THE WESTERN PtCTICIL A RITHMETIC, WI EREIN THE RULES ARE ILLUSTRATED, AND THEIR PRINCIPLES EX PLAINED: A GRET VARIETY OF EXERCISES,'PARTICULARLY ADAPTED TO T-IE CURRENCY OF THE UNITED STATES. WITH AN APPENDIX: CONTAINING THE CANCELING SYSTEM, ABBREVIATIONS IN MIULTIPLICATION, MENSURATION, AND THE ROOTS. DESIGNED FOR THE USE OF SCHOOLS AND PRIVATE STUDENTS. CORIPILED BY JOHN L. TALBOTT. CINCINNATI: PUBLISHED BY E. MORGAN & CO, Stereotyped by J A. James. 1851, Price 371 Cents. -------- -.-. District of Oho, to wit: District Clerk's Office. BE IT irMErMBElRER, that on tile seventeenth day of January, Anl no Domini eighteen hundred and forty-five, E. MORGAN & Co., of the said District, have deposited in this office the title of a book, tile title of which is in the words following, to wit: "The Western Practical Arithmetic, wherein the rules are iilustrated, and their principles explained, containing a great variety of exercises, particularly adapted to the currency of the United States: with an Appendix, containing the Canceling System, Abbreviations in Multiplication, Mensuration and the roots; designed for the use of schools and private students; compiled by JOHu L. TALBOTT;"the right whereof they claim as proprietors, in conformity with an act of Congress,entitled" An act to amend the several acts respecting copyrights". WM. MINER, Clerk of District. PREFACE. I-, presenting tihis work to the public, the author makes no pretensions to having discovered any new spring by which to put the youthful mind into action, nor any new method of communicating a knowledge of Arithmetic. He has founded his work on the belief that labor and labor ozly, can insure success in any pursuit; and that labor should always be bestowed lupon those objects which proi duce the greatest useful result. In the selection and arrangement of matter, therefore, those rules that are of the most general use, have been presented first, and their exercises made extensive, that the pupil many early become familiar with their principles, and expert in their application. The explanations accompanying the rules, are designed to facilitate the progress of private students, and to diminish the labor of teache-rs, especially in large schools, where they are unable to give to each pupil the necessary explanations. The MEi-SURtATIP- of Caipenzers', Mason', Plasteeers' and Pavers' work, ze., will be found an acceptable part of Arithmetic, to every man of business, and a practical knowledge of it will contribute much to the securitv and satisfaction of both workmen and employers, in estimating amounts of work. This has been introduced in consequence of numerous applications to the author to measure various kinds of work, and for instruction in particular rules of Mensuration. The system of Boock Keeping, is thought to be sufficient for all the purposes of jrmErs', unechazSics and retailers, in that necessary branch of a business education. How far the author has succeeded in his attempts to compile a zsefull worle, particularly adapted to the circumstances of the WTestern People, remains for themz to judge, and for experience to determine. NOTICE. T II-l f\avorable reception of this treatise and the increasing demand for it, have induced the publishers to revise, enlarge, and otherwise improve the work. Such alterations and amendments have been made as the experience of the author and of other intelligent and successful teachers has suggested; it is therefore presumed, that the work will be found more useful, and consequently more acceptable than heretofore. Numerous testimonials to the merits of the work, have been received; but its general adoption without any efforts to force its introduction, and its intrinsic worth, are our main reliance; we have therefore given it a thorough revision, and now submit the result of our labors to a discerning public. February, 1841. F MORGAN & Co. '1 CG;CONTENT S''4~~~~~~~~ ~~~PAGE. F Numeratin,...~...e.......................o.. 5 Simple Addition,-...-........................... 12 Simple Subtraction,"*.*"* *0......o.........~..~. ~ 15 Simple Multiplication,...........'.'*.. — 18 Simple Division, -...........2....3...ro o...... C 23 Addition of Federal Money-,...... 30 Subtraction of Federal Money,........................ 32 Multiplication of Federal Money"..-.'o. ~~.....- 34 1 Division of Federal Money,.- " -......"."".........* * 36 Reduction,.~...............................- * 38 Compound Addition,................... 0.. 60 Compound Subtracticn,....................... 66 Compound Multiplication, 7................................ 72 Coinpound Division,..................S....... 0 g Simple Proportion,' -...-......".. -....o......... 88 Compound Proportion, ~.'...............1O.OO. i00 Interest,..:.,................ 112 Compound Interest,"" "....-................. 118 Insurance, Commission and Broka8.ge,'..........'"""" 121 Discournt, *# ~&~X*~e-e.. ~ o~.................... 122 Equation of Payments,...............;..... —... 1 24 Barter, - "....."...,.....". 126 Loss and Gain,-' -.... ~........................ 128 Fellowship,- ". ".".....~...............~- "".... 132 Vulgar Fractions,.......................... 135 Reduction of Vulgar Fractions.... 1....3.6............o Addition of Vulgar Fractions,............ ~ 144 Subtraction of Vulgar Fractions,'".... "".........."'"" 146 Multiplication of Vulgar Fractions,.'.. —'.~ 147 Division of Vulgar Fractions,.............. 148 Decimal Fractions,................. 149 Addition of Decnimals,'.................. 150 Subtraction of Decimals,..*"............ ib. Multiplication of Decimals,..........."- "- 151 Division of Decimals,.....l..................... C ib. Reduction of Decimals, ~ 1........................... 153 Proportion in Decimals:............................... 155 Compound Proportion in Decimals, " """....... 156 Mensuration,...... ~.... ib. Involution,....1.......................... 167 Evolution,................'. 1... *. 169 Square Root,.-...............l.. ib. Cube Root,............... 174 Roots of All Powers, 17................. 179 Arithmetical Progression......................... 180 Geometrical Progression.................. 184 Appendix................................ 194 Exchange......................, 89 P romiscuous Exercise........................:9 ___________~ ~ ~19 ARITHMETICo. I I I ARITIMiETIC is that part of MATHEMATICS which i I treats of numbers. It is both a science and an art; — l the science explains the nature of numbers, and the principles upon whigh the rules are founded, while the I art relates merely to the application of the various rules, All the operaations of aritlhmetic are condLict(ed by means of FIvn fundamental rules, viz., Punmerastio2,? (which includes Notation,) aldd'itioz, Sibdtrtction, I Mtulip1ication, and Division. NUMERATION AND NOTATION. I 1Numernation is the art of representinmg figures or hnumbers by words; Notation is the art of representing numbes by characters called figures. Al numbers are represented by the following cIharac trs, which are called fi7gires or diits,. 0, 1,, 2, 4, 5, 6 7, 8,. nought, one, two, thlree, four, five, six, seven, eight, nine. The one is often called a uenit, it signifies a whole th:ig of a lkin d; two signifies two unrits or ones; three s nifies Li ree un1 ts or ones, &c. The' value which the figures have when standing alone is called their simjple valiue; but in order to denote numbers hirgher than it is necessary to give them ano- ther value called a loccal value, wheIli depen s entirely on the order or place in which they stand. IThus, w n hen j we wish to write tie number ten2 in figures, wTe do it ~ by combining tle characters already lknown, placing a l I on the left hand of the 0, thus, 10, vewhicl is read zicn. I'his 10 expresses ten of the units denoted by I, but I as it is only a sigl'e ten it is callied a zmit, and the I being wri tten in i1 e second order or second place Ui I eom the right hand to express it, it is called a unit of I the second order, the first place being called Lte Jrplce t:1'. - -^.. ^...5r..>*az5. —- - -_,~._.-^``._ L.^' w_: 5.^ 1 h6 UX;MERATION AND NOTATIONo of o uents, and the second9 the place of tens; ten'units T of the first order making one uznit of the second order, IW'hen units simply are named, units of the first order are always meant, when units of any other order are intended, the name of the order is always added I Two tens or twenty, are written 200 Three tens or thirty, 66 6 30. I our tens or forty, 66 40 IFive tens or fifty, 66 50 i' I Six tens or sixty, y 6 60o Sev en tens or seventy, 6' 10. Eight tens or eighty, 66 80o i I Jine tens or ninety, 6S 6' 0 I Ten tens or one hundred, 6 66 Oo I0 The numbers between 10 and 20, between 20 al nd 30, bi twepn 30 and 40, &c. may easily be expressed by considering the tens and units of which they are conlmposed. Thus, eleve, being combposed of one ten l and one unit, is eaxpIressed thus, II ventyI-three being! composed of t vwo tens, and thrlee units, is expressed i tltus 23' &Co S Sixteen being I ten and 6 units, is written thlus, I. TI hir lt-nine being 3 tens and 9 units, is written 39. x Si y- iour being 6 tens and 4, units, is written 64.0 jl Ninetiy-five beileng 9 tenls ad 5 units, is written 95 i Ten tens or oni he,1 cNr'ed forms a unit of the third li order; it is expriessed biy placing a I in the third plla e, -,I and filling the first and second places with cyphe 4, iths, 100. h tlundred is ex Spressed tlhu's 200 Three hnm dred tb-us, 300, &c,. i:ith thle o ide1rs of lnits, tens, and hlindr, 1l te0 aj numbers letwreen 0one and o0ne thounsand imay be readily ei epr essed. or example, in the number four hundred tt ai ailnd tentyi-sevRen, there are 4 h.undireds, 2e teis, aind 7 Ilunits. t: at 4i, 4 units of Ithe tihiri orde r, 2 ulits of the' t econd order, and 7 units of the firs 1t oder. I ence thle n.iumber is writtenl tlhu, 4 2 I iIn 1he anumber three iu1dred a-l d fivel t e tlre are s Ir ihuniidreds no tIes, nd 5 unLits or 3 u 1it of th.e 1thlir(-., NUMERATION AND NOTATION. 7 none of the second, and five of the first order, hence the number is written thus, n 7 3 0 5 Ten units of the order of hundreds, that is ten hundreds form a unit of the fourth order, called thousands, written thus, 1000. In the same manner ten units of the fourth order form a unit of the fifth order, called tens of thousands. The following may be regarded as the principles of Notation and Numeration. 1st. Ten units of thefirst or lowest order, make one i unit of the second olrder; ten units of the second order, make one unit of the third order, and universally ten units of any order nake a unit of the next higher order. 2d. ill numbers are expressed by the nine digits, and the cypher, and this is effected by giving to the sanme figure different values according to the place it occupies. Thus, 4 in the first place is 4 units, in the second place 4 tens or forty, and so on. This tenfold increase of value by changing the place of the same figure is usually expressed by saying that figures increase from right to left in a tenfold proportion. The names of the orders are to be learned from the NUMERATION TABLE.: U _''- ac: U2'-a S O ~ 02, ~m g 9 8 7 6 5 4 3 2 l The orders are likewise divided into periods of 0si0: [ places each, according to the following table. iTJ 8 NITUM ERATION AND NOTATION. of Billions. of'Jillions. of Units. 02 V TO he perods suceed those n th e table, are ions, O ctillions, and oillions, and analoical ames M-o tR.,o 0 a0 0the given numer int o hand, read the fig ures of each period as if they stood alone, and then add the name of the period. Thus, the nuber 04000508245, whe divided The periods succeedin those in the table, are Tis rd lons, Q;uadriltlions, Quintiilions, Sextillions, Septdi- l ons, Octbilions, antd hTonilliones, and analogical names might be formed for the succeeding higher periods. j From the preceding remarks the pupil will readily understand the reason of the Iolowingo rule for numer-e ating or expressing figures by words. RULLE.-Commlence at the right hand, and separate i the given number ilnto periods, then beginning at the left I hland, read the figures of each period as if they stood j alone, and then add tihe name of the period. TIhus, the number 8304000508245, wvheln divided into periods, becomes 8,304000,508245, and is read, Eight, billion, three huncerendand four thouslnd lmil- I lionz, five hzndred andm eight thoziscatnd t'IUO handir edi, and forty-Jive. The name un-it of the right hand period is commonly omitted in reading. EXERCISES IN NUMERATION. I Ex. 1. 35 10. 3700054 19..0031025 2. 204 11. 6130425 20. 68723145 3 3. 513 12. 2701030 21. 901023406 4. 2000 13. 3705423 a 22. 820302008 5, 305 4 14. 6803217 23. 310275603 i 6. 7428 15. 2003005 1 24. 60000501 7 10345 16. 70032004 25. 3000400230024 1{ 8. 40024 17. 62003005 26. 80000102051003 9. 61304 18S. 91010010 27. 50000021375604 1 28. 4000012000040250014 1 29. 1000982000375000482000354000271000032,561804.L & —.s-W,. —-ni}_eD1t'= sr~c3-a:..... S —-z NUMERATION AND NOTATION. 9 From the preceding tables and remarks, the pupil will likewise readily understand the reason of the following rule for notation, or expressing numbers by figures. RtuL: R.-tOMake a sufficient number of cyphers or dots, and divide them into periods, then underneath these dots write each figure in its prodper order and fill the vacant orders with cyphers. iNoT. —~-The object of the dots or cyphers, being to guide the learner at first, after a little practice he may dispense with them. I Ex. I Write down in figures the number twenty millions three hundred and four thousand and forty. Here millions being' the highest period named, we write cyphers to correspond with that, and the period of units, and then underneath these place the significant figures in their proper order, and afterwards fill the vacant orders with cyphers. 000000, 00000 2 3 0404.0 4 The pupil must recollect that cyphers being of no use except to fill vacant orders, are never to be placed I to tie lef et of whole numbers. IEXERCISES IN NOTATION. Express the following numbers in figures. I |E-S PISEXAMPLES. 2. Seventy-five. 3. Ninety. 4. One hundred and five. 5. Three hundred and twenty. 1. Nine hundred and four. 17. Eight hundred and ninety. 01 iTwo thousand three hundred and five. 9. Six thousand and forty. i 10. Seven thousand and four. I I. Eight thousand and ninety-five. i 12. Ten thousand five hundred and fifty-six i13. Forty thousand and forty. 14, Ninety-five thousand two hundred and sixty-seven. i 15. Eighty thousand one hundred and nine. tL....... -.^ ^ A.. 2...... A2 10 NUMERATION AND NOTATION. 16. One hundred and thirty-six thousand two hundred and seventy five. 17. Three hundred and seven thousand and sixty-iour. 1 18. Five hundred thousand and five. 19, One million, two hundred and forty-seven thousand, four hundred and twenty-three. 20. Ten millions, forty thousand and twenty. 21. Sixty millions, seventeen thousand and two. 22. One hundred and four millions two hundred and four thousand and sixty-five. 23. Five hundred and three millions, one hundred and two thousand and nine. 24. Ninety one thousand and two millions, and four. 25. Sixty billions, three millions and forty-one thousand. 26. One billion, one hundred million, one thousand and one. The Roman method of representing numbers, is by means of certain capital letters of the Rloman alphabet. Thus: I one XVII eighteen II two XIX nineteen ilI three XX twenty IV four XXX thirty V five XI, forty VI six L fifty VII seven LX sixty VIII eight LXX seventy IX nine LXXX eighty X ten XC ninety XI eleven C one hundred XII twelve CC two hundred XIII thirteen CCC three hundred XIV fourteen CCCC four hundred XV fifteen D five hundred XVI sixteen M one thousand XVII seventeen MDCCCXXXVIII 1838 NOTE 1. As often as any letter is repeated, so often is its value repeated. NOTE 2. A less character before a greater one, diminishes its value nTTE 3. A less character after a greater one, increases its value. EXPLANATION OF CHARACTERS. 1l QUESTIONS. I WYhat is Arithmetic? When is it a science? When is it an art? What are the fundamental rules of arithmetic? What is numeration? What is notation? What does a unit signify? What does two signify? Three, t &c.? What is meant by the simple value of a unit? What does the local value of a figure depend on? I-ow I do you write the number ten in figures? Why is the one in this case called a unit of the second order? How i many units of the first order does it take to make a unit of the second order? How many units of the second order does it require to form a unit of the third order? &c. Repeat the principles of notation and numeration. Repeat the names of each of the first nine orders as expressed in the numeration table. Repeat the name of each of the periods. Repeat the Rule for numeration. Repeat the Rule for notation. I EXPLANATION OF CHARACTERS.' Signs. Sig ficcations. -= equal; as 20s. ~ 1. - more; as 6 - 2 = 8. less; as 8 -- 2= 6. IX into, with, or multiplied by; as 6 X 2 = 12, b- y (i. e. divided by;) as 6 -2 = 3; or, 2)6(3.:::: proportionality; as 2: 4:: 6 12. or, g Square Root; as J 64 = 8. j/ Cube Root; as / 64 4. I / Fourth Root; as / 16 == 2, &c. l ~.... A vinculum; denoting the several quantities over which it is drawn, to be considered jointly as a simple quantity. tL - -- l12 SIMrPLE ADDITION. SIAIRhLE ADDI.TIO. SmrPLE ADDITION iS the art of collecting several nuinhers, of the same name, into olle sum. RULE. Place tlhe numbers with units lunder sznits, tens under tens, &c. Begin the addition at the units, or righlt hand column, and add together all the figures in that column; | then, if the amount be less than ten, set down the whole!sum: but if greater than ten, see how many tens there are, and set down the number above the even tens, and i carry one for each ten to the next colurn, and proceed with it as in the first. Prioof.-Be gin the addition at the top of each collumn, and proceed as before, and if the result be the same, it is presumed to be right-. I E.XAM1LES. 432 231 214 4 213 413 1221 2 - 1 1 1 2 31 o2 5 213 132 32 1 9 7 9 sum 8 9 7 suni 9 6 8 sumn 1 4 sumrri (5) Here 4, 2, 1, Sad 6 make 15. In fiftee thele 2 7 g 3 6 is one ten and five tunits. Set down the five unitsL under the units columnn, and carry one for the ten 7[ 9 ~ 92 to tthe next or tens column.!:3 8 9 4 I' 6 7 8 3 2 Then 1,4, 3, 4, 9 and 3 make 24: in 24 tllhere are I 5 9 ~ 4 4 two tens, and four over: set down the four undter0 5i 9 2 ~ the colnlmi of tens, and carry tco o the next or l..3.5... hundreds colvumn &c., to the last, where the whole 2 7 3 5 4 5 amount may be set down. 41. i[ — SIM'LPLE ADDITION. 13 (6) (7) (8) 4 7 3 8 6 9 9 7 8 6 7 2 7 5 2 1 2.9492 86937 3 7 2 3 8583 27849 7 8794 892 9 4 49878 23 567 2 8 887 7 2 9 3 7 9 83 72 74 3 92 4 8 7 3 2 1 2 3 4 5 i288034 386119 323653 2 4782 72683 84736 4 7 S 2 3 7 2 -6 8 3 - 4 7 3 6 j1 7 71 4 95 8 2 7 8928 1 27834 82783 27849 2 0 2 5 9 7 6 3 7 8 2 67883 7 689 2 28637 6 27 3 4 4 3987 73 862 (12) (13) (14) 73684 9376 7379 75 723 7463 4 7 3 87 729 6893 9 489 7 48 72 48 3 937 68 96 92 432 -,I ~ APPLICATION. 1. Add 224 dollars, 365 dollars, 427 dollars, and 784 dollars, together. oe $ Dol2l 2 4 D3 6 5 427 1g~~'~~~7 8 4.Answer. 11 8 00 Dollars2~r~ —--------------—,, 0 14 SI4MPLE ADMITIOX. 2. Add 3742 bushels, 493 bushels, 92'7 bushels, 643t bushels, and 953 bushels, together. Answer, 6758 bushels. 3. Add 7346 acres, 9387 acres, 8756 acres' 8394: acres, 32724 acres. Ans. 66607 acres. 4. Henry received at one time 15 apples, at another 115, at another 19. How many did he receive? Ans.. 149. 5. A person raised in one year 724 bushels of corn, in another 3498 bushels, in another 9872. How much in all? Ans. 14094 bushels. 6. A man on a journey, travelled the first day 37 miles, the second 33 miles, the third 40 miles, the fourth 35 miles. How far did he travel in the four days? Ans. 145 miles. 7. A has a flock of sheep containing 34. B has a flock of 47, and C of fifty-four. How many sheep are there in the three flocks? Ans. 135. 8. The distance from Philadelphia to Bristol is 20 miles; from Bristol to Trenton, 10 miles;,from Trenton to Princeton, 12 miles; from Princeton to Brunswick, 18 miles; fromn Brunswick to New York, 30 miles. HIow many miles from Philadelphia to New York? Ans. 90. 9. A person bought of one merchant, 10 barrels of flour, of another 20 barrels, of another 95' barrels. How many barrels did he buy? Ans. 125 barrels. 10. A wine-merchant has in one cask 75 gallons, in another 65, in a third 57, in a fourth 83; in a fifth 74, and in a sixth 67 gallons. How many gallons has he in allf Ans. 421 gallons. Quest ons. How many primary rules of Arithmetic are there? What are they called? What is addition? How do you place numbers to be added? Where do you begin the addition? Why do you carry one for ten, in preference to any other number? Ans. Because it takes ten ones to make one ten, ten tens to make one hundred, &c. (See table, pcge 9.) SIMPLE SUBTRACTION.. 15 SIMPLE $SUBTRACTION. SIMPE SUB'TRACTION is taking a less number from a greater, of the same name, to show the difference between them. The greater number is called the minuend. The less number is called the subtrahend. The dcliference, or what is left, is called the remainder RULE. Place the less number under the greater, with units under units, tens under tens, &c. Then draw a line under them; begin at the right hand or units place, and subtract each figure of the subtrahend from the figure of the minuend that is above it, and set the remainder below. When the figure in the subtrahend is greater than the one above it, borrow one (which is one ten) from the next figure, and add it to the figure of the minuend; then subtract from the Proof.-Add the remainder and the subtrahend together, and if the sum equal the minuend, the work is presumed to be right. EXAMPLES. (1) (2) 7 92527 4 3 Minuend 9 73 8 4 76 3 41 2 3 1 2 Subtrahend 2 14 25 3 45 1 3243 31 Remainder 71 2 4 2 23 (3) Here we cannot take seven from two; then we 7 2 6 3 9 8 2 must borrow one from the 8: that one is one ten; 6/ 4 2 5 r 3 7 then ten and two are twelve; now take seven from: - ~ o' twelve, and five remain. One is borrowed from the 8, leaving only 7; o 8 3 8 3 4 5 then take 3 from 7, and 4 remain: or, suppose 8.to i u.ltiminished; and to cancel the one which is borrowed from I tle 8., add one to the 3 below, making four; then four from eight and I firr'emain, as before, &c. 16 S-IMPLE SUBTRACTION. (4)'(5) 9273847 82703682 2641386 27341237 6632461 55362445 (6) (7) 783728 6 27368 3' 070 3273195 4321725 4564091 269361345 (8) (9) 68427362 593784 283 34613524 5 4 321432 (10) (11): 7 9 283684 9 20 37 842 24653128 413727 1 APPLICA ION i. 1. From 78 take 32 and what will remain? Answer, 46. 2. From 478 take 324. What will remain Ans. 154. 3. Charles had 723 apples, and sold 4:21. IHow man ny has he left? Ans. 302 4. James had 9768 dollars, and gave for a house and I lot 3453 dollars. How-many has he left?. Ans. 631 5, 5. A farmer had 3849 acres of land; he gave to his sons 2135 acres. How many acres has he left fobr himself? Ans. 14i. 6. There are two piles of bricks, one contains T7893 ii and the other 4389. HIow many more are;thre in the 1 oae than in the other? Ans. 3507. i I, __ SIMIPLE SUBTRACTION. 17 7. Bought 100 bags of coffee, weighing 145!0 Ibs., and sold thereof 63 bags weighing 6871 pounds; how many bags, and how many pounds remain unsold? Ans. 37 bags, and 7639 lbs. 8. A manl bought a chaise for 175 dollars, and to pay for it gave a wagon worth 37 dollars, and the rest in money. HIow much money did he pay? Ans. 138 dollars. 9. A mani deposited in bankl 8752 dollars, and drew out at one time 42i34 dollars, at another 1700 dollars, at another 962 dollars, and at another 49 dollars. How Imuch hadhe remaining in bank? Ans. 1807 dollars. 10. A merchant boulht 4875 bushels of wheat, and sold 2976 bushels. H-low any bushels remain in his possession? Ans. 1899. 11. A grocer bought 25 hogsheads of sugar, containing 250 hundred weight, alnd sold 9 hogsheads, containing 3 75 hundred weight. fHo many hogsheads and how many hundred weight had he left? Ans. 16 hogsheads, and 175 hundred weight. 12. A traveller who was 1300 miles from home, travelled homevard 235 miles in one week; in the next 275 miles; in the next 325 miles; and in the next 290 miles. How far had he still to go, before he would reach home? Ans. 175 miles. Questions. What is subtraction? W hat is the greater number called? What is the less number called? I What is the difference called?:How do you place numbers for subtraction? Where do you begin the subtraction? When the lower figure is greater than the upper one how do you proceed? Why: is the one you borrow, one ten. Anis. Because ten ones make one ten; and if I borrow one ten it will make ten ones again, &C. l How do you prove subtraction? ) 18 SIiPLE MULTIPLICATION. SIMPLIE MULTIPLICATION is a short method of performping, particular cases of addition. The number to be multiplied, is the multiplicand. The number to be multiplied by, is the multiplier. The number produced is the product. The multiplicand and multiplier are sometimes called factors. MUrLTIPLICATION TABLE. Twice 3 times 4 times' 5 times 6 times 7 times I make 2 1 make 3 1 make 4 1 make 5 1 make 6 1 make 7 2 4 2 62 8 I2 10 2 12 2 14 3 6 3 9 3 12 3 15 3 18 3 21 4 8 4 12 16 4 20 4 244 28 5 105 15,5 205 25 5 30 5 35 | 16 18 6 6 24 6 30 e 36 6 42 7 14 7 21 7 28 7 35 7 42 7 49 8 16 8 24 8 32 8 40 8 48 8 56 9 18 9 27 9 36 q 45 9 54 9 63 10 20 10 3010 40 10 50 10 6010 70 11 221 11 33 11 44111 55 11 6611 77 &112 2412 36 12 4812 6012 7212 84 8 times 9 times 10 times 11 times 12 times I make 8 1 make 9 1 make' 101 make 11 1 make 12 2 16 2 18 2 20 2 22 2 24 3 24 3 27 3 301 3 33 3 36 4 32 4 36 4 40 4 44 4 48 5 40 5 45 5 50 5 555 60 6 48 6 54 6 60 6 6 6 72 7 56 7 63 7 70 7 77 7 84 8 64 8 72 8 0 8 88 8 96 9 72 9 81 9 90 9 999 108 10 80 10 90 10 100 10 110 10 120 11 88 11 99 11 110 11 121 11 132 12 96 12 108 112 120 12 132 12 144 CASE 1. W hen the multiplier does not exceed 12. RI.tU. R.-Place the multiplier under the units i'iurTorfi he hmultiplicand; and multiply each figure of theo -:i^ I i dlicand in succession, and set down the amouitni3'd-i 1 arry, as in addition. Proof -Multiply the multiplier by the mlztpiFit'^d. i X,^ ~ ~ ~ ~ ~ I -- - I SIMPLE MIULTIP]LICATION. 19 EXAMPLES. 4 2 3 1 Multiplicand 3 42 53 7 3 4 2 2 Multiplier 3 4 8462 Product 1 02759 29368 36563 8375 4378 92-86 5 6 7 8 I 81 2 8 2 5 5 02 5 0 3 0 6 4 6 7 4 2 8 8 4375 7862 3724 7482 3 9 3 5 8 62 0 4 0964 897 4 EXERCISES 1 Mlultiply 44218 by 2 Product 8436 2 7321 by 3 2196 3 87602 by 4 3507wo 4 900078 by 9 8100702 5 ss8870 by 10 82687o00 o 278976 bvy l 3068736 7 5969 by 12 6837228 CASE 2.TlVaen the multiplier exceeds 12. ~RLE. —Place the multiplier as before, with units under units, &c. Then multiply all the figures of the multiplicand by the units figure of the multiplier, setting I down the product as before. i pc,,eed with the tens figure in the same manner, obiise rvg'ii 3 set the product of the first figure in the tens lDiaCe awid with the hundreds figure placing the first w h u~ct in hundreds place, &c., and add the several pro l.^ I?t..-: ether. 20 SIMIPLE 1ILTIPLICATION. XBZtMPLES. 43752 Here we multiply by the 6 or units figure 436 as before: then by the 3 or tens figurre, placing the first product in the second or tens 262512 place, immediately under the three, in the 131256 multiplier. In like manner we use the 4 175008 placing the first product in the third or hundreds place, immediately under the 4; after 19075872 which we add the several products together, and the work is done. 73684 37462 427 563 515788 112386 147368 224772 294736 187310 31463068 21091106 I MA.tiply 4736 by 34 Product 161024 2 5762 by 43 247766 3 6483 by 54 350082 4 7368 by 45 331560 5 4327 — y 56 242312 6 7382 by 67 494594 7 4728 by 76 359328 8 7584 by 87 659308 9 5678 by 78 442884 o1 7683 by 89 683787 11 4962 by 98 486276 I 12' 7384 by 87 642408 113 4376 by 97 4244172b 14- 7923 by 78 617994 1 5 6842 by 89 -:;3S 16 7648 by 523 39 s904 4 17 8473 by 456 338636SS 18 9372 by 567 i 1 SIMIPLE MIULTIPLICATION. 21 NOTE 1. —When either or both of the factors have noughts on the right hand, they may be omitted in the operation, and annexed to the product. Thus: 47 000 734 00 42 00 42 000 94 1468 188 2936 Product 197400000 Product 3082800000 NoTE 2.-When the multiplier is the exact product of any two factors in the multiplication table, the operation may be performed by separating the multiplier into its componelits, and multiplying first by the one, then its product by the other. Thus: 754 by 36 754 754 754 9 3 6 36 6786 2272 4524 4524 4 12 6 2262 27144 27144 27144 27144 1 9 and 4, or 3 and 12, or 6 and 6, multiplied together, produce 360 and by using either pair, according to the above note, th true reesult is obtained. E RCISES. I Multiply 756 by 42 Product 31752 2 645 by 24 15480 3 876 by 48 42048 4 963 by 56 53928 5 827 by 72 59544 6 946'by 81 76626 7 875 by 84 73500 -8 948 by 96 91008 9 795 by 108 85860 The pupil may work these by all the several pairs of comporents that he can find in the multiplier. 22 SIMIPLE MULTIPLICATION. NOTE 3.-~When the multiplier is not the exact pro duct of ally two numbers in the table, use two factors whose product is short of the multiplier, then multiply the sum by the number required to supply the deficiency and add its product to that obtained by the two factors. EXAMPLES. 4 7 3 23 2332 4081 1749 1749 5 3 6 1166 11660 12243 10494 13409 1749 1166 2915 13409 13409 13409 EXERCISES. 1 Multiply 846 by 26 Product 21996 2 784 by 29 22736 3 975 by 34 33150 4 859 by 43 36937 5 794 by 59 46846 PROMISCUOUS EXERCISES. 1. Charles has 24 marbles, and John has 13 times as many; how many has John? Ans. 312. 2. A gentleman owns 17 houses, for each of which he receives 250 dollars rent; how much does he receive for them all? Ans. 4250. 3. A laborer hired himself to a farmer for 11 years, at 150 dollars a year; how much did he receive? Ans. 1650 dollars. 4. A person wishes to purchase 26 shares of Bank stock at 75 dollars a share; what must he pay? Ans. 1950 dollars. 5. A mason having built a house, found that 98470 bricks were ill it; suppose he desires to build 19 such houses, how many bricks must he obtain for the purpose? Ans. 1870930. SIMPLE DIVISION. 23 SIMPLE DIVISION. SIMPLE DIVISION is a short method of performing several subtractions. The number to be divided is called the dividend. The number by which it is to be divided is called the divisor. The number of times that the divisor is contained in the dividend, is called the quotient. So many figures of the dividend as are taken to be divided at one time, is called a dividuaL. If any thing remain when the operation is completed, it is called the remainder. CASE I.-SHORT DIVISION. When the divisor does not exceed 12. RULE.-Place the divisor on the left hand side of the number to be divided. Consider how often the divisor is contained in the first figure or figures of the dividend, and set down the result below; observing how many remain, if any. If there be no remainder, consider how often the divisor is contained in the next figure: but if there be a remainder, call it so many tens, and add the next figure to it, and divide the sum, placing the result beneath, as before. Proof.-Multiply the quotient by the divisor; add the remainder, if any, and the product will equal the dividend. EXAMPLES. Dividend. Divisor 2)182 2)648 3)963 4)484 Quotient 241 324 24 SIMrLE DIVISION. 3)74 851 Here 3 are conta:ned in ______~__ 7 two times, and one re247'28 3- 2 Re a in er mains; plac the the uto uder the,seven, and suppose the o7one that remains to be one _~~.~ l ~ten, and add the next figProof 741851 ure (4) to it, which imakes fburteen. No 3 are contained in fourteen 4 times, and 2 remain. Set the 4 down under the 4 in the dividend, and suppose the tzoo that remain to be two tens, alnd add the next figure (1) to it, which make twentyone. Now 3 ilto 21 go 7 times, and no remainlder. Place the 7 under the I in the dividend, and proceed in the same manner with the other figures. 4)65270167 5)6572686 16317541+-3 Rem.n 1314537+1 R-em 4 5 Proof 65270167 Proof 6572G86 6)87390627 7)4873692 8)9273684 9)8379286 10)946873 11)893726 12)98796 180"i" ~ r~srz n> —~9 SIMPLE DIVISION. 25 EXERCISES. Divide 7893762 by 6 Ans. 1315627 9387984 by 7 1341140 —4 6928437 by 8 866054-s 9276874 by 9 1030763-7 8672934 by 10 867293-4 6873842 by 11 624894 —8 7369287 by 12 614107 —3 CASE 2.-LoNG DIVISION. When the divisor exceeds 12. RLE.-Place the divisor to the left hand of the dividend, as in case 1. Consider how often the divisor is contained in the. least number of figures into which it can be divided; and set down the result at the right hand of the dividend. Multiply the. divisor by the quotient figure thus found, and set the product under the dividual or figures supposed to be divided. Subtract the product from the dividual, and set down what remains. Bring down the next figure of the dividend, and proceed as before, till all the figures are' brought doTwn and divided. EXATPLES. Divisor. Dividend. Quotient. 27)984376(38458 Tlwenty-seven into 98 go 3 81 times: multiply the divisor -- (27) by 3 and set the product 174 under the dividual (98) and 162 subtract.'To the remainder (17) bring down the next fig123 ure (4) of the dividend. Now 108 27 into 174 go 6 times. Place - __ the 6 in the quotient and mul157 tiply (27) the divisor, by 6, 135 and set the product under 174 I ___- and subtract as before, &c. 226 216 -10 RIemainder.?r^irT~~li3~gir pT* 26 SIMPLE DIVISION. Divisor. Dividend. Quotient. 42) 98754 (2351 84 42 Divisor 147 4702 126 9404 -- 12 Remainder 215 - 210 98754 Proof 54 42 12 Remainder 32)789627(24675 65)1827538(28115 64 130 149 527 128 520 216 75 192 65 242 103 224 65 187 388 160 325 27 Rem. 63 Rem. EXEIRCISES. Divide 8'769 by 13 Quo. 674 Rem. 7 476 by 15 31 11 958 by 18 53 4 1475 by 28 52 19 5 t4277 by 31 137 30 25757 by 37 696 5 63125 by 123 513 26 t253622 by 422 601 is Illr —.^i SIMPLE DIVISION. 27 NOTE. —-Cyphers on the right hand of the divisor may be omitted in the operation, observing to separate as many figures from the right of the dividend, which must be annexed to the remainder. EXAMIPLES. 54 1 00)1463 1 40(27 32 0)7617 1 3(238 108 64 383 121 378 96 Rem. 540 257 256 Rem. 13 EXERCISES. Divide 40220 by 1900 Ans. -21 Rem. 320 137000 by 1600 85 1000 99607765 by 27000 3689 4765 2304108 by 5800 397 1508 NOTE 2. —-hen the divisor is the exact product of any two numbers in the multiplication table, the operation may be performed by dividing first by one of the component parts, and then the quotient by the other. To get the true remainder, multiply the last remainder by the first divisor, and add the first remainder. EXA.PLES, 1 98754 42 <. (6 14107-5 first remainder 2351-1 last remainder 7 first divisor 5 first remainder 12 true remainder 28 SI3PLE DIVISION. 27 3 984376 27 ) _ t9 3281255-1 36458-3 X 3 - 1 1 10 Rem ]EXERCISE]S. Divide 9756 by 35 Quotient 278 Rem. 26 8491 by 81 104 67 44767 by 18 2487 1 92017 by 5 1643 9 38751 by 48 807 15 74071 by 72 10195 31 APPLICATION. 1. Seven boys have 161 apples, which they divide equally among them. How many does each have? W Answer, 23. 2. What is the quotient, if 8736 be divided by 8, and that quotient by 4? Ans, 273, 3. If 350 dollars be equally divided among 7 men, what will be the share of each? Ans. 50. 4. How many times are 27 contained in 952?' Ans. 35 times and 7 over. 5. Suppose 2072 trees planted in 14 rows. How rmany trees will there be in each row? Ans. 148. 6. Several boys who went to ca.tiher nuts, collected 4741, of which each boy received 431. sHosw many boys were there? Ans. 11. 7. If the expense of erecting a bridge, which is 15036 dollars, be equally defrayed by 179 persons, what must each pay? Ans. 84 dollars. 8. Suppose a man receive in one year 2920 dollars; how much a day is his income at that rate; and sup- pose that his expenses for the year amount to 1769 dol- I lars. How much will he save in a year? Ans. His income will be 8 dollars a day;he will save 1151 dollars in a year. SIMPLE DIVISION. 29 Questions, What is division? What do you call the number that is to be divided? What do you call the number you divide by? What do you call the number obtained by division? What do you call that which is left when the work is done? When the divisor does not exceed 12, how do you perform the operation? When the divisor exceed 12, how do you proceed? How do you prove division? How may the operation be performed when there are cyphers at the right hand of the divisor? How may itbe performed when the divisor is the exact product of two numbers in the multiplication table? How do you obtain the true remainder in the last case? PROMIISCUOUTS EXERCISES iN THE PRECEDING RULES. 1. If the contents of five bags of dollars, containing $295, $410, $371, $355, and $520, be divided equally among 25 persons, how much is the share of each? Ans. $78.04 2. A man possessed of an estte of $30,000, disposed of it in the following manner: to his brother he gave $1500, and the balance to his 5 sons, to be equally divided among them. What was each one's share? Ans. $5700. 3. What number is it, which being added to 9709, will make 110901? Ans. 101192. 4. Add up twice 327, three times 794, four times 31196, five times 158S0, six times 95280, and once 33304. Ans. 812,344. 5. Three merchants have a stock of 14876 dollars, of which A owns 4963 dollars, B 5188, and C the remainder. How much does C own? Ans. 4725 dolls. il. ~.V. ~30 ~ FEDERAL MONEY. FEDERAL MONEY, OR MONEY OF THE UNITED STATES. TABLE. 10 mills make I cent 10 cents 1 dime 10 dimes I dollar 10 dollars 1 eagle These denominations bear the same relation to each other as those of units, tens, hundreds, &c. Federal money is therefore added, subtracted, multiplied, and divided by the same rules as Simple Addition, Subtraction, Multiplication, and Division. ADDITION OF FEDERAL MONEY. Rule. Place the numbers one under another, with mills on the right, cents, dimes, &c., in succession-; observing to keep mills under mills, cents under cents, &c. Then proceed as in simple addition. When halves or fourths of a cent occur, find their amount in fourths, and consider how many cents these fourths will make, and carry them to the column of cents. EXAMPrLES. Eagles. Dolls. Dies. s. Cents Mills. Dols..Ds. Cts. 3 7 8 9 5 7 8 9 7 9 8 7 9 7 8 2 3 8 7 9 6 8 4 8 9 8 6 6 3 7 4 7 8 9 8 4 8 2 E. 27 3 6 4 5 3 5 7 0 NOTE. —In common business transactions, eagles, dimes, and mills are not used: dollars, cents, and fractions of a- cent, are the only denominations kept in accounts. 1;.. ___ FEDERAL MONEY. 31 EXAMPLES. Ds. ct, Ds. cts. 34, 62 427, 68 56,31 342, 31 27, 82 427, 26 23, 68 793, 84 27, 42. 273, 42 169, 85 2264, 51 EXERCISES. (1) (2) ( Ds. ets. D cs. t. Ds. cts 468 31 927 24 273, 45 723, 62 768, 32 846, 37 845 92 427, 56 283, 75 736, 25 792, 34 846, 91 846, 31 587, 62 674, 75 428, 62 842, 27 273, 25 Ds. cts. One half is two-fourths; and one half more make 437, 62k four fourths, and three fourths more make seven 386, 814 fourths, and one fourth more make eight fourths, and, one half (or two fourths) more make ten fourths. 243, 18} Four fourths make one cent, then ten fourths make 427, 37i two cents, and leave two fourths, or one half cent. 428, [12 Set down the ~ cent, and carry the two cents to the next column. 1923, 12k Ds. cts. Ds. cts. Ds. cts. 274,81U 27, 681 56,064 362,87k 36, 81 32, 12i 421, 181 28, 62i 36, 25 625 314 37,934 42,62 241, 561 24, 62k 54, 81 32 rFEDEIRAL MONEY. A1PPLIC ATION. 1. Add 48 dollars 20 cents;' 14 dollars 58 cents; 100 dollars 25 cents; and 84 dollars 36 cents. Ans. 247 dollars 39 cents. 22. Add 6$7,c2, $34,31,, $72,06j., $41,31j, $25, 68k, and $87,431 together, and tell the amount. Ans. $268,431. 3. Bought a hat for $4,25 cents; a pair of shoes for $2,25; a pair of stockings for $1,25, and a pair of gloves for 75 cents. What is the cost of the whole? Ans. 18 50 cents. 4. If I buy coffee for $1, 18, tea for $2,50, cloves for 87, mlace for 931, cinnamon for $1,871, raisins for 2,68W, nutmegs for 37a, candles for 87k, and wine for 1 1,93, what must I pay for them? Ans. $13,25. Questions. What relation do mills, cents, dimes, &c., bear to each other? How are the addition, subtraction, multiplication, and division of Federal money performed? How do you place the numbers to be added? How do you proceed when halves, fourths, &c., occur? SUBTRACTION OF FEDERAL MONEY. RU-E.-Place the less under the greater, with dollars under dollars, and cents under cents; then, if there are 0 no fractions) proceed as in simple subtraction. If there is a fraction in the upper sum and none in the lower, set it down as a part of the remainder, and proceed as before. If there is a fraction in each sum, and the lower be less than the upper, subtract the lower from the upper, and set down the difference. If the lower fraction be greater than the upper one, borrow one cent, and call it four fourths, and add them to the upper fraction, and subtract the lower one from the sum. Proof —As in simple subtraction. FEDEIIIAL MONEYT 33 EXAMPLES. Ds, cis. Ds. cts. cts.. ci. 32 62 43 684 75 68 1 21 31 21,5 24, 4 12 -l. 1,31 $22, 434 51, 56 NOTE. —Three fourths cannot be taken from two.D, CetS. fourths: then borrow one cent from the two cents, 1 271,62 which has four fourths in it: add tihefour fourths to 93 the two fourths, this makes six fourths; subtract 13I 2 93i three fourths from six fourths, and three fourths (4) remain. Set down the 4 and add one to the next 138g 689 3, as in simple subtraction. EXERCISES. Ds. cts. Ds. cts. Ds. cis. 5 9 49 520, 31: 436, 31 35, i2 210, 2 243, 184 Ds. CD. Cts. D. D. cts. 2763, 62 237, 561 732, 31 124 37E 142, 875 261, 684'I~ JLI'APPLICATION.. Subtract $432,68S from 1000,931, I Ans. $568,25o 2. Subtraction shows the difference between two numbers; what is the difference between $37,62k and $939,87. Ans. 56,25. 3. Bought goods to the amount of $545,95, and paid at the time of purchase $350. How much remains unpaid? Ans. $195,95. 4. A merchant bought a quantity ofcolb'e, for which he paid $560. He afterwards sold it foi $610,87a Hlow much did he gain by the transaction? AIns. $50,87k..2 34 FEDERAL. MONEY. Questions. How do you place the numbers in subtraction of Federal Money? How do you perform the operation? If a fraction occur in the upper line or minuend, what do you do with it? If a fraction occur in each, how do you proceed? Suppose the lower fraction is greater than the upper one, how do you proceed? How do you prove subtraction of Federal Money? MULTIPLICATION OF FEDERAL MONEY. RULE.-Set the multiplier- under the multiplicand, and if there be no fractions, proceed as in simple multiplication; observing to separate the cents from the dollars in the product. If there is a fraction in the sum, multiply it, and see how many cents are in the product; set down the fraction that is over, and proceed as before. Or if the multiplier exceeds 12, multiply the sum, omitting the fractions; then multiply the fraction, and add the number of cents contained in the product, to the product of the rest of the sum. EXAIMPLES. Ds. cts. Ds. cts, Ds. cts. 12S 50 10, 564 23, 62A 4 2 5 $50, 00 $21, 12A $118, 12i Ds. cts. Ds. cts. 10,87& 125 times 4,181 24 times' are 125 one half make 24 72 fourths: four. —-- 125 halves: 2 - fourths are con5435 into 125 go 62 1672 tained 18 times in 2174 times, leaving 836 72 fourths, mak1087 one; that is, 18 ing 18 cents. 62A one half, mak-— ing 62i cents. $100,50 $1359,37A FEDERAL MONEY. 35. XERCCISES. 1 Multiply -$145,184 by 7 Ans. $1016,31i 2 7,87i by 47 370,12i 3 28,684 by 68 1950,75 4 42,314 by 58 2454,12k 5 137,62:k by 67 9220,87k ] 6 79,00k by 207 16354,03i 1':39~,6i APPLICATION. 1. What-will 8 pounds of cheese come to, at 18 cents a pound? Ans. $1 44 cts. 2. What is the value of 12 yards of linen, at 35 cents [a yard? Ans. $4 20 cts. 3. What cost 29 yards of cloth at $2 25 cts. a yard? Ans. $65 25 cts. 4. What will 213 barrels of flour cost, at $5 25 cents a barrel? Ans. $1118 25 cts. 5. Bought 321 barrels of cider at $1 25 cts. a barrel. What did it amount to? Ans. $401 25 cts. 6. What will 580 bushels of salt cost at $1 12i cts. a bushel. Ans. $652 50 cts. 7. What-is the value of 2 pieces of cloth, one containing 38 yards, and the other 26 yards, at $3 87k cts. a yard? Ans. $248. 8. What will be the cost of 132 pieces of linen at $17 37k cts. each? Ans. $2293 50 cts. 9. What will 8 cords of wood amount to, at 4 dollars 50 cents a cord? Ans. 36 dollars. 10. Sold 213 barrels of flour for 6 dollars 25 cents per barrel. Whaat is the amount? Ans. 1331 dols. 25 cts. 11. Bought 308 pounds of coffee at 21 cents a pound. Wh^t is the amount? Ans. 64 dols. 68 cts. 12. Bought 217 gallons of brandy at $1 184 cts. per gallon; and sold it for $1 37k cts. per gallon. What was the amount paid for the whole; the sum it sold for; and the gain'? Ans. Prime cost, $257 684: sold for $298 37k; gain, $40,681 36 IFEDERAL MONEY DIVISION. RULE.-Divide as in simple division. When a remainder occurs, multiply it by 4; and add the number of fourths that are in the fraction of the sum (if any) to its product: divide this product by the divisor, and its quotient will be fourths, which annex to the quotient. Proof.-As in simple division. EXAMPLES. Ds. fits. Ds. cts. Ds. cts. 2)45,22 3)63,184' 2)25,37i 22,61 21,061 12,681 Ds. cts. 25)629,684(25,1184 32)78800(24,62* 50 64 129 1481 125 128 46 200 25- 192 218- 80 200- 64 18 Here 18 cents rerriain; multiply 16 4 18 cents by four, brings them to 4 fourths of a- cent; add the 4, this makes 75 fourths: divide 75 fourths 75 by 25,. and i are obtained, which 32)64(2 or 25)75(3 place in the qiotielit: 64 EXERCISES. D. cts. Divide 56,15 by 10' Quotient $ 5,61^ 96,00 by 5 19,20 156,00 by 4 39,00 - 58,14 by 38 1,53 417,96 by 129 3,24 494,45 by 341 - 1,45, - 627,38 by 508 - 1,23t FEDERAL MIONEY. 37 APPLICATION. 1. If 7 pounds of-butter cost $189 cts., what is the value of 1 pound? Ans. 27 cts. 2. If 8 lbs. of coffee cost $2,04 cts., what is the price of one pound? Ans. 25& cts. 3. Bought 29 yds. of fine linen for $65,25 cts., what was the price per yard? Ans. $2,25. 4. Paid $58,75 cts. for 235 yds. of muslin, what was it per yard? Ans. 25 cts. 5. A piece of cloth containing 72 yds. cost $450, what was it per yard? Ans. $6,25. Questions. How do you perform division of Federal Money? How do you proceed when a remainder occurs? PROMISCUOUS EXERCISES IN THE PRECEDING RULES. 1. Bought 18 barrels of potatoes, each containing 3 bushels, at 25 cts. a bushel, what did they cost? Ans. $13,50. 2. A farmer sold 30 bushels of rye at 87 cts. a bushel, 30 bushels of corn at 53 cts. a bushel; 8 bushel of beans at $1,25 cts. a bushel; 2 yoke of oxen at $62 a yoke; 10 calves at?4 a piece; 15 barrels of cider at $2,37A a barrel, what was the amount of the whole? Ans. $251,62-. 3. What will be the price of four bales of goods, each bale containing 60 pieces, and each piece 49 yards, at 37a cents a yard? Ans. $4410. 4. Add $324,431 cts. $208,092 cts. and $507,90. cts. together, and divide the sum by 2, and what will be the result? Ans. $520,214. 5. Divide 400 dollars, equally, among 20 persons. WIhat will be the portion of each person? Ans. $20. 6. Divide 1728 dollars, equally among 12 persons. What does each one of them share? Ans. $144. 7. If 240 bushels cost 420 dollars; what is the cost of one bushel at the same rate? Ans, $1.75. 4 38 R:EDUCTION. REDUCTION. REDUCTION is the changing of a sum, or quantity, from one denomination to another) without altering the value. CASE 1. To reduce a sum, or quantity, to a lower denomination than its own. RULE. —Multiply the sum, or quantity, by that number of the lower denomination which makes one of its own. If there are one or more denominations between the denomination of the given sum, and that to which it is to be changed, first change it to the next lower than its own; then to the next lower) ahd. so on to the denomination required. DRY MEASURE. TABLE. 2 pints (pts.) make 1 quart, qt. 8 quarts - 1 peck, pe. 4 pecks - 1 bushel, bu. NOTE.-This measure is used for measuring grain, salt, fruit, &c. EXAMPLES. NOTE. —1. To reduce bushels to pecks, multiply by 4, because each bushel has 4 pecks in it. 1. Reduce 23 bushels to pecks. bu. 23 4 Amt. 92 pecks. 2. Reduce 35 bushels to pecks. Amt. 140 pecks. NOTE.-2. To reduce pecks to quarts, multiply by 8, because each peck has 8 quarts in it. i. — REDUCTION. 89 3. Reduce 27 pecks to quarts. Pe. 21 8 Amt. 216 quarts. 4. Reduce 43 pecks to quarts. Amt. 344 quarts. NOTE. —3. To reduce quarts to pints multiply by 2) because each quart has 2 pints in it. 5. Reduce 43 quarts to pints. qt. 43 2 Amt. 86 pints. 6. Reduce 32 quarts to pints. Amt. 64 pints. Reduce 34 bushels to pints. Xbu. 34 4 Multiply the busheJls by 4 to bring them to pecks. 136 8 Multiply the pecks by 8 to bring them to quarts. 1088 2 And multiply the quarts by 2 to bring them to pints. Amt. 2176 pints. EXERCISES. 7. Reduce 56 pecks to pints. Amt. 896 pints. 8. Reduce 47 bushels to quarts. Amt. 41504 qt. 9. Reduce 85 bushels to pints. Amt. 5440 pt. 10. Reduce 63 pecks to quarts. Amt. 504 qt. 11. Reduce 132 bushels to quarts. Amt. 4224 qt. 12. Reduce 234 bushels to pints. Amt. 14976 pt. NOTE. —4. When several denominations occur, reduce the highest denomination to the next lower one, and this again to the next lower, and so on; observing to add!the amount of each denomination, the number there is of that denomination in the given sum. 440 REDUCTION. EXAMPLES. 1. Reduce 23 bushels, 3 pecks, 5 quarts, 1 pint, to pints. bu. pe. qt. pt. 23 -3-5- 1 4 Multiply the bushels by 4 to bring them to pecks, and 92 add the 3 pecks to the amount, 3 which makes 95 pecks. 95 8 Multiply the pecks by 8 to bring them to quarts, and add 760 the 5 quarts, which makes 5 765 quarts. 765 2 Multiply the quarts by 2 to bring them to pints, and 1530 add the 1 pint which makes 1 1531 pints. 1531 amt. Or thus: bu. pe. qt. pt. 23 3-5-1 4 Multiply by 4 as above; add the 3, and set down the 95 amount, &c. 8 765 2 1531:Amt. as before. EXERCISES. 1. Reduce 13 bushels, 2 pecks, 7 quarts, 1 pint to pints. Amt. 879 pints. REDUCTION. 41 2. Reduce 24 bushels, 3 pecks, I quart to quarts. Amt. 793 qt. 3. Reduce 7 bushels, 3 pecks to quarts. Amt. 248 qt. 4. Reduce 3 pecks, 2 quarts to pints. Amt. 52 pt. 5. Reduce 7 quarts, 1 pint, to pints. Amt. 15 pt. 6. Reduce 32 bushels, 0 pecks, 1 quart. to pints. Amt. 2050. 7. Reduce 5 bushels, 1 peck, 0 quarts, I pint to pints. Amt. 337 pt. 8. Reduce 43 bushels, I peck to pints. Amt. 2768 pt. Questions. What is reduction? For what is case first used? How do you reduce a sum to a lower denomination than its own? How do you reduce bushels to pecks? Why do you multiply by 4? How do you reduce pecks to quarts? Why do you multiply by 8? How do you reduce quarts to pints? How do you reduce bushels to pints? AVOIRDUPOIS WEIGHT. TABLE. 16 drams (dr.) make 1 ounce, oz. 16 ounces - 1 pound, lb. 28 pounds - 1 quarter of a cwt. qr. 4 quarters, (or 112 lb.)* I hundred weight, cwt 20 hundred weight I ton, T. NOTE.-By this weight are weighed, tea, sugar, coffee, flour and other things subject to waste, and all the metals, except silver and gold. * The gross hundred weight of 112 pounds is nearly out of use; the decimal hundred weight of 100 pounds is taking its place. 4* 142 REDUCTION. EXAMPLES. 1. Reduce 23 tons to hundred weight. tons.'al~~~i ~23 20 Amt. 460 cwt. 2. Reduce 34 hundred weight to quarters. cwt. 34 4 Amt. 136 quarters. 3. Reduce 42 quarters to pounds. qrs. 42 28 336 84 Amt. 1176 pounds. 4. Reduce 73 pounds to ounces. lbs. 73 16 438 73 Amt. 1168 ounces. 5. Reduce 54 ounces to drams. oz. 54 16 324 54 Amt. 864 drams. REDUCTION. 43 6. Reduce 35 tons to drams. tons. 35 20 700 cwt. 4 2800 qr. 28 22400 5600 78400 lb. 16 470400 78400 1254400 oz. 16 7526400 1254-00 Amount. 20070400 drams. EXERCISES. 7 Reduce 24 pounds to drams. Amt. 6144 dr., 8. Reduce 36 hundred weight to pounds. Amt. 4032 lb. 9. Reduce 73 quarters to ounces. Amt. 32704 oz. 10. Reduce 2 tons to pounds. Amt. 4480 lb. 1 Reduce 4 tons to drams. Amt. 2293760 dr. 44 REDUCTION. 12. Reduce 3 tons, 13 cwt., 2 qu., 14 lbs., to pounds. T. cwt. qr. lb. 3 - 13 - 2 - 14 20 60 Or thus; 13 T. cwt. qr. lb. --- 3 - 13 - 2 - 14 73 20 4 - 73 292 4 2 _ ___ 294 294 28 28 ~ - 2366 2352 588 588 8246 pounds 8232 14 8246 pounds. 13. Reduce 2 tons. 15 cwt. 2 qr. to quarters. Amt. 222 qr. 14. Reduce 3 tons. 25 lb. to pounds. Amt. 6745 lb. 15. Reduce 5 cwt. 3 qr. 14 lb. to ounces. Amt. 10528 oz. 16, Reduce 2 cwt. 2 qr. 14 ounces to drams. Amt. 71,904 dr. TROY WEIGHT. TABLE. 24 grains (gr.) make 1 pennyweight, dwt. 20 pennyweights - ounce, oz. 12 ounces - 1 pound, lb. NOTE.-By this weight, jewels, gold, silver, and liquors, are weighed. REDUCTION. 45 E3XAMLPLES. 1. Reduce 32 pounds to ounces. lb. 32 12 Amt. 384 ounces. 2. Reduce 23 ounces-to pennyweights. oz. 23 20 Amt. 460 dwt. 3. Reduce 43 pennyweights to grains. dwt. 43 24 172 86 Amt 1032 grains. 4. Reduce 53 pounds to grams. lbs. 53 12 636'20 12720 50880 25440 Amt. 305280 grains. EXERCISES. 1. Reduce 24 ounces to grains. Amt. 11520 gr. 2. Reduce 32 pounds to pennyweights. Amt. 7680 dwt. 3. Reduce 132 pounds to ounces. Amt. 1584 oz. 4. Reduce 234 ounces to grains. Amt. 112320 gr. A46 REDUCTION. 5. Reduce 463 pounds to grains. Amt. 2666880 gr. 6. Reduce 47 pounds, 10 ounces,,15 pennyweights to pennyweights. Amt. 11495 dwt. 7. Reduce 5 pounds, 6 ounces, 4 pennyweights, 20 grains to grains. Amt. 31796 gr. APOTIHECARIES. WEIGHT. TABLE.. 20 grains (gr.) make I scruplej. sc. } 3 scruples - I dram, dr. 3.8 drans - 1 ounce, oz. 12 ounces - l pound, lb. NOTE.-By this weight apothecaries mix their medicines, but they buy and sell by Avoirdupois Weight. EXERCISES. 1. Reduce 32 pounds to ounces. Amt. 384 oz. 2. Reduce- 43 ounces to drams. Amt. 344 dr. 3. Reduce 27 drams to scruples. Amt.. 81 se. 4. Reduce 37 scruples to grains. Amt. 740 gr. 5. Reduce 28 pounds to drams. Amt. 2688 dr. 6. Reduce 36 ounces to scruples. Amt. 864 sc. 7. Reduce 27 drams to grains. Amt. 1620 gr. 8. Reduce 23 pounds to grains. Amt. 132480 gr. 9. Reduce 3 pounds, 5 ounces, 2 scruples to scruples. Amt.. 986. sc. 10. Reduce 7 ounces, 5 drams,. 14 grains to grains. Amt. 3674 gr. 11. Reduce 27 pounds, 7 ounces,- 2 drams,. 1 scruple, 2 grains, to grains. Amt. 159022 gr. CLOTH MEASURE.I TABLE. 4 nails (na.) make 1 quarter of a yard,. qr. 4 quarters - 1 yard, yd, 3 quarters - Ell Flemish, E. F1. 5 quarters - I Ell English, E. E. 6 quarters - I Ell French, E. Fr. NOTE.-~By this measure cloth, tapes, linen, muslin, &c., are measured. R EDUCTIO.o 47 I BEXSCISS. 2! 1. Reduce 24 yards to quarters. Amt. 96 qr 2o. Reduce 32 quarters to nails. Amt-. 128 na. 3. Reduce 27 yaLrds to nails. Amt. 432 na. R 4 ie duce 46 Flemis els ls to quart-ers. i At. 138 ir3 qr 5. Reduce 27 English ells to quarters. I Yf~ll~l~~rtiAmnt. 135 qr. I 6, Reduce 34 French ells to quarters, Amt. 204 qro 7o Reduce 45'Flemish ells to nails. Amt. 540 na.. Reduce 38 English ells to nails. Amt. 720 na. 90 Reduce 54 French ells to nails. Anmt. 1296 na. I10o Reduce 13 yards, 3 quarters to quarters. Amt. 55 qro i 11I lReduce 3 quarters, 2 nails, to nails. Aint. 14 na. 12., Reduce 24 yards, 2 nails to nails. tAmt. 388 na. 13. Reduce 13 E. ells, 2 qrs., 3 nails to nails. Amt. 271 nao 1 LONG IF tSURZ. TABLE. i 12 inches (in.) make I fot, ti. 3 feet 1i yarld, yd. 5, yards 1I RodPoleorPe.rch po 40 poles 1 Furlrong. 8 Furlong 1 Mile, 3 Miles 1I League. 60 Geographic or c I 69e Statute Msles degree.I Norr.E.-This mseasure s - used,for length and dis - tances, A Hand is a measure of four inches, and is used in measuring the height of horses. A Fathom is 6 feet, and is chieflyused in measuring the depth of water. 48 REDUCTION. i EXERCISES. 1. Reduce 23 leagues to miles. Amt. 69 m. 2. Reduce 43 miles to furlongs. Amt. 344 f. 3. Reduce 27 furlongs to poles. Amt. 1080 p. 4. Reduce 56 poles to yards. Amt. 308 yd. 5. Reduce 132 yards to feet. Amt. 396 ft. 6. Reduce 76 feet to inches. Amt. 912 in 7. Reduce 24 miles to poles. Amt. 7680 p. 8. Reduce 32 furlongs to yards. Amt. 7040 yd. 9. Reduce 86 poles to inches. Amt. 17028 in. 10. Reduce 26 leagues to yards. Amt. 137280 yd. 11. Reduce 52 miles to feet. Amt. 274560 ft. 12. Reduce 5 leagues to inches. Amt. 950400 in. 13. Reduce 24 degrees to statute miles. Amt. 1668 m.. 14. Reduce 12 miles' 3 furlongs, 25 poles to poles. Amt. 3985 po. 15. Reduce 14 leagues, 2 furlongs to poles. Amt. 13520 po. 16. Reduce 3 leagues, 2 miles, 6 furlongs, 18 poles to yards. Amt. 20779 yds. LAND, OR SQUARE MEASUREL TABLE. 144 square inches make I square foot, ft. 9 square feet - 1 square yard, yd. 30A square yards - 1 square perch, p. 40 square perches - rood, r. 4 roods - I acre, a. NoTE.-This measure is used to ascertain the quantity of lands, and of other things having length and breadth to be estimated. EXERCISES. 1. Reduce 27 acres to roods. Amt. 108 r. 2. Reduce 53 roods to perches. Amt. 2120 p. REDUCTION. 49 3. Reduce 28 perches to square yards. Amt. 847 sq. yds. 4. Reduce 36 square yards to square feet. 324 ft. 5. Reduce 27 square feet to square inches. Amt. 3888 in. 6. Reduce 34 acres to perches. Amt. 5440 p. 7. Reduce 42 roods to square yards. Amt. 50820 sq. yds. 8. Reduce 24 square perches to square feet. 6534 ft. 9. Reduce 32 roods to square feet. Amt. 348480 ft. 10. Reduce 23 acres to square inches. Amt. 144270720 sq. in. 11. Reduce 11 acres, 2 roods, 19 perches to perches. Amt. 1859 p. 12. Reduce 17 acres, 3 roods to perches. Amt. 2840 p. 13. Reduce 12 acres, 2 roods, 12 perches to square yards. Amt. 60863 sq. yd. CUBIC, OPR SOLID MEASURE. TABLE. 1728 cubick inches make 1 cubic foot 27 feet 1 cubic yard 40 feet of round timber, or 1 Ton or load 50 feet of hewn timber, 128 solid feet I Cord of wood NOTE.-This measure is employed in measuring solids, having length, breadth, and thickness to be estimated. EXERCISES. I Reduce 29 cords of wood to cubick feet. Amt. 3712 c. f. 2 Reduce 32 cubic yds. to feet. Amt. 864 c. f. 3 Reduce 23 cubic feet to inches. Amt. 39744 c. in. 4 Reduce 32 cubic yds. to inches. Amt. 1492992 c. in. 5 Reduce 2 cords of wood to inches. Amt. 442368-c. in. 6 Reduce 3 cords, 10 feet to feet. IAmt. 394 ft 7 Reduce 1 cord, 3 feet, 136 inches to inches. Amt. 226504 Ra' o50 REDUCTION. LIQUID MEASURE. TABLE. 4 gills make 1 pint pt. 2 pints'(pts) I quart qt. 4 quarts 1 gallon gal. 42 gallons 1 tierce te. 63 gallons 1 hogshead hhd. 2 hogsheads 1 pipe or butt pi. 2 pipes I tun. TNOTE.-This measure is employed in measuring cider, oil, beer, &c. EXERCISES. I Reduce 23 tuns to pipes. Amt. 46 pi. 2 Reduce 43 pipes to hogsheads. Amt. 86 hhd. 3 Reduce 34 hogsheads to gallons. Amt. 2142 gal. 4 Reduce 27 tierces to gallons. Amt. 1134 gal. 5 Reduce 53 gallons to quarts. Amt. 212 qt. 6 Reduce 724 quarts to pints. Amt. 1448 pt. 7 Reduce 37 pints to gills. Amt. 148 g. 8 Reduce 12 pipes to gallons. Amt. 1512 gal. 9 Reduce 4 hogsheads to quarts Amt. 1008 qt. 10 Reduce 32 gallons to gills. Amt. 1024 g. 11 Reduce 2 tuns to gills. Amt. 16128 gills 12 Reduce 32 gals 3 qts. to pints. Amt. 262 pt. 13 Reduce 2 hogsheads, 27 gals. 3 qts to quarts. Amt. 615 qt. 14 Reduce 3 tons, I hogshead, 15 gals. I qt to pints. Amt. 6674 pt. MOTION, OR CIRCLE MEASURE. TABLE. i60 seconds ("sec) make 1 minute min. 60 minutes I degree o deg. 30 degrees 1 sine. sin. 12 sines (or 360 degrees) 1 revolution NOTE. —This measure is employed by astronomers, navigators, &c. DL __ ^^ ^ -. REDUCTION. 51 EXERCISES. 1 Reduce 5 sines to degrees. Amt. 1500 2 Reduce 8 degrees to minutes. Amt. 4801 3 Reduce 6 minutes to seconds. Amt. 360 sec. 4 Reduce 12 sines to seconds. Amt. 1296000 sec. 5 Reduce 3 sines 15 degrees to minutes. Amt. 6300 min. TIME. TABLE. 60 seconds (see) make I minute min. 60 minutes 1 hour II. 24 hours I day 7 days 1 week 12 months (or 365 days) 1 year. NOTE.-The true year, according to the latest and most accurate observations, consists of 365 d. 5 h. 48 m. and 58 sec: this amounts to nearly 365i days. The common year is reckoned 365 days, and every fourth or leap year one day more on account of the fraction omitted each year, which being put together, every fourth year is added to it, making leap year 366 days. The year is divided into 12 months as follows. The fourth, eleventh, ninth and sixth, Have thirty days to each affixed, And every other thirty-one, Except the second month alone, Which has but twenty-eight in fine, Till leap year gives it twenty-nine. OR THUS: Thirty days hath September, April, June, and November, February hath twenty-eight alone, And each of the rest has thirty one. When the year can be divided by four, without a remainder, it is bissextile, or leap year. 1 -52 REDITCTION.:EXERCISES. Reduce 42 years to months. Amt. 504 m. 2 Reduce 23 days to hours. Amt. 552 h. 3 Reduce 36 hours to minutes. Amt. 2160 min. 4 Reduce 25 minutes to seconds. Amt. 1500 sec. 5 Reduce 14 days to minutes. Amt. 20160 min 6 Reduce 52 hours to seconds. Amt. 187200 sec. 7 Reduce 13 weeks to hours. Amt. 2184 h. 8 Reduce 12 weeks to minutes. Amt. 120960 min. 9 Reduce 3 years to minutes, allowing 365 days to each year. Amt. 1576800 min. 10 Reduce 15 years and 6 months to months. Amt. 186 m. 11 Reduce 4 weeks, 3 days, 22 hours, to hours. Amt. 766 h. 12 Reduce 7 years, 24 days, 43 minutes, to seconds. Amt. 222828180 sec. STERLING MONEY. TABLE. 4 farthings (qr).make I penny d. 12 pence 1 shilling s. 20 shillings 1 pound Farthings are usually written as fractions of a penny, thus i one farthing two farthings or a half penny. i three farthings. EXERCISES. I Reduce 14 pounds to shillings. Amt. 280 s. 2 Reduce 23 shillings to pence. Amt. 276 d. 3 Reduce 34 pence to farthings. Amt. 136 qr. 4 Reduce 4 pounds to pence. Amt. 960 d. 5 Reduce 13 shillings to farthings. Amt. 624 qr.. 6 Reduce 16 pounds to farthings. Amt. 15360 qr. 7 Reduce 13 pounds 14 shillings, to pence. Amt. 3288 d. 8 Reduce 3 pounds 15 shillings 6 pence to farthings. Amt. 3624 qr. REDUCTION. 53 FEDERAL MONEY. TABLE. 10 mills make 1 cent 10 cents 1 dime 10 dimes 1 dollar 10 dollars 1 eagle EXERCISES. 1 Reduce 5 eagles to cents. Amt. 5000 ct, 2 Reduce 3 dollars to mills. Amt. 3000 m. 3 Reduce 15 dimes to cents. Amt. 150 ct. 4 Reduce 3 eagles, 5 dollars to cts. Amt.3500 ct. 5 Reduce 7 dollars, 3 dimes, 6 cents, to mills. Amt. 7360 m. As eagles, dimes and mills are not used in accounts, they will generally be omitted in the subsequent exercises of this work. 4 fourths, or 3 thirds, or 2 halves, make 1 cent. 100 cents - - - 1 dollar. 6 Reduce 125 cents to halves of a cent. Amt. 250 halves. 7 Reduce 32 cents to fourths of a cent. Amt. 128 fourths. 8 Reduce 23 dollars to cents. Amt. 2300 ct. 9 Reduce 25 dollars 15 cents to cents. Amt. 2515 ct. 10 Reduce 15 dollars 37A cents to halves of a cent. Amt. 3075 halves. 11 Reduce 21 dollars 15 cents to thirds of a cent. Amt. 6345 thirds. 12 Reduce 5 dols. 37i cents to fourths of a cent. Amt. 2150 fourths. 13 Reduce 15 dollars 33& cts. o thirds of a cent. Amt. 4600 thirds. NOTE. To reduce dollars to cents annex two cyphers: thus 53 dollars are 5300 cents. To reduce dollars and cents to cents, place them to5* 54 REDUCTIOUN. gether without any separating point, and the amount will be cents. Thus 35 dollars 24 cents are 3524 cents. Questions. For what purpose is Dry measure used? For what is Avoirdupois weight used? For what is Troy weight employed? For what is Apothecaries weight employed? For what is Cloth measure employed? For what is Long measure used? For what is Land or Square measure used? For what is Cubick measure employed? For what is Liquid measure employed? For what is Sterling currency used? For what is Federal currency used? CASE 2. To reduce a sum or quantity to a HIGHER denomination than its own RULE.-Divide the sum or quantity by that number of its own denomination which makes one of the denomination to which it is to be changed. When there are one or more denominations between the denomination of the given sum and that to which it is to be changed; first change it to the next higher than its own, and then to the next higher, and so on. Remainders are always of the same denominations as the sums divided. DRY MEASURE. EXAMPLES. I Reduce 25 pints to quarts. pts. NoTE.-Divide by 2, because every 2 pints 2)25 make one quart. In 25 are 12 two's and - 1 over, that.is 12 quarts and 1 pint. qt.l2-lpt 2 Reduce 43 quarts to pecks. qt. Divide by 8, because every 8 qts. make 8)43 1 peck. In 43 are 5 eights and 3 over, that is 5 pecks and 3 quarts. pe. 5-3qt: -..,-.e -, -...~,,- _.p- t REDUCTION. 55 3 Reduce 26 pecks to bushels. bu. Divide by four because every 4 pecks 4)26 /make I bushel. In 26 are 6 fours and 2 over; that is 6 bushels and 2 pecks. bu. 6-2 pecks 4 Reduce 359 pints to bushels. pt. Divide pints by 2, brings them 2)359 to quarts; divide quarts by 8, brings them to pecks, and divide pecks by 8)179-1 pt. 4 brings them to bushels. 4) 22 —3qt. 5 b. 2 p. 3 qt. I pt. 5 Reduce 81 quarts to bushels. A. 2 bu. 2 pe. 1 qt. 6 Reduce 134 pints to pecks. 8 pe. 3 qt. 7 Reduce 194 pints to bushels. 3 bu. 0 pe. I qt. Questions. What is reduction? For what is case second used? How do you reduce a sum to a higher denomination than its own? When there are one or more denominations between the denomination of the given sum and the one to which you wish to reduce it, how do you proceed? Of what denomination is the remainder always? How do you bring pints to quarts? How do you bring quarts to pecsis? How do you bring pecks to bushels? How do you bring pints to bushels? AVOIRDUPOIS WEIGHT. 1 Reduce 65 cwt. to tons. Result 3 tons 5 cwt. 2 Reduce 27 quarters to cwt. Res. 6 cwt. 3 qr. 3 Reduce 109 pounds to qr. Res. 3 qr. 25 lb. 4 Reduce 123 ounces to pounds. Res. 7 lb. 11 oz. 5 Reduce 234 drams to ounces. Res. 14 oz. 10 dr. 6 Reduce 4274 drams to pounds. Res. 16 lb. II oz. 2 dr. 7 Reduce 175 quarters to tons. Res. 2 tons 3 cwt. 3 qr. 8 Reduce 6745 pounds to tons. Res. 3 tcns 25 lb. Res. 3 tens 25 lb 56 RE]DUCTION. TROY WEIGHT. 1 Reduce 378 ounces to pounds. Result, 31 lbs. 6 oz. 2 Reduce 235 pennyweights to ounces. Res. 11 oz. 15 dwt. 3 Reduce 748 grains to pennyweights. Res. 31 dwt. 4 grains. 4 Reduce 678 pennyweights to pounds. Res. 2 lbs. 9 oz. 18'dwt. 5 Reduce 732 grains to ounces. Res. 1 oz. 10 dwt. 12 grains. 6 Reduce 14752 grains to pounds. Res. 2 lbs. 6 oz. 14 dwt. 16 gr. APOTHECARIES WEIGHT. I Reduce 432 ounces to pounds. Res. 36 lbs. 2 Reduce 782 drams to ounces. Res. 97 oz. 6 dr. 3 Reduce 91 scruples to drams. Res. 30 dr. 1 scr. 4 Reduce 192 grains to scruples. Res. 9 sc. 12 gr. 5 Reduce 256 scruples to ounces. Res. 10 oz. 5 dr. 1 scr. 6 Reduce 12660 grains to pounds. Res. 2 lb. 2 oz. 3 drs. CLOTH MEASURE. 1 Reduce 60 quarters to yards. Res. 15 yds. 2 Reduce 60 quarters to English ells. Res. 12 E. ells. 3 Reduce 60 quarters to French ells. Res. 10 Fr. ells. 4 Reduce 60 quarters to Flemish ells. Res. 20 Fl. ells. 5 Reduce 52 nails to quarters. Res. 13 qr. 6 Reduce 123 nails to yards. Res. 7 yds. 2 qr. 3 na. 7 Reduce 543 nails to English ells. Res. 27 ells. 0 qr. 3 nails. LONG MEASURE. 1 Reduce 36 miles to leagues. Res. 121. 2 Reduce 75 furlongs to miles. Res. 9 m. 3f. REDUCTION. 57 3 Reduce 295 poles to furlongs. Res. 7 f. 15 p. 4 Reduce 286 yards to poles. Res. 52 p. 5 Reduce 365 feet to yards. Res. 121 yds. 2 ft. 6 Reduce 759 inches to feet. Res. 63 ft. 3 in. 7 Reduce 253 inches to yards. Res. 7 yds. 0 ft 1 inch. 8 Reduce 2792 poles to leagues. Res. 2 1.2 m. 5 f. 32 p. SQUARE MEASURE. I Reduce 287 roods to acres. Result 71 a. 3 r. 2 Reduce 245 perches to roods. Res, 6 r. 5 p. 3 Reduce 756 square feet to yards. Res. 84 yds. 4 Reduce 4731 square yards to perches. Yds. Res. 156 p. 12 yds. 301 4731 Bring the 30J yards and the 4731 yards 4 4 both to fourths, and divide. The remainder 48, is fourths of a yard; divided by four, 121 18924 156 p. brings it to yards, the true remainder. 121 18924 156 p. 121 682 605 774 726 4 i 48 Rem. 12 yards. 5 Reduce 3575 square inches to feet. Res. 24 feet 119 inches. 6 Reduce 1728 square perches to acres. Res. 10 a. 3 r. 8 p. CUBIC MEASURE. 1 Reduce 789 cubic feet to cords. Result, 6 c. 21 ft. 2 Reduce 343 cubic feet to yards. Res. 12 yds. 19 ft. 3 Reduce 9386 cubic inches to feet. Res. 5 ft. 746 in. 4 Reduce 70353i cubic inches to cords. Res. 3 c. 23 ft. 243 in. 2 58 REDUCTION. LIQUID MEASURE. I Reduce 25 pipes to unRes. 12 T. 1 P. 2 Reduce 34 hogsheads to pipes. Res. 17 P. 3 Reduce 1575 gallons to hogsheads. Res. 25 hhds. 4 Reduce 163 quarts to gallons. Res. 40 gal. 3 at. 5 Reduce 6048 pints to tuns. Res. 3 tuns. MOTION. 1 Reduce 1440 seconds to minutes. Result, 24 min. 2 Reduce 720 minutes to degrees. Res. 12 deg. 3 Reduce 342 degrees to sines. Res. 11 sines 12 deg. 4 Reduce 443907 seconds to sines. Res. 4 sines 3 deg. 18 min. 27 sec. TIME. I Reduce 1800 seconds to minutes. Result, 30 m. 2 Reduce 720 minutes to hours. Res. 12h. 3 Reduce 744 months to years. Res. 62 yrs. 14 Reduce 4649 minutes to days. Res. 3d. 5h. 29 min5 Reduce 48888 minutes to weeks. Res. 4w. 5d. 22hrs. 48 min, STERLING MONEY. 1 Reduce 78 shillings to pounds Res. 3~. 189. 2 Reduce 93 pence to shillings. Res. 78. 9d. 3 Reduce 39 farthings to pence. Res. 9d. 3qr 4 Reduce 656 pence to pounds. Res. 2' 14s. 8d. 5 Reduce 781 farthings to shillings. Res. 16s. 3d. lqr. 6 Reduce 6529 farthings to pounds. Res. 6X. 16s. Od. 1 qr. FEDERAL MONEY. 1 Reduce 250 halves to cents. Res. 125 cts. 2 Reduce 128 fourths to cents. Res. 32 cts. i Reduce 2343 cents to dollars. Res. 23 dol. 43 cts. 4 Reduce 1537i cents to dollars. Res. 15 dol. 37& cts. 5 Reduce 6150 half cents to dollars. Res. 30 dol. 75 cts. NOTE.-To reduce cents to dollars, separate two figures oi lthe right hand for cents; those on the left will be dollars, REDUCTION. 59 PROMISICUOUS EXERCISES. 1 How many bushels in 738 quarts? Ans. 23 bushels 2 quarts. 2 In 7 bushels) how many pints-? Ans. 448 pints. 3 How many cwt. in 5356 ounces? Ans. 2 cwt. 3qr. 261b. 12 oz. 4 How many drains in 3 qr. 23 lb. 14 oz.? Ans. 27616 drams. 5 Iow many grains in 9 oz. 14 dwt. 3 gr.? Ans. 4659 gr. 6 How many pounds in 7432 dwt.?Ans. 301b. Il oz. 12dwt. 7 How many scruples in 15 lb I1 oz. 6 drams? Ans. 4362 scr. 8 How many ounces in 216 scruples? Ans. 9 oz. 9 How many furlongs in 2346 yards? Ans. 10f. 26p. 3yd. 10 How many poles in 3 leagues? Ans. 2880 poles. 11 Iow many yards in 84 nails? Ans. 5 yd. 1 qr. 12 How many perches in 4719 square yards Ans, 156 perches. 13 How many square yards in one acre? Ans. 4840 sq. yds. 14 How many hogsheads in 9728 gills? Ans. 4 hhds. 52 gal. 15 How many pints in 2 pipes? Ans. 2016 pints. 16 H-ow many minutes in 3 days 6 hours? Ans. 4680m. 17 How many hours in 2 weeks and 4 days? Ans. 432 hours. 18 How many shillings in 27 four-pences? Ans. 9 s. 19 How many cords of wood in 93G4 cubic feet? Ans. 73 cords 20 feet. 20 How many cubic feet in 9 cords? Ans. 1152 feet. 21 How many inches round the globe, which is 360 degrees of 69t miles each? Ans. 1,585,267,200 inches, Enumerate the answer. 60 COMPOUND ADDITION. COMPOUND ADDITION. COMPOUND ADDITION is the art of collecting several numbers of different denominations into one sum. RUILE. Place the numbers so that those of the same denomination may stand directly under each other, observing to set the lowest denomination on the right, the next lowest next, &c. Then add up the several columns beginning with the lowest denomination: divide the sum by as many of the number of that denomination as it takes to make one of the next; and so on. Proof.-As in Simple Addition. DRY MEASURE. EXAM1PLES. bu, pe. qt. pt. The first column on the right makes 3 2 7 1 five pints. Five pints make two quarts 3 XL^ t ^and leaves one pint. Set down the one pint under the column of pints and carry 3 1 3 1 the two quarts to the column of quarts. 6 3 2 1 The column of quarts with the two quarts 2 2 6 1 added makes twenty four quarts. Twenty-four quarts make three pecks and leave ~ 2 ~ - no quarts. Set down 0 under the column l24 2 0 1 of quarts and carry the three pecks to the -........' column of pecks. The column of pecks with the three pecks added makes fourteen pecks. Fourteen pecks make three bushels and leave two- pecks. Set down the two pecks under the column of pecks and carry the three to the column of bushels. The column of bushels with the three bushels added makes twenty-four bushels. Here set down the whole amount. COMPOUND ADDITION. 61 bu. pe. qt. pt. bu. pe. qt. bu. pe. qt. pt. 23 3 7 1 4 3 7 3 7 1 34 26 1 5 2 6 26 1 42 3 5 1 6 1 5 0 5 0 51141 70 4 1 11 23 2 3 1 8 3 3 33 1 141 2 1 4 1 2 0 2 1 11 3 4 1 3 2 4 2 1 1 202 3 2 1 40 3 7 3 2 7 0 APPPLICATION. I Add 2 bu. 3 pe.; 7 bu. 3 qt.; 4 bu. I pe. 1 pt.; 6 bu. 4 qt. 1 pt.; and 3 pe. I qt. together. Amount 21 bu. 0 pe. 1 qt. 0 pt. 2 Add 3 bu. 2 pe. 3 qt. I pt.; 7 bu. 7 qt. 1 pt.; 3 pe. 1 pt.; 4 bu. 5 qt.; 4 bu. 3 pe.; 8 bu. 3 pe. 7 qt. 1 pt. together. Amt. 29 bu. 2 pe. 0 qt. 0 pt. 3 Add 7 bu. 1 pt.; 3 pe. 7 qt. I pt.; 6 qt. 1 pt.; 9 bu. 3 pe. 6 qt. 1 pt.; 3 bu. 3 qt.; 4 bu. 1 pe. Amt. 25 bu. 2 pe. 4 In a wagon load of grain contained in seven sacks, viz: in the first 4 bu. 3 pe. I qt.; in the second 5 bu. 7 qt. 1 pt.-the third 3 bu. 1 pe. 1 pt. —fourth, 3 bu. 2 pe. 6 qt.-fifth, 5 bu.-sixth, 4 bu. 1 pe. 1 pt.:-and in the seventh 6 bu. I pe. 1 pt. How many bushels? Questions. What is Compound Addition? How do you place the numbers to be added? Do you place the greater or smalle denominations in the right hand column? Where do you begin the addition? When the first column is added, how do you proceed with the sum? When you divide the sum by as many of that denomination as make one of the next; which do you set down, the remainder or the quotient? What do you do with the quotient? 6 62 COMrOUTND ADDITION. In what particular does compound addition differ from simple addition? Do you carry one for every ten in compound addition? Since you do not carry one for every ten, how many doyou always carry? A. One for as many of anydenomination as make one of the next. Here the pupil will have something with which to compare simple addztion, in which he carries one for every ten. This comparison will improve and correct his understanding of the "elementary rules.AVOIRDUPOIS WEIGHT.. T. cwt. qr. lb. T. cwt. qt. lb. oz. dr. (1) 15 3 2 15 (2) 7 11 216 413 4 8 3 9 15 7 3 8 1 7 82 1-9 110 1 38 19 1 12 8 13 163 8 3 17 -42 8 3 19 12 4 34 15 224 357 6 2 8 3 3 300 16 1 19 APOTHECARIES' WEIGHT. r S: 3 si 33 gr.(1) 6 3: i2 -2)) 84 7 6 0 12 19 9. 5 1 132 5 0 2 182 7 3 2 16 2 2 2 28 576 1 0 1427 6 7 0 19 40 5 0 0 14 0 6 1 9 306 7 3 2 TROY WEIGHT. lb. oz. dwt. lb. oz..dwt. gr. (1) 47 10 12 (2) 185 2 19 20 38 8 6 56 9 15 616-11 4 1472 11 2 17 7 2 16 385 0 8 5 13 911 10 8 7 12 124 6 9 COMPOUND ADDITION. 63 CLOTH MEASURE. yds. qr. a. E.E. q. a E.. qr. na. (1) 75 3 2 (2) 72 3 2 (3) 19 2 3 163 1 3 536 2 1 728 1 2 245 2 0 847 1 3. 142 0 1 738 3 1 1453 0 2 816 0 0 1785 2 3 41 2 0 32 1 2 3009 1 1 LONG MEASURE. L. M. fur. P. yd. ft. in. (1) 5 2 4 17 (2) 3 211 16 1 3 10 1 1 9 72 0 5 24 2 0 8 526 03 12 3 110 834 2 6 34 2 0 4 38 0 3 12 6 2 7 1493 2 2 29 SQUARE MEASURE. A.. R.P. A. R. P. (1) 39 2 37 (2) 487 2 17 62 1 17 25 3 28 68 038 a67 0 32 129 3 12 45 1 16 532 118 _26 0 29 832 2 2 11I 64 COMPOUND ADDITION. CUBIC MEASURE. yd. ft. in. cords. feet. in. (1) 75 22 1412 (2) 37 119 1015 9 26 19i 9 110 159 3 19 1091 48 127 1071 28 15 1110 8 111 956 49 24 218 21 9 27 18 17 1225 9 28 1091 186 18 67 135 122 863 3 In four piles of wood; the first containing 32 feet 149 inches; the second 121 feet 1436 inches; the third 97 feet 498 inches; the fourth 115 feet 1356 inches; how much did the whole amount to? Ans. 2 cords, 110 ft. 1711 in. 4 In six boat-loads of wood: the first containing 22 cords 114 feet, 987 in.; the second 18 cords, 121 feet, 1436 in.; the third 21 cords, 109 feet, 1629 in.; the fourth 15 cords, 82 feet, 1321 in.; the fifth 16 cords, 98 feet, 1111 in.; the sixth 24 cords, 89 feet, 987 in. H-ow much did they contain? Ans. 120 c. 105 ft. 559 in. LIQUID MEASURE. T. hhd. gal hhd. gal. qt. pt. (1) 18 2 54 (2) 385 42 3 1 62 1 39 27 36 2 0 327 0 4 132 17 0 0 46 1 19 729 250 0 285 3 28 173 47 2 1 740 1 18 COMPOUND ADDITION. 65 MOTIOCYN, b 1 t SIN. 0 (1) 17 55 48,2) 1 25 49 51 1 37 51 2 421 36 28 19 45 4 19 47 18 19 19 37 1 25 25 39 67 13 I 115 24 24 3. Add 5sin. 10o 46' 38, 110 37' 18"; lsin. 17012' 18"; 2sin. 52"; lsin. 15~ 12' 23"; and 110 57' 29" together. Ans. 11sin. 60 4658". 4. Add 45'; lsin. 90 18"; 14~ 21' 34"; 2sin. 8~ 13' 54:"; sin, 70 12' 19; and 47' 32" together. Ans. 8sin. 10~ 20' 37". TIME. Y. M. we, d. h. HI. min. sec. (1) 71 11 (2) 3 5 20 (3) 20 52 40 172 9 2 3 17 122 12 35 35 7 3 6 22 68 9 17 410 0 4 16 135 17 12 6 0 3 19 24 35 28 231 7 11 3 22 371 7 12 STERLING MONEY. s. d. ~ s. d. ~. d. (1) 2 3 4 (2) 7 9 4^ (3) 4 6 4 7 1 2 13 7 6 47 19 7 9 7 3 4 5 2 159'5 3 5 2 2i 1018104 78 6 114 23 13 11l 4 Add ~763 7s. 4d.; z39 4s. 9d.; ~162 17s. 2d; ~459 15s.; ~473 12s. 8d together. Ans. ~1898 16s. lid. &* " 66 COMPOUND SUBTRACTION. 5 Add the following sums: viz. ~69 18s. 7d.; ~175 2s. 6d.; ~1582 19s 4d.; ~175 13s. 9d.; ~143 13s. 8d.; and ~212 Os. 7d Ans. ~2359 8s. 5d COMPOUND SUBTRACTION. COMPOUND SUBTRACTION is the art of finding the difference between two numbers consisting of several denominations. RULE. vL..... Place the numbers, as in compound, addition with the less under the greater: then begin at the right hand denomination and subtract the lower number from the upper, and set down the remainder. If the upper number of any denomination be less than the lower one, add to the upper one as many as it takes of that to make one of the next; subtract the lower number from the amount and set down the remainder as before. Proof.-As in simple subtraction. EXAMIPLES. bu. pe. qt. pt bu. pe. qt. pt. 7 2 4 1 42 3 6 1 3 1 2 0 31 2 3 1 4 1 2 1 11 1 3 0 bush. pe. qt. 9 2 5 We cannot take 7 quarts from 5 quarts; then 2 3 7 borrow 1 from the 2 pecks. One peck has 8 quarts in it: 8 quarts added to the 5 quarts, 6 2 6 make 13 quarts. Take 7 qts. from 13 qts. and 6 qts. remain. Set down the 6 qt. Because I borrowed 1 from the 2 quarts, I must add one to the 3 below it, which makes the lower figure 4. Now 4 pecks from 2 pecks we cannot take: then borrow one bushel from the 9; that bushel has 4 pecks in it; 4 p. and 2 p. make 6 p. Now 4 p. from 6 p. and 2 pecks remain, which set down. Because I borrowed 1 from the 9, I must add 1 to the figure below it. I to 2 make 3. Take 3 from 9 and 6 remain. Set down the 6, ad the work is done. m~ COMPOUND SUBTRACTION. 67 bu. pe. qt. pt. bu. pe. qt. pt. 8 7 1 8 1 3 0 4 3 6 1 4 2 5 1 3 3 1 0 3 2 5 1 bu. pe. qt. bu. pe. qt. pt. 95 3 2 28 2 2 0 22 0 1 14 3 5 1 APPLICATION. 1. From a granary containing 94 bushels, 2 pecks, 7 quarts, have been taken 43 bush. 3 pe. 5 qr. How much remains? Ans. 50 bush. 3 pe. 2 qt. 2. From a wagon load of corn containing 63 bushels, 3 pecks, 4 qts., have been sold 27 bush. 3 pe. 7 qt. 1 pt. How much remains unsold? Ans. 35 bu. 3 p. 4qt. 1 pt. Questions. What is compound subtraction? In what particular does it differ from simple subtraction? How do you place the numbers in compound subtrac* tion? Where do you begin the operation? When the upper number of any denomination is less than the lower one, how do you proceed? Do you borrow one from the next? Do you call the number you borrow, one ten, as in simple subtraction? What do you call it? Ans. I call it one peck, or one yard, or one mile, as the case may be? What do you do with it then? Ans. I reduce it to quarts, or to feet, or to furlongs, &c. according to the nature of the case; then add these 68 COMIPOUND SUBTRACTION. to the upper figure on the right, subtract the lower figure' from the sum, and set down the remainder. When you borrow one from the upper figure, why do. you add one to the figure below it? NOTE.-Upon a clear conception of the principles involved in these questions, depends the pupil's correct knowledge of the science of Arithmetic. AVOIRDUPOIS WEIGHT. tons cwt. qr. tons cwt. qr, lb. cwt. qr. lb. oz. dr. From 45 11 3 52 12 3 15 17 0 0 0 0 Take 15 10 2 24 10 0 26 6 3 21 15 9 Rern. 30 1 1 28 2 2 17 10 0 6 0 7 1. Subtract 76 tons, 18 hundred weight, 3 quarters, from 195 tons, 2 hundred weight, 2 quarters. Ans. 118 tons, 3 cwt. 3 qr. 2. Subtract 14 pounds, 6 ounces, 3 drams from 20 pounds, 2 ounces. Ans. 5 Ibs. 11 oz. 13 dr. APOTHECARIES WEIGHT.;B ~ 3 gt 3 D gr. 1090 1 6 48 9 6 1 4 106 2 7 1 10. 0 2 8 983 10 7 3. From 59IS 13 23 take 5315 7g 53. Ans. 5S 53 53. 4. Subtract 14f &3 13 from 6911. Ans. 5411 23 73. TROY WEIGHT. 1b. o. dwt.gr. lb. oz. dwt. gr. lb. oz. dwt. gr. 10 6 18 0 8 3 0 2 106 0 0 15 4 0 2 20 2 11 8 6 106 2 20 6 6154 4. Subtract 141b. 6oz. lldwt. from 221b. 12dwt. 6 gr. Ans. 71b. 6oz. ldwt. 6,r. 5. From 161b. take 12lb. 11oz. iOdwt. 11gr. Ans. 31b. Ooz. dwt. 13gr. iN~~~~~~~~~~~~~~~~I C0OPOtFND SUBTRACTION. 69 CLOTH MEASURE. yds. qrs.na. E.q.... rs na.. qrs.. qrs. na From 71 3 1 42 0 2 51 2 2 Take 14 2 3 19 2 3 42 2 1 Rem. 57 0 2 22 2 3 9 0 1 4. Subtract 95 yards, 3 quarters, 2 nails, from 156 yards, 2 quarters,: 3 nails. Ans. 60 yds. 3 qr. 1 nail. 5. Subtract 14 English ells, 1 quarter, 2 nails, from 52 English ells, 3 quarters, 2 nails. Ans. 38 yds. 2 qr. LONG MEASURE. L. lM.fur. L. M.fur. P. yds.ft. in. From 24 7 56 1 0 19 6 2 10 Take 18 2 4 10 0 7 20 3 2 7 Rem. 5 2 3 46 0 0 39 3 0 3 4. Subtract 45 miles, 5 furlongs: 20 poles, from 320 miles, 3 furlongs, 36 poles. Ans. 274 M. 6 F. 16 P. 5. Subtract 15 yards, 2 feet, 6 inches, from 36 yards, 1 foot, 11 inches. Ans. 20 yds. 2 ft. 5 inches. LAND, OR SQUARE MEASURE. A. R. P. A. R. P. Yds. ft. in. From 96 3 36 195 2 2 25 2 72 Take 25 2 39 36 3 1 14 7 10 Rem. 71 0 37 158 3 1 104 62 4. Subtract 36 acres, 2 roods, from 900 acres, 3 roods, 16 perches. 864 A. 1 R. 16 P. 5. Subtract 72 acres, from 360 acres, 2 roods, 29 perches. 288 A. 2 R. 29 P. CUBICK MEASURE. yds. ft. in. cords. ft. in. 79 11 917 349 97 1250 17 25 1095 192 127 1349 61 12 1550 156 97 1629 .70 CONMOvND SUBTRACTION. 1. From a pile of wood containing 432 cords, 27 feet, and 1432 inches, have been hauled 156 cords, 92 feet, 946 inches: how much remains? Ans. 275 cords, 63 feet, 486 in. 2. From a bank of earth containing 2984 yards, 18 feet, have been taken 1436 yards, 21 feet: what remains? Ans. 1547 yds. 24 feet. LIQUID MEASURE. T. hhdgal. qt. pt. T. hhd.gal. qt. pt. 2 3 50 1 0 100 1 19 2 1 1 2 16 3 1 99 1 28 3 1 I 1 33 1 1 3. If I purchase 2hhd. of wine, and to oblige a friend send him 29gal., what quantity have I left? Ans. 17hd. 34gal. 4. Bought I pipe of wine, 4hhd. of brandy, 2 barrels of beer; I have since sold 93 gallons of wine, 29 of brandy, 1 barrel of beer: how much of each have I remaining? Ans. 33gal. of wine, 223gal. of brandy, and I barrel of beer. MOTION. 0,,, Sin. 0,, 79 21 31 6 10 12 48 41 41 52 3 8 39 29 37 39 39 3 1 33 19 A circle being 12 sines, how far has the hand of a watch to pass, after having gone through 4 sines, 23U.5' 29"? Sin. o 12 0 O 0 4 23 15. 29 COkMPOUvND SUBTRACTION. 71 2 A person residing in latitude 27c 32' 45" north, wishes to visit a place 52~ 24' 18'. north. How many degrees, miniutes, and seconds northward must he travel?' Ans. 24:0 51' 33'. TIME Y. M. w. d. ho. min. sec. H.min.sec. Y. Al. 6 9 3 1 3 40 20 16 29 33 18 11 1 6 2 6 2 57 36 7 36 44 9 10 5 3 2 0 42 44: 4. From 900Y. take 1 Y. 6m. Ans. 788 Y. 6m. 5. If I take 1 Y. 1M. from EY. what space of time will still remain? Ans. 4Y. 111 1. NOTE.-To ascertain the amount of time passed between two events, set down the year, month, and day of the latter event, and place those of the former below it, and subtract. 6. A bond was given 24th July, (7th month) 1809, and paid off 13th August, 1821. yrs. mo. ds. 1821 8 13 1809 7 24 12 0 20 7. The declaration of independence of the United States passed Congress, 4th July, (7th month) 1776; and the declaration of the late war with Great Britain, 18th June, (6th mO.) 1812. How many years, &c. between them? Ans. 35yr. llmo.. i4d. STERLING MONEY. ~. s d. ~ s. d.. s. d. 146 19 10 47 6 71 419 7 6 7 19 94 28 5 10 227 8 9& 139 0 04 72 COMOUNiB MULTIPLICATION. 4. Subtract ~200 9s. from ~1000 11s. 11d. Ans. ~800 2s. 11.d. 5. I have a purse of money containing ~1000 2s. 4id.: it I take out ~60 7s. Sd. what suim will be left? Ans. ~939 14s. 74d. COMPOUND MULTIPLICATION. ConrOUND MULTIPLICATION is the art of multiplying numbers composed of several denominations. CASE 1. When the multiplier does not exceed 12. RULE. Place the number to be multiplied as directed in compound addition; and set the multiplier under-the lowest denomination. Multiply as in simple multiplication, and divide the product of each denomination by as many as it takes of that to make one of the next greater; set down the remainder (if any) and carry the quotient to the product of the next denomination. Proof.-Double the multiplicand and multiply by half the multiplier. EXAMPLES. Bu. pe. qt. pt. 7 times 1 pint make 7 pints; 2 pt. 7 2 5 1 make 1 qt.; then 7 pt. make 3 qt. and leave 1 pt. Set down the 1 pt. and carry the 3 qt. to the product of the ~~~ ~~. —- ~ next figure. 53 2 6 1 7 times 5 qt. make 35 qt. to which add the 3 qt. which make 38 qt.; 8 qt. make one peck; then 38 qt. make 4 pecks and leave 6 qts. Set down the six quarts and carry the 4 pecks. 7 times 2 pecks make 14 pecks; add the 4 pecks, makes 18 pe.; 4 pecks make one bushel; then 18 pe. make 4 bushels and leave 2 pecks. Set down the 2 pecks and carry the 4-bushels. 7 times 7 bushels make 49 bushels; add the 4 bushels, makes 53 bushels, which set down, and the work is done. COMPOUTND MULTIPLICATION. 73 Bu. pe. qt. pt. Bu. pe. qt. pt. 9 3 6 1 23 2 5 1 5' 8 49 3 0 1 189 1 4 0 1. In one vessel are contained 29 bushels 2 pecks and 5 quarts: how many in 9 such vessels? Ans. 266 bu. 3 pe.5 qt. 2. If one tub will contain 8 bu. 3 pt. 5 qt. how much will 11 such tubs contain? Ans 97 bu. 3 pe. 7 qt. CASE 2. When the multiplier exceeds 12, and is the exact product of two factors in tile multiplication table. RULE. Multiply the given sum by one of the factors, and the product by the other factor. Proof.-Change the factors. EXAMPLES. 1. Multiply 3 bushels, 2 pecks, 7 qt. by 24. Product. bu. pe. qt. bu. pe. qt. 3 2 7 3 2 7 6 4 22 1 2 14 3 4 4 6 89 1 0 Proof 89 1 0 OR THUS: bu. pe. qt. bu. pe. qt. 3 2 7 3 2 7 3 8 11 0 5 2" 3 0 8 3 89 1 0 89 1 0 2. Multiply 7 bushels, 3 pecks, 5 quarts, by 36. Product, 284 bu. 2 pe. 4 qt. 3. Multiply 19 bushels, 2 pecks, 3 quarts, bv 42 7 D 74 COMPOUNPD WMULTIP LICAT.ION. CASE 3. When the multiplier exceeds 12, and is NOT the product of any two factors in the multiplication table. RULE. Multiply by the two factors whose product is the least short of the given multiplier; then multiply the given sum by the number which supplies the deficiency; and add its product to the sum produced by the two factors. -EXAMPLES. 1. Multiply 21 bushels,.l peck, 7 quarts, by 23. Prod. bu. pe. qt. bu. pe. qt. 21 1 7X3 21 1 7X2 5 3 OR THUS: 107 1 3 64 1 5 4 7 429 1 4 product of 20 450 3 3 product of 21 64 1 5 product of 3 42 3 6 product of 2 493 3 1 product of 23 493 3 1 product of 23 2. Multiply 19 bushels, 3 pecks, 7 quarts, by 34. Product, 678 bu. 3 pe. 6 qt. 3. Multiply 7 bushels, 3 pecks, 4 quarts, by 59. 4. Multiply 9 bushels, 3 pecks, 2 quarts, by 47. 5. Multiply 15 bushels, 1 peck, 7 quarts, by 78. 6. Multiply 12 bushels, 2 pecks, by 92. 7. Multiply 17 bushels, 3 quarts, by 98. 8. How many bushels in 104 sacks, each containing 7 bushels, 2 pecks, 3 quarts? 9. How many bushels of wheat on 125 acres, containing 21 bushels, 3 pecks each? '~ COMrOU^ND~ BELf'tiPLICATION. r5'c'o CASE 4. I Whlen the multiplier is greater thant the product (f any I two numbers in the multiplication table. RTULE. Multiply the given number by 10, as many times less one as there are figures in the multiplier. Multiply that product by the left hand figure of the multiplier. Multiply the given sum by the units figure of the multiplier; the product of the first 10 by the tens figure of the multiplier; the hundreds product by the hundreds figure of the multiplier, and so on, till you have multiplied by all the figures of the multiplier except the left hand one. Add all the products together, and you have the product required. EXAMIPLES. 1. Multiply 3 bushels, 3 pecks, I quart, by 456. bu. pe. qt. Product 1724 bu. 1 pe. 3 3 IXG 10 Because there are 3 figures, multiply __________ 2 times by 10. Multiply that product by 37 3.. the eft hand figure (4) of the multiplier. ^ 3 2X 5 Multiply the given number by the units 10 figure (6) and set the product beneath. Multiply the 10's product by the tens 378 0 4 figure (5) of the multiplier. Add the several products. 1512 2 0 22 2- 189 0 2 1724 1 0 Product 76 COMBPOUND SULTIPMLCATIONo 2. Multiply 53 bushels 2 pecks, 7 quarts, by 23450 hbu pe, qt. 53 2 7X5 1 ~~34 O 10 l5 537 0 6+,4 F 10 1 5371 3 4+3 I Mi....3I2.1107437 2 0 product of the 2000 161 8 5 2 4 300 2148 3 0 G6" " 40 208 2 3 " c 5 125970 1 7 Product of the 2345 NOTE. —Let the pupil try experiments, by multiplying simple num11 bers in this way. M3. Multiply 72 bushels, 1 peck, 2 quarts, by 4723. Product, 341531 bu. 3 peo 6 qt. 4. Multiply 13 bushels, 2 pecks, 4 quarts, by 5124. I~u Q~stQuestions. What is compound wmultiplication? In what does it differ from simple multiplication? When the multiplier does not exceed 12, how do you proceed? How many do you always carry? I How do you prove compound multiplication? How do you proceed when the multiplier exceeds 12, and is the exact product of two numbers in the multi" plication table? When the multiplier exceeds 12, and is not the exact product of any two numbers in the table, how do you proceed?. 1~-3 —— U,_______ COMIPOUND MULTIPLICATION. 7 How do you proceed when the multiplier is greater hian the product of any two numbers in the table? AVOIRDUPOIS WEIGHT. tons. cwt. qrs. cwt. qr. lb. oz. dr. 23 12 3 7 3 14 9 6 4 6 94 11 -0 47 1 3 8 4 tons cwt. q'. ewt. qr. lb. oz. dr. 7 15 3 7 3 24 12 14 8 9 5. Multiply 7 tons, 16 cwt. 3 qr. by 24. Product, 188 T. 2 cwt. 6. Multiply 3 cwt. 2 qr. 21 lb. 14 oz. by 30. Product, 110 cwt. 3 qr. 12 lb. 4 oz. 7. Multiply 3 tons, 7 cwto 2 qr. by 34. Product, 114 tons 15 cwt. APOTHIECARIES WEIGHT. s.3 3 D 3 B gr. f.5r 3 Bgr. 48 2 1 53 10 0 2 1 17 5 6 1 4 5 9 12 23 5 3 2 ATROY 2WEIGHT. 1 lb. oz. dwt. lb. oz. dwt. g.. Ib. oz. dt. gr. 6 6 5 16 43 0 8 10 113 6 0 6 2 4 6 134 11 12 4. Multiply 41 b. 6 oz. 18 dwt. 2 gr. by 7. p01~~ ~ Ans. 291 lb. 0 oz. 6 dwt. 14 gr. 5. Multiply 91 lb. 4 oz. 14 dwt. 16 gr. Ly 8. Ans. 731 lb. 1 oz. 17 dwt. S gi 7'' 78 COMPOUND MULTIPLICATION. CLOTH MEASURE. yd. qr. na. E. E. qr. na. E.F1. qr. na. E.Fr. qr. na. 20 2 3 37 4 2 18 0 3 14 1 3 6 8 12 9 124 0 2 5. If 19 yd. 1 qr. 2 na. be multiplied by 5r what number of yards will there be? Ans. 96 yds. 3 qr.2 na. 6. Multiply 56 Ells Eng. 3 qr. by 9. Ans. 509 Ells E. 2 qr. LONG MEASURE. deg.m.f...fur. p. m.fur.. yd. ft. in. 8 1 336 4 2 29 18320 1 2 10 12 7 5 96 17 6 32 4. Multiply 6 deg. 40 m. 7 fur. by 10. Ans. 65 deg. 61 m. 2 fur. 5. Multiply 44 m. 6 fur. 20 p. by 7. Ans. 313. m. 5 filr. 20 u LAND, OR SQUARE MEASURE. a. r. p. a. r. p. a. r. p. 49 2 17 19 3 20 10 0 33 2 6- 9 99 0 34 4. How many acres will 10 men reap in one day, allcwing them 1 acre 3 roods 11 perches each? Ans. 18 A. 0 R. 30 P. 5. Multiply 63 acres 3 roods 18 perches, by 11. Ans. 702 A. 1 R. 38 P. 6. How many acres in 15 lots, containingl7acres, 2 roods, and 20 perches each? Ans. 264A. 1R. 20P. COXPOUND MIJLTIPLICATION. 79 CUBIC MEASURE. cords. ft. in. d. dft. in. 7 28 1327 19 23 1421 6 8 43 44 1050 159 1 1000 cords. ft. in. yd. ft. in. 21 56 1432 27 13 1291 7 9 5 In a pile of wood are 14 cords 92 feet; how much in 24 such piles?,Ans. 353 c. 32 ft. 6 In a cellar, are contained 42 yards 25 feet; what are the contents of 23 such cellars, Ans. 987 yd. 8 ft..... LIQUID MEASURE. lihd.gal. qt. T.hhd gal. qt.pt. pi. hhd.gal. qtpt. 843 2 1 2 16 3 1 4 119 3 1 4 10 5 34 48 0 4 Multiply 3 T. 2 hhd. 50 gal. 2 qt. by 8. Ans. 29 T. 2 hhd. 26 gal. 0 qt. 5 Multiply 4 hhd. 41 gal. 1 pt. by 10. Ans. 46 hhd. 33 gal. 1 qt. 0 pt. MOTION. sin. o sin. o 3 27 48 1 24 48 25 7 9 27 14 36 16 13 1545 3 If a planet move through, 2sin. 15 ^23' of its orbit in one day; how far will it advance in 8 days. Ans. 20sin. 30 4' 8O Cin:)POUND DIVISION. TIME. iyears. mo. wceeks d.. d'. mi. mnL. sec. j r S ~),$i65 3 14 25 36 7 S 9 5. 8 30 5 16 321 9 50 24 4 lf a man can perform a piece of work in 2 yr.; mo., how long would it take him to perform 5 such? Ans. is yr. 3 mo, 5 if a laborer dig a drain in 2 weeks, 3 clays, how loni~ a time would he require to dif 9 such drains? Ans. 21 weeks 6 days. STERLING MONEY. ~ s. a. X~ s. d. ~ s. d. 246 13 34 14 6 04 11I 11 10k 11 9 10 2713 6 54 ~ s. d. ~.s d. 4 Multiply 37 6 9k by 5 Prod. 186 13- Ilk, 5 - 56 8 74 by 9 507 17 94 COMPOUND DIVISION. COMPOUND DIVISION is -the art of dividmg a sum which consists of several denominations. CsSE I. Whern the divisor does not exceed 12:. RULE. Divide the several denominations of the given sum,, one after another, beginning with the highest, and set their respective quotien.ts underneath. When a remainder occurs, reduce it to the next lower denomination, and add it to the number of lhe next denmination, and divide the sum as before. CO'MPOUND7u DIVISION. 8 EXAMrPLES. I bu. 2e. qt. pt. 7) P25 2 1 P Here 7 into 25 buh. 3 times and 4 remain. Set down the 3. Reduce the 4 bushels to pecks, 3 2 5 1 wlhici makes 16 pecks: add 16 pecks to 2 pecks, which make 18 peoks. Now 7 into 18 pe. 2 times, and leave 4. Set downu the 2. Reduce the 4 pecks to quarts, which makes 32 qcts. Add 32 qt. to 6 qt. —makes 38 qt. 7 into 38 qt. 5 times, and 3 remain. Set clown tile 5. Reduce- the 3 qt. to pt.-nakes 6 pt., add 6 pt. to I pt. makes 7 pints. 7 into 7, 1 time. Set down the 1, ard. the work is completed. bu. pe. qt. bu. pe. qt. 2) 8 2 6 3) 9 3 6 4 1 3 4 Divide 34 bu. 3 pe. 6 qt. between 9 persons. Ans. 3 bu. 3 pe. 4 qt. 5 92 bu. 3 pe. belong equally to 7 persons;: what is the share of each? Ans. 13 bu. 1 pe. CASE 2. When the divisor exceeds 121 and is the. exact product of two numbers in the multiplication table. RULE. Divide the sum by one of the factors, and the quotient by the other. Multiply the last remainder by the first divisor, and add the first remainder for the true remainder, as in simple division, note 2. EXAMPIXES. 1 Divide 89 bu. 3 pe. 7 qt. by 28. Quotient 3 bu. 6 qt. I pt. —18 kem. 82 COMIPOUND DIVISION. bu. pe. qt. 2(4 89 3 7 28 _ ~ (7 22 1 7-3 Rem. 3 0 6-5 Rem. 4 First divisor 20 3 First rem. True rem. 23 Quarts 2 28)46 pints Pt. 1 and a rem. of 18 pints undivided. 3. Divide 78 bushels, 3 pecks, 4 quarts, among 32 persons; what will be the share of each? Ans. 2 bu. 1 pe. 6 qt. I pt. and a remainder of 24 pints undivided. CASE 3. When the divisor is more than 12, and is NOT the exact product of any two numbers in the multiplication table. RULE. Divide the highest denomination of the given sum, as in case 2, simple division; and reduce the remainder, if any, to the next lower denomination; add the number of that denomination to the result, and divide as before. EXAbMPLES. 1. Divide 77 bushels, 1 peck, 7 quarts, by 23. Quotient,,3bu. 1 pe. 3 qt. 1 pt.13 rem* COMPOUND DIVISION.3 I I|~~ ~ EXAMPLES. bu. pe. qt. bu. pe. qe. pq, t. 23)79 1 7(3 1 "6 1+3 pint remainingo 69 23)516 1 10 0 1 123)21(1 pect 23 Rem N3 pin S Ans. 169 ha. 3 Fp. S ct. I pt0 and a rein. of pint. 1. A boat toad 0o corn, containing. 40'27 bushels, 3,', pe a ra is der oned qually what do pe onsdo wt is the I sha rhe viofor exceeds, but is pruc o Anumers in the tabe,ho do you perfot. a the. opera pinto,i Qsuestons. I What i s compound division? W hen the divisor does not exceed 12, h&ow do you i' I proceed? I/ When a remainder occurs; what do you do with it? i''Where the divisor exceeds 12, but s the product of two I l numbers in the tablelhow do you perform the operation?' I -ow do you find the true remzaitnder in the latter case? I When the divisor is more than 129 and is not the pro duct of any two numbers in the table, how do you per I orm the operation? L __ _ _ _ __ _ _~ _.... _~~.... \84 COMPOU'ND DIVISION. AVOIRDUPOIS WEIGHT. tons cwt. qr. lb. lb. oz. dr. 6)37 17 3 27 7)46 12 14 6 6 1 9-1 rem. 6 10 15-5 R. tons cwt. qr. cwt. qr. lb. oz. dr.'8)92 3 3 9)75 3 23 14 12 5. A quantity of iron weighing 473 tons, 19 cwt., 3 quarters, is owned equally by 22 persons; what is the share of each? Ans. 21 T. IO cwt. 3 qr. 161b.8 oz. 11 dr. Rem. 14dr. APOTHECARIES WEIGHT. I 5.3 3B 3 3 gr. 4)23 7 5 1 5)41 6 7 2 14 5 10 7 1 8 3 6 1 2-4rem. &I 5 39 l 3 5 gr. 6)46 9 1 2 7)93 7 5 2 14 5. Divide 127Ib 33 63 into 17 equal parcels: how much in each parcel? Ans. 71b 5 63 29 16gr. 8 rem. TROY WEIGHT. lb oz. dwt. lb. oz. dwt. gr. 8)34 10 15 7)45 11 16 22 lb. oz. dwt. lb. oz. dwt. gr. 9)78 9 16 8)82 7 14 21 COMPOUND DIVISION. 85 CLOTH MEASURE. yd. qr. ( na. Ells E. qr. na. 5)27 3 i 6)37 3 2 5 2 1 6 1 1+4 iem. yd. qr. na. Etll. qr. na. 7)45 3 2 8)37 3 1 LONG MEASURE. 1. m. f. n.. p. yd. 6)37 2 2 7)46 7 17 3 6 0 7 6 5 25 2 yd. ft. ni. f p. yd. ft. 6)53 2 9 7)S7 6 23 4 2 5. A traveller has ajourney of 946 miles, 6 furlongs, to perform in 26 days; how far must he travel each day? Ans. 36 m. 3 f. 12 p. 8 rem. LAND, OR SQUARE MiEASURE. A. R. P. A. R. P. yds. 7)37 3 -27 9)423 3 28 2 5 1 26 —5 Rem. 47 0 16 13+5 Rem. 3. A farm containing 746 acres, 3 roods, 29 poles, is to be divided equally between 9 heirs; what is the share of each? Ans. 82 A. 3 R. 38 P. and 7 rem. CUBIC MEASURE. cords ft. yd. ft. in. 8)97 48 9)148 16 493 12 22 16 13 1398+7R. 139s+". 86 COMPOUND DIVISION: 3. A boat load of wood, containing 92 cords 87 feet, is to be divided between 3 persons; what is the share of each? Ans. 30 c. 114 ft. 1 rem. 4. A quantity of earth, containing 6987 yards, 25 feet, is to be removed by 29 carters; how much must each remove? Ans. 240 yd. 26 ft. LIQUID MEASURE. tuns.hhd. gal. hhd. gal. qt. pt. 5)37 3 45 6)57 36 3 1 7 2 21+3 Rem. 9 37 2 1+1 Rem, tuns iahd. gal. hhd. gal. qt. pt. 7)84 2 32 8)93 43 3 1 5. A quantity of liquor owned equally by 27 persons, the whole quantity being 431 hhd. 47 gals; what Ii the share of each? Ans. 15 hhd. 62 gal. 1 qt.; 17 rem. MOTION. Sin.' " Sin. ~ 8)9 16 45 36 9)11 23 48 54 1 5 50 42 TIME. yr. mo. we.^ da. ho. min. see 11)848 10 12)24 6c 20 32 24 77 2 2 0 13 42 42 vr. mo. da. ho. min. sec. 4)375 8 7)37 16 28 32......,............................ COMPOUND DIVISION. 87 STERLING. MONEY. ~ s. d. ~ s. d. 6)82 14 6 8)143 7 10 13 15 9 17 18 54 s. d. ~ s. d. 7)78 10 11 9)98 17 1 ~ s. d.~ s. d. 19)36 16 3(1 18 9 6 Divide 113~ 13s. 4d. by 31. What is the quotient? Ans. 3~ 13s. 4d. 7 Divtde 189~ 14s. by 95. Quotient, 1~ 19s. lld.+ PROMISCUOUS EXERCISES 1 In 35 dollars-how many cents? Ans. 3500. 2 How many miles are there in 98 furlongs? Ans. 12M. 2fur. 3 How many weeks are there in 365 days? Ans. 52we. 1 da. 4 In 84 half cents how many cents? Ans. 42 cts. 5 In 8 tons 15 cwt., how many hundred weight? Ans. 175 cwt. 6 How many perches are there in 63 roods? Ans. 2520 square per. 7 How many pounds in 157s.? Ans. ~7 17s. 8 In 175 pecks how many bushels? Ans. 43bu. 3pe. 9 In 7642 cents how many dollars? Ans. $76 42cts 10 In 103 pints how many quarts? Ans. 51qt. lpt. 11 How many minutes are there in 720 seconds? Ans. 12min. 12 In 7 hogsheads, 33 gallons, how many gallons? Ans. 474 gal. -88 S~SINGCLE RULE OF THREE. PROPORTION; OR, THE SINGLE RULE OF THREE. PROPOR N is a ality of RATIONS;n That is, four numbers are proportional, when the first has the same ratio to the second as the'rTiaI has to the FOURTH. Thus, as 12: 4:: 24: 8; or as 4 12:: 8: 24. l The ratio of 12 to 4 is 3 and the ratio of 24 to 8 is 3. Or, the ratio of 4 to 12 is 3, and the ratio of 8 to 24 is 3. Then Four numbers are proportional, when the first is as many times the second or the same part of the second, as the third is of the fourth. Or, when the ratio of-the first to the second equals the ratio of the third to the fourth. The two quantities compared are called the TERMS of the ratio: the first is called the ANTECEDENT, and the second the CONSEQUENT. n1 any series of four proportionals, the first and fourth terms are called the EXTREMES, and the second and third the MEANS. The product of the Means, equals the product of the Extremes. Thus in either series above, 12X8=96, and 24X 4=96. Now suppose we have the three first terms of a series in proportion, and wve wish to find the fourth. Divide the product of the second and third terms by the first, and the quotient will'be the fourth term. In this manner let the fourth term be found in each of the following series. 2: 4: 8 is to what?.-ns. 16. 3: 11: 9 is to what?./ns. 33. 4: 6: 6 is to what? Ans. 9. 2 9:: 8 is to what? dins. 36. 5 7:: 15 is to what'?.ns. 21..RULE. Set that number which is of the name or kind in which the answer is required, in the third place: RATIO iS the relation of one thing to another of the same kind in regard to magnitude or quantity. SINGLE RULE OF THREE. 89 And, if the answer must be greater than the third term, set the greater of the remaining two terms in the second, and the less in the first place; but, if the answer must be less than the third term, set the less in the second, and the greater in the first place. When thefirst and second terms are not of the same denomination, reduce one or both of them till they are; and, if the third consist of several denominations, reduce it to the lowest, then Multiply the second and third terms together and divide the product by the FIRST, and the quotient will be the fourth term or answer. NOTE.-The answer will be of the same denomination as the third term; and, in many instances, must be reduced to a greater denomination. EXAMPLES. 1. If four pounds of sugar cost 50 cents, what will 24 pounds cost at the same rate? Ans. $3,00. 1st Term. 2d Term. 3d Term. L,__ ^~~~~^^ ~ ^ ~ ^In this question the answer is lbs. lbs. c-s. required to be money: therefore As~ 4' 24 ~ money (the 50 cts.) must be in As 4: 24:: 50 the third place. Because 24 50 pounds will cost more than 4 pounds-the greater (24 Ibs.) 4)1200 must occupy the second place: 4 _____ and the remaining term (4 lbs.) the first 3,00 2. If 24 pounds cost 300 cents (or 3 dollars;) how many pounds may be purchased for 50 cents at the same rate? Ans. 4 lbs. cts. cts. lbs. In this question the answer is reA.s 300: 50:: 24 quired to be in pounds; therefore 50 pounds (the 24) must be in the third place. Because 50 cts. will purchase less 300)1200 than 300 cts.-the less (50 cts.) -~- must occupy the second place: and 4 the remaining term (300 cts.) the first. _________________'_______________ 90 SINGLE RULE OF TIREE. 3. Bought a load of corn containing 27 bushels, 3 pecks, at 50 cts. per bushel, what did it cost? Ans. $13,87k. bu. bu. pe. cts. I: 27 3:: 50 4 4 Because p6cks occur in the second — 4 - t/erterm, the first and second are reduced 4 111 to pecks. 50 4)5550 $13,872 4. What are 42 gallons worth, if 3 gallons 2 quarts cost $1,20? Ans. $14,40. gals. qts. gals. D. cts. 3 2 42:: 1,20 4 4 As quarts occur in the first term, - ~- the first and second are reduced to 14 168 quarts. 120 14)20160 $14,40 5. If 8 bushels 2 pecks cost $4,25, how many bushels can I purchase with $38,25? Ans. 76'bu. 2 pe. D. cts. D. cts. bit. pe. 4,25 38,25:: 8 2 34 4 As two denominations occur _______ _ in the third term, it is reduced to the less; hence the result is 15300 34 pe. pecks, which must be reduced 11475 to bushels. 425)130050(306 pe. 1275 pe. 2550 4)306 2550 76 bu. 2 pe. SINGLE RULE OF THREE. 91 6. What will 5 lbb 6 oz. 5 dwt. of silver-ware cost at $1,50 per ounce? Ans. $99,371, oz. loz. z. dwt. D. cts. I: 5 6 5:: 1,50 20 12' o2 i2 As dwts. are in the second term, the first and second must be reduced to dwts. 20 66 20 1325 150 66250 1325 20)198750 $99,377. When 3 yards and 8 feet of plastering cost $1,40, what will be the cost of 16 yards? Ans. $5,76. yds. ft. yds. D. cts. 3 8 16:: 1,40 9 9 35 144 140 5760 144 35)20160(5,76 cts. S. How many yards of cloth can be purchased for 95 collars, if 4 yd. 3 qr. cost $9,50? Ans. 47 yd. 2 qr.; or 47' yd. $ ct. $ ct. yd. qr. qr. Ae 9,50: 95,00: 4 3 4)190(47 yd. 2 qr. 92 SINCLE RULE OF THREE. NOTE. —The operation may, in many instances, be contracted by dividing the second or third term by the first; or the first by either of the others, or by any number that will divide the first and either of the others without a remainder; and, using tho quotients instead of the original numbers. 9. If 24 yards cost 896, what will 8 yards cost? Ans. $32. yds. yds. D. yds. yds. D. 24a: 8a:: 96c or 24a:8:: 96a _- 4. 3c Ans, 32. - 4 32 10 If 36 bushels cost 1T72; what will 12 bu. cost? Ans. $24, bu. tb. D. buc. bu. D. 3,a: 12a:: 72c 12)36: 12:: 72a 3c 24 3a 1 24 APPLICATION. 1 When 4 bushels of apples cost $2 25, what must be paid for 20 bushels? Ans. $1-,25. btu. bu. D. cts. 4: 20:: 2,25 2 H-ow many yards of cloth can I buy for $60, when 5 yards cost $12? Ans. 25 yds. D. D. yds. 12: 60:: 5 3 If 6 horses eat 21 bushels of oats in a given time; how much will 20 horses eat in the same time? An3. 70 bu. 4 If 20 horses eat 70 bushels of oats in a certain time; how much will 4 horses eat in the same time? Ans. 14.bu. 5 If a family of ten persons use 7 bushels 3 pecks of wheat in a month; how much will serve them when there are 30 in the family? Ans. 23 bu. 1 pe. 6 If 14 lbs. of sugar cost 75 cents, how many pounds can be bought for three dollars? Ans. 56 lbs. 7 If 4 hats cost 12 dollars, what will 27 kats cost at the same rate? Ans. $81. SINGLE RULE OF THREE. 93 8 If 20 yards of cloth cost $85, what will 324 yards cost at the same rate? Ans.. $1377. 9 If 2 gallons of molasses cost 70 cents, what will 2 hogsheads cost? Ans. $44,10 10 If 1 yard of cloth cost $3,25 cts., what will be the cost of 6 pieces, each containing 12 yds. 2 qrs.? Ans. $243,75 cts. 11 If 3 paces or common steps of a person be equal to 2 yards, how many yards will 160 paces make? Ans. 106 yds. 2 ft. 12 If a person can count 300 in 2 minutes, how many can he count in a day? Ans. 216000. 13 What quantity of wine at 60 cts. per gallon can be bought for $37,80 cts. Ans. 63 gal. 14 If 8 persons drink a barrel of cider in 10 days, how many persons would it require to drink a barrel in 4 days? Ans. 20. 15 If 8 yards of cloth cost $12, what will 32 yards cost? Ans. $48. 16 If 3 bushels of corn cost $1,20, what will 13 bushels cost? Ans. $5,20. 17 If 9 dollars will buy 6 yards of cloth, how many yards will 30 dollars buy? Ans. 20. 18 If a man drink 3 gills of spirits in a day, how much will he drink in a year? Ans. 34 gal. 1 pt. 3 gi. 19 If 12 horses eat 30 bushels of oats in a week, how many bushels will serve 44 horses the same time? Ans. 110. 20 If a perpendicular staff 6 feet long, cast a shadow 5 feet 4 inches, how high is that tree whose shadow is 104 feet long at the same time? Ans. 117 feet, 11 EXERCISES. 1 If 12 acres, 2 roods, produce 525 busheli of4 corn, i how many bushels will 62 acres, 2 roods produce? Ans. 2625 bu. 2 If 7 men plough 6 acres, 3 roods in a certain time, I how many acres will 96 men plough in the same time? Ans. 92 A. 2 R. 11 Per. 12 yd.-,jj 94 SINGLE RULE OF THREE. 3 Suppose 3 men lay 9 squares' of flooring in 2 days; how many men must be employed to lay 45 squares in. the same time? Ans. 15 men. 4 If 7 pavers lay 210 yards of pavement in one day; how many pavers would be required to lay 120 yards in the same time? Ans. 4 pavers. 5 If 2 hands saw 360 square feet of oak timber in 2 days; how many feet will 8 hands saw in the same time? Ans. 1440 feet. 6 An engineer having raised a certain work one hundred yards in 24 days, with 5 men; how many men must be employed to perform a like quantity in'15 days? Ans. 8 men. 7 If 3 paces or common steps be equal to 2 yards; how many yards will 160 such paces make? Ans. 106 yd. 2 ft. 8 If a carriage wheel in turning twice round, advance 33 feet 10 inches; how far would it go in turning round 63360 times?.Ans. 203 miles. 9 Sound flies at the rate of 1142 feet in 1 second of time; how far off may the report of a gun be heard in 1 minute and 3 -seconds? Ans. 13 miles, 5 fur.,0 poles, 2 yd. 10 If a carter haul 100 bushels of coal at every 3 loads; how many days will it require for him to load a boat with 3600 bushels, suppose he haul 9loads a day?' Ans. 12 days. bu. bu. da. da. As 300: 3600:: I 12. 11, If 8 men can reap a field of wheat in 4 days; how many days will it require for 16 men to do it? Ans. 2 days. 12 Sold 10 yards of linen at 5 dollars 50 cents; what was it a yard? Ans. 55 cents. *A SQUARE is 10 feet long and 10 feet wide, or 100 square feet. This measure is employed in estimating the quantity of floorng, roofi ng, weather-boarding, &c.:I SINGLE RtULE 0o THREE. 95 13 If 7 pounds of cheese cost87k cents; what must I pay for 122 pounds? Ans. 15 dol. 25 ct. 14 If 1 ounce of silver cost 72 cents; what will 3 pounds 5 ounces come to? Ans. 29 dol. 52 ct. Why do you multiply the second and third terms to. gether and divide by the first? What will 24 pounds If 4 pounds cost 50 cents, divide the of bacon cost at 50 c. 50 cents by 4, gives the price of o er 4 po s 1^ 1I pound: thus 4 into 50 122. If /or every 4 po Unds?. 1 pound cost 121 cents, 2 pounds lb. lb. ct. ct. will cost twice that;. three pounds As 4: 24:: 50: 300 three times, and 24 pounds will cost 24 times 12k ct.; that is 300 cts. or 3 dol. cts. If the second and third terms be 4) 50 multiplied together, and their product 4!)l ^ z~ ~ divided by the first, the result will be the same as it is when. the third is di124 the price of 1 Ib. vided by the first, and the quotient 24 multiplied by the second. 50 25 300 ct. the price of 24 lb. 15 If 15 yards of broad cloth cost 80 dollars; what will 75 cost. Ans. 400 dollars. 16 A man bought A. yards linen for 82 50 cts.; what is the worth of 1 qr. 2 na. at the same rate. Ans. 62i ct. 17 If 321 bushels of salt cost $240 75 cents; what was it per bushel? Ans. 75 cents. 18 If the moon move 13 deg. 10 min. 35 sec. in one day; in what time does it perform one revolution? Ans. 27da. 7 hrs. 43 min. deg. min. sec. deg. da. As 13 10 35:360::1 96 SINGLE RULE OF THREE. 19 If a staff 4 feet long, cast a shade 7 feet on level ground; how high is a steeple whose shade is at the same time 198 feet. Ans. 1131 20 If a man's annual income be 1333 dollars, and lie spend 2 dollars 14 cents a day; what will he save at the end of one year? Ans. $551 90 ct. 21 Suppose A. owes B. 791 dols. 60 ct., and can pay only 37& cts. on the dollar; how much must B. receive? Ans. $296 85 ct. 22 Bought 3 casks of raisins, each containing 3 cwt. 1 qr. 14 lb.; how much did they cost at $6 21 ct. pet cwt.? Ans. $62,87i PROPORTION-DIRECT AND INVERSE. Hitherto, proportion has been treated in general terms; it now remains to consider the two kinds, DIRECT and INVERSE. DIRECT PROPORTION is that in which more requires more, or LEss requires LESS. Thus: yd. yd. dol. If 2 yd. cost 4 dol., 124 yd. being As 2: 124:: 4 more than 2 yd., will cost more than 4 dol. yd. yd. dol. And, if 124 yd. cost 248 dol.; 2 yd. As 124: 2: 248 being less will cost less. That is, more yards require more money, and less yards cost less money INVERSE PROPORTION is that in which more requires less; and less requires more. Thus: If 12 men built a wall It is supposed that 12 men perform in 4 davs; how many a piece of work in 4 days: a like c o. ay piece of work is to be done in 8 days; men can do it in 8 days? this will require a less number of men: that is, more days require less men. STATED. Here it is supposed that 12 men daa. a. m. m. performed a piece of work in 8 days: As 4: 8 inversely:: 12. 6 a like piece is to be done in 4 days; this will require more men. That is, Ca: more days require less men, and less da. da. m. m. days require more men. As8: 4 directly:: 12. 6 SINGLE RULE OF THRE3SE 9 All the past exercises in proportion are Direct-the following will be INVERSE PROPORTION. Questions in Inverse Proportion, may be stated and solved by the same rule that is given for Direct Proportion. EXERCISES. 1 If 6 mowers mow a meadow in 12 days; in what time will 24 mowers do it? Ans. 3 da. 2 If a man perform a journey in 6 days, when the days are 8 hours long; in what time will he do it when they are 12 hours long? Ans. 4 da. 3 If, when wheat is 83 cents a bushel, the cent loaf weighs 9 ounces; what ought it to weigh when wheat is $1 24A cts. a bushel? Ans. 6 oz. 4 If 100 dollars principal in 12 months gain 6 dollars interest; what principal will gain the same interest in 8 months? Ans. $150. 5. If 12 inches long and 12 inches wide, make I square foot; how long must a board be that is 9 inches wide, to make 12 square feet? Ans. 16 ft. 6 A. lent B. 500 dollars for 6 months; how long must B. lend A. 220 dollars to be equivalent? Ans. 13 months, 19 days.+ 7 There is a cistern having a pipe that will empty it in 12 hours; how many pipes of the same capacity will empty it in 15 minutes? Ans. 48 pipes. 8 A certain building was raised in 8 months by 120 workmen, but the same being demolished, it is required to be rebuilt in 2 months; how many workmen must be employed? Ans. 480 men, 9 If for 48 dollars 225 cwt. be carried 512 miles; how many hundred weight may be carried 64 miles for the same money? Ans. 1800 cwt. *A month is estimated at 30 days, unless a particular month t referred to. 9 E 98 SINGLE RULE OF THREE. 10 If 48 men can build a wall in 24 days; how many men can do it in 192 days? Ans. 6 men, 11 How many yards of carpeting 2 ft. 6 in breadth., will cover a floor that is 27 feet long and 20 feet wide? Ans. 72. 12 What quantity of shalloon that is 3 quarters wide, will line 7A yards of cloth that is 1 yd. wide? Ans. 15 yd, 13 How many yards of matting that is 3 quarters wide, will cover a floor that is 18 feet wide and 60 feet long? Ans. 160. 14 In what time will $600 gain the same interest that $80 will gain in 15 years? Ans. 2 years. Questions. What is proportion? When is the proportion direct? When is it inverse? Why is the proportion inverse in the last question? A. because it is more money requiring less time. Why is the proportion inverse in the 11th question? A. because the shalloon is narrower than the cloth; that is less width requiring more length. Why is the 10th question inverse? PROMISCUOUS EXEICISES. 1 A certain steeple standing upon level ground, casts a shadow to the distance of 633 feet 4 inches, when a staff 3 feet long, perpendicularly erected, casts a shadow of 6 feet 4 inches; what is the height of the steeple? Ans. 300 ft. 2 A ship's company of 15 persons is supposed to have bread enough for a voyage, allowing each person 8 ounces a day, when they take up a crew of 5 persons, with whom they are willing to share; what will be the daily allowance of each person now? Ans. 6 oz. 3 Bought 215 yards of broad cloth at 6 dollars a yard; SINGLE RULE OF THREE. 99.tE~1S~~ 599 what was the prime cost, and how must I sell it per yard to gain $135 on the whole. Ans. prime cost $1290,00; to be sold for $6,621 per yard. 4 If 100 men can complete a piece of work in 12 days; how many can do it in 3 days? Ans. 400 men., -f a board be 4A inches wide,; how long a piece will it take to make I square foot? Ans. 32'in. 6 A pole, whose height is 25 feet, at noon casts a shadow to the distance of 33 feet 10 inches; what is the breadth of a river which runs due East at the bottom of a tower 250 feet high, whose shadow extends just to the opposite edge of the water? Ans. 338 ft. 4 in. 7 A plain of a certain extent having supplied a body of 3000 horses with forage for 18 days; -how long would it have supplied 2000 horses? Ans. 27 da. 8 A piece of ground 1 rod wide and 160 rods long, makes 1 acre; how wide a piece must I have across the end of a farm 32 rods wide to make an acre'? Ans. 5 rods. 9 I have a floor 24 feet long, and 15 feet wide, which I wish to cover with carpeting that is 3 quarters of a yard wide. how many yards must I buy. Ans. 53 yards, 1 foot. 10 How much land at $2,50 an acre must be given in exchange for 360 acres worth $3,75 an acre? Ans. 540 acres. 11 What is the weight of a pea to a steel-yard, which is 39 inches from the centre of motion, will balance a weight of 208 lbs., suspended at the draught end'3 quarters of an inch? Ans. 4 lb. 12 If $28 will pay for the carriage of 6 cwt. 150 miles; how far should 24 cwt. be carried for the same money? Ans. 37k miles. 100 DOUBLE RULE OF THREE. COMPOUND PROPORTION; OR THE D3OUBLE RULE OF THREE. DIRECT AND INVERSE. COMPOUND PROPORTION is two or more series of proportionals combined. Five, seven, nine, or other odd number of terms, is always given to find a sixth, eighth, or tenth, &c., or answer. Rule for the Statement. Place the numbers that is of the denomination in which the answer is required to be, in the thirdplace. Then: Consider separately each pair of similar terms and place them agreeably to the rule for SIMPLE PROPORTION. OR,Work by two separate statements in simple proportion'. Rule for the Solution. Reduce the several pairs of terms to similar denominations as in single proportion, and the last to the lowest denomination given: Then Multiply the two initials, or left hand terms together for a DIVIsOR, and the other three for a DIVIDEND. Divide the latter by theformer, and the quotient wilt be the answer, in that denomination to which the thirdj Ierm was reduced. EXAMPLES. 1. If 6 men in 8 days build 40 rods of wall, how mucd will 18 men build in 20 days? Ans. 300 rods. * It would be well for the pupil to work each sum both ways. DOUBLE RULE OF THREE. 101 Men men The answer is required to be As 6: 18 rds given in rods: then rods must be' A j the third term. If 6 men build da.. da.': 40g 40 rods, 18 men will build more,; AS 8: 20 then more (18 men) must occupy -, - ~the seconds and less (6 men) the 48 3860 first place. 40 If 8 days produce 40 rods, 20 days will produce more; then more (20 days) must occupy the 48)14400(300 second, and less (8 days) thefirst 144 place. N. B. The first pair, or two upper terms must be alike. Also 00 the lower pair must be alike.That is, both must be men or both days, both hours or both bushels, &c. 2. If 6 men in eight days eat 10lb, of bread, how much will 12 men eat in 24 days? Ans 60. men 6: 12) 10 lb. days 8: 24) Contracted. 6:12 2 288 8 24 3' lO b. 10 6 48)2880(60 Ans. 10 288 60 Ans. 0 3. Suppose 4 men in 12 days mow 48 acres, how many acres can 8 men mow in 16 days? Ans. 128A. 4. If 10 bushels of oats be sufficient for 18 horses 20 days, how many bushels will serve 60 horses 36 days, at that rate? Ans. 60bu. 5. Suppose the wages of six persons for 21 weeks be 288 dollars, what must 14 persons receive for 46 weeks? Ans. $1472. 6. If the carriage of Scwt. 128 miles cost $12.80, what must be paid for the carriage of 4cwt. 32 miles? Ans. $1.60. 7. If 371b. of beef be sufficient for 12 persons 4 days, how many lb. will suffice 38 men 16 days? Ans. 4681b. 102 oz. 9* 102 DOUBLE RULE OF THREE. 8. If a man can travel 305 miles in 30 days, when the days are 14 hours long, in how many days can he travel 1056 miles, when the days are 12- hours long? Ans. 116 days.-2540. 9. If the carriage of 24cwt. for 45 miles be 18 dollars, how much will it cost to convey 76cwt. 121 miles? Ans. $153 26 cts.+720. 10. A person having engaged to remove 8000cwt. in 9 days; removed 4500cwt, in 6 days, with 18 horses: how many horses will be required to remove the balance in the remaining 3 days? Ans. 28 horses. 11. If 3 men reap 12 acres 3 roods in 4 days 3 hours, how many acres can 9 men reap in 17 days? Ans. 153 acres. men men 3: 9 a. r. Analysis. d. h. d.:: 12 3 If 3m. reap 12 a. 3 r. 4 3:17 4 1m. rea 4a. r. and 12 12 9 m. reap 38 a. 1 r. -_- i51 51 204 If 4 d. 3h. reap 38 a. 3 9 I r. I d. reap 9 a. and ~ — - --- 17 d. reap i53 a. Ans. 153 1836 51 1836 9180 153)93636(612 roods, or 153 acres 918 183 153 306 306 *The day is here estimated at twelve hours' PRACTICE. 103 12. If 40 men build 32 rods of wall in 8 days, working 10 hours each day; in how many days will 60 men build 48 rods, working 12 hours a day? Ans. 6 days, 8 hours. Men men Multiply all the initial 60: 40 terms (or 60, 32, and 12) rods rods days together for a divisor: and 32: 48: 8 the other four for a divihours hours dend. 12: 10 13. If 36 men dig a cellar 60 feet long, 24 feet wide, and 8 feet deep, in 16 days, working 16 hours per day, how many men can dig a cellar 80 feet long, 40 feet wide, and 12 feet deep, in 20 days, working 12 hours per day? Ans. 128. Questions. What is compound proportion? How do you state questions in compound proportion? Which terms do you multiply together for a divisor? Which for a dividend? What other method is there? PRACTICE. PRACTICE is a short and expeditious method of performing various calculations in business. CASE 1. When the given price is LESS than one dollar. RULE.-Set down the given number as one dollar, and take such aliquot part* or parts of that number, as the price is of one dollar, for the answer. *An aliquot part of a number is any number that will divide.t without a remainder; thus 4 is an aliquot part of 20; and 8 of 40; and 25 cents is an aliquot part of 100 cents, (or $1.) because 25 cts. are contained in 100 cts., an even number of times, without a remainder. 104 PRACTICE. TABLE OF ALIQUOT PARTS. CTS. CWT. ROODS. 50 L 10 roods 33- 1 5 2 7 3 41 1 3 25 1 4' a on i 4 8f 20 206 t 2 8 a2 12 1 1 J J perches 10 g qr. 20 6" 21 10 J 4. 1 Ibs. 10 5 T o 10 8 1 4 a bushel? Ans. 50 cts. cts. $ 25 - 826 82 a bushels, at one dollar a bushel, will cost 86 dollars: at 25 cents, or of a $ 206,50 dollar, it will cost one fourth as much. 0 2:'am aa00 2. What will 934 gallons of molasses cost, at 50 cts. $ 467I 3. What will 1832 bushels of salt cost, at 5 cents a bushel? Ans. $1374. cts.ts the cost will be 5'1 S126 82 as much as-at one dollar. 5 1 At 25 cents the cost will be 2 25 ~ as much as-at 50: cts. 916 cost at 50 c. 458 cost at 25 c.cent of a $ 1374 cost at 75 c. cts. $1374 cost at 75 c. PRACTICE. 105 cts. As before the cost at 50 cts. will 50 I 1832 be 1 as much as-at 1 dollar. a25^ 12~ ~At 25 cts. it will be i as much 425 as-at 1 dollar. 916 ct. at 50. 458 ct. at 25. $1374 ct. at 75. 4. What wili 680 pounds of sugar cost, at 1 0 cents a pound? $68. 5. What will 742 pounds of pork cost, at 6- cts. a pound? Ans. $46 37- cts. 6. What must I pay for 371 pounds of bacon, at 121 cts. a pound? Ans. $46 37i cts. 7. How much will 8750 bushels of rye cost, at 62cts. a bushel? Ans. $5468 75cts. S. How much must be paid for 4360 square feet of marble, at 87~ cents a foot? Ans. $3815. 9. What will 468 square yards of plastering cost, at 181 cents a yard? Ans. $87 75 cts. 10. How much will be the cost of laying 856 perches of stone, at 931 cents a perch? Ans. $802 50 cts. 11. What will the digging of a cellar, containing 180 cubic yards, cost, at 20 cents a, yard? Ans. $36. 12. What will be the cost of hauling 248 cords of wood, at 31i cents a cord? Ans. $77 50 cts. 13. What must be paid for 432 perches of stone, at 37i cts. a perch? Ans. $162. 14. How much must be given for 724 days labor, at 564 cents a day? Ans. $407 25. 15. What will 742 bushels cost at 10 cts. Ans. $74 20 16. 732 15 109 80 17. 732 20 146 40 18. 475 25 118 75 19. 684 30 205 20 20. 756 35 264 60 21. 927 40 370 80 22. 824 50 412 00 23. 682 55 375 10 24. 341 60 204 60 25. -784 70 548 80 r 2 106 PRACTICE. 28. What will 352 busnels cost at 61 cts.? Ans.$22 00. 29. 436 12k 54 50 30. 724 18 135 75 31. 956 31i 298 75 32. 742 37k 278 25 33. 274 438 119 87 34. 732 56t 411 75 35. 845 62i 528 12i 36. 684 681 470 25 37. 274 81t 222 62i 38. 386 931 361 87 CASE 2. When the given price is MIoRE than one dollar. RULE.-Multiply the given sum by the number of collars, and take the aliquot part or parts for the cents, as in Case 1. EXAMPLES. 1. What will 342 cords of wood cost, at 3 dollars 75 cents a cord? Ans. $1282 50. cts. $ 50 I 342 342 cords at $1, will cost $342; at $3 3 it will cost 3 times $342; at 50 cts. it will cost A as much as it will at $1.; and at 25 cents, k as much as it will at 50 cts.; 1026 which added together, will be the cost at 25' 171 $375. 85 50 1282 50 2. What will 250 acr. cost at $4 62k Ans. $1156 25 3. 435 5 87k 2555 621 4. 273 6 12k 1672 12k 5. 942 7 37i 6947 25 6. 846 3 681 3119 62i 7. 957 5 75 5502 75 8. 236 6 934 1637 25 9. 754 3 56* 2686 126 10. 932 27 25 25397 00 PRACTICE. 107 CASE 3. Witen the given quantity consists of several denominations. R-ULE.-Multiply the given price by the number of hundred weight, acres, yards, or pounds, &c. and take the aliquot parts for the quarters, roods, feet, or ounces, &c. EXAMPLES. 1. What will 240 acres, 2 roods, 10 perches, cost at $ 5 25 cents an acre? Ans. $3668 578 cts. 2 r. - 1525 240 61000 3050 10 p. 1 762k 95i+-2 rem. 3668571 2. What will 29 yards, 4 feet, of stone pavement cost, at $2 25 cents a yard? Ans. $66 25 ots. 3 square feet 1 225 29 2025 450 1 1 75 25 6625 108 PRACTICE. 3. What will 32 pounds 8 ounces of silver cost, at $15,62k a pound? Ans. $510 41k. 6 oz.Troy $1562k 32 16 3124 4686 2 oz. j 781: 2604+2Rem. 510 41z 4. What will 27 cwt. 3 qrs. cost, at $23 50 cts. a cwt.? Ans. $652 12a. 5. What will be the cost of 47 lb. 10 oz. (Troy) at $1 25 cts. Ans. $59 79. 6. What will 64 yds. 3 qrs. cost, at $2 25 a yard? Ans. $145 681. 7 Sold 83 yards 2 qrs. of cloth at $10 50 a yard; what does it amount to? Ans. $876 75. 8. What will the laying of 28 squares, 75 feet of flooring cost, at $2 25 cts. a square? Ans. $64 681. 9. What is the cost of 27 cords, 96 feet of fire wood, at $3 75 a cord. Ans. $104 06A 10. What is the value of 428 gals. 3 qts. at $1 40 cts. a gallon? Ans. $600 25 cts. 11. What is the value of 765 gals. 3 qt. 1 pt. at $2 184 cents a gallon? Ans. $1675 344 cts. 12. What is the value of 5 hhds. 31k gals. at $47 12 cts. a hogshead? Ans. $259 16 cts. 13. What is the value of 17 hhds. 15 gals. 3 qts. at $64 75 cts. per hogshead? Ans. $1116 93 cts. 7m. 14. What is the value of 120 bu. 2 pecks, at 35 cents a bushel? Ans. $42 17 cts. 5 m. 15. What is the value of 780 bu. 2 pecks, 2 qts. at $1 17 cts. a bushel? Ans. $913 25 cts.+ 16. What is the value of 1354 bu. I peck, 5 qts. 1 pt. at 25 cts. a bushel? Ans. $338 60 cts. 5m.+ 17. What is the value of 35 acres 2 roods 18 perches, at 1 doll;ars 35 cts. an acre? Ans. $1935 53 cts. 9m. ITARE AND TRET. 109 Questions What is practice? What is the rule for the solution of questions in practice? What is an aliquot part? Are 50 cts. an aliquot part of 100 cent-s What part of t1 is fifty cents? What part of $1 is 331 cents? What part of $1 is 25 cents? What part of,1 is 12i cents.? What part of $1 is 10 cents? What part of $1 is 20 cents? What part of $1 is 5 cents? What part of $1 is 4 cents? What part of' 1 is 61 cents-? TARE AND TRET. TARE AND TRET are allowances made on the weight of some particular commodities. Gross weight is the weight of the goods, together with the vessel that contains thefm. Tare is an allowance for the weight of the vessel. TretI is an allowance of 4 lb. for every 104, for waste &-c. Neat weight is the weight of the goods, after all allowances are made. RULE. Subtract the tare from the gross, and the remainder is the neat weight. EXAMPLES. 1 Bought a chest of tea, weighing gross 63 lb., tare 8 lb.-what are the neat weight and value, at 85 cents per lb? *'As tret is never regularly allowed in this country; no account of it is taken in this work. 1.0 ^ 110 TARE AND TRET. lb. 85 ct. 63 gross-or, weight of the chest and tea 55 lb. 8 tare-or, weight of the chest 425 55 neat-or, weight of the tea itself 425 $46,75 value. 2 Bought 5 bags of coffee, weighing each. 97 lb. gross, tare of the whole 7 lb.-what are the neat weight and value, at 25 cents per lb.? Ans.478 lb. neat —119S50. 3 The gross weight of a hogshead of sulphur is 1344 lb.; the tare 138 lb.-what are the neat weight and its value, at $4,75 per 100 lb.? Ans. 1206 lb. neat —$57,28. 4 Bought 3 barrels of sugar, weighing as follows, viz: 236 lb. gross, 23 lb. tare-217 lb. gross, 22 lb. tare225 lb. gross, 23 lb. tare-what are the neat weight and value, at $8 per 100 lb.? Ans. 610 lb. neat —$48,80. 5 Sold 3 hogsheads of sugar- weighing each 2 cwt. 2 qrs. 14 lb. gross; tare 2 cwt. I qr. 27 lb.-what are the neat weight and value, at $11,50 per cwt..? Ans. 35 cwt. 1 qr. 15 lb. neat —$406 91 cts. 6 What is the neat weight of 15 tierces of rice, weighing 48 cwt. 3 qrs. 12 lb. gross;, tare 6 cwt. 12 lb, and what is the value, at $5,25 per cwt.? Ans. 42 cwt. 3 qrs. neat —224,434. 7 What is the neat weight of 28 hogsheads of tobacco, weighing 201 cwt. 3 qrs. 12 lb. gross; tare 3140 lb.; and what does it come to at $5 per cwt? Ans. 173 cwt. 3 qrs. 8 lb —869 10, cts. 8 Bought 17 bags of grain, weighing 3561 lb, gross; tare 2 lb. per bag-what is the neat? Ans. 3527 lb. 9 What is the neat weight of 16 bags of pepper, each weighing 65 lb. gross; tare la lb. per bag-and what is the amount at 30 cents per lb.? Ans. 1016 lb. neat —$304,80. TARE AND TRET. 111 10 In 14 hogsheads of sugar, weighing 89 cwt. 3 qrs. 17 lb. gross; tare 100 lb. per hhd.-how much neat weight, and what is its value, at $9 per cwt.!9 Ans. 77 cwt. 1 qr. 17 lb. neat —,696,61i. 11 What are the neat weight and value of 16 hhds. of tobacco, each weighing 5 cwt. I qr. 19 1b. gross; tare 101 lb. per hhd., at -2 Os. IOd. per cwt.? Ans. 72 cwt. I qr. 4 lb. neat-. 169. 5s. 4id. 12 Bought 6 hhds. of sugar, each 1126 lb. gross,; tare 117 lb. per hhd. —what are the neat weight and value at $8,75 per cwt.-? Ans. 6054 lbs. $52972i. 13 What are the neat weight and cost of a hogshead of sugar weighing gross 986 lb.; tare 12 per cent, (or 12 lb. for every 100 lb.) at $8 per neat hundred pounds? lb. lb. lb. ib. i. lb. lb. ib. As 100.: 986,:.: 12: 118 Or as 100: 88::986.: 868, lb. lb. 986 gross. 868 118, tare.. 8 dol. 888, neat weight. $69,44 the value. 14 What are the neat weight and value of 4 hhds. of sugar weighing gross 45001b. tare 12 lb. per cent. at $8, 75 per cent.? Ans. 3960 lbs. neat -$346,50. 15 Bought 10 hhds. of sugar, each 920 lb. gross; tare 10 lb. per cent.-what are the neat weight and value at $9,25 per cwt.? Ans. 8280 lb. neat- $765,90. 16 Sold 3 casks of alum, each 675 lb. gross.; tare 13 lb. per cent.-what are the neat weight and value at $4, 25 per cent. Ans. 1762 lb. neat-$74,87.4375. Or, 1762 lb. neat-$74,88. nearly. 17 What is the neat weight of 48001b.gross: tare 12 lb. per cent.? Ans. 4224 lb. 18 What are the neat weight and value of 4 hhds. of sugar, each 12 cwt. 1 qr. 14 lb. gross; tare 12 lb. per cwt. at $12,20 per cwt.? Ans. 44 cwt. 22 i1 neat-$539 19I cts. 112 INTEREST. 19 Bought 17 hhds. of sugar, weighing 201 cwt 2 qrs. 13 lb. gross; tare 10 lb. per cwt —what are the neat weight and value at-$14 per cwt.? Ans. 183 cwt. 2 qrs. 13 lb. neat- $2570 62i cts. INTEREST INTEREST is an allowance made for the use of money. Principal is the sum for which interest is to be cor- puted. Rate per cent. per annumr is the interest of 100 dollars for one year. Amount is the principal and interest added together. CAsE 1. When the time is one year and the rate per cent. is any; number of dollars. RULE.-Multiply the principal by the rate per cent., and divide by 100; the quotient will be the interest for one year. EXAMPLES. 1. What is the interest of 500 dollars for 1 year, at 6 per cent. per annum? $500 6 100~ $30100 Ans. 2. What is the interest of 225 dollars for I year, at 7 dollars per cent. per annum? Ans. $15 75. 3. What is the interest of 384 dollars 50 cents, for I year, at 5 dollars per cent. per annum? Ans. $19 22,. 4. What is the amount of $275 for 1 year, at 6 per cent. per annum? Ans- $291 50. $275 6 16,50 interest 275,00 principal $291,50 amount INTEREST. 113 5. What is the interest of 1654 dollars 81 cents for 1 year, at 5 dollars per cent. per annum? Ans. $82 74-. 6. What is the interest of 1500 dollars, for I year, at I dollar per cent. per annum? Ans. $7 50. 7. What is the amount of $736, at 6 per cent. per annum, for 1 year. 780 dols. 16. 8. What is the interest of 524 dollars, for 1 year, at 5a dollars per cent. per annum? Ans. $27 51. 9. What would be the interest of 842 dollars, for I year, at 5. dollars per cent. per annum? Ans. $46 31. CASE 2. When the interest is required for several years. RUILE.-Find the interest for one year, and multiply the interest for one year by the number of years. EXAMPLES. 1. What is the interest of 500 dollars, for 4 years, at 6 dollars per cent. per annum? $500 6 30 00 4 $12000 Ans. 2. What will be the interest of 540 dollars, for 2 years, at 5 dollars per cent. per annum? Ans. $54 00. 3. What would be the interest of 482 dollars, for 7 years, at 6 dollars per cent. per annum? Ans. $202 44. 4 What is the amount of $736 81t with 7 years, nine months interest due on it, at 6 per cent. per annum? Ans. $1079 431. Note.-Ifthe interest is required for years and months, multiply the interest for 1 year by the number of years, and take the aliquot parts of the interest for 1 year, for the months. 10* 114 INTEREST. $736 814.~~ G~6 m'ro. 1 4420,874 interest I year. y7 3 mo. - 30946,121 interest 7 years 2210.431 interest 6 months 1105,213+ 34261,78 interest for 7 yr. 9 mo. 73681,25 principal ~_-[_ —- 107943 03 amount 5. What is the amount of $362 25 for 4 years 6 mo. at 6 per cent. per annum? Ans. $460 05i. CASE 3. When the interest is required for any number of months, weeks or days, less or more than one year. RULE.-Find the interest of the given sum for one year Then, by proportion, As I year Is to the given time, So is the interest of the given sum (for 1 year) To the interest for the time required. Or take the aliquot parts of the interest for one year, for the given time, as in note, Case 2. EXAMPLES. 1. What is the interest of $560 for 2 years and 6 mo. at 5per ct. per annum? Ans $70. 568 5 6 mo. i 2800 interest for 1 year 2 years 5600 1400 $70 00 interest for 2 years 6 months. INTEREST. 115 2, What is the interest of 325 dollars, for 4 years and; 2 months, at 4 dollars per cent. per annum? Ans. $54 16 cts. 6m.. 3. What is the interest of 840 dollars for 5 years and 3 months, at 4 dollars per cent. per annum? Ans. $176 40. 4. What is the interest of 840 dollars, for 5 years and 4 months, at 7 dollars per cent. per annum? Ans. $313 60. 5. What is the interest of 560 dollars, for 4 months, at 6 dollars per cent. per annum? 1560 i( 6 5 m. m. acts. $ c ts., cts. 100)33 60 As 12:4::33 60: 1120 Ans. 6. What is the interest of 1200 dollars, for 15 weeks, at 5 dollars per cent. per annum? Ans. $17 30. 7. What will be the interest of 240 dollars, for 61 days, at 4} dollars per cent. per annum? Ans. $1 90.+ 8. What is the interest of $1000, for 14 months,. at 7 per cent. per annum? Ans. $81 66g. 9. What is the interest of 450 dollars, for 6 months and 20 days, at 5i dollars per cent. per annum? Ans. $13 75. 10. What is the interest of 375 dollars 25 cents, for3 years 2 months 3 weeks and 5 days, at 6 dollars per ct. per annum? Ans. $72 92..+ 11. What is the amount of $736 for 28 weeks, at 10 per cent. per annum? Ans. $775 63. CASE 4.: To find the interest of any sum for any number of days, as computed at banks. RULE.-Multiply the dollars by the number of days, and divide by 6; the quotient will be the answer in mills. The interest of any number of dollars for 60 days, at 6 per cent. will be exactly the number of cents; and if any other rate per cent. is required, take aliquot parts, and add or subtract according as the rate per cent. is more or less than 6. 116 INTEREST. EXAMPLES. 1. What is the interest of 563 dollars, for 60 days, at 6 per cent. per annum-and likewise at 7 per ct. per an.? $563 Ans. $5,63 at 6 per cent. 60 $6,56.8 at 7 per cent. 6)33780 tin.at) - 6per 5630 mills, I 1 $5630 cent.) 938 Interest at 7 per cent. 6568 mills. 2. What is the interest of 854 dollars, for 30 days, at 6 per cent. per annum? Ans. $4 27. 3. What is the interest of 1100 dollars, for 48 days, at 6 per cent. per annum? Ans. $8 80. 4. What is the interest of 3459 dollars, for 75 days, at 6 per cent. per annum? Ans. $43 23 cts. 7 m.+5. What is the interest of 1500 dollars, for 60 days, at 5 per cent. per annum? Ans. $12 50. CASE 5. Thie amount, time, and rate per cent. given, to find the principal. RULE.-Find the amount of 100 dollars for the time required, at the given rate per cent. Then, by proportion, as the amount of 100 dollars for the time required, (at the given rate per cent.) is to the amount given, so is 100 dollars to the principal required. 3EXAMPLES. 1. What principal, at interest for 8 years, at 5 per ct. per annum, will amount to 840 dollars? Ans. $600. 5 dollars 8 years 40 Int. of $100 for 8 yr. 100 $ $ $ — ~~- 140: 840::100:600 140 Amt. of $100 for 8 yr. INTEREST. 117 2. What principal, at interest, for 6 years, at 4 per cent. per annum, will amount to $1240. Ans. $1000 3. What principal, at interest for 5 years, at 6 per ct. per annum, will amount to 2470 dollars? Ans. $1900. CASE 6. The principal, amount, and time given, to find the rate per cent. RULE.-Find the interest for the whole time given, by subtracting the principal from the amount. Then, as the principal is to 100 dollars, so is the interest of the principal for the given time, to the interest of 100 dollars for the same time. Divide the interest last found by the time, and the quotient will be the rate per cent. per annum, Or by compound proportion. EXABIIPLES. 1. At what rate per cent. per annum,v will 600 dollars amount to 744 dollars, in. 4 years? Ans. 6 per cent. $ $ $ $ $744 amount As 600: 100:: 144: 24. 600 principal yr. $ -- 4) 24 (6 rate per cent. 144 interest Or by compound proportion: $ $ As 600: 100 $ $ yr. yr.:: 144: 6 rate percent. 4: 1 2. At what rate per cent. per annum, will $1200 amount to $1476, in 5 years and 9 months? Ans. 4 per cent. 3. If 834 dollars, at interest 2 years and 6 months, amount to 927 dollars 82i cents, what was the rate per cent. per annum? Ans. 4^ per cent. 118 COMPOUND INTEREST. CASE 7. To find the time, when the principal, amount, and rate per cent. are given. RULE,3.-Divide the whole interest by the interest of the principal for one year, and the quotient will be the time required, or by proportion. EPAMIPLES. 1. In what time will 400 dollars amount to 520 dollars, at 5 per cent. per annum? Ans. 6 years' $ $ 400 520 5 400 20100 20)120(6 20: 120:: l: 6Ans. 2. In what time will ~1600 amount to ~2048, at 4 per cent. per annum? Ans. 7 years. 3. Suppose 1000 dollars, at 4i per cent. per annum, amount to 1281 dollars 25 cents, how long was it at interest? Ans.6Y. 3mo. COMPOUND INTEREST. Compound interest is that in which the interest for one year is added to the principal, and that amount is the principal for the second year; and so on for any number of years. RULE. —Find the amount of the given sum for the first year by simple interest, which will be the principal for the second year; then find the amount of the principal for the second year for the principal for the third year; and so on for any number of years. Subtract the first principal from the amount, and the remainder will be the compound interest required. EXAMPLES. 1. What is the compound interest of 150 dollars for 5 years, at 4 per cent. per annum? Ans. $32,49 COMPOUND INTEREST. 119 $150 $150 4 6 inst 1st year 6100 int. 1 yr. 156 amount 1st year 6,24 int. 2d year $156 162,24 amount 2d year 4 6,48.9 int. 3d year 6124 168,72.9 amount 3d year 6,74.9 int. 4th year. 162,24 175,47.8 amount 4th year 4 7,01.9 int. 5th year 6148.96 182,49.7 amount 5th year 150,00.0 principal 32,49.7 compound int. for 5 years. 2. What is the compound interest of 760 dollars, for 3 years, at 6 dollars per cent. per annum? Ans. $145 17 cts. 2 m.3. What is the compound interest of $242 50 cts., for 4 years, at 6 per cent. per annum? Ans. $63 65 cents. 4. What is the amount of 1300 dollars, for 3 years, at 5 dollars per cent. per annum, compound interest? Ans. $1504 91 cts. 2 m.+ 5. How much is tho amount of 3127 dollars, for 4 years, at 4i dollars per cent. per annum, compound interest? Ans. $3729 OOcts. 5m. Questions. What is interest? What is the principal? What is the rate per cent. What is the amount? How do you proceed when the interest for several ynars is required? 120 COMPOUND INTEREST. What is to be noted if the interest is required for years and months? When the interest is required for any number of weeks or days, less or more than one year, how do you perform the operation? How do you proceed to find the interest, at 6 per cent. for any number of days, as computed at banks? What is to be observed when the interest is at any other rate than 6 per cent.? How do you proceed, when the principal, amount, and time are given,, to find the rate per cent.? How do you find the time, when the principal, amount, and rate per cent. are given? What is compound interest? How is compound interest computed? PROMIISCUOUS EXERCISES. 1. What is the interest of 620 dollars 25 cents for 5 years, at 5i per cent. per annum? Ans. 1170 56 8 m - 2 What is the interest of $420, for I year, at 7 per cent. per annum? Ans. $29 40. 3 What is the interest of 1450 dollars, for 60 days, at 6 per cent. per annum? Ans. $,14 50 cts. 4 What is the compound interest of $626 25, for 3 years, at 54 per cent. per annum? Ans. $103 91.+ 5 What is the interest of $1659 for 3 weeks, at 4 per cent. per annum? Ans. $3 82^.+ 6 In what time will 500 dollars amount to 1000 dollars at 8 per cent. per annum, simple interest? Ans. 12 years, 6 months. 7 What principal, at interest for 6 years and 6 months, at 2 per cent. per annum, will amount to 250 dollars? Ans. $221 23 cts. 9 m. 8 At what rat. per cent. per annum, will $300 amount to $450, in 5 years? Ans. 10 per cent. 17NSURANCE, COMMISSION, AND BROKAGE. 121 INSURANCE, COM1UISSION, AND BROKAGE. INSURANCE, Commission and Brokage,are allowances made to insurers, factors, and brokers, at such rate per cent. as may be agreed on between the parties, RULE. Proceed in the same manner as though you were required to find the interest of the given sum for one year. EXAMPLES. 1 What is the commission on 625 dollars, at 4 dollars per cent? $625 4 Ans. $25,00 2 What is the commission on $1320, at 5 per cent.? Ans. $66. 3 What is the commission on 3450 dollars, at 4i dollars per cent.? Ans. $155,25. 4 The sales of certain goods amount to 1680 dollars: what sum is to be received for them, allowing 2j dollars per cent. for commission? Ans. $1633,80. 5 What is the insurance of $760, at 6k per cent.? Ans. $49,40. 6 What is the insurance of 5630 dollars, at 71 dollars per cent.? Ans. $436 32 cts. 5 m. 7 A merchant sent a ship and cargo to sea, valued at 17654 dollars: what would be the amount of insurance, at 18J dollars per cent.? Ans. $3310 12k cts. 8 What is the brokage on 2150 dollars at 2 per cent.? Ans. $43. 9 When a broker sells goods to the amount of 984 dollars 50 cents, what is his commission, at 1 dollar per cent.? Ans. $12 30i cts.+10 If a broker buys goods for me, amounting to (650 11 F 122 DISCOUNT. dollars 75 cents, what sum must I pay him, allowing him 11 per cent.? Ans. $24 76 cts. 1 m.+Questions. What are Insurance, Commission, and Brokage? How do you proceed to find the Insurance, Commission, or Brokage? In what does this rule differ from interest? It takes no account of time. DISCOUNT. DISCOUNT is an abatement of so much money from any sum to be received before it is due, as the remainder would gain, put to interest for the given time and rate per cent. RULE. Find the interest of 100 dollars for the given time at the given rate per cent. Add the interest so found to 100 dollars, then by prcportion, As the amount of 100 dollars for the given time, Is to the given sum, So is 100 dollars, To the present worth. If the discount be required, subtract the presentworth from the given sum, and the remainder will be the discount. NOTE. —When discount is made without regard to time, it is foudr precisely like the interest for one year. EXAMPLES. I What is the present worth of 420 dollars, due in 2 years, discount at 6 per cent. per annum? Ans. $375. DISCOUNT. 123 6 112: 420:: 100.375 2 12 100 112 2 What is the present worth of 850 dollars, due in 3 months, at 6 per cent. per annum? Ans. $837 434 cts.+ 3 What is the discount of 645 dollars, for 9 months, at 6 per cent. per annum? Ans. $27 77i cts. 4 What is the present worth of 775 dollars 50 cents, due in 4 years, at 5 per cent. per annum? Ans. $646,25. 5 What is the present worth of 580 dollars, due in 8 months, at 6 per cent. per annum? Ans. $557,69.-96 What is the present worth of 954 dollars, due in 3 years, at 4& per cent. per annum? Ans. $840 52 cts. 8 m.7 What is the discount of 205 dollars, due in 15 months, at 7 per cent per annum? Ans. $16 49 cts. 5 m.8 Bought goods amounting to 775 dollars, at 9 months' credit: how much ready money must be paid, allowing a discount of 5 per cent. per annum? Ans. $746 98 cts. 7 m. 9 I owe A. to the value of 1005 dollars, to pay as follows: viz. 475 dollars in 10 months, and the remainder in 15 months; what is the present worth, allowing discount at 6 per cent. per annum? Ans. $945 40 cts. 4 m. 10 What is the difference between the interest of 2260 dollars, at 6 per cent. per annum, for 5 years, and the discount of the same sum for the same time and rate percent.? Ans. $156 46 cts. 2m.4 124 EQUATION OF PAYMENTS. 11 What is the discount of 520 dollars, at 5 per cent.? $520 5 $26,O0 Ans. 12 How much is the discount of $782, at 4 per cent.? Ans. $315 28 13 What is the discount of 476 dollars, at 3 per cent.? Ans. $14,28. 14 Bought goods on credit, amounting to 1385 dollars: how much ready money must he paid for them, if a discount of 6 per cent. be allowed? Ans. $1301,90. 15 I hold A.s note for 650 dollars; but I agree to allow him a discount of 4L per cent. for present payment: what sum must I receive? Ans. $620,75. Questions. What is discount? What is first to be done? After having found the interest of 100 dollars, at the given time and rate per cent., what is next to be done? After having added the interest so found to 100 dollars or pounds, by what rule do you work to find the discount? When discount is made without regard to time, how is it found? EQUATION OF PAYMENTS. EQTJATION is a method of reducing several stated times, at which money is payable, to one meant or equated time, when the whole sum shall be paid. RULE. Multiply each payment by its time, and divide the sum of all the products by the whole debt, the quotient will be the equated time. EQUATION OF PAYMENTS. 125 Proof. —The interest of the sum payable at the equated time, at any given rate, will equal the interest of the several payments for their respective times. EXAMPLES. I C. owes D. 100 dollars, of which the sum of 50 dollars is to be paid at 2 months, and 50 at 4 months;, but they agree to reduce them to one payment; when must the whole be paid? Ans. 3 months. 50X2- 100 50 X4==200 100)300(3 months 2 A merchant hath owing to him 300 dollars, to be paid as follows: 50 dollars at 2 months, 100 dollars at 5 months, and the rest at 8 months; and it is agreed to make one payment of the whole; when must that time be? Ans. 6 months. 3 F. owes H. 2400 dollars of which 480 dollars are to be paid present, 960 dollars at 5 months, and the rest at 10 months; but they agree to make one payment of the whole, and wish to know the time? Ans. 6 months. 4 K. is indebted to L. 460 dollars which is to be discharged at 4 several payments, that is A at 2 months, \ l at 4 months, i at 6 months, and 4 at 8 months; but they I agreeing to make one payment of the whole, the equaI ted time is therefore demanded? Ans. 5 months. 5 P. owes Q. 420 dollars, which will be due 6 months hence, but P. is willing to pay him 60 dollars now, provided he can have the rest forborn a longer time: it is agreed on; the time of forbearance therefore is required? Ans. 7 months. 6 A merchant bought goods to the amount of 2000 dollars and agreed to pay 400 dollars at the time of purchase, 800 dollars at 5 months, and the rest at 10 months; but it is agreed to make one payment of the whole; what is the mean or equated time? Ans. 6 months. 11*~. I s126 XBARTER. BARTER. BARTER is the exchanging of one kind of goods for another, duly proportioning their values, &c. RULE. The questions that come under this head, may be done by the compound rules, the Rule of Three, or Practice, as may be most convenient. EXAMIPLES. 1 A country storekeeper bought 150 bushels of salt, at 56 cents per bushel; and is to pay for it in corn, at 331 cents per bushel; how much corn will pay for the salt? ct. ct. bu. bu. As 33: 56::150: 252 OR -ct. 56 33s)8400 150 3 3 2800 1100)252o00 56 ~~ ~~- ~ 252 bushels of corn. Cost of the salt. 8400 cts. 2 How much wheat, at 1 dollar 25 cents per bishel, will pay for 35 sheep, at 2 dollars 25 cents a piece? Ans. 63 bush. 3 How much sugar, at 9 cents per lb. will pay for I dozen pair of shoes, at I dollar 75 cents per pair? Ans. 233k lbs. 4 How much tea, at 80 cents per lb. will pay for 560 lbs. of pork, at 5 cents per lb.? Ans. 35 lbs. 5 Bought 4 hats for 3 dollars 50 cents; 4 dollars; 4 dollars 50 cents; and 5 dollars-how much corn at 32 cents will pay for them? Ans. 53 bush. 4 qts. 6 A. has 420 bushels of corn, which he barters with B. for cats, and is to receive 4 bushels of oats for 3 of corn — iow many bushels of oats must A receive? Ans. 560 bush. BARTER. 127 7 A boy bartered 735 pears for marbles, giving 5 pears for 2 marbles-how many marbles ought he to have received? Ans. 294 marbles. 8 A boy exchanges marbles for pears, and gives 2 marbles for 5 pears-how many pears should he receive for 294 marbles? Ans. 735 pears. 9 A farmer bartered 3 barrels of flour, at 5 dollars 25 cents per barrel, for sugar and coffee, to receive an equal quantity of each-how much of each must he receive, admitting the sugar to be valued at 9 cents per lb. and the coffee at 14 cents? Ans. 681 lb. nearly. 10 A bartered 42 hats, at 1 dollar 25 cents per hat, with B. for 50 pair of shoes, at 1 dollar 121 cents per pair-who must receive money, and how much? Ans. B. $3,75. 11 Sold 75 barrels of herrings, at 2 dollars 75 cents per barrel, for which I am to receive 75 bushels of wheat, at I dollar 8 cents per bushel, and the residue in moneyhow much money must I receive? Ans. 125 dolls. 25 cts. 12 Sold 35 yards of domestic, at 20 cents per yard, and am to receive the amount in apples, at 25 cents per bushel-how many bushels must I have? Ans. 28 bush. 13 Gave 35 yards of domestic for 28 bushels of apples, at 25 cents per bushel-what was the domestic rated at per yard? Ans. 20 cts. 14 What is rice per lb. when 340 lb. are given for 4 yards of cloth, at 4 dollars 25 cents per yard? Ans. 5 cts. 15 Gave in barter 65 lbs. of tea for 156 gallons of rum, at 33i per gallon-what was the tea rated at? Ans. 80 cts. per lb. 16 Q. has coffee worth 16 cents per pound, but in barter raised it to 18 cts.; B. has broad cloth worth 4 dollars 64 cents per yard-what must B. raise his cloth to, so as to make a fair barter with Q? Ans. $5,22. '128 LOSS AND GAIN. 17 B. had 45 hats, at 4 dollars per hat, for which A. gives him 81 dollars 25 cents in cash, and the rest in pork, at 5 cents per lb; how much pork will be required? Ans. 1975 lb. 18 Two merchants barter; A. receives 20 cwt. of cheese, at 2 dollars 87 cents per cwt.; B. 8 pieces of linen, at 9 dollars 78 cents per piece; which of them must receive money, and how much? Ans. A. $20,84. 19 If 24 yards of cloth be given for 5 cwt. 1 qr. of tobacco, at 5 dollars 7 cents per hundred; what is the cloth rated at per yard? Ans. $1.109. 20 A. barters 40 yards of cloth, at 98 cents per yard, with B. for 28s lbs. of tea, at 1 dollar 53 cents per lb.; which must pay balance, and how much? Ans. A. $4,405. 21 A has 7k cwt. of sugar, at 8 cents per lb., for which B. gave him 12A cwt. of cheese, what was the cheese rated at per lb.? Ans. $. 048. 22 What quantity of sugar, at 8 cts. per lb. must be given in barter for 20 cwt. of tobacco, at 8 dollars per cwt.? Ans. 17 cwt. 3 qrs. 12 lb. 23 P. has coffee, which he barters with Q. at 11 cts. per lb. more than it cost him, against tea, which stands Q. in 1 dollar 33 cents the lb., but he puts it at 1 dollar 66 cents; query, the prime cost of the coffoe? Ans. $. 443+ LOSS AND GAIN. By Loss AND GAIN, merchants and dealers compule their gains or losses. RULE. Work by the Compound Rules, by Proportion, or in Practice, as may be: most convenient. LOSS AND GAIN. 129 EXAMPLES. 1 Bought 1234 lbs. of coffee, at 129 cts. per lb., and sold the whole for 160 dollars; did I lose or gain by it, and how much? Ans. gained $5,75. 2 Bought 120 dozen knives, at 2 dollars 50 cents per dozen, and sold them at 184 cents a piece; did I gain or lose, and how much? Ans. lost $30. 3 Bought 1234 yards of muslin, for 17k cents, and sold it at 20 cents per yard; what was the gain? Ans. $30,85. 4 Bought 10 chests of tea, each 63 lbs. neat, for 600 dollars, and retailed it at 87k cents per lb.; did I gain or lose, and how much? Ans. lost 48 dol. 75 ct. 5 Gave 285 dollars 25 cents for 4564 lbs. of bacon, and sold it for 365 dollars 12 cents; what was the gain per lb? Ans. $. 14 cts. 6 Bought 1234 yards of muslin, for 246 dollars 80 cents, and sold it for 215 dollars 95 cents; what did I lose per yard? Ans. $. 2k cts. 7 Gave 25 cts. per bushel for corn, and sold it at 28 cents; what is the gain per cent.? Ans. 12 dolls. per 100 dolls. 8 Sold corn at 25 cts. per bushel, and 4 cts. loss; what was the loss percent.? Ans. $13,79. 9 Bought 13 cwt. 25 lbs. of sugar, for 106 dollars, and sold it at 91 cts. per lb.; what did I gain per cent.? Ans. 32 dolls. 73 cts. 10 Bought 126 gallons of wine for 150 dollars, and retailed it at 20 cts. per pint; what was the gain per cent.? Ans. 34 dolls. 40 cts. 11 Sold a quantity of goods, for 748 dollars 66 cents, and gained 10 per cent; what did I give for them.? Aiis. 680 dols. 60 cts. F2 130 LOSS AND GAIN. dols. dots. dols. 100 110:0::748,66 10 100 110 110)74866,00($680,60 12 Sold goods to the amount of $1234, and gained at the rate of 20 per cent.; what was the prime cost? Ans. $1028933i 13 Sold a quantity of goods, for $475, and at a loss of 12 per cent.; what did I give for them? dols. dols. dols. 100 88: 100: 475 12 100 88 88)47500(539,77+Ans. 14 Sold hats to the amount of $136, at 20 per cent. loss; what was the first cost? Ans. $170. 15 Laid out $755 in salt; how much must I sell it for, so as to gain 12 per cent.? 12 100 As 100:: 112: 755::845,60 Ans. 16 Bought 32 yards of mole skin for 128 dollars; what must I sell it for per yard, so as to gain 20 per cent.? Ans. 4 dols. 80 cts.-+ 17 Bought 17 yards of silk for 21 dollars; how much per yard riust I retail it for, and gain 25 per cent.? Ans. 1 dol. 54 cts.-+ 18 Bought 64 yards cf muslin for 13 dollars 50 cents, but proving a bad bargain, I am willing to lose 8 per \cent; what must I sell it at pez yard? Ans.19cts.4m.+ 19 When hats are bought at 48 cents, and sold at 54 ents; what is the gain per cent.? Ans. 12 LOSS AND GAIN. 131 20 If, when cloth is sold for 84 cents per yard, there is gained 10 per cent.: what will be the gain per cent. when it is sold for 1 dollar 2 cents per yard? Ans. 33 dols. 68 cts.-+ 21 Bought a chest of tea-, weighing 490 lbs. for $122 50ct. and sold it for $137 20 cents; what was the profit on each lb.? Ans. 3 cts. 22 Bought 12 pieces of white cloth, for 16 dollars 50 cents per piece:; paid 2 dollars -87 cents a piece for dying; for how much must I sell them each, to gain 20 per cent.? Ans. 23 dols. 244. 23 If 28 pieces of stuff be purchased at p dollars 60 cents per piece, and 10 of them sold at 14 dollars 40 cents, and 8 at 12 dollars per piece; at what rate must the rest be disposed of, to gain 10 per cent. by the whole? Ans. 5 dols. 568. 24 Sold a yard of cloth for 1 dollar 55 cents, by which was gained at the rate of 15 -per cent.; but if it had been sold for 1 dollar 72cents: what would have been the gain per cent.? Ans. 27 dols. 69+25 If, when cloth is sold at $. 935 a yard, the gain is 10 dollars per cent.; what is the gain or lops per cent, when it is sold at 80 cents per yard? Ans. 5 dollars 88+loss. 26 A draper bought 100 yards of broad cloth, for which he gave $56-I desire to know how he must sell it per yard, to gain $19 in the whole? Ans. 75 ct. per yard. 27 A draper bought 100 yards of broad cloth for $56; I demand how he must sell it per yard, to gain $15 in laying out $100? Ans. 64 ct. 4 m. 28 Bought knives at 11 cents, and sold them at 12 cents; what will I gain by laying out 100 dollars in knives? Ans. 9 dols. 09 — 29 Bought knives at 11 cents, and sold them at 12 cents; what did I gain by selling to the amount of 100 dollars? Ans. 8 dols. 333+ I 132 FELLOWHIIP. 30 If by selling 1 lb. of pepper for 10( cents' there are 2 cents lost, how mich is the loss per cent.? Ans. 16 dols 31 A merchant receives from Lisbon, 180 casks of raisins, which stands him in here 2 dollars 13 cents each, and by selling them at 3 dollars 68 cents per cwt.,- he gains 25 per cent.; required the weight of each cask, one with another? Ans. 81 lb, FELLOWSHIP. FELLOWSHIP is a method by which- merchants and others adjust the division ofproperty, loss, or gain, &c., inproportion to their several claims. CASE 1. SIMPLE FELLOWSIPr. When the claims are in proportion to the amount of stock, labor5 &c., without regard to time. lRULE. (By Proportion.) As the whole amount of stock or labor, Is to each man's portion, So is the whole property, loss, or gair, To each man's share of it. Proof —The sum of all the shares must equal the whole gain, &c. EXAMPLES. 1 Twq men bought a stock of goods for 480 dollars, of which A. paid 320, and B. 160. They gained 128 dollars by the transaction; what was the share of each? Ans. A. received 85 dols. 33t cts. and B. 42 dollars 661 cts. $ $ $ $ ct. Proof A's. stock $320 As480:320:: 128:85,331 $85,331 B's.stock 160 42,661 -.- $.$ 4$ $ ct. Whole st'k 480 As 480:160::128: 42,668 128,00 FELLOWSHIP. 133 2 Three workmen having undertaken to do a piece of work for 275 dollars, agreed to divide their profits in proportion to the amount of labor each one performed. M. labored 50 days, N. 65 days, and 0. 85 days: what was the share of each? Ans. M. received 68 dols. 75 cts.; N. 89 dols. 37A cts.; and 0. 116 dols. 87k cts. 3 A merchant being deceased, worth 1800 dollars, is found to owe the following sums: to A. 1200 dollars, to B. 500 dollars, to C. 700 dollars: how much is each to have in proportion to the debt? Ans. A. 900 dols., B. 375 dols, and C. 525 dols. 4 Three drovers pay among them 60 dollars for pasture, into which they put 200 cattle, of which A. had 50, B. 80, and C. 70: I would know how much each had to pay? Ans. A. 15 dols., B. 24 dols., C. 21 dols. 5 A man failing, owes the following sums: to A. 120 dollars, to B. 250 dollars 75 cents, to C. 300 dollars, to D. 208 dollars 25 cents; and his whole effects were found to amount to but 650 dollars: what will each one receive in proportion to his demand? Ans.A. $ 88.73.+ C. 221.84.+ B. $185.42.+ D. $153.99.+ 6 A bankrupt is indebted to A. 500 dollars 37i cents-to B. 228 dollars-to C. 1291 dollars 23 cents- to D. 709 dollars 40 cents; and his estate is worth 2046 dollars 75 cents: how much does he pay per cent., and what does each creditor receive? Ans. He pays 75 per cent., and A. receives 375 dollars 274 cts.; B. 171 dols.; C. 968 dols. 424 cts.; and D. 532 dols. 5 cts. 7 If a man is indebted to A. 250 dollars 50 cents, to B. 500 dollars, to C. 349 dollars 50 cents, but when he comes to make a settlement, it is found he is worth but 960 dollars, how much will each one receive, if it be in proportion to their respective claims? (A. $218 61 cts. 8 m.+ Ans.. $436 36 cts. 3 m.+ C. $305 01 ct. 8 m.+ 12 134 FELLOWSHIP. CASI 2. CO(MPOUMN FELLOWSHIP. When the respective stocks are considered with relation to time. RULE. (By Proportion.) Multiply each man's stock by its time; add the several products together; then: As the sum of the products Is to each particular product, So is the whole gain or loss To each man's share of the gain or loss. EXAMPLES. I Three merchants traded together; A put in 120 dollars for 9 months, B. 100 dollars for 16 months, and C. 100 dollars for 14 months, and they gained 100 dollars; what is each man's share? $ mo. A's. stock 120 X 9 = 1080 B's. stock 100 X 16 = 1600 C's. stock 100 X 14 = 1400 Sum 4080 Sum. Prod. $ As 4080: 1080::100: 26,47-+ A's. share. As 4080:1600::100: 39,21 — B's. share. As 4080:1400::100: 34,31+ C's, share. 2 Three men traded together; L. put in 88 dollars for 3 months, M. 120 dollars for 4 months, and N. 300 dollars for 6 months; they gained 184 dollars: what will each man receive of the gain? L. $ 19 09 cts. 4 m. Ans. M. $ 34 71 cts. 6 m. N. $130 18 cts. 8 m. VULGAR FRACTTTOS. 135 3 Two merchants entered into partnership for 16 months: A. put in at first $600, and at the end of 9 months put in $100 more; B. put in at first $750, and at the end of 6 months took out $250, at the close of the time their gain was $386, what was the share of each? Ans. A's. share was $200,791; B's. share was $185,20. 4 A., B., and C., made a stock for 12 months; A. put in at first $873,60, and 4 months after he put in $96,00 more; B. put in at first $979,20, and at the end of 7 months he took out $206,40; C. put in at first $355,20, and 3 months after he put in $206,40, and 5 months after that he put in $240,00 more. At the end of 12 months, their gain is found to be $3446,40; what is each man's I share of the gain? (A's. share is $1334,82i Ans. B's. - - $1271,61+ C's. - - $839,96 Questions. What is Fellowship? By what rule are its operations performed? When is Fellowship simple? When is it compound? In what respect is Fellowship compound? Ans. The proportion is compound: that is, the division of property, gain, &c., is founded on the compound proportion of the stock and time. VULGAR FRACTIONS. A VULGAR FRACTION is a part, or parts of a unit expressed by two numbers placed one above the other with i line between them. As l,, &c. The number below the line is the denominator, the iumber above the line is the numerator. The denominator denotes the number of parts into vhich the unit is divided. 136 VULGAR FRACTIONS. The numerator shows how many of those parts are to be taken. Fractions are either proper, improper, or compound. A proper fraction is one whose numerator is less than its denominator, as - or V. An improper fraction is one whose numerator is greater than its denominator, as 8 or. A compound fraction is a fraction of a fraction, as 2 of 2 Of 3 3 or of. A mixed number is a whole number and a fraction. REDUCTION OF VULGAR FRACTIONS. CASE 1. To reduce afraction to its lowest terms. RULE. Divide the terms by any number that will divide both without a remainder, and divide the quotient in the same manner, and so on till no number greater than one will divide them: the fraction is then at its lowest terms. EXAMPLES. 1. Reduce -48 to its lowest terms. 4) r-o _7- result. 2. Reduce 3-L to its lowest terms. Res. 4 3. Reduce -O to its lowest terms. Res. 4. Reduce 4 4 to its lowest terms. Res. 7 NOTE.-When a divisor cannot readily be found, divide the denominator by the numerator, and that divisor by the remainder, and so on, till nothing remain: the last divisor is the common measure of the two numbers; with which proceed as before. 5. Reduce 3- to its lowest terms. Res. VULGAnI FRACTIONS. 137 5 Reduce 5-% to its lowest terms. Res. 8. 85 85)136(1 Here 17 being the last divisor, 85 is the common measure of 85 -- and 136. 51)85(1,8 (51 34)51(1 85- 534 17 - -~ | ~~136 (8 17)34(2 34 6. Reduce 7-P to its lowest terms. Res. 7. 7.: Reduce 4 4 to its lovest terms. Res. 2. 8. Reduce 316- 1 to its lowest terms. Res. -. CASE 2. To reduce a mixed number to an improperfraction. RULE. Multiply the whole number by the denominator, and add the numerator to the product for the numerator of the improper fraction, and place the denominator under it. EXAMIPLES. 1. Reduce 12 4 to an improper fraction. 12 ~ Res. 12. 9 112 Nine 12's are 108; add ----- 4 makes 112 ninths. 9 2. Reduce 17 3to an improper fraction.. Rs. 2. 3. Reduce 45 - to an improper fraction. Res. 17. 4. Reduce 24 1-3 to an improper fraction. Res. 4 12 138 VULGAR FRACTIONS. -CASE 3. To reduce an improper fraction to its proper value. RULE. Divide the numerator by the denominator, and the quotient will be the whole number; the remainder, if any, will be the numerator of the fraction. EXAMPLES. 1 Reduce L7 to its proper value. Res. 3 a. 17 5)17 i 2 Reduce 1 2 to its proper terms. Res. 12 4. 3 Reduce 127 to its proper terms. Res. 17. 4 Reduce 22 to its proper terms. Res. 24 -. CASE 4. To reduce several fractions to other fractions having a common denominator, and retaining their value. RULE. Multiply each numerator into all the denominators but its own, for the respective numerators; and all the denominators together, for a common denominator. EXAMPLES. 1 Reduce 2 3 and 5 to a common denominator. Res. - 7, and -. 2X4X6-48) 3X3X6=54 Numerators 5X3X4=60) 3X4X 6=72, common denominator. Then we have 47 8for -; 4 for 3 and 6 for 5 Reduce each new fraction to its lowest terms, and the result will prove the work to be right. 2. Reduce 4, -, and —, to a common denominator Res. 26 24o and 18 3. Reduce 1, 2-I 5, and, to a common denominator. Res. 6 28 and.252 4. Reduce 7, 3, -, and -, to a common denominator. Res. 630 480 432 and 60 VULGAR FRACTIONS. 139 NOTE. —It is often convenient to use the least possible common denominator; to find which, divide the denominators by any number that will divide two or more of them without a remainder, setting down those that would have remainders; then multiply all the divisors and all the quotients together. 4 2 3 4 5 6 7 8 9 3 2 3 1 5 6 7 2 9 2 2 1 5 2 7 2 3 1 5 1 7 1 3 4X3x2X5X7X3 —2520 common denom. Which may be divided separately by 2, 3, 4, 5, 6,7, 8, and 9, without a remainder. EXAMPLES. 5 Find the least common denominator for 3, 71 5, and -9 and compute their equivalent fractions. TQ-Zi5. -. =_ = _