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Las diagrammes suivants illustrent la mAthode. irrata to pelure. 3 32X 1 2 3 1 2 3 4 5 6 *l^ 70 k ti' ■-fi^i r . / .'cHa ■V,-- ADAmS' NEW ARITHMETIC, SUITED TO HALIFAX CURRENCY; ':"\„'^ '•'■!;*-."" IN WHICH THt -' PRINCIPLES OF OPERATING BY NUMBERS v«, ■' , . - ► •-it*- ■■/P. ANALITICALLY EXPLAINED, AND SYNTHETICALLY APPLIED ') THUS. COMBINING THE ADVANTAGES TO BE DERIVED BOTH FROM 'THE INDUCTIVE AND SYNTHETIC MODE OF INSTRUCTING : rHE WHOLE MADE FAMILIAR BY A GREAT VARIETY OF USEFUL AND INTERESTING EXAMPLES, CALCULATED AT ONCE TO ENGAGE THE PUPIL IN THE STUDY, AND TO GIVE HIM A FULL KNOWLEDGK OF FIGURES IN THEIR APPLICATION TO ALL THE > ' PRACTICAL PURPOSES OF LIFE. ^ ^!'- :• .1' -^ '■•'>■• ^ . - V ■ » '• 1 . -^ DESIGNED FOR THE USE OF SCHOOLS & ACADEMIES IN THE BRITISH PROVINCES. .f', •'■ *:^-^-« '< BY DANIEL, ADAMS, M. D. ■ ■ -^r :. ..■^■■ SHERBROOKE,C..E. PUBLISHED BY WILLIAM BROOKS. .A V*N-i'' ,- U:>^ft^A ^• .-^^^t. idrnmi"^: '-"■■ ■,:'*'■:. «■- ?w PRINTED BY J. S, WALTON, SHERlQnoOKE, CANADA EAST» •WWWB^ ^%%^- a; t&0(. :^0 «^" •:^ '^ .';^?"'f ^t ;^'« Wr, te'C- '-? ? HCiencQ i« -^^ ■iv.i .-(^^ KtO I Ar;r 4* fi-rf^i' HU 1^ it i: j^ A € i: 4 Therm afe two xnehods of teach ing-: the synthetic, and th6 ^nalytici In the synthetic methud,the pupil is tirst presented urith a general view of the science he is siudyins. and after-^ wards with the particulars of vrhich it consists. The analytic method reverses this order t the pupil is first presented with the particulars, from which he is lea, by certain natural and easy gradations, to those views \^'hieh are more general and conv;;^ prehensive. The Scholar's Arithmetic pilblished in 1801, is synthetic. If that is a faolt of the work, it is a fault of the times in which it appeared. The analytic or inducftive ' method of teaching, as how applied to elementary instruction^ is among the improve*' ments of later years^. Its introduction is ascribed to Pcstaloz" zi, a distinguished teacher in Switzerland. It has been applied v to arithmetic, with gre^t ingenuity, by^ Mh CotBOmi, in our own country. The analytic is unquestionably the be^ method of acquiring J^nowledge ; the Synthetic is the belt method of recapitulating or reviewing it. In a treatise designed for school education, both methoila are useful. Such is the plan of the present un- dertaking which the author, occupied as he Is wim other ob- jects and pursuits, would \villingly have forborne, but that, the demand fofr the Scholar's Arithmetic still continuing,* an obli" gation, incurred by long-couthiued and extended patronage, did not allow him to tiecline the labor of a revisal, which should adapt itto the present lAore enlightened views of teaching this science in our schools. In doing this, however, it has been necessary to make it a ne«^ work* ' » In the exeCutW of this design, an analysis of each rule is iii-st given, containing a familiar explanation of its various principles ; after which follows a synthesis of these principles, with questions in form of a supplement. Nothing is taught dogmatically i no technical term is used till it h fined, nor any principle inculcated without a previous develope* ment of its trdth ; and the pupil ismade to understand the xe^ 6on of each process as he proceedst The examples under each rule are mostly of a practical na- ture, beginning with those that are Very easy, apd gradually advancing to those more difficult, till one is introduced cob- taining larger numbers, and which is not easily solved in the •knind>; then in a plain, familiat manner, the pupil is showa 4 ■" PREFACE. how thB solution may be facilitated bv figures. In this way he iff made to see at onoe their use anu their application. At the close of the fundamental rules, it has been thought advisable to collect into one clear, view the distinguishing prop- erties of those rules, and tomve a number of examples involv- ing one or more of them. These exercises will prepare the pupil more readily to understand the application of these to the succeeding rules ; and besides, will serve to interest him in the science, smoe he will find himself able, by the application of I a very few principles, to solve many curious^uestions. The arrangement of the subjects is that, which to the author has appeared most natural. Fractions, haye received all that consideration which their importance demands. The^ princi- ples of a rule ci^Ued Practice are exhibited, but its detail of cases omitted, as unnecessarV, since the adoption and general use of federal money. The Kule of Three^ or Proportion, is re- tained «nd the solution of questions invdlving th^ principles of proportion, by analysis, is distinctly shown. The articles Alligation, Arithmetical and Geometrical Pro- gression^ Atmuities and Permutation, were prepared by Mr. Ira YouNO, a member of Dartmouth College, from whose knowledge of the subject, and experience in teaching, I have derived im- portant aid in other parts o{^ the work.' f The numerical paragraphs are chiefly for the purpose of ref- erence ; these references the pupil should not be allowed to neglect. His attention also ought to be particularly directed, bv his instructor, to the illustration of each particular prinoi- ple, from which general rules are deduced j for this purpose, recitations by classes ought to be instituted in every school where arithmetic is taught. ' 11 le supplements to the rules, and the geometrical demon- strations of the extraction of the square and cube roots, are the only traitSnOf the old work preserved in the new. DANIEL ADAMS. ',. ?u ;.iT y-" ; J1 •■11' J ,. '• PUBLISHER'S PREFACE. The author of the following practical treatise upon [Arithmetic, has made himself favourably known in the United States, and to a considerable extent in the Canadas, I for a great number of years, by his works, designed for the use of Academies and primary schools. The " Scholars' Arithmetic," published in the year l801, continued in al- most universal use, until within a very short time past. — JButjuster views beginning to prevail, and sounder princi- Iples becoming established in the public mind, apon the sub- iject of elementary education, a revision of the work seem- led necessary. At this time, " Adams' New Arithmetic," I was published. This seems evidently to have been pre- Ipared with much care. The author has recognised in it Ithroughout, this .important kw in relation to the mind, that Jit must first be made acquainted with particular facts, or [there will be no ability to arrive at correct general conclu-* Isioas. Particular examples are therefore given upon eaeh jsubject, and from them, in a manner obvious to the ypung lind, all the general rules, are deduced. In other wQrds, the author has care^lly and prudently pursued, in his book, rhat is called the antUytic method. ' 'The care used in cle- ining necessary terms, which might not be quite clear, the )rdctical character of the examples given under each rule, the methodical disposition of the different parts of each subject, and of the different subjects, the general per- spicuity, simplicity and accuracy of the work, render it '.n^ |valuable to the pupil. It is due the author to observe, that ^Adams' New arithmetic," for its adaptation to the capacities of young ind ordinary minds, is justly considered the best practical [reatise which has been offered to the public. * In the present edition, the main purpose in view was to idapt Adams' work to the clirrency of the British Provin- ces. No separate article, as in the original, has been allott- d to Federal Money ; for this the pupil has been referred ^o Decimal Fractions, in which also almost all tbe exam- )les will be found in the money of the United States. Ad- litional examples in the compound rules have been given, A2 ^ l^tiBiisltit^A ^Util^ilbK: and the old ones r^ained, under the titl^ of tialifrft elih Irency ; and generally throUffhout the book, where denom-' initiotiit of tttoney oceurj IlalifaX currency has been sub< «tituted fof Federal money. The tules and eitamples in R,eduction of CutteMiea have bUen es^ntially changed j tod in Reduction, after the Table of £ngli9h Mon^y. which is called the Table of Halifax (JUrreney, A list of the Gold and Silver Coins cur- rent in the I^rovinc^, has been inserted. This may be dc ][)ended upon as entii'ely Accurate. The tables of f^rench, and Dry, Loilg, Square, ftnd Solid Measure, have been giv<< en-^and wh&t a^e the weights and measures establisheaby Uw in this Province i? also stated. Th6 most novel feature in the book will be found in the (ui^df Interest, (jertainly an innovation, but it is believed, an intprovement, has be6n made. The pounds in any giv^ en sum upon which interest is to be cast, are left to stand Bi the nnits^ i^nd the shillings and pence are reduced to decimal paHs ot a pounds The interest is then obtained the same as in. I^^deral Money, and the decimal parts in the tesult teducdd to shillings and penca It is considered that this method is ttior^ simple and concise, and will b^ found in practice to be more conveilient than any othet.^~ ijut setting asi^e considerations of temporary eonvenience^ if thi^ change and attempted amelioration, shall assist in Mttit ^ery jsHght degree in turning men's minds toward the Decinial AatiPi ana ihducing them to look forward to a period when all the denominations of money, weights and measures, throughout the world, shall be expressed in dec- hiAT.s, it cannot be affirmed that no benefit has been obtained. The importance of the principal and essential alteration \A we book, viz ; the adaptation of it to the ('utrency o'f I the cdunt!fy» will not fail to be observed by et^ery one* It I is indeed singular, that hitherto, no Canadian Arithmetic in the tenglish language, has been published* Mercantilr, HgricuitUfal, and gerterallvthe business men of the country, \yill be aWaTe of a benefit to Ije realized, and it is consider- ed that something also bearing a relation to political advan:| Jtage, maybe in the result ^ ;. ; , ^hcrkraakc, L, C, June 6^ 1849, I -I .<,.,.-5f i^^ ♦-..'•v 1, <•:.'« '*s I, ^ index; ■•» iini • »7r I*' . .yjitioh, i. - * Alliffation, - ■* Arithmetical Projrression, '■ Compound Numbert, • Addition of, I/' ..-f*. -Subtraction ofj * * -Multiplication audi Division of, > 194 '* • 58 80 85 89 154 59 213 '^ 39 • * 199 * 173 204 - 64 * 98 - 100 C'Om mission, Coins, Table of, - * Cube Root, Extractten of, Division, * ^ -» Duodecimals, * * -^-^ *- — - — Multiplication of, Kquation of PuymentSj - Evolution, ".'''' Federal iVloncy, - » ^'ructionfi, - « a Proper and Improper, " . " •* •* !f(i\ i' To change an Improper Fraction to a whole or '<»t"- TM '^■'■^ mixed number, - - - 100 To reduce a whole or mixed numbel' to an Im- proper Fraction, i. - ^ 101 TiJ redude a Fraction to its lowest terms, - 102 To divide a Fraction bv a Whole humbet) '^-^ *• 105 107 109 no 112 114 116 121 128 ( ,'•( ■■PI To multiply " «' " ii J.^ whole niimber by a Fraction, - — ^*-^ ^one Fractioti by another, - T'j divide a whole number bv a Friuclioti, ^ — ■■ — ^♦-^one fraction by anptiier. Addition and Subtj-aclion of Fractions, ){eduction of Decimal Fractions, a -*• Addition and Subtraction of, 132 —Multiplication of, - ^Division of. « * * 135 137 ■ -^*^* 143 llrfduciion of Vulgar Fractions to Decimals, 140 1*eno\vship, - - - - - - 190 ilalifax. Currency, * ^ * * - $3 -Reduction of. 8 INOBX. Insurance, - > "^ ' Interest, - - . l-Compound, - Involution, - • ,j(t,r> Multiplication, Simple, * Numeration, - - - Proportion, or the Rule of Three, Compound, - IM 151 - 166 203 - 28 9 - 176 184 r 234 58 Permutation, Reduction, Tables of Money, Weights, Measures, &c. 59—78 ——of Currencies, - - - - 149 Ratio of the Relation of Numbers, - - - 174 Subtraction, Simple, - ^ ^uaie Root, Extraction of, ■ iv^ 1 «■- V < :f''' • v^:/ », -■ ,-.. 21 205' 44 f % ^1 .r I : MISCELLANEOUS EXAMPLES. .;'^■ •■1' 1 , Baiter,ex. 20— 31. '^ | Position, ex. 88— 107. I'o dnd the area of a Square or Parallelogram, ex. 147 — 153. — a triangle, ex. 154 — 158. Having the Diameter of a Circle, to find the Circumference ; or having the Circumference, to find the Diameter, ex. 170 — 174. To find the Area of a Circle, ex. 175 — 178. .. ^ a Globe, ex. 179, 180. 'i'o rind the Solid contents of a Globe, ex. 181, 183. -— * Cylinder, ex. 184 — 186. . Pyramid, or Cone, ex. 187, 188. . any Irregular Brtly, ex. 201. Guaging, ex. 189, 190. j Mechanical Powers, ex. 191—200. ;•:• 1- <■ ^I't^/'t, '■■-. • r - ■ -■■■ .-. Forms of Notes, Receipts, Orders an J Bills of Parcels, page 260. Book Keepinij, - - ^- - - 261. I I NUMERATION. . ^ f 1. A BiNOLK or individual thinj; is called a unit, unity or one ; one and one more are called two ; two and one more are called three , three and one more are called four ; four and one more are called five ; five and one more are called six ; six and one more are called seven ; seven and one more are called eight ; eight and one more are called nine ; nine and one more are called ten, 6lc. These terms, which are expressions for quantities, are called numbers. There are two methods of expressing numbers shorter than writing them out in words ; one called the Roman method by letters,* and the other the Arabic method by figures. The latter is that in gjeneral use. ^n the Arabic method, the nine first numbers have each an appropriate char/icter to represent them. Thus, *In the Roman method by letiert, I repreientii one, \ Jive, X ten, L fifty, C one hundred, D ftbe hundr^, and M one thoueand. Aa onen aa any letter ia repeated, ao many timea ia ita value repented, unleaa H be a letter represenlini; a lesa number, placed before one rep- reaenting a greuter , then, the ieaa number is taken from the greater, thus| IV ropreaenU four, IX nine, Acfia will be aeen in the follow- ing TABLE :— Ninety One hundred Two hundred Three hundred Four hundred Five hundred Six hundred Snven hundred Eight hundred Nine hundred One thousand Five Thousand XXXX.or XL Ten thousond L Fifty thousand LX Hundred thousand LXX One million LXXX Two millions *IC> is used instead of D to represent five hundred, and for every ad- ditional Q annexed at the righ hand, the number is increased ten times. fClQ ia used to represent one thousand, and for every C and q put at each endj the number is increased ten timea. I A line drawn over any number increases its value t thousand ti*es. One Two Three Four Five Six Seven Eighi Nine Ten Twenty Thirty Forty Fifty Sixty Seventy Eighty I II III nil, or IV V .,„.,.., . VI VII VIII Vim, or IX X XX XXX LXXXX, or XC C CO >i;'.J'/'vv CCC ;''.'i'' .;V: cccc 1 D, orlD* f DC iy >. DOC iA DCCC ■*'' DCCCC .1 M, or Clot > t ' lOD. or Vt_ CCIo3,orX 1 000 ■%'• ■ CCCIooa, or C ' M M M ^S 10 NtJifEitAtloKi fli A tiAii^ unity f or ontt is tepteaenttA by this chatactei^j 1 Fiv4 Six ■iieoeA Eight Nine \ ■%>> B 1 •''^=U. u '"fa Ik •»; u ■ b ■*'■'' 4 Ten has "ho a{iipro^iate chai^abter ^o i^^^t ft t ^*^ is consider fid as forming a u&it of a second of higher order, consisting of tens, represented by the same character (I) as a unit of the first br lowe^ order, but is written in the second place ffom the tight hand, that is, on the left hand side of Units ; and as, in this caLe^ there are no units to be written with it, we write in the place of units, a cipher, 0, ^ff^'Whichof itself sjgnifies 'nothing; thus, Ten One ten and one iXnit a^e called ^r^^x^Mi^i ^Eleven two " '•.,- ,/M>y'im0i' three " :#»-#'?.^0.i^'p\rl^ Seventy :v,:^vi;^»'**||^. . ■■■■ ^ Fighti/ ^* Ninety Ten tens aife called a hundred. Which, fofmS a Unit of a still higher order, consisting of hundileds, represented by the same thaifacter (1) as a unit of ea«;hof the ibriftgoing orders, but is written one place further toward the left hand, that is on the left hand side of tens ; thus, - - ^ one hundred • 100. One hundred, one ten and one unit, arei called One hundred and eleven 111. ■<*•■•. ^■■■•■■■:S m NUMBHATiOIf, II «f, 1 .'>-v' . 2 J.- rf a - i i. m^ 5. ;^it*5^T 6. i^^ll^i! '7. m/^-n 8. a 3utis igher same irder, tight « ; and Titten ler, 0, fi "' io. n IL at 12. teeii 13. 'teen 14. 'trt 15^^ .en 16; aeeh 17. iteen la. t tecri 19> nty 20; t!f 30. 40. 60. ; ' 60. ntt/ 70. it^ 80. Bty 90> it of a still y the same iers» but is ■it is on the 1 M .100 yen ^ 9l ' There are three hiindre4 sixty-Ave days in a year, 1(1 this number are contained all the orders now described^ viz. units, tens, and hundreds. Let it be recollected, units occupy the first place on the right hand ; tens^ the second place from the right hand ; hundreds^ the third place, ^ This number may now be decomposed^ that is, separated into parts, exhibiting each order by itself, as fbllows i-^The highest or.> der, or hundreds, are tkreCy represented by this character, 3 ; but, that it may be made to occupy the third place, counti. ing from the right hand, it mustbe followed by Jtwo ciphers^ thus, 300, (three hundred.) The next lower order, or tens^ are six, (six tens are sixty,) represented by this character, 6; but, that it .may occupy the second place, which is the place of tens, it must be followed by one cipher, thus 60, (sixty.) The lowest order, or MM2^s, are five, re{^resented by a single character, thus, 5, (five.) We may now combine all these p^irts together, first writ-t ing down the five units for the right hand figure, thus, 5 ; then the six tens (60) on the left hand of the units, thus 65 ^ then the three hundreds (300)' on the left hand of the sij; tens, thus, 365, which number, so written, may be read three hundred, six tens, and five units ; or, as is more usual, three hundred and sixty-five. i. 4 :rf. ^ « f tl 3. Hence it appears, that figures have % different value according to the place they occupy, counting fron^ the right hand towards the left, V M^vi-> ''\]U3' A\v:}^ i^^l-*!^ m-i- s'!^'^ Ill Take for example the number 3 3 3, made by the same figure three times repeated. The 3 on the right hi^nd, or in i\vei first place, signifies 3 units ; the same figure, in the sec- ond place, signifi^ 3 tens^ or thirty ; its value is now in-> creased ten times. Again, the same figure in the ^Airc? pi ace, signifies neither 3 units^ nor 3 tens, but 3 hundreds, which is ten times the value of the same figure in the place immen diately preceding, that is, in the place of tens ; ^nd this ia a fundamental law in notation, that a removal of one plcu^e towards the left increases the value of a figure ten times. Ten hundred make a thousand, or a unit of the fourth ormn^- 6. Forty-nine hundred and sixty. •:,--- 7. Ninety-two thousand and forty-five, "^^ryi^ ts^oirt: '; 8. One hundred thousand, vw^-^fc^^' .'^•;J'te^'V!.'?.• •''^ 9. Two millions, eighty thousands, and seven hundreds. 10. One hundred millions, one hundred thousand, one hundred and one. 11. Fifty-two millions, six thousand, and twenty. 13. 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 billions, and seven millions. Addition of Siiii^ci UTtiinbers. 51 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 ? 3. Peter bought a wagon for lO shillings, and sold it so as to gain 4 shillings ; how many shillings did he get for it ? 4. Frank gave 15 walnuts to one boy, 8 to another, and had 7 left ; how many walnuts had he at first ? 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 ? : 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? 7. Samuel bought an orange for four pence, and some walnuts for three pence ; then he bought a knife for 1 shil- ling, and a book for 4 shilling ; how many shillings did he spend, and how many pence ? If 4. ADDITION OF SIMPLK NUMBERS). 15 8. A man had 3 c^ves worth 10 shillings each, 4 calves worth 15 shillings each, and 7 calves worjlh 2 pounds each; how may calves had het ■i'^>^ f (?* tv^r ,.i-^' ? . ., m, -^ l 9. A man sold a cow for 4 pounds, some corn for 5 poands, wheat for 7 pounds, and butter for 2 pounds; how many pounds must he receive t The putting together two or more numbers, (as W the foregoing examples,) so as to make one wJiole number, is called Addition, and the whole number is called the sum, or amount. ^- ' ' ' - — •■ - •- •-' ■■■■■• ; ^.'i. --.>;: r<^ , 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 ? 11. What is the amount of 3, 7, 2, 4, 8, and 9 pounds? 12. In a certain school 9 study grammar, 15 study arith- metic, 20 attend to writing, and 12 study geography ; what is the whole number. of scholars? . , , SiG^s. A cross, +, one line horizontal and the other per- pendicular, is the sign of addition. It shows that numbers, with this sign between them, are to be added together. It is sometimes resLdj^lus; which is a Latin word, signifying more. : .... ;. „-.^ ■■. /. Two parallell, horizontal lines, =, are the sign of equality. It signifies that the number before it is equal to the number after it. Thus, 5-f-3=8 is read 5 and 3 are 8 ; or, 5 plug (that is, more) 3 is equal to 8. -../// •^^ n In this mani following 2-f0= 2 2+1= 3 2+2= 4 2+3= 5 2+4= 6 2+5— 7 2+6= 8 2+7= 9 2+8= 10 2+9= 11 ICC I.11C ^u^ix uc mail uuic ADDITION TABLE. 3+0= 3 4+0= 4 5+0= 6 ,, ' '"t 3+1= 4 4+1= 5 5+1= a ; r 3+2= 5 4+2— 6 5+2— 7 - .'.y 3+3= 6 4+3= 7 5+3= 8 .,, ,|. 3+4= 7 4+4= 8 5+4= 9 .. 3+5— 8 4+5= 9 5+5= 10 ^.'.ift 3+6= 9 4+6=10 5+6= 11 3+7= 10 4+7= 11 5+7= 19 , ■y- 3+8— 11 4+8— 12 5+8=13 '-•/, 3+9= 12 4+9= 13 5+9b=14 ■■■'-'■.<' II 5. ADDITION OF SIMPLE NUMBERS. 16 ^ 6-f-0= 6 6+1= 7 64-2= 8 . 6-1-3= 9 6+4=10 , 6H-6= 11 , 6+6= 12 , 6+7=13 6+8=14 6-1-9= 15 7+0= 7 7+1= 8 7-f-2= 9 7+3=10 7+4=11 7+5= 12 7+6= 13 7+7= 14 7+8= 15 7-f-9=16 8- 8- 8- 8- 8- 8- 8- 8- 8- 8- -0: -1: -2: -3- -4: ■5: ■6: -7= •8= 9= 8 9 10 11 12 13 14 15 16 17 9+0= 9 9+1= 10 9+2= 11 9+3= 12 9+4= 13 9+6= 14 94-6= 15 9+7= 16 9+8= 17 9-j-9=18 6+9=: how many ? 8-1-7= how many ? ii_i_aj_Q — ' .m^-f' 4 ^ . . 8-1-7= how many ? J 4-j-3+2= how many? 6+4+5=: how many ? 2-f-0-j-4+6= how many? 7-f-8+0+8= how many? 9+3+3+4=: how many? • ' 8-j-2-f-8-f-3+5= how many? ;" 54.7+6+1+8= how many? *'^- 8-f_9_[.7+0+5+6= how many ' 4+1+0+4+4+5= how many 2-|-5-|-2-f-3-f-7-[-3= how many A ■ I .. i. ' ' * . s- • > " <' ■ 51 S. When the numbers to be added are small, the ad- dition 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 bopks in his little library, his father gave him 25 volumes more; how many volumes had he then ? One of these numbers contains 4 tens and 3 units. The other number contains 2 tens and 5 units. To unite these two numbers together into one, write them down one under the other, placing the units of one number directly under units of the other, and the tens of one number directly un- der tens of the other, thus : ' 43 volumes. Having written the numbers in this 25 volumes, manner, draw a line underneath. ■^-'■■■■■^'f- ■ ADDITION OF SIMPLE NUMBERS. ilM 43 volumes, 25 volumcSy 8 it''*.; 4;3 volumes, We thon begin at the right hand, and add the 5 units of the lower number to the 3 units of the upper number, making ; 8 units, which we set down in unit's place. We then proceed to the next column, and add the 'Z tens of the lower number 25 volumes, to the 4 tens of the upper number, mak- — ing 6 tens, or 60, which we set down in [j'is.68 volumes, ten's place, and the work is done. It now appears that Harry's whole number of volumes is |0 tens and 8 units, or 68 volumes ; that is, 43-4-25=68.. ■^f? . 14. A gentleman bought a carriage for 214 pounds, a Ihorse for 30 pounds, and a saddle for 4 pounds ; what was It lie whole amount? Write the numbers as before directed, with units uncler units, tens under tens. &/C. . ; ; »ition ? G. What is a fundamental law in notation 1 7. What is addition ? 8. What is the rule for addition 1 9. What is the result, or number kought, called ? 10. VVhat is the sign of addition ? 11. ——of equality 7 12. How is addition proved '? .*«*>-^f' "" ' EXERCISES. t. Washington was born in the year of our Lord 1 /32 ; he was 67 years old when he died ; in what year of our Lord did he die ? 2. The invasion of Greece by Xerxes, took place 481 years before Christ ; how long ago is that this current year 1849? 3. There are two numbers, the less number is 8671, the difference between the numlsers is 597 ; what is tlie 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 ? . 6, 6. SUBTRACTION OF ilMPLE NUMBES. 1 8 :j'i (5 I 3 2 r. () 7 9 3 2 670 3 1 457 8-i ^G pounds? 3re are Hi) Baches ; in lerries : in nd^^o bear 6. There are four numbers, the first 317, the second 812, ihe third 1350, and the fourth as much as the other three; ^hat is the sum of them all ? 6. A gentleman lefl his daughter 16 thousand, 16 hun* Ired and 16 pounds ; he left his son 1800 more than his laughter ; what was his son's portion, and what was the lount of the whole estate ? Ans. i Son's portion,19,416. ) Whole estate,37,032. 7. A man, at his death, left his estate to his four chil< iren, who, after paying debts to the amount of 1476 pounds, Received 4768 pounds each ; how much was the whole es> ite? ^«5. 20548. 8. A man bought four hogs, each weighing 375 pounds ; low much did they all weigh? Ans. 1500. 9. The fore quarters of an ox weigh one hundred and ^ight pounds each, the hind quarters weigh one hundred pd twenty-four pounds each, the hide seventy-six pounds, id the tallow sixty pounds ; what is the whole weight of le ox? Ans. 600. 10. A man, being asked his age, said he was thirty-four llears old when his eldest son was born, who was then fif- 3n years of age ; what was the age of the father ? ^ 11. A man sold two cows for five pounds each, twenty ^^*' ^ushels of corn for three pounds, and one hundred pounds ' tallow for two pounds; what was his due? . .:. . ■ . i >A.:) V :i\ '%. ISubtraction or Simple IViimbei's* T[ 6. 1. Charles, having 11 pence, bought a book, for [hich he gave 5 pence ; how many pence had he left ? 2. John had 12 apples; he gave 5 of them to his brother ; )w many had he left? 3. Peter played at marbles; he had 23 when he began, It when he had done he had only 12; how many did be se? <'2' %l. SUBTRACTION OF MMPLI HUMBBES. tl«. 4. A man bought an article fi^r 17 shillings and sold it again for 33 shillings ; how many shillings did he gain ? 5. Charles is 9 years old, and Andrew is 13; what is the difference in their agcs?{ 6. A man borrowed 50 pounds, and paid all but 18; how many pounds did he pay? that is, take 18 from 50, and how many would there be leil? 7. John bought several articles for 10 shillings ; he gave for 4 books 6 shillings; what did the other articles coet him? ■>. ' ' 'i.» ■:;' , «■■ ; .;,. 8. Peter bought a trunk for 17 shillings, and sold it for 23 shillings ; how many shillings did he gain by the bar* gain? «0. Peter sold a wagon for 32 shillings, which was 5 shilU ings more than he gave for it ; how many shillings did be give for the wagon ? 10. A boy, being asked how old lie was, said that he was S5 years younger than his father, whose age was 33 years; how old was the boy ? •i.U A'- 1. The taking of a less number from a greater (as in the foregoing examples) is called Subtraction. The greater number is called the minuend, the less number the suhtror hmdy and what is leil after subtraction, is called the differ' enu QX remainder. . o > , .'. ; r v > 11. If the minuend be 8, and the subtrahend be 3, what j is the remainder ? 13. If the subtrahend be 4, and the minuend 16, what is| the remi^inder ? , 13. Samuel bought a book for 11 pence; he paid downj 4 pence ; how many pence more must he pay ? Sign. A short horizontal Ijine, — , is the sign of sub- traction. It is usually read minus, which is a Latin word I signifying less. It shows that the number after it is to be I taken from the number fi^ore it. Thus, 8—3=5, is read! 8 minus or less 3 is equal to 5 ; or 3 from 8 leaves 5. The! latter expression is to be used by the pupil in committing! the following :.^^^iM'i^- HV. SUBTRACTION OP SIMPLE NUMBERS. that he was s 33ycai8; r (as in the ^he greater the suhtra- i the differ- 2—2: 3—2: 4—2: 6—2: (5—2: 7—2: 8—2: 0—2: 10^2: SUBTRACTION TABLE. ^t' :0 :1 :2 :3 :4 :i> :() :7 :8 3—3: 4—3: 6— iJ: 7—3= 8—5= 9—4= 12—3= 13—4= -J) -A 6—3=3 7—3=4 8—3=5 9—3=6 10—3=7 4—4=0 5—4=1 6—4=2 7—4=3 8—4=4 9—4=5 10—4=6 5—5=0 (;-5=i 7—5 2 8—5=3 10—5=5 7—7=0 8—7=1 9 7=2 10^7=3 8—8=0 9—8=1 6—6=0 7— ()=1 8 6-2 9 6—3 10— &4 SUPPEEMENT TO SUBTRACTION. !I7. 15 shillings. 7 shillings. Then, proceeding to the next column we sny 1 ten from 3 (tens,) and there remain 2 (tens,) which we set down in ten's place. Proceeding to tlic next column, we say, 1 (hundred) from 2 (hundreds,) and there remains 1, (hun- dred,) which we set down in hundred's place, and the work is done. It now appears, that the numher of sheep left was 123: that is, 237— 114r=123. After the same manner are performed the following ex- amples : 15. There are two farms ; one is valued at 073 pounds, and the other at 421 pounds; what is the difference in the value of the two farms I 16. A man's property is worth 2170 pounds, but he has debts to the amount of 1110 pounds ; what will remain after paying his debts ? 17. James having 15 shillings bought a book for which hfe gave 7 shillings ; how many shillings had he left? OPERATION. A difficulty presents itself here; for we 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 oth- er articles for 27 pounds ; what did the former cost him more than the latter? OPERATION. The same difficulty meets us here as inl Pirst articles, 85 the last example; we cannot take 7 fronij Other articles, 27 5 ; but in the last example the larger num — ber consisted of 1 ten and 5 units, which! Difference, 58 together make 15; we therefore took 'ij from 15. Here we have 8 tens and 5 units. We can nowj in the mind, suppose 1 ten taken from the 8 tens, whic 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, a^ before, and there will remain 8, which we set down. Tli taking of 1 ten out of 8 tens, and joining it with the 5 units, is called borrowing ten. Proceeding to the next higher or der, or ticns, we must consider the upper figure 8, from whici we borrowed, 1 less, calling it seven ; then, taking 2 (tens from 7 (tens) there will remain five (tens,) which we sej down, making the difference 58. Or, instead of making tin j>i 7- To subtrah csqual to Top the amo duce it, Thus and 7— From (be folio I. Wj pJacing lino und( II. Be the loiDer mainder III. V§ wt over '"■'■ ■■'"' "'T. '-'• ■1 ' ' ■ f 7, 8. BHBTRACTION OF BIMPLB NUMBEBB. 26 en from 3 : down in we say, 1 s 1, (him- 1 the work op left was lowing ex- jounds, and n the value I , but he has I cmain after i for which! ;left? lere; for wel e can take 7| n 8. ds, and oth-j lost him morel IS here as inl take 7 froinl klarger numT units, whici ;efore took 1 ^Ve can now, tens, whicli ose joined tij ' from 15, aj down. TM h the 5 unitsJ jxt higher orj ;, from whiclj ,ving 2 (tensi [vliich we sej if making tlij ingthe upper figure, IJess, calling it 7, we^^Qr make the lower figure 1 more, cdling it 3, and the result will be tbtt 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 owet 713 — 471= how many? Ans. 242 pounds. 20. 1612—465=howmany? Ans. 1147. 21. 43751— 6782=how many? Ans, 36969. ^ 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 left ? 40 — IS^how many ? Ans. 22 sheep. 23. A man jold 18 sheep, and had 22 left; how many had he at first ? 18 -|-22 =:how many ? 24. A man bought some articles for 75 pounds, and otk" ers for 16 pounds ; what was the difference of costs ? 75 — 16=how many ? Reversed, 59-|-16=how many ? 25. 114— 103 = how many? Reversed, ll-f,103= how Dany? 27. 143—76 = how many ? Reversed, 67+76 = how many ? Hence, subtraction may be proved by addition, as in tht Cbregoing examples, and addition by subtraction. To prove subtraction, we may add the remainder to the subtrahend, and, if the work is correct, the amount will be equal to the minuend. To prove addition^ we may subtract, successively, from the amount, the several numbers which were added to pro- duce it, and if the work is right, there will be no remainder. Thus 7+8+6 = 21; i>roo/, 21—6 = 15, and 15—8=7, and 7—7=0. From the remarks and illustrations now given, we deduee fbe 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 re- mainder directly below. III. When the figure in the loweilr number exceeds the fi|^ I ve over it, suppose 10 to be added to the upper figure ; but C m HK m ■Mil SUPPLEMENT TO SUBTRACTION- US. in this case we must add 1 to the lower figure in the next ciyiumn bej^ Subtracting. This J^^alled borrowing 10. EXAMPLES FOTl PRACTlCEv 27. If a farm and the buildings on it, be valued at 3000 pounds, and the buildings alone be valvcd at loOO pounds, what is the value of the land ? 28. The population of Lower Canada, at the last census, was 690782, at the census previous the census was 511917 ; what was the difFerence in the two enumerations ? 29. What is the difference between 7,748,203 and 928,671 ? 30. How much must you add to 353,642 to rriako 1,487,945? " 31. A man bought an estate for 3798 pounds, and sold it for 4137 pounsd ; did he gain or lose by it? arid how much ! :32. From 354,931,347,543 take 27,412,507,543. 33. From 824,264,213,909 take 631,245,653,3;j6. 34 . From 127,245,775,075,635 take 978,567,076,250. SUPPLEMENT TO SUBTRACTION , /■ QUESTIONS. 1. What Is subtraction? 1. What is the greater number call- ed ? 3. the less number? 4. What is the resvJt or answer called? 5. What is the sif^n of subtraction? 6. What is iIih rule? 7. What is understood by borrowing ten 1 8. Of what is subtraction the reverse? 9. How is subtraction proved ? 10. How is addition proved by subtraction ? » FXERCISES. 1. How long from the discovery of" America by Colum- ];us, in 1492, to the period of the cession by France of all li'.^r possessi(>ns in North America toGreat Britain in 1763? 2. Supposing a nmn to have been born in the year 1773, j h.jw old was he in 1H48? 3. Su])posing a man to have been 105 years old in the! TP.^r io4iS, in v;hat year was he born? 4. T'.ori? arc two numbers, whose difference" is 8 O'i; tiioj ^raat.cr number is 156&7; I demand the less? tl 8- SUPPLEMENT TO SUBTRACTION. 27 5. What number is that which taken from 3794, ileaves 865? 6. What number is that to which if you add 789, it will become 63o0 1 7. In a certain city, there were 123707 inhabitants ; in another 43,940 ; how many more inhabitants were there \v. one than in the other? 8. A man possessing an estate of twelve thousand pounds, jjave two thousand five hundred pounds to each of his two daughters, and the remainder to his son; what was his son's share? 0. From seventeen million take fifty-six thousand, and what will remain ? 10. What number, together with these three, viz, 1301, 2501, and 3120, will make ten thousand? 11. A man bought a horse for 35 pounds, and a chaise for 47 pounds ; how much more did he give for the chait-fe than for the horse ? 12. A man borrows 7 ten dollar bills, and three one dol- lar bills, and pays at one time 4 ten dollar bills and 5 one dollar bills ; how many ten dollar bills and one dollar bills must he afterwards pay to cancel the debt? Ans. 2 ten doll, bills and 8 one do! 13. The greater of two numbers is 24, and the less is 16 ; what is the difterence ? 14. The greater of two numbers is 24, and their differ- ence 8 ; what is the less number ? 15. The sum of two numbers is 40, the less is 16; what is tiie greater ? 10. A tree 08 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 ? 17. Elizabeth became Queen of England in 1558; how many years since? 18. A man carried his produce to market; he sold his pork for 14 pounds, his cheese for 11 pounds, and his but- ter for 9 pounds ; he received, in pay, salt to the value of pounds, 3 pounds worth of sugar, two pounds worth cf molasses, and the rest in money; how much money did he receive ? Ans. 23 pounds. 19. A boy bought several sleds for 13 shillings, and gave shillings to have them repaired ; he sold them for 18 shili- ^■J IIULTIPLICATION OF SIMPLB NUMBERS. 51 ^i 9l II i^ I 't- I :., p I Iv ings ; did he gain or lose by the bargain ? and how much t . 20. One man travels 67 miles in a day, another man fol- lows at the *rate of 42 miles in 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 first day ? of the second ? of the third? of the fourth? 21. One man starts from Toronto 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? Ans. .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 ? Ans. 82 pounds. 23> A man has property to the amount of 34764 pounds, but there are demands against him to the amount of 14297 pounds ; huw many pounds will be left after the payment of his debts ? 24. Four men bought a lot of land for 482 pounds ; the first man paid 274 pound, 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 ? { The second paid 80. Ans. < The third paid 60. , t The fourth paid 68. 25. A man, having 10,000 pounds, gave away 9 pounds; how many had he left ? Ans. 9991. JHultiplication of Sfiniple Jluinber§. ff 9. 1. If one orange cost 2 pence, how many pence must I give for 2 oranges ? how many pence for 3 or- anges ? for 4 oranges ? 2. One bushel of apples cost 3 shillings ; how many •hillings must I give for 2 bushels ? for 3 bushels ? 3. One gallon contains 4' (juarts ; how many quarts in 8 gallons ? in 3 gallons ? in 4 gallons ? 4. Three men bought a horse; each man paid 6 pounds V y^\ V6. IIUPTIPLICATION OF SIMPLE NUMBERS for his share; how many pounds did the horse cost them? 5. A man has 4 farms worth 95 pounds each ; how ma> ny pounds are they all worth? 6. In one pound there are 20 shillings ; how many shil* lings in 5 pounds ? 7. How much will 4 pair of shoes cost at 9 shillings ft pair ? 8. How much will 3 pounds of tea cost at 5 shillings a pound ? 9. There are 24 hours in 1 day ; how many hours in 3 days? 10. m 3 days' m 4 days' io 7 days ? 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,) wher« the same number is to be added to itself several times, tb9 operation may be , facilitated by a rule, called Multipli- cation, in which the number to be repeated is called ths multiplicand, aad the number which shows how many times the multiplicand is to be repeated is called the multiplier. The multiplicand and multiplier, when spoken of collectively are called the factors, (producers,) and the answer is called the product. 11. There is an orchard in which there are 5 rows of treei and 27 trees in each row ; how many trees in the orchard f In this example, it is In the first row, 27 trees, second " 27 <( a *t <( third (( 27 fourth (( 27 fifth <( 27 (( (( (( evident that the whoid number of trees will ba equal to the amount of five27's added together. In adding, we find that 7 taken five times In the orchard 135 trees. amounts to 35. We write doi^n 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 tb« 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 we need write the number 27 but once, and then, setting the multi- plier under it, we may say, 5 times 7 are 35, writing down e 2 ,4V- 30 UULTIPLICATION OF SIMPLE NUMBERS. ^ 9, 10. « '■ B'H i the 5, and reserving the 3 (tens) as in addition. Again .'j times 2 (tens) are 27 trees in each row. 5 rows. Multipticand, Multiplier, ^ Product, 135 trees, Ans. 10 (tens,) and 3, (tens,) which w«i reserved, make 13, (tens,) as before. * « iy 10, 12, There are on a board 3 rows of spots, and 4 spots in each row ; how many spots on the board ? A slight inspection of the figure will show that the number of spots may be found either by taking 4 three times, (3 times 4 are 12,) or by taking 'i four tijties, (4 times 3 are 12;) for we may say therf are throe rows of 4 spots each, or 4 rows of 3 spots each : therefore, we may use either of the given numbers for a multiplier, as best suits our convenience. We generally write thp numbers as in subtraction, the larger number up- permost, with units under units, tens under tens, &.c. Thus, Multiplicand, 4 spots. Note. 4 and 3are the /ac/cri-, 3Tultiplicr, 3 rows. which produce the product 12 Product, 12 4«^' it ■ ■ • Hence, — Multiplication is a short tfo/y of pctformin^ mam/ additions ; in other words — It is the method of repea- ting any number any given number of times. Signs. 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, th*? following table must be committed perfectly to memory. 'rnW'^ ..i..i^,;.»-iil . ;. 51 9, 10, ■ 5[ 10. MULITPLICATION Of SIMPLE NUMBGRJ^. 31 Again .'S (tens) are s,) and 5{, which we [,niakc IJJ, IS before. )ots, and 4 fiaure will ots may he 'c times, {'i four tijiiff, y say tlierf spots each ; nbors for a I generally lumber uj>- &.C. Thus, ihe far furs, roduct 12 pcrfnrminsi d of rrpta- in the form rhus, 3X4 r 4 times U lis rule, th*? memory. MULTICATION TABLE. 3X0 1 o 4 5 t) X 1 X 'IX 2 X ii X 1 X « ■J X 7 ^2 X 8 !2X 9 2X10 I 2 Xll 2 X12 : : 2 : 4 : f) = 8 = 10 = 12 = 14 = 16 — 18 = 20 — 22 1=24 4 X 10 4 X 11 12 ^ 40 44 48 \X 3 X 1 J X 2 ^ X 3 :3 X 4 ;j X r, 3X6 3 X 7 3X8 i X i) :i xio Xll Xl2 3 6 9 12 15 18 21 24 27 30 33 36 o X 5 X 5 X 5 X o X r> X ."> X 5 X •'S X 5 X 1 2 3 4 ^ 6 7 8 9 = 10 15 20 X 10 X II X 12 25 30 35 40 45 50 X X X X X X 7= 8: 0= 10: 1[= 12: 4X0 14 X 1 4X2 4X3 |4 X 4 4X5 4X6 |4 X 7 4X8 |4 X 9 4 8 12 16 20 24 28 32 36 i 2 3 4 6 X 6 X OX OX OX OX OX OX OX OX OX 10 OX 11 ox 12 8X 8X 8X 8X 8X 8X 8X 8X 55 8 X 00,8 X -^8X 8X 8X 5 6 7 8 9 6 12 18 24 30 30 42 '48 54 00 60 0z= \- o — 3::= 4= 5= 0= 1- 8= 9= 10= 11 = 12= 49 50 03 70 77 ^ 8 H) 24 32 40 48 50 04 72 80 88 9i X X X X 1 2 3 4 5 9X 9 X 9 X 9 " 9 " 9 " 9 " 9 " 9 " 9 " , 7 14 21 28 35 42 0= 1= o- 3= A 5= 0= 7= 8r 9= 10= 11= 12= ^ 10" 10" 1 10" 2= 10" 3 = 9 = 18 = 27 ^- 30 = 45 -- 54 = 03 = 72 = 81 = 90 -- 99 = 108 1 = = 10 = 20 -• 30 X 4= X 5= X 6= X 7::= X 8= X 9= xio= Xll= X12=: 40 55 00 70 80 90 100 110 1'^ X 0= X 1 = X 2zr: 3= 4= 5= 6= X X X X X 7= X 8= X 9= XIO- Xll= 12= <( 0=: 1 = 2= 3— 5=: 6= 7= 8^ 9= 10= 11= 12= f 11 2y 33 44 55 66 / 4 88 99 no 121 ija 12 24 36 48 60 72 84 96 108 120 133 144 MVLTIPLICATION OF SIMPLE NUIIBERI. )ly by each le multipR. msists of 3 ind 3 units. multiply- by the 3 gives in )ounds ths , writing :hat is, d> now ap- ime figures fferenc^— noved oma alue is in- the pri«« of 30 acres is evidently ten times as much as the price of 3 acres, that is, 70380 pounds ; and it is plain that theas two products, added together, give the price of 33 acres. These examples will be sufficient to establish the follow- ing * RULE. » I. Write down the multiplicand, under which write thi multiplier, placing units under units, tens under tens, &c., and draw a line underneath. II. When the multiplier does not excd^d 12, begin at the right hand of the multiplicand, and multiply each figurt contained in it by the multiplier, setting down and carrying the same as in addition. III. When the multiplier exceeds 12, multiply by each figure separately, first by the units, then by the tens, &c., remembering jdways to place the first figure of each pro- duct directly under the figure by which you multiply. Having gone through in this manner with each figure io the multiplier, add their several products together, and tha •um of them will be the product required. EXAMPLES FOR PRACTICE. 18. There are 320 rods in a mile ; how many rods aa there in 57 miles ? 19. Suppose it to be 706 miles from Halifax to Quebec-; how many rods is it ? 20. What will 784 chests of tea cost, at 17 pounds a chest? 21. If 1851 men receive 758 pounds apiece; how many pounds will they all receive ? Ans. 1403058 pounds. 22. There are 24 hours in a day ; if a ship sail 7 mile» ill an hour, how many niiles will she sail in 1 day, at thai rate ? how many miles in 36 days ? how many miles in 1 year, or 365 days ? Ans. 61320 miles in 1 year. 23. A merchant bought 13 pieces of cloth, each pieca containing 28 yards, at 2 pounds a yard ; how many yards were there, and what was the whole cost 1 Ans. The whole cost was 728 pounds. 24. Multiply 37864 by 235. Product, 889804a 25. « 29831 " 952. " 28399M3. 26. ♦* 93956 **8704. " 817793024. 34 CONTRAC^riOXS I.V MULTIPLICATION. IT 11. -^r' CONTRACTIONS IN MTTLTIPLICATJON. 1. When the multiplier is a composite number. ^ 11. Any number, which may hi prochiccd by the multiplication of two or more numbers, is called a rompos- ite number. Thus, 15, which arises from the multiplica- tion of 5 and 3, (;»X'i-^i«'5,) is a composite number, and the numbers 5 and W, which, multiplied together, produce it, are called component parts, or factors, of that number. So, also, 24 is a composite number ; its component parts, or factors riKiy be '2 and 1*2, ('iX VH^^'IA ;) or they niay be 4 and 0, (4X0=24;) or they may be 2, 3, and 4. (2X3 X4-z:24.) What will 15 pieces of cloth cost, at 4 pounds a piece ? 15 pieces are equal to 5X3 pieces. The cost of 5 pieces would be 5X4=20 pounds; and be- cause 15 pieces contains 3 times 5 pieces, so the cost of 15 pieces will evidently be 3 times the cost of 5 pieces, that is, 20 pounds X3=-(iO pounds. Ans. 60 pounds. 1. 4 6 %) 60 Wherefore, If the multiplier he a composite number, we may, if we please, multiply the nr'OiplicandJirst by one of the component parts ; that product by the other, and so on, \{ tJje component parts be more than two; and, having in this fe way multiplied by each of the component parts, the last pro- ' duct will be the product required. 2. What will 130 tons of potashes come to, at 24 pounds ^ per ton ? 6X4=24. It follows, therefore, that 6 and 4 are cora- p ' . ' , , , , 1080283 ,if '(/if rt 'i' 9. Multiply 543 by 206. ,, ,.. ,'..,,, i- ' ^ ' 10. " 1620 •• 2103. 11. J, i" 36243 " 32004. :Un 11. "^ •' / n* ' 11' ^UP^lfeAlENf h^b MtfLtlPLte'ATtbN. 'Xt. QUESTIONS. 1. What is Tnultiplicsition ? 2. What is the number /o &« mvltiplied ici^Wtid 1 3. to multiply by called? 4. .What is the result or answer called ? .5. Taken iiM^divcly, what arc the multiplicand and multiplier called ? 7. What ia the sign of I multiplication? 7. What does it show? 8. In what orcfcr m^sl [the given numbers be placotl for multiplication ? 6. How do I you proceed when the multiplier is less than 12 ? 10. When it \vxeeeds 12, what is the method of procedure? 11. What is u \com})Osite nurahet V 12* What is to be understood by the corn - lf)on«n^ partSf or /actors, of any number? 13. How may yoti Iproceed when the multiplier iasi composite number f 14. To Imultiply by 10, 100, 1000, &c., what suffices? 15.. Why? 116. When there are cipher's on the righi hand of the mullipii- cand, multiplier, either or both, how may we proceed? 17. When there are ciphers between the significant figures of the tiulti plier, how are they to be treated ? EXIJRCISES. 1. Ar army of 10700 men having pliindercJ a city, took o much money, that, when it waa shared among them, each an received 46 pounds ; what was the sum of money tak^n ? 2. Supposing the number of houses in a certain town to 145, each house, on an average, containing two famiHc»> nd each family 6 members, what vi ould be the number of nhabitants in that town? ;', \ Am. 1740 3. If 46 men can do a piece of work in CO days, how any men will it take to do it in one day ? D • ■:!..', 7 •( ,! . »f ^ SUPPLEMENT TO MtLTIPLICATIOIf. II 13. > iX 4 Two men depart from the same place, and travef in opposite directions, one at the rate of 27 miles a day, the other 31 miles a day ; how far apart will they be at the end of 6 days ? Ans. 348 miles, 5 Whet number is that, the factors of which are 4, 7,6, 1 and 20? iln*. 3360. 6. If 18 men can do a piece of work in 90 days, how | long will it take one man to do the same? ' ^ 7. What sum of money must be dividtsd be«ween 27 men, | so that each man may receive 1 15 pounds ? 8. There is a certain number, the factors of which are| 89 and 265 ; what is that number ? 9. What Is that number, of which 9,^12, and 14 are fac« tors? IP. If a carriage wheel turn round 346 times in running! 1 mile, how many times will it turn round in the distance] from Quebec to Montreal it being 180 miles. '■/':■::,_ ".\. ''"''' .i^-^.-''i--'*>i- ' -^n*. 62280.1 II. In one mihute ate 60 seconds; how many secondsj in 4 minutes? -^^i— in 5 minutes?—^ — in 20 minutes i| * in 40 minutes ? 12 In one hour are 60 minutes; how many seconds inl ,«n hour ? — — in two hours ? How many seconds firoin| nine o'clock in the morning till noon? 13. In one pound are 4 dollars ; how many dollars in 3| pmmds ? - — " in 300 pounds ? in 467 pounds ? 14. Two men, A and B, start from the same place at thel same time, and travel the same way ; A travels 52 miles al day, and B 44 miles a day ; how far apart will they be at| the end of 10 days? 15. If the interest of 100 pounds, for one year^ be sixl pounds, how many pounds will be the interest for 2 years!! for 4 years? for 10 years ? for 35 years ?| for 84 years ? 16. If the interest of one hundred pounds, for one year,! |>e six pounds, what is the interest for two hundred pound/ the same time ? 7 hundred pounds? 8 hundred! pounds ? 5 hundred pounds ? 17. A farmer sold 468 pounds of pork, at 3 pence ij pound, and 48 pounds of cheese, at 4 pence a pound ; ho>v| hiany pence must he receive in pay ? * 18. A boy bought 10 oranges ; lie kept 7 of them, and sold ^1'- Y ■- " ''['^-V the otheril for 5 peiice a piece; how muiy pence did he riv ceiveT '•■■.;■•/'■ -a-'' 10. The ccunponent parts pjf a certain huqaber ieure 4| 5, 7, 6,9, 8, and 3; what is the number? SO. In 1 hogshead are^3 gallons.; how inany gallons in 6 hogsheads? In 1 gallon are 4 quarts; how many quarts in 8 hogsheads.' In 1, quart are 2 pints; how manj piats in8h<^ea48^ .no ^^.. hm .qu'xi:- ni; bu. ;m.;,A Ml .-{A;? »'!. '^\- ' ' . j^^^^ of Siiuple numbers, 5| 14. 1. James divided 12 apples undng four boys; : how many did he. give each boy ^ ^. 'James would divide 12 apples fmppg three bo^rsi ^w many >must he giye each boy .' ';'!m ^oif h^''^ m v» ' •< i' ^>^rfi 3. John had 15 apples, and gave them to his playmates, who received 3 appl^each; ^ow many boys did he g^ve thew to/ 4. If you had ^ pence, how piany cakes could you buy at i2 pence a piec^? "..:...;;.=,. ..;.. ..■>,>..,. n \^.v.^■^y \>jv-->>, 5. How many yards of cloth could you buy for ^ pounds, at 2 pounds a yard .' 6. If you pay 250 shillings for 10 yards of cloth, what is one yard wprth 7 7. A man works 6 days for 42 shillings ; how many shil- lings is that for one day ? '• 8. How many quarts in 4 pints? - ■ in 6 pints,' — .— in 10 pints? 9. Ilow many times is 8 pooLtaioed in 88.' - < :' < h, ■-; 10. If a man can travel 4 miles an hour, how many hours would it talo him to travel 24 miles? 11. In an orchard ther^ ajre 28 trees standing in rows, ;, and there are 3 trees in a row ; how many roivs are there ? Remarh. When any one thing is divided into two equal , I parts, one of those parts is called a half; if into 3 equal, parts, one of those parts is <^s^pd a third t if into four e- qual parts, oiie part is called a quarter of n. fourth; if into . Uivie, one part is called, aj(/ii(ft, and so on. 12. A boy had two apples, and gave one hi^f an Apple to Neacb of his compapioi^; how man^ were his companioopf DIVISION OF SIMPLE NUMBERS. ^J 14, 15. Ti. A boy divided f£itiV appres amcHig*liiis companions^ 1}j ' ' ffiving ihem one third of ai) appje each : amoa^ bow msLny iid'hlaiViiie^M^lpiJMv •'•'• /^'^"- '- ^^^ ■ -"^^^ ; 14.. H-.V many quarters in three oranses ? ^ ,.,. ' ".' 15. How many oranges would it take to give J^ boyv one 19. A man had 30 sheep, and sold one finn oif ttiem ; how. many of them did he sell/ 20. A man purchased shipep fo^* 13 shillings api^tc^, and paid for theitt siR lift slnirings;' what Was theif rtiiinber ? 21. How many oranges, at 3 pep.ce each, mav be bought for i^'^##c^ ?;.'■>'■■""'' -^hl" -f fc^;'^-''t-' • ; -J-'- ^ -' i ' It is plain, that as many tlm^^ '9 p^c^''c'^*B^'t|(k^^' frdfei 13 p^ce, S^ many "oranges mky be bbti^ht if tfiti bbiect therefore, is to find how many tiiioe^ 3 is ' cbntain^ in ISl ,»otn/i!7t;[<[ -■ ]^>^ene^; ''^''' - ' ^'^ t^'-'^i^V *-' '*^'' "*^'^*' •*■ Ftm'of^^^l' ^.•ptn(^/''- ' "^e'sfee m'«fiisfe^aih^fe;^^ . — we may take 3 from 12 /our vf,d ffor l,fi(»-9? >•> 7t tlme^, «ftfer'wto there 'is lio ^ec£m(/ 0ra»^e, 3 pence. remainder; consieiqiiently,^ sti^' ,H>»arjoq ih. iui.vjM' mrjAiji^ctidn alone is sufficient for i . 6 ^ the operation Voiit«^eiflky|c;bme Tkird oi^titt^e, ^ 3 pfettcie.^ ^ id* the same' result by a" jirofces^, ~^ — in most pases ' ]6iiudi shorted, -luh vrrnm v;r.il : ^nJlal. galled DmHon:''-. '■' -; '■: ■ ^ Jfowrth orange,J^ pence. v y^|,i> ,,j;<» mt buli j^i ^^, ■^ eii.iu tjQr:^ \ otfUT I- a; • •(; fi) (iijau T'^H '."' •fy 15, It is plain, that the cost of one orange, (3 jjjfeticie^) multiplied by the number of oranges; ^4,) is'equaf to the cdW of al! 'tlie orangfes,' (12 pence j) 12 is, th^ttefore, ijiro- duct, and 3 one of its factcfl^s*; 4nd' t6 ftnd^hoir nr&ny iiAies 3 i^'Cfc^ntainfed iii 12 is to 'find fhtt iyihef factorVlvhichJ p\rl- ti^HM'Aito 3, will produce 12. .This factor W0 find by Wi^, to be 4,* |[4><8==12;) consequently, ii^contiitied in 12, 4 iiihHl' '•'^''"' ■' '■■■'■^ .i:i '■.;':> ^» ^^-'^'V'^ns. 4 oranges. i82,jA^niari^w6uld divide' liibrdng^'equaily artiohr 3 childteiir'hd^ inany orah^te wcj'uld e^ch chrl4 have? i Here the object is to dime the 12 orange^ jnt'o 3 eqpal paMSjr ami' id iisbertdin the '^mberof pranj[to i^ eacii o/ / fi 14, 15. H ^ J5. DIVISION OF SIMPLE NUMBERS. 41 those parts. The operation is evidently as in the last ex- ample, and consists in finding a number, which, multiplied by 3, will produce 12. This number we have already found to be 4. Ans. 4 oranges apiece. As, therefore, multiplication is a short way of performing many 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 hoto many times one number is contained in another; and also o{ dividing a number into any number of equal parts. In all cases, the process 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 multiplication The number given to divide by is called the divisor, and answers to one of the factors in multiplication. The result, or answer sought, is called the quotient, (from the Latin word quoties, how many ?) and answers to the other factor. Sign. The sign for division is a short horizontal line between two dots, -r. It shows thjit the number before it is to be divided by the number after it. Thus, 27-r9 =3, is read, 27 divided by 9 is equal to 3 ; or to shorten the expre*-' sion, 27 by 9 is 3; or 9 in 27 3 times. In place of the dots, the dividend is often writteii over the line, and the divisor under it, to express division ; thus, y =3, read as before. The reading used by the pupil in committing the follow- ing table may be 2 by 2 is 1, 4 by 2, &c., or 2 in 2 one time, 2 in 4 two times, &c. , ' DIVISION TABLE. J=l * =1 t -1 i=l 1=1 f = 1 1-2 f =^2 1=2 V»=2 V=2 V*_2 ^r =3 t =3 ^2 = 3 ¥-=3 ¥=3 V - 3 f -4 ^=4 ^^=4 ¥==4 ¥=4 ^8 = 4 y>=5 ¥=5 ^0=5 ¥=5 ¥-5 V = 5 ^ = 6 V«=6 V— 6 ¥=6 ¥ « v=« V_7 ¥=7 2^8—7 V~7 V 7 V = 7 V«=8 ¥=8 ^2 -—8 ¥ -8 V-8 V — H V« =9 V =9 3i« = 9 V ~9 ¥-9 V = 9 D2 m V V DIVISION OF SIMPLE NUMBERS. f| 15, 16 ■ i[| 16. DIVISION TABLE— CONTINUED. if=l 11=3 tf=4 if =6 ff=8 W=9 =1 1=1 i«=l. H-1 ==a 1^=2 fe— 2 ff=2 =3 V=3 i^=3 f*=3 =4 3^=4 H-4 H=4 =5 V=5 f^=5 H-5 =6 V-6 H=6 ff=6 =7 V=7 H='7 H-7 ==8 V-8 f^-8 f!-8 =9 V-9 H-9 ff-9 28- 42- 54- 32- 33 -7, or ^^= how iriany ? 49-^-7, or 4y9=how many? -6, or ^= how many? 32^-4, or 3^=how many? -9, or ^^= how many ? 99-^11, or ff=how many? -8, or ^— how many? 84-^-12, or f|=how many ? -11, or If =how many ? 108-^12, or V^^^how many? H 10. 23. How many yards of cloth, at 4 shillings a yard, can be bought for 856 shillings? 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 readily overcome, if we decompose this number, thus : 856=800-1-40+16. Beginning with the hundreds, we readily perceive that 4 is contained in 8 2 times ; consequently, in 800 it is contain- '"•GO times. Proceeding to the tens, 4 is contained in 4 1 time, and consequently in 40 it is contained 10 times. Lastly, in 16 it is contained 4 times. We now have 200-f-10-j-4=:214 for the quotient^vor the number of times 4 is contained in 856. Ans. 214 yards, We may arrive to the same result without decomposing the dividend, except as it is done in the mind, taking it b) parts, in the following manner : Dividend. For the sake of convenience, we| Divisor, 4 ) 856 write down the dividend with the di visor on the left, and draw a line be Quotient, 214 tween them ; we also draw a linei underneath." Then, beginning on the left hand, we seek how often the divisor (4) is contain^ U 15, 16 I m 16. DIVISION OF SIMI>LE NUMBERS. 43 ed in 8, (hundreds,) the left hand figure ; finding it to be 2 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 5 (tens) 1 time, which we set down in ten's place, directly under the 5 (tens.) But aft;er taking 4 times ten ouLof the 5 tens, there is 1 ten left. This 1 ten we join to tR 6 units, making 16. Then, 4 into 16 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 be perceived, is car- ried on partly in the mind, which is always easy to do when the divisor does not exceed 12. RULE. From the illustration of this example, we derive this gen- eral rule for dividing, when the divisor does not exceed 12: I. Find how many times the divisor is contained in the first figure, or figures, of the dividend, and, setting it di- rectly 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, (^ 15,) that the divisor and quo- tient are factors, whose product is the dividend, and we I 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 i other. To prove division, we may multiply the divisor by the quo- I 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- Uient^ and, if the work is right, the result will be the same las tne divisor. To prove Multiplication, we may divide the product by lone 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 ponnds would each receive? 44 DIVISION OF SIMPLE NUMBERS. II 16, 17. ■ fl 17. OPERATION. Dividend. Divisor, 5)13,462,725 Quotient, 2,692,545 Proof. Quotient. 2,692,515 5 divisor. In this example, as we cannot have 5 in the first figure, (1) we take two figures, and say 5 in 13 will go 2 times, and there are 3 over, which, joined to 4, the next figure, makes 34 ; and 5 in 34 will go 6 times, &-c. In proof of this example, we multiply the quotient by the di- visor, and, as the product is the same as the dividend, we con- elude that the work is right. — 13,462,7515 From a bare inspection of the above example and its proof, it is plain, as before stated, that>division is the reverse of multiplication, and that the two rules mutually prove each other. 25. How many yards of cloth can be bought for 4,354,560 shillings, at 2 shillings a yard? at 3 shillings? at 4 shillings ? at 5 shillings ? at 6 shillings ? at 7? at8?— T-at9?atl0? Note. Let the pupil be required to prove the foregoing, and all of the following examples. 26. Divide 1005903360 by 2, 3,4, 5, 6, 7, 8, 9, 10. 11, and 12. 27. If 2 pints make a quart, how many quarts in 8 pints? in 12 pints ? in 20 pints ? in 24 pints ? in 248 pints ? in 3674 pints? in 47632 pints? 28. Four quarts jnake a gallon ; how many gallons in 8 quarts ? in 12 quarts ? in 20 quarts ? in 36 quarts ? in 368 quarts ? in 4896 quarts ? in 5436144 quarts? 29. A man gave 86 apples to 5 boys ; how many apples would each boy receive ? Dividend. Here, dividing the Divisor, 5)86 number of the apples — (86) by the number of j Quotient^ 17 — 1 Remainder. boys, (5,) we find, that each boy's share would be 17 apples j but there is 1 apple left. 1] 1 7. 5)86 In order to divide all the apples equal- ly among the boys, it is plain, we must 17^ divide this one remaining apple into 5 DIVIBION OF SIliPLE NUMBERS. 45 tqMl parts, and give one of these parts to each of the boys. Tl}«n ,^aph boy'ssljare vWQujd.be 17,ppples, and o^^ fifth part pf WPthpr applq ; ;,^hiph is jyritten thus.^ ,lf7ff(appJle*:A ,,i, ; ,;.,;; 1 -4i»^. 17^ appleaeach. Tfjet^ 17, expressing pohof^ , ;app|es„ . ^ue caUi^d . int^^rs^, ;> (that i^j vH! tu .'> (Tsr H iV(^0. A nuniber like 17-^, composed of integers (17) I and a fraction^ {-^y) is cadled a mixe^ number, i> ^^< i< {'« ^ lo the pTfecedin^ example, the dn^ apple, whi6h W^s len I after eawyirtgitlife divisioh ai^ 'far as ooiild be by whole ndiii-''^ bers, i6 cdied the f*(^^iniEfer,'and is evidiently ja part of the ^ /fiw'ffiiitf yet lindivided. ; In order td c6rtii)Jete the divisiprj^^" this remain i6r, a» w^ before remai'ked, must be divide!^ into * 5 equal purta; biit thtt' divisor itself ex]f)resses the numbef , of parts.' If, now, we examine the fraction, we shall se6;' ' that it consists of the remaihdier (l)'f6r its nvmeraior. and the divisor <(&) for ltd a^ndmina^er' .. Therefore, if there' be i remainder, mi ii down,' at file right )iand of the quotient for iiie numerator of a fraction, under^ which write the divisoir for it^ ■denominator. .,' In proving this example, we fiud it necessary to, multiply our fi-action by 5 ; but this is^ ^^}^}(\ ^53 pounds from goV-j, jrnment, pow ihany pciuiids 'did each, irecg;v.e?;j,\ a,-, , oi^cn ^ \ Proof of last example. . ... m ' -■•' •'■.3- 46 DIVIfllON OF SIMP1.E NUMBKRB. . 51 l^i l^'. > " ' Dividend. ' Here, after earrying- the division aii' ' I>im)>M', 8)463 far aa possible hy whdle numhetB, we ' have a remainder of 5 pounds, which, Qutftient^' 66| written as above directed, giVes for the ans#er 66 pounds and f (five eighths) of another potind, fo «aeh man, , , ' » 5T 18. Here we may notice, that the eighth part of 5 pounds is the same as 5 times the eighth part of I pound, that iSf the eighfth part of 5 pouhds is | Of a pound. Henee, f ei^reaees the quotient of 5 divided by 8. Proof. f is 5 parts, and 8 times 6 is 40, thait \k, V=^« S6| wtiieh, resei^ed and added to the prbdnc^t oif 8 8 times 6, makes 53, &c. Hence, to muHipty a' —— fraction^ we may mattiply by the numerator, 4«'>3 and divide the product by the denominator. Qr, in proving division, we may multiply the whole num* ber in the quotient only, and to the product adx) the remain- der | asd this, till the pt^pil shall be more particularly taught in fractions, will be more easy in practice. Thus, 56X6;== 448, and 4484-^> the remainder, =453, as before. 31. T'lere ^re 7 ,day» i|»>a weftt; hpw many weeks in 365 days? ..(,... ,. ilras. 52^ weeks. 32. When flour is worth 2 pounds a barrel, how many barrels may be bought for 25 pounds? how many for 51 pounds t for 487 pounds ? for 7631 pounds ? 93. Divide ^40 pounds among 4 men, '' 640-7-4, or «f 0=160 pounds, ilna. / ',.r Am. 113. ■*\ liH! 'W \ Ana. 3d4f, 34. 678^6 or 6|8=howmany? 35. *'^«>=how many? 36. Ty*=^howmany1 37. »^*=how many? 38. «|f *=how many! 39. 4o|oi— howmany? ' ' '" 40. »o ij4jt> 1 2^how many f il 19. 4L Divide 4<370 pounds equity among 21 men. When, as in this example, the dfi visor exceeds 12, it is evident that the computation cannot be readily carried on in the mind, as in the foregoing examples. Wherefore, it is more convenient to write down the computation at length Vi^ the following manner : 1119. DIVISION OF SIMPLE NVMBCRS. 47 " OPERATION. We may write the divisor Divisor, Dividend, Quotient, and dividend a» in short di- ,. 21 )4370(208/r. vision, but instead of writing 42 ., , the quotient undtr the divi- ^, " ' dend, it will be found more 170 ( convenient to set it to the 168 ,! right hand, 'A 2 Taking the dividend by parts, we seek how oflen we can have21 in 43 (hundreds;) finding it to be 2 times, we set down 2 on the right hand of the dividend for the highest figure in the quotient. The 43 being hundreds, it follows, that the 2 must be hundreds^ This, however, we need not regard, for it is to be followed by tens and units, obtained from the tens and units of the dividend, and will therefore, at the end of the operation, be in the place of hundreds, as it should be. It is plain that 2 (hundred) times 21 pounds ought now to be taken out of the dividend ; therefore, we multiply the divisor (21) by the quotient figure 2 (hundred) now found, making 42 (hundred,) which, written under the 43 in the dividend, we subtract, and to the remainder, 1, (hundred,) bring 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 tho quotient, and bring down the next figure, making the whole 170. We then seek how often 21 can be contained in 170, and, finding it to be 8 times, we write 8 in the quotient, and multiplying the divisor by this number, we set the product, 168, under the 170 ; then subtracting, we find the remain- der to be 2, which, written as a fraction on the right hand of the quotient, as already explained, gives 208^ pounds, for the answer. This manne'- of performing the operation is called Long- Division. It jnsists in writing down the tcAo/ie 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 piany figures^ on the left of the dividend, a» d I.. <\i S.'f.', 0(1 1 DIVISION OF, SIMPLE NUMBERS. ^ 19, 'f'cbntttiritliediv'iaoi' oti■' 'III. Multii^ty the t'ivii^r by this quotient figure, and '- 'Write the prodiict under that part of the dividend taifen. IV. Subtract the product from the figures above, and to I ' tlie remainder brin^ down the next figure in the dividend, V anddiVid^ th^tiumber it makes up, as before. So continue ^tod<>;till all the figures in the dividend shall have been ''■ b!*ought doWn and dividedl'^ ;« •^•"» ' ^^ ;'''*,' ^\ >";';" 1 i'* ' i Note 1 . Having bi*6ught AowA a figibtb t(Nhe remainder, 'if thenurriber it makes up be less than the divisor,' write a| 'cipher in the quotient^ and bringdown the next figure. ' '■ 'iVo*i?2. ' If the product of thfe divisor, by any quotient I .' fi^ure^ be ^rgft^fer than the part of the dividend taken, it is an evidence that the quotient figure is too ?ar^^, knd must ^' be diittini^^di it the remainder tit any time be gf eaten ■ than, the divisor, or equal to it, the quotient figure is too\ '' ii/ia/f^ a^d mtist be increased. ' :>:i! .U bl fJji) V.'l'l EXAMPLES 'TOR fAACnCE/'); ''^ ,-''^"; ■!',|;* (, 1) j^l , 'll^W' maliy hrtgshet ds of molasses, at 7 poun^^' a hogs- head, may be bought for 6318 pounds ? ."^' i • Ans. 903| hogsheads. ;•>'*' ;8. If ti Wah's tTiciome be 124B poiinds a year, how much | h that pet' week^ there beirig ^2 weeks in a year f ' ^•'; V^/ j.'ki rn !.,.;■. u\ .- '•■.; ir: i 1 - i.-.'M. ' j^j^g. 24 fiounds perAveek, i>nf- .». What Will be the '(^utrtient of 153598, divided by 29 ?l .'•"'*•■>' ■■■'■■ ' '• i4»5.'5296f|, - ii!' '4. MtiW mariy tim^s is 6? coritairied in 30131 ? ^"'-Urtsi478|f times; tha| is, 478 time^, and^J of anbtherl 5. What will be the several qitotientsof 7652, divided hv \ 16"; SJ3, 34, 86, and 92? '"6. If a farm, containing 256 acres, be worth 1850 Jjpunds J what Is that per acre ?''^' '^ , , aiz: i.-<^ , i; . . 7. What will be the quotient of 97403^, divided by 365 f ■^^^•^- {- .' ' '' ' ^»s. 2671^V6 ' '^- 8. Divide 3228242 potinds equally among 563 men ; howl many pounds must each man i-eceive 1 Ans. 5734pQtinds,[ *f ■'^. tf 67624 be divided into 216' 586, arid 976 equal! parts, what will be the magnitude of one of each of thesej equal parts ? fl 20, 21. C0?iT9ACT10MB IN DIVMION, times they md of the igure, and I taken. Bve, and to I B dividend, o continue have been rema,inderJ ibr,' write a| figure, ny quotient I taken, it is I ^,knd must be gfeaten igure is too\ Ans. The magnitude of one of the last of these equal parts will be 59^^'. 10. How many times does 1030603615 contain 3215 ? Ans. 320561 times. 11. The earth' in its annual revolution round the sun, is said to travel 596088000 miles ; what is that per hour, there being 8766 hours in a year? - » / ' ■'• •' 12. »2 34^f|8 90;_ how many? i^V'** ;» ; vA,nv.Hm\f, 13. 40|§-fJ20:s= how many? ■/} M"^''^'^'-. '^M >'*>*/** "^ 14. 98^^^8»=how many? -^i^. 4^^ ^' -^^ 11^ J«a, :;-^% ,_v iti'T. ''j^''\vv5{ :.'"•*',:! ■.je:ih-: .■^''^■^ ^ - '•^'''.'j^ ' -'-''■ '•'}**'i'. '** J, divided!)) .ill . I [SSO founds J led by 365 1 IS. 2671:jVi l3 men ; howl I734 pounds! 1 976 equall Ich of these! 'i CONTRACTIONS IN DIVISION, -i* .^' 1. Tf^en fAe DIVISOR 15 a composite number. v«t,j|.f.| f[ 30. 1. Bought 15 yards of cloth for 30 pounds ; how much was that per yard? 15 yards are 3X5 yards. If there had been but 5 yards, the cost of one yard would be ^B=Q|t pounds ; but as there are 3 times 5 yards, the cost of one yard will evidently *- e but one third p irt of 6 pounds ; that is, ^«b:2 pounds, Ans, Hence, when the divisor is a composite number, we may, if we please, divide the dividend by one of the component I parts, and the quotient, arising uom that division, by the I other ; the last quotient will be the answer. V'*j» -v "*< '•■' ^m 2. If a man can travel 24 miles in a day, how many days I will it take him to travel 264 miles? It will evidently take him as many days as 264 contains 24. OPERATION. |24=6X4. 6)264 -w^ ^ or, ■m^^ jH^x ,'.: 4)44 1 i'<. ,H.', .'\:: 24)264(11 days, An5. 24 ...... U.„.,...^ 24- .---'^'••■: •4ms '^■i'^:' '11 days.' / 3. Ditide 576 by 48=(8x6.) 4. Divide 1260 by63=:(7X9.) 5. Divide 2430 by 56. IT. To divide by 10, 100, 1.000, fcc. 't.'fr ;^^ , ^31. 1. A note of 2478 pounds is owned by 10 me|^ hat is each mah's share ? **^' •- -M >; Jl ■J y^i M)' CONTBACTIONS IM DIVISION. 1I2f,22. I rv i I II' Each man's share will be equal to the number of tens con- tained in the whole sum, and, if one of the figures be cut off at the right hand, all the figures to the left may be con- sderedso many tens; therefore' each man's share will be ^147^^0 pounds. It is evident, also, that if 2 figures had been cut off from the right, all the remaining figures would have been so ma- ny hundreds ; if 3 figures, so many thousands, &c. Hence, we, derive this general Rule for dividing hy 10, 100, 1 000, 6lc. : Cut off from the right of the dividend so many figures as there are ciphers in the divisor ^ the figures to the leftof the point will express the quoiitnt^ and those to the right, the remainder. ' . 2. How many 100 in 424001 .4it5. 424. 4<24I00 Here the divisor is 100; we therefore cut off 2 ' figures on the right hand, and all the figures to the i hft (424) express the number of Ijundreds. a. How many 100 in 34567 7 Ans. 345^Vir- 1 4. How many huiMlreds in 4567640 hundreds ? >7 ... ... 5. How many hundreds in 345600 hundreds? 6. How many 100 in 42604 hundreds ? Ans, 426^^7. , 7. How many thousands in 4Q00? in 25000? 8. How many thousands in 6487 thousands ? Ans. 6^j^. 9. How m^ny thousands in 42863 thousands? in 368456 thousands? in 96842378 thousands? 10. How many tens in 40? in 400? in 20? in 468? in 487? in 34640? III. When there are ciphers on the right hand of the divisor. | ^ tt^. 1. Divide 480 pounds among 40 men ? OPERATION. 4[0)48|0 In this example, our divisor,! f^>) is a composite number, 12 pounds, tIw.?. (10x4=:40;) we may there^ fore, divide by one component part, (10,) and that quotient by the other, (4 ;) but to divide by 10 we have seen, is but to cut off the right hand figure, leaving the figures to the left of the point for the quotient, which we divide by 4, and the work is done. It is evident, that, if our divisor hadj been 400, we should have cut off 2 figures, and have divi- ded in the same manner; if 4000, 3 figures, &c. Hence,! this general Rule : ^Vhen there are ciphers at the rightl "'.i . -. i:' Wt _' .:' '.t'uJ. :. .^ . ='*..»,''- . . V. 5122. ■UPPLBMENT TO DtVItlOIf. 51 ftht divisor. hand of the divitor, cut them oflf, and also as many places in the dividend ; divide the remaining figures in the dividend, by the remaining figures in the divisor ; then annex the fig- urea out oflT firom the dividend, to the remainder. 2. Divide 748346 by 8000. » ^^ «' Dividend. • * > « Di»i««r,8|000)748)346 \ ' 1. 1 ' I . 5;^ Quotient, 93.-4346 Remainder. Ans. 9^Uh 3. Divide 46720367 by 4200000. Dividend. -i'-rr.t. ; \. J ... ' . 42|00000)467|20367(U^,«y|yQMO plier be 11, what is the multiplicand ? Ans. 64. 7. If the product be 704, ai|d the multiidicand 64, what is the multiplier ? '^^ ^^^-'* '»^^'''>. ^'- ^'^--i^^'^''-'^ Ms. ■ 11. 8. The divisor is 18, aiid the dividend 144; what is the quotient? 9. The quotient of two numbers is S, and the dividend 144 ; what is the divisor ? 10. A man wishes to travel 585 miles in 13 days ; how many miles must he travel each day ? 11. If a man travels 45 miles a day, in how many days will he travel 585 miles? 12. A man sold 140 cows for 560 pounds; how much wasthat for each cow ? ^ *; ' V ; , ' 13.. A man, selling his cows for 4 pounds each, received for all 560 pounds ; how many cows did he sell ? 14. If 12 inches make a foot, how many feet are there in 364812 inches? 15. If 364812 inches are 30401 feet, how many inches make 1 foot ? Id If you would divide 48750 pounds among 50 men, how many pounds would you give to each one ? 17. If you distribute 48750 pounds among a number of | men, in such a 'manner as to give to each one 975 pounds, 'how maiiy men receive a share? 18. A man has 17484 pounds of tea in 186 chests ; ho? many pounds in each chest ? '..'^i.^,: .'^i».,*i..':.j.K'/iilk.,fcSif4:;iX ,l,..ji'«i';.;,^j -';.^". , , ^ ,J .M%&': 1120.' MISCELLANEOUS QUESTIONS. 53 19. A man would put up 17484 pounds of tea into chests containing 94 pounds each ; how many chests must he have? 20. In a certain town there are 1740 inhabitants, and 12 persons in each house ; how many houses are there ? in each house are 2 families, how many persons in each fam- ily? 21. If 2760 men can dig a certain canal in one day, how many days would it take 46 men to do the same ? How many men would it take to do the work in 15 days ? in 5 days? in 20 days? 40 days? in 120 days? 22. If a carriage wheel turns round 62280 times in run- ning from duebec to Montreal, a distance of 180 miles, how many times does it turn in running 1 mile ? Ans. 346. 23. Sixty seconds make 1 minute ; how many minutes in 3600 seconds? in 86400 seconds ? in 604800 sec- onds ? in 2419200 seconds ? 24. Sixty minutes make one hour ; how many hours in 1440 minutes ? in 10080 minutes? ., in 40320 minutes ? in 525960 minutes ? 25. Twenty-four hours make a day ; how many days in 168 hours ? in 672 hours ? in 3766 hours ? 26. How many times can I subtract forty-eight from four hundred and eighty ? 27. How many times 3478 is equal to 47854 ? 28. A bushel of grain is 32 quarts ; how many quarts must I dip out of a chest of grain to make one half (^) of a bushel ? for one fourth (^) of a bushel ? for one eighth (^ ) of a bushel ? 29. How many is ^of 20? 247? ^ of 847? Ans. to the last, 4 quarts. ^of 48? ^ of 4 of 345878.^ 1 of 204030648 ? Ans. to the last, 1 02015324. 30. How many walnuts are one third part (^) of 3 wal- nuts 7 i of 6 walnuts 7 ^ of 12 walnuts ? ^ of 30.^ 1-^ of 45 .' ^ of 300 7- of 478.^ 31. What is I 4 of 3456320? of 4.^ i of 7843 7 Ans. iof to the last, ^0? 1152l06f. i of 320.? Ans. to the last, 196Gf . MISCELLANEOUS QUESTIONS, Involving the principles of the preceding rttles. Note. The preceding rules, viz. Numeration, Addition, E 2 4 M MISCELLANEOUS QtfeSTIONS. IT 51 ^Subtraction, Multiplication, and Division, are called the JFundcunmtal Joules of AritkmetiCf because they are the ^*' % ^^ I IT ^^» ^^' MISCELLANEOUS aUESTIt)Nli. 56 sailed the f are the L horse for ;he chaise the horse eater nunr- ss number «nd 3481, 4361, and ii^v^^ ':":' ahend are low do you given, how (me of them , and their ;r? their rf{^cr- the num- leir difer- difference is ids a piece ; ft pieces of t? id a yoke of I le, worth 31 >ney did he jutter. at 7 11. How many bushels cif wtat 6an you buy for If70 shillings, at 8 shillings per bushel? t[ ^4« When the price of on« pound, otie bushel, &/C. of any commodity -is given, how do you find the cost of any number of pounds, or bushels, &/C. of that commodity ? If the price of the 1 pound, &-c. be in shillings, in what will the whole cost be? If in pence, what? When the cost of ani/ given number of pounc^, or bushels, fee. is given, how do you find the price of one pound or bushel, &c. In what kind of money will the answer be ? When the cost of a number of pounds, &.c. is given, and also the price of one pound, &c. how do you find the num- ber of pounds, &-C. 12. When rye is 4 shillings per bushel, what will be the cost of 948 bushels ? 13. If G48 pounds of tea cost 173 pounds, (that is 4152© pence) what is the price of one pound ? When rh'^ 'motors are givfen,howdoyoufindthe product? When tri . Juct and one factor are given, how do you find the oti .aCtor ? When the divisor and quotient are given, how do you find the dividend ? When the dividend and quotient are given, how do you find the divisor ? '- > ^ 14. What is the product of 754 and 25? , 15. What number, multiplied by 25, will produce 18850 ? 16. 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, (365 days,)? how iiiany in 45 years ? How many cows could he buy with the money, ^t 742 pence each? 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. • Ans. 40 apples. 19. There is a certain number, to which, if 4 be added^ I and 7 be substracted, arid the difference be multiplied by 8, and the product divided by 3, the quotient will be 64 ; what is that number ? ' Ans, ^7. i «$ MISGBLLANEOtTSr QUSSTIONS« I IT 25. 4te B 20. A board has 8 rows of 8 squaifes.each; how many squares on the board ? ' * l> jg 4?^ . ; ^ .^ U 3«S. 21. There is a spot of ground 5 rods long, and 3 rods wide ; how many square rods does .it contain 1 Note. A square rod is a square (like one of those in in the annexed figure) meas- uring a rod on each side. By an inspection of the fig- ure, it will be «een, that there are as many squares in a row as rods on one side, and that the number of rows is equal to the number of rods, on the other side ; therefore, 5X3=15, the number of squares. Ans. 15 square rods. A figure, like A, B, C, D, having its opposite sides equal and parallel, is called a parallelogram or oblong. 22. There is an oblong field, 40 rods long, and 24 rods wide ; how many square rods does it contain ? j 23. How many square inches in a board 12 inches long, and 12 inches broad ? Ans. 144. 24. A certain township is six miles square ; how many square miles does itcontain ? Ans. 36. 25. A man bought a lot of land for 2246 pounds ; he sold one half of it for 1175 pounds at the rate of 3 pounds per acre ; how many acres did he buy 7 and what did it cost him per acre ? ■ 26. A boy bought a sled for 56 pence, and sold it again for 8 quarts of walnuts ; he sold one half of the nuts at 8 pence a quJTrt, and gave the rest for a penknife, which he sold for 18 pence ; how many pence did he lose by his bar- gains.^ 27. In a certain school-house, there are 5 rows of desks; on each row are six seats, and each seat will accommodate 2 pupils ; there are also two rows, of 3 seats each, of the same size as the others, and ^ne long seat where 8 pupils , may sit; how many scholars will this house accommodate? Ans. 80, ,28. How many square feet of boards will it take for the % ?5^.;> >m9^h^mm^iSv^fim, 67 floor of a room 16 feet long ti^ 15 feet wide, if we allow 12 square feet for waste .' ; 39. There is a room 6 yards long ai»d 5 yards wide ; how many yards of carpeting, a yard wide, will be sufficient to cover the floors, if the hearth and fireplace occupy 3 square yards? ,, '.^^khr^^n- n h: -r^iM' .I;:- 30. A board 14 feet long, contains 28 square feet ; what is its breadth ? |l^31. How many pounds of pork, ijirorUi 4 pence a, pQund, can be bought for 144 pence ? , ,i|,,{,.,f .vu <• ;;;^nMPTi; '.'>:>;;. /ixj i ^?: i ••ni i(/i ■■••/ i^iri.! '^f ■ > .- ( 4- f,- ''^l> COMPOUND NUMBERS. ^ < f^' > IT 36* 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\ number ; thus, 46 pounds 7 shillings and 6 pence, is a com* pound number ; so 4 years 6 months and 3 days, 4^ dQllarsI 525 cents and 3 mills, are compound numbers. ^ '[^ .' * Note. Different kinds, or names, are usa ally called differ- ent denominations. ..•..,i., Reduction. tr 27. In this Province as in England, money is reckoned in pounds, shillings pence and farthings. In the United! States, money is reckoned in dollar: 1127. HALIFAX CURRENCY. HALIFAX CURRENCY. 59 M-tf The present currency of Lower Canada, is called Halifax currency, having been introduced into this 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. e. pounds, shillings, pence, and farthings ; and the ratios of the different denom- inations to each other are the same as in English mon^v. i. e., the shilling is one twentieth of the pound, the peh - ,.ne twelfth of the shilling, and the farthing one fourth of the penny. In value Uiey are different, as will be seen in the II upon reduction of curjrencies; where the ratio of each to the other, ,and of both to Federal Money is exhi- bited, with the method of ascertaining them in practice, for particular sums. . Ik ■ ■ \ 2 farthings (qrs.) make 4 « 12 pence 20 shillings « « « .'I.'".? half-penny, marked ^d. ; penny, " d, , shilling, " 8. '? 1 pound, J « j€. Note. Farthings are often written as the fraction of a penny ; thus, 1 farthing is written ^d.^ 3 farthings, ^d., 3 farthings, |d. , , . ...■■^.... ^ . ;. It will be proper here to insert an abstract from the Pro- vincial statute passed in 1842, fixing, the value at which the gold and silver coins of other countries shall pass current in this Province. ,,..,,,. ; The values assigned to the several coins by law in Cana- da, are not arbitrary, but are proportioned (except in the I case of British silver) to the quantity of pure gold or silver in each. The £ currency was and is equal to 4 dollars of account ; and a note for $100 either in Upper or Lower Canada, is now, as it has always been payable by ^^25 cy., in any ccins equivalent by law to that sum. By the curren- cy Act the Provincial dollar of account is made equal in value to that of the United States. . ; i.. / ^>. 60 HAUPAX CVERINCY. The coins, current by lav, are : ^/ ^'f British gold coins at the rate of ;C1 4s 4d cy. to £\ stg. American Eagles coined before 1st July 1834, at ;^ 13s 4d cy — Do. coined between 1st July, 1834, and 1st Janua- ry, 1841, at J^ 10s, — aftd at the same r"tes for half Ea- gles, &,c. The above ar^ a legal tender by tale if within two grains of full weight, deducting ^ cy. for each 4^ of a grain want- British gold and Am^rlc^n gold rf''/ '■^'ni^i'?t;H'9.-io .'f??fiiwr*r-»'p -iuiK^'-y^.^. j Coined f Gold coiii of Prance, at OSs Id cy. per oz. troy.| before Apr. 26 1841. (( <( (( cur- Do. of Laplata & Columbia, at 89s 5d Do. of Portugal & Brazil, at 94s 6d '■' Sp. Mex. &j Chilion Doubloons at 89s 7d — if offered respectively in sums of not less than df ' rency at one time. « British silver as above stated. •. The dollars of Spain, United States, Peru, Chili, Central I America, States of South America and of Mexico, coined I before 1841, at 5s Id currency, and half dollars at 2s 6^dl currency. ClUarters.at Is 3d. Eights at 7^ and sixteenths! at 3^d, if legal weight. The parts less than halves being al tender at the said rates by tale to the amount of <£2 10s inl one payment, until they have lost one twenty-fifth of theirj weight, and not aftewards. ,'**t*i *ii ^' f French 5 franc silver pieces, coined^ before 26th April 1842,| at 4s 8d each. ) Gold and silver coins of the same nations of later dates! may be made current by proclamation to be issued as afore{ said.-rJ'^i }'>''- ■',<• t*?(iiv)n'i-.»".{>i!i,^ ■»»,;; >*>^i, ,V;trl1«iMi. -'^ ^ • Ciopper coiiiiof the United Kiiigdom, (or aiiy tobecoinj ed by Her Majesty of not less than five-sixths the weight o^ fuch coin) at th^ir nominal rates. < The least legal weight of a Sovereign is, 5dwts. 2^ grsj —of an Eagle coined before 1834, 11 dwts. 6 grs., aftei[ 1834, 10 dwts. 18 grs.— of a Dollar, 17 dwts. 4 grs.— of 5 franc piece 16 dwts. REDUCTION. 61 ••t:(; • / w '\ ,.Jj The £ sterling, in any act or contract made after the pas- sing of the Currency Act, [proclaimed 26 April, 1842] is to be understood as equivalent to £1 is Id cy., but in any act or contract made before that time, the word sterling is to be construed according to the intention of the Legislature or of the parties. .... .,;,,., i How many farthings in one penny? in 2 pence ? in 3 pence? in 6 pence ? in 8 pence? in 9 pence? in 12 pence ? in 1 shilling ? in 2 shil- lings? ,! i How many pence in 2 shil- lings? in 3 s.? in 4s. ? in 6s. ? in 8s. ? — - in 10s. ? — - in 2 Shillings and 2 pence ? in 2». 3d. ? in 2s. 4d. ? in 4s. 3d. ? How many shillings in 1 pound? in 2 ]f in 3^ ? in 4 £ in 1 4^ 6s. ? in 6£ 8s. ? in 3^ 10s.? - ore 1834, at I U,at93scy. >er 01. troy, d 98 7d " m£' cur- Chili, Central sxico, coined! ars at 28 e^dj nd sixteenths! lalves being al of .£2 10s inl -fifth of theirL The changing of one kind, or denomination, into another th April 1842,1 kind, or denomination, without altering their value, is call- ed Reduction. (^ fS7.) Thus, when we 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 6y divsion ; it is then called Reduction Ascending. Thus, to reduce pounds -in 2£ 15s. ? How many pence in 4 far- things? in 8 farthings? in 12 farthings ? in 124 farthings ? in 32 far- things? in 36 farthings? — in 48 qrs. ? How many shillings in 48 qrs? — — in 96 qrs? How many shillings in 24 pence ? -= — in 36d. ? in 48d.? m72d.? in 96d. ? in 120d. ? ■ in 26d. ? in 27d.? in 28d. ? in 30d. ? in42d.? inSld.? How many pounds in 20 shillings ? in 40s. ? in 60s. ? — — in 80s. ? in 86s. ? ^ in 128s; ? in 70s.? in 55s.? of later dates! sued as aforej iny tobecoinj the weight oi Sdwts. 2i grsj 6 grs., afterj I. 4 grs. — of ' REDUCTION. 51558. to shillings, it i» plain we must multiply by 20. And again, to reduce shillingH to pound?, we must di' ide by 20; It follows, therefore, that reduction , descending and ascending reciprocally prove each other. 2. In 1697Marthings, how many pounds? OPERATION. Farlhings in a penny 4)16971 3q»". Pence in a shilling, 12)4242 <)Vj 17 13 6 3 '■•*v, ■ 20s. jm't i!t- :•? • IgI 353s in 17^. I3s. ;tf - 4342d..' *^ 4q. i?. -1, i'< ).' < Farthings will be reduced to pence i if we divide them by 4, because every 4 far- things make 1 penny. There- fore, 16971 farthings, divided by 4, the .quotient is 4242 10971 5r5. the Ans. In the above example, be- cause 20 shillings make pound, therefore we multiply 17^. by 20, increasing the' p;^;;";;^ 7^ V;;;;f„"^^^^^^^^ product by the addition of the J^ ^^^^^ -^ forthings, of the given shillings (13,) which, it is evident, must always be done in lik^ cases; then, be- cause 12 pence make 1 shil ling, we multiply the shillings (ti53) by 12, adding in the given pence, (6.) Lastly, because 4 farthings make 1 penny, we"^ multiply the pence (4242) by 4, adding in the given farthings, (3.) We then find, that in 17<£. 13s. 6fd., are contained 16971 same name as the dividend. We then divide the pence (4242) by 12, reducing them to shillings ; and the shillings (353) by 20, reducing them to pounds. The last quotient 17 J?., with the several re- mainders, I3s. 6d. 3qr8. cqi>- stitute the answer. Note. In dividing 353s. by 20, cut off the cipher, &.C., as taught^ 22. furthings. .. 51 2?^. The process' in the foregoing examples, if care- fully examined, will will be found to be as follows, viz. To reduce high denominations To reduce low dimmtinotions to higher. — Divide the lowest denomination given by that to lower, — Multiply the high- est denomination by that num- r y, 5128. 3d again, y 20: It isccndittg ings,ho>«r 16971 3qr. 5)4242 Od. 0)351313* € 13s 6|<1. be reduced Lvide them fexy 4 far- ny. There- ngs, divided mt is 4242 tmainder . of ings, of the dividend, the pence iucing them he shillings cing them ast quotient several re- 3qr8, CQi>- '" . >■' ig 353s. by )her, &.C., [es, if caie- /s, viz. linotions the lowest in by that REDUCTION. 63 number which it takes of th^ same to make I )0f the next higher. Proceed in the «ame manner with eagh^ucceeding denomination^ until you have brought it to the denomina- tion required,. . - . , •.( , : kH '(■< a "•''>* yi>HA ill ' 5128. her wliich it takes of t1ie next less to make 1 of this higher, (increasing the product by the number given if any of that less denomination.) Pi'O'ceed in the same manner with each succeeding denomination, un- til you have brought it to the denomination required. In the two examples, from which the above general rules arc deduced, the denominations are pounds, shillings, pence and farthings, considered as in Halifax Currency ; but it is obvious that these rules can be applied to all currencies where the den minations are the same ; or to currencies in which the denominations are different; and in general to all compound numbers. EXAMPLES FOR PRACTICE. 3. Reduce 20<£- i^&- '^d- to pence. 4. 5. 7. 8. .!>i. 9. <( > i'--! •" .ys':A"' I !JI , l> •''■$■■" ; >!f ff ;'t 64 REDUCTION. y . OLD CURRENCY. <*; .! 12 deniers make ' i- -«*j.'« '¥ i sou. '^ - 20 sous " vmW.:^^ i Uy^e, or franc. ' The livrd^ lOd Halifax currency. In 32 livres lO sous how many sous? , . ;, . ; :' In 97 livres 11 sous, how many sous? -;^.,. ,; ,/>; In 650 sous, how many livres? , - > ^ n -! In 1951 sous, how many livres? In 10 livres 6 sous 9 deniers, how many deniers ? ' How many pounds currency in 96 livres ? !I20. .1'. If \\ ■ :>, '7 • #«, ft-. ■'.,•; .)■ 3 4 6 5 2 may be read, 34652 mills ; or 3465 cents ahd 2 mills ; or, reckoning the eagles tens of dollars, and the •The eagle is a gold coin, the dollar and dime are silvey coins the centis a copper coin. The mill is only tmagmary, there being no Coin of that denomination. There are half eiagles, half dol- lars, half dimes, and half cents, real coins. 1120. . ■;!' .;, -.M rr ; V ; i:./i tic. 1 1. H' !i • .> rst •;;T'kr. ' til f ■ • t ■ ' •.1".ll . ited States, urs, dimes, 1. • )i =1 dime. =1 dollar. =1 eagle* iber, shows dimes a nills, cents, rs of units, 3, they may artly in a hu?: . '! I • -.15 .1- ..^ sents aiid 2 rs, and the I I ~ [silvey coins I there being Ss,halfdol- ^ 29. ' REDUCTION. ' ' * (15 dimes tens of cents, which is the usual practice, the whole may be read, 34 dollars 65 cents and 2 mills. For ease in calculating, a point, (') called a scparatrh,'^ is placed between the dollars and cents, showing that all the tigitres at the left hand express dollars, while the two jirst fffures at the right hand express cents, and the third, mills. Thus, the above example is written $34'G52; that is, 34 dollars 65 cents 2 mills, as above. As 100 cents make a dv/llar, the cents may be any number from 1 to 99, often re- quiring two figures to express them; for this reason, two places are appropriated to cents, at the right hand of the point, and if the number of cents be less than ten, requiring but one figure to express them, the ten's place must be filled with a cipher. Thus, 2 dollars and 6 cents are written 2'Gt}. 10 mills make a cent, and consequently the mills never ex- ceed 9, and are always expressed by a single figure. Only one place, therefore, is appropriated to mills, that is, the place immediately following cents, or the third place from the point. When there are no cents to be M'riiten, it is ev- ident that we must write two ciphers to fill up the places of cents. Thus, 2 dollars and 7 mills are written 2'007'. Six cents are written, 06, and 7 mills are written *007.. Note. Sometimes 5 mills =3^ a cent is expressed frac- tionally: thus, *125 (twelve cents and five mills) is ex- pressed 12^ (twelve and a half cents.) ^ . 17 dollars and 8 mills are written, 17*008 , \ 4 dollars 5 cents, - - - - 4*05 , ' 75 cents, , - '75 24 dollars, 24* ^^■"- 9 cents, '09 4 mills, '004 6 dollars 1 cent and 3 mills, - 6'Or? Write down 470 dollars 2 cents ; 342 doll,?r;- 40 cents- and 2 mills ; 100 dollars, 1 cent and 4 mills ; 1 mill ; 2 mills ; 3 mills ; 4 mills ; ^ cent, or 5 mills ; 1 cent and 1 I mill ; 2 cents and 3 mills ; six cent^' and one mill ; sixty cents and one mill ; four dollars and one cent ; three cents ; I five cents ; nine cents. *The character used for the scparatrix, in the " Scholars' A- jrithmetic," was the comma, the comma inverted is here adopted^ (to distinguish it from the comma used in punctuation. F 2 6a REDUCTION OF FEDERAL MONEY. ^30. REDUCTION OF FEDERAL MONEY. tf SIO. How many mills in one cent? — in 2 cents? — in 3 cents ? — in 4 cents ? — in 6 cents ? — in 9 cents? — in 10 cents? — in 30 cents? — in 78 cents? — in 100 cents, (=1 dollar)? — in 2 dollars? — in 3 dollars? — in 4 dollars? — in 484 cents? — in 563 cents ? — ^in 1 cent and 2 mills ? — in 4 cents and 5 mills ? How many cents in 2 dollars ? — in 4 dollars ? — in 8 dollars? — in 3 dollars and 15 cents? — in 5 dollars and 20 cents ? — in 8 dollars and 20 cents ? — in 4 dollars and 6 cents? How many dollars in 400 cents ? — in 600 cents ? -r- in 380 cents ? — in 40765 cents ? Hmv many cents in 1000 mills? How many dollars in 1000 mills ? — in 3000 mills ? — in 8000 mills ? — in 4378 mills ? — in ^46732 mills ? As there are 10 mills in one cent, it is plain that cents are changed or reduced to mills by multiplying them by 10, that is, by merely annexing a cipher, (^ 12.) 100 cents make a dollar ; therefore dollars are changed to cents by annexing 2 ciphers, and to mills by annexing 3 ciphers. Thus, 16 dol- lars =1600 cents =16000 mills. Again, to change mills b ick to dollars, we have only to cut off the three right hand figures, (IT 21 ;) and to change cents to dollars, cut off the two right hand figures, when all the figures to the left will Jae dollars, and the figures to the right, cents and mills. Reduce 34 dollars to cents. Ans. 3400. Reduce 240 dollars and 14 cents to cents. Ans. 24014 cents. Reduce $748'143 to mills. Reduce 748143 mills to dollars. Reduce 3467489 mills to dollars. Reduce 48742 cents to dollarrs. ileduce 1234678 mills to dollars. Reduce 3469876 cents to dollars. Reduce $4867/467 to mills. Reduce 984 mills to dollars. Reduce 7 mills to dollars. Reduce $ *014 to mills. Reduce 17846 cents to dollars. Ans, 748143 mills. J.ns. $748443. Ans. 3467*489. Ans. $487*42. Ans. $ *984. A^v.z. $ 007. !13L RfiDrCTION. 67 Reduce 984321 cents to mills. */ Reduce 9617^ cents to dollars. Ans. $9G'17^., Reduce 2064^1^ cents, 503 cents, 106 cents, 921^ cents, 500 cents, 726 J^ cents to dollars. Reduce 86753 mills, 96000 mills, 6042 mills, to dollars. TROY WEIGHT. 11 31. It is established by law, that the pound Troy, with its parts, multiples, and proportions, shall be the standard weight for weighing gold* and silver in coin or bullion, drugs, and precious stones. The denominaticms of Troy weight are pounds, ounces, pennyweights and grains. TABLE. I pennyweight, marked pwt. 1 ounce, - - - - oz. 1 pound, - - - - lb. 24 grains (grs.) make 20 pennyweights - - 12 ounces - - - 1. How many grains m a silver tankard weighing 3 lb. 5oz. ? 3. Reduce 210 lb. 8 oz. J2 pwts. to pennyweights. 5. In 7 lb. 11 oz. 3 pwt. 9 grs. of silver, how many grains? o In 19680 grains many pounds, &c. how 4. In 50572 pwt. how ma- ny pounds ? 6. Reduce 45681 grains to pounds. • APOTHECARIES' WEIGHT. Apothecaries' weightt js used by apothecaries and phy- sicians, in compounding medicines. The denominations are poundts, ounces, drams, scruples, and grains. TABLE. ' 20 grains, (grs^) make 1 3 scruples - - - 1 8 drams - - - - 1 12 ounces - - - - 1 scruple, marked g. dram, - - - 3. ounce, - - - §. pound, - - - lb. •The fineness of gold is tried by fire, and is rerkoned in carats, by which is understood the 24th part of any quantity ; if it lose noth- ing by the trial, it is said to be 24 carats fine ; if it lose 2 carats, it is then 22 carats fine; which is the standard for gold. Silver which abides the tire without loss is said to be 12 ounces fine. The standard for silver coin is 11 oz. 2 pwts. of fine silver, and 18 pwts. of copper melted together. IThe pound and ounce apothecaries' weight and the pound and ounce Troy, are tho sainc» only differently divivideiJ, and subdivided. 68 REDUCTION. t[31. 7. In 9 fb. 8 §. 1 3. 2 9| 8. Reduce 55799 grs. to 19 grs., how many grains. (pounds. I .,-t'. AVOIRDUPOIS WEIGHT.* It is established by law that the pound Avoirdupois with Jts parts &c. shall be considered as the standard for weighing every thing commonly sold by weight, except those articles, in weighing which, Troy weight is used. The denominations are tons, hundreds, quarters, pounds, ounces, and drams. TABLE. 16 drams, (drs.) malce 16 ounces - - - - 28 pounds - - - - 4 quarters * 20 hundred weight ounpe, pound, - marked oz. lb. I quarter, ----- qr. 1 hundred weight - - - cvvt. 1 ton, T. Note 1. In this kind of weight, the wordls gross and net are used. Gross is the weight of the goods, together with the box, bal«, 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 1*12 lb. Merchants at the present time, in the principal sea ports of the United States, buy and sell by the 100 pounds. 9. A merchant would put 109 cwt. qrs. 121b. of rais- ins into boxes, containing 26 ib. each ; how many boxes will it require? 11. In 12 tons, 15 cwt. 1 qr. 19ib. 6 oz. 12 dr. how many drams? 13. In 28Ib. avoirdupois, how many pounds Tro) ? 10, In 470 boxes of raisins, containing 26 lb. each, how many cwt. ? 12. In 7323500 drams, how many tons? 14. In 34 Ib. oz. 6 pwt. 16 grs. Troy, how many pounds avoiadupois? •175 oz. Troy-192 oz. ayuirdupois, and 1751b. troy=144lb avoin pois, lib. troy=5760 grains, and 1 ib. avoirdupoi8=70U0 grains troy -,-■•^, ■ v^p-^. yr-Tj^-iT. 1131. REDUCTION' CLpTiI MEASURE. 69 "• ■■.•'•(-.';• 'fir-'. "^V^-. ' f\^i Cloth medifiure is \ised in selling cloths and other goods sold by the yard, or eJI. It is established by law that the English yard with its parts &:-c. shall be the standard for measuring all kinds of cloth or stuffs made of wool, flax &.c. the English ell, when there is a special contract for it may be used with its parts. The denominations are ells, yards, quarters and nails. ''TABLE. 4 nails, (na.) or 9 inches make 1 quarter, marked qr, 4 qiiaftlers or 36 inches, - - 1 yard, - - - yd. 3 quarters •--.--- 1 ell Flemish, - - E. Fl. 5 quarters " - -' i - - - 1 ell Engli'sh, - - E. E. 6 quarters '•- - i /-•'•-'•- 1 ell French, - - E. Fr. 17. In 9173 nails, how- many yards I 19. * In 188f yards, how many ells English ? ?.' 16. In 573^ yds.! c^: t^. how many nail^.i^^ ,'^ ' ' 18. ih 151 ells "Eng. ho^V many yards.? '•"/' Note. €6lis\ilt^ 2S ex. 16. i . , . \ . » ..^^;\),;:tir.^'U-i;!LONG MEASURE. :'■>;;!',' .^ ;-\ '^^ •■■ ' ■ Lon^ measure is us^ in measuring distances, or other things, where length is considered without regard to breadth. The denominations ate degrees, leagues, miles, fur]dngs, rods, ysirds^ feet, inches, and barley-corns. •.■, :.i>:*'r /It' f' •■•/, -^ — " •\ '.■'■. \ 1 •■ ,-. ■ ■ s> iv •. ] '■ ■ TABLE. '-* .,:. ..::.„•. . marked d barly-coins, (bair.) make 1 inch. 12 inches 3 feet 5^ yards, or 16^ f^et, - 40 rods, or 220 yards, - 8 furlongs, or 320 rods, - ' 3miles, .,-,..j; V - - 60geographicki„or G9|J^ ) st,atute miles, . - i 360 degrees, - - - | 1 foot, 1 yard, - - 1 - - 1 rod, perch, or pole, 1 furlong, - - - 1 mile, - - -• - 1 leauge, - - - . m. ft. yd. r. p. fur. M. L. 1 degree. deg. or ° a great circle, or circumfer- ence of the earth. 1131 RBpUCTION. .Ul 70 It is established by law, tliAtthe JrM*5 foot with its parts, &c. shall be the standard meaiiure of length, for measuring land, wood, timber, stone, masons', car'^enters', and joiners' work. The English foot may be used wlien there is a spe- cial contract for it. , ./ ," , •/ "^ TABLE. 12 lines make 1 inch. 3 toises make 1 rod. 12 inches - 1 foot. ' 10 rods - . - 1 arpent. 6 feet - 1 toise. 84 arpents - 1 leauge 1 French foot :^i-^-^fj English feet. 20. How many barley-corns will reach round the globe, it being 360 degrees ? Note. To multiply by 2 is to take the multiplicand 2 times ; to multiply by 1 is to take the multiplicand 1 time ; to multiply by ^ is to take the multiplicand half a time, that ia, the half of it. There- fore, to reduce 300 degrees to statute miles, we multiply first by the whole number, 09, and to the product add half , the multiplicand. Thus : 4)360 . eoi 3240 2160 180 half the multiplicand. 25020 satute miles in 360°. 22. How many inches iVom Quebec to Three Rivers, sup posing it to Be 90 miles ? > 24. How many tim^s will a wheel 10 feet and inches in circumference, turn round in the distance from Quebec to St. Annes, supposing it to be 60 miles ? 21. In 4755801600 barley, corns, how many degrjses? -i'i Note. The barley-corns be- ing divided by 3, and ihat quotient by 12r, we have 13210a500 feet which are tof be redu,qe|^ to rods. We c?in- not easily divide by 16^ on account of the fraction ^ ; but 16^ feet = 33 kalffeet, in 1 rod; and 132105600 /ccf = 264211200 half feet, which divided by 33)-giveB 8006400 rods. . . '^^.v,.' .> i»f »;'■ : .■■■''}.ii I Hence, when the divisor Is encumbered with a fraction, ^ or ^, &c., we may reduce the divisor to halves or fourths &,c., and reduce the dividend to the sarft^; then the' qho- tient will be the true answer. 23. In ' 30539520 inches, how many miles ? 25. If a wheel 16 feet in. in circumference, turh round 19200 times in gbiftjg from Quebec to St. Aanes, what ii the distaijce ! A^- .-70 th its parts, measuring ind joinej-s' •e is a spe- 1 rod. 1 arpent. 1 leauge. IGOObarley- legr^esT ley-corns be- 3, and ihat , we have vhich are to| IS. Wec^n- ! by 16^ on ictjon^; but ilffeet, in 1 )5600/ece = feet, which ves 8006400 the divisor is a fraction, reduce ox fourths |the dividend en the'qtio- Irue answer. |520 inches, 1132. REDUCTION. 71 may 16 feet ^ in. turib round rbitfg from mes, what it 26. In 28 leagues, 43 ar- pents, how many feet ? how many toises ? how many rods ? 27. In 7000 feet how many rods? how^many.arpents? t LAND OR SQUARE MEASURE. Square measure is used in measuring land, and any other thing, where length and breadth are considered. The de- nominations are miles, acres, roods, perches, yards, feet and inches. fl «IS. 3 feet in length make a yard in long measure ; but it requires 3 feet in length, and 3 feet in breadth, to make a yard in square measure ; 3 feet in length and 1 foot wide, make 3 square feet; 3 feet in length, and 2 feet wide, make 2 times 3, that is, 6 square feet; 3 feet in length and 3 feet wi(^e make 3 times 3, that is 9 square feet. This will clearly appear from the annexed figure. jj; „ 3 feet =-1 yard. ... ... - , . 1 . It is plain, also that a square foot, that is, a^^quare 12 inches in length and 12 iriches in breadth, must con- tain 12X12=B»144 square inches. 1- TABLE. 144 square inches=:12Xl2; that is, \ 12 inches in length and 12 inch- > make 1 square foot, es in breadth, ----- j 9 quare feet==3X3; that is, 3 feet ) in length and 3 feet in breadth } 1 30^ square y ai:ds=5^ X 5 1, or 272| V square feet=3l 64X16^ - - f 40 square rods, -------- 4 roods, or 160 square rods, - • i()40 acres, -------- Note. Gunter's chain, used in measuring land is 4 rods lin length. It consists of 100 linkf^, each link being '7-f^xs linches in length; 25 links make 1 rod long measure and |625 square links make 1 scpiare rod. 1 squa»re yard. ( 1 square rod. \ perch or pole 1 rood. 1 acre. 1 .square mile. n RXiDUCTION. FRENCH SaUARE MEASURE. 1131. . 144 squafe inches make 1 square foot. 36 - feet - - 1 toise. 9 - toises - - 1 rod. 100 - rods - - - 1 arpent. 7056 - arpents - - 1 league. 62500 French feet ^71289 English feet. Reduce 16 leagues to feet, to toises, to rods. Reduce 98764321 feet to toises, to rods, — —to ar- pents, — —to leagues. 29. In 776457 square feet, how many acres? Note. Here we have 776457 square feet to be divided by 272^. Reduce the divisor to take a fourth part of the mu\-\fourths, that is to the low- 28. In 17 acres 3 roods 12 rods, how many square feet ? Note. . In reducing rods to feet, the multiplier will be 272^. To multiply by |, is to tiplicand. The principle is the same as shown in fl 28, ex. 20. 30. Reduce 64 square miles to S4uare feet. ? 32. There is a town 6 miles square ; how many square miles in that town? how many acres ? est denomination contained in it ; then reduce the dividend to fourths, that is, to the same denomination, as shown H 31, ex. 21. . 31. In 1,784,217,600 sq. feet, how many square miles? 33. Reduce 23040 acres to square miles. 'jiri II SOLID OR CUBIC MEASURE. < ■ < i f I 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 are cords, | tons, yards, feet and inches. U 33. It has been shown, that a square yard contains I 3x3=9 square feet. A cubic yard is 3 feet long, 3 feet wide, and 3 feet thick. Were it 3 feet long, 3 feet wide and one foot thick, it would contain 9 cubic feet ; if 2 feet thick, it would contain 2X9=: 18 cubic feet; and, as it is 1I3i; 1 1183. , RBBUOTION. im; 3 feet thick, it does contain 3Xda37 cubic feet. This will clearly appear fir make one solid foot. 12 in breadth, 12 in thickness, } 27 solid feet,=3x3X3 40 feet of round timber, or 50 \ feet of hewn timber, | 128 solid feet,=8x4X4, that is, i 8 feet in length, '4 feet in > width, and 4 feet in height, j Note. What is called a cord foot, in measuring wood, is 16 solid feet ; that is, 4 feet in length, 4 feet in width, and 1 foot in height, and 8 such feet, that is 8 cord feet make 1 cord. FRENCH SOLID MEASURE. 1728 solid inches make 1 solid foot. 216 - - feet make 1 toise. 1000 French feet =1218,186432 English feet 1 solid yard. 1 ton or load. 1 cord of wood. 32. Reduce 9 tons of round timber to cubic inches. 34. In 37 cord feet of wood how many solid feet? 36. Reduce 64 cord feet of wood to cords. 38. In 16 cords of wood, how many cord feet? how many solid feet ? 40. In 12 toises how many inches? G 33. In 622080 cubic inch- es how many tons of round timber.' 35. In 592 solid feet of wood, how many cord feet ? 37. In 8 cords of wood, how many cord feet ? 39. In 2048 solid feet of wood, how many cord feet ; how many cords ? 4L In 834692773 inches how many feet; how many , toises T / 'I. \ ■.,■■. ^ -l.:; iitjA.i' :a y '',1 i f ^i')i f a- REDUCTION. -" WINE MEASURE. ' ' 1133,1 < It is established by law that the wine gallon with its {)arts, &c. shall be the standard liquid measure, for measurJ ing wine, cider, beer^ and all other liquids commonly sold by gauge,' or measure of capacity. The denominations are tuns, pipes, hogsheads, barrels, gallons, quarts, pints, ai^dj gills. ?■> J^;tv> -A^' tf-i 4 gills (gi.) 2 pints * 4 quarts 31^ gallons 03 gallons 2 hogsheads '■•■l:.ui TABLE. ^ -■ make * 1 pint, marked - 1 quart, * 1 gallon, 1 barrel, *• 1 hogshead, 4 pipe. '- i^!ip» . 2 pipes, or four hc^sheads 1 tun. pt. qt- gal, bar. hhd. P. T. Note. A gallon wine measure, contains 231 cubic inches. 42. Reduce 12 pipes of wine to pints. 44. In 9 P. 1 hhd. 22 gals. 3 qts. how many gills ? 46. In a tun of cider, how many gallons ? ^ 43. In 12096 pints of winej how many pipes 1 45. Reduce 39032 gills to| pipes. 47. Reduce 252 ctallons to| tuns. -.i. ALE OR BEER MEASURE. Ale or beer measure is used in measuring aie, beer, and] milk. The denominations are hogsheads, barrels, gallensj quart!4> and pints< m V TABLE. 2 pints (pts.) make 1 quart, marked 4 quarts - ; * i- >/ 1 gallon, - * 36 gallons - -• - 1 barrel, r»4 gallons - - - 1 hogshead, - qt. gal, bar, hh(i,| ntfir. A gallon beer measure, contains 282 cubic indie? V Reduce 47 bar. 18 gal of ale to pints. 50, In 29 hhds. of beer, how manyi pints ? 49. In 13680 pints of ale] how many barrels? 51. Reduce 12528 pints ttj hogsheads. 252 £rallons tol i -ivf.'ioKn.' V . a 'J.^ "I bu. ^^ ' fl33. ' REDUCTION.* a ' 7i^' DRY MEASURE. ^ t Dry measure is used in measuring all dry goods, such as grain, fruit, roots, salt, coal, &.c. The denominations are chaldron^, bushels, pecks, quarts, and pints, - TABLE, 2 pints (pts.) make - 1 quart, •• marked 8 quarts - »• - 1 peck, - 4 pecks - - - 1 bushel, •> r 36 bushels - - - 1 chaldron, Note. A gallon dry measure, contains 268f cubic inches^ A Winchester bushel is IS^ inches in diameter, 8 inches deep, and contains 2l50f cubic inches. It is established bj law that the Canada Minot, with its parts, multiples, and proportions, shall be the standard io Dry Measure, 1 pot==116'94569 English 'cubic feet 20 pots malyB one minot, OLD MEASURE. 16 litrons - make - 1 . . * * . boisseau. 3 boisseaux --- 1 ^-.-, minot. -.. 2 minots ---- 1 ,--,, mine.' 2 mines --■-* 1 ...,,.• setier. 12 setiers ---- 1 muid. 40 French cubic inches=::l litron. The mandard measure for the sale and purchase of coal, for this Province, is the chaldron of 36 minots, each minot to be heaped up. 52. In 75 bushels of wheat how many pints ? 54. Reduce 42 chaldrons of coal to pecks. 53. In 4800 pints, how mar ny bushels ?ii 65. In 6048 pecks, how , many chaldrons .' ' '.i. . '■ . .■v; - TIME. The denominations of time are years, months, weeks, days, hours, minutes, and seconds. ' ' ^ TABLE. 60 seconds (s.) - make - 1 minute, marked m. 60 roiuutes , - - 1 hour, - - h. *A i fi :,1 hi I 76 SM'Hiours RBDUCTION. / V day, week, - - month, - common, or ) Julian year, ) !I34. d. mo. yr- February, 2d, March, 3d, - 28 - 31 April, 4th, May, 5th, June, 6th, . 30 - 31 - 30 July, 7th, August, 8th, September 9th, October 10th, - 31 - 31 - 30 - 31 November 11th, . 30 December 12th, - ^ - 31 7 dtys '4 weeks 13 months, 1 day and 6 hours, or 365 days and 6 hours, 51 S4» The year is also divided into 12 calendar months, which in the order of their succession are numbered as foU lows, viz. January, 1st month, has 31 days. Note. When any year can be divided by 4 with- out a remainder, it is cal- led leap year, in which February has 29 days. The number of days in each month may be easily fixed in the mind by committing to memory the following lines : ' "• Thirty days hath September, April,, June and November, . February twenty-eight alone ; ' / All the rest have thirty-one. The first seVen letters of the alphabet. A, B, C, D, E, F, G, are used to mark 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 2>omtntca/ letter for that year. The Dominical letter being known, the day of the week on which each month comes in may be readily calculated from the following couplet: At Dover Dwells George Brown Esquire, Good Carlos Finch And David Fryer. These words correspond to the 12 months of the year, and the first Utter in each word marks the day of the week on ■; ;• J 1?31 I "/I -If mEOUCTIOlV. k»i.i> 77 which each corresponding month conies in; whence any other day may be easily found. For example, let it be required to find on what day of the week the 4th ai July falls, in the year 1827, the Dominical letter for which year is G. Good answers to July ; consequently, July comes in on a Sunday; wherefore the 4th of July falls on Wednesday. Nott. There are two Dominical letters in hap years, ont for January and February, and another for the rest of the year 56. Supposing your age to be 15y. 19d. J lli. 37m. 45s., how many seconds old are you, allowing 365 days 6 hours to the year / 58. How many minutes from the 1st day of January to the I4th day of August, inclu sively ? 60. 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 Jan. 20th, 1783/ 57. Reduce 475047465 se- conds to years. 59. Reduce 335440 minutes to days. 61. In 4079160 how many years ? mmutcs, \ CIRCULAR MEASURE, OR MOTION, Circular measure is used in reckoning latitude and lonifi- tude; also in computing the revolution of the earth and other planets rouna the sun. The denominations are cir> cles, signs, degrees, minutes and seconds. TABLE. 60 seconds (") make 1 60 minutes - - - 1 30 degrees - - - 1 12 signs, or 360 degrees, - 1 Note. Every* circle whether into 360 equal parts, called degrees. minute, marked '' degree, .. - t» sign, - - s circle of the zodiac, great or sni ill, is divisible 62. Reduce 9s. second.s. G2 130 'Zry to 63. In 1020300 , how many degreca ? itPPLEMKNT to RBBUCTION. ^M. II u 1 dozen. 1 gross. 1 great gross. 1 score. .;>: t "^ The following are denominations of things not includcU in the tables: — ^' .> 12 particular things make )' • 13 dozen .... ' 12 gross, or 144 dozen, Also, 20 particular things make 6 points make 1 line, ) used in measuring the length of 12 lines - 1 inch ) the rods of cluck pendulums. 4 * hA 1 h H I ^^^^ '" measuring the- height of *" ' an ^ liorses. 6 feet - 1 fathom used in measuring depths nt sea. 112 pounds make - 1 quintal offish. 24 sheets of paper make 1 quire. 20 quires - - - 1 ream. SUPPLEMENT TO REDUCTION. QUESTIONS. 1. What is reduction 'i 2. Of how many vnri^lies is reduction ? 3. what is underatood by different denominations, as of money, weight, mensure, &cc. ^ 4. How are high denominations brouj^ht into lower i 5. How are low denominations brought into higher I 6. What are the denominations of Halifax currency ? 7. What name is given to the currency of this Province l 8. And why? 9. Are the ratios of the diffi^rent denominations to each oilier the same as in English mo- ney ? 10. Will the rule for reduction of one denominetion to another in Halifax currrency; apply to all currencies in which the denominn- lions arc of the same name? 11. What is the usd of Troy weight and, what are the denominations'? 12. ^avoirdupois weight '{ > the denominations ? 13. What distinction do you make between gross and net weight 1 14. What distinction do you make between long, square, and cubic measure ? 15. What nre the denominations in long mensure 1 16. — ■— square measure? 17. — —in cubic measure '{ 18. How do you multiply by 1-2? 19. When the divisor contains a fraction how do you proceed ? 20. How is the superficial 'contents of a square figure found? 21. How is the solid contents of any body found in cubic measure? 22. How many solid or cubic feri 'of wood maiiC a cord 1 23. What is underslocd by & cnrd foot ? 24. I^owrminy such feet make a cord ? 23. What are the dennmin'itionH ■of dry measure 1 26. of wine measure ? 27. of time ? 28. ■■ of circular measure 1 29. For what is circular measure used / 30. How many rods in length is Gunter's chain? of how many links does it consist 1 how many)inkf> make a rod? 31. How many rods in a mile 1 '32, How many square rods in an acre 1 33. How many pounds make 1 cwt, ? 1IS4.'. •' I SUVPLVMENT TO REDUCTION. at gross. re. the length of eiitlulums. 16" height of lepths at sea. of fish. N. IS reduction ? 3. money, weight, Hht into lower 1 1 6. What ore me is given to re llie ratios of in Enftllsh mo- lion to another the denoniinu- j( Troy weight upois weight l make between make between denominations — ._ in cubic jen the divisor the superticial ilid contents of id or cubic ferl cord foot ? 24. dennminxtiuhii of time? 28. [measure used / ow many linU ow many rods . How many EXERCISES. 1. In 154 dollars, at 6s. each, how many pounds, 6lc. Ans. d8£. lOs. 2. In 36 guineas, at l£. 3s. 4d each, how many crowns, at 5s. 6d. ? Ana. 131 crowns and Ss. lOd. over. 3. How many rings, each weighing 5pwt. 7grs., mfey be made of 3Ib. 5oz. IGpwt. Sgrs. of gold ? Ans. 158. 4. Suppose a bridge to be 212 rods in length, how many times will a chaise wheel, 18 feetGinches in circumference, turn round in passing over it 1 Ans ISOj^ times. 5. In 470 boxes sugar, each 261b., how many cwt.? 6. In 101b. of silver, how many spoons, each weighing loz. lOpwt. ? 7. How many shingles, each covering a space 4. inches one way and (5 inches the other, would it take to cover 1 square foot? How many to cover a roof 40 feet long, and 24 wide? {See ^ 25.) Ans. to the last, 5760 shingles. 8. How many cords of wood in a pile 26 feet long 4 feet wide, and 6 feet high ? Ans. 4 cords, and 7 cord feet. i). There is a room 18 feet in length, 16 feet in width, and 8 feet in height ; how many rolls of paper, 2 feet wide, and containing 1 1 yards in each roll, will it take to cover the walls? ylw^-. 8^g. 10. How many cord feet in a load of wood 6-^ feet long, 2 feet wide, and 5 feet high? Ans. i-^^r cord feet. 11. If a ship sail 7 miles an hour, how fur will she sail, at that rate, in 3vv. 4d. 16h? 12. A merchant sold ]2 hhds. of brandy, at 83 a gallon; how much did each hogshead'come to, and to how much in currency did the whole amount ? 13. How much clot^ at 7s. a yard, may be bought ior 29.£. Is? 14. A goldsmith sold a tankard for 10,£8s. at the rate of 5s. 4d. per ounce ; how much Hid it weigh ? 15. An ingot of gold weighs 21bs. 8oz. 16pwt. ; how much is it worth at 3d. per pwt. ? 16. At 11 pence a pound, what will 1 T. 2cwt. 3qr.s. 161b. "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 sides ; if 3 shingles in width cover one foot in length, how many V, so ADDITION OF COMPOUND NUMBERS. t[ 34, 35. fihinglea will it take to lay one course on this roof? if 3 courses make one foot, how many courses will there be on 0ne side of the' roof? how many shingles will it take to cover one side ? to cover both sides ? Ans. 16020 shingles. 19. How many steps, of 30 inches each, must a man take in travelling 54^- miles ? 20. How many seconds of time would a person redeem in 40 years, by rising each morning ^ hour earlier than he now does? 21. If a man lay up ^ a dollar each day Sundays except- ed, how many pounds would he lay up in 45 years ? 22. If 9 candles are made from 1 pound of tallow, how many dozen can be made from 24 pounds and 10 ounces ? 23. If one pound of wool make 60 knots of yarn, how many skeins, of ten knots each, may be spun from 4 pounds 6 ounces of wool ? Addition of Compound J¥unibers. ^ 93, 1. A boy bought a knife for 9 pence, and a comb for 3 pence j how much did he give for both ? A7is. 1 shil- ling. 2. A boy gave 2s. 6d. for a slate, and 4s. 6d. for a book ; how much did he give for both ? 3. Bought one book for Is. 6d., another for 2s. 3d., an- otlier for 7d, ; how much did they all cost ? Ans. 4s. 4d. 4. How many gallons are 2qts.-j-3qts.-f-lqt. ? 5. How many gallons are 3 qts. -f- 2 qts. -f- 1 qt. + 3 qts. -j- 2qts. ? 6. How many shillings are 2d.-f 3d.-i-5d.-f 6d.+7d ? 7. How many pence are Iqr. -J- 2 qrs. -\- 3 qrs.-j-2 qrs. -flqr.? 8. How many pounds are 4s. -|- 10s. -|- 15s. -{- Is. ? 9. How many minutes are 30sec. -f- 45sec. -f- 20sec ? 10. How many hours are 40 min. -f- 25 niin. -{- (Jmin. ? 1 1. How many days are 4h. 4-8h. -f lOh. -f 20h. ? 12. How many yar<.\. in length are If. -j" 2f. t|- 1^- I ill ■•»■"- ,,,,.-, , ^,-., _.J.^,,,J 5135. > •- ■ ^ ■ * ADDITION or COMPOUND NUMBERS. 'tt^" £. s. d 15 14 6 20 2 8 5 6 4 13. How many feeet are 4 in. -f- 8 in. -f" ^^ >n. -)- 2in.4*> 1 inch? ■: |. • ', ': • • ,,,- •-.■(i' ■ i ■• • .- 14. How much \» the amount of 1yd 2ft. 6in' -{-^ yda. Ifl. 8 inches? 15. What is the amount of 2s. 6d.-|.48. 3d.-|-7s. 8d. ? 16. A man has 2 bottles, which he wishes to fill with wine ; one will contain 2 gal^ 3 qts. 1 pt. and the other 3 qts. ; 4iow much wine can be put in them ? 17. A man bought ahorse for 15c£ 14s. 6d., a pair of oxen for 20£. 2s. 8d., and a cow for 5^. 6s. 4d. ; what did he pay for all ? 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 column of pence, we find the amount to be 18 pence, which being = Is. 6d., it is plain that we may write Ans.^ 3 6 down the 6d. under the column of 1 — pence, and reserve the Is. to be add- ed in with the other shillings. Next, adding up the column of shillings, together with the Is. which we reserved we find the amount to be 23s. =1^. 3s. Setting the 3s under its own column, we add the 1£. with the other pounds, and, finding the amount to be 41^. we write it down, and the work is done. Ans. 4l£. 3s. 6d. Note. It will be recollected, that, to reduce a lower into a higher denomination, we divide by the number which it takes 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- tween shillings andl pounds, for 20, &.c. The above process may .be given in the form of a general KvLK for the Addition of Compound Numbers. I. Write the numbers to be added so that those of the same denominatiun may stand directly under each other. II. Add together the numbers in the column of the lowest denomination, and carry for that number which it lakes of er t'. *i ADDITION OF COMOPUND NUMBERS. '' tl 86. the (fuhe.to tttiike 1 of the next higher denominiation. Pro- ceed in this manner with all the denominations, till you come to: the last, whose amount is written as in simple numbers. : -ii ' Proof, The same as in addition'iof simple numbers, ' ^ V ' HALIFAX CURRENCY. £ s. d, qr. £ s. d. £ s. d. 46 11 3 2' 72 9 U 183 19 4 16 7 4 4 18 10^ 8 17 10 538 19 7 1 36 16 6| 15 4 ' £ 8. d. ' > £ s. d. .£ s. d. " 14 7+ 37 15 8 61 3 2^ 8 15 3 14 12 9f 7 16 8 63 4 7 17 14 9 29 13 10^ 4 17 8 23 10 9i * 12 16 2 23 4f 8 6 7 5f 6 6 7 14 51 24 13 91 10^ 54 2 7J^ 5 lOf 1 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 Federal Money, since that consists of different denominations. In Federal Money the denominations increase and decrease in a dedmal ratio. The pupil is therefore referred to the rules for the Addition, Subtraction Multiplication and Division of Decimals, which are the same absolutely with the rules for the addition, subtraction, multiplication and division of Federal Money. . ' TROY WEIGHT. tb. oz. pwt. ^^• oz. pwt. jn. 9z. pwt. gr. 3(3 7 10 11 6 14 9 18 42 6 9 13 9 G 16 13 16 81 7 16 15. 3 11 10 3 7 4 1135. -3Wf ADDITION QF gOMJPOtND NUMBEAS. 83 Bought a silver tankard, weighing 21b. 3 oz., a silver cup, weighing 3 oz. 10 pwt. and a silver thimble, weighing 2 pwts. 13 grs. ; what was the weight of the whole *? , AVOIRDUPOIS WEIGHT, 2r, T. cwt. qr. lb. oz. dr. 14 11 1 16 5 10 2 11 8 15 7 18 25 11 9 cwt. qr. lb. oz. dr, 16 3 18 6 14 3 16 8 12 22 11 10 / A pnan bought 5 loads of hay, weighing as follows, viz, 23 cwt (=: 1 T. 3 cwt.) 2 qrs. 17 lb. ; 21 cwt. 1 qr. 19 lb. ; 19 cwt. qr. 24 lb. ; 24 cwt. 3 qr.; 11 cwt. qr. 1 lb. ; how many tons in the whole ? CLOTH MEASURE. ^ yds. qr. ??. 36 1 2 41 2 3 (}5 7 E. F. qr. na. 41 1 2 57 5 8 57. 3 EE qr. nn: 75 4 2 35 7 C 28 3 1 There are four pieces of cloth, which measure -s follows, viz., 37 yds. JJ-qrs. 1 na. ; 18 yds. 1 qr. 2na. ; 40yt's. 3qrs. 3 na. ; 12 yds. qr. 3 na. ; how many yards in the whole ? LONG MEASURE. deg. mi. fur. r. ft. in. bar. ;>9 46 6 29 15 10 2 246 39 1 36 14 6 I 678 53 7 24 9 7 1 mi. fur. pol. 3 7 8 6 27 iia; m ADDITION OF COMPOUND NUMBERS. Tras. LAND OR saUARE MEASURE. Pol. ft. in. A. rood. pol. ft. in. 36 179 137 56 3 37 245 228 19 248 119 29 1 28 93 25 12 96 75 416 2 31 128 119 There are 3 fields which measure as follows, viz. 17 A. ar. 16p. : 28 A. 5r. 18p. ; UA. Or. 25p.; how much land 4in the three fields ? ^^ SOLID OR CUfiC MEASURE. Ton. ft. in. yds. ft. in. cords, ft. 29 36 1229 75 22 1412 37 119 12 19 64 9 26 195 4 110 8 11 917 3 19 1091 48 127 WINE MEASURE. hhds. gal. qts. pts, 50 53 1 7 27 39 3 9 13 1 tun. hhd. gal. qts, 37 3 44 5 19 1 50 1 28 2 Sll ![3 for Is. 2. A how m 3. A how m 4. bJ A merchant bought two casks of brandy, containing as follows, viz. 70 gal. 3 qts. ; 67 gal. Iqt. j how many hogs- heads of 63 gal. each in the whole? DRY MEASURE. Bush. p. qt. pt. 36 2 5 1 19 3 7 Ch. bus. p. qt. 48 27 3 5 6 29 1 7 r ■ \:-} 3i 1135. ^ S6. SUBTRACTION OP COMPOUND NUMBERS. TIME. Y. mo. IP. d. Ji. m. s. 75 11 3 6 23 55 11 84 9 2 16 42 18 32 6 5 5 18 5 27 8S Y. mo. w. d. 40 3 1 5 16 7 4 2 nz. 17 A. nuch land cords, ft. 37 119 4 no 48 127 5 1 Intaining as lany hogs- qt. 5 7 Siibtraetion of Compound JVnmbcris ^ 36. 1. A boy bought a knife for 9 pence, and sold it for Is. 4d. ; how much did he gain l-y the bargain? 2. A boy bought a slate for 2s. 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. 6d. ; how much did he then owe him? 4. Bought two books; the price of one was 4?;, 6d.. the price of the other 3s. 9d, ; what was the difference of their costs? 5. A boy lent 5s. 3d.; he received in payment 2s. 6d. ; how much was then due ? 6. A man has a bottle of wine containing 2 gallons and 3 quarts ; after tumiug out 3 quarts how much remained • 7. How much is 4 gal, less 3 gal.? 4 ga^. — (less) 2qt. ? 4 gal.— Iqt. ? 4 gal. — 1 gal. Iqt. ? 4 gal. — 1 gal. 2qts ? 1 4 gal, — 1 gal. 3qts ? 4 gal. — 2 gal. 3qts? 4 gal. 1 qt. — 1 gal. 3 qts.? 8. How much is 1ft. — (less) 6in? 1ft. — 8 in 7 6ft, 3 [inches, — 1 ft. 6 in. 7ft. Sin. — 4ft. 2in? 7ft. Sin. — 5ft, lOin?. 9. What is the difference between A£ Qi. and 1^ ^'s. ? 10. How much is 3oe~(less) Js.? 3i:— 2s. 3i:— 3s. ? \U —15s ? Z£ 4s.-^2^ 6s ? 10^ 4s. -^£ 8s .' 11. A man bought a horse for 30=^4'^. 8d., and a cow IforS.^ 14s. 6d. ; what is the difference of their costs? H ^ ' ■^ti m m 86 STTBTRACTION OF COMPOUND NUMBERS. 1136. js ■ ,\iiU% OPERATION. £. s. d. Minuend, 30 4 8 Subtrahend y 5 14 G As the two numbers are large, it will be conveniertt to write them down, the less under the greater, pence under pence, shil- lings under shillings, &c. We Ans. 24 10 2 \ may now take 6d. from 8d., and there will remain 2d. Proceeding to the shillings, we can- not take 148 from 48., but we may borrow as in simple num- bers, one from the pounds,=20s., which joined to the 4s, makes 24s. from which taking 14s. leaves 10s, which we iset down. We must now carry 1 to the 5^ making Q£ which taken from 30=£ leaves 24<£ and the work is done. Note. The most convenient way in borrowing is, to t»ub- tract the subtrahend from the figure borrowed, and add the difference to the minuend. Thus, in the above example, 3 4 from 20 Jeaves 6, and 4 is 10, The process in the foregoing example may lie presented in the form of a Rule for the Subtraction of Compound Numbers. I. Write down the sums or quantities, the less under the greater, placing those numbers which are of the same de- nomirlation directly under each other. II. Beginning with the least denomination, take succes- isively the lower number in each denomination from theup- ))er, and write the remainder underneath, as in subtraction of simple numbers. III. If- the lower number of any denomination be greater than the upper, borrow as many units as make one of the next hiijher denomination, subtract the lower number there- from, and to the remainder add the upper number, remem- bering always to add one to the next higher denomination for tl;.at which you boirv^wef^. Proof Add the vemainder and the subtrahend together, as in subtraction of simple numbers; if the work be right, the amount will be equal to the minuend. ;examples for practice. HALIFAX CURRENCY. £. s. d. £. s. d. 79 17 8 103 3 2 35 12 4 71 12 5 fl36. SUBTRACTION OP COMPOUND NUMBERS. 87 £ s. d. 81 10 11^ 29 13 3 tB s. d. 245 12 27 9 4^ 520 U 3 109 17 4 631 14 7 6 19 9 MISCELLANEOUS EXAMPLES. 1. A merchant sold goods to the amount of 136.£ 7s. 6|d , and received in payment 50<£ 10s. 4f d ; how much remain- ed due ? Ans. 85<£ 17s. If d. 2. A man bought a farm for 1256<£ 10s, and, in selling it, lost 87.£ 10s. 6d ; how much did he sell it for ," ^ws. 1168^ 19s. 6d. 3. A man bought a horse for 27^ and a pair of oxen for 19c£ 12& 8^d J how much was the horse valued more than the oxen 1 4. A merchant drew from a hogshead of molasses, at one time, 13gal. 3qts; at another time, 5gal. 2qts' Ipt; what v riON. OPERATION, £ s. d. qr. G )7 14 4 'Z cost of (S yards. £ s. d. qr. 15 8 fS price of \ yard 6 number of yds. Ans. 7 14 4 2 cost of 6 yards 6 times 3 qrs. are ISqrs.rr 4d. and 2qrs. over; we set down the two qrs; then, G times 8d. are 48d, and 4 to carry makes 52d. = 4s. and 4d. over, which we write down ; a^ain 6 times 5s. are 30s. and 4 to carry makes 34s. = 1<^ and 14s. over; G times 1^ are 6£, and one to carry makes 1£, which we write down, and it is plain, that 15 8 'ii price of 1 yarc\ Proceeding after the man- ner of short division, 6 is con- tained in 1£ 1 time, and l£\ over; we write down. the | quotient, and reduce the re- mainder {]£) to shillings, (20s,) which, with the given I shillings, (I4s,) make 34s: G in 343. goes 5 times, and 4s. over ; 4s. reduced to pence] :48d, which with the giv- en pence, (4d,) make 52d ; d in 52d. goes 8 times, and 4d,( the united products arising|over ; 4d, = 16 qrs. which, ION. ^^- n^- /,.. OP COMPOUND NUAIBSRS. 91 Ts, what is that per each measuring 10 e? I 32yus. 1 qr., how itaining 2 gal. 1 qt. aii^? ine, which he would ch must he put into les 8d?— '3X9(1? lOXDd? ings? — ^ of 29 3(11 . ^of3s. 0(1?— T^o ' G yards of cloth cost 4^d, what is the price I? ill be most convenient and dividing. TION. d. qr. 4 2 cost of 6 yards. 8 ^price of 1 yar(\ eding after the man- lort division, G is con- 7^ 1 time, and li ! write down, the and reduce the re- (Ji:) to shillings, hicb, with the given (I4s,) make 34s; goes 5 times, and 4s. reduced to pence [which with the giv ;, (4d,) make 52d ; ' roes 8 times, and 4(1. 'td. = 16 qrs. which, from the several cknomina- tions is the real product aris- ing from the whole compound number. . ,; , .. . U. Multiply ^£ 4s. Gd. by 7. 13. What will be the cost of 5 pairs of shoes at lOs. Gd. a pair ? 15. In 5 barrels of wheat, , each coritaining 3 bus. 3 pks. Gqts, how many bushels ? 17. IIoW many yards of cloth will be required for 9 coats, allowing 4 yards Iqr. 3na. to each ? 19. In 7 bottles of wine, each containing 2(|ts. Ipt. 3 gills, how many gallons ? 21. What will be the weight of 8 silver cups, each weighing 5oz. 12pvvt 17grs? 23. How much sugar in 12 hogsheads, each containing 9cwt. 3qrs.211b? 25. -In 15 loads of hay, each weighing IT. 3cwt. 2qrs. how many tons ? with the given qrn. (2) =s= 18 ([rs; G ^n IBqrs. gies 3 times and it is plain, that the unit- ed (quotients arising from the several denominations, is the real quotient arising from the whole compound number. 12. Divide 22je lis. 6d. by 7. 14, At2£ 12h 6d. for 5 pairs of shoes, wli • is that a pair : IG. If 14bu8.'^^ -,. Gqts.of wheat be e(iually divided into 5 barrels, how many bushels will each contain ? 18. If 9 coats contain 39 yds. 3qrs. 3na, what does 1 coat contain ? 20. If 5 gal. 1 gill of wine be divided equally into 7 bot- tles, how much will each con- tain ? 22. If 8 silver cups weigh 3lb. 9oz. Ipwt. IGgrs., what is the weight of each I 24. If 119cjvt. Iqr. of su- gar be divided into 12 hogs- heads, how much will each hogshead contain? 26. If 15 teams be loaded with 17T. 12cwt. 2qrs. of hay, how much is that to each team? When the multiplier or divisor, exceeds \2, the operations of multiplying and dividing are not so easy, unless they be composite nnmbers ; in that case, we may make use of the component parts, or factors, as was done in simple numbers. Thus 15, in the example 15 being a composite num- above is a composite number, ber and 3 and 5 its compo- ptoduced by the multiplica- nent parts, or factors, we miy I IMAGE EVALUATION TEST TARGET (MT-3) 1.0 1.1 U£i28 |2.5 •^ IM III 2.2 2.0 U! ■ 4.0 1.8 1.25 1 ,.4 , ,.6 ^ 6" ► vl /: ^ei ^J ■/A ^5^V# '/ Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716)872-4503 'r.^^r ■J MULTIPLICATION AND DIVIBION 1138. w tkm of 3 and 5, ( 3X 5 = 16.) We may therefore, multiply IT. 3cwt. 2qrs. by one of those component parts, or factors, and that product by the other, which will give the true answer, as has been al ready taught, (H 11.) OPERATION. T. evDt. qK ' / 1 3 2 3 one of tkef actors. 3 10 2 5 the other factor. divide 17T. 12cwt. 2qrs. by one of these component parts or factors, and the quotient thence arising by the other, which will give the true an- swer, as already taught, (1120.) 17 12 2 the answer. 27, What will 24 barrels of flour cost, at 2jS, 12s. 4d. a barrel ? 29. What will 1121b. of su- gar cost at 7:^d. per lb? Note. 8, 7, and 2, are fac- tors of 112. 31. How much brandy in Omfactor, OPERATION. T. ciot. jr 3)17 12 2 The other factor ^S)& 17 2 MSf 1 3 2 28. Bought 24 barrels of flour for6S^ 16s; how much was that per barrel? 30. If Icwt. of sugar cost . 3jB, 7s. 8d., what is that per ' lb? 82. \ Bought 84 pipes of 84 pipes, each containing 112jbrandy, containing 9468 gal. gal. 2qts. lpt^3g? 33. What wUl ISQyds. of cloth cost, ait 3£, 66. 5d. per yard? 139 is not a composite num< ber. We may, however, de- compose this number thus, 139^100+304-9. We may now multiply the price of 1 yard by 10, which will give the price of 10 yards, and this product again by 10, which will give the price of 100 yards. Iqt. Ipt ; how much in a pipe ? 34. Bought 139 yards of cloth for 461£ lis. lid; what was that per yard ? When the divisor is such a number as cannot be produced by the multiplication of small numbers, the better way is to divide after the manner of long division, setting down the work of dividing and re* ducing in manner as follows : -.i4t'ii-"-i-i. ; ^k'yiiijj-"-. .,.Ji'.s.i. ■; ,,jj(i*.,:i.j»^^-.j-.,'.'4- i^'.w.j 1138. OP COMPOUND NUMBERS. d3 We may then fnultiply the price of 10 yards by 3, which will give the 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 evidently give the price of 139 yards ; thus : £ s. d. 3 6 5 price of J t/ard, 10 33 4 2 price of 10 yards. 10 332 1 8 price of \00yds. 99 12 6 price of QOyds. 29 17 9 price of 9 yds. 461 11 U price of \2» yds. 30 yards, and in multiplying the price of 1 yard {3£ 6s. 9 yards, the multipliers, 3 and 9, need not be written down, but may be carried in the mind. £ 5. d. 139)461 11 11(3.^ 417 lit 30 , ■•;) ,b9\(Q 834 57 695 {5d. 695 i . • r?. Note. In multiplying the price of 10 yards (33^ 4s 2d.) by 3, to get the price of]6 times, (6s,) and a remainder The divisor, 139, is contain- ed in 461^ 3 times (3^,) and a remainder of 44^, which must now be reduced to shil* lings, multiplying it by 20, and bringing in the given ^hil* lings, (lis,) making 891s, in which the~ divisor is contained of 57s, which must be reduc- ed to pence, multiplying it by 5d.) byfi, to get the price of 12, and bringing in the given pence, (lid,) together mak- ing 695d, in which -the divi- sor is contained 5 times, (5d,) and no remainder. The several quotients, 3£ I. 5d. evidently make the answer. The processes in the foregoing exaropWs mayjaow be pre- sented in the form of a ' KvLEfor the Multiplication ofKvhEf/tr the Division of Com' Compound Numbers. 1. When the multiplier does not exceed 12, multiply suc- cessively the numbers of each denomination, beginning with pound Numbers. 1. When the divisor does not exceed 12, in the manner of short division, find how many times it is contained in ;■■+' 04 MULTIPLICATION AND DIVISION 1138. the least, as in multiplication of simple numbers, and carry as in addition of compound numbers, setting down the whole product of the highest denomination. II. If the multiplier exceed 12, and be a composite num- ber, we may multiply first by one of the component parts, that product by another, and s5 on, if the component parts be mere than two; the last product will be the product re- quired. III. When the multiplier exceeds 12, and is no^ a com- posite, multiply first by 10, and this product by 10, whic|i will give the product for 100 ; and if the hundreds in the mul- tiplier be more than one, mul- tiply the product of 100 by the number of hundreds ; for the tens, multiply the product of 10 by the number of tens ; for the units, multiply the multi- plicand ; and these several pro- ducts will be the product re- quired. the highest denomination, un- der which write the quotient, and if there be a remainder, reduce it to the next less de- nomination, adding thereto the number given, if any, of that denomination, and divide as before; so continue to do through all the denominations and the several quotients will be the answer. II. If the divisor exceed 12, and be a composite, we may di- vide first by one of the com- ponent parts, that quotient by another, and so on, if the com- ponent parts be more than two, the kst quotient will be the quotient required. III. When the divisor ex- ceeds 12, and is not a compos- ite number, divide after the manner of long division, set- ting down the Work of divid- ing and reducing. I 'examples for PRACTICE, \ « E^ALIFAX CURRENCY. ' £ ,S. d. £ 5, d. Multiply 81 6 5 93 4 11 by f 17 » ) ' 48 Vi*^--.^:.-: 1138. 1 ^38 39. OF COMPOUND NPKBERS. 95 )mination, un- 5 the quotient, a remainder, next less de- ling thereto the if any, of that and divide as tntinue to do denominations i quotients will jrisor exceed 12, site, we may di- le of the com- tiat quotient by on, if the com- be more than iiotient will be quired. the divisor ex- s not a compos- 1 iivide after the ig division, set- Work of divid- ing. 5. d. 4 11 48 L- , £ s. d. Mult.p]y98 3 10 by^ 78 £ s. d. 64 11 2 986 11 4 73 93 93 ' ' , 892 5 3 145 Divide 77 11 9 by 18. « 140 2 3 " 21. " 360 5 2" 133. " 7856 8 9" 197. £ s. d. 143 2 3 by 21. 1950 7 4" 98. 47 9 6 " 11. 562 8 3" 20. MISCELLANEOUS EXAMPLES. at 4s. 7^d. per 1. What will 359 yards of| 2. Bought 359yds. of cloth for83.£0s /•J-d; what was that a yard ? 4. If 441r.wt. 131b. of flour be contained in 241 barrels^ how much in a barrel ? 6. If 37lbu. Ipk. of wheat be divided equally into 135 bags, how much will each bag contain ? 8. At 759^ 10s. for 35cwt. of tobacco, what is that per lb? 10. If 14 men build 92 rods 12 feet of stone wall in 74 days, how much is that per day? cloth cost, yard? , 3 In 241 barrels of flour, each containing Icwt 3qr. 91b; how many cwt? 5. How many bushels of wheat in 135 bags, each con taining 2 bu. 3 pks ? 3X9X5=135. 7. What will 35cwt. of to- bacco cost, at 38 lO^d. per lb? 9. If 14 men build 12 rods 6 feet of wall in one day, how many rods will they build in 7^ days ? II 39. 1. At 10s. per yard, what will 17849 yards of cloth cost ? Note. Operations in multiplication of pounds, shillings, pence, or of any compound numbers, may be facilitated by I 96 MULTIPLICATION AND DIVISON, &,C. TI39. N taking aliquot parts of a higher denomination. Thus, in this last example, if the price had been 20s. i. e. \£ per yard, it is clear, the price of the whole would havq^been equal t» the whole number of' yards in pounds, 17849; but the price is 10s. i. e. ^£ per yard, and so the price of the whole will be equal to ^ the number of yards, ^'^f*^ in pounds; 8924i£, or 8924^: 10s. When one quantity is contained in another exactly 2, 3, 4, 5, &;C. times, it is called an aliquot or even part of that quantity ; thus 6d. is an aliquot part of a shilling, because 6d.X2=l shilling; so 3d. is an aliquot part of a shilling; 3d. X4=ls. ■ So 5s. is an aliquot part of a pound, for 5s. X4=lc£: and 3s. 4d. is an aliquot part of a pound, for 3s. 4d.XG=loe, &c. ' From the illustration of the last example it appears, that, when the price per yard, pound, &,c. is one of these aliquot parts of a shilling, or a pound, the cost may be found by dividing the given number of yards, pounds, i^c. by that number which it takes of the price to make Is. or 1£. If the price be Gd. we divide by 2 ; if 5s. we divide by 4 ; if 3s. 4d. by 6, fee. &:-c. This manner of calculating by ali- quot parts, is called Practice. 2. What cost 34648 yards ^f cloth, at 10s. or ^i^ per yard I at 5s.=r^^ per yard ? at 4s.=^,£ per yard? at 3s. 4d.=^^ per yard? at 2s.:=j^^£ per yard ? Ans. to last, 3464c£ 1 6s. 3. What cost 7430 pounds of sugar, at 6d.=: J^s. per lb ? at 4d.==^s.per lb? at 3d.=;^s per lb? at 2d.=^s. per lb ? at l^d.=|s. per \h? Ans. to the last, 7 4^0s.=928s. 9d.=46^8s. 9d. 4. At 3i? 16s. per cwt, what will 2qrs.=J^cwt. cost? what will lqr.:fc^cwt. cost? what will ]61b.= |cwt. coat ? what will 141b.=^cwt. cost ? what will 81b.=^\cwt. cost? Ans. ^o the last, 5s. 5|d. 5. What cost 340 yards of cloth, at 12s. 6d. per yard ? 12s. 6d.=10s. {=^£) and 2s. 6d. (=i£) ; therefore, ^)l)340 170<.€ := cost at 10s. per yard. 42^ 10s.=at 2s. 6d. per yard. Ans. 2\2£ M)s.=at I2s. 6d. per yard. 1139. 'hus, in this £ per yard, len equal t» •ut the price ■ the whole in pounds ; exactly 2, 3, part of that ng, because f a shilling ; »und, for 5s. a pound, for ippears, that, these aliquot be found by &>€. by that is.orl^. If vide by 4 ; if lating by ali- )s. or ^^per 4s.=|^ per at ^s.=j^xj£ ,3464.£ 16s. zr J^s. per lb 1 lb? -A6£ 8s. 9d. ^cwt. cost? will 161b.= ;? what J last, 5s. 5|d. |6d. per yard ? ^erefore, rd. rd. ird. 1139. 8VPPLBMENT TO COMPOUND NtJMBESfl, t Or, ,, , ...V.-..V 97 iOB.=^£)MO 2g. 6d=i of I0f,)170j6' at 10s. per yard. 42^ lOg. at2s. 6d. perybd. Ans. 212^. 10s. at 12s. 6d. per yard. SUPPLEMENT TO ^COMPOUND NUMBERS. . QUESTIONS. 1. What dbtinetioti do you make between slmpltf nnd eompwM numbers 1 (P 26.) 2. What is the«ule for addition of coDspouiid numbers ? 3. -—— for subtraction of, &,e. 4. There are three roii'* ditiona in the rule giten for tRuItipticaliiiii of cc^pdund htimlfer« i what are they, and the methods of procedure under each? 6. The same questions in respect ttf the dttision of (Sompound numbers f 0. Whan the multiplijpr or divisor is encumbered with a fraction, how d& you proceed t 7. How is the distance of time from one date ttf another found ? 8. How many degrees dbes the earth re^^olva frotn Wfest to cast in 1 hour t 9. In what tiine does it revolte ]e 1 Where Is the time or hour of the day earlier-^at the place most easterly of most westerly ? 10. The difference in longitude between two placet) b^ingl known, how ii the difference in time calculated? 11. How nfay operations^ in the fhultiplieation of compound numbers be fa« eilitated 1 12. What are some of the aliquot parts of £l f «•— of Is. 1 " of Icwtl 13. What is this manner of operating usually called 1 EXERCISES. 1. A gentlennan 13 possessed of l^dciz«tl of silver, spodtts, each weighing 3oz. 5pwt; 2 doz. of tea spoons^ eaeh weigh* ing ]5pwt. 14gr; 3 silver cans, «ach 9oz. 7pwt ; 2 «il?#r tankards, each 21oz. I5pwt ; and 6 silver .porrmgefrs^each Uoz. ISpwt; what is the weight of the whol^? >4rt5; I8Ib. 4o«. 3pwt. Note. Let the pupil he required to reverse and prove the following examples : 2. An English guinea should #eigh ^pwf. Cgr,' a p\eee' of gold weighs 3pwt. 17gr ; how buch is that short of the tveight of aguinea? 3. What is the weight of 6 chests of iei, each wtfighiitf dcwt. 2€(rs. 91b ? 4. In 35 pieces df cloth, eaek ffleafufing S7 j^rdSfhcw many yards ? 9d SUPPLEMENT TO COMPOUND NUMBERS. H ^f ^0. 5. How much brandy in 9 casks, each containing 45 gal. 3qts. Ipt? 6. If 31cwt. 2qrs. 201b. of sugar be distributed equally into 4 casks, how much will each contain .' 7. At 4^. per lb. what cost Icwt. of rice ? 2cwt ; 3cwt? Note. The pupil will recollect that 8, 7, and 2 are fac- tors of 112, and may be used in place of that number. 8. If 800cwt. of cocoa cost 18i£ 13s. 4d. what is that per cwt 1 what is it per lb. ? 9. What will 9:|cwt. of copper cost at 5s. 9d. per lb ? 10. If 6j^cwt. of chocolate cost 72ntreal and 1140. FRACTIONS. 99 expressing rs; bat that jak a whole td fractions, It will be recollected, (f[ 14, ex. 11,) that when a thing or unit is divided into 3 parts, the parts or fractions are call- ed thirds ; when into four parts, fourths ; when into six parts, sixths ; that is, the fraction takes its name or denomination from the number of parts into which the unit is divided. Thus if the unit be divided into 16 parts, the parts are called six- teenths, and 5 of these parts would be 5 sixteenths, expressed thus, ^. The number below the short line,. (16,) as before taught, (1[ 17,) is called the denominator, because it gives the name or denomination to the parts ; the number above the line is called the numerator, because it numbers the parts. The denominator shows how many parts it takes to make a unit or whole thing; the numerator shows how many of these parts are expressed by the fraction. 1. if an orange be cut into 5 equal parts, by what frac- tion is 1 part expressed ? 2 parts ? — — 3 parts ? — — 4 parts? ^ 6 parts ? how many parte make unity or a whole orange ? 2. If a pie be cut into 8 equal pieces^ and 2 of these pieces be given to Harry, what will be his fraction of the pie? iif 5 pieces be given to John, what will be his fr^tion t what fraction or part of the pie will be left ? It is important to bear in mind, that fractions arise from division f (If 17,) and that the numerator maybe considf^eda dividend, and the denominator a divisor, and the ralue of the fraction is the quotient ; thus, ^ is the^. quotient of 1 (^. numerator) divided by 2 (the denonainator ;) ^ is^tke quo- tient arising from 1 divided by 4, and f is 3 times as much, that is, 3 divided by 4 ; thus, one fourth part of ^ is the same as 3 fourths of 1. ,; Hence, in all cases a frftotjon is dtways expressed by the sign of division. f expresses the quotient, ( 3 ittlte4iyidend, omumerator, of which ( 4 is the divisor or denomiiiator. 3. If 4 oranges be equally divided among 6 boys, what part of an orange is each boy'-s share ? A sixth part of an orange is ^, and a sixth part of 4 oranges is 4 such pieces,=s5;|. Ans. | of an orange. 4. If 3 apples be equally divided among 5 boys, what part of an apple is each boys share 'f if 4 apples, what ? if 2t apples, what ? if 5 apples, what ? 100 FRACTIONS. H 40, 41. 5. What is the quotient of 1 divided by 3 ?-^— of 2 by 3 ? ^— K)f 1 by 4? ;of 2 by 4? of 3 by 4? of 5 by7? -of 6 by8? of4by5?— of2 by 14? 6. What part of an orange is a third part of 2 oranges ? one fourth of 2 oranges ? ^ of 3 oranges ? ^of threeoranges? j of 4? Jof 2? -- — f of 5? -^|of3? iof2? A proper fraction. Since the denominator sho\(rs the num- ber of parts necessary to make a whole thing, or 1, it is plain that when the numerator is less than the denominator, the fraction is less than a unit, or whole thing ; At is then called ^proper fraction. Thus, 1, |/&'C. are proper fractions. An improper fraction. When the numerator equals or ex- ceeds the denominator, the fraction eaualsor exceeds unity, or 1 ,' and is then called an imprmer fraction. 'J^hns, f , ^^ f , V^, are improper fractions. A milled number ^ as already shown, is one composed of a whole number and a fraction. Thus, 14j, 13|, d&c. ate ^ixed numbers. * 7. A father bought 4 oranges, and cut each orange into 6 equal parts ; he gave to Samuel 3 pieces, to James 5 pieces, to Mary 7 pieces, and to Nancy 9 pieces ; what was each one's inaction ? Was James' fraction proper or improper ? WJiy? ''^^ Was Nancy's fraction proper, or imprq>er? Why? To change an improper fiaction to a whole or mixed number. IT 41. It is evident that every improper fraction must contain one or more whole ones, or integers. 1. How many whole aisles are there in 4 halves (f) of ^ apple? inf?-: — Uif.^ y? ? in ^ in V? in »f«? in»f*? — in I of a yard ? in V? in\^ 2. How many yards in f of a yard ? — -inf? inf» in^? — ^ — in ^r I in 9^ ? in V ? 3. How many bushels in 8 pecks? that is, in f of a bush* el f in V> ? — in V ? in V* *? in \* ? int^o? inV? This finding how many integers, or whole things, are contained in any improper fraction is called reducing an improper fraction to a wMe or mixed number. S! '< y SdT-^ri^-. 1141,42. FRACTIONS. 101 4. If I give 27 children j- of an orange each, how many oranges will it take ? It will take ^/ ; and it is evident, OPERATION. that dividing the numerator 27, (=: the 4)27 number of parts contained in the frac- tion,) by the denominator 4, (= the Ans. OJ oranges, number of parts in 1 orange,) will give the number of whole oranges. Hence, To reduce an improper fraction to a whole or mixed number, — Rule : Divide the numerator by the dei|ominar tor; the quotient will be the whole or mixed nnmber. EXAMPLES FOR PRACTICE. 5. A man, spending ^ of a pound a day, in 83 days would spend ^ of a pound ; how many pounds would that be ? Ans. l^£. 6. In *|^^ of an hour, how many whole hours? TheGOthpartof an hour is a minute; therefore the ques- tion is evidently the same as if it had been, in 1417 min- utes, how many hours? Ans. 23g^ hours. 7. In ^{^^ of a shilling, how many units or shillings.^ • Ans. 730^ shillings. 8. Reduce *m® to a whole (w mixed number. 1). Reduce f^, '^,Uh iiU, \W, to whole or mixed numbers. To redtkcc a whole or mixed number to an improper fraction. IT 4:^8. We have seen, that an improper fraction may be changed to a whole or mixed number ; and it is evident that by reversing the operation, a wholc^ or mixed number may be changed to the form of an improper fraction. 1. In 2 whole apples, how many halves of an apple ? Ans. 4 halves ; that is f . In 3 apples how many halves ? in 4 apples? in 6 apples? in 10 apples? in 34? in 60? iu 170? in 492? 2. Reduce 2 yards to thirds. Ans. f , Reduce 2f yards to thirds. Ans. f . Reduce 3 yards to thirds - 3§ yards. 5 yards. 5f yards.- Reduce 2 bushels to fourths. — ^ — 2# bu. — G bush- 3 els. 4 -3^ yards. -G^ yards. •6^ bushels. — -7f bushels. 25f bushels. In IGy^ pounds, how many y'^ of a pound ? make 1 pound : if therefore, we multiply 16 by 12, that is, multiply the whole number by the deaominatcr, the J.2 12 lOS PRACTIONt. IT 42, 43 product will be the number of ISths in IQ£ : 16X 12=1U3 and this, increased by the numerator of the fVaction, (5,) evidently gives the whole number of 12ths ; that is, t^ut' a pound, Ans. OPBRATION. 16 Vb pounds 12 192=12th8 in 16 pounds, or the whole number. 5aal2th8 contained in the fraction. \97=z\^., the answer. < Hence,^ Te reduce a mied number to an improper frac- tion, — Rule : Multiply the whole number by the denomin- ator of the fraction, to the product add the numerator, and write the result over the denominator. i E^XAMPLES FOR PRACTICE. 5. What is the improper fraction equivalent to 23^^ hours? ilni. »|^7 of an honr. , 6. Reduce 730^^ shillings to 12ths. As ^ of a shilling is equal to 1 penny, the question id evidently the same as, in 730s. 3d., how many pence ? Ans. ^ff ^ of a shillintr ; that is 8763 pence. 7. Reduce l^f , 17|f , 8/^, 41^,^^^, and K^ to impro- per fractions. ' ^ 8. In 1562j^|^ days, how many 24ths of a day ? Ans. 3Jfi =3701 hours. 9. In 342f gallons, how many 4ths of a gallon ? Ans. ^^^ o{ a gallon==1371 quarts. To reduce a fraction to its lowest or most simple terms. ^ 4«l. 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 j . The effect on the fraction is evidently the same as if we had multiplied both of its terms by 2. In either case, theparta are made two times as many as they were before ; but thy are only half as large ; for it will take 2 tiflies as many fourths to make a whole one as it will take halves ; and hence it is that f is the same in value or quantity as 4. 1^ 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 : t a-'^is-'.^A-iiA'- f ^i-^ 1143. nUCTlONB. 103 multiplied by 3, it becomes | . Hence, Jf=faB=|, and the reverse of this is evidently true, that |=>|=^. It follows therefore, by n^ultiplying or dividing both terms of the fraction by the same number , toe change its terms without alttringits value. Thus, if we reverse the above operation, and divide both terms of the fraction | by*3, we obtain its equul, ^ ; divid- \Vi\t again by 2, we obtain ^ , which is the most simple form of the fraction, because the terms are the least possible by which the fraction can be expressed. The process of changing | into its equal y, is called redu- cinjr the fraction to its lowest terms. It consists in dividing both terms of the fraction by any number which will divide them both without a remainder , and the quotient thence aris' ing in the same manner, and so on, till it appears that no number greater than I will again divide them. A number which will divide two or more numbers with- out a remainder, is called a common divisor, or common meas- ure of those numbers. The greatest number that will do this is called the greatest common divisor. • . J. What part of an acre are 128 rods ? One rod is t^tt <>** an acre and 128 rods afc -f jJg of an acre. Let us reduce this fraction to its lowest terms. We find, by trial, that 4 will exactly measure both 128 and 160 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 fraction to its equaj ^, which is now in its lowest terms, for no greater number than 1 will again measure them. The operation may be presented thus : ■• ■' " S) ■ vl28 32 4 ^ *) Too = 40=^5 ''^ ^" ^'''y '""''• 2. Reduce 1^^, /^V, j|g, and j^fff to theii^ lowest terms. ^«3. h i, }, ii"d l Note. If any number ends with a cypher, it is evidently divisible by 10. If the two right hand figures 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 divisible by 5. • 3. Reduce |^^, ^^^, ^f ^, and §^ to their lowest terms. hi 104 PftACTlONS. !I44. IT 4-i • Any fraction may evidently be reduced to its low- est terms by a single division, if we use the greatest common divisor of the two terms. ^ The greatest common measure of any two numbers may be found by a sort of trial easily made. Let the numbers be the two terms of the fracticii ^ff. The common divisor cannot exceed the less number, for it must measure it. We will try, therefore, if the less number, 126, which measures itself, will-also measure or divide 160. 128)160(1 128 '■'flVr'^ Ci* 128 in 160 goes 1 time, and 32 re- main; 128, therefore, is not a divisor of — 160. We will now try whether this re- 32) 128(4 mainder be not the divisor sought ; for if 128 32 be a divisor of 128, the former divi- sor, it must also be a divisor of 160, which consists of 128 -|^32. 32 in 129 goes 4 times, without any remainder. Consequently, 32 is a divisor of 128 and 160. And it is evidently the greatest common divisor of these numbers; for it must be contained at least once more in 160 than in J 28, and no number greater than their difference, that is, greater than 32, can do it. > '• -^^ v .^;# Hence, the rule for finding the greatest common divisor of tido 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 divisor will be the greatest common divisor required. Note. It is evident, that,^when we would find the great- est common divisor of more than two numbers, we may first find the greatest common divisor of two numbers, and then of that common divisor and one of the other numbers, and so 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 the fraction f |, and, by it, reduce the .fraction to its lowest terms. OPERATION. ^^ 21)35(1 ^ . ^ 14)21(1 14 Greatest divis. 7)14(2. 14 Then, 7)H=|in5. -^.i;, f|44. ■ fl44,46. PKACTIOIffl. lOS r4. /*' ••».J' 4>w. ^. . Ans. J 6. Reduce ^ to its lowest terms. *^^»^'^*^^« :4„5. j^. iVb^e. Let these examples be wrought by both methoas; bj several divisprs^ and also by finding the greatest common divisor. ' ' V ■ V I s- 6. Reduce -^^ to its lowest terms., 7. Reduce ||f to its lowest ^rms. ! 8. Reduce ^^ to its loweil terms. ' 9. Reduce j^ff | to its lowest terms. Tq divide afrtution by a vfkoh number. />^^ •. 5| 45. 1. If 2 yards of cloth cost f of & pounds what does 1 yard cost? how much is :§ divided by 2i ' 1'^'^''^ 2. If a cow consume f of a bushel of meal in 3 days^ 1)0W much is that per day .' f H- 3= how much ? 3. ff a boy divide | of an braqge among 2 boys, how much ^ill^he give each one ? f-?- Se^how mueh / 4. A boy iKrtight 5t cakes for f} of a shilling; ivhat did I cake cost? •fjJ-J-Srshow much? -^ ■ ,..;?' ij * 5. If 2 bushels of apples cost -^ of apouhd, what latitat per bushel ? 1 bushel is the half of 2 bushels; the half ^ is ^, »> ^^mff «;»;«'.,H:.-^mn«^-l»,;. ^.*■UH* -mi ^ b.v. ■ j^^ -ji^ pouiid. 6. If 3 horses consume |f of a ton of hay in a months vAiti will 1 horse consume in tl^^aan^e tJm^} •ff are 13 parts ; if 3 horses consume 12 such parts hi a month, as many times as 3 are contained in 12, so many parts 1 horse wUl consume. '^- ''« *« r ^ Ans. -^ of ato»n. 7. If f § of a barrel of flour be divided equdly among 5 families, how much will eaph family receive ? || is 25 parts ; 5 into 2£»^goes 5 times. Ans. ^ of a barrel The process in the foregoing examples is evidently di< viding a' fraction by a whole number; and consists, as may be seen, in dividing the numerator ^ (when it can be done without' a remainder,) and under the quotient writing the denominator. But it not unfirequently happens,^ that the numerator will not contain the whole number without a remainder. • /'■''•' '•"'■ '^j--^ i--> ■ t _ ^ r'^H 8. A maq divided ^of a pound equally amppg 2 persons; what part of a pound did he give to each '} fr ', ^ of a pound divided into 2 equal parts Vjl be 4tNi Ans. He gave I of a pound to each. '''■'^'''^T' 106 FRACTIONS. tl 45, 46. 1 fl 46, 47 9. A mother divided ^ a pie among 4 children ; what part of the pie ''id she give to each ? ^ -r 4 = how much ? . id. A boy divided ^ of an orange equally among 3 of his companions ; what was- each one's share ? ^ -7- 3 = how much? 11. A man divided f of an apple equally between 2 chil. dren ; what part did he gi||e to each ? f -7- by 2 = what part of a whole one ? > . . .. f is 3 parts : if each of these piu'ts he divided into 2 equal parts, they will make 6 parts. He may now give 3 parts to one, and 3 to the other : but 4ths divided into 8 equal parts become 8ths. The parts are now twice so mantff but they are only half so large; consequently, § is only half so much as f . Ans. f of an apple. In these last exainples, the fraction has beqn divided by 'multiplying the denominator, without changing the numera* tor. The reason is obvious; for, by multiplying ]the denom- iinator by any number, the parts ate made so many tinies I 4nia//«r, since it will take so many mpte of them to m;^a whole one ; and if no more of these smaller parts be taken | than were before taken of the larger, that is, if the aumer- ator be not changed, the value of the fraction is evidently | made so many times less. c ,..;.,;..% a. ,.:>•,( t? If 4L6. Hence, we have two ways to divide a fraction \ hy a whole number. I. J^ivide the numerator by the whole number, (if it will I contain it widjout a remainder,) and under Ui« quotient | "write the denomiaator. Otherwise, , ,, ^ .,. - -u II. Multiply the denominator by the whole number, and| over the product write the numeratcu*. , , ,. ; ^ ..^ EXAMPLES FOR PRACTICE, > ;, 1. If 7 pounds of tobacco cost f^ of a pound, what is | that per pound ? f|-f-7=how much 1 Ans. ^o(vi lb. ,,jt 2* If ^ of an acre produce 24 bushels, what part of ao| acre will produce 1 bushel ? ^$-7-24=:how much ? 3. If 12 yards of silk cost ^^ of a pound, what is that a | yard? |f-4-I2=howmuch? ,..,4» Divide f by 16. Note. When the divisor is a composite number, the in* telligent pupil will perceive, that he can first divide by on«| component part, and the quotient thence arising by the oth- ■ ■ ..^i. U 45, 46. ■ fl 46, 47. FRACTIONS. 107 ^» er; thus he may frequently shorten the operation. In the last example, 16=8X2 and |-;-8=s^, and ^-r2— i ^. . Ans, 1^. 5. Divide ^^ by 12. Divide ^ by 21. Divide f |>y 24. 6. If 6 bushels of wheat cost £1% what is it per bushel ? Note. The mixed number may evidently be reduced to an improper fraction, and divided as before. Ans. it=2T ^^ ^ pound, expressing the fraction in its lowest terms. (H 43.) 7. Divide ^4|4 by 9. Cluot. ^^^ of a pound, a Divide 12f by 5. Q«o^ V^==2f. 9. Divide 14f by 8. Qmo^ If^. 10. Divide 184^ by 7. Quot. 26^. Ab^c. 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 quotient of the whole number ; thus, in the last example, dividing 184^ by 7, as in whole numbers, we obtain 26 integers, with 2^ [ s=^ remainder, which divided by 7, gives -^ and 26-|-f^ =26/j,i4Ms. 11. Divide 2786^ by 6. Ans. 464f . 12. How many times is 24 contained in 7646^^ ? Ans. 318ff^. 13. How many times is 3 contained in 462^ 1 Ans. 154^. To multiply a fraction by a whole number. t[ 47. 1. If 1 yard of cloth cost ^ of a pound, what will 1 2 yards cost? -}X2=:how muctt? 2. If a cow consume ^ of a bushel of meal in 1 day, how I much will she consume in 3 days .^ ^X3=:how much? 3. A boy bought 5 cakes, at f of a shilling each; what : did he give for the whole ? ^^X5=how much? 4. How much is 2 times ^ ? ■ > 3 times ^ ? _ 2 times f? 5. Multiply ^ by 3. tby2. iby7. 6. If a man spend f of a shilling per day, how much will he spend in 7 days ? f is 3 parts, u he spend 3 such parts in 1 day, he will [evidently • spend 7 times 3, that is, \* = 2| in 7 days. ^. "^ 108 fl&ACTIONS. U 47, 4g, Hence, we perceive, a fraction is multiplied hy multipli/ing the numerator, without changing the denominator. But it has been made evident, (tl 46,) that multiplying the denominator produces the same effect on the value of the fraction, as dividing the numerator : hence, also, dividing the denominator virill produce the samd etTect on the value of the fraction, as m.uttiplying the numerator. In all cases, therefore, vrhere one of the terms of (he fraction is to be multiplied tH^ same result will be effected by dividing the other ; and where one term is to be divided, the same result may be effected by multiplying the other. This principle, borne distinctly in mind, will frequently etiable the pupil to shorten the operations of fractions. Thus, in the following example : , At ^ of a pbund, for 1 pound of sugar, what will 11 pounds cost ? Multiplying the numerator by 11, we obtain for the pro* duct f f =i of a pound for the answer. tf 4^. But by applying the above principle, and dlvid' ing the denominator, instead of multiplying the numerator we at once come to an answer, f in much lower terms, ll^nce, there are two ways to multiply a fraction by a whole number, I. Divide the denominator by the whole number, (when it can be done without a remainder,) and over the quotient write the numei'atof'. Otherwise, II. Multiply the numerator by the whole number, and an* der the product write the. denominator. If then it be an improper fraction, it may be reduced to a whole or mixeji li umber. EXAMPLES FOft PAACTlCE. 1. tf 6ne man consume ^of a barrel'of flour in a months how much will 18 men consume in the same time 1 — — 6 ftien ? — 9 men 1 Ans, to the last, 1^ barrels. 2. What is the product of ^ multiplied by 40? ^^X 40=equal how much ? Ans. 23f . 3. Multiply T^^T by 10. by 20. ■ by 18. -— by 36. by 48. — by 60. ^ote. When the multipllei' is a composite number, ik pupil will recollect (IT 11)« that he may multiply first by one component part, and tjiat product by the other. Thtts^ U 47, 48. I ^ ^^ 49. FHACTIONI. 109 in the last example, the multiplier 60 is equal to 12X5; therefore, TVVXl2=if, and jfX5= fj«»A , Ans. 4. Multiply 5i by 7. Ans. 40^. Note. It is evident that the mixed number may be re>- duced to an improper fraction, and multi[died, as in the pre- ceding examples; 'but the operation will usually be shorter, to miutifriy the fraction and whol^ number separately^ and add the results together. Thus, in the last example, 7 times 5 are 85; and 7 times f are ^ss^^ 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 pro* duct of the whole number. 5. What will 9^ tons of hay come to at ^ per ton ? Ans. 2a£ 198. 6. If a man travel 2 ^ miles in one hour, how far will he travel in 5 hours ? in 8 hours ? — — in 12 hours ? ■ in 3 days, suppose he travel 12 hours each day ? Ans. to the last, 77f miles. Note. The fraction is here reduced to its lowest iermsy the same will be done in all the following examples. To multiply a whole number by a fraction. tl 49. 1. If 36 pounds be paid for apiece of cloth, what cost f of it.' 36Xf^how much? f of the quantity will cost f of the price; | of a time 36 pounds, that is, f of 36 pounds, implies that 36 be first di- vided into 4 equal parts, and then that one of these parts be taken 3 times ; 4 into 36 goes 9 times, and*^3 times 9 is 27. Ans. 27 pounds. From the above example it plainly appears that the object in multiplying by a fraction, whatever may be the multipli' cand, is to take of the multiplicand a part, d^oted by the multiplying fraction] and that this operation iff composed of two others^ viz. a division by the denominator of the maIti*-~ plying fraction, and a multiplication of the quotient by the numerator. It is a matter of indifference, as it respects the result, which of these operations precedes the other, for 36 X3-^4=37, the same as 36-M X 3=27. Hence, — To multiply by a fraction, whether the multi- plicand be a whole number or a jTrcicfion,— Rule : Dividt the multiplicand by the denominator of the multi- N m FRACTIONS. V V N ^ use. fAy'mg frtotion, andjpiukiply the quotient by the numerator; or, (which will oftenbe found more convenient in practice,) first muitiply by the numerator, and diyide tl)e;prp4uc( by .thjC denominator. .<,^ ',..r.?F* ,..,■! i- ».:;^ ^i.ti.. -. ■» .,^.j' Muitiplipatioo, therefore, when implied to fractions, does ■ot always imply aogmeiitatMm, or incr-ease, as in whole Viumbers; for, v^aesi the muh^>Iier is less than vnitfft it will Always require the |M;oduct to be less than the multiplicaa4i to'which it would 'he oi^yequaljif the multiplier wei« ). We have seen, (IJIO,) that, when two numbers are mul> Aiplied together, either c^ them maiy be made the multiplier without i^e<^ii|g the iresult. In the last example, therefore, Instead of multiplying 16 by ^ we m%y milMipIy ^ iby 161, (^ 47,) and the result w,iU be the same, ^^t i j <> ^%1iMfi.9» FOR PRApTlOE. % What will 40 bmweis of meal come to at | of a poun4 per^bMcel? 4(lX|*^PW mficht 9. What will Ayaxde of cloth cost at f of a pound per jvti'i ^X|7=h0wmuQh cWe ^ fraction by lOnoiAen. i -> . ' IT «1^0. 1. A man owning ]|- of a farm, sold ^ of his share; -what part 'of the H^hote »f^m lUd he 'Sell t i of ^ is how muchl We have just seen, (^ 40,') that to tmultipjl^y byajftaictian, is to divide the muUij^Uediitd^hy ihe^enaii^iHator, and to mul' t^ly the quotient by the numerator. ^ divided by 3, the de- nominator of the mult^yui^ £raction, (YT 46,) is ;^, which, multiplied by 2, the numerator, (^ 4^,) is :^, Ans. The process, if carefiiUy considered; will befound to con- sist in multiplying togMhefthettpomumertdw^.fer a new m' meratoTy nndike two denwniilteiiorsjor a new dtnomindtitr. BXAMPL£8 FOR PRACTieE. % A man, having f of a pound, gave ^V of it for a din- ner what did the dinner cost him? Ans. ^ pound. 3. Multiply i by f Multiply i^ by f FrodMct.^y 4. How much is f of § of |- of |^? un* v« •, > v^t.j, . Note. Fracticms like the above, connected by the word H 5». ■ ^ 50, 6i; FRACTIONS* 111 of^ are sometimes ealled compound ftnctitms. The word of implies their continual multiplietUion into each other. When there are several fractions to be imritiplied eonthi- ually together, as the severai numefators tae factors of the ticw ntnneratov, and the stverai (knominator^ wt^ factors of the new deacnHinator, the operation ma^ be shortened k^ droppiiig those factors which are the same in hath terms j on the principle eiqdained in f[ 4^ Thus, in the \dst exMnpIe, I, f , {, ^f we And a 4 and a 3 both among the numerators afed uiKHig theidteiloRiinfrtors ; tJi^Peibre we drop them multi* plying tog)etlM» oiAf the vemainuiif tivtmemUnt^f 9XVsrl4, for ft BOW nuBMrator, and theremaininf d«Donmnter», SXo 3540, for a new denemhiater, making jfsE^, Ans, ae before. 5. I of I of f of I of ^^ of ^of ^howmuch? Am. ^. 6. WlMt is the ooMiiioal prt^ec of T, f , f of | and 3| ? Nine. The mt^r 7 may be reduoed to the form of an impropeip Jhietion, ij writing a nnit nnder it ibr a den<«ii- nator, thus, |. Ans.H^. 7. At ^ of a pound a yard, what wiR ( of a j^ard of 8. At If pounds per barrel for floor, what wilt -^ of a barrel eostf lj2sV then VX^=^!M' ^^' ' •' ' d. At f of a pound, per yard, what eeet 7| yards? - ^ '^ •■'"■'•■ • . Ans. \ 10. At i9| per yard, what ooit d| yardis.' Ana. f ll 11. What li the continued prodnet of 0, |« ^ of j, ^, and U of f of I f -4n». Mf . 5T ol« The Rous for the ninltiplieation of frat^tlens may now be presented at one view : ''' "^ " • I. 7*0 mk^iph ttf inaction by m lekole numher, at a whole numhet fry a fiiutum.'^Divide the denonnnator by fh^ whole number, when it can be done without a remainder ; otherwise, multiply the numerator by it, and under the pr»- duet write the denominator, which may then be reduced to a whde or mixed number. II. To muU^ly it mixed ntcmfrer frj^ a whok xtimfrer;— Multiply the fraction and integers, separately, and add their products together. \Wk Tq multiply one fraction by another^ — ^Multiply to* 119 PRACTI0N8. U Hh «2. ftther the numeraiors for a new numerator, and the deno- minators for a new denominatcM'. Note. If either or both are mixed numbers^ they may first be reduced to improper fractions. EXAMPLES FOR PRACTICE. 1. At fj( per yard, what cost 4 yafds of cIotH ? 5 yds? 6 yds? . u 8 yds? -r— «0 yds? 2. Multiply 148 by ^ -, by | Product, f Product, i&i. Product, 2|. Product, I. Product 3. i4n5. to the last, 15^. ^byA bytlr. Ztost product, 44 •^. 3. If 2 1^ tons of hay keep 1 hoyse through the winter, how much will it take to keep 3 horses the same time ? 7 horses? .13 horses ? Ans. tq the last, 37-^ tons. 4. What will 8^ barrels of cider come ^^ ai 7 shillings per barrel? ,m,? ^ii^i^.r.'^ 'ut ] 'i , ^/-'iM >(,-i' } . 5. At 14|;f per cwt. what will be the cost of 147 cwt ? 0. A owned f of a note ; B owned -^ of the same ,* the note amounted to.lOQQ^; what was each 9n9> share of the mcNiey f 7. Multiply ^ off by ^ off,. ; ^ ni: 8. Multiply 7i by 2^. 9. Multiply ^ by 2J. i ;,. hi vIh 10. Multiply! of 6 by f 11. MuKiply J of 2 by ^ of 4. . m ,. 12. Multiply continually together ^ of 6, f of 7, | of 9. and I of 10. Product,^. 13. Multiply 1000000 by |. Product, 55555d|. 7*0 divide awkah number hy a fraction. ^ «I3. We have already shown (5T 46,) how to ^xt\6fi a fraction by a whole number ', we now proceed to shpyr how to divide a whole number- by a fraction. 1. A man divided Q£ among some poor people, giying them I of a pound each ; how many were the pejrs(»\s who received the money ? 9~£g=4iow many ?vS--*.>4^ }«»??. ^4. 1 pound is 1^, and 9 poi^nds is 9 times as many, that is, ^/ ; then f is contained in ^ as many times as 3 is con- tained in 3G. Ans. 12 persons, That is, — Multiply the dividend by the denominator of the dividing fraction, (thereby reducing the dividel;^l to parts of the same magnitude as the divisor) and divide ^Ae pro\ duct by the numerator. ,1. m --.s.*! . , tF52,63. FRAvrONP.. m Il3i 2. How many time« 19 f contained in $ 7r i-H^^^w .;. V OEBRATION. _.. . ■ y,) 1; s,f,i . jU .1; ,f .,,,;;■.. . .8 Dividend. - '.- :..,r,!U> n^rft ;; ii .i 6 Denominator ati i-i,' itoi' i^ •wp« S'li./i 'i > »5f<:jit..ii J lit ii V' (V- IfflftJJl' Y« nr. Numeifatar, 3 ) 4j0 vrr^^J^- A- ^.'l»f. .^! Quotient, 13^ times the answer. < - '1> •• To mnlti|>ly by a fraction, wie 1)av$ Sjsen, {^ 4d>) implies two operations— a division and a multiplicaiim i 90 {^fK>, to divide by a fraction inipiiestwo opera|ion4-a ffiM/%A'ca/^ .> *.■ i' tient by the numerator. Note, In either case, it is matter of indiflierence, as it respects the result, which of these operations precedes the other ; but in practice it will frequently be more conveni> .ent, that the multipdiplication precede the division. 12 multiplied by |, Uie pro- duct is 0. In multiplication,, the mul- tiplier being ies5 than unity, or 1, will require the product to be less than the multipli cand, (51 49,) to ,^hich it is only equal when the multipli- er is 1, and greater when the multiplier is more than 1. 12dividle(iby f, the quo- tient is 16. In division, the divisor be- inipless than unity, or 1, will be contained a greater num- ber of times; consequently will require the quotient to be greater than the dividend, to which it will be equal when the divisor is 1, and less when the divisor is more than 1. EXAMPLES -FOR PRACTICE. , 1. How many times is ^ contained in 7 1 7-j-^=How many? • ,• ,:i. . . .^^^ , ■ '■ ■ \ ^:..-. K2 114 FRACnONf. 1F«3,54. 2. How many timeB eui I draw |^ of a gallon of wine out of a cask containing 26 gallons ? 3. Divide 3 by f 8 by f . 10 by ^ 4. If a man drink -^ of a quairt of rum a day, how long will 3 gallons last him I / ^>.*i»v* ,* 5. If 2f bushels of oats sow an aere, how many acres will 22 bushels sow? 22-r-2f=:how many times? v Note, Reduce the mixed number to an improper frac* tion, 2|=y. .Vi.v.ii:! '■>"* '''^"li/ t^'t v^ Ans. 8 acres. 6. At IpS a yard, how many yards of cloth may be bought for S7£ ? Ans. 26f yards. 7. How many times -^/^ contained in 84 ? Ans. 90-^ time». 8. How many times is ^ contained in 6 ? Ans. ^ of I time. 9 How many timeftia 8f contained in 53 ? Ans. 6}^ time;^. 10. At f of a pound for building 1 rod of stone wall, how many Fodi may be built for 67 jf ? 87-rfeBhow ma-^ ny times? " •• Tq divide one Jr action hy another . ^54. 1. At § of a pound per parrel^ how much rye n^iay be bought for f of a pound X f is contained in ^ how ma* ny times? -'^ Had the rye been 2 whcik pounds per barrel, instead of f of a pound, it is evident, that f of a pound must have been divided by 2, and the quotient wouki have been ^ ; but the divisor is 3ds, and 3ds will b» contained 3 times where a like number of whole ones are contained 1 time ; conse- quently the quotient ^ is 3 times too smally and must therefore in order to give tlie true answer, be multiplied by 3, that is, by the denominator of the divisor ; 3 times ^%= -^fl- barrel, answer. The process is that already described/ ff 52 and 53. If oarefully considered;, it will be perceived,' that the numerator of the divisor is multiplied into the denominator of the di- viidcnd, and the denominator of the divisor into the numer^ ator of 1}he dividend ; wherefore in practice, it will be more convenient to invert the divisor ; thus, §^inverted becomes f; (hen multiply together the two upper terms for a numerator uifKd tM ttp(k lower terms for a denominator ^^hs in the mul.ti» \U-:J:. ^»M. ■ ^54,55/ PItACTlONff. 115 e : conse- plication of one firttetion by wiother. Thus, in the %bofe example, 3X8 9 ^ -ku .-u.um .. . s - . w .. ...;.*. - .-=:—, M before. '' •':'-•<■.•>•■;■■'-•>'-; '" *'* KXAMPLCS FOR PRACTICE. ' ' ' 2. At I of a pound per bushel for wheat, how many bushels may be bought for | of a pound? How many times is ^ contained in 1 1 Ans. 3^ bushels. 3. If I of a yard of cloth cost j^ of a pound, what is that per yard ? It will be recollected (tf 24) that when the cost of any quantity is given to find thepme of a unit, we diftrtde the cost by the quantity. Thus, f (the cost) divided by { (the quantity) will give the price of 1 yard. Ans. f ^ of a pound per yard. Proof. If the work be right, (^ 16, " Proof,") the pro- duct of the quotient into the divisor will be equal to the dividend ; thus, MXi=i. This, it will be perceived, is multiplying the price of one yard (ff) by the quantity (l). to find the cost (^ ;) and is, in fact, reversing the question ; thus, if the price of one yard be f | of a pound, what will | ol a yard cost ? Ans. f of a pound. Note. Let the pupil be required to reverse and prove the succeeding examples in the same manner. 4. How many bushels of wheat at ^ of a pound per bushel, may be bought for f of a pound ? Ans. 4f bushels. 5. If 4^ pounds of butter serve a family 1 week, how many weeks will 36^ pounds serve them ? The mixed numbers, it ^dl be recollected, may be re- duced to improper fractions. Ans. Syf ^ weeks. 6. Divide ^ by i, Quot. 1 Divide ^ by ^ Quot. 2. 7. Divide | by }, Quot. 3 Divide | by ^^ Quot ff. 8 Divide 2^ by 1^, Quot. If Divide lOf by 2^ Quot. 4if . 9. How many times is ^ contained in f ? Ans. 4 times. 10. How many times is ^ contained (n 4^ ? ^ns. 11 f times. 11 Divide f off by J off Quot. 4. « t[ titl. The RuLEyor division of fractions may now be presented at one view : — I. To divide a fraction hif a whole number ^-r-DivideHlaSi 1W FRACTIONS. 1f«6,W numerator by the whole number, wh«n it otn be done with- out ti remainder, and under the quotient write the denoinin^ tor ; otherwise, multiply the d^oHfinaKar by it, and qver the product write the numerator. , , * II. To divide a yikole number hy afractioHf — Multiply the dividend by the denominator of th§ fraction, and divide the product by the numerator, III. To divide one fraction by another, ^-Invert the divisor and multiply together the two upper terms for a numerator^ and the twdlower^rms for ^ denominator. Note. If either or both are inixed numbers, they may be reduced to improper fractions. EXAMPLES FOB FHACTICE. 1. If 7 tb of tobacco cost -^ of a pound, what is it per pound ? ^^-7-7=how much '^ | of -^ is how mudi t 2. At|A for I of a parrel of cider, what is that per bar- rel? 3. If 4 pounds of sugar cost ^ofi pound, what does 1 pound cost? , ^.;[ ,- 4. If I of a yard cost 13s. what is the price per yard? ^ If 14 1 yards cost 43<£, what is the price per yard ? Ans, 2Hi. 6. At 4^ pounds for 10^ bafrels of cider,, what is that per barrel ? Ans. ^£. 7. How many times is f contained in 746 T Ans. 1969^. 8. Divide j^ off by ^. Quot. |. 9. Divide i of ^ by f off. 10. Divide ^ of 4 by ^. 11. Divide 4f by f of 4. 12. Divide ^ of 4 by 4^. Divide I by ^ off. Quot. ^^. Quot. ^. Quot. 3. » ' Quotil^. -hrf^=i6^w . nucb I II — f^— ^» 6. A boy htring | of an tpple, giT6 ^ of it to hit tiater ; what part of the apple had he left ? f--^how much t When ike denominators of two or more firactiona are alike f (aa in the foregoing examples) they are said to hare a common denominator. The parts are then in the same denomination, and, consequently, of the same magnitude or value. It is etldent, thereibre, that they may be added or subtracted, by adding or subtracting their uumeraion, that is, the number of then: parts, oare Uiing taken to write un- der the' result their piopf r denominatof , ThM, ^^ff^^sz TT» 8 l"*i' *' 6. A boy having an orange, gave f of it to hls^iste^ and ^ to his brother ; what part ^ the orange did he gire away f 4ths and 8ths being parts of diferent Magnitudes, or value, cannot be add^ together. We must therefore firsi reduce them to parts of the same magnitude, that 'is, to a common denominator, f are three parts. If each of these parts be divided into 2 equal parts, that is, if we multiply both terms of tl^ fr-ac^ion ^ by 2, (tl 43) it will be changed to f ; then f and ^ are |. Ans. |^ of an ; ^h iU-' ^'U? •«iI7* I -mi vf.I U Si7. From the very process of dividing each of the parts, that is, of increasing the denominators by multiplying them, it (bllows that ettch denominator must be k factor of the common denominator ; now, multiplying all the denomii payors together will evidently produce such, a numb^r^ >; 118 MACTIONt. 1167. ' Henee,-^To ^eihtte fractions ofdiftnnt demnmnnt^rs tn equivalent fractions, having a common denominator ^-^KvLf: : Multiply togetbeir all the denomfaiators for a common deno- minator ; and as by this process each denominator is mul- tij^lied by all the others, so, to retain the value of each flrae- tion, maltiply each numerator by all the denominators, ex- cept its own, fer a new numerator^ and under it, write the common denominator. m,': fiXAMPLBt FOR PRAC^IClft. 1. Reduce ^, | and ^ to fraetioii» of equal value, having ft common dCTiominator. 3X4X5£=:60, the common denomittator. <' '- ' « 2x4X^=^» tJie new numerator for the first fVaetiom. 3X3X^=45, the new numerator for the second fraction. 3X4x4aBs48, the new numerator for the third fraction. The new fractions, therefore, are |^, |4, and ff . By an inspection of the operation, the pupil will perceive tliaf the numerator and denominator of eaeh fVaction have been mul- tiplied by the same numbers : consequently, (51 43) that their value has not been altered. 3. Reduce to equivalent fractions of a common denomi" nator, and add together ^^ | and|, ' ^ni. U-¥U+ U ^ U * ^ Hh amount. 4. Add together f and f. Amount, l^f. 6. What is the amount ofj-j- j -f j-f-^ ? Ans. ft^l^. 6. What are the fractions of a common denominator equivalent to f and | ? Ans. ^ and ^, or ^ and ^. We have already se^n (^ 56, ex. 7,) that the common de- nominator may be any number, of which each given deno- minator is a factor, that is, Imy humbei which may be divi- ded 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 leaet com- mon multiple ; therefore, the least common denominator of any fractions is the least common multiple of all their denom- inators. Though the rule idready given will always find a commim multiple of the given denominators, yet it will not always find their /eojt common multiple. In the last ex- ample, 24 is evidently a common multiple of 4 and 6, for it fl «7, 58. niACTIONt. il9 minai&rs tft rr,— Rule : Bmon deno- ator 18 mui- if each fVae- linCitors, ex- it, write the alue, having Vaetioii. ad inaction, fraotioni' 1 1*. By Ml leive that the ive been mul- (^ 43) that mon denomi' will exactly measutie both of them ; but 12 will do the saitie, and as 12 is the least number that will do this, it i* thft least oommoB muhi|^ of 4 and 6. It will theref(n 5| 8 and 10 will measure? Ans. 120. 11. There are 3 pieces of cloth, ;. '.•;(.■■.;'. Before adding;, reduce the fractional parts to their least con^mon denominator ; this being done, we shall have, * Ackling together all the 24ths, viz. 18-|-20 7f= 7^1 ) H-21, we obtain 69, that is, ^1=2^. We 13|=1^^ > write down the fraction ^ under the other 15|3sl5|^ ) fractions, and reserve the 2 integers to be — carried to the amount of the other integers, Ans. 97^ making in the whde 37^ An84 12. There was a piece of 'doth ooniaihing 34f yards, from which were taken 12§ yards ; how much was there left ? We cannot take 16 twenty-fourths (]j^) from, 9 twenty-fourths, (^) we must theriefiH'e borrow 1 integer7:24 twenty-fourths, (f^) which, with ^, makes ^ ; we can now take ^ from f|, and there will remain ^ ; but as 121=12^1 ' 4ns. 21^ yds. fl 59, 60. REDUCTION OF FRACTIONS. 121 ions of the i0t common »Bt we need I this would many IStbs ' 12, thus : ■i T^, and -^. be least com^ 1^, amount, oommon de- ji ^i difference. 3 and 10 will *y Ans. 120. ling T| yards, j; how many to their least dl have, vi2. 18+20 L=:2ii. '^^ ier the other ntegers to be ther integers, ig 34f yards, uras there left? wenty-fourths irths, M I] 1 integer5c24 iich, with A, take ^ from iin^;buta8 we borrowed, so also we must carry 1 to the 12, which makes it 13, and 13 from 34 teaves 21. Ans. 21^|. 13. What is the amount of ^ of f of a yard, f of a yard, and I of 2 yards ? Note. The compound fraction may be reduced to a sm- pk fraction ; thus, ^ of f =| ; and I of 2=f ; then, f-|-§ -f-f=Tfo=lT¥(T yds., anstocr. ff 59. From the foregoing examples we derive the fol- lowing Rule : — To add or subtract fractions, add or sub- tract their numerators, when they have a common denomina- tor ; otherwise, they must first be reduced to a common de- nominator. Note. Compound fractions must be reduced to simple fractions before adding or subtracting. EXAMPLES FOR PRACTICE. 1. What is the amount of f, 4§ and 12? Ans. 17^|. 2. A man bought a farm, and sold f of ^ of it ; what part of the farm had he left ? Ans. |. 3. Add together ^, |, I, -^u, } and |^? Amount. 2f|. 4. What is the difierence between 14y\ & ^^liu ' ^ns. From 1^ take f . From 3 take ^. From 147^ take iSi 1 16 Remainder, f Remainder, 2f Rem. 981 Retn. 5. 6. 7. 8. From i of j\ take J- of ^j. 9. Add together 112.^, 31 1§, and lOOOf. 10. Add together 14, 11, 4f, ^V and 11. From I take ^. From ^ take f. What is the difference between ^ and ^ ? -j and I ? 37 h 12. and 13. V and f ? f and f ? f and 3 ? 4 How much is 1—^ ? 1— J ? 1— f 2—4? 2^—1? 3f— J^? 1000- . 1 ? 1 5 REDUCTION OF FRACTIONS. ^ 60, We have seen (1127,) that integers of one de- nomination may be reduced to integers of another denomi- nation. It is evident that fractious of one dcnotnimition, after the same manner, and by the same rules, may be re- duced io fractions of another denomination; that is, frac- tions, like integers, may be brought into lower denomin;;- tious by multiplication, and into liigher denonlinati<•ll^^ l»\- (livision. ]2-;3 REDUCTION OF FRACTIONS. IT 60. To reduce higher into LOWEp denominations. (Rule. See H 28.) 1 . Reduce ^^(^ of a pound to pence, or the fraction of a penny. Note. Let it be recollect- ed that a fraction is multipli- ed either by dividing its de- nominator, or by multiplying its numerator. jlTy^.X20=,JjS. X 12 = ^d. Ans. Or thus: -.jU of "i^ of '/= a|o=|. of a penny, Ans. ;}. Reduce xaVir of a pound to the fraction of a farthing ? TaW*X20=:j§^^:iy s. X 1'* =i¥^dx4=f¥8%=fq. Or thus : Num. 1 20 s. in I^. 20 12 d. in 1 s. 240 4 q. in 1 d. ' 960 Then Wxi^—H ^«>'- 5. Reduce 2^^^ of a guin- ea to a fraction of a penny. 7. Reduce f of a'guinea to the fraction of a pound. Consult 11 28, ex. 12. 1>. Reduce f of a moidore, at i£. 10s. to the fraction of a guinea. II. Reduce 2T of a pound, Troy, to the fraction of an ounce. To reduce lower into higher ^Icnominations. (Rule. See H 28.) 2. Reduce f of a penny to the fraction of a pound. Note. 'Division is perform- ed either by dividing the nu- merator, or by multiplying the denominator. f d. -i- 12 = tVs- -^ 20==: ^1^^. Ans, Or thus : f of -^^ of s^-~ 4. Reduce f of a farthing to the fraction of a pound. -|-20 — ^^xs — r^ViF'^- Or thus : Dcnom. 4 4 q. in Id. 16 12d. in Is. 192 208. in ^l. 3840 Then ■s^\js='£t^- Ans* 6. Reduce f of a penny to the fraction of a guinea. 8. Reduce | of a pound to the fraction of a guinea. 10. Reduce f J of a guinea to the fraction of a moidore. 12. Reduce | of -an ounce to the fraction of a pound Troy. -/ « 1160. fl 60, 61. REDUCTION OP FRACTIONS. 123 HIGHER 'S. 128.) 1 penny to tund. 5 perform- ig the nu- plyingiiiG of 2^ = ' a farthing L pound. .12=Tt^s. 71* • a penny to ruinea. a pound to uinea. of a guinea |a moidore. >f "an ounce »f a pound V3^ Reduce ^'g of a pound avoirdupois, to the fraction of an ounce. 15. A man has j}w of ^ hogshead of wine ; what part iri that of a pint ? 17. A cucumber grew to the length of ijiyVti "^ ^ ^^^^^ 5 what part is that of a foot ? 19. Reduce f of l of a pound to the fraction of Is. 21. Reduce i of ^j of 3 pounds to the fraction of a penny. !I 61. It will frequently be required io find the value of a fraction^ that is io re- duce a fraction to integers of less denominations. 1. What is the value ©f f of a pound ? In other words, reduce f of a pound to shil- lings and pence. |ofa.£ is^o--i3i shil- lings; it is evident from ^ of a shilling may be obtained some pence ; ^ of a shilling is i-2=:4d. — that is, multiply the numerator by that num- ber which will reduce it to the next less denomination, and divide the product by the denominator ; if there be a remainder, tnuitiply and di- vide as before, and so on ; the several quotients, placed one after another in their order, will be the answer. 14. Reduce f of an ounce to the fraction of a pound avoirdupois. 16, A man has {^^ of a pint of wine ; what part is that of a hogshead ? 18. A cucumber grew to- the length of 1 foot 4 inches =ifi.=| of a foot ; what part is that of a mile? 20. f ^ of a shilling is f of what fraction of a pound '? 22. ^f of a penny is ^ of what fraction of 3 pounds? W ^^ ^ penny, is ^ of what part of 3 pounds ? ^^ of a penny is ^ of -^y of how many pounds ? It will frequently be re- quired to reduce integers to the fraction of a greater de- nomination. • 2. Reduce 13s. 4d. to the fraction of a pound. 13s. 4d. is 160 pence ; there are 240 pence in a pound? therefore, 13s. 4d. is ]J^f8=| of a pound, ^hat is, reduce the given sum or quantity to the least denomination men- tioned in it, for a numerator ; then reduce an integer of that greater denomination (to a fraction of which it is re- quired to reduce the given sum or quantity) to the same denomination, for a denomi- nator, and they will form the fraction required. 124 REDUCTION OF FRACTIONS. ^01. EXAMPLES FOR PRACTICE. 3. What is the value of f of a shilling ? OPERATION. Numer. 3 12 «Denom. 8)36(4d. 2q. Ans, 32 4 4 16(2q. 16 5. What is the value of f of a pound Troy? 7. What is the value of f of a pound avoirdupois 1 9. I of a month is how ma- ny days, hours and minutes ? 11, Reduce f^ of a mile to Its proper quantity. 13. Reduce -^^ of an acre to its proper quantity. 15. What is the value of ■{% of a dollar in shillings, pence, &:.c. ? 17. What is the value of ^g of a yard? 19. What is the value of fr^ of a toil. 4. Reduce 4d. Scj. to the fraction of a shillitio-. OPERATION. 4d. 2q. Is. 4 12 12 4 48 Denom. 18 Numer. il=l Ans. 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 avoirdupois. 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 ? 12. Reduce 4 fur. 125 yds. 2 ft. 1 in. 2| bar. to tne frac- 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. 18. Reduce 2 ft. 8 in. l^b. to the fraction of a yard. 20. Reduce 4 cvvt. 2 qr. 12 lb, 14 oz. 12/^ dr. to the fraction of a ton. Note. Let the pupil be required to reverse and prove the following examples : ^ 61. SUPPLEMENT TO FRACTIONS. 125 '21. What is the value of y\ of a guinea? 22. Reduce 3 roods, 17^ poles to the fraction of an acre. 23. A man bought 27 gal. 3 qts. 1 pt. of molasses; what pirt is that of a hogshead ? 24. A man purchased fV of 7 cwt. of sugar ; how much sugar did he purchase ? 25. 13h. 42m. 51 fs. is what part or fraction of a day ? SUPPLEMENT TO FRACTIONS. 1. What are fractions f 2. Whence is it that the parts into which iuiy Ihinn cr any number niay be divided, t;ika their name'? 3. How arc fractions represented by figures ? 4. Wh»l is the aumher above the line called ?— Why is it so culled i 5. What is th i numbor below the line called ?— Why is it so called ?— What does it show 1 T.. What is it which determines the mu^nilude of tliB purls ? — Why ? 7. What is a simple or proper fraction ? — — an improper fraction a mixed number ? S. How is an improper fraction reduc«d to a wiiole or mixed number? 9. How is a mixed number reduced to an improper frac- tion t a whole number ? 10. What- ib understood by the terms of thefra.'tion ? 11. Haw is a fraction reduced to its most smp/c or l)west ierms? 12. What is understood by a coniman divisor? by the greatest common divisor? 13. How is it found ? 14. How miay ways are there to multiply a fra;-tio.i by a \vhole number ? 15, How does it appear, that dividing the denominator multiplies the frac- tion? 16. How is a mixed number multiplied ■? 17. What is implied ill niulliplying by a fraction'? 18. Of how many operations docs it consist?— What arf! they? 19. When the multiplier is k'is than a unit, what is the product compered wiih the muiliplicaud ? 20. How (io you multiply a whole number by a fraction "? 21. How do you multiply one fraction by another ? 22. How do you multiply a mixed number by a mixed number ? 23. How does it api>ear, that in nf^ulti- plying both terms of the fraction by the same number the valuj of the fraction is not altered 1 24. How many ways are thereto divide a ftaclion by a whole nnmlier 1 What are they ? 25. How does it appear lh:il a fraction is divided by multiplying its 'Jcnominalor ? 20. How does dividing by a fraction differ from multiplying by a fraction (" 27. When the rftuisor is /.!sa* thin a uiiil, what is ihj qtjoiienl compared with the dividend ? 118 What is understood by a fo/;i,7io;i dunoiuiua- lor ? the least comimm donomiuator .'' 2\i, Huw dur-s il appear llial each g-jrendtiiioniinalor must be a factor of the co;!ur<'jii diinomin- alor '{ 3D. How is the common denoniitntor to two or more fraeiions found? 31. What i.s understood by a multiple.^ by a comman niull>;,'ii:? by the least common muliipti^ ? Uiiit is tiie pro- Cfist. of iMidingil'? '.]2. How are frin-Lions adJed and .subtracted.'* 3'i. How is a faction cf a gieiUiir dtinomiuaiion reduced to oiie of a less ? of a less to n gieater? 34. How are fractions o( a ujreatrr Ac- nomination rjduced to integers of a Icbd ? io'c^or-^ <)\ a les* iic!ii);niua'iori to Itie fraction of agrcu'er? L2 126 SUPPLEMENT TO FRACTIONS. !IQ1.62. EXERCISES. 1. What is the amount of ^ and f 7 of ^ and f ? of 12^, 3f and 4f ? Ans. to the last, 20^. 2. To 1^ of a pound add f of a shilling. Amount, 18|s. Note. First reduce both to the same denomination. 3. I of a day added to f of an hour, make how many hours ? what part of a day ? Ans. to the last, f f d. 4. Add ^ lb Troy to /g- of an ounce. Amount, 6 oz. 11 pwt. 16 gr. 5. How much is | less ^ ? i^^i of f of |^ ? Ans. to the last, ^|3. 6. From ^ shilling take f of a penny. Rem. 5^d. 7. From ^ of an ounce take |- of a pwt. Retn. 1 1 pwt. 3 gr. 8. From 4 days 7^ hours, take 1 day 9^g hours. Rem. 2 days, 22 hours, 20 min. 9. At £^ per yard, what costs f of a yard of cloth ? 1] 69. The^me of unity, or 1, being given to find the cost of any quantity, either less or more than unity, multi- ply the price by the quantity. On the other hand, the cost of any quantity, either less or more than unity, being given, to find the price of unity, or 1, divide the cost by the quantity. Ans. £±ii 1, If 1^ ib of sugar cost ^5 of a shilling, what will ff of a pound cost ? This example will require two operations : first, as above, to find the price of 1 lb ; secondly, having found the price of 1 lb, to find the cost of |f of a pound. /-jS.-^l^ {\f of t7^s. H 54)=t9^Vs. the price of 1 ib. Then, tV^s.XU (f! of TV5S. 11 50)z=:f 9-Hs-=4d. m-m- the answer. Or we may reason thus : first to find the price of 1 lb ; J-^ ib costs -/-s. li we knew what y'vj lb would cost, we might repeat this 13 times, and the result would be the price of 1 lb, -\^ is 11 parts. If y^^ lb costs -j\s. it is evi- dent y^^ lb will cost j\ of T\=Ti5S- ^"^^ it ^ ^^il^ cost 15? times as much, tliat is, ^^g'^s.=the price of 1 lb. Then, f 4 o^^^-=nU^- the cost of -H of a pound, f ^H^.^^cl. '^f ff5&4' ^^ before. This process is called solvfinr the ques- tion by analysis. After the same manner, let the pupil solve the folUmin? questions : 1IQ1,62. fl62» SUPPLEMENT TO FRACTIONS. 127 ,d f ? t, 20i^. mnt, I8^s. tion. how many St, If d. pwt. 16 gr. le last, ^f ^. Rem. 5^d. 1 pwt. 3 gr. irs. irs, 20 min, cloth ? 1 to find the inity, multi- mi, the cost being given, he quantity. ' ns. £^^. X will f f of st, as above, id the price ■H (if of >s.xe(e ■ :e of 1 tb; [Id cost, we )uld be the js. it is evi- Ivill cost V^ Then.e ];if tke qifcs- io folUm'in? 2. If 7 lb of tobacco cost J of a pound, what is that a pound ? I of f =how much ? What is it for 4 tb ? | of |=how much ? What for 12 lb ? Vof f =how much ? Ans. to the last, £lf. 3. If G^ yards of cloth cost £{i, what cost 9^ yards ? Ans. £4. 5s. 4^d. 4. If 2 oz. of silver cost lis. 3d. what costs ^ of an oz 1 Ans. 4s. 2d. 2^q. 5. If f oz. costs 4s. Id. what costs 1 oz ? Atis. 5s. 8|d. 6. If + ib less by } costs 13^d. what costs 14 lb less by 1 of 2 tb ? Ans. £4. 9s. 9^\d. 7. If J yard costs £l, what will 40.i yards cost. Ans. £50. Ig. 2fd. 8. If 1^5^ of a ship costs c£25l, what is ^^ of her worth ? Ans. £5^. 15s. 8^d. 9. At ^£3| per cwt. what will 9f tb cost ? 10. A merchant owning | of a vessel, sold f of his share tor c£39. 5s. what was the vessel worth? Ans. .£448 lis. 104d. 11. Iff yards cost £^, wliat will f^ of an ell Eng. cost. Ans. 1 7s. Id. 2fq. 12. A merchant bought a number of bales of cloth, each containing 129^|- yards, at the rate of £7 for 5 yards, and sold them out at the rate of <£ll for 7 yards, and gained of200 by the bargain ; how many bales were there ? First find for what he sold 5 yards ; then what he gained on 5 yards — what he gained on 1 yard. ^Then, as many times as the sum gained on 1 yard is contained in <£200, so many yards there must have been. Having found the num- ber of yards, reduce them to bales. Ans. 9 bales. 13. If a staffs^ feet in length, cast a shado.-/ of G feet, how high is that steeple whose shadow measures 153 feet ? Ans. 144i feet. 14. If IC men finish a piece of work in 28^ days, how Icmg will it take 12 men to do the same work ? First find how long it would take 1 man to do it ; then 12 men will do it in j\ of that time. Ans. 37^ days. 15. How many pieces of merchandise, at 20^s. apiece, mast be given fjr 240 pieces, at 12^s. apiece? Ans. 149^^-^. 16. How many yards of booking that is Ijyd. Wide will 1)0 sutficient to line 20 vds. of camlet that is } of a vard wide ? 123 DECIMAL FilACTIONS. !I 62, 63. First find the contents of the camlet in square measure ; then it will be easy to find how many yarda in length of booking that is 1^ yd. wide it will take to make the same quantity. Ans. 12 yardti of camlet. 17. If 1] yd. in breadth require 20^ yds. in length to make a cloak, what in length that is ^ yd wide will be re- (juired to make the same? Ans. {Ml yds. 18. If 7 horses consume 2| tons of hay in ti weeks, liow many tons will 12 horses consume in 8 weeks ? If we knew how much 1 horse consumed in I week, it would be easy to find how much 12 horses would consume in 8 weeks. 2^='j-' tons. If 7 horses consume y tons in 6 weeks; one horse will consune } of iji=J^^ of a ton in weeks; and if a horse consume ^^ of a ton in 6 weeks, he will con- sume } of ^i=iVV of a ton in I week. 12 horses will con- sume 12 times t'\5'a=f^| in 1 week, and in 8 M'eeks they will consumes times |§|='^\2—-(j^ tons, answer. 19. A man with his family, which in all were 5 persons, did usually drink 7f gallons of cider in 1 week ; how much will they drink in 22^ weeks when 3 persons more are added to the fiimily ? " yl«5. 280| gallons. 20. If 9 students spend oflO^ in 18 days, how mucli will 20 students spend in 30 days ? Ans ct'39. 18s. 4|5^(l fle'ciitial Fractions. ff 63* We have seen, that an individual thing or num- ber may be divided into any aumber of equal parts, and that these parts will be called halves, thirds, fourths, fifths, sixtlis, &.C., according to the number of parts into which the thinor or number may be divided; andthateachof these parts may be again divided into any other number of equal parts, and so on. Such are called common or vulgar fractions. Their denominators are not uniform, but vary with every varying division of a unit. It is this circumstance v/hich occasions the chief difficulty in the operations to be performed on them; for when numbers are divided into difterent kinds or parts, they cannot be so easily compared. This difli- culty led to the invention o{ decimal fractions, in which an individual thing or number is supposed to be dividfMl first into ten equal parts, which will be tenths, and each of tlic-ic <1 fi4. DECIMAL FRACTIONS. 129 parts to be again divided into ten other equal parts, which will be hundredths ; and each of these parts to be still fur- ther divided into ten other equal parts, which will be thoU' sandths ; and so on. Such are called decimal fractions, (from the Latin word decern, which signifies ten,) because they increase and decrease in a tenfold proportion, in the same manner as whole numbers. ^ 61. In this way of dividing a unit, it is evident, that the denominator to a decimal fraction will always be 10, 100, 1000, Of 1 with a number of ciphers annexed ; consequently, the denominator to a decimal fraction need not be expressed, for the numerator only, written with a point before it, (') called the separatrix, is sufficient of it- self to express tlie true value. Thus, 6 TJ 685 TOdff are written '6. . • • • • /^ • • '685. The denominator to a decimal fraction, although not ex- pressed, is always understood, and is 1 with as many ci- phers annexed as there are places in the nuinCia'or. Thus, '3705 is a dacimal consisting of four places ; consequently, I with four ciphers annexed, (10000) is its proper denomi' nator. Any decimal may be expressed in the form of a common fraction by writing under it its proper denomina- tor. Thus, '3705 expressed in the form of a common frac- tion, is TV.iVb- When the whole numbers and decimals are expressed to- gether, in the same immber, it is called a mixed number. Thus, 25'63 is a mixed number, 25', or all the figures on the left hand of the decimal point, being whole numbers, and *63, or all the figures on the right hand of the decimal point, being decimals. The names of the places to ten-millionths, and, generally, how to read or write decimal fractions, may be seen from tiie following 130 DECIMAL FRACTIONS. TABLE. !I 64. t5 ill -^f orca oZzir C(«r*r 55 3 ^ cr =p n o ^'^ ^O *■« iS p a w ^1 («l (V CO o B •a ad 2(1 Ist place.>o» II II II II II place. M 4* place. )^;jf -^ o II Hundreds. Tens. Units. Ist place. © C5 o QL Ci © o 2d place. © M o c: gc oi ii ;Jd place, o o o 01 o ^ 4th place, o oom 5tli place. ^ cjr, Oth place. ^ ^ 7tli place. ^ J,, Tenths. Hundredths. Thousandtlis. Ten-Thousandths. Hundred-Thousandths Millionths. Teu-Millionths. c: is ^f gp . p ss s a. ■ji From the table it appears, that the first figure on the right hand of the decimal point signifies so many tenth parts of a unit ; the second figure, so many hundredth parts of a unit ; the third figure, so many thousandth parts of a unit, 6lc. It takes 10 thousandths to make 1 hundreth, 10 hundredths to make 1 tenth, and 10 tenths to make I 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 a starting point, from whence whole numbers proceed, con- tinually increasing in a tenfold proportion towards the left fl(>5, 00. DECIMAL FUACTIONB. 131 hand, and decimuls conUnuaWy derrr as in ^ in the same pro- portion, towards tlic right hund. But iia decimHKs docrcnHO towards the right hand, it foUows of course, that they in- crease tovvardH the left hand, in the same manner as whole mnnhcrs. ^ 0*1. The value of every figure is determined by its place from unifn. Consequently, ciphers placed at the ri'^'/i^ li;uid of decimals do not alter their value, since every sig- nificant figure contiinies to possess the same place from unity. Thus, '5, '50, '500, are all of the same value, each being ecjual to -^f*^ or ^. But every cipher placed at the frft hand of decimal frac- tions diminishes them tenfold, by removing the significant figures further from unity, and consequently making each part ten tin»es as small. Thus, '5, *05, '005, are of diflTer- etit value, *5 being eqal to y^, or }, '05 being equal to ^^g, or v},f, and '005 being eqal to jr/ij!T> <>r -j^xj' Decimal fractions, having different denominators, arc rea- dily reduced to a common denominator, by annexing ciphers until they are equal in number of places. Thus, '5, *00, mi may be reduced to '500, *0G0, '234, each of which has 1000 for a common denominator. 11 00. 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, '0853 (the lowest denomination, or right hand figure, being ten-thousandths) is read 0853 ten-thousandths. Any whole number may evidently be reduced to decimal parts, that is, to tenths, hundreths, thousandths, &c,, by annexing ciphers. Thus, 25, is 250 tenths, 2500 hun- dredths, 25000 thousandths, &c Consequently, any mixed number may be reAd together giving it the name of the low- est denomination or right hand figure. Thus, 2ral'03 may be read 2503 hundredths, and the whole may be expressed in the form of a common fraction, thus, ^'^V- The denominations in federal money are made to cor- respond to the decimal divisions of a unit now described, dol- lars being units, or whole numbers, dimes tenths, cents hun- dredths, and mills thousandths of a dollar; consequently, 132 DECIMAL FRACTIONS. Tf t)6, 67. the expression of amj sum in dollars, cents and mills, is sim- ply the expression of a mixed number in decimal fractions. Forty-six and seven tenths=4e/y==46'7. Write the following numbers in the same manner : Eighteen and thirty-four hundredths. Fiily-two and six hundreths. Nineteen and four hundred eighty-seven thousandths. Twenty and forty-two thousandths. One and five thousandths. 135 and 3784 ten thousandths. 9000 and 342 tep thousandths. 10000 and 15 ten-thousandths. 974 and 102 millionths. 320 and 3 tenths, 4 liundredths and 2 thousandths. 500 and 5 hundred thousandths. 47 millionths. Four hundred and twenty-three thousandths. ADDITION AND SUBTRACTION OF DECIMAL FRACTIONS. If 07. As the value of the parts in decimal fractions in- creases in the same proportion as units, tens, hundreds, &:,c., and may be read together, in the same manner as whole numbers, so, it is evident that all the operations on decimal fractions may he performed in the same manner as on lohole numbers. The only difficulty, if any, that can arise, must be in finding lohere to place the decimal point, in the result. This, in addition and subtraction, is determined by the same rule ; consequently, they may be exhibited together. I. A man bought a barrel of flour for $8, a firkin of but- ter for $Jd*50, 7 pounds of sugar for 83^ cents, an ounce ol pepper for G cents; what did he give for the whole 1 Note. See the table of Federal Money, ^ 27. Let the pupil go back now and read carefully all that is said respect- ing Federal Money in Reduction. From wtiat is there stated it is plain, that we may readily reduce any sums in federal money to the same denominations, as to cents or mills, and '*•"■•'■<>''.■ '"^•■'v'-'-y''^ ''V ' ;>■'■' f*f. DECIMAL FRACTIONS. 138^ Is, is sim- ctions. ler : indths. dthe ECIMAL fractions in- kdreds,&'C., Ir as whole on decimal \as on whok larise, must Let the laid respect- Ithere stated in federal mills, and and add or subtract them as simple numbers. Or, what is the same thing, we may set down the sums, taking care to write dollars under dollars, cents under cents, and mills un- der mills, in such order that the sepaifating points of the several numbers shall fall directly under each other, and add them as simple numbers, placing the separatrix in the amount directly under the other points. OPERATION. $g' = 8000 mills, or lOOOths of a dollar. 3<50 = 3500 mills, or lOOOths. «835=: 635 mills, or lOOOths. . ^06 = , 60 mills, or lOOOths. ^ An$. $12<395=12395 mills, or lOOOths. As the denominations of fe/ieral money correspond with the parts of decimal fractions, so the rules for adding add subtracting decimals are exactly the same as for the s^me operations; in federal money. 2. A man owing $375, paid $17575; how much did be then owe 1 OPERATION. ^ $375' == 37500 cents, or lOOths of a dollar. 175'75= 17575 cents, or lOOths of a dollar. $199'25= 19925 cents, or lOOths. Wherefore, — In addition and subtraction of decimal fractions, — Rule : Write the numbers undqr each other, tenths under tenths, hundredths under hundredths, according to the value of their places, and point off in the result as many places for decimals as are equal to the greatest num- ber of decimdl places in any of the given numbers. EXAMPLES FOR PRACTICE. 3. Bought 1 barrel of flour for 6 dollars and 75 cents, 10 lb. of coffee for 2 dollars 30 cents, 71b. of sugar for 92 cents, 1 lb. of raisins for 12^ cents, and 2 oranges for 6 cents ; what was the whole amount? Ans. $10455. 4. A man is indebted to A, $237'62 ; to B, $360 ; to C, $8642^; to D, $9'62Jf ; and to E, $0*834; what is the amount of his debts? vlns. $684*204. 5. A man has three notes specifying the following suiiiii^ viz. three hundred dollars, fifty dollars sixty cents, and nina M i\ i^-' 1&4 ADDITION AND SUBTRACTION OF DECIMALS. ff 67. Sip dollars eight eents; what ia^the amount of the three notes? CV-MV.i;-^;tM.i?'i .*-<*;*'-. n> . atfrii.!-;-. ;, ... - Jns. $359*68, 6. A mail gave 4 dollars 75 cents for a pair of boots, and 2 dollars 12^ cents for a pair of shoes ; how much did the boots cost more than the shoes? OPERATION. OPERATION. ' ;.,N 4750 mills. 2125 mills. w, 75 2425 / 2625 mills= $2*625 Arts. 82'625 Ans. 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. 3'37o. S. A man bought a horse for 82 dollars, and sold hiin •ngftin for seventy-nine dollars seventy-five cents ; did he !gainorlo8e? and how much ?"' * : m , 9. A man sold wheat at several times as follows, viz, 13'25 bushels ; 8'4 bushels ; 23*«5l bushels ; 6 bushels, and *75 of a bushel; how much did he sell in the whole? Ans. 51*451 bushels, 10. What is the amount of 429, 21^^^^^, 355, j^kxiy hh, and 1 /^ ? ;' ^- ; ' Ans. 808^*/-":■. :.h Vi- 136 MULTIPLICATION OF DECIMALS. H 68. M, kuLE. la the multiplication of decimal fractions, multiply as in yivhxM numbers, and from the product point off so many fig- ures for decimaJis as there are decimal places in the multi- plicand and multiplier counted together, and, if there are ,not so many figures in the product, supply the deficiency by prefixing ciphers. As 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. EXAMPLES FOR PRACTICE. « 4. At $5'47 per yard, what cost 8*3 yards of cloth ? Ans. 45'401. • 5. At $'07 per pound, what cost26'5 pounds of rice ? •v Ans. $V855 cwt. 6. If a barrel contain 1*75 cwt. of flour, what will be the weight of '6^ of a barrel ? Ans. 1*1025. 7. If a melon be worth $0*9 what is *7 of a melon worth ? ^ Ans. 6^ cents. 8. Multiply five hundredths by seven thousandths Product, *000a>. 9. What is *3 of 116.' , >4n5. 34'8. 10. What is *85 of 3672! . Ana. 3l2r2. * 11. What is *37 of *0563t Ans. *020831. : 12. Multiply 572 by *58. 13. Multiply eighty-six by four hundredths. Product, 3'44. 14. Multiply *2062 b, *0008. 15. Multiply forty-seven tenths by one thousand eighty- six hundredths. 16. Multiply two hundredths by eleven thousandths. 17. What will be the cost of thirteen hundredths of a ton of hay, at $11 a ton? 18. What will be the cost of three hundred seventy-five thousandths of a cord of wood at $2 a cord ? 19. If a man's wages be seventy-five hundreths of a dol- lar a day, how much will he earn in four weeks, Sundays excepted? 20. What will 250 bushels of rye come to at $0*88^ per bushel f ilns. $22r25. 24. What is the value of 86 barrels of flour, a^ *'*'^'^ ' barrel? >u!*/^..,ight for 631 / In this example, there are decimals in the divisor, and none in the dividend. $4*75:^^475 cents, and $31, by an- nexinrr two ciohers r=:3100 cents ; that iy, reduce the divi- dead to parts of the same denomination as the divisor. — M2 138 DIVISION OF DECIMALS. U 69. itik' Then, it is plain, as many times 475 cents are contained in 3100 cents, so many barrels may be bought. 475)3100(6fi^^ -barrels, the answer; that is, 6 barrels 2850 and f ^§ of another barrel. But the remainder, 250, instead of bp. 250 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 divisor continued, placing the decimal point after the 6, or wlioio ones 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)3r00(6'526+barrels, the answer, that is t> barrels 2850 and 526 thousandths of anotlier bar- rel. 2500 By annexing a cipher to the first 2375 remainder, thereby reducing it to lOths, and continuing the division, wo 1250 obtain from it '5, and a still further 950 remainder of 125, which, by anuex- V ing another cipher, is reduced to 3000 lOOths, and so on. 2850 The last remainder, 150, is |-^ft ot a thousandth part of a barrel, wiiicli 150 is of so trifling a value, as not to merit notice. If now we count the decimals of the dividend, (for overv cipher annexed to the remainder is evidently to be counted a decimal of the dividend,) we shall find them to he Jim, which corresponds with the number of decimal 'plnce.s in the divisor and quotient coui^ted together. • 3. Under II 68, ex. 3, it was required to multiply 'l'J5 In *03 ; the product was '00375. Taking thi« product for a dividend, let it be required to divide *00375 by *125. One operation will prove the other. Knowing that tlic nu'inbor of decimals in the quotient and divisor, counted together, will be equal to the decimal places in the dividelid, we may divide as in whole numbers, being careful to retain tiie de- cimal points in their proper places. Thus : «1 69, 70. DIVISION OF DECIM.4LS. 139 reduced to OPERATION. 'l-25)*00375('03 The divisor, 125, in 375 goes 3 375 times and no remainder. We have only to place the decimal point in 000 the quotient and the work is done. There are five decimal places in the dividend ; conse- quently there must be five in the divisor and quotient count- ed together ; 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 prefixiniT a cypher. The operation by vulgar fractions will bring us to the *=!ime result. Thus, '{'^5 is ^0^)%, and '00375 is xaWrra •• "ow, ^y3^7^^^-i-^ij3y5jy=-j..^i-Sc)oo^ =^3 ^='03 the same as betbre. *j 71>. The foregoing examples and remarks are .suffi- cient to establish the following . RULE. In the division of decimal fractions, divide as in whole niuribers, and from the right hand of the quotient point off as many figures for decimals as the decimal figures in the dividend, exceed those in t) j divisor, and if there are not .so many figures in the quotient, supply the detficiency by prefixing ciphers. If at any time there is a remainder, or if the decimal iigurcs in the divisor exceed those in the dividend cyphers may be annexed to the dividend or the remainder, and the quotient carried to any necessary degree of exactness ; but 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 .^ Ans. 83(),375. 5. At $'75 per bushel, how many bushels of rye can be bought for $141 ? Ans. 188 bushels. ; 6. At Ql cents apiece, liow many oranges may be bought for $8 ? Ans.. 128 oranges, 7. If '6 of a barrel of flour cost $5, what is that per bar- rel ? Ans. 8'333+ Divide 2 by 53'1. Qmt. '037-]- 140 REDUCTION OP VULGAR FRACTIONS. &-C. ^ 70, 71-. 9. Divide *012 by '005. 10. Divide three thousandths by fl^ur hundredths. Quot. '075. 11. IIow many times h *I7 contiined in 8.^ 12. If I pa" $40S'75 for 75 ) pounds of wool, what is the value of I pound ? Am^. $0'625; or thus aO'624 13. If a piece of cloth, mer.saring I25yards^, costf I8l"i5 Wh:it is that a yard ?, Ans. 81*45. 14. If 536 quintals of fish cost $1913,52, how much is that a quintal ? An:;. $3*57. 15. Bought a farm, containing}^;! acres, for $^3213 ; what did it cost me,per acre ? ^Ins. $38*25. 16. At $954 for 3816, yards of flannel, what is that per yard.' Ans. $0*25. REDUCTION OF COMMON OR VULGAR FRAG- TIONS TO DECIMALS. •»" fj 71, 1. A man hns '^ of a barrel of flcur ; what is that expressed in decimal parts ? As many times as the denominator of a fraction is con- tained in the numerator, so many whole ones are contained in the fraction. We can obt lin no whole ones in f , be- cause the dc.-iominator is not contained in the numerator. We may, however, reduce the numerator to tenths, (1]69, ex. 2,) by annoxing a cipher to it, which, in effect, is mul- tiplying it by 10, making 40 tenths, or 4*0. Then, as ma- ny times as the denominator, 5 is contained in 40, so ma- ny tenths, are contained in the fi'action. 5 into 4-0 goes 8 times and no remainder. »,^ns. *8 of a bush. 2. Express ^ of a dollar in decimal parts. The numerator, 3, reduced to tenths, is f ^, 3*0, which, divided by the denominator, 4, Tlie quotient is 7 tenths, and a remainder of 2. This remi^inder must new be reduced to hundredths by annexing another cipher, making 20 hun- dredths. Tilth, as many times as the denominator 4, i.s contained in S), so many hiindredihs also may be obtainec'. 4 into 29 goes 5 times, and no remainder. ^ of a dolljf, therefore, reduced to decimals, is 7 tenths and 5 hundredth?, that is, *75 of a dollar. f -.^-i V » If 71. RBOUCTIOliI^ or VULGAR FRACTIONS, 6i^C. Ul The operation may be presented in form as follows :- Num. Dtnom. 4)3*0('75 of a dollar, the answer, 28 20 *J0 3. Reduce ^^ to a decimal fraction. The numerator must be reduced to hundreths by annex- ing two ciphers, before the division can begin. 66)4'00('0606-f-, the answer. 396 400 396 As there can be no tenths, a cipher must be placed in the quotient, in tenths place. 4 Not^. -^g cannot be reduced exactly ; for, however lonj; the division be continued, there v.-ili stxJ be a remainder.* it is sufficiently exact for most putpoies, if the decimal be ex tended to three or four places. • Decimal figures which continually repp.at, like *06, in this ex- ample, are called Repdends, or Circu'.aivr.g Decimals. If only CM figure repeats, as '3333 or '7777, &c, it is called a single re- petend: If two or more figures circulate alternately, as *060606, '234234234, &c. it is called a compound repetend. If olhor fi*;- ures arise before those which circulate as '743833, '1430I010I, kc. the decimal is a mixed repetend. A single repetend is denoted by writing: only the circulating fig- ure, with a point over it thus : '3, signi es that the 3 is to hn I'ontinually repeated, forming an infinite or never ending series dVa. A compound repetend is denoted b' i point over ihefirnt ami laat repeating figure : thus, 234 sigi les that 234 is to be contin- ually repeated. It may not be amiss, here to show how the value of any repe- tend may be found, or in other words, how it may be reduced lo its equivalent vulgar fraction. If we attempt to reduce |^ to a decimal, we obtain a con- tinual repetition of the figure 1 : thus, *lllll, that is, the repetend *i The value of the repetend *i then is ^; the val- ue of '22,2, &,c. the repetend '2 will httwice as much; that 142 REDUCTION OF VULGAR F^ATIONS. II Tl. From the foregoing examples we may deduce the follow- ing general Rule : To reduce a common to a dccimnl frac- tion : — Annex one or more ciphers, as may he necessary, to the numerator, and divide it by the denomiaator. If then there be a remainder, annex another cipher, and divide as before, and so continue to do so long as there shall continue is, f. In the same manner, *3=f, and '4^|, and *5=^, and so on to *9, which=f=:l. 1. Whatis the value of *^? Ans. |. 2. What is the value of 'Gt^w-s. |=f . What is the val- ue of *3 of *7 ? of 4 ? of 5 ? of '9 ? ' 1? If g'g be reduced to a decimal, it produces '010101, or ■ • • • the repetend'Ol. The repetend '02, being 2 times as much, • • • • must b*e /^ and *03=:^\, and *4 8, being 48 times as much, must be |f , and '74=Jf , &c. If g^y be reduced to a decimal, it produces '001 ; conse" quently, *002==5f^, and '037 = ^/^, and 425==|f f , &c. As this principle will apply to any number of places, we have this general Rule, for reducing a circuhting 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. 3. What is the vulgar fraction equvialent to '704 ? Ans. ffl. 4. What is the value of '003? 014? '324? ' 01021 ? —'2*463 ; —'002103 ? Ans. to the last,r^^^^\^. i. What is the value of '43 ? In this fraction, the repetend begins in the second place, or place of hundredths. The first figure, 4, is -^xf, and the repetend, 3, is ^ of ^, that is, /(, ; these two parts must be added together. T*(T-|-^iy=M=^^, cms. 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? ^■i'^%Tf^U=N^xj^ns. 7. What is the value of • 138 ? ' 16 ? ' 4123 ? It is plain, that circulates may be added, subtracted, multiplied, and divided, by first reducing them to their equivalent vulgar fractions. fl 71,72. REDUCTION OP DECfMAL FRACTIONS. 148 to be a remainder, or until the fraction shall be reduced to any necessary degree of exactness. The quotient will be the decimal required, which must consist of as many deci- mal places as there are ciphers annexed to the numerator ; and if there arc not so many figures in the quotient, the de- ficiency must be supplied by prefixing ciphers - • , : ■ • ■ • f EXAMPLES FOR PRACTICE. 4. Reduce ^, l, 5^, and tt^^tj to decimals. Ans. *5; '25; '025; '00797-f 5. Reduce f ^, j-^^xf, TT^a, and ^ttV\)f *« decimals. Ans. '602-1- ; '003 ; '0028-}- ; 'OOOlSS-f- 6. Reduce' ^^1, -^^ft b^oo to decimals. 7. Reduce |, g^, -^h* i. f» tV. 'h> ^h to decimals. 8. Reduce i, g, |, ^, f, f , ^, ^, ^l, VV to decimals. REDUCTION OF DECIMAL FRACTIONS. 1] 73. Fractions, we have seen, (^ 60) like integers, are reduced from iow to higher denominations by division, and from high to lower denominations by multiplication. To reduce a compound num- ber to a decimal of the high- est denomination. I. Reduce 7s. 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 7s, making 7'5s, and divided by 20, is '375<£, the answer. The process may be pre- sented in form of a rule, thus : Divide the lowest denomina- tion given, annexing to it one or more ciphers, as may be necessary, by that number which it takes of the same to make ojic of the next higher denomination, and annex the To reduce the decimal of(j^ higher denomination to inte- gers of lower denomination.^. 2. Reduce '375<£ to inte- gers of lower denominations. ft '375<£redacedto shillings, that is, multiplied by 20, is 7'50s. ; then the fractional part, *50s, reduced to pence, that is, multiplied by 12, is 6d. Ans. 7s. 6d. That is, multiply the given decimal by that number which it takes of the next lower de- nomination to make owe of this higher, and from the right hand of the product point ofiT as many figures for decimals as there are figures in the given decimal, and so con- 144 REDUCTION OF DECIMAL FRACTIONS. u n. ■4 •y ^•j quotient, as a decimal to that higher 3. A farmer bought 7 yards of broadcloth for ^^8^, a barrel of flour, for £^^^, a cask of lime for ^^If, and 7 lbs. of rice for £^ ; he paid 1 ton of hay at £S^^, I cow &i ^6f , and the balance in pork at £-^ per tb ; how many were the pounds of pork ? Note. In reducing the common fractions in this example, it will be sufficiently exact if the decimal be extended to three places. Ans. 108f lb. 4. At 12^ cents per lb, what will 37f lbs of butter cost ? Ans. $4'718^. 5. At $17'37 per ton for hay, what will U^ tons cost ? Ans. $201'92|. 6. The, above example reversed. At $20l*92f for 11 §■ tons of hay, what is that per ton ? Ans. il7'37. 7. If '45 of a ton of hay cost $9, what is that per ton? Consult ^ 62. Ans. $20. 6. At '4 of a dollar a gallon, what will '25 of a gallon of molasses cost? Ans. 4 of a dollar. t 148 SUPPLEMENT TO DECIMAL FRACTIONS. ^ 74. 9. At 9 dollars per cwt. what will 7 cwt. 3 qrs. 16 lbs. of sugar cost 1 . Note. Reduce the 3 qrs. IC lbs. to the decimal of a cwt. extending the defl^al in this, and the examples which fol* low to /owr places. ',^- - v • -"^ Ans. 7r035-|- 10. At $69*875 for 5 "cwt. i qr. 14 lbs. of raisins, what is that per cwt. Ans. $13. U. What ^ill 2300 Ib^ of hay come to at 7 mills per lb? 12. What will 765^ lbs. of coffee come to at 18 cents per lb? ^ ^w.s. $137'79. 13. What will 12 gals. 3 qts. 1 pt. of gin cost, at 28 cents ^ quart ? Note. Reduce the whole. quantity to. quarts and the de- cimal of a quart. ; ' .^ i4«5. $14*42. 14. Bought 16yds. 2qrs. 3na. of broadcloth for $100*125. what was that per yard ? Ans. $6, 15. At $1*92 per bushel, how much wheat may be pur- chased for $*72 ? Ans. 1 peck 4 qts. : 16. At $92*72 per ton, how much iron may be purchased ■for $60*268/,:.,,; ,...,^ ,.;:"^^;«.^,P IV-;,.. 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 ? Ans. 1 1 cwt. 1 qr. 23 lbs. . 18. At $302*4 per tun, what will 1 hhd. 15 gals. 3 qts. of wine cost? . v • . .<:, . wi ^»5. $94*50. 19. The above reversed. At $94*50 for I hhd. 15 gals. 3 qts. of wine, what is that per tun ? 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 sufficiently exact for all practical purposes. If, however, ^rcafcr exact- ness be required, the decimals must be extended to a greater number of places. 20. At $1*80 for 3^ qts. of wine, what is that per gallon ? 22. If f of a ton of potash cost $60*45, whai; is that per ton? 21. At $2*215 per gallon, what cost 3^ qts ? 23. At $96*72 per ton for potash, what will |^ of a ton cost ? 1174. REDUCTION OF CURRENCIES. t4d Reduction of Currencies. In the United States, since the act of Conf ress in 1786, establishing Federal money, calculations in aioney hare generfllr been niAde ill dollars, cents and mills. In England, the denomtoations, though the same in name as the currency of this Province, are diSbrent in value. In the United States, previous to the act of Congress, it was the cus- tom to reckon in pounds, shillings &c. ; and now, though all accounts are kept in federal money, snnall' sums are mentioned frequently ia these denominations. There are different currencies of the same nunc in different parts of the United States. It may be necessary often in commercial dealings, and in the course of ordinary business, to change Tttues in foreign currencies into the currency of the Procinces. Supposing there is a sum in federal money— $21 '604* We find by the table of coins, IT 27, that 1 dollar is equal to 5 shillings, and of course 4 dollars are equal to 1 pound, there being 4 times 5 snillings in 20 shillings. The faiue of pounds then, it is clear, is 4 times that of dollars, and of course dollars are reduced to pounds by dividing tht given sum, by 4. 4) 24 dollars. < h ii Is. There remain, however, $^604, 6 " . w and 4 mills to be changed to Halifax currency. By reterrin; to Decimal Fractions, V 66, we 8e» that dollars are the units in federal money, and cents and mills decimal parts : cents hundredths, and mills thousandths. We have then simply to divide these decimals of a dollar by 4, and the quotient will be In decimal parts of a pound, thus : 4) '604 of a dollar. *151 of a pound. This an be reduced to shillings, pence and farthidgs by inspection, (see IT 73) as follows : £451 equal to 3j. Od. Iqr. We find that $24<604 is equal io £6 3s. Od. Iqr. jins. The following then, is the general rule to reduce federal money to Halifax currency— (fit;t(2e the given sum by 4^ and the quotient will be in pounds and decimal parts of a pound, which can be reduced to shil- lingSf pence and farthings by inspection. EXAMPLES FOR PRACTICE. % Reduce $500 to Halifax currency, Jtns £125 do 27'304 do do ' 6 lis 6d !qr. do I18'25 do do 29 lis 'M ■ do 236'50 do do 59 23 i'A Reduce $490 to Halifax currency $56'03 $93 81 1 $836:^ i^l977'642 respectively to Halifax currency. To reduce Halifax currency to federal money, we must reverse i!ie process in the above examples. The rule is as follows ; Reduce thf; siiillings, pence kc. if any, to the decimal of a pound, by inspection, N2 150 REOUCTIOif Of CURRENCIES. 11 74. and multiply the given turn by 4, the product will be in the denomina- tiont of federal money. EXAMPLES FOR PRACTICE. Reduce £125 Halifax curreney to federal money. Jins. $500 do 69 9i 6d do do do 236'50 In order to change English sterling money, and the currencies in some degree in use in the different parts of the United States, into Halifax currency : since the denominations are the same in namci it will be necessary te take some other currency, the denominations of which are different, as a common object of comparison for these currencies, and for HaliAui currency. By this means we shall be able to ascertain the ▼ftlues of the former relaUvely to those of the latter. We will take federal money as this common object of comparison, and will compare with a unit of one of ite denominations, the dollar ^ one or more units of a denomination of Halifax currency, and the before mentioned cur* rencies, the thiUing. In Halifax currency - 5s. =$1./ In English, or sterling money* '(4s. 6d.)4^s.=$l. In New England currency - • - - 6s. =$1. In New York currency 8s. =$1. Iiv Pennsylvania currency (7s. 6d. ) - - 7^s.= $ 1 . In Halifax currency 5s. are equal to $1, and in English sterling money, 4^s. are equal to $1. Sterling money, then, is to Halifax currency as 5 to 4J-, or to avoid the fraction » as 10 to 9, since 2X5=10, and 2x4^=9. Therefore, to change sterling money into Halifax currency, multiply by *^, or, take once the given sum, and add ^, thus — 9) <^48 13s 9d sterling money. 5 8 1 54 10 Halifax currency. Reduce J£56 17s. 6d. sterling to Halifax currency. Ms. £C3 3s. lOd. do 92 4s. 6d. do do do In New England currency, 6s. are reckoned to the dollar. New England currency, then, is to Halifax currency as 5 to 6. There foie, to reduce New England currency to Halifax currency, take live'sixths of the given sum, thus — 6) J£14 5 4 New England currency. 2 7 5f 5 <£M 17 4^ Halifax currency. * Without Premium, which varies from 5 to 8 per cent. -'•an =;J>T,-,^jip^W. n 74, 75. INTEREST. 151 Reduce J^60 4s. lOd. New England currency to Halifax currency. Ans. £50 4s O^d. To reduce Halifax currency to sterling money, or to re* duce Halifax to New England, it is only necessary to re- verse the process in the foregoing operations. To reduce sterling to Halifax, we multiply by ^, there- fore, to reduce Halifox currency to sterling tmoney,---divide the given sum by \p, or, what is the same, multiply by -^^y that is, take -^ of the given sum ; e. g. 10) ^54 10 Halifax currency. 5 8 1, • ' ■ :■ 9 jB48 12 9 sterling money. In the same manner, to reduce Halifax currency to New England, — take f of the given sum, or, add \ to the given sum. From the foregoing rules and illustrations the pupil him- self will be able, by pursuing a similar course, to reduce, with facility, any currency, the denominations of which are pounds, shillings, &c. to any other in which the denomina- tions are the same. The following is the general rule for finding a multiplier to reduce any currency to the par of anotlier : Make the number of shillings that are equal to a dollar in the cur- rency to he reduced^ the denominator of a fraction ; and over this, for enumerator, write the number of shillings that are equal to a dollar in the currency to which the given sum is to be reduced. Let the pupil find multipliers to reduce New York and Pennsylvania curreficies to Halifax, and then Halifax cur- rency to those. lil[TER£8T. U TiS. 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 fgr a longer or shorter time. 15a INTEREST. 1175 '01. '005. The number of pounds so paid for the use of a hundred pounds, one year, is called the rate per cetU or per centum ; the Yfotda per cent, or per centum signifying 6^ the hundred. The highest rate allowed by law m the Canadas is 6 per cent,* that is, 6 pounds for 100 pounds, 6 shillings for 100 shillings ; in other words, j%ij of the sum lent or due is paid for the use of it one year. This is called legal interest, and will here be understood when no other rate is mentioned. Let us suppose the sum lent or due to be one pound, The hundredth part of one pound or ^jj of a pound is, de- cimally expressed, thus, '01, and yf^ of a pound, the legal interest, written as a decimal fraction, is '06. So of any rate per cent. 1 per cent expressed as a common fraction, is t^jy ; decimally, 1 per cent is a half of one per cent, that is, i per cent is a fourth of one per cent, that is, - '0025. ^ per cent is three times a quarter per cent, that is, '0075. Note. The rate per cent is a decimal carried to two places that is, th hundredths ; all decimal expressions lower than hundredths are parts of one per cent. |- per cent, for instance, is '625 of I per cent, that is, '00625. Write 2^ per cent as a decimal fraction. 2 per cent is '02, and ^ per cent is '005. Ans. '025. Write 4 per cent asLa decimal fraction. 4^ per cent — 4f per cent. 5 per cent. 7^ per cent. 8 per cent. 8f per cent. 9 per cent. 9| per cent. 10 per cent. (10 per cent is ^V^; decimally '10) 10^ per cent. 11 per cent. 12 J per cent. 15 per cent. / 1. If the interest of one pound for a year be '06 of a pound, what will be the interest on ^5 for the same time? It will be 25 times 6 or 6 times 25, which is the same thing : — 25 '06 1*50 answer; that is, ^1 and 5 tenths The 5 tenths must be reduced to shillings, pence and fart i. as by th e rule • In the New England States the legal ra' ; is the same as in the Canadas. In England it is 5 per cent. 1175. INTEREST. 153 '01. •005. •0025. , •OOTS. places oer than ent, for for the reduction of decimals; or with sufficient exactness by inspection. See U 73. '50, or 'o of a pound equal 10 shillings. The interest of <£25 for a year is then £1 10s. To find the interest on any sum for one year, it is evi- dent 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 multiplication of decimal fractions, will be the interest required. Note, Principal is the money due, for which interest is paid. Amount is the principal and interest added together. 2. What will be the interest of of 32 3s. for one year, at 4^ per cent ? Vie are to multiply the principal by the rate per cent, 4 j-, expressed '\h the form o^a decimal '045 ; we must therefore reduce the 3s. in the principal, to decimal's by inspection. We find 3s. equal to '15. There being five decimal places £2li^\^ principal. in the multiplicand and mul- •045 rate per cent. , tiplier, 5 figures must be point- ed off for decimals from the • 16075 product, which gives the ans- 12860 ^ wer 1 pound and 44675 hun- ' dred thousandths. Anything <£1'44675 less than thousandths need not be regarded ; hence, .£1*446 is sufficiently exact for the answer. 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 2, because one 36 equals 44 farthings. In 44 qrs. there are lid. ^*446=8s. lid. The interest, then, of ^32 3s. for one year, at 4^ per cent, is .£1 8s. lid. answer. Always, then, if there are shillings, pence and farthings, or either denomination, in the given principal, reduce them to the decimal of a pound by inspection, before multiplying hy the rule. After obtaining the answer in decimals, reduce the tenths, hundredths and thousandths to shillings, pence and farthing A, by inspection. The method of effecting each reduction, is exhibited in U 73 and 74, and must be made perfectly familiar to the pupil's mind. < 3. What will be the interest of £\ 1 3s. 4d. for one year, at 3 per cent ? at 5^ per cent ? -' at 6 per cent ?— ■ — at 1\ per cent ? at 8^ per cent 1 at 9f per cent ? ,> "M^r- 154 INTEREST. 1175,76. I fl76. at 10 per cent? at 10^ per cent' at 11 per cent? at 11^ per cent? at 12 per cent? at 12^ per cent ? ' 4. A tax on a certain town is £406 ISs. 102d. on which tKe collector is to receive 2^ per cent for collecting ; what will he receive for collecting the whole tax at th -' rate ? In this example, the shillings, &lg. reduced to the decimal of a pound equal '795. Multiply therefore, <£406'795 by the rate 2^, thkt is '025. The answer, in decimals, is <£10'169; the tenths, d&c. reduced to shillings, &c. equal 3s. 4^ The answer then, is i^lO 3s. 4d. Note. In the same way are calculated commission, insu> ranee, buying and selling stocks, loss and gain, or anything else rated at so much per cent without respect to time. 5. What must a man, paying 37^ per cent on- his debts, pay on a debt of ^£132 5s. ? Ans. <£49 Us. iO^d. 6. A merchant having purchased goods to the amount of <£5d0, 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 £12 10s. ; whole amount, £652 10s. 7. A merchant bought a quantity of goods fojr £173 15s. how much must he sell (hem for to gain 15 per cent ? - Ans. £199 16s. 3d. 51 76* Commission is an allowance of so much per cent to a person called a correspondent , factor^ or hroher^ for assisting merchants and others in purchasing and selling goods. 8. My correspondent sends me word that he has pur- chased goods to the amount of £1286 on my account ; what I will his commission come. to at 2^ per cent ? Ans. £32 3s. 9. What must I allow my correspondent for selling goods to the amount of £2317 9s. 2^. at a commission of 3^ per | cent? -4ns. £75 6s. 4d. Insurance vis an exemption from hazard, obtained by the payment of a certain sum, which is generally so much [ per cent on the estimated value of the property insured. Premium is the sum paid by the insured for the insurance. i '.^ 'isc- -N^,'. -A; ' ■ H 75, 76. I II 76. INTEREST. 155 Policy is the name given to the instrument or writing, by which the contract of indemnity is effected between the insurer and insured. ^ 10. What will be the premium for insuring a ship from Montreal to Liverpool, valued at 9450jP, at 4^ per cent ? « Ans. <£425 5s. 11. What will be the annual premium for insurance on a house against loss by fire, valued at 875^ at f per cent '? By removing the separatrix 3 figures towards the left, it is evident, th6 sum itself may be made to express* the pre- mium at 1 per cent, of which the given rate parts may be taken; thus, one per cent on 875x is 8*75 and f of 375^ a&5m£. Ans. 6£ lis. 3d. 12. What will be the premium for insurance on a ship and cargo valued at 6310<£ at ^ per cent ? at f-per cent 1 at ^ per cent ? at f per cent .^ at | per cent? Ans. at | per cent the premium is d9£ 7s. 8^d. Stock is a general name for the capital of any trading company or corporation, >or of a fund established by gov- ernment. The value of stock is variable. When 100 pounds of stock sells for 100 pounds in moneys the stock is said to be atjpar, which is a Latin word signifying equal; when for more^ it is said to be above ^^ar; when- for less, it is said to be below par. 13. What is the value of 756^ of stock, at 12^ per cent ? that is, when 1 pound of stock sells for 1 pound 12^ hundredths in money, which is 12j- per cent above par, or 12^ per cent advance, as it is sometimes called. Ans. 850ir lis. 14. What is the value of 3700^ of bank atock, at 95^ per cent 1 that is 4^ per cent below par ? Ans. 3533£ lOe. 15. What is the value of I20je of stock, at 92^ per cent ? at 8QI per cent? at 67| per cent ? at 104^ per cent? at 108^ per cent? at 115 per cent? at 37^ per cent advance ? Loss AND Gain. 16. Bought a hogshead of molasses for 15<£; for how much must I sell it to gain 20 per cent ? Ans. 18je. 17. Bought broadcloath. at 12s. 6d. per yard; but, it be- r ill 156 INTEREST. TT 76, 77. I lag damaged, I am willing to sell it so is to lose 12 per cent; how much will it be per yard? Ans. Us. 18. Bought calico at Is. per yard ; how must I sell it to gain 5 per cent ? 10 per cent? 15 per cent ? . to lose 20 per cent? Ans. to the Iast,9J^(l. U 77m We have seen how interest is cast on any sum 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 6 per cent, )eiiig *06, is *01, one hundredth for 2 months, *005 five thousandth (or^ a hundredth) for 1 month of 30 *. ) days, (for so we reckon a month in casting inter- -liff" -; 'f est,) and ' ; ' ■ ^vul >■ :^;' »<■ . > *001 one thousandth for every 6 days ; 6 being contained 5 times in 30. Hence, it is very easy to cast in the mind, the interest on 1.^, at G per cent for any given time. The hundredth^ it is evident, will be equal to half the greatest even num- ber of months ; the thousandth will be 5 for the odd month, if there be one, and 1 for every time 6 is contained in the given number of the days. Suppose the interest of 1=£, at 6 per cent, be required for 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 (3X6=18) days, will be *008, which, united with the hundredths (*048) give 4 hundredths and 8 thousandths; 4 hundredths, and 8 thousandths, or, *048j6 redticedz=:lld. Ans. lid. 1. What will be the interest on \£ for5 monthsGdays? G months 12 days? 7 months 1 8 months 34 days ? 11 months 6 9 months 12 days? days ? > 12 months months 6 days ? — ;— 16 months? 10 months? 18 days? — 15 Odd Days. — 2. What is the interest of £1 for 13 months 16 days .5' ' ''■ ■ The hundredths will be 6, and the thousandths 5, for the odd month, and 2 for 2 times 6 === 12 days, and there is a remainder of 4 days, the interest for which will be such ;. i^.^i ^ 76, 77. I fl 77. INTERKST. w )se 12 per Ans. lis. I sell it to cent ? e last, 9^(1. Ml any sum 1 frequently cent, )eing nonth of 30 isting inter- r contained the interest hundredth, t even num- odd month, ained in the be required number of [redths'04; |h, and 3 for united with ;housandths; ced=lld. Arts. lid. |nths6days? 8 months .nths? ■ s? 15 13 months IS 5, for the Id there is a IviH be such part of 1 thousandth as 4 days is part of 6 daya, that is, | ss $ of a thousandth. Ans. '067f . 3. What will be the interest of £1 for 1 month 8 days ? 2 months 7 days ? 3 months 15 days ? ~<— 4 months 22 days ? 5 months 1 1 days ? 6 months 17 days ? 7 months 3 days? 8 months U days ? 9 months 2 days? 10 months 15 days ? 11 months 4 days ? 12 months 3 days ? Note. If there is no odd months and the number of days be less than 6, so that there are no thousandths, it is evident, a cipher must be put in the place of thousandths ; thus, in the last example, — i2 months 3 days, — the hundreths will be '06, the thousandths 0, the 3 days } a thousandth. Ans. Is. ^d. 4. What will be the interest of £\ for 2 months 1 day ? . 4 months 2 days ? 6 months 3 days ? 8 months 4 days? 10 months 5 days ? for 3 days? for 1 day ? for 2 days ? for 4 days ? for 5 days ? 5. What is the interest of £56 2s. 7f d. for 8 months 5 days.' The interest of £1, for the given time, is '040^; therefore, i) and ^)jf5643 principal. '040| interest of £1 for the given time. 224520 interest for 8 months. > 2806 interest for 3 days. v , 1871 interest for 2 days. * ^'29197 =.£2 5s. 9fd. . 5 days=3 days -f-2 days. As the multiplicand is taken once for every six days, for 3 days take ^, for 2 days take ^, of the multiplicand. ^-|- ^= |^. So also, if the odd days be 4 = 2 days -f-2 days, take ^ of the multiplicand twice ; for 1 day, teike |. Frcwn the illustrations now given, it is evident, — Tojin^d the interest of any sum in Halifax currently, or any other cur- rency of which-the denominations are pounds, shillings, &c. at 6 per cent, it is only necessary to multiply the given prin- cipal, after having reduced the shillings and pence in it to the decimal of a pound by inspection, by the interest of \£ for the given time, found as above directed and written as 15H INTEREST. ■ II 77. a decimal fraction ; after pointing off as many places for decimals in the product hh there are decimal places in both the factors counted together, these can be reduced back ttgain to shillings and pence by inspection^ ^ iout '— • EXAMl'IiES FOR I'RACITICE. liUI"., 6. What is the interest of .^'87 3s. Of d. for 1 year 3 months? Ans. £6 lOs. 9^d. 7. Interest of .€116 Is. 7f for 11 mo. 19 days? Ans. cf6 158. 0|d. Interest of .£200 for 8 mo. 4 days ? £8 2s. Tfd. of 17s. for 19 mo. ? Is. 7|d. of £8 lOtf. for 1 year 9 mo. 12 days ? 18s. 2fd. of ^675 for 1 mo. 21 days ? £5 14s. 8|d. 9. 10. .«( 11. 12. 13. H. 15. 16. 17. of .£8673 for 10 days? of 14s. 7id. for 10 mo. ? of i:96 for 3 days ? of .£73 10s. for 2 days ? of ^180 158. f rSdays? of jeiSOOO for 1 day / Handth, the pounds themselves express the interest in thou- sandth for six days, of which we may take part" Thus, 6)15000 thousandths, <( (t t( (( £U 9s. l^d. 8jd. 1 Note. The I interest oi'£\ J for 6 days be- J ing 1 thou. 2'500, that is, £'2 10s. Ans. 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 £0)00 for 120 years? Ans. .£7200. 19. What is the interest of ^520 Os. Of d. for 30 years and 6 months? Ans. .£951 13s. 5fd. 20. What is the interest on .^400 for 10 years 3 months and 6 days? ^W5. .£246 8s. 21. What is the interest of .£220 for 5 years ? for J 2 year^ ? 50 years ? Ans. to the last, .£660. i^2. What is the amount of £"86, at interest 7 years ? Ans. £1^2 2s. 4|d. 23. What is the interest of $48*30 for 1 year? It must be clear to the pupil's mind, that to obtain the II 77. I H 78, 79. INTEREST. IfiO interest upon any sum in federal money, fur any time, we proceed just as wo do in Il.ilifax currency ; only' we are not coiupclled to reduce any part of the given sum to decimals, since all the denominations of federal money arc in a deci- mal ratio. T.ie answer to the last example is $!2,899. What is the interest of IG4 for 2 years ? Ans. $VGS. Whit is the interest of $i)8'50 for 7 years, months and 10 days ? Ans. >I44'489. 5T «&i. 1. What is the interest of 36 pounds for 8 months, at 4j- per sent ? _■,. . • • i i . ^ m Note. When the rate is any other than six per cent, first find the interest at six per cent, then divide the interest so found by such part as the interest, at the rate required, ex- ceeds or falls short of the interest, at six per cent, and the quotient added to or subtracted from the interest at six per cent, as the case may be, will give the interest required. £m '04 i^ per cent is f of six per cent ; therefore from the interest at six per cent subtract ^ ; ^)144 the remainder will be the interest at 4^ per '36 cent. . , ,, , £ViM £i Is. 7id. answer. 3. Interest of <^54 16s. 2|d. for eighteen months, at tive per cent ? Ans. £4 2s. 2J^d. 3. Interest of ^500 for nine months and nine days, at ciorht per cent? Ans. £Sl. \ Interest of ^62 2s. 4Jd. for one month and twenty days, at four per cent ? Ans. 6s. lO^d. 5. Interest of .£85 for ten months and fifteen days, at 12} per cent ? Ans. £9 Ss. lOfd. 6. What is the amount of .£53 at ten per cent for seven months? Ans. £5& Is. 9^d. The time^ rate per cent and amount given, to find the principal. ff 7d» 1. What sum of money, put at interest at 6 per cent, will amount to £61 Q3. 4f d. in I year 4 months ? The amount of ^l at the given rate and time i& <^r08; hence ^6l*02-i-£l'08=.56'50, the principal required; that is, find the amount of ^1 at the given rate and time, by which divide the given amount ; the quotient will be the principal required. Ans. <£56 lOs. 160 INTEREST 1IT9. 2. What principal, at 8 per cent, in 1 year 6 months, will amount to .£85 2s. 4f d. 1 Ans. .£76. 3. What principal, at 6 per cent, in 11 months 9 days, will amount to .£99 6s. 2f d. ? An&. ^94. 4. A factor receives .£988 to lay out after deducting his commission of 4 per cent ; how much will remain to be laid out 1 It is evident he ought not to receive commission on his own money. This question, therefore, in principle, does not differ from the preceding. NoU. In questions like this, where no respect is had to time, add the r.att to i^l. Ans. .£950. 5. A factor receives J61008 to lay out after deducting his commission of 5 per cent ; what does his commission amount to .' Ans. .£48. jj^ llii'i Discount. — 6. Suppose I owe a man .£397 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 is 6 per cent ? I ought to pay him such a sum as, if put at interest, would, in one year, amount to jf397 10s. The question, therefore, does not differ from the preceding. Ans. if 375. Nott. An allowance made for the payment of any sum of money before it comes due, as in the last example, is called discmtnt^ 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 worth. 7. What is the present worth of .£834 payable in 1 year, 7 months and 6 ()ays, discounting at the rate of 7 per cent ? Ms. .£750. 8. What is the discount on .£321 12s. 7^d. due 4 years hence, discounting at the rate of 6 per cent ? Ans. ^62 5s. 2f d. 9. How much ready money must be paid for a note of £'18, due fifteen months hence, discounting at the rate of 6 per cent.' Ans. £16 14s. lO^d. 10. Sold goods for .£650, payable one half in 4 months, and the other half in 8 months ; what mqst be discounted for present payment? 4w5, £\S. The fl 80, 81, 82/ INTEKEST. •161 11. What is the present worth of jf56 4s. payable in 6ne year eight months, discounting at 6 per cent? at 4^ per cent ? "at 5 per cent ?- — at 7 per cent 7 at 1^ per ^ cent ? at 9 per cent ? Ans. to the last £A% 178. 4|d. The time, rate per cent, and interest being given to find the , principal ^ 80. i. What sum of money put at interest sixteen months, will gain ^10 10s. at 6 per cent ? £{ at the given rate and time, will gain *08 ; hence, £lO'50-^Je'03=^l3^25, the principal required; that is— find the interest of £\. at the given rate and time, by which divide the given gain or interest ; the quotient will be the •principal required. Ans. <£131 5s. 2. A man paid £i 10s. 4fd. interest at the rate of 6 per cent at the end of 1 year 4 months ; what was the principal ? Ans. £5& 10s. 3. A man received for interest on a certain note at the end of one year ,£20; what was the principal, allowing the rate to have been 3 per cent t Ans. .£333 6s. 8d. The principal, interest and time being given, to find the rate per cent./ ^81. 1. If I pay ^3 15s. 7^^. interest for the use of £36 for 1 year 6 months, what is that per cent ? The interest on £36 at one per cent, the given time, is £'54 ; hence £3*78-^-£'54='07, the rate required ; that is, find the interest on the given sum, at one per cent, for. the given time, by which divide the given interest ; the quotient will be the rate at which interest was paid. Ans. 7 per ct. 2. At £2 6s. 9^d. for the use of £468 for a month, what is the rate per cent ? Ans. 6 per cent. 3. At £46 16s. for the use of £520 for two years, what is that per cent ? Ans. 4] per cent. The prices at which goods are bought and sold, being given^ to find the rate per cent o/'gain or loss. ^ 89. 1. If I purchase cloth at £1 2s. a yard, and sell ii at £1 7s. 6d. per yard ; what do I gain per cent ? This question does not differ essentially from those in the foregoing paragraph. Subtracting the cost from the price 02 16^ INTEREST. ■'^' !F82,83. w 'io \ at sale» it is evident I gain ^'275 on a yard ; that is f ,{& of the first cost, f '^^='25 per cent, the answer. That is,— 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 JS550 ; what per cent profit must he make ^o gain <£66 ? • * • ' Ans. 12 per cent. 3. What p^r cent profit must he make on the same purchase to gain ^38 10s. ? to gain j£24 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, (IF 75, note,) is a decimal carried to two places, or hundredths ; all decimal expressions lower than hundredths are parts of one per cent. 4. Bought a hogshead of liquor, containing 114 gallons, at .£'96 per gallon, and sold it at £\ Os. Od. 3|qrs. per gal. what was the whole gain, and what was the gain per cent ? Ans. £4c 18s. 5f d. whole gain. — 4^ gain per cent. 5. A merchant bought a quantity of tea for £365, which, proving to have beeij, damaged, he so!d for .^332 3s. ; what did he lose per cent ? 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 if^Tf V^ to be reduced to one common dene- minator. , Neglect the ciphers in the denominators — 6 X 2 X 6=60; 1 4-2 -|- 2=5, the number of ciphers. The common denominator is then 60 and 5 ciphers. 6 X 2 X 6=72 ; this with 4 ciphers is first numerator. 5 X 6 X 6=180 ; this with 3 ciphers is 2d numerator. 5 X 2 X 1=10 ; this with 3 ciphers is 3d numerator. Each numerator has as many as 3 ciphers; cut offthtee from each, and three from the common denominator ; ^^^^ '\'hU-^M^^x^u\=^i^' Then ^100, the given princi- pal, multiplied by ^j^=£^^=:£\5 3s. 4d. The reasons of the different steps in the foregoing pro- cess will appear : when the rate, as in the above example, is 6 per cent, it is obvious that the interest of ^ny given principal for one year is yf^y or -^^ of that principal. For any number of years, the interest must be as many times ^^y of the principal as there are units in the given number of years. In the example, 2 is the given number of years ; multiply then ^^ by 2 ; or multiply the lowest terms of a fraction, tht numerator of which is the given rate, and the denominator 100, by the given number of years, -^jj of the given principal then is the interest for 2 years, y^jyof the given principal is the interest for 1 month; for there are 12 months in a jear, and y^^ X iV = t^ott or g-^^. ^^^^^ of the given principal is the interest for 1 day ; for there are 30 days in I month, and y/j^ X i^if = ^^xTu =FTrVir- We have then -^^ of given principal, as the interest for 1 year ; 2"^ J of sam3, for 1 month, and ^a^jyjy for 1 day. For 2 years, we have ^^y X 2 = ^(y: for 6 months ^^^ X f»=^ishl for ten days, ^^Vt X 10=^^^^^=^^^- ^^, j^^ and ^^^ then of the given principal are the interest of ,£100 for 2 years, 6 months and' 10 days. It is clear now, why we reduce these several fractions to one common denominator, add them together, and by the resulting fraction multiply the given principal. Find the interest upon £78 4s for 3 years, 9 months and 6i.diys, by this method, at 6 per cent and also at 5 per cent. ■, -i-x-—--.-^' 1184. • •: ' INTEREST. m To find the intereet due on Notes, S^c. when partial pay- ments have been made. IT 84. There is no statute in this Province, prescribing any particular form or method of casting interest upon notes or other obligations. It is believbd the following method is f^enerally allowed before the courts of the country, and also is that which has obtained to the greatest extent in mercantile transactions. Rule. — Compute the interest upon the value for which the note or other instrument was given, tp the time of pay- ment, which add to the principal ; find the amount also of each endorsement to the time of payment, which several amounts add together, and the sum subtract from the amount of the value upon the face of the note, or other instrument. , : . 1. For value received, I promise to pay Louis Rousseau, or order, one hundred pounds fifteen shillings, with interest. iflOO 15s. . John Burton. May 1, 1822. On this note were the following endorsements . • Dec. 25, 1822, received ^10 July 19, 1823, " Sept. 1, 1824, " June 14,1825, " 1 4s. 3 6s. 21 15s. April 15, 1826, " What was due Aug. 3, 1827.1' 54 9s. Ans. £U 3s. Id. The whole time is, from May 1st, 1822, to Aug. 3, 1827, which is 5 years, 3 months, 2 days. The interest of .£100 153. for this time is ^31 15s. 4f d. This added to the value for which the note h as given i8 £100 15s.-f-<£31 15s. 4|d.= £iS2 10s. 4fd. which is equal to the amount of the value for which the note was given. The first endorsement is £10; the date of this endorsement is Dec. 25, 1822; the time of payment is Aug. 3, 1827. The time, therefore, for which interest is to be cast upon this endorsement, is 4 yrs, 7 mo. 8 ds. The interest for this time is £'2 15s. 3d. which, added to the endorsement, makes its amount £\^ 15s. 3d. In the same way find the amount of each other endorsement, by casting the interest upon it from the day of its date to the day of the payment of the note, and add this interest to the principal, that is, the endorsement. . ,, ■■ ( 16^ COMPOUND INTEREST. The 2d endorsement is 3d • 4th " : 5th The time for which interest is to be cast upon the 2d endorsement is - -4 years, months, 23 days 3d " - - - 2 " 11 " 2 " 4th 'f - - - 2 " 1 " 19 " ^84,85. I ^^^' £ 1 4s. 3 68. 21 15s. 54 9s. 5th - 1 (( (( t( it (( (( The interest upon the 2d endorsement is " '• 3d 4th 5th The amount of the 2d endorsement is »3d " 4th '" 5th The amount of 1st endorsement we found to be 12 The sum of the ^mounts of all the endorsements 101 The value upon the face of the note is - 100 The amount of this value is - - - 132 Subtract the sum of amounts of endorsements 101 (< (( (( £ 2 4 1 3 24 58 18 " s. d. 5 10 11 6f 15 8| 4 11^ 9 10 17 GJ 10 8| 13 l\i 15 3 7 3f 15 10 4f 7 Si- Balance due Aug. 3d 1827, ^ 31 3 1 2. For value received, I promise to pay Thomas Wilson, or order, two hundred thirty-eight pounds eighteen shillings, with interest. £238 18s. Charles Stewart. Jan. 6, 1820. On this note were the following endorsements, viz : April 16, 1823, received April 16, 1825, Jan. 1, 1826, " • What was due July 11, 1827 ? £ s. rf 23 10 19 4 87 19 COMPOUND INTEREST. Tf S5» A. promises to pay B. ^256 in three years, with interest annually ; but at the end of one year, not finding it convenient to pay the interest, he consents to pay interest IF 84,85. I ^^^' TEWART. COMPOUND INTEREST. Ki7 on the interest from that time, the same as on the principal. Note. — Simple Interest is that which is allowed for the principal only ; compound interest is that which is allowed for both principal and interest, when the latter is not paid at the time it becomes due. Compound Interest is calculated by adding the interest to the principal at the end o£ each year, and making the amount the principal for the next succeeding year. 1. What is the compound interest of £256 for three years, at 6 per cent ? i , '. ' • * £256 given sum or first principal. '06 15 .^6 interest, > . u jj j * ^l n-atnix • • i ? to be added together. 2oo*00 prnicipal, ) ° 27r36 amount or principal for second year. *06 16'2816 compound interest 2d year, ) added 271*36 principal, , do | together 2S7'6416 amount or principal for 3d year. ♦06 17*258496 compound interest 3d year, ) added 287*641 principal, . do « j together 304*899 amount. 256 first principal subtracted. ^'48*899 compound interest for three years. Ans. £48 17s. llfd. 2. At 6 per cent, what will be the compound interest, and what the amount of £1 for two years 7 what the amount for 3 years ? for 4 years ? for 5 years ? for 6 years ? for 7 years? for 8 years ? Ans. to the last, £1 lis. lO^d. It is plain that the amount of £2, for any given time, will be two times as much as the amount of ill ; the amount of £3 will be three times as much, &c. 168 COMPOUND INTEREST. 1185. Hence, we may form the amounts of one pound, for seve- ral years, into a table of multipliers for finding the amount of any sum, for the same time. TABLE, Showing the amount of One Pound or One Dollar ^c.for any number of years not exceeding 24, at the rates of 6 and 6 per cent Compound Interest. Years 1 2 3 4 W0 o 6 7 8 9 10 11 12 5 per cent 1<05 1*1025 1*15702+ r215504- l'27628-l- l'34009-[- 1*40710-1- 1*47745-1- 1*55132-1- 1*62889-1- 1*71033-1- 1*79585-1- 6 per cent 1*06 V1236 1*19101+ 1*26247-1- 1*33822-}- 1*41851-1- 1*50363-1- 1*693844- 1*68947-1- 1*79084-1- l*89829-f- 2'01219-|- Years 13 14 15 16 17 18 19 20 21 22 23 24 5 per cent 1*88564+ 1*97993+ 2*07892-1- 2*18287-1- 2*29201+ 2*40661-1- 2*52695 2*65329+ 2*78596-f- 2*92526-1- 3*07152+ 3*22509-1- 6 per cent 2*13292+ 2*26090-}- 2*39655-1- 2*54035-f. 2*692774- 2*85433-}- 3*02559-}- 3*20713-}- 3*39956+ 3*60353-}- 3*81974+ 4*04893-}- Note 1. Four decimals in the above numbers will be sufficiently accurate for most operations. Note 2. When there are months and days, you may first find the amount for the years, and on that amount cast the interest for the months and days ; this added to the amount, will give the answer. 3. What is the amount of X600 10s. for 20 years at 5 per cent, compound interest ? j- at 6 per cent ? £1 at 5 per cent by the table is ^£2*65329 ; therefore, 2*65329X600*50=£l593*30+is £1593 6s. Ans. at 5 per cent; and 3*20713X600*50= £l925*881+is £1925 17s. 7^d. ans. at 6 per cent. 4. What is the amount of £40 4s. at 6 per cent com- pound interest, for 4 years .^ for 10 years? for 18 years ? for 12 years ? • for 3 years and 4 months? for 24 years, 6 months and 18 days ? Ans. to the last £168 2s. 8|d. Note. Any sum at compound interest will, double itself in 11 years, 10 months and 22 days. i < H 85. 1 ff 85. COMPOOND INTBRE8T. 160 ibers will be From what has now been advanced, we deduce the fi^ lowing general ....;..,; RULE. I. To find the interest when the time is one year, or, to find the rate per cent on any sum of money, without respect to time, as the premium for insurance, commission, &c. — Multiply the principal or given sum, after having reduced the shillings and pence in it to the decimal of a pound, by the rate per cent, written as a decimal fraction ; afler point- ing off as many places for decimals in the product as there are decimals in both the factors, and reducing these deci- mals back to shillings and pence, we shall obtain the inter- est required. II. When there are months and days in the given time, to find the interest on any sum of money at 6 per cent, — Multiply the principal, reducing the shillings and pence by inspection, by the interest on one pound for the given time found by inspection, and the product, as before, will be the interest required, taking care ^o reduce the decimal parts to shillings and pence by inspection. III. To find the interest on one pound at 6 per cent, for any given time by inspection, — It is only to consider that half the greatest even number of months will denote hundredths of a pound, and that there will be five thou- sandths of a pound fo. he odd month, (if there be one) and one thousandth for every six days. IV. If the sum given be in federal mone^, — The deno- minations bef ng in a decimal ratio, we are saved from the necessity of effecting the reciprocal reductions, at the be- ginning and end of the process, otherwise proceed precisely as in Halifax currency. V. If the interest required be at any other rate than six per cent, (if there be months, or months and days in the given time,) — First find the interest at six per cent ; then divide the interest so found by such part or parts, as the in- terest, at the rate required, exceeds, or fsdls short of the interest at six per cent, and the quotient, or quotients, ad- ded to or subtracted from the interest at six per cent, as the case may require, will give the interest at the rate required. Note. The interest on any number of pounds, for 6 days at 6 per cent, is readily found by cutting off the unjit or P 170 .1 INTIREST. %i:or) !185. 3^! right hand figure ; those at the left hand will «how the in- terest in hundredths for 6 days. '>') EXAMPLES FOR PRACTICE. 1. What is the interest of J^1600 for 1 yoar 3 months? ^ Ans. £120. ^S.JW'hat is the interest of i£5 16s. for 1 year IL months? Ans. 13s. 4d. 3. What is the interest of <£2 5s. 9^d. for 1 month 19 days, at 3 per cent f ■ Ans. 2^d. 4. What is the interest of <£18 for 2 years 14 days at 7 per cent ? Ans. £'2 1 Is. 4^d. 5. What is the interest of i^l7 13s. 7^d. for 11 months 28 days? Ans. £1 Is. Id. 6. What is the interest of .£200 for 1 day? 2 days? 3 days ? 4 days ? 5 days ? Ans. for 5 days, 3s. 3f d. 7. What is the interest of half £'001 for 567 years? . , i. ; , Ans. 4d. 8. What is the interest of £81 for 2 years 14 days, at } per cent ? | per cent ? | per cent ? 2 per cent ? ■ 3 per cent ? I j- per cent ? 5 per cent ? per cent ? 9 per 7 per cent ? cent ? 10 per 7^ per cent ? 8 per cent ? 12 per 12^ per cent? Ans. to last, £20 12s. lO^d. What is the interest of £*09 for 45 years, 7 months, 11 days? ^»5. 48. 10|d. 10. A.'s note of £175 was given Dec. 6, 1798, on which was endorsed a year's interest; what was due 1st Jan. 1803? Note. Consult Ex. 16, Supplement to Subtraction of Compound Numbers. Ans. £207 4s. 4fd. 11. B.'s note of £56 15s. was given June 6, 1801, on interest after 90 days; what was there due 9th Feb. 1802? Ans. £58 3s. d^d. 12. C.'s note of £365 was given Dec. 3, 1797 ; June 7, 1800, he paid £97 3s. 2id. ; what was there due 11th Sept. 1800? " ^ns. £327 Os. 7id. 13. Supposing a note of £422, dated July 5, 1797, on which were endorsed the following payments, viz. Sept. 13, 1799, £208 4s. ; March 10, 1800, £96; what was there due Ist Jan. 1801 ? 1185/ ,;r, 7«.^'T tom.iMCli¥'Tb iilYBiietT. Snpplement to Interest. QUESTIONS. 171 1 . Whdt ii interest f 2. How ia it computed 1 3. What is uodor- (tood by rate per cent 7 4. by principal t 5. — — > by amount f 6. by legal Interest ? 7. by commiisionl^ 8.— —Insur- ance f 9. premium t 10. policy ? 11. Stock t 12. What is understood by stock bein); at par ? 13. ahovo par T 14. below par? 15. The rate per cent is a decimal carried to bow many places I 16. What are decimal expressions lower than hun- dredths 1 17. How is interest (when the time is one year) eommis- sian, insurance, or anything else rated at so much per cent without respect to time, found? 18. When the rate is one per cent, or less, how majf the operation be contracted t 19. How is the interest on one pound at 6 per cent, t>r any givsn time, found by inspection ? 20. How is interest cast at 6 per cent, when there are months and days in the given time 1 21. When the given time is less than 6 days, how is the interest most readily found ? 22. If the sum given be in federal money, how is interest cast 1 23. When the rate is any other than 6 per cent, if there be months and days in the given time, how is the in- terest found ? 24. What is the rule for casting interest on notes, Ac. ffhen partial payments have been made ?.. 25. How may the principal be found, the time, rate per cent and amount being given ? 26. What is understood by discount ? 27, — — by present worth 1 28, How is the principal found, the time, rate per cent and interest being given? 29. How is the rate per cent of gain or loss found, the prices at which ^oods are bought and sold bein^ given ? /30. How is the rate per cent found, the principal, interest and time being given ? 31. How ia the time found, the principal, rate per cent and interest being given 1 32. How may interest be cast by vulgar fractions ? 33. What is the reasoning in regard to this rule f 34. What is simple interest ? 35. compound interest 1 36. How is compound interest computed ? EXERCISES. ' 1. What is the interest of £279 10s. ^d, for 1 year 10 days, at 7 per pent? v s ; ' " 't Ans. i£19 13s. 6^. 2. What is the interest of J?486 for 1 year 3 months 19 days, at 8 per cent ? Ans. £50 138. 4Jd. 3. D.'s note of ^6203 was given Oct. 5, 1808, on interest after 3 months; Jan. 5, 1809, he paid ^50; what was there due 2d May, 1811? Ans. .£175 7s. 2d. 4. E.'s note of ^870 was given Nov. 17, 1800, on inter- est after 90 days ; Feb. II, 1805, he paid ^186 ; what was there due 23d Dec. 1807 ? Ans. ^1009 lis. 6fd, 5. What will be the annual insurance, at f per cent, on a house valued at ^1600 ? Ans. £\9, m ii< '>:f m uiinum$s:^ to i»TiREfi'» 1186. 6. What will be the insurance of a thip and cargo, valued at £5643 at 1^ per cent ? at ^ per cent ? at fg per cent ? at -f^ per cent ? at ^ per cent ? Note. Consult II 76, ex. 11. Arts, at ^ per cent i^42 Gs. ^d. 7. A man having compromised with his creditors at (i'2^ per cent, what must he pay on a debt of <;fl37 9s. 2jd. ? Ans. ;£85 188. 3d. 8. What is the value of £800 Montreal Bank stock, at 1 12 j- per cent / Ans. .£900. 9; What is the value of <£560 15s. of stock, at 93 per cent? iltt5. Je521 9s. ll^d. 10. What principal, at 7 per cent, will, in 9 months 18 days, amount to <£422 8s. ? Ans. .£400. 11. What is the present worth of ^426, payable in 4 years 12 days, discounting at the rate of 5 per cent ? In large sums, to bring out hundredths and thousandths correctly, it will sometimes be necessary to extend the de< cimal in the divisor to five places. Ans. £354 10s. l^d. 12. A merchant purc|iased goods for £250, ready money, and sold them again for £300, payable in 9 months ; what did he gain, discounting at 6 per cent ? Ans. £37 1 s. 7^d. 13. Sold goods for £3120, to be paid one half in three months, and the otheV half in six months ; what must be discounted for present payment ? Ans. £6S 9s. lOd. 14. The interest on a certain note for 1 year 9 months was £49 17s- 6d. ; what was the principal ? Ans. £475. 15. What principal, at 5 per cent, in 16 months 24 days, will gain £35? ilns. £500. 16. If I pay £15 lOs. interest for the use of £500, nine months and nine days, what is the rate per cent ? 17. If I buy candies at $'167 per Id, and sell them at 20 cents, what shall I gain in laying out $100 ? ^115. $19*76. 18t Bought hats at 4b. a-piece, and sold them again at 48. 9d. ; what i9> the profit in laying out £100 ? Ans. £18 15s. 19. Bought 37 gallons of brandy at $1'10 per gallon, and sold it for $40 ; what was gained or lost per cent ? 20. At 4s. 6d. profit on one pound, how much ib gained in laying out £100, that is, how much per cent ? j/l^ - ilns. ^2108. W; I4VATI0N OF rATMBNTt. .^'«' 173 go, valued at ^, It? 2 68. 5{d. ^rs at (J2i . 2id. ? L88. ad. stock, at f. ^900. at 93 per s. U^d. [nonths 18 . ^400. irable in 4 mi? lousandths nd the de- Os. l^d. Ldy money, iths ; what 17 Is. 7id. If in three it must be 9s. lOd. 9 months s. ^^476. IS 24 days, ^'500. '500, nine ;11 them at $19'76. again at 18 153. er gallon, cent.' ib gained |22 lOs. 21. Bought cloth at $4*48 per yard ; how muit I idl it to gain 13^ per cent? -f v Ans. $(i*Oi. 22.' Bought a barrel of powder for £4t; for how much must it be sold to loae 10 per cent ? Ans. £B 128. 23. Bought cloth at 16i. per yard, which, not proving so good as I expected, I am content to lose '17^ per cent ; how must I sell it per yard? Ans 128. 4jd. 24. Bought 50 gallons of brandy at 92 cents per gallon, but by accident, ten gallons leaked out ; at what rate must I sell the remainder per gallon, to gain upon the whole cost at the rate of ten per cent ? i4ns. $1*265 per gal. 25. A merchant bought ten tons of iron for I960 ; the freight and duties came to $145, and his own charges to $25 ; how must he sell it per lb, to gain twenty per cent by it? Ans. 6 cents per lb. Equation of Payment!. ^ HS* Equation of Payments is the method of finding the mean time for the payment of several debts due at different times. 1. In how many months will one pound gain as much as five pounds will gain in six months? 2. In how many months will one pound gain as much as forty pounds will gain in fifteen months ? " Ans. 600. 3. In how many months will the use of five pounds be worth as much as the use of one pound for forty months ? 4. Borrowed of a friend one pound for twenty months ; aflerwards lent my friend four pounds ; how long ought he to keep it to become indemnified for the r ,e of the one pound ? 5. I have three notes against a man ; one of <£12, due in three months ; one of £9, due in five months ; and the other of £6, due in ten months; the man wishes to pay the whole at once ; in what time ought he to pay it ? £12 for 3 months is the same as £1 for 36 months, 9 5"" 1 45 6 10 " " 1 60 <( 27 141 P2 174 RATIO, OE TnB RBLiVnOIf OF NVVBER8. 1186,87: I ^87^ He miglil thtf efore have one ptmnd 141 inotith0;!and'he may fc«^ twenty-seven pounds ^V part es k»^g; that U, yy* 3= five months- 6-f-day»,,i4»i. IrnrW 'it-,:: :t . ^,:f , ...1 Ratio : or Kclation of TVumbers. U 87. 1. What part of a gallon is three quarts.' oi gallon is four quarts, and three qu, :ts is f of four quarts. Ans. f of a gallon. ,::..« ' ,.;. ....,, ,.p,„. tl 86, ST. I If 87; ratio; or tbb ublavioh of numbers. 175 tha^^and'he gr that w, r aymenla,— int, and dh 3 payments, n that ivhat er it is due, time before lently equal tvhile in the ount c^ the he interest ; )r, however, business, as ) be paid as ths, and the ►ne payment to be? 1 months, ths ; £m to n 4 months ; whole ? ' days+. paid at the [ht n]ionths ; whole? months. ^ in three )nths ; what months, crs. [juarts ? one )ur quarts, a gallon. 2. Whatt part of 3 quarts ia ofie gallon? 1 gallon being 4 quarto, is ^ of 3 quarts ; that is, 4 quarts is 1 time 3 quarts and I of another time. ^ns. ^=s\^. 3. What part of five bushels is twelve bushels ? Finding what part one number is of another, is the same as finding what is called the ratio or relation of one number to another ; thus, the question, What part of five bushels is twelve bushels ? is the same as What is the ratio of five bushels to twelve bushels ? The answer is ^=2f . RatiOf therefore, may be defined the number of -times one number is contained in another ; or, the number of times one quantity is contained in another quantity of the same kind. 4. What part of eight yards is thirteen yards? or, What is the ratio of 8 yards to 13 yards.' 13 yards is -^ of 8 yards, expressing the division /rrtc- tionally. If now we perform the division, we have for the ratio 1|; that is, 13 yards is one time 8 yards, and f of another time. ; . We have seen (51 15, sign,) that division may be expres- sed /rac<8ona%. So also the ratio of one number to another, or the part one number is of another, may br expressed fi-actionally ; to do which, make the number which is called the jpar/, whether it be the larger or the smaller number, i\iQ numerator of a fraction, under. which write the other number for a denominator. When the question is. What is the ratio, &-c.? the number last named ts the /)ar< ; con- sequently it must be made the numerator of the fraction, and the number j^rs^ named the denominator. 5. What part of 12 pounds is 11 pounds? or, 11 pounds is what part of 12 pounds? 11 is the number which ex- presses the part. To put this question in the other form, viz. What is the ratio, &:.c., let that number which expresses the part, be the number last named ; thus, What is the ratio ol 12 pounds to 1 1 pounds ? Ans. \^. 6. What part of i6'l is 2s. 6d. ? or, What is the ratio of £\ to 2s. 6d. ? £1:=240 pence, and 2s. 6d.s=30 pence; hence, ^W^gj is the answer. 7. What part of 13p. 61 is ^1 lOs. ? or, What is the ratio of 13s. Od. to £i lOs. ? Ans.^. 176 .^?;m SULK or thrbb. * i 1188. fl88, 89 il' 8. What is the ratio of 3 to 5 ? of 5 to 3 ? of 7 to 19? of 19 to 7? of 15 to 90? of 90 to 15? — - of 84 to 160 ? of 160 to 84 ? of 615 to 1107 ? of 1107 to 616? Ans. to the last f. PROPORTlOnr: t OR THE SINGLE RULE OF THREE. ^ 88, 1. If a piece of cloth 4 yards long, cost ^12, what will be the cost of apiece of the same cloth seven yds. long ? Had this piece contained twice the number of yards of the first piece, it is evident the price would have been twice as much ; had it contained three times the number of yards, the price would have been three times as much ; or had it contained only half the number of yards, the price would have been only half as much; that is, the cost of seven yds. will be such part of -^12 as seven yards is part of four yards. Seren yards is | of 4 yards ; consequently, the price of 7 yards must be J of the price of 4 yards, or I of £12 : I of ^12, that is, 12X|=V=^21, answer. 2. If a horse travel 30 miles in 6 hours, how many miles will he travel in 1 1 hours at that rate ? 11 hours is y of 6 hours, that is, 11 hours is one time 6 hours, and | of another time ; consequently, he will travel, in 1 1 hours, Ltime 30 miles, and | of another time ; that is, the ratio between the distances will be equal to the ratio between the times. V of 30 miles, thatis,30X V=^F=55 miles, no error has been committed, 55 miles must be miles. This is actually the case ; for ^^=: V Ans. .;5 miles. Q,uantilies which have the same ratio between them are said to ht proportional. Thus, these four quantities — HOURS HOURS. MILES. MILES. If, Uien. V of 30 the first i is, the ra ratio bet\ proportio bination numbers Tode bers 6, 1 which is same par as many or relatio the antec portion t viz. the h the consie the propc the cons( Thecc numerate fraction, first ratio tios are e Thetv by reduci of the on and, cons same pro case, for if four n and last, the secor Hence given, to ing that times or tion thus 6. 11, 30. 55, written in this order, being such, that the second contains !I88. 11188,89. RULE OF THRES. 177 ? Of -of 90 to — of 615 last |. EE. ;ost ^12, even yds. ■ yards of een twice of yards, or had it ce would seven yds. pur yards, jricc of 7 f 12 ; I of any miles ne time 6 ill travel, me ; that the ratio If, Uien. V of 30 miles, them are lies — contams the first as many times as the fourth contains the third; that is, the ratio between the third and fourth being equal to the ratio between the first and second, form what is called a proportion. It follows, therefore, that proportion is a com- bination of two equal ratios. Ratio exists between iwn numbers ; but proportion requires at least three. To denote that there is a proportion between the num- bers 6, 11, 39, 55, they are written thus — 6 11 : : 30 : 35 which is read, 6 is to 11 as 30 is to 55 ; that is, is the same part of 11 that 30 is of 55 ; or, 6 is contained in 1 1 as many times as 30 is contained in 55 ; or, lastly, the ratio or relation of 11 to 6 is the same as that of 55 to 31). fl 89. The first term of a ratio, or relation, is^( tiled the antecedent, arid the second the consequent. In a pro- portion th§re are two antecedents, and two consequents, viz. the antecedent of the first ratio, and that of the second ; the consequent of the first ratio and that of the second. In the proportion 6:11 :: 30 : 55, the antecedents are 6, 30 ; the consequents 11, 55. The consequent, as we have already seen, is taken for the numerator, and the antecedent for the denominator of the fraction, which expresses the ratio or relation. Thus, the iirst ratio is y, the second ^^=y ; and that these two ra- tios are equal, we know, because the fractions are equal. The two fractions ^ ^"d ^^ being equal, it follows that by reducing them to a common denominator,^ the numerator of the one will become equal to the numerator of the other, and, consequently, that 11 multiplied by 30 will give the same product as 55 multiplied by 6. This is actually the case, for 1 1 X 30=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 the 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 hours occupied in travelling, we write the pi'opor- tion thus — JIOURS. 6 HOURS, 11 MILES. 30 MILKS. Win' 178 RULE OF THREE. Now, since the product of the extremes is equal to the product of the means, we multiply together the two mean8, Jl and 30, which makes .330, and, dividing this product by the known extreme, 6, we obtain for the result 55, that is, 55 miles, which is the other extreme or term sought, 3. At ^54 for 36 barrels of flour, how many barrels mat be purchased for £1861 In this question, the unknown quantity is the number of barrels bought for .£186, which ought to contain the 36 barrels as many times as ^186 contains £54 ; we tlius get the following proportion : Pounds, Pounds. Barrels. Barrels. . ' 54 : 186 :: 36 : : 36 1116 558 64)6696(124 barrels, answer. 54 129 108 "2T6 216 The product 6696 ®* the t\Vo means, divided by 54, the known extreme, gives 124 barrels for the other extreme, which is the term sought, or answer. Any three terms of a proportion being given, the opera- tion by which we find the fouTth, is called the Rule of Three. A just solution of the question will some times re- quire that the order of the terms of proportion be changed. This may be done, provided the terms be so placed, that the product of the extremes shall be equal to that of the means. 4. If 3 men perform a certain piece of work in ten days, how long will it take 6 men to do the same ? The number of days in which six men will do the work, being the term sought, the known term of the same kind, viz. ten days, is made the third term. The two remaining terms are 3 men and 6 men, the ratio of which is t. But the more* men there are employed in the work, the less time will * The rule of three lias sometimes been divided into direct and in' vtrse, a distinction which ia lolally useless. It may not hotvever be aniitts to explain, in this place, in what this distinction consists. The Rule of Three Direct is when more requires more, or /ess re- RULE OF THREE. 179 be required to do it ; consequently the days will be less in proportion as the number of men is greater. There is still a proportion in this case, but the order of the terms is in- verted ; for the number of men in the second set being two times that in the first, will require only one half the time. The firjtt number of days, therefore, ought to contain the second as many times as the second number of men con- tains 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 ratio of the number of days. With a view, therefore, to a just solution of the question, we reverse the order of the two first terms, (in do- ing which, we invert the ratio,) and instead of writing the proportion 3 men : 6 men (f ) we write it ii men : 3 men, (g) lluit is, men. men. days. days. ■ 6:3 :: 10 Note. We invert the ratio when we reverse the order ol the terms in the proportion, because then the antecedent takes the place of the consequent, aud the consequent that of the antecedent ; consequently, the terms of the fraction which express the ratio are inverted ; hence the ratio is inverted. Thus, the ratio expressed by f=2, being inver- ted, is ^=^: Having stated the proportion as above, we divide the product of the means, (10x3=30,) by the known extreme 6, which gives 5, that is, 5 days, for the other extreme or term sought. w4ns. 5 days. From the examples and illustr.ii; ions now given, we de- duce the following general quires lesSy as in this example .— if 3 men dig a irench 48 feet long in a certain lime^ how many feel will 12 men dig in the same time ? Here it is obvious that the more men there are employed, the more work will be done ,* and therefore, in this instance, more requires mor;;. Again — if 6 men dig 48 feet in a given time, how much will 3 men dig in the same time ? Here less requires less, for the less men there are employed, the less work will be dune. The Rule of Three 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 12 men to dig the same quantity ? Here more requires loss ; that is, 12 men being more than ti, will require less time. Again — if 6 men perform a piece of work in seven days, how long will three men be in performing the same work/ Here less requiros more j for the number of men being less, will require more time. ''-m ■'■.m ■m m 180 RVLE OF THREE. RULE. Of the three given numbers, make that the third term which is of the same kind with the answer sought. Then consider, from the nature of the question, whether the answer will be greater or less than this term. If the answer is to be greater J, 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, wlJch will always be of the same denomination as the thi a* term. Ni*te, L If the first and second terms contain different d!"n(»ur, :'a. jns, they must both be reduced to the same de- noiwia:!.*^. If 8 ; 'itth of cloth cost £\ 4s. what will li64 qrs. cost ? yds. qrs. 8 : 364 : : £1 43. Reduce 8 yards and 364 quarters to the same denomina- tion, by dividing the 364 quarters by 4, which will bring it into yards. 3|*=91. yds. yds, 8 : 91 :: ^1 4s. Note 2. If the third term be a compound nur.iber, 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. ydi. yds. 8 ' : 91 : : ^1 4s. 20 24s. ♦^ Now multiply the 24s. by 91, and divide the product by 8 ; the answer will be shillings, which can be reduced to pounds ; or, the 4s. can be reduced to the fraction of a pound, 4S.-7-20, that is, ^Tr=i of a pound ; so ^1 4s.=^l|. Or, we can reduce the 4s. to the decimal of a pound ; 20)40 which, annexed to the £\y is equal to £V2. — '2 .■* ,. «ULB OP THEfcB, m The first method is most usually practised. ' ' 'i' ' ' . Note 3, The same rule is applicable, whether the giv.en KjuaDtities he iIH^gra], fractjional, or decimal, EXAMPLES FOR PRACTICE, 5. If 6 horses consume 21 bushels of oats in three v/efii(.ii, fiov inany bushels will serve 20 horses the same time ? Ans, 70 bushels, 6. The above question reversed. If 20 horses consume 70 bushels of oats in 3 weeks, how fn^jiy bushels will servj^ 6 horses the same time ? 4w«. Ql bushels, 7. If 365 men consume 75 barrels of provisions in niyi^ rrioriths, how piuch will 500 me^ consume in the sam^ time ? Ans, J02^^ barrels, 8. Tf 500 men consume 102^^ barrels of" provisions iij j) lYionths, how much will 365 mei) consume in the saniit^ time 1 "^ Ans, 75 barrels, 9. A goljlsmith sold a tankard for ,£10 12s. at the rait« of 5s, 4d, per ounce ; I demand the weight of it, Ans. 39 oz, 15 pw^t, 10. If the moon move 13 ^ 10' 35" in a day^ iij wi)ajt time does it perform one revolution ? Ans, 27d, 7h. 43ui, 11. If a person whose rent is £33, pay £8 28. parivslj taxes, how much should a person pay whose rent is ^£"97 ? Aiifi. £9 2s. 2|f d, 12. If I buy 7 fcs, of sugar for 3s, Od, how many poundi^ ean I buy for £1 10s. ? , Ans, 56 tha, 13. If2 lbs. of sugar cost Is, 3d., what will 100 lbs, ot coffee cost, if 8 }bs of sugajr are worth 5 Jbs, of coffee ? Ans, ot',5/ 14. If I give £6 for the use of £100 for 12 months, whai jDust J give for the use of £983 the same time ? 15. There is a cistern which has 4 pipes ; the first will fill it in ten minutes, the second in twenty minutes, the vT4 in forty minutes, the fourth in eighty minutes ■ in wha.t Uff^ yvill all four, running together, fill it ? iV + 5V +5V + ffV ^ ih ^^istern in ! minute, Ans,5\- m'iTiiitf«^ 16. If a family of 10 persons spend 3 bushela of mBlf. i(j a month, how many bushejs will serve thgnji when thera ttf^ 30 in the famWy ? An.s, 9 hu«t)f 5«; ,..■,),. 1 m 182 RULE OF THREE. TI89. i I Note. The rule of Proportion, although of frequent use, is not of indispensable necessity ; for all questions under it may b9 solved on general principles, without the formality of a proportion ; that is, by analysis, as already shown, jj 62 ex. 1. Thus, in the above example, — If 10 persons spend 3 bushels, 1 person, in the same time, would spend -j^y of 3 bushels, that is, -^ of a bushel ; and 30 persons would spend 30 times as much, that is f$=9 bushels, as before. 17. If a staffs feet 8 inches in length, cast a shadow of 6 feet, how high is that steeple whose shadow measures 153 feet ? , Ans. U4h feet. 18. The same by analysis. If 6 feet shadow require a staff of 5 feet 8 inches=68 inches, one foot shadow will re- quire a staff of '^ of ^ inches, or ^^ inch; then 153 feet shadow will require 153 times as much; that is, ^^ X 153 -_io|.o4_i734 inches=144^ feet as before. 19. If i£3 sterling be equal to JG3^ Halifax, how much Halifax is equal to .£1000 sterling ? Ans. .£1 1 11 2s. 2§d. 20. If £1111 23. 2§d. Halifax be equal to jeiOOO sterling, how much sterling is equal to £3^ Halifax? Ans. £3. 21. If £1000 sterling be equal* to £1111 2s. 2§d. Hali- fax, how much Halifax is equal to £3 sterling ? Ans. £3^. ' 22. If j£3 sterling be equal to £3;^ Halifax, how much sterling is equal to £1111 2s. 2fd. Halifax ? Ans. £1000. 23. Suppose 2000 soldiers had been supplied with bread sufficient to last them 12 weeks,' allowing each man 14 oz. a day ; but, on examination, they find 105 barrels, contain- ing 200 lbs. each, wholly spoiled ; what must the allowance be to each man, that the remainder may Ir.^t them the same time 1 Ans. 12 ounces 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? . i The whole weight, 147000 lbs. '^"*' ( Spoiled, ^i"«" " 21000 25. — 2000 soldiers, having lost 105 barrels of bread, weighing 200 lbs. each, were obliged to subsist on 12 oz. a day for J 2 wetk?^ : had none been lost, they might have had long ? 31. days a 51 80. ■ ^ 89.^^ RULE OF THREE. Ids Ans. 14 oz. a day ; what was the whole weight, including what was lost, and how much had they to subsist on I i Whole weight, 147000 lbs. ( Left to subsist on, 126000 " 26. 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 ? ) Whole weight, 147000 lbs. Ans. SLoss, 21000 " j 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 re- quired to be built in 2 months ; I demand how many men must be employed about it. Ans. 480 men. 28. There is a cistern having a pipe which will empty it in ten hours; how many pipes of the same capacity will empty it in 24 minutes ? JJns. 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, aJt the same allowance, if the garrison be reinforced by four hundred men ? Ans. 6f months. 30. If a piece of land, 40 rods in length and 4 in breadth, make an acre, how wide must it be when it is but 25 rods Ans. 6% rods. 31. If a man perform a journey in 15* (lays are 12 hours long, in how many will the days are but 10 hours long ? 32. If a field will feed 6 cows 91 days, how long will it feed 21 cows.-* Ans. 26 days. 33. Lent a friend £292 for 6 months ; sonle time after, he lent me .^806 ; how long may I keep it to balance the favor ? Ans, 2 months 5-f-days. 34. If 30 men can perform a piece of work in 1 1 days, how many men will accomplish another piece of work, four times as big, in a fifth part of the time ? Ans. 600 men. 35. If -f^ lb. of sugar cost ^^ of a shilling, what will |§ of a pound cost ?, Ans. 4d. 3f g^4^ q. Note. See tl 62, ex. 1 , where the above question is solved by analysis. The eleven following are the next succeeding examples in the same paragraph. long ? days when the he do it when Ans. 18 days. A •% (' m ktht OP tHIttii flsd.oo. I ^ v>>*'. -4' &6i If 7 lbs. of sugar cost f of 5s. what tiost 12 lbs. Ans. ()^s; 87. if 6^ yards of oloth eost i^3, what cost 9^ yards ? .?:?'/.. . : ' Ans. ^4 5s. 4^(1. 88. If 3 oz. of silver cost lis 9fd- what costf oz. ? A715. 4s. Sfd. 80. If ^ osjj cost 4-/2S., what costs 1 Oz. 1 Ans. Gs. 5d. 40. If ^ tbi less by I ib eost 13|d., what cost 14 lbs. less l)y I of 2 Ibs; Ans. £i 9s. 9ir\d. 41. If f of a yafd cost £^i what will 40^ yards cost ? Ans. £59 Is. 2^d. 42. if f^p of a ship cost £'iiolf what is -^ of her worth ? ilns. ^^53 15s. 8^d. 43. At <^3| per cwt., what will Of Ibs; cost ? Ans. 6sv 3^d 44. A merchant owning 4 of a vessel^ sold § of his share for .£957; what was the vessel worth ? Ans, .^. .^0-:: IMAGE EVALUATION TEST TARGET (MT-S) 1.0 I.I 1.25 y^l^S 12.5 2.2 1^ 1^ It? i^ 2.0 1.4 m I 1.6 V vl Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 873-4503 • . I 189 COMPOUND PROPORXION. and that seven hours has to ten hours. Stating this in the form of a proportion, we have 13 days :.12days ) ^^^ •, ., 7 hours : 10 hours \ ' ' ^^^ "^^^"' * "^'^^^ by which it appears that 273 is 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, being done, gives 360 miles for the fourth term, or answer, as before. - > In the same manner, any question relating u 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 deep, in how many days of 9 hours each, will 24 men dig a trench 420 yards long, 5 wide and 3 deep ? Here the number of days, in which the proposed work can be done, depends on five circumstances, viz. the num- ber of men employed, the number of hours they work each day, the length, breadth and depth of the trench. We will consider the question in relation to each of th^se circum- stances, in the order in which they have been named^ — L2.0 5. Ji'i Bit:: 1st. The number of men employed. Were all the circum- stances in the two cases alike, except the number of men and the number of days, the question would consist only in finding in how many days 24 men would perform the work which 248 men had done in 5 days ; we should then Jiave 24 men : 248 men : : 5 days : days. 2d. Hours in a day. But the first laborers worked 11 hours in a day, whereas the others worked only 9 ; less hours will require more days, which will give 9 hours : 1 1 hours : : 5 4ays : days. 3d. Length of the ditches. The ditclies being of unequal length, as many more days will be necessary as the second is longer than the first ; hence we shall have 230 length : 420 length : : 5 days : days. 4th. Widths. Taking into consideration the widths, which are different, we have 3 wide : 5 wide : : 5 days days. 5th. Depths. Lastly, the depths being different, we have 2 deep : 3 deep 5 days days. •«iiK'„ii,.! fWI COMPOUND PROPORTIOK. 187 It would seem, therefore, that 5 days has to the fourth term, or answer, the same proportion that 24 men has to 248 men, whose ratio is ^^, 9 hours " 1 1 hours, the ratio of which is V, 230 length " 420 length 3 width " 5 width 2 depth " 3 depth ail of which, stated in form of a proportion, we have (( (( ((. Men, 24 . 248} Hours, 9 11 Length, 230 420 y Width, 3 . 5 Depth, 3 : 3J common term. : : 5 days : ■ davs. Tf 91. The continued product of all the second terms •348X11X420X5X3, multiplied by the third term, 5 days, and this product divided by the continued product of the first terms, 24X9X230X3X2, gives 288^%*^^^^% days for the fourth term, or answer. 288^jj^, But the first and second terms are the fractions y-^, y, If g, ^ and f , which 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 numbe^of days given to the number of days sought, is equal 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 them, thus — 248X11X420X5X3 17186400 1718G400 = and this frac. 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 manner from the multiplication of several ratios, is called a com- pound ratio. From the examples and illustrations now given, we de- duce the following general RULE 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 ...%;>; .•J«\:rfeAVWi'i:i.';. i ■'^^/.'\ii^i .■y.'Vf .^^V*^' *■■" 188 COMPOUND PROPORTION. 1I9I. the remaining numbers, take away two that are of the same kind, and arrange them according to the directions givea in simple proportion : then any other ,two of the same kind, and so on till all are used. Lastly, multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms, and the quotient will be the 4th term, or answer required. EXAMPLES FOR PRACTICE. 1. If 6 men build a wall 20 feet long, 6 feet high, and 4 feet thick in 16 days, in what time will 24 men build one 200 feet long, 8 feet high and 6 feet thick? Ans. 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? ^ns.^92 lis. lOfd. 3. If 56 ibs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days? Ans. 36 Ids. T/ie same by analysis. If 7 men consume 56 lbs. of bread, 1 man, in the same time, would consume f of 56 lbs.= V^ lbs. ; and if he consume ^ lbs. in 14 days, he would consume ^^ of ^6=|| lbs. in one day. 21 men would con- sume 21 times so much as 1 man ; that is, 21 times |f= *il^ lb.s. in i day, and in 3 days they would consume 3 , times as much; that is, ^^^^z=ZQ lbs. as before. Ans. 36 lbs. 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. ? Ans. 7 men. 5. If 7 oz. 5 pvvt. of bread be bought for 4fd. when corn is 4s. 2d. per bushel, what weight of it may be bought for Is. 2d. when the price per bushel is 5s. 6d. 1 Ans. I lb. 4 oz. 3|^f pwts. 6. If <^100 gain £Q in 1 year, what will ^400 gain in 9 months ? Note. This and the three following examples reciprocally prove each other. ^■jt -. of the same itions given same kind, ed product i continued i be the 4th high, and 4 n build one «s. 80 days, i^eighing 12 r the freight i? 5 lis. lOfd. en 14 days, ins. 36 lbs. bs. of bread, >f 56 lbs.= , he would would con- times 11= consume 3 ins. 36 lbs. )les by the o the same days' work, 5s. 2^d. ? ins. 7 men. when corn bought for 3|Jf pwts. gain in 9 eciprocally If 01. eL'ppLiJMErit tcr Single ri/Le of 'tHiUEji, . 180 7. If .^100 gain £Q in I'yeary \ti what time will ^400 gain .£18? 8. If £400 gain .£18 in 9 mottths, what is the rate per sent per annum ? 9. What principaly at C per cent per annum will gain jf 18 in 9 tnonths ? 10. A Usurer put out $75 at interest, and at the end of 8 months, received, for principal and interest, $79; I de- mand at what fate per cent he received interest. Ans. 8 per cent. IL If 3 men receive £Q^^ for 19;i^ days' work, hoW Much must 30 men receive for 100^ days 7 Ans. £305 Os. 8d. "?»|;i( (^iipplrment to Sliig'lc Rule of Three. QUESTIONS. 1. What is propdrlioiA ? 2^ ^a^v many nuniibeiCs are required io form a ratio f 3< HoMi many to form a proportion / 4. \Vh;il ii> tlie iirst term of a ratio called / d. ^ ttte second term 1 6< Which is taken for the numerator, and which f6r the denominator of the fraction elpressing the ratio ? 7. How may il be known when 4 numbera are In proportion 7 8. Hating three terms m the prdportion given, how (iiay the fiurlh term be found 7 9. \Vhiil is the opurdtion, by which the fourth ternd is found; calleid 1 lO. How does a ratio beco:ne in' terted % 11. What is the ruld in proportion % \%, In what denomi- httion will the 4ih term or anstter be found 1 19. If the fust and sc cohd terms contii.i different denominations, what is to b« done 1 II. What is compound proportion^ or double rule of three 1 15< Hale % m EXfiRCISE^ 1. if I buy 76 yards of cloth fcr £28 6s.' lOdr f^ qrs/ What does it cost per ell English 1 Jins. 9s. 3^d/ 2. Bought 4 pieces of Holland, each containing -24 ells English for .£24 ; how much was that per yard? Ans 4s. 2. A garrison had provisions for 8 months, at the rate of 15 ounces to each person per day ; how much must be al- lowed per day in order that the provisions may last 9^ months? Ans. 12|f oz* 4. HovV milch land at 12s. 6d. per acre, must be given in exchange for 360 acres, at 188. 9d. per acre ? Ans, 540 acree. f i90 '.J, TELLOWSHIP. ! t/ !I 01,92. It! 92. 6. Borrowed 185 quarters of corn when the price was 19s. ; how much must I pay when the price is 17s. 4d. 1 Ans. 202f| 6. A person owning f of a coal mine, sells f of his share for iifflTl ; what is the whole mine worth? Ans, £380. 7. If I ot a gallon cost |^ of a pound, what cost ^ of a tun? _ Ans. £UQ. 8. At ^l^per cwt. what cost 3^ lbs.? Ans. lO^d. 9. If4J-cwt. can be carried 36 miles for 35 shillings, how many pbunds can be carried 20 miles for the same money ? Ans. 907^ lbs. 10. If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour ? in each minute ? in each second ? Ans. to the last, 15" of a deg. 11. If a family of 9 persons spend c£ 1 12 lOs.. in 5 months, how much would be sufficient to maintain them 8 montjis if 5 persons more were added to the family ? Ans. .£280. Note. Exercises 14th, I5th, 16th, 17th, 18th, 19th and 20th, •' Supplement to Fractions" afford additional exam- ples in single and double proportion, should more examples be thought necessary. FEIiLOWl^HIP. ^ 9^. 1.. Two men own a farm ; the first owns ^, and i\\9 second owns f of it ; the farm is sold for ,^40 ; what is each man's share of the money .^ 2. Two men purchase a horse for 20 pounds, of which one pays 5 pounds, and the other 15 pounds ; the horse is sold for 40 pounds ; what is each man's share of the money ? 3. A- and B. bought a quantity of cotton ; A. paid 100 pounds, and B. 200 pounds : they sold it so as to gain 30 pounds ; what were their respective shares of the gain ? The process of ascertaining the respective gains or losses of individuals engaged in joint trade, is called the rule of Fellowship. The money, or value of the articles employed in trade, is called the capital or stock ; the gain or loss to be shared is called the dividend. ^91,92. In92. FELLOWSHIP. 191 It is plain that each man's gain or loss ought to have the same relation to the whole gain or loss, as his share of the stock does to the whole stock. Hence we have this Rule : — As the whole stock : to each man's share of the stock : : the whole gain or loss : his share of the gain or loss. 4. Two persons have a joint stock in trade ; A. put in i:-250, and B. .£350 ; they gain £400 ; what is each man's share of the profit ? OPERATION. A.'s stock, £250 ) Then, B.'s " 350 C 600 : 250 :: 400 : £166 13s. 4d. A.'s Whole stock, £600 ^ 600 : 350 :: 400 : 233 6s. 7d. B.'s The pupil will perceive that the process may be con- tracted by cutting off an equal number of ciphers from the first and second, or first and third terms ; thus, 6 : 250 :: 4 : ^166 13s. 4d. &c. It is obvious, the correctness of the work may be ascer- tained by finding whether the sums of the shares of the gains are equal to the whole gain ; thus, £166 13s. 4d.-|- £233 6s. 7d.=£400, the whole gain. 5. A. B. and C. trade in company ; A.'s capital was £175, B.'s £200, and C.'s £500 ; by misfortune they lose £250 ; what loss must each sustain? r£ 50 A.'s loss. Ans. I 57 2s. lO^d. B.'s " C.'s i 57 2s. lO^^d. 142 17s. lid. /" 6. Divide $600 among 3 men, so that their shares may be to each other as 1, 2, 3, respectively. ^ws. $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. ; how much must each sustain of the loss ? ylHs. A. 70, and B. 30 hhds. J?. A. and B. companied; A. put in £45, and took out f of the gain; how much did B. put in ? Ans. £30. Note. They took out in the same proportion as they put in ; if f of the stock is £45, how much is f of it? 9. A. and B. companied, and trade with a joint capital of If ilri«' m m yl i !SIi'<:H m PELLOWSHIV. 1IW,W.|f93. ti- i^400; A;, receiv.ftf for his sh^re of the gain, } as muchM ^ ; what was the stock of ea«h ? ., )rfJ33 6ii, 74. A's stock, ^»«- f £266. 13s. 4d. B's stock, 10. A banlfrupt is indebted to B $780, to C $4|80, and to .P $760 ; bis e»tate is worth only $($00 ; bow must it be di» vided? , : . Note. The question evidently involves the priociples of fellowship, ana may be wrought by it, Ans. B $234, C $138, and P $228, 11. B and C venture e(}ual stocks in trade, and clear <^164 ; by agreement, B was to have 5 per cent of the prof* its, because be managi^ the concerQs; C was to have but 2 per cent 4 what was <}ach one's gain? and bow muob did B receive for his trouble ? Ans. B.'s gain was <£U7 2e, lO^d. and C.'a£40 li7s, l^d, and B. received ^7Q 5s. 8^d, for his trouble, 12. A cotton factory, viUued at ,£12000, is 4iviHed into 100 shares ; if the promts amount to 15 per c;ent yearly, what will be the profit accruing to J share ,^-'-.— to 2 shares? — ^to 25 shares ? Ans, to the last £450, 13. In the above-mentioned factory, repair^ are to be made which will cost ^340 ; what will be the tax on each ^are, necessary to raise the wm ? ^" ' on 2 shares ? «• — ' on 3 shares ? on 10 shares? Ans, to the last, ^£34, 14. If a town raise a tax of <£1850, and the whole town he vajued at ^37000, what will that be on ^1 ? What will be the tax of a man whose property is valued at ^1780? Ans, Is. on a pound, and £8^ on 4^1780, 51 93. In assessing taxes, it is necessary to have au inventory of the property, bcih real and personal, of the ^vhole town, and also of the whole number of the polls; an4 ;as the poIJs are rated at so much each, we must jirst take out from the whole tax what the polls amount to, and the remainder is to be assessed on the property. We may then .find the tax upon one pound, and make a table pontainipg the taxes on one, two, thr^ep, &.c, to ten pounds j then on twenty, thirty, &-c, to a hundred ; then on 100, 200, &.c, to 1000 pounds. Then knowing the inventory of any indivi* 4u^.l, l\ is easy to find the tax upon his property* 15. A ^2259 1 what is real ests ^874, ai It will taxes to process exactnes will be r will be n final ans 540X 2259*90 property, tax on o] Tax. on Now,! by the ta The tax In like m Two pol] i67'62= 1FW,l».|ff93. FELLOVrSUIP. 198 . A's stock, I. B'9 stock, MjSO.andto lusjt it be di» riocipleji of nd P 1228, , and clear of the proff 10 have but V iQuoh did *6 173. l^d, divided into pent yearly, -to 2 shares? e Jast ^450, s are to be taK on eajch liares ? "• — ' le last, ^34, )vhole town What will ^1780? on 4PJ780, to have an onal, of the e polls ; an4 ist ^rst take to, and the ^e may then i containing Js^hen on 200, &e, to any indJvi' 15. A certain town, valued at jf 64530, raises. A tax of i2259 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 estate is valued at jf 1340, his personal property at f 874, aad who .pays for two polls. ^ w It will be better in questions relating to the assessment of taxes to use decimals, as we have done in interest. The process will be shorter, and the result will be obtained with exactness. The shillings, therefore, in the given values, will be reduced to the decimal of a pound, and the table will be made out decimally, and the decimal parts in the final answer can be reduced to shillings and pence, 540X'60"(3s.)=v£324, amount of the boll taxes, and 2259'90 (.£2259 I8s.)— je324c=1935'90, to be assessed on property. .£64530 : 1935'90 ;; ^l'03i or »ff|t^='08 tax on one pound, . . i TABLE. Tax, on £ £ £ £ 1 is '03 Tax on 10 is I '30 2 '06 SO '60 3 '09 30 «90 4 '12 id 1*20 5 '15 50 1'50 6 '18 50 1*80 7 '21 . 70 2'10 8 '24 80 2*40 9 '27 90 2'70 Tax on jg.,:' i) 100 iE ( 3' 200 6* 300 9' 400 11' 500 15' 600 18' 700 21' 800 24' 900 27' [000 30' Now, to find B.'s tax, his real estate being £1340, 1 find by the table that £ £ * The tax on - - 1000 is - - 30' 300 9' 40 1'20 The tax on his real estate - . - 40'20 In like manner I find the tax on his personal property to be - • - - - ^ 26*22 Two polls at '60 each, are . , - 1*20 .£67'62r=£67 12s. 4^d. ansmr. Amount, 67'63 VH tmtLOWMtt. 16. What will C/b tax amoam to whoso inrentary is 874 dollars retd^ and 210 doUars personcU property, nnd who pagra for three polU ? ^ ^fi5. $34'%i. 17. What will be the tax of a man paying for one polU whose property is valued at $34*8* ? at $768 ? — -| at $940 T ' at $4657 ? ^4^.9. to last, 1 140*91 . 18. Two men paid $W for the use of a pasture 1 motith;! A. kept in 24 cows, and B^ 10 cows; how much shouldj each pay / 19. Two men hired a pasture for $10; A. put in 8 cowsl 9 months, and B. put in 4 cows 4 months ; how much| should each pay ? • ^ 04* The pasturage of 8 cows for 3 months is thel same as S^ cotws for 1 month ; and the pasturage of 4 com'sI for 4 months is the same as of 16 cows for one month. Thel shares of A. and B. therdfore^ are 34 to 10, as in, the formerl question. Hence, when time is regarded in fellowship,— I Multiply ;each one's stock by the time he continues it inl trade, and use the product for his share. This is calledl l^ouble Fellowship. Ans. A. $16, and B. $4 20. A. and B. enter into partnership ; A. puts in £100 sixl months, itnd then puts itt <£50 more; B. puts in £200 fourl months, and then takes out £80 ; at the close of the year,! they find that they hiame gained £95 ; what is the profit of] each? J ( £43 14s. 2^d. A.'s share. 21. A. with a capital of $500, began trade Jan. 1, 1826, and meeting with success, took in B. as a partner, with a capital of $600, on the 1st March following ; four months after, they admit C. as a partner, who brought $800 stock at the close of the year, they fmd'the gain to be $700; how must it be divided among the partners ? r $250 A.'s share, Ans. I 250B.'s *< ( 200C.'s " 3. ^ QUESTIONS. ■/'■•' '.'■ *# ' ■ 1. What is fellowship 1 2. What is the rule for operating'? When time is regarded in fieiiowship, what is it called? 4. What is the method of operating in double fellowship ? 5, How are taxes assessed ? 6. Flow is fsllowship proved 1 iOXlOATION. Its ALLI«AT10]¥. a I!! rr. ^ IKS. Alligation is the method of mixing two or morr himples, of different qualities, so that the compoeitioa may* I be of a mean or middle quality. . . Whep the quantities and prices of the simples are gives I to find the mean price of the mixture compounded of them, I the process is called ^Alligation Medial^ 1. A farmer mixed together 4 bushels of wheat, worth 66 I pence per bushel, 3 bushela of rye, worth 32 pence per bushel, and 2 bushels of corn, worth 28 pence per bushel ; ' what is a bushel of the mixture worth ? It is plain that the cost of the whole, divided by the nui»> I bar of bushels, will give the price of one bushel. ; .:C' < • 4 bushels, at ^ pence, cost 264 pence. 3 " .: . 88 "96 2 " .1 m " 56 «< d bushels cost u .'(-' .11 416 pence. ? *^6=i46f pence, Am. 2. A grocer mixed 5 lbs. c^ sugar, worth lOd. per lb. 8 IS. worth 12d. 20 lbs. worth 14d. ; what is a poand of the mixture worth? i4n«. 12|^d. 3. A goldsmith melted together 3 ounces of gold 20 ca>* rats fine, and 5 ounces 22 carats fine ; what is the fioeness of the mixture ? Ans. 21^. 4. A grocer puts 6 gallons of water into a cask contain- ing 40 gallons of rum, w«rth 2s. 7d. per gallon ; what is a gallon of the mixture worth?) ! i ; v < Ans. 2s. 2||d. 5. On a certain day the inercilry was observed t6 9tand in the thermometer as follows :—^ hours of the day it stood k 64 degrees ; 4 hour^at 70 degrees ; 2 hours at 75 degrees, aad 3 hours at 73 degrees ; what was the nuaa, tempNature forthal'day? .-vl ,.■';'■><)■ '\m\: -f <, t(. 'i'--^- .■ ;„ It is plain this question does not dif&i', in the mode of its operation .from the former. Ans. 69-j^ degrees. il HO* When the mean price or rate, and the prices or riitOs of the several simples ,are given, to find the propor- tions or quantities of each simple, the process is caiUed alU-> gation alternate i alligation iQternate is, therefore, the re- verse of alligation mediq^l, and may be proved by^k. . 196 ALLIGATION. HOe 1. A man has corn worth 40d. per bushel, which he winhes to mix with rye worth 60d. per bushel, so that the mixture may be worth 42d. per bushel ; what proportions or quantities of each must he take 1 Had the price of the mixture required erceeded the price of the corn, by just as much as it fell short of the price of the rye, it is plain he must have taken equal quantities of corn and rye ; had the price of the mixture exceeded the price of the corn by only half as much as it fell short of the price of the rye, the compound would have required twice as much corn aa rye ; and in all cases the lexs the difierence between the price of the mixture and that of one of the sim^ pies, the greater roust be the quantity of that simple, in pro- portion to the other ; that is, the quantities of the simples must be inversely as the difF:irences of theif prices from the price of the mixture ; therefore, if these differences be mu» tually exchanged, they will directly express the relative quantities of each simple necessary to form the compound required. In the above example, the price of the mixture is 4^d. and the pricie of the corn is 40d. ; consequently the 'diflfbrence of their prices is 2d. ; the price of the rye is 50d. Mrhich differs from the price of the mixture by'8d. There' fore, by exchanging these differences, we have 8 bushels of corn to 2 bushels of rye for the proportion required. Ahs. 8 bushels of corn to 2 bushels of rye^ or in that pro- portion. The correctness of this result may now be ascertained by the last rule; thus, the cost of 8 bushels of com at 40 pence is 320 pence ; and 2 bushels of rye at 50 pence is 100 pence ; then, 320r|-100^^20, and 420 divided by the number of bushels, (84*2):zrl0, gives 42 pence for the price of the inixturei< : • I ;■' '" ■' • ' '' '• -.'■' " " ■' 2w A merchant has several kinds of tea • some at 8s. 'some' at 9s. some at lis. and some at 12s. per lb. ; what propor- tions of each must he mix, that he may sell the compound at lOs. pet lb. Here we have 4 simples ; but it is plain that what has just been proved of ^«>o will apply to any number ofpaifs, if in each pair the price of > how do you liiid the quantity of each bimplu ? . ^ Oy. Duodecimals are fractions of a foot. The word is derived from the Latin word duodecim, which signifies twelve. A foot, instead of being divided decimally into ten equal parts, is divided duodecimally into twelve equal parts, called inches, ox 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 be divided into twelve equal parts, called thirds ('"); each third into twelve equal j/arts called /owr^As, {"") and so on to any extent. 200 ^ • MULTIPLICATION OP DUODECIMALS. TI97. In this way of dividing a foot, it is obvious that 1' inch or prime is - - - - tV of a foot, 1" second is ^ of ^ — xii (( (( I'" thirdis^Vof-i^of-jJ^ - . =^^^^ i"" fourth is -^ of ^^ of ^ of ^^ . = Yuis^ V"" fifth is tV of tV of rV of tV of tV = ^f^Viij " Duodecimals are added and subtracted in the same man- ner as compound numbers, 13 of a less denomination making one of a greater, as in the following TABLE. 12"" fourths make 12"' thirds 12" seconds 12' inches or primes Note. The marks, ', ", "', "", &c. which distinguish the different parts, are called the indices of the parts or deno- minations. V" third, 1" second, I' inch or prime, 1 foot. MULTIPLICATION OF DUODECIMALS. Duodecimals are chiefly used in measuring surfaces and solids. 1. How many square feet in a board 16 feet 7 inches long, and 1 foot 3 inches wide ? Note. Length Xbreadth=superficial contents, (^ 25.) OPERATION. 7 inches or primes= j^ of a foot and 3 irtches=f'j of a foot ; consequently, the product of 7' X^'=-h\ of a foot, that is, 21' = 1' and 9", wherefore, we set down the 9", and reserve the 1' to be carried forward to its kns. 20 8' 9" proper place. To multiply 16 feet by 3' is to take ^^2^ of tP = A|^ that is 48'; and th. '' which we reserved makes 49',=4 feet 1'; we therefore set down the 1', and carry forward the four feet to its proper place. Then, multiplying the multiplicand by the one foot in the multiplier, and adding the two products together, we obtain the answer, 20 feet, 8', 9". The only difficulty that can arise in the multiplication of duodecimals is, in finding of what denomination is the pro- ft- Length 16 7' Breadth 1 3' 4 1' 16 7' 9" 197. duct o as abbi the _prfl denomi the ab( 16 4 ■■7t:r^'; T[97. hat of a foot, (( e same man- enomination J, ►r prime, itinguish the Its or deno- lALS. surfaces and jet 7 inches (!I 25.) es=y^ of a ^2 of a foot ; oduct of 7' that is, 21 ' ore, we set reserve the ward to its multiply 16 and th( V herefore set its proper he one foot ogether, we )lication of is the pro- .,.^„. ■■-... fld7; »i^ v; .! MttLTlPLTCATION OT DUODECFBIAtli. 28f duct of any two denominations. This may be ascertained as above, and in aU cases it will be found to hold true that the product of any two denominations will always he of the denomination denoted by the sum of their indices. Thus, in the above example the sum of the indices of 7'X3' is " ; consequently, the {Product is 21" ; and thus primes multi- plied by primes will produce seconds ; primes multiplied by seconds produce thirds ; fourths multiplied by 5ths produce ninths, &/C. It is generally most Convenient, in practice, to multiply the multiplicand first by the feet of the multiplier, then by the inches, &c. thus : — ft- 16 7' 1 3' 16 4 7' I' 9' 16 feet X 1 foot = 16 feet ; and 7'X 1 foot = 7'. Then, 16 feet X 3'=48'^ 4 feet, and 7'X3'=21"=1' 9'. The two products added together, give for the answer, 20 feet 8' 9", as before. ii 20 8' 9" ' 2. How many solid feet in a block 16 feet 8' long, 1 foot 5' wide, and 1 foot 4' thick ? ft- Length, 15 8' Breadth, 1 5' t I. ,it 15 8' 6 6' 4//' • 22 2' 4" s 1 4' 1 22 2' 4" 7 4' 9" 4'" The length multiplied by the breadth, and that pro- duct by the thickness, gives the solid contents. (1133.) :!.:• . i • • ^ Ans. 29 7' 1" 4'" ■ From these examples we detive the following Rule :— Write down the denominations as compound numbers, and in multiplying, remember that the product of any two de- nomiiiations will always be of that denomination denoted by the sum of their indices. im m: MVLTtPLlCATlON OF DUODECIMALS. H 97, 98. -i BXABIPL'KS for FRAOTICIi!. 3. How many square feet in a stock of 15 boards, 12 feet S' in length, and 13' wide ? Ans. ^5 feet 10', 4. What is the product of 371 feet 2' G" multiplied by 181 feet 1' 9" t n ^ ; Ans, 67243 feet 10' 1' 4 " 6" '. Note. Painting, plastering, paving, and some other kmda 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. ' A 5. A man painted the walls of a room 8 feet 2' in height, and 72 feet 4' in compass ; that is, the measure of all its sides ; how many square yards did he paint ? Ans. 65 yards 5 feet 8' 8". 6. How many cord feet of wood in a load 8 feet long, 4 fee^ wide, and 3 feet 6 inches high ? Note. It will be recollected that 16 solid feet make a itord foot. Ans. 7 cord feet. 7. In a pile of wood 176 feet in length, 3 feet 9' wide, and 4 feet 3' high, how many cords? Ans. 21 cords, 7-i^ cord feet. 8. How many cord feet of wood in a load 7 feet long, 3 feet wide, and 3 feet 4' high ; and what will it come to at 2s. per cord foot ? Ans. 4^ cord feet, and will come to 8s. 9d. 9. How much wood in a load 10 feet in length, 3 feet 9^ in width, and 4 feet 8' in height ? and what will it cost at $1*92 per cord ? Ans. 1 cord and 2 j| cord feet, and it will come to $2'62^. T 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 foot into ten equal parts. On one end of the rule for 1 foot, let each of these parts be divided into ten other «qual parts. The former division will be tenths, and the latter hundredths of a foot. Such a rule will be found very convenient for surveyors of wood and lumber, for painters, joiners, 6i,c. ; for the dimensions taken by it being in feet and decimal parts of a foot, the casts will be no other than :^o many operations, in decimal fracticms. 10. How many square feet in a hearth stone, which, by 3 ^0'»^' Iff 98,99. INTOLUTiaN. 003 \A ards, 12 feet 205 feet 10', luhiplied by 1" 4'" 6"". other kinds contents in evident, will 2' in height, re of all its 5 feet 8' 8". feet long, 4 feet make a 7 cord feet, feet 9' wide, f^ cord feet, feet long, 3 coine to at ne to 8s. 9d. gth, 3 feet 9' ill it cost at le to $2*GQi, rood, diinen- For this pur* le it into feet rd of the rule nto ten other iths, and the >e found tery for painters, being in feel other than , which, by a rule, as abore described, measures 4'5 feet in lehgth, and 3^6 feet in widt^ ? And what will be its cost, at 75 cents per st^re foot T Ans. 11'7 feet ; and it will cost $8*7T5. 11. How many cords in a load of wood 7*5 feet in length, 3*6 feet in- width, and 4'8 feet in height ? Ans. 1 cord 1 ^ ft. 13. How many cord feet in a load of wood 10 feet long, 3*4 feet wide, and3'5 feet high? > - Ans.J^^, V , QUESTIONS. 1. What are ^liotlecimats ? 2. From what is the word tier ired ? 3. Into how many parts is a Toot usually divided, and what are the parts called? 4. What are the other denominations ? 5. What is understood by the indices of the denominations ? 6. In what are duo- decimals chiefly used f 7. How are the contents of a surface bounded by straight lines found 'I 8. How are the contents of a solid found 1 9. How is it known of what denomination is the product of any two denominations t 10. How may a scale or rule be formed for taking divaenaions in feet and decimal parts of a foot 1 :m ... IJVYOIilJTIOJV. ; f[ 99. Involution, or the raising of powers, is the mul- tiplying any given number into itself continually a certain number of times. The products thus prodaced are called the powers of the given number. The number itself in called 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 first power of 6, 5X5= 25 is the 2d power, or square of 5, =5^ 5X5X5=.125 3d " cube, of 5, =53 5X5X5X5=625 4th " biquadrate, of 5,=5* The number denoting the power is called the index, or exponent; thus, 5* denotes that 6 is raised or involved to the 4th power. ■■ ^s',. ,. 1. What is the square or 2d power of 7.^ Ans. 49. 2. 3. 4. 5. 6. (( ({ <( of 30 ? An3 900. « of 4000? ' Ans. 16000000. cube or 3d power of 4 ? Ans. 64, " of 800.? ilns. 512000000. 4th power of 60 ? Ms. 12960000. 204 EVOLUTION. !:/ 7. What is the square of 1 ? of 4? 8. What is the cube of 1 ?- of 4? !I 99, 100. of 2? of 3? Ans. 1, 4, 9, and 16. .of 2? of 3? . ^101 Ans, 1, 8, 27, and 64. f|? of|?. ^ws. I, ^, If, 10. What is the cube of § ? -, — of | ? of | ? 9. What is the square of §?- ^ns. ^, /A, and ^f f . • the 5th power of ^ ? Ans. ^, and -j^. — the cube ? 11. What is the square of ^ ? — 12. What is the square of 1'5? Ans, 2'25, and 3'375. 13. What is the 6th power of r2'? Ans, 2*985984. 14. Involve 2^ vo the 4th power: Note, A mixed number like the above may be reduced to an improper fraction before involving : thus, 2^=J ; or it irtay be reduced to a decimal j thus, 2i=:;:2*25. Ans. m^=25ifi. 15. What is the value of 7*, that is, the 4th power of 7 .^ Ans. 2401, 16. How much is 9^ ? 6& ? 10* ? ^«s. 729, 7776, 10000. 17 How much is 2^ ? 3^ 1 4^ ? 53 ? — ^ 6* ? 103 1 jing^ to the iggt^ 100000000. The powers of the nine digits, from the first power to the fifth, may be seen in the following . TABLE. Roots 1 1 2| 3| 4| o\ 6| 7| 8| 9 Squares 1 4| 9| 16 1 25 1 36 1 49 j 64| 81 Cubes 1 «l 27 i 64 1 125 1 216 1 343 1 512 1 729 Biquadratsl |16| 81! 256 1 625 1 1296 1 2401 1 4096 1 6561 59049 Stursolids 1 1 32 1 243 1 1024 1 3125 1 7776 1 16807 1 32768 | CV0LUTIO]¥, ^ 100. Evolution, or the extracting of roots, is the method of finding the root of any power or number. ' The root, as we have seen, is that number which, by a continual multiplication intd itself, produces the given power. The square root is a number which, being squared, will U 99, 100. of 3? ,9, and 16. f3? 27, and 64. [)f|? *• t» ir* 1 4"' ,\, and Iff. iver of^? s. ;J, and 7^. , and 3'375. s, 2*985984. e reduced to 2^=1 ; or it VV=25^fi- lower of 7 ? Ans. 2401. r776, 10000. — 53? — - 100000000. irst power to fllOl EXTRACTION OF THE SQUARE ROOT. 265 8| 9 64| 81 512 1 729 4096 1 6561 {2768 1 59049 produce the given number ; and the cube, at third root, is a number which, being cubed or involved to the third power, will produce the given number ; thus, the square root of 144 is 12, because 12^=144 ; and the cube root of 343 is 7, be- cause 73, that is, 7X7X7=343 ; and so of other numbers. Although there is no number which will not produce a perfect power bv involution, yet there are many numbers of which precise roots 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 surd numbers, and those whose roots can be exactly obtained, are called rettional 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 \/16 ; and the cube root of 27 is 3 5 expressed v27, and the 5th root of 7776, V 7776. When the power is expressed by several numbers, with the sign -j- or — between them, a line, or vinculum, is drawn from the top of the sign over all the parts of it ; thus the square root of 21 — 5 is \/21 — 5, &c. roots, is the iber. which, by a given power, squared, will Fxtraction of the Square Root. IT 101. To extract the square root of any number is to find a number, which, being multiplied into itself, shall produce 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, the Soor of which the carpeting will cover ? that is, what is one side of a square, which contains 625 square yards ? We have seen (IT 32) that the contents of a square sur- face is found by ipultiplying the length of one side into itself, that is, by raising it to the second power ; and hence, hav- ing the contents (625) given, we must extract its square rout to find one side of the room. This we liiust do by a sort of. trial, and ist. We will endeavour to ascertain how many figures ■H.. 806 EXTRACTION OF THE 8QUABE ROOT. nioi. OPERATION. 625(2 4 225 Fig.X, 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 as many, or one figure k»s than twifce as many figures, as are in the root ; of which truth the pupil may easily satisfy himself by trial. Pointing off the number, we find that the root will consist of two figures — a ten and a unit. 2d. We will now seek for the first figure, that is, for the tens of the root, and it is plain that we must extract it from the left hand period 6, (hun- dreds ) The greatest square in 6 (hundreds) we find, by trial, to be 4, (hundreds) the root of which is 2, (tens==:20) therefore, we set 2 (tens) in the root. The rooty 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 20x20=400 yards, now disposed of, and which, consequently, are to be deducted from the whole 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, (hun- dreds) and bringing down the next period by the side of the remainder 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 two 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 sides c d and 6 c of the square A. ■a o a !1 101. do, by point- f two figures ins just twice ly figures, as easily satisfy ; find that the of two figures nit. low seek for hat is, for the ind it is plain [tract it from iriod 6, (hun- eatest square we find, by tundreds) the 2, (tens=20) t 2 (tens) in ooty it will be me side of a then, form a ,) each side i supposed 2 expressed obtained. ) yards, now )e deducted 225 yards. subtracting •6 of 2, (the iod 6, (hun- e side of the the addition it the figure ddition must e divided by quotient will tothesidcft TlOl. EXTAACnON OF THE SQUARE ROOT. 907 But our root already found, =2 tens, is the length of one side of the figure A ; we therefore take double this root,s=4 tens, for a divisor. OPERATION CONTINUED. 625(25 4 45)225 225 Fig.^. 20 yds. 5 yds. m 20 5 •a B 5 5 o a— D— 100 25 1 c A ' c ■ / •o ^ § 20 20 20 5 490 100 a b 20 yds. 5 yds. The divisor 4 (tens) is in reality 40, and we are to seek how many times 40 is contained in 225, or, which is the same thing, we may seek how many times 4 (tens) is contained in 22, (tens) rejecting the oi right hand figure of the 'S, dividend, because we * have rejected the ciph- er in the divisor. We find our quotient, that is, the breadth of the g addition, to be 5 yards; ^ but if we look at}?^. 2, S* we shall perceive that this addition of 5 yards to the 2 sides does not complete the square; for there is still want- ing in the comer D, a small square, each side of which is equal to this last quotient, 5; we must therefore add this quotient 5, to the divisor 40, that is, place it at the right hand of the 4 (tens) making it 45 ; and theh the whole divisor, 45, multiplied by the quotient, 6, will give the con- tents of the whole addition around the sides of the figure A, which, in this case, being 225 yards, the same as our dividend, we have no remainder, and the work is done. Consequently, J?^. 2 represents the floor of a square room, 25 yards on a side, which 625 square yards of carpeting will exactly cover. The proof may be seen by adding together the several parts of the figure, thus : — m EXTRACTION OF THU SQUARE ROOT. !l 10! , r. " <( (( (( (I Or we may prove it by involution, thus :— 25X25=:GJ3, as be- fore. l^be square A contains 400 yards. * figure B " 100 " C " 100 ** D « 25 "; vf,-sf' ■. , . • r- — ^ ..v|,n. .» iVoo/ 625 ,.. From this example and illustration, we* derive the follow- ing general , . _ RULE *^ ' Fbf the JSxtraction of the Square Root. 1. Point off the given number into periods of two figures each, by putting et, dot over the units, another over the hun- dreds, and 80 on^ These dots show the number of figures of which the root will consist. • I II. Fiitd the greatest square number in the left hand pe- riod, and write its root as a quotient in division. Subtract the square number from the left hand period, and to the, remainder bring down the next period for a dividend. III. Double the root already found for a divisor ; seek liow many times, tjie divisor is contained in the dividend, excepting the right hand figure, and place the result in the root, and also at the right hand of the divisor ; multiply the divisor, thus augmented, (by the last figure of the root, and subtract the product frorn the dividend; to the remainder bring down the; neict period for a new dividend. IV. Double tUe root already tbund for a new divisor, and continue the operation as before, until all the periods arc brought downf Note 1. IC We double the right band figure of the last divisor, we sh^U have the double of the root., Note'i. As the value of figures, whether integers or de- cimals, is determined by their distance from the place of units do we must always begin at unit's place to point off the givea number, and,.if it be a mixed number, we must point it off both ways £rom units, and if there be a deficiency in any pe- riod of decimals, it may be supplied by a cipher. Tt is plain the root must always consist of so many integers and decimals 3ks there are periods belonging to each in the given number. ,« ;■" EXAMPLES FOR PRACTICE. %. What i^ the sq^uane root of 1,0342^6 ? If 101 may prove it tion, thus :— :6'i5, as be- ve the follow- 90t. )f two figures over the hun- ber of figures left hand pe- •11. Subtract , and to the. lividend. divisor ; seek the dividend, result in the multiply the the root, and e remainder I. divisor, and periods are ! of the last egers or de- )laceofunitB off the given t point it off ;y in any pe- r. Tt is plain ind decimals en number. T 101, 103. BXTR ACTION OP THE ■QVAM HOOT. ^ ; OPERATION. ' t ■ 1 l6342t)56 (3316 Am. , • .' ' , ■ 9 ! 1 t 63) 134 134 ( 641) 1036 t What is 641 ': 6436) 38556 38550 5. the square root of 43364 ? 1 1 OPERATION. What is 43264 (308, An$. 4 408) 3264 3364 4. the square root of 993901 ? Ans. 999. 5. <( 234' 99? Ans. ]5'3. 6. <( 984*5192369241? ^«5.31'05671. 7. tt '001296? Ans. '036. 8. «' , •3916? Ans. '54. 9. « 36373961 ? Ans. 6031. 10. « " " 164 ? Atis. 13'8-f- IT I OS. In this last example, as there was a remain((er 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 to any assigned degree of exactness; but the pupil will re iidily perceive that he can never in this manner obtain the precise root ; for the last figure in each dividend will always be a cipher, and the last figure in each divisor is the same as the last quotient figure; but no one of the nine digits u>i>ltiplied into itself. S3 9tit •UrriiBMBNT TO Till SQUARB BOOT. Uioa 1102 produces a number ending with a cipher ; therefore, what- ever be the quotient figure, there will still be a remainder. 11. What is tiie square root of 51? -- Ans. 1'73-|-. 12. " •♦ •• 10 7 ylwi-. 34«4-. 13. " " " 184*2? AnitA^'57-\-. 14. " " " ^? Note. — We have seen (IT 99, ex. 9,) that fractions are squared by squaring both the numerator and the denomina- tor. Hence it follows, that the square root of a fraction is li)und by extracti.ig the root of the nuir»erator and of the denominator. The rcx)t of 4 is 2, and the root of 9 is 3. Ans. ^. 15. What is the square root of -J^? Ans. ^. 1«. " " •' t'd^? Ans.^^. 17. •• ♦* ♦• j\^T Ans. ^2=1 la " " " 2dir Ans. A}. When the numerator and denominator are not exad squares, the fraction may be reduced to a decimal, and the approximate root found, as directed above. 1 9. What is the square root of f =75 ? Ans. 'SfiG-f. 20. " •• " JJ^? Ans. *9\2'\-. SUPPLEMENT TO THE SQUARE ROOT. QUESTIONS. I. Whnt ia involution ? 2. What is understood by a power if 3. the first, the second, Ur.; third, the fourth power 1 4. What is the index, or exponent? 5. Hoiv do you involve a number to any le- (|uircd power ? 6, Whiit is cvotulion ? 7. What ia a root 1 8. Can Kie precise raot <>f nil numbera be found ? d. What is a seration 1 19, Why can we never ob- tain the precise root of »uid numbera ? 20. How do we extract the square root of vulgar fractions 'i EXERCISES. 1> A general has 409i> men; how many must he place in tank and tile, to form theiu into a square 1 .<4»s. 04. 2. If J rods doe 3. H( tainini^r 4. Th IS 5184 of equal 5. A. other CO field con how mai 6. If how mu( iiig 4 tin 7. Ift of one 4 times as large ? 8. Iti of a part length ai Note. two equa 9. Iw that my how mat each rov 10. T square r required 11. T is the di Note. to the sc Therefo diamete iquare i l± 1 IT 103 sforc, what- remainder. [ns. 1'73-|-. Iwi. 3'1«4-. u. 13'57-|-. actions are ; denomina- 1 fraction is and of the of 9 is 3. Ans. ^. /1ms. 4. Ans. -^ff. Ahs. -^7=^. Ans. 4^. e not exact lal, and the Ins. '866-1-. Ins. '9I2-I-. lOOT. power ? 3. 4. Whxlis )er to any re- wtl 8. Can s'lrd number f re root of any sriods of two divisor? 14. the dividend f id of tbd divi- xie point off i is a rtimain- we never ob- re extract the he place in .<4»s. 04. 1102 •UPPLEMBNT TO THE IQUilFV: ROOT. 311 2. If a square field contains 2025 square rods, how many rods does it measure on each side ? Anf, 45. 3. How many trees in each row of a square orchard con- tainin^jT 5625 trees ? Ans. 75. 4. There is a circle wliose area^ or superficial contents, is 5184 fc>et ; what will be the len^h of the side of a square of equal area? \/5184-=72 feet, i4n.," /no-Ji^-iijijiiw operation. 1860867 (123 Ms. ^":'' ' - 1 M ...v.. 12x300=300) 860 first Dividend. 600 ..«;•!.: i : , 22 X 1x30= 120 ;;, .. ^ 23 = 8 ..; .,;/;;■ .:,■• '; .M'[i/t;i uii i ;? 728 first Subtrahend. 122^X300=43200) 132867 second Dividend. 82X12X30= ., .. .33= 129600 3240 27 the >i .1 000000 7. What is the cube root of 373248? ' 8. " " " 21024576? 132867 second Subtrahend. 9. 10. 11. 12. . ■■>■ « 84*604519? '000343? 8 7 Ans. 72. ilns. 276, Ans. 4'39 il»5. '07. ^ns. 1'25-f. ilH5. ^ Note. See tf 99, ex. 10, and H 102, ex. 14. 13. " " " ^ff? ^4«s.f M4. " , f, " « t5?^^? .; ilns.-jV 15. " V, " " sixf^ ^«s.'125+. 16. " ..V " T^? . . Ans.i SUPPLEMErjfT TO THE CUBE ROOT. QUESTIONS. 1. What is a cube ? 2, What is understood by the c^le root? 3. What is it to extract the cube root ? 4. Why is the square of the quo tivo.nt multiplied by 300 Tor a divisor 1 5, Why, in finding the subtra hend, do we multiply the square of the last quotient figure by 30 times the former figure of the root 1 6, Why do we cube the quotient figure ? 7. How do we prove the operation 1 ^105. 1. Wl eet long 2. Th many sol 3. Ho what woi 4. Th. what woi — 64 5. Th( what is i — 64 cube heir conl iroportioi all solid f 6. If a ill be thi lbs.? 7. Ifa will be th( 8. Ifa what is thi 12 9. The er, and tl mailer gl 10. Ift he diatne ivould it U 11. Ift earth, and diameter c 12. Th ftf the less ■ I 11104 s. end. rahend. 1 Dividend. If 105. SUPPLEMENT TO THE CUBE ROOT. 319 '.•t \ Arts. 72. ilns.276, Atts. 4<39. Ans. *07. Ans. 1'25-f, ./Ins. ^, ilns. y'j, ^ns.'125-i- Ans. i lOOT. c-l e j-oot ? 3. lare of the quo ling the subtra- ire by 30 times [uotieni figure ? 5. v: ^'"-^ ".'? • EXERCISES.!' "^'^ -- • «' • V 1. What is the side of a cubical mound, equal to one 288 feet long, 216 feet broad, and 48 feet high ? Ans. 144 ft. 2. There is a cubic box, one side of which is 2 feet ; how many solid feet does it contain ? Ans 8 feet. 3. How many cubic feet in one 8 times as large ; and what would be the length of one side? > y i upo''tioii is true of the similar sides, or of the diameters of all solid figures of similar forms. 6. If a ball weighing 4 lbs. be 3 inches, in diameter, what vill be the diameter of a ball of the same metal, weighing )2 lbs. ? 4 : 32 : : 3* : 63 . Jlns. 6 inches, 7. If a ball, 6 inches in diameter,, weigh 32 pounds, what will be the weight of a ball 3 inches in diameter ? Ans. 4 lbs. 8. If a globe of silver, one inch in diameter, be worth $6, what is the value of a globe one foot in diameter ? ^ns. $10368. 9. There are two globes ; one of them is 1 foot in diame- ter, and the other 40 feet in diameter ; how many of the mailer globes would it take to make one of the laiig^r ? Ans.^Aim. 10. If the diameter of the sun is 112 times as much as the diameter of the earth, how many globes like the earth would it take to make one as large as the sun ? Ans. 1404928. 11. If the planet Saturn is 1000 times as large as the eartl^ and the earth is 7900 miles in diameter, what is the diameter of Saturn ? Ans. 79{X)0 miles. 12. There are two planets of equal density ; the diameter of the less is to that of the larger as 2 to 9 ; what is the ratio t)i" th>#tolidities ? Ans. , f ^ ; or, as 8 to 729. Ih' :i m ■ ■ lift,, mi 220 ARITHMETICAL PROGRESSION. ^ 105, 106. f|106. Note. The roots of most powers may be found by the square and cube root only : thus, the biquadrate or 4th root is the square root of the square root ; the6th root is the cube root of the square root ; Ihe 8th root is the square root of the 4^h root ; the OtKroot is the cube root of the cube root, d&c. Those roots, viz. the 5th, 7th, 11th, &x. whith arc not resolvable by the square and cube roots, seldom occur; and when they do, the work is most easily performed by logarithms ; for if the logarithm of any number be divided by the index of the root, the quotient will be the logarithm ef tlie root itself. . - ARlTHi«IETl€AL PR0«R£SS10iV. II 106, Any rank or series of numbers more than two, increasing- or decreasing by a constant difference, is called an Arithmetical Series, or Prf^ression. 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 5t/6^rac^}0» of the com* mon difference, they form a descending series, rnt i 3, 5, 7, 9,11,13,15, &c. is an ascending series, ^""^' ) 15,13,11, 9, 7, 6, 3, &c. is a descending " The numbers which form the series are called the tema of the series. The first and last terms are the extremes^ and the other terms are called the means. There are five things in arithmetical progression, any three of which being given, the other two may be found :— 1st. The /frs* term. 2d. The last term. 3d. The number of terms. 4th. The camTnon difference. 5th. The sum of all the terms. 1. A man bought 100 yards of cloth, givilig 4d. for the first yard, 7d. for the second, lOd. for the third, and so on with a common difference of 3d. ; what was the cost of the last yard ? As the common difference, 3, is added to every yard ex- cept the last, it is plain the last yard must be 99X3; = 297 )>ence more than i\iQ first yard. Ans. 301 |ii^nce, 5T 105, 106. found by the te or 4th root tot is the cube iquare root of he cube root, >c. yrhith arc eldom occur ; performed by »er be divided the logarithm ssiorv. lore than two, mee, is called ja) addition of ng series ; but on of the com- ending series, ending " led the termi the extremes, gression, any be found :— f|106. ARITHMETICAL PROGRESSION. 231 g 4d. for the rd, and so on e codt of the [very yard ex- |9X3;=297 U.aOl^nce. Hence, when the first term, the common difference, and the number of terms are given, to find the last term, — Mul- tiply the number of terms, less one, by the common differ- ence, and add the first term to the product for the last term. 2. If the first term be 4, the common difference 3, and the number of terms 100, what is the last term ? Ans. 301. 3. There are ' in a certain triangular field, 41 rows of corn ; the first row, in one corner, is a single hill ; the se- cond contains three hills, and so on, with a common differ- ence of 2 ; what is the number of hills in the last row ? Ans. 81 hills. 4. A man puts out <£! at 6 per cent simple interest, which in one year amounts to £i-^(f, in two years to iSf l-^o, and so on, in arithmetical progression, with a common difference of'£-^ ; what would be the amount in 40 years? /• ■ Ans. £^^. Hence we see, that the yearly amounts of any sum, at simple interest, form an arithmetical series, of which the principal is the first term, the last amount is the last term, the yearly interest is the common difference, and the rfumber of years is one less than the number of terms. 5. A man bought 100 yards of cloth in arithmetical pro- gression ; for the first yard he gave 4d., and for the last 301 pence; what was the common increase of the price, on each succeeding yarfl ? This question is the reverse of example 1 j therefore, 301 —4=297, and 297-f-99=:3, common difference. Hence, when the extremes and number of terms are given to find the common difference, — Divide the difference of the ejctremes by the number of terms, less 1 , and the quo- tient will be the common difference. 6. If the extremes be 5 and 605,, and the number of terms 151, what is the common difference ? Ans. 4. 7. If a man puts out £1 at simple interest, for 40 years, and receives at the end of the time ^3f^, what is the rate ? If the extremes be 1 and 3|3-, and the number of terms 41, what is the common difference ? Ans. -^. 8. A man had 8 sons whose ages differed alike; the youngest was 10 years old, and the eldest 45; what was the common difference of their acres ? Ans. 5 years. 9. A man bought 109 yards of cloth in arithmetical series; T2 mfi ARITHMETICAL PROQRESSION, IT 106 he give 4 pence for the first yard, and 301 pence fur the last yard; what was the average price per yard, and what was the amount of the whole ? ' Since the price of each succeeding yard increases by a constant excess, it is plain the average price is as much less than the price of the last yard as it is greater than the jwice of the first yard ; therefore, one half the sum of the first and last price is the average price. ^One half of 4d -f 301d. =: 152Jd. = average ) price ; and the price, 152^d.Xl^^=!li>^0d,= V Ans. jC^i 10s. lOd., whole cost. ) Hence, when the extremes and the number of terms arc j^iven, to find the sum of all the terms,, — Multiply half the sum of the extremes by the number of terms, and the pro- duct will be the answer. 10, If the extremes be 5 and C05, and the nun. 2r of terius be 151, what is the sum of the series ? An,i. 4G05o. Jl. What is tl:9 sum of tlie first 100 numbers, in their natural order, that is, 1,2, 3, 4, &-c. Ans 5050. 1*2. How HKiny times does a common clock strike in 12 hours ? Ans. 78, 1 3. A man rents a house for £o^ annu ally, to be paid at the •-.lose of each year ; what will the rent am« i^int to in 20 years, .illowing G per cent simple interest for the use of the money ? The last year's rent will evidently be c€50 without inter^ f:.st, t!ie last but one will be the amount of .£50 for 1 year, the liist but two the amount of .£50 for 2 years, and so on, \n arithmetical Series, to the first, which will be the amount of ^'51) for 19 years==£l07. Iftlie first term be 50, the last term 107, and the number oC terin^ 20, what is the sum of the series ? Ans. ^1570, 14. What is the amount of an annual pension of ^£100, he'uis, in arrears, that is, remaining unpaid, for 40 years, allowiijcf 5^ per cent simple interest ? Ans. ,£7900. 1.3. There arc, in a certain triangular field, 4.1 rows of Cv)rn ; the first row being in one corner, is a single hill, and the 1 i-^t row, on the side opposite, contains 81 hills ; how many hills of corn in the field? Ans. 1631 liills. 10. If a triangular piece of land, 30 rods in length, be 2) rods wide at one end, and come to a point at the other, v/lHt auinber of square rod.s does it contain ? Ans. 300 17. arithi what the s( ••' IT 106 pence for tlic ard, and what increases by a s as much less than the jwice of the first and Ans. ^i 5r of terms arc dtiply half the , and the pro- he nuu. 3r of An.. 40055. ibers, in their Ans 5050. -k strike in 12 Ans. 78, be paid at the to in 20 years, nf the money ? without inter- 50 for ] year, rs, and so on, )e the amount d the number Ans. ^1570. sion of o£:iOO, for 40 years. Ans. ,£7900. d^ 4,1 rows of ngle Jiill, and il hills ; how 16S1 hills. in length, be at the otlier, Ans. 300. ^ 106, 107. GEOMETRICAL PROUReSSION. 223 17. A debt is to be discharged at 11 several payments, in arithmetical series, the first to be £5, and the last .£75 ; what is the whole debt .' common difference between the several payments ? Ans. whole debt X440 ; common difference £7. 18. What is the sum of the series 1, 3, 5, 7, 9, &lc. to lOOU ^115.251001. Note. By the reverse of the rule under ex. 5, the differ- ence of the extremes 1000, divided by the common differ- ence 2, gives a quotient, which, increased by 1, is the num- ber of i,-~— when descending? 4, What are the iiurn* bers Ibrminj^ the progression rallmi? n, What are the first and last term? called '? 6, What are the other terms called '{ 7, When the first term, common diflerence, and number of tcrnis are g'ven, how do you find the last term ? 8, Hotv may urithmelical progression be ap- plied to simple interest ? 9, When the extremps and t.urn'or of torus Hre given. Iiuw do you find the common diin^rcnce ? 1 0, — — how do you lind the sum of all the terms 1 CiiEOnETRICAL PRO(>iUE88IO.\. IT 107. Any series of numbers, continmllySncreasing by a constant multiplier, or decreasing by a constant divi- sor, is called a Geometrical Progression. Thus, 1,2,4,8,16, tSic. is an increasing geometrical series, and 8, 4, 2, 1, J-, -|, &c. is a decreasing geometrical series. As in aritlun€tical, so also in geometrical progression, there are five things, any three of which being given, the other two may be found : — 1st. The first term; "Zd. The /rt.sf term ; 3d. Tht number of terms ; 4th. The ratio ; 5th. The sum of all the terms. 234 OEOMETRICAl. PROOREStlON. 11107 i ill ■i' The ratio is the multiplier, or divisor, by which the series is formed. 1. A man bought a piece of silk, measuring 17 yards, and, by agreement, was to give what the last yard would come to, reckoning 3 pence for the first yard, G pence for the second, and so on, doubling the price to the last ; what did the piece of silk cost him? 3X2X2X2X2X2X2X2X2X2X2X2X2X2X2X2 X2=196608 pence,=:;£819 4s. Ans. In examining the process by which the last term (106608) h-ys been obtained, we see that it is a product of wiiich the ratio (2) is sixteen times a factor, that is, am time less than the number of terms. The last term, then, is the sixteenth power of the ratio, (2) multiplied by the first term, (3.) Now, to raise 2 to the 16lh power, we need not produce all the intermediate powers; for 2*=2X2x2X2=16, is a product of which the ratio 2 is 4 times a factor ; how, if 16 be multiplied by 16, the product, 256, evidently contains the same factor (2) 4 times-|-4 times,=8 times; and 256X 256=65536, a product of which the ratio (2) is 8 times -j- 8 times,=16 times, factor; it is, therefore, the 16th power of 2, and, multiplied by 3, the first term, gives 196608, the last term, as before. Hence, When the first term, ratio, and number of terms, are given, to find the last term, — ' I. Write down a few leading powers of the ratio with their indices ,over them. II. Add together the most convenient indices, to make an index less by one than the number of the term sought. III. Multiply together the powers belonging to those in- dices, and their product, multiplied by the first term, will be the term sought. 2. If the first term be 5, and the ratio 3, what is the 8th term ? Powers of the ratio with 1 1, 2, 3, -j- 4=7* their indices over them V 3, 9, 27, X 81=2187 X 5, first J term,= 10935, Ans. 3. A man plants 4 kernels of corn, which, at harvest, produce 33 kernels ; these he plants the second year ; now, fluppi woull nelni 4.1 interl allo« »>| 5. yard] the ■<*v . 11107 hich the series '> :^'-^ .,"(:;; Ing 17 yards, St yard uould d, (i pence for the last ; what <2X2X2X2 erm (19G608) t of which the time less than the sixteenth term, (3.) J not produce X2=1G, is a ►r ;'how, if 16 ntly contains (s; and 256 X is 8 times -j- e 16th power s 196608, the 3f terms, are le ratio with ces, to make m sought, to those in- st term, will lat is the 8th 37 X 5, first at harvest, year; now, 11107. GBOMRTAICAL PKOOKEmOlV. supposing the annual increase to continue 8 fold, what would be the produce of the 16th year, lUlowing 1000 ker- nelntoapint? Amj). 2199023355*552 buaheli. 4. Supposing a mar had put out one penny at compound interest in 1620, what would have been the amount in 1624/ allowing it to double once in 12 years ? 2> 7 S131072. ^115. je54G 2s. 8d. 5. A man bought 4 yards of cloth, giving 2d. for the first yard, 6d. for the second, and so on in 3 fold ratio ; what did the whole cost him ? S-(-6-|- 18+54=^80 pence '' i4n5. 80 pence. In a long series, the process of adding in this manner would be tedious. Let us try, therefore, to devise some shorter method of coming to the same result. If all the terms, exceptiig the last, viz. 2-f-^H'l^> ^ multiplied by the ratio, 3, the product will be the series 0-f-18-f-54, sal>> tracting the former series from the latter, we have for the remainder, 54 — 2, that is, the last term less the first term^ which is evidently as many times the first series (2-f-&-f-18) as is expressed by the ratio, less one ; hence, if we divide the difference of the extremes (54 — 2) by the ratio, less 1, (3 — 1) the quotient will be the sum of all the terms, except the last, and, adding the last term, we shall have the whole amount. Thus, 54— 2=52, and 3—1=2 ; then 52-^-2= 26, and 54 added, makes 80. Ans. as before. Hence, when the extremes and ratio are given to find the sum of the series, — ^Divide the difference 6f the extremes by the ratio less I , and the quotient, increased by the greater term, will be the answer. 6. If the extremes be 4 and 131072, and the ratio 8, what is the whole amount of the series ? , 131072—4 I l-f 31072=149796. Ans. 8—1 7. What is the sum of the descending series 3, 1, ^, ^, ]>;■, &c. extended to infinity ? It is evident the last term must becone 0, or indefinitely near to nothing ; therefore, the extremes are 3 and 0, and the ratio 3. Ans. 4^. 8. What is the value of the ihfinite series 14'iH~y6"f"BV» &.C.? Ans, 1^. .:.-i„ \.J..Mt^,-' !nL-\- :■•-. t.::!f. /Ml 1 n Jl 226 GEOMETRICAL PROGRESSION. ^ 107. 9. What is ♦he value of the infinite series, i^.-i-i^ "h TT^ ^^t ^^^ ^^^ ^^^ sauie, the decimal *ll 111, &c. coutinually repeated ? Ans. ^. 10. What is the value of the infinite series, T^Ty~l~Tir§TrTj> &,c., descending by the ratio 100 ; or, which is the same, the repeating decimal '020203, &c. Ans. ^^. 11., A gentleman whose daughter was married on a new year's day, gave her ^1, promising to tripple it on the first day of each month in the year ; to how much did her por- tion amount .' Here, before finding the amount of the series, we must find the last term, as directed iu the rule after ex. 1. Ans: £265720, The 2 processes of finding the last term, and the amount, may, however, be conveniently reduced to one, thus : — When the first term, the ratio, and the number of terms, are given, to find the sum or amount of the series^ — Raise the ratio to a power whose index is equal to the number of terras, from which subtract 1 ; divide the remainder by the ratio, less 1, and the quotient, multiplied by the first term, will be the answer. Applying this rule to this last example, 3 V^ss531441 and o31441— I — : Xl=c£265720. ilMs. as before, i 3—1 12. A man agrees to serve a farmer fojty years, without any other reward than 1 kernel of corn for the first year, 10 for the second year, and so on,, in 10 fold ratio, till the end of the term ; 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 ? IQio— 1 1^ 1,111, 111,111, 111,111,111, 111, HI, — —- XI— ^ 111,111,111,111,111, kernels. iln-s. j^:2,170,13i3,388,888,888,8i6S,888,888,886,888,a^, !7s. 9^d. 13. A gentleman dying, left his estate to his 5 sons, to the youngest .£1000, to the second o£'J500, and ordered that each son should exceed the younger by the ratio of H; what was the amount of the estate ? ^ is !I 107. I fl 108. OEOMETRICAL PROGRESSION. 227 the decimal Ans. ^. is the same, Ans. ^g. ied on a new t on the first did her por- ies, we must BX. 1. '.s\ ^65720. the amount, thus : — ber of terms, riesj — Raise le number of linder by the ne first term, =531441 and lars, without first year, 10 till the end fes, allowing his corn for 1,111,111, kernels. 5 sons, to ordered that atio of 14 ; Note. Before finding the power of the ratio 1^, it may be reduced to an improper fraction=s^y or to a decimal, V^. ^^—1 1'5^— 1 — X 1000 =r: 13187^; or, X 1000 = <£13187'50 = ^^13187 10s. Ans. 1*5—1 I Compound Interest hy Progression. "■ ' " ' ^ 108. 1. What is the amount of <£4 for 5 years, at 6 per cent compound interest ? We have seen (fl 86) ihtiX compound interest is that which arises from adding the interest to the principal at the close of each year, and, for the next year, casting the interest on that amount, and so on. The amount of £\ for one year is 1'06; if the principal, therefore, be multiplied by r06^ the product will be its amount for one year ; this amount multiplied by TOO, will give the aniount (compound ir>ter- est) for two years ; and this second amount multiplied by r06, will give the amount for three years ; and so on. Hence, the several amounts arising firom any sum at com- pound interest, form a geometrical series, of which the prin- cipal is the first term ; the amount of ^1 or $1, &;C. at the given rate per cent, is the ratio ; the time, in years, is one less than the number of terms ; and the last amount is the last term. The last question may be resolved into this : If the first term be 4, the number of terms 6, and the ratio .1'0(), what is the last term ? 1'065=1«338, and I*338x4=je6'362-|-. Ans. £b 7s. ^A. Note 1. The powers of the amounts of <£1, at 5 and at 6 per cent, may be taken from the table under IT 85. Thus, opposite 5 years under 6 per cent, you find 1*038, &.c. Note 2. The several processes may be conveniently exhi- bited by the use of letters, thus :-^ Let P represent the Principal. R " Ratioortheamountof.^ Ij&c.for 1 yr. T " Time in years. A " Amount. When two or more letters are joined together, like a word, they are to be multiplied together. Thus, PR. im- plies, that the principal is to be multiplied by the ratio. When one letter is placed above another, like the index of •5«3 GEOMETRICAL PR0VRE6SI0N. 1I108li|i09. W'Am i terest ? JUns. £4Q. 7. At what rate per cent will £6 amount to <£8'7846 in 4 years ? If the first term be 6, the last term 8'7846, and the num- ber of terms 5, what is the ratio ? A 8'7846 — = Rt. that is, =z 1'4641 = the 4th power of P 6 the ratio ; and then, by extracting the 4th root, we obtaio I'lO for the ratio. Ans. 10 per cent. 8. In what time will ^6 amount to .£8'7846, at 10 per cent compound interest ? A 8'7846 — =Rt. that is, z=r4641=l'10T ; therefore, if P 6 we divide 1*4641 by 140, and then divide the quotient thence arising by TIO, and so on, till we obtain a quotient that will not contain 140, the number of these divisions will be the number of years. ^ JJns. 4 years. «1109. GEOMETRICAL PROGRlJpSION. 229 and the num- 9. At 5 per cent compound interest, in what time Will £40 amount to £68 8s. ? Having found the power of the ratio 1*Q5, .is belbre, which is 1*71, you may look for this number in thie table under the given rate, 5 per cent, and against it you will find the number of years. »^ns. 11 years. 10. At 6 per cent compound interest, in what time will £A amount to £o 7s. O^d. Ans. 5 years. Annuities at Compound Interest, , , . IT 109* It may not be amiss, in this place, briefly to show the application of compound interest, in computing the amount and present worth of annuities. An annuity is a sum payable at regular periods of one year each, either for a certain number of years, or during the life of the pensioner, or for ever. When annuities, rents, &c. are not paid at the time they become due, they are said to be in arrears. The sum of all the annuities, rents, &c. remaining un- paid, together with the interest on each, for the time they have remained due, is called the amount. 1. What is the amount of an annual pension of ^100, which has remained unpaid 4 years, allowing 6 per cent compound interest ? The last year's pension will be ^100, without interest ; the last but one will be the amount of .£100 forgone year ; the last but two the amount (compound interest) of £W{) for two years, and so on ; and the sum of these several amounts will be the answer. We have then a series of amounts, that is, a geometrical series, (51 108]^ to find the sum of all the terms. If the first term be 100, the number of terras 4, and th<» ratio 1*06; what is the sum of all the terms ? Consult the rule under ^ 107, ex. 11. 1*06*— 1 . *06 ■X 100=437*45. Ans. c€437 ys Hence, when the annuity, the time, and rate per cent, are given, to find the amount — Raise the ratio (the amount of £1, &,c. for one year) to a power denoted by the num- ber of years ; from this power subtract 1 , then divide the U 230 GEOMETRICAL PROGRESSION. 51110 'li m remaindfr by the ratio less 1, and the quotient multiplied by the annuity, will be the amount. Not«. T\\e powers of the ainounts, at 5 and 6 per cent up to the 24th, may be taken from the table under 11 85. 2. What is the amount oi' an annuity of .£50, it being in arrears 20 yeajrs, allowing 5 per cent compound interest ? Ans. of 1653 5s. 9id. 3. If the. annual rent of a hcJuse, which is .£150, be in arrears 4 years, what is the amount, allowing ten per cent compound interest ? A ns. <£69(> 3s. 4. To how much would a salary of .£500 per annum amount in 14 years, the money being improved at six per cent compound interest 1 in 10 years ? in 20 years ? '-in 22 years ? in 24 years ? Ans. to the last, £25407 15s. If 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 worth. 5. What is the present^ worth of an annual pension of £100, to continue for four years, allowing 6 per cent com- pound interest ? The present worth is evidently a sum which, at six per cent, compound interest, would, in four years, produce an amou t equal to the amount of the annuity in arrears the same time. By the last rule we find the amount=£437'45, and by the directions under 1| 108, ex. 4, we find the present worth =£34651. ^ Ans. £346 10s. 4^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 givg the present worth. 6. What is the present worth of an annual salary of £100 to continue twenty years, allowing five per cent ? ^«s. .£1246 4s. 4fd. The operations under this rule being somewhat tedious, we subjoin a ^ 110. GEOMETRICAL PROGRESSION. 231 TABLE Showing the present worthorjCl or $1 annuity, at 5 and 6 per cent, compound iulerest, for any number of years from 1 to 34. 653 5s. 9id. £25407 15s. Vears 1 2 3 4. 5 7 8 9 10 11 12 13 14 15 16. 17 5 per cent. 0'95238 1*85941 2*72325 3'54595 4'32948 5*07569 5'78637 6'4632l 7*10782 7*72173 8'3064l 8*86325 9*39357 9*89864 10*37966 10'83777 11*27407 6 percent. 0*94339 1*83339 2*67301 3*4651 4*21236 4*91732 5*58238 6*20979 6*80169 7*36008 7*88687 S'38384, 8*85268 9*29498 9*71225 10*10589 10*47726 Years 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 5 per cent. 11*68958 12*08.532 12*46221 12*82115 13*163 13*48807 13*79864 14*09394 14'37518 14*64303 1489813 15*14107 15*37245 15*59281 15*80268 16*00255 16*1929 6 per cent. 10*8276' 11*15811 11*46902 11*76407 12*04158 12*303538 12*55035 12*78335 13*00316 13*21053 13*40616 13*59072 13*76483 13*92908 14*08398 14*22917 14*36613 It is evident that the present worth of £2 annuity is two times as much as that oi£l ; the present worth of <£3 will be three 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 £1 annuity, and multiply it by the given annuity, and the product will be the present worth. 7. What ready money will purchase an aniiiuity of .£150, to continue 30 years at 5 per cent compound interest ? The present worth of .£1 annuity, by the table, for thirty years, is 15 37245 ; therefore, 15*37245X150=^2305*867 =£2305 17s. 4d. Ans, 8. What is the present worth of a yearly pension of £40, to continue ten years at 6 per cent compound interest? at 5 per cent ? to continue fifteen years ? ^20 years ? 25 years ? 34 years ? Ans. to the last, £647 14s. 3fd. When annuities do not commence till a certain period of time has elapsed, or till some particular event has taken place, they are said to be in reversion. I 4iliil 111' •'} ¥','.' m GEOMETRICAL PROGRESSION. Tf 111 . 9. What is the pfesent worth of <£100 annuity, to be con tinued four years, but not to commence till two years hence, allowing 6 per cent compound interest ? The present worth is evidently a sum which, at 6 per cent compound interest, would, in two years, produce an amount equal to the present worth of the annuity, were it to commence immediately. By the last rule, we find the present worth of the annuity, to commence immediately, to be ^346'dl, and by directions under IF lOS, ex. 4, we find the present worth of i.'346'51 for two years to be c£308'393. Ans. £308 7s. lO^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. JO. What ready money will (iurchase the reversion of a lease of £G0 per annum^ to continue 6 years, but not to commence till the end of three years, allowing 6 per cent compound interest to the piirchaser ? The present worth to commence immediately, we find lo 295'039 be 295'039, and — ^ =247*72 Ans. .£247 14s. 4|d. 1'063 . 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 is 34-6=9 years, and then subtracting from this sum the present worth of the annuity, continuing for the time of the reversion, 3 years. Or, we may find the present worth of £1 for the 2 times by the table, and multiply their difference by the giveii annuity. Thus, by the table, The whole time, 9 years=6'80169 The time in reversion, 3 " =2'67301 Difference, 442868 60 11. ..5e247'72080 ^47*72080=^247 14s. 4|d. Ans. What is the present worth of a lease of if 100, to con» ^flll. tinue ailowii ■ 81 ml which per ceil In t| tate is equal tl tiplied] (lividec m and '.V''-\^-.;'->:jifc'-^'Lj-;i^-, ;y, to be con- years hence, ch, at 6 per produce an luity, were it we find the mediately, to X. 4, we find )ede308'393. 508 7s. lO^d. uity taken in the present 1, divided by in reversion, « aversion of a I, but not to ig 6 per cent y, we find lo •47 14s. 4|d. finding the mmediately, is 3-f-6=9 resent worth reversion, 3 l&l for the 2 )y the giveii 169 60 |80 4^d. Ans. [00, to con» 11 111. GEOMETRICAL PROGRESSION. 233 tinue 20 years, but not to commence till the end of 4 years, allowing 5 per cent ? — ^what, if it be 6 years in reversion ? 8 years? 10 years ? 14 years? 11 111. 12. What is the worth of a freehold estate of which the yearly rent is ^€60, allowing to the purchaser 6 per cent? In this case, the annuity continuesybr ever, and the es- tate is evidently worth a sum of which the yearly interest is equal to the yearly rent of the estate. The principal w/w/- tiplied by the rate gives the interest ; therefore, the interest divided by the rate will give the principal; 60-^*06=: 1000. Ans. .£1000. Hence, to find the present worth of an annuity, continu- ing for ever, — Divide the annuity by the rate per cent, and the quotient will be the present worth. \ Note. The worth will be tlie same, whether we reckon simple 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 annuity, whether they be employed at simple or com- pound intere' t. 13. Wha. is the worth of .£100 annuity, to continue for ever, allowing to the purchaser 4 per cent? allowing 5 per cent? 8 per cent? 10 per cent 7 15 per cent ? 20 per cent ? Ans. to the last, .£500. 14, Suppose a freehold estate of .£60 per annum, to com- mence two years hence, be put on sale ; what is its value, allowing the purchaser 6 per cent ? Its present worth is a sum which, at 6 per cent compound interest, would in two years produce an amount equal to the worth of the estate if entered on immediately. 60 — =£1000=the worth, if entered on immediately, *06 £1000 and ==i:889'996=£889 19s. lid. the present worth. r062 The same result may be obtained by subtracting from the worth of the estate, to commence immediately, the pre- sent worth of the annuity 60, for two years, the time of reversion. Thus, by the table, the present worth of £1 for U2 "<"►, I'M ^:. 'ji 'i I *J:J4 PEllMttATION'. 11 la' two years is 1 '83339 X 00=1 1 0'0034=:Dresent worth of JGliO Tor two years, and Jei000—n0'0034 =^ .£809*9960=^889 19:>. lid. Arts, as before. 15. Wliat is the present worth of a perpetual annuity of •£ii)\), to commence G years hence, allowing the piwchaser •> })er cent compound interest ? what, if 8 years in re« version ?— -10 years ? — ^4 years ? 30 years ? Ans. to the last, .£462 15s. l^d. The foregoing examples in compound interest have been confined to i/earli/ payments ; if the payments are half-yearly; we take half the principal or annuity, half the rate per cent and twice the number of years, and work as l)efore, and so for any other part of a year. QUESTIONS. i. What is a geometrical progression or series ? 2. What is the I'utio ? 'i. When the lirst term, the ratio and the number of terms, arc given, how do you find the last terAi ? 4. When the extremes and ra- tio are given, how do you find the sum of all the terms 'i 5. When the first teriii, the ratio, and the number of terms are given, how do you find the amount of the series i G. When the ratio is a fraction, ho\' do you proceed I 7, What is compound interest 1 8. How does it appear that the amounts arising by compound interest, form a geomc- triciil scries f 9. What is the ratio in compound interest ? the number of terms ?-^^— the first term 1 — — the last term? 10. When the rate, the time and the principal are given, how do you find the amount i 11. Wlien A 11 and T are given, how do you find P / 12, VVheii A P ana T are given, how do you find R ? 13. When A P and 11 are given, how do you find T ? 14. What is an annuity ? lb. When are annuities said to be in arrears? 16. What is the amount? 17. In u geometrical series, to what is the amount of an annuity equi- valent? 18. How do you find the amount of an annuity, at compound interest? 19. Wha*; is the present worth of an annuity? iiow eotnputed at compound interest '? ■ - ■ ■ • ' how found by the table 1 20. What is understood by the term reversio'ii '( 21. How do you find the present worth of an annuity, taUen in reversion? by the table? 22. How do you find the present worth of a freehold estate, or a per- petual annuity ? — ^ tlie same taken in reversion ? -by the table I fER.TlIJTATJOJV. ^f lllS. Permutation is the method of finding how many different ways the order of any number of things may iie varied or changed. 1. 1 they c how m Hac only ii and b m IX with a next bi and b that is, o i worth of jCOO lal annuity of the piir chaser S years in re* ears? tG-J 15s. l^d. est have been •e half-yearJv; rate per cent. )efore, and so 2. What is the er of terms, art xlremcs and ra- i 5. When the n, how do you !i fraction, ho»» 8. How does it , form a geomc- erest ? the |n? 10. When you find the t)u find P ? 12, Wiien A P and annuity ? 1.5. 9 the amount? !» annuity equi- f^ at compound ty ? iiow he table 1 20. do you find the • by the table '; Late, or a per- y the table l inding how ■ things may M13 Miscellaneous examples. J235 1. Four gentlemen agreed to dine together, so long as they could sit every day in a different order or position ; how many days did they dine together 7 Had there been but two of them, a ahd h, they could sit only in 2 times 1 (1x2=2) different positions, thus, a b, and b a. Had there been three, a b and c, they could sit in lX2X3=(i different positions; for, beginning the order with a, there will be two positions, viz a b c, and a c b; next beginning with 6, there will be two positions, b a c^ and bca; lastly, beginning with c, we have c ab^ and cba, that is, in all, 1X2X3=6 different positions. In the same manner if tlvtre be four, the different positions will be 1X2X8X4=24. y|«5. 24. Hence, to find the number of different changes or per- mutations, of which any number of different things are ca* pable, — Multiply continually together all the terms of the natural series of numbers, from one up to the given number, and the last product will be the answer. 2. How many variations may there be in the position of the nine digits ? Ans. 862880. 3. A man bought 25 cows, agreeing to pay for them one penny for every different order in which they could all be placed ; how much did the Cows cost him ? Aas. i:6463004184T2l2441600000. 4. A cei'tain church has 8 bells ; hoAv many changes may be rung upon them ? Ans. 40320. MlSCELLANE^OUS EXAMPLES. 51 113. 1; 44- ;x7— 1=^=60. A line, or vinculum, drawn over several nun\bers, signi- fies that the numbers under it are to be taken jointly, or as one whole numbi Ans. 30. 2. 9— «-|-'4 X 8-f-4— 6=how many ? Ans. 230. 3. 7-1-4— 2 4- 3-f40 X5— how many? 3-J.6— 2X4— 2 4. 2X2 =howmany? Ans.^^. 5. T^iere are 2 numbers ; the greater is 25 times 78, and their difference is 9 times 15 j their sum and product are I 236 MISCELLANEOUS EXAMPLES. 'f 113 113. t)3. B t come t vill it tu 24. IIS pay le recei 25. required. Ans. 3765 is their sum, 3539250 their product 0. Wliat is the difference between thrice five and thirty.^ ii and thrice thirty-five ? 35-|-3— 5 X 3-}-30=60, Ans. '^[\\ \yQ 7. Wliat is the differeiice between six dozen dozen, and half a dozen dozen ? Ans. 792. 8. What number divided by 7 will make 6488? 1). What number multiplied by 6 will make 2058 ? 10. A gentleman went to sea at 17 years of age; 8 years after, he liad a son born, who died at the age of 35 ; after whom the father lived twice 20 years; how old was the father at his death? Ans. 100 years. 11. What number is that which, being multiplied by 15, the product will be ^ ? ^_^l5=^ig., Ans. 12. What decimal is that which, being multiplied by 15, the product will be *75 ? '75-i-I5='05, Ans. 13. What is the decimal equivalent to t^l ? Ans. '0285714 14. What fraction is that, to which if you add f , the sum will be I ? ylns. fg. 15. What number is that, from which if you take f , the remainder will be ^ ? Ans. |^. 10. What number is thal^, which being divided by f , the quotient will be 21 ? Ans. 15f, 17. What number is that, from which if you take f of itself, the remainder will be 12 ? Ans. 20, 18. What number is that, to which if you add f of ^ of itself, the whole will be 20 ? Ans. 12. 1 9. What number is that of which 9 is the § part ? Ans. 13 J . 20. A farmer carried a load of produce to market ; he sold A )U9hel, leived ushels 26. nust be H 27. H n excha Note. ire giver he same 28. H n exchai 29. A rave him low man 30. A 7801t}s of pork, at 3d. per lb ; 250ft)s of cheese, at 5d. per lb ; )arterin£ 1541bs of butter, at lOd. per lb. In pay he received 601bs of sugar, at 7d. per lb; 15 gallons of molasses, at 2s. 3d. per gallon; -^ barrel of mackerel, at 18s. 9d. ; 4 bushels of salt, at 6s. 4d. per bushel ; and the balance in money ; how much money did he receive ? Ans. ^15 14s. 8d, 21. A farmer carried his gram to market, and sold 75 bush- els of wheat at 7s. 3d. per bushel ; 64 bushels of rye at 4s. 9d. per bushel ; 142 bushels of corn, at 2s. 6d. per bushel. In exchange, he received sundry articles : — 3 pieces cloth, each containing 31 yds. at 8s. 9d. per yd. ; 2 quintals fish, I Is. 6d. per quintal ; 8 hhds. salt, £l Is. 6d. per hhd. and the balance in money ; how much money did he receive ? Ans. £9 14s v'orth IS he broai 30d. : ■s. lO^d ctly alii 31. If It 28. 6d 3xchangi 32. if 33. If ushel ? 34. II '[ 1I3| their product ve and thirty, JO, Ans. ill dozen, and Ans. 792. 88? ! 2058 ? f age ; 8 years e of 3f> ; after old was the ns. 100 years. Itiplied by lo, 15=^'^, Ans. Itiplied by 15, 5='05, An!<. Ins. '0285714 dd f , the sum Ans. U- 3u take f , the Ans. il ded by f , the Ans. 15f. you take f of Ans. 20. add f of ^ of Ans. 12. irt?Ans. 13^. irket ; he sold at5d. per lb, ceived 60Ibs at 2s. 3d. per jshels of salt, how much ^15 14s. 8d. 113. MIHCELLANEOUS EXAMPLES. 237 sold 75 bush- rye at 4s. 9d. bushel. In Es cloth, each Ash, lis. 6d. d the balance Ans. ^9 14s A man exchanges 760 gallons of molasses, at 2s. per allon, for 064- cwt. of cheese at £i per cwt. ; how much trill be the bahin($e iu his favor ? Ans. £0 10s. 23. Bought 84 yds. of cloth at (5s. 3d. per yd. ; how mUch did t come to ? how many bushels of wheat at 7s. 6d. per bushel, vill it take to pay fufr it ? Ans. to the last, 70 bushels. 24. A man sold 342tbs of beef at 4d. per lb, and received )is pay in molasses at 2s. per gallon ; how many gallons did le receive 7 ^ Ans. 57 gallons. 25. A man exchanged 70 bushels of rye at 4s. 6d. per ushel, for 40 bushels of wheat at 7s. per bushel, and re- eived the balance in oats at 2s. per bushel ; how many )ushels of oats did he receive? Ans. 17^. 26. How many bushels of potatoes at Is. Od. per bushel, nust be given for 32 bushels of barley it 2s. 6d. per bushel ? Ans. 53^ bushels. 27. How much salt, at $1*50 per bushel, must be given n exchange tor 15 bushels of oats, at 2s. 3d. per bushel ? Note. It will be recollected that when the price and cost ire given to find the quantity, they must both be reduced to he same denomination before dividing. Ans. 4^ bushels. 28. How much wine, at $2*75 per gallon must be given n exchange for 40 yards of cloth at 7s. 6d. per yard 7 Ans. 21y^Y gallons. 29. A. had 41 cwt. of hops at 30s. per cwt. for which B. ave him <£20 in money, and the rest in prunes at 5d. per ib. ; low many prunes did A. receive? Ans. I7cwt. 3qrs. 4tb. 30. A. has linen cloth worth 2s. 6di per yard; but in )artering he will have 2s. 9d. per yard ; B. has broadcloth ivorth 18s. 9d. per yard, ready money ; at what price ought he broadcloth to be rated,_ih bartering with A. ? * 30d. : 35d. :: 225d. : 262^d. ans. Or, ,^^ of 225d.=c£l s. lOJ^d. ans. The two operations will be seen to Be ex- ctly alike. 31. If cloth worth 2s. per yard, cash, be rated in barter tSs. 6d., how should wheat, worth 8s. cash, be rated in xchange for the cloth ? Ans. lOs. 32. If 4 bushels of corn cost $2, what is it per bushel ?' 33. If 9 bushels of wlieat cost =£3 7s. 6d. what is that per ushel.'' 'w* -,}*■• Ans. 7s. 6d. 34. If 40 sheep cost £25, what is that per head? "•■;'> 338 MI8CELLANB0US EXAMPLES. 5I11J M' I , V m I: ' r- 3o. If 3 bushels of oats cost 7s.. Gd. how much are the per bushel? Ans. 2s. CdL 36. If 23 yards of broadcloth cost <£21 Os. what is th( price per yard ? Ans. lOs (jd 07. At 2s. 6d. per bushel, how much corn can be bough for lOs. Ans. 4 bushels 3^. A man haying £%5, would lay it out in sheep, at 12s Cd. aw much (lid h Is? .£888 2s. 9|(i )ds lit Montreal rtation ; he soli ; cost ; for ho» .£673 6s. 4|d , but for cash the books cost r IS. £56 6s. 4fd vas marked £ 11 it at 33i pel Ans. 15s MISCeLLANBOUS EXAMPLBf. 239 49. I bought tt cask of liquor, containing 120 gallons, for jff42; for how much myst I sell it to gain 15 per cent? how much per gallon 1 Ans to the last, 48. 0|d. 50. Bought a cask of sugar, containing 740 pounds, for £59 4s. ; how -must I sell it per pound, to gain 25 per cent? Aifs. 2s. 51. Wlmt is the interest at 6 per cent, of j£71 Os. 4^d. for 17 months 12 days ? Ans. .£6 33. 6jd. 52. What is the interest of .£487 Os. OJd. for 18 months? Ans. i:43 16s. l^d. 53. What is the interest of $8*50 for 7 months? - Ans. $'297;^. 54. What is the interest of .£1000 for 5 days ? Ans. 16s. 8d. 55. What is the interest of lOs. for ten years ? Ans. 68. 66. What is the interest of $84'25 for 16 months and 7 days, at 7 per cent ? Ans. $7*486-|- 57. What is the interest of $164'01 for 2 years, 4 months and 3 days, at'5 per cent? A^ . $18'032. 58. What sum put to interest at 6 per c( .it, will in two years and 6 months, amount to $150 ? , Ans. 8130*4344- Note. See H 79. 59. I owe a man .£476 10s. to be paid in 16 months with- out interest ; what is the present worth of that debt, the use of money being worth 6 per cent ? Ans. .£440 5s. G^d. 60. What is the present worth of JElOOO payable in four years and 2 months, discounting at the rate of 6 per cent ? 61. Bought articles to the amount of j£500, and sold them for £575, how much was gained ? What per cent was gained? that is, how ,many p'ounds were gained on each .£100 laid out ? If .£500 gain .£75, what does j6^100 gain? Ans. 15 per cent. 62. Bought cloth at .£3 10s. per piece, and sold it at £4; 5s. per piece ; howmuch was gained per centum ? Ans. 21 ^. 63. A man bought a cask of liquor, containing 126 gal- lons for £283 10s. and sold it out at the rate of £2 15s. per gallon? how much was his whole- gain ? how much per gal- lon ? how much per cent ? Ans. His whole gain .£63; per gallon 10s. which is 22f per centum. 64. If .£100 gain .£6 in 12 months, in what time will it gain £il .£10 ? £U ? Ans. to the last, 28 months, 240 MISCELLANEOUS EXAMPLES. 11113 65. In what time will ^54 10s. at 6 per cent, gaiA £i 3s. 7^ d. .4ns. 8 months 66. Twenty men built a certain bridge in 66 days, but i being carried away in a freshet, it is required how man men can re-build it in 60 days ? ' :.; .J days, days. men. 50 : 60 : : 20 : 24 men. Ans 67. If a field will feed 7 horses 8 weeks, how long will if! feed 28 horses ? Ans. 2 weeks. 68. 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 IG rods in length, to contain the same 7 Ans. IQ rods, [[113. 76. sions su part, th; 77. I be wort value o 78. I I among khan 3 79. water 69. If I purchase for a cloak twelve yards of plaid | of a yard wide, how much bocking 1^ yards wide must I buy to line it .^ * Ans. 5 yards. 70. If a man earn £18 15s. in 6 months, how long must he work to earn ^115 ? Ans. 30§ months. 71. B. owes C. .£540, but B. not being worth so much money, ,C. agrees to take 15s. on a pound ; what sum must C. receive for the debt 1 . Ans. <£405. 72. A cistern whose capacity is 400 gallons, is supplied by a pipe which lets in 7 gallons in 5 minutes ; but there is a leak in .the bottom of the cistern which lets out 2 gallons in 6 minutes. Supposing the cistern empty, in what time would it be filled ? In one minute ^ of a gallon is admitted, but in the same time I of a gallon leaks out. Ans. 6 hours 15 minutes. 73. A ship has a leak which will fill it so as to make it sink in ten 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 what time will it sink ? * In one hour the ship would be ^ filled by the leak, but in the same time it would be -^ emptied by the pump. Ans. 30 hours. 74. A cistern is supplied by a pipe which will fill it in 40 minutes ; how many pipes of the same size will fill it in five minutes ? Ans. 8. 75. Suppose I lend a friend .£500 for foti? months, he promising to do me a like favour ; some time afterward, I have need of .£300 ; how long may I keep it to balance the former favour ? , Ans. 6§ months. -■•-•:i tliia 113. MISCELLANEOUS EXAMPLES. 241 cent, gaift £i ins. 8 months 60 days, but red how many li men. Ans low long will it Ans. 2 weeks. rods in width t be a field IC Ans. ip rods of plaid I of a I must I buy to AnS' 5 yards low long mus f. 30f months 'orth so much i'hat sum must Ans. ^405 IS, is supplied 3 ; but there is out 2 gallons in what time t in the same sis minutes, as to make it trill clear it in legins to leak, the leak, but le pump. ns. 30 hours. will fill it in will fill it in Ans. b. ' months, he afterward, I 3 balance the 6f months. 76. Suppose 800 soldiers were in a garrison with provi- sions sufficient for 2 months ; how many soldiers mu it de- part, that the provisions may serve them 5 months ? Ans, 480. 77. If my horse and saddle are worth <£21, and my horse be worth six times as much as my saddle, pray what is the value of my horse ? - Ans. £18. 78. Bought 45 barrels of beef at 17s. 6d. per barrel, among which are 16 barrels whereof 4 are worth no more than 3 of the others ; how much must I pay ? Ans. £S5 178. 6d. 79. Bought 126 gallons of rum for ^27 10s. how much water must be added to reduce the first cost to 3s. 9d. per gallon ? Note. If 3s. 9d, buy one gallon, how many gallons will £^7 10s. buy ? • Ans. 20^ gallons. 80. A thief having 24 miles start of the officer, holds his way at the rate of 6 miles an hour ; the officer pressing on afler him at the rate of 8 miles an hour, how much does he gain in one hour ? how long before he will overtake the thief? Ans. 12 hours. 81. A hare starts 12 rods before a hound, but is not per- ceived by him till she has been up 1^ minutes ; she scuds away at the rate of 36 rods a minute, and the dog, on view. makes after at the rate of 40 rods a minute ; how long will the course hold, and what distance will the dog run ? Ans. 14^ minutes, arid he will run 570 rods, 82. The hour and minute hands of a watch are exactly together at 12 o'clock ; when are they next together ? In 1 hour the minute hand passes over 12 spaces, and the hour hand over one space ; that is, the minute hand gains upon the hour hand eleven spaces in one hour ; and it must gain twelve spaces to coincide with it. Ans. Ih. 5m. 27-,\s. 83. There is an island 20 miles in circumference, and 3 men start together to travel the same way about it ; A. goes two miles per hour, B. four miles per hour, and C. six miles per hour ; in what time will they come together again ? Ans. 10 hours, 84. There is an island 20 miles in circumference, and two men start together to travel round it ; A. travels two miles per hour, and B. six miles per hour ; how long betoro they will again come together ? r W v, :C. ;-' * if 242 MUCRLLANE0U8 EXAMPLES. mil (113. B. gams 4 miles per hour, and must gain twenty miles ti earn to overtake A. ; A. and B. will therefore be together once ii every five hours. ach? Ans. 85. In a river, supposing two boats start at the sami 12 learn time from^ places 30U miles apart ; the one proceeding u| 93. A stream is retarded by the current two miles per hour, whili mother, that moving down stream is accelerated the same ; if boti reese ;" b«» propelled by a steam engine which would move them i addition miles per hour in still water, how far from each startiiij have, an place will the "boats meet 7 Ans. 113^ miles from the lower place, and 187^ milei from the upper place. 86. A man bought a pipe ( 1 26 gallons) of wine for .£275 he wishes to fill 1 bottles, 4 of which contain two quarts and 6 of them 3 pints*each, and to sell the remainder so !u to make 30 per cent on the first cost; at what rate per gal Ion must he sell it ? ^ Ans. ^65*936+ 87. Thomas sold 150 pine apples at Is. 3d. apiece, ar« received as much money as Harry received for a certain number of water-melons at 9d. apiece ; how much moni j did each receive, and how many melons had Harry ? Ans. £9 7s. 6d. and 250 melons, 88. The third part of an army was killed, the fourth part taken prisoners, and 1000 fled , how many were in this army! 'This and the 18 following questions are usually wrought by a rule called Position, but they are more easily solved on general principles. Thus, ^-j-i=T^2 of the army ; there- fore, 1000 is -^ of the whole number of men ; and if-j^ b« 1000, how much is 12 twelfths, or the whole ? Ans. 24000 men 89. A farmer being asked how many sheep he had, ans- wered that he had them in 5 fields ; in the first were { of his flock, in the second |, in the third | in the fourth jVi and in the fifth 450 ; how many had he ? ^ins. ISOO, 90. There is a pole, ^ of which stands in the mud, ^ the water, and the rest of it out of the water ; required the part out of the water. ^ Ans. -f^, 91. If a pole be \ in the mud, f in the water, and 6 feet out of the water, what is the lengtli of the pole ? Jins. 00 feet, 92. The amount of a certain school is as follows : j'^ ot the pupils study grammar, f geography, -^ arithmetic, /j in many ha 100- 94. Ii I pears, how mai 95. Ii \ red, a ed ; ho\ 96. a the sam ber. 97. \^ the sum 9S. V the sum 84 = 99. A and f ol niucli 7 The 2A tilTl I 100. and ' 101. was on twice a their aj 102. .v;* mi! [115. )gether once i rt at the sami proceeding uj per hour, whil e same ; if hot d move them 1 each startiii md IST^ mile MISCELLANEOUS EXAMPLES. 243. twenty miles ti earn to write, and 9 learn to read ; what is the numbix ef ivine for cf 275 ain two quarts, emainder so i at rate per ga ins. je5*936-f | red, and i > Jd. apiece, am cd ; how man} d for a certa ' much mon( Harry ? id 250 melons the fourth par e in this armj sually wrought ; easily solved B army ; there- ; and if^^ be J. 24000 men. p he had, aiis rst were ^ of the fourth ^, ^tns. 1200. the mud, | in ; required the Ans. T^j, er, and 6 feet >/his. 90 feet. bllows : j'^ ot trithmetic, ^'j ach? Ans. 5 in grammer, 30 in geography, 24 in arithmetic ; 2 learn to write, and 9 learn to read. 93. A man, driving his geese to market, was met by nother, who said, "Good morrow, sh^ with your hundred says he, "I have not a hundred ; but if I had, in ddition to my present number, one half as many as I now ave, and 2^ geese more, I should have a hundred :" how many had he ? 100 — 2^ is what, part of his present number ? Ans. He had 65 geese. 94. In an orchard of fruit trees, ^ of them bear apples, { pears, } plums, 60 of them peaches, and 40, cherries ; how many trees does the orchard contain ? Ans. 1200. 95. In a cf ri.. i village, ^ of the houses are painted white ' 3 are painted green, and 7 are unpaint- 38 in the village ? Ans. 120. 96. Sl2ven eighths of a certain number exceed four fifths of the same number by 6 ; required the number. I — t==i^ ; consequently, 6 is ^xf o^ 'he required num- ber. ^n.5. 80. 97. What number is that, to which if |of itself be added, the sum will be 30 ? Ans. 25. OS. What number is that to which if its ^ and I be added, the sum will be 84? 84 = 1 -f-i "|~i=|^ times the required number. Ans. 48. 99. What number is that, which, being increased by f and f of itself, and by 22 more, will be made 3 times as much ? The number, being taken 1, f , and f times, will make 2/^ times and 22 is evidently what that wants of 3 times, Ans. 30. 100. What number is thit, which being increased by f , I and I of itself, the sum will be 234f ? Ans. 90. 101. B, C, and D; talking of their ages, C said his age was once and a half the age of B, and D said his age was twice and one tenth the age of both, and that the sum of their ages was 93 ; what was the age of each ? Ans. B 12 years, C 18 years, D 63 years old. 102. A schoolmaster being asked how many scholars he i 244 MISCELLANEOUS EXAMPLES. U 113.1 H li3. had, said, "If I had as many more as I now have, f as ma- ny, ^ as many, ^ and ^ as many, I should then have 435 ;" ^hat was the number of his pupils? Ans. 120. 103. B and C commenced trade with equal sums of money ; B gained a sum equal to ^ of his whole stock, and C lost £2Q\) ; then B.'s money was double that of C's ; what was the stock of each ? By the condition of this question, one half of f . that is, f of the stock, is equal to ^ of the stock, less £200 ; consequently, J£200is f of the stock. Ans. .£500. 104. A man was hired 50 days on these conditions, — that for every day he worked, he should receive 3s. 9d., and for every day he was idle, he should forfeit Is. 3d. : at the expiration of the time, he received £2 17s. 6d. , how many days did he^ work, and how many was. he idle? Had he worked every day, his wages would have been 8s. 9d.X50=£9 7s. 6d. that is £2 10s. more than he received ; but every day he was idle lessened his wages 3s. 9d,-}-ls. 3d.=5s. ; consequently he was idle 10 days. Ans. He wrought 40, and was idle 10 days. 105. B and C have the same Income; B saves ^ of his ; butC, by spending <£30 per annum more than B, at the end of 8 years tinds himself <£40 in debt ; what is their income, aud what does each spend per annum ? Ans. Their income, £200 per annum ; B spends £175, and C £205 per annum. 106. A man, lying at the point of death, left his three sons his property ; to B ^ wanting £20, to C ^; and to D the remainder, which was £10 less than the share of B ; what was each one's share ? A ns. £80, £50, and £70. 107. There is a fish, whose head is 4 feet long ; his taij is as long as his head and half fine length of his body, and his body is as long as his head ^nd tail ; what is the length of the fish ? y The pupil will perceive thaj i^. length of the body is ^ the length of the fish. /^ Ans. 32 feet. 108. B can do a certain ^iece of work in 4 days, and C can do the same work in £f days ; in what time would both working together, perform it ? Ans. l^ days. 109. Three persons can perform a certain piece of work in the following manner : B and C can do it in 4 days, C U 113.1 !1 l»3. MISCELLANEOUS jBXAMPLES. 245 ive, ^ as ma- i have 435 ;" Ans. 120. [ual sums of )le stock, and that of C's ; of f . that is, less £200 ; Ans. .£500. conditions, — eiye 3s. 9d., t Is. 3d. : at 7s. 6d. , how idle? lave been Ss. I he received ; 3s. 9d.-f Is. idle 10 day.s. J saves ^ of 3 than B, at what is their jends £175, ft his three ^; and to D share of B ; »0, and £70. [>ng ; his tail is body, and the length he body is ^ Ans. 32 feet, days, and C ) would both Ins. l^^ days, ece of work 3 4 days, C and I^ in 6 days, and B and D in 5 days : in what time can they all do it together ? Ans. 3^ days. 110. B and C can do apiece of work in 5 days ; B can do it in 7 days ; in how many days can C do it ? .■4 ns. 17^. 111. A man died, leaving £1000 to be divided between his two sons, one 14 and the other 18 years of age, in such proportion that the shire of each, being put to interest at 6 per cent, should amount to the same sum when they should arrive at the ago of '21 ; what did each receive ? Ans. The elder £546 3s. 0|d.+ ; the younger £453 1 6s. lid. 112. A house being let upon a lease of five years, at £15 per annum, and the rent being in arrear for the whole time, what is the sum due at the end of the term, simple interest being allowed at 6 per cent ; Ans. £84. 1 13. If three dozen pair of gloves be equal in value to 40 yards of calico, and 100 yards of calico to three pieces of satinet of 30 yards each, and the satinet be worth 2s. Gd. per yard, how many pair of gloves can be bought for 20s. ? Ans. 8 pair. 114. B. C. and D., would divide £100 between them, so that C. may have £3 more than B. and D. jC4 more than C ; how much must each man have ? Ans. B. jC30, C. £33, and D. £37. 115. A man has pint bottles, and half-pint bottles; how much wine will it take to fill one of each sort ? now much to fill two of each sort ? how much to fill 6 of each sort ? 116. A man would draw off 30 gallons of wine mto one pint and two pint bottles, of each an equal number ; how many bottles of each kind will it take to contain the thirty giillons ? Ans. 80 of each. 117. 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 qis. 12 lbs. of tea? Ans.m. 1 18. If 18 grains of silver make a thimble, and 12 pwts. make a tea-spoon, how many, of each an equfil number, can be made from 15 oz. 6 pwts. of silver ? Ans. 24 of each. 119. Let sixty pence be divided among three boys in such a manner that, as often as the first bus three, the second shall have five, and the third seven pence ; how many pence will each receive .' Ans. 12, 20 and 23 pence. W T'y 246' MISCELLANEOUS EXAMPLES. IT U3 120. A gentleman having fifty shillings to pay among his labourers for a day's work, would give to every boy 6d., to every woman 8d., and to every man 16d. ; the number of boys, women and men was the same ; I demand the num- ber -of each? ^«.! 17s. 6d. to pay among his la- borers : to every boy he gave 6d., to every woman 8d., and to every man 16d. ;. and there were for every boy three women, for every woman two men ; 1 demand the number of each ? Ans. 15 boys, 45 women, and 90 men. 122. A farmer bought a sheep, a cow, and a yoke of oxen for i;20 12s. 6d. ; he gave for the cow 8 times as much a& for the sheep, and for the oxen three times as much as for the cow ; how much did he give for each ? Ans. For the sheep, 12s. 6d. the cow i^5, and the oxen -£15. 123. There was a farm of which B. owned f , and C. ^\ ; the farm was sold for £>^A\ ; what was each one's share of the money ? Ans. B,'s £126, and C.'s i:315. 124. Four men traded together on a capital of £3000, of which B. put in ^, C. i, P. i,,and E, -j^; at the end of 3 years they had gained £2364 ; what was each one's share of the gain? ^ii5. B.^si:il82,.C.'s<£591,D.'s.£394,E.'s£l97. 125. Three merchants companied ; B. furnished f of the capital, C. §, and D. the rest ; they gain t" D 's cloth per yard, and what was B, and C.'is share of the gain? 600 1200 48 Nvtc. D.'s gain being 430, ig = == — of the 1662^ 332& 133 nholo gain ; hence the gain oCBt aijd C is readily found ; also the price at w *> D.'s cloth was valued, per yard. Ans. D.'s cloth pe ard, £1, B.'s share of tlie gain, £31 55., C.'.s share, £21 i. 6d. 127. Three gar eners, B. C. and D; having bought a c.ecc of ground, f id the profits of it amount to £120 per. m ann in £,1 mu of H 113 nii3. MISCELiANBOUtr EXAMPLE*. 247 pay among hia ery boy 6d., to the number of Hand the num- Jlns. 20. among his la- :oman 8d., and ^ery boy three lid. the number J, and 90 men. a yoke of oxen les as much a& as much as for Ans. For the 5. If, andC. ^f ; one's share of md C.'s £315. al of £3000, of at the end of 3 I one's share of J94,E.'s£l97. lished f of the 150 ; what part h onej's sliare .'s share of the 8158. put in c€l25, iy gained ^'83 wa& the value C.'s share of bo 48 =— of the 55 133 eadily found ; per yard. ne gain, of 31 ing bought a to £120 per. annum. Now the sum of money which they laid dY)wn wa» in such proportion, that, as often as B paid £o, C. paid £7, and as often as C. paid £4, D. paid ^£6 ; I demand how much each man must have per annum of the gain ? Note. By the question, so often as B paid ^5, Di paid f of o^7. Ans. B. ^26 13s. 4d., C. £37 68. 8d ; D. £56. 128. A gentleman divided his fortune among his son-s, giving Bw £d as often as C. £o^ and D. £2 as ^ ^ js C. £7 ; D.'s dividend was 1537| ; to what did the ./hole es- tate amount? An^. .£11583 88. lOd. 129. B. and C. undertake a piece of work for .£13 lOs., on which. B. employed 3 hands 5 days, and C. employed 7 hands 3 days ; what part of the worjc was done by B., and, what part by €. ? what was each one's share of the money ? Ans. B. -^ and C ^"^ ; R.'s money £G: 12s. 6d , C.'s «£7 17s. 6d. 130. B. and C. trade in company for one year only ; on the 1st of January B. put in £300, but C. could not put any money into the stock until the 1st of April ; what did he then put in to have an equal sliare with B. at the end of the year? Ans. .£400. 131. B. C. D* and E. spent 35s. at a- reckoning, and be- ing a little dipped, agreed that B. should pay §, C. ^, D. ^, and E. ^ ; what did each pay in this proportion ? Ans. B. 13s. 4d., C. 10s., 1). 6s. 8d. and E. 5s. 132. There are 3 horses belonging to 3 men, employed to^draw a load of plaister from Montreal to Stanstead, for £6 12s. 2d. B: and C.'s horses together are supposed to do f of the work, R. and D.'s -^q, C. and I> 's ^ J ; they are to be paid proportionally; what is each one's share of the money ? > r B.'s £2 Hs. 6d. (=^5) Ans. I C;'s I 8s. 9d. {==^) ( D.'s 2 68. Od. (s^Jj) Proof,. £6 128. 3d. 133. A person who was possessed of f of a vessel, soldf^ of his share for £375 ; what was^ the vessel worth ? Ans. £1500. 134. A gay felloM; soon got the better of f of his fortune ; he then gave £1500 for a commission, and his ptofusion eontin^ued till he had! but £450. left, whicji he fotmd to be- 248 MISCELLANEOUS EXAMPLES. 11113. I ^ 113. just J of his money after he r ad purchased his commission ; what was his fortune at first? Ans. X3780. 135. A younger brother received £1560, which was just ^j of his elder brother's fortune, and 5^ times the elder brothel's fortune was J as much again as the father was worth ; what was the value of i.is estate ? Ans. £19165 Hs. 3fd. 136. A gentleman left, his son a fortune, -^\ of which he spent in three months; ^ of | of the remainder lasted him 9 months longer, when he had only -C537 left ; what was the 8um beciueathed him by his f ither ? Ans. £2082 18s. 2-j2^-d. 137. A cannon ball, at the first discharge, flies about a mile in 8 seconds ; at 4his''rate, how long would a ball be in passing from the earth to the sun, it being 95173000 miles distant ? Ans. 24 years, 40 days, 7 h. 33 min. 20 sec. 138. A general, disposing his jirmy into a square bat- talion, found he had 231 over and above, but inc iing each side with one soldier, he wanted forty-four to fill up the squ.are ; of how many men did his army consist ? Ans. 19000. 139. B. and C. cleared by an adventure at sea, 45 gui- neas, which 'was £35 per cent upon the money advanced, and with which they agreed to purchase a genteel horse and carriage, whereof they were to have the use in proportion to the sums adventured, which was found to be 11 to B. as often as 8 to C* ; what money did each adventure? Ans. B. £104 4s. 2f3d., C £75 !5s. 9f^d. 140. Tubes may be made of gold, weighing not more th in at the rate of jj^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? Ans. 1 Ife 8oz. Gpwts. 3^^^ grs. 141. A military officer drew up his soldiers in rank and file, having the number in rank and file equal ; on being re- inforced 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 was again reinforced with three times his whole number of men, and after placing them all in the same form as at first, his num- ber in rank and file was 40 men each ; how many men had he at first ? _ Ans. 100 men. 142. Supposing a man to stand 80 feet from a steeple, and tha 100 fee high ab of the of the 143 east, at south, apatt w 24 hour 144. rods ; w 145. posite c of each 146. tance ol of the fi A 147. of carpe of it ? 148.) how ma Whei area or Whe given, 1 149. 20 rods E ■ V- 1IH3. ^113. MISCELLANEOUS EXAMPLES. J?40 nmission ; IS. £3780. ^l was just the elder father was I.. 148. 3fd. which he asted him at was the 8s. 2^jd. s about a a ball be 95173000 n. 20 sec. [uare bat- nr i'lng fill up the s. 19000. a, 45 gui- idvanced, horse and roportion to B. as 5s. 9Ad. lot more would be cross the distance 3A grs. rank and being re- le placed in rank 'as again iicn, and lis num- men had 00 men, steeple, and that a line reaching from the belfry to the man is just 100 feet in length, the top of the spire is three times as high above thf: ground as the steeple is ; what is the height of the spire //and the length of a line reaching fr«jm the top of the spire to the man 1 See fl 103. Ans. to the last, 197 feet nearly. 143. Two ships sail from the same port- one sails directly east, at the rate of 10 miles an hour, and the other directly south, at the rate of 7^ milea an hour ; how many miles apart will they be at the end of 1 hour ? 2 hours? 24 hours ? 3 days ? Ans. to last, 900 miles. 144. There is a square field, each side of whi^h is 5:) rods ; what is the distance between opposite corners 1 Ans. 7U'71-f rods. 145. What is the area of a square field, of which the op- posite corners are 70*71 rods apart ? and what is the length of each side ? Ans. to last, 50 rods nearly. 146. There is an oblong field, 20 rods wide, and the dis- tance of the opposite corners is 33^ rods ; what is the length of the field ? its area ? Ans. Length 2Gf rods ; area 3 acres, 1 rood, 13^ rods. 147. 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 feet=30 yards. Ans. 148. If the floor of a square room contain 30 hquare yds. how many feet does it measure on each side .' Ans. IH feet. When one side of a square is given, how do you find its area or superficial contents ? When the area or superficial contents of a square is given, how do you find one side ? 149. If an oblong piece of ground be 80 rods long and 20 rods wide, what is its area ? Note. — A parallelogram, or ohlong, D c has its opposite sides equal and parallel, but the adjacent sides unequal. Thus, A. B. C. D. is a parallelogram, and also E. F. C. D. and it is easy to see that the con- tents of both are equal. Ans. 1600 rods=10 acres. i 2.>0 MISCRLLANKOUS EXAMPLES. 1 113." \50. Whut is tlie length of un oblong, or parullelogram, whose areii is ten acres, and whose broadth is 12U rods ? Ans. 80 rods. 151. If the area be ten acres, and the lenirth 80 rods, whiit is the other side ? When the length and breadth arc given, how do you fuid the area of an oblong or parallelogram ? When th« area and one side are given, how do you find the other side ? ' 152. If a board be 18 inches wide at one end, and ten inches wide at the other, what is the mean or average width of the board? " Ans. 14 inches. When the greatest and least width are given, how do you find the mean width ? 153. How many square feet in a board 16 feet long, 1*8 fuet wide at one end, and 1*3 at the other? r8-fi'3 Mean width, =1*55 ; and r55X 16=24'8 feet, Ans. 2 154. What is the number of square fee( in a board 20 feet long, 2 feet wide at one end, and running to a point at the other ? * Ans. 20 feet. How do you find the contents of a straight edged board, when one end is wider than the other ? If the length be in feet, and the breadth in feet, in what denomuiation will the product be ? If the length be feet and the breadth inches, what parts of a foot will be the p/oduct ? 155. There is an oblong field, 40 rods long and 20 rods wide ; if a straight line be drawn from one corner to the opposite corner, it will be divided into two equal right- angled triangles ; what is the area of each ? ' Afis. 400 square rodszr.2 acres 2 roods. 156. What is the area of a triangle, of which the base is 30 rods, and the perpendicular 10 rods? Ans. 150 rods. 157. If the area be 150 rods and the base 30 rods, what is the perpendicular ? Ans. 10 rods. 158. If the perpendicular be 10 rods, and the area 150 rods, what is the base ? Ans. 30 rods. When the legs (the base and perpendicular) of a right- angled triangle are given, how do ydu find its area ? f lis/ I 51 113. •nSCELLANEOUS EXAMPLKI. 251 llelogram, rods ? t. 80 rods. I 80 rods, i> you find you find 1, and ten age width 14 inches. DW do you ; long, 1'8 : 16=24*8 1 board 20 a point at '/.5. 20 feet. fed board, t, in what ivhat parts id 20 rods ler to the ual right- s 2 roods. he base is 150 rods. ods, what . 10 rods. area 150 . 30 rods. if a right- a? B When tlie area and one of the legs are given, how do you find the other leg I Note. Any triangle may be divided into two right-angled triangles, by drawing a perpendicular from cue corner to the opposite side, as may be seen by the annexed figure : c Here, A. B.C. is a triangle, divided into two right-angled triangles, A. d C. and d B.C.; therefore, the whole base A. B. multiplied by one half the \perpendicular, d C, will give the area A of the whole. If A. B.«==60 feet, and d C=16 feet, what is the area ? Ans. 480 feet. 159. There is a triangle, each side of which is 10 feet ; what is the length of a perpendicular from one angle to its opposite side 1 and what is the area of the triangle ? Note. It Is plain the perpendicular will divide the oppo" site side into two equal parts. Ans. Perpendicular, 8*66-|-feet; area, ^3'3-f-feet. 100. What is the solid contents of a cube measuring six feet on each side ? Ans. 216 feet. When one side of a cube is given, how do you find its solid contents ? When the solid contents of a cube are given, how do you find one side of it ? 161. How many cubic inches in a brick which is 8 inches long, 4 inches wide, and 2 inches thick ? in 2 bricks ? in 10 bricks ? Ans. to the last, 640 cubic inches. 16*1. How many bricks in a cubic foot ? in 40 cubic feet ? in 1000 cubic feet.' Ans. to the last, 27000. 163. How many bricks will it take to build a wall 40 ft. in length, 12 feet high and 2 feet thick ? Ans. 25920. 164. If a wall be 159 bricks,=100 feet in length, and 4 bricks,=16 inches in thickness, how many bricks will lay one course ? 2 courses ? 10 courses ? If the wall be 48'courses,=:8 feet^high, how many bricks will build it? 150X4=600, and 600X48=28800, Ans. 165. The river Po is 1000 feet broad, and 10 feet deep, and it runs at the rate of ^ miles an hour ; in what time will it discharge a cubic mile of water (reckoning 5000 feet to the mile) ijito the sea ? Ans. 26 days, 1 hour. '•• I 252 MISCELLANEOUS EXAMPLES. una 166. If the country which nupplies the river Po with water be 380 miles long, and 12U broad, and the whole land upon the surface of the earth be 62,700,000 square miles, and if the quantity of water discharged by the rivers into the sea be everywhere proportional to the extent of land by .which the rivers are supplied, how many times greater than the Po will the whole amount of the rivers be ? Ans. 1375 times. 167. Upon the same supposition, what quantity of water, altogether, will be discharged by all the rivers into the sea in a year, or 365 days ? Ans. 19272 cubic miles. 168. If the proportion of the sea on the surface of the earth to that of land be as 10^ to 5, and the mean depth of the sea be a quarter of a mile ; how many years would it take, if the ocean were empty, to fill it by the rivers running at the pre- sent rate? Am. 1708 years, 17 days, 12 hours. 169. If p cubic foot of water weighs 1000 oz. avoirdupois, and the w%ight pf mercury be 13^ times greater than water, and the height of the mercury in the barometer (the weight of which is equal to the weight* of a column of air on the same base, extending to the top of the atmosphere) be thirty inches ; what will be the weight of the air upon a square foot ? a squaremile? and what will be the whole weight of the atmosphere, supposing the size of the eiUth as in questions 166 and 168? Ans. 2I09'375 lbs. weight on a square foot. 52734375000 " " " mile. 10249980468750000000 " " of whole atmosphere. 170. Ifa circle be 14 feet in diameter, what is its cir- cumference ? Note. It is found by calculation, that the circumference of a circle measures about 3if times as much as its diameter, or more accurately, in decimals, 3*14159 times. Ans. 44 feet. 171. Ifa wheel measure 4 feet across from side to side, how many feet around it ? Ans. 12^, 172. If the diameter of a circular pond be 147 feet, what is its circumference ? Ans. 462 feet. 173. What is the diameter of a circle whose circumfer- ence is 462 feet ? Ans. 147 feet. una ^113. MISCELLANEOtS EXAMPLES. 353 jr Po with whole land uare miles, rivers into t of land by ;reater than 1375 times, ty of water, into the seu mbic miles, of the earth h of the sea t take, if the r at the pre- s, 1*2 hours. ivoirdupois, than water, (the weight f air on the Ire) be thirty on a square lole weight eiUth as in ire foot, mile, tmosphere. is its cir- cumference ts diameter, ns. 44 feet, ide to side, Ans. 12f f feet, what s. 462 feet. circumfer- s. 147 feet. 4ns. dif. 174. If the distance through the centre of the earth, from side to side, be 7911 miles, how many miles around it ! 7911 X3* 14159=24853 square miles, nearly. An^ 175. What is the area or contents of a circle whose meter is 7 feet, and its circumference 22 feet ? Note. The area of a circle may We found by multiplying half the diameter into half the circumference. ,Ans. 38^ square feet. 176. What is the area of a circle whose circumference is 176 rods ? Ans. 2464 rods. 177. If a circle is drawn within a square, cont^.ning one square rod, what is the area of this circle? Note. The diameter of the circle being one rod, the cir cumference will be 344159, Ans. *7854 of a square rod, nearly. Hence, if we square the diameter of any circle, and mul- tiply the square by '7854, the product will be the area of the circle. 178. What is the area of a circle whose diameter is ten rods ? 102X*7854=78'54. Ans. 78*54 rods. 179. How many square inches of leather will cover a bail 3^ inches in diameter? Note. The area of a globe or ball is 4 times as much ns the area of a circle of the same diameter, and may be found, therefore, by multiplying the whole circumference into tlu^ whole diameter. Ans. 38^ square inches. 180. What is the number of square miles on the surface of the earth, supposing its diameter 7911 miles? 7911 X24853=196,G12,3?r?, Ans. 181. How many solid inches in a ball 7 inches ii diame- ter? Note. The solid contents ol a globe are fov.nd by multi- plying its area by ^ part of its diameter. Ant;. i79| solid inches. 182. What is the number of cubic miles in the earth, supposing its diameter as above ? Ans. 259,223,031,435 miles. 183. What is the capacity, in cubic inches, of a hollow globe 20 inches in diameter, and how much wine will it contain, one gallon being 231 cubic inches ? Ans. 4188*8-j-cubic inches, and 18*13-j-gallons. X 2o4 MISCELLANEOUS EXAMPLES. T 113.1^ 113 •J ; :'» V i ; -I ' 184. There is a round log, 'all the way of a bigness • the areas of the circular ends of it are each 3 square feet; how ^any solid feet does one foot in length of this log contain -^ feet.in length ? 3 feet? 10 feet ? A solid o thii^ form is called a cyKnder. - '■ ■' fifow do you find the solid content of a cylinder, when the area of one end and the length are given ? 1S5, What is the solid content of \ round stick 20 feet long and 7 inches through, that is, the ends being 7 inches in diameter ? Find the area of one end, as before taught, and multiply it by the length. . ' Ans. 5'347-j-cubic it^x. If you multiply square inches by inches in length, whar parts of Q.foot will the product be 1 if squai:^ inches by feet in length, what part ? / 186. A Winchester bushel is 18'5 inches in diameter, . and 8*inches deep ; how many cubic iilches does it contain ? ^/li. 2150'4-f . It is plain, from the above, that the solid content of all bodies, which are of uniform bigness throughout, whatever may be the form of the ends, is found by multiplying the area of one end into its height or length. Solids which decrease gradually from the base till they I ''«'^ p»ir come to a point, are generally called Pyramids. If the base I ^f^t in be a square, it is called a square pyrdkid ; if a triangle, a ^iangular pyramid ; if a circle, a circular pyramid or a cunc. The point at the top' of a pyramid is called the vertex, and a line, drawn from the vertex perpendicular to the hascy called the perpendicular height of the pyramid. The solid cemtent of any pyramid may be found by multi- plying the area of the hast by ^ of the perpendicular height. 187. What is the solid content of a pyramid whose base i.s 4 feet square, and the perpendicular height 9 feet ? ''■^ ' • 42X1 = 48. Ans. 48 feet. There is a cone, whose height'is 27 feet, and whose Note. addinor i tenths, < tors, to 1 to a cyl Now, '7854, ( and that sjolid cor 231, (no wine gal will give In thi eter will Ions, by only mul by '0034 for decii .Ions, mu Hence Multiply multiply I In the the head 2.> in.-j- isi 188; base is 7 feet in diameter ; what is its content? Ans. 346 } feet. 189. There is a cask, whose head diameter is 25 inches, bung diameter 31 inches, and whose length is 36 inches: how many wine gallons does it contain? how many *beer gallons ? . .^ ^ Th fin. 190. mater is inches 191. prop, on pounds aiiced the pro; Note. faet froi inches, b .(i_.-;,. ■ ■; li )igness ; the re feet ; how log contuiii ? A solid of Under, when itick 20 feet ing 7 inches md multiply j-cubic feet. cngtJi, wh;if i^ inches by u diameter, 3 it contain ? s. 2150'4-|-. ntent of all It, whatever ijjlying the ise till they If the base a triangle, a lid or a cunc. vertex, and ;he base, is iid by multi- jular height. whose base feet? Ins. 48 feet. :, and whose 346 j} feet. 5 25 inches, 36 inches ; how many MISCELLANEOUS EXAMPLES. w:if Note. The mean diameter of the cask may be found by adding 2 thirds, or, if the staves be but a little curving, 6 tenths, of the difference between the head and bung diame- ters, to the head diameter. The cask will then be reduced to a cylinder. Novv, if the square of the mean diameter be multiplied by '7854, (ex. 177) tjie product will be the 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 '231, (note to table, wine meas.) will give the content in wine gallons, and, dividedby 282, (note to table, beer meas.) will jjive the content in ale or beer measure. In this process, we see that the square of the mean diam- eter will be multiplied by *7854, and divided, for wine gal- lons, by 231. Hence we may contract the operation by only multiplying by their quotient, •^/;/'-*=:'0034) that it^, by *0034 (or by 34*, pointing off 4 figures from the product for decimals.)' For the same reason we may, for beer gal- ,lons, multiply by ('vW^'O^^S, nearly) '0028, &c. Hence this concise Hvle for guagiKg or measuring caska : Multiply the square of the mean diameter by the length; multiply this product by 34,/i/r wine, or by 28 for beer, and pointing off four decimals, the product will be the con- trnt in gallons and decimals of a gallon. In the above example, the bung diameter, 31 in. — 25 in. the head diameter=5 in. difference, and % of 6=4 inches; *2> in. -{-4 in.=i9 in. mean diameter. Then 292=341, and 841 X36 in.=30276. ( 30276X34=1029384. Ans. 102*9384 wine gals. \ 33726X28==S47728. Ans. 84'272S beer gals. 191). How many wine gallons in a cask whose bung dia- «n3ter is 3t) inches, head diameter 27 inches, and length 45 inches ? Ans. 166'617. 191. There is a lever 10 feet lonp, and i\\e fulcrum, or prop, on which it turns is 2 feet froi.i one end ; how many pounds weight at the end, 2 feet trom the prop, will be bal- anced by a power of 42 pounds at the other end, 8 feet from the prop. Note. In turning around the prop, the end of the lever 8 faet from the prop will evidently pass over a space of eight inches, while the end 2 feet from the prop passes over a Then. r:> 256 MISCELLANEOUS EXAMPLES. ■f 11113, 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 •ver. Hence, in this example, 2 pounds, 8 feet from the prop^ will balance 8 pounds 2 feet from the prop ; therefore »if we divide the distance of the 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 four times as much as the power 42X4=168. Ans. 168 lbs. 192. Supposing the lever as above, what power would it require to raise 1000 pounds 1 Ans. ' <^°=250 lbs 193. If the weight to be raised be 5 times as much as the power to be applied, and the distance of the weight from the prop be 4 feet, how far from the prop roust the power be applied 1 Ans. 20 feet 194. If the greater distance be 40 feet, and the less half of a foot, and the power 175 lbs., what is the weight? Ans. 14000 pounds 195. 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. I14f lbs. and 85f lbs 196. There is a windlass, the wheel of which is 60 inches 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 ? Note. The spaces passed over are evidently as the diam- ofers or the circumferences ; therefore, ®^^'=10, ratio. Ans. 2400 pounds, 197. li 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 .' Ans, 6 inches, Note. This calculation is on the supposition that there is no friction, for which it is usual to add 1^ to the power which is to work the machine. 19*8. There is a screw whose threads are 1 inch asunder, which is turned by a lever 5 feet=60 inches long ; what is the ratio of the weight to the power ? Note. The power applied at the end of the lever will de tf 113. principle in ctly balance I'-es they pass et from the »p ; therefore prop by the lent will al- tower ; f =4, s the power, ns. 168 lbs. wer would it p 0=250 lbs. much as the weight from St the power Ans. 20 feet. the less half tveight ? iOOO pounds. pounds ; the I feet 6 inches Im the hands ir. , and 85i^ lbs. h is 60 inches )pc coils, is 6 axle will be as the diam-\ ), ratio. 3400 pounds. inches, what of the weight ins. 6 inches, I that there is ; power which inch asunder, long ; what is lever will de- !I 113. MISCELL. NK I US EXAMPLES. 257 scribe the circumference of a circle 60X2=120 inches in diameter, while the weight is raised I inch ; therefore, the ratio will be found by dividing the circumference of a circle whose diaYneter is twice the length of the lever, by the dis- tance between the threads of the screw. 120X3|=377| 377f circumference, and =377|, ratio. Ans. 1 199. There is a screw, whose threads are ^ of an inch asunder ; if it be turned by a lever 10 feet long, what weight will be balanced by 120 lbs. power? Am. 30171 fts. 200. 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 ? Ans. 120. 201. A rough stone wis put into a vessel, whose capacity was 14 wine quarts, which was afterwards filled with 2^ quarts of water ; what was the cubic content of the stone ? Ans. 664^ inches. /. •%j X2 258 FORMS OF NOTES AND RECEIPTiT^ / ( t^ Forms of JVotes^ Receiptor Orders and ^r,,v.>ftjit*0»y,; Bills oi Parcels* «.?.;;.' *-r! n.;. i'*-.-'«r"f :;■*■■->■ M^'^W'M" ■™.*''^' NOTES. No. 1. Montreal, Oct. 22, 1849. For value received, 1 promise to pay to Oliver Bountiful, or order, two pounds, ten shillings and sixpence, on demand, V'ith interest. William Trusty. Attest, Timothy Testimony. No. IL ^^ Kingston, Oct. 10, 1849. For value received of A. B. in goods, wares, and mer- chandize, this day sold and delivered, I promise to pay him or bearer, pounds, shillings^ and pence, in ten davs from date, with interest. C D . No. III. By two Persons. Stanstead, Oct. 1, 1849. For value received of , in this day sold and delivered, we jointly and seTeraJly promise to pay him, or order, pounds, shillings and peace in days irom date, with interest. B C . I>- E- I' : ti ' «.'. RECEIPTS. , Montreal, Oct. 20, 1849. Received from Mr. Durance Adley, ten pounds, in ful[ of all accounts. OrvaNd Constancy. Receipt for Money received on a Note, York, Nov. 1, 1849. Received of Mr. Simon Eastly (by the hand of Mr. Titus Trusty) sixteen pounds, ten shillings and sixpence, which is endorsed on his note of June 3, 1831. Samson Snow. '^i;;,*;'- " ■■■■■ ■■^■'?^• j ORDCIIS AND BILLS OF PARCEL?. , 259 crs and 22, 1849. r Bountiful, on demand. Trusty. 10, 1849. s, and mer- e to pay him — pence, in - D . 1, 1849. lay sold and pay him, or nice in C . E . 20, 1849. ands, in fuH ■NSTAN'CY. e, . 1, 1849. )f Mr. Titus encc, which )x Snow. « 'ks' Receipt for Money received on Account. ' ' > Stanstead, June 2, 1849; Received of Thomas Dubois, twenty pounds, on account, Orlando Prompt. Receipt for Money received for another Person. Sherbfooke, June 4, 1849. Received from P. D. twenty-five pounds for account of J. T. Eli Trueman. Receipt for Interest due on a Note. s Quebec Dec. 18, 1849. 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 Gray. •Receipt for Money paid before it becomes Due. Prescot, May 3, 1849. Received of T. Z. fifteen pounds, advanced in full for one year's rent of my farm, leased to the saiJ T. Z. ending the first day of April next, 1850. John Honorus. ORDERS. Belville, Nov 3, 1848. 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 Mann. Montreal, Sept. 1, 1848. Mr. Timothy Titus. Please to deliver to Mr. L. D. such goods as he may call Tor, to the amount of seven pounds,, and place the same to the account of your obedient servant, NicANOR Linus. BILLS OF PARCELS, It is usual, when goods are sold, for the seller to deliver to the bjyer, with the goods, a bill of the articles, and their "y: ' ■J 1260 BILLS OF PARCELH. ^i^^MV prices, with the amount cast up. Such bills arc sometimes called Bills of Parcels. ^ Montreal, 6th May, 1849. Mr. Abel Atlas, Bought of Benjamin Buck, £ . s. d. 12:i yards tii^ured Satin, at I2s. 6d. per yard, 7 10 il 8" " Sprigged Tabby, at (Js. 3d. " t> 10 Received Payment, jeio y « Benj. Buck. 'tr-- Montreal, 14th May, 1849. Mr. John Burton, 'if , ' Bought of Geo. Williams, !5 hhds. new Rum, IIS gallons each, at Is. 6d. per gallon, 2 pipes French B.andy, 126 & 132 gal. 5s. 7d. 1 hhd. brown Sugar, 9f cwt. at £2 lis. 9d. per cwt, 3 casks Rice, 269n> each, at 3d. *' tb. 5 bags Coffee, 751b each, at Is. 2d. " " 1 chest hyson Tea, S61b, at 43. 8d. " " Received Payment, For George Williams, : ] Thomas Rousseau. Ol Wilderness, 8th Feb. 1849. Mr. Simon Johnson, Bought of Asa Fullum, r)(>S2 feet Boards, at ^1 lOs. per M. . ' 2 Is. Hd; " 3 3s: 2d. " / 1 Os. Od. " 1 lOs. Od. " 12s. 6d. " 13s 9d. " 2000 830 " ^Stuff, 15!>0 " I.ithin;:. 6;'>0 " Plank,' 879 " Timber, 23G " OJ 00 CO c: »p 00 ^ Cs • £iS 8?. 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