14 ^-■^ i^ : L ■ !-■■ M ^ 10 0-4-0 = ADDITION OF SIMPLE NUMBERS. IT 4, 5, 0- - 1 0-1 h2 6-j ka 0- -4 0- -r> (i-\ ^0 OH h7 0-1 h^ OH ho (» / 8 9 10 11 12 13 14 15 7-fO 7+1 7-1-2 4-3 + 7 7-1-8 7 4- 9 7 8 9 10 11 12 IJJ 14 15 10 8 8 8 8 8 8 8 8 8 8 1 2 3 4 5 7 8 9 8 9-1 9 9- 10 94 11 9- 12 9-} 13 9-1 14 9- 15 9H 10 9H 17 9- 1 2 3 4 5 7 8 9 --7=^=10 I 9 10 11 12 13 14 15 10 17 18 •'IwikUli *r..'; va lai'tii-K* *i Ihl'.i'i) 5-f 9 8-f7 4-1-3 6-1-4 2-f-Q 7-j- 1 3-f-O 9-J-2 1+3 1 -t-2 8-1-9 6-1-2 == how = how + 2 = --5 = --4-f --0-- --9-- --6 -5 io^ r. I. ,•!.-<: •kj: .UN': many % many 1 ■ how many 1 how many 1 6 = how many 1 8 = how many 1 5 = how many 1 . 4 -|- 5 = how many % "^ ' 7 -f- 8 = how many 1 4 - - 5 -f- 6 = how many 2 - - 4 -f- 5 = how many -|- 8 -|- 3 = how many \\ \ i\' ,-5i-.!-' 1 >;■■' ^■idvatm «>»}.* I; f r^-.'.t*'-. .*>> ':^Ukmi! -Mlt :*«;*^ *'irtlfi'i,11// II i^'H 5. When the numbers to be added are small, the addi- tion is readily performed in the mind ; but it will frequent- ly be more convenient, and even necessary, to write the numbers down before adding them. 13. Harry had 43 books in his little library, his father gave him 25 volumes more ; how many volumes had he then % r, : b ^s.- I -^p , *'• -xs i ■^- S;\4i ;■-, i • ?• *• One of these numbers is father had he IT 5. . -■ ADDTTIOK or tlMPLE irVMBKKS. 17 is. The ite these one un- ctly un- directly in this 43 volumes, 2.3 volumes, .' ■ St I asi 43 volumes, 25 vo/umes, We then begin at the right hand, and add the 5 units uf the lower number to the 3 units of the upper number, making 8 units, which we set down in unitV place. Wc then proceed to the next column, and add the 2 tens of the lower number to the 4 tens of the upper number, mak- ing () tens, or 60, which we set down in ten's place, and the work is done. It now appears that Harry's whole number of volumes is C tens and 8 units, or 68 volumes ; that is, 43-|-25=68. 14. A gentleman bought a carriage for 214 pounds, a horse for 30 pounds, and a saddle for 4 pounds ; what was the whole amount 1 Write the numbers as before directed, with units under units, tens under tens, &c. OPERATION. ' ' . H.' ^W Carriage, 214 pounds, Add as before. The units will Horse, 30 pounds, be 8, the tens 4, and the hundreds Saddle, Apmmds. 2, that is, 214+30+4=248. .iW ■K-'^ /Answer, ''24S potmds. ,.- ,;» ^. .> • . After the same manner are performed the following ex- amples : J?' 15. A man had 15 sheep in one pasture, 20 in another pasture, and 143 in another ; how many sheep had he in the three pastures 1 15 + 20 + 143 = how many 1 16. A man has three farms, one containing 500 acrci), another 213 acres, and another 76 acres ; how many acres in the three farms 1 500 + 213 + 76 = how many 1 17. Bought a farm for 625 pounds, and afterward sold it so as to gain 150 pounds ; what did I sell the farm for 1 625 + 150 = how many 1 .' «- --tKJ •. * ^ s> > wijy^ Hitherto the amount of any one column, when added up, has not exceeded 9 ; consequently has been expressed by a single figure. But it Avill frequently happen that the amount of a single column will exceed 9, requiring two oi more Jig- urcs to express it. 18. There are three bags of money. The first contains jgttte " IH -r.',.--*.'-'??*-^-* ADDITIOH OP BIMPLK A'VUBERM. f 5. 970 pounds, the second 65i) pounds, the third 524 pounds ; what is the amount contained in all the bags 1 OPERATION. Firat bag, 870 Second bag, 05li Third bag, 524 Jmount. 2053 Writing down the numbers as. al- ready directed, we begin with the right hand, or unit column, and find the amount to be 13, that is, 3 units and 1 ten. Setting down the 3 units, or right hand figure, in unit's place, directly under the column, we reserve the 1 ten, or left hand figure, to be added with the other tens, in the next column, saying, 1, which wc reserved, to 2 makes 3, and 5 are 8, and 7 are 15, which is 5 units of its oum order, and 1 unit of the next higher order, that is, 5 tens and 1 hundred. Setting down the 5 tens, or right hand fig- ure, directly under the column of tens, we reserve the lejl hand figure, or 1 hundred, to be added in the column of hundreds, saying 1 to 5 is 0, and are 12, and 8 are 20, which being the last column, wc set down the whole num- ber, writing the 0, or right hand figure, directly under the column, and carrying forward the 2, or left hand figure, to the next place, or place of thousands. Wherefore, we find the whole amount of money contained in the three bags to be 2053 pounds— the answer. Proof. We. may reverse the order, and, beginning at the top, add the figures downward. If the two results are alike, the work is supposed to be right. From the examples and illustrations now given, we de- rive the following fi, ,, ,^ ;. -fw RULE. ]. Write the numbers to be added, one under another, placing units under unit:^, tens under tens, &c. and draw a line underneath. II. Begin at the right hand or unit column, an^fadd to- gether all the figures contained in that column : if the amount does not exceed 9, write it under the column ; but it the amount exceed 9, so that it shall require two or more ti'^ures to express it, write down the unit figure only under the column ; the figure or figures to the left hand of units, being tens, are so many units of the next higher order, which, being reserved, must be carried forward, and added to the first figure in the next column. •• -.,- III. Add each succeeding column in the same manner, and set down the whole amount at the last column. 19. A I 1817 poui pounds, a jf hay p^ 20. A r-pounds, 1 21. A 150 bushi els of bar amount 1 22. St sterling ; |sion-Hog pounds ; lonume bums '! 23. If lingcoun j Dorches Stanstea |iW,871; I what w [ties at 1 24. 1 [from E; iniore *, the bui '' kings fi 363 ye christia to the 286; 310 •1*a^*'ili. ir 5. 5. ADDjnrioN or bimplb rcbibubs. ««« " 19 founds ; a&al- ith the nd find 3 units [3 units, place, reserve T tens, |d, to 2 8 of its , 5 tens and fig- tbe l^ umn of are 20, e num- der the :ure, to we find >ags to EXAMPLES FOR PRACTICE. 10. A man bought four loads of liay ; one load weighed 1817 pounds, another weighed 195(li pounds, another '2\'A\ )ounds, and another 2210 pounds ; what was the amount )f hay purchased 1 20. A person owes A 100 pounds, B 522 pounds, C 785 f*pounds, D 92 pounds ; what is the amount of his debts 1 21. A farmer raised in one year 1200 bushels of Avheat, 1.30 bushels of Indian corn, 1000 bushels of oats, 1086 bush- els of barley, and 74 bushels of peas ; what was the whole lamounti ufns. 4210. 22. St. Paul's Cathedral, in London, cost 800,000 pounds sterling ; the Royal Exchange 80,000 pounds ; the Man- (sion-House 40,000 pounds ; Black Friars Bridge 152,840 )ounds ; Westminster Bridge 3^(9,000 pounds, and the lonument 13,000 pounds; what is the amount of these lurasl *, Jf -fns. 1,474,840 pounds. 23. If at the census in 1831 , the population of the follow- 1 ing counties was as follows : — Lower Canada : Gaspe, 4, 171 ; [Dorchester, 11,946; Nicolet, 12,504; Sherbrooke, 0,814; fStanstead, 8,272: Upper Canada : Gore, 23,552 ; Home, i ;I2,871 ; Niagara, 21,974 ; London, 20,180 ; Ottowa, 4,45t) ; iwhat was the whole number of inhabitants in these coun- [ties at that time 1 jins. 152,740. 24. From the creation to the departure of the Israelites [from Egypt was 2513 years ; to the siege of Troy, 307 years [more ; to the building of Solomon's Temple, 180 jears ; to the building of Rome, 251 years ; to the expulsion of the kings from Rome, 244 years ; to the destruction of Carthage, 363 years ; to the death of Julius Cesar, 102 years ; to the christian era, 44 years ; required the time from the creation to the christian era. .^.-., ,,. .„...r.,. ,- -^f^s. 4004 years. 'A:!'' -if''' i-i'^tT'-f' 35. ^ 26. 286370542 106 1 310742931 5638 6 2 53034792 2 4 7 13 5 8 67 3 .*f*' '^ 43675 8 3021463 17 5 2349713620 6081 27530621 7 5 652174630128 870326347201 3 # 20 KUPPLBMEJIT TO NUMERATION AND ADDITIOlf. If oSf 5, l>] { 11 27. ■ ^ ':A".y'j*r ■". 28. r, a 6 4 2 7 (i 3 I 2 3 2 8 12 34 r, (» 7 2 i) 4 8 « r> 7 4 2 8 7 {) U 4 . 3 1 (i 2 8 3 r» 9 7 18 2 3 7 r, 4 8 2 1 3 ,^1 28 3 4 00 7 3 2070 , 3 (Hi 3 4 2 1 7 3 2 2 3 .-» 4 7 H 2 4 3 7 4 2 8 5 3 7 8 2 8 5 0007 HO 901^ M T^ 29. What is the amount of 4672:?, 0742, and 980 founds ' 30. A man has three orchards -, in tho first there are 140 trees that bear apples, and 04 trees that bear peaches ; in the second, 234 trees bear apples, and 73 bear cherries ; in the third, 47 trees bear plums, 30 bear pears, and 25 bear cherries ; how many trees in all the orchards 1 5. [the tbil [what i| T). Ired ai laaught imouni 7. who, |receiv< Itatel 8. 'IS*! ItVi ,? ■r' "»' Hi'',, SUPPLEMENT '• « TO NUMERATION AND ADDITION. ^^^ ^ v.-,w • "7». QUESTIONS. j^r' '^ .1 1. What ia a single or individual tiling called ? 2. What is nota- tion ? 3' What are tho incthuds of notation now in use ? 4. How many are the Arabic chnractcfB or figures ? 5. What is numeration ? <>. What ia a fundamental law in notation P 7. What is addition? b. What ia tho rule for addition? 9. What is the result, or number sought, called? 10. What is tho sign of addition ? 11. — — of cnuality ? 12. How is addition proved ? . .^ ., -■ f . ..vy. i^ >ji"6''^')ti/-V •:<■? EXERCISES. *' 1. Washington was born in tho year of our Lord 1732; he "was 67 years old when he died ; in what year of our Lord did he die 1 2. The invasion of Greece by Xerxes took place 481 years before Christ ; how long ago is that this current year 1833 \ " 3. There are two numbers, the less number is 8671, the difference between the numbers is 597 \ what is the greater number. 4. A man borrowed a sum of money, and paid in part 684 pounds ; the sum left unpaid was 876 pounds , what was the sum borrowed \ [years Ueen yi 11. [bushel [of tall *\ \ whi( 2 er; 3 but l06( .;^r^,-, - ..-.^*-M«. **-^ . »i>iTiow. iToj i 4 I ;j -* I <) 7 U i> w -2 4 a (j ' Oi^f) ^ollnds ' tJiere are J 40 f peaches ; in f cherries ; in and 25 bear If 5, 0. SUBTHACTION OF SIMPLE NUMBKRS. 21 ON. .•Vj. What is nota- 'SO? 4. How < numorntion ? i» ndtljtion ? "» or number 11. of ^ord 1732; I ear of our » 481 years ear 1833 1 8671, the he greater ti in part tJs, what 5. Then; are four numbers, the first 317, the second 812, I the third I'.V'A), and the fourth as much as the other three ; Iwhat is the sum of them alii (i. A gentleman left his daughter 1(> thousand, 1<> hun- Ired and 10 pounds ; he left his son l^M) more than his lughtcr; what was his son's portion, and what was the imount of the whole estate 1 ^ ^ Son's portion, 19,416. ^"^*' I Whole estate, 37,05)2. 7. A man, at his death, left his estate to his four children, rho, after paying debts to the amount of 1476 pounds, [received 4768 pounds each ; how much was the whole es- Itatel i^t. 26548. 8. A man bought four hogs, each weighing 375 pounds ; how much did they all weigh 1 jtna. 1500. 9. The fore quarters of an ox weigh one hundred and »ight pounds each, the hind quarters weigh one hundred ind twenty-four pounds each, the hide seventy-six pounds, ind the tallow sixty pounds; what is the whole weight of [the 0x1 ^s. 600. 10. A man, being asked his age, said he was thirty-four [years old when his eldest son was born, who was then fif- [teen years of age ; what was the age of the father 1 11. A man sold two cows for five pounds each, twenty [bushels of corn for three pounds, and one hundred pounds [of tallow for two pounds ; what was his due ! ^iWv 5 1^ a I ■! in i iT. w. I ll i> i lti i [ M Mr i^^mmm^^ vi^o. le tg Qp sii^pLE NUMBERS. ^: ''p'*^ IF 6. 1. Charles, having 11 pence, bought a book, for which he gave 5 pence ; how many pence had he left 1 2. John had 12 apples; he gave 5 of them to his. broth- er ; how many had he left 1 3. Peter played at marbles ; he had 23 when he began, but when he had done he had only 12 ; how many did he 1086 1 , 22 ■UBTRACTIOI* or tlllPLB RUMBBIIf. f 6. 4. A man bought an article for 17 ihiilinf^ and sold it again for 22 •hilllnf(s; how many ahillinKii ^'^ '>° K*"^^ />. CharlcH in 9 years old. and Andrew is 13; what in th« difierence in their afi^esl 6. A man borrowed 50 poundii, and paid all but 18 ; how many pounds did he pay 1 that is, take \H from 50, and how many would there be left 1 7. John bought several articles for 10 shillings ; he gave for 4 book* 6 shillings ; what did the other articles cost himi •'! * ' I •* '>• •>' 8. Peter bought a trunk for 17 ihillingt, and sold it fot 22 shiUinga ; how many shillings did he gain by the h.n- gaini 9. Peter sold a waggon for 22 shillings, which Wh.t o shillings more than he gave for it ; how many shiilings did he give for the waggon 1 10. A boy, being asked how old he was, said that he was 125 ysars younger than hit lather, whose age was 33 ye«rs ; how old was the boy 1 The taking of a less number from a greater (as in the foregoing examples) is called Subtraction. The greater number i§ called the •nmticnd, the less number the aubtra- h«ndf and what is ieil aAer subtraction is called the differ- «nee, or rtmainder. 11. If the minuend be 8, and the subtrahend 3, what is the difference or remainder 1 jfnt. 5. 12. if the subtrahend be 4, and the minuend 16, what is k.: on word '»i«' nifying /«««. It shows that the numly.i^ tifirr i\. ^ to be taken from the number htfore it. Thus, 8~os=5, is read 8 mi- nus or less 3 is equal to 5 ; or 3 from 8 leaves 5. The lat- ter expression is to be used by the pupil in committing tho foll>wiD| 5- fUir -V. .n t>ti ^ ■ML' • VBTRACTIO* or tlMiTt! f 7. and Kid it ho gain t w bat ih the SUBTRACTION TABLK. ',» ; he gavo irticles coit -f d sold it fot ^^ by the har- ^hich Wk!» 6 hiilinge did that he wm • 38 jtmn ; <^ (a« in the 'he greater the Mubtra- the difer- 3, what is ;j ^n$. 5. 16, what it paid down o( subtrac- word '>'«» 'O be taikon read 8 mi- Thelat- lilting tho 0—3=3 ! 7-3==4 8--3=r> 9~3=:«J 7~5c=>2 8— r>=;i 9-5=4 10-.' 7—7=0 «-7=«l 9-7=3 10-7=3 6—6=0 7-ft=l 8-6=«3 9-6=3 10-6=4 j.< >,■ ii-.'o pv— ncsiO "HI 10-fea^ "9-9=0 10— 9i»l *:«^ : •■• 7 8 9 12 13 -3 = — 4 = — 3 = -4 = how manyl how many 1 how many 1 how many 1 how many 1 18 28 22 33- 41 7 = how many 1 7 s=: how many 1 13 = how many 1 - 5 Bs how many 1 15 = how many 1 If 7. When the numbers are ntnall, as in the foregoing camples, the taking of a less number from a greater is readi- done in the mind ; but when the numbers are large, the aeration is most easily performed part at a time, and there- ^re it is necessary to torite the numbers down before per- ling the operation. 14. A farmer having a flock of 237 sheep, lost 114 of lem by disease ; how many had h« left 1 Here we have 4 units to be taken from 7 units, 1 ten to taken from 3 tens, and 1 hundred to be taken from 2 indreds. It will therefore be most convenient to write ^e less number under the greater, observing, as in addition, place units under unit^, ten9 under tens, &c. thus : OPERATION. 237 the mmutHd, Vake 114 the tuitrmkmd, 123 tkt remainder. We now begin with the units, saying, 4 (units) from 7, (units,) and there remain 3, (unit8,)which we set down directly under the column in unit's place. 34 SUPPLEMENT TO SVHTRACTION. ir 4 . It i, Then, proceeding to the next cohiinn we say 1 (ten) from 3, (tens,) and there remain 2 (tens,) which we set down in ten's place. Proceeding to the next cohimn, we say, I (hundred) from 2, (hundreds,) and there remains I, (hun- dred,) wliich we set down in hundred's place, and the work is done. It now appears, that the number of sheep left was 123: thatis, 237— 114==1-2:J. . ' After the same manner are performed the following ex amples : , 1 5. There a re two farms ; one is valued at 973 pounds, and the other at 421 pounds; what is the difference in the value of the two farms 1 K). A man's property is worth 2170 pounds, but he has debts to the amount of 11 10 pounds; what will remain af- ter paying his debts. 17. James, having 15 shillings bought a book for which he gave 7 shillings ; how many shillings had he left 1 OPERATION. 15 shillings. A difficulty presents itself here; for we 7 shillings. cannot take 7 from 5 ; but we can take — 7 from 15, and there will remain 8. 8 shillings left. 18. A man bought several articles for 85 pounds, and other articles for 27 pounds ; what did the former cost him more than the latter 1 OPERATION. The same difficulty meets us here as in First articles, 85 the last example ; we cannot take 7 from Other articles, 27 5 ; but in the last example the larger num- — her consisted of 1 ten and 5 units, which Difference, 58 together make 15 ; we therefore took 7 from 15. Here we have 8 tens and 5 units. We can now, in the mind, suppose 1 ten taken from the 8 tens, which would leave 7 tens, and this 1 ten we can suppose joined to the 5 units, making 15. We can now take 7 from 15, as before, and there will remain 8, which we set down. The taking of 1 ten out of 8 tens, and joining it with the 5 units, is called borromng ten. Proceeding to the next higher or- der, or tens, wo must consider the upper figure 8, from which we borrowed, 1 less, calling it seven ; then, taking 2 (lens) from 7 (tens) there will remain five (tens,; which we set down, making the difference 58. Or, instead of mak- ri }.» H 1. m f 7,8. gUBTRACTIOW OF SIMPLE WCMBERS. '25 I (ten) from set down in , we say, 1 ins I, (hun- nd the work leep left was llowing ex E pounds, and in the value I, but he has II remain af- 3k for which i left 1 4 here J for we ye can take main 8. )ound8, and ler cost him •'-..it -,:^ IS here as in take 7 from I larger num- units, which efore took 7 iVe can now, tens, which ose joined to from 15, as Jown. The I the 5 units, i higher or- II re 8, from then, taking tens,; which tead of mak- ing the upper figure 1 less, calling it 7, we may make the loicer figure 1 more, calling it •!, and the result w ill be the same ; for 3 from 8 leaves 5, the same as 2 from 7. 19. A man borrowed 713 pounds, and paid 471 pounds ; how many pounds did he then owe 1 7 1 3 — 47 1 = how jQaDY 1 jins. 242 pounds. 20. 1012 — 465 = how many l yfns. 1 147. 21 . 43751 — 6782 = how many 1 /Ins. 36969. «r 8. The pupil will readily perceive, that subtraction is the reverse of addition. 22. A man bought 40 sheep, and sold 18 of them; how many had he lefi 1 40— 18 = how many 1. ^ns. 22 sheep. 23. A man sold 18 sheep, and had 22 l«ft ; how many had he at first 1 18x22= how many 1 24. A man bought some articles for 75 pounds, and otliers i for 16 pounds; what was the diflFerence of the costs 1 '«?«?i' '75 — 16 =i how many 1 Reversed, 59 -|- 16 = how many 1 25. 114— 103e=4iow many 1 Reversed, ll-}-103= how many 1 26. 143—76= how many 1 Reversed, 67+76= how many 1 - " Hence, subtraction may be proved by additian, as in the foregoing examples, and addition by subtraction. 7b prove aubtractum^ we may add the remainder to the subtrahendy and, if the work is right, the amount will be [«qual to the ininuend. s i^ « jH^j; " * To prove addition, yre may sw5frac^ successively, from the imount, the several numbet^s which were added to produce J^it, and, if the work is right, there will be no remainder. Thus 74-84-6=21 ; proof, 21—6=15, and 15—8=7, and 7—7=4). From the remarks and illustrations now given, we deduce the following RULE. I. Write down the numbers, the less under the greater, placing units under units, tens under tens, &c. and draw a line under them. II. Beginning with units, take successively each figure in the lower number from the figure over it, and write the remainder directly below. ... ....,.!. ., U<.i-.« .... . III. When the figure in the lower number exceeds the figure over it, suppose 10 to be added to the upper fignie , c SUPPLSMBUT to tUBTmACTIOH. f 8. but in this case we raust add 1 to the lower figure in the iMxtcolumn before vuhlracling. Thii is caUe4 borrowing 10. EXAMPLES FOB PRACTICE. 97. If a faroi and the buildings on it be valued at 3000 pounds, and the buildings alone be valued at 1500 pounds, what is the value of the land 1 96. The p )pulation of the two Canadas, at the last cen- •us, wu 739,(MN), at the census before, the population was 048,000 ; what was the difference in the two enumerations! 99. What is the difference between 7,648,203 and 938,671 1 30. How much must you add to 358,649 to make 1,487,9451 3h A man bought an estate fur 3798 pounds, and sold it again for 4137 pounds ; did be gain or lose by it 1 and bow muchl 39. From 364,710,825,193 take 97,940.386,574. 33. From 831,025,405,270 take 651 '308,604,789. 34. From 197,368^7.916,843 take 978,654,887,3^. ■% m H SUPPI^EMElfT TO SUBTRACTiaN. QUESTIONS. I. What in subtraction T 3. Wh«it ii the grader nvmlier AsUed ^ 3. the less number ? 4. What is the residt or ansuer call«4 • 5. What is the n^ of subtraction? 6. What is thenrfef 7. \yhat 18 uiiderstuod by borrowing ten? 8. Of what is subtraction the re- vf.rsiif 9. How is suhtiaction proved ? 10 How is addition praved hv subtraction P EXERCISES. 1. How long from the discovery of America by Colifm- bus, in 1492, to the period of the cession by France of ^11 her possessions in North America to Great Britain, in 17G3. 9. Supposing a man to have been born in the year 1773, how old was he in 1827 \ 3. Supposing a man to have been 80 yearfi old in the year 1826, in what year was he born 1 4. There are two numbers, whose difference is 8764 ; the greater number is 15687 ; I demand the le^s 1 ' I .j.A^li f 8. ■^ tUPPLKMClIT TO SOBtRACTfON. igure in the orrotoing 10. led at 3000 SCO pounds, 2 to make , and sold it b 1 and how ,574. ^ 4,78a. i '• J • .' wwercalM? if 7. m^l tctinn the re- dition proved by Coliim- ance of ^U n, in 1763. year 1773. old in the 8764; the .5. What number is that which, taken from '.1794, leavci 8651 0. What number is that to which if you add 789, it will become 6360 1 7. In a certain city, there were 123,706 inhabitants ; in another 43,940 ; huw many more inhabitants ivcre there in one than in the other 1 8. A man possessing an estate of twelve thousand pouttdi, gay* two thousand five hundred pounds to each of his two daughter, and the remainder to his ten ; what wat his son's share Y 9. From sevettteen tBillion take filty-sik thoa8*nd» and what will remain 1 10. What number, together with tkeee three, tis. 1801, fllMl) and aidO, will make ten thousand 1 11. A ma* bought a hoMe for 95 pounds, and a chaise for 47 pounds ; how muah Aiove did Ike fire for the ohaiie than Ibv the horse 1 12. A roan borrows 7 ten dollar bills and three one dolkr bills, and pays at oiie time 4 fen dollar bills and 5 one dol- lar billM ; he# mafiy ten dollar bills and one dollar bifb itttist he tfterwatds pay to tnneel the debt 1 i^M, % ten doll, bills and 6 one doH. lis. The gteater of two numbera is 24, and the less is 141 ; what is their diffbrtencel l4i The greater ol two numbers is 24, and their diflfer* «i^e 8; what Is the lessnnmberl 15. The sum of two numbers is 40, the l«ss is 16 ; what is the greater 1 16. A tree, 68 feet high, was broken off by the wind; the top part, which fell, was 49 feet long ; how high was the stump which was left 1 17. Elizabeth became Queen of England in 1558; how many years since 1 18. A man carried his produce to market ; he sold his pork for 14 pounds, his cheeKC for 11 pounds, and his but- ter for 9 pounds ; he received, in pay, salt to the value of 6 pounds, 3 pounds worth of sugar, 2 pounds worth of taio- lasses, and (he rest in money ; how much money did he ft- ceive \ Jn: 23 pounds. 19. A hoy bought several sleds for 13 shillings, and gave 6 ahillings to have them repaired ; he sold them for 18 shill- 'W^4 MULYIPLICATIOS OF SIMPLE NUMBERS. IT H, 9. V ^ g iiigft ; (lid he gain or lose by the bargain 1 and how much \ '2(K One man ti-avcls 07 miles in a day, another man fol- lows at the rate of 42 miles a day ; if they both start from the same place at the same time, how far will they be apart at the close of the tirst day \ of the second \ of the third? of the fburthl >;<11'^ - : , '21. One man starts from York Monday morning, and travels at the rate of 40 miles a day ; another starts from the same place Tuesday morning, and follows at the rate of 70 miles a day ; how far are they apart Tuesday night 1 ^na. 10 miles. 22. A man owing 379 pounds, paid at one time 47 pounds, at another time 84 pounds, at another time, 27 pounds, and at another 143 pounds ; how much did he then owe % f bttcMuuli oat ^Am^ »;•» J^na. 82 pounde. 23. A man has property to the amount of 34764 pounds, but there are demands against him to the amount of 14297 pounds ; how many pounds will be left after tbe payment of his debts 1 - <, r --• -! . • - 24. Four men bought a lot of land for 482 pounds ', th« first man paid 274 pounds, the second man 194 pounds less than the first, and the third man 20 pounds less than the second ; how much did the second, third, and the fourth man pay 1 ft- ,tV. ^;^ !•,-♦*<''*'? - C The second paid 80. ^n8.< The third. paid 60. C The fourth paid 68. 25. A man, having 10,000 pounds, gave away 9 pounds ; how many had he left 1 ^ns. 9991 . ra' 03 ! r ,f,.t K 1 OF SIMPLE NUMBERS. v. :i tl IT 9. 1. If one orange cost 2 penee, how many pence must I give for 2 oranges 1 how many penee for 8 oranges % —— for 4 oranges 1 «i 2. One bushel of apples cost 3 shillings ; how many shillings must I give for 2 bushels 1 — — for 3 bushels 1 3. One gallon contains 4 quarts ; how many quarts in 2 gallons 1 — ^ in 3 gallons 1 in 4 gallons 1 4. Three men bought a horse -, each man paid 6 pounds for hi 5. ny pi 6." Unp;9| 7, pair H.I ■1S Ilr BERS. f 8, 0. i how much ! )thei man fol- uth start from they be apart ond 1 of norning, and )r starts from at the rate of lay night 1 ^na. 10 miles, le 47 pounds, pounds, and lowe'? i. 82 pounds. 1764 pounds, mt of 14297 the payment pounds; th« r pounds les6 less than the \ the fourth ond paid 80. rd.paid 60. rth paid 68. y 9 poulida ; /Tns. 9991. nany pence penee for 3 how many )ushels 1 quarts in 2 I 6 pounds t ^9. MULTIPLICATION Or SIHPLB NUMBBBS. 2» for his shaie; how many pounds did the horse c6«t th'^T 5. A man has 4 farms woiJli 9.> pounds each ; luow ma- ny pounds are they all worth \ 6. In one pound there are 20 shillings; how many shil- lings in r> potindsl 7. Huw much will 4 pair of shoes cost at sliillings a pair"? 8. How much will 3 pounds of tea cost at 5 shillings a pound 1 .; 9. There are 24 hours in I day ; how many hours in 2 days 1 in *.{ days \ in 4 days 1 in 7 days \ 10. Six boys met a beggar and gave him 9 pence each \ how many pence did the beggar receive \ When questions occur, (as in the above examples,) where the same number is to be added to itself several times, the operation may be much facilitated by a rule, called MuUi- plication, in which the number to be repeated is called the multiplicand, and the number which shows how many times the multiplicand is to be repeated is called the multiplier. The multiplicand and multiplier, when spoken o( collectively, are called the factors, (producers,) and the answer is called the product. 1 1 . There is an orchard in which there are .5 rows of trees, and 27 trees in each row ; how maby trees in the orchard 1 In this example, it is In the first row, 27 trees. evident that the whole ** second " 27 " number of trees will be " third " 27 " equal to the amount of " fourth " 27 " five 27's added together. •« fifth " 27 " In adding, wo find that 7 taken five times In the orchard 135 trees. amounts to 35. Wc write down the five units, and reserve the 3 tens ; the amount of 2 taken five times is 10, and the 3, which we reserved, makes 13, which, written to the left of units, makes the whole number of trees 135. If we have learned that 7 taken 5 times amounts to 35, and that 2 taken 5 times amounts to 10, it is plain wc need write the number 27 but once, and then, setting the multi- tiplier under it, we may say, 5 times 7 are 35, writing down C2 4 •^*' il M ■I 30 W^ fT MULTIPMCATION OP HIMPLR NUMBKKS. T 0, lO. tlifi^^ and reserving the '.\ (tens; as in addition. Again 5 **r<' > times 'i (tens) are Multiidicajid, 27 /recs in each row. 1(1 (tens,) and ;{, (tens,) which we reserved, make 13, (tens,) as before. JUuUiplier, Product, ■> rotes. 135 trees, /fns. If lO. 12. There are on a board') rows of spots, and 4 .spots in each row ; how many spots on the board 1 ^ 3s ?R ?fi ^ slight inspection of the figure will show that the number of spots may be B^ ^ 96 S found either by taking 4 three times, (3 times 4 are 12,) orby taking 3 /our Sign. Two short lines crossing each other in the form of the letter X, are the sign of multiplication. Thus, 3X4 = 12, signifies that 3 times 4 are equal to 12, or 4 times 3 are 12. Note. Before any progress can be made in this rule, tlie following table must be committed perfectly to memory. . ^ ..; , , . . .. . _ :. . ti. J»:;3 '"■' ' \' ' '- i ,■: ; •** ;• . di'ii-n i\ ■>': ')^i '[' \if %.■• '«"!'.. •':',: - ■■■< ;• . ^- '-.iMj/ni-' r V'.t 1 :i^ •J Z' t. '# tHm-.: ■ v^' '..|i jij''l.-r' >" d^- 'ji». K' t'lm' Urn- X 4 X s. ^ 0, 10. Again 5 (tens) are ,) and ;{, which we make V3, « before. )ots, and 4 figure will s may be times, C.i four times, say there >ots each ; nbers for a generally umber up- fcc. Thus, he factors, luct 12. •■ ' .1 ning many eating any the form Js, 3 X 4 4 times 3 rule, tlie nory. vv, Hi-- 10. MVLTIPLICATION OF SIMPLE "VUMBRRS. MlLTIPLICATIOiV TABLE. M ore 04 X 10 24 X 1 1 4} 4X \'i ^^>X = 2 3 4 .'i G 8: 9: 10: 11 12 407 44 7 4K7 X X X = 101-, s^- = 14o X 24;r> X 1^7 X ''X = ld7_XJI2=J 10 11 12 i= 0|'>x 1= f f>x 12 = != mx = t = 12|(> X 1 = i = 15;()X i2 1=18,«X 3 = 21Gx 4: i = 24«x 5 = 27|G X = 306 X 7 = 33;Gx 8: = 300 X 9: ^T>« X 10 .-jjP X = 208 X = '2riS X = 3018 X :3ii8X = 408X : 458 X : 508 X = 5^8 X 0(^8 X "O'^X 128X 1 •> /v 3 4 5 G 7 8 9 10 = 11 = 12 = = 49110 X 4= 40 = 50 10 X 5= .50 = (i;no X 0= m = 70 10X 7= :o 77 10X 8= S> 8410 X 9= m (, 10 X iO= 100 ^10 X 11=110 10 X 12=120 10 24 11 32|ll 40 48 = 5(> X X IlX 11 X 11 X 4 = 1 5> . 3: 4 X = G4jll X 5 = = 7211 X G = 80111 X T = HHll X 8 = 4OX 11 = 8 X 12 = 189 X 249 X 309 X 369 X X 48|9X 549 X 609 X 669 X 729 X = 127 X = 167 X = 207 X = 24 7 X = 287x = 327X ~ 3(^7 X = 1 = 2 = 3 = 4 = 5 = 6 = = 1 = 2 = 3 = 4 = 5 = G = 7 = 8 = 9 = X X 96'! 1 X ~oii 9^1 18^ X X X 9 10 11 12; II 22 :J3 44 55 00 88 99 110 {21 132 27|12 X 3612 X 4512 X 5112 X 6^12 X 7-^12 X 81 12 X 90,12 X 9912 X 1^9 X 12= 10812 X ^ 012X 09X10 = 9X 11 = 2110X = 2810X 1 = 3510X2 = 42!l0X 3 = 1 2: 3 4 5 : 6 1^ I •■ 8: 9: 10; 11: 10,12 X 20I2 X 12 12 24 30 48 (»0 72 84 96 lOe 120 132 144 If- ■H:l it *:« >» rV '>*%} '•'1 ;:f i'T*i •'>5»'if 0'^^ MULTIPLICATIOM Of 81MPLIS NUMBERS. HI 10 9 ^^ ^^' 9X2 s=how many 1 4X0 =Iiow many 1 H X 9 =l>ow many 1 3x7 =l»ow many \ 5 X «* =l)OW many ? 4 X 5i X 2 = '24. iJ X '2 X •'> = l»"W many 1 7 X 1X2 = lioxT many 1 H X 'J X 2 = liow many 1 IJ X 2 X 4 X i> = l»ow many ' 13. What will 84 barrels of flour cost at 2 poimrls a bar rel 1 J^iiH. 108 pounds. 14. A merchant bought 12 dozen hats at the rate of Vi pounds per dozen ; what did they costl y/ns. 144 pounds. 15. How many inches are there in 253 feet, every foot being 12 inches 1 OPERATION. The product of 12, with each of the signiti- 25!) cant figures or digits, having been committed 12 to memory from the multiplication table, it : is just as easy to ranltiply by 12 as by a sin- jfna. 3036 gle figure. Thus, 12 times 3 are 30, &c. 16. What will 470 barrels of fish cost at 3 pounds a bar- rel 1 ^ns. 14-28 pounds. 17. A piece of very valuable land, containing 33 acres, was sold for 246 pounds an acre ; what did the whole come tol As 12 is the largest number, the product of which, with the nine digits, is found in the multiplication table, there- fore, when the multiplier exceeds J2, we multiply by each figure in the multiplier separately. Thus : OPERATION. 246 pounds, the price of one acre. 33 number of acres. 738 pounds, the price of three acres. 738 pounds, the price of thirty acres, The multipli- er consists of 3 tens and 3 units. First, multiply- ing by the 3 "~~~"^~~ units ffives us Ans.SllS pounds, the price of 31^ acres. ,j.qq pounds the price of 3 acres. We then multiply by the 3 tens, writing the first figure of the product (8) in ten's place, that is, di- rectly under the figure by which toe multiply. It now appears that the product by the 3 tens consists of the same figures as the product by the 3 units ; but there is this difference— the figures in the product by the 3 tens are all lemoved one place further to the left hand, by which their value is in- creased tenfold, which is as it should be, because the price :RH. H 10 V "^ 1^' MVLTEPLICATION OF SIMPLE IfUMBSUM. :» many ? many 1 many 1 iiuw many ! oiinrls a bar 08 pounds, e rate of V2 44 poundfl. every foot f the signifi- n committed ion tabJe, it as by a sin- e 3(>, &c. ounds abar- 4:28 pounds. Iff 33 acres, lolecometo'? which, with table, there- ply by eaclj he multipli- onsists of 3 and 3 units, t, multiply- by the 3 * gives us pounds the ns, writing that is, dj- 3W appears ime figures lifference— moved one ^alue is in J the price of 30 acres is •TideDtly ten times as much as the price of 3 acres, that is, 738<) pounds ; and it is plain, that these two products, added together, give the price of 33 acres. These examples will be sufficient to establish the fol- lowing • 1 RULE. ■ '■-'-' I. Write down the multiplicand, under which write the -multiplier, placing units under units, tens under tens, &c., land draw a line underneath. II. When the multiplier does not exceed 12, begin at the ^ right hand of the multiplicand, and multiply each figure contained in it by the multiplier, setting down, and carry- ^ing the same as in addition. ,^,^ . III. When the multiplier ixaeda 13, multiply by each igure separately, first by the units, then by the tens, kc, remembering always to place the first figure of each pro- rauct directly under the figure by which you multiply .^ Having gone through in this manner with each figure in ' the multiplier, add their several products together, and the :4um of them will be the product required. , , EXAMPLES FOR PRACTICE. ^^/''^ ^'^ Id. There are 320 rods in a mile ; how many rods are [there in 57 miles 1 19. Suppose it to be 706 miles from Halifax to Quebec ; how many rods is it 1 20. What will 784 chests of tea cost, at 17 pounds a- ' chest 1 21. If 1851 men receive 758 pounds apiece; how many pounds will they all receive ? yfna, 1403058 pounds. 22. There are 24 hours in a day ; if a ship sail 7 miles in an hour, how many miles will she sail in 1 day, at that rate 1 how many miles in 36 days 1 how many miles in 1 year, or 365 days 1 j^s. 61320 miles in 1 year. 23. A merchant bought 13 pieces of cloth, each piece containing 28 yards,, at 2 pounds a yard ; how many yards, were there, and what was the whole cost 1 jins. The whole cost was 728 pounds. 24. Multiply 37864 by 235. Product, 8898040. 25. «' 29831 " 952. " 28399112. 26. » 93956 « 8704. «« 817793024. COHrtRACTiniVt III MVLTIPLfCATIOR. f 11 ^111 I I I I _ ;I51 1$ P 'I CONTRACTIONS IN MULTIPLICATION. I. If hen the multiplier is acomponite number. f 11. Any nuraber, vrhich may be produced by the multiplication of two or more nuinhuri, is called a compoBttt number. Thus, 15, which ariiies rrom the multiplication of r> and 3, (5X')=1''>.) ^^ ^ composite nuinher, aitd the num bert 5 and 3, whicb, multiplied to(;ether, produce it, are billed component porta or factora of that ntimber. So, alio, 34 M a composite number ; its component pcrte, or Jhdor§, mky be 2 and 12, (2X 12=34 ;) or they may be 4 and 6, <4X&=^4 ;) or they may b« 2, 3, and 4. (2x3x4=^.) 1. What will 1& pieces of elolh cost, at 4 pounds a piecel 15 pieces are equal td 5x3 piece*. The oost 4 df 5 pieces would be 5x4==20 pounds ; aud be- 8 cause i^ pieces contain 3 tidies 5 pieces, so the -^ cost of 15 piecel Vrill «videntlr be 3 times the ^ cost of 5 piecei, tbAt ii, 20 pounds X 3es66 pound!. 3 j^». 00 poun^. Wherefore, Jff" the multiplier be a co^npoaite number, we may, 'if we please, multiply the multiplicand first by one of thecom- f&nmt paiiir, that product by the other, emd ao M, if the com- ponent parts be more than two; and, hatinjt in this Way multiplied by each of the Ctittpenent patts, the iMf product will be the product required. 9. Whal will 186 totti of potaihes Come to, at 24 pounds |>er ton 1 ' "■ 6x43i«24. It follows, th«refofe, that and 4 are etm- peihtn% parts or factors of 24. Hence, 18^ tons. ' ^' 6 one of the component parts, or factors. it **' 816 pdUftds, fhe price of 6 tdilS. 4 the other component part, or factor. ],i- Jns. 3264 pounds, the price of 136 tons. " ' < 9. Supposing 342 men to be employed in a certain piece of Work, for which they are to receirc 28 pounds, each, how much will they all receive 1 7X4»28 if AS. 9576 pounds. f II 12, 13. OOVTBAOTIOVt IK MULfirLICATlO*. 36 'ION. piced by the I a compoBttt iplication of Hi the num |duce it, are So, also, , ot /kcton, be 4 aad 6, nda a piece! ' Theooft Is ; and be- eoes, go the I timni the ^60 pound!. 00 poundi. er, we may, t of the com- if the corn- in this tray (Mf product t S4 pounds 4 are com- factomi. tor. ?'^ 'tain piece wis, each, 6 pounds. 4. Multiply :)G7 by 48. lOHIi •• 72. 6. Product, 17010. 4776H. " THIIW 11. ffAen tkt multiplier i$ 10, 100, 1000, &c. IT 13. It will be recollected, (If M.) that any figtire, on ll>ciDg removed one place towards the left hand, has its vai- tie increased //>n/u/J; hence, to multiply any number by 10, |t is only necessary to write a cypher ^)n the right hand of it. 'hus, 10 times 2.) are 250 ; for the 5, which was units be iTore, is now^ made ten$, and the 2, which was ten$, before, (s now made hundreds. So, also, if any figure be removed (too places towards the left band, its value is increased 10ft times, &c. Hence, When the muttiplitr is 10, 100, 1000, or 1 with any num- \er of ciphers annexed, annex as many ciphers to the multi- licand as there arc ciphers in the multiplier, and the mul- tiplicand, so increased, will be the product required. Thus, Multiply 46 by 10, the product is 460. 83 " 100, " '• 8300. 05 " 1000, " " 95000. EXAMPLES FOR PRACTICE. 1. What will 76 loads of corn cost, at 10 pounds a load T 2. If 100 men receive 32 pounds each, how many pound* |wm tbey all receive! j. ,,v 3. What will 1000 piece* of ^oadcloth coft, estiwatin^ Bach piece at 78 pounds 1 4. xMultiply 5682 by 10000. 5» „ 82134 „ 100000. *I 13. On the principle suggested in the last f , il fol- lows, When thure are cipktra on the right hand of the muUipii- \eand, miUHpHer, either or kstk, we may, at first, neglect these ciphers, multiplying by the aignifieaM Jiguren only ; after which we must annex as many ciphert to the prodoot as there are ciphers on the right hand oi the multiplioand and multiplier, counted together. :■'-([&. -■> •' '. *><■'* Mi .15 rOltTRACTIORB IKT HULTIPLICATtOIT. \i\iul< ,-^10*^' EXAMPLES FOR PRACTICE. I. If I:MN) men receive 4(M) pounds apiece, how many puundH will they all receive 1 OPERATION. '^^^ ciphers in the multiplicand 4(iO ,•; and multiplier, counted together, i'WO Arc three. Disregarding these, we write the 8ign\/icant figures of the 13H multiplier under the significant fig- '40 uk-es of the multiplicand, and mul- — tiply ; after which we annex three y/n«, 508000 pounds, ciphers to the right hand of the product, which gives the true answer. 3. The number of distinct buildings in New England, appropriated to the spinning, weaving, and printing of cot ton goods, was estimated, in 1826, at 400, running, on an average, 700 spindles each ; what was the whole number of spindles 1 3. MulUply 257 by 6300. 4. " Sim " 17. 5. " 9340 •' 6. . *' 5200 " 7. " 378 '* '■*- 1' "'..{ '(■'',• OPERATION. 378 204 460. 410. 204. > ■ ' '. ( ,; JJ 1512 000 756 77112 In the operation it will be seen, thaf multi- plying by ciphers products nothing. There- III. Whin there are ciphers between the 8ign\ficarU Jig ures of the multiplier, we may omit the oiphers, multi- plying by the significant figures only, placing the first figure of each product directly under the figure by which we multiply. ,..»j- «)i^;f ■%! i^si^s'i ,.i.ii*i- 'twvf .v *»/>?'«■{•■■ ';i»'. ->'^«if<' »?» EXAMPLES FOR PRACTICE. , 8. Multiply 154326 by 3007. *r 13 •orrLKMKMT ro multipmoatioit. 37 OPERATION. 4(W.r7H Product, Amm»Z&Z 9. Multiply 54a by 206. 10. " ICaO " 2103. 11. " 3(>243 " ;i'2004. England, ing of cot ■ ling, on an >le number M tha( multi' [ Therc- SUPPLEMENT TO MULTIPLICATION. QUESTIONS. I > I. What is multiplication P 2. What ii the numbor to be multipli- ed called P :^ to multiply by called P 4. What is the result or answer called P 5. Taken collectivelif, what nre the multiplicand and multiplier called ? 0. What is the «t^n of multiplication P 7. What doefi It show P 8. In what order must the civen numbers bo placml for multiplication ? 0. How do you proceed when the multiplier is less than 12 ? 10. When it exceeds 12, what is the method of proce- dure P 11. What IB u composite numhcr ? 12. What is to be under- stood by the component parts, or factors, of any number? 13 How may you proceea when the multiplier is a composite number ? \A. To multiply by 10, 100, 1000, &c., what suffices P 15. Why ' lb When there are ciphers on the right hand of the multiplicand, multi- plier, either or both, how may wo proceed P 17. When tliere nre ci- phers between the significant figures of the multiplier, how nrr they to be treated ? i EXERCISES. 1. An army of 10700 men having plundered a city, took so much money, that, when it was shared among them, each man received 46 pounds ; what was the sum of money taken 1 2. Supposing the number of houses in a certain towtt to be 145, each house, on an average, containing two families, and each family 6 members, what would be the number of inhabitants in that town 1 ^ns. 1740. 3. If 46 men can do a piece of work in 60 days, how many men will it take to do it in one day 1 ^ t •» * » SUPPLEMENT TO MULTIPLICATIOW. IT i:j. P* 4. Two men depart from the same place, and travel in opposite directions, one at the rate of 27 milei a day, the other 31 miles a day ; how far apart will they be at the end of days 1 y/ns. 348 miles. ii. What number is that, tlie factors of which are 4, 7, 6, and 20 1 jfns. 33G(). 6. If 18 men can do a piece of work in 90 days, how long will it take one man to do the same 1 7. What sum of money must be divided between 27 men, so that each man may receive 115 pounds 1 8. There is a certain number, the factors of which are 89 cnid 265 ; what is that number I 9. What is that number, of which 9, 12, and 14 are fac- tors \ 10. If a carriage wheel turn round 346 times in running I mile, how many times will it turn round in the distance from Quebec to Montreal it being 180 miles. Ans. 62280. 11. In one minute are 60 seconds ; how many seconds, in 4 minutes 1 in 5 minutes'! — — in 20 minutes'! in 40 minutes 1 12. In one hour are 60 minutes ; how many aeconda ' an hour '! in two hours 1 How many seconds from nine o'clock in the morning till noon 1 13. In one pound are 4 dollars ; how many dollars in ;J pounds '! in 300 pounds '! in 467 pounds '! 14. Two men, A and B, start from the same place at the snme time, and travel the same way ; A travels 52 miles a o six pounds, what is the interest for two hundred pounds the same time '! 7 hundred pounds 1 8 hundred pounds 1 95 hundred pounds '! J 7. A farmer sold 468 pounds of pork, at 3 pence a pound, and 48 pounds of cheese, at 4 pence a pound ; how many pence must he receive in pay 1 1 8. A boy bought 1 oranges ; he kept 7 of them, and sold f 13, 1 Itheothe 19. |6, 9. 8, 20. I [in 8 h( Iquarts i '^^|)ints in ir 14 ow m: 2. Jj ow m: 3. J( ho re( em tc 4. U ;t 2 pe 5. H t2po 6. If one 3 7. A illing 8. H IT Vl. and travel IS a day, the be at the . 'MS miles, are 4, 7, 6, ^ns. :J.3G0. s, how long sen 27 men, hich are 89 14 are fac- ia running le distance m. 62280. seconds, in lUtes 1 ' seconds londs from dollars in undsl )lace at the 52 miles a they be at iar, be six )r 2 years 1 sarsi one year, red pounds 8 hundred 3 pence a. und; how n, and8old hr 13, 14. DIVISION or SIMPLE NUMBEBS. ftt» [the Others for 5 pence a piece ; how many pence did he receive! 19. The component parts of a certain number are 4, 5, 7, \G, 9, 8, and 3 ; what is the number 1 20. In 1 hogshead are 03 gallons ; how many gallons fin H hogsheads 1 la 1 gallon are 4 quarts ; how many [quarts in 8 hogsheads 1 In 1 quart are 2 pints ; how many 'yipints in 8 hogsheads 1 OF SIMPLE NUMBERS. ( r. ^ 14. 1. James divided 12 apples among four lx>ys -, low many did he give each boy "! M. 2. James would divide 12 apples among three boys ; llow many must he give each boy 1 3. John had 15 apples, and gave them to his playmates, rho received 3 apples each ; how many boys did he give lem to 1 4. If you had 20 pence, how mafiy cakes could you buy |;t 2 pence a piece 1 5. How many yards of cloth could you buy for 30 pounds, ^t 2 pounds a yard 1 6. If you pay 250 shrHings for 10 yards of cloth, what one yard worth 1 7. A man works •€ -days for 42 shillings ; how many lillings is that for one day 1 8. How many quarts in 4 pints '? -— in 6 pints \ in 10 pints 1 9. How many times is 8 contained in 88 1 10. If a man can travel 4 miles an hour, how many hours ould it take him to travel 24 miles 1 11. In an orchard there are 28 trees standing in rows, d there are 3 trees in a row ; how many rows are there 1 Remark. When any one thing is divided into two equal arts, one of those parts is called a half; if into 3 equal arts, one of those parts is called a third ; if into four equal arts, one part is called a quarter or a fourth; if into five, ne part is called a fifth, and so on. 12. A boy had two apples, and gave one half an apple to WiU of hi|i companions i how manr were his companions 1 ♦ 4. * DIVISION OF SIMPLE XITMBBRS. ^•1 of them ; [piece, and miuber 1 be bought k be taken the object, ed in 12. imple, that n 12 four here is no 9ntly, sub- tent for the ay come to process, ia rter, called !«t ■ n Vr tliose parts. The operation is CTrflently fts in the last ex- ample, ai)(I consists in linding a number, which, multiplied by ;], will produce I*i. This number we hare already found to l)c 4. ^118. 4 oranges apiece. As, therefore, multiplication is a short way of performing "iiiany additions of the same number; so division is a short way of performing many subtractions of the same number; and may be defined. The method of finding how many times one 7iumber ia contained in another ; and also of dividing a number into any number of equal parts. In all cases, the pro- cess of division consists in finding one of the factors of a given product, when the other factor is known. The number given to be divided, is called the dividend, and answers to the product in multi])lication. The num- ber given fo divide by h called the divisor, and ijinswers to one of the factors in multiplication. The result, or answer '«ought, is called the quotient, (from the Latin word quoties. how many 1) and answers to the other factor. SiGX. The sign for division is a short horizontal line between two dots, -J-. It shows that the number before it is to be divided by the number after it. Thus 27 -^ 9 = I:£ is read, 27 divided by 9 is equal to 3 ; or to shorten the ex- pression, 27 by 9 is 3 ; or 9 in 27 3 times. In place of the \dots, the dividend is often written over the line, and the di- visor under it, to express division ; thus, %^ = 3,' read as [before. * .p '- . ' The reading used by tho pupil in committing the follow- ing table may be 2 by 2 is 1, 4 by 2, &«. or 2 in 2 one ftime, 2 in 4 two times, &c. (3 pence,) f^^^K J. uvi&iu: \ TAJBl- m. ual to the 9 ^ =^ f =1 1-1 f =1 f t)r«, a pro- B ^ =2 f-2 f-2 V_2 1 v^ any times ^1=3 f =3 ^--3 '^ liioh, mul- ^^ i =4 V" 4 i/> = 4 ^•'^••^ ¥ ndi by tri- Wm \p=5 V=5 ^/ = -> ^-^ ¥ d in 12 4 ^m i^ s=(i ^8=0 ¥ = « 3 6 1 oranges. w ^ "^ '' V 7 2^8 __ 7 ¥ = 7 among 3 9 -'^ '^^ ^ 21=8 3JJ=8 V - ^^ 4 8 ▼el 1 '/ — ^ V-9 .^6 ==9 Y=9 ¥ 3 •qual s D2 in each of ■ •m = 1 :2 :3 4 ;5 G 8 9 ¥ ¥ 42 r 4 7 ;. C 7 3 7 3 4 r> = 8 = 9 4)1 OIVIIIOH or tlMPLK NUMBBRI. ^ 15, 10. Hi l^. DIVISION TABLE-CONTINUED. '.« ¥ = V v = 'it V 1 3 4 5 6 7 8 9 1=1 ¥=3 |g = l 1 j[ = l 5 V=6 V=8 18 = 7 = 9 I fJJ=:9 H=2 H = 4 1^ = 5 ^»=9 i! — 1 L^ — 2 f^ — 3 tl = 4 n — 5 E = ti fi = 7 !f — 8 li)J = 9 .28 -^ 7, or ^^=. how many \ 42 -^ 0, or ^^= how many 1 54 -^ 9, or ^^= how many % 32-^8, or 3^= how many 1 _. , ._, -. ^^ ^ *J:3 -h ll.orf J=how many 1 IO8-7-I2, ot\9^=\\oyi many 49 _^ 7, or y= how many i 4, or \^= how many 1 — ll,orf^= how many 1 •rl2, or II =how many 1 32 99 84 •1 l«. 23. How many yards of cloth, at 4 shilhngs a yard, can ho bought for 856 shiiUngsl Here the number to be divided is 856, which therefore is the dividend ; 4 is the number to divide by, and therefore the divisor. It is not evident how many times 4 is contain- ed in so large a number as 856. This difficulty will be readilv Qvcrcome, if wo decompose this number, thus : 856=800-1-40-1-16. lieginuini^ with the hundreds, we readily perceive that 4 is contained in 8 2 times ; consequently, in 800 it is contained 200 times. Proceeding to the tens, 4 is contained in 4 1 time, and consequently in 40 U is contained 10 times. Lastly, in 16 it is contained 4 times. Wo now have 2lH>-f-10-f-4=2l4 for the quotient, or the number of times 4 is contained in 856. Ane. 214 yards. We may arrive to the same result without decomposing the dividend, except as it is done in the mind, taking it by parfs, in the following manner : Dividend. For the sake of convenience, we /Hvisor, 4 ) 856 write down the dividend with the divi- O of 1 -m"! sor«n the left, and draw a line between * ^ ' ^ them; we also draw a line underneath. Then, beginning on the left hand, we seek how often the Idiviso^d) [finding it (which fall iProceedin which we But aflor ten left. Then, 4 work is This H Division. on partly the divi)^ From rulcfor I. Fi tirst figv ly unde next fi? II. F divideni all the Pro< tient ai have a merely Hen other. 7b] tient, as the tient, as th< 7b factoj other men If 1. DIVISIOIf or SIMPLK aVMBBRS. m many I many'! V many 1 y many 1 many \ lillings a erefore is therefore I contain- ' will be bus : that 4 is ontained (1 in 4 1 times. >w have of times 4 yards, mposing ng it by Llivisoi- (1) is contained in 8, (hundreds,) the left hanti tiguro ; ] finding it to bo 'i times, we write 2 directly under the 8 which falling in the place of hundreds, is in reality '200, [Proceeding to tens, 4 is contained in .'> Ucns) 1 linic^ I which we set down in ten's place, directly under the 5 ^tcuM.) But after taking 4 times ten out of the ."» tens, there is 1 [ten left. This I ten we join to the (> units, making I(>. [Then, 4 into 1(» goes 4 times, which we set down, and the work is done. This manner of performing the operation is called Short Division. The computation, it may bo perceived, is carried on partly in the mind, which it is always easy to do when tlie divisor does not exceed VZ. RULE. From the illustration of this example, we derive this general rule for dividing, when the divisor does not exceed 1*2 : I. Find how many times the divisor is contained in the first figure, or figures, of the dividend, and, setting it direct- ly under the dividend, carry the remainder, if any, to the next figure as so many tens. II. Find how many times the divisor is contained in this dividend, and set it down as before, continuing so to do till all the figures in the dividend are divided. Proof. We have seen, (IT 15,) that the divisor and quo- tient are factors, whose product is the dividend, and wc have also seen, that dividing the dividend by one factor is merely a process for finding the other. Hence division and multiplication mutually prove each other. 7b prove division, wo may multiply the divisor by the quo- tient, and, if the work be right, the product will be the same as the dividend ; or we may divide the dividend by the quo- tient, and, if the work is right, the result will bo the same as the divisor. 7b prove multiplication, wc may divide the product by one factor, and, if the work be right, the quotient will be the other factor. • ■ EXAMPLES FOR PRACTICE. 24. A man would divide 13,462,725 pounds among 5 men ; how many pounds would each receive 1 f m DIVIHTOW OF BIMPI-K NVMBRRN. ^ 16, K. OPERATION. Dividend. Divisor, r»)V\,Ui'2,7'irt Quotient, i.im/A^ Proof. Quotient. d{ visor. i;j,40»,7r> In this examplr, as wo cinnoi have .■> in the first ligure, ( I ) W( take two figures, aiul say, ;> in }'•] will go 'J times, atul there are '>] over, which, joined to i, the next figure, makes JM ; an«i 5 in tM will go ti times, &c. In proof of this example, we mul tiply the quotient l)y the divisor, and, as the product is the same as the dividend, wc conclude that the work is right. From a hare in- spection of the ahove example and its proof, it is plain, as before stated, that division is the re- verse of multiplication, and that the two rules mutually prove each other. 25. How many yards of cloth can be bought for 4,J{o4,o00 shillings, at 2 shillings a yard 1 at 3 shillings 1 at t shillings 1 at 5 Shillings 1 at shillings 1 at 7 1 at 8 \ at 9 1 at 10 \ Note. Let the pupil be required to prove the foregoing, and all of the following examples. 2(). Divide 10059033G0 by 2, 3, 4, o, G, 7. 8, 9, 10, 11, and 12. 27. If 2 pints make a quart, how many quarts in 8 pints! in 12 pints 1 i— — in 20 pints 1 in 24 pints 1 in 248 pints 1 in 3074 pints 1 in 47G32 pints 1 28. Four ({uarts make a gallon ; how many gallons in 8 quarts 1 in 12 quarts 1 in 20 quarts 1 in 30 quarts 1 in 308 quarts 1 in 4890 quarts 1 in 5436144 quarts 1 29. A man gave 80 apples to 5 boys ; how many apples would each boy receive 1 Dividend. , Divisor, 5) 86 17 — 1 Remainder. Here, dividing the number of the apples (80) by the number of boys, (5,) we find, that Quotient, each boy's share would be 17 apples ; but there is 1 apple left. IT 17. 5)86 In order to divide all (he apples equal- ly among the boys, it is plain, we must divide this one remaining apple into 5 17. 'hen cai )art of Thel lis, whole ing pari (that is Fract )ne dir< Bhowinj 'i m 17. DIVIBIOlf or 8IKPLR NVMBERS. ^ wc en n not irv, (I) w< ^y, '> in in there arc .*) i, the next ^ in JM will le, wo mul he divisor, he same as do that the a bare in- cainple and m is the re- nuitually fa ] at ings 1 foregoing, 9,10, 11, n 8 pints 1 ints 1 pints 1 lions in 8 — in3G uarts 1 in »y apples ding the e apples umber of ind, that I 1 apple fqualparts, and give one of these parts to eacA of the boys, 'hen each boy's share would be 17 apples, and one fifth aartof another apple ; which is written thus, 17^ apples. jfns. 17^ apples each. The 17, expressing whole apples, are called integers, (that is, whole numbers.) The ^ (one fifth) of an apple, express- 'Ing part of a broken or divided apple, is called a fraction, (that is a broken number.) Fractions as we here see, are written with two numbers, )ne directly over the other, with a short line between them, showing that the uppit number is to be divided by the tr. The upper number, or dividend, is, in fractions, called the numerator, and the lower number, or divisor, is Balled the denominator. Note. A number like 17|, composed of integers (17) knd a fraction, (^,) is called a mixed number. In the preceding example, the one apple, which was left ifter carrying the division as far as could be by whole num- >er8, is called the remainder, and is evidently a part of the Idividend yet unatviaea. in unim tu comiilote the divinon, [this remainder, as we before remarked, must be divided into [5 equal parts ; but the diviior itself expresses the nuiirt>er }f parts. If, now, we examine the fraction, we shall see, [that it consists of the remainder (.1) for its numerator, and [the divisor (5) for its denominator. Therefore, if there be a remainder, set it down at the kight hand of the quotient for the numerator of a fraction, ^under which write the divisor for its denominator. proof the last example. m 5 86 In proving this example, we find it necessary to multiply our fraction by 5 ; but this is easily done, if we consider, that the fVaotion | express- es one part of an apple divided into 5 equal parts ; hence, 5 times | is |=3l, that is, one whole apple, which we reseive to be added to the units, saying, 5 times 7 are 35, and one we reserved makes 36, &.c. 30. Eight men drew a bounty of 463 poonds from gov* itrnment, hd^ many pounds did each reoeirel flW* liffiAi^-i: %* \*, i ■i s . u I / 40 ■ w y.»- - - - • • < vj DIVISION or IIMPLE KUMBERB. IT IR, 19. Divisor, 8 ) l.'ilJ Here, aAer carrying; the division as far as possible by whole numbers, we have a remainder of 5 pounds, which, written as above directed, gives for 453 the answer 5tf pounds and | (five eighths^ of another pound, to each man. IT 18. Here we may notice, that the eightii part of 5 pounds h the same as 5 times the eighth part of 1 pound, that is, the eighth part of 5 pounds is | of a pound. Hence, I expresses the quotient of 5 divided by 8. I is 5 parts, and 8 times 5 is 40, that is, V='^> which, reserved and added to the product of 8 times 0, makes 53, fcc. Hence, to multiply a fraction, we may multiply the numerator, and divide the product by the dmomina/or. Or, in proving division, we may multiply the whole num> ber in the quotient only, and to the product add the remain- der ; and this, till the pupil shall be more particularly taught in fractions, will be more easy in practice. Thus, 56x8= 448, wd 448-|-5, the remainder, e=s453, as before. 31. There are 7 days iu a week; how many weeks in ^65 days 1 jina. 52} weeks. 32. When flour is worth 2 pounds a barrel, how many barrels may be bought for 25 pounds 1 how many for 51 pounds 1 for 487 pounds 1 for 7631 pounds 1 .^33. Divide 640 pounds among 4 men. 640-^4, or 6|o=:=i60 pounds, Am. Ans. 113. 34. 678-^6 or 6^8= how many 1 35. 5 0^0= how many 1 36. 7:^*= how many 1 37. ='V®*= how many 1 38. a|^4= how many 1 39. ^''f « '= how many 1 40. 2 i^^o 12— how many 1 Ans. 384f . 4^ ^ >:. f 19. 41. Divide 4370 pounds equally among 21 men. When, as in this example, the divisor exceeds 12, it is evident that the computation cannot be readily carried on in the ihind, as in the foregoing examples. Wherefore, it is more convenient to write down the computation at len^h^ in the following manner : 10. OPF. >tt'isor, / 'l\ ) nrts, we tiuding it )f the div 13 being ?his, \\o\ )y fens a lividend, in the ^ It is io be tak livisor (* laking lividcnd, ing do^ We th hens-,) 1 kuoticnt, 170. A^ ^nd, find juUiply 168, urn ler to b« 5f the q \ox the c Thisi Yvtision From r 1. P them b Idividen 11. bM \^ IH, 19. fivision as >bcr«, we fs, "Which, gives for ker pound, [part of 5 I pound, Hence, }duct of 8 tultiply a H otor, and hole nuni« remain - ly taught 56X8= • weeks in 1^ weeks. ow many ny for 51 ndsl ids, ^ns, fns. 113. s. 384|. (1 men. 2, it is ried on fore, it length, 19. DIVI8I0!T or SIMPLE ^riMBERII. ft OPF.HATION. ' '■""" hvisor, Dividend, Quotient. 11 ) 1370 ( -iO-Vi. . 42 # - 170 168 •A- "^.V : may write the divisor anu dividend as in short di- vision, iHit, instead of writing the Quotient tmdtr the divi- dend, it will be found more convenient to set it to the right hand. Taking' the dividend by 2 Remainder. \arts, we seek how often we can hare 21 in 43 (hundreds ;) inding it to be 2 times, we set down 2 on the right hand >f the dividend for the highest figure in the quotient. The 13 being hundreds, it follows, that the 2 must be hundreds. This, however, we need not regard, for it is to be followed ^y tens and units, obtained from the tens and unuij of the lividcnd. and will therefore, at the end of the operation, in the place of hundreds, as it should be. It is plain that 2 (hundred) times 21 pounds ought now |lo be taken out of the dividend ; therefore, we multiply the livisor (21) by the quotient figure 2 (hundred) now found, [naking 42, (hundred,) which, written under the 43 in the lividcnd, we subtract, and to the remainder, 1, (hundred,) >ring down the 7, (tens,) making 17 tens. We then seek how often the divisor is contained in 17, [tens ;) finding that it will not go, we write a cipher in the luoticnt, and bringdown the next figure, making the whole 170. We then seek how often 21 can be contained in 170, ind, finding it to be 8 times, we write 8 in the quotient, and, liultiplying the divisor by this number, we set the product, 108, under the 170 ; then subtracting, we find the remain- ler to be 2, which, written as a fraction on the right hand )f the quotient, as already explained, gives 208^ pounds, for the answer. Km f:.i^. This manner of performing the operation is called Long twision. It consists in writing down the vahole computation. From the above example, we derive the following •" *^ ^.,. RULE. I. Place the divisor on the left of the dividend, separate them by a line, and draw another line on the right of the [dividend to separate it from the quotient. II. Take as many figures, on the left of the dividend; as J.:.l ■x \ : 1 * I > 48 <*■' DITIMIOIf OP 81MPLR NUMBERS. f 10.^1 20,21 contain the diviiior once or more ; seek how many times they contain it, and place the answer on the right hand of the dividend for the first figure in the quotient. III. Multiply the divisor by this quotient figure, and! write the product under that part of the dividend taken. \y. Subtract the product from the figures above, and to! the remainder bring down the next figure in the dividend, and divide the number it makes up, as before. So continue to do, till all the figures in the dividend shall have been brought down and divided. Note 1 . Having brought down a figure to the remainder, if the number it makes up be less than the divisor, write a cipher in the quotient, and bring down the next figure. Note 2. If the product of the divisor, by any quotient figure, be greater than the part of the dividend taken, it h an evidence that the quotient figure is too large, and must be diminished. If the remainder at any time be greater than the divisor, or equal to it, the quotient figure is too small, and must be increased. ^ EXAMPLES FOR PRACTICE. 1 . How many hogsheads of molasses, at 7 pounds a hogs- head, may be bought for 6318 pounds 1 ^718, 903f hogsheads. 2. If a roan's income be 1248 pounds a year, bow much is that per v(eek, there being 52 weeks in a year 1 jina. 24 pounds per week. 3. What will be the quotient of 153598, divided by 29? J^s. 5296^. ft » 4. How many times is 63 contained in 30131 1 jfns. 478^ times ; that is, 478 times, and |^ of another time. 5. What will be the several quotients of 7652, divided by r 16, 23, 34, 86, and 92 1 6. If a farm, containing 256 acres, be worth 1850 pounds, what is that per acre % 7. What will be the quotient of 974932, divided by 3651 Ans. ^71^. 8. Divide 3^228242 pound! equally among 563 men ; bow many pounds must each man receive 1 Jhi8. 5734 pounds. 9. If 57624 be divided into 216,586, and 976 equal parts, what will be the magnitude of one of each of these "* equal parts 1 parts wil It). Ill !.>/ ml 3. ] 4. 1 6. ] 6. ] what Y ^ '^-J? 20, f?l. COHTBACTl'oifll IK DITIIIIOV. 40 times (hc\ and of the guro, and 1 taken, ve, and to dividend, ■•o continue lave been remainder, or, write a figure. »y quotient taken, it i^ ) and must be greater gure is too ads a hogs- hogsheads, bow much Pi per week, led by 29? ». 6296^. of another divided by >0 pounds, i by 3651 i«n; how 4 pouDdf . •76 equal k of these ^M. The inu);nitudu o( one of tUu last of these e9g\"(;. ,., , 10. lluw manv times ducii lOGOtM).^!^ contain .')2151 yinn. :)>iU54il times. 11. The earth in its annual rcvulution round tiaid to travel 5tH>0H.^(MH) miicH ; what is that per hour, there being HTtki houri* in a year 1 , > 12. '^:'if|gf8 0"= how many 1 . l',\. *ofJf^2»= how many 1 14, 9«vV'V^'= **o^ ™*"y ^ 1. CONTRACTIONS IN DIVISION. ■ -I fVhen the divisor ti* a compositb wvmBbb. H 20. 1. Bought 15 yards of cloth for 30 pounds ; how iliiuch was that per yard 1 15 yards are 3x5 yards. If there had been but 5 yards, the cost of V nc yard would be ^=6 pounds ; but as there arc 3 times 5 yards, the cost of one yard will evidently be but one third part of 6 pounds ; that is, |^=2 pounds, ^nn. Hence, when the divisor is a composite number, we may, if we please, divide the dividend by one of the component arts, and the quotient, arising from that division, by the other ; the last quotient will be the answer. 2. If a man can travel 24 miles in a day, how many davit will it take him to travel 264 miles 1 It will evidently take him as many days as264 eontains24. OPERATION. .^i, 24=6X4. 6)264 4)44 , 3. Divide 576 by 48 = (6x6.) 4. Divide 1260 by 63 === (7x9) 5. Divide 2430 by 81. 6. Divide 448 by 56. ■ W,...l . t-' 11 days. 24)264(11 days, Jin*. 24 24 , r, (it! -.f U V-'. .-iV .!) II. 7b dimde by 10, 100, 1000, &c. H ^1. 1. A note of 2478 pounds is owned by 10 men ; what is each man's fl^are 1 f # rONTRACTIO!«t llf nTviHinir, f2l,'w *■ 424|(N) Each man'ii share will he equal to the niimhorof /«n. How many 100 iu 42604 hundreds 1 Jns. 426Tf<) 7. How many thousands :u 4000 1 in 25000 1 S. How many thousands in 6487 thousands 1 jfns.^^^. \K How many thousands in 42863 thousands 1 in 368456 thousands 1 in 96842378 thousands 1 10. How many tens in 40! in 400 1 in 20 1 in 468 1 in 47841 in 346401 " -■ "'": '^ ' ' ' • ■ ''>••• • ■ - ' III. When there are ciphgrs onthe right hand of the divisor, f 3ii{, 1. Divide 480 pounds among 40 men 1 OPERATION. 4!0)1H|0 14 In this example, our divisor, TTnounds ^u ^^^'^ '^ ^ composite number. Impounds, ^is. (10X4=40;) we may there- tore, divide by ane component part, (10,) and that quotient by the other, (4 ;) but to divide by 10, we have seen, is biit 10 cut off the right hand figure, leaving the figures to the left of the point for the quotient, which wedivide by 4, ^nd 1 lie work is done. It is evident, that, if our divisor had bocu 400, we should have cut off 2 figures, and have divi- ded in the same manner ; if 4000, 3 figures, &c. Hence, I his <^eneral Rulk : IVhtn there are ciphers at the right hand Divisitr 3. Divi .H <• tti >?• k t«. •VrPLBMKIIT TO OIVIMIOK. /mi* con en bo cui y be con re will hi It off from en NO ma- Hence, (M), 1000. \y figureM the left of the right. //1 8. 424. e cut off 2 figures to ■ 1/. • 426t4o 01 \ in 4681 in K dtrt«or. lo/ /A* i«or, cut theiu off, and also art manj places in the dividend ; divide the remaining figiireti in the dividend by [the remaining ftgurc '42 ;'i'i'i<- ! li . ' ' "~520367 /Jmainifcr. }^ 4. How many pieces of cloth can be bought for 346500 Jbounds, at 20 pounds per piece 1 , , f r>. Divide 76428400 by 900000. •• . m - > 6. Divide 345006000 by 84000. ' i 7. Divide 4680000 by 20, 200, 2000, 20000, 3000, 4000. j60, 600, 70000, and 80. , .,,, ,,,\; „ .... i " ■ : -.-.\.' / U; ilhl SUPPLEMENT TO DIVISION. (MJESTIONS. , I" '' 1. What ii division ' 3 In what does the;>roe««5 ofdivision con- list .' 3. Division i« the reverse of what .'' 4. What is the number to be divided caWed, and to what does it answer ia multiplication P 5. IWhat is the number to divide by called, and to whnt doon it answer. I&c. .' 6. What is the result or answer called, &c. ? 7 What is the sign of division, and what does it show ? 8. What is the other way of lexpressing division .' 9. What is short division, and ho\v is it pcr- jformed P 10. How is division proved? 11. How is v.tutiplication [proved.^ 12. What are mt«^erjr, or whole nunibtrs ? l.'». What ari- /rac(ton5, or broken numbers .'' 14. Whnt is a mixed number ? 15. When there is any thing left after division, whnt is it called, and [how is it to b« written.' 16. How are fVactions irrtffen >* 17. What I is the upper number called.' 18. — — ihe lower number P 19. How do you multiply a fraction .' 20. To what do the numerator and jthedenominntor of a fraction answer in division ? 21. What islong I division ? SS. Role .' S3. When the divisor is a composite number, how may we proceed .<* 24. When the divisor is 10, 106, or 1000, Ac. how may the operation be contracted.' 25. When there are ciphers at the ri|ht band of the divisor how may we proceed .' M •VrPLCMBlIT TO DlTltlOK. H^i, •«i. < !» EXERCISES. 'I , * V> *f H. 1. An army of 1500 men, having plundered a city,tuok '26'^5000 pounds ; what was each man's share 1 %.-.,. i,.- 2. A certain number of men were concerned in the pay- ment of 18950 pounds, and each man paid 25 pounds ; what was the number of men 1 3. If 7412 eggs be packed in 34 baskets, how many in a basket 1 4. What number must T multiply by 135 that the product may be 505710 1 5. Light moves with such amazing rapidity, as to pass from (he sun to the earth in about the space of 8 minutes. Admitting the distance, as usually computed to be 95000000 miles, at what rate per minute does it travel 1 6. If the product of two numbers be 704, and the multi- plier be 11, what is the multiplicand'! jins. 64. 7. If the product be 704, and the multipIicaDd 64, what is the multiplier'! ^s. 11. 8. The divisor is 18, and the dividend 144 ; what is the quotient 1 9. The quolient of two numbers is S, and the dividend 144 ; what is the divisor 1 10. A man wishes to travel 585 miles in 13 days ; how inany miles must he travel each day 1 11. If a man travels 45 miles a day, in how many days will he travel 585 miles 1 12. A man sold 140 cows for 560 pounds ; how much was that for each cow 1 13. A roan, selling his cows for 4 pounds each, received for all 560 pounds ; how many cows did he sell 1 "'"^ "' 14. If 12 inches make a foot, how many feet are there in 364812 inches 1 15. If 364812 inches are 30401 feet, how many inohes make 1 foot 1 17. If you would divide 48750 pounds among 50 men, how many pounds would you give to each one *! 16. If you distribute 48750 pounds among a number of men, in such a manner as to give to each one 975 pounds^ how many man receive a share 1 ' ' •^18. A BMtn has 17484 pounds of tea in 186 chests ; how OMoy pounds fn«aoh chest 1 • •' - ...^41^ jM*s<'.''j .«i ,!•' t+j V«J'' -■'. -^"'a W»Jt ^ 2-2. MI8CflLLA?rC0US QUKSTIUIfS. 5:1 1!). A man would put up 171S4 pounds of tea into chests containing !M pounds each ; how many chestt> nuiflt he havel 'iO. In a certain towu thcrt! arc ]711) inhabitants, and I'i persons in each house ; how many houses arc there T in each liouse are '2 famihes, how many persons in each family 1 '2\. If '27('A) men can dig a certain canal in one day, how »nany days wouUI it take 4t) men to do the samel How many men would it iake to do the work in io days 1 in '> days I in -^0 daysl 40 days 1 in 120 days? 20. If a carriage wheel turns roimd ()2280 times in run- ning from Quebec to Montreal, a distance of IHO miles, how many times does it turn in running 1 mile 1 u^ns. 'VUi. 21). Sixty seconds make 1 minute ; how many minutes in :t(»0() seconds 1 in 86400 seconds 1 in 004800 seconds 1 in 2410200 seconds. 1 24. Sixty minutes make one hour; how many hours in 1440 minutes? in 1(K)80 minutes 1 in 40;i2n minutes 1 in ^.59()0 minutes ^ ' ' ^ 25. Twenty-fouKpours make a day ; how many days in ^i)S hours ] in 072 hours 1 in 3700 hours 1 20. 'How many times can I subtract forty-eight from tour hundred and eighty 1 27. How many times Ji478 is equal to 4785 1 1 28. A bushel of grain is 32 quarts ; how many quarts must I dip out of a chest of grain to make one half < ',) oi a f»ushel 1 for one fourth (}) of a bushel 1 — : for one j/ns. to the last 4 quarts. I of 48 1 4 ot •.'r of 3458" rt ! f nt" oightli (I) of a bushel 1 21). How raanv is 4 of 20 1 •247 1 }, of 847 1 •,>040300{S'» y//is. to tholusi, 102015324. 30. How many walnuts are one third part ( vV) of 3 wal- nuts 1 \ of (i walnuts 1 '^ of 12 walnuts'? 1 of 30 ? 1- of 45 ? i of 3(10 ? . r- -?- of 478? 31 What is 1 of 4 ? ; of 78 13? ] of 34 5( )320 ? u'/n s , lo the inst , 1 1 52 1 < s? . .| of 20 ? — r- >: of 320 / y//i8. !o the Ua-f, VUmi MISCELT.ANEOI S QUESTIOV.S, Jnvolvuig the prbic'ples of the preccJine; rules. - Xott. The preceding rules, viz. NunT^ratinn. Addiiion E 2 54 MISCBT.TMIfKOirs ({UKHTIONM. IT 2.^ Subtraction, Multiplication, antl Division, are called the JTandamental Rules of yfrithmetic, because they are the foun- dation of all other rules. ] . A HKin bought a chaise for 57 pounds, and a horse for 34 pounds ; what did they both cost ? '2. If a horse and chaise cost {H pounds, aiul the chaise cost 57 poinds, what is the cost of the horse 1 If the horse , cost '24 pounds, what is tlie cost of the cliaise ? i 3. If the sum of '2 numbers be 487, and the greater num bcr be 34H, what is the less number ? If the less number be 139, what is the greater number? - "■■' 4. If the minuend be 7842, and the subtrahend 3481, what is the remainder? If the remainder be 4301, and the ' minuend be 784'i, what is the subtrahend ? *J *13. When the minuend and the subtrahend are given, how do yoii find the remainder? * When the minuend and remainder are given, how do you find the subtrahend ? Wlieu the subtrahend and the remAier are given, how do you lind the nunuend ? When you have the sum of two numbers, and one ot them t^ivcn, how do you find the other 1 ;. Wlii'n you have tiie greater of two numbers, and theii difference given, how do you find tlie less number? When you have the less of two numbers, and their fH^'er- ciicc given, how do you find the greater number? - 5. The sum of two numbers is 48, and a?je of the num- bers is 19; what is tlie other! r,. The s;rcatcr of two numbers is 29, and their difference 1!» ; what is the less number! 7. The less of two numbers is 19, and their difference is H) ; what is the greater ! IVi ' ' 8. A man bought 5 pieces of cloth at 44 pounds a piece ; '.^74 (lo/.on of shoes, at 3 pounds a dozen; 600 pieces o< calico, at (5 pounds a piece ; what is the amount 1 1). A man sold six cows at 5 pounds each, and a yoke of oxen, for 19 pounds ; in pay, he received a chaise, worth 31 pounds, and the rest in money ; how mucii money did he receive 1 H). Wljpt will be the cost of 15 pounds of butter, at ', pence per pound! ' ' ' ^ '* \ ' •> * ■m 4ii •T '23, 24. MI8CELLANKOUS QITB9TI0NS. • >•> 1 1 . How many bushels of wheat can you buy for if^TO shillingH, at H shillings per bushel! U 14. When the price of oiie pound, one bushel, &c. of any commodity is given, how do you find the cost of any number of pounds, or bushels, &c. of that commodity 1 If thft price of the 1 pound, &c. be in shillings, in what will the whole cost be 1 If in pence, what 1 When the cost of any given number of pounds, or bushels, &c. is given, how do you find the price of one pound or bushel, &c. In what kind of money will the answer ho 1 When the cost of a number of pounds, &c. is given, and also the price of one pound, &c. how do you find tbo num- ber of potinds, &c. 1'2. When rye is 4 shillings per bushel, what will be tli'. cost (11 ■ ! ' bushels'! 13 ^ \H pounds of tea cost 173 pounds, (that is Jlo'^O pence; ivnat is the price of one pound"! " * When the factors are given, how do youfind the productT Wlien the product and one factor are Jiiven, how do you find the other factor \ When the divisor and quotient are given, how do you find the dividend ! When the dividend apd quotient are given, how do you find the divisor 1 11. What is the product of 754 and 25 1 15. What number multiplied by 25, will produce 18850' 1(). What number, multiplied by 754, will produce 18850! 17. If a man save 5 pence a day, how many pence would he save in a year, (30.5 days,)! how many in 45 years ! How many cows could he buy with the money, at 74fi pence each 1 , * ,> ' , 18. A boy bought a number of apples ; he gave away ten of them to his companions, and afterwards bought thir- ty-four more, and divided half of what he then had among four companions, who received 8 apples each ; how many apples did the boy first buy ! Let the pupil take the last number of apples, 8, and re- verse the process. j^ns. 40 apples. 19. There is a certain number, to which, if 4 be added, and 7 be subtracted, and the difference be multiplied by 8, and the product divided by 3, the quotient will be 64 ; what is that number ! /Inn. '21 . »=•. *. ■. 21. There is a spot of {ground 5 rods long, and :> i^us vfiTh ; how many square rods does it contain 1 Note. A square rod is a square (like one of those in the annexed figure) meas- uring a rod on each side. ])y an inspection of the figure, it will he seen, that there are as many squares in a row as rods on one side, and that the number of fows is equal to the nui/iber of rods, on the other side ; tliere- lore, 5X'J==l'>j the number ofs(juaros. Ans. 15 square icds. A figure, like A, B, C, D, having its opposite sides equal ijnd parallel, is called a \)araIlelogravi or oblong. 22. There is an oblong fioM, 40 rods long, and 24 rods wide ; how numy square rods does it contain 1 23. How many square inches in a board 12 inches long, and 12 inches broad 1 j^ns. 144. 24. How many square feet in a board 14 feet long and 2 feet wide 1 25. A certain township is six miles square ; hov,- many square miles does it contain 1 • y/ns. 3(). 2(). A man bought a lot of land for 224(> pounds ; he sold one half of it for 1175 pounds at the rate of 3 pounds per acre ; how many acres did he buy 1 and what did it «ost him per acre 1 27. A boy bought a sled for 5() pence, and sold it again Tor 8 quarts of walnuts ; he sold one half of the nuts at 1^ pence a quart, and gave the rest for a penknife, which he sold for 18 pence ; how many pence did he lose by his bar- gains 1 28. In a certain school-house, thtre are 5 rows of desks ; in each row are six seats, and each seat will accommodate 2 pupils ; there are also two rows, of 3 seats each, of the same si xe as the others, and one long seat where 8 pupils may sit; how many scholars will this house accommo- date! ' . ,. - ^j^g po 20. How many square feet of boards will it take for the H 25. mtCELLANKOVt qUEBTIONK. $7 floor of a room \(\ feet long, and 15 feet wide, if we allow \'Z square feet for wa«te 1 Sii. There is a room yards long and 5 yards wide ; how many yards of carpeting, a yard wide, will be sufficient to cover the doors, if the hearth and fireplace occupy 3 square yards 1 31. A board, 14 feet long, contains 28 square feet ; what is its breadth 1 3*2. How many pounds of pork, worth 4 pence a pound, can be bought for 144 penfjel : > > . ^ 33. How many pounds of butter, at 9 pence per pound, must be paid for 25 pounds of tea, at 38 pence per pound ? 34. 4-f5-|-6-(-l-f 8= how many 1 35. 44-34-10—2—4+6—7= how many 1 30. A man divides 30 bushels of potatoes among 3 poor men ; how many bushels does each man receive 1 What is ^ of thirty 1 How many are § (two thirds^ of 30 1 37. How many are one third (^) of 3 1 of 6 1 of 9 1 ■ of 282 % of 45674312 1 38. How many are im thirds (^) of 3 1 — - of 6 •? of 9 1 of 282 1 of 45674312 1 39. How many are i of 40 1 f of 40 1 ^ of 60? fof60? iof80? of 124? of 246876 ? f of 246876 1 ., . 40. How many is ^ of 80 ? — ^of80? — foflOO* 41. An inch is one twelfth part (^) of a foot ; how many feet in 12 inches ? in 24 inches ? in 36 inches ! in 12243648 inches? 42. If 4 pounds of tea cost 128 pence, what does 1 pound cost ? 2 pounds 1 3 pounds ? 5 pounds ? 100 pounds ? 43. When oranges are worth 4 pence apiece, how many, can be bought for 1464 pence ? ^ ' ' , ' " 44. The earth, in moving round the sun, frarels at the rate of 68000 miles an hour ; how many miles does it travel in one day, (24 hours ?) how many miles in one year, (365 days?) and how many days would it take a man to travel this last distance, at the rate of 40 miles a day ? bow many years ? Jlna. to the Jasf, 40800. 45. How many pence can a man earn in 20 weeks, at 35 p«nc0 per day, Sundays excepted ? 46. A mtn married at the age of 23 ; ht lived with hit r>s COMPOVIfD NUMBERS — RKDVCTION. IT 20,37. r wife It years; she then died, leaving liira a daughter, VZ years of age ; H years after the daughter was married to a nan 5 years older than herself, who was 40 years of age w'-hOH the father died ; how old was the father at his death * ■.'»■.."•■' ►.-j'fi:, t'lu. ■'■ i.:M' '-: ■ ./■,->•,' uins. iHi. 47. There is a field 20 rods long, and 8 rods wide ; how iTiany square rods does it contain ? Ana. 160 rods. 4H. What is the width of a field, which is 20 rods long^ and contains 160 square rods. « 49. What is the length of a field, 8 rods wide, and con- taining 160 square rods 1 50 What is the width of a piece of land, 25 rods long, and containing 400 square rods ? ■'U- .■..! / ■^} ,#■»' IT ^/6, A number expressing things of the same kind is called a simple number ; thus, 100 men, 56 years, 75 cents, are each of them simple numbers ; but when a number ex- presses things of different kinds, it is called a compound num- ber ; thus, 46 pounds 7 shillings and 6 pence, is a compound number ; so 4 years 6 months and 3 days, 43 dollars 25 cents and 3 mills, are compound numbers. Note. Different kinds, or names, are usually called dif- ftrent denominations . Tl curn Its C( Scoti dcnoi pence nomii ney, pernj fourth be see ratio exhibi for pa] I 2f ft % 37. In this Province as in England, money is reck- 'oned in pounds, shillings, pence and farthings. In the United States, money is reckoned in dollars, cents and mills. These are called denominations of money. Time is reck- oned in years, months, weeks, days, hours, minutes, and «econds, called denominations of time. Distance is reck' oned in miles, rods* feet, and inches, called denomina|^ons of measure, &c. ,.. ,,,,., , , n ^' The relative value of these denominations is exhibited in tables, which the pupil sauat commit to memory, {n ^- ^ 27. HALIFAX rURHBlVCY. 59 HALIFAX CURRENCY. The present currency of Lower Canada, is called Halifax currency, having been introduced into thi« Province, after its cession to Great Britain, by France, in 1763, from Nova Scotia. The denominations are the same in name as the denominations of English money, i. o. pounds, shillings, pence, and farthings ; and the ratios of the different de- nominations to each other are the same as in English mo- ney, i. e. the shilling is one twentieth of the pound, the perny one twelfth of the shilling, and the farthing one fourth of the penny. In value they are different, as will be seen in the H upon reduction of currencies ; where th« ratio of each to the other, and of both to Federal Money is exhibited, with the method of ascertaining them in practice, for particular sums. 2 farthings (qrs.) uiako 1 half-penny, marked .jd. 4 '« " " 1 penny. " d. 12 pence " 1 shilling, •* s. 20 shillings " 1 pound, ' " £. Note. Farthings are often written as the fraction of a penny ; thus, 1 farthing is written ^d., 2 farthiiigs, ^d., 3 farthings, -^d. It will be proper to insert here a list of the gold and sil- ver coins, made current by law in the Province of Lower Canada, with their several weights and values annexed. J J GOLD COINS. '-■■ o:*i- i u. 'f 'H English, J/merican, and Portuguese. DBIVOMINATIOir. WEIGHT. VALUK i ■ • . ;.-J //v- >' f^ -''■■■■ ■ pwts. V«- £. ». i. A Guinea, I "nf ifi '^ ■ 5 6 1 3 4 A Half Guinea, - 2 15 11 8 A third Guinea, ^^w*^ 1» - .. 1 18 7 9 A Johannes, - ,• JsiU <^i 18 4 A half Johannes, - \fi '^'^ 9 2 A Moidore, --mi-jvI 6 18 1 10 An Eagle, ij ^Wiawv ^i^vt-^ 11 a 2 10 A half Eagle, /--^fctr ^'<- | 5 15 1 5 e nf,. r' i IJ, lIALirAX CUMRKNCY. GOLD COINS— CONTINUED. French and Upaniah. IT 27. DKNOMINATIOIf. A Doubloon, ., •. .,.- A half Doubloon, A Louis d'Or, coined before I A Pistole, coined before llO'i. The 40 Francs piece, coined before iT9-2. The 40 Francs piece, coined since 1792. The 20 Francs piece, coined before 1792 The 20 Francs piece, coined since 17i)2. /93. WEIGHT. puis. 17 8 12 r> 4 4 4 8 6 8 6 4 3 4 3 V ALUK . 7. rfT 14 6 17 3 3 1 1 2 18 8 3 1 10 2 1 16 2 18 I 18 1 SILVER COINS. A Crown, 5 6 An English shilling, 1 1 A Dollar, 5 A Pistareen, - - - /rt^. >• .»j; 10 A half Pistareen, - - - - 5 The American Dollar, 5 ^ French piece of 4 livres 10 solsTournois, 4 2 A French piece of 6 livres, 5 6 The piece of 5 Francs Tournois, coined ? since 1792, S 4 8 The French piece of 36 sols Tournois, 1 8 The French piece of 24 sols Tournois, 1 1 The statute provides, that '^all the higher and lower de nominations of the above gold and silver coins, shall alsc pass current, and be deemed a legal tender in payment o; all debts, and demands whatsoever, in this Provtnoe, in thej same proportions, respectively." 7\oo pence farthing is allowed for every grain over, ori under the legal weight, on English, American, and Portu guese gold ; and two pence and one fifth, for every grain ovei | or under the lega'i weight, on French and Spanish gold. Payments in gold above twenty pounds shall be made in buQc, where one of the parties making or receiving the IT 21 H 27. StRDUOTION. 61 ▼ALUS. ) 14 6 17 3 2 8 18 3 I 10 2 1 16 2 18 1 18 1 5 6 1 1 5 10 5 5 4 2 5 6 1 1 8 8 1 id lower de , shall aisc payment o: rinoe, in the] lin over, or] and Portu y grain ovei ! lish gold, be made in eeiving the same Hhall require it ; Enj3;liffh, American, and Portuguese gold at eif^hty nine 8hillinf:,8 per ounce TVoj/ ; French, and Spanifili ^old at eigfUy-aeven shillings and eight pmce half- penny per otince Troy : — and a deduction Khali be made of ^ l^rain Troy, for each piece of gold coin ho weighed, as a compensation t(» the receiver for the loss that may accrue in afterwards paying away the same hy the single piece. In copper coin no person is obliged to receive mure than one shilling at one payment. •. t How many farthings in 1} How many pence in 4 far- penny 1 in 2 pence 1 things 1 in 8 farthings 1 in '\ pence ? in 6 in 12 farthings 1 in pence 1 in H pence? — 24 farthings'? in J?2 far- in 9 pence 1 in 12 pence ? things ? in 36 farthings ? in 48 qrs. 1 How many shillings in 48 qrs. ? in in 1 shillins: 1 m2 shillings 1 „ . f, , ., 96 qrs. 1 How many pence in 2 shil-i ^ ^ « , lings 1 in 3 s. 1 inj How many shillings in 24 4 s. 1 8 8.1 in 6 s. 1 - in pence ? in 2 in 48d. 1 in 10 s.? — shillings and 2 pence 1 in 96d. 1 ■ in 2s. 4d. 1 in 26d. ? in 4s. 3d. 1 . : lin 28d. 1 in 42d. 1 in 2s. 3d in 36d. 1 — in 72d. ? — in I20d. 1 - in27d. 1 — in 30d. I — in 51d. 7 How many shillings in 1 t ;i)ound 1 in 2J^. 1 How many pounds in20shil- in {i£. 1 in 4ofc'. ? lings 1 in 40s. 1 in in 4£. 6s ? in ii£. 8s. lOOs. ? in 80s. ? in in {i£. 10s. ? in|86s. 1 in 128s. ? — I 2£. 158. ? . I70h. ! in 55s. 1 in ,»*5 The changing of one kind, or denomination, into another kind, or denomination, without altering their value, is call- ed Reduction. (H 27.) Thus, when wo change shillings into pounds, or pounds into shillings, we are said to reduce ' them. From the foregoing examples, it is evident, that, when we reduce a denomination of greater value into a de- nomination of less value, the reduction is performed by mul- tiplication ; and it is then called Reduction Descending. But when we reduce a denomination of less value into one of greater value, the reduction is performed by division; it, h then called Reduction j^scending. Thus, to reduce pounds F ,t,iwv ,«J RBDVCTtOir. f 28. I) . f hH H toflhillingH, it Im plain, vte must multiply by 30. And again, t<) riMhice »hillinp;« to ponndti, wo must divide by 20. It follows, thort'fore, that reduction descending and ascending reciprocally prove each olher. 1. In nX. V,h. <)^d. hoW| ^2. In 10071 farthings, how loany fartiiing.s ? OPKRATION. <£. 8. d. qrs. 17 13 a 208. * ' 35JJs. in \1£. 13». I2rf. 4242J. 4g. 16971 given pence, (O.) ^J^stly,^^ ^^^^^ The last quotient. because 4 larthmes make H^^^^ ^.^^ ^,,^ ^^^^^^ ,^. mainders, 13s. 6d. 3qrs. con- igs penny, we multiply the pence' (4242) by 4, adding in the!" . ", ,°' " •"; ■ ^ c \x ' /5v \xT stitute the answer, given farthmgs, (3.) We then find, that in 17.£. 13s. '6^d., are contained 16971 •arthinss. ilM; Note. In dividing 353s. by 20, we cut off the cipher, kc, S.9 taught IT 22. 7 28. The process in the foregoing examples, if care- I'ully examined, will be found to be as follows, viz. To redtice low denominationn to higher, — Divide the lowest denomination given by that 7b reduce high denominaiiona to lower, — Multiply the high- est denomination by that num- i» If 28. again, h rending r», how ?2S. ■ ■DVCTlOlf. ^'M' 69 reduced dc them 4 far- There s divided is 4242 aindcr of gs, of the dividend, he pence ;ing them e shillings ;ing them t quotient, iveral re- iqrs. con- r 353s. by pher. &c., 8, if care- 56' ominationf the lowest n by that 1 ber which it takett of the nextnuiiiber which it takeit oi the less to make 1 of (his higher, same to make 1 of (he nent (increaHing (he product by the higher. Proceed in the Hami- number given, if any, of that'manner with each succecdinic less denomination.) Procec i'» i« «u til you have brought it to the i vn i ^* I denomination required. . > . r In the two examples, from which the above general iul«» are deduced, the denominations are pounds, shillings, pence and farthings, considered as in Halifax Currency ; but it is obvious that these rules can be applied tu all currencies where the denominations are the same ; or to currencies in which the denominations are difiereat i and in general to all compound numbers. - . -^ - ^ EXAMPLES FOR PHACTICK. I Reduce 20<£. 14s. 2d. to pence. (C 24<£. to farthings. 66,£. 6s. 6d. to pence. 158£. to farthings. " 1234;^. 15s. 7d. to farthings. 337587 farthings to pounds, &c. f ^ns. 351^. 13s. Ud, 3q. 1185388 farthings to poundb, &c. utna. 1234^:. J 58. 7d. jfna. 4070. jfns. 23040. Jfna. 15913. jin$. 15168U. ifns. 11853H8. 10. Reduce 32^. 15$. 8d. to farthings. „|» . 12. In 29 guineas, at 1£, 3s. 4d. each, bow many qrs. 1 14. Reduce $163, at 6s. each, to pence 1 16. In 15 guineas, how many pounds 1 Note. We cannot reduce guineas directly to pounds, but we may reduce the guineas to thillinga, and then the shil- lings to pounds. 11. Reduce 314,72 tarthings to pounds. ^.it) J, .,>,, 13. In 38976 farthings, how many guineas % 15. Reduce 11736 pence ta dollars. 17. Reduce 21• i n i In 1951 sous, how nwiny livres 1 i ,, ., , . . .. m In 10 livres 6 sous 9 deniers, how many denieri^ How many pounds currency in 96 Uvresi ,,„ , , I IS ^ 1 FEDERAL MONEY. IT d9* Federal money is the coin of the United States. The kinds, or denominations, arc eagles, dollars, dimes, centf, and mills. TABLE. ID mills - - • are equal to - 1 cent. 10 emts, (=100 mills,) 'isi* - - - =1 dime. 10 dimes, (=100 cents=1000 mills,) - =1 dollar. 10 dollars, (=100 dimos =1000 eenta =10000 mitia) ^=1 eagle.* Sroir. This character, $, placed before a number, shows it to express /ed«ra2 money. As 10 mills make a cent, 10 cents a dime, 10 dimes a dollar, &c. it is plain, that the relative value of mills, cents, dimes, dollars and eagles corresponds to the orders of units, tens, hundreds, &c. in simple numbers. Hence, they may be read either in the lowest denomination, or partly in ft higher, and partly in the lowest denomination. Thus : ilS •1.^ S2JJ sAii«/-vy i'» .;■ t 3 IS 5 2 may be read, 34652 mills; or 3465 cents and 2 mills ; or, reckoning the eagles tens of dollars, and the * The eagi« i%» gold com, iho dollar and dime are tilvfr ooinn, (he ceat is ?:MM»;Vi ; that is, 'M (iullarN (m cents 'i inilld, as above. As HM) cents make a dollar, tho centn may be any number Irnm I to'.M), often ro- quirin;; two lifriiros in express them ; lor this reason, (mh» places are apfirofiriitetl to cents, at tho right band of the • point, and il the number of rents be less than /♦/., requiring * but one figure to express thorn, the tcn'n place must be filled ^ with a cipl er. Thus, 'i dollars and cents arc written 'Z'Wi lU mills make a cent, and consequently tho milh i.ever e „- 4;cd 9, and aie always expressed by a single figure. Only #• )/te place, tlioroforc, is appropriated to mills, that is, the ilaco immediately following cents, or the third place (i 'n [the point. When tlierc are no cents to be written, it is ev- ident that we must write tioo ciphers to fill up the places v>f ^ :ents. Thus, '2 dollars and 7 mills are written 'i'tW* . Six Jonts are written, 00, and 7 mills arc written '007.. ^ , > Note. Sometimes 5 mills s=j^ a cent is exprensod frac-' *^j^ ionally : thus, '125 (twelve cents and five mills) is .x- fe '*' )ressed 12^, (twelve and a half cents.) ♦^^^ *"* / 17 dollars and 8 mills are written, I7'(Mlh * 4 dollars o cents, 4'0.5 '.-I - * . 75 cents, - - - - - - -, , «76 „»,.j * , "^: 24 dollars, 2i« ^'; ^ 9 cents, '{;•' * ^" 4 mills, ----..- *OiM . 6 dollars 1 cent and :J mills, - (i'OUt ' * Write down 470 dollars 2 cents ; 842 dollars H\ crv^ md 2 mills ; 100 dollars, I cent and 4 mills ; 1 mill ; 'J nils ; 3 mills ; 4 mills ; 4 cent, or 5 mills ; I cent .Tnd I Imill ; 2 cents and '.\ mills ; six cents and one miil : sixt% Irents and one mill ; four dollars and one cent ; three c^nts ; jfive cents ; nine cents. t The character iwd for ihc^eparatrix. ift the " Scholars' Aii«hm»>lic,"' wa» jtliP comma ; ihe comma inyerte,! \ cents 1 — in 9 cents 1 — in 10 cents 1 — in 30 cents 1 — in 78 cents ^ — in 100 cents, (=1 dollar) 1 — in 2 dollars ? — in 3 dollars 1 — in 4 dollars 1 — in 484 cents 1 —in 503 cents 1 — in 1 cent and 2 mills 1 — in 4 cents and 5 mills'? How many cents in 2 dollars ? — in 4 dollars .' — in 8 dollars ? — in 3 dollars and 15 cents 1 — in 5 dollars and 20 cents ? — in 8 dolli.rs and 20 cents ? — in 4 dollars and cents ? How many dollars in 400 cents? — in 600 cents? — in 380 cenls 1 — in 40765 cents? How many cents in 1000 mills'! How many dollars in 1000 mills 1 — in 3000 mills? — in 8000m.ills? — in 4378 mills 1 — in 840732 mills? , . = ..: «. , .. . .* ' As there are 10 mills in one cent, it is plain that cenls are changed or reduced to mills by multiplying them by 10, thai is, by merely annexing a cipher, (IF 12.) 100 cents make a dollar ; therefore dollars are changed to cents by annexing il ciphers, and to mills by annexing 3 ciphers. Thus, 10 dol Jars = I.GOO cents = 10000 mills. Again, to change mills back io dollars, we have only to cut off" the three ri^ht hand figures, (H 21 ;) and to change cents to dollars, cut off the two right hand figures, when all the figures to the left will be dollars, and the fi/^ures to the right, cents and mills. Reduce 34 dollars to cents. - v.; Am. 3400, Reduce 240 dollars and 14 cents to cents. jins. 24014 cents. Reduce $748' 143 to mills. Reduce 748143 mills to dollars. Reduce 3467489 mills to dollars. Reduce 48742 cents to dollars. >^ Reduce 1234678 mills to dollars. Reduce 3460^176 cents to dollars. Reduce |4867'467 to mills. Reduce 984 mills to dollars. Reduce 7 mills to dollars. Reduce $'0114 to mills. ' Reduce 17846 cents to dollars. Jns. 748143 mills. Jns. $748' 143. Jns. 3467'489. Jina. |487'42, Ann. % '984 Jna. I 007 . IV.VK' w »aidto b the .sian Silvci iiii'li<'d t Tlie !>( V, at • \ '2 cents ? - in 9 8 cents ^ -^ in a Hn 50n 5 mills l ! — in o dollars — in 4 ints ? — cents in - in 3000 1. -- in cents arc y 10, that ts make a iinexing il 18, lOdol- ange mills right hand cut oflF the le left will mills». ^ns. 3400. 1014 cents. ^43 mills. $748' 143. 3467«489. .. |487'4'i. ^ V J* ♦ , 1 1 . « i,*-." Jr. . »' ■ns. I '084 'n*. I OOT ? 31 f * 1^^ W* REDACTION. 67 Reduce 9^4;?21 nnts to milU. v ' *♦ t Reduce OfH?} cc.its to dollars. i//i». ?06«17^. Reduce "JiMiij ci.nts, 503 cents, 106 cents, 921 j^ ceols, .'>00 cents, 7i(li <*ents, to dollars. Reduce 8t37.'>3 n ill.s, 9(i!)00 mills 6042 mills, to dollars. TROY WEIGHT.* ^[31. It isostahlisiicd by law, that the pound Troy, with lis parts, multiples, and proportions, shall be the standard weight for weifjhing gold* and silver in coin or bullion, drugs, and preciotis stones. The denominations of Troy weight are pounds, ounces, pennyweights and grains.. TABLE. •24 grains (kgrs.) make 1 pennyweight, marked pwt. 20 pennyweights - 1 ounce, oz. 12 ounces 1 pound, - lb. t I. How many grains in a' 2. In 19680 grains hour silver tankard weighing 3 ib.jmany pounds, &c. r> oz. 1 3. Reduce 210 lb. 8 oz. 12 pwts. to pennyweights 1. In 50572 pwt. how ma- ny pounds 1 5. In 7 lb. 11 oz. 3 pwt.j 6. Reduce 45681 grains to 9 grs. of silver, how many pounds. , . .. ; grains'! I APOTHECARIES' WEIGHT. N Apothecaries' weightf is used by apothecaries and phy- sicians, in compounding medicines. The denominations arc pounds, ounces, drams, scruples, and grains. TABLE. ' ' ' . 20 grains, (grs.) make 1 scruple, marked 3 scruples - - - 1 dram, - - - 8 drams - - - - 1 ounce, - 12 ounces - - - - i pound, - - - 5. ft, a *Ttie fineness or gold is tried by fire, and isreckone'i in carats, by wbieb i.s understood (he 21ih partof any quainily ; ifii lose nothing in the inal, it i» kaidto be 24 caruis (ine; it it luso 2 caraia, it is then 22 carats fine, which is (he .Htandard for gold. Silver which abides the lire without loss is said to be 12 ounces fine. The '\iudHrd forsilver coin is 11 oz. 2 pwts. of fine silver, and 18 pwis. of coppei iiiel (i>d together. 1 1'lie pound and ounce apothecaries' weight and (he pound and oviio* I'rt y, are the same, only differently (/tvtWf<^, and tubdivided. w -**♦ ft^ RKDr€TIOI». * IT 31 r 7. In 91b- H5. I 3. 2 3. 19 grg., how many grains. H. Reduce 55799 gre. to pounds. "* AVOIRDUPOIS WEIGHT.* it is established by law that the potind AvoinUipuii» trith its parts, &c. siiall be considered as the standard for weighing every thing commonly sold by Aveight, except those articles, in weighing which, Troy weight is used. The denominations are tons, hundreds, quarters, pounds ounces, and drams. , TABLE. 16 drams, (drs.) make 16 ounces . - . - - '28 pounds - •j4 quarters - - - - 120 hundred wcisrht - - 1 ounce, - marked - oz. 1 pound, ib. I quarter, qr. 1 hundred weight, - - cwt. - - ... T. 1 ton, Note I. In this kind of weight, the words prons and net are used. Gross is the weight of the goods, together with the box, bale, bag, cask, &c, which contains them. Net weight is the weight of the goods only, after deducting the weight of the box, bale, bag, or cask, &c., and all other al- lowances. ' Note 2. A hundred weight, it will be perceived, is 112 lb. Merchants at the present time, in the principal sea ports of tfie United States, buy and sell by the 100 pounds. 0. A merchant would put 10. In 470 boxes of raisins, 109 cwt. qrs. 12lb. of rais- containing 26lb. each, how ins into boxes, containing 26 many cwt. 1 lb. eac' ; how many boxes ..«ij.t: >««:.' f will it require 1 11. In 12 tons, 15 cwt. 1 qr, 19lb. 6 oz. 12 dr. how many drams'! ^ ' ^ '^* ' IIJ. In 28lb. avoirdupois, how many pounds Troy 1 12. In 7323500 drams, how many tons 1 14. In 34 lb. Ooz. 6 pwt. 16 grs. Troy, how many ipounds avoirdupois \ ,^ ] 4 nails, 4 quart > 3 quart [5 quart [6 quart 16. i !how m 18. i ^many ] Note. Lon ^things, Thede rods, y • 175 o« Troy=192 02. avoirdupois, and 17611). Trover 1441b avoirdii- poit, 1 >b. Troy=6760 grains, and 1 lb, avoirdupoit-^7000 grains Troy. t ^r 31. gr«. to >ir(lupoio idard for , except is used. , pounds oz. rb. qr. cwt. T. and net her with m. Net cting the I other al- isll2tb. a ports of s. of raisins^ ach, how ; 'A ' anas, how . 6 pwt. w many ! llbavoiniti- mi 131, /' nviovcTion. CLOTH MEASURE. eo Cloth measure is used in seUing cloths and otTier s;oo;t?li|?.» J»JS_ .. ?■■; LONG MEASURE, t ♦ Long measure is used in measuring distances, or other things, where length is considered without regard to breadth. The denominations are degrees, leagues, miles, furlongs, rods, yards, feet, inches, and barley-corns. TABLE. 3 barley-corns, (bar.) make 1 inch, - marked - in. 12 inches, 1 foot, ----- ft. 3 feet, 1 yard, - - - - iJ. fd. 5^ yards, or 16^ feet, - 1 rod, perch, or pole, r. p. 40 rods, or 220 yards, 1 furlong, - - . . - fur. 8 furlongs, or 320 rods, - 1 mile, - - - - ^ - M. 3 miles, 1 league, - - - - L. 60 geographical, or 69^ ? , , - I'lt <^ .*>T. statute miles, W I ' *'^«'*"' " " ^^S- or» 360 degrees \^ K^eat circle, or circumfoi- ' --..^..Mi^ (■ ence of the earth. ■ f 70 ■ KDUCTION. \ MM It 18 establif-hed by Uw, that the Parts foot with its parts, tu, shall .b« the standard measure cf len£;th, fur measuring l&Dd, wood, timber, stone, masons', carpenters', and join- ert' work. The Englihh foot may be used when there is a special contra<^ for it. ,, , , . v TABLE. 12 lines m^ke 1 inch. 3 12 inches 6 feet - foot, toise. toises make 10 rods 84 arpens - - 1 French foot = l^xf English feet. rod. arpent. league. 20. How many barley-corns will reach round the globe, it being 360 degrees 1 Note. To multiply by 2 is 21. In 475;5801600 barley corns, how many degrees ! Note. The barley-corns be to take the multiplicand 2,ing divided by 3, and that times ; to multiply by 1 is to'quotient by 12, we have take the multiplicand 1 time ;|l3210(>500 feet, which at? to to multiply by ^ is to take be reduced to rods. We can- j. the multiplicand half a time, that is, the half of it. There- fore, to reduce 360 degrees to statute miles, we multiply first by the whole number, 69, and to the product add half the multijtlicand. Thus r ^)360 3240 2160 180 half the multiplicand. 25020 statute miles in dOO''. 22. How many inches from Quebec to Three Rivers, sup- posing it to be 90 miles t. 24. How many times will not easily divide by 16^ on account of the fraction ^ ; but 16^ feet = 33 half feet, in 1 rod; and 132105600 feett=: 264211200 half feet, which, divided by 33, gives 8006400 rods. ' ■ Hence, when the divisor is encumbered with a fraction, ^ or ^, Sic, we may reduce the divisor to halves or fourths &c., and reduce the dividend to the same ; then the quo- tient will be the true answer. 23. In 30539520 inches, how many miles 1 ,*».. f'» 25. If a wheel 16 feet 6 in. a wheel 16 feet and 6 inches in circumference, turn round in circumference, turn round,19200 times in going from in the distance from Quebec Quebec to St. Annes, what is to St A ones, supposing it to be 60 miles 1 c* '*1. t ; .^j^a the distance 1 •- 32 26. Ir pens, ho many toi Squar JLhinji;, w inatiuns Hnchcs. ir S'Z. tt rcqtiin .K^ yard ii fnake 3 %Bake 2 JP teet W! ^ill clea 3 font 344 squi 1 12 1 ®^ ' 1 9 squ: 1 ''^ pO^ squi sq 40 squ 4 ^40 Note. n lengt nches ii HT) squ # « V, . r jk: ■ BDCCTIOir. »9 n m. In 08 Icaprues, 43 art 27. In 7000 fe«>t how msDj pens, how many feet 1 huw rods 1 how many arpens "^ njaoy toises \ liow many rods'i fT barley - jgrees 1 -corns be and that we have lich at? to We can- by 16^ on ion ^ ; but feet, in 1 00 /«rft=: ety which, s 8006400 1 divisor is \ fraction, lay reduce i or fourths >e dividend 1 the quo- 110 answer. i 20 inches, 3 feet 6 in. urn round ;oing from es, what \% LAND OR SQUARE MEASURE. Square measure is used in measurinjj land, and any other Ihing, where length and 6read/A arc considered. The denom- inations are miles, acres, roods, perches, yards, feet anJ inches. \ IT 3:i. 3 feet in length make a yard in long measure ; lut it requires 3 feet in length, and 3 feet in breadth, to maW % yard in square measure ; 3 feet in length and 1 foot wide, ^ake 3 square feet ; 3 feet in length and 2 feet wide, ■lake 2 limes 3, that is, (i square feet ; 3 feet in length and JP teet wide make 3 times 3, that is 9 square feet. This |iirill clearly appear from the annexed figure. 3 foot— lynnl. It is plain, also, that a square foot, that is, a square 12 inches in length, and 12 inches in breadth, must con- tain 12X1^2=144 square inches. TABLE. 1 square yard. ^ 1 square rod, C perch orpole 144 square inches=12Xl2 ; that is, 12 inches in length and 12 inch- ^ make 1 square foot. 4 es in breadth, M square feet=:3X3 ; that is, 3 feet 7 % in length and 3 feet in breadth ) fO} square yards=5AX5^. or 272^ 7 ^ square feet=16AXl6j - - S '40 square tods, - 1 rood. 4 roods, or 160 square rods, - - 1 acre. •" '■ MQ acres, ..... , . 1 square mile. yote. Gunter's chain, used in measuring land is 4 rods In length. It consists of 100 links, each link being 7^ jnches in length ; 25 links make 1 rod, long meaiure, anc^ s5 square links make 1 tquare rod. f # ' i'i ' k-< 72 BEOUCTIOn. ^ 31,32 '!,t. I I '! ii 'f*^ u U^' '? %■?•'«. FR1ENCH SQtJARE MEASURE. 144 SMjarc inches make 1 Kqiiar? (uot. 30 - feet - - 1 toi:( . 9 - toises - - ] rod, iOO - lodu - 1 arpr ii». 7056 - arpcis - I Iea<;\i '. (J25l)0 french fe-t cr=712t?!5 Engl .h (eet .1^ Hflducc H) »eafrut^ to feet, to toises, to rods. ^ Reduce iH701.i!4 1 feet to toises, to to Is, to a; pew, lo leagues, •^S, In n acres 3 roods I2| 2(>. In 7-64.'i: sq'^re ftot, '.«-;r* • t^- OJLr,'9r "'iti-ufr [ 1 ■ ■■■(,uir.>f. ,M SOLID OR CUBIC MEASURE. - 'J »■'■ Solid or cubic measure is used in measuring things that have length, breadth, and thickness ; such as timber, wood stone, bales of goods, &c. The denominations ^re cords tons, yards, feet, and inches. IT 38. It has been shown, that a square yard contain* 3X3 ==9 square feet. A cubic yard is 3 feet long, 3 feet wide, and 3 feet thick. Were it 3 feet long, 3 foet wide and one foot thick, it would contain 9 cubic feet ; if 2 feet thick, it would contain 2X9= 18 dubic feet ; and, as it i> f Nl^ r> ^ 31,32 f 33. RBDVOTIOH. I yd— 3ft lon^. ive 776457 divided bj he divisor to the low- ontaincd in le dividend to the same hown 1131, 17,600 sq. uare miles ^ l40 acres tc I'r.fr < ■ ■ things that oher, wood. !^re cords rd contaiD^ long, 3 feet J foet wide it; if 2 feet ind, as it i:> U feet thick, it does contain 3X^ = 27 cubic feet. This will clcarljr appear from the an nexed figure. It is plain, aJHO, that a cubic foot, that is, a solid 12 inches in leuf^th. 12 inches in breadth, and 12 inches in thickness, vvill con- tain l2Xl2Xl''i=n28 solid or cubic inches. I make 1 solid foot. 1 solid yard. 1 ton or load. I cord of wood. '^m- TABLE. 1728 solid inchcs,=12X 12X12, that is, 12 inches in length, 12 in breadth, 12 in thickness, 27 solid feet,=3X3X3 - - - 40 feet of round timber, or .50 > ♦ feet of hewn timber, - 5 , 128 solid feet,=8X4x4, that is 8 feet in length, 4 feet in pridth, and 4 feet in height, Ao/«. What is called a cord foot, in measuring wood, in 6 sulid feet; that is, 4 feet in 'ength, 4 feet in width, and foot in height, and 8 such feet, that is, 8 cord feet make cord. FRENCH SOLID MEASURE. 17!. olid inches make 1 solid foot. 216 - - feet - I toise. 1000 French feet = 1218' 1864^2 English feet. 32. Reduce 9 tons of rouad >13. In 622080 cubic inches mbcr to cubic inchvs. how many tons of round tim- herl ;J5. In 592 solid feet of wood, how many «ord feet 1 36. Reduce 64 cord feet of 37. In 8 couis of wood, how ood to cords. many cord feet ^ ;J8. In 16 cords of wood,' 30. In 2048 solid feet of ow man> cor%i feet 1 how wood, how many cord feet; how many cords 1 41. In 834692773 inches how many feet 1 how manf toises 1 34. la 37 cord feet of wood, w many solid feet 1 lany solid feet 1 40. In 12 tones how many inches 1 74 REDDCTIOW, V33. i' I \ WINE MEASURE. It is cstaMiKVo'l by law that the wine gallon Mrilh iU parts, &c. sliiill be (he Rtandard liquid inea»ure, for mcasur- inf( wine, cir, heer, and all other liqiiidH conunonly Nuld by gauge, or measure of capacity. The denoininutinns are tuiiK, pipes, hogiitieads, barrels, gallons, quarU, pints, and gills. TABLE. 4 gilts (gi.) ' make - - 1 pint, marked Pt 2 pints - - 1 (piiiri, - qt. 4 quarts - - ■> i gallon, jjal bar. 31 J gallons ' - - 1 barrel, Ofl gallons - - - 1 hogshead, hhd. ^ 2 hogsheads - - - 1 pipe. P. 2 pipes, or four hogsheads 1 tun, T. M^ote. A gallon, wine measure, contains 231 cubic inches. 42 Reduce 12 pipes of wine 43. In 124I1H) pints of wine. to pints. how many pipes ? 44. In 9 P. 1 hhd. 22 gaU. 45. Reduce 39(KJ2 gills to 3 qts. how many gills 1 pipes 1 46. In a tun of cider, how 47. Reduce -^52 gallons to many galluiisl tuna. ALE OR BEER MEASURF. Ale or beer measure is used in measuring ale, beer, and milk. The denominations are hogsheads, barrels, gallons, quarts, and pints. TABLE. make 1 quart, marked qt. 1 gallon, - . - gal. 1 barrel, - bar. 1 hogshead, - hhd. A gallon, beer measure, contains 282 cubic inches. 2 pints (pts.) 4 quarts 'Mi gallons - 54 gallons - Piote. i 48 Reduce 47 bar. 18 gal. of ale to pints ''t 50. In 29 hhds. of beer, 1|0W many pintgl 49. In 13680 pints of ale, how many barrels 1 51. ReduLe 12528 pinti to hogsheads 1 f 83. with iU mcasur ily Null) ions aru ut8, and HfW , beer, and s, gallons, d qt. g*l bar. hhd. ibic incheft. inta of ale, 38 pints to RBDrOirioii. K DRY MEASURE. ^ n\ pt ch. Dry ineaiinrc is used in meaituring all dry ^oods, such as Krain, fruit, routs, («a)t, coal, &c. The denouiinationi are chaldruns, bushels, pecks, quarts, and pints. . - ^' . TABLE. 2 pints (pts.) make - 1 quart, - marked H quarts, 1 peck, . . - - 1 pecks 1 bushel, . . - - 'M\ bushels 1 chaldron, . - - Note. A gallon dry measure, contains 2G8| cubic inches. "^ A Winchester bushel is 18^ inches in diameter, 8 inches deep, and contains 'il^Oif cubic inches. It is estabhshed by law, that the Canafa Minot, vrith its parts, multiples, and propurtions, shall be the standard in Dry Measure. ^ ■■•#' 1 put = 116'94589 English cubic feet. ' '^ 20 pots make 1 minot. ' ''^ OLD MEASURE. ^ 16 litrons - make - 1 . . . boisseau. i] boisscaujt - - - 1 . . - . minot. ' 2 minots - w? ^-^- 1 - - - - mine. '\^ . 2 mines -^ * *i™- 1 - ^* jp. - setier. 12 setiers -... !-.._ muid. 40 French cubic inched = 1 litron. The standard measure for the sale and purchase of coal, for this Province, is the chaldron of 36 minots, each minot to be heaped up. -'-mh' -inj^. • i^ff^- -i* iauiri^i- 52. In 76 bushels of wheaf how many pints 1 54. Reduce 42 obaldrons of coal to pecks. ^ p^ ' 53. In 480D i^to, how ma- ny bushels 1 55. In 6048 pecks, bow ^ imany ebaldrons 1 The denominations of time are years, months, we«kt, days, hours, misyites, and seconds. * > . "^^^rv- TABLE. •*'* **^i^« 60 seconds 's.) - make - 1 minute,' 60 mimites *' v* #•*<••«»*«*' *- marked 1 hour, ■f Xrmj -C'^fif ^ ^ n KBDUOTIOV. 24 hours 7 days 4we«ks 13 months, 1 day and 6 hours, ) 1 or 'Vi5 days and hours, 5 I day, - - 1 week, - - 1 month, common, or Julian year J f 34. d. mo. y*-- V. ml January, February, March, 1st month, 2d, - - 3d, - - has 31 - 28 - 31 April, May. June, . 4th, - - 5th, - - 6th, - - - 30 - 31 - 30 July. August, September, October, 7th, - - 8th, - - 9th, - 10th, - - 31 - 31 - 30 - 31 November, 11th, - - 30 December, 12th, - 31 H 34. The year is also divided into 12 calendar months, which in the Older of their succession are numbered aa fol lows, viz. • , J , daya.n ■^f'*;**!- Note. When any year can be divided by 4 with- out a remainder, i( is call- ed leap year, in which February has 29 days. The number of daya in each month may be eaaily fixed in the mind by committing to memory the following lines : Thirty days hath September, ^ ,. April, June and November, " i» February twenty -eight alone ; V ' ' % •^*' *^® '••** '^*^® thirty-one. The first seven letters in ihe alphabet, A, B, C, D, £, F, 6, are used to otark the several days of the week, and they are disposed in such a manner, for every year, that the letter A shall stand for the 1st day of January, B for the 2d, &c. In pursuance of this order, the letter which shall stand for Sun- day, in any year, is called the Dominical letter for that year. The Dominical letter being known, the day of the week on which each month « '>mes in may be readily calculated from the following couj[>.et : * . - . - * 4 At Dover Dwells George Brown, Esquire, > • >f-; Oood Carlos Finch And David Fryer. These words correspond to the 12 months of the year, and tfie fit9t Ittter in eaeh word piarks the day of the week on 4 1914. IBDUCTIOa, *.- which each correiiponftin|i( month comet in ; whence tinjr other day may be ea«ily foiimt. For example, let it We required to find on whitl daj of the week the 4th of July falls, in the # year 1K27, the Dominical letter for which year is G. Ci/ood amtwem to July ; coi)!ie(|ijently, July comes in on a Sunday ; whtrefore thr 4lh of July fallii on Wi'iJnesday. yiUe. There are tioo Dominical letters in leap yuarti, one for January and February, and another tor the rust uf , the year. 5fJ. Supposing your ago to| r»7. Reduce 475tM74H5 se- be I5y. I9d. II h. :<7m. 459.,cond8 to years. , how many seconds old are you, allowing JMWi days <> • _ Lours to the year 1 ' » 5H. How many minuteH from! 50. Reduce 3^'>440 minutes the 1st day of January to the to days. 14th day of Aueust, inclu- • "< i- sively ? GO. How many minutes from the commencement of the war between America and Eng- land, April 19th, 1775, to the settlement of a general peace, which took place .fan. 20th, 1783? 61. In 107!Ht}() how many years ? minutes, k CIRCULAR MEASURE. OR MO HON. Circular measure is used in reckoning latitude and 1( ijgi tude ; also in computing the revolution of the earth and other planets round the sun. The denominations arc- a- cles, signs, degrees, minutes, and seconds. TABLE. •^ 6<1 seconds (") make I minute, market' 60 minutes - * - 1 degree, - - • JIO degrees . - . | sign, . . n 12 signs, or 360 degrees, I circle of the zodiac. ^^ote. Every circle, wiicther great or small, is divisible into U(»() equal parts, called degrees. 62. Reduce 98. 13« 25' tol 63. InlO-iO.'kK) howawnsiT seconds. Idegreesi SUPPLCMaHT TO BBDUCTIO; 5 M I" ! Tho fohowiui; ar« (UnomtnattonM of tbingii not incii Jed id the table* ;— « 12 pnrlicular thing* 12 tluztii 1*2 grotM, or 144 doztfii, mako Alio, make 1 dozen. 1 f^rosii. 1 threat grusit. 1 scuro. 30 particular things 6 pointN uiaku 1 line, { uned in mcaiiuring the length ct 13 lines - 1 inch, c thr nnlM of chM-k pendulums. J . . « 1 J ^ used in meahurinn; the heisht ot 4 Inches 1 hand, { . • * ^* ». . c horses. 7 ^ v. ' feet 1 fathom, used in uipasuring depths at sea 11*! pounds niak*' 1 <{uintal of hsih, '24 sheets ot paper mako I quire. 30 quiren - - 1 ream. «'•- * 4' fl SUPPLEMENT TO REDUCTION. UUE5tis of Halifax currency? 7. What name is given t< tijo ciirrettr V "f this Province r 8. And why? *V Are the rntion <.) the JifTi-reni denoniinnlions to each other tiie same as in KngliHh mo- ney :' 10. Will tho riiU; for reduction of one denoininntion to unothei III Halifjix currency, n|)ply to all currencies m whicli the denoniin.-i- tionH Mro of llie same name ? 11. What is the use of Troy weight MuU uhat iiio ihi^ dfenoa)iiialion8 ? 12. avoirdupoise weight-' the denomiiialions ? 13. What distinction do you mnke between ^TOi'^ an(i nrMvolght ? 14. Whnl distinctions df> you make between long, squaio, nnd cubic rneuHure .' 15. What aro the denomination.* in lonw meiiM;ro? IG. in square measure? 17. in cubii measure? IH How do you multiply by J? 19. When the divisoi contain.'* a fracticm, how do you proceed? 5iO. How in the superficial contents of .-i square figure found ? 'i\. How is the solid content* of any body found in cubic measure \ ^. How many soiid or cubic feel of wood malio a ct»rd ? '-J3. What is undetsiood by a cord/oof.'' 24 How many such feet make a cord r 25. What nro tlw denomination.i of dry measure ? 20. of wine measure ? 27. of time? 2H of circular measure ? 21). For what is circuI.Tr moaKuie u«ed 30. How many rods in length is Gunter's chain ? of how many link-; does a consis>t? how many links mako n rod ? 31. How iMny rod* in a mite? 32- How many square rods in an acre ? 33. How many pounds make 1 cwt. ? -^ if- 'J^ 134, VUrrLBMBMT TO RMOVCTIOR. 79 I KXERCIr^KS i In l-')t (lallant, ut Th. each, haw many potinHn, kc. Ann. X^. Hid. 2. In 30 g)jin<;aM, at \£ !)ii. 4(1. Aach, how many crowns. at 6«. 6d. \ -^Z""- IIM froum« umi 'in. lOd. ortr. 3. How many rintj* each woi^hinf^ Apwt. Tjfr*., m.iy he made oi 311). ■>oz. Hipwt. 'igr«. of goldl ^n». I5H. 4. Suppoae a hrifi);c tn he ''IVl ruda in len;;th, hnw many tiiMl will a chaise wheel, IH feel <( inches in circumference, turn round in pasiiing over it ? Am. \^^i limes. 5. In 470 hoxeH Riifi;ar, each 2(>lb., how many cwt. \ 6. In lOlh. of silver, how many spoons, each weigluii^ loz. lOpwt. 1 7. How many shinf^lcs, each covering a space 4 inches one way and inches the other, would it take to cover 1 square foot \ How many to cover a r\ide, and 6 feet hip;h 1 Ans. 4 cords, and 7 curd feet. 9. There is a room 18 feet iu hmgth, 10 tect in width, and 8 feet in height ; how many rolls oi paper, 2 feet wide, and containing 1 1 yards in each roll, will it take to cover the walls 1 Ans.H^f^. 10. How many cord feet in a load of wood OJ feet long. I^ feet wide, and 5 feet high 1 Ans. 4^ cord feet. 11. If a ship sail 7 miles an hour, how far will sho sail, at that rate, in 3w. 4d. lOh ? 12. A merchant sold 12 hhds. uf bvandy, at 83 a gallon ; how much did each hogshead come to, and to how much in currency did the whole amount 1 13. How much cloth, at 78. a yard, may be bought for Q9£ Is? 14. A goldsmith sold a tankard for 10<£ 86. at the rate of 5s. 4d. per ounce ; how much did it weigh 1 15. An ingot of gold weighs 21b. 8oz. lOpwt ; how much is it worth at 3d. per pwt ? 10. At 11 pence a pound, what will 1 T. 2cwt. 3qrs. iOlb. of lead come to ? 17. Reduce 14445 ells Flemish to ells English. 18. There is a house, the roof of which is 44^ feet in length, and 20 feet in width, on each of the two tides ; if 3 shingles in width cover one foot in length, how many 11 m ADDitioar OP coMPouvD ivvMBBBa. T 34, 35. Khinglet will it take to lay one course on this roof? if :} courses make one foot, how many cotii.ses will there be on one side of the roof 1 how many shingles will it take to cover one side ? to cover both sides 1 Jina. 16020 shingles. 19. How many nteps, of 30 inches each, must a man take in travelling .'>4j^ miles 1 20. How many seconds of time would a person redeem in 40 years, by rising each morning ^ hour earlier than he now does 1 21. If a man lay up i dollar each day, Sundays except «d, how many pounds would he lay up in 45 years 1 22. If 9 candles are made from 1 pound of tallow, how xjnany dozen can be made from 24 pounds and 10 ounces 1 ^ 23. If one pound of wool make (iO knots of yarn, how many skeins, of ten knots each, may be spun from 4 pounds I) ounces of wool 1 I > OF COMPOUND NUMBERS. '- IF 35. 1. A boy bought a knife for 9 pence, and a comb for 3 pence ; how much did he give for both 1 ^ns.'A shil- ling. 2. A boy gave 2 s. 6 d. for a slate, and 4 s. 6 d. for a book ; how much did he give for both 1 3. Bough! one book for I s. 6 d., another for 2 s. 3 d., an- other for 7 d. ; how much did they all cost 1 y/ns. 4 8. 4 d. 4. How many gallons are 2 qts. -|- 3 qts. -\- \ qt. 1 5. How many gallons are 3 qts. -f- 2 qts. ~j- 1 qt. -j- 3 qts. -|- 2qts. 1 6. How many shillings are 2d. -{- 3d. -f- 5d. -j- 6d. -j- 7d. \ 7. How many pence are Iqr. -j- 2qrs. -j- 3qr8j -|- 2qrs. -flqr.'' 8. How many pounds are 4s. -f- lOs. -j- 15s. -j- Is. 1 9. How many minutes are 30 sec. -}- 45sec. -f- 20sec. ! 10. How many hours are 40 min. -\- 25 min. -f- (5min. 1 11. How many days are 4h. -f 8h. -\- lOh. -\- 20h. 1 12. How mtny yards in length are If. -|- 2f. -f- If. 1 f 35. ADDITION or COMPOUITD IIUMBBRK. 81 shingieti. st a man s except •si ow, how ounces 1 am, how 4 pounds nd a comb rts.l shil- or a book , 13. How many feet are 4in. -\- 8in. -|- lOin. -|- '2in. -f- lin. 1 14. How much i& the amount of 1yd. '2ft. Gin. -}- ^yd«. 1 ft. 8 in. I 15. What is the amount of 2s. 6d.-f-4s. 3d.-f-78. 8d. ! 16. A man has 2 bottles, which he wishes to fill with wiue ; one will contain 2 gal. 3 qts. 1 pt. and the other 3 qts. ; how much wine can be put in them ? 17. A man bouo^ht a horse fur 15£, 14s. Ocl., a pair of oxen for 20c£. 2i. 8d., and a cow for 5c£. 6s. 4d. ; what did he pay for all 1 When the numbers are large, it will be most convenient to write them down, placing those of the same kind, or de- nomination, directly under each other, and, beginning with those of the least value, to add up each kind separately. OPERATION. In this example, adding up the cohimn of pence, we find the amount to be 18 pence, which being = Is. 6d., it is plain that we may write down the 6d. under the column of pence, and reserve the Is. to be add- i ed in with the other shilling;^. || Next, adding up the cohimn of shillings, together with Hhe Is. which we reserved, we find the amount to be 23s. c=l^. 3s. Setting the 3s. under its oum column, we add |the l£. with the other pounds, and, finding the amount to }e 41<£. we wiite it down, and the work is done. j^ns. 41.£. 3s. 6d. yote. It will be recollected, that, to reduce a lower into higher denomination, wa divide by the number which it lakes of the lower to make one of the higher denomination. in addition, this is usually called carrying for that number : thus, between pence and shillings, we carry for 12, and be- Jtween shillings and pounds, for 20, &c. The above process may be given in tne form of a general JRule/m* the .Edition of Compound Numbers. I. Write the numbers to be added so that those of the lame denomination may stand directly under each other. II. Add together the irumbers in the column of the lowest lenominatioD, and carry for that number which it takes of Jlna. £. s. d. 15 14 6 20 8 5 6 4 41 3 6 82 ADniTIOH OF COMPOUND IfUMBRRt. f J«. the rame to make 1 of the next higher denomination. Pro ceed in this manner with all the denoiuinations, till you come to the last, whose amount is written as in simple nuiu bers. Proof. The same as in addition of simple numbers. EXAMPLES FOR PRACTICE. HALIFAX CURRF.NCY. £ 8. d. qr. £ s. d. £ 46 11 3 2 72 9 6^ 183 16 7 4 4 18 10^ 8 538 19 7 1 36 16 5j 8. d. 19 4 17 10 4 15 £. 14 8 62 4 23 6 91 8. 15 4 17 6 d. 3 7 8 7 10^ £. 8. d. 37 15 8 14 12 9f 17 14 9 23 10 9i 8 6 14 5^ 54 2 7^ £. s. d. 61 3 2^ 7 16 8 29 13 10^ 12 16 2 7 5J 24 13 5 102 No examples in Federal Money are here introduced, al though the general rule for the addition of all compound numbers is precisely applicable to the addition of Federa. Money, since that consists of different denominations. In Federal Money the denominations increase and decrease in a decimal ratio. The pupil is therefore referred to the rules for the jiddition, Subtrartion, Multiplication and Division q; Decimals, which are the same absolutely with the rules for the addition, subtraction, multiplication and division of Fed eral money. TROY WEIGHT. lb. ex. pvot. gr. 36 7 }0 11 42 6 9 13 81 7 16 15 oz. pwt. gr. 6 14 9 8 6 16 3 U 10 ot. pwt. gr. 18 13 16 3 7 4 IT 35. ADDITIO* or COMPOUND MU] 1 1. d. 19 4 17 10 15 4 t d. 2i 8 5 lOi 2 7 3 5f Bought a silver tankard, weighing 21b. 3 o«., a tiWer cup, weighing 3 oz. 10 pwt. and a silver thimble, weighing 'Z pwt. 13 grs. ; what was the weight of the whole 1 AVOIRDUPOIS WEIGHT. T. nrt. qr lb. oz. dr- 14 11 1 10 5 10 2 11 8 15 7 18 25 11 9 25 cwt. qr. lb. oz. ir 16 3 18 6 14 2 16 8 12 22 11 10 A man bought 5 loads of hay, weighing as follows, viz. [23 cwt. (=1 T. 3 cwt.) 2 qrs. 17 lb. ; 21 cwt. 1 qr. 161b ; jlO cwt. qr. 24 lb. ; 24 cwt. 3 qrs. ; 11 cwt. qr. 1 lb. ; [how many tons in the whole? S 11 84 9 2 16 42 18 32 6 5 5 18 85 V. m. ic. // 4U 3 \ .'» 16 7 1 27 5 2 0. i« .n r 119 i 110 3 127 fl(3 2 1 taiuing as any hogs A QiFmwmjKs^ump OF COMPOUND NUMBERS. IT 36. I. A boy bought a knife for 9 pence, and sold it for Is. 4d. ; how much did he gain by the bargain 1 2. A boy bought a slate for 28. 6d., and a book for 3s. 6d. , how much more was the cost of the book than of the slate ? 3. A boy owed his playmate '2s. ; he paid him Is. f>d. . how much did he then owe him 1 4. Bought two books; the price of o)l^ was 4s. (id., thf- price of the other 3s. 9d. ; what was the difference of then costs 1 5. A boy lent 5s. 3d^ ; he received in payment 2s. 6f) , how much was then due 1 6. A man has a bottle of wine containing 2 gallons and 3 quarts ; after turning out 3 quarts, how much remained ! 7. How much is 4 gal. less 3 gal ? 4 gal. — (less) 2qts ? 4gal. — Iqf! 4 gal. — 1 gal. Iqtl 4gal. — Igal. 2qts ? 4 gal. — 1 gal. 3qts1 4gal. — 2gal. Sqtsl 4gal. Jqt. — 1 gal. 3qts. 1 8. How much is 1ft. — (less) 6in ? 1ft. — 8in.? m. :{ inches, — 1ft. 6in I 7ft. Sin. — 4ft. 2in ? 7ft. Sin. — oi^ . lOJnl 9. What is the difference between 4£ 6s. and 1^' 8s ■> 10. How much is 3£ — (less) Isl 3ofc' — 2sl tii— :Js.' 3i: __ )5s 1 M 4s. — 2£ 6s ? 10=fc' 4s. — ^£ 8s f 11. A man bought a horse for 30^' 4s. 8d., and a cow for 51' 14s. 6d. ; what is the difference of their costs ^. ' H » ^t f'i fh I ^<) NUBTnACTION OF COM I'Ol'IV P M'MBERS. IT :i<; OI'KKATION. jt. s. ft. Minuend, J 10 4 "^ Sj^'trnhe.nd, ;> 14 f> •I /7nH. 24 10 :! As tlio two mimln-rsarc largi , it will lio convenient ti) >vrit<> them clown, the less under the greater, pence under jx^nco, shil- lings under shillin;;s, &c. Wt, may now lake (id. Ironi >A., and Proceeding to the shillings, we can itier(' will remain :^d. n »t take 14s. from 4^"., hut wc may hurrow, as in simple numljcrs, I from the poundSj='20s., which joined to the Is. in decs 21s, from whicli taking 14s. leaves 10s, which we »et down. AVe must now carry I to the of making OX' winch taken from JJOX' loaves 'iA£ arut the work is done. Note. The most convenient way in borrowing is, 1o sub 'ract the subtrahend from the figure borrowed, ; nd add the lilVcrence to the minuend. TJius, in the above example, (4 ;ioai ')() leaves 6, and 4 is 10. The process in the foregoing example may be presented III the form of a Rule for the SuhtracHon of Compound Num- \. *\V'rite down the sums or quantities, the less under the :.',ifater, placing those numbers which are of the same denom- . nation directly under each other. \l. Ik'gining Avitji the least denomination, take succes- ivcly tj|je lower number in each denomination from the up- j)ci\, and write the remainder underneath, as in subtraction tCf-'iDijilo numbers. ' ■■■■■ HI. If the lower number of any denomination be greater iliaif the upper, borrow as many units as make one of tht- next higher denomination, subtract the lower number there- Moju, ai-d to the remainder add the upper number, vemem- icring always to add I to the next higher denomination for (hat vliich you borrowed. Provf. Add the remainder and the subtrahend together, js iii subtraction of sinjple numbers ; if the work be rights amount will be equal to the minuend. ' if EXAMPLES FOR PIIACTICE. HALIFAX CF-RHEIVCY 79 17 H f ! ' ■>■ 103 .} 35 12 4 •: ...V 71 12 ,\\ «f ;i#t ♦* srBTRACTlON OF COMPOUJID XUMBERS. X largi *«.f *29 I. J .'J i7 9 I- r>-20 1 1 :J nm 17 4 mi 14 7 l\i H rf. 1. MISCELI.ANEOUS EXAWl'LLS. ■* A merchant sold goods to the amount of l:j()i,* 7s. Old. iiul received in payment 5()i.' 10s. 4fd ; how much remain d due 1 ^ns. 85£ 17s. I'd. 'i. A man bought a farm for 1250=£ 10s, and, in sellin*; it, lost 87<:t* 10s. fid ; how much did he sell it fori ^ns. IIOH^ ll>s. ■ = 'jit ^t *^, fT f0 KVSTAACTfOS) OF COMPOI'NU NtMBKRI. IT 96, 37 acr^s (I roods ; the otiter, 14 acres 3 rood* he has two pastures also ; one containing 'it) A. 'Z r. "27 p ; the other, i't A. 't r. ^''^ p : how much more has he of pasture than of mowing; 1 12. From OIA. 2r. lip. '2Wt. take ZV)\. .>r. 31p. 13'2f(. 13. From a pile of wood, containing 21 cords, was soM, di one time, H cordit 70 cubic feet ; at another time, 5 cords 7 cord feet ; what was the quantity of wood left '\ \i. How many days, hours and minutes of any year will he future time on the 4th day of July, '20 minutes past '■) o'clock, P. M 1 /Ins. 180 days, 8 hours, 40 minutes. 15. On tho same day, hour and minute of July, given in the ahovo example, what will he the difference between the past and future time of that month 1 10. A note, bearing date Dec. 28th, 1820, was paid Jan, 2d, 1827 ; how long was it at interest 1 The distance of time from cite date to that of another may be found by subtracting the first date from the last, observ ing to number tho months according to their order. (IT 34.) OPERATION. . 1^ W827. Istm. 2d day. Note. In casting in- d 1826. 12 28 terest, each month is "T~j reckoned 30 days. yins. 17. A note, bearing date Oct. 20th, 1823, was paid April 25th, 1825 ; how long was the note at interest ? 18. What is the difference of time from Sept. 29, 1810 to April 2d, 1819 1 Ans. 2y. Cm. 3(1 - 19. London is 5lo 32', and Montreal 45o 30', N. lati | tude ; what is the difference of latitude between the tw l[ places 1 Ans. 6« 2 20. Montreal is 73*' 20', and the city of Washington i«|i 77* 43' W. longitude ; what is the difference of longitiKlt|| between the two places ? ./fns. 4<* 23 J 21. The island of Cuba lies between 74« and 85o V longitude ; how many degrees in longitude does it extend i * L ^ 37. 1. When it is 12 o'clock at the most easterly extremity of the island of Cuba, what will be the hour a| the most westerly extremity, the difference in longitude be ing 11" 1 Note. The circumference of the earth being 360^ ^ ani Millie earth performing one entire revolution in 24 hours, ii ^s" " ;r?. i^ MVi.rivi.tr \rioy wn niviRiojf, Aic M» follows, that tiic motion ol the earlii, on its Mirtar« iri n. vv'st to oast, is \ |i>" of motion in I honr of time ; ronscqucnttv. i" of motion in I njintites of time, ami I of motion in I seconds of time. frdui these premises it follows, that, when there i» a (lit JrTcnce in longitmie between two places, there will l)c :i orresponding ilifTercncc in the hour, or time of the day. The ditFerencc in longitude being l.">'', the difference in timt will be one hour, the place faslcrly having the time oi iht ilay I hour earlier than the place westerly, which must i • particularly reganied. If ihe difference in longitude be 1", the dilferonco in (oik will bo I minutes, &c. Hence, — If tlie difference in longitude, in degrees am', minutes, Ijotween two places, bo multiplied by i, the |»rt- luct will bo the difference in time, in minutes and ser( nds nhich may be reduced to hours. Wc are now prepared to answer the above questiun 11* Hence, when it is 1*2 o'clock ai tia . 4 most easterly extremity of the islanci. 77 . , it will be IG minutes past 11 o'( b ( U 44 minutes. , ,, , . ' at the most western extremity. ■l. Montreal being 73" 20' W. longitude and Washifi;^ut. 77" 43' ; when it is 3 o'clock at the city of Washington I what is the b.our at Montreal 1 y/ns. 17 minutes 32 seconds past 3 o'< b tk 3. Lov.er Canada being about 73", and the Sandwic}. Islands about 155" W. longitude, when it is 98 minuter pa.st i<) o'clock, A. M. at tiic Sandwich Islands, what will be tbt [hour in Lower Canada ? y/rts. 12 o'clock at noon, lackinaf 4 niijiwt**: •^ ■"■■ .# ^miFJ^WJllPlLIl^JlWA^^ ^ J)2^ I ■'«...'i' OF COMPOUND NUMBEKS. >■* . f • ■ * ♦ ' ^ m c# T 38. 1. A man bought 2 yards of cloth, at I.«. Od. jjci ,'* • ! vard ; what was the cost 1 - *< u <- " ' H2 '»0 Mtfl^TI PLICATION AND DIVIB10M uto %> ^ k '4. II 2 yar of cloth cost ;J shillini^s, wli^/ In tiiat pei yard ? :{. A man liao three 'ieces of cloth, each measuring 10 yi\h. Itqrs ; how many yardti in whole 1 1. If '.i equal piece* of cloth contain '.W yds. 1 (jr., how inucli docs each piece contain ^ •'*. A man ha^* five bottles, each containing Q gal. I (jt. I pt ; how much wine do they all contain 1 (». A man has II gal. iJqts. Ipt. ol wine, which he would divide equally into 5 bottles ; how much must he put into each bottle 1 T. How many shillin;fs are 5? times 8d 1 — 3xUd "^ - :JXlOd? — 1X*o 1.V ; tl'f m down before multiplying and dividing. MJTRVTION. I OPERATION. .f s. d. (jr. £ s. d. qr. I 5 8 3pnceo/l yard.\G)7 14 4 2 cost of yards., imumberof yards. YTs^ price of 1 yard. Proceeding after the man- ner of short division, 6 is con- tained in 7£ 1 time, and \£ over ; we write down the ■^> quotient; and reduce the re- mainder {l£) to shillings, (20s,) Avhich, with the given (14s,) make 34s; goes 5 times, and f A. y/ii'i.'i 14 4 'Z cost of ii yards. (j times 3qrs. are 18qrs. = td. and 2qrs. over ; we set times "'cT. are 4Sd, and 4 to carry makes ,">2d. = 4s. and 4cl. :iver, which we write down ; a2:ain, 6 times 5s. arc 30s. shillings, .\nd 4 to carry makes JMs =6 in 34s \£ and 14s. over ; G times 4s. over ; 4s. reduced to penct ii' are f5£, and 1 to carry = 48d, Avhich, with the giv- inakes 7£, winch wc write en pence, (4d,) make .>2d ; <> 'lown, and it is plain, that in 52d. goes 8 times, and 4d. the united products arisinglover ; 4d. = Ifiqrs, which. r 1 ;». or dOBtPOUXD XUMBRR*. \n from the several u nomina-iwith the i;iven qn*. ^*j) =£ l^ tiuns is the real prcMluct aris-lqrB; o the cost 14 A oi' 5 pairs of shoes at lOs. Od. '.pairs of si. 1. for o ^ c is that i\ a pair ? jpair 1 15. In r> barrels of wheat,! IG. If I4hus. vipks. (iqts. ol each containing '2 bus. 3 pks. 'wheat he c<|uiilly divided into *) qts, how many bushels 1 |.) barrels, how many bushels Iwill each contain 1 17. How many yards of! IH. If 9 coats contain 39 cloth will be required for 9 yds. 3qrs. 3na, what docs 1 coats, allowing 4 yds. Iqr.jcoat contain 1 3 na. t • each ? | 19 In 7 bottles of wine,j 20. If o gal. I gill of wine be divided equally into 7 bot- tles, how much will each con- tain 1 •22. If 8 silver cups weigh 3lb. 9oz. Ipwt. lOgrs, what is the weight of eacli ? 24. If li9cwt. Iqr. of su- gar be divided into 12 hogs- heads^ how much will eacli. hogshead contain ? 26. 1i 15 teams be loaded with 17T. 12cwt. 2qrs. of hay, how much is that to each team? each containing 2qts. Ipt. 3 gills, how many gallons 1 21. What will be the weight of 8 silver cups, each weighing 5oz. 12pwt. 17grs''. 23. How much sugar in 12 hogsheads, each containing 9cwt.3qrs. 211b 1 25. In 15 loads of hay, each weighing IT. 3cwt. 2qr8, how many tons 1 JVhen the multiplier, or divisor, exceeds 12, the operations of multiplying and dividing are not so easy, unless they be composite numbers ; in that case, we may make use of the romponent parts, or factors, as was done in simple numbers. 15 being a composite num- ber, and 3 and 5 its compo- nent parts, or factors, we may Thus 15, in the example above, is a composite number produced by the multiplica- • f « IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I Uit2^ 12.5 ui lU i: ^ 114 II. ^ ti& 1110 12.2 1.25 1.4 1.6 6" ► ff %. '/] ^yi % %' ^> Photographic Sciences Corporation 33 WIST MAIN STREET WEBSTER, N.Y. 14SS0 (716) S72-4S03 ^ *\^ 1^^^ ^ ^ 1 <\ j OS 4^1IULTIPI.tcfATI01l AlfD I^IVltlOIV f 88 w '< tiun of .3 and 5, (3X5 s^divide 17T. 12cwt. '2qrt: by 15.) We may, there(ore,'ono of these component parts, multiply IT. 3owt. Sqn. byjor factors, and the quotient one of thoee component parts J thence arising by the other, or factors, and that product byi which will give the tme the other, which will give the answer, as already taught, true answer, as has been al-(1f 20.) ready taught, (f 1 1 .) OPERATION. T. cwt. qr. I ^^^ II one of the factors . OPERATION. T. cwt. qr l^One factor, 3) 17 12 2 8 10 2 5 the other factor. 17 12 "2 the answer. The other factor, 5) G 17 2 jfM.l 3 2 27. What will 24 barrels 28. Bought 24 barrel of of flour cost, at ^£ 12s. 4d .'flour for 62^ 16s ; how much a barrel f {was that per barrel 1 29. What will 1121b. ofj 30. If 1 cwt. of sugar cost sugar cost 7|d. per lb 1 fi£ 7s. 8d, what is that per Note. 8, 7, and 2, are fac-lb 1 tors of 112. 31. How much brandy in 32. Bought -84 pipeo of H4 /tipes, each containing 112hrandy, containing 9468 gal. gal. 2qts. Ipt. 3g1 |lqt. Ipt; how much in a pipe 1 :33. What will ISOyds. ofi 34. Bought 139 yards of cloth cost, at 3<£ 6s. 5d. percloth for 461^ lis. lid; yard 1 139 is no||i composite num- ber We may, ho;wever, de compose this number thus, 139=100-1-304-9. what was ttiat per yard 1 When the divisor is such a number as cannot be produced by the multiplication of small numbers, the better way is to We iBay now multiply the^divide after the manner of price of 1 yard by 10, whichlong division, setting down will give the price of 10 yards, the work of dividing and re- and this product again by 10,'ducinK in manner as follows : which will give die price of *r' ♦ 100 yards. % ov coMroysD vviibb«% . W« may th«B multiply the price of KKyardt by 3, wbieh will give tba price of 30 yards, and the price of 1 yard by 9, which will give the price of 9 yards, and these three pro- ducts, added together, will CTidently give the price of 139 yards ; thus : £. 9. d. 3 6 5 priet of I yard. 10 93 33 4 2 price of 10 yardt 10 £ 130)461 417 "44 20 f. 11 d. 11 (3£ :)32 99 29 1 12 17 8 price itf 100 yds. 6 price of 30 yds. 9 price ^ 9 yde. 461 11 11 price ({f 139 yds. 891 ( 6e. 834 "67 606 (5d. 695 The divisor, 139, is contoiff- ^ sd in 461;f 3 times, (3;f ,) and a remainder of 44£, which must now be reduced to shil- lings, multiplying It by 20, and bringing in the given shil- i lings, (Us,) making 891s, in ' which the divisor is contained ,, Noit, In multiplying the B times, (6s,) and a remainder price of 10 yards (33£ 4s. of 57s, which must be reduc 2d.) by 3, to get the price of sd to pence, multiplying It by 30 yards, and in multiplying 12, and bringing in the giveu the price of 1 yard (3;^ 6s. pence, (lid,) together mak- 5d.) by 9, to get the price of ing 095d, in which the divi- 9 yards, the multipliers, Sior iscontained 5 times^ (5dJ and 9, need not be written tnd no remainder. ^ down, but may be carried in The several quotients, 9£ the mind. |0a. 5d. evidently make th% lanswer. '^ The processes in the foregoing examples may now be ^Ih sented in the form of a '" *" RuLv far th€ MuUiptication of^vhrnftrthe DiehUm qTOm-^ 4 ■*« compound Numhen I. When the multiplier does fud exceed 12, multiply suc- cessively the numbers of each potmd yumberf. 1. When the divbor does not exceed 12, in the manner of short division, find how denomination, beginning withmany timee it is oontainsd. iik % * 94 uvifTtwviCArtom amd division f^€ 4 the Icait, at in iniiltiplicatioii,the highMt denominatioD, uii- of aimple numben, and carrjider which write ||ie quotMnt;, z* in addition of compound and if there be « remainder, numbers, settin^^ down thejreduce 'it to the next less de- whole product of the highest^nominatiun, adding thereto the ividc MISCELLANEOUS EX AMptE8.4 . j 1. What will 359 yardf ofit 2. Fought 359yds. of oloth ;Ioth cost, at 48. 7id. per rardl 3. In 241 barrels of flour, Meh containing Icvrt. 3qr. Mb : how mauy cwt 1 How manv bushels of 5. for Sd£ Os. 4id ; what w^s (hat a yar*1 *►• . ^ 4i If.44}ewt. 131b. of flour be contained in 241 barrels, how much in a barrel 1^ 6. If 371bu. Ipk. offbeat rheat in 135 bags, each con -Ibe divided equally tnto*^ 135 #•# m lining 2 bu. 3 pks 1 3x9X5s=135; bags, how much will each^- bag contain % / 7. What will 36cwt. of to- 8. At 759^ lOs. for 35c wf. ceo cost, at 3«. lOid. pel of tobacco, what is that p^ Ibt . 10. If 14 men build 92 rmfs^ 12 feet of stone wall in 7h any rods will they biNl#^1iBi|dayf, hew much is that per 9. If 14 men build li feet at wall in one day, h days day W a,^* 'I IF 89. 1. At 10|.i)«jr ntdr what will 17849 y^its of » loth cost 1 < i Note. Oporationi in ijiultipUcaUoD of pound^ shillings, ?nce, or of any compound numbers, may be facilitated by % * # I ; « ^f t i R 96 MVLT/VLIOATIOfr AND DITIIIOW, &C. 1491 taking olifuot part* of a higher ienomituUion. Thus, in thia last exao^le, if tha price had bten 208. i. e. Uf per jrard, it is cleai:, the price of the whole would have been equal to the whole number of yards in pounds, 17840 ; but the price Is 191, i. e. i£ per yard, and so the price of the whole will be equal to | the number of yards, I'f** in pounds ; 8924ii:, or 8924^ lOs. ^When one quantHy is contained in another exactly 2, 3, 4, 5, &c. times, it is called an aUqxu/t or tvm part of that quantity ; thusiSd. is an ali^ot part of a shilling, because Q|^X2=1 shilling ; so 3d. it an aliquot part of a shilling ; 3d.X4=sls. So 5s«ia an aliquot part of a pound, for 58. X4^1<;^ ; and 8t. 4d. la and aliquot part of a pound, for 3a. 4d.X&=>li^* &c. Wtom th^ illu«tfeatiou of the last example it appears, that, when the price per yard, potpBd, &o. is one of these aliquot parts of a shillingi or a pound, the cost may be found by dividing the giikn miMMr of yards, pounds, lie. by that number which it tajkes of the price to make Is. or l£. It the prMe be €i||. we divide by 2 ; if 58. we divide by 4 ; if 3s. 4d. by 6, ftp. &c. Thia manner of calculating by ali- quot parte, is called Practice. 2. What cQft 34l648 yardi of cloth, at lOs. or l£ pet -ard> — — at 5s.bbs4;£ par yard 1 at 4s.B=r^ per yard? at 3s. 4d.aBf£ per yard 1 at2s.=^ per yard % Ana. to but, 3464^ 16s. 3. What coat 7430 pounds of sugar, at 6d.sBi8. per lb ? — ^ at 4d.s:|8. per lb? at dd^B per lb 1 at 2d,s=^. per lb 1 at l|d.=s4s. per lb 1 jtn». totkt latt, 7 V''**^^^'^* ^•^ 4^ @"« ^^' i 4. At 3j? 16fl. per cwt, what will 2qrsjssicwt. cost 1 -vr- what will lqr.=B^wt. coat 1 what will 161bj: iewt. cost 1 what will 141b.=slcwt. cost 1 what j rfll 8lb.s^wt. coat 1 ills, to the hut, 5s. &f140 Or, •2ii. M —i of 108.)n0<£ at IOb. per yard. 42i: lOs. at 28. 6d. per yard. Ans. 212^ lOs. at 128. 6d. per yard. ^ 8UPPI^EMENT TO COMPOUND NUMBERS. QUESTIONS. ^' 1. What distinction do you make bftweon linple and compound numbers P (H 2C.) 2. What is the rule ibr addition of compound numbers ? 3. for subtraction of, A^c. ? 4. There are Ukree con* ditions in the rule given for multiplication of compound numteri; what are they, and the methods of procedure under each ? 5. Thn same questions in respect to the division of compound numbers ? 6. When the multiplier or divisor is encumbered With a fraction, how do you pioceed ? 7. How is the distance of time from one date to another found ? 8. How many degress do«8 the earth revolve from west to east in 1 hour P !). In what time does it revolve 1^ } Where is the time or hour of the da^ earlier — at the place moet easterly or most westerly ? 10. The difference in longitude between two places being known, how is the difference in time calculated .' 11. How may operations, in the multiplication of compouad numbers, be fa- cilitated ? 12. What are some of the aliquot parts of 1£ P of Is. ? of Icwt. ? 13. What is this manner of operating usually called ? EXERCISES. 1. Agentleinan is possessed of 1^ dozen of silver spoons, each weighing 3oz. 5pwt ; 2 dos. of tea spoons, each weigh- ing 15pwt. 14gr; 3 silver cans, each 9or.. 7pwt; 2 silver tankards, each 21oz. 15pwt ; and 6 silver porringers, each lloz. 18pwt ; what is the weight of the whole 1 Jns. 181b. 4oz. 3pwt. Note. Let the pupil he required to reverse and prove the following examples : 2. An English giUBBB should weigh 5pwt. 6gr ; a piece of gold weighs 3pwt. 17gr ; how much is that short of the weight of a guinea ? 3. What is the weight of 6 chests of tea, eaodweighinf 3cwt. 2qrs. 91b? 4. In 36 pieces of cloth, each ioeasuring27 yards, how many yards 1 -.-m-:-^ ^^ . ^ I ._. . % V'- 98 ~ SVPPLBMKNT TO COMPOUHD MUMBBBI. If .')9, 40. 5. How much brandy in 9 casks, each containing 45gal. :iqt8> lpt.1 (). If 31cwt. 2qrs. 201b. of sugar bo distributed equally into 4 casks, how much will each contain ? ' 7. At 4^d. per lb., what costs Icwt. of rice? 2cwt.; 3cwt1 Note. The pupil will recollect, that 8, 7 and 2 are fac- tors of 1 12, and may be used in place of that number. 8. If 800cwt. of cocoa cost 1S<£ \Ss. 4d. what is that per cwt 1 what is it per lb 1 « 9. What will O^cwt. of copper cost at Ss. Od. per lb ? 10. If Gj^cwt. of chocolate cost 72c6' 16s. what is that per lb 1 ^ # 11. ^hat cost 4^ bushels of potatoes, at 2s. 6d. per bttiliell Note. 2s. 6d. is | of 1^ (See V 39.) 12. What cost 86 yards of broadcloth, at los. per yard 1 Note. Consult IT 39, •x. 5. 13. What cost 7^46 pounds of tea, at 7s. 6d. per lb ^ , at 14s. per lb 1 ISs. 4d 1 ' 14. At^94*25 per cwt. what will be the cost of 2qrs. ol tea ? of 3 qrs 1 of 14 lbs. 1 of 21 lbs ! of 16 lbs 1 1 of 24 lbs 1 Note. Consult IF 39, ex. 4 and 5. 15. What will be the cost of 2 pks. and 4qts. of wheat, at Hs. 6d. per bushel 1 16. S^jpposing a meteor to appear so high in the heavens as to be visible at Montreal, 73" 20', at the city of Wash- ington, 77» 43', and at the*Sandwich Islands, 155« W. lon- gitude, and that its appearance at the city of Washington 1)0 at 7 niinute« past 9 o'clock in the evening ; what Will he the hour and minute of its appearance at Montreal and »t the Sandwich Islands 1 1 ., IT 40, * We have seen, Ct 17,) that numbers oxpftTftHing ^ole thitigs are calleil integers, or whole numlwrs ; but that, in division, it ib often necessary to divide of brenk a whole tiling into parts, and that these parts are called fractiens, or broken nuvtihsrs. iwjx* ■ i ■ T40. rBACTiom. It will he recollected, (f 14, ex. 11,) that when a thing or unit is divided into 3 parts, the parts or fractions are call- ed third» ; when into four parts, fourth* ; witen into six parts, $\xtk» ; that is, the fraction takes its iMfiM or denomination from the number of parity into which the unit is divided. Thus, if the unit be divided into 16 parts, the parts are called stx- teentha, and 5 of these parts would be 5 aixteenthe, expressed thus, y^. The number below the short line, (l(i,) as before taught, (IT 17,) is called the denominator, because it gives the name or denomindion to ine parts ; the number above the line is called the numerator, because it numbers the parts. The denominator ithows how many parts it takes to make a unit or whole thing ; the numerator shows how many of those parts are expressed by (he /roc/ton. 1. If an orangB bo cut into 5 equal parts, by what frac- tion is 1 part expressed 1 2 parts 1 — - 3 parts 1 4 parts 7 5 parts 1 how many parts make unity or a whole orange 1 2. If a pie be cut into 8 equal pieces, and 2 of these I pieces be given to Harry, what will be his fraction of the pie 1 if 5 pieces be given to John, what will be his fraction 1 what fraction Or part of the pie will be left 1 It is important to bear in mind, that fractions arise from iliviaion, (T )7,) and that the numerator may be consideivd a \ dividend, and the denominator a dtrtsor, and the vahte of the Traction is the quotient; thus, ^ is the quotient of 1 (the 1 numerator) divided by 2, (the denominator ;) | is the quo- Itieut arising from 1 divided by 4, and j is 3 times as much, [that is, 3 divided by 4 ; thus, one fourth part of 3 is the same as 3 fourths of 1. Hence, in all cases, a fraction is always expressed by liie \fiign of divition. ^ "- expresses the quotient, of which \ ^ ,,,y^^i,,^^ „ denominator. 3. If 4 oranges be equally divided among 6 boys, what lart of an oMnge is each boy 's share 1 . " •/ o n i 5 ; A sixth part of 1 orange is \, and a sixth |Uirtt}f 4 oranges |i8 4 such pieces, =s |. jtns. | of an oiange. 4. If S apples be'equally diTided amofi^5 boys, what part 3f an apple is each boy'f ilMre 1 if 4 apples, what 1 if 2 ipples, what ? if 6 applet, what 1 a vi .viju^^ ,\ '%■ ..z;&.' ,^,-. rAAOTIORt. f 40. 41. 5. What n the quotitiit of 1 dif ided by 3 1 — ofS by 3 f — oflby4r — of 2 by 41 — of 3 by 41 —of 5 by 7 1 — of 6 by 8 1 — of 4 by 6 1 — of 2 by 14 1 6. What part of an orange it a third part of 2 oranges ! — one fourth of 2 orangee 1 — ^ of 3 orangea 1 — ^ of three oranges 1 — iof41 — ^of'il — |of.>1 — «| of 3 1 — I of 2 ? jt proper fraction. Since the denominator showi the num- ber of parts necessary to make a whoU thing, or 1, it it plain, that, when the fiumtrator if Ittn than the denominator, the fraction is less than a unit, or whole Mug ; it is then called a proper fraction. Thus, L^, fltc. are proper fractions. j/n improper fraction. When the numerator equate or ex- ceeds tho denominator, the fraction equah or exceeds unity, or 1, and is then called an improper fraction. Thus, |, J, f , y, are improper fractions. A mixed num6«r, as already shown, is one composed of a whole number and a fraction. Thus, 14^, 13]^, tic. arc mixed numbers. 7. A father bought 4 oranges, and cut each orange into | 6 equal parts ; he gure to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces ; what was each one's fraction 1 Was James' fraction proper, or improper 1 Why 1 Was Nancy's fraction proper, or improper 1 Why 1 lb change an improper fraction to a tohoU or mixed number. K 41. It is evident, that every improper fraction must] contain one or more whole ones, or integera. How many whole apples are there in 4 halves (|) o(| an pie 1 — infl — inf? — in y 1 — rinV'I — in V '• — «n *f •* '^ '« 'f * '^^ .i^«2. How many yards in f o( a yard 1 — in f of a yard 11 — infl — infl — inVfl — in yi — in^] — in V ^ — »n V ^ — « V ^ 3. How many bushels in 8 pecka 1 that is, in f of a bu8h| tjM — in y ^ — »n V ^ — in y 1 — in V '■ — in »^» 1 — in V 1 rw*;^ This finding how many integers, or whole things, arel contained in any improper fraction, is called rsducuig «»\ in^^n^^ fraction to a whole pr mixed nufn6er. 1 41, 4a. rBACTIO.<«S. 401 4. If I give *i7 children { of an oning* tucli, huw many oranges will it take 1 It will take y ; and it ie tTidsBt, UPKRATION. 4 yn Jn». i\l orangt». thai dividing the nunMrator, t27, (■> thm number of parts contained iti the frac- tion,) by the denominator, 4, (sk thf number of parts in I orange,) will giv<- the number of voholt oranges. Hence, 1\t reduc* an improptr fraction to a whole or mxjeed number,— RvLi: : Divide the numerator bj the denomina- tor; the quotient will be the whole or mixed number. EXA1VIPLE8 l-XIR PRACTICE. .'>. A man, spending ^ of a pound a day, in Kl days woiHd spend "g3 of n pound ; how many pounds would that bo 1 W> ♦» whole or mixed numbers. 7b reduce a whole or mixed nun^er to an improper fraction. * If 43. We have seen, that an improper fraction may b( changed to a whole or mixed number ; and it is evident, that, by reversing the operation, a wliole or mixed number may be changed to the form of an improper fraction. 1. In 2 whole apples, how many halv>>9 ?t an apple 1 Jns. 4 halves ; that is, ^. In 3 apples, how many halves \ in i apples 1 in 6 apples 1 in 10 apples 1 in 24 1 in 00 ? in 170 1 in 492 1 3. Reduced yardir tofAtrds. An»- f. Reduce 2§ ya/d.s to thirds. Am. §. Reduce 3 yards to thirds —- 3^ yards. — 34 yards. — 6 yards. — 5f yards. — 6 § yards. 3. Reduce 2 bushels io fourth*. — 2|bu. — 6 bush- els. — 6|^ bushels. — 7^ bushels. — 25f bushels. 4. In 16^ pounds, how many ^ of a pound 1 \^ make 1 pound : if, therefore, we multiply 16 by 1'^, that i.s, multiply the whole number by the denominator, the 12 .. - „ ,■ lot rBACTIOMt. K 42, 4.1. i 1 product will bo tlie number of I'illii in Uk£ lOX Vi^l^i, ukI thif, inorMMd by the numerator of the fraction, (H,) •▼identlj givea tbe whole number oS IStlii ; that it, ^ oi a pound, Jfnt, OPERATION. 10,1^ poundu » 192 Bs 12thi in 16 pounda, or the tohoU nurobor. 5=r I2thi contained in ihe fraction. 197 = t^ , the answer. Hence, 7b reduce a mixed number to an improper fr act ion,— RvLB : Multiply the whole number by the dunominatur oi the fraction, to the product add the nurqerator, and write the raault over the denominator. EXAMPLES FOR PRACTK K. 5. What it tbe improper fraction equivalent to t23^^ hours'! Jins. '|,^^ of an hour. (]. Reduce 730^^ shillings to l^ths. , V At ^ of a shilling is equal to 1 penny, the question is evidently the same as, in 730s. 3d., how many pence 1 yfna. ^\^^ of a shilling ; that is 8703 pence. 7. Reduce l^Jg, 17f5,8i7o&^, 1/,;^^^, and 7^^ to improp- er fractions. 8. In 156^1^ days, how many '24thii of a day 1 *• Ant. 3iJ ' = 37«1 hours. ^ 9. In 342 j gallons, how many 4ths of a gallon 1 * " Jtnt. >\7i of a gallons: 1371 quarts. 7b reduce a fraction to its lowest or most simple teriUH. K 48. The numerator and the denominator, taken to- gether, are called the terms of the fraction. If ^ of an apple be divided into 2 equal parts, it becomes f . The effect on the fraction is evidently the same as if we had multiplied both of its terms by 2. In either case, the parts are made 2 times as mary as they were before; btU they are only HALF AS LAROB ; for it will take 2 times as many fourths to make a whole one as it will take halves ; and hence it is that ^ is the same in value or quantity as }. f is 2 parts ; and if each of these parts be again divided into 2 equal parts, that is, if both terms of the fraction be 1 49. rBACTiom. 103 aiuUiplitd by % it becomeii ^ . Ilcnco, ^ ac | k } , and tli« ^ reverM of this is efidcnUy true, that ^av } as ^. .« ^< It followi therefore, 6y multipljing or dividing both ttmt itf the fraction by tht game nun^r, vte changt ita ttrma witk- • >u< ttUering it$ vahu. Thus, if we reverse the above operation, and divide both torniM of the fraction | by 2, we obtain its equal, '{ ; divid- int; again by 2, we obtain ^, which is the moat simple form oi the fraction, because the terms arc the least possible by which the fraction can be expressed. ,. The process of changing | into its equal X, is called re~ ducing thefhction to itn lowttt tnma. It consists in dividing ttoth ferma of the fraction by any namher which will divide them lioth without a renuHnder, and the quotient thence artHtng in the aame mannir, and ao on, tillit apptara that no number great-^ er than 1 will again divide them. A number, which will divide two or more numbers with- out a remainder, is called a common diviaor, or common mean are of those numi)HrH. The greatest number tliat will do this is called the greatest common diviaor. , u^,,, « 1 . What part of an acre are 128 rods 1 One rod is j^ of an acre, and 128 rods are jg^ of an acre. Let us reduce this fraction to its lowest terma. We find, by trial, that 4 will exactly measure both 128 and Kit^ and, dividing, we change the fraction to its equal ||^. Again, we find that 8 is a divisor common to both terms, and, di- viding, we reduce the fracti<^n to its equal ^, which is now in its lowest terms, for no greater number than 1 will again measure them. The operation may be presented thus i , *) 8 160' y 32 40* 3^ of an acre, anawer. 5 H^ 2. Reduce |^, ^, j|^, and j4|| to the«r lowest terms. -^«. i. h h an<* i^ Note. If any number ends with a cipher, it is evidently divisible by 10. If the two right hand iigares are divisible by 4, the whole number is also. If it ends with an even number, it is divisible by 2 ; if with a 5 or 0, it is divisi ble by 5. 3. Reduce |gS, ^, j^, and f| to their lowest teitns. '1* H rm ♦ mi FBACTI0K8. ir44. =?■ 32)128(4 128 ^ f "% 44* Any fraction may evidently b< reduced to its low- %!Oiii terms by a single division, if we use t'-.e greatest common divisor of the two terms. The greatest ^ommon measure of any two numbers may be found by a sort of trial easily made. Let the numbers be the two terms c( the fraction fgj. The common divisor cannot exceed the /««« number, for it must measure it. We will try, therefore, if the leas number, 128. which measures itself, will also divide or measure 160. 12'^) 100(1 128 in 160 goes 1 time, and 32 re 128 ' ^ vMvn; 128, therefore, is not a divisor ot 160. We will now try whether this re mainier be not the divisor sought ; for il 32 be a divisor of 128, the former divi sor, it must also be a divisor of 160, which consists of 12<^ -f-'^2. 32 in lis goes 4 times, witftotU any remainder. Consequently, 32 is a divisor of 128 and KK). And it is evidently the greatest common divisor of these numbers ; for it must be contained at least once more in 160 than in 128, and no number greater than their difference, that is, greater than 32, can do it. * <* Hence, the rule for finding tfie greatest common divisor of two numbers : — Divide the greater number by the less, and that divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remain. The last dtotsor will be the greatest common divisor required. Note. It is evident, that, when we would find the great est common divisor of more th^n two numbers, we may first lind the greatest common divisor of turn numbers, and then of that common divisor and one of the (4her n'lmbers, and iio on to the last number. Then will the greatest common divisor last found be the answer. 4. Find the greatest common divisor of the terms of th( fraction §J , and,)bv it, reduce the fraction to its lowest terms OPERATION. _§iiA«ftch«i« *;■ # m FUAOTIOlfS. ir45,40. ^fe- |v I?' 9. X Biother divided ^a pie among 4 children ; what part of the pie did she give to each 1^-4-4 = how much 1 10. A hoy divided ^ of an orange equally among 3 of his companions; what was each one's share 1 ^-4-3= how much 1 11. A man divided f of an apple equally between 2 children ; what part did he give to each ? ^ divided by 2 = what part of a whole one 1 j^ is 3 parts : if each of these parts be divided into2 equal parts, they will make parts. He may now give 3 parts to one, and 3 to the other : but 4ths divided into 2 equal parts, become 8ths. The parts are now twice so many, but they are only hay so large ; consequently, f is only half so much as f . Ana. f of an apple. in these last examples^ the fraction has been divided by multiplying the denominator, without changing the numera- tor. The reason is obvious ; for, by multiplying the de- nominator by any number, the parts are made so many times smaller, since it will take so many more of them to make a whole one ; and if no more of these mialler parts be taken than were before taken of the larger, that is, if the numerator be not changed, the value of the fraction is evi- dently made so many times less. "f 46. Hence, we have two ways to divide a fraction by o whole number : I. Divide the numerator by the whole number, (if it will contain it without a remainder,) and under the quotient write the denominator. Otherwise, ' II. Multiply the denominator by the whole number, and over the product write the numerator. EXAMPLES FOR PRACTICE. 1. If 7 pounds of tobacco cost f^ of a pound, what is that per po^*nd 1 ^—Tssshow much 1 jhis. >^ of a po«nd. f»{! 2. If ^^ of an a«re produce 24 bushels, what part of an acre will produce 1 bushel 1 ^§---24=: how much 1 3. If 12 yards of silk cost |f^ of a pound, what is tbat a yard ; |^-i-12= how much ? * »/4. Divide f by 16. i biuKV; k \o I h^lnvtiintsm A .h Note. When the divisor is a composite mimber, th« in- telligeDt pupili will ipereaive, tliat he can irst divide by one component part, and the qootaent thence arising by the oth- i t- lon IS evi- f 46. 47. Ar. FRACS < -Jf er; than he may frequently shorten tlie operation. In the last example, 16=8X2, and | -7-8=^, and | -^2 = tV- -,. Divide T*j by 12. Divide ^^ ^T 2J • Divide J| by 24. 6. If 6 bushels of wheat cost Jflf what is it p^r bushel? Note. The mixed number may evidently be reduced to an improper fraction, and divided as before. jfns. ^^1=^2^ of a pound, expressing the fraction in its / owest terms. (IT 43.) 7. Divide £A^ by 9. Quot. /^ of a pound. 8. Divide 12f by 5. Qmt. ^=24. 9. Divide 14^ by 8. Quot. 1§J. 10. Divide 184^ by 7. ^is, 26tV- ybte. When the mixed number is large, it will be most convenient, first, to divide the whole number, and then re- duce the remainder to an improper fraction ; and, after di-. viding, annex the quotient of the fraction to the quotientjof the whole number ; thus, in the last example, dividing 184j by 7, as in whole numbers, we obtain 26 integers, with 2^ 1=1 remainder, which, divided by 7, gives -^ and 26 -f- j^ =^(j^, Am. 11. Divide2786|by 6. ^ns. 464g. 12. How many times is 24 contained in 7646^^ 1 Ana. 318f4f 13. How many times is 3 contained in 462^1 . AuB. 1.54^ \ ' 7\) multiply a fraction by a whole nunUier. IT 47. I. If 1 yard of cloth cost ^ of a pound, what will 4 |2 yards cost 1^x2:= how much 1 2. If a cow consume ^ of a bushel of meal in 1 day, how [much will she consume in 3 days ? ^X3 as how much 1 3. A boy bought 5 cakes, at ^ of a shilling each ; what 1 [did he give for the whole 1 ^X5 = bow much 1 4. How much is 2 times ^ 1 3 times ^ 1 2 f himesf 1 5. Multiply ^ by 3. f by 2. i by 7. 6. If a man spend f of a shilling per day, how much will I he spend in 7 days? f is 3 parts. If he spend 8 such parts in 1 day, he will {evidently spend 7 tines 8, that is, V^=2| in 7 days- i^ 108 rRACTIOlft. H 4T, 48. # M % Hence, vre perceive, a fraction is muUiplied by multiplying the numerator, without changing the denominator. But it has been made evident, (f 46,) that multiplying the denominator produceti the tame effect on the value of the frac tion, 38 dividing the numerator i hence, also, dividing the de- nominator will produce the same effect on the value of the fraction, as multiplying the numerator. In all cases, there- fore, where one of the terms of the fraction is to be multiplied the same result will be effected hy dividing the other ; and where one term is to be divided, the same result may be effect- ed by multiplying the other. This principle, borne distinctly in mind, will frequently enable the pupil to shorten the operations of fractions. Thus, in the following example : At ^ of a pound, for 1 pound of sugar, what will 11 pounds cost 1 Multiplying the numerator by 11, we obtain for the pro- duct |§&=^ of a pound for the answer. IT 48. But by applying the above principle, and divid- ing the denominator, instead of multiplying the numerator vie at once come to an answer, §, in much lower terms. Hence, there are tyro ways to multiply a fraction by a whole number: 1 . Divide the denominator by the whole number, (when it can be done without a remainder,) and over the quotient | write the numerator. — Otherwise, ;- II. Multiply the numerator by the whole number, and un- der the product write the denominator. If then it be an improper fraction, it may be reduced to a whole or mixed number. EXAMPLES FOR PRACTICE. 1. If one man consume ^ of a barrel of flour in am9nth, how much will 18 men consume in the same time ? —-— 61 men 1 9 men 1 Ana. to the last. 1^ barrels. 2. What is the product of -^ multiplied by 40? ^X\ 40 c= how much ? Am. ^. 3. Multiply ^ by 10. by 18. by 21. -,— by 36. by 48. by 60. Note. When the multiplier is a compoeite number, the I pupil will recollect (IT U,) that he may multiply first byj pne component part, and that product by the other. Thus, *tT't i therefore, tWX12=|J, and 13X5=^^=^, An*. 4. Multiply 5^ by 7. Ann. 40 j. yote. It is evident, that the mixed narober may be re- duced to an improper fraction, and multiplied, as in the pre- ceding examples ; but the operation will uraally be shorter, to multiply the fraction and whole number Beparattly, and add the results together. Thus, in the last example, 7 times 5 are 35 ; and 7 times f are ^=a5^, which, added to 35, make 40|, Ans. Or, we may multiply the fraction first, and, writing down the fraction, reserve the integers, to be carried to the product of the whole number. 5. What will m tons of hay come to at 3;^ per ton T Ans. 28i:. 19s 6. If a man travel 2^^ miles in 1 hour, how far will he travel in 5 hours 1 in 8 hours 1 — : in 12 hours ■» "V«. in 3 days, supposing he travel 12 hours each day ? Ans. to thtlast. 77f miles. Note. The fraction is here reduced to its lowest terrott the same will be done in all the following examples.^ .«■ 7\) multiply a whole number by a fraction: ' '^ " IT 49. 1. If 36 pounds be paid fur a piece of cloth, what costs ^ of it 1 36x^how much ? ^ of the quantity will cost f of the price ; } a time 36 pounds, that is J of 36. pounds, implies that 36 be first di- vided into 4 equal parts, and then that 1 of tliese parts be taken 3 times ; 4 into 36 goes 9 times, and 3 times 9 is 27 . Ans. 2? pounds. From the above example it plainly appears that the ob- ject in multiplying by a fraction, whatscer may 6« the multipli- cand, is to take qjf the puiHpUcm^ a part, deiwted by the mul- tiplying fradion ; andthat this operation is composed, of two others, viz. a division hy the denominator of ttie multiply- ing fraction, and di muUiplieation of the quotient by the nu- merator. It is a matter of indifference, at it respects the retidt, which of these operations precedes the other, for 36 X3-r4=27, the same as 36-7-4x3=^7. Hence, — 7b muttipli by afratHon, ichether ths multiplicand be a whole number or a fraction, — Rule. Divide the multiplicand by the denominator of (he multi- K s 110 FRACTIOIVS. f 60. #" f plying fraction, and multiply tho quotient by the numera- tor ; or, (which will often be found more convenient in prac- tice,) Ant multiply by the numerator, and divide the product by the denomii)ator. Multiplication, therefore, whenappplied to fractions, does not always imply augmentation or increase, at in whole numbers} for, when the mutliplier is lessthaniim/y, it will always require the product to be less than the multiplicand, to which it would be only equal if the multiplier were 1 . We have seen, (ir 10,) that, when two numbers are mul- tiplied together, either of them may be made the multiplier, without effecting the result. In the last example, there- fore, instead of multiplying IG by ^, we may multiply ^ by 16, (IT 474) and the result will be the same. >ri iwf(>m. EXAMPLES FOR PRACTICE. { . v 2. What will 40 barrels of meal come to at ^ of a pound per barrel 1 40Xf=bow much 1 .3. What will 24 yards of cloth cost at f of a pound per yaid 1 24Xii:=:how muchi mfdm Hi^iii^di^:^;^:' ;.:^-^^^. 4. How much is ^ of 90 1 — -f of 3091 ^rjOdSt r». Multiply 45 by /^i Multiply 20 by ^. ,^ . . * . Jb multiply one fraction by another. IT 50, 1. A man owning ^ of a farm, sold § of his «ihare ; what part of the whole farm did he sell 1 § of | is h»w much 1 5w We have just seen, (IT 49,) that to multiply by a fraction, is to divide the multiplicand by the denominator, and to mul- tiply the ^quotient by the numerator. ^ divided by 3, the de- nominator of 4he multiplying fraction, (IT 46,) is -^, which, multiplied by 2, the numerator, (IT 48,) is -j^, j^8. The process, if carefully eonsidered, will be found to con- sist in multiplying together the two numerators for a new nu- merator, and the two denominators for a new denominator. ■^,ai m^fMy EXAMPLES FOR PRACTICE, ii , .i^t: 2. A man, having ^ of a pound, gave ^ of it for a din- ner what did the dinner cost him 1 jtns. -^ pound. 3. Multiply I by f . Multiply ^^ by f Product, ^^ 4. How much is f of § of 1^ of 1 1 Note. Fractions like tho above, connected bv the word 150,51. 7^ CTIOllt. Ill word ^- oft are sometimes called oompottni/ fraction*. The or implie$ their continual multiplication into tack otktr. When there are several fractions to be multiplied con- tinually together, as the $ev*ral numerator* wea factor* of the new numerator, and the *everal denominator* *lvo factor* oi the new denominator, the operation may be shortened by dropping those factor* which are the tame in both terms, on the principle explained in IT 43. Thus, in the last example, ^, f , h h ^^ ^nd ^ 4 ^^^ ^ 3 ^^^^ amon^ the numerators and among the denominators ; therefore we drop them multiply- ing together only the remaining numerators, 2X7 = 14, tor a new numerator, and the remaining denominators, 5XS = 40, for a new denominator, making ^^ = ^, An*, as before. 5. f of f of f of f of T^V of I off == how much 1 An*. ^. 6. What is the continual product of 7, ^, f of f and 3^1 Not*. The integer 7 may be reduced to the form of an improper fraction, by writing a unit under it for a denomin- ator, thus, |. An*. H^. 7. At ^ of a pound a yard, what will j- of a yard of cloth cost 1 S. At If pounds per barrel for flour, what will /^^ of a barrel cost 1 > 'i'-; -sis"!'.. ?•■ ' ■a^mul^. m ■ l|B=y then V X-ft^iVff^- Ans. r 9. At f of a pound, per jiA, what cost 7f yards t An*. 6^^'. 10. At $2^ per yard, what cost 0| yards 1 An*. $14§|. 11. What is the continued product of 3, f, f of |, 2^, and +Aof£of 41 „ ,^ , ,' , _. , ^na- iik- IT 51, The RuLK for the roultiplicatioifoffraotioni may now be presented at one view : I . 7b multiply a fraction by f tohole number ^ or a whole number by a/roctton.-MDivide the denominator by the wkole number, when it can be done without a remainder ; other- wise, multiply the numerator by it, and under the product write the denominator, which may then be reduced to a whole or mixed number. II. 7b multiply a mixed nund>er by a whole number, — Multi- ply the fraction and integers, separately, and add their pro- ducts together. III. /b multiph/ one fraction by a7to/A«r,— Maltifily togeth- y*t '^- # m lis rHACTt •V- K61,G2. er the numerdiori for a new numerafbr, and the derwrnina- tort for a new denominator. Note. If either or both are mixed numbera, they may, first be reduced to improper (Vaetions. <^ -*'^ ^^ EXAMPLES FOR PRACTICE. 1. At f jff per yard, what cost 4. yards of cloth 1 5 "y Js. l 6 yds. 1 8 yds. ? 20 yds. 1 Ans. to the last, 16£. I 2. Multiply 148 by j. by J. by ^. by ^. " :'■'. ,„,. ; ,, J^f^ prodMty 44^. ^. If 2 ^9 tohs of hay keep I horse through the winter how much will it take to keep d horses the same time 1 7 horses 1 13 horses 1 Ann. to the last, 37^,y tons. 4. What will 8/j barrels of cider come to, at 7 shillings per barrel 1 5. At 14f<£ per cwt. what wilt be the cost of 147 cwt.l ' 6. A owned f of a hdte ; B owned ^ of the same ; the note amounted tb l000<£ ; what was each one's share of the money % . i "•^** 7. Multiply i of § by f of |. ^rt&c^ f Product. ^. 8. Multiply 7^ by 2 ^, , Product. 16^. 9. Multiply I- by 2§. tstiitf ji^i^jtiooq Product. 2f 10. Multiply f of 6 by f. "' Product, 1. 11. Multiply I of 2 by I of 4. Product, 3. 12. Multiply continually together ^ of 8, § of 7, f of 9, and I of 10. ' Product, 20. 18. Multiply lOOOeOOby f*rtw .im-^ Product, 555555f. Jb dt«{(2« a tbhoU number fey a/rocfion. *''r^ * T 5l2. We have already shown, (IT 46,) how to divide a fractionlby a #ho)e number ; we now proceed to show how- to divide a whole number by a ifhietion. 1. A man diTided ft£. ^m<»ig.totne poor people, giving them ^ of a pound eadi ; llew maiiff were the persons who received the money % %^^tsss how many 1 1 pound is ^, and 9 pounds is 9 tiiiies as many, that is, 1^, ; then f is oovltoined in ^as many times as 3 is con- tained in 36. Ana. 12 persons, Hi Tiktit M,-^MuUijphf the iwitkndhy the lUnoininator of the dhiiing fraction, (thereby reducing the dividend to parts of the same magnitude as the divisor,) and diind* the pro- i^ 6y Uit mmiruitor. •', 4- JP- * 'rii^' to show how 2. How ntoy tiims is ^ conUintd in 8 1 8-r^ how many 1 „t.,^ OPERATION. 8 Dividend. 5 Denomlnatof. 1 J t^'- i Numerator, 3)40 ^fV-U* i i'. ;t. *^ "K SB, M. *i. How many tiiuea can I draw ^ of a gtllon of wine out of a cask containing '20 gallons 1 3. Divide 3 by }. by jf. 10 by f . 4. If a man drink j^^ of a quart of rum a day, how long will 3 gallons last hlni 1 5. If 2^ bushels of oats sow an acre, how many acre» will 22 busheli sow 1 22 -7- 2^ =s how many times 1 Note. Reduce the mixed number to an improper frac tion, 2}t=si y . ifn«. 8 acres. 6. At l^jfc' a yard, how many yards of cloth may bo bought for 37 £ 1 An$. 26^ yards. 7. How many times ^^j contained in 84 1 <* t 4* /fna. 9()| times. 8. IIow many times is ^ contained in tl 1 /ins. I of 1 time. 0. How many times is 8f contained in 53 1 u> hct.ij, .u^K:;>i ^1^9' ^k times. 10. At g of a pound for building 1 rod of stone wall, how many rods may be built for 87 j€ 1 87 — | == how many times'! ^ '^■. «■??•,'--!, 7b divide one fraction by another. ^ o 1, 1 . At ^ of a pound per barrel, how much rye may ko bought for £ of a pound ? | is contained in |. how ma- ny times ! ' .,*,..-., ..,-i... , <• Had the rye been 2 whole pouifids per barrel, instead of 5 of a pound, it is evident, that ^ of a pound must have been The process is that already described, IF 52 and H 53. If r.arefully considered, it will be perceived, that the numerator of the divisor is multiplied into the denominator of the divi- dend, and the denominator of the divisor i^to the numerator of the dividend ; wherefore, in practice, it will be more con venient to invert the divisor ; thus, -f inverted becomes ^ : then multiply together the two upper terms for a numerator <* and the two lower terms for a denominator, as in the inulti- f 54,15. # * rilAOTIOIIS. K 115 plication oT odc fraction by another. Tbut, iu the above example, » X «)^ ^ 2 X 5 '^ 10' at before , EXAMPLES FOR PRACTICE "2. At I of a pound per bushel fnr wheat, how many bushels may he bought fur | of a pound % How many times is I contained in j^l jfnn. !U bushels. n. If ^ of a yard of cloth cost I of a pound, whnt is that per yard? It will be reeollected, (f 24,) that wh«n tlM cost of any quantity is given to find the priet of a unit, we divide the cost by the quantity. Thus, | (the cost) divided by I (the quantity) will give the price of 1 yard. j£n9. §1 of a pound per yard. Proof. If the work be right, (If 10, " Proof,") the pro- duct of the quotient into the divisor will be equal to the dividend ; thus, §| X & = ?• This, it will be perceived, is multiplying the price uf one yard (§|) by the quantity {}) to find the cost (^ ;) and is, in fact, reversing the question, thus, if the price of 1 yard be §| of a pound, what will I of a yard cost 1 Ana. ^ of a pound. ybte. Let the pupil be required to reverse and prove the succeeding examples in the same manner. " 4. How many busheU of wheat, at yV of a pound per bushel, may be bought for ^ of a pound 1 Ans. 4$ bushels. •>. If 4^ pounds of butter serve a family 1 week, how many weeks will 36| pounds serve them 1 The mixed numbers, it will be recollected, may be re- duced to improper fractions. 0. Divide ^ by ^. Quot. 1 7. Divide | by J. Quat. 3 8. Divide 2^ by 1^. uins. Syjy weeks. Divide^ by 1^. Quot.'i. Divide f by T?^. Quot.^. 4)ivide lOf by 2|. quot. 4|^. 9. How many times -^ contained in f 1 Ans. 4 times. 10. How many times is ^ contained in 4^ 1 ^%. Ana, llf times. 11 . Divide f of f by I of |. *f f^l ^ *^k <^? ^^~* quot. 4. H 55. The Re LB for dttn«to»i of fraetinns may now be presented at one view :-~ I. 7b divide a fraction by a whole n%mU»er,— Divide the 116 rBACTfom. ii,5n. numirator by the whole number, when it can ha done iritb- iitit a remainder, and under tho quotient write the denomi- nator ; otherwise, multiply the dtnominator bjr it, and over the product write the numerator. II. 7b divitU a idAo(c numbir by a fraction, — Multiply the dividend by the d$nomina^r of the fraction, and divide the product by the numerator, III . 7b dividt one fraction by another, — Invert the divinr, and multiply together the two upper term* for a numerator, And the two lower termi for a denominator. yott. If either or both are mixed numberi, they may be reduced to improper fractions. , EXAMPLES FOR PRlCTICE. ' I. If 7 lb. of tobacco cost ^^.j ^^ * pound, what ii it per pound 1 ^-^7c= how much ? | of ^(y\r is how much 1 '2. At ^£. for f of a barrel of cider, what is that per bar rel? 3. If 4 poundt* of sugar cost ^V of a pound, what does 1 pound cost 1 4. If ^ of a yard cost 13s. what is the price per yard 1 o. If* 141 yards cost 42£. what is the price per yard 1 6., At 4|} pounds (or 10^ bars els of cider, what is that per barrel 1 yfn$. f£. 7. How many tiroes is | contained in 74G? jfns. 1989^. 8. Divide jj of f by |. Divide J by | of § quot. |. Quot. 3^1. :_, 9. Divide } of f by | off ^ - QiwL ^f. ■ 10. Divide^ of 4 by ,*5. ' '* * -'^ Quot. 3. H. Divide 4f ?jy f of 4. Quot. 2^. J 2. Divide ^ of 4 by 4 1. quot. |f . -v %?' <;■ %V. ADDITION AND SUBTRACilON OF FRArT^ IT 56. 1 . A boy gave to one of his companiou* f of an orange, to anoth^ |, to another | ; what part of an orange did be give to all 1 |-(-|-j-^rs how muchi jiks. |. 2. A i.'^vr con:iumes, in 1 month, -^ of a tonuof hay ; a horse, in the same time, consumes -^ of a ton ; and a pair of oxcL 1^-;^ hov much <.o they all consume 1 how much more dwri thft t\orse contv^ne than the cow 1 — - the oxen no with* deoomi- Kod over AfuUiply nd divide • diviwr, jinerator, J may bo itii it per much 1 it per bar hat does 1 er yard I or yard 1 at is that rns. 1989i. \uot. m (iuot. if. Qiiot. 3< Quo*, f^. ^T^;T^S. f 50, r»7. mACTIO!!*. in » of an I an orange Am. l- hay; a land a pair low much the oxen than the horsol i^j-f l^-^^^-« how muc . / ^—S^ how much 1 1*5— A"" **°^ "****** ^ 3. ^^^-^^=a how muchi f — |^s: huNV muehl 4. 3S-h^+A-fi«H-:^^T«= bow much 1 jj -^^-b how much 1 rt. A lioy, having J of an apple, garo | of it to his tis* ter ; whtt part of Iho appio had he left 1 J — i= how much 1 -'-nMTi.-.' -nT When ;t< : ^ ^o ninators of twoor more fractions are alike (as in t . fo.igom^'' examples,) they are said to havoacom- mon do n tor. The parts are then in the same denom- itii> "1 ] 1. Reduce f, f and f to fractions of equal value, having a common denominator. J)X4X5=s60, the common denominator. '**" '"*' '" t>X4X«^=40, the new numerator for the first fraction. 8X 3X^=^45, the new numerator for the second fraction «)X4X4=:49i the new numerator for the third fraction. Tho new fractions, therefore, are |^, |^, and |§. By an insr^ctioDof the operation, the pupil will perceive, (hat the uirmerator and denominator of each fraction have been mul- tkplied by the same numbers ; consequently, (IT 43,) that their value has not been altered. 3. Reduce to equivalent fractions of a common denomi- nator, and add together, ^, f , and ^. '-^f"^^-- ' ■ • "":. -: ^«. fa-fH-l-4*=H=lH. amount. 4. Add together f and f. Amownt^ \\\. 5. What is the amount of ^^^-f.|-)-| 1 Ans, ffJ«=lj^Y^. 6. What are the fractions of a common denominator equivalent to \ and \ 1 Ans. ^\ and f ^, or -^ and \^. We have already seen, (IT 56, ex. 7,) that the comtMm denominator may be an^ number, of which each givm de- nominator is a factor, that is, any number which may be di- vided by each of them without a remainder. Such a number IS called a common multiple of all its common divisors, and the least number that will do this is called their least com- mon multiple ; therefore, the least common denominator of any fractions is the least common multiple of all their denominators. Though the rule already given will always find a common multiple of the given denominators, yet it will not always find their least common multiple. In the last example, 24 k evidently a common multiple of 4 and 6, for it will exactly measure both of them ; but 12 will do the same, and as 12 is of 4 K 67, 58. rRACTions. 119 a til. i ittS 12 is the Uaat number that will do this, it is the l*a»t common multiple 0(4 and 6. It will therefore be convenient to haT* a rule for finding this least common multiple. Let the num- bers be 4 and ti. It is evident, that one number is a multiple of another, when the former contains all the factors of the latter. The factors of 4 are 2 and 2, (2x2== 4.) The factors of 6 are 2 and Ji, (2X3 = 6.) Consequently, 2X2X3 = 19 contains the factors of 4 that is, 2X2 ; and also contains the factors of 6, that is, 2X3. 12, then, is a common multiple of 4 and 6 and it is the least common multiple, because it does not contain any factor, except those which make up the numbers 4 and ; nor either of those repeated more than is necessary to produce 4 and 6. Hence it follows, that when any two numbers have a factor common to both, it may be once omitted ; thus, 2 is a factor common both to 4 and 6, and is consequently once omitted . ^m, < ' H 58. On this principle is founded the Rvlw: for finding the least common multiple of two or more numbere. Write down the numbers in aline, and divide them by any number that will measure two or more uf them ; and write the quo- tients and undivided numbers in a line beneath. Divide this line as before, and so on, until there are no two numbers that can be measured by the same divisor ; then the contin- ual product of all the divisors and numbers in the last line will be the least common multiple required. Let us apply the rule to find the least common multiple of 4 and 6. *•: • ^"-. - >. < ; • i, 4 and 6 may both be measured by 2 ; the 2)4 . 6 quotients are 2 and 3. There is no number great- er then 1, which will measure 2 and 3. There- 2 .3 fore, 2X2X3 s= 12 is the least common mul- . ft' ,.^^ tiple of 4 and 6. If the pupil examine the process, he will see that the di- visor 2 is a factor common to 4 and 6, and that dividing 4 by this factor gives for a quotient its other factor, 3. In the same manner, dividing 6 gives its other factor, 3 . Therefore the divisor and quotients make up al) the factors of the two numbers, which, multiplied together, must give the com- mon multiple, u ■^ri' mtt tit*^'- . - - /'ifTwH'T *i»'*? 'rfTffk' -vf**} •»« 120 rUACTIONS. IT 58. 7. Reduce ^, ^, ^ and ^ to equivalent fractions of the least common denominator. OPERATION 2 )4 ■ 2 . 3 . fi 3)2 . 1 . 3 . a 2 . 1 1 Then, 2X3X2 = 12, least common denominator. It is evident we need not multiply by the Is, as this would not alter the number. To find the new num* rators that is, how many 12th8 each fraction is, wo may take ^, ^, §, and I of 12, Thus: J of 12=9 >^ jjg^ numerators, which, C t^^ written over the com- ^ ^ t ^of 12=6 f of 12=8 I of 12=i2 iV<.' mom denominators, give / . gCZi ]I2 — 5 jins. ^, T^v, ,^, and -j^. 8. Reduce ^, f , and ^ to fractions having the least com- mon denominator, and add them togethei. jins. ^f-|-/^-ff^=||==l||, amount. 9. Reduce ^ and ^ to fractions of the least common de- nominator, and subtract one from the other. ,,- .^«^ ,^W -.,-- ^8. Y^ — ^=^, difference. 10. What is the least number that 3, 5, 8 and 10 will measure 1 jins. 120. 11. There are 3 pieces of cloth, one containing 7| yards, another 13^ yards, and the other 15|^ yards ; how many yards in the 3 pieces. -^.r '\ ^ Before adding, reduce the fractional parts to their least common denominator ; this being done, we shall have, Adding together all the 24ths, viz. 18-f 20 7J= 7^f ^ -j-21, we obtain 59, that is, ^1=^. 13^=13^$ ? We write down the fraction ^ under the i5|=15f^ ) other fractions, and reserve the 2 integers /t 0711 ^^ ^® carried to the amount of the other in- j3n». «57^^ tegttn, making in the whole 37^ jfns. 12. There was a piece of cloth containing 34f yards, from which were taken 12f yards ; how much w^s there left 1 We cannot take 16 twenty-fourths 34| = M^ (^) from 9 twenty-fourths, c^ ;) we 12§ 3= 12^1 must, therefore, borrow 1 integer,ssr24 jffiB 2m lids twenty -fourths, (J|,) which, with 3^, jsns. 4iii yas. j^Mk^^^ . ^e can now take ^ from }f , and there will remain ^l ; but, aa we borrowed, ao also t» f 59, 60. BEDCCTIOlf OP FBAGTIOSB. 131 we must cany 1 to the 12, which makes it 13, and 13 from 34 leaves 21. ^n«. 21^. 13. What is the amount of ^ of ^ of a yard, ^ of a yard, and I of 2 yards 1 Jfote. The compound fraction may be reduced to a «m- jile fraction ; thus, ^ of | = f ; and ^ of 2 = ^ ; then, | -f J -I- f r= ijg = 1 ^^ yds., anmoer. T 59. From the foregoing examples we derive the fol- lowing Rule : — To add or mMr act fractions ^ add or subtract their numerators, when they have a common donominator ; otherwise, they must first be reduced to a common denom- inator. Note. Compound fractions must be reduced to simple fractions before adding or subtracting, j \,, .h^^^^ ,.^^, EXAMPLES FOR PRACTICE. ^vv h 'i 1. What is the amount of f , 4§ and 12 1 Jtna. Y1^{ 2. A man bought a farm, and sold ^ of ^ of it ; what part of the farm had he left 1 Aruu |. 3. Add together ^, ^, ^, -jZ^, ^ and ^^ 1 Amownt. i|| 4. What is the difference between 14^ and 16i^ 1 5. From 1^ take f Jr mH '*■ v , 6. From 3 take ^. ^ tid^ .♦< 7. From 147^ take 48|. i = 8. From \ oi ^ take \ of ^. 9. Add together 112^, 31 If, and lOOOf . 10. Add together 14, 11, 4f , ^ and ^. 11. From I take ^. From | take f . 12. What is the dUference between ^ and ^1 ^ and A 1 I and § 1 ^ and f 1 f and ^ I f and f 1 13. How much is \—\ ? 1— A 1 l—f 1 1— f 1 2—^? 2-41 2i~i1 3|-:jV'? 1000-^1 Remainder, Remainder^ 2§ .; /2em. 98 f Rem 37 - jt'*.' '. . til' "■jj **"v f -?■, We have seen (f 27,) that integers of one de- 4^ REDUCTION OF FRACTl&NS. ■\'}itt ireo. nomination may be rbduced to integers of another denomt natum. It is evident, that Jractions of one denomination, after the same manner, and by the same rules, may be re- ^ duced to fradions of another denomination ; that is, frac- tioMt like integers, may be brought into lower denoroina • J. ...J,. ...._. . (jS.iv& ■ ti 122 ■ EDUcrioir or FBACTioifn. If 60. fl- i. tions by multiplication, and into higher deifbininationt by division. 7\j reduce higher into lowbr denominations. "■^' ■ (RyLE. See II 28.) 1. Reduce ^^^ of a pound to pence, or the fraction of a penny Note. Let it be recollected ed either by dividing the nu- ihat a fraction is multiplied ei- ther by dividing its denomin ator, or by multiplying its nu erator. -^ ' " ; '^ i d. y/ns. Or thus : aizy of Y o( V= IIB=f of a penny, jina J. Reduce j^srs of a pound he fraction of a farthing *! =T^^ d.X4=-^^\j==f q. V Or thus : Num. 1 ■•<»•!- . 20 8. inl.£. 20 12 d. in 1 s. 240 4 q. in 1 d. i«r /j 7b reduce totcer into higher denominations. '^ (Rule, See H 28.) 2. Reduce f of a penny tc the fraction of a pound. Note. Division is perform- 900 merator, or by multiplying the denominator. fd.-:- 12==^ 8.-20 = ^^^£.Ans. <'■-' ■'^; Or thus: f of -j^ of 5^ == 1% J. 4. Reduce f of a farthing to the fraction of a pound. i q.-r4=^d.H-12=T.t3r8. Or thus: >«^ .u. Denom. 4 .f^ ,.r 4 q. in 1 d. «'5 >.t 12 d. in Is. 5rt I rt ' Then, y^V^^f q- -<^«' 5. Reduce ^g^w of a guin ca to the fraction of a penny. 7. Reduce |^ of a guinea to "* the fraction of a pound, ji Consultir 28, ex. 12. 9. Reduce f of a moidore, A .1 m;vci\£M ^\: 3840 ' * i'f««> * Then TrATJ=TaVir^' •^*- 6. Reduce f of a penny to the fraction of a guinea. 8. Reduce f of a pound to the fraction of a guinea. 10. Reduce ^ of a guinea at 1<£. 10s. to the fraction of to the fraction of a moidore. a guinea. 1 1 . Reduce t^ of a pound, Tro}-. to the fraction of an ounce. JTroy. J #t 12. Reduce | of an oance to the Iracti^D of a pound •160, (il. HKOVCTIO?! Of VRAOTIOMt. ^ 123 13. Reduce 3»g of a pound,! 14. Reduce f of an ounce avoirdupois, to the fraction of to the fraction uf a pound an ounce. avoirdupois. 15. A man has y^t of a 16. A man has -^ of a pint hogshead of wine ; whatpartlof wine ; what part is that ot is that of a pint 1 "a hogshead ? 17. A cucumber grew to the IH. A cucumber grew to length of ^g'^xT of* »n>l«i what the length of 1 foot 4 inches part is that of a foot! j=|§=4 of a foot ; what part 19. Reduce f of ^ of s pound to the fraction of a shii ling. 21. Reduce ^ of tSt o^ 3 pounds to the fraction of a ir 61. It will frequently is that of a mile 1 20. f^ of a shilling is § of what fraction of a pound 1 22, ^ of a penny is ^ of what Cniction of 3 pounds 1 ^ of a penny, h ^ of what part of 3 pounds ? i^Bf of a penny is ^ of ^ of how many pounds 1 i:- It will frequently be requir- he required to^nd the vahte c/led to reduce integers to the afractum^ that is to reduce afraction of a greater denomin- fraction to integtes of lest de- ation. . .•> •Uaom « m -j^ M nominations. ,; ,i.,[h ?.j."«Atuuua t'ib> ^maaA ^--fiub v., 1 . What is the value of | % Reduce 13 a. 4 d .' to the of a pound 1 In other words, fraction of a pound. Reduce § of 9 pound to fhil- 13s. 4d. is 160 pence ; there lings and pence. , >i .] t are 240 pence in a. pound? § of a <;e is y '^'Idi ^^ therefore, 138. 4d. is ^f^^ lings; it is evident from ^ of of a pound. Th»t is,— Re- a shilling may be obtained duce the given sum or quan- 6ome pence ; ^ of a shilling istity 1|p the least denomination 1^=4 d. That is, — Multiply mentioned in it, for a numer the numerator by. t^at nua-fitor; then reduce an integer ber which will reduce i|t to of that greater denomination the next less denoqaination, (to a fraction of which it is and divide the product by the required to reduce the given denominator ; if there be a re- sum or quantity) to the same mainder, multiply and divideidenomination, for a denomi- as before, and so on ; the sev-nator, and they will form i^n eral quotients, placed one af-iraction required., ctr*/ il teranother, in their order, wiill I ^Tf xtooifiwob^^ tl. be the answer. mi «• 124 I) II t', ! ^* iTBOtrOTIO!! or rRACTIOlf*. EXAMPLES FOR PRACTICE. IT 61. 3. What is the value of } 4. Reduce 4d. 2q. to the or a shilling 1 fraction of a shilling. OPERATION. Numer. 3 12 * ir- • DeDOiii.8)30(4d. 2q. i^s. 32 , •j ■'■*• rt. ^*r'*"j hi 16(2q. 16 ;s;fw -luiA a \o .f.-.tj 5. What 18 the value of f of a pound Troy 1 7. Whatii the value off of a pound avoirdupois 1 9. f of a month is how ma ny days, hours, and minutesi 11. Reduce f of a mile to its proper quantity . "Q > ^'^ t8. R»dtf«e ^ Of tt^ aen to its proper quantity. 15. What is the value of If of a dollar in shillings, pence, ke. 1 17. What is the value of ^<, of a yard 1 19 1^ of a ton. »"»>j«u fe OPERATION. 4 d. 2q. 1 s. 4 12 rH 18 Numer 12 4 M,' 48Denom. 6. Reduce 7 oz. 4 pwt. to the fraction of a pound Troy. 8. Reduce S oz. 14f dr. to the fraction of a pound a- voirdupois. Note. Both the numerator and the denominator must be reduced to 9ths of a dr. 10. 3 weeks. Id. 9h. 36m. is what fraction of a month t 12. Reduce 4 fur. 125 yds. 2ft. 1 in. 2| bar. to the fhtc- tion of a mile. 14. Reduce 1 rood 30 poles to the fraction of an acre. 16. Reduce 4s. 8^d. to the fraction of a dollar. i»U«r 18. Heduee 2 ft. 8 in. l^b. to the fraction of a yard. What is the value of 20. Reduce 4 cwt. 2qr. 12 lb. 14 OB. 12^ dr. to the frac- Ition of a ton. 1^ ,.:i. Jfot: Let the pupil be required to reverse and prove the following examples :,^ ,ioi«rt;V9* ^m ^ «« gf JjHii ^witMy^; xj. 21. What is the value of ;^f of a guinea 1 22. Reduce 3 roods 17^ poles to the fraction of an acre. T61. •CPPLBMBIIT TO rBACTI«»l. 135 * IT 61. ■■••'t ■•■■ Iq. to the 1 «. If. - 4 rSDenom. , .,» 4 pwt. to md Troy. . 14f dr. pound a- numerator ir must be I dr. 9h. 36ni. I month t 125 yds. the frac- 90 poles acre, ^d. to the in. l^b. ^ard. 2qr. 12 I the frac- Irove the in acre. 11 23. A man bought 27 gal. 3 qts. 1 pt. or molasses ; what part is that of a hogshead 1 24. A man purchased ^^ of? cwl. of sugar ; how much sugar did he purchase 1 )., ^ ,. 25. 13h. 42m. 51 ^s. is what part or fraction of a day ! 1-^ /W»r.fl 1 .w "•'■■"J '*^ SUPPLEMENT TO FRACTIONS. .•J B i. What are fractions f 2. Whence is it that the parts into whicii any thing or any number may be divided) take their name ? 3. How •ire fractions represejited by figures ? 4. What is the number above the line called ? — Why is it so called ? 5. What is the number below the lino called ? — Why is it so called ? — What does it show ? 6. Wha t is it which determines the magnitude of the parts ? — Why ? 7. What is a simple or proper fraction ? an improper fraction a mixed number ? B. How is nn improper fraction reduced to a whole or mixed numbei i 9. How is a mixed number reduced to an improper frac- tion ? u whole number P 10. What is undeistood by the terms of the fraction? 11. How is a fraction reduced to its most simple or lowest terms ? 12. What ia understood by a common divisor ? by the greatest common divisor ? 13. How is it found ? 14. How many ways are there to multiply a fraction by a whole number.' l-'i. How does it appear, that dividing the denofttinaJLor multiplies the frac- tion? 16. How is amixed number multiplied ? 17. Wtiatis implied in multiplying by a friction? 18. Of how man 3^ operations does it consist ? — What are they ? 19. When the multiprier is less timii n unit, what is the product compared with the multiplicand f 20. How do yuu multiply a whole number by a fraction .-' 21. How do you multiply one fraction by another .' 22. How do you multiply a mi \ 23. How does it appear, that in multi- plying both terms uf the fraction by the same number the value of the fraction is not altered .' 24. How many ways are there to divide .1 fraction by a whole number? — What are they.' 25. How docft it appear that a fraction ts divided by multiplying Us denominator ;' '^{,. How does dividing by a fraction differ from multiplying by a fruction ' 27. When the divisor is less than a unit, what is the ({uoticnt com- pared with the dividend? 2H. What ia understood by a common de- nominator ? the Least common denominator? W. How docs it appear, that each^teen dcnominatior must be a factor of the common denominator ? 20. How is the common denominator to two or morc^ fractions found ? 31. What is understood by a multiple ? by u common mxdtiple ? -.— — by the /east common multiple 1 — Whnt is the. Srocess of finding it? 32. How are fractions added and subtracted ^ 3. How is a fraction of a greater denomination icduced to one of :i less ? of a less to a greater ? 34. How are frnctions of a greater denomination reduced to integers of a less? integers of n loss de- nomination to the fraction of a Ereatcr ? L2 " ^ 196 ■ •*! T, ■tlP»LBMBl«T TO VIIACTI0K8. V 61, 62. ill
  • ii^Q^ «IV li ^ f 68. We have seen, that an individual thing or num- ber nay be divided into any number of equal parts, and that these parts will be called halves, thirds, fourths, fifths, sixths, &c., accoiding to th« number of parts into which the thing or number may be divided : and that each of these parts may be again divided into any other number of equal parts> and so on. Such are called common or tmjgar/racttons. Their denom- inators are not uniform, but vary with every varying diviiion of a unit. It is this circumstance which oeoasions the chief difficulty in the operations to be performed on them ; for when numbers are divided into di£ferent kinds or parts, they cannot be so easily compared. This difficulty led to Uie in- vention of decimal fractions, in which an individual thing, or number, is supposed to be divided first into tm equal parts. f 04. DBCIMAL rRACTIOIVS. ^M • n them ; for which will be ttnth» ; ami Mch of these parts to bo a^ain di- vided into ten other equal partN, whicli will bo hundredth$ ; and each of these parts to be still further divided into ten other equal parts, which will be thou»andtha ; and so on. Such are called decimal fr action* , (fronn the Latin word decern, which signifies ten,) because they increase and decrease, in a tenfold proportion^ in the same manner as whole numbers. T 64. In this way of dividing a unit, it is evident, that the denominator to a decimal fraction will always be 10, 100, 1000, or 1 with a number of ciphers annexed ; conse- ^quently, the denominator to a decimal fraction need not be [expressed, for the numerator only, written with a point before it (') called the »eparatrix, is sufficient of itself to express the true value. Thus, ;; - .^->«cj ,1 lt.i:':!ji:' '» m. •^ are written '6. 'fW '27. <• AiiU The denominator to a decimal fraction, although not ex- i pressed, is always underttood, and it 1 with as many ci- ! phers annexed as there are placee in the numerator. Thus, I '3765 is a decimal consisting of four places ; consequently, I I with four ciphers annexed (10000) is its proper denomina- I tor. Any decimal may be expressed in the form of a com- imon fraction by writing under it its proper denominator. Thus, <3765 expressed in the Ibrm of a common fraetkni, is TTTtf When whole numbers and decimals are expressed to- {gether, in the same number, it is called a mixed nun^er. Thus, 25'03 is a mixed number, 25*, or all the figures on the left hand ^ f the decimal point, being whole numbers, and l'63, tar all the figures on the right hand of the decimal point, [being decimals. The names of the places to ten-millionths, and, generally, low to read or write decimal fractions, may be seen from the following .; ;,;_,,*.;/* ,.k^ u ..^ju ^„,.^,., .^.^4^,;. - • 1!M DBLMMAL rRACTIOXB. ir(M. T» hii M \ . "Ip- ■in mf ^»t» hau , h ^ % TABLE. Sid ^2 :<«:^1.1,:^ . »-1- ^< 3d place. 2d place. c 3 3 (TV lit place. 1st place. 3d place. 3d place. 4th place. 5th place. Gth place. 7th place. it K> tl-Htfv 1.1 ...I S c. II II II II II II II Hundred*. M »« Tons. i»» V\ -i 9 ,' Units. " * Tenths. Hundredths, Thousanths >i r Vl6\ O CC O 9i X Cr< 01 o e o cno e o o M , © © w 'T c «s . . . © a o. >4» ? •13 ii- III Ten-Thousanths. . llundred>ThoussDdlli». MiUionths. > Ten-ilillionths. \ y/Un?'Jj M^-frio ,•»■»'.«. Mg§ S JS §. * • -life' c viuto .<: ^ 8 g-^8 \fl^>iv. From the table it appears, that the first figut'e on the.right^ hand of th« decimal point signifies so many tmtk parts of a unit ; tVie second figure, so many htmdredth parts of a uliit the third figure, so many thousandth parts of a unit, &c. 1:1 Cakes 10 thousandths to make 1 hundredth, lU htkidredthe to mak^^l tenth, and 10 tenths to make 1 unit, in the same manner as it takes 10 units to make 1 ten, 10 tens to make^ 1 hundred, &c. Consequently, we may regard unity as aj starting point, from whence whole numbers proceed, con linually increasing in a tenfold proportion towards the lefti « ft'i, «o. DsriMAL wuACtiofi: i:)l V %l 4 . .h»r' ■>»■ ^r >• ■ i| /•( etllhs. ij, . aDths. .u.n 'houtantlii. )d-ThouBandtli». nthi. Aillionths. ulrc on iheit-Ight tnUk parts of a parts of a uAit a unit, &c. I>' 10 bubdredths; init, in the same I tens to make ^rd unity as a^ proceed, con Lowards the lefti hand, and decimals continiiilljr dttrtatimg, in tho same pro- portion, towards the ri^ht hand But as docimah d«ereas« towards the ri((lit hanJ, it lullows uf coursv, that they in- crease towards tho iett lund, in the same ntaiinur an wholt* numbers. IT 0.». The vahin of every fif^urc is determined hy its place troni units. Conticquentty, ciphers plactd at the rif^ht hand of decimals do not alter their ralue, since every signifi- cant figure continues to possess the same place from unity Thus, *5, *i'A), '5(M) aro all of the same value, each being equal to y'^, or j. Rut every cipher, placed at the left hand of decimal frac- tions, diminishea them tenfold, by removing the significant figures further from unity, and consequently making each part ten times as small. Thus, '5, *05, '005, are of different value, '5 being equal to yV^ ^i" ^ > '^'^ heing equal to y^^, or 2^ ; and '^ *>e»«»K eq"** *» Tj/Vrr. o«" yin • Decimal fractions, having different denominators, are read- ily reduced to a common denominator, by annexing ciphers until they are equal in number of places. Thus, '5, '06, "234 may be reduced to '500, 'OOO, '234, each of which has 1000 for common denominator. If QQ, Decimals are read in the same manner as whole numbers, giving the name of the lowest denomination, or right hand figure, to the whole. Thus, <6853 (the lowest denomination, or right hand figure, being ten-thousan^fiM) is read, 6853 ten-thousandths. ^,<, ,^^,,, , Any whole number may evidently be redaci^ to decimal parts, that is, to tenths, hundredths, thousandtlis, kc. by an- nexing ciphers. Thus, 25 is 250 tenths, 2500 hundredths, 25000 thousandths, kc. Consequently, any mixed numbsr nay be read together giving it the nams of the lowest de- nomination or right hand figure. Thus, 25T are fnadtio cor- rtspond to the decimal dipi8U>M of a vnit now deteribed, dol- lar $ being unite, or toAole numbere, dimes tenth$f unts ftim- dredth»\ and mUU thoutandthe <^ a dqUar ; conawufnttv, the % 1311^ DBOIMAL VRACTIOaS. T66, 67. # txpTfnum of any 8Um in dottart^ centH, and milU, it simply the expreasion of a mixed number in decimal fractions. Forty -six and seren tenthf=46/,f=46'7. Write the following numbers in the same manner : ^^,.,^ Eighteen and thirty-four hundredths. Fifty -two and six hundredths. - Nineteen and four hundred eighty-seven thousandths, t Twenty and forty-two thousandths. ' #:'>Kitv> i< >* ! One and five thousandths. ^ - |i 135 and 37B4 ten-thousandths. ' •' 9000 and 342 ten-thousandths. 10000 and 15 ten-thousandths. 974 and 102 millionths. 320 and 3 tenths, 4 hundreths and 2 thousaodthi. H 500 and 5 hundred-thousandths. ^^ '-^^^!;i^i>,' - u- ^: '■'•'. - i}i(*"-:- .'"i ,■■ .A'-xr. ■^*'^. a '..-.iy^* ■ 1 ;.H ■k *^y-^i 47 millionths. fMttt 'fis -.=(««;■**.> ,^' j-^rijit ■ Four hundred and twenty -three thousandthi. f V< ^ ; ri'v t_^ '. "?:».* Pii-'t .» " ^ i,^v-t'^- ,^*-»» -' uin%. .^'*»»^'.' ^^11% ' fy. I M ' 'jf Tint I ■. ».;-•.• in s. . 6 v<4v • * ^-f f ^^e;-i_-»i ttt^Kt 1-, . fl ; it f .* * ^ADDITION AND SUBTRACTION OF DECIMAL FRACTIONS. _.' W'ifia»Hi--j ijlJ-t l.p/'ij IT 67. As the vahie of the parti in decimal fractions in- creases in the same proportion as units, tens, hundreds, 8tc. and may be read together ^ in the same manner as whole nottbers, so, it is erident that aU the operationB on deeimal fractions may he performed in the eame manner as on whole nimben. The only difficulty, if any, that can arise, must be in finding where to place the decimid point in the result. This, in addition and subtraction, is determined by the same mle ; consequently, they may be exhibited ti^thef . 1 . A man bought a barrel of floor fbr $8, a firkin of Sut- ter fbr 13*50, 7 pounds of sugar fbr 83} cents, an ounce of pipper for 6 cents ; #bat did he give for the whole 1 > ;iVa(«. See the table of Fedeial Money, % 27. liCt the pupil go back now and read carefully all that is said lespecfe- ing federal money, in Reduction. From what is there statfid it is plain, that we nay readily veduce any sums in fedeial Money to the same dieneminations, as to cents or mills, *■', f 67. DECIMAL rRACTIOt\8. !;):{ and add or subtract tiiein as simple niiinbcrs. Or, what Ik tlio sanv thinp, wc may set down the .sums, taking care to writo dollars uuder dollars, cents under cents, and mill.s uit dcr mill;^, in such order, that the separating points of tlie Severn! numbers slinll fall directly under each other, and add them as simple numbers, placing the scparatrix in tht; amount directly under the other points. OIT.UATION. -„ ^8' = MMJO mills, or 10(M)ths of a dollar \ „?^ , , 3'5() = :j:iOO mills, or IWMJths. ;, ., -, siiJ.'S = !^35 mills, or KMMHhs. ,,, r '00 = (M) mills, or lOOOths. Am. ei2'395= 1-2395 mills, or IfiOOths. As the denominations of federal money correspond with the parts of decimal fractions, so the rules for adding atid subtracting decimals arc exactly the same as for the same operations in federal money. r&r ; 2. A i».an owing $37o, paid 8175'7/) ; liow much he then owe ? OPERATION. -5' = 37500 c«nts, or lOOths of a dollar. 175'75 = 17.j75 cents, or lUOths of a dollar. , did 8375- •* ■' * 8iy9'*'25 = 1U1)"25 cents, or lOOths. , ^,, . Wherefore, — In the addition and subtraction of dcciniai fractions, — Rvle : Write the numbers under each oilier, tenths under tenths, hundredths under hundredths, accordiiip, to the value of their places, and point off in the rusult.s as many places for decimals as are equal to the greatest niiiu ber of decimal places in any of the given numbers, ^v,, *«*>«. EXAMPLES FOR PRACTICE. *' ' ^'' • ' 3. Bought 1 barrel of flour for dollars 75 cents, 10 lb. ^ of coffee for 2 dollars 30 cents, 7 lb, of sugar for 92 cents, 1 lb. of raisins for 12 J^ cents^ and 2 oranges, for G cent.-^ , what was the whole amount 1 y/n^. •iflO'l.i.j j| 4. A man is indebted to A, $237'62 ; to B, 83.^0; toC, $8C'12^ cents ; to D, $9'02^, and to E, .'!!0'83i ; what ia the amount of his debts 1 Ans. .'!'684'204 5. A man has three notes specifying the following sums, viz. three hundred dollars, fifty dollars sixty cents, and M # . 134. AI>DITIO> AM) SUBTBACTIOIf OP DECIMLS. 1[ 67. li. r\ nine dollars eight cents ; wliat is the amount of the three notes 1 jfiiB. >?:Jo9'68. 0. A man gare 4 dollars 75 cents for a pair of hoots, and •Z dollars 12^ cents for a pair of shoes ; how much did the boots cost more than the shoes 1 OPERATION. ^ ^5 OPERATION. ' 4750 mills. or, ^ $4''J5 iu. 2125 mills. $<2'125 ^ 2G2o mills = )!<2-625 yins. e2'625 Jns. 7. A man bought a cow for eighteen dollars, and sold her again for twenty-one dollars thirty-seven and a half cents ; how much did he gain! Ans. 83'375. ^^. A man bought a horse for 82 dollars, and sold him again for seventy-nine dollars seventy-five cents ; did he gain or lose 1 and how much 1 9. A man sold wheat at several times as follows, viz. 1 3'25 bushels < 8<4 bushels ; 23,051 bushels ; (i bushels, and '75 of a bushel ; how much did he sell in the whole 1 j/ns. 51 '451 bushels. 10. What is the amount of 429, 21-i^V. 355,t^^tt. It^o andl-j^-^l ^ns. 8()8-iViT=\r. or808'143. 1 1. What is the amount of 2 tenths, 80 hundredths, 89 thousandths, 6 thousandths, 9 tenths, and 5 thousandths 1 An8. 2. 12. What is the amount of three hundred twenty-nine, and seven tenths ; thirty -seven and one hundred sixty-two thousandths, and sixteen hundreths 1 13. A man, owing $4316, paid $376'865 ; how much did he then owe 1 Ans. $3939«I35. 14. From thirty-five thousand take thirty-five thou- i>andths. ' Jns. $34999<965. 15. From 5'83 take 4'2793. Ans. 1'5607. 16. From 480 take 245'0075. Ana. 234'9925. m 17. What is the diflference between 1793*13 and 817' 056^31 ifns 976,07307. 18. From 4jf ^ take 2-^^- Remainder , l^^^ or 1*98. 19. Wlmt it the amount of 29^, ^74j^r,^j,j,^, 97 ,^%, ^^SjT^jjTj, 27, and 100^ 1 Ans. 942*957009. ■••f 1^' Li. H 67. f the three . H:i59'ti8. hoots, and tch did the ON. ("((■ 15 yJns. and sold her L half cents ; 'rts. g3'3:5. id sold him its i did he ollows, viz, ', (i bushels, the whole 1 451 bushels. , or808a43. ndredths, 89 iousandths 1 wenty-nine, ed sixty -two how much $3939'135. ;y-five thou- |f34999'965. fns. 1«5607. i8.234'9925. 13 and 817* 976,07307. f, or 1*98. 142*957009. f 68. MCLTIPLICATtt>r«r Of T)KCtMAT.H. 13.1 MULTIPLICATION OF DECIMAL FRACTfONJ^ IT 68. 1. How much haf itt 7 loads, each containing a3'571 cwtl. , , * OFERATJON. ^ 2:}'571 cwt. = 2iJ571 lOOOths of a cwt. j, ,'.f Ans. 104,997 cwt.=r 104997 lOOOths of a cwt. We may here (II 66) consider the multiplicand so many thousandths of a cwt., and then the product will evidently be thousandths, and will be reduced to a mixed or wfioir num- ber by pointing off 3 figures, that is, the same number as are ^ in the multiplicand ; and as either factor may bo made the multiplier, so, if the decimals had been in the muUipUer, the same number of places mutt have been pointed off for deci- ^ mals. Henoe it follows, we must altcays point off in the pro duct as many places for decimol* as there are decimals places it^, both factors. 2. Multiplys *75 by *35. ^t:H ^t^ ^M-l^fW- > OPERATION. •75 '25 375 150 ' *% In this example, we have 4 de- cimal places in both factors : we Qsust therefore point off 4 places for decimals in the product. The reason of pointing off this num- ber may appear still more plai'H, if we consider the two factors as *1875 Product. common or vulgar fractions. Thus, *75 is -j^, and *25 is -^ : now, ^ X T^« TWff^ = *1975, jh^. same as be fore. H .|j^ 3. Multiply *125 by *03. OPERATION. *125 *03 f-'Wl fcl-JV S • % •00375 Product, Here, as the number of significant figures in the product is not equal to the number of decimals in both fac- tors, the deficiency mtist be supplied by prefixing ciphers, that is, placing th«in at the left hand. The correctness of the rule may appear from the fbllowing process : '125 is ^rt^, and «03 is t^it ' n^^. iWrX-r^n^ TTf'rt^^nT*=^66375, the same as before. These exampAs will be sufficient to establish >ilhe follonr- ing * % \m MULTIPLIOATIOV OP PECfMALI. ir 08. 'm ,> *»ri i t t. ■ RULE. ir.trir In the multiplication of decimal fiactions, multiply as in whole number!, and from the t>ro(]uct pointy so many fig- ures for decimals as there are decimal places in the multipli- cand and multiplier counted together, and, if there are not so many figures in the product, supply the deficiency by » prefixing ciphers. An the denominations of federal money correspond with the parts of decimal fractions ; the rules for the multiplica- tion and division of both are the same. f * ■ -^4. EXAMPLES FOR PRACTICE. ^ ' s^^ > ^^^^h^^V 4. AT^5'47 per yard, what cost 8'3 yards of cloth I Ans, $45'40I . 5. At $'07 per pound, what cost 2G'5 pounds of rice ? jfns, $VS55> 6. If a barrel contain V75 cwt. of flour, what will be the weight of '03 of a barrel 1 Ann. 1'1025 cwt. m 7. If a melon be worth ^'09, what is *7 of a melon worthl Ann. O^V xents. 8. Multiply five hundredths by seven thousandths. Product, *00035. y. What is '3 of 1101 ^ Ans. 34'8. ^ 10. What is '85 of 3072 ? Product, 3'44. 14. Multiply '2002 by '0008. 15. Multiply forty -seven tenths by one thousand eighty- six hundredths.' 16. Multiply (wo hundredths by eleven thousandths. 17. What will be the cost of thirteen hundredths of a ton of hay, at $1 1 a ton ? ' * IHi- What will be the cost of three hundred seventy-five thousandths of a cord of wood at $2 a cord 1 19. If a mftn's^ wages be seventy-five hundredths of a dollar a day, horn much will he earn in 4 weeks, Sundays #xoeptad 1 * -tfim sfs-' >iii 1 20. What will 250 burhels of rye come to, at 90'88^ per bushel 1 A An«.$221<25. 21. Wfcat is the value of 87 barrels of Hour, at $6'37^ a Wrell T69. niVlBIUK OF DKCIUAL9. 137 22. What yr\l\ be the cost of a hophcad of molasstfe,' containing (>:} gallons, at 28^ cents a gallon 1 An0.$17int i>t\' tkrse figures for deciiB&Is in the quotient, equal to Uie num- ber of decimals in the dividend, which brings us to the same result as before. 2. At $4*75 a barrel for cider, how many barrels luav be * bought for $31 ? In this example, tl)ere are decimals in the divisor, and jnone in the dividend. $4'75 = 475 cents, and .'*31, by annexing two ciphers, = 3100 cents ; that is, reduce the di'- [vidend to parts of the same denomination as the divisor. * J M 2 # %m ' % t I»8 DIVIttlOIV or DCCIMALB. ira9. Then, it is plain, as many tunes as 475 cents are contained in 3100 eents, so many barrels may be bought. 47.5)3IOO(6f^^ barrels th* antxcer ; that is, <» l>arrels and 2860 f a^ of another barrel. 2o00 2375 ''-iS'< ,1? r\i ~^Q But the remainder, 2.>0, instead of be- ing expressed in the form of a common fraction, may be reduced to lOths by annexing a cipher, which, in effect, is multiplying it by 10, and the division continued, placing the decimal point after the (>, or whole fines already obtained, to distinguish it from the decimals which are to follow. The points may be withdrawn or not from the divisor and dividend. OPERATION. ; 4'75)3l'0()(()'526-f barrels, the answer, that is, (J barrels 2850 ' .1? and 520 thousandths of another bar- ■:'' -• rel. By annexing a cipher to the first remainder, thereby reducing it to lOths, and continuing the division, wc obtain from it '5, and a still fur- ther remainder of 125, which, by an- nexing another ciplier, is reduced to lOOths, and so on. The last remainder, 150, is l^f^ of a thousandth part of a barrel, which h of so trifling a value, as not to merit notice. If no'.v we count the decimals of the dividend, (for every cij-her annexed to the remainder is evidently to be counted a decimal of the dividend,) we shall find them to be Jive, Tvhirh corresponds with the number of decimal places in the divisor apd quotient counted together. }. Under ^i t>S, ox. 3, it was required to multiply '125 by '0^5; the product was '00375. Takipg this product for a dividend, let it be required to divide *00375 by •125. One operation will prove the other. Knowing that the number of decimal places in the quotient and divisor, counted to- gether, will be equal to the decimal places in the dividend, we may divide as in whole numbers, being careful to retain *the decimal points in their proper places. Thus, J^:- 1250 950 3000 2850 150 « V ^_l|;?iP^l^, \ ^ in»«^^7 yi!> mwt^•>ui^n'm'^n^i> "ymm ^'.nm m hr If 09. f (HI, TU. f^ . DIVISIOJf or DECIMALS. 139 iro contained ■ l> l>arrels and • nstoad of be- ef a common ing a cipher, 1 the division s (>, or whole the decimals idrawn or not t is, barrels another bar- er to the first educing it to the division, ind a still fur- which, by an- is reduced to 150, is 4-^^ of barrel, which nd, (for every to be counted •ui to be five, imal places in iltiply '125 by product for a 3y '125. One at the number r, counted to- tbe dividend, areful to retain Thus, ■■a oj hb:4-^ orERATION. '12.>)'0();t75('():J The divisor, 125, in 375 Roes 3 375 times and no remainder. We have — > ' only to place the decimal point in 000 the quotient, and the w ork is done. Tiiere are five decimal places in the dividend ; conse- quently there must be five in the divisor and quotient count- ed togeiher ; and, as there are three in the divisor, there must be two in the quotient ; and since we have but one figure in the quotient, the deficiency must be supplied by prefixing a cypher. The operation by vulgar fractions will bring us to the same result. Thus, '12^5 is j'^^^^^, and '00375 ii y0Vof^,,T . »ow, i.TW5a-HTWff=Tiia888(j=T3<7='«'5 tl^^ same as before. f H 70. The foregoing examples and rcmavks are suffi- cient to establish the following ^, c. , , ,;. .» RULE. In the division of decimal fractions, divide as in whole numbers, and from the right hand of the quotient point oft as many figures for decimals, as the decimal figures in the dividend exceed those in the divisor, and if there are not 80 many figures in the quotient, supply the deficiency by prefixing ciphers. ^ If at any time there is a remainder, or if the decim&l figures in the divisor exceed those in the dividend, ciphers may be annexed to the dividend or the remainder, and the quotient carried to any necessary degree of exactness ; bm the ciphers annexed must be counted so many decimals of the dividend. EXAMPLES FOR PRACTICE. 4. If $472'875 be divided equally between 13 men, how much will each one receive 1 Ans. $36'375. 5. At 8'75 per bushel, how many bushels of rye can be bought for $141 ? Ans. 188 bushels. 6. At 6^ cents apiece, how many orangei may be bought for $8? Ans. 128 oranges. 7. If '6 of a barrel of flour cost $5, what is that per bar- ren ilsv>m*fenf) An«.8'333-f 8. Divide 2 by 63'1 i^r ^ ^ QwJt. '0374- 11 n. i 140 REDUCTION or VULGAR rRAOTIORS, &C. IT 70, 71. 9. Divide '012 by 005. . i Quot. 2'i. 11. Divide three thousandths by four hundredths. Quot. '075. 11. How many times is '17 contained in 8 1 12. If I pay $468*75 for 750 pounds of wool, what is the value of 1 pound t Ana. 90'fiQ5 ; or thus $0*62^. 13. If a piece of cloth, measuring 125 yards, cost .^181 '^ what is that a yard 1 Ans. $1'45. 14. If 530 quintals of fish cost $1913'52, how much is % that a quintal 1 Ans. 83'57. 15. Bought a farm, containing 84 acres, for $3213 ; what did it cost me per acre 1 An0; 4'00( «Or>0(i-|-, the an«tt'cr. ?* 390 ,'.: 7 400 396 v^ », As there can be no tenths, a cipher must be placed in the quotient, in tenth's place. Note. ^ cannot be reduced exactly ; for, however lonp the division be continued, there will 6till be a remainder.* It is sufficiently exact for most purposes, if the decimal Tm extended to three or four places. ""Decimal figures which continually repeat, like '00, in this exam- ple, are called Repetenda or Circulahng Decimals. If only one /^re repeats, as '3333 or '7777, &c. it is called a single renetend. If tvio or more figures circulate alternately, as '060(500, '1^4234234, &c. it is called u compound repetend. If other fieures arise before those which circulate, as '743333, '143010101. &c. the decimal is called a mixed repetend. .1 single repetend is denoted by writing only the circulating figure ^ with a point oyer it : thus, '3, signifies that the 3 is to be continually repeated, forming an infinite or never ending series of 3's. ^ compound repetend is denoted by a point over the first and Zort re- peating jifure ■ thus, '234 signiOes that 234 is to be continually r«-v pcatcd. * It may not be amis*, hern to show how the value of any repetend may be found, or in other words, how it may be reduced to its equita- & lent vulgar fraction. If we attempt to reduce ^ to a dtdmtU, we obtain a con- tinual repetition of the figure 1 : thus, *llin, that is, the repetend'i . The value of the repetend *i, then is | ; the valu» of '22^, &c. the repetend '2 will be twka as much ; that !»» ^ 149 RBOvcvioN or vtTLOAn fractionh, &c. T7I. f.*< I !.' k Hr From the foregoing examples wc may ileduce the follow- ing general Rulr : — 7b reduce a ammon to a decimal frac- tion : — Ann«x one or more ciphers, a* may he nscesNary, to the numerator, and divide it by the denominator. If then there* bo a remainder, annex another cipher, and divide as before, and so continue to do so long as there shall continue ^. In the same manner, 'i}=J, and'4=|, and '5=^, and soon tol), which=|=l. ^ r 1. What is the value of «8? Ans. |. 'Z, What is the value of «G 1 Ans. §==J. What is the value of «3 1 of '7 ? '4 1 '5 1 '1) 1 'V. If g^ be reduced to a decimal, it produces '010101 , or the repetend «6i. The repetend '62, being 2 times as much, must be ^ and ♦63=^!'g» *nd '4^i being 48 times as much, must be ||. and '74= J|, &c. ^ If ^l^ be reduced to a decimal, it produces 'OOi ; conse- quently, '002=y§y. and '037=^, and 425=i§|, &c. As this principle will apply to any number of places, we have this general Rulk for reducing a circulating decimal to a vulgar fraction. — Make the given repetend the numer- ator, and the denominator will be as many 9s as there are repeating figures. .«v 3. What is the vulgar fraction equivalent to *704? 4. What is the value of '003 1 *6l4 1 '3241 *6l02i 1 '24631 '002103 ? jfns. to the last, iniWaTr 5. What is the value of '431 In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is ^j, and the repetend, 3, is ^ of -,\r, that is. ^ ; these two parts must be added together, ^-f-^^yss aggi — |g , ans. Hence, to find the value of a mixed repetend, — Find the value of the two parts separately, and add them together. 6. What is the value of '153? TV^-i-7§,r=i5§=-i^, «ns. i 7. What is the value of '138 1 '16 1 '4123 1 It is plain, that circulates may be added, subtracted, mul- tiplied, and divided, by first reducing them to their equiva lent vulgar fractions. ■i ; f |5 7I,T2. RKnrrTiow or decimal rHACviofts. 14U , and '5e=4> ^^^ ing 'i times as eing 4d times as , to bo a remainder, or until tl'.c fraction ihall be reduced to any necessary degree of exactness. The quotient will be the decimal required, which must consist uf as many deci- mal placcH as there are ciphers annexed to the Dunurttor ; and i there are not so many ti(i;nres in tlie quotient, the de- ficiency must be supplied by prefixing ciphers, EXAMPLES FOR PRACTICE. 4. Reduce J, |, ^j^g^j^, and j^^rj to deciroah. yinn. '5 ; 'i>r> ; «t)-i> ; ^iMVM^ 5. Reduce ^, j^\„, j^^^, and c^VVg to decimals. (). Reduce ^^, ^\j'V, ^y^,^ to decimals. 7. Reduce 3^, ^, ^l-^, i, J, j\, ^\, 5,^ to decimals. 8. Reduce |, f , |, j^, f , f , |, tj\j ^^, vJ^ to decimals. REDUCTION OF DECIMAL FRACTIONS. IT 7^. Frt; Uions, we have seen (H (50,) like integers, are reduced from low to higher dcnonminations by division, and from high to lower denominations by inultiplication. To reduce a compound num- ber to a decimal of the highestMgher denomination to integers denomination. 1. Reduce 78. 6d. to the decimal of a pound. 6d. reduced to the decimal of a shilling, that is, divided by 12, is <5s., which annexed to the 7^. making 7'5s., and div-'^cd by 20, is «375^ the anawer. The process may be pre sented in form of a ruht thus:— Divide the haest denumiBa- tion given, annexing to it one or more ciphers, as may be 7b reduce the decimal of a of lower denominations. 2. Reduce «375£ to inte- gers of lower denominations. '375ir reduced to shillings, that is, multiplied by 20, is 7'508. ; then the fractional part, '50s., reduced to pence, that is, multiplied by 12, is 6d. jins. 78. 6d. That is,-Multiply the given decimal by that number which it takes of the next loteir de- nomination to iiiiake ofMof this higher, and from the right necessary, by that number band of the product point off I which it takes of the same tojas many figures forwiecimalii make one of the next higher.as there are igufes in the denominatioo, and annex thelgiven decinud, and so cqntin* '*«■ • f ■ RBDUCTIOIV UV nilClMAI. rHACTIfllV!!. T72. quotient, as a decimal to thaluc lo il(> throuf^h all thu du lii;;lier (li'iiominatioi) ; ko cun iioiinnatiunt; tlicNoveraliiuni titiue to do, until the \ilioli'l)crii nt tlx; leU hand ul th« •hall he reduced to the deci dcciuiai poinlii will I o the mal rctjuircd. ' value of tho fraction in the proper dcnoniinations. EXAMIM.KS FOR PRACTICK i:X AMI'I.KS FfHf l'KA<'TICE. .'{. JU'ducc I oz. KIpwt. to| 4. Kediico 'I'i'ilhs. Troy t(» iiitfj^ors of lower irenouiina- tionti. . OI'KKATION. tho fraction of a pound. OPKRATION. •,»0)IO'O |)wt. 1'2)I'5 oz. 'ia"» 11). y/Ais. o. Reduce 4cwt. 2^qrs to the decimal of a ton. Note. ^ = 'Z'ii. T. Reduce 38{;als. Jl'.'S^qts. of beer, to the decimal of a hhd. ^ 9. Reduce Iqr. '2n. to the decimal of a yard. 11. Reduce 17h. 6m. 438. to tho decimal of a day. 13. Reduce 2l8. lO^d. to the decimal of a guinea. 15. Reduce 3cwt. Oqr. 71bs. 8oz. to the decimal of a ton. Ib.^ 'I'r> o/. I'.'iOO pwt I0MM»0. yZ/i.-;. Ioz.l0p\v( (). What is the value ci '23'^> of ton 1 8. What is the value of '72 hogshead of beer 1 10. What is the value of '375 of a yard 1 12. What is the value of '713 of a day 1 14. What is tho value of '78125 of a guinea 1 1(). What is the value of 'I5;334821 of a ton 1 Let the pupil be required to reverse and prove the fol lowing examples : 17. Reduce 4 rods to the decimal of an acre. 18. What is the value of '7 of a lb. of silver 1 19. Reduce 18 hours, 15m. 50'4sec. to the decimal of a day. 20. What is the value of <67 of a league 1 •iti.'fi^S'-)^ .«J4±^ ^l. Rtduce lOs. 9^d. to the fraction of a pound. f 78. There is a method of reducing shillings, pence, IMS. 5 72. f 73, 74. RBOVCTioii or decimal riAorroira. 145 t a\\ tliu do hoveraliium hand of th» will lo the ittion in the Uionii. If piiA<:Tici:. I'AhH. Troy to er trciiomina fas. loz.l()p>N' tho value ft ho value of '7'2 the value of I 3 the value of s tho Talue of iinea 1 the value of toni the fol .i 1 prove pre. ■-' :v iverl le decimal of a Ipound. lillingi. pence, and farthinKB to the decimal of a pound, hj tnfjMdion, noort •imple and cnnciiic than the foregoinfc. The reasoning in relation to it is an follows . ^ of 'iOs is 'is. ; therefore exery '2i. is ^^, or '\£. Er- try sliillinjc is j^r =* T&ii» "^ 'OAri*. Pence aie readily re- duced to farthings. Kvery farthing is ^^nJ^. Had it so happened, that KHM) farthings, instead of INM), had made a pound, then every farthing would have been in\)o< or '001^. 960 -f 40 =: 1000 ; that is, ^V of 9(10 added to MM) is 1000. Taking ./^ of a number, and adding to that number, is the game as multiplying the number by unity and the fraction 3V> 'i'c- ^nppose you have the fraction ^i^. K you mul- tiply both the numerator, and denominator by Ir^^, you do not change the value of the fraction. Do this, and you ob- tain ygg^ ^ then is equal to ^^g^^. ^\f£ is 24 far- things ; of course it follows that 24 farthings is equal to t9$^' therefore, if the number of farthings, in th« given pence and farthings, bo more than 12, ^ part will be more than ^ ; therefore add 1 to tlj^cm : if they be more than 'V^, 2V r^i'^ ^i" ^® '"o''* ^^^^ H * therefore ad^Q'^to them : then call them so many thousandths, and the result will be correct within less than ^ of jj^xj ^^ ^ pound. Thus, 17s. 5|d. is reduced to the decimal of a pound as ft>l- lows : 168. = 'S£ and Is. = «05rf. Then, 6Jd r=23 far- things, which, increased M 1, (the number being more than 12, but not exceeding 36,) is '024<;f, and the whole is 'ft74£ the antwer. Wherefore, to reduce ahillinga, pence and farthinga to the decimal of a pound f>y tnapection, — Call every two fthillings one tenth of a pound ,- every odd shilling, five hundredths; and the number of farthings, in the given pence and farthings, so many thousandths, adding one, if the number be more than twelve and not exceeding thirtynMX, and tw . if the number be more than thirty -six. * T 74. Reasoning as above, the result, or the three first figures in any decimal of a pound, may readily be rediMed hack to shillings, pence and farthinn, by inspection. Double the first figure, or tenths, for shillnlgs, and, if tlM second figure, or hundredths, be^o« or more than five, reckon anoth- er shilling; then, after the five is deducted, call the figures N 446 ■OPPLKJIBIfT TO DECIMAL PRACTIOHt. T74. i« the stcond and third place so many rarthings, abating ont when they are aI>ove twelve, and ttco when above thir- ty -six, and the result will be the answer, Rufficit>ntly exact for all practical purposes. Thus, to find the value of '876 i.' by inspection : — '8 tenths of a pound ... <06 hundredths of a pound '026 thousandths, abating 1, = 25 farthings ♦876 of a pound 16 shillings. 1 shilling. Os. 6^d. = 17 ». 6id. ^flS. i^,?^ tV.i V EXAMPLES FOR PRACTICE. '^ 1. Find, by inspection, the decimal expressions of 9s. 7d. and 128. Ofd. ^ns. '479 iT, and '603 jf. 2. Find, by inspection, the value of '523 .£, and '694^. jfns. 10s. 5^d., and 13s. lO^d. 3. Reduce to decimals, by inspection, the following sums, and find their amount, viz ; Ids. 3d.; 8s. llii^d.; 10s. 6|^d.; Is.JSid.; id. and 2^. . ^^ . , ,j,^; Jmmnt, ;f 1'833. If Find the value of '47*. NoU. When the decimal has but (too figures, after taking Ji^,^ q^i the shillings, the remainder, to be reduced to thousandths will require a cipher to be annexed to the right hand, or 'supposed to be so. jina. 9s. 4^d. 5. Value the following decimals, by inspection, and find their amount, viz. n^y£.; '357«£.; '9ia£.; '74^.; 'S^.; •25iS.; <09je.; and '008^. An«. 3<£. 12s. lid. SUPPLEMKNT TO DECIMAL. FRACTIONS. ^ ^ QUESTIONS. 1. Wlint me decimal fraction*? 2. Whence is the term derived? 3. [Tow (In decimals differ from comnton fractions .' 4. How are deci- mal fr«pti(ina written ? 5 How can the proper denominator to a dec- imal fraction be known, if itbe not expressed.' 6. How is the value v,i>C every figure determined .' 7. What does the first figure on the right hand or the decimal point signify i the second figure i —— the third figlire .' — — fourth ^ure .' 8. How do ciphers, placed at t)>e right handikf deeimnis nfllet their value' 9. Placed nt the left hand liow do they iiflTect their value .' 10. How are decimals read .'11. How .#r« decimal fractions, having difierant denomiuators, reduced to a lOHt. f 74. lings, abnting len aboTC thir- icionlly exact ^alueof<876;fc' 16 shillingi. 1 shilling. Os. 6^d. = \7». 6id. if fit. sions of 98. 7d. £, and «603^. £, and «e94 £. and 13s. lO^d. rollowing sums, Id.; lOs. 6^d.; mount, .£1'833. cs, after taking d to thousandths right hand, or Jns. 9». 4f d. ection, and find •74£.; '5£.', lACTIONS. ihe 4 term derived .' How are deci- nminator to a dec- How is the value figure on the right 1 figure ? the TB, ptuced at t^e id nt the left hand la read r 11. How tors, rvduced to a ir74. MUPPLBMBXT TO DECIMAL rRACTIOllS. 147 «ommnn denominator? 12. What is nmixrd number ? 1$. How may any wholo nutubcr hs reducril to decimnl partu ? 14. How can any mixed number bo rttad togetlivr, nnd the whole expresiied in the form of a common traction ' 15. VVIi.nt is observRd rcspcciing the denom- inations in fedorat money ? 10. What is the rule fur nddition and sub- traction of diicimnis, particularly ns respects placing the decimal point io the results?^— multiplicution .' division.' 17 How is a com- mon or vulgar fraction reduced to a decimal i' 18. What it the rule ibr /educing a compound number to a decimal of the highest denomin- ation contained in it .' 1!K What is the rule for finding the value of any given decimal of a higher denomination in teims of n lower.' 20. What is thn rule for reducing shillings, pence and farthings to the de- cimal of a pound, by inspection .' 21. What is the reasoning in rela- tion to this rulei* 22. liow may the three first figures of any decimal of a pound be reduced to shillings, pence and futthiiigs, by inspections '% #% VI % 3,^ (( EXERCISES. 1 . A merchant had several' remnants of cloth, measuring as follows : "^ **- ^ «»«!> » ip we«^ 7 f yds. '^ How many yards in the wholo, and what i would the whole come to at $3*67 per yard 1 I iVbfe. Reduce the common fractions to deci- r nials. Do the same wherever thly occur in the I examples which follow. ^ ' ] Ans. 36'475 yards. $133'863X, cost. 2. From a piece of cloth, containing 36|yds. a merchant sold, at one time, 7^ yds. and, at another time, 12| yds. ; how much of the cloth had he left ? Ans. ]6'7yds. 3. A farmer hought 7 yards of broadcloth for 8^pf ., a barrel of flour for 2-^ £., a cask of lime for lf<£., and71bs. of rice for ^£.', he paid 1 ton of hay at 3-^^., I cow at 6§ £. and the balance in pork at j^. per lb.; how many were the pounds of pork 1 ■^, NoU. In reducing the common fractrons in this Sample, it will he sufficiently exact if the decimal be extended to three places. Ans. 108| lb. 4. At 12^ cents per lb. what will 37} lbs. of butt«r cost \ Ana, |4<718j. 5. At $17*37 per ton for hay, what will 11^ tons coat 1 Ans. $201<92|. 6. Th€ above examph rnerseiL At $201 '92| for II | tons of hay, what is that per ton 1 Ans. $17*37. 7. If '45 of a ton of hay cost $9, what is that per ton 1 QmsuU IT 62. Ant. $20. 148 ■VPPLBMKST TO DECIMAL FBACTIOHt. f 74. ' nw I 6. At <4 of a dollar a galloo, what will '25 of a gallon of molasses cost t Ant. i< 1 . 9. At $9 per cwt., what will 7 cwt. 3 qra. 16 lbs. of su- gar cost f Not*. Reduce the 3 qrs. 16 lbs. to the decimal of a cwt. exteodiog the decimal in this, and the examples which fol low tof&ur places. An*. 71'035-|-. 10. At $e9'875 for 5 cwt. Iqr. 141bs. of raisino, what is that per cwt. Ans. $13. 11. What will 23001bs of hay come to at 7 mills per lb. Ans. $16' 10 12. What will 765^lbs. of coffee come to at 18 cents per lb. Ans. 1137*79. 13. What will 12 gallons, 3 qts. 1 pt. of gin cost, at 28 eents a quart 1 ybie. Reduce the whole quantity to quarts and the deci- mal of a quart. Ans. $14*42. 14. Bought 16 yds. 2qrs. 3na. of broadcloth for $100<125; what was that per yard 1 Ans. $6. 16. At $1<92 per bushel, how much whtat may be pur- oliMed for 9'7t % Ans. 1 peck 4 qts. 16. At $92*72 per ton, how much iron may be purchas- ed for $60*268 1 Ans. 13cwt. 17. Bought a load of hay for $9'17, paying at the rate of $16 per ton ; what was the weight of the hay 1 Ans. llcwt. Iqr. 23lbs. 18. At $302*4 per tun, what will 1 hhd. 15 gals. 3 qts. of wine cost % Ans. $94*50. 19. The above reversed. At $94*50 for 1 hhd. 15 gals. 3qts. of wine, what is that per tun 1 Ans. $302*4. Note. The following examples reciprocally prove each other, excepting when there are some fractional losses, as explained above, and even then the results will be sufficient- ly exact for all practical purposes. If, hower greater exact- ness beVequired, the decimals must be extended to a great- er numbepof places. ^. At $1*80 for ^ qts. of] 21. At $2*215 per gallon, wine, what is that per gal. 1 what cost 3^ qts.1 22. If I of a ton of potash^' 23. At $96*72 per ton for woott $60*45, what it thatpot-ashes, what will | of i^ ton per ton 1 cost 1 Y 74. RKDUCTlOlf or CURRENCIES. REDUCTIOIV OF CURRENCIES. 149 In ihe United SuitM, nince tlie act of Congrcm in 1786, ettabliahing Fed * 4) '0U4 of a dollar. * .» '151 of n pound. This can be reduced to nhillings, pence And farthings by inspecliun, (See H 73,) ns foj^lows : '151£ oqual Ss. Od. Iqr. We find that$24 G(M is equal to G£ 3s. Od. Iqr. .tns. The following then, is the general rule, to reduce federal money (•• Halifax currency, — Divide the given sum by 4, and tlic quotient will be inpounds and decimal parts of a pound, which can be reduced to shiU- inji^s, pence and farthings b'l inspection. EXAMPLES FOR PEACTICE. Reduce $500 to Ualiflix currency, Jins ia5£ " $27'304 " 6£ lis fid for " «n8'2.'. «' 2a£ii8.:w. " $23(5'.'jO •' fin£ 2s Cd. Reduce $400 to Halifax cnrroncy. S.'hJ'03, |tO:V8l4, $85'G3, $1977' IMioni^'tQdcral money. EXAMPLES FOR PRACTICE. f- Jiti ^. Therefore, to reduce New-England currency toHalifix currency, — Take ^ o( the given sum, thus, — G) 14i! 5s. 4d. New-England currency. 2 7 ^ ^ 5 11 17 4^ Halifax currency. Redtice 6Q£ 4fl. lOd. N. E. currency to Halifax eurr«*ncr JnM. 50.t* 4«. O^d' J ^ m m. ♦ 1174, 76. INTEREST. !50 i*.r>3jC3a. lOd. To reJuco Halifax currtncy to sterling monejr, or to re- •lucu Halifax to New-England, it is only necessary to ft- verse the process in the iorcgoing operations. To wduce sterling to Halifax we multiply by ^, there- fore, to reduce Halifax currency to sterling money, —Divide the given sum by 'g^, or, what is the sartle, multiply by ^, that is, take -^y of the given sum ; e. g. 10) (>4£ Os. lOd. Halifa^currency. 4, 8 1 * % 48 li) 9 ilerling money, ' In thtf same manner, to reduce Halifax currency to New- England,— Take | of the given sum, or, add { to Uie gir en sum. From the foregoing rules and illustrations, Ibe pupil him- self will be able, by pursuing a similar cor.rse, to reduce, with facility, any currency, the denominations of which are pounds, shillings, &c. to any other, in which the denoniina tions are the same. -The following is the general rule for finding a multiplier 4 tc reduce any currency to the par of another : Make the number of shillings that are equal to a dollar in the curren- cy to be reduced, the denominator of a fraction ; and over this, for a numerator, write tHe number of shilliDgs that are e- qual to a dollar in the currency to which the given sum it to be reduced. " ^ Let the pupU find multipliers to reduce NeW-York and Pennsylvania, currencies to Halifax, and then Halifax cur- . rency to those. ^t' V :i^f:mwmi3QW<, It! H 75. Interest is an allowance made by a debtor to a creditor for the use of money. It is computed at a certain number of pounds for the use of each hundred pounds, or so many dollars for each hundred dollars, &c. one year, and in the same proportion for a greater or less sum, or tor a longer or shorter time. The number of pounds to paid for the use of a hundred # INTEREST. t75- |, •01. '005. '0625. •0075 poundi, one year, is called the rate, per cent or per ctntum; th0 wordii per cent, or per centum, sygnifyint; hy the hundred. The hip;hpst rate allowed hy law in the Canadan is 6 ptr etnt* that is, pound.s for KM) pounds, G shitlin^rs jfor 100 nhillings ; in oilier words ^^^ of the sum lent or due ta paid far the ' < if it ofM year. This is called legal interest, and will h I'C understood when no other ralo is mentioned. Let vH suppose the sum lent, or due, to he \£. The KMUh part of li^ Or j^ j of a pound is, decimally expressed, ^thus, <01, and j§j of a pound, the legal interest, written u a decimal fraction, is -- «06. So of any rate per cent. 1 per cent, expressed as a common fraction, is f^jji decimally, -. ,~ " " ^ per cent, is a half of 1 per cent that is, - I per cent is a fourth of 1 per cent that is, 1 per cent is 3 times ^ per cent, that is, Note. The rate per cent is a decimal carried to two pla- te$, that is, to hundredths \ all decimal expressions lower than hundredths are parti of 1 per cent. | per cent for instance, is '025 of 1 per cent that is, '00025. Write 2^ per cent as a decimal fraction. ^ 2 per cent is '02, and \ per cent is '005. Ans. '025. Write 4 per cent as a decimal fraction. 4^ per cent. 4f per cent. .5 per cent. 7^ per cent. 8 per cent. 8£ per cent. 9 per cent. Oi per cent. 10 per cent. (10 per cent is jV'a i decimally, '10,) 104 per cent. 11 per cent. 12^ per cent 15 per cent. 1. If the interest on \£ for 1 year, be '06 of a pound, Hvhat will be the interest on 25£ for the same time 1 t^^ It will bo 25 times 0, or 6 times 25, which is the tamo "^hing : — ».^ --vv 25^:. : # «96 ■si:^,''-^i 1'50 aniwer ; that is, \£ and 5 tenths. The 5 tenths must be reduced to shillings, pence and farthings by the rule *In the Now-Englnnd States the logal rat« it tb* •itme as in tb« Canail«i. In England it in 5 per coot f?5- }r per ctntum ; by the hundred. madan is 6 per lillings for 100 ! or due is paid 1/ interest, and IS mentioned. he \£. Th« tally expressed, rest, written at I - - '06. f 75. IllTBRCST. 161 >n, IS J8, •01. •005. •0625. •0075 irried to two pla- xpressions lower I per cent for i25. Ans. 'QfHS. 4i per cent. per cent. »t. 91 per decimally, '10,) JJ per cent '06 of a pound, lame time 1 'hich is the same The 5 tenths Ithingsby the rule] U)« ciitntt aa in tli«l & for the reduction df Uecintals ; or with sufiicienl exact' ness by inspection. See T, 7'3. '50, or '5 of a pound equal 10 shillings. The interest of 2.>j6'. for a year is then \£. lOs. To find the interest on any sum for 1 year, it is evideat we need only multiply it by the rate per cent written as a decimal fraction. The product, observing to place the point as directed in raultiplieation of decimal fractions, will be the interest required. Aote. Principal is the money due, for which interest if paid. AaiovNT is the principal and interest added tugctlier. 2. What will be the interest of 3*2^;. ;Js. 1 year, at 4j per cent? We are to multiply the principal by the rate per cent, 4^, expressed in the form of a decivial, '045 ; we must therefore reduce the 3s. in the ))rincipa!, to decimals by iuspecticn. We find 3s. equal to '15. There being five decimal places ^'•i'l5£. principal. in the multiplicand and mul '045 rate per cent. 1()075 12860 P44675<£. ♦ ♦" tiplif 5 figures must be point- ed ca lor decimals from the product, which gives the an- swer 1 pound and 44675 hun- dred thousandths. Any thing less than thousandths need not be regarded, hence, 1<446<£ is sufficiently exact for the an- swer. The '446 must be reduced to shillings, pence and farthings by inspection. Double the '4 for shillings, equals 8s.; call the '046 so many farthings, deducting two because one 36 equals 44 fai things. In 44 qrs. there are 11 d. '44a£=8s. lid. The interest, then, of 32c£. 38. for I year, at 4^ percent, is \£. 8s. lid. answer. Always, then, if thore are shillings, pence and farthings, or either denomination, in the given principal, reduce t^m to the decimal of a pound by inspection, before multiplying by the rule. After obtaining the answer in decimals, reduce the ^tenths, hundredths and thousandths, to shillings, pence and far- things, by inspection. The method of effecting each reduc- tion, is exhibited in IF 73, and 74, and must be made per- fectly familiar to the pupil's mind. 3. What will be the interest of U£. Ss. 4d. for 1 year, at 3 per cent 1 at 5^ per cent? at 6 per cent 1 — - at 7} per cent 1 at 8^ per cent 1 • at 9^ per cent 1 — *- at 10 per cent 1 at 10^ per ceat V — — at 11 per m I IWTBREflT. % 75, 76. * eenti at 11 J per cent 1 at 12 per cent ! at ]2i per centi 4. A (ax on a certain town is AWi£. i5.H. 10|il., on which the collvctor iii to receive 2^ per cent (or culluctiii^ ; what will he receive fur collecting the whole tax at tliat rate \ In this example, the shillings, &c. reduced tothedecional of a pound equal '795. Multiply tiArefore, 4i)CrWa£. by the rate 2^, that is, '025. The answer, in decimals, is i0'160c£.; the tenths, &c. reduced to shillings, &c. equal 3s. 4^d. The answer, then, is, 10^. lis. 4d. yolc. In »he same way arc calculated commission, in- surance, buying and selling stocks, loss and gain, or any thing else rated at so much per cent without respect to titne. 5. What must a man, paying 37^ percent on his debts, pay on a debt of r^'2£ 5s. 1 Ana. 49^:. lis. lO^d. 6. A merchant, having purchased goods to the amount of 580c£. sold them so as to gain 12^ per cent, and in the same proportion for a greater or less sum ; what was his whole gain, and what was the whole amount for which he sold the goods ? Ans. His whole gain was 72c£. 10s. whole amount, 652c£. lOs. 7. A merchant bought a quantity of goods for 173i2. 15s. bow much must he sell them for to gain 15 per cent 1 Ans, 199<£. 16s. 3d % f 76. Commission is an aliowance of so much p«r cent to a person called a correnpondent, factor, or broker, for assisting merchants and others in purchasing and selling goods. i 8. My correspondent sends me word that he has pur- ehased goods to the amount of 1286£, on my account; what jrill his commission come to at 2^ per centl j^s. tXi£. SsJU 9. What must! allow my correapondent for 'selling goods gto the amount of 23174? 9a. 2fd. at a commission of S^- per ccDtl , i^tt. 76^ 6i. 4d. InavKAzrcB is an exemption from haxard, the payment of a certain sam, which is genera' ptr csfil. on the estimated Talue of the property Prbbhom is the sum paid by the insured for tbeitwi Kedby f 76- lllTBBBtT. 153 Policy is the name (griveii to the instruincnl or writing, by which (he contract of indemnity ii effected between the insurer and insured. 10. What will be the premium for insuring; a ship from Montreal to Liverpool, valued at 945Ul£, at 4^ per cen^ 1 v/ns. i25jt 5s. 11. What will be the annual premium for insurance od la house against loss by fire, valued at 875^' at ^ per centi By removing the separatrix 2 figures towards the left, it fit evident, the sum itself may bo made to express the pre- (roium at 1 per cent, of which the given rate parts may taken ; thus, 1 per cent on 875;^ is 8'75^', and £ of r5£ is 6<562.£. Ans. &£ 1 Is. 3d. 12. What will be the premium fur insurance on a ship ind cargo valued at 6310<£, at ^ per rent 1 — at § per jnt 1 — at f per cent ? — at | per cent 1 — at ^ >er cent 1 Ans. At | per cent the premium is39j6! 7s. 8}d. » Stock is a general name, for the capital of any trading company or corporation, or of a fund established by govern- nent. The value of stock is variable. When 100 pounds of |tock sells for 100 pounds in money, the stock is said to be |t par, which is a Latin word signifying equal ; when for re, it is € per yard f jf/tM. Ill 18. Bought calico at la. per yard ; how must 1 sell it to gain 5 per cent. ? 10 per cent. 1 15 per cent. 1 to luM 20 per cent. 1 v/ns. to the lant, Ofd. per yard. V T7. We hive seen how interest iH cast on any aum of money when the time is one year ; but it is frequently necessary to cast interest for months and days. Now, the interest on !<£' for 1 year, at per cent, being *06, is *01, one hundredth for 2 months, 'OO/) five thousandths (or ^ a hundredth) for 1 month of 30 days, (for so we reckon a month in casting inter- est,) and '001 one thousandth for every 6 days ; 6 being contained | 5 times in 80. Hence, it is very easy to cast in the mind, the interest! on l£, at per cent for any given time. The hundredth$,\ it is evident, will be equal to half the greatest even num ber of months ; the thousandths will be 5 fur the odd month,! if there he one, and 1 for every time 6 is contained in the| given number of the days. Suppose the interest of \£, at per cent, be required furl 9 months, and 18 days. The greatest even number of the| months is 8, half of which will be the hundredths, '04 ; the thousandths, reckoning 5 for the odd month, and 3 for the 18 (3 times 6 = 18) days, will be <008, which, united withl the hundredths ('048) give 4 hundredths and 8 thousandths,! 4 hundredths, and 8 thousandths, or, '048<£ reduceds=ll(l/ .V ' \ Ans. lldj 1. What will be the interest on \£ for 5 months 6 daysj 6 months 12 days 1 7 months 1 8 monthj 24 days? months 12 dayRt — 10 months 1 11 months 6 days 1 12 months 18 days 1 ^ 1^ months 6 days ? 16 months 1 Odd DATS. 2. What is the interest of li^ for 13 moutlij 16 days ? The hundredths will b« 6, and the thousandths 5, for odd month, and 2 for 2 times 6= 12 days, and there isj renuiinder of 4 days, the interest for which will be sue ^% t 77. lIVTRHMaT. 166 teing contained! l£ for 13 moutlJ part of 1 titoitsandth m I daji is part of days, that is, | ss } of a tliousnndtli. Ans. '(M^Tf . U. What will he the interest of \£ for 1 munth 8 dajrs 1 — - 2 inoiilhs 7 days 1 J) months 15 days 1 — 4 months tW days 1 — 6 months 1 1 days ? — 6 months 17 days 1 — 7 months :) days 1 — 8 months 11 daysl — 9 months 2 days 1 — 10 months 15 days 1 — U months 4 days 1 — 1*2 months 'A dk.ye 1 JVb/«. If thert ia n) odd month, and the number ofday» be haa than (i, ao that there are no thouaandtha, it is evident, a cipher must bo put in the place of thousandths ; thus, in the last example,— r2 months 3 days,— the hundredths will be '06, the thousandths 0, the 3 days ^ a thousandth. Ana, Is.2jfd. 4. What will be the interest of l£ fur 2 months 1 day 1 — 4 months 2 days ? — months 3 days "! — 8 month* 4 days 1 — 10 months 5 days 1 — for 3 days 1 — for 1 day ? — for 2 days 1 — for 4 days 1 — for 5 days 1 5. What is the interest of 5&£ 2s. 7f d. for 8 months 6 days 1 The interest of \£, for the givun time, ii *040^ v therefore, , , , ^) and ^)56«13;f . principal. '040| interest of l.£ for the given time. 224520 interest for 8 months. 2806 interest for 3 days. 1871 interest for 2 days. ' 2'29197£ = 2£ Ss. 9f d. 5 days s= 3 days -|- 2 days. As the multi plicand it taken once for every 6 days, for 3 days take ^, for 2 days take ^, of the multiplicand. ^ -|- ^ =-. |. So also, if the odd dayt be 4 =:2 days -f- 2 days, take ^ of the multiplicand twice; for 1 day, take |. From the illustrations now given, it is evident, — 7b find the interest of any sum tn Halifax currency, or any other currm- ey of tohich the deru>minationa are pounds, shillings, fcc. at 6 per cent, it is only necessary to multiply the given principal, after having reduced the shillings and pence in it to the de cimal of a pound by inspection, by the interest of l£ for the given time, found as above directed and written m a deci- IM IVTKRBfT. T77. mal frtction ; iner pnin(in); off* «■ manjr place « for decimals in the prcMiucl as there aro (ieciinal phcp« in liotli the fac- tors counted together, these can be reduced back again to ahilltngs and pence by iniipection. EXAMPi.ES FOR FRACTICE 6. What is the interest of H7£ 3i. Ojd. for I year 3 months 1 yJna. tU' lOs. 9|d. 7. Interest of \Ul£ Is. 7^d. for II mu. 19 days? jina. 6£ ir>8. O^d. !^. Interest of 2(H)£ for 8 mo. 4 days 1 S£ 2s. 7;d. % '¥ 11. 19. 13. U. 15. m IT. 5£ 14s. 8^. 14i: 9s. ijd 8fd. A'ote. The interest of l£ for 6 days be ing 1 thou- 28. 7f d. of 17s. for 19 mo. 1 Is. 7td. of 8£ lOs. for 1 year 9 mo. 12 daysl 18s. 2fd. of 675.£ for 1 mo. 21 days 1 - of mVU for 10 days 1 of 14s. 7Jd. for 10 mo. 1 of m£ for 3 days 1 of 73^ 10s. for 2 days 1 of 180^ 15s. for 5 days 1 of 15000.€ for 1 day ? ■andth, the pounds themselves express the interest in thou 9ondih9for six days, of which wo may take parts. Thus, 6)15000 thousandths, "'"- ' 2<500, that is, 2<£ lOs jfns. to the last. When the interest is required for a large number of years, it will be more convenient to find the interest for one year, and multiply it by the number of years ; after which find the interest for the months and days, if any, as usual. 18. What is the interest of lOOOiS for 120 years 1 Jlns. 7200je. ^ 19. What is the interest of 520iS Os. 9f d. for 30 years and 6 months 1 Ans. 951.£ ISs. 5f d. 20. What is the interest on 400iS for 10 years 3 months and 6 days 1 Jns. 246^6 8i. 21. What is the interest of 220^6 for 5 years 1 — for 12 years 7 — 50 years 1 Ans. to the last 660^6. A 9S2. What is the amount of 86j6, at interest 7 years 1 Ans. 122jS2i. 4fd. % 23. What is the interest of |48«30 for 1 year ? -i It must be clear to the pupil's mind, that to obtain the t 78,79. ihtbrkiit. 157 8. to the last. the int«ro!it upon any nuin in federal nionty, for any time, we pruceeii juiit as we ilu in liahrax currency ; unly we are nut cuinpellod to reduce any part ot the given hudi to decimald, ftincu all l\w deiiouiiuHtions of fedeml money are iDadecimul ratio. The answer to the last exniuple is ^V!'H(M). What in the intcresl of )i«t>4 for 'Z yearn \ Jan. ;j*?'tiH. What is the lutereiit of 'if'JH'.'tO for 7 years, <> niunths and JOdaysl . .Vn«. )*M'IH1» f 7H. 1. What is the interest of 30 pounds fo» 8 months, at A.^ per cent 1 A'o/c. When the rate is any other than «tx per cent, first find the interest at G per cent, then divide the interest r i found hy sucli part as the interest, at the rate re per ^'cnt, as the case may be, will give the interest required. 4^ per cent is j^ of G per cent ; therefore, from the interest at G per cent subtract | ; the remainder will be the interest at 4j pur cent. l£ Is. 7|d. anatoer. 04 i)T44 '36 l(H£. *2. Interest of 54£ His. 2f d. for 18 mouths, at 5 per cent \ Ans. A£ 28. 2f d. t). Interest of 50()<£ for 9 months 9 days, at 8 per cent ? Jin9. 3U . 4. Interest of 62<£ Ss. 4fd. for 1 mo. 20 days, r^ 4 per centi Jm. Os If'^d. 5. Interest of ^£ for 10 months 15 days, at Vl^ per cent! AnH.{)£ .'">.«. lO^d. 6. What is the amount of 53^ at 10 per cent f^r 7 nio.? ^ " Ah8. 561' Is. 9Jd. Tht time, rate percent, and amount given, to find the prin- cipal. f 79. 1. What sum of money, put at interest at G per cent, will amount to 61^ 4^d. in i year 4 months 1 The amount of l£ at the given rate and time, is 1 '08i>'; hence. 61<02£-ri'06£=s56<50, the principal required; that is, — Find the amount of \£ at the given rate and time, by which divide the given amount ; the quotient will be the principal required. Jhv. 5i\£ 10s O ^ 158 IlTTtSHRST. T79. 2. What principal, at 8 per cent, in 1 year 6 months, will amoiini o SoM 2s. 4f I. 1 ' ^/n«. 76iB. 3. \Vh»t principal, al 6 per cent, in 11 months days, will amount to \i9£ (is. 2}(l.1 jfns. MiS. 4. A factor receives 98i^jS to lay utit after deducting bis commfssion of 4 per cent ; how much will remain to be laid out 1 It is evident, he ought not to receive commi8sion on his tnon money. This question, therefore, in principle, does not diflfer from tiie preceding. Note. In questions like this, where no respect is had to tiiM, add the rate to l£. jfns. 950i£. 5. A factor receives lOOSiS to lay out after deducting bis eommission of 5 per cent, what does his commission amount to ? JtiB. 48£. ;J»* li tiiU li t DiscoiiifT. 6. Suppose I owe a man 397iS 10s. to be paid in 1 year, without interest, and I wish to pay him now ; how much ought I to pay him when the usual rate is6 perct.1 1 ought to pay him such a sum, as put at interest, would, in 1 year, amount to 397£. 10s. The question, therefore, dues not differ from the preceding. ^b. 375JS. A'ote. An allowance made for the payment of any sum of money before it becomes due, as in the last example, is called Dhcount. The sum which, put at interest, would, in the time and at the rate per cent, for which discount is to be made, amount to the given sum, or debt, is called the present toorth. 7 . What is the present worth of QQA£, payable in 1 year 7 months and 6 d^ys, discounting at the rate of 7 per centi jfn$. 75Si£. f^. What is the discount on .^l.^ 12s. 7^d. due 4 years henco, discounting at the rate of 6 per cent 1 jfnB. Gii£ 5s. 2fd. 0. How much ready money must be paid for a note of i^£, due 15 months hence, discounting at the rate of 6 per oenti ^n$. 16£ 14d. lOfd. 10. Sold goods for 650;^, payable one half in 4 months, and the other half in 8 months ; what must be discounted for present payment 1 Ant, IS£. f 88,81,82. IHTBRCIT. 159 11. What it the present worth of li6£ 4i. payable in 1 year 8 months, discounting at per cent 1 at 4| per eent 1 at 6 per cent 1 at 7 per cent 1 at 7 J per cent 1 at 9 per cent 1 jfna. to the last 4a£ 17s. 4{d. The time, rate per cent., and interest being given, to find the principal. IT 80. 1. What sum of money, put at interest 16 nronths, will gain lOi: lOs. at 6 per cent 1 1;^ at the given rate and time, trill gain '08 ; hence, 10<5a£ -r '08,*' = 131'25,^ the principal required ; that is, — Find the interest of l£, at the given rate and time, ky vohich divide the given gain, or interest ; the (, lotient will be Ms principal required. ^jSfns. \^\£ 5ii. 2. A man paid4cf lOs. 4^d. interest at the rate of 6 per cent at the end of 1 year 4 months ; what was the principal 1 Jns. 56j& lOs^^ 3. A man received, for interest on a certain note, at the end of 1 year, 20j£ ; what was the principal, allowing the rate to have been 6 per cent 1 Ans. 333iS 6s. 8d. The principal, interest and time betng given, to find the rate per cent. IT 81, 1. If I pay 3jS 15s. 7|d. interest, for the use of Q0£ foi 1 ypar and 6 monthR, what is that ppr cent ? The interest on 36£ at I per cent, the given time, is *5A£ ; hence, 3<7SiSH-'54i&=;<07, the rate required ; that is, Find the fnterest on the given sum, at 1 per cent, for the given time, by which divide the given interest ; the quo- tient will be the rate at which iiitefeit was paid. Ans. 7 per cent. 2. At 2£ 6s. 9fd. for the use of 468^ 1 month, what is the rate per cent 1 Ans. o per cent. 3. At 46iS 16s. for the use of 520j&, 2 years, what is that per centi Ans. 4^ per cent. The prices at which goods are bought and sold, being given, to find the rate ptr cent, of oain or loss. IT 82. 1. If I purchase cloth at l£ 2s. a yard, and sell it at l£ 7s. 6d. per yard ; what do I gain per' centi This question does not differ essentially from those in the foregoing paragraph. Subtracting the cost firom the price t 160 IBVBRBgT. f 83, 83. at lale, it is evident I gain "21.'t£ on a yard, that ii, /|i^ c»r tlie first co«t. ^.^^=='25 per cent the annoer. That ii. —Make a common fraction, writing; the gain or loss for the numerator, and the price at which the article was bought for the denominator ; then reduce it to a decimal. 2. A merchant purchases goods to the amount of 'SSOiS ; what per cent profit must ho make to gain 60i£ 1 ^na. 12 per cent. 3. What per cent protit must he make on the same purchase to gain ti8£ lOs. ? to gain 2i£ 15s. ? to gain 2£ 15s. ? Note. The last gain gives for a quotient <005, which is ^ per cent. The rate per cent, it will be recollected, (IT 75, i^iNr^laote,) is a decimal carried to two places, or hundredths ; all * decimal expressions lower than hundredths are parts of 1 per cent. -*"ii1 4. Bought a hogshead of liquor, containing 114 gallons, at '96<£ per gallon, and sold it at l£ Os. Od. S^qrs. per gal- lon; what was the whole gain, and what was the gain per cent 1 Ans. A£ 18s. 5f d. whole gain. 4i gain per cent. 5. A merchant bought a quantity of tea for 365<£, which, proving to have been damaged, he sold for 332j£^ 38. what did he lose per cent 1 Ans. 9 per cent. 6. If I buy cloth at 2£ per yard, and sell it for 2£ 10s. per yard, what should I gain in laying out 100<^. Ana. 25£ ^ 7. Bought indigo at 6s. per lb. and sold the same at 4s. 6d. per lb.; what was the loss per cent ? Ans. 25 per cent. 8. Bought 30 hogsheads of liquois at 000c£ ; paid in du- ties 20jf 13s. 2f d. ; for freight, 40o^ 15s. l^d. ; for porter- age, &£. Is. ; and for insurance, 30^ 16s. 9fd. ; if I sell them at 26£ pe. hogshead, how much shall I gain per cent! Ans. 11 '695 per cent. Th* principal, rate per cent, and interest being given ^ to find the titM. T 83. 1. The interest on a note of 36£, at 7 per cent, was 'S£ 15s. 7^d., what was the time 1 The interest on 96£ 1 year, at 7 per cent, is 2£ 10s. 4|d. 8'78£4-2'52£=l'5 years, the time required ; that is,—Find the iat«re«t for 1 year on the principal given, at the gtvta /.'■ % 83. iivTrastT. 161/ rata, by which divide the given interest ; the quotient will be the time required, in years and decimal farts of a yP4r , the latter may then be reduced to months and days. An«. I year <> u)untfi» 2. It' iH£ 14s. 2|d. interest be paid on a note of i'ZtU 106. what was the time, the rate being (» per cent 1 Ana. 2':W4=*i years 4 months. .1. A note of 60()-i*, paid interest '2H£, at *< per rent . what was the time 1 Ana. '410-f"=='^ raontlis so nearly as to be cAlled .», and would be exactly 5 but for the fraction lost. 4. The interest on a note of 217;£ 5s. at 4 per cent, wa^ 2fi£ 4s. lOd. ; what was the time ! " Ans. ii y. ;J mo. Xote. When the rate is 6 per cent, wc may dividr t^ie interest by ^ the principal, removing (he scparatrix /:) denominator ; add them together, ari'I h\- the resolling fraction muUiply the given principal. 2 'm Itt2 ISXailKIT. f 83. If- at I Find the inUreit of 10C£ for 2 ycart, 6 moDlht and 10 ilayi, at 6 p«r Mnt. ^ (orrT*») X 2 years c= ^^ T^TJ («»■. T^ffn) X <^ months = yg^. ^, 3^g,f, ^^ are to be re<.'.)ced to one common deuorain- ator. Neglect the ciphers i' the denominators — 6X2X6 = 60; 1 -f 2-|-2=5, the number of ciphers. The common denominator is then 60, and 5 ciphers. ■ ; 6 X 2 X 6s=3:72 ', this with 4 ciphers is 1st numerator. %, 5 X 6 X 6= 180 ; this with 3 ciphers is 2d numerator. 5 X 2 Xl =^- 10 i this with 3 ciphers is 3d numerator. Each numerator has as many as 3 ciphers ; cut off 3 from each and 3 from the common denominator ; ^,f^a -|~ M9 + o'JV = Wts *= ^' Then 100£, the given principal, multiplied by ^^V = V-^ == ^^ ^s. 4d. The reasons of tb^ different steps in the foregoing process will appear — When the rate, as in the above example, is 6 per cent it is obvious, that the interest of any given prin- cipal for one year is -j-J^* of -^a of th?t principal. For any number of years, th? interest must be ?.s many times ^ of the principal, as there are units in the given number of years. In the example 2 is the given number of years ; mu'iiply then t^j by 2 ; or multiply the lowest terms of afrac- tiiyi. the numerator of which is the given rate, and the denom Inr.tor 100, by the given nuinber oj years. ^^ of the given principal then is the interest for 2 years, xaffff of *^*^ &^' en principal is the interest tur 1 month ; for there are 12 inoatlis ip a year, ar»d -j^^ X tV = tj\ttj» <>«* s-ia- j#or 10 .lay., ^^^j X 10= ^'-^^ == ^h- -^a. wh and ^^o then of the given principal are the interest of lOO^t^ for 2 y«?ars, G months and 10 days. It is clear now, why we re- duce these several fractions to one common denominator, add them together, and by the resulting fraction multiply tht! given principal. Find the interest upon 78i.' 4s. for 3 years, <) montli!. aud6d.^ys, br this mcthod,'at6per cent, andalsoatfi prceut. ^ f '-m-j-^^ 164. f INTKKEtT. 103 7h find the interest due on iwti't. V'*- w/**?'* pmii^l pajfmmt» have been iiiAi--U. ^ 84. Tlicrr is no statute in tint IVovince, prescnbinp; any particular ionn or inetlMnl uf rksiiu^ interest upon noten or other oltli^ations. it is biliuVf t the following metliod is generally alU.wed btfore the cotirtit ( f the cQun- try, and also i:» thut which has obtained to the greatest ox- tent in mercantile transacliuns. Rule. — Compute the interest upon the value for which the note, or otiier instrument was given, tu the time of payment, which add to the principal ; find the amount also of each endorsement to the time of payment, which sever al amounts add together, and the sum subtract from the a mount of the value upon the face of the note, or other in- strument. 1. For value received, I promise to pay Louis Rousseau, or order, one hundred pounds fifteen shillings, with interest 100£ 15s. JOH)* BURTOH. May 1, 1822. On this note, were tlie following endorsements : ^ "^ Dec. 25, 1822, received lOig. "' *^- July 10, 1823, " 1^ 4s. Sept. 1, 1824. <« 3^68. June 14, 1825, " 21.£ 15s. * ^ April 15, 1826, " 54i: 98. 1 What was due Aug. 3, 1827 1 Ann. di£ Ss. Id. The whole time is, from May Ist, 1822, to Aug. 3, 1827, which is 5 years, 3 months, 2 days. The interest of 100.1 I5s. for this time is 31<£. I5s. 4fd. This added to the value for which the note was given is, 100<£ 15s.-|-31<^ 158. 4|=: 132£ lOs. 4f d. wiiich is equal to the amount of the value for which the note was given. The first endorsement is \(i£ ; the date of this endorsement is Dec. 25, 1822 ; the 'iime of payment is Aug. 3, 1827. The time, therefore, fur which interest is to be cast upon this endorsement, is 4 yrs. 7 mo. 8 days. The interest for this time is 2i£ 15a. 3d. which added to the endorsement makes its anumnt \2£ 15s. 3d. In the same way find the amount of each other endorse ment, by casting (he interest upon it from the day of its I date to the day of the payment of the note, and add this lu- tereit to the principal, that is, the tndorsement. >:'ii 164 COMPOITM) IffTHKBCT. 9I| ^f 84, R'l The'jd endorscinont is ---->-•- ]i£ 4s. '• ^ The time for whi* !i inti rest iit to bo cast tipon th < 2nd endorsement is - - - - 4 years rrinthi ti^'il »^a^ 3d " - • - - 2 " 11 '« ^? ♦• 4th *' ♦ • . « 2 '« 1 .« I9 «c 5th " ^ .. - - 1 ' 3 .. 18 " The interest upon the 2nd endorsomciit h 0£; '^s. lOd ^' - " ' ;Jd •' (U: lis, 6^'l. V .•/->^-''" '• 4th " '■":- 2=f i;;s. Rjf(t. i " '* 5th " \£ \%. \\U\. The amou.iL of Cfv^ 2ud r(i t)ui interest from that time, the tianic as on the princi* pal. . ^ iV»M. Simple interest is that which is allowed for th« pnn:.p<'ii only ; compound interest is that which is allowed for boLh principal and interest, when the latter is not paid ac the time it becomes due. Compound interest is calculated by adding the interest to th principal at the end of each year, and making tho arar^unt tho principal for the next succeeding year. \ . What is the compound interest of 256bC for 3 years, at 6 \^r cent ? 25fy£ given sum or first principal. '06 15*36 interest, ? . i j i i * «u or^.nn • • i f 'O be added together. 256'00 principal, 3 * 271 '36 amount, or principal for 2d year. '06 16'2RI6 compound interest 2d year, ) addded to- 271 '36 principal, do. S gethtr. 287*6416 amount, or priiicJpal for 8d year. '06 17'258496 compo interest 3d year, 287*641 principal, do. > added U^ S gHher. 304'899 256 48,899c£ amount. * first principal subtracteii. * compound interest for 3 years j^s.'iS.*: 1 7s. Il|d. 2. At 6 per cent, what will be the compound interest, nd what the amount, of 1^ for 2 years ? what the mount for 3 years'! for 4 years t for 5 years 1 ■ 6 years 1 for 7 years 1 for 8 years 1 lor ufHs^ to the last, t^ lis. 10-^d. It is plain that the amount of 2c€, for any given tiin«, rill be 2 times as mvich as the amount of 1^ ; the amount )f 3^ will be .*^ times as much, &c. * ^ Hence, we may form the amounts of 1<£, for aeTtialyeari, m 4 160 C!OMPOC«» HVTBREBT. f 86 i« into a table of maUipliert for Gnding the amount of any turn for tho same time. ^ TABLE, Showing the amount of It', or 14^, &c. for any number of years; not exceeding 24, at the rates of 5 and per cent compound interest. Teart 1 II 4 r> () 7 H 9 10 11 12 5 'ppi- cent. 1*05 1 • 15702- - I '2 1550- - l'27r.28-- 1'34()09 1'407I04- 1'47745-- l'ft,'»i;J2-- V 1 1033- - 1 '79585-- fi per c«nl. I '00 11230 I'MMOl-- 1'20247-- 1 '33822- - l'4ri51-- i'6«J03-- 1 '59384- - 1'0H947-- 1 '02889- - 1'7'.W84 1 '89829+ 5>-'012194- Yr*. 13 14 15 10 17 18 19 20 21 22 23 24 5 |irt fcni 1 '88504 - l'97993- 207892- 2182874- 22920 1 2=400(JI-- 252095 2'e5329-f 2'785904- 2'<)252()+ 3071524- i3'22o09-i- fi |>or cent. 2' 13299-^ •J'2<)090- - 2'39065- - 254036-- 2(i9277-- 2 85433- - 3'02559-- 320713-- 3'39950- - 3'(H)^i5i}- - 3'81974-- 4'04893-4- Uote I. Four decimals in the above numbers will be suf | ciently accurate for most operations. ybte 2. When there are months and days, you may Arsll And the Amount for the ffsars, and on that amount cast thel interest for the months and day* ; this, added to the amount,f will give the answer. 8. What is the am( int of G00£ 10s. for 20 years, at a| per cent, compound interest ? at 6 per cent ? 1<£ at 5 per cent, by the table is '2'05329£ ; therefore I 2'05329X600*50=1593'30£-|- is 1593<£ Os. Ana. at 5 pe{ cent ; and 3'20713X600'50e»192d'881<£-f- is 1925t£ &7s| 7|d. jfn». at 6 per cent. 4. What is the amount of 40^;^ 4s. at 6 per cent, cor pound interest, for 4 years 1 for 10 years 1 for 11 . years 1 — — for 12 years 1 for 3 years and 4 months | for iM years 6 months and 18 days 1 \: Am. to the last 16a£ 2s. 8^il Note. Any sum at compound interest will doiible itsef in 11 years 10 months and 22 days. From what hat low been advanced we deduce the id lowing general f 85 COMPOVND INTER KIT. rs will be tuf you may fifsll jount cast the! o the amountJ 20 yearu, at 5| cent? 9£; therefore Ana. at 5 pet is 1925^ m Irsi - — - i"«^ and 4 months I jt 16a£ 2s. 8f iU doable itsel deduce the fo RTLE. I. To fin'l the interest when the time is I year, or, to find the rate per cent on any Hum of money, without respect to time, as the prpmiiim for insurance, commission, &c., — Multiply tl.e princi|al, or given sum, after havinf; reduced the shilliti^s, and penro in it to the decimal of a pound, by the rate per cent, writleu as a decimal fraction ; after point- ing off as many iilaeiH for derimals in the product as there are decimals in lK)lh the factors, and reducing; these deci- mals hack to RJiillings an terest in hundredths for 6 days. ;. ^ M- t 164 INTRRKST. , >, .,. What is the interest of 17.i' 138. 7jd. for II months f2H days ^ yfns. 11 Is. Id. <». What is the interest of 'JOOf for 1 day \ — 2 days? ;i days 1 4 days 1 5 days 1 » > #; ^ns. for 5 days, Hs. ftjd. 7. What is the interest of half '001 ci" ;or 507 years 1 jfnH. 4d, 8. Wliat is the interest of 8l£, for 2 years 14 days, at ^ per cent 1 | per cent 1 | per cent ? 2 per cent 1 3 per cent ? — 4| per cent 1 — 5 per cent ^ 6 per cent I 7 per cent ? 7^ per cent 1 8 per cent 1 — 9 per cent 1 — 10 per cent 1 — 12 per cent 1 121 per cent 1*i} Ann. to last, 20c£ 12s. lO^d 9. What is the interest of '09<£ for 45 years, 7 months. 11 daysl Am. 4s. lO^d 10. A's note of llb£ was givt.n Dec. 6, 1798, on which was endorsed one year's interest ; what was there due Jan. 1, 18031 xfvtt. Consult ex. 16, Supplement to Subtraction oil Compound Numbers. An9. 207j6' 4s. 4|d.| 1 11. B's note of 56j6' 15s. was given June 6, 1801, on in tereft after 90 days ; what was there due Feb. 9, 1802 1 Ans. 58.;^ 3s. 9fd. 12. C's note of 365i,' was jiut«!«l ■" 3. Wliat in iiiidur- rtttMid by ra'.o per ceni. ? 4, ti. -_ by i«gal ilitercsl: !). *'pri-niiiifn ? in iintlprirtood > nnco : by priiicipiil .' Ti. . 7. by coiiiniiiiitiuii 10. policy ' II, - by anidiini ' 8. iniur- — »tock ' 12 Whnt in iintlprirtood bv xtork bt-ing nt par / I'.V above par' 14. — — bolow pnr ? If*. Tbe rnte per tont. in n dccinml mrriod to liow ninny pincns ^ It!. VV brtt aru flcrimal oxprcisionB /oterr thnn liundretltbs :* 17. IIuw ia interest, (wbon llio tiinr is I year,) cum- inisnion, innurancr, or any thing olso rutnd at m> murb p»T cent, with- out rospocl to time, found ? IH. When the r'.\\\- \* oiu; per cent., or lc uer cent., whi ii there arc inuuthn and davH in tin- flivcn time ? yi When the givni liun; is less tlian (1 days, liow is thr interest most readily fbund ? 22. If the sum given be in I'imIci ;il mo- ney, how is interent cntt ' 'JA. Yi lien the nito is any other than per cent., if there be months and days in the given time, how is the in- terest found .' 24. What w the rule for casting inturest on notcK, &,c when piirlial payments liave been made? 2.'*. flow may the principjil bo found, the time, rate per i;ent. nnd amount being given ? 2ti What is understood by tliscovnt'' 27. by present uorth? 2.-^. How in the principal found, the time, rate per cent, and lutorcflt biiing given Sf>. IIow 18 the rate uer ce " goods arc bought ami iiold being given .' 3(1. found, the principal, inti-rcst, mid time being given ' 'M, Howjh tht time found, the principal, lale per cent and intere.-^t being given - '.V2 How may interest be cast by vulgar frartiotu f il^t. What is the reii soning in regard to this rule ? 'M, What, is simple interest ' IJ5. compound •ntercBli' 'Ml. liow is compound interest computed ' -ent of gHin or loss found, the pi ices ul wliinii llow is the rJite pcrceiii EXERGISES 1. What 'm ttie interest of ^T^U 10«. 2fd. for 1 y«ar iO i. days, at 7 per centl Kns. l^£ IIJs. ()jd 2. What is the intere»^ of 486^" lor 1 year, 3 months, iU days, at 8 per cent? Ans. ii(\£ 15?». 4Jd 3. D'snote of 203i,' was given Oct. .'5, JSOs, .in iiitere»»t . after 3 months ; January 5, 180(>, he paid o{)£ ; what wai^ there due May 2, 1811 1 Ans. Vr^C Ts. 2d 4. E's note of 870X, was given Nov. 17, 18, he paid 186£ ; what was there due Dec. 23, 1807 •». Ans. 101I9^- Us. ti^.l 5. What will be the annual insurance, at ^ per fcent, on a house valued at 1600^? Ans. l^Ji' 6. What will be the insurance of a ship and cargo, valu«d X P % inftt «t •trPPLBWSVT TO TllTrilKST. f K i. at rtiVi'.U: »t I i per cetU 1 — - at J p«r cent ? at /^ per cent ! at f ^ per cent 1 at J per cent ? .Vor«. ('(.Msultr 7j(l. 7. A man having com prom iied with liiN cmditors at (^i) per cent, what must he pay on a debt of t.')7»^' Os. *2fdA ^nt. fJ'ii. If^a. ;}d ^. Wttat ia the value uf HOOil Moutrcai Bank stock, at 11*24 per cent ? //n$. 9()0£. - 9. What in the vahic of rAV)£ !5s. of stock, at IKJ per reijt! jf/n. o-.Mi,'!)*. Il^d. 10 What principal at 7 {tvt cent, will, in 1) monthN i^ dayi auount to \)tl£ Hs. ^is. 4UO;6'. 11. What is the present worth of 420;i', payable in 4 years and 12 uays, discounting; at the rate of 5 per cent % In large sums, to bring out the hundredths and thousandths iurrectly, it will sometiuaes he necessary to extend the deci- mal in the divisor to five places. * jins. '<^A£ 10s. Ifd. V2. A merchant purchased goods for i25Uj£, ready money, and sold them again for 300^^, payable in 9 months ; what c'td he gain, discounting at (i per cent 1 Ans. d7£ Is. 7fd. IM. Sold goods for 3120j6', to be paid, one half in » nioutliS, and the other half in li nM>ntIis ; what must bedis- • ounteJ for present payment 1 Ans. 68*f 9s. lOd. 11. Thu interest on a certain note, for 1 year 9 months, was 49J,' 17s. 6d., what was the principal ? Ans. Alo£. 15. What principal, at 5 per cent., in 16 months 34 days, will gain 35^1 Am. SOOjf. i(i. If 1 pay l.>Jt' lUs. interest for the ose of 50a£, 9 *^ months and 9 days, what is the rate percent 1 17. If I buy candles at $ '167 per lb., and sell theni at *^() cents, what }». 21. Bought cloth at $4*4)!< per yard; how must I sel^ it to gain I2A per cent? ^^ Ans. |!5'b4. 4 » ^ Y M, m. B^VATtON or PAVMBNV«» «| i ♦•!*• 2*2. Bought a barret ol powUwr fur 44.' ; tor bow much uiUMt it be sold to lose 10 |>cr Mnt J jfM. \V£ Vln %\. Bought eloth at l<>i. per yanl, which not proving to good a« 1 exp«€tcd, 1 am coutent to lose '17| p«r cent ; how muit I Mil it per yard/ j1n». 12f. 4 id *i4. BoughtoO galioua ol' brand}', at 92 centaper gallon, but hj accident lU gallona leaked out ; at what rate must I tell the remainder per gallon, to gain upon the whole co«t at the rate of 10 per cent 1 Ant. 91'2(V> per gal. 25. A merchant bought 10 tont of iron for ^950 ; the freight and duties came to $145, and hia own charges to if'i-'>; how must he sell it per lb. tu gain "20 per cent hy it ! Aim. 6 eents per lb. • " 9' %i^ m EQUATION OF PAYMENTS. f Jb' V 86. Equation of payments is the method of finding the mean tiose for the payment of several debts due at dif- ferent times. 1 . In how many roonthf will 1£ gaio as much m /» pounds will gain in G months 1 2. In how many months will l£ gain as much as 40i^ will gain in 15 months 7 Aru. 600. 3. In how many months will the use of 5£ be worth as much as the use of l£ for 40 months 1 4. Borrowed of a friend 1;^ for 20 iqpnths ; afterward)! lent my friend 4£ ; how long ought he to keep it to become indemnified for the use of the l£ 1 •>. I have three notes against a man ; one of 12i£, due [in ') months ; one of 9£, due in 5 months ; and the other I of Ci£, due in 10 months ; the man wishes to pay the whole jat once ; m what time ougl^ be to pay it 7 12£ for 3 months is the same as l£ for 36 mouths, and 9£ for 5 months is the same as l£ for 45 months, and G£ for la months is the same as l£ for 60 montfie. / '^-^ 27 UT He mighf. therefore, hare 1^ 141 months, and he may »jt Jii' L • A.f f^V 9 ♦.1 .% 17^ "^ATIO ; OR THE RBLATIOR Or »VMBEKS. f 86, 87. koep 27 puundft ^V I^c^ "^ 1<^S '> ^^^^ '<^> W^^^*^ months ti-\- days I An«»cer. # Hence, — To find the mean time for leTeral payments, — KuLE : Multiply each Kum by its time of payment, mnd di- vide the hum of the products by the sum of the paytMntB, and the quotient will be the answer. Note. This rule is founded on the supposition, that what is gained by keeping a debt a certain time after ii is due, is the same as what is lost by paying it an equal time before it is due ; but, in the ticst case, the gain is evidently equal to thj interest on the debt for the given time, while, in the second case, the Iobb is only equal to the ditcount of the debt for that time, which is always leaa than the intereet ; therefore, thp rule is not exactly true. The error, howev- er, is so trifling, in most questions that occur in business, as scarce to merit notice. ^v G. A merchant has owing to him 300i£, to be paid as fol>- lows : 5Q£ in 2 months, lOOiS in 5 months, and the rest in S months ; and it is agreed .to make one payment of the ' whole : in what time ought that payment to be 1 4»j-;k*^ , r ^. An«. 6 nMMiths. 7 A OWDs B 136^, to be paid in 10 months ; 96^ to be paid in 7 months ; and 200<£ to be paid in 4 months ; what is the equated time for the payment of the whole 1 ,, : . ;;{/ Ans. 6 months, 7 days-f-. 6. A oires B 60a£, of ^hich 200j^ is to be paid at the present time, 200 in 4 months, and 200 i^ 8 months ; what » the equated time for the payment of the whole 1 * :T ; 'f^ i '; ^^ Arm. 4 months. "^ ' 91 A owes B 300i?, td be paid as follows : ^ in d-taapnths, 1^ in 4 months, and the rest in 6 months : what is the equa- ted time 1 Ans..4^ months « J^£d^ 'r*^. ■'? f, \tji 'It «i RATIO ; i ^i- OR ^ TH1E Ii£lL.ATION OP NtJMBiSlKS. IT 87. 1 . What part of 1 gallon is 3 quarts 1 1 galton is 4 quarts, and 3 quarts is | of 4 quads. Ans } of a gal -..k (^7. mATi« ; on tmb ui^Amfiu or arvMiBRi. 173 'i. What pari of :l qflarU it t gallon ? 1 galloo, \)t'tng t quarts, IB I of 3 quarti ; th?t if, 4 quarts it 1 time 3 quarts and ^of anotbar time. Ans. ^csJ:^ 3. What part of 5 bushels is I'i bushels 1 Finding what part one number is of another is the saino as finding what is called the ratio, or relation of one num- ber to another ; thus, the question, What part of 5 bushels is 12 bushels ? is the same as What is the ratio of •'> busheh to 1:;! bushels 1 The answer is V=^- Ratio, therefore, may be defined, the nt^ber of times ub«; number is contained in another ; or, the number of times one quantity is contained in another quantity of the saint- kind. »i , V « 4. What part of 8 yards is 13 yards ? oi, What is the ra- tio of 8 yards to 13 yards 1 13 yards is ^ of 8 yards, expressing the division /radton- ally. If now we perform the division, we have for the ratio 1 ^ ; that is, 13 yards is 1 time 8 yards, and | of another time. r We have seen, i^ 15, sign,) that division may be expresse/minator. ; " L " 1 V ;?" • ' ": * 5. What part of 1'^ pounds is 11 pounds 1 or, 11 puund.s is what part of 12 pounds'! 11 is the number which ex presses the part. To put this question in the other forui. viz. What is the ratio, &e. let that number, which expresA- os the part, be the number last named ; thus. What is the ratio of 12 pounds to 11 pounds 1 Ans. -f^. U. What part of l£ is 2s. Cd. ! or, What is the ratio ot IJ? to 2s. (id.? 1<£= 240 pei^ce^ and 2s. Od e=30 pence i hence, ^^qs=>^. is the Answer. .}^«^ ., -^ . 7. What part of 13s. (id. is l£ lOs. or. What is the ra- tio of !*U. Gil. to l£ lOs. ? An«. 2^". 8. What is the ratio of 3 ic 5 1 -? — of 5 to 3 1 of T P-2 >: 174 to 19 T «i;lb or t^rke. 7 88. ':» >vflO6 of \m to 84 1 of«jr,tonirr? of 1 107 to HI 5 1 Am. to the last ^. ^^JK' 'd .^v> V/ Mm ■ ' \ Tlf< RUI^E OP THREE. **1I HH. I. If a piece of cloth, 4 yards lonj^, cost l'Z£ , what will be the cost of a piece of the same clotli 7 yards ^f Had this piece coulaincd twice the uumbor of yards of the first piece, it ig evident the price would have been twice as> much ; had it contained 3 times the uumboi of yards, the l)rice would have been 3 times as much ; or had it contained only half the number of yards, the price would have been '>nly half as much ; that is, the cost of 7 yards will be such part of 12 pounds as 7 yards is part of 4 yards. 7 yards is I of 4 yards ; consequently, the price of 7 yatds must be | of the price of 4 yards, or J of 12 pounds, } of 12 pounds, that is, 12X4=^ V'^^'^^ P**""^^' *"*^*''*''' "^ , !?2. If ahorse travel 30 miles in 6 hoursj, how many miles wilt lie travel in 11 hours at that rate 1 ^ '...,: V n liours is V*^*^ ^ hours, that is, 11 liour^ I'ii'l fim<^ 6 !)o(irs, and ^ of another time; consequently, he will travel, in 1 1 hours, 1 time 30 miles, and | of another tiiae, that is, the ratio between the distai/ces will bo equal to the ratio be- twcfjti the times. 'J of 30 miles, that is, 'JOX V="t^'*«**''» "''es. If, then, no error has been commvltied, 55 miles must be y of 30 miles This is actually th<* case ; for ^§==:i(>. ' ■ 'ti ' Ans. SA miles. <^uantitiep which have the same ratio between therti are said to he proportional. Thns, these four quantities!, ' bo'irs. liours. miles. niite«. '* 'f^ ' d;'' H, 30, 55, 4 written in this order, beiniijj such, that the second cunSaffi** (he first as manv times as the fourth cnntains the third, that Jim -""V^ .spg'W^ ? 8b. f S8, 89. ri«E OF TIlRfit:/ « m is, the ratio between tfte third and fourth bcmg equal to th» ra*io between the first and second, form u hat is called a pro- portion, It follows, therelore, (hat propurtton in a combination of two equal ratios. /?a/io exisis between /mjo numhurs ; but pro- j)ortib>ti requires at leastt three. To denote that there is a proportion between the num- bers 0, 11, 30, and 55, they are written thus : — 6:11::30:*> which is read, is to 1 1 as 30 is to 55 ; that is, (i is th* same part of 11 that 30 is of 55 ; or, (i is contained in 11 ;»s many times as 30 is contained in 55; or, lastly, the ratic «>r relation of 11 to 6 is the same as tliat of 55 to 30. ''• 89. Tiio first terra of a ratio, or relation, is called thr antbcedtnt, and the second the comefjuent. In a proportion there are two antecedents, and two consequents, viz. the antecedent of the tirst ratio, and that of the second ; the con- sequent of the first ratio, and that of the second. In the pin f^A i^l^ 55, the antecedents are G, 30 J tho f' portion G : 11 :: 3i) : consequents, 11, 55. The consequent, as we Iiave already seen, is taken for tho numerator, and the antecedent for the denominator of the- fraction, which expresses the ratio or relation. Thus, the lirst ratio is 'g', tho second ^i;==V ; and that the.ie tv» ,u- tios are equal, we know, because the fractions are equal. The two fractions y and ^^ being equal, it follows that, by reducing them to a common denominator, the numerator of the one will become equal to the numerator of ♦Me ^ther. and, consequently, that 11 multiplied by 30 wiil gi the sanie product us 55 multiplied by 6. Tliis is actual, tho cane'; for 11 X30=:330, and 55X6=330. Hence it follows, —If four numbers be in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or of th« two means. Hence it will be easy, having three terms in a proportion given, to find the fourth. Take the last example. Know- ing that the distances travelled are in proportion to the times or hou^-s occupied in travelling, we write the proper- tion,|thus ; hotiM. hours, miles, miles. V ; 11 :: 30,; '-^ « J 70 # " liULB or Tiimite. f «9 •'■ m t m % Nuw, Hincc the product of the extreme* is equal to the |iro^, which ought to contain the 36 bar rcls as many times as IWijf contains 541* ; we thus get the following proportion : jMiundj. pounds. Iiarr*?!* barrels. 54 180 ;3« :ivof Three Dirtc{ i« when vu/re requires inore, or Irtg w .^Jtii^ '^^. t m '"■HMi'!' 599 MVLE or TllRBtC. 177 \ie required (o do it , consequently, tho days will he Una in proportion a« the number uf men is greater. There is stilt a proportion in this case, but the order of the terms is in- verted ; for the number of men in the ttecond set, )«tiin|; two times that in the first, will require only one half the tinte. The first number of days, therefore, ought to contain tho second as many times as the second number of men contains the first. This order of the terms, being the reverse of that assigned to them in announcing the question, we say, that the number of men is in the inverse raftoof the number of days. With a view, therefore, to a just solution of the question, wo reverse the order of the two first terms, (in doing which we invert tho ratio,) and, instead of writing the proportion, 3 men : men, (f ,) we write it, 6 men : 3 men, (§,) that is, . v*t-* ». fvuki * « •"*"• "»*"• ^"^■*- '^^y^- V ; «> ; ; lU t.. Note. We invert the ratio when we reverse the order of the terms in the proportion, because then the antecedont takes the place of the consequent, and the consequent that of the antecedent ; consequently, the terms of the fraction which express the ratio are inverted ; hence the ratio is in- verted. Thus, the ratio expressed by fatsS, being inverted, is f=4. Having stated the proportion as above, we divide the product of the means, (10x3as=30,by) the known extreme^ 6, which gives 5, that is, 5 days, for the other extreme, or term sojght. >m¥H M jfn8.6 d^ys. From the examples and illustratioos now given, we de- duce the following general quires Ze55, as in this example: — If 3 'men djg a trench 48 ibet loni; in a certain time, how many reet will 12 men dif; in the sntne timeP Hero it is obvious, that the more men there are employed, liie more work will be done ; and therefore, in this instance, more requires more. Again: — If U men dig 48 feet in a given time, how miuch will 3 men dig in the snme time i Here less requires lessy for the less men thero are employed, the lesi work wilt be done. The Rule of Thrte Inverse, is when more requires less, or less requires more, as in this example : — If 6 men dig a certain quantity of trench in 14 hours, how many hours will it require 19 men to dig the same quantity ? Here more requires less ; that is, 12 men being Tnore-tlmii 0, will require less time. Again:— If (> men perform a piece of work in 7 days, how long will 3 men be in perfbrming the same work r Here less requires more ; for the number of men, being less will re- quire more time. .,»^&>|s,. n^ I 'T lU HVLC OP THRB£. RULE. f 80. or the three jriren numbers, make that the third term which is of the .lame kind with the answer sought. Then consider, from the nature of the question, whether the an- swer will be greater or less than this term. If the answer is to be greater, place the greater of the two remaining num- bers for the second term, and the less number for the first term ; but if it is to be less, place the less of the two re- maining numbers for the second term, and the greater for the first ; and, in either case, multiply the second and third terms together, and divide the product by the first for the answer, which will always be of the saror denomination as the third term. yote 1. If the first and second terms contain tujerent denominations, they must both be reduced to the mmt de- nomination. Jf 6 yards of cloth cost l£ 4s. what wiil 364 qrs. cost 1 yds. qrs. > m' 8 : 364 :: !<£ 48 ^ta Reduce 6 yards and 364 quarters to the same denomina- tion, by dividing the 364 quarters by 4, which will bring it into yards. 3|4-«91. yds. yd«. . .'*?^'> - mi, 8 : 91 :: 1^ 48. iH :^ui mil yote 2. If the third term be a compoutid number, it must either be reduced to integers of the lowest denomination, or the low denominations must be reduced to a fraction of the highest denomination contained in it. r^^Hf '*W"f i 'vh')ua^4}. h yards. 8 ynrds. 91 1^48. 24 8. Now multiply the 24s. by 91, ^ad divide tho product by 8 ; the answer will be shillings, which can be reduced to pounds. **■ ,j Or, the 48. can be reduced to the fraction of a pound ; 4s. -1.20, that is, jjV^^ of ajiouwl; so l£ 48.=lf£. Or, we can reduce the 4s. to the decimal of a pound; ^^^ which, annexed to the l£, is equal to 1'2^. .* o1 «1 -H t<} » el', pay S£ 2s. parish taxes, how much should a person pay whose rent is 97 j£ ? jfns. 9£ 28. 2ffd. 12. If I buy 71bs. of sugar for 3s. 9d., how many pounds can I buy for i£ 10s. jfns. 5(i Ib^. .13. If 2 lbs. of sugar cost Is. 3d., what will 100 lbs. ot col3ee cost, if S lbs. of sugar aie worth 5 lbs. of coffee ! jfns. !xi'. 14. If I give (W for the use of lOOi^ for 12 months, wbat must I give for the use of 9S3<£ the same time 1 Ana. 5S^%£. 15. There is a cistern which has 4 pipes ; the lirst will fill it in 10 minutes, the second in 20 minutes, the third irt 40 minutes, the fourth in 80 minutes ; in what time will all four, runnipg together, fill it 1 : *^'^' • • tV -t 7TT + iV -f bV = i§ cistern in 1 minute. j^». 5^ minutes. IG. If a family of 10 persons spend 3 bushels of malt in a month, bow many bushtls will serve them wben there are 30 in the family 1 i^»4. 9 bushels. im RLLB or TIIKKR. T W). if 1 ( m J! Note. The rule of proportion , allliougfi nf Irequent iifie, is nut o( indispeniiuble necetnity ; for ail ()ii«stionN under it may be noIvim! on general principles, without the formsility of a proportion ; that is, by unalysiff, m already shown, • d'i, ox. 1. Thiis, in the above cxaqpple.—ir 10 persons sp«n(i JJ bushels, 1 persfiH, iu the paiue time, ^ould spend ^,, of :] buHholN, that is, f\j of a IhihIicI i and ,W persons would spend ']0 tin)cs as much, that is, |^ =-0 bushels, as before. 17. h a slaflF, 5 ft. R in. in length, cast a shadow of (i feet, how hif^h i.s that steeple whose shadow measures 153 feet 1 /fns. 144^ feet. 18. The same by analysifi. If fl ft. shadow require a staff of 5 ft. 8 in. s= f}8 in., 1 ft. shadow will require a staff of ^ of ()8in. or f^^f in. ; Mien 153 ft, shadow will require 153 times as much ; that is, 'jf X 153= '»|o* = 1734 in. = 144i ft., »s before. 11). If ',\£ sterling be equal to ^^£ Halifax, how much Halifax is equal to lOOOci' sterling! " '^" *' Ans. lini'28. 23d. 20. If 1111£ 2s. 23d. Halifax, be equal to lOOOil stcr ling, how much sterling is equal to 3^<£ Halifax ? Ans. ^£, 21. If lOOOf sterling be equal to 1111^28. 2§d Hali- fax, how much Halifax is equal to 9£ sterling 1 Ans. 3^:£. 22. If 3£ sterling be equal to ^£ Halifax, how much sterling is equal to lllliS 2s. 2§d. Halifax. Aiis. 100«U\ 23. Suppose 2000 soldiers had been pupplied with bread sufficient to last them 12 weeks, allowing each man 14 ounces a day ; but, on examination, they find 105 barrels, containing 200 lbs. each, wholly spoiled ; what must the allowance bo to each man, that the remainder may last them the same time 1 Ans. 12 ex.. a day. 24. Suppose 2000 soldiers were put to an allowance of 12 oz. of bread per day for 12 weeks, having a seventh part of their bread spoiled ; what was the whole weight of their bread, good and bad, and how much was spoiled 1 ^ The whole weight, 147000 lbs. «i|iik/»iJ6. 2000 soldiers, after losing one seventh part of their bread, had each 12 oz. a day for 12 weeks ; what was the whole weight of their bread, including that lost, and how much might they have had per day, each man, if none had been lost ? C Whole weight, 147000 lbs. ^718. \ Loss, - - 21000 lbs. V 14 oz. per day, had none been lost. 27. There was a certain building raised in 8 months by 120 workmen ; but, the same being demolished, it is requir- ed to be built in 2 months ; I demand how many men mu!>t be employed about it. Ans. 480 men. 28. There is a cistern having a pipe which will empty it in 10 hours, how many pipes of the same capacity will empty it in 24 minutes ? jins. 25 pipes. 29. A garrison of 1200 men has provisions for 9 months, at the rate of 14 oz. per day ; how long will the provisions last, at the same allowance, if the garrison be reinforced by 400 men 1 jina.^ months. 30. If a piece of land, 40 rods iu length and 4 in breadth, make an acre, how wide must it be when it is but 25 rods long? ^ns. Of rods 31. If a man perform a journey in 15 days when the days are 12 hours long, in how many will he do it when the days are but 10 hours long 1 Ana. 18 days. 32. If a field will feed 6 cows 91 days, how long will it feed 21 cowsl j§na. 26 days 33. Lent a friend 292i£ for 6 months ; some time after, he lent me &iQ£ ; how long may I keep it to balance the ftivor 1 jfna. 2 months 5 -{- days. 34. If 30 men can perform a piece of work in 11 days, how many men will accomplish another piece o( work, 4 time« as big, in a fifth part of the time ? jfns. 600 men. 35. If 1^ Ih. of sugar cost ^ of a shilling, what will 2| of a lb. cost 1 ^8. 4d. 3^UH- Note. See IT 62, ex. 1, where the above queaticn is solv* cd by analysis. The eleven following are the next suc- ceeding examples in the same %^ IW RULE or TMMKC f 89, 90 • If 'if ; IpI M. If 7 lU. of sugar cott | oCm., what c i)^ ; '29^1^ yds., at the rate of 7£ for 5 yds., and sold thoJT 71 a; the rate oi lliC for 7 yds., and gained W0£ by the uaru'aiu ; how many bales were there 1 Arts. 9 bales. ' 47. At 'J£ for 6 barrels of flour, what must be paid for 178 barrels 1 An«. 367^^. 48. At 98. 6d. for 3 cwt. of hay, how much is that per ton? Ans. a£36. 4d. 49. If 2'5 lbs. of tobacco cost 75 cents, how much will IS'5 lbs. cost 1 V*"^'.' ^'iff}r'H Ane. $5'55 50. What is the value of '15 of a hogshead of lime, at lis. ll^d per hhd. 1 An«. Is. 9fd. 51. If *15 of a hhd. of lime cost Is. 9fd.,'wbat is it per hhd. 1 «» f "^ Atis. lis. 11^. limwu,u, COMPOUND PROPORTION. ff^;^; T d whicl Nc that a dayl loill aI Tl{ This WJ answl T voo. C0MP0V9ID pnoroRTioir. 1?!^ 7 houni in a Hay, h w manr milcfi will he travel in \H days, if he travel 10 hourn in a day 1 'I 4 queation may be aolved aevcral wayH. First by ancUystH : — If we knew how many luilea the nan travelled in 1 hojr it is plain, we mi/^ht take thif number 10 timox, which would he the number of miles he would travel in li hours, or in L lies, would bf ■'. travelling ^ of 273 id^of V.-' '«ilp«. in 1 day of 7 h. miles is y^ miles, the distance he travels in 1 hour : then, 10 times y^ = zjp miles, the distance he travels :n 10 hours ; and 12 times 2j^o = ^ ^t^o __ 350 miles, the dig- tanco he travels in 12 days, travelling 10 hours each day. Ana. 360 miles. But the object is to show how the question may be solv- ed by proportion : — First ; it is to be regarded, that the number of miles tra- velled over depends upon two circumstances, viz. the num- ber of days the man travels, and the number of hours he travels each day. Wo will not at first consider this latter cinumstance, but suppose the number of hours to be the same in each case : tli« question then will be, — If a man travel 273 miles in 13 days, how many miles mil he travel in 12 days 7 This will furnish the following proportion : — .„.*.. 13 days : 12 days : : 273 miles : miles > *^ ■"•'•■•-»■• which gives for the fourth term, or answer, 252 miles. Now, taking into consideration the other circumstance, or that of the hours, we roust say, — If a man, travelling 7 hours a day for a certain nun^er of days, travels 252 vUles, Itoto far mil he travel in the same time, if he travel 10 hours in a day ? This will lead to the following proportion :— • ».* (1 i 7 hours : 10 hours : : 262 miles : miles./ . , This gives for the fourth term, or answer, 360 aiiles. We see, then, that 273 miles has to the fourth term, or answer, the same nroportion that 13 days has to 12 days, IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I l^|2£ 12.5 1-25 ■ 1.4 III 1.6 "A V] >:5 /. Hiotographic Sciences Corporation 23 WfST MAIN STREET WEBSTER, N.Y. US80 (716)872-4503 m \ :\ \ [V i ^ .A^ 1 ^i It tl {l 184 COMPOUHD PROPOETIOir. f 90. and that 7 hours hat to 10 houn. Stating thia in the form of a proportion, we have. 13 day. : 12 day. > , . 273 „,,*. . mil«< 7 hours : 10 hours S by which it appears, that 973 i. to be multiplied by both 12 and 10 ; that is, 273 is to be multiplied by the product of 12X10, and divided by the product of 13X7, which, be- ing done, give? 360 miles for the fourth term, orannrer, a. before. In the same manner, any question relating to compound proportion, however complicated, may be stated and solved. 2. If 248 men, in 5 days, of 11 hours each, can dig a trench 230 yards long, 3 wide, and 2 d(iep, in how many days, of 9 hour, each, will 24 men dig a trench 420 yanl» long, 5 wide and 3 deep 1 Here the number of days, in which the proposed work can be done, depends on five circumstancesy viz. the number of men employed, the number of hour, they work each day,, the length, breadth, and depch of the trench. We will con- sider the question in relation to each of these circumstances, in the order in which they have been named : — l.t. The numher of men employed. Were all the circum- stances in the two case, alike, except the number of men and the number of days, the question would consist only in finding in how many day. 24 men would perform the work which 248 men had done in 5 days ; we should then have ^. ^^> , 24 men : 248 men : : 5 day. : days. 2d. Hours in a day. But the first laborers worked 11 hours in a ^ay, whereas the others worked only 9 ; less hours will require more days, which will give 9 hours : 11 h6urs : : 5 days : days. ^**" - 3d. Length of the ditches. The ditche. being of unequal length, as many more days will be necessary as the second is longer th^n the first ; hence we shall have a \,i' ^0 length : 420 length : : 5 days : days. 4th. Widths. Taking into consideration the widths, which are different, we have 8 %ide : 5 wide :: 5 days : days. .^^ ' ^ Qth. Depths.. Lastly, the depths being different, we have 2 vleep : 3 deep : : 5 days : da^s. * ( f 90. M form l«i both 12 oduct of lich, be- iwer, at Mnpound i solved. an dig a )W many 20 yard* work can umber of »acb day^ vrilt con- mstances, e circum- f men and in finding irk which re ys. -^ forked 11 y 9 > Uss >f unequal he second daya^ ' le widths, |s. It, we have If. .Pi'^m.^ If 91 COMPOVHP PROPORTION, 185 ' It would seem, therefore, that 5 daya haa to the fourth term, or answer, the same proportion that 24 men has to 248 men, whose ratio is y/ , that 9 hours has to 1 1 hours, the ratio of which is y , that 230 length has to 420 length, - • . . - . ||g, that 3 width has to 5 width, that 2 depth has to 3 depth, - - -'"'- ' - all which stated in form of a proportion, we have Men, 24 : 248-1 -.v,.; days. Hours, 9 : 11 I commoateroi. Length, 230 : 420 V : : 5 days : Width, 3 : 5 Depth, 2 : 3 J If 91. The continued product of all the second terms 248 X 11 X 420 X 5 X 3, multiplied by the third term, 5 days, and this product divided by the continued pro- duct of the first terms, 24 X 9 X 230 X 3 X 2, gives '^^^%^TT (la^ys for the fourth term, or answer. 288^. But the first and second terms are the fractions ^/^, y , ^i%} ^ ^^^ h ^hich express the ratios of the men, and of the hours, of the lengths, widths and depths of the two ditches. Hence it follows, that the ratio of the number of days given to the number of days sought, is equa* to the product of all the ratios, which result from a comparison of the terms relating to each circumstance of the question. The product of all the ratios is found by multiplying to- gether the fractions which express th^.m, thus, 2 48X11X420X5X3 _1 7186400 .... 1718fM00 24 X 9 X230X3X2 298080 ' »«'*""'""""'"' 298080 ' represents the ratio of the quantity required to the given quantity of the same kind. A ratio resulting in this man- ner, from the multiplication of several ratios, is called :i compound ratio. From the examples aiid illustrations now given we de- duce the following general for solving questions in compound proportion, or double rule of three, viz.— Make that number which is of the same kind with the required answer, the third term ; and, of the remaining numbers, take away two that are of the same Q 2 . 186 COMPOVlirD PROPORTION. IT 91 kind, and arrange them according to the directions given in f imple proportion ; then, anj other two of the aame kind, and 10 on till all are used. Laatly, multiply the third term by the continued product of the second terma, and divide the result by the continued product of the first terms, and the quotient will be the fourth term, or answer required. EXAMPLES FOR PRACTICE. 1. If 6 men build a wall 20 feet long, 6 ft. high, and 4 ft, thick in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft thick 1 Ana. 80 days, 2. If the freight of 9 hhds. of sugar, each weighing 12 cwt. 20 leagues, cost 16£, what must be paid for the freight of 50 tierces, each weighing 2^ cwt. 100 leagues 1 An$,92£ lis. lOfd. 3. If 56 lb. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days 1 Ana. 36 lbs. The same by analyns. If 7 men consume 56 lbs. of bread, 1 man, in the same time, would consume i^ of 56 lbs. = y lbs. ; and if he consume y lbs. in 14 days, he would con- sume -^ of yE=||lb. in 1 day. 21 men would consume 21 times so much as 1 man ; that is, 21 times ||=i^|61bs. in 1 day, and in 3 days the ould consume 3 times as much ; that is, 3||8s_equ&' ibs. as before. Ans. 36 ilbs. ■' Note. Having wrought the following examples by the rule of proportion., let the pupil be required to do the same by analysis. 4. If 4 reapers receive 2£ 15s. 2^. for 3 days' work, how many men may be hired 16 days for 25<£ 15s. 2^d. 7 . -(jjiit •■:h-;-' ■'; A«8» 7 m^n. 5. If 7 oz. 5 pwt. of bread be bought for 4^d. when corn is 4s. 2d. per bushel, what weight of it may be bought for Is. 2d. when the price per bushel is'59. 6d.,1 Ans. 1 lb. 4 oz. 3|^pwt8. 6. If lOae gain &£ in 1 year, what will 400;^ gain in 9 months 1 Note. This and the three following examples reciprocal- ly prove each other. 7. If 100^ gain 6^ in 1 yea.-, in what time will 400iS ■ 'rijn' ■ ■'^' IT 91. IS given in latne kind, ed product continued the fourth !i, and 4 ft, A one 200 t. 80 days, igbing 13 the freight lis. lOfd. days, how ns. 36 }bs, i. of bread, 56 lbs. = ivould con- 1 consume =4f6Ibs. 3 times as n«. 36 ilbs. les by the ) the same ys' work, M. 2^d. ? ns. 7 inen. when corn bought for 3|^pwt8. \£ gain in recijMTOcal- (viil 400iS H 91. aOPPLBMBIfT TO ataOLB MULB OF THBBB. 187 8. If 40(K£ gain 18i& in 9 montbe, what it the rate per cent per annum 1 9. What principal, at 6 per cent per annnm, will gain 18i;f in 9 months 1 10. A usurer put out $75 at interest, and at the end of 8 mouths, received, for principal and interest, $79 ; I demand at what rate per cent, he received interest. Ana. 8 per cent. 11. If 3 men receive 81^^^ for 19^ days' work, how much must 20 men receive for 100^ days 1 Ans. ii05c£ Os. 8d. ^'I^^'*' !»«. SUPPLEAUSNT TO TH£ SINGLE RULE OF *^'^- THREE. QUESTIONS. A> .■:4 ;K:^.^M^>-j.: :^4 1. Whnt is pronortion } 2. How many numben are required to tbrm a ratio } o. How laany to /brm a proportion .' 4. VVlmt is liis first term of a ratio called ^ 5 the second term .^ G. Which is taken lor th^ numerator, and which for the denominator of the fhic-' tion expressing the ratio } 7. How may it be known when 4 num- ' bers are in proportion .' 8. Having three terms in a proportion given^ how may the fourth term bo found.' 9. What is the operation, by which tne fourth teri^ is found, called .'' 10. How does a ratio he> come inverted .•* 11. What is the rule in proportion.' 13. In what denomination Will the 4tb term or answer, be found .' 13. If the firs; aod second terms contain difierent denominations, what is to be doner 14. What is compound proportion, or double rule oi' three P 15. Ruk.^ £X]iiRCIS£S. '<-.3^..r^ 1. 1(1 buy 76 yards of cloth for 28c£ 5s. lOd. ^^qrs. what does it cost per eli English 1 Ans. 9s. 3f d". 2. Bought 4 pieces of Holland, each containing 24 ell^ English for 24jf ; how uueh was that j)er yard T * ^* :" Ans. 49. 3. A garrison had prarision Air & months, at the rate of 15 ounces to each person pa* day; how much must be al- lowed per day, ii| order that the provision^ may last 9j- months 1 Ans. 12|f oai. 4. How much hind at Ifs. 6d. per acre, must be given in exchange for 360 acres, at ISs. 9d. per acre 1 Ans. 540 acres. '« •%-* # IHH FELL0W»||IP. f 9i,!hi 5. Borrowed 185 quarterg of com when the priee wa« 1!)8 ; how much mait I pay when the price is 17i. 4d. 1 Aim. 2(»f j. {\. A person owning f of a coal mine, selk f of hii share lor n\£ ; what is the whole mine worth 1 Ans. 380i^. 7. If ^ of a gallon cost ^ of a pound, what costs | of a tun 1 An«. 140i:. 8. At 1^ per cwt. what cost 3J lbs. 1 Ans. lO^d. 0. If 4^ cwt. can be carried :)6 miles for 35 shillings, how many pounds can be carried 30 miles for the same money ? An8.907ilbs. 10. If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour I—— in each minute 1 in each second 1 Ans. to the la§t, 15" of a deg 11. If a family of 9 persons spend \12£ lOs. in 5 months, how much would be sufficient to maintain them 8 months if 5 persons more were added to the family 1 Ans. *280i&. Note. Exercises 14th, 15th, lOth, 17th, 18th, 19th and *30th, *' Supplement to Fractions,'' afford additional examples in single and double proportion, should more examples be thought necessary. «ii si^ .:u,!^ >_, > ti * U^mitILss. .%vft c*^ t^>, .«i , ( 142ie 178. Ifd. P'stoss. 6. Divide $600 among 3 men so that their shares may* be to each other as 1, 2, 3, respectively. An*. $100, $200, and |300. 7. Two merchants, A and B, loaded a ship with 500 hhds. of rum ; A loaded 350 hhds. and B the rest ; in a storm the seamen were obliged to throw overboard 100 hhds.; low nkuch must each sustain of the loss 1 Ans. A 70, and B 30 hhds. 8. A and B companied ; A put in 45:£, and took out f of the gain ; how much did B piit in ? Ans. 30jd. M Nrtion as they pvt in i if 3 fifths of the stock is 45£ how much is 2 fifths of it ? 9. A and B companied, and trade with a joint capital ot »;'* ^ V ••* ^ •A 190 FELLOWSHIP. IT 9-2, 98 V 400<£ ; A receives, for his share of the gain, } as much as B ; what was the stock of each 1 . , 5 iai£ 68. 7d. A 's stock. ^^•' J 26ft£ 13s. 4d. B's stock. 10. A bankrupt is indebted to B $760, to C $400, and to 1> $7(W ; his esUte is worth only .^OOO ; liow must it be di- vided 1 Xote. The question evidently involves the principles of fellowship, and may be wrought by it . Afis. B $234, C^138, and D $328. 1 1 . B and C venture equal stocks in trade, and clear l(ii£ ; by agreement, B was to have 5 percent of the prof- its, because he mana^g^ed the concerns ; C was to have but 2 per cent ; what was each one's gain 1 and how ituch did B receive for his trouble 1 Ana. B's gain was IITjS 28. lO^d. and &» 46£ 17s. 1 jd. and B received 70k£ 5s. Sfd. for his troubl0. 12. A cotton factory, valued at 12000^, is divided into 100 shares ; if th« profits amount to 15 per cent, yearly, what will be thf profit accruing to t share 1 to 2 shares 1 to (15 shares 1 Ana. to the last 450ig. 13. In the a^ove-raentioned factory, repairs are to be made which wil/ cost 340£ ; what will be the tax, on each share, necessary to raise the sum 1 on 2 shares 1 — — on 3 shares 1 on 10 shares 1 Ana. to the last 34<^. 14. If a town raise a tax of 1850<£, and the whole town he valued at 37000^6, what will that be on !£% What will be the tax of a man whose property is valued at 1780^ I .. j Ans. Is. on a pound, and 89£ on 1780>£. , u IT $3. In assessing taxes, it is necessary to hava an in- ventory of the property, both real and personal, of the whole town, and also of the whole number of the polls 1 and, as the polls are rated at so much each, we must first take out from the whole ioit what the polls ^tnount to, and the remainder is to be" assessed on the property. We may ihen find the tax upon 1^ knd make a table containing the tlxes on 1, 2, 3, &i;. to \0£ ; then on 20, 30, &c. to lOOjf; and then on 100, 200, &c. to lOOOf. Then knowing the inventory of any individual, it is easy to find the tax ui>on his property. ^ ^'H^^H' -■^■;; ''|^s*-#-- '"-;'' ' 'W^ ':*-'-■" *^^%.'4 15. A certain tawn, value'd at''645dOl;f, raises a tax of «* i .# •F K 92, 05} I much ii A 's stock. B't stock. GO, and to •t it be di- Qciples of ID $228. and clear f the prof- have but i1|iuch did I i7s lid- vided into t. yearly, to 2 Iast450ie. are to be X, on each real ■ last 34£. hole town What will I78a£? on 1780<£. ava an in- lal, of the the polls 1 must first ntto, and We may aining the J. tolOOjf; owing the ) tax upon ill tax of IT 03. t'DLLOWHUIP. 191 2259i£ 18s. ; there are 540 polls, which are taxed 3s. each ; what is the tax on a pound, and what will be B's tax, whose real trttUe is valued at ]340i','his personal property at 874^', and who pays for 2 polls. It will be better in questions relating to the assessment of taxes to use decimals, as we have done in interest. The procera will be shorter, and the result will be obtained with exacliess. The shillings, therefore, in the given values, will te reduced to the decimal of a pound, and the table will he male out decimally, and the decimal parts in the Anal answe^ can be reduced to shillings and pence. 540 X <60 (3s.) = 324^, amount of the poll taxes, and 2259'9rt (2259jf 18s.)-324ies=1935'90, to be assessed on propertj^, 64530i: '03, tax \n}£. 1935'i)0 :: l£ 'm ; or, yV^s^^ == Tax on^ is (( '!..' *t J' IS 'im is '09 is '12 ^is <15 fiis <] 7 is 'S 15 18 21 Sis '24 9 is '27 TABLE. £ £ Tax on 10 is " 20 is " 30 is 40 is 50 is 60 is ■»• m »*. '03 06 (( (( pertytobe - - , -^^^..-,. - - 2 ppUs at '60 each, are jt<>«| 4j4*^ xi^ . . - 40'20 26'22 - - - 1'20 ■•^' Amount, 67^ i^ ♦'4,^^ f 67'02£==67je 12s. 4^d. Jm. 16. iWhat will C'ltax amount to, whose inventory is 874 dollars rco/, and 210 dollars personal property, and who pays forSpttllsl , .™ - ^,^ .^^^^ ^^.$34'32. *>; r ■" m rSLLOWBHIP. T 94, 95. ^ 17. What will be the Ux of a maft, paying for 1 poll, whoee property is valued at $34*82 ? at $768 « at $9401 at $4(i57 1 jIm. to the lait, $140'3h 18. Two men paid 2£ lOs. for the use of a pasture 1 month ; A kept in 24 cows, and B 16 cows ; how much should each pay 1 19. Two men hired a pasture for 2i£ IDs. ; A ptt in 8 cows 3 months, and B put in 4 cows 4 months ; hoir much should each pay ? / IT 04. The pasturage of 8 cows for 3 moDth» is the same as 24 cows fi>r I month, and the pasturage of 4 cows for 4 months is the same as of 16 cows for 1 rooni). The shares of A and B, therefore, are 24 to 16, as in lie former question. Hence, when /tme is regarded in fellowslip, — Mul- tiply each one's stock by the time he continuesit in trade, and use the product for his share. This is ctlled Double FelUnoshtp. Am. A l£ 10;. and B 1^. 20. A and B enter into partnership ; A pujs in 100£ 6 months, and then puts in 50£ more ; B puis in 200i£ 4 months, and then takes out 80;^ ; at the clow of the year, they find that they have gained 95<£ ; what I the profit of «ach 1 A.. H3«^ 14s. 2fd. A»s shaue. •^■•d61<£ 58.91. B»s share. 21. A. with a capital of 500;^, began tradi Jan. l^ 1826, and meeting with success, took in B as a jartner, with a ; capital of 600i£, on the first of March following ; 4 months after, they admit C as a partner, who brougit 800i£ stock ; at the close of the year, they find the gaia to be 700i£ ; how must it be divided among the partners 1 C 25a£, A>i share, ; ' <^^ i^8.<250i6, B'l share. < i 200i£, C'« share. aUESTIONS. 1. What ii fellowship? 2. What ii the rale for operaing r 3. When txTM ii regarded in fellowship, what is it called ? 4. What ii the method of operating in double fellowship ? 5. How aretax^s as- eessedPC. How is fellowship proTed ? .^ t 95. Alligation is the method of mixing two or mon simples, of different qualities, so that the composition may b of a mean, or middle quality. ' ^"t ' r th T 94, 96. fing for 1 poll. at $768 • s last, tHO'SF. of a pasture 1 its; bow much )s. ; A pit in 8 th8 ; ho« much moDth» U the irage of 4 cows 1 roonii. The as in /he former Uowslip,— Mill- nuesit in trade, is ctlled Double 10;. and B l£. pu;s in lOOf 6 pi^ in 200<£ 4 \oai of the year, it k the profit of .2^d. A'sshai^e. . SB. B*s share. td4Jan. I, 1826, . fartner, with a nring ; 4 months sit 800;e stock ; 1 to be 700i£; ■ ? \hii S0£, A*| share^ S0£, B*i share. m£, C*s share. for operainc r Mled? 4. What is How are^u^i ai- .1 ^- ing twoormor npositioi may b< f O/i, 06. ALLIGATIOIV. 103 When the quantUie$ and prictt of the simples arc giren, tu find the mean price of the mixture, compounded of them, the process is called Alligation Medial. „ « 1. A farmer mixed together 4 bushels nf whrat, worth (id pence per bushel, :) bushels of rye, worth tV2 pence per busiiel, and 2 bushels of corn, worth 28 pence per bushel ; what is a bushel of the mixture worth 1 It is plain, that the coat of the whole, divided by the number of butheltt will give the price of one bushel. -^ ' 4 bushels, at G() pence, cost 264 pence. * 3 Q <( at 32 at 28 (( <( 96 56 ■f; 9 bushels cost 416 pence. *|« s= 46§ pence, Ans. 2. A grocer mixed 5lb8. of sugar, worth 10 pence per lb. 8 lbs. worth 12 pence, 20 lbs. worth 14 pence ; what is a pound of the mixture worth 1 Jim. I2|^d. 3. A goldsmith melted together 3 ounces of gold 20 carats fine, and 5 ounces 22 carats fine ; what is the fine- ness of the mixture? /Ins. 21 j^. 4. A grocer puts 6 gallons of water into a cask containing 40 gallons of rum, worth 2s. 7d. per gallon ; what is a gal Ion of the mixture worth 1 Ans. 28. 2||d. 5. On a certain day the mercury was observed to stand in the thermometer as follows ; 5 hours of the day, it stood at 64 degrees ; 4 hours, at 70 degrees ; 2 hours, at 75 degrees, and 3 hours, at 73 degrees ; what was the mean temperature for that day 1 It is plain this question does not differ, in the mode uf its operation, from the former. jlns. 69^^^ degrees. T 96. When the mean price or rate, and the prices or rates of the several simples are given, to find the proportions or quantities of each simple, the process is called Alligatim. jtUemate ; alligation alternate is, therefore, the reverse of alligation medial, and may be proved by it. ;; "' . 1 . A man has corn worth 40 pence per bushel, which he wishes to mix with rye worth 50 pence per bushel, so that the mixture may be worth 42 pence per bushel ; what pro- portions, or quantities of each, mutt he ta^e ? R 194 ALLlOATIOir. T WJ. lit Had the price of the mixture required excndtd the price of the corn, by juat a$ much ai it fell thort of the price of the rye, it ii plain, he must have taken equal 100!=420, and 420, divided by the num- ber of bushels, (8-{-2,) = 10, gives 42 pence for the price of the mixture. 2. A merchant has several kinds of tea ; some at 8 shil- lings, some at 9 shillings, some at 11 shillings, and some at 12 shillings per pound ; what proportions of each must he mix, that he may sell the compound, at 10 shillings per pound 1 Here we have 4 simples ; but it is plain, that what has just been proved of ttoo will apply to any number of pairs, if in eacli pair the price of one simple is greater, and that of the other less, than the price of the mixture required. Hence we have this RULE. ^i^-j^*B^jf The mean rate and the several prices being reduced to the same denomination, — connect tcith a continued line each f IHJ. f 96. ALLIOATIOir. 196 td the price the price of ^ontitieM of ceeded (he hort of the required 2 he le$n the that of one uf that nm- ities of the their prices ae differen- expreM the m the corn- rice of the 40 pence ; pence; the D the price aging these hels of rye, or in that ertained hj at 40 pence ce, is 100 y the num- the price of e at 8 shil- nd some at each must illings per t what has r of pairs, md that of required. reduced to f line each price that in less than tkt mean rate Ufith on* or mor>' that m OR RATER, and each prici orbatbm than the metiii, rat-.toUh one or more that i$ less. fVrite th« difference between the uu ah ratr. or pnu, and the price qf bach siMrLB oppotite the pric- 'vith tnhitU it ts connected ; (thus the difierence of the two prn es m each pair will be mutually exchanged ;) then the mim of tUt differ- encee, etanding againet «ny price, toiU expreee tltn kri. vrivi; QVAN TiTT to be taken of that price. By attentively considering the rule, the pupil will per- ceive, that there may be at many diflferent ways of mixing the simples, and consequently as many diflfcront aiiNwers, as there are different ways of linking the several prices. We will now apply the rule to solve the last question;— At 10«^ 8». — 9».— , 11«.— 1 12» lbs. -1 -I -2 OPERATIONS. Or, ^jina. lOi. -2-1-1=31 1 =iU -l-|.2=:3f'^ 2 =2j Here we set down the prices of the simples, one directly under another, in order, from least to greatest, as this is most convenient, and write the mean rate, (10s.) at the left hand. In the first way of linking, we find, that we may foke in the proportion of 2 pounds of the teas at 8 and 128. to I pound at 9 and lis. In the second way, wc find for the answer, 3 pounds at 8 and lis. to I pound at 9 and 12s. 3. What proportions of sugar, at 8 pence, 10 pence, and 14 pence per pound, will compose a mixture worth 12 pence per pound 1 jina. In the proportion of 2Ibs. at 8 and 10 pence to6Ibs. at 14 pence. Note. As these quantities only express the proportions of each kind, it is plain, that a compound of the same mean price will be formed by taking 3 times, 4 times, one half, or any proportion, of each quantity. Hence, i When the quantity of one simple is given, after finding the proportional quantities, by the above rule, we may say, jfa the PROPO&TiONAL ^^ntity : it to the oi vbk quantity : • 80 ia each of the other proportional quantities : to the »B- <}Uibed quantities of each. ^4 i-}% ,^ii*« ,i** f;> •- i .nv»^VK* 196 ALLEGATIOir. ir96 4. If a man wishes to mix 1 gallon of brandy worth 168. with rum at Os. per gallon, so that the mixture may be worth lis. per gallon, how much rum must he use 1 Taking the differences as above, we find the proportions to be 2 of brandy to 5 of rum ; consequently, 1 gallon of brandy will require 2^ gallons of rum 1 j4n$. 2^ gallons. 5. A grocer has sugars worth 7 pence, 9 pence, and 12 pence per pound, which he would mix so as to form a com- pound worth 10 pence per pound ; what must be the pro- portions of each kind ? Ans. 21bs. of the first and second to 41bs. of the third kind. 6. If he use lib. of the first kind, ho\« much must he take of the others 1 if 4Ibs. what 1 if eibs. what \ if lOlbs. what ? if 201bs. what ? Ans. to the last^ SOlbs. of the second, and 40 of third. 7. A merchant has spices at 16d. 20d. and 32d. per pound ; he would mix 5 pounds of the first sort with the others, so as to form a compound worth 24d. per pound ; bow much of each sort must he use 1 . . n ■^^\' jina. 51b8. of the second, and 7^1bs. of the third. 8. How many gallons of water, of no value, must be mixed with 60 gallons of rum, worth 48 pence per gallon, to reduce its value to 42 pence per gallon 1 Ans. Sf gallons. 9. A man would mix 4 bushels of wheat, at 90 pence per bushel, rye at 70 pence, corn at 70 pence, and barley at 30 pence, so as to sell the mixture at 48 pence per bush- el i how much of each may he use ? ^ 10. A goldsmith would mix gold 17 carats fine with some 19, 21, and 24 carats fine, m that the compound may be 22 carats fine ; what proportions of each must he use 1 Ans. 2 of the 3 first sorts to 9 of the last. 11. If he use loz. of the first kind, how much must he use of the others 1 What would be the quantity of the compound 1 Ans. to last, 7^ ounces. 12. If he would have the whole compound consist of 15 oz., how much must he use of each kind 1 if of 30oz., how much of each kind 1 if of 37^k., how much 1 ^ Ans. to lasty 5oz. of the 3 first, and 22^oz. of the last. Hence, when the quantity of the compound is given, we may say, As tke sum of the proportioital quantities fiund by the ABOVE RULE, t« to the qiuirUity required, so is eadi proportional quantity, found by the rule, to the requir- ed quantity of each. 7 96 ' worth 16». ire may be usel proportions 1 gallon of 2^ gallons. Qce, and 12 form a com- be the pro- > third kind, lust he take (That 1 40 of third, per pound ; B others, so bow much r the third. I, must be per gallon, S| gallons. 90 pence and barley I per bush- with some d may be ) use 1 f the last, must he ty of the ^ ounces, sist of 15 of 30oz., much t r the last, iven, we Aesjbund so ia ew^ tBQUIR- t 90. 97. liVOOBCIMALS. 197 13. A man would mix 100 pounds of sugar, some at 8 pence, some at 10 pence, and some at 14 pence per pound, no that the compound may be worth 12 pence per pound ; how much of each kind must he use ? We find the pruportions to be, 2, 2, and 6. Then, 'Z-\-'i -f.()=:10, and r 2 : 201bs. at 8d. ) 10 100 : ^ 2 : 201bs. at lOd. [Ans. C 6 : eOlbs. at 14d. ) 14. How many gallons of water, of no value, must be mixed with brandy at 120 pence per gallon, so as to fill a ves- sel of 75 gallons, which may be worth 92 pence per gallon \ Ans. 17} gallons of water to 57^ gallons of brandy. 15. A grocer has currants at 4d., 6d., 9d. and lid per lb. ; and he would ni'.ke a mixture of 240lbs., so that the mixture may be sold at 8d. per lb. ; how many pounds of each sort may he take 1 . Ans. 72, 24, 48, and 961bs., or 48, 48, 72, 72, &c. Note. This question may have five different answers. QUESTIONS. » 1. What is alligation ? 2. medial.' 3. the rule fur oper- ating .' 4. What is aliisation alternate P 5. When tho price of the mixture, and tho price of the several simples, are given, how do you find the proportional quantities of each simple.' H. When th? quan- tity of one simple is given, how do vou find the others ? 7. Vvhtri the quantity of the whole compouncj is given, how do you find th« quantity of each simple ? .; . ^ui IT 97. Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of bein^; divided decimally into ten equal parts, is divided duodecimally into twelve equal parts, called inches, or primes, marked thus, (')• Again, each of these parts is conceived to be divided into twelve other equal parts, called seconds, {"). In like manner, each second is conceived to bo divided into twelve equal parts, called thirds, ('") ; each third into twelve equal parts, callod/oj;rMs,("") : and so on to any extent. In this way of dividing a foot, it is obvious, that il 2 196 MVLTIPLltATlOH Or DCODECllfALt. IT 97. r inch ot prime if- -^ of a foot. 1" tecond is -^ of ^ - - - = ^^^ of a foot. I" third is tV o^ tV of tV ' * = ttV^ ©^ » ^o*- 1 "" fourth is tV of tV o^ tV of tV = julire of a foot. 1'"" fifth i» tV of iV of tV of tV of T^ zTiVw of a foot,&c. Duodecimals are added and subtracted in the same man- ner as compound numbers, 12 of a less denomination mak- ing 1 of a greater, at in the following TABLE. V 12'"' fourths make 1'" third, '. 12'" thirds - - - 1" second, ' '' 12" seconds - - 1' inch or prime, 12' inches, or primes,! foot. Xote. The marks, ', ", "', "", &c. which distinguish the different parts, are called the indices of the parts or denomi- nations. .'ii: '^f.h> MULTIPLICATION OF DUODECIMALS ''M Duodecimals are chiefly used in measuring surfaceB and solids . 1. How many square feet in a board IG feet 7 inches long, and 1 foot 3 inches wide ? ' - > w >.* ' Note. Length Xbreadth=superficial contents, (^ 25.) OPERATION. 7 inches, or primes = ^^ of a foot, and 3 inches=^^ of a foot; consequently, the product of 7' ' X3'==T^V of a foot, that is, 21" ==r and 9" ; wherefore, weset down the 9", and reserve the r to be carried forward to its ft Length, 16 Breadth, 1 T 16 r 3' F r 9' ^7/18.20 8' 9" proper place. To multiply 16 feet by 3' is to take ^^of ^ =4|, that is, 48' ; and the 1' which we reserved makes 49',=s4 feet 1' ; we therefore set down the T, and carry forward the 4 feet to its proper place. Then, multiplying the multiplicand by the 1 foot in the multiplier, and adding the two products together, we obtain the answer y 20ft, 6' 9". The only difficulty that can arise in the multiplication of duodecimals is, in finding of what denomination is the pro- IT 97. fW. MVTIPLIOA'TIOII or DUODBCIMALS. 190 foot, foot, foot, foot. I foot^&c. me man- ion mak- me, tguish the >r denomi' LS. rfacea and 7 inches (^ 25.) = T^u of a ^ of a foot; oduct of 7' that is, 21" ore, "weset eserve the ard to its ce i^of \^ ed makes and carry ultiplying ind adding 20ft, 6' 9' . lication ol is the pro- duct of any two denominations. This may be ascertained as above, and in all cases it will be found to hold true, that the product of any Uno denominations toiU always he of the de^ nomination denoted by the sum of their mvicsin. Thus, in the above example, the sum of the indices of 7'X3' is" ; consequently, the product is 21" ; and thus pn'm^A multiplied by primes will produce seconds ; primes multiplied by second* produce Mtrds ; fourths multiplied hy fifths produce ninth9,kc. It is generally most convenient, in practice, to multiply the multiplicand ^rst by the feet of the multiplier, then by the inches, &c. thus : — 16 r 16ft.Xl ft.=lCft., andrxlft =r. Then, 16 ft.x3'=48'r=4 ft., and rx3'=3r'=l' 9". The two products added together, give for the answer, 20 ft. 8' 9", as before. 20 8* 9" ^'' ^'' ' ''' ■ 2. How many solid feet in a block 15 ft. H' long, I ft. 5' 1 3' ♦■. 16 r 4 r 9" wide, and 1ft. 4' OPRRATION. Length, 15 8' Breadth, 1 5' thick 1 The length mi breadth, and that thickness, givs th (IT 33.) • iltiplied by the product by the e solid contents y ' : 15 8' ■ •' 6 6' •- !■' >' _^.. .._ 4" 22 2' Thickness 1 4' 4" . ■-» u ' ,.ni 22 2; ,fu.-:. m 7 4' 4" 9" 4' - '5r v. iJtm- 1 Jns. 29 r i" K" From these examples we derive the following Ki;;.i: :— Write down the denominations as compound numbers, and in multiplying remember, that the product of any two de- nominations will always be of that denomination denoted by the sum of their indices. 200 MULTIPLICATION OF DVOf^ECIMALt. f 97, U8. .'4 EXAMPLES FOR PRACTICE. „< r2 3. How many square feet in a atock of 15 boards, 12 ft. 8' in length, and 13' wide 1 Ana. 205 A. 10'. 4. What is the product of 371ft 2' (V multiplied by 181 ft. 1 9" 1 Ans. 67242 (t. 10' 1" 4 "6"". Note. Painting, plastering, paving, and some other kinds of work, are done by the square yard. If the contents in square feet be divided by 9, the quotient, it is evident, will be square yards. 5. A man painted the walls of a room 8 ft. 2' in height, and 72 ft. 4' in compass ; (that is, the measure of all its sides ;) huw many square yards did he paint 1 Ans. 65 yds. 6 ft. 8' 8" 0. How many cord feet of wood in a load 8 feet long, 4 feet wide, and 3 feet 6 inches high ? Note. It will be recollected, that 10 solid feet make a eordfoot. Ans. 7 cord feet. 7. In a pile of wood 176 ft. in length, 3 ft. 9' wide, and 4 ft, 3' high, how many cords 1 Ans. 21 cords, and 7-^'^ cord feet. 8. How many cord feet of wood in a load 7 feet long, 3 feet wide, and 3 feet 4 mches highl and what will it come to at 2s. per cord foot 1 ■:. Ans. 4| cord feet, and will come to 8s. 9d. 9* How much wood in a load 10 ft. in length, 3 ft. 9' in width, and 4 ft. €' in height ? and what will it cost at $1'92 per cord 1 AtM. 1 cord and 2-ff cord feet, and it will come to $2'62^. V 98* Remark. By some surveyors of wood, dimen- sions are taken in feet and decimals of a foot. For this pur- pose, make a rule or scale 4 feet long, and divide it into feet and each footinto 10 equal parts. On one end of the rule for 1 foot, let each of these parts be divided into 10 other equal parts. The former division will be lOths, and the lat- ter lOOths of a foot. Such a rule will be found very con- venient for surveyors of wood and of lumber, for painters. Joiners, &c. ; for the dimensions taken by it being in feet and decimal parts of a foot, the casts will be no other than so many operations in decimal fractions. 10. How many square feet in a hearth stone, which, by a rule, as above described, measures 4'5 feet in length, and 2,6 feet in width 1 and what will be its cost, at 75 cents per ir 97, 98. irds, VZ a 205 ft. 10'. ied by 181 1"4"' 6"". other kinds contents in vident, will r in height, -e of all its ,. 6 ft. 8' 8" feet long, 4 feet make a 7 cord feet. [)' wide, and ^g cord feet. ' feet long, 3 -will it come me to 88. 9d. :h, 3 ft. 9' in !Ostat$l'92 ,e to $2'62^. rood, dimen- I For this pur- de it into feet ]d of the rule Into 10 other and the lat- |nd very con- fer painters, [being in feet 10 other than [, which, by a length, and It 75 cents per IT 98,99. IlfTOLITTIOir. 201 square foot 1 ^$. 11*7 feet; and it will cost f8'775. 1 1 . How many cordp in a load of wood 7<5 feet in length, d<6 feet in width, and 4<8 feet in height 1 Ant. 1 cord l.^^ ft. 12. How many cord feet in a load of wood 10 feet long, 3'4 feet wide, and 3<5 feet high 1 An«.7^^. QUESTIONS. lj .V 1. What are duodecimals.' 2. From what is the word derived f 3. Into how many parts is a foot usually divided, and what are the parts called .' 4. What are the other denominations .' 5. What is understood b^ the indices of the denominations f 0. In what are du- odecimals chiefly used .' 7. How are the contents of a surface bound- ed by straight lines found .' 8. How are the contents of a solid foMiid ■* 9. How is It known of what denomination is the product of any two denominations^ 10. How may a scale or rule be formed for taking dimensions in feet and decimal parts of a foot ' 'S 99, Involution, or the raising of powers, is the mul- tiplying any given number into itself continually a certain number of timesw> The products thus produced are called the powers of the given number. The number itself is call- ed the first power or root. If the first power be multiplied by itself, the product is called the second power or square : if the square be multiplied by the first power, the product is called the third power, or cube, &c. thus, 5 is the root, or 1st power of 5. 5x5= 25 is the 2d power, or square, of 5, =52 5x5X5=125 is the 3d power, or cube, of 5, =5* 5X5X5X5=625 is the 4th power, or biquadrate, of5=5* The number denoting the power is called the index, or exponent; thus, 5* denotes that 5 is raised or involved to I the 4th power. 1. What ii the square, or 2d power of 7 1 An«. 49. 2. What is the square of 30 1 An*. 900. 3. What is the square of 4000 1 r Ans. 16000000. 4. What is the cube, or 3d power, of 4 1 An«. 64. 5. What is the cube of 800 1 An«. 512000000. 6. What is the 4th power of 60 7 Ans. 12960000. 7. Whatis the squar* of 1 ? of 2 / of 3 1 Bf 4 ? ^f '-i w -vt < . Ans. 1, 4, 9, and 16. 202 KTOLUTIOir* H 99, 100. 8. What is the cube of 1 \ of 4? 9. What is the square of f 1 10. What is the cube of $ 1 -of 2? of 3? An«. 1, 8, 27, and 64. — oft? of J1 An». |, M, if . _ cf 1 1 of J ? ^s**- A» ■^' *"^ Ma- ll. What is the square of ^ 1 the 5th power of ^ 1 Am. ^, and ^. 12. What is the square of P5 1 the cube 1 .^8. 2'25, and 3*375. 13. What is the 6th power of 1'2 ? Jns. 2'985984. 14. Involve 2^ to the 4th power. Note. A mixed number, like the above, may be reduced to an improper fraction before involving : thus, 2|=f ; or it raav be reduced to a decimal ; thus, 2|=2<25. jins. ^^=25^*. 15. What is the value of 7^, that isj the4th power of 7 1 - -A.' r*fo. ;•. ■ ;. -\^^fi. jna. 2401. 16. How much is 9^ 1 6* ? 10* ? ^n«. 729, 7776, 10000. 17. How much is V \ 38 1 4^ 1 53 1 65 1 103 t j^8. to the last, 100000000. The powers of the nine digits, from the first power to the fifth, may be seen in tho following TABLE. :Ro«is "o r ls> "Powprsfl \ Z\ 3 | 4 | 5! 6| 7| 8L Squares lor Sd PowersI I | 4| 9 1 16| 251 36> 491' 64| "811 <^unc H l orSd Fowers ll | ftl 87 > (5 41 125) 21ti| :t43l 6i2| Bii quadratp s ,()r 4trywi;f9l 1 jl6| 81 | 256| (t25TI196r >401l 40961" SGriiotids |or5ih Powits|I \:H\'m lltyi4|3l"i5nT76|tb807ia!nbril "6561 oUOii* EVOL.UTION. *ifH*' 1. 1 1 > IT 100. Evolution, or the extracting of roots, is the me- thod of finding the root of any power or number. The roof, as we have seen, is that number, which, by a| continual multiplication into itself, produces the given pow I «r. The square root is a number which, being squared, will produce the given number ; and the ctAe, or third root, is J number, which being cubed or involved to the 3d poweij will produce the given number : thus, the square root of 14{ find a duce tl 1. wide, floor of one sid We face is self, th having root to This Ist. there w ingoflF each : ( )9, 100. 3?- , and 64. . il. If- 1? and m- vex of i '*. [, and i- I ind 3*375. 2985984. be reduced 2i=| i or ^^=^^; power ot 7 5 Jns. 2401. 776, 10000. ToOOCMOOOO. IT 101 ■XTBACTIOn OF THE aQCARB ROOT. 203 power to the 81 >t8, is the mo )er. , whieh, by al le given po>y I r squared, will hird root, is >| he 3d powetj lareroot of 1*^ is 12, because 12'ssl44 ; and the cube root of 343 is 7, be- cause 7 3, that is, 7x7X7=s343 ; and so of other riumbers. Although there is no number which will not produce a perfect power by involution, yet there are many numbers of which prtci$e root$ can never be obtained. But by the help of decimals, we can approximate, or approach, towards the root to any assigned degree of exactness. Numbers, whose precise roots cannot be obtained, are called $urd numbers, and those whose roots can be exactly obtained, are callcMi rational numbers. The square root is indicated by this character /^ placed before the number ; the other roots by the same character with the index of the root placed over it. Thus, the square root of 16 is expressed /v/16 ; and the cube root of 27 is expressed -^27 1 and the 5th root of 7776,>C^7776. When the power is expressed by several numbers, with the sign -{-or — between them, a line, or vinculum, is drawn from the top of the sign over all the parts of it ; thus, th& square root of 21—5 i»\/21— 5, &c. EXTRACTION OF THE SQUARE ROOT. IT 101. To extract the square root of any number is io find a number, which, being multiplied into itself, shall pro- duce the given number. 1. Supposing a man has 625 yards of carpeting, a yard wide, what is the length of one side of a square room, tho floor of which the carpeting will cover 1 that is, what is one side of a square, which contains 625 square yards 1 We have seen, (Tf32,) that the contents ot a square sur- face is found by multiplying the length of one side into it- self, that is, by raising it to the second power ; and hence, having the contents, (625) given, we must extract its square root to find one side of the room. This we must do by a sort of trial : and, 1st. We will endeavor to ascertain how many figures there will be in the root. This we can easily do, by point- ing off the number, from units, into periods of two figures each ; for the square of any root always contains just twice ^ 204 CXTB ACTION OF THE IQUARE BOOT. H 101. OPERATION. 625(2 !r **, as many, or one figure Uis than twice as many figures, as are in the root ; of which truth the pupil may easily satis- fy himself by trial. Pointing off the number, we find, that -the root will consi&t of ttoo fig- ures, a ten and a unit. 2d. We will DOW seek for the first figure, that is, for the tent of the root, and it is plain that we must extract it from the left hand period 6, (hundreds.) The greatest square in 6 (hundreds) we find, by trial, to be 4, (hun- dreds,) the root of which is 2, (tens, s= 20 ;) therefore, we set 2 (tens) in the root. The root, it will be recollected, is one side of a square. Let us, then, form a square, (A. Fig. 1.) each side of which shall be supposed 2 tens, = 20 yards, expressed by the root now obtained. The contents of this square are 20 X 20=400 yards, now disposed of, and which, consequently, are to be deducted from the wliole number of yards, (625,) leaving 225 yards. This deduction is most readily performed by subtracting the square number 4, (hundreds,) or the square of 2, (the figure in the root already found,) from the period 6, (hundreds,) and bringing down the next period by the side of the re- mainder making 225, as before. 3d. The square A is now to be enlarged by the addition of the 225 remaining yards ; and, in order that the figure may retain its square form, it is evident, the addition must be made on ttoo sides. Now, if the 225 yards be divided by the length of the two sides, (20 -|- 20 = 40,) the quotient will be the breadth of this' new addition of 225 yards to the i^des c d and 6 c of the square A. But our root already found, = 2 tens, is the length of one side of the figure A ; we therefore take doubU tliis root, = 4 tens, for a divisor. >» The parts The sqi The fie '7 tx V j(\^i^ •."-?^*iV "-iA^f,. -'V H 101 f 101. Kurc§, as ily satis- ind, that r tvx) fig- t. geek for ,t is, for and it is extract it period 6, greatest reds) we > 4, (hun- hich is 2, sfore, we )ot. The llected, is Let us, (A. Fig. h shall be 20 yards, root now rards, now 9 deducted 225 yards, acting the (the figure lundreds,) of the re- le addition the figure lition must divided by le quotient ards to the ngth of one lis root, =5 BXTRACTlOll or TM« 8QVAIIE HOOT. 206 OPE R ATION-CONTI N UEiy. 625(25 4 «'>''..'}. '<;> 1 45)225 225 Fio. 20 yds. »H H. fi vds. o B 30 6 100 D. 9» '-i a A Vi^:1?i- »>jNT«i/.m# «» >» a 20 •if:n'» 400 » 100 The dit^iiior, 4, (tens) is in reality 40, and we are to seek how many times 40 is cuntaincd in 225, or, which is the samu thin^, wc may seek how many times 4 (tens) is contained in 22, (tens,) rejecting the right hand figure of thd dividend, because we have rejected theciphcr in the divisor. Wcfind our quotient, that is, the 6rea* .. . j.^..^-. ^- i„,v,! * Note 1. If we double the right hand figure of the latt divisor, we shall have the double of the root. Note ^2. Ai) the value of figures, whether integer! or decininls, is determined by their distance from the place of units, so we must always begin at unit's place to point off the given number, and, if it.be a mixed number, we must point it off both ways from units, and if there be a deficien- fiy in any period of decimals, it may be supplied by a cipher. It is plain, the roof must always consist of so many integers f nd decimals as there are periods belonging to each in the given number. 11. EXAMPLES FOR PHACTICE. 1% What it th# |<|nare root of 10342G56T, >-Vr 0^ a 1^ -4 T 101 follow 9 figures over tlio r of fig hand pe- Subtract o the ro- I. or ; seek dividend, lit in the Itiply the root, and ■enainder V divisor, le periods the la$t iteg«rs or place of point ; 1 10342G5(i(:»10, Jn» 9 r- 62) 134 124 .1 m 'If,-' '-I , 'ft 641 ) 1026 641 6426)38556 38556 •»*j 3. What is the square ruot of 43264 ^ OPERATION,,;,' . i.Mr." 2(r • ii. :• I 43264 C 20H, Jn$. 4 408)3264 3264 > ' . '-■IT.,'.\^,i •i-U-. •iv r4i >>; ^ 10. What is the square root of 164 1 Jn». 12'8-|>. 1 T i02. In this last example, as there was a remainder, after bringing down all the figures, we continued the opera- tion to decimals, by annexing two ciphers for a new period, and thus we may continue the operation (o any assigned de- gree of exactness ; but the pupil will readily perceive, that be can never in this manner, obtain the precise root ; for the last figure in each dividend will ali^ays be a cipher, and tb* last figure in each divitor is ^9 tame as the last qtiotiefU Ji^»f but no ooepf the nine digilf^ multiplied into HmU, \^ . m "I «06 •VPPLCMBItT TO TIIS iqCABB BOOT. f lO^. pro74-- 14 Wh:it is the H(|uaro root of J 1 Jfnte. Wo have seen, (IT 91), ox. !).) that fractions ar' $qu(irc(l by s(iurtrint; both the numerator and the (lonoin'i.i- tor. Ilonco it f()Ih)ws, that the square root of a fracdun «» found by extracting the root of tho numerator and c*" iho Ue- nominator. The root of 4 is 2, and tho root o. 9 i? •?. 1.'*. What is the 8<|uaro root of /,; jfna. '(. Jtna. ^. Jns. ^ =■ £. \{\. WlKit is tho square root of ^t^^ 17. What IS the s(]uare root of f-^^ \ 18. What is tho square root of 'iO^ 1 Tt'hen the iMPacrator and denominator are not exact aqiiarea, the fraction may he reduced to a uecimal, and lh« approximate root found, as (hrectod above. 10. What is tho squaro root of J =3 «75 1 ^n». ♦800-}-. 20. What is t!rj square root of J^ ? An$. '912-1-. n ti is I a erl col ma SlTPPr.EMENT TO THE SQUARE ROOT. ^^ ^^^tU^n.K.i.iUr QUESTIONS. -•——"- ■' __j*^ 1. Wl'at is jr. volution ? 2. What is understood by a power? 3. -*— • tho fiist, fh(5 second, the third, the fourth power? -1. What is tho index, or exponent? 5. How do you involve a number to nny required powor .' G. What is evolution P 7. What is a root ? 8. Can tho precise root uf all numbors be ''in. Wliy dp we placo the quotient figure lo the right hand of the divisor ? 10. How may we prove the work ? 17. VVhy do we point off mixed numbers both ways from units ? 18. When there ie « n«fnainder, how may we continue the operation ? 10. Why can we i^evei obtain iba precise root of surd number* ? 20. Hotv do we ei- ^a form tii , into a square 1 ^n§. Oi. 2 If a square liuld contains :2(|'2o a'|u»re ruds, huw uiaoy rods does it measure on e.icU itide 1 jfna. 45. 3. How many trees in each row ui' a square orchard con taining 5025 trees ? jina. 75. 4. There is a circle whoso arta, or superAcia' contents, is 51d4 feet ; what will be the length of the aid* of a «(|uare a equal area 1 \/5184=-T''J feet, Jn$. 6. A has two fields, one containing 40 acruit, and the oth- er containing 50 acres, for which B offers him a at^uare field containing the same number of acres as buth of tht.'iiu ; how many rods must each side of thix lield mtsasuru ! jfna. \*Mi rods. G. If a certain square field measure, 30 rodit u each side, how much will the side of a square field measure, contain ing 4 times as much ? \/20X'iOX4=4i) rods' Ans 7. If the side of a square be 5 feet, what will . o the side of one 4 times as large 1 9 times as large ' l^ times as large 1 25 times as largo .' Mt times as large 1 ^ns. 10ft. 15ft. 2011. 25ft. andlWft. 8. It i* required to lay out 288 rods of land in the furm of a parallelogram, which shall be twice as many rods in length as it is in width. Note. If the field be divided in the middle, it will form two equal squares. j^ns. 24 rods long, and 12 rods vide. 0. 1 would set out, at equal distances, 784 apple ( l>os, so that my orcliard may bo 4 times as long as it is br id ; how many rows of trees must I have, and how nuny t ces in ear^i row 1 y/zts, 14 rows, and5G in eacli r w. 10. There is an oblong" piece of land, containing i'*2 square rods, ofwhich the width is ^ ^^ nmch as the I >ngl i ; required its dimensions. y/ns. 1(1 by I*-'. 11. There is a circle, whoso diameter is 4 inches ; what is the diiiineter of a circle, times as large 1 Xittc. T!ie areas or contents of circles are in proportion tt) the 'iqttar.'s of their diameters, or of their clrciin\ferences. Therefore . to lind thi; diameter required, sqaaro the given diameier, multiply tiie square by the given ratio, and the square root of tlic pro Uict will l>e the diameter rerfuired. \/^X4xy=12 inches, jfna. 12. There nn* two circular ponds in a gentleman's picas- tio •VPPLBMBITT TO THB •«»▲&■ BOOT. f i03. ■re ground ; the diameter of the less is 100 feet, aod the {Toater is 3 times as large ; what is its diameter 1 uins. 173'2-f ft. 13. If the diameter of a circle be 12 inches, what is the diameter of one { as largo i j^ns. G inches. H 103. 14. A carpenter has a large wooden square ; one part of it is 4 feet long, and the other part 3 feet long ; what is the length of a pole that will just reach from one •nd to the other 1 A . Note. A figure of 3 sides is called a triangle, and if one of the corners be a square corner, or right angle, like the angle at B in the annexed figure, it is called a right-angled triangle, of which the square of the longest side A C, (called the hypot- enuse, is equal to the sum of the squares of the other two sides, A B and B C. 42=16, and 32=9 ; then, ^^+16=5 feet, Ans. 15. If, from the corner of a square room, 6 feet bo meas- ured off" one way, and 8 feet the other way, along the sides of the room, what will be the length of a pole reaching from point to point 1 Ans. 10 feet. 10. A wall is 32 feet high, and a ditch before it is 24 feet wide ; what is the length of a ladder that will reach from the top of the wall to the opposite side of the ditch 1 jins. 40 feet. 17. If the liiddcr be 40 feet, and the wall 32 feet, what is the width of the ditch 1 Jins. 24 feet. 18. The ladder and ditch given, required the wall. //ns. 32 iGfi. 10. The distance between the lower ends of two equal rafters is 32 feet, and the heiijht of the ridge, above the beam on which thoy staml, is 12 feet ; required the length of each rafter. Ans. 20 feet. 20. There is a building; 30 feet in length and 22 feet in width, and tlie eaves project beyond the wall 1 foot on ev- #ry side ; the roof terminates in a point at the centre of iba contaj Not 4. fcetl feclll H ioa. lOd the 2-f ft. t is the inches. e ; oDo t long ; om OQo re of 3 riangle, corner* or right gle at B igurc, it t -angled ich the rest side ! hypot- > the sum / . eet, Ana. bo mt'at- the bides reaching . 10 feet, it is 24 irill reach lie ditch 1 J. 40 feet, set, what J. 24 feet, all. J. 32 left. two equal above the he length s. 20 feet. 22 feet in ot on ev- itrc of tbv> T 104. BXTBAOTIOV OV THB CCBB BOOT. 311 building, and is thete supported by a post, the top of which is 10 feet above the beams on which the rafters rest ; what is the distance from the foot of the post to^the comers of th« eaves 1 and what is the length of a rafter reaching to the middle of one aide 7 a rafter reaching to the middle ot one endl and a rafter reaching to the comers of the eaves 7 Jnswera, in order, 20ft. ; 15'62-f-ft. ; IS'SG-fft. ; and 22«364-ft. 21. There is a field 800 rods long and 600 rods wide ; what is the distance between two opposite curncrs ? jtiis. 1000 rodfl. 22. There is a square field containing 90 acres ; how ma- ny rods in length is each side of the field 1 and how many rods apart are the opposite corners 1 Answers, 120 rods ; and 169'7-|-rods. 23. There is a square field containing 10 acres ; what dis lance i« the centre from each corner 1 ^ns. 28'28-j- rods. ■) 'Ollijt: •ti ■'■■^ EXTRACTION OF THE CUBE ROOT. IT 104. A solid body, having six equal sides, and each of the sides an exact square, is a cube, and the measure in length of one of its sides is the root of that cube ; for the length, breadth, and thickness of such a biwly are all alike ; consequently, the length of one side, raised to the third power, gives the solid contents. See I 33. Hence it follows, that extractinjj the cube root ot any ntim ber of feet is finding the length of one side of a cubic body, of which the whole contents will be equal to tlie given number of feet. 1. What are the solid contents of a cubic block, of which each side measures 2 feet? jJns. 23=2x2X2=8 feet. 2. How many solid feet in a cubic block, measuring 5 feet on each side 1 Jins. .'>-5=12,5 feet. 3. How many feet in length is each side of a cubic block, containing 125 solid feet 1 Jns. v'l25=5 feet. Note. The root may be found by trial. 4. What is the side of a cubic block containing 04 solid feetl 27 solid feet! 216 solid feet? 512 solid fceti ^rw. 4ft. ; 3 ft. ; 6 ft. ; and 8 ft. I ml \ ji 312 exTBAOTioir or thb cobb root. T 104. I 5. Supposing a man has 13824 fe«t of timber, in sepa- rate blocks of 1 cubic foot each ; he wishes to pile them up ID a cubic pile ; what will bo the length of each side of such a pile 1 It is evident, the answer is found by extracting the cube root of 13S24 ; but this number is so large, that we cannot so easily find the root by trial as in the former examples ; Wfr will endeavor, however, to do it by a sort of trial; and, 1st. We will try to ascertain the number of figures, of which the root will consist. This we may do by pointing ths number off into periods of /Aree figures each (If 101, ex. 1.) OPERATION. 13824(2 8 5824 FiR. c 20 20 ^-; ^ffl I ;i 'l||n JP 20 i: Pointing off, we see, the root will consist of 2 figures, a ten and a unit. Let us, then seek for the first figure, or tens of the root, which must be extracted from the left iiand period, 13 (thousands.) The greatest cube in 13, (thousands) we find by trial, or by the table of powers, to be 8, (thousands,; the root of which is 2, (tens ;) therefore, we place 2 (tens) in the root. Tiie root, it will be recollect- ed, is one side of a cube. Let us, then, form a cube, (Fig. I.) each side of which shall be supposed 20 feet, expressed by the root now obtained. The contents of this cube are 20 X 20 X 20 = 8000 solid foot, wiiicii are now disposed of, and which, consequently, are to be deducted from the whole number of feet, 13824. 8000 taken from 13824 leave 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from the period 13, (thousands,) and bringing clown the next pe- riod by the side of the remainder, making 5824, as before. 2d. The cubic pile A D is now to be enlarged by the ad- dition of 5S24 solid feet, and, in order to preserve the cubic form o( the pile, the addition must be made on one hvK of itii Ides, that is, on 3 sides o, b, ;ind c. Now, if tlio.V.24 20 400 2U sooo"- Coiiieji's. f 104. , in sepa- i them up ti side of r the cube we cannot ixamples ; trial; and, figures, of r pointing 01, ex.1.) e see, the 2 figures, iBt us, then [ figure, or vhich must m the left thousands.) be in 13, ul 61/ trial, moers, to be lie root of ) therefore, in the root. )e recoUect- 1 cube. Let be, (Fig.l.) ch shall be expressed obtained, lis cube are 8000 solid nsequently, ^eet, 13824. leduction ia number, H, ound,) from he next pe- as before, d by the ad- vc the cubic one b;;:f el if the oVZi H 104. BXTBACTION OP THE CUBE ROOT. •2i\i solid feet be divided by the square contents of these ^ cquai aides, that is, by 3 times, ['2OX'2O=t(M)]=:120O, tho quo- tient will be the thickness of the addition made to each of the sides a, b, c. But the root, '2, [tens,] already found, is the length of one of these sides ; we therefore s(|uaro the root, 2, [tens,]=--20X20==:400, forthe»7«rtrc contents o{ om side, and multiply the product by 3, the number of sides*, 400X3=1200, or, which is the same in effect, and nioro convenient in practice, we may square the 2, (ten.s,) and mul- tiply the product by 300, thus, 2X2=4, and 4x300= 1200, for the divisor, as before. • , \^ -. ^ Tho divisor, 1200, is coi»- OPERATION-CONTINUED. tained in the dividend 4 times; 13824(24 Hoot. consequently, 4 feet is the i2- 8 nivis. 1200)5824 Dividend. 4800 ' 960 _ 5824 0000 f. 'J Fig. II. thickness of the addition made to each of the three sides, a, b, c, and 4X1200=4800, is the solid feet contained iu these additions ; hut if wo look at Fig. if., we shr.!! ;>fir. ceive, that this addition to the 3 si«Jes does not coicplcro tho cube; for there are deficiencies in the 3 corners n, n, n. Now the length of each of these de- ficiencies is the same as the length of each side, that is, 2 [tens] =20, and their width and thick7iess are each equal to the last quotient figure, [4]; their contents, therefore, or tho number of feet required to*' ^^fill these deficiencies, will bo found by multiplying the square of the last quotient fig- ure,[42]=lG, by the length ; ^ of all the deficiencies, that is, by 3 times the length of each le, which is expressed by the former quotient figiire, 2 fns.] 3 times 2 [tens] are 6 [tens] =00 ; or what is tha ]nie in efTect, and more convenient in practice, we may 1, id ^A ♦,*-! 214 KXTUACXION or THK OUBC BOOT. H 104. multiply the quotient figure, 2, [tent,] by 30, thus, 2x30 s=s60. as before ; then, G0Xl6==960, C4)ntents of the three deficiencies, n, n, n. Looking at Fig. 111., we perceive there is still a defi- ciency in the corner where 1 the last blocks meet. This deficiency is a cube, each side of which is equal to the last quotient figure, 4. The cube 20 of 4, therefore, [4x4x4=^ 64,] will be the solid contents of this corner, which, in Fig. IV. is seen filled. Now, the sum of these sev- eral additions, viz. 4800-|- 960-1-64=5824, will make the subtrahend, which, sub tracted from the dividend, leaves no remainder, and the work is done. Fig. IV. shows the pile which 13824 solid blocks o one foot each would make when laid together, and th« root 24, shows the length o one side of the pile. Th( correctness of the work ma be ascertained by cubing th side now found, 24^, thus, 2 X24X24=13824, the give number ; or it may may be proved by adding together tl contents of all the several parts, thus, >v ;' ■ ■ Feet. ' ^ '■'■'■' '-'■. R000=content8 of Fig. I. 4800=addition to the sides a, b, and c. Fig. 1. -)- 960=addition to fill the deficiencies n, n, n, Fig. II. 64=addition to fill the corner e, e, e, Fig. IV. L\ Fig. IV. 24 feet. 24 feei. /^ * 13824=contents of the whole pile, Fig. IV. 24 feet i each side. From the foregoing example and illustration we den th« following f 104. .^ 104 KXTRACTION Or TUB CUBE ROOT. 215 lis, 2X30 the tbr«e ;. Ill, we till a defi- ner where ,eet. This e, each side to the last The cube [4X4X4=^ )lid conteDts lich, in Fig. ofthesesev- riz. 4800-i-l will make] which, sub he dividead,! ^der, and th«| 0W8 the pile solid blocks ol would inakel sther, and ^H the length oj \e pile. Th4 \xe work mai jy cubing th" 24^, thus,2| 824, the givcj (r together tlil Fig. I- r»,n, Fig. 11. Fig. IV. IV. 24 feet tion we deri^ RULE For extracting the Cube. Hoot. I. Separate the given number into periods of three figures each by putting a point over the unit figure, and every third figure beyond the place of units. II. Find the greatest cube in the loft hand period, and put its root in the quotient. III. Subtract the cube thus found from the said period, and to the remainder bring down the next period, and call this the dividend. IV. Multiply the square of the quotient by 300, calling it the divisor. V. Seek how many times the divisor may be had in the dividend, and place the result in the root ; then multiply the divisor by this quotient figure, and write the product under the dividend. VI. Multiply the square of this quotient figure by the former ^gure or figures of the root, and this product by 30, and'place the product under the last ; under all write th« cube of this quotient figure, and call their amount the aub- irahend. ^ , VII. Subtract the subtrahend from tho dividend, and to the remainder bring down the next period for a new divi- dend, with which proceed as before; and so on, till the whole is finished. vfi} * ut o , '^ • - Note 1 . If it happens that the divisor is not contained in the dividend, a cipher must be put in the root, and the next period brought down for a dividend. Note 2. The same rule must be observed for continuing the operation, and pointingoff for decimals, as in the square root. Note 3. The pupil will perceive that the number which we call the divisoTy when multiplied by the last quotient fig- ure, dues not produce so large a number as the real subtra- hend ; hence, the figure in the root must frequently b« smaller than the quotient figure. EXAMPLES FOR PRACTICE. ' , ' .J 6. What it the cube root of 1860867 \ ' ' i <■ '. i 4 o 16 MUrPLKMENT TO THB CUBE BOUT. 11 504. •''iV > OPERATION. 1806807 (123 Ads. "" 1 r » V> -1 .iia).-<'i'^" 12x300=:J()(>)HOO first Dividend. 000 •J-'X1XJJ0= 120 23 = 8 •^r If '•■i la 728 first Subtrahend. »M «V«, 12-' X 300=43200) 132867 second Dividend. < ...... ■■ , ,,,j, •■ ■ u ■■ .7. 32X12X30= 3«= 129600 3240 27 ■.'.i»'i • ' 132867 second Subtrahend. 000000 7. What is the cube root of 373248 1 8. What is the cube root of 21024576 1 9. What is the cube root of 84'604519 1 10. What is the cube root of '000343 1 11. What is the cube root of 2 1 ' ^ 12. What is the cube root of j^ 1 i" ' ■ Ans. 72. Ans. 276. Ans. 4'39. Ans. '07 Ans. 1'25-f-. Ans. f . Note. See 1[ 99, ex. 10, and IT 102, ex. 14. '' '*' 13. What is the cube root of ^f^l <-',ifnir • * ^^^' h 14. What is the cube root of ^^ 1 ..,;i "-- Ans. t^. 15. What is the cube root of -^ 1 , ,. ;; .^ Ans. '125-f-. 16. What is the cube root of -j^-y 1 ^ .-^ Ans. |. ,<,; SUPPLEMENT TO THE CUBE KOOT. aUESTIONS. , , .. 1. What is a cube' 2. What is understood by the cube root? 3. What is it to extract the cube root.' 4. Why is tlie square of the quotient muhiplied by 300 for a divisor.' 5. Why, in finding the Kubtrahend, do we muhiply the square of the Inst quotient figure by 30 times the former fisure of the rpQt ? 6. Why do we cube the quo- tient figure ? 7. How do wc prove the operation ? _ , .rt^ H 604. IT 105. SUPPLEMENT TO TH^^ CUBE ROUT. 217 1 . . exi:rcise«. ;/ ' .<'fKi /•• ••5' f^i'^t \f . , : ' ' ■, ■ »■ 'Ui d. .,4 '."* » idend- K trahend. 53* Ans. 72. Ans. 276. Ans. 4'39. Ans. '07 18. l'25-l-. Ans. f. Ans. |. Ans. ^. 18. '125-f. Ans. |. >OT. ibe root i 3. luare of tho finding the ent figure by ibo tTie quo- 1. What is the side of a cubical mound, equal to one i288 loet long, 210 feet broad, and IH feet bij^h 1 Jna. Ill It. 2. There is a cubic box, one side of wbicli ib 2 feet ; how- many solid feet does it contain y y/ns. 8 feet. 3. How many cubic feet in one 8 times as large \ and what would bo the lenj^th of one side J jliis. i'A solid feet, and on« side is I feot. 4. There is a cubical box, one side of which is 5 feet ; what would be the side of one containing 27 timo^i as much .' 01 times as much \ 125 times as mucli 1 /ins. 15, 20, and "25 feet. 5. There is a cubical box, measuring 1 foot on each side ; what is the side of a box 8 times as largo ? 27 times 1 64 times 1 y/ns. 2, 3, and 4 feet. IT 105, Hence wo see, that the sides of cubes are as the cvhe roots of their solid contents, and, consequently, Ihoir contents art; as the cubes of their sides. The same propor- tion is true of the similar sides, or of the diameters of all solid figures of similar forms. 0. It a ball, weighing 4 pounds, be 3 inches in diameter, what will be the diameter of a ball of the same metal, weigh- ing 32 pounds? 4 : 32 : : 33 : O^ .<^/is. inches. 7. If a ball, 6 inches in diameter, weigh 32 pounds, what will be the weight of a ball 3 inches in diameter 1 y//is. 4 lbs. 8. If a globe of silver, 1 inch in diameter, be worth ^ifO, what is tho value of a elobe 1 foot in diameter 1 '^ ^^■'•- •••;;- ^^5^ §10308. 9. There are two globes ; one of them is 1 foot in diam- eter, and the other 40 feet in diameter ; how many of the smaller globes would it take to make 1 of the larger ? .Ms. fvlOOO. 10. If the diameter of tho sun is 112 times as much as tho diameter of the earth, how many globes like the earth would it take to make one as large as the sun 1 Aa's. 1404928. 11. If the planet Saturn is 1000 times as large as the earth, and the earth is 7900 miles in diameter, what is the diameter of Saturn 1 A*a8. 79000 miles. 12. There are two planets of equal deiisity ; the diame- ter of the less is to that of the larger as 2 to 9 ; what is the ratio of their solidities I Jn%. ^1^^ ; or, as 8 to 729. T 11 218 ARITHMETICAL PROGRKBSIOK. If 105, 106. Note. Tlie roots of lajst powers may be found by the square and cube roo'. only : thus, the biquadrate, or 4th root, is the square root nl tho f jiiare root ; the (Jth root is the cube root of lln' MHi.rf root ; the 8th root is the square root of the 4th r(-()t ; tin Dth not is the cube root of the cube root, &c. Those rooi?), viz. the 5th, 7th, 11th, &c., which are not rcsolvaMo bv ti.c s'jiiare and cube roots, seldom oc- cur, and, when thov ?", the work is most easily performed by logarithms ; fur^ if the logarithm of any number be di vided by the iiuii x. .)f ti.e root, the quotient will be the log- arithm of the root itwlf. ARITHMETICAL. PROGRESSION* \ "ft/ IT 106. Any rank or series of numbers, more than two, increasing or decreasing by a constant difference, is called an Arithmetical Series, or Progression. When the numbers are formed by a continual addition of the common difference, they form an ascending series ; but when they are formed by a continual subtraction of the com tnon difference, they form a descending series. «,, 5 S, 5, 7, 9, 11, 13, 15, &c. is an o5cmding seriei. ''""' I 15, 13, 11, 9, 7, 5, 3, &c. is a descending series. The numbers which form the series are called the termt of the series. The^rsf and last terms are the extremes, and the other terms are called the means. There are five i.hings in arithmetical progression, any three of which being giveO, the other two may be found :— 1st. Thc^irsf term. rJd. Tlie last term. ^kI. The number of terms. . 1th. The tonwwn difference. ' "ith. Tlie sum of all the terms. !, A man bought 100 yards of ctoth, giving 4 pence for the first yard, 7 pence for the second, 10 pence for the third, and so on, with a common difference of 3 pence ; what was the cost of the la«t yard ! As the comuton difference, 3, is added to every yard except the last, it is plain the last yard must be 99 X 3, = 'i97 pence, more than the^/ir»t yard. ^dns. 301 pence. [)5, 106. by the Uh root, it is the I a re root the cube ., Avhich doin oc- jrformed er be tli the log- ^ 106. ARITIIMKTICAL PROOREfSION. 219 [lan two. is called idUion of rtcs ; but the com r^g seriei. ng series. the term* ernes, and any three pence for the third, IV hat was •I iji-i ' rd except )i pence. '' Hence, when the first term, the common difference, and the number of terms, are given, to find the last /erw,— Multiply the number of terms, less 1 , by the common difference, and add the first term to the product for the last term. 2. If the first term be 4, the common difference ',i, and the number of terms 100, what is the last term 1 jfns. 1)01 . 3. There are, in a certain triangular field, 41 rows of corn ; the first row, in 1 corner, is a single hill, the second contains 3 hills, and so on, with a common difference of 2 ; what is the number of hills in the last row 1 jfns. 81 hills. 4. A man puts out l£, at 6 per cent simple interest, which, in 1 year, amounts to 1^^<£, in 2 years to l^ nrd ; what was the average pric« per yard, and what was the amount of the whole \ Since the price of each Huccccding yard increases \>y a constant excess, it is plain, the average price is as much leas than the price of the last yard, as it is greater than th« price of the first yard ; therefore, one lialf the sum of the first and last price is t!ie average price. One half of -Id. -f 301 d. = ir>-2^d. = average ^ price ; and the price, I.W^d. X 100 = 15250d. = ? Ant. m£ 10s. lOd., whole cost. 5 Hence, when the extremes and the number of terms Are given, to find the sum of all the /crms,— Multiply J the sum of the extremes by the number of terms, and tho product will be the answer. 10. If the cxtremei be 5 and 605, and tho number of terms be 151, what is the sum of the jieries 1 Ana. 46055. 11. What is the sum of the first 100 numbers, in (heir natural order, that is, 1, 2, 3, 4, &c. 1 Ana. 5050. 12. How many times does a common clock strike in 12 hours 1 Ans. 78. 13. A man rents a house for 50£, annually, to be paid at the close of each year ; what will the rent amount to in 20 years allowing 6 per cent, simple interest, for the use of the money 1 The last year's rent will evidently be 50i^ without in.> terest, the last but one will be the amount of 50<£ for 1 year, the last but two the amount of 50^ for 2 years, and so on, in arithmetical series, to the first, which will be the amount o( 50£ for 19 years = 107<£. If the first term be 50, the last term 107, and the number q( terms 20, what is the sum of the series 1 Ans. 1570iJ. 14. What is the amount of an annual pension of 100^, being in arrears, that is, remaining unpaid, for 40 years, allowing 5 per cent simple interest ? Ans. 7900£. 15. There are, in a certain triangular field, 41 rows of corn ; the first row, being io 1 corner, is a single bill, and the last row, on the side opposite, contains 81 hill» ; how oiany hills of corn in the field T Ans. 1681 hills, 16. If a triangular piece of land', 30 rods in lengtli, be 20 rods wide at one end^ and «on)j^ to ai ^^oint at the other, what 5 100. t 106, 107. nCOMBTRICAL PROGRRSRIOIV. Wl jncc for h\, and c« l»y a 11 ch leas han th« ) of the \ Ant. je given, [li of the ; v^ill be umber of >. 46055. in (heir la. 5050. ike in 12 Ans. 78. |0 be paid unt to in le use of hout in- r 1 year, BO on, in amount 3 number 1570iJ. 3f 100£, 10 years, 7900ie. rows of hill, and 1ft ; how 1 hills. ,be2a r, what number of nquaro rod.i does it contain ? \n^. 300. 17. A debt ii to bo discharged at 11 .several paymenti, in arithmetical series, the first to be 't£, and the last 75il ; what is the whole debt ? the common difference be- tween the several payment.s ? Ana* whole debt, 440jC ; common difference, 7.1', 18. What if the sum of the series 1, 3, 5, 7, 0, &o. to lOOn An«. ^'ilOOf. Note. By the reverse of the rule under ex. 5, the differ- *nce of the extremes 1000, divided by the common dlffermce 2, gives a quotient, which, increased by 1, is the nunihtr of terma = 501. 19. What is the sum of the arithmetical series 2, '2^, 3, 34, 4, 4^, &c., to the 50th term inclusive 1 Ans. 7 V2\. 20. What is the sum of the decreasing scries 30, 'i9i, idif, 29, 28J, &c. down to 01 Note. 30 -r 1^ 4- 1 = 91, number of tcrm^. Ann. 13tM QUKSTIONS. 1. What is an arithinotical progroRsion .' 2. When is ihi: series called ascending? 3. when descending T 4. What arc the iiumbera, forming the progression, called ? 5. What arc the first and last terms called ? 6. What are the other terms called ? 7. When xhf first term, common diflference, and number of terms, aro given, how do you find the last term ? 8. How may arithmetical progrcssioti tu applied to simple interest'' 0. When the extremes and niirnb«r of terms are given, how do you And the common difTcrcnro ? 10 — - bow do you find the sum of all the terms P - > ' GEOMETRICAL PROGKESSIO^. t It 107. Any series of numbers, continually incrcaMn;: by a constant multiplier, or decreasing by a constant (livi!>ui. is called a Geometrical Progression. Thus, 1,2, 4, H, 1 d, &c. is an increasing geometrical series, and 8, 4, 2, 1, .\, ,[, ke. is a decreasing geometrical series. As in arithmetical, so also in geometrical progression, there are five things, any tkr»€ of which beinj; given, the other fipo niav bo found :— Ist. Theirs/ term. 2d. The last term. ' • ' - T^> I' 39^ ai;uMt:TRinAL pnor.RCHKioM f 407. :td The numher of terniH. j nult, y^ Itli. Tliu rofio. J i. jjj*,|, ^ I .'>lh. Th(.' HUin of all the terms. Tliu ratio i!« the tnaltiplin, or divivor, hy whicU iho t»crics M formed. ,j, * I. A man l)oii};lit a pificc of fcilk, measuring H yards, lud, by aj;rt!t'mrut, was to {xivc wli;it tlie last yard would coino to, rt'ckoiiing :{ pence for the first yard, (J p«Mue for the second, ami tio ou douhlitii; the price to tlic la:)t ; what (lid the piL'Cu of hilk coht hiui ! X"2=—:l!Mt()0S pence, ==H|iJ£; 4s. answer. nAn^ In c\aniinin<; the process by wiiich the last term fI(M»fl(l.s) iia« heen ohtained, we see, that it is a product, of which the r itio \'l) is sixteen times a factor, that is, one time Ichh than the nunilHT ol terms. The last term, then, is tlio sixteenth power of the rntio, i'Z,) multiplied by l\ic first ti.rm (I^.) Now, to raise '2 to the KJt'.i power, we need not produce all tliu i.itenncdinte yuwcrs ; fur '2^ = '2 X '^ X i X -= Hi, is A pr.Kluct of whicli the ratio '2 is 1 tinies a factor ; now, if Kj \)ii muUiplied by 1(1, the pnnluct, !io(), evidently con- t.iiiis ilic same factor {'2) 1 times -j- 4 times, = 8 times ; and '^.iti x :2'><5 = ()•>"); J<), a product of which the ratio (2) is S times -|- '^ times, = h) ti:nes, factor ; it is, therefore, tho nUh power of -2, and, multiplied by 3, the fust term, givos I'.XKiUH, iho last term, as before. Hence, fK/ic/t tlie first term, ratio, and number of terms, un git'i''*, to find the lust term, — ' ' -' f. Write down a few leading powers of the rutin with tlieir indices over thorn. H. Add together the most convenient indices, to make an index /I'.vs by one than the number of the term souji,ht. in. Multiply together the potoers belonp;in^ to tliose tn- li/cfs, and their product, multiplied by the first term, will be tlie term sought. •2. If the lirst term be "i, and the ratio 3, what is the htb term \ Powers of the ratio, with > ., ,. ^^. ' ^^, ,^.ry ^ - their indices over them. > ,• . . iot»>r ,-»„„ C lirsit term, = lUyoo, j'/ns. /. 1 lo; H 107. OeitMKTHICAf. FHOfillKSSIAN. ni to hcriiN ' yardf, (1 \VO\ll>\ t i wlral vliich the /tss than sixteenth it protUicc witli f> mnke au light. ' h those iv!- Iterm, will is the bth !lb7 X •■> .1. A man plants 4 kernels of corn, which, at harvest, pru(h)cc '-Vl lurneN , tht»i«' l»e plantn thr stion'l year ; now, nupposin); tho annual inrri'nsti to continue H fold, \t|iat would he the proU\ ratio ; what did the whole cost him ? ., •J-j-ii-j-ijS-f-T)! = MO pence. ' •' An«. HO'pence In a lont; series, tho process of adiiiii^ in thi-^ niannot would 1)0 tedious. Let us try, thcrrfure, to device some, shorter method of cominp to tho same result. If all thr U)ims,cxceptiiit^ the Idst, viz, '2-|-(>-|-|S, ho nuiltiplicd by the ratio, ;3, the product will he the series O-j-IS-j-.V! sufitracl ing tho former series from the latter, we have, f«r the remain- der, r>4— 'i, that is, the last term, less the Jirst hrm, which is evidently as many times the first series ('«i-(-()-|-l>^) as i% expressed by the ratio, lesH 1 : hence, if wo divide the difler cnce of the extremes (54 — '-i) hy the ratio, /.'xs I, {^^ — 1, 5 the quotient will he the sum of all the terms, exccj'tinc; thi last, and, addin^j; the last terni, vfv shall have the whoh amount. Thus, ol, — •* = 5-2, and 13 — 1 = ri ; then, ,Vi -^ '2 = "2<), and 54 added, makes 80, Ans. as before. Honci!, when the extremes and ratio are given, to lind th** sum of the series, — Divide thof/i/ZtrfTJCt^of the extremes by the ratio hsa I, and the (juotient, increasad by the f^rcat.:r t>:rm, will bo the answer. {). If the extremes be 4 and 131072, and the ntio ^, what is th« whole amount of the series ^ 153107^ i .-. . - .- ~ jr£' 1 +1^1072 = 149790 jJnstca, 7. What is the sum of t!ic descending series 3, 1, ^.7 Jj, kc, extended to infinity! •ju.^.i.' / It is evident the last term must become 0, or inc^^fini/ no;;r to nolhinj^ ; therefore, the extremeji are 3 and Oy" the ratio 3. An/ % •^24 REOMETRICAL PROGRESSION. ir 107. I rViTTr + TTT*ffff. «c., or, '11 11 1, &c., continually repeated? 8. What is the value of the infinito series 1 -^ ^ ~|-^-|- ^, &c. ? Ans. ij. 9. What is the value of the infinite series, ^ -f- j^^ -|- &c., or, what is the same, the decimal Ans. }f. 10. What is the value of the infinite series, T^jy-f-TtjjTTff' &c., descending by the ratio 100, or, which is the same, the repeating decimal '020202, &c. 1 Ana. ^. 11. A gentleman, whose daughter was married on a new year's day gave her 1 pound, promising to triple it on the firit day of each month in the year ; to how much did her portion amount 1 Here, before finding the amount 720ie. Ans. as belore. ;j — 1 12. A man agrees to serve a farmer 40 years without any other reward then I kernel of corn for the first year, 10 for the, second year, and so on, in 10 fold ratio, till the end of the time ; what will be the amount of his wages, allowing 1000 kernels to a pint, and supposing he sells his corn for 30 pence per bushel 1 lO^"-! ^ ,_ f 1,111,111,111,111,111,111.111,111. ____Xi--^ 111,111,111,111,111 kernels. \ \ns 2,170,138,388,888,888,888,888,888,888,888,888£. VTs. 9|d. v3. A gentleman, dying, left his estate to his 5 sons, to he youngest 1000<£ to the second loOOiS. and ordered, )a( each son should exceed the younger by the ratio of 11 ; hat was the amount of the estate 1 Note. Before finding the power of the ratio U, it may IT 107. IT 108. GEOMETRICAL PROORESSION. 225 ns. 1^. lecimal Ans. If. me, the ns. ^. n anew t on the did her ve must 5572(«*. ) amount s :— rrms, are the ra; ,i Consult the rule, under IT 107, ex. 11. i^"Zl X 100=437'45. Ans. 437^6 9i . Hence, when the annuity, the time^ and rate per cent, are given, to find the amount, — Raise the ratio (the amount of \£, &c. for 1 year) to a power denoted by the number of years ; from this power subtract 1, then divide the re- mainder by the ratio less 1, and the quotient, multiplied by the annuity, will be the amount. f^ Note. The powers of the amounts, at 5 and 6 per cent 228 GEOMETRICAL PROGRESSION. IF 109. % I t up to the 24th, may be taken from the tabio, under ^ 85. 2. What is the amount of an annuity of !H)£, it beinv in arrears 20 years, allowing 5 per cent compound interest 1 Ans. l().53i:5s. 9»d. 3. If tlic annual rent of a house, which is loOi.', be in ar- rears 4 years, what i« the amount, allowing 10 per cent compound interest \ Ans. G9Cc€ 3s. 4. To how much would a salary of 500i^ per annum amount in 14 years, the money being improved at per cent compound interest ? in 10 years 1 in 20 years 1 in 22 years 1 in 24 years 1 Ans. to the last, 25407c£ ISs. ^ 110, If the annuity is paid in advance, or if it be bought at the beginning of the first year, the sum which ought to be given for it is called the present icorth. o. What is the present worth of an annual pension of 100£, to continue for 4 years, allowing 6 per cent compound interest \ The present worth is, evidently, a sum, which, at 6 per cent, compound interest, would, in 4 years, produce an a- mount, equal to the amount of the annuity in arrears the same time. By the last rule we find the amount=43.7*45i!, and by the directions under 11108, ex. 4, we find the present worth= 346'51£. : I ' Ans. 346ie 10s. 4f d. Hence, to find the present worth of any annuity,— First find its amount in arrears for the whole time ; this amount, divided by that power of the ratio denoted by the number of years, will give the present worth. 6. What is the present worth of an annual salary of 100.^ to continue 20 years, allowing 5 per cent ? Ans. 1246^ 4s. 4fd. The operations under this rule being somewhat tedious, we subjoin a ^ " ''^' "' ...P' TABLE, Showing the present worth of 1<£ or $1 annuity, at 5 and 6 per cent compound interest, for any number of years, fror 1 to 34. 4^: I - . ; /, r: ■i''» *M >r'i' -,.,^,, ir 109. it bein^ interest 1 '5s. 9^d. be inar- per cent 09Ci: 3s. ;r annum at per ~ in 20 107<£ 158. r if it be um yrhich )en!>ion of compound li, at 6 per duce an a- irrears the and by the it worth= lOs. 4f d. lity,— First lis amount, number ryoflOO^ 48. 4fd. lat tedious, at 5 and of years, . *•-;>; '.. Years. 5 per cent. I 6 piT cm'. Years. 5 percent. 1 0'952:58 0'943:W 18 1168958 1 '85941 1«83339 19 12'08,>32 3 2'72325 267301 20 1246221 4 3'54595 346.51 21 1282115 5 4'32948 4'21236 22 13' 163 6 5'075r>9 491732 23 13'48807 1^ 5'78G37 5'58238 24 13'79864 8 6'46321 6'20979 25 14'09394 9 740782 680169 26 1437518 10 7'72173 7'36008 \ 27 14'64303 11 8'30641 7'88687 28 14'S9813 12 8'86325 8'38384 29 15'14107 13 9«39357 8«85268 30 15'37245 14 9-89864 9-29498 31 ) 559281 15 10*37966 9'71225 32 15'802aS 16 10*83777 1040589 33 16'00255 17 11«27407 10'47726 34 16' 1929 ^ 110. GEOMETRICAL PROfiKESSION. 229 Gpfr cpiil. 108276 11'15811 11 '46902 11 '76407 12'04158 12'30338 I2'55035 12'783;}5 13'00316 1321053 13'40616 13'59072 13'76483 13'92908 1408398 1422917 1436613 It is evident that the present worth of 2^ annuity is 2 times as much as that of l£ \ the present worth of 3jt' will be 3 times as much, &c. Hence, to find the present worth of any annuity, at 5 or 6 per cent,— Find, in this table, the present worth of l£ annuity, and multiply it by the given annuity, and the product will be the present worth. 7 What ready money will purchase an annuity of 150iJ to continue 30 years, at 5 per cent, compound interest 1 The present worth ot l£ annuity, by the table, for 30 years, is 15«37245 ; therefore, 15'37245xl50=2305'867c£ =2305ie 17s. 4d. Answer. 8. What is the present worth of a yearly pension of 4(U" to continue 10 years, at 6 per cent compound interest ? at 5 per cent 1 to continue 15 years ? 20 years ? 25 years 1 34 years ? Ans. to the last, 647i: Hs. 3:;'d. When annuities do not commence till a certain period oi time has elapsed, or till some particular event has taken place, they are said to be in rkversion. , ,-, t,,-., .. 9. What is the present worth of lOOil annuity, to be continued 4 years, but not to commence till 2 years hence, allowing 6 per cent compound interest? The present worth is evidently a sum which, at 6 per cent compound interest, would in 2 years produce an amount u *i,*M> GEOMETRICAL I>ROORES8I09r. f 111 equal to the present worth of the annuity, were it to com- mence immediately. By the last rule, we find the present worth of the annuity, to commence immediately, to be 'MO^rAjO, and, by directions under If 108, ex. 4, we lind the present worth of ti4(')'5l£ for 2 years to be ii{)8'\i9iU'. Ans. 308iE 7s. 10^ d Hence, to find the present worth of any annuity taken in reversion, at compound interest, — First, find the present worth, to commence immediately, and this sum, divided by the power of the ratio, denoted by the time in reversion, will give the answer. ,' , ^ ..I' ." 10. What ready money will purchase the reversion of a lease of 60£ per annum, to continue G years, but not to commence till the end of 3 years, allowing 6 per cent com- pound interest to the purchaser ? The present worth to comnence immediately, we find to 295'039 bo, 29r>'039, and -pQ^=:247'72. Ans. 247.^6 14s. 4|d. It is plain, the same result will be obtained by finding the present worth of the annuity, to commence immediately, and to continue to the end of the time, that it 3-|-0=i^ years, and then subtracting from this sum the present worth vi' the annuity, continuing for the time of the reversion, 3 years. Or, we may find the present worth of 1£ for the two times by the table, and multiply their difference by the ;,ivcn annuity. Thus, by the table, The whole time, 9 years,=:6'80169 The time in reversion, 3 years,=2'67301 Difference,=> ^ - •,. 12868 60 ' ' 247*7208ft^ • 247'72080^=247^ 148. 4fd. Ans. I i . What is the present worth of a lease of IOOjB to con- i'um^ 20 years, but not to commence till the end of 4 years, allowing ."» per cent 1 what, if it be 6 years in rever- sion 7 8 years 1 10 years 1 14 years I T 111. 12. What is the worth of a freehold estate, of which, tlio yearly rent is GOi^, allowing to the purchaser 6 per cent 1-" ■^''* " ••^/usmvcf ^t ;'1\-}-h rn-'^-^- ; m;!? - In this case, the annuity continuesybrerer, and the estate id versif Th f HI. CROMETRICAL PROCRKBtilOlV. .»• ]1 f 111 t to com ■ le present ;|y, to bo t, we find 7s. lOJd y taken in he present divided by rercrsion, irsion of a but not to cent coni- we find to J 148. 4|d. finding the mediately, , 3-|-0=i> sent worth sversion, 3 £ for the Dce by the 59 )1 60 m£ 4fd. Am. 0£ to con- of 4 years, s in rever- arsl estate, of purchaser the estate is evidently worth a sum, of which the yearly inten.'st is equal to the yearly rent of the estate. The principal mul- plied by the rate gives the interest ; therefore, the interest divided by the rate will give the principal ; G()-f-'0<)=1000. Ans. HKXXf. Hence, to find the present worth of an annuity, continu- ing forever,— Divide the annuity by the rate per cent, and the quotient will be the present worth. Note. The worth will be the sanje, whether we reckon isimple or compound interest ; for since a year's interest of the price is the annuity, the profits arising from that price can neither be more nor less than the profits arising from the an- nuity, whether they be employed at simple or compound interest. 13. What is the worth of lOOjf annuity, to continue for- ever, allowing to the purchaser 4 per cent 1 allowing .'i per cent 1 8 per cent 1 10 per cent 1 15 per cent 1 20 per cent 1 jfns. to last, 500.£. 14. Suppose a freehold estate of 60£ per annum, to com- mence 2 years hence, be put on sale ; what is its value, al- lowing the purchaser G per cent 1 Its present worth is a sum which, at 6 per cent, compound interest, would, in 2 years, produce an amount equal to the worth of the estate if entered on immediately. — ; = 1000<£ = the worth, if entered on imn^ediately, 1000<£ and j-qw.2~"=889'99Gcf=889je 19s. Ud. the pre-:ent worth. The same result may be obtained by subtracting from the worth of the estate, to commence immediately, the pres- ent worth of the annuitj 60, for 2 years, the time of re- version. Thus, by the table, the present worth of 1£ tot 2 years is 1 '83339X60=1 10'0034r=present worth of (iOJ for 2 years, and 1000<£— 110'0034=889'9966i.'=889je 198. lid. jfns. as before. ., u /; . 15. What is the present worth of a perpetual annuity of 100j6', to commence G years hence, allowing the purchaser a percent compound interest 1 what, if 8 years in re- version 1 10 years ? 4 vears? 30 years'! - u/ns. to the last, 462i^ 15s, l^d The foregoing examples, in compound interest, have been \ i»Z PBRMVTATIOlf. If 11^' confied to yearly payments ; if the payments are hal/ year- ly, we take half the principal or annuity, half the rate per cent, and twice the number of years, and work as before, and so for any other part of a year. ' ' ' - QUESTIONS 1. What is a gnomotrienl progression or series? 2. What is the ratio i* 3. When tlie first term, the ratio and the number of terms, are given, how do you find the last term? 4. Wiien the extremes and ratio are given, liow do you find the sum of all the terms? 5. When the first term, the ratio, and the number of terms, nre given, how do you find the amount of the series? G. When the ratio is a fraction, how do you proceed? 7. Wimt is compound interest? 8. How does it appear that the amounts, n rising by compound interest, form a geometric'il series ? 0. What is the ro/to in compound inter- •st ? the number of terms? the^r^t term ? the last term ? 10. When the rate, the time, and the principal, are given, how do you find the amount? 11. When A. R. and T. are given, how di> you find P.? 12. When A. P. and T. are given, how do you find R. ? 13. Wiien A. P. and R. are given, how do you find T. ' 14. What is in annuity? 15. When are annuities said to be in arrears? 1(3. What is the amount ? 17. In n geometrical series, to what is the a- mount of an annuity, equivalent ? 18. How do you find the amount of an annuity, at compound interest ? 19. What is the ■present worth of an annuity ? how computed at compound interest ? | how found by the table ? 20. What is understood by the term rever- sion.^ 21. How do you find the present worth of an annuity, taken in rct>er*t(m ? by the table? 22. How do you find the present worth of a freehold estate , or a perpetual annuity ? the samo taken in reversion ? by the table ? . , , . ., , IT lift. Permutation is the method of finding how many different ways the order of any number of things may be varied or changed. 1. Four gentlemen agreed to dine together so long as they eould sit, every day, in a different order, or position ; how many days did they dine together 1 Had there been but ttoo of them, a and b, they could sit only in 2 times 1 (1x2=3) different positions, thus, a b, and& a. Had there been three, a band c, they could sit in 1X2 X3=6 different positions ; for, beginning the order with a, there will be 2 positions, viz. ab c, and ac b ; next begin- ning with 6, there will be 2 positions, bac, and b c a; last- tati bloi natl aiui ;r;. .# H 112 I are halfyedit- ftlf the rate per work as before, 2. What is tiio number of terms, len the extremes all the terms ? 5. terms, are given, ^hen the ratio is a >und interest ? 8. impound interest. I compound intor- i? the last term * ;iven, how do you ire ffiven, how di> w do you find R. f T. > 14. What is in arremrsf 16. , to what is the a- i find the amount I the present worth interest ? • )y the term rever- an annuity, taken u find the present ^ the saroo ding hoiiv many things may be her so long as er, or position ; jy could sit only bus, a b, and b uld sit in 1X2 »e order with a, ; next begin - and b c a ; last- T 113. mSCRLLANEOUS EXAMPLES. ^ ly, beginning with c, we have cab, and c b a, that is, in all, 1x2X3==6 different positions. In the same manner if there be /our, the different positions will be IX^X^X 4=24. jins. 24. Hence, to tind the number of different changes or permu- tations, of which any number of different thingi* are capa- ble,— Multiply continually together, all the terms of the natual series of numbers, from 1 up to the given number, and the last product wiil be the answer 2. How many variations may there be in the position of the nine digits 1 ^ns. 3C2880. 3. A man bought 25 cows, agreeing to pay for them t penny for every different order in which they could all be placed ; how much did the cows cost him 1 jfnt. G4630041847212441600y00i: 4. A certain Church has 8 bells ; how many changes may be rung upon them. , jins. 40320. MISCELLANEOUS EXAMPLES. If 118. 1. 4-1-6X7-1=60 A line; or vinculum, drawn over several numbers, signi- fies, that the numbers under it, are to be taken jointly, or as one whole number. 2. 9— 8-f 4 X8-f 4— 6=how many 1 Ant. 30. //ns. 230. 3. 7-f-4— 2-f-3-|-40Xo=how mauy? 4. ?i:?Z?^lfZ?=how many 1 v Jns.iU. 2X2 ^ .5. There are 2 numbers ; the greater is 25 ffmes 78, and their difference is 9 times 15 ; their sum and product are required. Ana. 3765 is their sum, 3539250 their product. 6. What is the difference between thrice five and thirty, and thrice thirty-five \ 35f 3-5X3+30= 60, Ans. 7. What is the difference between six dozen dozen, and half a dozen dozen 1 Am. 792. 8. What number divided by 7 will make 6488 ? 9. What number multiplied by 6 will make 2058 1 U 2 ^ ' -• ■V. ^ .*r 2.14 MIRCELhAKEOUB EXAMPLK8. ir im. 1 10. A gentlemen wont to nea at 17 years of age ; 8 years tfter Ik had a sun born, who died at the agv of liTt ; after whom the father lived twice *^() yearit ; how old wao the fa* thor at his death 1 A/is. 100 yoarn. 11. What iitimhrr is that, which being multiplied by 15 the product will he J 1 2-i-l''»=L^» Ans. 12. What decimal \a that, which being multiplied by 15, the product will be '75 1 «75-^-l.'i=«0r> Ans. 13. What is the decimal cquivaleut to Vr ^ ^Ans. '02H5714. 14. What fraction is that, to which if you add '■^, the sum will bo ^ 1 Ans. y^. 1,'». What number is that, from which if you take I, the remainder will be I 1 Ans. J§, 1(1. What number is that, which bein«;divi(!ed by J, the quoti(;nt will be '21 1 Ans. 15|. 17. What number is that, from which if you take j of itself, the remainder will bo i'Z \ Ans. 20. 10. What number is tliat, to which if you add f of | of itself, the whole will be '10 1 Ans. 12. 20. What number is that, of which is the § part 1 Ans. 13^. 21. A fanner carried a load of produce to market; he sold 780 lbs. of |)ork, at 3d. per pound; 250 lbs. of cheese, at.5d. per lb. ; 151 lbs. of butler, at lOd. per lb. : in pay he received 00 lbs, of sugar, at7d. per lb. ; 15 gallons of molasses, at 2s. 3d. per gallon ; \ barrel of mackerel, at 18s. IKl.; 4 bushels of salt, at 6s. 4d. per bushel ; and the balance in money; how much money did he receive 1 Ans. 15£ 14s. 8d- 22. A farmer carried his grain to market, and sold 75 l)ushels of wheat at 7s. 3d. per bushel ; 04 bushels of rye, at 4s. Od. per bushel ; 142 bushels of corn, at 2s. (Jd. per bushel : in exchange he received sundry articles: — 3 pieces cloth, each containing 31 yards, at 8s. Od. per yard; 2 quintals of fish, lis. Od. per quintal; 8 hluls. of salt, l£ Is. G;l. perhhd. and the balance in mouey : how much money did he receive 1 Ans. 0£ 14s. 23. A man exchanges 7G0 gallons of molasses, at 2s. per gallon, for 004- cwt. of cheese, at l£ per cwt. ; how much will be the balance in his favor 1 Ans. Oi,' 10s. 24. Bought 84 yards of cloth, at (is. 3d. per yard ; iiovf «*»»<*:■'• v.. IT 113. ; 8year» ;jr» ; attcr UH the fa- 00 yoarn. ietl by 15 =2^, Ans. iod by: 15, =«05 Ans. •0285714. J, tlio sum 'Ans. ig. ake j^, thu Ans. f§. 1 by J, the Ans. 15|. take f of Ans. 20. 1 ^ of I oC Ans. 12. part \ Ans. 13^. arket ; he of cheese, : in pay gallons of ckerel, at ; and the ivel £ 14s. 8d. fid sold 75 els of rye., IS. (jd. per — :J pieces r yard ; 2 It, l£ U ch money 9£ 1 4s. es, at 2s. wt. ; how . Di: 10s. ard ; hovf ^ 111. MIICKLLANBOV* EkAMPLRS. W.1 much did It como to ! IIow many busheh of wheat, at 7». (5d. per bushel, will it take to pay for it 1 Ant. to the larit, 70 btishels. 2.>. A man sold .'112 pfunids of beef, at M. por poiniif, and received his pay in molasseK, at 2s. pt r gallon ; how illons many ^ did I 10 receive \u%. in illons. 2(>. A man exchaufjjfd 70 bushels of rye, at 1?*. (! I. prr bushel, for 40 bushels of wheat, at 7«. per bushel, ami rr- ceiTed the balance in oats, at 2s. per bushel; how nianj' bushclsof oats did he receive 1 An.s. Itf. 27. How many busluls of potatoes, at Is. (Id. per bush- el, must be given for \Vl bushels of barley, at 2.i. tid. ])er bushel 1 Ans. ^'Sl bushel.jC, wouM lay it out in uheop, at ly*. Vn\. apii'ce, how many lavi hv. buy ? Aiis. 40 40. U 'i\) rows cost 7.')t', what is the price of I cow f of "Z coWH 1 of o coNvs \ of 15 cows ? Ans. to the last, 5()£5s. 41 . If 7 men consume '21 lbs. of njoat in one week, how nuich would I man consunu! in the same timel '2 men 1 r> men ? l(» men I Ans. to the last,34f lb. Note. Let the pupil aUo perform these questions by the lulo of proportion. i'i. It I pay l-X" 10s. for the use of 'J.ji,', how much must I pay for the use of IfiX I.'):).! Ans. If. '2$. 6d. 43. What premium must I pay for the insurance of my house aj^ainst loss by fire, at the rate of J per cent, that is, I pound for UK) pounds, if my iunise bo valued at '247') -t' ^ Ans. l^ 7s. 6d. 44. What will bo the insurance', per annum, of a store and contents, valued atyS7GX' bs. at 14 per centum 1 "Ans. 14a£iJs. Ud. 4.">. What «ommission must I receive for selling 47Bi' worth of books, at M per cent ! Ans. \'\S£ 4s. 9^d. 1(). A merchant bought a (|uanlity of goods for 734<£, and sold them so as to gain '21 per cent ; how much did he gain 1 and for how muc\t did he sell his goods 1 Ans. to the last, 888c£: 2s. Ofd. 47. A merchant bought a quantity of goods at Montreal, for 500i,', and paid 4^£ for their transportation ; he sold ti»em so as to gain "24 per cent, on the whole cost ; for how much did he sell theml Ans. GTM fis. 4|d kS. Bought a quantity of books for G4c£, but for cash a discount of 12 per cent, was made ; what did the books cost? Ans..5Gi:6i. 42d. 49. Bought a book, the price of which was marked 1£ *2s. Od. but for cash the bookseller will sell it at 33^ per cent discount ; what is the cash price 1 Ans. 15s. 50. I bought a cask of liquor, containing 120 gallons, for A''Z£ ; for how much must I sell it to gain 15 per cent '^. how much per gallon 1 Ans. to the last, 4s. [d. f 113. what in the m IDs. Vn\. ifi 1)0 botiglit i. 4 biinhcls. n sheep, nt A lit. 40 of t cow ! ows ? ast, ')(i£ 5s. > week, how '2 men 1 last, 34 f lb. tions by the f much must . It*, lis. 6(1. ranee of my :cnt, that is, \ at 247o.i; ^ 12i: 7s. Gd. u, of a store ntura 1 18^28. Ud. celling 4781' iS£ 4s. 9^d. s for 734jf , much did he k£ 2s. Ofd. at Montreal, [)n ; he sold ost ; for how (3c^r)S. 4|d it for cash a id the books )6£ Gs. i^d. marked 1£ t at 33^ per Ans. 15s. ) gallons, for 5 per cent 1 last, 4s. (d. ^ 113. MjSCfiLLANroLf KXAMPI.r,. 231 o-.^ What ig the interest at « .> An* 2i ror 17 ^,, ,, ^,^^.^ ^ est, at per cent of 7U ().. .,/;• '>J What i. th. interest of 4H7<-o. 2 i^T' ?^" •^'- 4' -.J w, '•*"'• H'"'!'^ months 1 •'^- What .. the interest of ,8.50 for'^mc^^L'^ ^^"• J"' ^»'»t '• the intemst of 1000,^ for r. daj^Hi^' ^'^^'^ ^ What is the interest of 10s. for lo yearti" ""' ^''' ?«' ivl ^^' ""' ^ * '""'»^'"' «"'! 7 -nd 3 day/a'tV'; '"^'^^^* "^ ^'^^'Ol for -^Tm f''-^' /rq ^■' ^' «> per cent ? * »or ^ yrs. 4 months years ami r^ *"'" P"* *« interest at fi n "'• •^»^'<>3«4. ZTlT^%^: '"°""' *° ^'^«' ""' "'"' " *' "0. I owe a man d7'i^ in ^ . A"«. •'^I.'t0'4ij_i ""Of »o„ey being .onnZ'T^ZT" "^ '"=" '"*<. "'^"«^ Ram 7o£, 'M. Bought cloth at 3^ 10« n ^'^ !•> per «ent - P" P- . how »„cf ;»:-S-; -a^^o^a*:. .. T^ £ro?S^;r:„r;jo,-, An^^|_. Ans. His whole gain H^' . per centum. ^ '« ^ ; per gal. iQs. ^hieh j, oo. ?5. If 100=^ gain 6^ in 10 ^ k».n 4^1 _ f Jf^n 12^n,onths, i„ ,,,at time Mill it Ans. to the last, 28 month.. 'apif ! k 238 MISCELLANEOUS EXAMPLES. IT 113. (i<>. I;i what time will oiX 10s. at G per cent, gain, 2£ 3s. 7;|^d. Ans. 8 months. C)7. '20 men built a certain bridge io 60 days, but, it be- ing carried away in a freshet, it is required how many men can rebuild it in 50 days. . ■,. : . da vs. dajrs. men. 50 : GO :: 20 : 24 men, ans. GS. If a iield will feed 7 horsss ^ weeks, liow long will il feed 28 horses 1 Ana. 2 weeks, (J9. If a field, 20 rods in length, must be 8 rods in width to contain an acre, how much in width must be a field, lt> rods in length, to contain the same ? Ans. 10 rods. 70. If I purchase for a cloak 12 yards of plaid | of a yd. wide, how much booking 1^ yards wide must I buy to line it ! Ans. 5 yards. 71. If a man earn T8c£ 15s. in 5 months, how long must he work to earn ll.'ii^l Ans. 30f months. 72. B owes C 540i^, but B, not being Avorth so much money, C agrees to take 15 s. on a pound; what sum must C receive for the debtl Ans. 405<£. 70. A cistern, whose capacity is 400 gallons, is supplied hy a pipe which lets in 7 gallons in 5 minutes ; but there (s a leak in the bottom of the cistern which lets out 2 gal- Ions in () minutes, supposing the cistern empty, in what time would it be filled / In 1 minute I of a gallon is admitted, but in the same time 2 of a gallon leaks out. Ans. G hours, 15 minutes. 74. A ship has a leak which will fill it so as to make it sink in 10 hours; it has also a pump which will clear it in 15 hours; now, if they begin to pump when it begins to leak, in Avhat time will it sinkl In one hour the ship would be j\ filled by the leak, but tn tbe same time it would be y^j emptied by the pump. jfns. 30 hours. 75. A cistern is supplied by a pipe which will fill it in 40 minutes ; how many pipes, of the same bigness, will fill it in 5 minutes '' ^ns. 8, 70. Suppose I lend a friend 500<£ for 4 months, he prom- ising to do me a like favour ; some time afterward, I havej need of 300^^ ; how long may I keep it to balance the for- mer favour 1 w/ns. 6f months.] 77. Suppose 800 soldiers were in a garrison with pro 83. togethc In faour on the 12 spac 84. three m goes 2 per houi 35. •] two men per hour, •gain eoi B gain ir 113. gain, 2£ i months, •ut, it be- lany men men, ans. long will .2 weeks, s in width a field, lt> jj. 10 rods. 1-1 of a yd. buy to line js. 5 yards. long must ,0§ months. :th so much it sum roust Ans. 405i^. , is supplied } ; but there ts out 2 gal- y, in what le same time ir> nainutes. to make it fu clear it in it begins to Ihe leak, but le pump. s. 30 hours, ill fill it in ess, will fill Jns. 8. ,hs, he prom- rard, 1 have ince the for- 6| months. m with pro- 113. MISCELLANEOUS EXAMPLES. 239 visions suflicient for 2 inonths ; how many soldiers must de- part, tluit the provisions maj serve them .I months 1 j^iis. 480 78. \f my iTorsc and saddle are worth '2l£, and my hf»rse b« worth (3 times as much as mj saddle, pray what is the value of my horse ? y/;js. 18.,.. Ans. 24000 men. 00. A farmer, being asked how many sheep he had, an- swered, that he had them in 5 fields ; in the first were ^ of his flock, in the second ^, in the third ^, in the fourth ^, and in the fifth 450 ; how many had he 1 Ans. 1200. 91 . There is a pole, j- of which stands in the mud, ^ in the water, and the rest of it out of the water ; required the part out of the water. Ans. -^. 92. If a pole be ^ in the mud, ^ in the water, and 6 feet out of the water, what is the length of the pole 1 Ans. 90 feet 93. The amount of a certain school is as follows : -^ o the pupils study grammar^ f geography, ^^ arithmetic, learn to write, and 9 learn to read : what is the number eachi •*% ij'.^"' H 113. 113. MISCELLANEOUS KXAMPLES. 241 :r every same time up stream that moT- th be pro jm 8 miles ; place will 187^ miles 3f wine for . contain 2 e remainder it what rate 5'936je f. apiec*^, '< for a ce'-:^ ; Dduch moafcy larry 1 250 melons. le fourth part |in this army? ig are usually e more easily ^ of the jber of men ; ir the whole ? 24000 men. he had, an- irst were ^ of I ie fourth t^, Ans. 1200. fhe mud, ^ in required the Ans. -^I [er, and 6 feel Ans. 90 feet follows : tV ^\ lithmetic, ^pJ le number of Ans. 5 in giainincr, IW in geography, -^4 in arithmt?lic ; 12 Icaru to Avrite, anci \) learn to read. J)4. A man, driving his geese to market, was met bv another, who said, "Gotnl moiiow, sir, with your hundred gccse ;" says he, "I have not a hundred ; but it I had, in addition to my present numl)er, one half as many as I now have, and 2^V geese more, I should have a hundred :" how many had he \ 100 — 2^ is what part of his present number 1 Ans. He had ti/i geesp. 95. In an orchard of fruit trees, \ of them bear apples, ] pears, Jl plums, (K) of them peaches, and 40, cherries : how many trees does the orchard contain 1 Ans. 12(M) 9fi. In a certain village, ^ of the houses arc painted white, I red and, ^ yellow, 3 are painted green, and 7 are \mpaint- ed ; how many houses in the village 1 Ans. 120. 97. Seven eights of a certain number exceed four lifths v( the same number by • repuired the number. ^ — * = ^!\y ; consequently, G is^j''^- of the required num- ber. * Ans. 80. 98. What number is that to which if |, of itself be added, the sum will be 30 1 Ans. 2;">. 99. What number is that, to which if its A and -^ be added, the sum will be 84 1 84 = I +i+i= 1 times the repuired number. Ans. 48. 100. What number is that, which, being increased by ^- and ^ of itself, and by 22 more, will be made 3 times as much 1 The number, being taken I, §, and ^ times, will make 2-^T. times and 22 is evidently Avhat that wants of 3 times. Ans. 30. 101. FFhat number is that, which being increased by ^, I and I of itself, the sum will be 234J 1 Ans. 90. 102. B, C, and D, talking of their ages, C said bis ago was once and a half the age of B, and D said his aoe was twice and one tenth the age of both, and that the sum of ther ages was 93 ; what was the age of each 1 Ans. B 12 years, C 18 years, D 63 years old. 103. A schoolmaster being asked how many scholars he had, said, ♦' If I had as many more as I now have, J as ma- ny, ^ as many, \ and Jas many, 1 should then have 435;" what was the number of his pupils'! Ans. 120. w i4i> MlSCELLAWKOfS EXABIPLES. IF 11.']. 104. B and C commenced trade with C(|nal sums of mon- ey ; n gained a sum equal to ^ of liis whole stock, and C lost 'JOOi,* ; then B's money was douhle that of C's ; what was the stock of each ? JBy the condition of this question, one half of >;, that is, ^ of the stock is equal to | of the stock, less 20(h£ ; con- sequently, 200uf is * of the stock. Ans. />fW)£. lO.j. A man was hired 50 days on these conditions, — that for every day he worked, he should receive 'is, 9d., and for every day he was idle, he shoi/ld forfeit Is. 3d. ; at the c' piration of the time, he received 'Z£ 17s. Od. ; how many days did he work, and how many was he idle 1 Had he worked every day, his wages would have been 'is. 9d.Xi>0=9i^ 7s. Gd. that is 2£ 10s. mcro than he receiv- ed ; but every day he was idle lessened his wages 3s. 9d. -f-ls. 3d.=5s. ; consequently he was idle 10 days. Ans. He wrought 40, and was idle 10 days. lOG. Ti and C have the same income ; B saves ^ of his ; Init C, by spending 30i£ per annum more than B, at the ond of 8 years finds himself 40=;^ in debt ; what is their in- rome, and what does each spend per annum ? Ans. Their income, SOOci" per annum ; B spends 175:^, and C '20')£ per annum. i()7. A man, lying at the point of death, left his three sioiis iiis property ; to B ^ wanting SOi^, to C ^, and to D the remainder, which was 10<£ less than the share of B; what was each one's share 1 Ans. 80i^, 50<=f, and70»f. 10?^. There is a fish, whoso head is 4 feet long ; his tail is as long as his head and half the length of his body, and his l)0(ly is as long as his head and tail; what is the length of the fish 1 Tlic pupil will perceive that the length of the body is J^ the Ic^ngth of the fish, p^: > Ans. 32 feet. lO'J. Bean do a certain piece of work in 4 days, and C ran do the same work in 3 davs ; in what time would both woj icing together, perform it 1 Ans. 1^ days. 110. Three persons can perform a certain piece of work in the following manner: B and C can do it in 4 days, C and D in days, and B and D in 5 days : in what time can they all do it together 1 Ans. 3/^ days. 111. B and C can do a piece of work in 5 days ; B can do it in 7 days ; in how many days can C do it ? ^n9. 174- J, V 113. MISCCLLArVROVS CXAMI'LRS. '24{i ir ii;^ IS of mon- k, and C ;'s ; what 0., that is, ){)£; con- ns. 5(M)i.'. idilions,— re :js. l)d.. Is. 3d.; at Gd. ; how lie 1 have been he receiv- es 38. 9d. s. e 10 days. ^ of his ; B, at the is their in- ds 175£, his three and to D are of B ; and 70.£. his tail lody, and length of |body is ^ 32 feet". TS, and C 3uld both I^ days. of work ,s, C and Itime can )-#7- days. "b can fnii. 174- ll'i. A man died, leaving l(MM)dC to be diridet'i l)ctwoen his two sons, one 11, and the otiier IH years of a^c, in such |)roportion, that the share of each, being put to interest at <» per cent, should amount to the same sum when they should arrive at the ajje of '21 ; what did each receivo ' ^ns. The elder .jltit' 3s. O^d. -j- ; the younger, |.'j.'J> IGs. lid. 113. A house being let upon a lease of o years, at lij.t per annum, and the rent being in arrear for the whole timt, what is the sum due at the end of the term, simple interest being allowed at per cent 1 y/ns. ^■lc£'. 1 14. If 3 dozen pair of gloves be equal in value to 10 yards of calico, and 100 yards of calico to thVee pieces of satinet of 30 yards each, and the satinet be worth 2s. tUl per yard, how many pair of gloves can be bought for 20s.? y/ns. 8 pair, 115. B C and D would divide lOOjE between them, so as that C may have S£ more than B, and D 4£ more than € ; how much must each man have 1 Ans. B 30c£, C 33^', and D 37^. 116. A man has pint bottles, and half pint bottles ; bow much wine will it take to fill one of each sort ? how much to fill 2 of each sort ? how much to fill G of each sort 1 117. A man would draw off 30 gallons of wine into i pint and 2 pint bottles, of each an equal number ; how many bottles will it take, of each kind, to contain the JO gallons'! Ans. 80 of each. 118. A merchant has canisters, some holding 5 pounds, some 7 pounds, and some 12 pounds ; how many, of each an equal number, can be filled out of 12 cwt. 3 qrs. 12 lbs. of tea 1 y/ns. GO. 119. If 18 grains of silver make a thimble, and 12 pwts. make a teaspoon, how many of each an equal number, can be made from 15 oz. G pwts. of silver 1 jins. 24 of each. 120. Let GO pence be divided among three boys, in such a manner that, as often as the first has 3, the second shall have 5, and the third 7 pence, how many pence will each receive 1 p %' Ans. 12, 20, and 28 pence. 121. A gentleman, having 50 shillings to pay anong hia laborers for a day's work, would give tq every boy (id., to every woman 8d. and to every roan 16d. ; the number of ^^m^' '^^^ 244 MISCELLANEOUS EXAMPLES. ir 113- 'I hoys, woinon And men, was the same ; 1 demand the num- btT ol' cacli. ^ns. '20. {'Z'Z. A gentleman had 7.£ 17s. (nl. to pay among, his la- borers; to every hoy he gave (nl., to every woman Hd., and to every nian Kid.; and there were for every hoy three wo- men, for every woman iwo men ; I demand the niuuher of each 1 .//i". \'i hoys, 45 women, and 90 men I'iU. A farmer hoii^ht a sheep, a cow and a yoke of ox- en for '20ij l*2s 0(1.; he gave for the cow H times as much as for the sliecp, and for the oxen 3 tinjes as mucli as for the cow ; how much did he give for each 1 j/ns. For the shcop, l'2s. Gd., tlie cow 6£, and the oxen I'il. There was a farm, of which B owned ^, and C 4.^; the farm was sold for 441cl' ; what was each one's share of the money 1 y/ns. B's \'Z{'y£, and C's 'Jl/ii,'. 125. Four men traded together on a capital of 3000ofc', of which B put in }, C |, 1) ^, and E ^W; at the end of 53 years tliey liad gained 2'J04^t' ; what was eac!» one's share of the gain 1 Jns. B's 118'2X, C's 591 £, D's 394^, E'a 197£. 12t). Three merchants companicd ; B furnished § of the capital, C | , and D the rest ; they gain 1250<£; what part of the capital did D furnish, and wliat is each one's share of the gain 1 jins. D furnished ^^(yof the capital ; and B's share of the gain was 500i,', C's 4G8i: 15s., and D's 281=£ Ss. 127. B, C, and D, traded in company ; B put in 1254^, C 87c£ 10s., and D 120 yards of cloth ; they gained 83^ 2s. Gd., of which D's share was 30£ ; w^hat was the value of D's cloth per yard, and what was B and C's shares of the gain 1 r ^, . , . on/" 600 1200 48 Note. D's gain heing 30jl is of the l6G2t 3325 133 whole gain : hence the gain of B and C is readily found ; also the price at which D's cloth was valued per yard. Ann. D's cloth, per yard, l£, B's share of the gain, 31.£ 5s., C's share, 2\£ 17s. 6d. 128. Throe gardeners, B, C, and D, having Imught a piece of ground, find the profits of tt amount to 120£ per an- num. Now the sum of money which they laid down was in such proportion, that, as often as B paid o£ C paid 7i2, of givl D'sk moil II on hanJ part) sion IF 113- he nnni- jinn. ii). 1^ his la- jH(I., and three wo- iiiiul)er of 190 men ike of ox- s niucit as as for the the oxen and C 4-^; 's share of 3000^', of ! end of ;i ue's share E'8 197£. edf of the what part ne's share liare of the in 125^, lined 83^ the value shares of of the 133 lly found ; lyard. |gain, 31 € rht a piece per an- Idown was paid 7£, •1 113. MISCKLLANROCS KXAMI'LKS. '245 W^' i\ and a.s often as C.paid A£, V) paid i\£; I demand how oiuch «;ach man must have per annum of the };ain. Note. Bv the questtiun, so often a.s B paid ii£, J) ftaid ^ of HX. Am. B -JtiX* 13s. 4d., C \ViX (Is. Hd., D 5(>£'. 129. A gentleman divided his fortune among his son.i, giving B 9£ as often a» C 5X, and D 3 1' as ol'ten as C TX '; D's dividend was 1.537^; to what did the whole estate a- niount 1 Ans. U.lsai,' Ss. I()d 130. B and C undertake a piece of work ibr 13^ HJs. on which B employed 3 liands 5 days, and C employed 7 hand83days; what part of the work was done by B \ what part by C 1 what was each one's share of the money \ Alls. B ^^ and C /^- ; B's money ')£ I'is. fJd. C's IX 178. (jd. 131. B and C trade in company for one year only ; on the 1st of January, B put in 300X, hut C could not put any money into the stock until the 1st of Aprrl ; what did he then put in to have an equal share with B at the end of tlie year 1 Ans. 40(U'. 132. B, ( , f , and E, spent .'Jos. at a reckoning, and, be- ing a little dipped, agreed that B should pay j, C 4, 1^ \, and E \ ; what did each pay in this proportion \ Ans. B 13s. 4d., C 10s., D 6s. 8d., and E .".s. 133. There are 3 horses, belonging to 3 men, employed to draw a load of plaister from Moutreal to Stanstead, for ♦»<£ 12s. 3d. B and C's horses together are supposed to dn % of the work, B and D's ^^^ C and D's ^% ; they are to be paid proportionally; what is each one's share of the niot\- ey 1 C B's 2je I7s. 6d. (=4P.) Ans.^C'slcf 8s. 9d.(=:3;.) .. . ,; , , C D's2i; 6s. Od. (=4^.) 'v ^ Proof, G^ 12s. 3d. 134. A person, who was possessed of f of a vessel, soM \ of his share for 37.ji^; what was the vessel worth \ . i| Ans. ir>OO.i*. 135. A gay fellow soon got the better of f of his fortune; he then gave 1500,£ for a commission, and his profusion continued till he had but'4."»0<£ left, which he found to be just ■] of his money, after he had purchased his commis- sion ; what was his fortune at first ? Ans. 37H04.'. 136. A youn^^er brother received I.560.;f, which was just 246 MISCEI.LANKOUS KXAMPLF.S. IT 113. ^r^ofl . brothfi worth ; 137. lis elder brother's H fortune was ^ forhiite, and Hj times the elder as much a^ain an the father was what was the vahie uf his VHtatc \ Ans. VMi\ri£ 14s. 3^d. A {;entlenian left his son a fortune, /(^ of which ho spent in three months ; ^ of ji of the remainder lasted him 9 months longer, when lie had only /iSTdC left ; what was the sum bequeathed him by his father 1 Ans. 208'2£ IBs. '2^^d. 13H. A cannon ball, at the first discharf;e, flies about a mile in H seconds ; at this rate, how long would a ball be i.» passing from the earth to the sun, it being 1)^5173000 miles distant \ Ans. '2{ years, 40 days, 7 hours, 33 minutes, 20 seconds. 130. A general, disposing his army into a square battal- ion, found he had '231 over and above, but, increasing each side with one soldier, bo wanted 44 to fill up the square ; of how many men did his army consist 1 Ans. 19000. 140. B and C cleared by an adventure at sea, 45 guin- as, which was 35£ per cent, upon the money advanced, nd with which they agreed to purchase a genteel horse nd carriage, whereof they were to have the use in proportion *o the sums adventured, which was found to be 11 to B as often as 8 to C; what monev did each adventure 1 Ans. B 10"4ir 4s. 2j^d., C 75c€ ISs. 9-^d. 141. Tubes may be made of gold, weighing not more than at the rate of j-^j-^ of a grain per foot; what would be the weight of such a tube, which would extend across the Atlantic from Quebec to London, estimating the distance at 3000 miles 1 Ans. lib. 8oz. opwts. Sj% grs. 142. A military officer drew up his soldiers in rank and file, having the number in rank and file equal ; on being reinforced with three times his first number of men, he placed them all in the same form, and then the number in rank and file was just double what it was at first ; he wa» again reinforced with three times his whole number of men, and, after placing them all in the same form as at first, his number in rank and file was 40 men each ; how many men had he at first 1 Ans. 100 men. 143. Supposing a man to stand 80 feet from a steeple, and that a line reaching from the belfry to the man is just 100 feet in lerigth the top of the spire is 3 times as high above •jj^V, IT 113. ir 113. MISCF.I.LANROIJ.i KXAMPt.F.f.. 217 the elder ither was 14s. ^d. which he isted him what was 88. '2^^d. s about a a ball be ir»l73000 seconds, are battal- i8ing each le square ; ns. 19000. , 45 guin- advanced, iteel horse proportion ill to B as 1.58. 9^d. not more »at would aicross the istance at . 3-r\ gre. rank and on being men, he umber in he was r of men, first, his lany men 100 men. eple, and just 100 2h above the ground as the steeple is ; what is the hciglit of the spire \ and the length of :\ line reaching from the top of the spire to the man \ See ^ 103. Ans. to last, Il»7 feet nearly. 144. Two ships sail from tite same port ; on** sails direct- ly east, at the rate of 10 miles an hour, anti the other direct- ly south, at the rate of 7} miies an hour ; how many railps apart will they be at the end of 1 hour 1 '2 hours ! 24 hours 1 3 days 1 Ans. to last, 000 miles. 145. There is a square field, each side of which is 5(» rods ; what is the distance between opposite corners 1 Ans. 70' 7! -f- rods. 146. What is the area of a square field, of which the op- posite corners are 70'7l rods apart 1 and wiiat is the lengt!'. of each side 1 ^/ns. to last, ."iO rods nearly 147. There is an oblong field, :iO rods wide, and Ihe dis- tance of the opposite corners is 33^ rods ; what is the length of the field 1 its area ? ^ns. Length, 2Gf rods ; area, 3 acres, 1 rood, 13^ rods. 148. There is a room 18 feet square ; how many yards- of carpeting, 1 yard wide, will be required to cover the floor of it % 182 = 324 ft — ;j6 yards, .^ns. 1,49. If the floor of a square room contain 36 square yards, how many feet does it measure on each side ? j^na. 18 feet. When one side of a square is given, how do you find its ars 24H MIKl'KV.LAXROL'S EXAMPLCH. i un. I I Ml Ie»:gtli H» rod8, liow do yuu tind how do you find 1.1*2, If the area lvido at one end, and iO in- ches wide al the other, \vhat is the mean or average width of the hoard 1 Ana. 14 inches. When the i^rcntest and least width are given, how do you find tliu mean width \ lot. How many square feet in a hoard IG feet long, 1'.^ feet wide at one end, and I'JJ at the other! Mean width, ilMiJ-i? == l'o5; and 1'55X 1<> = ''i4'H feet, An». * [\ ' I't'}. What i.s the numher of K|uare feet in a hoard 20 feet long, 2 feet wide at one end, and running to a point at the other \ Ans. 20 feet. How do you iind the contents of a straight edged board, when one end is wider than the other ? If the length he in feet, and the breadth in feet, in what *Ienomination will the product be 1 If the length he feet, and the breadth inches, what par^s of a font will be the product 1 15f». There is an oblong field, 40 rods long and 20 rods wide ; if a straight line be drawn frein one corner to the op- posite corner, it will be divided into two equal right-angled triangles ; what is the area of each ' Ans. 400 square rods = 2 acres 2 roodt. 157. What is the area of a triangle, of which the base is 30 rods, and the perpendicular 10 rods ? ^ns 150 rods. 158. If the area be 150 rods, and the base 30 rods, what is the perpendicular 1 Ans. 10 rods. 159. If the perpendicular be 10 rods, and the area 150 rods what is the base 1 Anf>. 30 rods. When the legs (the base and perpendicular) of aright-an- gled triangle are given, how do you find its area 1 When the area and one of the legs are given, how do you find the other leg? Note, Any triangle may be divided into two rig^^t-ar^gle4 triangles, by drawing a perpendicular from one corner to tho opposite side, as may be seen by the annexed figure. i im. {) rodM, ruu find you fin«' re wicUli 4 inches, w t\o you ionp, 1'^ board 'i(> a point at s. 20 feet, red board, t, in what at parts of »d 20 rods othoop- ht-angled 8 2 roods. he base is 150 rods. rods, what . 10 rods. area 150 . 30 rods. aright-an- ow do you IglU- angled rnerto tbo ;ure. 113. M I »<.'r I. L A jr r.ovn r.WMVvr.n. to A H C is a Irianijlr, (hvided ri:;lil annl«*t| triaii;:lis, A •/ / n (' ; tliiri«f(;rc, tin- whoh' base, A II, nuilliplic'l ly one half'ihv prrj)em!irnltir il (', will ^ivo thu area d A ot the whole. If A «=()() feet, and J 0=10 feet, what iii the area '. Ans JH) fert. 1()0. There is a trianpic, eat h siilr of wliirli is ll^ JVet ; what is the length of a perpeniiicular from one ans^li' lo its opposite side 1 and what is the area of the triiin;rl(i ! iV»/c. It is plain, the perpenchcniar will (livid'' tlio opju) site side into two r0. 165- If a wall be li>() biicks, = 100 f«et, in l("no;tb, and 4 bricks, =: 16 inches, in thickness, how many bricks will lay'one course ''. 2 courses ? 10 courses 1 If thfr wall be 48 courses, = H feet high, how manv brick'j will bijild it 1 150X 4 =600, and OOOX. 4s ='2H-'0(), Ans. 166. The river Po is 1000 feet broad, and 10 fort deep, and it runs at the rate of 4 miles an hour; in what time- will it discharge a cubic mile of water (reckoning otJuO feet to the mile) into the seal Ans. 26 days, 1 hour. 107. If tlie country, which supplies the river Po with Water, be 380 miles long, and 120 broad, and the whole land upon the surface of tho earth be 62,*(M),000 square miles, and if the quantity of water discharged by the rivers into the sea be every where proportional to the extent of land by which the rivers are uuppUed ; how many timcR Ui 350 MI«i('RLI. \!VROt'8 RXA>II*LCS. T 113. I grcalcr than llio Fo will tli<.' wludo amount nr the rivers be! And. I:i7'i tiiiioM. I(W. Tpon the same Miipposition, what quantiiy of wa- ter, a ltoK«' '>*•*'. will be discharged by all the rivem into the Nca ill a year, or 'Hl't days. Aris. l!K27v! cubic miles. UV.i. If the proportion uf the sea on the Kurface of the earth to that of land l)o as HI I to.>, and the mean depth of i.ie sea ho a quarter of a ujiie ; liow many years would it take, if the ocean were ompty, to till it by the rivers runnin;j;at the present rate 1 Ans. I70H years, 17 days, 12 hours. ITO. If a cubic foot of water weigh 1000 oz. avoirdu- pois, and the woij;ht of mercury be lti\ times (greater than of water, and the heip;iil of the mercury in the barometer (the weiG^ht of which is <»(|ual to the weight of a column of air on the same base, oxteiidinp; to the top uf the atmos- phere) be IJO inches ; what will be the weight of the air ypon a stpiare foot 1 '■ a stpiare mile ] and what will bo the whole weight of the atmosphere, supposing the size of the earth as in cpiestions 100 and 1(18.' Ans. 'J101)'875 lbs. weight on a square foot. o-J734:{75000 " " mile. 10-24iK)804G87.j0U00000 " " of the whole atmosphere. 171. If a circle be 1-1 feet in diameter, what is its oircuin- ference ? Note. It is found by calculation, that the circumference o( a circle measures about 3| times as much as its diameter, or, more accurately, in decimals, ;M 1151) times. Ans. 44 feet. 17*2. If a wheel measure 4 feet across from side to side, how many feet around it 1 Ans. 12f . 17Jj. If the diameter of a circular pond be 147 feet, what is its circumference? Ans. 462 feet. 174. What is the diameter of a circle, whose circumfer- ence is 402 feet 1 Ans. 147 feet. 175. If the distance through the centre of the earth, from side to side, be 7911 miles, how many miles around iti 7911X344159=2485:J square wiiles, nearly, Ans. 170. What is the area or contents of a circle, whose di- ameter is 7 feet, and its circumference 22 feet 1 Note. The area of a circle may be found by multiplying ^ the diameter into ^ the circuiuference. Ans. 334 sq. feet. 177. What is the area of a circle, whose circumference is 1 7C rods! . Ans. 2464 rods. .•^^■■■^ ..^ ' - 1-^ .it >kij ■ ■-■ - .■i*,^' -■- rivers be! iT'i timcH ly of wa- ll into the bic niilc!*. ICO of the depth of irs would the rivers VZ hours. ,. avoirdu- 2nter than barometer column of he atmos- of the air at will bo the size of ot. le. mospherc. ts oircum- nferenee of imeter, or, 44 feet. e to side, Ans. 12f . eet, vrhat 462 feet. rcumfer- 147 feet. earth, ound if^ rly, Ans. whose di- iltiplying 1 sq. feet, mference 464 rods. :i 111 ll.'i 17!-. h mSCELIiANrOlH F.XAMPMM. :U di ith contai ^51 » square square rod, what i% llto artu «>f this rirclr ? \()fi\ The diaiijcti r cf lli(.> rirrh- brinp: ' rod. tlit'cir fumferiMico will be .{ I I I.V.J, Ans. 'T.*^.'*! ol a >q. ro«l marly. llen<-c, if wo N(|(iari> the diainttfr < f any circli*, and mul- tiply the .'jqiiaro by '7S'>I, the j»ro(hict will be the area of the circlt'. 17H. What is the area of a circle whose diumc.Li- iy 10 rodsl l()-X*7.s"'>l=7rt'.')l. Ans. :8«.>4 rods. ISO. How many square inche" of leather will cover a l>all lil inchrs in dianx ter ] Note. The area of a };lobe or ball is I tinjcs as much as the area of a circle of the same diameter, ami may be found, therefore, by multiply int^ the whou. circu...ferem». ,nto tljc whole diameter. Ans. liSj^ squa.* nches*. 181. What i-s the number <>( scjuure miles on ik hurfaci- of the earth, supposing its diameter 7!)ll mIcs 1 791 1 X'^^^on=] ;>(>, 51-2,08Jl, Ans. 182. How many solid inches in a ball 7 inches in diam eterl Note. The ftol'ul contents ol a globe are found by mulli plying its area by ,1 part of its diameter. Ans. ^79j5 solid inches. 183. What is the number of cubic miles in the earth, supposing its diameter as above 1 Ans. 250,2;J:J,()31,4ll.j miles. 184. What is the capacity, in cubic inches, of a hollow globe 20 inches in diameter, and how much wine will it contain, 1 gallon being 231 cui : inches 1 Ans. 4188'8-f- cubic a ■ iies, and 18*13-|- gallons. 185. There is a round log, all the way of a bigness ; the areas of the circular ends of it are each i\ square feet; how- many solid feet does 1 foot in length of this log contaiti? 2 feet in length ? 3 feet 1 10 feet ? A solid of this forrn is called a Cylinder. How do you iind the solid content of a cylinder, when the area of one end, and the length are given ? 186. What is the solid content of a round stick, 20 feet long and 7 inches through, that is, the ends being 7 inches in diameter 1 Find the area of one end, as before taught, and multiply it by the length. ., * > , Ans. 5'347-f- cubic feet. /:^' #f •**««, / •^52 MISCELLANKOUS KXVMPI.ES.' IT 113. If you nmlliply s(|u;ire inches by inrhet in lrnp;th, what parts of m foot will Iho pioduct he 1 if square indies by feet in len{!;tl», wliat parti 1S7. A \Viiich(;sl(r Inishel is IH'5 inches in diameter, and H inches deep; how many cubic inches does it contain? Ans. '2ir,0' l-f. U is pi.\in, from the above, tliat the solid content of all bodies, which are of uniform bif^ness tliroughout, whatever may bo the form of the ends, is found by mulliphjins the area (fonc end into its height or length. Solids wliicli decrease tenths, of the difference between the head and bung diame- ters, to the head diameter. The cask will then be reduced to a cylinder. >^ ^- Now, if the square of the mean diameter be multiplied by *78o4, (ex. 177,) the product will be tlae area of one end, and that, multiplied by the length, in inches, will give the solid content, in cubic inches, (ex. 185,) which, divided by ^1, (note to table, wine meas.) will give the content in — •"» nrtllons, and, divided by 282, (note to table, beer meas.> i the content in ale or beer measure. k process we see, that the square of the mean diam- \ f .' ^ 113. til, what liameter, contain? l.-SO'l-f. :nt of all tvhatever lying the till they " the base r-ianglf, a or a cone, •rtex, and le base, is by muUi- \dir height. hose base ct? IS 48 feet, nd whoso }46^ feet. 2.5 inches, () inches ; ow >many found by urving, Dg diarae- »e reduced Itiplied by one end, 1 give the divided by content in >eerineas.^ lean diam- ir 113. MISCEIXANTOUS FXAMPLF.S. 249 eter will be multiphed by '78.'>4, and divided, for wine gal Ions, by 231 . Hence we may contract the operation by only multiplying by their quotient, { ''.f^'>^z= '0034 ;) that is by '0034, (or by 34, pointing off" 4 figures from the product for decimals.) For the same reason wc may, for beer gal- lons, multiply by ('r^^./= •OO'ivS, nearly.) «(>0*^H, &c. Hence this concise Rv lk, fvr guaging ormeasuring caska, Multiply the square of the mean diameter by the length; mul- tiply thi in. the head diameter = (i in. difference, and | of = 4 inches ; 25 in.-j-4 in.=29 in. mean diameter. Then, 292 = ,S41, and 841 X36 in- = 3027(5. -,, c 30276X34 = 1029384. Ans. 1029384 wine gals, men, j 3070(5x28 = 847728. Ans. 84'7728 beer gals. 191. How many wine gallons in a cask whose bung di- ameter is 36 inches, head diameter 27 inches, and length 45 inches 1 > Ans. 166'617. 192. There is a lever 10 feet long, and i\\c fulcrum, or prop, on which it turns is 2 feet from one end ; how many pounds weight at the end, 2 feet from tlie prop, will be bal- anced by a power of 42 pounds at the other end, 8 feet from the prop 1 Note. In turning around the prop, the end of the lever 8 feet from the prop will evidently pass over a space of 8 inches, while the end 2 feet from the prop passes over a space of 2 inches. Now it is a fundamental principle in mechanics, that the weight and power will exactly balance each other, when they are inversely as the spaces they pass over^ Hence, in this example, 2 pounds, 8 feet from the prop, will balance S pounds 2 feet from the prop ; therefore, if we divide the distance of i.He power from the prop by the distance of the weight from the prop, the quotient will al- ways express the ratio of the weight to the power ; |=4, that is the weight will be 4 times as much as the power, 42x4= 168. ..,, . ,,. , Ans. 168 lbs. 193. Supposing the lever as above, what power would it require to raise 1000 pounds ? Ans. » <^o=250 \h§. 194. If the weight to be raised be 5 times as much as the power to be applied, and the distance of the weight fromlht ^ii^ ^» MISCRLLANBOUB EXAMPLSS. IT 113. 1% prop be 4 feet, how far from the prop must the power be ap- plied ? Ads. 20 feet. 195. If the greater distance be 40 feet, and the less ^ of a foot, and the power 175 poundH, what is the toeight ? Ans. 14000 pounds. 190. Two men carry a kettle, weighing '200 pounds ; the kettle is suspended on a pole, the bale being 2 feet 6 inches from the hands of one, and 3 feet 4 inches from the hands of the other ; how many pounds does each bear. Ans. 114f lbs. and85|lb8. 197. There is a windlass, the wheel of which is 60 inch- es in diameter, and the axis, around which the rope coils, is 6 inches in diameter ; how many pounds on the axle will be balanced by 240 pounds at the wheel 1 Note. The spaces passed over are evidently as the diam- ters, or the circumferences ; therefore, ^ = 10, ratio. ^ns. 2400 pounds. 198. If the diameter of the wheel be 60 inches, what must be the diameter of the axle, that the ratio of the weight to the power may be 10 to 1 1 Ans. 6 inches. Note. This calculation is on the supposition, that there is no friction, for which it is usual to add ^ to the power which is to work the machine. 199. There is a screw, whose threads are I inch asunder, which is turned by a lever 5 feet, = 60 inches, long ; what is the ratio of the weight to the power 1 Note. The power applied at the end of the lever will de- scribe the circumference of a circle 60X^2=120 inches in diameter, while the weight is raised 1 inch ; therefore, the ratio will be found by dividing the drcwnference of a cirdey rohose diameter is twice the length of the lever, by the distanet between the threads of the screw. 120X3^ ==3774- circumfer- enec, and 377|^ 377|, ratio, Answer. 200. There is a screw, whose threads are | of an inch a- sunder ; if it be turned by a lever 10 feet long, what weight will be balanced by 120 pounds power 1 Ans. 30171 lbs . 201 . There is a machine, in which the power moves over 10 feet, while the weight is raised 1 inch ; what is the power of that machine, that is, what is the ratio of the weight to the power 1 Ans. 120. 202. A rough stone was put into a vessel, whose capacity »*.; ir 113. ir b«ap- 20 feet, less ^ of t? pounds, ids ; the 6 inches te hands 185^ lbs. 60 inch- 3 coils, is le will be the diam- tio. ) pounds, es, what le weight 6 inches, hat there he power I asunder, ig; what r will de- inehet in sfore, the ' a cirel»t \a distanct sircumfer- n inch a- lat weight )0171 lbs . oves orer the power ight to the Ans. 120. capacity roRMaOF N0TB9 A«0 RECKIPTl. mi was 14 wine quarts, which was afterwards filled with 2^ i\{n of water ; what was the cubic content of the stone 1 ,j- ,, Ans. Ot)4| inches. T-><^rf- '/.,''■ %iii'fi FORMS OF NOTES, RECEIPTS, ORDERS \SD BILLS OF PARCELS. 1&:'- , ^1^:' ^<0{t^Q^ No. I. Montreal, 0« t. *22, IKW. For value received, I promise to pay to Oliver Bountiful, or order, two pounds, ten shillings and sixpence, on demand, with interest. William Trustv jfttest, Timothy Testimony. No. III. '' Kingslon, October 10, I8*.V.1, For value received of A B, in goods, wares and met chandize, this day sold and delivered, I promise to pay him, or bearer, pounds, shillings and pence, in ten days from date, with interest. C D . No. III. . ' "* By two persona. Stanstcad, October 1, 1^33. For value received, of , in this day sold and delivered, we jointly and severally promise to pay him, or order^ pounds, shillings and pence in days from date, with interest B C , D- MB^T' ^ s Montreal, October, 10, 1833. Received from Mr. Durance Adley ten pounds, in full oi all accounts. Oavand Constancy . Receipt for mormf received on a Note. ' "• York, November 1, iS'Xi. Received of Mr. Simon Eastly (by the hand of Titus Trus- ty ,) sixteen pounds ten shillings and six pence, which is en* dorssd on his note of June 3, 1^1. X 2 Samsov Sirow.. %2 ORDERS, AND BILLS OF PARCELS. ^ Rtceipt for money rtceived on ufecount. Stanitead, June 3, 1833. Receired of Thomas Dubois twenty pounds, on account, Orlando Prompt. Rictipt for money rtceived for another person. Sherbrooke, June 4, 1833. Received from P. D. twenty-five pounds for account of J. T. Eli Truman. •M ,J'»V Receipt for interest due on a Note. Quebec, December 12, 1839. Received of I. S. fifteen pounds, in full of one year's in- terest of 250<£, due to me on the day of last, on note from the said I. S. , Solomon Grat. , . "v ,' Receipt for Money paid before it becomes due. ■^ • Proscol, May 3, 1833. Received of T. Z. fifteen pounds, advanced in full for ona year's ren; of my farm, leased to the said T. Z. ending th« first day of April next, 1834. John Honor vs. pnee calle mi 8 3 hofs 2 pipes 1 hogsl 3 casks 5 bags 1 chest d^mwamQ^ ».v Belviile, November 3, 1833. Mr. Stephen Girard. For value received, pay to A B, or order, five pounds and six shillings, and place the same to my account. Saul Ma^tn. Montreal, September 1, 1830. Mr. Timothy Titus. Please to deliver to Mr. L. D. such goods as he may call for, to the amount of seven pounds, and place the same to the account of ,1 our obedient servant, ._^,, Nicanor Linus. '1 ji'l'. Vi BLLS OF PARCELS. V < ■ - It is usual when goods are sold for the seller to deliver to the buyer, with the goods, a bill of the articles, and their 5682 feel 2000 '* 800 " 1500 " 050 " 879 " 236 " » f BILLS or PARCELS. '^53 ». D. leven bdient ret to I their priMi, with the amount cast up. Such biils are sometimes caliad bills of parcels. Montreal, 6th May, IWW. Mr. Abel Atlas. Bought of BcNj. lic'CK. -' • '■ .t*. 8. d. 12^ yds. figured Satin, at 12s. 6d. per yd. 7 • 10 • 3 8 " Sprigged Tabby, - 6s. 3d. - - 2 • 10 3 . Received payment, JL'IO- Bcprj. Buck. 6- a - Montreal, 14th May. 1833. Mr. JoHw RiTRToif, Bought of Georgr Williams, 3 hogsheads new Rum, 118 gal. each, at Is. 6d. per gal. 2 pipes French Brandy, 126 & 132 gal. " Ss. 7d. 1 hogshead brown Sugar, 9^ cwt. " 2j& lis. 9d. ewt. 3 casks of Rice, 269lfo. each, <« 3d. tb. 5 bags Coffee, 751b. each, «« Is. 2d. " 1 chest hyson Tea, 861b. ^ »« 48. 8d. "^ Received payment, lUi^ 68. 7£d. For Gborok Williai^s, Thomas Rousseau. Wilderness, 8th Feb. 1833. Mr. SiMOir Johnson, Bought of Asa Fullvm. 5682 feet Boards, at 1£ lOs. per M. 2000 •* •' " 2£ Is. 8d. " 800 '* Stuff, "33-2 1500 " Lathing, " 1 • 10- 12-6 236 " «' • ** 13 • 9 660 •* Plank, " 1 879 " Timber, " <( (C IC (( 18<£ 8s. fd. "W £ ii'^ m- !< O s c — o s > •^ ' ccountf several n o *-?. 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