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Tous les autres exemplaires originaux sont film^s en commenpant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. The last recorded frame on each microfiche shall contain the symbol — ♦> (meaning "CON- TINUED"), or the symbol V (meaning "END"), whichever applies. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbole — ^ signifie "A SUIVRE", le symbole V signifie "FIN". Maps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent Stre filmds & des taux de reduction diffdrents. 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McMillan, printers. 78 prince william street. 1869. ! i i i PREFACE »■» THOuan elc'inentary works on Arithmetic Are ia abandaneo, ye\ it seems desirable that there should bo added to this an cztensivo treatise on the commercial rules, and commercial laws and usages. It is not enough that the school-boy should bo provided with o course suited to his age. There must be supplied to him something higher as ho advances in ago and progress, and nears tho period when he is to enter on real business life. Tho Author's aim has, therefore, been to combine these two objects, and to produce a work adequate to carry tho learner from the very elements up to the highest rules required by those preparing for business. As tho work proceeded, it was found necessary to extend the original programme considerably, and, therefore, also the limits of the book, so as to mako it useful to all classes in the community. In carrying out this plan, much care has been taken to unfold the theory of Arithmetic as a science in as concise a manner as seemed consistent with clearness, and at tho same time to show its applications as an Art. Every effort has been made to render the business part so copious and practical as to afford the young student ample information and discipline in all the principles and usages of commercial intercourse. For tho same reason some articles on Commercial Law have been introduced, as it was a prominent part of the Author's aim to produce a work which should be found useful, not only in the class-room, and the learner's study, but also on the merchant's table, and the accountant's desk. The Author begs to tender his best thanks to J. Smith Ilomans, Esq., New York City, Editor and Proprietor of tho "Banker's Magazine and Statistical Register/' for the able manner in which he supplied this part of the work. Throughout the work particular care has been taken not to enunciate any rule without explaining the reason of the operation, for, without a knowledge of the principle, the operator is a men. calculating machine that can work but a certain round, and is almost sure to be at fault when any novel case arises. The ezplanatiooe IV. rREFACE. arc, of course, more or less llic result of reading, but, nevertheless, they aro mainly derived from personal study and experience In teaching. The great mass of tlic exercises arc likewise entirely new, though the Author has not .scrujilcd to make selections from somrfol the most apjtDwd works on the subject; but in doing so, he has confined hims^ell" almost entirely to such (|uestions as arc to be found in nearly all popular books, and which, therefore, are to be looked upon as the common property of .science. Algebraic fijrni.s have been avoided as much as possible, as being unsuitcd to a largo proportion of those for whom the book is in- tended, and to many altogether unintelligible, and besides, those who understand Algebraic modes will have all the less difficulty in understanding the Arithmetical ones. Even in the more purely mathematical parts the subject has been popularized as much as possible. In arranging the subjects it was necessary to follow a certain logical order, bui the intelligent teacher and learner will often find it necessary to depart from tliat order. (See suggestions to teachers.) Every one will admit that rules and definitions should be ex- pressed ill the smallest possible number of words, consistent with pcr.spicuity and accuracy. Great pains have been taken to carry out this principle in every case. Indeed, it might be desirable, if practicable, not to enunciate any rules, but simply to illustrate each case by a few examples, and leave the learner to take the j^jrm- cl2^lc into his mind, as his rule, w'ithout the encumbranco of words. Copious exercises arc appended to cac'.. rule, and especially to the most important, such as Fractions, Analysis, Percentage, with its applications, &c. Besides these, there have been introduced extensive collections of mixed exercises throughout the body of tho work, besides a large number at the end. The utility of such miscellaneous questions will be readily admitted by all, but tho reason why they are of so much importance seems strangely over- looked or misunderstood even by writers on the subject. They arc spoken of as review exercises, but their great value depends on some- thing still more important. An illustration will best serve hero. A class is working questions on a certain rule, and each member of the class lias just heard tho rule enunciated and explained, and therefore readily applies it. So far one important object is attained, viz., freedom of operation. But .something more ia necessary, The rREFACE. V. learner must bo tau^'ut to discern ivJiat rufc la to he, applied for the solution of cac'a quostion proposod. The pnpil, under careful teach in;.', may bo nLlo lo uudcr,-,taud fully every rule, and never con found any one \Yith any otlio;-, and yet bj doubtful what rule i.:i to bo applied to an intlivldu:d case. The miscellaneous problems, therefore, are intended not s-.o much as exereirfcs on the oinnitioiif of the J'lTcrent rulci as on the ^nuJc ('/(■tj>2i^ijiiij those rules; or, iu other worda, to practice the pupil ia perceiving of what rule any proposed (jucstion 13 a particular case. Great importance should be uttached to this by the practical educator, not only as regardj rcadi nesa iu real business, but also as a mental exercise to the younp Btudcnt. The Author is far from supposing, much less asserting, that the work is complete, especially as the whole has been prepared in less than the short space of six months. It is presented, however, to the public in the confident expectation that it will meet, in :i great degree at leas^, the necessities of the times. "With thi:j view, there arc given extensive collections of examples and exereiscj, involving money in dollars and cents, with, however, a number iu pounds, shillings and pence, sufficient for the purpose of illustration. This seems necessary, as many must have mercantile transactions v.-ith Britain and British America. The llulc for finding the Greatest Common Measure, though not new, is given in a new, and it is hoped, a concise and convenient form of operation, The llulc for finding the Cube Root is a modification of that :.;iven by Dr. Hinds, and will be found ready and short. In treating of Common Fractions, Multiplication and Division have been placed before Addition and Subtraction, for two reasons. First, — In Common Fractions, Multiplication and Division present much less difficulty than Addition and Subtraction ; and, secondly, ■xa in "Whole Numbers Addition is the Rule that regulates all others ; 50 in Fractions, which ori;:'inato from Division, w^ sec, in liko icd, L^hc manner, that all other operations result from Division, and, in con- nection with it. Multiplication. Several subjects, commonly treated of iu works on x\rithmctic, have been omitted in order to leave space for more important matter bearing on commercial subjects. Duodecimals, for example, have been omitted, as that mode of calculatic n iy now virtually rapcr.'cdod n. PREFACE. by that of Decimals, liartcr, too, has been passed by, as questions of that class can easily be solved by tho llulo of Proportion, which ban been fully explained. Tho subject of Analysis has been gone into at considerablo length, and it is lioped that tho now manner in which tho explana- tions and solutions arc presented, and tho extensive collection ol exorcises appended, will contribute to make thi^ a valuable part oJ the treatise. The view given of Decimal Fractions seems the only true one, and calculated to give the student clear notions rci^arding tho nature of the notation, as a simple extension of the common Arabic system, and also appropriate to show the convenience and utility of Decimals. The distinction becween Decimals and Decimal Fractions has been ignored as being " A distinction without a dificrence." Decimals ia merely a bhort way of writing Decimal Fractions ; thus, .7 is merely a convenient mode of writing /y. These differ in form only, but otherwi.se are as perfectly identical as J and ^V The contracted methods of Multiplication aud Division will bo found, after some practice, extremely useful and expeditious in Decimals expressed by long lines of figures. The averaging of Accounts and E(|uation3 of Payments, Cash Balance and Partnership Settlements, have been introduced aa essential parts of a commercial education, and, it is hoped, will form a most important and useful study for those preparing for business, and probably a safe guide to maoy in buciucss who have not sys* tematicallv studied the subieot. r^% SUGGESTIONS TO TJiACIIEIlS. The author wonld first refer to the remark made m tho Prcfaco that ho tloc3 not expect that tho Teacher will follow tho logical order adopted in tho book, and even advises that he should not do bo in many caacs. lie knows by experience that the same order doc3 not suit all btudents any more than the same medical treat- ment suits all patients. Tho courc requires to bo varied according to age, ability and acquirements. Tho greatest difEcultics generally present themselves at the earliest stages. What more serious diffi- culty, for example, has a child to encounter than tho learning of tho alphabet ? Though this is perhaps tho extreme case, yet others will bo found to bo in proportion. For beginners, therefore, wo recom- mend the following course. Let tho elementary rules be carefully explained and illustrated by si;n/j?c' examples, and tho pupil shown how to work easy exercises ; this done, let tho whole bo reviewed, and exercises of a more difficult kind proposed. Tho decimal coinage should tlien be taken up. In ex- plaining this part of the subject tho teacher ought to notice carefully that the operations in this case diflfer in no way from those already gone through in reference to whole numbers, except in the preserving of the mark that separates the cents from the dollar:?, usually called the doeinal point. Tho next step ought to be tho whole subject of denominate numbers, and in illustration and application, tho rule of practice. After a thorough review of all the ground now gone over, Simple Proportion may bo entered upon, using such {questions as do not involve Fractions. Then, after a course of Fractions has been gone through. Proportion should bo reviewed, and questions which involve Fractions proposed. After this it will generally be found de- sirable to study Percentage, witli its applications. The order in which the rest of the course shall be taken is com- paratively unimportant, as the student has now realized a capital on which he can draw upon for any purpose. The author would, in tho strongest manner possible, impress on tho minds of teachers the great utility of frequent reviews, and espcciallv of constant CKcrcisc ia tho addition of money columns Vlll. KUOOr.STTOXS TO TKACHKn'*. To mnlio the oxnrci.so-t unli r »v:o!i rrili? <>f pro^jrcshivc diflficulty, fti far as po.s.sibl"\ li:»s Ikoii an n))ji'ci k<>_ t cnnst.intly in viow, a.s also to givo each cxiTci.so tlio soiubl mcj ufa rcil (^^l(•^ti^>ll, i'nr all [)crsous, especially the youDL', tako irroator intcro-t in exercises tliat assume the form of reality tliaii iit hucli i\^ uro merely uL^truct ; anJ, bcsiJos, this is a preparatory exereliio t.) tho application of t!ic ruloJ aflerwardi". At every htau^o llio jtroatcst caro blwulJ bo takca th.it iho learner thoroughly unJer.standij the meuniiv^of caeli rule, and the conditions of each (luc.ilion aud the terms iu whicli it ii csprcsacd, before bo ftttcini)t.< to Kolvo it. The Teacher should not always bo talking or working on the blaek-board ; lie should rerjuiro tho pupils to speak a good deal in answer to (luostious, and also work much on their elates, and each in his turn on the board for illustration to the rest. Finally, it i^ .snp:gcsted to every Teacher to keep constantly be- fore Ills miud Imth of the two chief works ho has to accomplish. Fii'J, tho developcmcnt of tho mental powers of his pupil ; and, secondly, imparting to him Buch knowledge as he will require to use when he enters upon life, either as a professional man, or a mcr- ehant or clerk, f^omo Kccm to consider these two objects incompati* ble, as if taking up time in mental training left insufficient time for tho imparting of actual knowledge. This is a palpable crrror, for tho more the mental powers are cultivated, the more readily and rapidly will any Bpccles of knowledge bo apprehended, and the more surely, too, will ii be retained when it has been mastered. Mental culture is at once tho foundatioit and the means ; the other is the super- structure raised on that foi;ndation and by that means; or it may bo compared to a great capital judiciously embarked in trade, and often turned, and therefore yielding good proGts. It frequently happens, however, from tho peculiar circumstances of individuals aud families, and even communities, that young men require to be hurried into business, ho as to bo able to support themselves ; but even in .luch ca.scs tho desired object will bo much more readily and securely attained by such u course than by what is usually and not inappro- priately called " Cramming'' Every effort has been made to give to this book tho character hero rccommcndod. csDO^iolly in tho ezplanvill, it is bnpcd, bo found bigbly profitable. A few additiunul biiitd uro fubjuiiicd for tho bencGt of thoso seeking a libcrul uud iiructicul commcrciul education. As in all branchcH, so in Arithmctio, it is of the utmost conso* ([uencc tu digest tho rules of the art thoroughly, and store theiu iu the inciuory, to bo reproduced when required, and applied with accuracy. Hut this is not enough ; something more is needed by tho Student. To be an eminent accountant he must ucijuiro rapidity of operation. Accuracy, it is true, should be attained first, especially u it is the direct means of arriving at rcadine!:S and rapidity. Accu- racy may be called the foundation, readiness and rapidity tho two wings of the supcrhtructure. Either of these acquirements is indeed valuable in itself, but it i.s the combination of them that constitutes real eflectivc skill, and makes the possessor relied upon, and looked up to in mercantile circle.s. Some one may ask, '• IIow arc these to be acquired? " Tho answer is as simple as it is undeniably true; only h>f extensile practice, not in the count ing-houso or warehouse, indeed, t'lough the.so will improve and mature them, but in the school and college, so that you may tafcc them with you to the busi- ness ofiiee when you go to your first day's duty, d'o prrjniral is a maxim that all iiitLlIIgrnt business men will alllrm. Bo (o prepared that you will not keep your customers waiting rcstle.s.sly in your office or warehouse while you arc puzzling through the account you are to render to him, but strive rather to surprise him by having your bill ready so soon. Another important help to the attaining of this rapidity, as no- ticed iu the note at foot of page IS, is not to use the tongue in calcu- lating but the rj/c and tho miml Nor should the course of self-discipline end here. To bo an ex- pert accountant even, is but one part, though an important one of a qualification for busincBS. Study Commercial geography— commer- cial and international rcUtions—political cconomy'—tariffs, &c , &c. X. SUGGESTIONS Ti) rOMMERCIAL RTUDENTS. Study even ])oUtirs. not for their own sake but ou account of the manner in whloli thoy afl'oct tnulo anil commerce. Do not, except in Iho c:)sc of some f^orious difficulty, indulge in the indolent haljit of I'.skin.^' your teacher or fellow Htudent to work the question for you ; work it out yourself— rely upon your .self, anil aim at the freedom and correctness which will give you confiJcncc in yourself, or rather in your powers and acquirements. Another cau- tion will not bo out of place. 3Iany students follow the practice of keeping the text book beside them to see what the answer is; this has the .S'lme effect as a /t' fJ//(y (/Hos^'ou in an examination, being a ^uido to the mode by seeing the result. Study and use the mode tc come at the result ; gain that knowledge of principles and correctness of operation that will inspire the confidence that your answer is cor- rect without knowing what answer the text book or the teacbcr may assign to it. Tlujre arc two things of such oonstaut occurrence and requiring such extreme accuracy thut they must be specially meutioned, — thcj ore the addition of money columns and the making of Bills of Par- cels. Too much care and practice cau scarceiv be bestowed on theso. P TABLE OF CONTENTS. 250 117 PAOE. iVrithmotlo „ i;$ Addi tion.««^ »< 1 1 1 1 1 1 1 1 17 Alli^iitioa Analysis „ Aimuitios o\0 Avi'rage, Goneral 'J 1 'J Accounts and Invoiica 1-3 Avemgia;,' jf Accou uts 211 Ajcioms 17 Account of Salea 251 T {ankin;,' 179 Hankrcp tcy 2 P) 2 I5rokcrago 190 Bank Discount 181 Book-keeping Exercises IVl'i Coins, Gold V.C,0 Coin3, Silver 851 Cental System 800 Commission 180 Currency, Paper 203 Currency of Canada 204 Commercial Paper 155 Custom 1 louse Business 215 Cubolvoot 2S9 Decimal Coinage 02 Denominate Numbers 45 Subtraction of !'>0 llulti plication of 50 • — Division of 53 Discount 177 — Bank 181 Division, Simple 20 Equation of Payments 234 Exercises, Set I CI • Set II 113 Bet III 103 Bet IV 812 Excliango 2G5 ' Arbitration of 275 • American 200 Sterling 271 Evolution 283 Fractions Gi. Fractions, Common 61 • Division of 70 Multiplicatiou of CD Addition of „ 72 Subtraction of 74 ■-» Denominate 75 Xll. INDEX. Fractions, Dociraal 77 Iicduction of 83 Additinix and Subtraction lit' SL Multiiilication of 87 • Contra-jtLd Jlctliod 89 Division of 92 Contracted Method 94 Denominate 06 Foreign Moneys 274 Creatcst Common Pleasure 58 Interest 110 Simple 141 Compound 174 Insurance 195 Lilb 201 Involution 280 Interest Laws, United .States 352 CanaS0. — Again, if I find that I have (ico cne-hundrcd-dollar bills, six one- dollar bills, but no ten-dollar bills, and I write only 20, this would bo plainly incorrect, for the 2 would stand for ten-dollar bills r.idy, but by inserting n.scro mark between the figures I thruw the 2 into the place of hundreds, and $20G represents correctly that I have two onc-liundred dollar bills, and sic one-dollar bil!s, but no tcu-doliar bills. Tho superiority of this simple system over the cumbrous Roman one will be manifest from its simplicity and broviiy by writing cighty-ciglit according to both systems — thus : LXXXVIII, and 88. RtTLE FOR NOTATION. WritO tne significant figures of the first period (lamod h\ th ir proper places, filling up any places not named with ciphc.*^, just as it you were writing the units period with nothing to follow ; then, to indicate that something is to follow, place a coiunvi to the right, and do the same for every period down to mnU, inclusive. For example, teacher says : " Write down one hundred and y'lx millions; ' pupil writes lOG and pauses; teacher adds, "ninety th(>usand;' pupil fills up thus : 100,090, and pauses; teacher concludes: "aut eighteen ; " pupil completes 100,090,018. If tho teacher k!iou1(J say sixteen millions and the pupil write 010. the cipher woult be manifestly superfluous, as it has no effect on figures placed to th' right of it, but only on those placed to tho left. EXERCISES. Wiite in figures and read tho following quaiUitie:-: : 1. Ten millions, seven thousand and eleven. 2. Ninety billions, seven thousand and ten. 3. Eighteen millions, sixty thousand and nine hundred. 4. Forty thousand and nine hundred. ADDniON. 17 5. Eighty-seven milliona and one. C. Ninety thousauJ, seven hundred and eight. 7. Eleven millions, eight hundred thousand imd twenty-four. 8. Six hundred and seven thousand and ninety-seven. 9. Eight hundred and seventy billions, sixty thousand and eighteen. 10. Eleven billions, eleven millious, eleven tliousand and eleven. AXIOMS. 4. — Axioms used in the sequel : I. Things that arc equal to the same thing, or to equals, are equal. II. If equals be added to equals, the wholes are equal Corollary. — If ecjuals be multipUed by the same, the producta are equal. III. If equals be taken from equals, the remainders are equal. Cor. — If equals bo divided by the same, the quotients arc equal. IV. The whole is greater than its part Cor. — The whole is equal to all its parts taken together. V. Magnitudes which coincide, or occupy the same or equal spaces, are equal. N. B. — This axiom is modified by, but still is the principle of, all business transactions, jiurchascs, sales, barters, exchanges, &c., &a, where the articles traded in are not iqaaU, but equivakn^s. ADDITION. 5. — Addition is the mode of combining two or raoro numberg into one. The operation depends o:: axiom II. The result is called the sura. Thus : $S-fi;9-]-$G=$L'3. The sign plus (+) indi- cates addition. To illustrate the operation, let it be required to find the sum of the live imnibcrs of dollar.s noted in the margin. First, the numbers are placed so that those of the same name arc in vertical columns, i. c, units under units, tens under tens, &c. Next, we find that tho sum of the units' column is (Ax. IV., Cor.) 27, i. e., two tens and seven units. Next, we find that the sum of the tens' column is 35, but, as it is the teM* $287G54 758287 012873 494768 830195 v 18 ATilTIDIETIO. 27 350 2400 27000 2G000O 2700000 $2989777 column, we -write (Art. 3) 350 ; in the c.nno maiv ncr wo find \\\q sum of the hundreds' column to 1)0 2400 ; the sums of the others will bo soon hy inspection. Having thus obtuined the sum of each column, each being snmnud as if itnlfs, but placed in succession towards the hft f by Arts. 2 and 5), vrc now take the sum of the partial results, whieli (Axiom IV. Cor.) is '.lie sum of the whole, viz.: $2,909,777. In practice the operation is much abbreviated in the following manner : — When the units' column lias been added, and we find the sum to be 27, I. c, 7 units and 2 tens, we write down the 7 units under the units' column, and add up (Art. 3) the 2 tens with the tens' column, and we find the sum ia 35 tens, L c, 5 hns and 3 liundruh, and we placo the 5 /( ns under the tens column, and add u;) tho 3 hnmlredtt with the JiiauJrejJs' column, and so on. The transferring of the tens, obtained by adding tho units' column to the tens column, and the hundreds obtained by adding the tens' column to the hundreds' column, (tc, &c., is called carrying. In all such operations the learner should carefully bear in mind tho principle explained in Art. 3., that every figure to the left is ten times the value that it would have if one place farther to tho right.* EXERCISES. Find tli'j sums of tho foUorvin.g <(uan titles ; $287054 758287 012873 494708 830195 $2989777 (1) (2) (3), (4) 895703 9987(5 • 49170 987054231 03879 89705324 283527 1234507S9 r.43S7 42350798 059345 903700504 789 50798423 7984 89070 500G 137508 23507989 31059 7590S0391 278052 79S4235G 908438 07091)8707 85945 05324897 2S90392 4340001745 721090 357055787 • We wouUl strongly rocommond every one who wishes to become an expert .accountant, to avoid the common practice of dr.iwliui^iip a column of figuroH in the manner that may be suiEck'ntly iiluslratiMl by ibo aildiug of tlie units' column of the above example. Never say 5 and b are i;> ; 13 and J are i6 : l** and 7 are 2a ; 23 and -i arc 27 ; but rua up vouv column thus : 5. 13, IS, I'd. nno man- iilunin to ) seen by 11 of each ut placed : ami 5), Its, ^vIlicl^ ic whole, ibroviiitod s' cohnim to bo 27, ,0 7 unifa 3) the 2 10 sum is we placo J up tho id so ou. Jdini; tiio gained by is called ly bear in .0 the left icr to tho CO rG5324 i5G798 '98-123 )G79S9 $-12:]5C !24S97 )55787 an expert. I ol figuroH the units' I 3 are iG : 13, IS. is. AX^Dirroy (») (6) 785G3 4798G 12345 5798 07890 . 19843 •8705 r)G479 43 J ID 28795 876di 807 32109 1984 7891:3 G8195 G51:j9 3879 93. G5 G98 43JS3 5879 . 77S77 17985 nfi.OAo 19 (7) 33G981 (0) 189 976 85 73 338 793 49 75 218 305 113 279 67 76 84 1379 5159 S053I1' (10; 98 89 76 C7 281 592 078 58 G7 98 149 G7 54 72 298 2744 918273 (i5U)23 ."> 74859 203748 597485 98o879 9S7G5 987G 087 45{)S79 o45G78 '1705J57 47 96 S3 50 74 82 97 68 76 49 76 54 78 69 37 1044 738 C59 471 39; GSy 8JC 7>'!i 'J7S lot 99!j S8S 777 Glifi 5") 5 h97 978 124G0 (12) 1298 7G4 5837 &i95 7S9 638 546 98 475 394 82 157 638 594 789 114 19715 rmnk ^,11 ,p,,l,,- ,qnally lo nn.liiplicatiou, an,l therofuro to ov(- arl hn^ The quick operator uses tho oyc, and not the toaguu. r 20 ARlTmiETIO. There is no method of proving the corrcotnoes of any addition witli positiro ccrtjiinty, but a very cotTvcnicnt mode of chcei.mg is to add each cdluiim both upwards and downwards. Another mode is, to add by parts and take tlie sum of those. This is a very securo method in the case of long columns, but not u) ready aa the former. If the same losult is found by each method, tlie en m may bo acoountcd correct. 578G43957 235412712 3-13231245 I SUBTRACTION. 6. SrSTRACllON is the convcr.sc of addition, i, e^ it is the mode of linding the dlfforcnce between two numbers, or, in other words, the excess of ono nunibcr above another. The number to bo subtracted is calltd the subtrahend, and that from which it is to bj taken the minuend, and the result is called tlie re- mainder, dilTcrcncc or excess. The sign used for subtraction is a line ( — ) culled minus, or less. Let it bo required to find the difference between $rj7SG430j7 and $235412712. Having placed them in vertical columns, as in addition, it is obvious that 2 units taken fruui 7 units will leave 5 units, and that 1 ten taken fruui 5 tens will leave 4 tens, and so on. But if it is required to find the excess of .$513074208 above 8347695319, we find that cacli figure of the subtrahend, except the last, count- ing from rigbt to left, is greater than the corresponding one of tho minuend, and therefore, to find the correct difference, we have recourse to a simple artifice, which is deduced from the principle of tho notation, and may be illustrated in the following manner : — Taking Iho ([uestion in the margin, we arc first required to subtract 7 units from 3 units. Now, though tlic algebraic notation furnishes the means of noting tho diflfercncc directly, the ordinary arithmetical form doea not, but still it furnishes the means of doing it indi- rectly. By Art. 3 each figure to tho left is ten times the value of the next to its right, thereibre we take one of tb.e 3 tens and call it ten units, and add it to the 3 units, and thus vre have 13 units, which let Us enclose in a parenthesis or bracket, thus : (13), to indicate that the whole quantity, 13, is to occupy the units' place ; when one of tho three tens haa been thus transferred to the unit«' 333,333 177,777 155,550 2(12)(12)(12)(12)(13) 17 7 7 7 7 1 5 6 SUBTRACTION. 21 (Idition ng 13 to node is. J Bccuro former, may Ix* t is the in other nunilxir 11 which the rc- ition is a find the ig placed s obvioU3 5 5 units, re 4 tens, he CXCC83 Ifind that count- ono of wc havo inciplc of 'oUowing lire first , thoug!i oting the onu docs it indl- thc value and call 3 units, which let bracket, the whole ho units' tens has he unit%' 200000 rjoooo 12U00 IL'OO 13 o33o33 place, only itco tens remain in the place of tens, and wo are now rcfjulrcd to take 7 tens from 2 tens ; to do this wo have recourse to the same artifice, by calling one of the hundreds tens, which j/ivcs 10 tens and 2 tens, and so on to the end, the last 3 necessarily becoming 2. Wo can now subtract 7 from 13, &o,, &c. This mode of resolution depends on the corol- lary to Axiom IV. The parts into which the whole is virtually resolved are shown in the margin. This artifice is popularly called borrowing. In practice the resolution can bo efl'octcd mentally as we proccol, and as each figure from which we lorrow is diminished hy unity, it is usual to count it as it stands, and to compensate for this to increase the one below it by one, for, as in the example, 7 from 12 is the same as 8 from 13, and 2 from 3 is the same as 1 from 2. We arc now prepared to answer the pro;,oscd question, as annexed, and wo say 9 from 8, wo cannot, and there are no tens to borrow from, we therefore take one of the hundreds and call it 10 tens, and one of the tens and call it 10 units, which with 8 units makes 18 units, and wo take from 18 and 9 remain. We have now only 9 tons left, but we reckon ihcm as ten, and to compensate for the surplus ten, we reckon the 1 below as 2, and say 2 from 10 and 8 remain. Wo proceed thus to the cud, and find the whole rcmaiuder to be ^105778880. $513074208 r^347895319 §165778889 EXET. (JISES. RE.MAINDEFS. 1.— From 847G^.902i take 4rG584350=3710WGG2. 2. 3. 4. 5. " 1010305001 " 070085093—339019968. " 59G3S743 '•' 1870G854— 40S-l'8S9. " 7S13-57 " " 111111111 ' r.7i5079-. 40G7578. 03057293^ 124.;;]818. In Subiraclion, as in Addition, wo have no method of proof that arrives at positive certainty, but cither of the two following methods may be generally relied upon. 1. — Add the remainder and subtraliend, and if the sum is equal to the minuend, it is to bo presumed that the work is correct. 2. — Subtract the remainder from the minuend, and if this second remainder la the came as the iiubtrahe&d, the work may be accounted correot. 7 'I" li! 22 ARITHMEnC. M U L T I P 7. 1 C A T I O N . 7- — MuLTiPL'c.VTioM may be fimply dtriiicd by 9ny\n^ thnt it U u Blioit method orp(!rii)rmin;j iulJition, when all llic (|uaniitic9 to be added uro tho H:inu! or »Mjuul. Thu.s : G ;-0 :-G i-G-|-G-j-G-j-U-|-C, means that ci,;ht hixos arc to bj added tiycthci , or that kix i:* to bo repeated as ol'tiii as there are units in tii,'ht, and wo say that 8 times G is IS, and write it thus: 8xC^-i3. So also 8 j- 8-i-.S ;-8-,-S-i-8 yivos 48. .^0 that G.8_S.G— 18, and thus wo can construct a niultiplieition tabic. The number to bi^ repeated is :'ailed the mulliplieaud, and the one tlr.it shows how often it is to bo ropcated is called the multiplier, and the result is called tho protluot, or what is produced, and hence the multiplier and multi- plicand arc also called the factors or uiakors, or producers, and tho operation may be called findinj a product when the factors arc given. IIcuco also the modo of carrying is tho uamc in raultipiication as in addition. MULTIPLICATION TABLE. I ■ i I • i'wico } times 4 times .1 times G tujcv t t timed I is 2 1 is li 1 is 4 1 is 5 1 is G 1 ia 7 2 — 4 2—0 2 — 8 2 — 10 2 — 12 2 — 14 3 — G 3 — n 3 — 12 • > 15 3 — 18 3 — 21 4—8 1 — 12 4 — 10 4 — 20 4 — 24 4 — 28 G — 10 r, — i:. f) — 20 r» — 25 5 — 30 5 — 35 f) — 12 G — Jo C. — 24 li — 30 • G — 3G r. — 42 7 — 14 7 — 21 7 — 28 7 — 35 7 — 42 7 — 49 8 — k; 8 — 2 \ 8 — 32 8 — 40 8 — 48 8 — 56 y — 18 9 - 27 9 oC — 45 9 — 54 9 — 63 JO — 20 10 — ;j() 10 — 40 10 — 50 10 — CO 10 — 70 n - 22 11 — ;j;} 11 — 44 11 — 55 11 — GG 11 — 77 12-21 1 2 — v.'', 12 — 48 12 — GO 12 - 72 12 — 84 Vi limes 9 tiim JS 10 timos ' 1 times 12 times 1 is 8 1 is 9 1 is 10 I is IL I ia IJ li— IG 2 V6 2 ~ 20 •> __ •'•> 'J. — irl :!— 21 » — 27 3 — ."O ;; — 33 3 — 3(J 4 — 32 4 — .•id 4 — 10 4 — 44 4 —48 40 f) — 45 — i'jU " — 55 n —GO <; — 48 G — fil G — GO (; — CG (• 72 '; — f)6 7 — 03 7 — 70 7 — 77 7 —84 '6 — f.4 8 — 72 8 — 80 6 — h3 8 — 96 f)— 72 9 — 81 D — 90 9 — 99 !) —108 10 — 80 10 — 90 10 —100 10 —110 ; 10 —120 11 — 88 11 — 99 11 —110 11 —121 11 —132 12 — 9f) 12 — 1 C'S 12 —120 12 -132 ( 12 — M4 MULTirUCATIOX. timed ia 7 — 14 — 21 — 28 - :j5 -42 - 49 - 56 - (i3 -70 -n - M timc3 — U -J(i — 48 — GO — 72 — 84 — 96 —108 —120 —1:52 — I'W: Regarcliiig tiio fulluuing pnrt of this t;tM<>, gee puggcgticns to 'f e;iclicr.< : 1 .1 liun.' 1 1 1 1 tiiiU'sl l.> tiiui's M '» * IH li2 ;{ _.. ■\>- 1 ~ r.i ^ _.. 8(1 (J — 9(; 7 .— 11'. H — ii;.'- 9 11 ! i; tlUK < \H tliiit'H 19 UUU>.i u :ti 2 iM :iii 2 id ^8 :i — 51 :J - ;•)! ;) — 67 I --- Ci> 4 - 72 4 — 7G 5 -- &:, '» -- W n - 95 ('• — 10 (i — l')8 — lU 7 - - 1 1 'J 7 - rjij 7 — 133 H "- i;;(i b - 1 1 1 H — 152 9 -- ij.j 9 — IC'J 9 — 171 Uf liuvo ill (ho above J.ihlo corrccU'il ibo i^toin gMinmuiicil hliimliT at.' roiinv)!! o;'s.iyInc,' n'^Ul ;Iiiii'< l\v>» aiiu bixluca. Wlioii more tliaii two factors aro jjiveii, the operation Ls cUlcJ continu'Jil nuiUijilifitlon, as Gx'i)v-^(r)--180. "VVIku the I'.ictors cousist of more (i ,Mirci :)ian one, the most convenient nioilu of operation is that .'•hown by the anu.'XoJ example, whore the niuhiplieaud is first repcatei S time.x, then 00 tiuics, or whieh is the same thini; limes w'.ien the lir.-t li;,'ur; of the second hno is placet] under the; secoiul fiu'ure of the lir?t line, *'. r. (art. 2,) in the plaeu < f tens, and then the partial products :iri> added, whieh (A.r. IV. Cor.) ^rivco ihe full product. H"nce wo deduce the 'J4r>18G .03 L'7t;i488 JOTiJO'.G *iyOJ72 IIULK FOll MULTIPLICATION. yiiodVdis PLoe the lauitiplier under the nuiltiplioand, nnita under unit;-, ten^i under tens, &c., &,^., — conjuicncin{» .\t the ridit, multiply e:ich fl-rure of the nniltiplicand hy eaeli fi;iure of I lie multiplier in fujccr-sion, jihieing llie reMtltii in j arallel lines, and units, tens, &c., iu vcrfi'.r! coluustis, — :.dd ill the lines, and the Hum of all the partial pro.hiet3 will (Ax. T\'. Cor.) be tlie wliole product required. A.4 far aS tlic learner has eimniit'.ed a niultiplicati(»n table to mc- i.iory, Bay to 12 times 12, the worV e-n Le done by a f-indc operation. V\'|;en a!)y number i.< iimltiplicd by it.^e'f, t';e Mrodnet is called the f-(ju:ne or second power • if tiiat nund)er, ;ind the product of three c<(U."! f'.ctors is called a cube (»r third power, the p-ro- duet of iour equal factors the f -urth power, &c., &e. The terms fqnarc and cube arc derived from .superficial and tolid measurement. Th;* annexed square has eaehcf its sides divided into 5 equal pails, and it will be louud on inspection that the wLolc %aiic eoutiuna 5 rrrr 24 ARITHMETIC. 25 (=5X1")) small squares, all equal in area, and having all their sides equal. — II ciico because 5x5 represents the whole area, 25 is called thu t:(juarc o[' ;"», or the second power of 5, because it is, the product ol' tlio two equal factors 5 and 5. A cube is a solid bodj, the k'iii;th, brcadtli and thickness of which arc all equal, and hence, if these dimensions be each represented by 5, the whole solid will be represented l)y 5X5X5--125, which is therefore called tho cube or thh'd jiower of 5. The terms square and cube are often used without any reference to supcrfl iial and solid measure, i'or exanqile, in lineal measure an expression for distarcina straight line hi often called the square and cube of a certain number, thus : bl is called the square, and 729 the cube of 0, although these are only used to show that the distance is not 9 in cither case, but in the one 9)<9, and in the other 9x9X''^- I'l such cases tho terms Bccond and third power arc therefore to be preferred, and since no solid can have more than three dimensions, we have no term corres- ponding to square and cube for the product of four or more equal factors, and therefore wc arc obliged to use the words fourtli pow^r, fifth power, &c., &c. j1 ■ill 1 t CONTRACTIONS AND PROOF. There are many cases in which multiplication may bo performed, by contracted methods, but the utility of those, for the purposes of accuracy, is, at least, doubtful. The most secure method in the great majority of cases, is to follow the general rule. Multiplication by 10, 100, &.C., is effected at once by adding a cipher for ten, two for 100, &c., &c. Tl:e following is, next to the above, the most uafo and useful contraction that can be adopted. It is exhibited in the subjoined exanqiles, but purposely without explanation, as an oxer cise for the learner's reflection : Oudixahy MKTiion. 35097X17 17 CO>TnACTEO SIETHuD, 3o0.)7xl7 2-4987U Oi.viI.NAltY MKTUOO, 35097X71 71 CO.NTliACTKD METTOI* 35097x71 249879 249879 35G97 G00849 35G97 249879 2534487 G0G849 2534487 Tho only practically useful proof of the correctness of the i;ro- duct, is tiae one subjoined, but even it, though it scldon\ fails, docs not secure positive certainty . MULTirUCATIOX. 25 all their ;a, 25 is asc it ii i a solid ^uul, and olo 8oUd ailed the :iro often ,re. l^or , straight icr, tlms : these arc ic, hut in the terms i since no ni corres- orc equal performed, urposcs of led in the |llpUcation I- ten, two most mfo cd in the an oxer ht.D METnoi> iT9 U87 If the rro- fails, doe,j Add together all the figures of each factor separately, rejecting 9 from all sums that contain it, and multiply the remainders together, rejecting cvciy from the result, — add the figures of the product in the same manner, and if the two remainders arc equal, the work mai/ be accounted as correct, but if they arc not equal, the work must be wronir. The reason of this proof depends on the propcrt}' of the number 0, that if any number be divided by 9, the remainder will be the same as if the sum of its digits were divided by 9. — Thus: 7l221j3-:-0^.S24GS3+f., and the sum of the digits is 24, and 24-:-9-^2-|-G, /. c. 9 is contained ia 2-4 twice with a remainder 0. Every 9 is rejected because 9 is contained in itself once evenly, and tlicrelbre cannot aU'ect the remainder. Let it now be required to multiply 122 by 21. Now, 122^9Xl3-|-5, and 24^9X2-1-0, and if we multiply together the two factors thus resolved, wc get 9Xl3X9x2-i-9x2xJ>H-9Xl3XC+Gx5, and since 9 is a factor of all but the last, the last only will give a remainder when divided by 9, and therefore the whole product will give the same remainder when divided by 9, as GXij-:-9, which gives the remiainder 3, for 6X5— -30 and3(>-:-9 gives 3 with a remainder 3. To test this by trial, we find 122-:-9=::13 with a remainder 5, and 24-:-9:i:^2 w'ltli a remainder G, and the product of these remainders is Gx5=:30, and 30-:-9^3 with a remainder 3. Again, 122x24=2928, and 2928-T-9=:-325 with a remainder 3, as in the case of the factors. EXERCISES. 9387 1. 7890X5=39480. 2. 581907X8=4055730. 3. 4. 5. G. 7. 10X4=3754984. 193784X7=1350488. 391870X9=3520884. 987450x0=5924730. 400783x52=25832710. 8. 719804X43=30954152. 9. 375907X04=24001888. 10. 27859X29=807911. 11. 079854X83=50427882. 12. 759084X187=142000908. 13. 0372X1034=8777848 14. Find tiio second power of 389 ? Ans. 151321. 15. Find the third power of 538 ? Ans. 155720872. 10. Find the fourtli power of 144 ? Ans. 429981096. 17. Find the cidje of 09 ? Ans. 970299. 18. 5790 seamen liavc to bo paid 109 dollars cch ; what is the amount of the treasury order for that purpose ? Ans. $979,524. 19. A block of buildings is 87 feet long; 38 fc^eJ: deep, pad 29 feet high ; how many cubic yards docs it contain ? Ans. 3550 ^ cubic yards; 'fTr 26 ARITHMETIC. 20. If 29 oil wells yield 19 gallona an hour each; how much will they all yield iu a year ? Ans. 201115 gals. 21. If the rate on each of 1597 houses bo $19 ; what is the whole assessment ? Ans. $30343. 22. If 1297 persons have paid up 9 shai'c. cacli in a railway company, and each share is 015 j what is the working capital of the coniDany ? Ans. S172095 ,(; I.' ir DIVISION. 8. — Dn''IsiON is the converse operation tO multiplication. It is the mode of finding a required factor when a product and another factor are given. It bears the same relation to subuviction tliat multiplication does to addition, as will be seen below. Ly Ax. IV. Cor. we may resolve any complex quantity into its component parts; 80 division is resolving a certain quantity called the dividend into the number of parts indicated by another quantity called the divisor, (divider,) and the result is called the quotient (how often.) Let it be required to find how often 8 is con- tained in 279,856. We can resolve 279,- 85G as in the margin ; then dividing tho lines separately by 8, we obtain the partial quotients, the sum of which is tho whole quotient. But this resolution may bo done mentally as we proceed. Wc first see that 8 is not contained in 2, therefbro we take 27, and find that 8 is contained in it 3 times, with a remainder 3 ; next combining this 3 with the next figure 9, we get 39, in which 8 is contained 4 times, with a remainder 7 ; combining tliib 7 with the next figure 8, wc have 78, in which 8 is contained 9 times, with a remainder G ; combining this with the 5 following, we obtain 65, and 8 is contained in it 8 times, with a remainder 1, which combined with the G makes 16, and 8 is contained twice in IG. The correctness of the result may be tested by multiplying the quotient by tho divisor. AVhen the divisor consists of more than one figure, tho learner must have recourse to a trial quotient, but after some prao- tice he will have little difficulty in finding each figure b^ insi)Qvtioii. 8 240000 30000 32000 4000 7200 900 640 80 16 2 8 279856 34982 DIVISION. 27 ow inucli at is the a railway tal of the ion. It is nd another iction taat ly Ax. IV. Qcut parts-, vitlcoJ iuto tlio divisor, 'ten.) Let 1 8 is con- .solvo 279,- lividiug tlio the partial tho whole m may ho Wc firsti therefore contained Icr 3 ; next IwhicU 8 is 7 with the lines, with a obtain 65, Ih combined correctness Int by tho Ifiguro, tho I some prafr 1 insi)QgtiiOB« KJr Let it bo ro4uir<(l ti find liow often 298 is coi.tainod in 431 70G. — The numbers Ijcin^' arrany;cd in the convenient order indicated in the manrin, \vo mark off to the riixlit of tho dividend blank spaces for the trial :;iid triio (jiiotiont.-^. Wc readily ficc that 2 is contained twice ill •!. but caunnt .so oas>ily sec whether the \vhok' divl.-^or 208 i.s contaiuod twice in tho name number of fijnins of the dividend, (viz. ■JiJl.) v,e therefore make trial, and ])lace the 2 in the trial quotient, and multiply the divisor by 2 to fmd how much we .shall have to .subtract from 431 . We find 298 )< 2 ^=590, larger than 431 , and therefore we reject 2 and try 1. Now 298x1-^-298, less than 43T, BO WC subtract and find a remainder of 133, and as this proves correct^ wc place the 1 obtained fa 298)43176U( 2.1.5.4.5.4.9.8 trials. 298 i44y true quotient. 1337 1192 1456 1192 2G46 2384 262 298 the true quotient. Wc find our next partial divdcnd by writing 7, the next figure of the dividend after the re- mainder 133. Our experi- ence of the first case sug- gests to us that though 2 is contained G times in 13, yet on multiplying something will have to bo carried from the 98 which wc expect will make the result too large, and therefore wc at onco try 298x5:^^-1490, larger than 1337, and so we try 4, and find 298X4=1192, which being less than 1337, wc subtract and find a remainder of 145 ; and liaving placed tho 4 in the true quotient, we bring down the next figure of the dividend, giving a partial dividend 145G. Ey in- spection, as before, wc see that G would be tco large, owing to the carrying from 98, wc try 5 and find 293x5—1490, v.hieli is larger than 145G; wc try 4, and find 293x4 1192, which i.s less tiian 145G, so we subtract and find a remainder of 2G4. Having placed this 4 after the other 4 in the true quotient, we bring down G, the last figure of the dividend, wc try 9, and find 298x9=2682, which is greater than our last partial dividcrid, 234G ; we try 8, and find 298X8=2384, and this being lesss than 2646, we subtract it from , but we find that which is 28 AEirHMETTC. ]\' > 298 298000 119200 11020 23S-1 202 that number, and find a final remainder of 2G3, and close the question by entering 8 in the true quotient. The mode adopted to indicate that the remainder 2G2 still remains to be divided, which cannot bo actually done, as it is less than the divisor, is to write the 293 below the 2G2, and draw a lino between them, thus H!;*, as also is scon in the margin. The resolution into partial dividends is also shown in the margin, where it will be f^oet that the partial dividends, includ ing the remainder, make up the whole original dividend. So alst the partial quotients are exhibit- ed, making up the whole true quoti- ent. That the trial quotient is not a single number, like the true quo- tient, but merely a succession of detached numbers, used as separate trials, is indicated by placing a full point between each pair. "When we have multiplied the divisor by any figure in the trial quotient, and subtracted the product from the partial dividend, should the remain- der be greater than the divisor, we perceive that the trial figure is too small, and we must try a larger. From these illustrations we can deduce a Remainder Dividend 4317GG =1000 ^ 400 == 40 r:. 8 1448 RULE FOR DIVISION (1.) Place the given numbers in the same horizontal line, put- ting the divisor to the left of the dividend, with a vertical line be- tween them, draw another vertical line to the right of the dividend, and enter the quotient, figure by figure as obtained, to the right of that line. (2.) Find by the principles of multiplication, how often the divisor is contained in the same number of figures of the divi- dend ; place the number thus obtained in the quotient, and multiply the divisor by it, and subtract tho product from the corresponding partial dividend. (3.) To the remainder annex the next figure of tho dividend, and proceed as before, and so on till all the figures of the dividend are exhausted. (4.) Should there be a remainder, write it and the divisor after the quotient, thus : ^l "]"^"!''""' The divisor is often written to the right of the dividend, an'l the quotient written below it, a horizontal line seoaratiDg the two. DIVISION. se the question ted to indicate licli cannot bo the 298 below I line between also is seen in •csoludon into alf-o bliown in it will be ycei dcnds, includ make up the lend. iSo iilsc :.s are exhibit- lolc true quoti- 3 the true quo- ed as fjcparatc pair. "When quotient, and ^d ihc remain- trial figure ia tal line, put- rtical lino be- tho dividend, the right of in, how often I of the di vi- and multiply orrcsponding ; figure of the Sgures of the ider, write it end, and the e two. EX-UIPLK OP FORM 1 470)8503504(18053,2 7 Q. 47G 3333 3808 2550 2380 1704 1428 27G 29 EXAMPLE OP FORU 2. 18G0904 1 87 174 21389^4 120 S7 339 2G1 780 69G 844 783 61 exeboises. 1. 15547G8-;-21G=7l98 2. 318S4470-:-779=40930. 3. 573S0G25-:-75-5=-^7575. 4. I2S1009S--732:::.17500."«-. 5. 0313702S59-:-4GS7310J:19S7 9 C. 44014841047G-:-73SS524G=G079!!''^^ 7. 100oSS2S2929-:-138G^.79024G8 .Vo'""""^'* B. 35G7G210S32-:-79094451 .7G4095'>'"^^o«7 9. 53uSlSS34-:-907.. 501802. '''^^^^-^' iO. 1700G49155Gl-:-750 ^ 2 '40G44479 fl. 5542702970Gl-:-75S4-:::7;;0S41G3^^'^<' 12. G0435G74G34520 -:-7G4095:..7D09laEro 7 n a . lo How many bag.., each containing 87 pounds' wHl "4 c.o .pA pounds of flour fill? \n'^ 035 079 " ^""^"9, will ^4,8 j3,4G4 A'>. -i-U,blu,2ol pounds of cotton are made ud in 2'!^ 87Q Kni how many pounds in each bale ? Ans 89. ' ^^'' ' DIVISION. 1. -^9087532-1-2=24843700. 2. 57980327-3=19328775.3 ^i nil )• !l (1 ; , 1 ; 30 ARITHMETIC. 3. 87905328-^4:.. 21991332. 4. 7963821-:-5.-ml5927G4V. G875324--G^-11-15SS7J 3987G54-:-7^509GG4f 5. 6. 7. 1987G532-:-8:^-24S45G6i. 8. 297G8j4-:-9.:^3307G1';;. 9. 49G7532-:-10::r-_49G753^. 10. 4G879352-:-ll- 11. 187G5314-:-12. :42G1759-j3p .15G3777J. 12. 13. 14. 15. IG. 17. 18. 19. 20. 7SG5424G-:-18=43G9G80-^. 75088-:-52=:1444. lG74918-:-189=88G2. 31884470-:-779=40930. 573S0G28---7575^7575;y33^^. 5542702921 98-: -7584r=73084163-nJ55. 887899S0979-:-9584=:9264397y'-V5. 102030429729-:-12345G=S2G452J!i5J|. 2G7S179-:G000--3G500=10077204.^ 21. 41*7 racn fell 1G3798 trees; liow many does each fell on an average ? 22. If 148 houses pay a tax of $7844 ; what is the rate on each on an average ? Ans. $53. 23. If $415143630 arc levied from 4455 townships ; what is the portion of each on an average ? Ans $93180. 24. How many lots of G754 each are contained in 3968091 51372 ? Ans. 587G3718. 25. What quotient will he obtained by dividing 9G1504803 twice by 987 ? Ans. 987. 9.— TABLES of MONEY, WEIGHTS & MEASURES. DECIMAL COrXAGE. 10 mill3 (M) aro 1 cent (ct.) 10 cents 1 dime (d.) 10 dimes, or 100 ccnta.. . 1 dollar (§) BIIITISU on STEUUXO MONEY. 4 farlhinga, or 2 half pennies, are 1 penny (d.) 12 pence 1 shilling (s.) 20 Bhillings 1 pound (£) AVOIRDUPOIS WEIGHT. TADLE. IG drams make 1 ounce, marked oi. IG ounces 1 pound, *• lb. 23 pounds 1 quarter, " qr. 4 quarters 1 hundredweight, " cwt. 20 cwt 1 ton, '• t. Note.— This weight ia used in weighing heavy articles, as meat, groceries, vegetables, grain, etc. TABLES OF MONEY, WEIGHTS AND MEASURES. 31 TROY AVEIGHT. TABLE. 24 grains (ffrs.) make ^ . 20 pciinvweiffhts J Pennpt-eight, marked dwt 12 ounces, ._. '. j """^ts u ^^ i pounJ, " Hj NoT..-Troy weight i3 usoJ in wei^^bing the preciou. metals and stones. ' APOTHECARIES' WEIGHT. TABLE. 2a grains (grs.) make i, , 3 ficriipics. ... I f't'i'iilJl". marked scr. 8 draws.... 1 ilram, '■ ar. 12 ounces 1 ounce, " oz. V . , ■ ■ ■ ." ' pound. '• lb. J.oTi.:.--Apollieoanes and Physicians mix their medicines by ihia weight but tliey Ijuy and .sell by Avoiruupois. weigiu, PRODUCE WEIGHT-TABLE. OKALV. }J^eat CO po!uids to the bushel. Oats " Corn Corn in cob Barley . . Rye Buckwheat. Peas Beans , Tares u • i i< 5 (J k( t. SO (1 i; 48 " li [i6 •'( 11 •18 H n GO (1 li CO 11 a CO (• li SEKDd. JJ'o'^"' CO pounds to the bushel. Ti.;othy Ileiup Bhii! grass . Red Top.... Hungarian / Ri-ass ... i Millet Rape -IS 51 11 8 48 48 50 (> « '• l< 11 l( •^■♦'3 1^6 jardb liuu cluth H--^^ 321.82 1 (10.) |157.'J9 2(;s.73 9S5.-15 197.0G 3^5.18 R7i;.75 795.85 CG713 (Jj9.(33 4893.07 (11.) Sold to S. Fulton, Aurora, 12 pairs of worsted storking.s $13. 18 " " flannel drawers --. 24 " " kid j^loves 8. 50 Kcliool books '^9. 29 yards of satin ^3, 90 selioul copy books 1 180 yards of ribbon ^9 84 yards of tickinj^ ''^^ 122 yards of hhccting ^^ 255 50 75 03 72 23 ,84 ,70 ,08 .18 QO 12. The shares in fin oil-well speculation are $5 each ; A. takca 15 shares; B. 25; C. 20; D. 1 ; E. 11 ; F. 37 ; G. 10; 11.18; I. 8; K. 21 ; L. 14 ; and 14 other persons take 10 shares eadi ; what is the capital of the company, and how many shares arc there ? Ans. Sl,030 and 320 shares. 13. If 17 vessels bring to the port of Boston cargoes of the follow- ing values ; what does the whole amount to ? $2305.75, 81793.87, $3815.25, $2718.03, $4180.50, $3179.13, $1023.88, $4311.75, $1987.38, $2975.75, and the other 7 average $2089.13. Ans. $47781.80. Subtraction of dx)Uars and cents. (1.) (2.) (3.) $507819.83 $83750.17 $17423 37^ 278950.89 70489.71 9G54.03| 288802.94 7206.46 7708.74 \ l)LCI>LU. COINAGE. 37 4. What '.s the dWroronci; bctv/ccii 27G0 dollan and 5') cental, and 987 dollars 87. Uvi its? Any. !;17vbl.G2i. f). Tlio debit t^idi* (.f :i lodi^'or is $1770.80, and tliu credit hido 8870.50 ; wliut i.s the balance V Ans. §S94.30. (J. The credit bide; (if ii cabh book is 8S795.88, and iIk' debit sido is 310358.1.S; what i.i tlu; balano;? .\ns. ;iJl5'J2.30. A linn owes ?227'.JGS.25, and (he cstutcj i.s worth !}9S7'J4.75; what i.s the fatato of tho affair.s of the firm? Ana. — The lirni in I liable to pay 8129,203.50 over and above the asaet.s. P. A ship and cargo were worth $27509.50, — the ship w;;a lost, lid only $0734.00 worth of the cargo saved ; what was the los.s ? Ans. $20724.90. MuUijjUailiuii of dollars ui\d ants. * Mi. $365.75 87 (2.) 81873.47 09 (3.) 1805.03 93 -4-) ♦24780.38 45 25i;025 292000 1080123 1124082 259089 779067 12393190 9914552 31820.25 129209.43 80503.59 1115387.10 Division of dollars and cents. 1. $28G42..4-:-29r:- 1987.06. 5. <^1 943243.55— 08: :=SlO70.S5 4. 8::713.3.34-:-87-:^T.e420.82. 6. |;U421.25-:-03-^«498.75. $00509.03-:-76-879G.18. 7. e2S479.75-:-7S-..305.12i. §43009.75H-98:^e438.87A. 8. e2595.37J-:-709^$3.37-J. 9. ^.2927.30 a year; how much per day? Ans. $8.02. 10. $3953.19 a year ; how much for every working day ? Ans. $12.03. 11. 209 persons have to pay a tax of $1312.72; what is the average tax on each ? Ans. $4.88. 12. Acollectionof ^'544. 04 is made by 187G persons; hownmcli did each irivo on an average ? Ans. 29 cents. * We inns! hiTf ciintion 11. o tyro against fnch modes of oxprossion as this, --•'ninltiply $.-.j by $12." Such an (xpie.s.sioa U s'mply tibsiird, for to pay $!2 fiines b-.i.miRht usi well nio;in 1200 times !»;8.'>, or 12l)0i) times 1 83, wliich wcnM all jrivc widely difTerent results. We miiy indeed have to mul- tiply iidi'noininatennn:ber rc'pre!5entini» i;SJ, by another deuominatr number reprt'.-^i-niinjf il2, as o.ten happens in ques.ioas involvin;; proportion, e. f/., iu interest ; but so soon a^ we use the number iJ.orany denominate number us ;i multiplier it ceases to be denomina.e, :tnd becomes abstract, and no louger re- presents i.nj' denomination,bntmerely tho number of time.^ tho other is to be re- pealed. Wo object uvea to tho puttiog of each qucstioua as "catch qucstiocs." li i gii- -m 38 uormiETio DECIMAL AND DUODECIMAL CtmBENCIES. I! As there is frequent intercourse between the United States and the Lower British American Provinces, it has been t)iought desira- ble to show the method of changing Decimal or Federal currency Lto Duodecimal or Halifax currency, and vice versa. The traffic between the coast lino of the States and that of the Lower British Provinces is very considerable. The trader, therefore, of either re. quires to be perfectly familiar not only with the comparative value of the currencies of both countries, but also with the coins and paper money used by both. Besides there is constant personal intercourse by travelling and migration, and this makes an intimate acquaint- ance with all the details of both currencies a most important acquire- ment. This applies more or less, though in diflferent degrees, to all the British Provinces, except to Canada, where the decimal system has been adopted, though, unfortunately, not universally followed j but it is highly probable that, if the proposed confederation of the British Provinces should be carried, the decimal coinage will be uni- versally adopted, and universally adhered to by the next generation at least, if not by the present. For the same reasons the mode of chandng Federal into Sterling money, and vice versa, has been explained under the head of Ster- ling Exchange. This seems quite as necessary as the preceding, because the traffic between the States and Britain is on an extensive Bcale, and the coming and going of passengers may now be reckoned by thousands, all of whom require to understand thoroughly both currencies and the circulating media of both countries. The in- creasing facilities of communication are progressively and rapidly extending the trade, including the passenger traffic, between the two countries, and hence the greater necessity that all porsSons engaged in business, or in any way exchanging operations, sb.o-.Ud intimately understand how to change the money of each cou y into that of the other. for tliele.inu'V is but too apt to look at the question just nHicBtanls, witliou ever thinking ol' the principle on which it i.s intended to try \\va. Tho ivbsurdity of the expression may bo shown by tho different lights in which 1 he long discuss- ed question, to multiply 2s. Cd. by 2s. (id. may be viewed. \\.) As 2s. Gd, is j of a pound, the question may bo taken as meaning that 2s, Cd. is to be divided Into 8 equal parts, and 1 of them tivkcn. which would bo 3|d. (2.) As 2s. Cd. li 22 elulliugs, tiie qiiestiou uight be tukcD as raeaoing that 2a. Cd. was to be DECi:\LVL COINAGE. 39 The origin of the mark (3) for dullurs n somewluit uucertaiti. Some .suppose it to bo a contraciion for I:. S., the initials of the Uuitcd (States, but it seems to have been in Udo in coutiuental Europe before the discovery of America, andtlicreforc must bo an importatiou. The following explanation of it-i origin seems more probable, for if the other were correct, wc should surely have some record of it. Ac- cording to an ancient fable or fancy, tiie pillars of Hercules marked tho limits of the world towards the west and were said to support the world. From their position at the entrance to the ^lediterra- neau Sea they were objects of interest to the Spaniards and were represented on one side of their coin called the real, and in the coin for 8 reals the 8 was warpcid around them, thus forming the mark. To reduce currency money to the denominations of tlio decimal coinage. Since 100 cents make 1 dollar, and 1 dollars make I pound, 400 cents make 1 pound currency, and therefore to find the numlxr of cents in any given number of pounds, wc must multiply the pounds by 400. Again, since 20 cents make 1 shilling or 12 pcncc^ to find the number of cents in any given number of shillings, wo must multiply the shillings by 20. Lastly, 5 cents are equal to 'A pence, and 12 farthings are also equal to .'] pence, and (Ax. L) things that are equal to the same thing, are equal to one another ; thcrcibro, 5 cents are equal to 12 farthings, and 1 farthing is th') ?i of 5 cents, or fn of 1 cent. Hence to find the number cf cents in any number of pence and farthings, we multiply the number of fartiiings in the given pence and farthings by 5. and divide tto product by 12. Having obtained the three rcbiilts, wc add them r.U together. Thus to change £48 ISs. 9:2-d. to dollars and cents, we nmltiply 48> 'i00=19200 18x 20^ 3G0 IGJ 19570,^ 48 by 400, 18 by 20, and take {>, of 9J, or 39 farthings, and add the three together, which gives us 19570^ cents, or'^$195.7GJ. repeated '1\ tiiuos, which would make (is. od. (ii.) The iaterpretatiou niigbt bo. that as 2s. Cd. u oO pence, that the other 2s. (Jd. is to bo repeated 30 limes, which would give £3 Ijs. Od. (1.) The phrase may also be interpret- ed as mcaniug that oOd. was to be repeated ol) tiaies, which would also give £3 15s. Od. The last two iuterpretatious are the same in two different forms, and give the same result. This is the only view in which the expression has any sense, and proves our statement, that whenever a denoraiiiato number is used as a nuiltiplier, it ceases to bo denomiaate, and becomes abstract. Tho same principle w'U apply to division. 40 AIUTHMETIO. EXsnaisEs. I Ml I'M! (1.) JE79 X400=-31600 16 X 20== 320 6idX {u= m (2) £117 X400=46800 17 X 20= 340 8-^dX 1%=- 14i^2 $319.30| 3. £87.14.10|=$350.97-1^. 4. dE29.19.9r=^119.95. 5. JE67.13.4£=.8270.67U. 6. £279.15.10^==$1119.17i 7. jell8.11.4i=-0474.27i. 8. £79.8.4=$317.GG§. 9. £37.18.8=0151.73^. 10. £57.8.11i=$229.79-^V 11. £49.7.6=$197.50. $471.54/, — ■ !■■■ »mf 12. £137.16.8=$551.33J. 13. £236.19.2i=$947£4i. 14. £19.16.8=$79.33|. 115. £98.1.1i=?392.22J. IG. £87.11.8=0350.33^. 17. £457.12.6=01830.50. 18. £219.4.7J=0876.92iA. 19. £49.9.4;i=$197.871i. 20. £287.18.10^=^1151.77^. i J, 400)19576K48 400 1' ; ' ■' 1 . : 3576|r 3200 ; 1 ; it !. ' ,1 !■■' 1 ' '1 ^ii • 2{)376K18 20 176 IGO IGi 12 To change dollars and cents to Halifax currency, we must i^- verse the above operation. Thus, to reduce $195.76^^ to £. e. d. — First, reduce the dollars and cents to cents, then divide by 400, which gives 48, the even number of pounds, with a remainder of 376| cents ; then divide this remainder by 20, which gives 18, the number of shillings, with a remainder of 1G|- cents, as in the converse operation, wo multiplied by 5, and divided by 12, so now wc multiply by 12, and divide by 5; thus, 10^X12=195, and 195-:-5=39, the number of farthings, and this being reduced to pence and farthings, gives 9|, so that |195.7Gi=£48.18.9|. Or the work may be shortened by the fol- lowing method. As $4 make £1, the number of £'s iu $195.7G}, will be the sumo as the number of times that 4 is contained in th« 195 dollars, which gives £48, and $3 remaii? 5)195 (39 »)ECIMAL COINAGE 41 $195— 76J 4)195 £48—300 ing. Now, these tlirec dollars are equiva- lent to 300 cents, which added to the rc- maining 7G^ cents, gives 3761 cents; this dmded by 20, will give the shillings, be- cause 20 cents are equal to one shilling, and it is self-evident that the number of shillings in 376J cents, will be tjie t^mi^ as the num- ber of times 20 is contained in that num- ber, which gives 18 shillings, and IG.} cents remaining. Lastly, as 5 cents i,ro equal to 3 pence, one cent will be equal to ?, of 3 pence, which is § of a penny; therefore, if . . . one cent IS equal to • of a penny the re- majning IGJ cents will be equal to ie\ times of a pen ^ whi h" m- i tence we have $195.76^ equal to £48.18.9|. ^ 20 3761- b18— 16J 3 5j48| « I m 9^, EXERCISES. 1. Keducc $119.95 to Halifax currency. 2. Reduce $270.67] | 3. Reduce $474.27^ 4. Reduce C197.50 5. Reduce $1119.17^ 6. Reduce ^551. 33J 7. Reduce $1830.50 8. Reduce $1151.77^ it « « « Ans. £-29.19.9. Aus. £t;7.13.4^. Ans. £118.11.-1^ Ans. £49.7.G. Ans. £270.15.101. Ans. £l;J7.1G.8. Ans. £-137.12.6. Ans. £287.18.10J. MIXED EXERCISES. 1. Reduce 2. Reduce 3. Reduce 4. Reduce 5. Reduce 6. Reduce 7. Reduce 8. Reduce 9. Reduce 10. Reduce £430.7.8^ to dollars and cents. Ans. $1745 54* f^Jfl r". "'^'^''^ '"''■'"'^- ^^'' ^136 ] 9.4 1! k^oJb.ho to Halifax currency. Ans 141 4 3 CGOG 10.8^ to dolhrs and cents. Ans. $2427 94-' " ^oio 99 to Halifax currency. An«. £93.19.11^" ;;• ^^^- '' ^f^'^' ^"^ <^^>»ts. Ans. 7;]|- cents.' r cents to Haliiax currency. Ans. 10 ' pence. 10^ pence to do.ar.s and cents. Ans. n,'.. cent* ^3 cents to old Canadian currency. Aus. 13^ pence 42 AKirnMETIC. I ,!;i! REDUCTION. 1 1 . — llEDUCnoN 13 the mode of expressing any givon ((uantity in terms of a liiglicr or lower denomiuation, c. g., expressing any given number of dollars as cents, and vice versa, any number of cents as dollars. When a higher denomination is changed to a lower (as dollars to cents), the process is called reduction tZcscending, and when ahnvcr is changed to a higher (cents to dollars), it is called reduction ascending. Beginners arc generally puzzled by the word reduction, which in its ordinary acceptation means maJcing less, v/hereas the learner finds that when dollars arc chana;ed to cents, the number denoting the amount is increased a hundred fold. The explanation lies in the original use of the word reduc, to bring hack, which would raig- gest that the dollars were originally cents and arc hroitjht htich to cents, or that the cents were originally dollars and arc hroajht hacli to dollars. Thus, by a transition common in all languages, the idcii of bringing back was gradually losf and the idea of changing from one denomination to another alone retained. Again, since one dol- lar is equal to one hundred cents, it is plain that the number repre- senting any amount in cents will be one hundred times greater, taken abstractly, than that representing the same in dollars, and so in all denominate numbers. Some explain the term reduction as taken originally from the changing of a liigher to a lower denomination, and afterwards applied to the converse operation. This seems satis- factory enough as regards the present meaning of the word but does not accord with its derivation. Either explanation will clear up tho young learner's conception of t'-c term. If we wish to express 17 cwt. 3 qrs. 20 lbs,, in terms of the lowest denomination,viz. lbs., we must first find how many quarters are equivalent to 17 cwt. 3 qrs. which we find by multiplying the 17 cwt. by 4 and adding in the 3 qrs. for 4 qrs. make 1 cwt, — and then since 25 lbs. make 1 qr. we multiply the 71 qrs. by 25 to find the number of lbs. which, with the 20 old lbs. added in, is 1795 lbs., and thus v/c see that 1795 lbs. are equivalent to 17 cwt. 3 qrs. 20 lbs. Tho proof depends on the converse operation, as in the margin, for, since the number denoting the pounds is, abstractly, 25 times the num- ber denoting the (juartcrs, we must divide tlie number denoting the pounds by 25 to obtain that denoting the quarters, and, in like manner, we must divide the number representing the 17 cwt. 3 qrs, 20 lbs. quarters bv 4 to find that denotinir the cwt. qrs. lbs. 17.3,20 4 71 25 375 142 1795 7^ 5)1795 i) 71 qrs. 20. DEDUCTION, 48 26 acres, 2 roods, 3G rtvb. 4 106 40 4276 rods.— Ans In the same manner 26 acres, 2 roods, 36 rods will be reduced 'to rods by multiplying the acres by by 4 and adding the odd roods, which gives 106 roods, and this multiplied by 40 with the odd rods added in gives the rods, for 4 roods make one acre and 40 rods 1 rood. Conversely the rods di- J o- J ^^^^'^^ ^y 40 will give 106 rood« m>d .1„ rod, „™-, u„d 106 rood. divided by 4 will give 20 acrjand SeTooL"'"' '™'' "' 'ho original ci„cstio„-20 acre, 2 rods, EXERCISES. 1 . How many dollars arc there in 47986 cents ? An - '479 S6 ^. llow many cents arc there in 187 dollars? Ans 1870o' lbs ? "'''"^ ^°''"'^' ""'^ *^''*' '" ^ ^°"' ^'^ ''^*- 2 ^i'"- ''"<^ 21 4. llow many pounds are there in 18 cwt. and 22 lbs. ? 5. Rcduco 14796 lbs. to tons, &c. ? ^"'- ■^^-^• . -,, ^"s- '^ t^'ns, 7 cwt. 3 ors 91 IKo (i. Reduce 7643 quarters to tons, &c. ? ' ^ ' ^^'* »5- TT ^^^- ^^ *""s> 10 cwt. 3 ors 7. How many drams are there in 18 lbs. 13 oz. and 15 dvs. ? ' 8. How many pounds are there in 2785 drams ? ^"'* '^^^^' n TT ^^^- 10 lbs., 14 oz 1 dr 22 glinsT ""' '"'" '" ""^ " '' ^^^^ ^^ ^^^ 18 dw't. and 10. How many lbs. in 46891 grs. ? '^"'' ^°^^^*' 11 TT -n . . '^'"- ^ ^K 1 f'z-, 13 dwt., 19 grs. 1. ow many gills m 4 tuns, 1 pipe, 1 hdd, and 52 gak ? 1-. How many tun,., &c. in 198462 drams ? 13. How many bushels in 89(54 lbs. of wheat? 14. llow many bushels in 14382 lbs. of barley ? lo. How many buhhols in 48028 lbs. of peas ? 16. llow many bushela in 4683 lbs. of timothy seed? 14 ARITHMETIC. '1 '1111 ' .13 :''\ m\ 17 Reduce 98 miles, 5 furlon-s iind 30 rods to rods ? Ans. 31590 rods. 18. How many inches from Albany to New York ' 150 miles). 19 Ilovr many miles are there ia 5271GS feet ? Ans. 09 miles, G fur., 29 pr., 3 yh^, I'r.. u in, 20. Reduce 57 acre ,, 3 roods and 2^t rods to rous ? Ai!s. 92('.l rnds. 21 How many square yards arc there in 17 r.cres, 2 rcuds r.nd » ,o Ans. 852-i-i ?j yards, ^sV Find the number of acres, &c., in 479G85971 square inches. Ans. a. 7G.1.35.19.2.119. 23 How many acres do 176984 square yards make ? ^ Ans. a. 3G.i.'.10.21L 24. How many square links are there in 37 acres ? ^ ^ Ans. 3,700,000 links. 25. How many acre*, &c.. ia 479,803,201 square links ? Ans. 4798 a., (> eh,, .^JUl. 26. 7 864,391 cubic inches; how many cubic yr.rds V ' Ans. yds. 1G8.1 5.263'. 27 9 cubic yards, 7 cubic feet, 821 cubic inches ; how many ,..',« Ans. 432821 cubic inches. 3ubic inches f 28. IIow many gills does a tun contain ? Ans. 80G4 gills. 29. How many gallons, kc, do 479S65 gills make ? Ans. gals. JlJJy.j.VM. 30. How many pints are there in 28 bu., 3 pecks and 1 gal. ?- Ans. 1848 pmts. 31. 27 yards, 3 qrs., 3 nails ; how many nails? Ans. 447 nails. 32. 286 nails; how many yards, &c. ? Ans. 17 yards, .^ qrs., 2 nls. 33 36° 40' 25"; how many seconds? Ans. l.V202.> . 34'. How many decrees, &e., in 4078G" ? Ans, 13° 49'.4G". 35. The area of x\ew York State is 29.440,000 acres; how many jquare miles? 3G. How long would it take a railway train to move a distance jquul to that of the earth from the .un (95 nyllion. of miles) at a ,peed of 52 miles an hour ? An». 208 years, 201 day-, 19/, hours. 37. The area of Pennsylvania is 47000 square miles ; how many jquare feet ? DENOMINATE KUMI3ERS. 45 38. Sound movos about llliO foot in a second of time; how long ■would it bo in movini^ from the earth to the ^5un ? Ads. i J^ears, 27 days, ;5 hours, 50 iiiln., b^^' sec. 39. How many s-cconds of this century had tiapscd at the end of 1864, countin;^ the day ut 24 hours ? Ans. 2,019,G8>],40()". 40. The great bell of Moscow weighs 127,836 lbs.; how many tons, &c., docs it weigh, the quarter being 28 lbs. ? Ans. 57t. Ic. Iq. IGlbs. 41. IIow many days from the 11th July, 1861, to the 1st of AjTril, 1864 ? Ans. 995 days. 42. A congregation of 569 persons made a collection of £40.6,1 ; how many pence did each give on an aYcragc ? Ans. 17d. 43. The British mint can strike off 20,000 coins in an hour; what is the value of all the pennies coined in one day of 12 liours* work? Ans. ,11,000. 44. 417 tons of fish were caught at Newfoundland in one season, and sold by the stone of 14 lbs., at an ayerage price of 42 cents a stone ; what did they bring ? Ans. $25020. 45. How many feet from pole to pole, the earth's diameter being 7945 miles ? Ans. 41949600 feet. DENOMINATE NUMBERS. 12. — When numbers arc spoken of in general, without reference to any particular articles, such as money or merchandise, they are called abstract, but when they are applied to such articles they arc Bomctimes called aj^pUcate, as being applied to some particular arti- cles to express their quantity ; sometimes they are called concrete^ (growing together,) as attached to some particular substances, and Bomeiimcs they are called denominate, as denoting quantities that consist of diifercnt denominations, as dollars and cents, — pounds, ounces, &c. The elementary rules of addition, subtraction, multi- plication and division, arc performed on denominate numbers, exactly in the f-ame way as on abstract numbers, with this single difFerence, that when a lower denomination is added, and gives a sum equal to one or more units of the next higher denomination, we carry that unit, or those units, to the next higher denomination. Thus : if the sum were 24 inches, we should call that two feet. Iw abstract and decimal numbers we always reduce, or carrj', hj tena. 46 ARITHMETIC. L I . 1 Ilcrc wc liiid the sum of the inches to be o4, and as 12 inches luako one foot, the number of feet in 34 inches will bo the same as the number of times that 12 is contained in 34, which is twice, with a remainder of 10, therefore we write the 10 under the column of inches, and add up the 2 feet with the column of feet, and obtain 11 feet, and as 3 feet make 1 yard, the number of yards will bo the same as the number of times that 3 is contained in 11, which is 3 times with a remainder of 2 ; we therefore write the 2 odd feet under the column of feet, and add up the 3 yards with the column of yards, and the whole amounts to 94 yds., 2 ft., 10 in. The same operation would be carried out if we had rods, Sec, given, and is applicable to all operations in denominate numbers of any kind. In the exercises on the addition of denominate numbers, one quea- tioa in abstract numbers is given to contrast with the denominate. VUs. ft. in. 12 2. 9 IG 1. 11 27 •> .o. 8 3G .3. 4 94.2.10 xxehcises. (1) 7865437 198675 847G154 1869538 4187643 5768299 28365746 (2.) $857.03 189.50 G84.87I 498.75 807.12^ 365.371 917.25 4380. 50 J (3.) £76.18. 4 17.11. 4| 99.19. 9 11.11.11 67.15.10^ 79.19. 9 28.12. 1 63. 8. 4^ 445. 17. 5i r4.) $1967. 87J 2075.75 3194. 62i 7658.50 8976. 37J 2873. 12| 1769.25 2481.92 30997.42 I .1 (5.) (6.) (7.) (8.) lbs. oz. dr3 t. cwt. qrs. Iba lbs. oz. (Iwt. grs. •bs. oz. (Irs. scr. pra. 13.14.10 26.17.3.21 3.11.16.21 5.11.7.2.19 15.11.10 18 11.0.19 5. 8. 7.11 4.10.4.1. 7 11. 4. 9 25.15.1.16 7. 9.18.23 3.11.6.2.14 8.12.13 13.17.2.20 11.10.15.17 1. 9.3.1.12 15. 7. 8 39. 4.1.23 12. 7. 9. 8 2. 4.5.0.10 10.13. 11 28.16.3.14 16.10.11.22 6. 7.2.2. 9 8. 9. 6 18. 8.19.18 2. 8.1.1.13 4.15.15 89.10. a 153. 3.2.13 77. 8. 0. 28. 4.0.1. 4 DENOMINATE NUMBERS. (9.) m. fur. roda yda •176.7.39.5 85.4.20.1 70. G. 29. 3 42.3. 8.2 07. 1.11. 2 118.3.10.3 81.2.31.1 79.0.21.2 18 .3 .OO .3 740.2. 6.0 (10.) y.10 for Fall goods, and expended for private purposes $Ij0.8() and ludgcd the rest ill the Uank, liow niucli have I bunked ? Ann. $9f:i;.2S 0. I bou-ht 47 tons, 17 ewt., 1 qr., 18 lbs. of firain, and iiuvc Bold 20 tons, 18 cwt., J qrs., 22 lbs. of it; how much liavo 1 in fitorc V Ans. 17 tons, 18 cwt., 1 (jr. 21 lbs, G. If the dist.incc from Washington to Dover be IGl miles, 1 furlong and 20 rods, and that of Baton llouge 1407 miles, 1 fur- long, 'M rods, Low much farther is Baton llougc from "Washington than Dover? Ans. 1245 m. 7 f. T.Gr. 7, A farmer possessed 1279 acres, 2 roods, 21 rods, and by his will left 789 acres, o roods, 3G rods to his oldest sou, and the rest to the second ; how much had the younger ? Ans. 489 acres, 2 roods, 25 rods. 5. The latitude of London (England,) is 01° .nO\49"N., and that of dihraltur ijG''.G\30" N. ; how many degrees is Gibraltar south oi London? Ans. 15''.24M9''* 9. The earth performs a revolution round the sun in about CCS days, 5 liours, 48 minutes and 48 seconds, and the planet Jupiter in about 4IJ32 days, 14 hours, 2G minutes and 55 seconds ; how much ioDgor decs it take Jupiter to perform one revolution than the earth.? Ans. 3907 days, 8 h., 38 min., 7 sec. 11). 1 bimght 54 tbs,, 10 oii. of tobacco, and 11 oz. of it were lost Ijy drying; and I S(jld 3G lbs., 12 oz. of it to A. ; and 11 lbs., 9 oz. to B. ; and u: I,:ULTIPLICATION. 1. $1 79 GX 47^-^88441 2. 2.J:^li^i X 14 l=^£42o.3.0. 3. $168.87iX 04=810808. 4. £1^.9x'225=dC255.18.9. .^v, 51 MULTirUCATION. Tx Find tlio duty on 97 consignments of mcrchrndiso at 6SG.G2;, ^''^'^^^ ''. Ans. $S402.G2i- It i.s often convenient to multiply denominate numbers by the /actors of the multiplier. Tims : to multiply by 81 is the same a.s to multiply by 7 and 12. Tims, in the annexed examples, since 12x^-84, 18 tons, 12 cwl., 2 qra., 11 Ibs.xSi. is tho same as 18 tons, 12 cwt., 2 qrs., 11 lbs, X 12x7, &c. (G.) (7.) (8.) toua cwt. qrs !bs. 18.12.2.11x84 12 nc, roods 27.2. . rdi 29. 8 X72 yds. ft. in. 11.3. 7X150 6 223.11.1. 7 7 221 1. 32 9 GO. 2. 11 6 15G4.19.0.24 1993. 0. 8 304. 2. 7G (9.) cwi. qrs. lbs. 23.3.22X49 7 lbs. 49. 1829.0. [10.) oz. drf. n.l2xG3 7 1G7.3. 4 7 348. 2. 4 9 1174.2. 3 31C3. 4. 4 yi!s n. ill. 11 .. 3 .. 7 150 1050 8 1 . . G 450 .. 537 .. 179 IG50 .. .. G .. .. 1829 .. .. f Tiius: 11 yds., 3 ft., 7 in., multiplied by 150, Avill give (1) 150 limes 7, wluJa is 1050 in., and divided by 12, is 87 ft., G in.,— 2) 150 limes 3, whicli is 450 ft., and added to the ST already found, gives 537 ft., and divi- ded by 3, gives 170 i't. without remainder,— (3) 150 times 11 is 1G50 yards, whieh, added to the 179 already found, gives 1829 ft., so that tho final result is 1829 yds., ft., G in., as already obtained bv the method of factors. 52 AEITmiETIC. cwt. qrs. ibs 9.3.22-f-8G 80 £2.13.1^ 125 h cwt. qrs. lbs. 1.2. 17+27 27 8577l^l7 X331.18.0J 45.0.9 G. How many seconds lias a person livca wuo nas completed his twentieth year, tlic year consisting of 305 days, 5 hours, 48 minutes, and 48 seconds? Ans. 031138500. 7. Bought 7 loads of hay, each weighing 1 ton, 3 cwt., 3 qrs., 12 lbs ; what did the whole weigh ? «. If a man can reap 3 acres and 35 rods per day, liow much wlU he reap in 30 days ? Ans. 90 acres, 90 rods. 9. If u steamboat ply across a channel, the breadth of which is cnual to 2^, 25\ 10^\ what angular space has she traversed at the end of 20 trips? 10. Hamilton, Ross & Co., of Boston, have charged me on an invoice of 00 tons, 17 cwt., 1 qr., and 20 lbs. of iron, at $55 per ton, and 1 pipe, 1 hhd., 34 gals, and 3 qts. of wine, at $3.00 per *»al. §4213.57, how much is this amount astray ? 11. If a man saves 45 cents a day, how much will lie save in the year, omitting the Sabbaths ? 12. If 12 gallons, 3 quarts, 1 pint of molasses be used iu a hotel in a week, how muck would be used in a year at that rate ? Ans. 10 hhds., 39 gals., 2 qts. 13. If a man can saw one cord of wood in 8 hours, 45 minutes, 50 seconds, iu what time will he saw 11 cords ? Ans. 4 days, 24 minutes, 10 seconds, 14. If 13 waggons carry 3 tons, 15 cwt., 1 qr., 15 lbs. each how much do they all carry ? Ans. 49 tons, cwt., qr., 20 lbs. 15. If a man travel 20 mllea, 5 furlongs, and 20 rods a day, how mufli would he travel at that rate in a year ? Ans. 7550 m., 7 fur., 20 rods. IG. There are 24 piles of wood, each containing 3 cords, 42 cubic feet ; what is the whole quantity ? Ans. 79 cords, 120 ft. 17. If 17 hhds. of sugar weigh 12 cwt., 1 qi'., 20 lbs. each, how much will the whole weigh ? Ans. 211 cwt., 2 qrs., 15 lbs. 18. Allowing 75 yards, 18 feet, for the surface of 9 rooms, how much paper would be required to cover the wall ? Aus. 603 sq. yards. DIVISION. .^3 19. Purchased from R. Bell 493 cwt., 3 qrs., 21 lbs. of iron a* f cents per lb. ; what docs it amount to ? 20. What must I receive for 2 lbs., 5 ozs., 1-1 dwts., 21 grs. o: gMjd, at 618.50 per oz. ? 21. DcJivCred James Grant 7 tuns, 1 pipe, 49 gals, of Poi't Win J - $2,75 per gal. j vrhat is tho {"..nount of tho invoice ? DIVISION. Ids. 42 ,ft. low lbs. Iiow rds. In Division, all remainders arc to bo reduced to tho next lowci denomination, and in that ^'""'i^ divided, to get tho unit.'* of that denomination. EXERCISES. 1. A silversmith made half-a-dozen spoons weighing 2 lbs,, 8 ozs, 10 dwts. ; what was the weight of each ? Ans. 5 ozs., 8 dwts., S grs 2. If 45 waggons carry GS5 bushels, 2 pecks, 4 quarts, how mucli does each carry on equal distribution ? Ans. 15 bushels, 7j quarts ii. If a labourer receives 140 lbs., 13 ozs. of meat as payment foi 2G days' work, how much is that per day, on an average ? Ans. 5 lbs., 12r;V o^^' 4. If a steamer occupies 48 days, 17 hours, and 40 minutes, iti making 121 trips ; what is tho average time ? Ans. 9 h. 40 min, 5. If 98 bushels, 3 pecks, and 2 quarts of grain, can be packed in 37 equal-sized barrels ; how much will there be in each ? Ans. 2 bush., 2 pecks, 5!Z qt.-t G. Tf a man Las an income of $75000 a year ; how much has he aniiour, allowing the year to consist of just 3G5 days? 7. An ]"]nglish nobleman has £124,GS5 a yeilr ; how nmch has he per minute, the pound being worth $4.84, and the year to consist of 3G5 days, 5 hours, 48 minutes, and 48 seconds? Ans. $1.14-[ 8. In a coal mine, 97 tons, 13 cwt., 2 qrs. were raised in 97 days ; how much was that per day, on an average ? 9. If $15.50 bo the value of 1 lb. of silver, what will b.; tlx weight of $500000 worth? Ads. 32253 lbs., oz., 15 dwts., 11^ grs. 54 AEITHMEIIC. 11. If 1246 bushels of wheat are produced in a field of 16 acres what is the yield per acre ? 12. A gardener pulled 13500 bushels of apples off GO trees; how many, on an average, were in each bushel ? 13. If 13 hogsheads of sugar weigh G tons, 8 cwts., 2 qrs., 7 lbs., what is the weight of each ? Ans. 9 cwt., 3 qrs., 14 lbs. 14. '.ruat is the twenty-third part of 137 lbs., 9 oz., 18 dwts., 22 grs. ? Ans. 5 lbs., 11 oz., 18 dwts., Sj^rj grs. 15. A shipment of sugar consisted of 8003 tons, 17 cwt., 1 qr., 12 lbs., 10 oz., net weight; it was to be shared equally by 451 gro- cers ; how much did each get ? Ans. 17 tons, 14 cwt., 3 qrs., 18 lbs. 14 oz. 16. If a horse runs 174 miles, 20 rods, in 14 hours, what is his speed per hour ? Ans. 12 miles, 3 fur., 19 rods. 17. A farmer divided his farm, containing 322 acres, 2 roods, 10 rods, equally among his seven sons and G sons-in-law ; what was the share of each ? Ans. 24 acres, 3 roods, 10 rods. 18. If 132 bushels, 3 pecks, 7 quarts ef corn be distributed equally among 23 poor persons ; how much does each get ? Ans. 5 bushels, 3 pecks, 1 quart. 19. A man having purchased 119 cwt., 3 qrs., 23 lbs of hay, and drew liome in G waggons ; how much was on each waggon ? Ans. 19 cwt., 3 qrs., 23 lbs. MIXED EXERCISES ON DENOMINATE NUMBERS. 20. A gentleman, by his will, left an estate worth $2490, to be divided among his two sons and 3 daughters in the following propor- tions : The widow was to receive one-third of the whole, less $345; the younger son $212 more than his mother; the older son as much as his mother and brother, lacking $335.50, and the three daughters were to have the remainder, share and share alike ; what was the share of each ? Ans. The widow got $484 ; the older son got $844^ ; the younger son got $G9G ; each daughter got $155J. 21. A gentleman left a property in land, consisting of 448 acrca, 3 roods, 24 rods, to be divided among his four children in the following proportions :— The youngest was to get 4 acres, 3 roods,, 6. rods more than the eiglith part ; the second youngest was to get one- fifth of the remainder ; the oldest but one was to get one-third of the remainder, and the oldest the residue; what was the share of each? DinSION. 55 r la le Le ? Ans. The youngest got GO acres, 3 roods, 24 rods ; the next got 77 acres, 2 roods, lli rods; the next got 103 acres, 1 rood, 345 rods ; the oldest got 20G acres, 3 roods, 29J rods. 22. A ship made the following headway on iiix successive days : On Monday, 3°, 8', 45" south, and 1°, 51' east ; on Tuesday, 2^, 30' south, and 2°, 1', 15" cast ; on Wednesday, 4°, 0', 52" s utli, and 1° cast; on Thursday, 1°, 4S', 52" south, and 3^ 10', 22" east ; on Friday, 1°, 10' south, and 4S', 29" east; and on Saturday, 50', 30" south, and 3^, 52', 11" cast; find her distances south and cast from tlic port of departure. Ans. South 13°, 52', 59"; East 12^ 49', 17" 23. A vintner sold in one week, 51 hogsheads, 53 gallons, 1 quart, 1 piut ; in the next week, 27 hogsheads, 39 gallons, 3 quarts; m the next week, 19 hogsheads, 13 gallons, 3 quarts; how much did he sell iu the three weeks ? Ans. 98 hogsheads, 43 gallons, 3 quarts, 1 pint. 24. In a pile of wood there are 37 cords, 119 cuhic feet, 76 cubic inches ; in another there are 9 cords, 104 cubic feet ; in a third there are 48 cords, 7 cubic feet, 127 cubic inches, and in 9 fourth there arc 01 cords, 139 cubic inches. Find the whole amount. Ans. 150 cords, 102 feet, 342 inches 25. The following cargo "was landed at Tortland from Liverpool 1 78 tons, 3 cwt., 2 qrs., 20 lbs. of Irish pork; 125 tons, 15 cwt., i qr., lbs. of iron; 90 tons, 12 cwt., 2 qrs., 20 lbs. of West oi England cloth goods; 225 tons, 9 cwt., 12 lbs. of Scotch coal, an3 lOG tons, 1 qr. of Staffordshire pottery; what is the whole amoua; of the consignment ? 20. If a man can count 100 one-dollar bills in a minute, and keep working 10 hours a day ; how long will it take him to couat a million ? Ans. IG5 days, 27. The earth's equatorial diameter is 4184742G feet ; how many miles ? Ans. 7925 and 3420 ieet 23. The earth's polar diameter is 7899 miles, 900 feet; hox* many feet ? Ans. 41707020 feet. 29. Sound is calculated to move 1130 feet per second ; how far off is a cannon, the report of which is heard in 1' 9"? Ans. 77970 feet. 30. If the circumference of a waggon wheel be 14^ feet ; how often will it turu round in a mile, : 5280 feet ? Ans. 300 times. 56 AEITHMETIC. PROPERTIES OF NUMBERS The term Integer, or VHiole Number, is used in contradistinction to the term Fraction. All numbers expressed by the natural series 1, 2, 3... 10... 20... 100, &c., are called integers, so that 3 and 4 arc integers, but f is a fraction. All numbers in the natural scries 1, 2 3, &c., that can bo resolved into factors, are called Composite, while those that cannot be 60 resolved are called Prime. Since 4=^2 X 2, it is called composite, and so G, 8, 9, 10, &c., but 1, 2, 3, 5, 7, 11, &c., arc called prime because they cannot bo resolved into factors. Tims, 11 can only be resolved into 11 Xl, or IXH, ^ad these are not factors in the strict meaning of the word. A Prime Factor is a prime number, which is a factor of a com- posite rumber. The factors of 10 arc 2 and 5, both prime numbers, A composite number may have composite factors, as 36, which has 4 and 9 as factors, and both of these are composite. AVhen any number will divide two or more others, it is culled a. Common Factor. Thus, 3 is called a common factor of 6, 9, 12, 15, &c. lumbers that have no common factor, as 4, 6, 9, are said to be pr,mc to each other. To resolve a composite number into its prime factors, divide it by the least possible factor that it contains, and repeat the procesa till a prime number is obtained. EXAMPLES. 2)06 2)48 2)24 2)13 2) 6 3 SO thai Jie prime factors of 9G arc 2X2X2X2X2X3* rnoi^Er.TiEs or numbees. 57 Also, because 5x7x11=385, wo see that 6, 7 and 11 are the prnne factors of 385. EXERCISES. 1. What are the prima factors of 2310 ? Ans. 2, 3, 5. 7, 11. 2. Wliat are the prime factors of 17G1? Ans. 2, 2, 3, 3 7 7' 3. What arc the prime factors of 180C42 ? , „ Ans. 2, 3, 7, 11, 17, 23. 4. What arc the prime factors of 95 ? Ans. 5 19. 5. What arc the prime factors of 51 ? Ans. s' 17.' C. What arc the prime factors of 99 ? Ans. 3 3' ll! 7. What are the prime factors of 651 ? Ans. s' 7 ai.* 8. "NV^hat are the prime factors of 3G2880 ? o ^n .. ^°'- ^' ^' ^' ^' ^' 2' 2' '^' ^' •'^' ^' ^> ^• J. What factors are common to 84, 105, and 147 ? Ans. 3 7. 10. What are the prime factors of 308 ? Ans. 4, 7, IK Whether a number is prime or composite cau only be found bv trial. ^ The only even prime number is 2; for 4, G, 8, 10, &c., are all multiples of 2. , The only prime number ending in the digit 5 is 5 units, and all other nuiubers ending in cither 5 or are multiples of 5. A. I) D 1 T I O N A L EXERCISES. 11. Is 101 prime or composite? Ans. Prime. 12. Is lOy prime or composite ? Ans. It has the factors 2, 3, 3, 11. 13. Is 171 pvimo or composite ? An.s. It has the factors 3, 3, 19, 14. Is 473 prime or composite , Ans. Prime. 15. Is 477 prime or com'posite ? Ans. Composite! IG. Is 5493.^>3250 prime or composite ? Ans. Composite. 17. Is G74041 prime or composite ? 18. Is 19[; pri'jic or composite? 19. Wliat arc the prime factors of 210? 20. What are the prime lactors of 51051 ? Ans. 3, 7, 11, 13, 17. NoTE.--Vv-e have tliougbt it siifcieut uuder tliia head to give only the kadinij auU most useful principles. Ans. Composite. Ans. Prime. Ans. 5, G, 7. 53 AWTHME": !:il ' j GREATEST COMMON MEASURE. 13 When any quantity is contained an even number of times in a greater, the greater is called a multiple of the less, and the less a submultiplc, measure or aliquot part of the greater. Thus : -IS is a multiple of 2, 3, 4, G, 8, 12, 16 and 24, and each of these is a sub- multiple of 48. _ . "When one quantity divides two or more 'others evenly it is called a common measure of those quantities, and the greatest num- ber that will divide them all is called the greatest common measure. Thus ; 7 is a common measure of 63 and 49, and it is also the greatest common measure, for no larger number will divide both evenly. When any quantity is measured evenly by two or more others, it is called a common multiple of them. Thus : 24 Is a common mul- tiple of 2, 3, 4, 6, 8 and 12. A number which can be divided into two Wiu«l integral parts is callcc: an evai number, and one which cannoi, DC so divided is called an odd number. Hence all numbers of the wries 2, 4, G, 8, 10, 12, &c., arc even, while those of the series 1, 3, 5, 7, 9, 11, &c., arc odd. Heucc the sum of any number of even quantities is even ; also, the sum of any even number of odd quantities is even ; but the sum oi any odd number of odd quantities is odd. This principle is of great use iu checking additions. A prime number is one which has no integral factors cxccpS itself and unity ; a composite number is one that has integral fac- tors greater than unity, and numbers which have no co.nmon factor greater than unity are said to ho prime to each other. Of the uv^t kind Mve 1, 2, 3, 5, 7, 11, &c., of the second, 4, G, 8, 9,^10, 12,i;c. ; also, 2 anu 7 a.o prime to each other, and so arc G and 7. If one quantity measure another it will measure any mulliplj (;1 it. Tims : since 3 measures G, it will also measure 12, IS, Ci, &c., because it is a factor of all these. If one quantity measure two or more others, it will also mciisur,' their sum and difference, and also the sum and different ol' anv :iiil GREATEST COMMON MEASUia;. 59 a^pSty."' *■""■"■ ^"^ " "■"'"''' *- "k™ «>^"« takoa ^Kl. - 1 anil 18 anJ so the otinr pnrt, C ; 9 divides 45 and 27 and r-..r.s, cac ■ of which has a common factor, ,l„,t factor will ilso .^asu..o the. sum. Thus : 9 .noasures IS.'^T J a^ald t,:': o vuiuiuuii measure oi two or more quantities. KULE. Divide the greater by the less, and then the less by the re- mainder, untd nothing is left, and the last divisor wilf bo t^^ greatest common measure. vi.ur win do tlie EXAMPLE. 2145 1326 819 507 312 195 117 78 39 3471 2145 1320 819 507 312 195 117 78 78 A concise form of the work is exhibited iu the margin. The quotion'.s are omitted as unnecessary. The last divisor, 39, is the Q CM., as may be proved by trial. If it is re-' quired to lia.1 the G. C. 31. of more than two numbers, first find the G. C. 31. of two of them, and then the G. C. M. of that and another, and so on. EXERCISES. Find the G. 0. M. of the following quantities: 1. 247 and 323. 2. 532 and 1274. 3. 741 and 1273. 4. 1041G and 257G1 5. 4GS and 1203. G. 285714 and 999999. 7. 15803 and 21489. 8. 8280 and 11385. 9. 17222 and 32943. 10. 19752 and G9133 Ans. 19. Ans. 14. Ans. 19. Ans. 93. Ans. 6. Ans. 142857. Ans. 29. Aos. 1035. Ans. 79. Ans. 9876. ^ • u » • • t) 9 45 2 90 3 270 o 540 Expunge all common factors and take the continued product of all the results and divisors. Thus, to find the L. C. M. of 2, 3, 4, G, 9, 18, 27,30, ar- range them in a horizontal line, and as 2, 3, G, 9 are all contained in 18, they may be omitted, as in the second line, then, as 2 i3 contained in 4, 18 and 30, it may be divided out, and as 9 in the third line is contained in 27, it inaj be omitted, as in tho fourth line ; and 27 and 15 be- ing botli divisible by 3, we ob- tain in the fifth line 2, 9, 5, all prime to each other, and the products of these and the divis- ors 3 and 2 ia the L. C. ^l* 540. GREATEST COMMON MEASUEE. EXERCISES Find tho L. C. M. of the foUomng quantities i 1. 8, 12, IG, 24, 33. 2. 35, 42, 45, 81, 100. 3. 2,4,8,10,32,04,128 4. 2, 3, 5, 7, 11. 5. 3, 9, 27, 81, 243, 729. G. 12, 10, 18, 30, 48. 7. 3, 4, 5, 0, 7. 8. 2, 3, 4, 5, 0, 7, 8, 9. 9. 2, 4, 7, 12, 10, 21, 5(3. 10. 2, 9, 11, 33. EXAMPLES FOR PRAOTrOE. 6] Alls. 528. An3. 50700. Ans. 128. Ans. 2310. Ans. 729. Ans. 720. Ans. 420. Ans. 2520. Ans. 330. Ans. 198. 1. "What will 320 caps cost at $7.50 each ? Ans. $2400. 2. Ifyou can purchase slates at 20 cents each; how many can you buy for $7.40 ? ^^ns. 37. 3. Ifyou can walk 4 miles an hour; how far can you go in '^4 ^^°"^^^ Ans. 9*0. 4. What will be the cost of 210 'barrels of pork at $7.50 per ^^^^^^^ Ans. $1020. 5. How many sheep can be bought for $500 at $3.50 per head ? Ans. 100. 0. If 825 pounds of beef are consumed by a garrison in one day ; what will be the cost for days at 11 cents per pound for beef? Ans. $544.50. 7. A farmer sold 185 acres of lanrl at $25 per acre, and received in payment 17 horses at $70 each, and 12 cows at $20 each ; how much remains due ? Ans. $3195. 8. A merchant bought 120 yards of American tweed at $1.15 a yard; GO yards of flannel at 95 cents per yard, and 13 dozen pairs of gloves at 35 cents per pair ; what was the amount of his bill ? Ans. $249.00. 9. At $2 per gallon ; how much wine can bo liought for $84? Ans. 42 gala. 10. A boy had $5.50, and he paid one dollar and five cents for a book ; how much had he loft ? Ans. $4.45. 11. What will 18 cords of wood cost ;it $4,75 per cord ? Ans. $85.50. 62 iUlITHMETIC. 12. Low many pounds of sugar can 6j bought ibr $0.35, at 11 cents per pound ? Ans. 85 lbs 13. Vn.iXt will a jury of 12 men receive for coming from Ivings- ton to Albany at 10 cents a mile each; the distance being 00 miles ? 14. A grocer bought a hogshead of molasses at 32 cents per gallon; but 18 gallons leaked out, and he sold the remainder at 55 cents per gallon ; did he make or lose, and how much ? Ans. He gained $4.59. 15. If a clerk's salary is $800 a year, and his personal expenses $320 ; how many years before he will bo worth $GGOO, if he has $1000 at the present time ? Ans, 20 years. 16. A speculator bought 200 bushels of apples for $90, and sold the same for $120 ; how much did he make per bushel ? Ans. 15 cents. 17. A person sells 15 tons of hay at $22 per ton, and receives in payment a carriage worth $125, a cow worth $45, a colt worth $40, and the balance in cash ; how much money ought lie to receive ? Ans. $120. 18. IIow many pounds of butter, at 20 cents per pound, must be given for 18 pounds of tea worth 75 cents per pound ? Ans. 07^ lbs. 19. A grocer bought 7 barrels offish at $18 per barrel ; but one barrel proved to be bad, which he sold for $5 less than cost, and the remainder at an advance of $3 per barrel ; did he gain or Jose, and how much ? Ans. Lost $13. 20. A man bought a drove of cattle for $18130, and after sel- ling ' 4 of them at $51 each, the rest stood him in $43 each ; how many did he buy ? Ans, 406. 21. What will 2 cwt. of cheese cost at 9^ cents per pound ? Ans. $19 00. 22. A. is worth $960, B. is worth five times as much as A., less $600, and C. is worth three times as much as A. and B. and $300 more ; what are B. and C. worth each, and how much arc they all worth ? Ans. B. $4200 ; C. $15780 ; all $20940. 23. A boy bought a dozen knives at 15 cents each, and after selling half of them at the rate of $2.22 per dozen, ho lost three, and Bold the balance at 25 cents each ; did he make or lose, and how much ? Ans. Gained 6 cents. 24. A lalx)urer bought a coat woi'tli $16, a vest worth $3, and a hous two farnil rciuii year | the and 31 3 ociJ GREATEST COMMON MEASURE. C3 ^y all 1)940. after ;, and how lents. and a pair of pantg worth 85.50 ; how many ilays had ho to work to pay for his suit ; his services being worth 50 cents per day ? Ans. 49 days. 25. "What will 14 hushcls of clover seed cost at 12^ cents per pound ? Ans. $105. 2G. A farmer sold a load of oats weighing 183G pounds, at 30 cents per bushel ; how much did he receive for the same ? Ans. $10.20. 27. A produce dealer bought at one time, one load of wheat wcigliing 1)240 pounds, at $1.05 per bushel; one load of barley weighing 2400 pounds, at 85 cents per bushel; one load of rye weighing 2S00 pounds, at 05 cents per bushel ; two loads of pease, each 2400 pounds, at GS cents per bushel ; three loads of buckwheat, each weighing 1400, at 55^ cents per bushel ; and a quantity of oats weighing 578 pounds, at 33 cents per bushel; what liad he to pay for the whole ? 23. A farmer has 12 sheep worth $3.50 each ; 9 pigs worth $4.05 each ; one cow worth $35, and a fine horse valued at $150, He exchanges them with his neighbour for a yoke of oxen worth $75 ; two lambs worth $1,925 each; a carriage worth $100, and takes the baknce in calves at $1.50; how many calves does he receive? Ans. 20. 29. A and B sat down to count their money, and found that they had together $225, but A had $15 more than B ; how much had each ? Ans. A $120, B $105. 30. A miller bought 250 bushels of oats for $85 and sold 225 bushels for $70 ; what did the remainder cost him per bushel ? Ans. GOc. 31. A widow lady has a farm valued at $0720; also three houses, worth $12530, $11324, and $9875. She has a daughter and two sons. To the daughter she gives one-fourth the value of the farm, and one-third the value of the houses, and then divides the rciuuinder equally among the boys , how nmch did each receive ? Ans. daughter $12923, each son, $137G3. 32. A man wont into busii 5 with a capital of $1500 ; the first year he gained $800, the second year $950, the third year $700, and the fourth year G25, when he invested the whole in a cargo of tea and doubled his money ; what was he then worth. Ans. $9150. 33. A boy paid out 30 cents for apples, at the rate of G for 3 cciita ; how many apples did he ')urchase ? Ans. 60. C4 AMTIIMETIO. ill!!!: 04. A Kchoolboy bought 12 oranges at .j cents each, and sold them for 12 cents more than he paid for them; how nu-oh did ho BcU them at each ? Ans. 4c. 05. A clorli's income is 82098 a year, and his expenses $ t.50 I)cr day ; how much will lie savo in two years ? Ans. $2111. 30. A speculator bought 200 acres of land at 6 15 per acre, and afterwards sold 150 acres of it for 611550 ; the balance ho sold at a gain of $5 per acre, and received in payment $250 cash, and the balance in sheep at $5 each , how many sheep did he receive ? Ans. 450 sheep. 37. A butcher bought 1) calves for $54, and 9 lambs for $lil.50 ; how much more did he pay for a calf than a lamb ? Ans. $2.50. IjS, a farmer sold to a grocer 380 pounds of pork, at 7 cents per pound ; 150 pounds of butter, at 17 cents per pound, and one cheese weighing 53 pounds, at 9 cents per pound ; and received in payment 22 pounds of sugar, at the rate of 11 pounds for a dollar; 150 pounds of nails, at G cents per pound ; 15 pounds of tea, at G5 cents per pound ; one half-barrel offish, at $18 per barrel, and one suit of clothes worth $27; did the farmer owe the grocer, or the grocer the farmer, and how much ? Ans. the grocer owed the farmer 12 cents. 30. A milkman sold 120 quarts of milk, at 5 cents per quart, and took in payment, one pig worth $1.50, and the balance in sheet- ing, at 10 cents per yard ; how many yards did he receive ? Ans. 45 yards. 40. How many poimds of cheese, at 9 cents per pound, must be given for 27 pounds of tea worth 80 cents per pound ? Ans. 240. FRACTIONS. 14. — Vulgar or Common Fractions. — When wo have di- vided any number by a less, and find no remainder, the quotient is called an integer, or whole number. When we have divided any number by a less as far as possible, and find a remainder still to be divided, but less than the divisor, and therefore not actually divisible by it, wc must have recourse to some method of indicating this. "We have seen already that the conventional sign of division is this mark (-—) ; thus, 3-:-4 means that 3 is to be divided by 4, and this being impossible, we indicate the operation either as above or by writing the three in the place of the upper dot, and the 4 ia the place of the lower, thus, f . or .-mi hut tl) Th, v.-'mt a fiM tho ,s;i Ai dcnoii, iilustr; ' < d Ml ^■inu r divIJii] iu-.iltip! obt:iiri obtiiii tion is donomij ■ I I TRACTIOKS. C5 di- nt is any to be sible Wo tliia 1 this or by a the Tlio n.ituro of a fraction may bo vicwcil in two wnys. First, wo iii;iy consiilor tli:it :i unit i.-i divided into ;i certain number of cqutd parts and :i certain nnmhti* of these parts taken; or, scco)uUi/, that a Miunber j^rcuter than unity is divided into certain equal parts, and our. oi' these p'U'ts taken ; thus, -^ means oitlicr tliat ;i unit is divided i;ito 1 (.'ipial parts and three of tliein taken, or that three is divided int ) i c'lpiai p:rts and one of them taken. Fur example, if a foot ])' divilod into l- ctiual parts, c:icli of tiic.se parts will be Ji inches, and (lu'ee of them will be nine inches; and uince 3 feet make 3G iii'.'hes, if we divide :» feet into 4 equal parts, each of these parts v»'ill bo 1) inches, and lionco ^ of 1 --.} of 15. The lower figure is called the di'iioiuinator, because it bhows the denomination or number of p;rts int(.) v.-hiel» the unit is suppescd to be divided, and the upper oii«.! i'S called the numerator, because it shows the number of those parts considered in any given (question. AVhcn both are spoken of to;,"jther ihey are called tho terms of the fraction. What may be considered the fundamental principle on vrhicli all t'le operations in fractions depend is this : that the form, but vj;\t t!io value of a fraction, is altered, if both tho terms are either nuihiplicd or di\-ided Ijy the ."^ame quantity. If we take the fraction J and mulriply its terms by 2, we get J;. Now, the -j!,- of a foot is an inch aad-a-lnlf, and therefore ^'. is G inches and G half-inches, or 9 inches ; l)ut we have seen that "^ of a foot is 9 inches, therefore J of a foot is the same as f; of a foot. So also ^ of £1 and ^ of £1 arc both 15s. The same will hold good whatever the unit of measure may bo, or whatever the fraction of tluit unit. Hence, universally the yonn of a fraction is altered if its terms be cither nmltiplicd or divided by the same number, but its value remains tho same. Again, if we multiply the numerator '.l by 2, but leave the denominator -1 unchanged, we obtain ',', and, keeping to our (irst illustration, " of a foot is G times three inches, or 18 inches, which ii dmblo of 9 inches, tho valuj of ;^. We should have obtained the same result by taking 1 and dividing its denominator by 2, without dividing its numerator. Hence, a fraction is multiplied by cither iUAltiplyin^ its numerator or dividing its denominator. In like manner, if we take the fraccioa ^ and divide its numerator by 2, wo obtain J, and if we multiply the denominator of its equal J by 2, wo obtain the same rusult, ^. Hence, ^ is ^ of f, and therefore a frac- tion is divided by cither dividing its numerator or multiplying its denominator. These princinlcs may also be referred to the obvious 66 AKITHJIETIC. fact that in divt Jmj; any qu entity tlio greater the divisor the Iiss ths quoHcnt, and the Ic^^ tlio divisor the greater the quotient. As it io ahyays desirable to liive tlic smallest nuiubers possible to handle, It; the operator ob^ierve this as a universal rule — dloidc when yon ca7i. Fractious are classliied iu four different ways, according to four difibrcnt circumstances. I. They are divided into Proper and Improper Fractions, A proper fraction is one whose numerator is loss than its denomi- nator. Iu strictness such alone is a fraction. An improper fraction is one whoso numerator is greater than its deno'ninator. Strictly tliis is not really a fraction, but only a certain quantity expressed in the fractional form. II. iSimplc and Compound Fractions. The term simple fraction, as opposed to compound fraction, expresses that the fraction is multiplied by unity alone, as -g-, whicli means either g- of 1 or -J,- of 5, or g-X 1=8X5. A compound fraction is one tliat is multiplied by some other quantity. A fraction is called compound it' either multiplier or mul- tiplicand, or both, bo fractional. Thus: ^^ of ;} and -^ of 11 arc both compound, and are written .^Xu ^"^"^^ 's^XH- III. Simple and Complex Fractions. The term simpb fraction, as opposed to complex fraction, means that there is only one division. Thus: ] [) means that a single number, 15, is divided by a single number, IG. A complex fraction is one of which either the numerator or de- nominator, or both, are fractional, that is, it indicates a division, when cither the given product or given factor, or both, arc fractional. Thus : f^-:- ,',-, or i and - and jj£ arc complex fractions and cx- Libit the only three posssible ibrms. IV. Vulgar, or Common, aad Decimal Fractions. Decimal fractions are those expressed with a denominator, 10, or apowcrof 10, r.y., V;,,-,'^"^,., -j'^i^. Any fraction not :;o expressed is ctdled vulgar or common. Thus : :2- would bo called a common fraction, but its equivalent, -^^^j^, would be called a decimal fraction, and is written -75, the denomina- tor being omitted, but its cxistoncQ being indicated by the mark (•)> called the decimal Doint. fhactions. 67 A Mixed quantity is ona expressed partly by a whole numbci and partly liy .1 fraction, ns '!^, 1)1\. This is not another kind ot fraction, but simply another niodo of writing an improper fraction •vhcn the Jisibion inilicated has been performed as fur as possible Thus: v^— 4^, and 'S'. -Uh. It is often taid that there are six kinds of fractions — proper i..iproper, simple, compound, complex, and mixed. This is logi (' lily incorrect, for a pro;ier fraction is simple, and a mixed quantitj IS :ui improper fraction in another form. 15. — QPEaATioNs IN Common Fractions. — From the prin cipks laid down (Art. 21,) wo can deduce rules for all the operations in fractions. I. An improper fraction in reduced to a mixed quantity by per- forming the division indicated, as ■'^,^^=24J. . ll. A mixed quantity is reduced to an improper fraction by multiplying the integral part by the denominator and adding in the numerator, as 12J--=;-'.^''. So also an integer may bj expressed in the fractional form by writing 1 as a denominator, and multiplying the terms by whatever number will bring it- to any required denomination. Thus: to reduce 7 to the same denamination as j], write ]• and multiply the terms by (5, and the result, ~\^, will bo equivalent to the integer 7, and of the same form as jj. 1: X E 11 c 1 s E a , IV de- Lsion, LOiial. cx- |lO,or imon. lomina- Irk (•)> 1. I'^xpress >> ] Express 0. l]x press 4. E\-/n>s 5. l^.cprcss C. ] Express 7. !■]>; press H. Express '.). Express 10. ]'].\p;'ess 11. J']xi)ress 12. I'Lvpress l.'}. Express 11. Express '!-,'— as a whole or mixed number. ';l as a whole or mixed nuud)er. 1 it -",'','- as a whole or mixed number. -'.,'-,'- as a whole or mixed nuiiiber. -^■P'^7i?- as a whole or mixed luunber ".."''' as a whole or mixed number. -•;'; as a whole or mixed number. '■'] us a whole or mixed number. ' ] :": < -IS a whole or mixeil number. '1','- as a whole or mix.^d number. ' .' ■' as a whole en- mixed number. --Jj-- as a whole or mi;;ed number. --,-y- as a whole or mixed number. >' as a whole or mixed number Ans. 49. Ans. 71. Ans. 5^;! Ans. fiV'i;. /Vns. n^\. Ans. 7;,S. Ans. 7^C Ans. 89. Ans. 10,",. Ans. 10;;. ^nf.. 12/,. Ans. 4j. 68 AEITHMETIC. 15. Express -'•/- as a Tvliolc or mixed number. Ans. 24J, IG. Express -Vy- as a whole or mixed number. Ans. SAy. 17. Express -^'l- as a whole or mixed number. Ans. Sj"-,-* 18. Express -V- as a whole or mixed number. Ans. 5^. 19. Express J-;^-L as a whole or mixed number. Ans. 30J. 20. Express -yj]- as a whole or mixed number, Ans. 83 j^^. 21. Express Jj^j'- as a whole or mixed number. Ans. O^'g. 22. Express 27! as an improper fraction. Ans. -~: 23. Express GG',- as an improper fraction, Ans. ^'^^. 2i. Express 15J-3 as an improper fraction. Ans. -Ytt- 25. Express 7.]- as an improper fraction. Ans. ^^-. 2G. Express 49 as a fraction with the same denominator tisf^. Ans. r'^".^. 27. Ex- .^ss 193. as a fraction of £1. Ans. AR. 28. Express 11 inches as a fraction of a foot. 29. Bring l, ^, J, J, j'^ to the same denomination Ans 30. Express 11 as a fraction having the same denominator as ^^-J^. in. To reduce a fraction to its lowest terras or simplest form, divide the terms by their greatest common measure. This is often readily done by inspection, as :j:T=: (!-=§, but in such questions as J-[Jv :^, the most secure and speedy method is to find the G. C. M. of the terms and divide them by it. Thus : the G. C. M. of the frac- tion 1-215 is 1092, and the terms of the fraction divided by tliis give J, the simplest fonn. EXERCISES. Ans. j|. 1. Reduce ■^^^;^g to its lowest terms or simplest form. 2. Reduce 'I'^H to its lowest terms or simplest form. 3. Reduce -^frj^ to its lowest terms or simplest form. 4. Reduce uVu"o"u'u *° '^^^ lowest terms or simplest form. 5. Reduce 4"!] jj to its lowest icrnvi or ^^-lulplenL form. 6. Reduce yrs^jb ^^ '^^^ lowest tei'ai'i or simplest form, 7. Reduce yvy-j-,- i-> its lowest terms or simplest form. 8. Reduce :f.|.|.| to its lowest terms or simplest form. 9. Reduce ji-^-^^-j to its lowest terms or simplest form. Ans. J. Ans. j^. Ans, f» Ans. ^L. Ans. g. Ans. 1. Ans. -,^. Ans, f . Ans f. FRACTIONS. by 10. Reduce 11. Ecduco 12. Reduce 13. Reduce Hi y I *° ^^^ lowest tenc? or simplest form. Ans. f . T'V'iWii ^^ ^^^ lowest terms or simplest form* Ans. -^Tj. 5li to its lowest terms or simplest form. Ans. j^. Ti/yWub ^^ '^^^ lowest terms or simplest form. Ana. §. 14. Reduce ^,1]'^^ to its lowest terms or simplest form. iins. 7-25. "i'YoS ^° ^^^ lowest terms or simplest form. Ana "ioio" *° ^^^ lowest terms or simplest form, 4 "3 5 ^'^ ^^^ lowest terms or simplest form. j'Wo ^'^ ^^^ lowest terms or simplest form. 15. Reduce IG. Reduce 17. Reduce 18. Reduce 19. Reduce 20. Reduce X iuluy ^^ ^^^ lowest terms or simplest form. CO 10 a ^° "^^^ lowest terms or simplest form. Ans- Ans. 12- Ans. 1%, Ans, f • Ans. 1 1 21. Beduca ■Hraiiro^^ij^o to its lowest toi-ms or simplest form. Ans, J. R |4U* 5 6' 1 a- |s|. I\'^. To vfiultiply one fraction by anotlier, multiply numerator by numerator and dcuohiiualor by denominator. 1 ... 1 ... 1 HI. ..111. ..Ill ^iius : I X ^^:>^. To illustrate that A of ^ is ^,, take a lino ;md let it bo divided into 3 parts, and oacli of those again into 3 parts, as in the margin, wc find that the result is 9 paits, each, of course, being -j of the unit. AVc Lave seen tliat a fraction is multiplied by multiplying the numerator or dividing the denominator. Now, if it were required to multiply £- by f', wo could not diviJo the dcuomlnator, as 5 is not contained in 1, and therefore Vv'e luuUiply tlio nuincrritoi' and obtain ';"', but v.'c have multiplied by a quantity equal to T limcs the given one, and tiierclbi'c wo umal divide the product by 7, i. c. (Art. 21,) wo must multiply the denominator -t '^y 7, which gives l^ for ihc correct product. EXEllCISES. 1. Multiply -j^ by 1^? 2. What i^ Jie product of ;J by -]-J ? 3. What is the product of -,^7 by [: ? 4. AVhut is the product of ^ by [^ ? 5. What is the product of -.l by A 2? Ans.^. Ans. l^. Ans. i,'2. Ans. gf. Aus. ^-g. 70 ArjTnjiETic. An .-, f> "s- Cos- G. Whr.t is t]i3 product of I by -,^^j ? 7. What is the product of -f^^ by -'j ? 8. What is the pvcduct of vf; by -/y ? 9. What is tlic product of !j by ^ ? Ans. fj. 10, AVhat is the product of ] ~ by ■/,- ? Ans. /|,^^. When the product has been obtained it should be reduced to its lowest terms. Tims : tho product of -,"j- by ]l is -^-^v,, the terms of ■which arc both divisible by 11, and eo we get the equivalent fraction •^Tj. But wc might as well have divided by 11 before multiplying^, for by this method wo should at once have I'ound the fraction ia its yimplcst form, viz., ,".. In the iuiuo manner any number or num- bers which arc factors of both numerator and denominator, may bii Emitted in the operation. This we call cancelling in preference to the c::cessivcly awkward terra " cauccU'itionJ' This method will bo clearly .seen in exercise 11. If either the multiplier or m\'.lt;;.'!:c.aid bo a mixed (juantity, it must be reduced to :iu improper fraction beibrc tlic multiplication ia performed. Thus: S-'X^^'l^X^S-^'^ll-'^'^Hv 11. What fraction u equal to ^ oi' ii of ^ of i of •} of ^ of ;^ of ? Ans. A. 12. What quautity is equal to 12^- multiplied by 7^ ? Ans. 97{4. 13. What quantity hi equal to 19^^- multiplied bv 1 rr? Ans. 36. 14. What is the value of ^ of ^ of ^^ of ^ ? 15. What is the value of ^r of ;^ of fj of -14 ? IG. What is the product of 27-^- ])y og ? 17. What is tho product of ] fj by H ? 18. What is the product of 5^ l)y 5-^- ? 19. rind the square and cube of ^| ? 20. What is the cube of;-;*? 21. JMultiply 27 by .J.-? Ans. ^VW- Ans. 107,fi. Ans. -^. Ans. 30J. An« .?S!> rind 1013 Ans. 1. V.-X)IVISiON OF FFvACTIONS. To divide one fraction by another, multiply by tho recipro- cal of the divisor ; or, in otlior words, invert the divisor and multi- ply. In the language of iscioico^ the reciprocal of a fraction ia tho fraction vrlth its terms inverted. Thus : 1} is the reciprocal of ^ ; ;| of -|. To find the rcciDrccul of a whoh number, we innst first isl '7 ^* 5.1. ;ipro- lulti- r . 1 fiwt DIVISION or FRACTIONS. 71 rcprcscDt it as Iv.iving a denominator 1, — thus 4==r'J. ; G^*;, and therefore the reciprocals arc :} and -J-. The rule for division may bo jn'ovcd ill two ■ways : First troof. — Let it be required to divide 7",- by (i- Ifwc liad been required to divide l)y the whoro number 5, we should tiiher have divided (Art. 14. j tlic numerator, or muUiplied the denoiuina- loij — as the numerator is not divi;.-ibL' by ;'>, wo multiply tlic dc- lioiuiuator, and obtain ■' ^ ; but Vv'C have div'ided by a (juantit^' equal to six times the ^Iveu one, and tliercforo, to ccfrDpcnsate, ^vo must multiply the result by G, which gives -^ j. Second riiooF. — Write the question in the complex fonn — 'i-i, ihcu Art. 14.) multiply both terms by 11. and ',',' is obtained ; ii and again multiply the teniio by G, and ^ j is the result as bcfi.ire. — 'Xhc two operations are virtu:illy the same, thou;.:h csln'jlLcd in dif- ferent forms, and both arc equivalent to the tcelmlcal rule, " Invert the divisor and multiply." IJiscd quantities mutt be reduced to improper fractions as in multiplication. The expressions muUlpUcaiion aud division, as ap- plied to fractions, arc cxtcn^:ions 01' the ordinary meanings of tliosc tcrm.j, for in their rrigiual meaning, the fornicr implies increase, o.nd the latter dicrease ; but v/hcn two proper I'ractions are iiiultipaed together, the product is less than cither of the factors, and v.'hcn oao proper fraction is divided by another, the quotient is greater r.i:.;i either the divisor or dividend. This will be scfu ]jy tho annexed examples : ;}X^=^A. I}uti^^H;! and l~-.^, both eroatcr than >^ '. •W.-o 7— •_ .•1—7 x/ .) ft But ■^■^^H.'j and 'x=^\'^, both lc;-s than .'2 f .. If two fractions Iiavc? ^ common denominator, their quotient is the quotient of their numerators. Wo have placed multiplication and division of fracttJns before addition and subtraction, because, as in "wholo immbcrs, multiplication and division ar;> deduced from addition and subtraction, so conversely in fractions, additicn aud subtraction are to bo deduced from uujllipiication and division, for a fraction is produced by division, aud (iic multiplication of a i'raction is merely the repeating of the divided unit a certain number of times, -Thus : ^ is a unit divi(lc(J into 8 equal parts, and § is that fraction repeated 7 times. W ii'^il iiiiiiii 72 ARITHMETIC. EXEUCISES. Ans, 1. Divide ,\ by §;fT-^-l-AXil. 2. ^Vlrat is the quotia^t of \:i divided by -. vy / 3. What is the quotient of ^\; divided by \ l~; ? 4. What is the quotient of p^, divided by ^'i 5. What is the quotient of l]^- divided by i| ? C. What is tlic quotient of 3G divided by 19^ ? 7*. What is the qTiotlent of ^ divided by 2-g-? 8. What is the quotient of 4?,- divided by 15 ? 9. What is the quotient of ^;? divided by 2}^ ? 10 What is the quotient of 75/^ divided by 9 ? 11. AVhat is the quotient of 0^'/ divided by 91 ? 12. What is the quotient of 5| divided by ^i^^ ^ ^ ^ . , 13. Divide tlie product of ^, ^ and ^ by the product of ^,4^ B_ r, Ans. -g — ij- ^" U. What is tlic quotient of -jV of {^-^ of jl of yV-W\j «^ «J i\.US. 'x.-, >}■• AnS. ■i^ry AA— 1-i! 11 — -^1 1 ivns. w.ioii Ans. {1,. Ans. ^l. Ans. 1|^ Ms. 1 f^-^. Ans. -j"\,. Ans. ^C^ Ans. S^. Ans. f;. •ir, R Aus. 1 1 Ans. 8;-2. Ans. l-j^y. Ans. 729. 15. How many ./.j are there in -/VjJ 16. What is the value of ^ of t-:-£- of ji '^ 17. Divide 27 by .^^ ? Elenee, any quantity divided by its reciprocal gives vhe square of that number, and exercise 21, of luurtlplication, sho;vs that any quantity multiplied 1)y its own leciprocul gives unity. 18. Divide ^^i by t, and the quotient by ^ ? 19. Divide ^^ by -^j, and the quotient by n ? 20. Divide ^.^ by 'jij? 21. Divide {J by ' Aus. Ij'^r. Ans 4'J* Ans. o-^ (j2t' 4» ? Ans. ^. VI -ADDITION OF FRACTIONS. We have seen *hao no quantities can bo added together except they are in the same denomination. Wd can add - ;,>, -^ and * 'as thoy are all of the same denomination, sevenths, and we Ta ._. We can easily see that to add -| and f we have % to alter the form of f to S, and we have both fractions of the same denommatlon, and therefore can ^^^^0+^^ be guided by some rule. To find the vduc of -i+u+.+y-h-^- h :cpt land wo liavc lions LT — « ' .10. a ■ bvast -1... ADDITION or FEACTIONS. 73 By Art. 13 vc find the L. C. I^I. of 4, 0, 8, 0, 12 to be 72 and the rest of the common operation is oqiiiralcnt to nmUiplying the terras of each fraction by 72. Thiis: if the terms of J be both multiplied by 72, wo get CaS^^r^x^— v^> ^^t wq might as v.-cU have divided 72 by 4 before nmltiplying, aud, to balance that, have multiplied the nuuK'rator o, not by 72, but by the fourth part o£ 72, viz., 18, giving ■']:\, as the following scheme will show: — -^xli^:Jxlb>^H4x!«=vJ- The other fractions being altered in the fcame manner, we get iv! -•-,'' n^-y:'!-}-;^?;-!-^;^:, and as these aro now all of the same denomination, though not i»ltered in value, wo can add thorn, and we fiod |Ta+5-iitl^=^ti-H^H^+■^+ 13=2.335. Hence the RULE . Fi'.id the L. C. M. of all the denominators, which will be the common denominator ; divide thiis common multiple by each dcnomi- ioator, and multiply the quotient by each numerator in puccessiou lor new numerators j add all these new numerators together, and place the common denominator below the sum, and the fraction thus ob- tained will be the sum of the given fractions. If the numerator, thus obtained, bo greater than the denominator, the resulting frac- tion may be reduced to a whole or a mixed number by di ;ision. £X EBCISE S, Ans. 14. Ans. 2|-. Ans. IS/y^. r. .1 1. Express i^-f-Aj-rA+A- ^^ ^ single fraction ? 2. Find the sum of ^. k, ^ and ^ ? 3. Add together J-|,*'l f;^, 2;j.y, 3£? and 5 J^ ? 4. What fraction is equal to -^-i-|-i-i+ J^-f -^L-f J.. ? ^Vns. D. What fraction is equal to U-i-25-[-3r2-[-4;jl5(;-}-0« ? Ans. 25] 7. J. G. Express h of f -j-j of ■•-|-| of i- as a single fraction ? Ans. IS^i^. Ans 7. Find the sum of m, 8--, 3 A and43 ? 8. Find the sum of \ 0*' ^ -\- -h^\- ;]of "? I I 9. Wliat single fraction is equivalent to \ of ^]--|-||( of ^-f-^ of { ? Anf 10. What single fi-action is equivalent to f of % of l-i-,^ of -^ H-|of^oi,' " Ans. -,V 11. What single fraction is equivalent to \ of % of ^-|~S '^'^ « °^ % 'i Ans. A*?!. m 74 AEITHMETIC, VZ. Simplify 2 , _, y -' ivns. l^j ao- -8-,' Ans, ^^, Ans. r, 1 •! 1 13. Find a sinslc fraction equivalent to •?. of £. of §-{-{3 f^^^ 14. Divide the sum of ^ and ? by the .urn .^f ^ nnd ^? 15. Simplify Hx^y±>tr^^:^'^fpM ' i4--i-i--> 16. Simplify ig^icf I ^ ' YII._SUBTRACTION OF FEACT'ONS. What wo havo said of aMition enable, us to gi« at once tl.o ,m;i.e fou subtraction. ^ nca«oe the "ivcu fractions, jf necessary, to new ones lumng a tnornhvitor as in addition, and subtract the numerator of common denominator as . , ^^^^^^ aonorainator Slwt:— :,rdC:;n4 f-aon wiii.. tUo di^erenco ''";,:' tr-(L*Tr;ubtract -, ^.m ,v no- >^^;'-;f- 12'V above 7&, wo find now fractions with a conunon dononunator ^ 1^ ^d - and wo writo 12^-7-. Now wo ayo required c';' W "^r-'from « but a3 vo cannot do this directly, we first to subtract ^i Iron, v..,, oui ^ f ti,» 1 o nropedin"- uwts, and call it ^. , (}ov ..i^-i, take one of he 1. P ccedin^^ ^ ^^^^^^^^^^^^ ^^^^ ^ ^^^^^^^ ^^^ ^.^_ ^^^^nTff • " i'i^^l^ic 'subtraction, 8 from 12, and we iii^ maming H , or as in b 1 convenient to sub- the total excess to bo 4^1. In praetic. 1 1. g_^c.^i7 ^nd tract 15 from 24, and add 8 ; thus 21-l;)-9, md 9 , b_i., the answer is 4;i|. EXERCISES. 5 What is the difference between g and ., r G What is the difference between ^li and ^--'^ 7. What is the difference between •;}{! and -,-u ( 8. What ia the excess of 20^ above d^^ ? 4. 1-^^=1^- Ans. J?. Ans, Ans. 2U' ^^l3. lOHil DENOMLXATE PKACTIUNS. IS 0. Fromn-;|tako3^? ^^^3_ 2JV^^ 10. Whatistlio difforcncobotwccnD-'^- nndG--'"^'-? Anq i^^'i' 11. What is the vuhicof M-[^_1J- i_i ?'"'" ■ Anf To" 1-. What IS tlio difforcnco between lOO-'g and 50'-'> ? Ans, 49i-''.2 I a. Whiit id the difforcnco between \ of J and J of ^i- ? ' Anl^Q 11. Wli.t ii tlic difference between § of /^ and [l of < ? Ans - 3 3 n. What i. the value of H-^-^.+ m ? 'Xi^^j; VIII.-DENOMINATE FRACTIONS. Hitherto ac Iiavc treated of fractions absfractlj, and wo mnst nov^r apply the pri ,piplc.s hud down to denominate numbers and sliow 1h)w a n^crion may bo transformed from one denomination to another of the .same kind, e. umcrator to the next lower denomiiui- tlon%nd divide the result by the denominator; if there be a rc- m-iinder reduce to the next denomination, and divide again, ani continue' the same operation till there is cither no remainder, or down to the lowest denomination by which the integer i^ counted. Thu^ since a ton is 20 cwt., ] of G tons is 120 tons divided by ., wlilcll irivcs 17 cwt., with a remainder of 1, which, reduced to (jrs , mW "ivo 4, in which 7 is not contained, and the 4 qrs. reduced to lbs., will give 100, and this divided by T produces 14^ ; so that, \- of a ton Is 17 cwt., qrs., 14| IbtJ. E X E 11 C I S E S . 1 AVliat is the value of 7. of a ton? Ans. 11 cwt., 2 qrs., IG-J lbs. 2. Wlvat is the value of /', of a yard ? Ans. 2 feet., 8i lu. 3. What is the value of ^ ;• of a mile ? 1. What is the value of ',;; of a shllUug Stg. ? Ans. 11 -d. 5 What is the value of ] of a ton ? Ans. 11 cx110> .087'. l/u 140 120 112 80 80 any 1 O10(.142S57 7 frac- 1 _ ^ get 1 30 same 1 28 L and 1 20 :cforc 1 14 ly 1)0 1 But 1 GO of 10 1 56 ; divi- 1 40 inator 1 35 ), i- <'"■> 1 phcrs, as "we 50 49 cxam- 1 a rule n ij L E . Divide tht! uumrntfor, vilh a riphci' or dphcrs nnnrxed, hi/ t/ie dinomiiujitor. Thus ] '. v-i/lyivc, we neo that there are eommon fractii'jns who.^c lernihi can bo multiplied by such powers of 10 as will make the numerator divisible by the denomina- tor without remainder, but it often happens that no power of ten will effect this, and that remain- ders occur which cannot be made divisible even- ly by the denominator, by the addition di" any number of ciphcrfJ. Such fractions will never terminate, and there- fore are called intcrminatc, and the connuon fraction ciin never bo expressed exactly in the decimal form, and all wc can do is to make an approximation more or less close, according to the number of de- cimal places to which wc carry it. Let us take the fraction ^. — First, n is not contained in 1, and thei-elbrc wc place the decimal point in the cpioticnt, and add a cipher to the numerator, and wc find that !) is con- tained once in 10, with a remainder 1, — annexing another cipher, we again obtain 1 in the quotient. and this will obviously continue ad infinitum. — This recurrence is marked by a dot or dash over the figure, thus: .1 or .1'. If we express ,1 as a decimal, wo find that after wo have got six figures in the (quotient, wo have a remainder 1, the same as the original numerator, and therefore we .should again obtain the s:imc quotient .142857, and hence this is called a circulating or periodic deci- mal, and the first and last of the recurring figures arc marked with a point or trait. Thus: .142857 or .1'4''S57'. Again, it often happens that some figure uo not recur whilst others following them do, as in the annexed cxamulc, after wc Lave ijot i ,.,. J I 80 ^vniTn:\n:Tic. five figures the 11500 -vhicli .uavo w. tlio third Ihaip: ;'. in the qnt*ti cut recurs, and by pursuing the division, wc should find 345 rccuvrincr 'wltliout •nu 3oJUJ)rilllO(.12k5. end. Wlicn all the fi:i;ures recur, the fraction is culled a j^urc periodic deci- mal; Avlien only sruic of thcni recur, it ia called mixed, and the term rapeater is applied when only one figure recurs, as -J.=.llll, &C.=.i or ,'',rrr.5S333, &c.= . 583. Since tlic denominator is always 10, or a power of 10, and since 10 has no factors but 2 and 5, and therefore jwwcrs of 10 no factors but 2 and 5, or powers of these, it follows that no deci- mal will terminate except the denomina- tor bo expressed by either or both of these, or some power or product of them. Hence all terminating decimals are deri- ved from common fractious Laving for denominator some figure of the series 2, 4, 8, IG, 32, &c., or 5, 25, 125, ic., or, 10, 20, 40, 50. 60, 80, 100, &c. 33300 TSfOO CGGOO 115000 99900 151000 133200 178000 1G5500 11500 EXERC I. 3. 1. llcduco the common fraction J to a decimal. 2. lleduce the common fraction ^ to a decimal. 3. lleduce the conunon fraction ^ to a decimal. 4. lleduce the connnon fraction l- to a decimal. 5. lleduce the common . aetion ,', to a decimal. G. Kcduce the common fraction ^ to a decimal. 7. lleduce the common fraction -J- to a decimal, r^. Reduce the common fraction ] to a decimal. 9. lleduce the common fraction ! to a decimal. 10. lleduce the common fraction y'j to a decimal. 11. Ileducj th , counuon iVaction ,'j- to a decimal. 12. rtcduce the common fraction /^ to a decimal. 13. lleduce the commou i'raction ?.- to a decimal. 14. lleduce the common fraction i to a decimal. Ans. .25. Ans. .5. Aiis. .75. Ans. .3. Ans. .i. Ans. .125. Ans. .10. Ans. .i4285t. Ans. .2. Ann. .1. Ans. .09. Ans. .083. An3. .G. Ans .8. to nui thl 15a £1 DECBLYL FRACTIONS. 81 15. IG. 17. IS. 19. 20. 21. 22. 23. 2-1. Reduce lloilucc IlcducG 11 educe Ivcducc Reduce lleduce lleduce Reduce Jleducc the common tlie common the common tlie counnon the common the common the common the common the common the common fraction fraction fraction fraction fraction fraction fraction fraction fraction fraction ;i to .1 decimal. ^ to a decimal. ;] to a decimal. ^ to a decimal. ,', to a decimal, i to a decimal. { 'I to a decimal. ] \ to a decimal. 1 '.: to a decimal. J ;[ to a decimal. jVns. .83. Ans. .375. Ans. .625. Ans. .875. Ans. .4. Ans. .7U285. An.s. .90. Ans .9115. Ans. .923073. 25. Reduce the common fraction ] ,'. to a decimal. Ans. .G875. 26. Reduce the common fraction .iV to a decimal. 27. Reduce the common fraction .'..', to a- decimal. Ans. .34375. 2S. Reduce the common fraction j^'r;- to a decimal. >' I .3. .1. o .1. 09. ■083. .S. 29. Reduce the commoD fraction ^l to a decimal. Ans. .46S354430370<. 30. Reduce the common fraction ■^-^r^ to a decimal. Ans. .0044. 31. Reduce the common fraction ^',j to a decimal. Ans. .620408103265306122448979591836734093877551. 32. Express -./.j decimally. Ans. .01. 33. Express rjl,j decimally Ans. .OOi. 34. Express tjJtj decimally. 35. Express -^/^-^-^ decimiUy. Ans. .00059994. To reduce a denominate nuraboi to the form of a decim;d IVac- tion, reduce it to the lowest denomination icJiich it amtdins ; reduce the intcrjral unit to the same denomination, and divide the former hy the latter. Thus, to eXj'^rcss 18s. 4d. as a dechnal of ;C1, \\q must reduce it to pence, the lowest denomination gi't'cn, and divide it by 240, the number of pence in XI, which gives the fraction !^;j|]::::^^;|=:J .^, and this reduced to a decimal, gives .916 or £.916. In like manner 15s. lOM. ib reduced to half-ponce, viz., 381, md the hulf-pcncc in £1 are 480, and -^^i"^- yij- which expressed dccimallv is .79375. "\l' 82 ATJiTiiMrnc. £XERCISEh, 1. What decimal of £1 is lis. 4M. ? Ans. .56875. 2. Express 15s. 9;2d. as a dccimai of £1. Ans. .790G25. 3. "What decimiil of a square mile is an acre? Ans. .0015G25. 4. Express 1 pound troy as a decimal of 1 pound, avoirdu- pois.* Ans. .82285714. 5. lleducc 17 cwt. to the dccimai of a ton. G. Express | ;} of a cwt. as a decimal of a ton. Ans. .85. Ans. .04G875. oz. ll-:-lG=.n.GS75 22.GS75-:-25=.9075 2L9075-T-4=.72G875 cwt. 11.72G875~20=::.58G34375 1G)11 :5)22.G875 4)2.9075 The operation nnncxed is often convenient in practice. To reduce 11 cwt., 2 qrs., 22 lbs., 11 oz., to the decimal of a ton. First, we divide the I"* oz. by IG, the num- ber of oz. in 1 lb., and then annex the 22 lbs., and divide by 25, the lbs in a qr,, and so on. The first form of the work is best suited for illustration, the .second is neater in practice. The prniciple is the same as that implied in the general rulo given above. 20)11.720875 .58034375 ADDITIONAL EXERCISES. 7. Reduce 10 drams to the decimal of 1 lb. Ans. .0390G25. 8. Reduce 11 dwt. to the dccimai of 1 lb. Ans. .04583. 9. Express 1 oz., avoirdupois, as a fraction of 1 oz., troy, (sec note.) Ans. .9114583. 10. Reduce 5 hours, 48 minutes, 49.7 seconds to the decimal of a day. * A caution .sccru.s necessary here, for since tlie pound (troy.) contains 12 ounces, and the pound (avoirdupois.) IG, the natural conclusion would be that lh<' pound (troy) U J^ or ^^ of the pound avoir dupois. This is not correct, for tlic ounce troy ex- ceeds tlie ounce avoirdupoi.s by 42J grains, though tlio pouud avoirdupois (7000 gr.^:.) o.\eeeds the pound Troy (5710 grs.) by 1210 grains. This will be manifest from the oj. .ration on the margin, where tlie standard weichts arc ' '^cn. 57G0-;-12 7000~1G =480 .437* uiffcrencc. 42^ cas( t07', tha tJierl DECDIAL FRACTIONS. 83 II. — Reduction op Decimals to Common Fraction-s. — To find the common fraction corresponding to any given decimal. — This involves three cases according as the fraction is a terminating decimal, a pure circulating decimal, or a mixed circulating decimal. The first case scarcely requires proof. AVe give it, however, in order to assist those unaccustomed to the algebraic notation, to under- stand more clearly the form of illustration used in the other cases. Let us take the fraction .9375, and use d for decimal. Wc no^s' write d::=.93T5, and multiplying both terms by 10000, wo obtain 10000 d=9375, and therefore f^^^ytj'ooo' ^^l^ich reduced to its low- est terms is [[;, the common fraction rcf|uircd. This is simply put- ting for denominator 1, followed by a cipher for each figure in the decimal. To find the value of a pure circulator, suppose .(J. Putd:;=.G, or d=3.GGG, and multiply by 10, which gives 10 d=:G.GG, and writing the former cxprcs- .sion beneath, and subtracting, we get 9 d=G, and consequently d= d=,666+ 9d=G or 5, the common fraction Bought. 70 100 d^72.72 99 d-r-.72 d=r.5G8i 10000 d-^5GS1.8i 100 d^ oG.Si 9900 d^5G25 Let us now seek the vulgar fraction correspond- ing to .72. Put d^z=.72, multiply by 100, and subtract as before, and there results a rcmain- mainder of 99 d:^72, or . EXERCISES Ans. 4,y 1 Find the vulgar fraction corrcpouding to M. 2. Find the vu-ar fraction covrespo.dins to .54 Ans^ ^ ^ 3. Find the vul:^av fraction corresponding to .^i^T. A-IS. ,.oo- iVns. J' Ans. J. Ans. -^. Ans. f . A, in SORR 4. Find the vulgar fraction corresponding to .i, 5, Tind the vulgar fraction corresponding to .3. G. Find the vulgar fraction corresponding to .T.^ -4 . Find the vulg-ar fraction corresponding to .75. _ a Find tlic vulgar fracUon corresponding to .47o43, 9. Find the vulgar Iraction corresponding to i6835443C.707.^ Ans. >jT}. 10. Find tbo vulgar fraction corresponding to M,^ 11. Find the vulgar fraction corresponding to .1G2. 12". Find the vulgar fraction corresponding to .14.^ 13 Find the vulgar fraction correspondhig to .0138. l-L F-ind the vulgar fraction corresponding to .068I. 15. Tind the vulgar fraction corresponding to .593. Thclast rule may be deduced from the other two m the follo.- ,,. .nnner-Let us take the mixed circulator .418, and th. be.ng :;;:,.pllcd by lO, the i^ur becomes a .hole number, and to^reservo the same value, 10 is put as a divisor, which ^^'r^^^^ ,utby rule II. .-c bavc .iM"^, -^ ^-^ ^^° f ^'^ '''''' ""'H ; , ,« ,or.,8_4u^|3 and thia result correspouda to nile lit. Ans. ^. Ans. -J/f. Ans. ig. Ans. 7-/2 • ns. 'ij-^. Ans. l^. i Tl ADDITION AND SUBTRACTION OF DECIMALS. 85 4» 1 fl ~10 > nr.-ADDITION & SUBTRACTION OF DECIMALS. From what has bccu said, it is plain that decimals can be added and subtracted just as whole numbers, care being taken to keep the decimal points in the same vertical line. In all operations into ■wliich rcpctcnds enter, it sliould be observed that in order to have a result true to any given number of places, it is generally desirable to carry out the repctcnd to one or two places more than ilic required ii;un"ber. It is often sufficient, however, to allov/ for what vroukl be oar/icd, ■which can usually be done by inspection. In all cases, res- pect should be had to the degree of exactness which the nature of the calculation requires. The figures beyond those required can be estimated and added in. Thus, if only five places arc required, and the calculation be carried to six places, and the seventh figure is rv iLU'gc one, it should be added to the sixth %urc. This may bo stated in the form of a RULE. Add and subtract cs in xcliole numhcrs, Jceejiing the decimal p'nnis in the same vertical line. s.^. EXERCISER. ih' (1) (2.) (3.) .13 1.78045 8.58333333-h 5i.?noofiOooo 1 3.9VSG3 17.74747474-f- 3. 1444-41444 -i- ' \ 'Z' 7.84303 lli08GoO?08-f- 2.5 • ''I'i' ■132T32 GA-roimoo .SHjoj^.' J5-J- . 1 (? 9.511T9 15,GGi1jGuG7 11.875G0(]0;,O '• '^iT 11,09857 ,7G9J9G9r 4. Si jS( :")Sr5-|- iUow- 5.48491 11.00000000 7.111111111-1- 1 • _ 44.GG213 171.97297079 00.0793J072-i In exercise 2, the eighth figure of each of the fifth and si.xth liues is made 7 instead of G, which renders it unue^jcssary to make any allowance for the ropotcnds that would follow;, but this change is not uiado on any of the last figures of cxeiciso 3, and therefore we add 2 for what would be carried from the tenth decimal place to the nintli. 4. Find in the decimal form the sum of '^> §, ^, |. Ans. 2.31G. 5. Find in tlio decimal form the Bum of i, f, f, -[ [;, {;••', J:.'^, | "I. Au3. 6.0078125. ■Ji Qg AltlTHMETIC. 6. Find in tlie decimal form the sum of ||, |, ^"V. 73- Ads. 2tu'iOt 7. Find in the decimal form the sum of 2f , 4|, 5/o^ 8 What is the sum of .786425, .975324, .17G009, .33, .62519375, .4? Ans. 3.2S295175, , ' 9. Add to 6 places 18.127G, 11 .349, 12.145, 8.G48, 15.23. f Ans. 05.504414. 10. Find to G places the sum of 15.7, 12.4, 18.387, -^^^ .9, .45, 10.45, .12345. .... ^^- 'f'>f- ll. What is the sum of .70, .416, .15, .648, .23 to five places Ans. 2.5-Uoi. of decimals? ^ ca 12. Reduce to decimals, and find the sum of §, V., 21' 55- Ans. 1.416. 13. Find the sum of 427, .410, 1.328, H.029, 5.47G to si. ^, . , Ans. 10.078037. places of decimals. . . -.^0.10^^ 14. Required the sum of 1.25, 1.4, 1.037, 1.885, 1.084, 1.9o7, .or Ans. 32.750458. 1.148 and 1.704085. .. ^''^- . 15. Find the sum of ^.6321, .81532, .154926, .7532 to true to Ans. 2.1bt)<. four nlaces. n ^ r. «ca 16. From 3.408 suhstract 1.2591, and you find the excess 2.2089. 17 What is the excess of 10.008576 above 5.789 ? Ans. 4.219576. 18. From 11.4 take 1.48, and there remains to six places 9.959596. 19. What is the excess of 7.8 above 1.3754658? Ans. 6.4245341. 20. What is the difference between 9.46574, and '±•18345? Ans. 5.282iy. 21 Exm'c« decimally, the difference between §+|+Ha+7S ,;,'*^" Ans. 2.34613+. 22 What is the difference, according to the decimal notation, between I and { •; true to six places of decimals ? Ans. .636363. MULTIPLICATION OP DECIILYLS. 87 23. What is the difference between ^ and :j expressed decimally true to six decimal places ? Ans. .071428. 24. "What is the diflfcrcncc between the vulgar fractions corres- ponding to Ad and .5 ? Ans. 0. 25. Find the value of .78G425-}-.97u324-f .17G009-i-.32+ .G2519375— 3.2S29517o-[-.4. Ans. 0. 20. What is the difference between 138.G012, and 128.8512 ? Ans. 9.75 27. What is the excess of 31.0322 above 5.G74-|-1.83+.3125-|- l8.C2+4.3+.395~.5. Ans. 1.0007. 28. What is the excess, expressed dcciniallj, of 5.83 above 4 J?-. Ans. 1.G582. 29. What is the difibvonce between 8.375 and 7^* true to six decimal places ? Ans. .94G428. 30. What is the value of GOl.050725-441.001— .00025— 3.818475—150.1-;-. 125. Ans. .25. f i V.-MULTIPLICATION OF DECIMALS. If we multiply a decimal by a whole number, the ^,.„jcss is pre- cisely the same as if the multiplicand were a whole number, but care must be taken to keep the decimal point in the same relative position. Thus, in the annexed example, as there are three dcciuial i)luccs iu the multiplicand, we make three also in the product. If we have to multiply a whole, number by a decimal, wo must mark off a deci- mal in the product for each decimal in the multiplier. — The reason of this M-ill be nianifest from the considera- tion that if we multiply 8 units by .G, or ,",,, wo'^308 3. Multiply 1.05 by 1.05, and the proclnet by 1.05. . p. , ,, . , Ans. 1.157025. 4. I'lnd the continual product of .2, .2, .2, .2, .2 .2. r ir , . , «. Aas. .000004. 5. Multiply .0021 by 21. ^„, ^^^^ C. Multiply 3.18 by ^11,7. ^,3. 132 COO." 7. Multiply .08 by .030. j,,,^ .00288. 8.MuUipy.3by.7. ^„,.,,1^ 9. Multiply .31 by .32 ;^,,^ Ojjj^^^ 10. Fmd the continual product of 1.2, 3.25. 2.1''5 ,, „ , . . Ans. 8.2875. 11. M„I„py 11.4 ty 1,14. ^^^ ^2^g^_ 1-. bind the continual product of .1, .1, .1, .1^ .1 .i. 1., ,T , . , , Ans. .000001. 13. Multiply 1240 by .008. ^ns. 9.92 14. Find the continual product of .101, .011, .11, l.l and 11."' ir Tif u- , r. ■ ■ Ans. .001478741. lo. Multiply 7.43 by .802 to six places of decimals. in Til , . , Ans. .64083:), 10. Multiply 3.18 by 11.7, and the product by 1000. 17 Ar u- , ,.. , Ans. 132606. 17. Multiply .144 by .144. ■ Ans. .020736. 15. What is the continual product of 13.825, 5.128 and .001 ? * in wt . • , ^"^- -^708946. iJ. What 13 the continual i)roduct of 4.2, 7.8 and .01 ? -u. What is the continual product of .0001, 0.27 and 15.9 ? Ans. .U099G93. Contracted iMETnoD.-In many instances ^\horc long lines of figures are to be multiplied together, the operation may bo very much siiortcned, and yet sufficient accuracy attained. We may instance wliat the student will meet with hereafter, calculations in compound interest and annuitir., involving sometimes most tedious operations. l>y the following method the results in such cases may be obtained v\ith great ease, and correct to a very minute fraction. 1»; we aro omputins doUars and cents, and extend our calculation to foiir 90 AMTHMETIC. pb.ces of (Icciwala, wo pre treating of the onc-lmndrcdtli part of a cent, or the ten-tliousandth part of a dollar, a quantity so minute as to become relatively valueless. Hence we conclude that ihrco or four decimal places arc sufficient for all ordinary purjrascs. Thcro arc cases, indeed, in which it is necessary to carry out the decimals farther, as, for instance, in the case of Logarithms to be considered hereafter. The principle of the contracted method will be best ex- plained by comparing the two subjoined operations on the same quantities. Let it be required to find the product of C.35G42 and 47.G'153, true to four places of decimals : EXTENDED OPERATION. COXTRACTED OPERATION. G.35G42 G.35G42 47.0453 3540.74 19;0G926 2542508 317 8210 444943 2542; 5G8 38138 38138 52 2542 444949 i 317 2542508 10 'H carried, 302.8535 37826 302.8535 RULE FOR THE CONTRACTED METHOD. Place the iinits' figure of the tchole mimher xindcr the last required decimal place of the inuI(ij)Ucancl, and the other integral figures to the right of that in an inverted order, and the decimal figures, cdso in an inverted order, to the left of the integral unit; multiply hy each figure of the inverted multiplier, beginning loith the figure of the multijflicaiid immediately above it, omitting all figures to the Tight, hut allowing for what would have been carried if the decimal had been carried out one place farther — place the first figure of each partial product in the same vertical column, and the others in, verti- cal columns to the left ; the sum of these columns will be the required product. Thus, in the above example, we arc required to find tbo product correct to four decimal places, therefore we set the units' figure, 7, under the fourth decimal figure, and the tens' figure, 4, to the right, and the decimal figures, 6453, to the left iu reversed order ; then wo pi col as I i s| wiJI liuj tinj labl MULTirLICATlON OF DECIMALS. 91 the imal ' each vcrti- luircd ■oduct riglit, icn v.«5 multiply tho whole lino by 4, ami tlicn we multiply by 7, omitting ilic 2 wliicli .stands to the riyht, but iillowio'^ 1 for what would havo been can led, that is, wc say 7 tinu's 4 is 28, and 1 i.-i 20, and wo Wi'itc the nine under the S, the first iigurc of the first partial product. By comparin;; the contracted method with the lii^urcs of the extend' cd Ibrm. which are to tlie Icl't of tho vertical line drawn after the fourth decimal figures, it will be seen that the figures of each column arc the .same but placed in reversed order, which makes no diCercnce in the .sum, as 5-1-0- -o-|-5---8. This is the f^anie principle as the contracted method of multipl)-ing by 17, 71, 853i> ,%^ IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 2.5 112 S Itf 12.0 1.4 1= 1.6 V 0^ 'c-1 "^ '.'> ►c>^ c% 4* Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 V ^v ^^ :\ \ ^^^-^v^o ;\ ^-^".^ '^U 'iy\. v-^ &x y"^ 92 ASITIQIETIC. ADDITIONAL EXERCISER. 21. Multiply .26736 by .28768 to four decimal piaocs Aas. .0769. 22. Multiply 7.285714 by 3G.74405 to five decimal places. Ans. 267.70665. 23. Multiply 2.656419 by 1.723 to six decimal places. Ans. 4.578932. 24. What decimal fraction, true to six places, will express the product of -i»j multiplied by /^ ? Ans. .113445. 25. What decimal fraction is equivalent to f ^ Xgf ? ' Ans. .46748. 26. What is the second power of .841 ? Ans. .707281. 27. What is the product of 1.65 by 1.48, true to five places ? Ans. 2.45975. 28. Express decimally 2-i%Xl' Ans. 2.393162. 29. What is the product of 73.637i by 8.143? Ans. 599.6272077. 30. .681472 X.01286, true to five places, will give .00876. In the last exercise it must l)e observed that since there is no whole number, and five decimal places are required, we must place a cipher under the fifth decimal figure, and write .01286 in reversed order. That the result is a sufficiently close approximation will be evident from the consideration that the last figure 6 is only six one*hundred>thousandtha of the unit, and consequently the next figure would bo oiCy oue-mil- lionth nart of the unit. .681472 68210.0 681 136 66 1 .00876 VI.-DIVISION OF DECIMALS. We have already seen (1) that we cannot perform any operation except the numbers concerned kte of the Same denomination, or one of them be abstract ; (2) that when a denominate number is used either as a multiplier or a divisor, it ceases to be denominate, and becomes abstract, and (3) that the rules for addition, subtraction, multiplication and division of integers apply equally to decimals, the only additional requirement being the placing and moving of the decimal point. DIVISION OF DECDIALS. 93 Deration or one lis used Ite, and fraction, lals, the of the Suppose tncu we are required to divide 1.2321 by 11.1, we must y (1) bring both quantities to the same denomination. Now tie dividend is carried down to ten-thousands for 1.2321=l-[— ,"jfjfg\j, and therefore we express 11.1 in the corresponding form, ten-thou- sandths or ll + ,'oooo> °^ 11.1000, so that wo change the form, but not the valu« of 11.1, the divisor. Again, by (2) the .1, whicli originally expressed a tenth of some unit, and therefore was in realitj denominate, now becomes abstract as one of the figures of the given factor of 1.2321, by means of which we are to find the other factor. Hence by (3) we can now divide 1.2321 by 11.1000, as if both were whole numbers, and this is the reason for omitting the decimal point when we have made the number of decimal places equal. Beginnera generally feel a difficulty in conceiving how a fraction divided by a fraction can give a whole number. The difficulty may be easily re- moved by noticing that J is contained twice in J for -|=J, e. g., a half dollar contains, or is equivalent to, two quarter dollars. Thus the fraction ^ divided by the fraction ^, gives the whole number 2. So, also, ^ is contained 4 times in ^, and therefore ^-i-^=4, a whole number. Hence, when we have reduced the divisor and dividend to the same denomination, tre may omit the decimal point, as we have only to find haw often the one is contained in the other. Hence the AU LE. t ' 7)r the nnmher of decimal places in the divisor and dividend he not equal, make them equal hy supplying ciphers or repetends, and then divide as in whole numbers, and the quotient so far will he a whole number, hut if there is a remainder, annex ciphers or repetends, and the part of the quotient thus obtained will he a decimal. The decimal places may be supplied as the work proceeds, as it is easy to see how many ciphers or repetends must be supplied ; for we have seen in multiplication that the number of decimal places in any product must be equal to all the decimal places in the factors, and, since a dividend must always be viewed as a produce, it follows that the difference between the number of decimal places in dividend and divisor will indicate how many ciphers or repetends must be supplied. EXERCISES. 1. Divide 47.58 by 26.175 to six decimal places. S. Divide 70.8946 by 13.825 to three places. Ans. 1.817765. Ana. 6.128. , ill!: '■ \ 94 ABITHIIIETIO. ! 3. Divide 468.7 by 3.365 to eix places of decimals. Ans. 139.309889. 4. Express decimally l-f-^f^. Ans. 233.3. 5. Express in the decimal form ^ of ^-^-f of | true to six places of decimals. Ans. 1.054687. 6. Divide the whole number 9 by the fraction .008. Ans. 1125. 7. What is the quotient of 5.09 by 6.2 ? Ans. .81 nearly. 8. Divide .54439 by 7777. Ans. .00007. 9. What decimal is obtained by dividing 1 by 10.473654 ? Ans. .09547766. 10. What is the difference between f-i-^ <^d I~^ti ^^ ^^^ ^^c^' mal form ? OONTRAOTED METHODi The work may often be much abbreviated in the manner exhibi- ted by the following example : (1.611 .14736).23748 14736 9012 8841 170 147 23 14 14736)23748(L611 • • • • 14736 40 36 040 736 9012 8842 170 147 8 304 23 15 "i Here it is required to divide .23748 by .14736. Since both divisor and dividend contain the same number of decimal planes, no alteration is needed, and so we 64637:43682(.7995 • • • • 38246 15354 12970 2384 2162 5436 4917 519 491 28 27 216 6 Divide 73.64 by .432. and .43682 by .54637 to 4 decimal ] aces each. To ehow that there will be three integral places in the 96 ARITHMETIO. 1 I quotient of Ex. 1, we must consider that there arc two places of whole numbers in the dividend and none in tlio divisor, and, therefore, j£ we divide 73 and 6, the first decimal place of the dividend by ,4, the first figure of the divisor, we get three integral places. Hence, since we are to have four decimal places, we shall have seven figures in all. This contraction is extremely useful when there axe many decimal places. 3. Find the quotient of 8.6134—7.3524 to four decimal places. Ans. 1.1715. 4. Divide .61 by 13.543516 to five decimal places. Ans. .04549. 5. Divide .53 by 77.482 to five decimal places. Ajm. .00756. 6. Divide .812.54567 by 7.34 to three decimal places. Ans. 110.649. 7. Divide 1 by 10.473654 to six decimal places. Ans. .0954.7. 8. Divide 7.126491 by .531 to six decimal places. Ans. 13.4208ST. 9. Divide 1.77975 by the whole number 25425. Ans. .00007. 10. Divide to ei^ht places .879454 by .897. Ans. .98043924. \ai.--DENOMINATE DECIMALS. To express one denominate number as a fraction of anotlier oftht same kindf reduce both to the lowest denomination contained t» eitJier, make the former the numerator and ilie latter the denomir nator of a common fraction, and reduce the fraction so fcyund to Q decimal in the manner already ^pointed out. EXAMPLES. To express 16 cents as a fraction of a dollar : Here the lowest denomination mentioned is cents, and we reduce a dollar to cents and write j'(fjj=25, and, dividing 4 by 25, wc get .16. To express lis. 4Jd. as a decimal of XI, we reduce both to half-pence, aod obtain |bB=iVd, which, reduced to a decimal, ia .56875. EXERCISES 1. Beduce Ss. 10|d. to the decimal of £1. Ans. £. .2937&. 2. Heduce lOJd. to the decimal of £1. Ans. £ .04375. 3. Reduce 15s. 9|d. to the decimal of £1. Ans. £ .790625. 4. Express 3 roods and 11 rods as a decimal of an acre. Ans. .81875. BEDUCTION OP DENOMINATIONS. 97 llowcst cento Ixprcsa \c, aod )4375. )0625. J1875. 6. Express 3 cwt., 1 qr., 7 lbs., as a decimal of a ton. Ans. . 166. 6 Ho.duce 37 rods to the decimal of a milo, Ans. .115625. 7. Kcduco 7 0Z3. 4 dwts., to the decimal of a pound. Ans. .0. 8. Reduce a pound troy to the decimal of a pound avoirdupois ; correct to six decimal places* ♦ Ans. .822857+- t). Reduce 5 hours, 48 minutes, 49.7 seconds, to the decimal of a day, taken as 24 hours. Ans. .2422419. 10. Express an ounce avoirdupois aa a decimal of a pound troy, Ans. .9114383. DEDUCTION TO DENOMINATIONS. To find the value of a fraction in the lower denominations, expressed as a decimal of any given denomination, multipljj in suc- cession hy the numbers which express the given and lower denomina- tions, and after each multiplication cut off from the right as many decimal figures as are contained in the given decimal, and the figures to the left of the d^'dmal point will give the required value. EXAMPLES. I, To find tlic value of .64379 of a pound (apothecary). We -12 multiply by 12, by 8, by 3 7.72548 8 5.80384 3 2.41152 20 and by 20, which gives 7 ozs., 5 drs., 2 scrs., and a little over 8 grs. Repetends must be reduced to comraou frac- tions, or found approximately. 8.23040 2. To find the value of ,7 : of a day, whicli is 18 hours, 39 min. and nearly 69|scos. .77777 24 V carry t. 311109 155555 18.66659 60 39.99540 GO 59.72400 *Tho Btandard pounds are meant here, viz. : troy, 57G() grains, "voltdupois 7(W0 gnUos. Taking tli RATIO AND PROPORTION. '" ' '• 17t^-ItAno is the relation which one quantity bears to anotbei nf the same kind with respect to magnitude, or the number of times that the less is contained in the greater. Thus, the ratio 7 to 21 is S, because 7 is contained 3 times in 21, or 21 is 3 times 7. The eame result is obtained if we divide 7 by 21, for we then find 3pp=J, which means that 7 is ^ of 21, and this expresses the very game relation as before ; for, to say that 7 is ^ of 21 is precisely the same as to say that 21 is 3 times 7. (See note under Inverse Pro- |K>rtion.) And, therefore, 3 is called the measure of the ratio. The kiumbers thus compared are called the terms of the ratio— the first Ibhe antecedent and the second the consequent, and the relation is written 7 : 21. The sign ( : ) originally indicated division^ That the magnitudes must be of the same kind will be obvious from the consideration that 7 bags of flour could have no ratio to 21 dollars, for multiplying 7 bags of flour by 3 would not make them 21 dollars, but 21 bags of flour, and multiplying 7 dollars by 8 would not make them 21 bags of flour, but 21 dollars. Hence, |iho less could not be increased to make the greater, except they are homogeneous, or of the same kind. Proportion is the equality of ratios. The ratio of 9 to 27 is 3, but we have seen that the ratio of 7 co 21 is ftlso 3, therefore the ratios of 7 to 21 and of 9 to 27 are the very the Pro- The first lion is bvious Ltio to make tirs by [ence, ley are the BiTIO AND PBOPOBTTON. 99 same, or 7~21=9-^'27, and these quantities arc, therefore, called proportionals. Tho sign ( : : ) was formerly used for equality, and is still retained for equality of ratios, and the sign (=) is used for tho actual equality of quantities, though occasionally used for equality of ratios. Ilcncc, the usual mode of writing the equality of two ratios is 7 : 21 : : 9 : 27. Such a statement is called a pro portion, or an analogy, and is read — 7 is to 21 as 9 to 27, t. e., 27 exceeds 9 as many times as 21 exceeds 7, and this is expressed by saying 27 is the same multiple of 9 that 21 in of 7, or that 9 is the same sub-multiple, measure, or aliquot part of 27 that 7 is of 21. The four quantities are called the terms of tho proportion ; the first and last are called tho extremes, and second and third the means ; also, the first and third arc called homologous, or of the same nam^ i, e., both are antecedents, and so tho second and fourth are homo- logous, for they are both consequents. The last term is called a fourth proportional to the other three, and we shall denote it by F. P. There are two simple ways of testing the correctness of an analogy. The first is to divide the second term by the first, and the fourth by the third, and if the quotients are equal, the analogy is correct. This is manifest from what has been already said. The second principle is, that, if the analogy be correct, the product of the extremes is equal to the product of the means. To prove this, let us resume the analogy, 7 : 21 : : 9 : 27. We have seen that 21-v-7=27~9, or 3=3. Now, if each be multiplied by 63, we have (by Ax. IL, Cor.,) 189=189. But 189 is the product of 27 by 7, the extremes, and also of 21 by 9, the means — these products then are always equal. From this simple principle wo readily deduce a rule for find- ing a fourth proportional to three given quantities. Let the quan- tities be 48, 96, and 132, written thus : 48 : 96 : : 132 : , the required quantity. Now, 132X96=12672, the product of the means are therefore equal to the product of the extremes. We have, therefore, a product, 12672, and one of its factors, 48 , hence, dividing this product by the given factor, we find the other factor to be 264, which is therefore the fourth proportional, or fourth term of tho prbportion, and wo can now write the whole analogy, thus:^ — 40 : 96 : : 132 ; 264, To prove tho correctness of the operation, multiply 264 by 48^ and 12672 is obtained, the same as before. Henoe, f 100 ABTTflMETIO. THE RULE. TXvide the product of the second and thud iermsbi/ ihejintf mvi the quotient will he the required fourth term. To show the order ia which the three given quantities are to be arranged, let it bo required to find how much 730 yards of linen will cost at the rate of $30 for 50 yards It is plain that the answer, or fourth term, must be dollars, ibr it is a price that is required, and in order that the third term may have a ratio to the fourth, the $30 must bo the third term. Again, since 730 yds. will cost more than 50 yds., the fourth term will be greater than the third, and therefore the second must be greater than the first, and therefore the statement is yds. Tila. t 50 : 730 : : 30 : 4th proportional, and by the rule i^^|pJ}= jnoo 6TJ =438, the fourth term, and we can now write the whole analogy, 60 yds: 730yds::e30: $438. This may be called the ascending scale, for the second is greater than the first, and the fourth greater than the third. If the ques- tion had been to find what 50 yards of linen will cost at the rate of S438 for 730 yards, wo still find that the answer will be dollars, and that therefore, as before, dollars must bo in the third place, but we see that the answer will now be less than 438, as 50 yards, of which the price is required, will cost much less than 730 yards, of which the price is given, and that therefore the second term must be less than the first. Hence the statement is 730 yds : 50 yds : : $438 : F. P., and by the rule —^■]^^—=-^0, the fourth proportional We now have the full analogy 730 yds : 50 yds : : $438 : $30. As the second is less than the first, and the fourth less than the third, this may be called the descending scale. If the first should turn out to be equal to the second, and therefore the third equal to the fourth, we should isay that the quantities were to each other in the ratio ox equdUty. nULE FOR THE ORDER OF THE TERMS. If the question invplies that the consequent of the second ratio must he greater tlian the antecedent, maJee the greater term of the first ratio the consequent, and the less tlie antecedent, and vice versa. The questions hitherto considered belong to what is called Direct Proportion, to distinguish it from another kind called Inverse PrO' portion ; because, in the former, the gieater the number given, tho less will be the corresponding number required, and vice versjij^ EATIO AND PEOPOETION. 101 rath \of the [versa* {Direct en, tliC' whereas, m tno lattcr, the greater the number given, the less will bo tho number required, and vice versa. To illustrate this, lot it bo required to find how long a stack of hay will iced 12 horses, if it will feed 9 horsea for 20 weeks. Here the answer required is time, and therCiOro 20 weeks will bo the antecedent of the seeond ratio ; but the greater the number of horccs, the tihorter time will tho hay last, and therefore tho fourth term will be less than tho third, and there- i'oro the statement will not be 9:12, but the reverse, 12 : 9 ; and ht'ucc the name Inverse, because the term 9, for which the time (20 weeks,) u given, and which therefore wo should expect to be in tho first place, has to bo put in the second ; and tho term 12, for which tho time is required, and which therefore wo should expect to bo in the second place, has to be put in the first, and thus the whole ana- logy is 12: 9:: 20: 15* The principal changes that may bo made in the order of the terms, will be more readily and clearly understood by the subjoined Bchemo, than by any explanation in words ; Original Analogy : 8 : 6: : 12 : 9 for 8X9=72=0X12. / >crnately : 8 : 12 : : 6 : 9 for 8 X 9=72=0X12. By Inversion : : 8 : : 9 : 12 for 0x12=72=8x9. By Composition : 8+G : : : 12-1-9 : 9 or 14 : : : 21 : 9 for 14X9=120=0X21. By Division : 8—6 : 6 : : 12—9 r 9 or 2 : 6 : : 3 : 9 for 2X9= 18=0X3. By Conversion : 8 : 8—0 : : 12 : 12—9 or 8 : 2 : : 12 : 3 for 8X3=24=2X12, Simple transposition is often of the greatest use. Lot us take an e>)sy practical example, in calcula- ting wh:it power will balanc3 a given weight, when the arms of the lever are known, lot P bo tho power, W the weight, A tho arm of power, and B p ^ the arm of weight. The rule is, that i W the power and weight are inversely as the arms. This solves all the four possible cases by transposition. • Inverse ratio is sometimes spoken of, but in reality there is no suol thing. It is true that Inveree Proportion requires the terms of one of th« ratios to be inverted, bat that is a matter of analogy, not of ratio, for we havt seen already that 7-»-21 expreesea the verv same relation as 21-(-7.--(See in. B fl 102 ABITHMTTnO. Jl A : B : : W : P, gives the power when the others are known, B : A : : P : W gives the weight when the others are known, W : P : : A : B gives the arm of weight when the others are known, P : W : : B : A gives the arm of power when the others are known. The work may often be contracted in the following manner : — Resuming our example 48 : 9G: : 132 : fourth proportional, we see that 96 is double of 48, and therefore the ratio of 48 to 9G is tho Bamo as that of any two numbers, the second of which is double tho first, and 48 : 9G is the same as 1 : 2, and we reduce tho analogy to the simple form of I : 2 : : 132 : 4th prop., and we have J-^jX^— 264, tho term required, as before. In tho example 50 : 730 : : 30 : 4th term, wo have -7-i|].Xi»J>=r.7jiXM=^iL3_xroi5^73><(j_438. This is equivalent to dividing the first and second by 10, and tho first and third by 5. Hence we may divide the first and second, or first and third by any number that will measure both. Tho samo principle will also be illustrated by the consideration that the second and third are multipliers, and the first a divisor ; and if wo first multiply, and then divide by the same quantity, the one operation will manifestly neutralize the other. Thus : 48 : 90 : : 132 : F. P. may be written 1X48 : 2X48: : 132 : F. P. ; where it is plain that since by first multiplying 132 by 48, and then dividing by the same, the one operation would neutralize tho other, both may bo omitted. In proportion, when the means are equal, such as 4 : 12 : : 12 : 36, it is usual to write the analogy thus — 4 : 12 : 36, and 12 is called a mean proportional between 4 and 36. To prove the correctness of this statement, we multiply 36 by 4 and 12 by itself, and as both give 144, the analogy is correct. Now, as 144 is the square or Becond power of 12, so 12 is called the second root, or square root of 144, or that which produced it, or the root from which it grew ; hence, to find a mean proportbnal between two given quantities, we have tho following BULE Multiply them together , and take the square root of theprodua Thus, in the above example, 4X36=144, the square root of which is 12. Again, to find a mean proportional between 9 and 49, we mul- troductory remarks.) The term Reciprocal Eatio is liable to tho samo objec- tion, for though 3 and ^ are reciprocals, yet they express the same relation. When the expression Inverse Eatio is legitimately used, it docs not refer to u single ratio, hut meaos (bat tv30 ratios are so related that one of them must bi- inverted. ia$l RATIO AND PROPORTION. 103 oduci mul- objcc- ulation. Ifer to a a\ist 111! liply 49 by 9, which is 441, tlie square root of which is 21, which is a mean proportional between 9 and 49, i. c, 9 : 21 : 49, or, writ- ton at fiUi length, 9 : 21 : : 21 : 49. Proof: 49x9=441 and 2lX21='441. As the learner is not supposed, at this stage, to know the method of finding the roots of quantities beyond tlio limits of the multiplication table, wo append a table of squares and roots at the end of the book. When each quantity in a series is a mean proportional between two adjacent quantities, the quantities are said to be continued, or continual proportionals. Thus : 2 : 4 : 8 : 16 : 32 : 64 : 128, and 3 : 9 : 27 : 81 : 243, are series in which each is a mean pro- portional between two adjacent ones. Let us take IG and the trwo adjacent ones, 8 and 32 — the analogy is 8 : 16 : : 16 : 32. Proof: 8X32=256, and 16x16=256. So also, 27 and the adjacent terms, 9 and 81. The analogy is 9 : 27 : : 27 : 81, and the proof, 9X81= 729, and 27X27=729. This subject will be treated of at length in a subsequent part of the work, but this expbnation has been introduced here to fill up die outline and let tlio learner understand the nature of continued proportiouals. EXESOISES. 1 If 6 barrels of flour cost $32, what will 75 barrels cost ? Ans. S400. 2. If 18 yards of cloth cost $21, what must be paid for 12 yards ? Ans. $14. 3. How much must be paii for 15 tons of coal, if 2 tons can be purchased for $1 o ? Ans. $112.50. 4. If you can walk 84 miles in 28 hours, how many minutes will you require to walk 1 mile ? Ans. 20^ 5. What will 14 horses cost, if 3 of the average value con bo bought for $270? Ana. $1260. 6. What must be paid for a certain piece of cloth, if g of it cost $9. Ans. $13.50. 7. If 5 men are required to build a wall in 5 days, how many men will do the same in 2i days ? Ans. 10. 8. If 16 sheep are f of a flock, bow many are there in the same ? Ans. 24. 9. What must be paid for 4^ oorda of wood, if the cost of 3 cords is $10? Ans. $15. Hf ' '" ■B ; : Hi ^ iH i 1 10^ ABITHMEnO. 10. What is the height ot a tree which casts a shadow of 125 Ibet, if a stake 6 feet high produces a shadow of 8 feet ? Aus. 93|. 11. How long will it take a train to run from Syracuse to Os- wego (a distance of 40 miles), at the rate of 5 miles in ISy'j minutes ? 12. If 15 men can build a bridge in 10 days, how many men will bo required to erect three of the same dimensions in ^ the time ? Ans. 90. 13. If a man receive $4.50 for 3 days' work, how many days ought lie to remain in his place for 025 ? 14. How much may a person spend in 94 days, if he wishes to tiave $73.50 out of a salary of $500 per annum ? 15. If 3 cwt., 3 qrs^ 14 lbs. of sugar cost $30.50, what will 2 qrs. , 2 lbs. cost ? Ans. $4,879+. IG. 5 men are employed to do a piece of woik in 5 days, but after working 4 days tlicy find it impossible to complete the job in Ici's thtin 3 days mojc, how many additional men must bo employed to do the work in the time agreed upon at first ? Ans. 10. 17. A watch is 10 minutes too fast at 12 o'clock (noon) on Mon- day, and it gains 3 minutes 10 seconds a day, what will be the timo by the watch at a quarter past 10 o'clock, A. M., on the following Saturday ? Ans. 10 h, 40 m. 3G,7g s. 18. A bankrupt owes $972, and his property, amounting to $607.50, is diatributed among his creditors ; what does one receive wliose deiii.nd is $11. 33] ? Ans, $7,083+. 19. What is the value of .15 of a hhd. of lime, at $2.39 per Uid.V Ans. $.3585. 20. A garrison of 1200 men has provisions for -^ of a year, at the Tate of ^^ of a pound per day ; how long will the provisions last lit the same allowance if the garrison be reinforced by 400 men ? Ans. G^ months. 21. If a piece of land 40 rods in length and 4 in breadth mako uii acre, how lonq must it be when it is 5 rods 5J feet wide ? Ans. 30 rods. 22. A borrowed of B $743, for 90 days, and afterwards would returii the favor by lending B $1341 ; for how long should he lend it? 23. If u man can walk 300 miles in G successive days, how many milca hoa ho to walk at the end of 5 duyt> ? Ans. 60. MTIO AND PBOPORTIOIT. 105 to 5+. pel )85. at I last Iths. lako rods. )uld llcnd lliow 50. 24. If 495 gallons of wine cost $394; how much will $72 pay for? Ans. 90 gal. 25. If 112 head of cattle consume a certain quantity of hay in 9 days ; how long will the same quantity last 84 head? Ans. 12 days. 2G. If 171 mcu can build a house in 1G8 days; in what time will 108 men build a isimilar house ? Ans. 2G6 days. 27. It has been proved that the diameter of every circle is to the circumfcrcuco as 113: 355; what then is the circumference of the moon's orbit, the diameter being, in round numbers, 480,000 miles ? Ans. l,507,064/j«3 ui. 28. A rouud table is 12 ft. in circumference ; what is its diameter ? Ans. 3 ft. 91 n in. 29. A was sent with a wuriunt ; after he had ridden 65 miles, B was sent after him to stop the execution, and for every IG miles that A rode, B rode 21 , How far had each ridden when B overtook A ? Ans. 273 miles. 30. Find a fourth proportional to 9, 19 and 99. Ans. 209. 31. A detective chased a culprit for 200 miles, travelling at the rate of 8 miles an hour, but the culprit had a start of 75 miles ; at what rate did the latter travel ? Ans. 5 miles an hour. 32. IIcw much rum may bo bought for $119.50, if 111 gallons eoi^t $89,625 ? Ans. 148 gallons. 33. If 110 3ards of cloth cose $18 ; what will $63 pay for ? Ans. 385 yards. 34. If a man ^7alk from Rochester to Auburn, a distance rf (say) 73 nnlc3 in 27 hours, 54 minutes ; in what time will he walk at the same rate from Syracuse to Albany, supposing the distance to be 152 miles ? *J5. A butcher used a false weight ll.J oz., instead of 16 oz. 1 ,.• ft pound, of how many lbs. did ho defraud a customer who bought 112 just lbs. fiuui him ? Ans. 9|^ lb.". 36. If 123 yards of muslin cost $205 ; how much will 51 yaiui cost? Ans. $85. 37. In a copy of Milton's Paradise Lost, containing 304 p-^-^q, the combat of Michael and Satan commences at the 139th page; at what pa<;e may it be expected to commence in a copy containing 328 pages? Ana. The fourth proportional is 149|^ ; and hcuco the passage will commence at the foot of page 150 38. Suppose n man, ))y travelling 10 hours a day, performs » II 106 AllITHMETIO. journey in four weeks without desecrating the Sabbath ; now maTiy weeks would it take him to perform the same journey, provided he travels only 8 hours per day, and pays no regard to the Sabbath ? Ans. 4 weeks, 2 days. 39. A cubic foot of pure fresh water weighs 1000 oz., avoirdu- pois; find the weight of a vessel of water containing 217^ cubic in. Ans. 7 lbs., 13 III oz. 40. Suppose a certain pasture, in which are 20 cows, is sufficient to keep them 6 weeks ; how many must be turned out, that the same pasture may keep the rest G months ? Ans. 15. 41. A wedge of gold weighing 14 lbs., 3 oz., 8 dwt., is valued at £514 4s. ; what is the value of an ounce ? Ans. £3. 42. A mason was engaged in building a wall, when another cnmc up and asked him how many feet he had laid ; he replied, that the part ho had finished bore the same proportion to one league whicli j^,y does to 87 ; how many feet had he laid ? 43. A farmer, by hi? will, divides his farm, consisting of 9T acres, 3 roods, 5 rods, between his two sons so that the share of tlie younger shall be f the share of the elder; required the shares. Here the ratio of the shares is 4 : 3, and we have shown that if four magnitudes are proportionals, the fiist term increased by the second is to the second as the third increased by the fourth is to the fourth. Now, 97 acres, 3 roods, 5 rods, being the sum of the shares, we must take the sum of 4 and 3 for first term, and cither 4 or 2 for the second, and therefore 7 : 4 : : 97 acres, 3 roods, 5 rods : F.P., t. e., the sum of the numbers denoting the ratio of the shares is to one of them as the sum of the shares is to one of them. This Rives for the elder brother's share, 55 acres, 3 roods, 20 rods, and the youiigcr's share is found either by repeating the operation, or by subtracting the share thus found from the whole, giving 41 aoreSj 3 roods, 2D rods. -44. A legacy of $398 is to be divided among three oiphans, in parts which shall be as the numbers 5, 7, 11, the eldest receiving tli<: largest share ; required the parts ? 23 : 5 : : 398 : 8612, the share of the youngest. 23 : 7 : : 398 : ISl^'g, the share of the second 23 : 11 : : 398 : 190^^3, the share of the eldest. 45. Three sureties on $5000 are to be given by A, B and 0, so that B's share may bo one-half greater than A's, and C's onft-balf greater than B's ; required the amount of the securitv of each ? IS COMPOtIND PBOPOBTION. 167 Ans. A'3 share, $1052.63/g ; B's, $1578.941^ ; C's, $2368.42/^. 46. Suppose that A starts from Washington and walks 4 miles an hour, and B at the same time starts from T5oston, to meet him, at the rate of 3 miles an hour, how far from Washington will they meet, the whole distance being 432 miles ? 47. A certain number of dollars is to be divided bfitween two persons, the less share being f of the greater, and the diLerenco of the shares $800 , what are the shares, and what is the whole sum to bo divided ? Ans. Less share, $1600 ; greater, $2400 ; total, $4000. 48. A certain number of acres of land are to be divided into two parts, such that the one shall be | of the other ; required the parts and the whole, the difference of the parts being 716 acres ? Ans. the less part 537 acres ; the greater, 1253 acres ; the whole, 1790. 49. A mixture is made of copper and tin, the tin being J of the copper, the difference of the parts being 75 j required the parts and the whole mixture ? Ans. tin, 37^ ; copper, 112J ; the whole, 150. 50. Pure water consists of two gasses, oxygen and hydrogen ; the liydrogen is ah out y\ of the oxygen ; how many ounces of water will there be whcu there are 764||oz. of oxygen more than of hydrogen ? Ans. 1000 oz. COMPOUND PROPORTION. Is, w 111'! 80 lalf Proportion is called simple when the question involves only one condition, and compound when the question involves more conditions than one. As each condition implies a ratio, simple proportion is expressed, when the required term is found, by two ratios, and com- pound, by more than two. Thus, if the question be, IIow many men would be required to reap 65 acres in a given time, if 9G men, working equally, can reap 40 acres in the same time? Hero there is but one condition, viz., that 96 men can reap 40 acres in the given time, which implies but one ratio, and when the question has been stated 40 : 65 : : 06 : F.P., and the required term is found to be 156, and the proportion 40 : 65 : : 96 : 156, we have the propor- tion, expressed by two ratios. But, suppose the question were. If a man walking 12 hours a day, can accomplish a journey of 250 miles in 9 days, how many days would he require walking at the 108 AfilTHMETIO. same rate, 10 hours each day, to travel 400 miles ? Here there are two conditions, viz. : first, that, in the one case, ho travels 12 hours a day, and in the other 10 hours ; and, secondly, that the distances are 250 and 400 miles. The statement, as we shall presently show, would be 10: 12 )n-.»^ Here each condition im- v- 9 : ITrv'^, 250 : 400 f -*' plies one ratio, 10 : 12 and 250 : 400, and when the required term, which is IT^f^, is found, there are four ratios, viz., the two aheady noted, and 9 : 17/^, gives two more, one m relation tQ 10 ; 12, and one in relation to 250 : 400. This will be evident, when we have shown the method of {Statement and operation. EXPLANATORY STATEMENT AND OPERATION. 11 1 33; 3 12 12 F.P. 36 PRACTICAL STATEMENT AND OPERATION. 11:33 18: 5 ]■■■ 12 : F. P. 18 1 5 :3G : 2 F.P. 10. 1 3 ]■■■ 2 : F. P. 1 : 5 } • *" •^"• Let the question be, How many men would be required to reap . o3 acres in 18 days, if 12 men, working equally, can reap 11 acres in 5 days ? Wc first proceed, as on the left margin, as if thcrt .v^ero only one condition in the question ; or, in other words, as if the number of days were the same in both cases, and the question were — If 12 men can reap 11 acres in a given time, how many men will be Re- quired to reap 33 acres in the same time. This, then, is a question in simple proportion, and by that rule we have the statement — 11 : 33 : : 12 : F. P., which, by contraction, becomes 1 : 3 : : 12 : F. P. ; and thus, we find F. P. to be 30, the number of men required, if the time were the same in both cases. The question is now resolved into this : How inany men will be required to reap, in 18 days, the same quantity of crop that 3G men can reap in 5 days? This is obviously a case of inverse proportion, for the longer the time allowed the less will be the number of men required, and hence the statement, 18 : 5 : 36 : F. P., which, by contraction, becomes 1 : 5: : 2 : F. P., which gives 10 for the number of men. The work may be shortened by making the two statements at once, as on the right margin. We first notice that the last term is to represent a a ai tl ec ta I sa sei N( on is ( CCSI thai COMPOUND PROPORTION. 109 :)n r w L8 I? lO 38 3S tie hi la certain number of men, and, therefore, we place 12 in the third place ; next, wo Bce that, other things being equal, it will take more men to reap 33 than to reap 1 1 acres, and that, therefore, as far as that is concerned, the fourth term will be greater than the third, and so we put 11 in the first place, and 33 in the second. Again wo sec that, other things being equal, a less number of men will be required when 18 days are allowed for doing the work, than when it is re- quired to be done in 5 days, and that therefore the fourth term, as far as that is concerned, will be less than the third, and therefore wc write 18 : 5 below the other ratio as on the margin. Then by con- 1 • 3 ) traction wo get ^ ! k [■ : : 2 : F. P. Now, as 3 in the first term is to be a multiplier, and 3 in the second a divisor, we may omit these also, and we obtain j^ '. ^ |- : : 2 : 10, the answer as before. The full uncontracted operation would be to multiply 18 by 11, which gives 198, then to multiply 33 1 y 5, which gives 1G5, then multiply 165. the product of the two second terms, by 12, and divide the result, 1980, by 198, the product of the two first terms, which gives 10 as before. Because in the analogy 198 : 165 : : 12 : 10, the first two terms are products, this kind of proportion has been called compound^ and the ratio of 19 to 165 is called a compound ratio. Wc can show the strict and original meaning of the term compound ratio more easily by an example, than by any explanation in words. Lot us take any scries of numbers, whole, fractional or mixed, say 5, ^, I, 19, 12, 1, 17, 11, \i, 25, then the ratio of the first to the last is said to be compounded of the ratio of the first to the second, the second to the third, tlie third to the fourth, &c., &c,, &c., to the end- Now the ratio of 5 to 25 is -['-=-^, and the several ratios are in this 11X18:33X5 198 165:: 12 165X12= 198 F. P. =10 1 15 2 5 order,|x|X-|.Xi5XT3X-V-Xl4XifX 14 which leaving finally ''^£-=b as before. If we took them in reverse order, viz., aV^^S) ^** is obvious that all therein could be cancelled, as each would in suc- cession be a multiplier and a divisor. Wo would also remark ihat compound proportion is nothing else than a number of questionu in simple proportion solved by ono opera- iiU AmTHMETIO. li' tion. This will be wident from our second example by comparing the tvpo operations on the opposite margins. Again, wo remarked that every condition in^plics a ratio, and that therefore the third and fourth terms of our firs'; exampb really involve two ratios, one in relation to each of the preceding. Hence univorsally the number of ratios, expressed and implied, must always Ic nn(] which 40 women would weed in 6 days, if 7 men can do as much as 9 women ? Ans. Rjfg days. 5. Suppose that 50 men can dig in 27 days, working 5 hours a day, 18 cellars which are each 48 feet long, 28 feet wide, and 15 feet deep; how many days will 50 men require, working 3 hours each day, to dig 24 cellars which are each 36 feet long, 21 feet wide, and 20 feet deep ? Ans. 45 days. 6. If 15 bars of iron, each 6 ft. 6 in. long, 4 in. broad, and 3 in thick weigh 20 cwt., 3 qrs., (28 lbs.) 16 lbs. ; how much will 6 bars 4 ft. long, 3 in. broad, and 2 in. thick, weigh ? Ans. 2 cwt., 2 qrs., 8 lbs. 7. If 112 men can seed 460 acres, 3 roods, 8 rods, in 6 days : how many men will be required to seed 72 acres in 5 days ? Ans. 21. 8. If the freight by railway of 3 cwt. for 65 miles be $11.25 ; how far should 35/;! cwt. be carried for $18.75 ? 9. If a family of 9 persons can live comfortably in Philadelphia for $2500 a year ; what will it cost a family of 8 to live in Chicago, all in the same style, for seven months, prices supposed to be | of what they would be in Philadelphia ? 10. If 126 lbs. of tea cost $173.25 j what will 68 lbs. of a differ- 3nt quality cost, 9 lbs. of the former being equal in value to 10 lbs. of the latter ? 11. If 120 yards of carpeting, 5 quarters wide, cost $60; what will be the price of 36 yards of the same quality, but 7 quarters wide ? Ans. $25.20. 12. If 48 men, in 5 days of 12J hours each, can dig a canal 139f yards long, 4| yards wide, and 2| yaids deep ; how many hours per day must 90 men work for 42 days to dig 491 j'g yards long, 4| yards wide, and 3 J yards deep ? Ans. 4. 13. A, standing on the bank of a river, discharges a cannon, and B, on the opposite bank, counts six pulsations at his wrist between the flaab and the report ; now, if sound travels 1142 feet per seoond. ■; 112 ABITHMEno. % 'I' M and the pulso of a person in health beats 75 strojces in a minute, what is the breadth of the river ? Ans. 1 mile, 201 g feet. 14. If 2G4 men, working 12 hours a day, can make 240 yardi of a canal, 3 yards wide, and 12 yards deep, in 5 days ; how long will ic take 24 men, working 9 hours a day, to raako another portion 420 yards long, 5 yards wide, and 3 yards di op ? 15. If the charge per freight train for 10800 lbs. of flour be $H for 20 miles; how much will it bo for 12500 lbs. for 100 miles? Ans. $92j. IG. If $42 keep a family of 8 persons for 16 days; how long, .t that rate, will $100 keep a family of G persons ? Ans. 50 jj" days. 17. If a mixture of wine and water, measuring G3 gallons, con- sist of four parts wine, and one of water, and be worth $138.60 ; what would 85 gallons of the same wine in its purity be worth ? A.QS. 6233.75. 18. If I pay IG men $G2.40 for 18 days work ; how much must I pay 27 men at the same rate ? 19. If GO men can build a wall 300 feet loc^, 8 feet hi^h, and G feet thick, in 120 days, when the days are 8 hours long; in what time would 12 men build a wall 30 feet long, 6 feet high, and 3 feet thick, when the days arc 12 hours long ? Ans. 15 days- 20. If 24 men, iu 132 days, of 9 hours each, dig a trench of four degrees of hardness, 337^ feet long, 5| feet wide, and 3^ feet deep; iu how many days, of 11 hours each, will 496 men dig a trench of 7 degree** of hardness, 465 feet long, 3| feet wide, and 2] feet deep ? Ans. 5J. 21. If 50 men, by working 3 hours each day, can dig, in 45 days, 24 cellars, which are each 3G feet long, 21 feet wide, and 20 feet deep ; how many men would bo required to dig, in 27 days, working 5 hours each day, 18 cellars, which are each 48 feet long, 28 feet wide, and 15 feet deep ? Ans. 50. 22. If 15 men, 12 women, and 9 boys, can complete a certain piece of work in 50 days ; what time would 9 men, 15 women, and 18 boys, require to do twice as much, the parts performed by each, in the same time, being as the numbers 3, 2 and 1 ? Ans. 104 days. 23. If 12 oxen and 35 sheep eat 12 tons, 12 cwt. of hay, in 8 days ; how much will it cost per month (of 28 days,) to feed 9 oxen and 12 sheep, the price of hay being $40 per ton, and 3 oxen being supposed to cat aa much as 7 sheep ? ^ Ans. $924 USJ^ MISCELLANE0UI3 EXERCISES. 113 24. A vessel, whoie speed waa 9;' ini:en ror hour, left Bellevilla nt 8 o'clook, a. m., for Gaaanoque, a disruucc of 7 1 miles. A second vessel, whose speed was to that of the fiist as C \s to 5, starting from the same place, arrived 5 minutes before the first ; 7. hat time did the second vessel leave Belleville ? Ans. 55 min. past 10 o'clocii, a. m, 25. If 9 compositors, in 12 days, working 10 hours eiich day, can compore 3G sheets of IG pages to a sheet, 50 lines to a page, and ^5 letters in a lino ; in how many days, each 1 1 hours long, can 5 com* positors compose a volume, consisting of 25 sheets, of 24 pages in a sheet J 44 lines in a page, and 40 letters in a line ? Ans. 16 days. MISCELLANEOUS EXERCISES ON THE FBECEDINO RULES. ' iys, feet jng [eet p. lain jind [cb, \ya. 8 :en ling 24 Ans. y§g. Ans. f . 1. What is the value of .7525 of a mile? Ans. 6 fur., ri, 4 yds, 1 ft., 2| in. 2. What is the value of .25 of a score ? Ans. 5. 3. Reduce 1 ft. G in. to the decimal of a yard. Ans. .5. 4. What is the value of 14 yards of cloth, at $3,375 per yard ? Ans. $47.15. 5. What part of 2 weeks is y\ of a day ? 6. What part of £1 is 13s. 4d ? 7. Keduce ^^ of a day to hours, minutes and seconds. Ans. 2 hours, 52 min., 48 sec. 8. Add I of a furlong to ^ of a mile. Ans. 7 fur., 31 rds, yd., 1 ft., 10 in. 9. What is the value of .857^ of a bushel of rye ? Ans. 48 pounds. 10. Reduce 47 pounds of wheat to the decimal of a bushel. Ans. .783J. 11. Reduce 9 dozen to the decimal of a gross. Ans. .75. 12. Add /(J of a cwt. to | of a quarter. Ans. 3 qrs., 10 lbs. 13. Subtract | of a day from f of a week. Ans. 4 days, 3 hrs. 14. From \^ of 5 tons take | of 9 cwt. Ans. 2 tons, 17 cwt., 1 qr., |f lbs. 15. Bow many yards of cloth, at $3 J a yard, can be bought for US)? Ans. 13.^?; yards. 16. A man bought | of a yard of cloth for $2.80 j what was the rate per yard ? Ans. $3.20. 17. How many tons of hay, at $16 J per ton, can be bought for «196^? a Ans. 11|| tons. m 114 AMTHMETIO. 18. At $17| jMjr week, how many wookfl can a family board foi $765§ ? Ans. 43,} weeks. 19. What number must bo added to 26§, and the Bum multipli- ed by 7 J, thnt the product may bo 496 ? Ans. 37 g. 20. A mail owns f of an oil well. Ho sells "J of bib share for $3500 ; what part of his "hare in the well has ho still, and what is it worth at the same rate 'f 21. How long will 119^ hhds. of water last a company of 30 men, allowing each man g of a gallon a day ? Ans. G27 days. 22. Ileducc ^ of 21, -^h of 1^,, and 3^ of 2|, to equivalent frac- tions having the least common denominator. Ans. |g, ^l, -Y'jj". 23. From '| of 21 of 4, take ^ of ^>fj "f tJ- Ans. 2^. 24. What is the sum of |, J, :Jr, I, J, 4, ], and -J? An3..l325o- 25. What is the sum of I /, of Sg+fl of 85 ? Ans. 22^1 5 J. 2G. How long will it take a person to travel 442 miles, if ho travels 3^ miles per hour, and 8^ hours a day? Ans. IG days. 27. Find the sum of 2^ of ~^q, ^ of I of {'>, of 4 J and J. Ans. G.^fj. 28. A has 2\ times 8^ dollars, and B G|- times 9| dollars ; how much more has B than A? Ans. $44 -|^. 29. If I sell hay at $1.75 per cwt. ; what should I give for 9^ tons, that I may make $7 on my bargain. Ans. $329. 30. If 7 horses eat 93J^ bushels of outs in GO days ; how many bushels will one horse cat in 87| days ? Ans. 19^'. 31. Bought 14 7j yards of broadcloth for $102.90- what was the value of 87^ yards of the same cloth ? Ans. $612. 32. How many bushels of wheat, at $2j per bushel, will it re- quire to purchase 168^^^ bushels of corn worth 75 cents per bushel? Ans. 47 ,\. 33. If in 82J- feet there are 5 rods ; how many rods in one milo ? Ans. 320. 34. Suppose I pay $55 for | of an acre of land ; what is that per acre ? Ans. $88. 35. If I of a pound of tea cost $1.GGJ^ ; what will -7j of a pound cost? Ans.$1.55|l. 36. Subtract the sum of 2 J and 1 ,'3, from the sum of -^y 7^ and 3^ and multiply the remainder by 3 j\. Ans. 24{^. 37. If I lb. cost 23/^ cents j what will 2}i cost ? Ans. 772^ cents. * MISCELLANEOUS EXEBQSES. 115 38. What is tho difference between 2y'(,X3^ and 2,JX3t\j ? 2? "a a* how •or 91 ini\ny l9^ as tho |$612. it rc- shclV nilo? 320. [at per $88. lound ^ and J. 1 cents I 39. If I lb. cost ti ; what will | ^ lb. cost ? 40. What is the difference between J of Ans. ^'j. Ans. 39^ ccnta. and iH+^ Ki+;xs, 41. If 4j\ yards cost $12^^ , what will 2^ yards cost ? Ans. 475 cents. 42. Bought ^ of 2000 yards of ribbon, and gold j of it ; how much remains ? Ans. 285!) yards. 43. Divide the sum of A^, J, I, J-j], 3^, {:,?, jH^E by tho sum of J, ^> fe» t'(7» Am c'n fiiij and divide the quoticntby 0,-5,, and multiply the result by f of [?. Ans. |. 44. I bought 1 of a lot of wood land, consisting of 47 acres, 3 roods, 20 rods, and have cleared J of it ; how much remains to be cleared ? Ans. 20 acres, 3 roods, 31|- rods. 45. What is tho difference between If^ and I'jg ? Ans. -J^. 46. If $[2 pay for a 1^ at. of flour; for how much will $j pay? Ans. l:j';j St. 47. Mount Blanc, the highest mountain in Europe, is 15,872 feet above the level of the sea ; how far above tho sea level is a clim- ber who is j^j of the whole height from the top, i. e., ^^^, of perpen- dicular bight ? Ans. 12890 feet. 48. What will 45.94375 tons cost if 12.796875 tons cost $54.64 ? Ans- $196.17. 49. If I gain $37.515625 by selling goods worth $324.53125 ; what shall I gain by selling a similar lot for $520.0635416. ? Ans. $60.1884. 50. If 52.815 cwt. cost $22.345 ; what will 192.064 cvvt. cost at the same rate ? Ans. $81,512+ 51. Required, tho sum of the surfaces of 5 boxes, each of which is 5^ foot long, 2^ feet high, and 3^7 feet wide, and also the number of cubic feet contained in each box. The box supposed to be made from inch lumber ? 52. If I pay ^-^^ for sawing into throe pieces wood that is 4 ft. long ; how mujch more should I pay, per cord, for sawing into pieces of the same length, wood that is 8 feet long ? Ans. 22J cents. 53. A sets out from Oswego, on a journey, and travels at tho rate of 20 miles a day ; 4 days after, B sets out from the same place, and travels the same road, at the rate of 25 miles per day ; how many days before B will overtake A ? Ans. 16. 116 ABTTHMETIO. 64. A farmer having 56^ tons of hay, sold | of it at $10g per ton, and the remainder at $9.75 per ton ; how much did he receive for his hay ? Ans. 8580^ jj . 65. If th ram of 87-14 and 117^4 ^^ divided by their difference ; what will be the quotient ? Ans. Gg^i- 56. If 8| yards of silk make a dress, and 9 dresses bo made from a piece containing 80 yards -, what will be the remnant left ? Ans. IJ- yards. 57. A merchant expended $840 for dry goods, and then had re- mtuning only f^ as much money as ho had at first ; how much money had he at first ? Ans. $3430. 68. If a person travel a certain distance in 8 days and 9 hours, by Iravelling 12 hours a day ; how long will it take him to perform the same journey, by traveling 8 J hours a day ? Ans. 12 days. 69. If 15 horses, in 4 days, consume 87 bushels, 6 qrts: of oats ; how many horses will 610 bushels, 1 peck, 2 qrts, keep for tho same time ? Ans. 105. 60. Beduce 1 pound troy, to the fraction of one pound avoirdu< pois. Ans. 141- 61. Eeduce to a Himplo fraction. Ans. §. 62. What will be tho cost of 8 cwt., 3 qrs., 12 J lbs. of beef, if 4 cwt. cost $34 ? Ans. $75/5. 63. If 4 men, working 8 hours a day, can do a certain piece of work in 15 days ; how long would it take one man, working 10 hours a day, to do the same piece of work ? Ans. 48 days. 64. Divide $1728 among 17 boys and 15 girls, and give each boy ■^Y ^ much as a girl ; what sum will each receive ? Ans. Each girl, $66f«-; each boy, $42^ f-. 65. If A can cut 2 cords of wood in 12^ hours, and B can cut 3 cords in 17 J hours ; how many cords can they both cut in 24^ hours? Ans. 8-j%\, 66. If it requires 30 yards of carpeting, which is f of a yard wide, to cover a floor ; how many yards, which is 1|- yards wide, will be necessary to cover the same floor ? Ans. 18, 67. A person bought 1000 gallons of spirits for $1500 ; but 140 gallons leaked out ; at what rate per gallon must he sell the remain- der go as to make $200 by his bargain ? €8. What must be the breadth of a piece of land whose length is 40^ yards, in order that it may be twice as groat as another pieoo of ANALYSIS AND SYNTHESIS. H7 iece of hours days. .boy cut 3 oura? yard le, \yill IS. 18. itUO bmain- land whose length is 14^ yards, and whose breadth is 13j^^ yards? Ans. 9J yards. 69. If 7 men can reap a rectangular field whose length is 1,800 feet, and breadth 900 feet, in 9 days of 12 hours each ; how long will it take 5 men, working 11 hours a day, to reap a field whose length is 800 feet, and breadth 700 feet ? Ans. 3^ days. 70. 124 men dug a trench 110 yards long, 3 feet wide, and 4 feet deep, in 5 days of 11 hours each; another trench was dug by one-half the numbor of men in 7 days of 9 hours each ; how many feet of water was it capable of holding ? Ans. 2208 cubic feet. 71. If 100 men, by working hours each day, can, in 27 days, dig 18 cellars, each 40 feet long, 30 feet wide, and 12 feet deep ; how many cellars, that arc each 24 feet long, 27 feet wide, and 18 feet deep, can 240 men dig in 81 days, by working 8 hours a day 'r Ans. 256. 72. A gentleman left his son a fortune, J of which ho spent in 2 months, J of the remainder lasted him 3 months longer, and f ol what then remained lasted him 5 months longer, when he had only $895.50 left; how much did his father leave uim ? Ans. $4477.50. 73. A farmer having sheep in two different fields, sold ^ of the number from each field, and had only 102 sheep remaining. Now 12 sheep jumped from the first field into the second ; then the num- ber remaining in the first field, was to the number in the second field as 8 to 9 ; how many sheep were there in earh field at first ? Ans. 80 in first field ; 56 in second. 74. A and B paid $120 for 12 acres of pasture for 8 weeks, with an understanding that A should have tlie grass that was then on the field, and B what grew during the time they were grazing; how many oxen, in equity, can each turn into the pasture, and how much should each pay, providing 4 acres of pasture, together with what grew during the time they were grazing, will keep 12 oxen 6 weekSj and in similar manner, 5 acres will keep 35 oxen 2 weeks ? J A should turn into the field 18 oxen, and pay $72. Ans. igth is Leooof ) B should turn into the field 12 oxen, and pay $48. ANALYSIS AND SYNTHESIS. Analysis is the act of separating and comparing all the different parts of any compound, and showing their connection with each other, and thereby exhibiting all its elementary principles. 118 ABITHMETIO. ■ 1 * I Tho converse of Analysis is Synthesis. The meaning and use of these terms will probably be most readily comprehended by reference to their derivation. They are both pure Greek words. Analysis means loosing up. The general reader would here probably expect loosing down, as employed in most popular definitions; but we may illustrate the Greek term, loosing vp, by our own everyday phrase, tearing up, which means rending into shreds, the English vj) conveying the same idea here as the Greek ana in analysis. The Greek synthesis means literally placing together ; that is^ the component parts being known, the word synthesis indicates the act of combining them into one. Wo might give many illustrationL=i, but one will suffice, and wo vhoose the one which will be most generally understood. When wo analyse a sentence, wo loose it vj), or tear it vj), into its component parts, iiud by synthesis wo write or compose, i. c, put together the part.?, which, by ;in:'lysis, we have found it to consist of. Whoa wo connncnce to analyse a prohlcm wo reason from a given quantity to its unit, imd then from this unit tu the required quan- tity ; hence, all our deductions are self-evident, and wo therefore require no rule to solve a problem by r.nalysis. Although this part of arithmetic hi usually called analysis, yet, a3 it is really both analysis and synthesis, we have given it a title in acoovdaucc with the principles now laid down. EXAMPLE. 1. If 12 pounds of sugar cost $1.80, what will 7 pounds cost ? SOLUTION. 12)1.80 .15 7 $1.05 If 12 lbs. cost $1.80, one pound will cost the ,1. of $1.80=:rl5 cents. Now, if 1 lb. cost 15 cents, 7 lbs. will cost 7 times 15 ccnts=to $1.05. Therefore, 7 lbs. of sugar will cost $1.05, if 12 lbs. cost $1.80. NoTE^-The work may bo somewhat shortened, especially in long -ques- tions, by arrauging it in the following manner, bo as to admit of cfiucelUug, if possible:— 15 W^ 1 ^i 1 15^ 2. If 5 bushels of pease cost $5.50, for whut csn you purchase 19 bushels? • Ans. $20.90. ■rnHMM ANALYSIS AND SYNTHESIS. 119 3. If 9 men can perform a ceitain piece of labor in 17 days, how long will it take 3 men to do it ? Ans. 51 days. 4. How many pigs, at $2 eacb, must be given for 7 aliecp, worth U a head ? Ans. 14. 5. If $100 gaiu 6G in 12 months, how much would it gain in 40 months ? "■ ' Ans. $20. C. If 4| bushels of apples cost $3J , what will be the cost of 7J bushels? SOLUTION. *" In the first place, 4| bushcls=V- bushels, and $3^=45^?. Now, since Y" bushels cost $--„"-, one bushel will cost $--if— h-V-= 3jf-X j\=$5, and 7J- or -V' bushels will cost J# times $5=3 K-'i-=^ $5, the value of 7^ bushels of apples, if 45 bushels arc worth $3-J. the 15 .05. 12 hueft- fUug, hase 1.90. OPERATION 7. If ^ of 3| lbs. of tea cost $1? ; what "will be the cost of 5 J pounds ? Ans. $4.1 2|. 8. 100 is g of what number ? Ans. 150. 9. If ^ of a mine cost $2800 ; what is the value of 3 of it ? Ans. $4200. 10. I of 24 is If times what number? Ans. 10. 11. I of 40 is ^3 of how many times J of § of 20 ? Ans. 9. 12. A is 16 years old, and his ago is ~ times 5 of his father's age ; how old is his father ? Ans. 36. 13. A and ^3 were playing cards ; A lost $10, which was I times g as much as B then had ; and when they commenced 3 of A's money wa.s equal to f of B's ; how much had each when they began to play ? Ans. A $45 ; B $40. 14. A man willed to his daughter $5G0, which was .', of f of what he bequeathed to his son ; and 4 times the son's portion was § kho value of the lather's estate ; what was the value of the estate ? Ans. $13,440. 16. A gentleman spent J of hia life in St. Louis, }^ of it in Bos- ton, and the remainder of it, which was 25 years, in Washington; irhat age was ho when ho died ? \.ns. 60 years. w If 'r 120 ABITHMETIO. 16. A owns ^, and B j\ of a ship ; A's part is worth $650 more than B's ; what is the value of the ship ? Ans. $15,600. 17. A post stands -^ in the mud, ^ in the water, and 15 feet above the water ; what is the length of the post ? Ans. 36 feet. 18. A grocer bought a firkin of butter containing 56 pounds, for $11.20, and sold f of it for $8| ; how much did he get a pound ? Ans. 20 cents. 19. The head of a fish is 4 feet long, the tail as long as the head and ^ the length of the body, and the body is as long as the head and tail ; what is the length of the fish ? Ans. 32 feet. 20. A and B have the same income ; A saves \ of his ; B, by spending $65 a yeur more than A, finds himself $25 in debt at the end of 5 years j what did B spend each year ? Ans. $425. "21. A can do a certain piece of work in 8 days, and B can do the same in 6 days ; A commenced and worked alone for 3 days, when B assisted him to complete the job ; how long did it take them to finish the work? SOLUTION. rif A can do the work in 8 days, in one day he can do the ^ of it, and if B can do the work in 6 days, in one day he can dp the ^ of it, and if they work together, they would do ^-{-^=25 o^ ^^^ work in one day. But A works alone for 3 days, and in one day he can do ^ of the work, in 3 days he would do 3 times ^=f oi" the work, and as the whole work is equal to | of itself, there would be § — ^=f of the work yet to bo completed by A and B, who, according to Ihe con- ditions of the question, labour together to finish the work. Now A and B working together for one day can do £^ of the entire job, and it will take them as many days to do the balance f as ■^'^ is contain- ed in f, which is equal f X^-— -2| days. 22. A and B can build a boat in 18 days, but if C assists them, thoy can do it in 8 days ; how long would it take C to do it alone ? Ans. 14| days. 23. A certain pole was 25^ feet high, and during a storm it was broken, when f of what was broken oif, equalled § of what remained '• how much was broken off, and how much remained ? Ans. 12 feet broken off, and 13i- remained. 24. There are 3 pipes leading into a certain cistern ; the first will fill it in 15 minutes, the second in 30 minutes, and the third in one hour ; in what time will they all fill it together ? Ans. 8 nuD., 34| seo. ANALYSIS AND SYNTHESIS. 121 03 itbe 3on- A md in- cni, fd'. 25. A. and B. start together by railway train from Buffalo to Erie a distance of (say) 100 miles. A goes by freight train, at the rate of 12 miles per hour, and B by mixed train, at the rate of 18 miles per hour, C leaves Erie for Buffalo at the same time by ex- press train, which runs at the rate of 22 miles per hour, how far from Buffalo will A and B each be when C meets them. 2G. A cistern has two pipes, one will till it in 48 minutes, and th(^ other will empty it in 72 minutes ; what time will it require to fill the cisterti when both are running ? Ans. 2 hours, 24 min. 27. If a man spends {'^ of his time in working, J in sleeping, -Jg in eating, and 1^ hours each day in reading ; how much time will be left? Ans. 3 hours. 28. A wall, which was to be built 32 feet high, was raised 8 feet by G men in 12 days ; how many men must be employed to finish the wall in C days ? Ans. 30 men. 29. A and B can perform a piece of work in 5/^ days ; B and C in 6§ days ; and A and C in G days ; in what time would each of them perform the work alone, and how long would it tako them to do the work together ? Ans. A, 10 days ; B, 12 days ; C, 15 days ; and A. B, and C, together, in 4 days. 30. My tailor informs mo that it will take lOJ sqUaro yards of cloth to make rac a full suit of clothes. The cloth I am about to purchase is 1 J yards wide, and on sponging it will shrink ..^g in width and length ; how many yards of this clotlimust I purchase for my "new suit?" Ans. G-jggg yards. 31. If A can do f of a certain piece of work in 4 hours, and B can do f of the remainder in 1 hour, and C can finish it in 20 min. ; in what time will they do it all working together ? Ans. 1 hour, 30 min. 32. A certain tailor in the City of Brooklyn bought 40 yards of broadcloth, 2J- yds wide ; but on sponging, it shrunk in length upon every 2 yards, -,'j of a yard, and in width, 1 J sixteenths upon every IJ yards. To line this cloth, he bought flannel IJ- yards wide which, when wet, shrunk ^ the width on every 10 yards in length, and in width it shrunk ^ of a sixteenth of a yard ; how many yards of flannel had the tailor to buy to line his broadcloth ? Ans. 71/3 yards. 83. If G bushels of wheat are equal in value to 9 bushels of bar- Icy, and 5 bu8b«la of barley to 7 boshek of oats, and 12 bushels of leo. ri 122 ARITHMETIO. oats to 10 bushels of jxiasc, and 13 bushels of pease to ^ ton of hay, und 1 ton of hay to 2 tons of coal, how many tons of coal are cquaJ in value to 80 bushsls of wheat ? SOLUTION. If G bushels of wheat are equal in value to 9 bushels of barley, or 9 bushels of barley to G bushels of wheat, one bushel of barley would bo equal to J of G bushels of wheat, equal to §, or § of a bushel of wheat, and 5 bushels of barley would be equal to 5 times §• of a bushel of wheat, equal to f X5=J3"-=3J- bushels of wheat. But 5 bushels of barley are equal to seven bushels of oats ; hence, 7 bushels of oats are equal to 3J bushels of wheat, and one bushel of oats would be equal to 3^-=-7=A^ bushels of wheat, and 12 bushels of oats would be equal to 12 times ^ J^^J^Y-^SI bushels of wheat. But 12 bushels of oats arc equal in value to 10 bushels of pease, hence, 10 bushels of pease are equal to 5| bushels of wheat, and one bushel of pease would equal ^-v-10=1^ of a bushel of wheat, and 13 bushels of pease would equal -4xl3=-\--=:7^ bushels of wheat. But 13 bushels of pease equal in value ^ ton of hay, hence, i ton of hay equals 7| bushels of wheat, and one ton would equal 7|X2= 14f bushels of wheat. But one ton of hay equals 2 tons of coal, hence, 2 tons of coal are equal in value to 14| bushels of wheat, and one ton would equal 14§-f-2=:7| bushels of wheat. Lastly, if 7| bushels of wheat be equal in value to one ton of coal, it would take as many tons of coal to equal 80 bushela of wheat, as 7| is coutained in 80, which gives lOfg tons of coal. Note. — This question belongs to that part of arithmetic usually called Conjoined Proportion, or, by some, the " Chain Rule," which has each ante- cedent of a compound ratio equal in value to its consequent. We have thought it best not to introduce such questions under a head by themselves, on account of their theory being more easily understood when exhibited by Analysis than by Proportion. Questions that do occur like this will most probably rclrtto to Arbitration of Exchange. Although they may all be worlied by Compound Proportion as well as by Analysis, yet tiie most expe- ditious plan, and the one generaly adopted, is by the following RULE. Place the antecedents in one column and the conseqv/mta in another, on the right, with the sign of equality between them. Di- vide the continued product of the terms in the column containing tJie odd term hi/ the continued product of the other column^ and the quotient will be the answer. ANALYSIS AND SYNTHESIS. 123 Let U3 uoTv tako our last cxaninlc (No. 33), and solve it by tbis tile 1 171 T\ * the \he C bushels of wheatr::=9 bushels of barley. 5 bushels of barley =-7 bushels of oats. 12 bushels of oats=:10 bushels of pcasc 13 bushels of pcasc=rj ton of hay. 1 ton of hay 1^2 tons of coal. — tons of coul^-^SO bushels of wh& 20 ^ S- %, 9, Ans. 34. If 12 bushels of wheat in Boston are equal in value to 12J bushels in Albany, and 14 bushels in Albany are worth 14^ bushels in Syracuse ; and 12 bushels in Syracuse are worth 12 J bushels in Oswego ; and 25 bushels in Oswego are worth 28 bushels in Cleve- land ; Low many bushels in Cleveland arc worth 60 bushels in "Boston ? , Ans. 75|f . 35. If 12 shillings in Massachusetts are worth 10 shillings in New York, and 24 shillings in New Yck are worth 22J shillings in Pennsylvania, and 7-^ shillings in Pennsylvania are worth 5 shillings in Canada ; how many shillings in Canada arc worth 50 shillings in Massachusetts? Ans. 41§. 36. If men can build 125 rods of fencing in 4 days, how many days would seven men require to build 210 rods ? SOLUTION. If 6 men can build 120 rods of ienciug in 4 days, one man could do J of 120 rods in the same time ; and ^r of 120 rods is 20 rods. Now, if one man can build 20 rods in 4 days, in one day he would build ^ of 20 rods, and J of 20 rods is 5 rods. Now, if one man can build 5 rods in one day, 7 men would build 7 times 5 rods in one day, and 7 times 5 rods=35 rods. Lastly, if 7 men can build 35 rods in one day, it would take them as many days to build 210 rods as 35 is contained in 210, which is ; therefore, if men can build 120 rods of fencing in 4 days, 7 men would require days to build 210 rods. 87. If 12 men, in 36 days, of 10 hours each, build a wall 24 feet long, 16 feet high, and 3 feet thick ; in how many days, of 8 124 ARITHMETIO. hours each, would the same lot of men build a wall 20 feet long, 12 feet high, and 2^ feet thick ? Ans. 23/g. 38. If 5 men can perform a piece of work in 12 days of 10 hours cacTi ; how many men will perfhrm a piece of work four times as large, in a fifth part of the time, if they work the same number of hours in a day, supposiui:; that 2 of the second set can do as much work in an hour as 3 of the first set ? Ans. 66^ men. .NoTK. — Sucb questions as Ibis, wliero tbe answer involves a fraction, may frequently occur, and it may be asked bow f of a man can do any work. Tbe answer is simply tbis, tbat it requires OG men to do tbo work, and one man to continue on working § of a day more. 89. Suppose that a wolf was observed to devour a sheep in ^ of nn hour, and a bear in ^ of an hour ; how long would it take them together to cat what remained of a sheep after the wolf had been eating J an hour ? Ans. IO/3 min. 40. Find the fortunes of A, B, C, D, E, and F, by knowing that A is worth $20, which is J as much as B and C are worth, and that C is worth J as much as A and E, and also that if 19 times the sum of A, B and C's fortune was divided in the proportion of f , ^ and ^, it would respectively give f of D's, ^ of E's, and J of F's fortune Ans. A, 20 ; B, 55; C, 25 ; and D, E and F, 1200 each. 41. A and B set out froD'. the same place, and in the same direc- tion. A travels uniformly 18 miles per day, and after 9 Jaya turns and goes back as far as B has travelled during those 9 days ; bo then turns again, and pursuing his journey, overtakes B 22^^ days after the time they first set out. It is required to find the rate at which B uniformly travelled. Ans. 10 miles pec day. 42. A hare starts 40 yards "before a gr^hound, and is not per- ceived by him until she has been mtraing 40 seconds, sho scuda away at the rate of 10 miles an hour, and the dog pursues her at the rate of 18 miles an hour ; how long will the chase last, and what dfa- tance will the hare have run ? Ans. GOn^ sec. j 490 yards. 43. A can do a certain piece of work in 9 days, and B can do the same in 12 days; they work together for 3 days, when A is taken sick aiid leaves, B continues on working alone, and after 2 days he is joined by C, and they finish it together in 1 J days ; how long would C be doing it alone ?^ Ans. 12 days. 44. A, in a scuffle, seized on f of a parcel of sugar plums ; B caught § of it out of his hands, and C laid hold on -^\ more ; D ran oft with all A had left, except 4 which E afterwards secured slyly for himself: then A aad C jointly set upon B, who* in the conflict, lei I'RACTICE. Jlat A „„,i B la^t acquired, „„"„/' ''f''- , ^ "'«" ^'mok J „f dfclj. recovered f „f it i^ IL^W '' '"'°*: '%, with J„o earned off J „ j,,^ ^^ » 'V-"! »iarcs aga,„, ^^^ ^^^ ° a-d agreed that ,he x „f fi?'^"- ^P/" '% ttey called a trufe equally dirideda„„4C,^tt:v ^" ^ °' '=-'■"■<'»«"' '.on, had each of the compoti orsT """^ ^^"-^ »f'<=-' »& dfatrib- A«s.Alad2863;B,6335C2x« ,^ , • "^•'. C, 2«8 ;D, 10294 and E, 4950 1(7.621 46* PRACTICE The rulo wliich Is riM^rj r> *•• P. Hero the fi«t term beiri !"""''' ""^ ^ •■ 2S: : 7 ■ "Vomuliiplicatio; burtr"'"" ^°°'^' ""^ "' "■^/ourth term of a„' aoaW """' «^»«' '^ -% ♦ '•o-ij per barrel wa ^ot, i "'*'^'^<^^s of flour, at ^y^^- In n.an''c:; r;:^ .f^^^^P'^ »^-C2i «' to multiply the 46 bv 7 t'l " °™™'- " Mar, theprLtf46 afso' ^""^ ""^'"S ''^' «»dI2Jceu,.beiugi'f';;,''-l""''°*''' contswill bo the fourth o^l^rT "" '^* »5Aandthe„h„iocor::ro.r'" length would bTTf , . ''"°"'°" ^'^'^'I "' P. P Ti: u ' •' *■> IS; : *t87*- J350.75 It350.75 126 ARITHMETIC. I! 4 *■ plication because the first term is unity, and divided by 1 vrovld noi alter the product of the other two terms. Thus : 2 qrs. 10 lbs. 6 " I of 1 cwt. I of 2 qrs. A- of 10 lbs. 4.87J 36 18 2922 1461 175.50 =: pncc of 3 owt., @ $4.87J percv/t. 2.437= " 2 qrs. " '' '< .487:= " 10 lbs. " " '« .243= " 5 " " " '* $178,607= '' 36 cwt., 2 qrs., 15 Ibs^ We would call the learner's special attention to the followin direction, as the neglect of it is a fertile source of error. Whenever you take any quarrtity as an aliquot part of a higher to find tho price of the former, be sure you divide the line whicJi is tlic^rice at therate of that higher denomination. To find the rent of 189 acres, 2 roods, 32 rods, at 64.20 jxsc acre. 2 roods=:^ of 1 aero, 20 rods— I of 2 voous, 10 rod3=^ of 20 rods, 2rod9=^oflOrods. 420 189 210 525 2625 625 3780 8360 A20 $696.74 Since the rent of 1 aero is $4.20, the half of it, 02.10, will be the rent of 2 roods, the rent of 20 rods will b© . 525, tho J of the rent of 2 roods, the half of that^ .262 5, will be the rent of 10 rods, and, lastly, .0525 will bo the rent of 2 rods, which is the J- of 10 rods. We then multiply by 189, and set the figures of the product in tho usual order, bo that the first figure of the product by 9 shall be under the units of cents, &c., and then adding all the partial results, we find the fimd answer, $796.74, the rent of 189 acres, 2 roods and 32 perches. EXERCISES. 1. What is the price of 187 owt at $5,37| per cwt. ? Ans. $1005.12^ PBACTTCE. 127 19 10, is, be f 3 25, bo 13 en ho Itho er we d 2. What is the value of 1857 lbs., at $3.87J per lb. ? Ans. 67195.87^ 3. What will 4796 tons amonnt to at $14.50 per ton ? Ans. $21582 I. What is the price of 29 score of sheep, at $7.62^ cacli ? Ans. $4422>50 5. Sold to a cattle dealer 196 head of cattle at $18.75 each, find the amount. Ans. $3G75 6. Sold to a dealer 97 head of cattle, at $16.12 J each, on the average ; find the price of all. Ans. $ 15G4.12|. 7. What is the price of IG tons, 17 cwt., 2 qrs. of coal, at $8.62^ per ton ? 8. What is the yearly rent of 97 acres, 3 roods, 20 rods, at $4,37^ per acre ? 9. If ih man has $12.50 per week ; how much has he per year ? Ans. $G50. 10. If a clerk has $2.12J salary for every working day in the year; what is his yearly income? Ans. $6G5.12^. II. If a tradesman earn $1.G4 per day ; how much does he earn in the year, the Sabbaths not being reckoned ? Ans. $513.32. 12. If an officer's pay is a guinea and a half per day ; how much has he a year ? Ans. £574 17s. Gd. 13. What is the price of 479 cwt. of sugar, at $17.90 per cwt. Ans. $8574.10. 14. Find the price of 879 articles, at $1.19 each. 15. Tindthe cost of 1793 tons of coal, at $7.87 J per ton. 16. What is the value of 2781 tons of hay, at $8.G2J per ton ? 17. Wliat is the rent of 189 acres, 2 roods, 32 rods, at $4.20 per acre? Ans. $795.74. 18. What is the price of 879 hogs, at $4.25 each? 19. "What will 366 tons of coal come to at $8.12J per ton ? 20. "What is the price of 118 acres, 3 roods and 20 rods of cleared land, at $36.75 per acre ? Ans. $4368.66. 21. "What is the price of 286 acres, 1 rood, 24 rods of uncleared land, at $7.25 per acre ? Ans. $2076.40. 22. A has 84 acres, 2 roods, 36 rods of cleared land, worth $24.60 an acre ; B has 298 acres, 3 roods, 24 rods of uncleared land, worth $4,40 an acre — they exchange, the difference of value to be paid in cash; which has to pay, and how much ? Ans. B $989.08. ^ !^ \ X28 ABITHMF.TIO. ACCOUNTS AND INVOICES. Accounts aro etatemonts from merchants to customers that haro pur chased gouds on credit, and aro generally made out periodically, unlesr specially called for. An invoice is simply a statement rendered by the seller to the buyer, a lime of purcbaeo, showiog the orticlda bought, and the prices of each. 1. New York, July let, 1866. Mr. Jamss ANDEB.soy, To Fbexch, TTmTE & Co , Dr. 18G6. 2.«* 1.35 2.»'> Jany. 4, To 2 lbs. tea, l.i«^ ; 3 Iba cofToo, 45c. ; 20 lbs. rice, 10c. . . ** 29, " 2J yds. Amer. twoed, l.**" ; 1 vest 1." 2.0i Feb. 10, " 14 lbs. Mus, sugar, 12^0. ; 10 lbs. cms. white sugar, 20c. COc. 28c. l.»<» " 22, " 1 lb. bk. Boda, ; 1 lb. car. soda, ; 4 lbs. coffee, 45c.. 3.00 87jc. Mar. 11, *' 10 yds. print, 30c. ; trimmiug", &a., per bill 1.8 85c. 2.0'i « 19, '< 2 lb?, tobacco, 90c. ; 1 gal coal oil, ; 2gals.BjTup, l.oo, US i.oo)i Apr! 12, " » yd. blk. sihi, 3 "o ; 4 yd. blk. velvet, C.oa?s 3.25 40e. 60c. May C, " 2 lbs. tea, l.«a'si ; l bottlo pickles, : 1 lb. pepper, 85c. l.oo 1.50 « 20, " 1 bag salt, ; 10 Iba. sugar, lOo. ; 3 lbs. raisins, 50c. . . . 75c. 2.S0 « 31, " 3 lbs. cunaiits, 25c. ; 10 lbs. white sugar, 25c l.«o 12 Jc. 2.00 JunelO, " 2 lbs. lobacco, 76o. 5 J lb. B. soda, 25o. ; 20 lbs. rice, 10c. . 40c. 10c. 30c. l."^» " 17, '• llb.cloves, ; J lb.nutmeg8, ; J cinnamon, jllb.tea, $47.61 2. BALTlMOIiE,Oct. IS^ISfiC. Mil. William PxTrERsoK, To Moffat &. Mukkat, Dr, 186G. July 3, To 14 yds. fancy print, 20c. ; 12 yls. col'd silk, £.''« " 11, " 2 ladies' felt hats, 2.oo; 2 prs. kid gloves, l.oo " 22, " 4 prs. cotton hose, 40c. ; 3 yds. red flannel, 80c Aug. 19, " 2 J yds. blk. cassimere, 2.0= ; 2 J yds. cotton, 20c " 27, " l| yds. white flannel, 75c. ; buttons, 10c. ; twist, 15c.. .. " " Sept. 1, " 2 suits boys' clothes, 9.oo ; 2 felt hats, l.oo "«} " 8, " 2 prs glove? 80c. ; 2 neckties, 62Ac " 22, " J doz. prd. cotton hose, 7.5o ; j doz. shirts, 2C.oo Contra. Cr. 20.00 15.00 \ug.l8,ByCash, ;27.Cash, $35.00 « 2i). " firkin butter, 95 lbs., at 22c 20.90 65.90 Bahvnce due. ..,.., $41.71^ Received payment io full, MOFFAT AMUUBAY. Oct. ACCOUNTS AND INVOICES. 120 » KociiESTER; Jan. 2ml, 18C6. Mr. John Deans. To Wood & Ftoger, Lr. " - t • 18CG. July 4, To 12 lbs. sugar, 10c. ; >T lbs. tea, l.o» ; 2 lbs. tobacco, 87Jc. " 11, " 1 bbl. salt, 2a3 ; 2 lbs. indigo, 25c. ; 11 lbs. pepper, 30o " 18, " 2 prs. socks, 45c. ; 1 neck-tlo, 75c. ; 2 scarfs, 25c " 25," 101bs.sugar,llc.; 201ba.dr'dnpplos,10c.; 21bs.coffoo28c " " " 18 lbs. dried peaches, 12Jc. ; 1 bush, onions, l^^a^i Aug. 4, " 12 lbs. rice, 7c. ; 2 gals, syriip, 75c. ; 14 lbs. sugar, 12c... « " " 13 lbs. mackerel, 12c. ; 2 lbs. ginger, 20c. ; 2 lbs. tea, l.»« '• 21. " 2 prs. kid gloves, las ; 2 boxes collars, 37 jc Sept. 12, " 10 lbs. sugar, 15c. ; 2 lbs. coffee, 35c. ; 1 lb. chocolate, 40c Oct. 4, " 2 felt hats, 13«; shoo blacking, 25c " 21. '.' 2 Ibs. pepper, 15c. ; soda, 40c. ; salpetre, SOc. ; Bait, 75c. Contra. Or. 10.00 6.00 Sept. 14. By Cash, ;0ct.4, Casli Oct. 17. " 2 bbla. winter apples, 2««...... ,... $ 18.42 ^7.61 |IG. Boston, Nov. 1st, 186C. Ma. "W'm. Reid. To Casipbell, Linn & Co,, Dr. Aug. 4, To 2 prs. kip boots, 3s« ; 2 pr.9. cobourgg, 2"* r . .-. » 7 yds. fancy tweed, 2^^ ; trimmings, loo ; buttons 25c.. 2 prs. gloves, 75c. ; 3 prs. Focks, 35c. ; 2 straw hats, 40c, " 10 yds. jmnt, 35c. ; trimmi'igs, l^^ ; ribbons, 75c 3 neck-ties, C2 J c; 2 prs. boys' gaiters, 2""!^) shootio3,12Jc. 1 business coat, 14oo ; 2 felt hats, l^s ; 1 umbrella, 2=0 2 flannel shirts, 42= ; i pr. pants, S^o | over-coat, IGoo. 2 lace 6cai'&, 225 j 3 prg. woollen mita 75o. | pins, 2^c. " 17. (1 Sept. 4, a " 26. « Oct. 11, u " 22, ii •• 27, K " 80. c In II Cordra. Cir, 1000 810 Sept. 12. By Cash ; Oct. 4, Cash, ^ Oct. 24. " 300 lbs. cheese, 10c. ; 75 lbs. butter, 25c Balance due $37.60 Received payment. » CAMFBELL, LINN & CO. 19U ▲BITHUETIOt ■ -'IJ Auburn, Sept. Ist, IPCC. Ms. S. Siirrs To WitflOM, Rat & Co., Dr. 'i> '• 1866. Jan. 15, To CydsB. cloth, 4. "O; 2 doz. buttons, 90(3.; ft ozfl-thrcaa, 15c. 20, '* 40 yds. fac. cot., ICc. ; 7 flpoola cot., 4c. ; l2 yds. rib., 35c. " 15 yds. B. silk, 2.3 0; IC yds. lining, 15c. ; Ssllk spools, llo. •' 3 yds. drill, 31c. ; 5 yds. cob'rg, 31c. ; 2 papiTs need. 18o. " 9 yds. coating,. 'i.'-O; IJ yds. voting, l.oo; 5 pr. Los(», 40o. " 21 yds. print, 20c.; lOJ yds. muslin, 30c.; 2 prs. gloves, l.-*** u It 30. Feb. 20. Har. 18, " 31. Apr.l5, " 25. May 29, J me 5, " i:. July 6. Aug. 10. " 4 prs. gloves, l.io; 10 yds. ribbon, 18c. ; G hand'k. 3Gc. " 3 prs. blankets C.^ »; 4 counterpanes, 3."°; 15 yds. cot.,25c. " 2 summer bals, l.osj o yds. ribbon, 40c. ; 2 leathers, 2="* " 4 prs. Blippors, 1.4°; 4 prs. hose, COc. ; 3 prs. hose, 40o. " 3 wool shawls, 5.3 0; 1 B. suit, 30.ooj 9 ozs. thread, I80. " 40 yds. cotton, 30c. ; 3 spools, 12c. ; 2 spools, 10c " It yds. flannel, 75c. ; 4 hand'ks., 35c. ; 12 yds. t;ipe, I3e< 15.00 10.00 Jan. 15. By Cash, ; 22. Cosh, Feb. 20. May 15. June 5. Or. 50 lbs. butter, 40c. ; G cwt. pork, 104o. " 6 geese, 80c. ; 14 fowls, 40c " 60 lbs. wool, 50c. ; 10 lbs. wool, 60c. . 30.00 10.00 July 6. " Cash, ; Aug. 10, Cash, Balance due. ^82.73 Bbooeux, July liHh. 1866. Mr. B. R. Eillts. To J. WiucTAMS, Dr. 1806. Jan. 10, To lOlbs.M.sugar.lSc; lClbs.W.8ugar,20c.; 121b9.C.8ng&T,l8o. " 30. Feb. 12. Mar. 30. Apr. 5, " 25. May 1. JunelS. July 12, « 29, « 31, €( (( « M 15 lbs. raisins, IGc; 13 lbs. raisiQ3,15c.; 10 lbs. raisins,! 8c. 9 lbs. cur'nts, 13c.; 12 lbs. cur'nts, 14c.; 6 lbs. cur'ut3,20c. CO lbs. salt, 2c.; 2 lbs. wash, soda, 23c.; 1 lb. bak. soda,2Jc. G lbs.D. apple.s,12c.; lOlbs-bisc'ts, 17c.; 5 lbs. bisc'ts,2Ic, 3 cwt. flour. 4.B0| 2 cwt. C. meal, 2.3 oj 3 ibg. butter, 25o. 10 lbs. pork, 20c. ; 19 lbs. cheese, 10c. ; 14 lbs. sugar.lSc. 5 lbs. tea, 1.3 3; 9 g^ig. molasses. 40c. ; G doz. eggs. 12c. 5 lbs,8ugar,l()C.; 9J lbs. raisins, ICc; 10 lbs.cur'nts,125c.. 14 lbs. bacon. 12c. ; 5 lbs. cheese, IGc. ; 4 lbs. butter,25c. 41bs. tea, 1.0 0; 2 lbs. tea, 1.3 o; 6 lbs. coffee, 3.50 10 lbs. salt, Ijc. ; 3 lbs. indigo, 90c. ; 1^ lbs. blue, 30o. 3 Itoi ealt Del re, 35c.; 4 doz. eggs,12Jc.; 6 lbs. butter, irv\ leceived payment $83.16 J. WILUAMS. HGGOUNTS KSD IKTOICES. 131 Albakt. Dec. 1, 1866 Hk. Geo. Sivpsosv, To TATLon & Grant, Dr. 18C6. Jnly 7, " 12, " 24, Axis- 4, ' J2. 6ept.21, Oct. 12, «' 20, Nov. 4, To 12 lbs. sngnr, 15c. ; 2 lbs. tea, l.«8 ; 3 lbs. co^'feo, 35o. . . . " 2 lb9.tobacco,87Jo.; 3 lbs.ralsinss30c.; 12 Iba.currnnts, 15o. " 3 lbs. gunpowder, G2 Jc; 6 lbs. shot, 18c. ; 2 lbs. glue, 25c. " 12 lbs. washing soda, 15c. ; 4 Ib.s. baking soda, 25c " 1 box mustard, l.«o; 2 lbs. filberts, 30c.; 2 lbs. alm'ds, 35c " 8 lbs. sugar, 14<5. ; I lb. tea, l.i»>4 ; 3 lbs. chocolato, 4Cc. " 4 lbs. Dg.<), 15c. ; 2 lbs. oraugo peel, 30c. ; spices 40o " 2 lbs. but. blue, 18p.; 2 lbs. aiUplmr, 20c.; 3 lbs. Boda, 35c. 18.00 " 21b3.emok.tobncco,90o.; 21bs.sauff,20c.ilbusinc8a6uU, in li Cordra. Or. 8.00 5.00 ATig.l2,B7Cn8b, ; Sept 21, Cosh, ;....» Oct. 20, " 100 Iba. dried apples, 15c. ; 50 Iba. peaches, 20c. . , BddoQce due. •7.91 DflTBOZT, Sept. 30th, 1866. ]Ur. S. Smith, To Ra7, HiLt & Co., Dr., 1866. Jan. 1, To " 10, " " 13, " Feb. 2, " " 8, " " 13, " Mar. 4, " " 15, " Apil. 6, " May 10, « June 12, « 5 lbs. tea, l.''" ; 15 lbs. sugar, 15c.; 1} lbs. ciuuamon, 2.»^ 18 lbs. rice, 10c. ; 16 lbs. salt, 4c. ; 34 lbs. oat meal, 6c.. . 12 lbs. raisins, 18c. ; 3 lbs. tobacco, 68c. ; J lb. snuff, .34c.. 10 lbs. cur'nts, 17c.; 10 lbs. ginger, 41c.; 5 lbs. mustard, 42c. C lbs. sugar, 18c.; 13 lbs rice, 8c.; 21 lbs. dr'd apples, IGc. 25 lbs. raisins, 18c. ; \ lb. B. soda, 30c. ; \ lb. nutmegs, 22c 12 lbs. coffee, 3Gc.; 6 Ibs.M. sugar, 15c.; 4 Ibs.W. sugar, 20c. 4 lbs. mustard,30c.; 3 lbs. tobacco, 30c.; 12 lbs. ginger,27c. 2 lbs. currants, 20c. ; 14 lbs. rice, 8c. ; 9 lbs. tur. seed, 45c. l|lbs. cin'mon, 70c.; 12 lbs. sago, 31c.; 14 lbs. sugar, 21c. 16 lbs. salt, 3*^. ; 2 lbs. indigo, 90c. ; 61 lbs. corn starch, 14c. 40 IbB. flour, 4c. ; 30 lbs. corn meal, 3c. ; 25 lbs. coffee, 3Sc. 1 $88.46 I ! 132 ABHEXETIO. 31. Chicago, Jaru 4th^ 1866. Mr. Elias 0. CoNKLirr, Bought of J. BuNTIN & Co., 12 J«ams of fooboap paper ..„.@ $3.25 15 dozen school books @ 4.50 23 slates. @ 1.30 7 " photograph albums ©15.00 3 " Bullion's grammar @ 7.00 8 " fifth reader @ 3.50 5 gallons of black ink @ 1.10 4 dcMsen American Oonunercial Arithmciio .@ 18.00 $361.90 Hcceivcd payment, BUi^TIN & Co. 32 ToBONTO, Jon^ 12tti, 1866. Me. James H. Burritt, Bought o/ Morrison, Taylor & Co,, 15 cwt. of cheese „@ $9.00 4 cwt. of flour @ 4.25 120 pounds of bacon @ 0.14 7 bushels of corn meal @ 0.75 12 firkins of butter @ 13.50 20 bushels of dried apples @ 2.25 13 " " peaches @ 4.00 11 cwt. of buck-wheat flour @ 5.50 | 15 cwt. maple sugar @ 8.00 | 25 bags of common salt i@ 1.15 i 67 barrels of mess pork @ 13.00 i 68 " beef. @ 9.73 ' 13 Jbnshels of oloTer seed »Q 7.50 $2143.80 Beceivod payment by note at 30 days. For MOHEISON. TAYLOE & Co., A. 0. QXNBT. 3 F< (( (I « For (( K( assm BILLS OF PARCELS. 133 83 Hamilton, January 2ad, 1866. Mr. M. McCullooii, To Joseph Light, Stationer, Dr. For 500 French envelopes. @ $3.00 per thousand. " 12 doz. British American copy books. ..@ 1.15 " 6 *' B. B. lead pencils @ .50 " 5 gross mournincr envelopes @ 1.05 " 2 reams mourning note paper @ 3.15 '* 4 " tinted note paper @ 3.15 ** 2J " Foreign note paper @ 1.40 " 3 " " letter paper @ 3.00 <» 1 doz. First Books @ .15 " 5 boxes Gillott's No. 303 pens @ .90 « 5 doz. Third Books @ 1.62J " 10 quires blank books, half tound (^ .35 " 2 paoks visiting cards ^ ,37j^ «71.98 KoTB^Billa should not be bigncd tuidl settled. l80 It. 34. Bbockville, Jan. 5th, 1866 N. D. Galbreaith, To R. FiTZSlMMONS & Co., J)r, For 24 lbs. Mackerel @ 05^ " 3 gallons Molasses @ 45 « 13 lbs. Young llyson Tea @ 87J " 13 lbs. brown Sugar , @ U <' 15 bushels of Potatoes 45 Or. For 10 lbs. Butter .» @ 17o» " 5doz. Eg^s @ 125- " 3 gallons Maple Molasses ,.^ @ 95 " Note at 20 d&ys, to balanw..* 17.05 122.23 R. FiTZSlMMONS & Co. Note.— Such a Bill as this would bo termed ft Barter Bill. ml 184 AlilTHMEHC. 35 KiNUoTON, Jan. :ind, 1866. •T/VMES Thompson, Esq., To A. Jaudine & Co., Dr For 3 doz. Buttons q 60.12 " 5J yards of black Broadcloth 5.50 *' 20 yards Shcctinir (k 15 " 1 chest Y. II. i -a, 83 lbs [95 " 18 yards French Print .20 " 2 skeins of Silk Thread @ .09 " 5 yards black Silk Velvet @ 3.50 " 20Jb3. Loaf Sugar .18 " 2 gallons MolassQs .40 " 1 bag of common Salt @ 1.15 " 25 lbs. Rice @ .09 " 3 sacks Coffee, 70 lbs. each @ .12 ^ , Cr. ei66.7'l By Cash..., 50.00 Balance due S116.7 I 36 Algonquin, Jan, 15th, 1865. W. rLEJIING & Co., Boughtof J. & A. WmonT, 1500 lbs. Canadian Cheese ^ g 09 300 bushels FaJl Wheat @ 1.25 9 brls Pot Ash, net 7050 lbs @ 5.75 per cwt. 150 bushels Spring Wheat (a I.15 200 " Potatoes ; X 45 600 " Oats @ .374 150 " Pease @ .65 50 " Indian Corn '. @ .50 60 " Apples @ .60 3 kegs Butter, UO lbs. e^-cK @ .18 60 bushels llyo , @ .70 40 " Barley @ .80 $1688.12 Kccoivcd payment, J. & A. WRIGHT. >iWi i fmiWJFiii FEBCEKTAGE. 135 Iwt. J.12 PERCENTAGE. 18. — Percentage is an allowance, or reduction, or estimate of a certain portion of each 100 of the units that enter into any given calculation. The term id a contraction of the Latin expression for one hundred, and means literally hy the hundred. In calculating dollars and cents, G per cent, means 6 dollars for every 100 dollars, or 6 cents for every $1, or 100 cents. If we are estimating the rato of yearly increase of the population of a rising village, and find that at the end of a certain year it was 100, and at the end of the next it was 106, we say it has increased 6 per cent. i. c, 6 persons have heen added to the 100. So, also, if a large city has a population of 100,000 at the end of a certain year, and it is found that it has 106,000 at the end of the following year, we say it has increased 6 per cent., which means that if we count the population by hundreds wo shall find that for every 100 at the end of the one year, there are 106 at the end of the next ; because one hundred thousands is the same as one thousand hundreds, and we have supposed the increase in every 100 to be 6, the total increase will be one thousand sixes or €,000, giving a total population of 106,000 as above, or an increase at the rato of 6 per cent. A decrease would be estimated in the same manner. Thus, a f$Jling off in the population of 6 persons in the hundred would be denoted by 100 — 6=94, as an increase of 6 in the hundred would be denoted by 100-f-6=:106. So, also, in our first example, a deduction of $6 in SlOO would be $100— 6=$94, and a gain would bo $100+$6=$106. The portion of 100 so allowed or estimated, is called the rate per cmt, as in the examples given, 6 demotes the rate per cent., or the allowance or estimate on every 100. Should the sum on which the estimate is made not reach 100, we can, nevertheless, estimate what is to be allowed on it at the same rate. Thus, if G is to be allowed for 100, then 3 must be allowed for 50, and 1^ for 25, &o. The number on which the percentage is estimated is called the basis. Thus, in the example given regarding the population of a city, 100,000 is the basis. When the basis and percentage are combined into one, the result is Bailed the amount. If the rat« per cent, be an increase or gain, it is to be added to the basis to get the amount, and if it is a decrease^ or loss, it is to be subtracted from the basis to get the amount. This latter result is sometimes called the remainder. i;i« 136 ABITHMETIO. E'rom what has been said, it is plain that percentage is nothing else than taking 100 as a standard unit of measure — (See Art. 1) — and making the rate a fraction of that unit, bo that G per cent, is yg^=(Art. 15, V.) .OG. Wc may obtain the same result by the rule of proportion. Thus, in our illustrative ei.ample of an increase of G persons for every 100 on a population of 100,000, the analogy will bo 100 persons : 100,000 persons : : G (the increase on 100) : 6,000, the increase on 100,000. It is manifest that the same result will be obtained whether wc multiply tlie third by the second, and divide by the first, or whether wc divide the third by the first, and multiply the result by the second ; or, which is the same thing, mul- tiply the second by the result. Now, we already found that 6s-100=j{;y==.0G, the same as before. So also, 7 per cent, of any loss is seven one-hundredths of it, i. c, jJyr^r.OT. It should be carefully observed that such decimals represent, not the rate per cent., hut the rate per unit. Though this is easily comprehended, yet we know by experience that learners are constantly liable to commit errors by neglecting to place the decimal point correctly. Wo w uld therefore direct parti- cular attention to the above caution, which, with the rule already laid down, under the head of decimal fractions, should bo sufficient to guide fmy one who takes even moderate pains. EXERCISES ON FINDING THE RATE PEK UNIT. At \ per cent., what is the rate per unit? Ans. .OOJ. At J per cent., what is the rate per unit? Ans. .OO-J. At 1 per cent., what is the rate per unit? Ans. .01. At 2 per cent., what is the rate per unit? Ans. .02. At 4 per cent., what is the rate per unit? « Ans. .04. At 7^ per cent., what is the rate per unit ? Ans. .07|-. At 10 per cent., what is the rate per unit ? Ans. .10. At 12i- per cent., what is the rate per unit ? Ans. .12f. At 17 per cent., what is the rate per unit? Ans. .17. At 25 per cent., what is the rate per unit? Ans. .25. At 33J^ per cent., what is the rate per unit ? Ans. .33J-. At GGj^ per cent., what is the rate per unit ? Ans. .66§. At 75 per cent., what is the rate per unit? Ans. .75. At 100 per cent., what is the rate per unit? Ans. 1.00. At 112J per cent., what is the rate per unit ? Ans. 1.12|. At 150 per cent., what is the rate per unit? Ans. 1.50. At 200 per cent,, what is the rate per unit ? Ans. 2.00. PEBCEKTAGE. 137 01. l02. |04. hi. ho. LT. i5. (5. )0. Ih )0. I. To find the percentage on any given quantity at a given rate; On the principles of proportion, -wc have as 100 : given quan- tity : : rate : percentage, and as the third term, divided by the first, gives the rate per unit, we have the simple RULE: Multiply the given quantity by the rate per unit, and the product will be the percentage. EXAMPLES. To find how much 6 per cent, is on 720 bushels of wheat, wc have 6-;-100=:.06, the rate per unit, and 720X 08=43^ bushels, the percentage. To find 8 per cent, of $7963-75, in like manner, we have .08, the rate per unit, and $7963.75 X -08 gives $637.10, the percentage. Instead o£per cent the mark (7o) is now commonly used. EXERCISES ON THE RULE. 1. What does 6 per cent, of 450 tons of hay amount to ? Ans, 27. 2. What is 10 per cent, of 6879.62J ? Ans. $87.96. 3. If 12 per cent, of an army of 47,800 men be lost in killed and wounded ; how many remain ? Ans. 42,064. 4. What is 5 per cent, of 187 bushels of potatoes ?, Ans. 9.35. fi. What is 2^ per cent, of a note for $870 ? Ans. 21.75. 6. Find 12^ per cent, of 97 hogsheads ? Ans. 12.12*. II. To find what rate per cent, one number is of another given number : — Let us take as an example, to find what per cent. 24 is of 96. Here the iiasis is 96, and we take 100 as a standard basis, and these are magnitudes of the same kind, and 24 is a certain rate on 96, and we wish to find what rdte it is on 100, and by the rule of proportion, we have the statement 96 ; 100 ; : 24 : r.P.=^-^§-a=25. Tlxcrefore 24 is 25 per cent, of 96. From this we can deduce the simple RULE.. Annex two ciphers to the given percentage^ and divide that hu tlie basis, the quotient will be the rate per cent, 7. Whai per cent, of 150 is 15 ? Ans. 10. 8. What per cent, of 240 is 36 ? Ans. 15. 138 AmTHliETIO. ». What per cent, of 18 is 2 ? Ans. 11^. 10. V/hat per cent, of 72 is 48 ? Ans. 66|. 11. What per cent, of 57G is 18 ? Ans. ^, 12. What per cent, is 12 of 480 ? Ans. 2 J. 13. Bought a block of buildings in King street for $1719, and sold it at a gain of 18 per cent, j what was the gain ? Ans. $309.42. 14. Vested $325 in an oil well speculation, and lost 8 per cent. ; whnt was the loss ? Ans. $26.00. 15. In 1841 the population of Cleveland was about 15,000, it is now about 50,000 ; what is the rate of increase ? Ans. 233^. 16. An estate worth $4,500 was sold j A bought 30 per cent, of it ; B, 25 per cent. ; C, 20 per cent. ; and D purchased the remain- der J what per cent, of the whole was D's share ? Ans. 25. 17. If a man walk at the rate of 4 miles an hour ; what per cent, is that of a journey of 32 miles ? Ans. 12J. 18. What is the percentage on 61370 at 2| per cent. ? Ans. 37.G7I-. III. Given, a number, and the rate per cent, which it is of another number, to find that other number, .400 is 40 per cent, of a certain number, to find that number. As 40 : 100 : : 400 : F. P.=i ±M>i^LDo^lfiQQ, Hence we derive the RULE. Annex two ciphers to tJte given number, and divide hy the rate •per cent. EXERGIBEB. i. A bankrupt can pay $2600, which is 80 per cent of his debts ; liow much does he owe ? Ans. $3250. 2. A clerk pays $8 a month for rent, which is 16 per cent, of his sularj' ; what is Jiis yearly salary ? ' Ans. $600. 3. In a manufacturing district in England, 40,000 persons died of cholera in 1832, this was 25 per cent, of the population ; what was the population ? Aus. 160,000. 4. Bought a certain number of bags of flour, and sold 124 of them, which is 12J per cent, of the whole. Required, the number of bags purchased. Ans. 992. 5. In a shipwreck 480 tons are lost, and this amount is 15 per cent, of the whole cargo. Find the cargo. Ana. 3200 tons. PEBCEKTAGE. 139 )ts; J50. t.of 1)00. lied Iwas ^00. of Lber |92. |per 6. A firm lost $1770 by the failure ot' another finn; tho loss was 30 pel' cent, of their capital ; what was their capital ? Ans. $5900. IV. To find the basis when the amount and rate are given : — Suppose a man buys a piece of land for a certain sum, and by selling it for $300, gains 25 per cent. ; what did he pay for it at first ? — Here it is plain that for every dollar of the cost, 25 cents arc gained by the sale, i. c, 125 cents for every 100, which gives us the analo- gy, 125 : 100:: 300 : F. P.; or, dividing the two terms by 100, 1.25 : 1.00 : : 300 : F. P., which by the rules for the multiplication and division of decimals, gives -^Jf)S-^=$240, the original cost. Again, suppose the farm had been sold at a loss of 25 per cent. This being a loss, wo subtract 25 from 100, and say, as 75 : 100 : : 300 : F. P.^iY5^=$400, tlie prime cost in this case. Hence we deiivc the RULE. Divide the given amount hi/ one increased or diminished hy the given rate per unit, according as the question implies increase or decrease, gain or loss. EXERCISES. 1. Given tho amount $198, and the rate of increase 20 per cent, to find the number yielding that percentage. Ans, $165. 2. A field yields 840 bushels of wheat, which is 250 per cent, on the seed ; how many bushels of seed were sown ? 3. At 5 per cent, gain j what is the basis if "the amount be $126 ? Ans. $120. 4. At 10 per cent, loss; what is the basis, tho amount being $328.5 ? Ans. $365, 5. A ship is sold for $1*2045, which is a gain of f per cent, on the sum originally paid for it ; for how much was it bought at first ? Ans. $12000. 0. A gambler lost 10 per cent, of his money by a venture, and had $279 left ; how much had he at first, and how much did he lose ? Ans. He lost $31, and had $310 at first. 7. A grocer bought a lot of flour, and having lost 20 per cent, of tho whole, had 160 bags remaining j how many bags did bo buy? Ans. 200. 8. A merchant lost 12 per cent, of his capital by a bankruptcy, and had still $2200 left ; what was his whole capital ? Ans. $2500. 140 AKITHMETIO. 9. Sold a sheep for $5, and gaiued 25 per cent. ; what did I pay for it? Ans. H- 10. Lost $1 2000 on an investment, which was 30 per cent. . f the whole ; what waa tho investment ? Ana, ^0000. INTEREST. From a transition common iii language, tho word interest has been inappropriately applied to the sum paid for the use of money, hut its original and trae moaning is simply the use of money. To illustrate this, we will suppose that A horrows of B $100 for ono year, an.l at the end cf tho year, when A wishes to settle the account, he gives B $107. Were we to ask the question of almost any per- son except an accountant, whether A or B received the interest, wo should undoubtedly receive for an answer that B received it. But such is not the case. A having had the use of that money for ono year, paid B $7 for that use or interest ; hence A received the inter- terest or use of that money, and B received $7 in cash for tho same. It is only by considering this subject in its true light that account- ants are able to determine upon the proper debits and credits that arise from a transaction where interest is involved. If an individual borrows money, ho receives the use of that money, and when he pays for that use or interest, he places the sum so paid to that side of his "interest account" which represents interest received, and if he lends money, he Jias parted with the use of that money, and when he re- ceives value for that use or interest, he places tl.e sum so received to that side of his " interest account" which represents interest de- livered. Wc think that this explanation is sufficiently clear to illustrate the difference between interest and the value received ox paid for it. It will also be noticed that we havo given many of the exercises in the usual form, c. ducc $4.50 interest in 1 year, 3 months, at 6 per cent. ? SOLUTION. If a principal of $1 is put on Interest for 1 year, 3 months, at 6 per ocnt., it will produce .075 interest. Now*, if In this example, .075 be the interest on $1, the number of dollars required to produce $4.50, \vill be represented by the number of times that .075 is corL*- tained in $4.50, which is CO times. Therefore, $60 will produce $4.50 interest in 1 year, 3 months, at 6 per ocnt. Hence the SIMPLE INTEBEST. RULE. 151 6 Divide the given xnt&M 6y ths interest of $1 for the given time, oi the given rate per cent. EXEBOISES. 98. What principal will produce 77 cents interest in 3 months, 9 days, at 7 per cent. ? Ans. $40. 99. What principal will produce $10.7] interest in 8 months, 12 days, at 7^ per cent. ? Ans. $204. 100. What principal will produce $31.50 interest in 4 years, at 3^ per cent. ? Ans. $225. 101. What sum of money will produce $79.30 interest in 2 years, 6 months, 15 days, at 6^ per cent. ? Ans. $480. 102. What sum of money is sufficient to produce $290 interest in 2 years and 6 mouths, at 7^ per cent. ? Ans. $1600. OASEIX. , To find the rate Feb cent., the principal, the interest, and the time being given. EXAMPLE. 103: If $3 be the interest of $60 for 1 year, wnat is the rate per cent.? SOLUTION. If the interest of $60 for 1 year, at 1 per cent, is .60, the re- quired rate per •■'.ent. will be represented by the number of times that .60 is contained in 3.00, which is 5 times. Therefore, if $3 is tho interest of $60 for 1 yeai', the rate per cent, is 5. Hence the RULE. Divide the given interest by the in 'est of the given principal at 1 per cent, for the given time. EXERCISES. 104. If the interest of $40, for 2 years, 9 months, 12 days, is $13.36 J what is the rate per cent. ? Ans. 12. 105. If I borrovr $75 for 2 months, tiud pay $1 interest ; what is the rate per cent. ? Ans. 8. 162 ABITHMETIO. 106. If I give 02^5 for the use of $30 for 9 moi 'i9 ; what rate per cent, am I paying ? Ans. 10. 107. At what rate per cent, will $150 amount to $165.75, in 1 year, 4 montiis, 24 days ? Ans. 7^. 108. At what rate per cent, must $1, or any sum of money, bo on Interest to double itself in 12 years ? Ans. Ans. 8 J. 109. At what rate per cent must $425 be lent to gain $11.73 in 3 months, 18 days? Ans. 9^. 110. At what rate per cent, will any sum of noncy amount to three times itself in 25 years ? Ans. 8. 111. If I give 814 for the interest of $125 for 1 year, 7 months, 6 days ; what rate per cent am I paying ? Ans. 7. CASE X. To find the time, the principal, the interest, and the rate per cent, being given. EXAMPLE. 112. How long must $75 be at interest, at 8 per cent., to gain $12? ' SOLUTION. The interest for $75, for 1 year, at 8 per cent., is $0. Now, if $75 require to be on interest for 1 year to produce $6, it is evident that the number of years required to produce $12 interest, will be icprescntel by the number of times that 6 is contained in 12, which is 2. Thereftre, $75 will have to be at interest for 2 years to gain $12. Hence the KULE . Divide the given interest by the interest of tlie prindjtal for one year, at the given rate per cent. EXERCISES. 113. In what time will $12 produce $2.88 interest, at 8 per cent ? Ans. 3 years. 114. In what time will $25 produce 50 cents interest, at 6 per cent. ? Ans. 4 months. 115. In what time will $40 produce 75 cents interest, at 6;^ per cent. ? Ans. 3 montlis, 18 days. SIMPLE TNTEBEST. 153 116. In what time will any sum of moacy double itself, at G per cent. ? Ans. 16 years, 8 months. 117. In what time will any sum of money quadruple itself, at 9 per cent. ? Ans. 33 years, 4 months. 118. In what time will $125 amount to $138.75, at 8 per cent. 7 Ans. 1 year, 4 months, 15 days. 119. Borrowed, January 1, 1805, $00, at G per cent, to be paid as soon as the interest amounted to one-half the principal. AVhen is it due ? Ans. May 1, 1873. 120. A merchant borrowed a certain sum of money on January 2, 1850, at 9 per cent., agreeing to settle the account when the in- terest eq^ualled the principal. When should he pay the same ? Ans. Feb. 12, 1367. merchants' table For showing in what time any sum of money will double itself, at any rate per cent., from one to twenty, simple interest. 1 — 1 Per cent. Yoiirs, Per cent. Years. Per cent. Yoarf-'. L*oi' cent. i'ears. 1 100 G IGff 11 Q I 16 6|r 2 50 7 14 l4 12 17 5;5 3 33^ 8 13 7v;, 18 5^ 4 25 9 Hi 14 7 19 h% 5 20 10 10^ 15 20 5 MIXED EXERCISE K. 121. What is the interest on $64.25 for 3 years, at 7 per cent.? Ans. $13.49. 122. What is the interest on $125.40 for G months, at G per cent. 7"^ Ans. 3.76. 123. What is the amount of S369.29 for 2 years, 3 months, 1 day, at 9 per cent. ? Ans. $444.1G. 124. What must bo paid for the use of 75 cents for G years, 9 months, 3 days, at 10 per cent. ? ^ Ans. 51 cents. 125. What will $54 amount to in 254 days, at 10 per cent. ?* Ans. $57.81. * Thia and the followiog exercised Cmarked with a *) are to be worked \>J Case VI. i/>4 ABITHMETIO. 126. What must be paid for the interest of $45 for 72 days, at 9 per cent. ?* Ana. 81 cents. 127. What is the interest of $240 from January 1, 1866, b June 4, 1866, at 7 per cent. ? Ans. $7.14. 128. What will $140.40 amount to from August 29, 1865, to November 29, 1866, at 6J per cent. ? Ans. $151.83. 129. What principal will give $4.40 interest in 1 year, 4 months, 15 days, at 8 per cent. ? Ans. $40. 130. In what time will $40 amount to $'14.40, at 8 per cent.? Ans. 1 yr., 4 mos., 15 days. 131. At what rate per ceat. will $40 produce in 1 yr., 4 mos., 15 days, $4.40 interest ? Ans. 8. 132. What must be paid mr the interest of $145.50 for 240 days, at 91 per cent. ?* Ans. $9.22. 133. Wliat will $160 amount to in 175 days, at 6 per cent. ?* Ans. $164.67. 1J4. At what rate per cent, must any sum of money be on interest to quadruple itself in 33 years and 4 months ? Ans. 9. 135. In wliat tunc will any sum of money double itself, at 10 per cent. ? Ans. 10 years. CASE X I. To find the interest on bonda, notes, or other documents draw- ing 7 i^u percent, interest. Since .07/^ or, .073 would be the rate per unit, or the interest of $1 for I year or 865 days, it follows that the interest for 1 day would be the -^^ part of .073 which is .0002, equal to two tenths of a mill, hence the RULE. Multiply the principal hj the nuniber of da}js, and the product by two tenths of a mill the result will he the answer in mills. EXAMPLE. What must be paid for the use of $75 for 36 days at 7-f^ per so LtJTlON. The interest on $75 for 36 days would bt the same as the inter- est on $75X36=$2700 for 1 day, and at fy of a mill per day would be $2700X-.0002=54 cents. 2. What would be the interest on $118.30 for 42 days at 7/^ j^bt cent. Ans. 99ctd. COmiEBCIAL PAPEB. 156 COMMERCIAL PAPER. iw- tuct Icr- \ay % ltd. OouuEiiOlAL paper is divided into two classes — iTzaOTlAfiLE aud NON-N£CK)TIABL£. NEGOTIABLE COMMERCIAL PAPER. Negotiable commercial paper is that which may bo freely trans- ferred from one owner to another, so as to pass the right of action to the holder, without being subject to any set-offs, or legal or equitable defences existing between the original parties, if transferred for a valuable consideration before maturity, and received without any defect therein. Negotiable paper is made payable to the payee therein named, or to his order, or to the payee or bearer, or to bearer ; or some similar term is used ; showing that the maker intends to give the payee authority to transfer it to a third party, free from all set-offs, or equitable or legal defences existing between himself and the payee. NON-NEOOTIABLE COM ME U CIA I. PAPER. Non-negotiahle commercial paper is that which is made payable to the payee therein named, without authority to transfer it to u third party. It may be passed from one owner to another by assign- ment, or by indorsement, but it passes subject to all set-offs, and legal or equitable defences existing between the original parties. HOW XIIE TITLE PASSES. The title to n^otiable paper passes from one owner to another by delivery, if made payable to payee or bearer, or to bearer. It passes by indorsement and delivery, if made payable to payee or order. The title to non-negotiable paper passes by a mere verbal assignment and delivery, or by indorsement and delivery. PRIMARY DEBTOR. In a promis&ory note there are two original parties — the maker .'.nd the payee. The obligation of the ma' r is absolute, and con- tinues until the note is presumed to have been paid under the Statute of Limitations. The maker is the primary debtor. In a bill of exchange there are three parties. AVhcn the drawee accepts the bill^ ho becomes the primary debtor upon the bill of exchange, PROMISSORY NOTE NOT PAYABLE IN MONEY. "When a promissory note is payable in anything but money, it docs not come within the Statute. There is no presumption that it is founded upon a valuable consideration. A consideration must bo \ 156 AniTinrETio. alleged in the complaint, .'nid proved on tho trial. The acknowledg- ment of n considerutiou in tuch promissory note, by inserting the vrords " value rccnrci/," i.s pufiicicnt to cast upon tho defendant the burden of proof that tlicro was no consideration. Tho acknowledg- ment of "vuluu received," raises tho presumption that tho note was given for value ; but this presumption may be rebutted by the do- I'cndiiut. A ncgotiublo instrumont i.s ;i written promise or request for the payment of a ccrtuin hum cf money to order or bearer. A negotiable instrument must be made payable in money only, and without any condition not certain of fulfillment. The person, to who.'^o order a negotiable instrument is made payable, must bo ascertainable at tho time the instrument is made. A negotiable instrument may j^ivc to tho payee an option between the i)aymcnt of the sum sp;K'ilied therein, and tho performance of another act. A negotiable instrument may bo with or without date; with or without seal ; and with or without designation of tho time or place of payment. A negotiable instrument may contain a pledge of collateral secu- rity, with authority to dispose thereof. A negotiable instrument must not contain any other contract than sucli as is f;pccified. Two different contracts cannot be ad- mitted.' Any date may be inserted by the maker ef a negotiable instru- ment, whether past, present, or future, and the instrument is not invalidated by his death or incapacity at the time of the nominal date. There are several classes of negotiable instruments, namely : — 1. Bills of Exchange; 2. Promissory Notes; 3. Bank Notes; 4. Cheques on Banks and Bankers ; 5. Coupon Bonds ; 6. Certifi- cates of Deposit; 7. Letters of Credit. A negotiable instrument that doss uut specify the time of pay- ment, is payable immediately. A negotiable instrument which docs not specify a place of pay- ment, is payable wherever it is held at its maturity. An instrument, otherwise negotiiiblc in form, payable to a person named, bu' adding the words, "or to his order," or " to bearer," or equivalent thereto, is in the former case ruyable to the written order of such person, and in the latter caisc, payable to the bearer. A negotiable instrument, made payable to the order of the maker, or of a fictitious person, if issued by the maker for a valid considera- tion, without indorsement, has the same effect against him and all other persons having notice of the facts, as if pajrablo to the bearer. A negotiable instrument, made payable to the order of a person obviously fictitious, is payable to the bearer. The eigaature of every drawer, acceptor (lud iadorser of a uego* COMMERCIAL PAPER. 157 @on » ten ler, Ira- lall ler. tiablo instrnmcnt, is presumed to have been xnaao for a valuable consideration, before the maturity of the instrument, and in the ordinary course of business, and the words "value received," acknowledge a consideration. Ono who '^rites h\s name upon a negotiable instrument, otherwise than as a maker or acceptor, and delivers it, with his name thcveon, to another person, is called am indorser, and his act is called an indorsement. One who agrees to indorse a negotiable instrument is bound to write his signature upon the back of the instrument, if there is sufficient space thereon for that purpose. When there is not room for a signature upon the back of a nego- tiable instrument, a signature equivalent to an indorsement thereof may bo made upon a paper annexed thereto. ' An indorsement may bo general or special. A general indorsement is ono by which no indorser is named. A special indorsement specifies the indorsee. A negotiable instrument bearing a general indorsement cannot be afterwards specially indorsed ; but any lawful holder may turn a general indorsement into a special one, bywriting above it a direction for payment to a particular person. A special indorsement may, by express words for that purpose, but not otherwise, bo so made as to render the instrument not negoti- able. Every indorser of a negotiable instrument warrants to every eubse- quent holder thereof, who is not liable thereon to him : 1. That it is in all I'espects what it purports to be ; 2. That ho has a good title to it; il. That the signatures of all prior parties are binding upon tbcm ; 4. That if the instrument is dishonored, the indorser will, upon notice thereof duly given unto him, or without notice, where it is excused by law, pay so much of the samo-as the holder paid therefor, with interest. One who indorses a negotiable instrument before it is delivered to tlic payee, is liable to the payee thereon, as an indorser. An indorser may qualify his indorsement with the words, " with- out recourse," or equivalent words; and upon such indorsement, he is responsible only to the same extent as in the case of a transfer without indorsement. Except as otherwise prescribed by the last section, an indorse- ment " without recourse" has the same effect as any other indorse- ment. An indorsee of a negotiable instrument has the same right against every prior party thereto, that he would have had if tho contract had been made directly be jn them in the first instance. An indorser has all the ri{2;hts Oi, * guarantor, and is exonerated from liability in like manner. 158 ARinWIET.C. ''I iii Ono who indorses r. negotiable instrument, at the request, and for tho " accoraniodation" of another party to tho instrument, haa all the rights of a Burcty, and i-. cxontv.itcd in like manner, in respect to every one having notice of tlio facts, except that ho is not entitled to contribution from subsequent iiidor.scrs. The want of consideration for tho undertaking of a maker, acceptor, or indorscr of a negotiable instrument, docs not exonerate him i'rom liability thereon, to j:n indorsee in good I'aith for a consid- eration. An indorsee iu due course la ono who in good faith, in tho ordi- nary course of business^ and for value, before its apparent maturity ur presumptive dishonor, acquires a negotiable instrument duly indorsed to him, or indorsed generally, or payable to the bearer. An indorser of a negotiable instrument, in due course, acquires an absolute title thereto, so that it is valid in his hands, notwith- standing any provision of law making it generally void or voidable, and notwithstanding any defect iu the title of the person irom whom ho acquired it. One who makes himself a party to an instrument intended to be negotiable, but which is left wholly or partly in blank, for the pur- pose of filling afterwards, is liable upon the instrument to an indorsee thereof in duo course, in whatever manner, and at whatever time it may be filled, so long as it remains negotiable in form. It is not necessary to make a demand of payment upon the jrineipal debtor in a negotiable instrument in order to charge him ; )ut if the instrument is by its terms payable at a specified place, and 10 is able and willing to pay it there at maturity, such ability and willingness arc equivalent to an offer of payment upon his part. Presentment of a negotiable instrument for payment, when necessary, must be made as follows, as nearly as by reasonable dili- gence it is practicable : 1. Tho instr\iment must be presented by the holder, or Lis authorized agent. 2. The instrument must bo presented to the principal debtor, if ho can ho found at the place where presentment should be made, and if not, then it must bo presented to some other person of discretion, if one can bo found there, and if not, then it must be presented to some other person of discretion, if one can be found there, and if not, then it must be presented to a notary public within the State ; 3. An instrument which specifies a ])lacc for its payment, must be presented there, and if tho place specified includes more than one house, then at the place of residence or business of the principal debtor, if it can be found therein ; 4. An instrument which does not tie bringen solche auf Eechnung law Bericht von Ilerrn ' London. Italun. Livorno, le 25 Seltonhre, 1848. Per £500 Sterlinc. A Trc mesi data pagate per qiicsia prima de Camhio (una sol voUa) alV ordinc . la somma di Lire cinque cento sterline valuta cam' hitta, e ptmeie in conto M. S. sccondo Vavviso Addio. Al Londra. Spanisu. Malaga, a 20 de Sdb^ de 1848. Son £300. A noventa diasfecha se serviran V' mandarpagar por csla ly/intra de cam- hio a la orden de los S'"'* ires cientas libras Esterlinaa en oro o plata valor recibido de dhos S"* que anotaran valor en cuenta segun aviso de ,____^— _^____^______ A los S'-" Londres. POBTrOtTESB. £C00 Esterlinas. Lisbon, aos 8 dc Dczemhro de 1848. A Sessenta dias de vista precizos pagard V por esia nossa unica via de Letra tSegura, d nossa Ordtm a quantia acima de Seis Centas Livras Esterlinas valor da nos recibido em Fazendas, que passera em Comta scgundo o aviso de . , Ao Sen __ Londres. T3irx OF Exchange on London'. £317 19s. ICd. rhiladeJphia, Oct. iHih, 18GG. Sixty days after slglit of this, my first Bill of Exchange (second and third of the same elate, and tenor unpaid), pay to the order of Williams & Mann, Three Hundred and Forty-seven Pounds, nineteen Shillings and Ten pence, Sterling, value received, with or without further advice. KERR. BROWN & Co. To II. n. Gladstone, Banker, London. Lnlanti Dr..iFT. $971 ^J^ CJiicago, Sept. Wh, 1866. Nindy days after sight, pay to the order of Manning and Munson, Mt.c Hundred and Seventy-one and ^ DoUara. vabie received, aiid charge Vu same to our account. SMITH & EVANS. Tu Samuel Sy.iLL & Co., BcMmortt M± 162 ABITHMETIO. Bills of Excliangc are the highest dass of commercial paper known to the law, and it has never been the ciierished object of the law merchant, — which has been permitted by the English courts to insinuate itself into the common law, till it now forms a part of that code, — to uphold them inviolate, as far as possible. While the la> mercatoria (or mercantile law) is deeply impregnated with tho principles of equity, those principles have been chiefly marked, to enable courts of law to enforce equitable rights, and upon this principle was the negotiability of bills of exchange insisted upon and finally maintained at the common law ; but when equitable principles have been invoked for the purpose of destroying the validity and security of bills of exchange, they have been iiat0UC(l to with great disfavor and only admitted as exceptional cwfi CHECKS. '•1\ A chock is snBstantially tho same as an inland bill of exchange : it passes by delivery, when payable to bearer, and the rules as to presentment, diligence of the holder, &c., which are applicable to the one, are gene ••>lly applicable to the other. 2. A chbv . is an appropriation of the drawer's funds, in tho hands of the banker, to tho amount thereof, and, consequently, tho drawee has no right to withdraw them before the check is pai(L 3. The characteristics which distinguish checks from bills of exchange are, that checks are always drawn on a bank or banker ; that they are payable immediately upon presentment, and without days of grace ; and that they are not presentable for acceptance, but only for payment. The want of due presentment of a check, and notice of the non-payment thereof only exonerates the drawer in so far as actual damages have thereby resulted to him. LETTERS OF CREDIT. In addition to the commercial paper before mentioned, there i3 an extensive business done by the issue of " Letters of Credit.^' These arc issued by prominent bankers in London, Paris, New York and other cities, to travellers who arc about to visit foreign countries, and who arc thus saved the risk and expense of carrying any large amount of cash about them. These Letters op Credit are addressed by the banker to his correspondents abroad, authorizing any one or more of them to pay to the person named, any portion of the sum mentioned in the letter. Thus a person leaving New York for the Pacific Ports, South America or Arctic Ports, or any city or place in Europe or other portions of the world, need carry very little cash. At the first port of arrival he is able to realize such funds as may bo Qfioessary to pay COMMERCIAL PAPER. ts to 163 hia expenses to a turthcr port by using his Letters of Credit. A traveller may go round the world, with the aid of such a Credit, and never have more than one hundred do'llars in his pocket. No loss from exchange need occur, in such cases : bills on London being iu demand throughout the civilized world. The usual charge by the bankers for such *' Letters of Credit," is one per cent, where the trader docs not pay the amount of tho Letter in advance. Where he pays in advance, no charge is made ; the use of the money ia tho banker's hands being an equivalent for the cost of tho credit. Letters of Credit are also extensively used by importers when travelling abroad for tho purchase of goods j also by supercargoes and captains of vessels for the purchase of cargoes in foreign ports ; also as remittances to distant ports in Asia, Australia, &c., for the purchase of cargoes of foreign goods. Before Letters of Credit were adopted or in circulation, it was the practice among American and other merchants to remit specie to remote parts for investment in foreign merchandize tho the ,A ».\^li3y I DAYS OF GRACE. 'I 1. In moBt countries, wten a bill or note is payublu nc p certain time after datC; or after sight, or after demand, it is not payable tho precise time mentioned in the bill or note, but days of grace are allowed. 2. The days of grace are so called, because they were formerly gratuitous, and not to be claimed as a right by the person on whom it was incumbent to pay the bill, and were dependant oa the inclina- tion of the holder ; they still retain the name of days of grace, thougli the custom of mcrekints, recognized by law, has long reduced them to a certainty, and established them as a right. 3. In England, Scotland, Wales, and Ireland, three days grace ure allowed ; in other countries they vary from three to twelve days. 4. The days of graco as allowed in England, are generally allowed ia the United States, at least no traces can be found of a contrary decision, except in the Stater of Massachusetts, where it has been held that no days of graco are allowable, unless stipulated in the contract itself. Jt is probable that a bill of exchange was, in its original, nothing more than a letter of credit from a merchant in one country, to hia debtor, a merchant in another, requesting him to pay the debt to a third person, who carried the letter, and happened to bo travelling; to the place where the debtor resided. It was discovered, by experi- ence, that this mode of making payments was extremely convenient to all parties: -to the creditor, lor he could thus receive his debt trithout troabloi risk or expenae— to the debtor, for the faoilitv of 16i ARITHMETIC. payment was an equal accommodation to him, and perhaps drew after it facility of credit to the bearer of the letter, who found himself in funds in a foreign country, without the danger and incumbrance of carrying specie. At first, perhaps, the letter contained many other things besides the order to give credit. But it was found that the original bearer might often, with advantage, transfer it to another. The letter was then disencumbered of all other matter ; it was opened and not sealed, and the page on which it was written, gradually shrunk to the slip now in use. The assignee was, perhaps, desirous to know beforehand whether the party to whom it was uddrcssed would pay, and sometimes showed it to him for that pur- pose; his promise to pay was the origin of acceptances. These letters or bills, the representatives of debts due in a foreign country, were sometimes more, sometimes less, in demand ; they became, by degrees, articles of traffic ; and the present complicated and cvbfitruse practice and theory of exchange was gradually formed^ \... PABTIAL PAYMENTS Tartitil payments, as the term indicates, arc the part payments of promissory notes, bonds, or other obligations. When these payments are made the creditor specifies in writing, on the back of the note, or other instrument, the sum paid, and the time when it is paid, and acknowledges it by signing his name. The method approved of by the Supreme Court of the United Stales, for casting interest upon bonds, notes, or other obligations, upon which partial payments have been made, is to apply the pay- ment, in the first place, to the discharge of the interest then due. If the payment exceeds the interest, the surplus goes towards discharg- uig the principal, and the subsequent interest is to be computed on the balance of the principal remaining due. If the payment be less than the interest, the surplus of interest must not be taken to aug- ment the principal, but interest continues on the former principal until the time when the payments, taken together, exceed the in- terest due, and then the surplus ia to be applied towards discharging the principal. RiJLi:. Tind the airmunt of the principal to the time of the first pay- meat ; subtract the payment from the amount, and then find the amount of the remainder to the time of the second payment ; deduct the payment as before; and so on to the time of settlement. But if any payment is less thar the interest then due, find the amount of the sum due to the time when, the payments, added to- gether, shall be equal, at least, to the interest already due ; then find the balance, ana proceed as before. PABTIAL PAIMSNIS. 166 EXAMPLE. 1. On the 4th of January, 1865, a note was given for $800. payable on demand, with interest ^t 6 per cent. The following pay* monts weie receipted on the back of the note : February 7th, 1865, received. $150 April 16th, **^ « 100 Sept., 30th, " " 180 January 4th, 1866. " 170 ^ March 24th, « " 100 June 12th, " " 50 Settled July 1st, 1867. How much was due • solution: Face of the note, or principal $800.00 Interest on the same to February 7th, 1865 (1 month, 3 days) 4.40 Amount due at time of 1st payment ► 804.40 First payment to be taken from this amount 1 50.00 Balance remaining due February 7th, 1865. 654.40 Interest on the same from February 7th, 1865, to April 16th, 1865 7.525 Amount due at time of 2nd payment 661.925 Second payment to be taken from this amount 100.000 Balance remaining due April 16th, 1865 561.925 Interest on the same from April 16th, 1865, to September SOth, 1865 15.359 Amount due at time of 3rd payment 577.284 Third payment to be taken from this amount 180.000 "Balance remaining duo Sept. 30th, 1865 397.284 Interest on the same from Sept. 30th, 1866, to January 4thj 1866 6.290 Amount due at time of 4th payment 403.574 Foaxthj)ajmentto be taken from thia amount 170.000 BaloDOo remaining due January 4tht 1866 233.574 166 AEirmiETia Interest on the same from Jan. 4th, 186G, to March 24tb, 1866 3.114 Amotint due at time of 5th paymeoi 236.688 Piftb payment to bo taken from this amount 100.000 Balance remaining due, March 24th, 1866 136.688 Interest on the same from 3Iarch 24th, 1866, to Juno 12th, 1866 1.799 Amount due at time of 6th payment 138.487 Sixth payment to he taken from this amount 50.000 Balance remaining due June 12th, 1866 88.487 Interest on the same from June 12Ui, 1866, to July 1st, 1867 5.589 Amount due on settlement 94.076 IP 1: 2. $1600. CHAKLEST0N,Februai7 16th, 1865. On demand, I promise to paij Jacob Anderson, or order, one tlwu&and six hundred dollars, with intereM, at 7 j^cr cent. Jon.N Fortune Jr. There was paid on this note, April 19th, 1865 $460 July 22nd " 150 August 25Ui, 1866 50 Sept. 12th, " 100 Dec. 24th. " 700 How much was duo December 31st, 1866 ? ' SOLUTION. Face of the note or principal , $1600.00 Interest on the same from Feb. 16th, 1865, to April 19th, 1865 19.60 Amount due at time of 1st payment. 1619.60 Fir£" payment to be taken from this amount... 460.00 Balance rcmoiuiu? due, April 19th. 1865 1159.60 )0 PABTIAL PAyiffENTS. l67 Interest on the same from April 19th, 1865, to July 22nd, 1865 20.969 Amount dae at time of 2nd payment 1180.5G9 Second payment to be taken from this amount 150.000 Balance remaining due, July 22nd, 1865 X030.569 Interest on the same from July 22nd, 1865, to Aug. 25th, 1866, greater than 3rd payment,* Interest on the same fro n July 22nd, 1865, to Sept. 12th, 1866 82.359 Amount due at time of 4th payment ', 1112.928 Third and fourth payments to be taken from this amount, 150.000 Balance remaining due Sept. 12th, 1866 962.928 Interest on the same from Sept. 12th, 1866, to Dec. 24th, 1866 19.098 Amount due at time of last payment 982.026 Last payment to be taken from this amount 700.000 Balance remaining due Dec. 24th, 1866 282.026 Interest on the same from Dec. 24th, 1866, to Dec. 31st, 1866 382 Amount due at time of settlement, Dec. 31st, 1866 $282,403 3. $350. " ' ^ BosTON,May 1st, 1864. On demand I promise to pay William Brown^ or order, three hundred and fifty dollars, with interest, at 6 per cent. James Weston. There was paid on this note, December 25th, 1864 $50 June 30th, 1865 5 * The interest on S1030.5G17, from July 22nd, 18G5, to August 25th, 18G6, Is $78,752, and the payment made at this dnte, is only $50, not enough to pay the interest, so if wo proceeded, as in the former case, to add the interest to the principal, and subtract ; est, $78,752, over the payment, $50, which would be in elTect interest interest or coir^'iund interest which the law doest not allow. |!|I I 168 ' ARITHMETIO. August 22nd, 1866 lo Juno 4th, 1867 100 flow much was duo April 5th, 1868 ? Ana. $251.67. 4. $609.65. Brantford, Juno 8th, 1861. Six montlu after date, toe jointly and severally promisi to pay John Anderson, or order, six hundred and nine -f'^'^ dollars, at the Royal Canadian Bank in Toronto^ with interest at per cent after maturity. S/MUEL Graham. T. B. Bearman. There was paid on this note, October 4th, 1862 $25.00 March 15th, 1863 16.25 August 24th, 18G4 3G.50 "What was due December 19th, 1865 ? AnS. 670.27. 6. $874.95. Kingston, May 9th, 1863. Three montJia after date, I jiromise to pay Harmon Cwmmings, or order, eight hundred and seventy four -f^ dollars, with interest after maturity at 6^jcr cent. Thomas Goodpay. Thero ivas paid on this note, / April 12th, 1864 $56.30 July 14th, 1865 24.80 . Sept. 18th, :866 240.60 What was due February 9th, 1868 ? Aus. $773.07. When the interest accruing on a uoto is to be paid annually sdopt the following RULE.* Compute tJic interest on (he principal to the time of settlementf and on each year's interest after it is due, then add the sum of tha * "When notes, bonds, or other obligations, are given, " with interest payable annually," the interest is due at the end of each year, and may be collected, but if not collected at that time, the interest due draws only simple interest, and the original principal must not be increased by any addition of yearly interest. If nothing has been paid until maturity on a note drawinjf annual interest, the amount due consists of the principal, the total annual iuterest, or the simple interest, and the simple interest on encli item of annnol intor<*Et from the time it bacame duo uutU paidi I'AKTIAL FAYMKNTO. Igg interests on the annual interests to the amount of tfie principal, ana from this amount take the paymtnts^ and the interest on each, from the time they wercpa,Ui (g th,Q time of settlement, the remainder will be the amounC due* 6. $500. Pbescott, May Ist, 1864. One yeur after date, for value received, /promise to pay Musgrovc & Wright, or order, Five Hundred Dollars, at their office^ in the city of Toronto, with interest at Qper cent., pay ahle annually. James J^Ianninq. There was paid on this note : May 4th, 1865 $150 Dec. 18th, " 300 How much waa due Juno 1st, 1866 ? S HT T I N . Fa«e of note, or principal $500.00 Interest on the same from May 1st 1864, to June 1st, 1866 62.50 Amount of the principal at time of settlement , C}Q2«50 First year's interest on principal $30 Interest on the same from May 1st, 1865, to June 1st, 1866 $1.95 Second year's interest on principal $30 Interest on the same from May 1st, 1866, to Juno 1st, 1866 15 Amotini; of interest upoa annual interest 2.10 Total amount of principal -$564*60 First payment, May 4th, 1865 $150.00 Interest on the same from May 4th, 1805, to June 1st, 1866 9.70 Second payment, December 18th, 1865 300.00 Interest on the same from December 18th, 1865, to June 1st, 1866 8.20 Pajnnents and interest oa the same. 467.90 Amount due June 1st, 1866, $96.70 1 70 ABITHMETIO. 7. $700. Cincinnati, Jauuary 2n(I, 1863. Eighteen months after date, I promise to pay to thi order of J. II. TV7?sow, Seven Hundred Dollars, for value rcctived, with interest at G^;cr cent., pai/uble annually. Thos. a. Bryce. There was paid on this note : January 15th, 1864 $350 July 2ntl, 18G4 300 What amount was due January 2od, 1865 ? Aus. $107 22. 8. $950. Indianapolis, Jan.Srd, 1863. Tioo years after date, I promise to pay A. R. Tennison, or order, Nine hundred and Fifty Dollars, loith interest at d per cent, payable annually, value received. James S. Paumenteb. The following payments were receipted on the back of this note: February Int. 1864, received $500 May 14th, " " 100 January 12, 1865, " 300 Wliat was due May 6th, 1865 ? Ans. $18^94. 9. $250. Mobile, January 2nd, 1863. Three years froui date, for value received, I promise to pay Michael Wright, or order, Two Hundred and Fifty Dollars, with interest, payable annually, ot 6 per cent. Calvin W. Pearsons. At First National Banh here. "What was the amount of this note at maturity ? Ans. $297.70. CONNECTICUT RULE. The Supreme Court of tho State of Connectioub has adopted tho following R IT L E . Compute the interest oil the principal to the time of the first pay- ment ; if that be one year or more from the time the interest com- menced,addit to the principal, and deduct the payment from the sum total. If there be after payments made, compute the interest on the balance due to the next jiaymmt, and then deduct the payment as dove., and in like nmnnarfronv onepaymmt to (nmther; V'H a?^ thr r lie m IS PAETIAL PAYMENTS. 171 payments are absorbed, provided the time between one payment and another he one year or more. If any payments be made be/ore one year's interest has accrued, then compute the interest on the principal sum due on the obligation for one year, add it to the principal, and compute the interest on (he sum paid, from the time it toas paid, up to the end of the year ; add it to the sum paid, and deduct that sum from the principal and interest, added as above. If any payments be made, of a less sum than the interect arising at the time of such payment, no interest is to be computed, but only on the principal sum for any period. Note.— If a year extends beyond the time of sei/Zfinicn<, find tbo amount of the remaining principal to the time oi scltkment ; And also tbo uiuoimt of tbo payment or payments, if any, from the time they were paid to the tiuic of suttlemeat, and subti'act Iheir sum from the amount of the i)riuclpal. i::i:amples. 10. $900, Kingston, Juno 1st, 18G2. On demand toe promise to pay J. R. Smith S Co., or order, nine hundred dollars, for value received, ioith interest from date, at 6 per cent. Jones & Wriqht. On the back of this note were receipted the following payments : Juno 16th, 1863, received $200 August 1st, 1864, " 160 Nov. 16th, 1864, " 75 Feby. 1st, 1866, " 220 What amount was due August 1st, 1866 ? SOLUTION. Face of note or principal $900.00 Interest on the same from June 1st, 1862, to Juno 16th, 1863 56.25 Amount of principal and interest, Juno 16th, 1863 956.25 First payment to bo taken from this amount 200.00 Balance due 756.25 Interest on the same from June 16th, 1863, to August 1st, 1864 51.046 Amount due August Ist^ 18&4 .••..« ......r.. 807.296 172 AEirmiETio. Second payment to be taken from this amount 160.000 Balance due 647.296 Interest on Ihe same for one year 38.837 Amount duo August 1st, 1805 686.133 Amount of 3rd payment from Nov. 16th, 1864, to August 1st, 1865 78.187 Balance due. 607.946 Interest on the same from August 1st, 1865, to August 1st, 1866 36.476 Amount duo August 1st, 1866 644.422 Amount of 4th payment from February 1st, 1866, to August 1st, 1860 226.600 Balance duo August 1st, 1866 » 0417.822 meroiiants'rule. It is customary among merchants and others, when partial pay ments of notes or other debts arc made, when the note or debt is Bcttle:! within a year after becoming due, to adopt the following im L£ . Find the amount of the principal from the time it became due until the time of settlement. Then find the amount of each payment from the time it was paid until settlement, and tubtract their sum from the amount of the principal. EXAMPLE. 11. $400, Maitland, January 1st, 1865. For value received, /promise to pay J. B. Smith & Co., or order, on demand, four hundred dollars, with interest at 6 per cent. A. E. Cassels. The following payments were receipted on the back of thia note : "February 4th, 1865, received $100 May 16th, « " 75 August 28tli " " 100 November 25th, '< " 80 What was due ut time of settlement, whioh was Deocmbcr 28th, 1865? PARTIAL rATMENTS. ^-g SOLUTION. Principal or face of note , $400.00 Interest on the same from Jan. 1st, 18G5, to Dec. 28tli, 1865 23.80 Amount of principal at settlement 423.80 First payment $100.00 Interest on the samo from Feb. 4th, 1865, to Dec. 28th, 1865 5.40 Second payment 75.00 , ' Interest on the same from May 10th, 1865, to Dec. 28th, 1865 2.77 Third payment 100.00 Interest on the same from August 28th, 1865, to Dec. 25th, 1865 2.00 Fourtli payment 80.00 Interest on the samo from Nov. 25th, 1865, to Dec. 28th, 1865 44 Amount of payments to bo taken from amount of principal ^ 365.61^ Balance due, December 28th, 1865 $58.18^ 12. ^500. Cleveland, January 1st, 1865. Three months after date, J promise to pay James Man- ning, or order, five hundred dollars, /or value received, at the First National Bank of Buffalo. Cyrus lilNG. Mr. King paid on this note, July 1st 1865, $200. What was duo April 1st, 1866, the rate of interest being 7 pci* cent ? Ans. $324.50. 13. $240. • Philadelphia, May 4th, 1865. On demand, I promise to pay A. K. Frost & Co., or orJer, two hundred and forty dollars, for value received, with in- terest at 6 2)cr cent, David Flook. The following payments were receipted on the back of this note : September 10th, 1865, received $60 January 16th, 1866, '* 90 What wna due at the time of settlement, which was May 4th, 1866? Ans. $100.44. 174 14. $340. ARITHMETIO. Lowell, Juno 16th, 1864. Three months after date, 1 promise to pay Thomat CulverweU, or order, three hundred and forty dollars, with intcrcsty at G per cent. William Manmnq. On this note were receipted the following payments ; October 14th, 18G4, received $86 • February 12th, 1865, " 40 What was duo at time of settlement, Aug, 10, 1865? Axis. $232.0.6 4!:: ■ COMPOUND INTEREST. When interest is unpaid at the end of the year, it may, by special ugrcenient, be added to the principal, and in its tv .-n bear interest, and so on from year to year. Wlien added to the principal in thia wr.y, it is said to be compound. A person may take compound interest and not be liable to the charge of usury, provided the person to whom he lends money chooses to pay compound interest, but he cannot legally collect it unless there has been a previous agreement to that effect. EXAMPLE. 1. What is the compound interest of $60, for 4 years, at 7 pot cent. ? SOLUTION. Principal $60.00 ^ Interest on the same for one year 4.20 New principal for 2nd year 64.20 Interest on the same for one year 4.494 New principal for 3rd year Interest on the same for one year. 68.094 4.S08 New principal for 4th 3^ear .. 73.502 Interest on the same for one year 5.145 Amount for 4 years » 78.647 Principal to be taken from same 10.000 Compound interest for 4 years $18647 The method of finding compound interest la usually much .sliort- cned by the following tabic, which shows the amount of $1 or :Cl for any number of years not exceeding 50, at H, 3^, 4, 5, 6 and 7 percent. The amount of $1 or £1 thus obtained, beiug multiplied by the given principal, will give the required amount, from whijh, it the principal'bc taken, the rcmuindcr will be compound interest : COMPOUND INTEBEST. 175 TABLE, *1f MOWWO Tins AJrOTTN-T 0» 0>n! DOtLAR AT COMPOtJXD RTnRMT FOU ANT RtniDBR OF TGAIU} NOT EXCEEnt>-0 FIPTI. No. T 3 par cent. 3,'i per cent. 4 per cent. 6 per cent per cent. 7 per cent. 1.030 UOJ 1.035 000 1.040 000 1.050 000 1.060 000 1.070 000 2 LOGO 900 1.071 225 1.081 COO 1.102 500 1.123 600 1.144 900 3 1.992 727 1.108 718 1.124 864 1.157 G25 1.191 016 1.225 043 4 1.125 509 1.147 523 1.169 859 1.215 506 1.202 477 1.310 796 C 1.159 274 1.187 686 1.216 G53 1.27G 282 1.338 226 1.102 :52 C 1.194 052 1.229 255 1.265 319 1.340 096 1.418 519 1.500 730 7 1.229 874 1.272 279 1.315 932 1.407 10 J 1.503 630 1.605 781 8 1.2CG 770 1.316 809 1.368 569 1.477 455 1.593 848 1.718 136 9 1.304 773 1.362 897 1.423 312 1.551 328 1.C89 479 1.838 459 10 1.343 91G 1.410 o99 1.480 244 1.628 895 1.790 818 1.967 151 11 1.334 234 1.459 970 1.539 454 1.710 339 1.898 299 2.104 852 12 J.425 701 1.511 069 1.601 033 1.795 856 2.012 196 2.252 192 13 1.4C8 534 1 563 950 1.665 074: 1.885 649 2.132 928 2.409 8I£ 11 1.512 590 1.G18 C94 1.731 G7G 1.979 932 2.260 904 2.578 534 15 1.557 967 1.675 349 i.800 944 2.078 928 2.396 55 S 2.759 032 16 1.604 706 1.733 986 1.872 981 2.182 875 2.540 3->- -'.952 164 17 1.C52 818 1.794 C75 1.947 901 2.292 018 2.C92 773 3.158 815 18 1.702 '^33 1.857 48(; 2.025 817 2.406 G19 i?8J4 339 3.379 932 19 1.753 506 1.922 501 2.106 849 2.52G 950 .,.025 600 3.616 526 20 1.806 111 1.989 789 2.191 123 2.G53 298 3.207 135 3.869 684 21 1.860 295 2.059 431 2.278 768 2.785 963 3.399 5l. 4.140 562 22 1.916 103 2.131 512 2.369 919 2.925 261 3.C03 537 4.430 402 23 1.973 587 2.206 114 2.464 71G 3.071 524 3.819 750 4.740 530 24 2.032 794 2.283 328 2.563 304 3.225 100 4.018 9.35 5 072 307 25 2.093 778 2.363 215 2.665 830 3.38G 355 4.291 871 5.427 433 2G 2.156 591 2.445 959 2.772 470 3.555 673 4.549 383 5.837 353 27 2.221 289 2.531 567 2.883 369 3.733 456 4.822 316 6.2i;j 868 28 2.287 928 2.620 177 2.998 703 3.920 129 5.111 687 CM IS 838 29 2.356 566 2.711 878 3.118 651 4.116 136 6.418 38S 7 114 257 30 2.427 262 2.806 794 3.243 398 4.321 942 5.743 491 7.612 255 31 2.500 080 2.905 031 3.373 133 4.538 039 G.0S8 101 8.115 113 32 2.575 083 3.006 708 3.508 059 J.7C4 911 6.453 387 8.715 271 33 2.052 335 3.111 942 3.648 381 5.003 189 G.810 590 9.325 3.*0 3t 2.731 9n5 3.220 860 3.794 31 G 5,253 318 7.251 025 9.973 1 14 35 2813 862 3.333 590 5.946 089 5.516 015 7.GS6 087 10.676 581 3() 2.890 278 3.450 266 4.103 933 5.791 816 8.147 252 11.423 942 37 2.98.-. 227 3.571 025 4.268 090 6.081 407 8.636 087 12.223 GI« 38 3.074 783 3.696 on 4.438 813 C.385 477 9.154 252 13.079 271 39 3.167 027 3.825 372 4.616 366 G.701 751 9.703 507 13.994 820 40 3.262 038 3.959 200 4.801 on 7.039 989 10.285 718 14.974 458 U 3.359 899 4.097 831 4.993 Oji 7.391 988 n.ooi 861 16.022 670 12 3.460 C96 4.241 258 5.192 784 7.761 588 11.557 033 17.144 257 43 3.564 517 4.389 702 5.400 495 8.149 667 12.250 455 18.311 35.5 V 3.G71 452 4.543 342 5.G16 515 8.557 150 12.985 4S2 19.628 460 ■i' 3.781 596 4.7()2 358 .5.841 176 8.985 coa i:].7(;4 611 21.0)2 •1.52 4(5 3.89.-. 01 1 4.866 911 G.074 823 9.434 258 14.500 487 22.472 623 47 4.011 895 5.0.37 284 G.317 816 9.905 971 15.465 917 M.0I5 707 46 4.132 252 5.213 589 G..57a 528 10.401 270 16.393 872 :5.728 907 49 4.2.) 6 219 .5.396 065 G.833 349 10.921 333 17.377 504 27.5:9 930 50 4.?,^:\ 906 .5.584 1-27 7.106 683 11.467 400 18.420 154 29.457 025 Koto.— If each of iho numbon iu tho tublQ bo diiaiuiiUod by 1- tlio romaindor will 4?no(c ttc lutoroal of |l. insioaa ofita ataoimt. 176 AEITHMETIC. EXEECISES. 2. What is the compound interest on $75, for 2 years, at 7 per cent. ? Ans. $10.87. 3. Wh' will $50 amount to in 3 years, at G per cent , compound interest ? Ans. $59.55. 4. WL..O is the compound interest on $G00, for 2 years, at 6 ;^yer cent., payable half-yearly ? Ans. §75.31. 5. What will $320 amount to in 2^ years, at 7 per cent,, com- pound interest ? Ans. $379.19. C. Wliat is the compound interest of $150, for 3 years, at 9 per cent. ? Ans. $U^b. 7. What is the compound interest on $1,000, for 2 years, at 3^ pertsent, payable quarterly? Ans. $72.IS. 8. What will $4G0 amount to in 3 years, 4 months, 10 days, at 6 per cent., compound interest ? Ans. $559.74. 9. What is the compound interest on 'slSGO, for 8 years, at T per cent.? ' Ans. $1335.83. 10. What will be the compound interest on $75.20, for 20 years, at 3^ per cent. ? Ans. $74.43. 11. How much more will $500 amount to at compound than simple interest, for 20 years, 3 months, 15 days, at 7 per cent.? Ans. $764.14. 12. What sum will $50, deposited in a savings "bank, amount to at compound interest, for 21 years, at 3 per cent, payable half-yearly i Ans. $173.03. 13. If a note of $60.60, dated October 25th, 1856, with the interest payable yearly, at 6 per cent., be paid October 25th, 1860 j what will it amount to at compound interest ? Ans. $76.51. 14. What remains duo on thQ followiDg notC; April 1st, 1863, at 7 per cent, compound interest ? $1,000. Cleveland, January 1, 1858. For value received, I promise to pay A B. Smith & Co., ot order, one thousand dollars on demand, with interest at 7 per cen* J. D. Foster. On the back of this note were receipted the following payments : June 10, 1858, received $70 Sept. 25, 1859, " 80 July 4, 1860, " 100 DISCOUNT AND PEESENT "WOETH. 177 Nov, 11, 1861, June 5, 1862, SO 50 Aus. $1022.34. DISCOUNT AND PRESENT WORTH. its: Discount heiug of the same nature as interest, is, strictly speak- ing, iLe use of money before it is due. The term is applied, liowever, to a deduction of so much per cent, from the face of a bill, or the ilcducting of interest from a note before any interest has accrued. This is the practice followed in our Banks, and is therefore called Bank discount, in order to distinguish it from true discount. The method of computing bank discount differs in no way from that of computing simple interest, but the method of finding truo discount is quite different, e. g., a debt of $107, duo one year hence, is considered to be worth $100 now, for the reason that $100 lot out at interest now, at 7 per cent., would amount to 6107 at the end of a year. In calculating interest, the sum on which interest is to bo paid is known, but in computing discount we have to find ic/iat sum must be placed at interest so that that sum, together with its interest, will amount to the given principal. The sum thus found is called the " Present Worth." We have already seen that $1.00 is the present worth of $1.07 due one year hence, at 7 per cent., therefore, to get the present worth of any sum due one year hence, at 7 per cent., it is only necessary to find how many times $1.07 is contained in the given sum, and wc have the present worth ; hence To find the present worth of any sum, and the discount for any time, at any rate per cent., we have the following RULE. Divide tJie given sum by the amount of $1 /or the given tim£ and rate, and the quotient will be the present worth. From the given sum subtract the present worth, and the remainder will be the discount. EXERCISES. 1. What ib the prcseat worth of $224, duo 2 years hence, at 6 per oeut. ? 12 Ads. |200. 1V» AmTHMETTO. 2. What is tho discount on $670, due i year and 8 months hence, at 7 per cent. ? Ans. $70. 3. What is the discount on $501, due 1 year and 5 months hcnco, at 8 per cent. ? ^ Ans. $51. 4. What is iho present value of a debt of $678.75, due 3 years and 7 months hence, at 7^ per cent. ? Ans. $534.97^. 5. What is the discount on $88.16, due 1 year, 8 month, , and 12 days hence, at G per cent. ? Ans. $8.16. 6. If the discount on $1060, for 1 year, at 6 per cent., is $60 ; what is the discount on the same sum for one-half the time ? Ans. $30.87. 7. llow much cash will discharge a debt of $145.50, duo 2 years, 6 months and 12 days hence, at G per cent. ? Ans. $126.30. 8. If I am offered a certain quantity of goods for $2500 cash, or for $2821.50, on 9 months credit; which is the best offer, and by how much ? Ans. Cash by $200, 0. What is tho difference between the interest and discount of $46.16, due at the end of 2 years, 6 months, and 24 days, at per cent. ? Ans. 95 cents. 10. A merchant sold goods to the amount of $1500, one-half to be pa*d in 6 months, and the balance in 9 months ; how much cash ought he to receive for them after deducting IJ per cent, a month ? Ans. $1331.25. 11. Suppose a merchant contracts a debt of $24000, to bo paid in four instalments, as follows: one-fifth in 4 months; one- fourth in 9 months ; one-sixth in 1 year and 2 months, and the rest in 1 year and 7 months • how much cash must he give at once to discharge the debt, money being worth per cent. ? Ans. 22587.65. 12. Bought goods to the amount of $840, on 9 months credit; how much money would discharge the debt at the time of purchasing the goods, interest being 8 per cent. ? Ans. $792,45. 13. A bookseller marks two prices in a book, one for ready money, and the other for one year's credit, allowing discount at 5 per cent. If the credit price be marked $9.80 ; what ought to bo tho price marked for cash ? Ans. $9.33. 14. A man having a horse for sale, offered it for $225, cash ; or, $230 at 9 months credit ; the buyer chose tho latter ; did tho seller lose or make by his bargain, and how much, supposing money to be worth 7 per cent. ? Ana, He lost $6>47. 15. A. B. Smith owe» John Mannl'ig as follows :^$365.37, to BANKS AND RANKINO. IVJ bo paid December 19tb, 1863; $101.15, to be paid July 16tb, 1864 ; $112.50, to be paid Juno 23rd, 1862 ; 696.81, to be paid April 19th, 1866, allowing discount at 6 per cent. ; how much cash should Manning roceivc as an equivalent, January 1st, 1862? Ans. $653.40. 16. I buy a bill of goods amounting to $2500 on six months' credit, and can get 5 per cent, off by paying cash ; how much would 1 gain by paying the bill now, provided I have to borrow the money, and pay 6 per cent, a year for it i Ans. S53.7.'). BANKS AND BANKING. or, ileu It. I to General Principles of Banking^ — Banks are coiumonly divided into the two great classes of banks of deposit and banks of issue. Thi:-', however, appears at first sight to bo rather an imperfect classi- fication, inasmuch as almost all banks of deposit arc r.t the same time banks of issue, and almost all banks of issue also banks of deposit. But there is in reality no ambiguity ; for by banks of deposit arc meant banks for the cuatody and employment of the money deposited with them or entrusted to their caro by tlieir customers, or by the public ; while by banks of issue are meant banks which, besides employing or issuing the money entrusted to them by others, issue money of their own, or notes payubiu c*-, demand. Tlie Bank of England is principally a bank of issue ; but it, as well as tho other banks in the different parts of the empire that issue notes, is also a great bank of deposit. The private banking companies of London, and the various provincial banks, that do not issue notes of their own^ are strictly banks of deposit. Banking business may be con- ducted indifferently by individuals, by private companies, or by joint stock companies or associations. UtiUti/ and Functions of Banhs of Deposit. — Banks of this class execute all that is properly understood by banking business ; and their establishment has contributed in no ordinary degree to give security a.id facility to commercial transactions. They afford, when properly conducted, safe and convenient places of deposit for the money that wculd otherwise have to be kept, at a considerable risk, in private houses. They also prevent, in a great measure, the necessity of carrying money from place to place to make payments, and enable them to be made in the most convenient and least expen- sive manner. The objects of hunJcing. — Correct sentiments beget correct con- duct. A banker ought, therefore, to apprehend correctly, tho objects of banking . They consist in making pecuniary gains for tho atQckholders ty legal oDerationa. The bnsmcss is emineatlj 180 AETTHMETIC. beneficial to society; but some bunkers have deemed the good of Bociety bo much more wort^iy of regard than the private good of Btockholders, that they have supposed all loans should be dispensed with direct reference to the beneficial effect of the loana on society, irrespective, in some degree, of the pecuniary interests of the dispen- Bing bank. Hudi a banker will lend to builders, that houses or ships may bo multiplied; to manufacturers, that useful iabrics may be increased ; and to merchants, that goods may be seasonably replen- ished. PIo deems himself, cx-officio, tho patron of all interests that Concern hia neighbourhood, and regulates his loans to these interests by tho urgency of their necessities, rather than by the pecuniary profits of the operations to the bank, or tho ability of the bank to sustain such demands. Tho late Bank of the United b^tatcs is a remarkable illustration of these errors. Its manager seemed to believe that his dutes comprehended the equalization of foreign and domestic exchanges, the regulation of the price of cotton, tho up- holding of State credit, and the control, in some particulars, of Congress and tho President — all vicious perversions of banking to an imagined paramount end. /^ When wo perform well tho direct duties of our station, wo need ' not curiously trouble ourselves to effect, indirectly, some remote duty. Results belong to Providence, and by the natural catenation of events (a system admirably adapted to our restricted foresight), u man can usually in no way so efficiently promote the general wel- fare, as by vigilantly guarding the peculiar interests cominitted to Lis care. If, lor instance, his bank is situated in a region dependent for its prosperity in the business of lumbering, the dealers in lumber will naturally constitute his most profitable customers; hence, in promoting his own interest out of their wants, he will, legitimately, benefit them as well as himself, and benefit them more permanently than by a vicious subordination of his interests to theirs. Men will not ongagc permanently in any business that is not pecuniarily beneficial to them personally ; hence, a banker becomes recreant to even the manufacturing and other interests that he would protect, if ho so manage his bank as to make its stockholders unwill- ing to continue tho employment of their capital in banking. This principle, also, is illustrated by the late United States Bank, for the stupendous temporary injuries which its mismanagement inflicted on society, are a smaller evil than the permanent barrier its mismanage- ment has probably produced against the creation of any similar institution. Bank of England Notes Legal Tender. — According to the law a» it stood previously to 1834, all descriptions of notes were legally payable at tho pleasure of the holder in coin of the standard weight and purity. But the policy of such a regulation was very question- able ; and we regard the enactment of the Stats. 3 & 4, Will. 4, c. 99j wliich makes Bank of Endand notes legal tender, everywhere BANKS AND BANKING. 181 0, U8 !l Z except at the Bank and its branches, for all sums above great improvement. Savings Banks have been in use in Europe over fifty year?, and in Canada and the United States, almost as long. They a. o established for tho purpose of receiving from people in moderato circumstances, Bmall sums of money on interest. In England tho deposits arc held by tho Government, and invested in the three per cent, funds. In New England, New York and other States, tho deposits aro generally loaned on bond and mortgcge at six or seven per cent, interest. Friendli/ Societies. — Friendly Societies aro associations, mostly in England, of persons chiefly in tho humblest classes fur the pur- pose of making provision by mutual contribution against those con- tingencies in human life, the occurrence of which can bo calculated by way of average. The principal objects contemplated by such Bocietics arc tho following: The insurance of a sum of money to be paid on the birth of a member's child, or on tho death of a member or any of his i'amily; tho maintenance of members in old ago and widowhood ; tho administration of relief to members incapacitated for labor by sickness or accident ; and the endowment of members or their nominees. Friendly Societies arc, therelbre, associations for mutual assurance, but arc distingushed from assurance societies, properly so called^ by tlie circumstance that the sums of monev which they insure arp comparatively small. BANK DISCOUNT. not lomes ^ould iwill- Thia r the id on lage- The Bank Discount of a note is the simple interest on the sum for which it is given from tho time it is discounted to tliP time it becomes due, including three days of grace. Suppose, for example, in getting a note of $200 discounted at a bank 1 am charged $12 for discount, which being deducted, 1 receive but $188, so that I pay interest on $12 which I did not receive. From this it is clear that I am paying a higher rate of interest in discounting a note at a bank, than I would pay were I to borrow money at the same rate, we derive the following As bank discount is the same as interest, ,w aa rally Jight Ition- c. Ihere RULE Find the interest on the sum specified in the note at the given rate^ and for the given time, including three days of grace, and thi'' will be the bank DISCOUNT. Subtract the discount from the face of the note, and the remain der will he the proceeds or present worth. f- 182 AEITHMETia EXERCISEd 1 . What 13 the bank discount on a note, given for GO days, for $350, at C per cent. ?* Ans. $3,67. 2. AVhat is the bank discount on a note of $495, for 2 months, at 5 per cent. ? Ans. 4.33. 3. AVhat is the present value of a note of $7840 discounted at a bank for 4 months and 15 days, at 6 per cent. ? Ans. $7659.08. 4. How much money should be received on a note for $125, payable at the end of 1 year, 3 months, and 15 days, if discounted at a bank at 8 per cent. ? Ans. $112. 5. A note, dated December 3rd, 1860, for $160.40, and having 6 months to run, was discounted at a bank, April 3rd, 1861, at 6 per cent. ) how long had it to run, and what were the proceeds ? Ans. 64 days; proceeds $158.71. 6. On the first day of January, 18G6, 1 received a note for $2405 at 60 days, and on the 12th of the same month had it discounted at a bank at 7 per cent. ; how much did I realize upon it. Ans. $237.61. 7. A merchant sold 240 bales of cotton, each weighing 280 pounds, for 12^ cents per pound, which cost him, the same day, 10 cents per pound ; ho received in payment a good note, for 4 months' time, which he discounted immediately at a bank at 7 per cent, j what will be his profits ? Ans. $1479.10. 8. I hold a note against Clemcs, Rice & Co., to the amount of $327.40 dated April lllh, 1866, having six montLs to run after date, and drawing interest at the rate of 6 per cent, per annum. "What arc the proceeds if discounted at the Girard Bank on the 10 th of August, at 7 ,^„ per cent. ? Ans. $332.99. NoTi:. AVbon ii note drawing interest, is discountecl at a bank, the interest is calculated on the face of the note from its date to the time of maturity, and added to the lace of the note, aud tiiis amount discounted for the length of time the note has still to run. 9. What will be the discount on the following note if discounted at the City Bank on the 17th of Nouember, at 6 per cent. (360 days to a year), * Throughout all the exercises, unless otherwise specified, the year is to be considered as consisting of 3G5 days. Since it is customary in business when a fraction of a cent occurs in and result to reject it, if le-s than half a cent, and if not less, to call it a :ent, we have adopted this principal through' ovit the book BAM DISCOU^T. 183 $527.fV5. Oberlin, Oct. 4, .866. Ninety aays after date for value received, ive promist to pay to the order of Smith, Warrcmi' Co., five hundred twenty- seven and ^(^^ dollars at the City Dank, Ohcrlin, with interest ai eight per cent. Thompson & Burns. 10. What will be the discount at Ty^^ per cent, on u note for $227.41, drawing interest at 8 per cent., dated May 1st, 1865, at 1 year after date, if discounted on March 7th, 1866 ? 11. What amount of money will I receive on the following note, if discounted at the First National Bank of Detroit on June 21st, at 9 per cent. ? $473.80. Detroit, May 17, 1866. Jlirce months after dale I promise to pay to the order «f J. i?. Sing & Co., four hundred and seventy-three and ['^^^^ DoUars, at the First National JBanJc, Detroit, for value received icith interest at 'il-f'Q per cent. Richard Dunn. 12. What must I pay for the following note en August lutli, 1866, so as to make at the rate of 30 per cent, interest per annum on the money I pay for it? Ans. $708.54. $746.75. Adrian, January 19, 1866. One year from date, for value received, we promise io pay James Ames, or order, seven hundred and forty-six -^^j^ dollars, at the Commercial Banh, Adrian, loith interest at ^-^^^^ per cent. per annum. Wilson & Cummings. 13. A holds a note against B to the amount of $478.92, dated May 10th, 1865 at 1 year after date drawing 7,^o per cent, interest. I purchase this note from A. on August 18th, paying for it such a Bum that will allow me 20 per cent, interest on my money. What shall I pay for it ? 14. I got ray note for $2000 discounted at a bank. May 20, 1862, for 2 months, and immediately invested the sum received in flour. June 7, 1862, 1 sold half the flour at 10 per cent, less than cost, and put the money on interest at 9 percent. August 13, 1802, I sold the remainder of the flour at 18 per cent, advance, and expended tho money for cloth at $1 per yard ; 12 days after I sold the cloth at $1.1 6 j per yard, receiving half the pay in cash, which I lent on interest at 7^ per cc^t. and a note for the other half, to be on inter- 184 ARITHMETIC. est from October 4, 1862, at 6 J per cent. When my note at the bank becanio due I renewed it for 5 months, and when this note became due I renewed it for 2 months, and when this note became due I renewed it for sucli a time that it bccume due July 20, 1863, at which time I collected the amount due me, and paid njy note at the bank. Required the loss or guin by the transaction. It is sometimes necessary to know the amount for which a note mast be given, in order that it shall produce a given sura when dis« counted at a bank. . EXAMPLE. 1. Suppose we require to obtain $236,22 from a bank, and that we are to give our note, due in two months ; for what amount must we draw the note, supposing that money is worth 9 per cent. ? SOLUTION. From the nature of this example wc can readily perceive that fiucTi a sum must be put on the face of the note, that when dis- counted the proceeds will be exactly $236.22. If wc were to take a one dollar note and discount it at a bank for the given time, and at the given rate, the proceeds would be .98425. Hence, for cvcri/ dol- lar wc put upon the face of the note wc receive .98425, and to re- ceive $236.22 wc would have to put as many dollars on tho face of the note as arc represented by the number of times that .98425 is contained in $230.22, which is 240. Therefore, we must put $240 on the face of a note due at the end of two months to produce $236.22 when discounted at a bank at 9 per cent. From this we deduce the following RULE. ^ Deduct the hanh discount on $1, for the given time and rate, from $1, and divide the destined amount hj the remainder. The quotient Kill he the face of the note required. 2. For what sum must a note be given, having 4 months to ran, that shall produce $1950, if discounted at a bank at 7 per cent. ? Ans. $1997.78. 3. What must be the face of a note, so that when discounted for 5 months and 21 days, at 7 per cent., it will produce $57.97, cash? Ads. $60. iit_ BANK DISCOUOT. 185 4. tetipposo your note for C months is discounted at a bank at 6 per cent., and $484.75 placed to your credit, what must have been the luce of the note ? Ans. $500. 5. A merchant bought a quantity of goods for $000. For wlijit sum must he write his note, to be discounted at a bank for G montlis, at G per cent. ? Ans. $G18.S8. G. A firmer bought a farm for $5000 cash, aud having only one- half of the £um on hand, he wishes to obtain the balance from tJio bank. For what sum must he give his note, to be discounted for moutlib, at G per cent. ? Ans. $2Glt).17. 7. If a merchant wishes to obtain $550 of a bank, for what sum must he give his note, payable in GO days, allowing it to be dis- counted at ^ per cent, per month ? Ans. $555.75. 8. I sold A. Mills, merchandize valued at $018.10, for which ho was to pay me cash, but being disappointed in receiving money ex- pected, he gave me his endorsed note at 00 days, for such an amount that when discounted at the bank at 7 per cent, it would produce the price of the merchandize. What was the face of die note ? G. I am owing R. Harrington on account, now duo, 8108.45 ; lie also holds a note against me for $210, due in o4 day.'?, iucluuJug days of grace ; he allows a discount of 8 per cent, on the note, and if I give him my note at GO days for an amount that wiil be sufficient if discounted at G per cent., to produce the amount of account and note. What will be the face of new note ? 10. Samuel Johnson has been owing me $274.48 for 84 days. I charge him interest at G per cent, per annum for this time, and he gives me his note at 90 days for such an amount that when dis- counted at the Girard Bank, at 8 per cent., the proceeds will equal the amount now due. W^hat is the face of the note ? Ti-om the many dealings business men have, in regard to dis- count and interest, it is frequently required to know what rate of injterest corresponds to a given rate of bank discount. E X A iM P L E . 1. What rate of interest is paid when a note, payable in 362 days, is discouuted »t 10 per cent. ? ^^ IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 fm m iMo 12.0 111= lA 11 1.6 V}. ^ n /a '<^. (T: .^^^ <$• V '5v y ^^ Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY 14580 (716' 872-4503 186 AniTHMETIO. SOLUTION. If WO discount $1 for the given time, and at tlio given rate, the proceeds will be .90, or 90 cents. Hence, tlio discount being 10 cents, we are paying 10 Ciits for the use of 90 cents. Now, if we pay 10 cents for the use of 90, for the use of 1 cent wo must pay ^j'y of 10 cents, or J of a cent, and for $1, or 100 cents, we must pay 100 times J of n cent, or J-gii^.H', and for $100, $11^, or llj per cent. Therefore, to find the rate of interest correeponding to a givcQ rate of bank discount, we deduce the following BULE. Divide the given rate per cent., expressed decimally, or the rait per unit, by the number denoting the proceeds of $1 for the giuen time and rate. The 'Quotient will be the rate of interest required. EXERCISES. 2. Wliat rate of interest is paid when a note, payable in 60 days, is discounted at 7 per cent. ? Ans. T-i^i^. 3. What rate of interest is paid when a note, payable in 3 months, is discounted at G per cent. ? Ans. Cy'tP/^. 4. A note, payable in 6 months, is discounted at 1 per cent, a month ; what rate of interest is paid ? Ans. 12^|J. 5. What rate of interest is paid, when a note of $200, payable in 70 days, is discounted at f per cont. a month ? Ans. 9/g j. 6. When a note of $45, payable in 05 days, is discounted at 7 per cent., to vhat rate of interest does the bank discount correspond ? An"V-'4^J5^'^ ^il'*'lJ«. .>,. . ■!^4k' ZziButtANOB. INSURANCE. 195 Insttbanoe is an engagement by v^hioh one pariy is bound, in eonsideration of' receiving a certain sum, to indemnify another for fiomethiag in case it should in any way be lost. The paity under- taking the risk is seldom, if ever, an individual, but & joint stock company, represented by an agent or agents, and doing business under the title of an " Iruurance Company," or " Assurance Contr pany," such as the "Globe Insurance Company," the "Mutual Insurance Company." Some companies are formed on the principle that each individual sTiareholder is insured, and shares in the profits, and bears his portion of the losses. Such a company is usually called a Mutual Insurance Comjpanxf. The sum paid to the party taking the risk is called the Premium •of Insurance, or simply the Premium. The document binding the parties to the contract, is called the Tolicy of Insurance, or simply the Policy. The party that undertakes to indemnify is called the Insurer, or underwriter after he has written his name at the foot of the policy. The person or party guaranteed is called the Insured. As there are many different kinds of things that may be at stake or irisked, so there are different kinds of insurance which mi^ bo classified under three heads. Fire Insurance, including all cases on land where property is ex- posed to the risk of being destroyed by fire, such as dwelling houses, stores and factories. Jifarine Insurance. — This includes all insurances on ships and cargoes. Such an insurance may be made on the ship alone, and Tn that case it is sometimes called ?mll insurance, and sometimes bot- tomry, the ship's bottom representing the whole ship, just as wa say fifty sail for fifty ships. The insurance may be made on the cargo alone, and is then usually called Cargo Insurance. It may be made onT)oth ship and cargo, in which case the general term Marins In- surance will be applicable. This kind, as the name implies^ insures against all accidents by sea. Life Insurance. — This is an agreement between two parties, that in case the one insured should die within a certain stated time, the other shall, in consideration of having received a stipulated sum annually, pay to the lawful heir of the deoeaeed, or some one mei^ i t 196 ABTTHUETia tioned in hifl mW, or some oilier party entitled thoroto, the Amount recorded in the policy. !For instance, a man may, on the occasion of his morriago, insure liis life for a certain sum, so that should ho die within a certain time, his vidow or children shall bo paid that sum by the other party. iLgain, a father may insure the life of his child, so that in cose of tho child's doath within a specified time, he shall bo paid the sum agreed upon, or that tho child, if it lives to a certain ago, shall bo entitled to tliat sum. One person may insure tho life of another. Supposing that A owes B a certain sum, i liero is tho lisk that A may die bcforo ho is able to pay B ; another party engages, for a certain yearly sum, to pay B in case A should fail to do so during his lifo time. Tu some instances, insurances aro effected to gain a oupport in case of sickness. Such a contract is called a Health Insurance. In- surances are now also effected for compensation in case of railway accidents. These we may call Railway Accident Insurances. A policy is often transferred i'rom ouo party to another, especi- ally OS collateral security for debt or some analogous obligation. If tlio payments agreed upon aro not regularly kept up, tho policy lapses, that is, becomes null and void, so that tho holder of it forfeits not only his claim to tho sum insured, but also the instalments pre- viously paid. In many companies a person can insure in such a way OS to bo entitled to have a share of tho profits. The date at which tho system of insurance began cannot be clearly ascertained ; but, whatever its date, its origin seems to have been protection against tho perils of the sea. Wo know that it was practised, in a certain way, by the ancient Greeks and Boman^. If & Ivoman merchant sent a cargo to a distant port, he made a contract ^th some one engaged iu such business, that he would advance a certain sum, to be repaid with interest, if the vessel reached her destination in safety, but should tho vessel or cargo, or both be lost, tho lender was to bear tho loss. This was termed respondentia, (a Tospondence) a term corresponding pretty nearly to tho English nord repayment. It was lawful to charge interest in such eases, above the legal interest in ordinary cases, on account of the great- uesfl of the risk. The lender of tho money usually sent an agent of bis own on board tho vessel to look after tho cargo, and receive the repajmient on tho SKfo delivery of tho goods. This agent corrca- pomlod pretty nearly to our more modern supercargo. As the art of navigation advanced, and the ecoarities aifordod by law becomo XNSmiANOE. 197 c amount more striugent, and also facilities of communioation iuorcased, this eystcm gradually gave way, and hai eventually been supplanted by communications by post, and tclegrapbio messages to agents ut tho ports of destination, With regard to tho equitablcncss of insurances, and their utility in promoting eommcrcial cxtcrprise, wo may remark that they make the interest of every merchant, tho interest of every other. To bhow this, we may c jmparo an insurance office to a club. Suppose tho merchants of a town to form a club, and establish a fund, out of which every member, if a loser, was to bo indemnified, it is plain that no loss would fall on tho individual, except his share as a mem- ber of the club. Even so tho insurance system causes that each speculator, by insuring his own stake, contributes so much to tho funds of a company, which is bound to indemnify each loser. On the other hand, the insurer or insuring company, gains in this way, that the profits accruing from cases where no loss is sustained, far exceed the cases where loss is sustained, and tho trifling expense of insuring is of no moment to tho insured, in comparison with tho damage of a disastrous voyage, or consuming conflagration. By thd Insurance system, loss is virtually distributed over a large commu- nity, and therefore falls heavily on no individual, from which wo draw our conclusion, that it is equivalent to a mutual mercantile indemnificxition club. Wo must now show the rules of the club, and principles on which its calculations are made. Tho principal thing to be taken into account, in all insurances, is the amount of risk. For example, a store, where nothing but iron is kept, would bo considered safe; a factory, where fire is used, would be accounted hazardous, and one where inflammable sub- stances arc used would bo designated extra hazardous, and tho rates would bo higher in proportion to tho increased risks. As, however, the degrees of risk are so very varied, only a rough scale can bo made, and hence the estimate is nothing more than a calculation of probabilities. In lile insurances, tho rates arc regulated chiefly by tho age, and general health of the individual, and also by the gen- eral health of tho family relations. Connected with thb is the cal- culation of the average length of human life. Almost all the calculations in insurance come under two heads. FiESX, to find the premium of insurance on a given amount, and at a c;iven rate ; and, becondlt, to find bow much must bo insured at a 198 ABITHMETIO. I given rate, so that in cace of loss, both the principal and premium may bo recovered. As the premium is reckoned as so much by the hundred, insur- ance is merely a particular case of percentage. Hence to find the premium of insurance on ary given amount at a given rate per cent., ^deduce the {bllowing R17LE. f AMttply ^he given amount hy the rate per unit." EXAMPLES. 1. To find the cost of insuring a block of buildings valued at $2688, at 6 per cent. ? Here we have .06 for the rate per unit, and $2688X.06=$161.28, the answer. 2. What will bo the cost of insuring a cargo worth $3679, at 3 per cent.? The rate per unit is .03, and $3f)79X.03=:$110.37, the answer. 3. A gentleman employed a broker to insure his residence and outhouses, valued at $2760, the rate being 8 per cent., and the bro- ker's charge 1^ per cent, ; how much had he to pay ? The cost of insurance is $2760X-08=$220.80, and the brokcrago $41.40, whioh added to f 220.80, will give $262.20, the answer. EXERCISES. What will be the premium of insurance on goods worth $1280, at B^ per cent. ? Ans. $70.40. 2. A ship and cargo, valued at $85,000, is insured at 2J per cent. ; what is the premium ? Ans. $1912.50. 3. A ship worth $35,000, is insured at 1^ per cent., and her cargo, worth $55,000, at 2J per cent. ; what is the whole cost ? Ans. $1900.00. 4. What will be the cost of insuring a building valued at $58,000, at 2J per cent. ? Ans. $1450.00. * It is plain that the rate can be found, if the amount and premium are given, and the amount can bo found if the rate and premium are given. In the case of insuring property, a pvofessional surveyor ia often employed to value it, and likewise in the cose of life insurance, a medical certiQcate is required, and in each case the fee must be paid by the person insured. As 100, the basis of percentage, is a constant quantity, when aoy tWO of the other quantities an; giveOi the third can be found. :i ii I I INSTTBANOE. 199 6. What most I pay to insure e, house valued at $898.50, at | per cent. ? 6. A village store was valued at $1180 ; the proprietor insured it for six years j the rate for the first year was 3J- per cent., with a reduction of ^ each succeeding year j the stock maintained an aver- ago value of $1568, and was insured each of the six years, at 2^ per cent. J how much did the proprietor pay for insurance during the six years ? Ans. $397.53. 7. A store and yard were valued at $1280, and insured at 1^ per oent. i the policy and surveyor's fee came to $2,25 ; what was the whole cost of insuring? Ans. $16.65. 8. W. Smith, Port Huron, requests R. Tomlinson, Toronto, to insure for him a building valued at $976 ; K. Tomlinson effects the insurance at 4| per cent., and charges | per cent commission ; how much has W. Smith to remit to R. Tomlinson, the latter having paid *he premium ? Ans. $46.36, 9. The cost of insuring a factory, valued at $25,000, is $125 what is the rate per cent. ? Ans. ^ 10. A 1^ per cent, insuring my dwelling house cost me $50 what is the value of the house ? Ans, $4000.00, To find how much must be insured for, so that in ease of loss, both principal and premium may be recovered. Here it is obvious that the sum insured for must exceed the value of the property in the same ratio that 100 exceeds tho rate. EXAMPLE. To find what sum must be insured for Oii property worth $600, at 4 per cent., to secure both property and premium, we have as $100— 4:=$96 : $100 : : $600 : F. P.=^<\,V— ==$625, the sum required. Taking the rate per unit we find •^ioo^'=tV*5=' • ^®' This gives tho RULE. Divide the value of the property by 1, diminiahed by the rate pet vnitf and the quotient will be the su7n required, EXA9IPLES. 1. A foundry is valued at $874 : for what sum at 8 per cent, must it be insured to secure both the value of the property and the premium ? One minm the rate or 1,00— .08=,92, and $874-H.92 =H950 the answer 200 ARrrHMirrrc. The premises of a gunsmith, who sells gunpowder, are valued at $2618.85: for how much, at 15 per cent., must they be insured in order to recover the value of the property and also the premium of insurance ? Subtract .15, the rate per unit, from 1, and the remaiit' der is .85 and $2618.85—.85 gives $3081, the sum required. EXERCISES Mi 1. A chemist's laooratory and appurtenances are Taluea ac $26,250, for what sum should he insure them at 6J per cent., to secure both property and premium ? $28,000. 2. A New York merchant sent goods worth $1,186 by water conveyance tc Chicago ; h089=$26.70, the premium, LIFE INStmAKOE. 203 From these explanations we can now derive a nile for finding the annual premium, when the age of the individaal and the mm to be insured ibr are knows. For Lifo. 2.75 2.85 2.^5 3.07 3.19 3.32 3.45 3.60 3.75 3.92 4.09 4.27 4.46 4.67 4.89 512 5.36 5.62 5.89 6.19 6.50 6.83 7.18 Find the age in tJie left hand column of the table, and opposite this in the vertical column for the given period will be found the fyremium on $100 for one year, and this divided by 100 will give the premium on $1 for one year, and the given sum multiplied by this will be the whole annual premium^ EXERCISES. 1. What will be the annual premium for insuring a person's life, who is 18 years old, for $1000 for 7 years ? Ans. $8.80. 2. What amount of annual premium must be paid by A. B. Smith, who wishes to insure his life for 7 years for $2000, his age being 25 years Ans. $19.60. 3. John Jones, 35 years of age, wishes to effect an insurance for life for $1500. What amount of annual premium must he pay ? Ans. $37.20. 4. A gentleman in Chicago, 32 years of age, being about to start for Australia, and wishing to provide for his family in case of his death, obtains an insurance for seven years for $3000. What amount of annual premium must he pay ? Ans. $33.30. 5. Amos Fairplay, 48 years of age, being bound on a dangerous voyage, and wishing to provide for the support of his widowed mother, in case of accident to himself, insures his life for 1 year for $2500. What amount of premium must he pay ? Ans. $42.75. 6. A gentleman, 50 years of age, gets his life insured for $3000, by paying an annual premium of $4.4G on each $100 insured; if he should die at the age of 75 years, how much less will be the amount of insurance than the payments, allowing the latter to be without interest ? Ana. $345 7. A gentleman, 4£ years of age, gets his life insured for $5000, for which he pays an annual premium of $180, and dies at the age of 50 years. Suppose we reckon simple interest at 7 per cent, on his payments, what ia gained by the insurance ? Ans. $3911. ler tliflti 204 ABTTHMETIC* PROFIT -AND LOSS. Tm the language of arithmetic, the cxprGssion Prc.fit and Loss in tisaally applied to something gaiiicd or something lost in mercantilo transactions, and the most important rule relating to it directs how to find at what increased rate above the cost price goods must be Bold to produce a fair remuneration for time, labour and expendi- ture ; or, in case of loss by unforeseen circarastanecs, to estimate the amount of that loss as a guide in future transactions. There arc other cases, however, which we shall consider in detail. CASE I. TVhcn the prime cost and selling price aro knoMm, to find the gain or loss. BTTLE . Find, hj the rule of practice, the price at the difference between the prime cost and selling price, which will he the gain or loss ac- cording as the selling price is greater or less than the prime cost; or, Find the price at each rate, and take the difference. EXAMPLES. To find what is gained by selling 4 cwt. of sugar, whicli cost 125 cents per lb., at 15 cents per lb. Here the difference between the two prices is 2^ cents per lb., and 400 lbs., at 2^ cents per lb., will give $10. Also, 400 lbs: at 15 cents per lb.=SGO, and at 12^ cents=$50, and $60 — $50=$10. Again, if 120 lbs. of tobacco be bought at 92 cts. per lb., and, being damaged, is sold at 75 cents per lb., the loss will be a loss of 17 cents in the pound, and 120 lbs., at 17 cents per lb., is $20.40 ; or, 120 lbs., at 92 cents, will come to $110.40, and at 75 cents, to $90, and $110.40— $90=$20.40, CXEROISES. 1. If 224 lbs. of i6h be bought at GO cents per lb., and sold at 95 cents per lb. ; how much is gained ? Ans. $78.40. 2. A grocer bought 24 barrels of flour, at $5.80 per barrel, and sold 12 barrels of it at $6.10 per barrel, 9 barrels at $6.20 per bar- rel, and the rest at $6.25 ; how much did he gain ? Ans. $8.55. 3. If a person is obliged to sell 216 yards of flannel, which cost Iiim $86.40, at 87^ cents per yard j how much does he lose ? Ans. $5.40. PROPTT AND LOSS. 205 4. If a dealer buys 78 bushels of potatoes, at 62J- cents per bushel, and retails them at 87^ cents per bushel; hov/ much does he gain? Ans. $19.50. 5. A wine merchant bought 374 gallons of wine, at $3,20 per gallou, and e-^ld it at $3.35 per gallon ; how much did he gain ? Ans. $5C.10. CASE II. To find at what price any article must be sold, to gain a certain rate per cent., the cost price, aud the gaiu or loss per cent, being known. RULE. Multiply the cost price hy 1 plus the gain, or 1 minus the loss, f EXAMPLE. If a quantity of linen bo bought for 75 cents a yard ; at what price must it bo sold to gain IG per cont. ? Since IG per cent, is IG cents for every dollar, each dollar in the cost price would bring $1.1 G in the selling price, so that we have |1.16X.75=.87; or 8 7 cents. EXERCISES. 1. Railroad shares being purchased for $2500, and sold at a gain of 20 per cent. ; for what amount were they sold ? Ans. 63000. 2. A property having been bought for $2000 was sold at a gain of 10 per cent. For what was it sold ? Ans. $2200. 3. A horse was bought for $50, but, proving lame, was sold .it a Joss of 15 per cent. At what price was he sold ? Ans. 842.50. 4. Bought a horse for $897 and sold it at a loss of 11 p^v cent ; for what sum was it sold? Ans. $798.33. 6. A merchant buys dry goods for $1562 and sells thorn at a profit of 22 per cent. For what were they sold ? Ans. $1905.04. CASE HI. To find the cost when the selling price and the gain per cent, are known. RULE. Divide the selling price by 1 plus the gain, or 1 minus the loss. To find what was the first cost of a quantity of flour which produced 8 per cent, profit by being sold for $127.44. Ill 206 ARTrHMETTO. 'i'l ! I Mii I 1 1 11 'imee the gain it> 6 per cent, of the oost^ it follows that each dollar laid out has brought in a return of $1.08, and therefore the cost must have been as many dollars as the number of times that 1.08 lb contained in 127.44, which is 118, and therefore the first cc&'u must have been $118. EXERCISES. 1. If flaxseed is sold at $17.40 per bushel, and 13 per cent, lost, what was the first cost ? Ans. $20.00. 2. A dealer bought 116 hogs for $580, and sold them at a gain of 25 per cent. ; at what price did he sell each on an average ? 3. If 13 sheep bo sold for $52.65, and 25 per cent, gained on the first cost, how much was paid for each at first ? Ans. $3.24. 4. If 16| per cent, bo lost on the sale of linen at $1.25, what was the first cost ? Ans. $1.50. 5. If a quantity of glass be sold for $4, tuid 10 per cent, gained, for what sum was it bought ? Ans. $3.64, nearly. CASE IV, To find the gain or loss per cent, when tho first oost and soiling price are known. BULB Divide th? gain or loss hy the first eos^, EXAMPLE. If a web of linen be bought for $20 and sold for $25, what is tho gain per cent ? Here $5 are gained on $20, and $20 is I of $100, therefore $25 will be gained on $100, i. e., 25 per cent. EXERCISES. 1. If a quMitity of goods be bought for $318.50, and sold for $299.39, how much per cent, is lost ? Ans. 6 per cent. 2. If two houses are bought, the one for $150 and the other for $250 ; and the first sold again for $100 and the latter for $350, what per cent, is gained on the whole ? Ans. 12^. A grocer buys butter at 24 cents per lb. and sells it at 30 oenis per lb., what does ho gain per oont ? Aus. 25;. PBOFIT AND LOSS. 207 4. If a cattle dealer buys 20 cows, at an average price of $20, iind pays 50 cents for the freight of each per railroad, what per oen';. docs he gain by selling them at $25.62^ each ? Ans. 25. 5. A tobacconibc bought a quantity of tobacco for $75, but a part of it being lost, he sold the remainder for $60 : what per cent, did ho lose ? Ans. 20. CASE V. Given the gain or Iocs per cent, resulting from the sale of goods at one price, to find the gain or loss per cent, by selling the same at anotlier price. RULE. ' Find hy case ill. the Jirst cost, and then hy case IV. the gain or Joss per cent, on that cost at the second selling price. EXAM FLE. If a farmer sells his ho^is at $5 each, and realizes 25 per cent. ; what per cent, would he realize by selling them at $7 each. "We find by case III., that the cost was $4, and then by case IV. what the aain per cent, would be on the second supposition, that is $3-7-4=.75, or 75 per cent. EXERCISES. 1. If a grocer sells rum at 90 cents per bottle, and gains 20 per cent. ; what per cent, would he gain by selling it at $1.00 per bottle ? Ans. 33J. 2. Tf a hatter sells hats at $1.25 each, and loses 25 per cent. ; what per cent, would he lose by selling them at $1.60 each ? Ans. 4. 3. If a storekeeper sells cloth at $1.25, and loses 15 per cent. ; would he gain or lose, and how much, by selling at $1.65 ? Ans. He would gain 12 per cent, nearly. 4. A milliner sold bonnets at $1.25, and thereby lost 25 per cent. ; would she have gained or lopt by selling them at $1,40 ? Ans. She would have lost IG per cent. 6. A merchant sold a lot of goods for $480, and lost 20 per cent. ; would he have gained or lost by selling them for $720, and how much ? Ans. He would have gained 20 per cent. 6. A quantity of grain was sold for $90, which was 10 per cent, less than the cost ; what would have been the gain per cent, if it had been sold fe; $150 ? Ans. 50. w 4 203 AMTHMETIO. 7. A gi'occr soid tea at 45 cents per pound, nnd thereby gained 12| per cent. ; what would ho havo gained per cent, if ho had sold tho tea at 54 cents per pound ? Ans. 35. S. A farmer sold corn at 05 cents per bushel, and gained 5 per cent. ; what per cent, would ho have gained if ho had sold tho corn at 70 cents per bushel? Ans. 13j^. UlSOELLAN^EOnS ESEROISSS. 1. If I buy goods amounting to $465, and sell them at a gain of 15 per cent. ; what are my profits? 2. Suppose I buy 400| barrels of flour, at $10.75 a barrel, and Bcll it at an advance of ^ per cent, j how much do I gain ? Ans. 625.14. 3. If I buy 220 bushels of wheat, at $1.15 per bushel, and wish to gain 15 per cent, in selling it ; what must I ask a bushel ? 4. A grocer bought molasses for 24 cents a gallon, which ho sold for 30 cents ; what was his gain per cent. ? Ans. 25. 5. A man bought a horso for $150, and a chaise for $250, and sold the chaise for $350, and the horse for 100 ; what was his gain per cent. ? Ans. 12J. 6. A gentleman sold a horso for $180, and thereby gained 20 per cent. ; how much did the horse cost him ? Ans. $150. 7- In one year the principal and interest of a certain noto amounted to $810, at 8 per cent, j what was the face of the noto? Ans. $750. •8. A carpenter built a house for $990, which was 10 per cent, leas than what it was worth ; how much ehould ho have received for it so as to have made 40 per cent. ? Ans. $1540. •j9. a broker bought stocks at $96 per share, and sold them at $102 per share ; what was his gain per cent. ? Ans. 6J-. 10. A merchant sold sugar at C^ cents a pound, which was 10 per cent, less than it cost him ; what was the cost price ? Ans. 75 cents per pound. 11. A merchant sold broadcloth at $4.75 per yard, and gained 12} per cent. ; what would he have guined per cent, if he had sold it at $5.25 pcv yard ? Ans. 24^^. 12. I sold a horse for $75, and by so doing, I lost 25 per cent. ; whereas, I ought to have gained 30 per centi } how much Vias he sold for under bis real value? Ans. |55« PROTIT AKD LOSS. *^eM^ 13. A watch which cost me $30 I have sold for (35, on a credit of 8 uonths ; what did I gain by ipy bar(;ain, allowing money to bo worth 6 per cent.? Ans. $3.65. 14. Bought 84 yards of broadcloth, at 85.00 per yard ] what must be my asking price in order to fall 10 per cent., and still mako 10 per cent, on the cost ? Ann. $(3.1 1|. 15. A farmer sold land at 5 cents per foot, and gained 25 per cent, more than it cost him ; what would have been his gain or loss per cent, if he had sold it at 3^ cents per foot ? Ans. 12^ per cent. loss. 16. What must I ask per yard for cloth that cost $3.52, so that I may fall 8 per cent., and still moke 15 per cent., allowing 12 per cent, of soles to be in bad debts ? Ans. 65. 17. A merchant sold two boles of cotton at $240 each ; for one he received 60 per cent, more than its cost, and for the other 60 per cent, less than its cost. Did he gain or lose by the operation, and how much ? Ans. loss $270. 18. Bought 2688 yards of cloth at $2.16 per yard, and sold one-fourth of it at $2.54 per yard ; one-third of it at $2.75 per yard, and the remainder at $2,90 per yard. Find the whole gain, ond the gain per cent. Ans. $1612.80 and ^T^ijh P^^ ^^^^* 19. A flour merchant bought the following lots : — 118 barrels at $9.25 per barrel. 212 « 9.50 " 315 " 9.12J " 400 " 10.00 « The expenses amounted to $29.50, besides insurance at ^ per cent. At what price must he sell it per barrel to gain 15 per cent ? Ans., $11.05. 20. Bought 100 sheep at $5 each ; having resold them at once and received a note at six months for the amount ; having got the note discounted at the Fourth National Bank, at tax per cent., I found I had gtuned 20 per cent, by the transaotion. What was the Gelling price of each sheep? Aub.. S6.19. H .[ I til I I 210 ABITHMETIOt STORAGE. When a charge is made for the aocommodation of having goods kept in store, it ia called ttorage. Accounts of storage contain the entries showing when the goods vrero received and when delivered, with the number, the description of the articles, the sum charged on each for a certain time, and the total amount charged lor storage, which is generally determined by an avorago reckoned for eomo specified time, usually one month (30 days). EXAMPLES. 1. What will be the cost of storing wheat at 3 cents per bushel per month, which was received and delivered as follows : — Received, August 3rd, 1865, 800 bushels ; August 12th, 600 bushels. De- livered, August 9th, 250 bushels ; September 12th, 350 bushels ; September 15th, 400 bushoL"), and Ootobcr 1st, tUo balance. i SOLUTION. i 18U. Bash. Days. Bcub. August 3. Received 800 x 6 = 4800 in store for one day. « 9. Delivered 250 Balance 550 X 3 = 1650 in store for one day. " 12. Received 600 Balance 1150 X 31 =35050 in store for ono day. Sept. 12. Delivered 350 Balance 800 X 3 = 2400 in store fcr one day. « 15. Delivered 400 Balance 400 X 16 = 6400 in store for one day. Oct. 1. Delivered 400 Total 4.... 50900 in store for one day. 50,900 bushels in store for one day would be the same as 50900-f-o0=rl096§ bushels in store for one month of 30 days, and the storage of 1697 bushels for one month, at 3 cents per month, would equal 1697 X. 03=650.91. It is customary, in business, when the number of articles upon which storage is to be charged, ns found, contains n fraction lesi 8T0RA0Z. 211 than a haff, to reject the fraction ; but if it is more tfuxn a JuzJf, to i-cgard it ns an entire article. From the solution of the foregoing example, we deduce the fol- bwin: RULE Miildphj the niimhcr of bushcJs, barrels, or other articles, by the uumbcr of days they are, in store, and divide the sum of the pro- Juch by 30, or the number of days in any term c.grccd vpon. The quolknt will give the number of bushels, barrels, or other articles or* tchich storage is to be. charged for that term. 2. ^Vhat will bo the cost of storing salt at 3 cents a barrel per month, which was put in store and taken out as follows : — Put in, January 2, 18G6, 4uO barrels ; January 3, 75 barrels ; January 18. 300 barrels ; January 27, 200 barrels ; February 2, 75 barrels. Taken out, January 10, UO barrels; January 30, 150 barrels; February 10, 190 barrels; February 20, 300 barrels; March 1, 250 barrels; and on March 12, the balance, 150 barrels? Ans. $39.44. 3. Received and delivered, on account of T. C. Musgrove, sundry bales of cotton, as follo'vs: — Received January 1, 1866, 2310 bales; January 16, 120 ba). ; February 1, 300 bales. Deli- vered February 12, 1000 bales ; March 1, GOO bales; April 3, 400 bales ; April 10, 312 bales ; May 10, 200 bales. Required the num- b)r of bales remaining in store on June 1, and tho cost of storage up to that date, at tho rate of 5 cents a bale per month. Ans. 218 bales in store ; 6321.18 cost of storage. 4. W. T. Leeming & Co., Commission Merchants, Albany, in account with A. B. Smith & Co., Oswego, for storage of salt and gunpowder, received and delivered as follows : Received, January 18, 1866, 400 kegs of gunpowder and 50 barrels of salt; January 25, 250 barrels of salt; February 4, 150 barrels of salt, and 50 kegs of gunpowder; February 15, 100 kegs of gunpowder ; March 5, 64 kegs of gunpowder; April 15^ 50 kegs of gunpowder, and 75 barrels of salt. Delivered, February 25, 15 kegs of gunpowder, and 40 barrels of salt; March 10, 150 kegs of gunpowder, and 285 barrels of salt ; April 20, 200 kegs of gunpow- der ; April 125, 50 barrels of salt, and 200 kegs of gunpowder. Required the number of barrels of salt and kegs of gunpowder in store May 1, and the bill of storage up to that date. The rate of 212 ARITHMETIO. Btorago for salt being ^ cents a barrel per month, and for gunpowder 12 cents a keg per month. Ans. In store, 50 barrels of salt and 99 kegs of gunpowder ; bill of storage, $20G.01. GENERAL AVERAGE, This is the term used to denote tlie contribiUion of all persons interested in a ship, fi-eight, or cargo, towards the loss or damage incurred by any particular part of a ship, or cargo, for the preserva- tion of tie rest. This sacrifice of property is called yc<wances are a or break* : of the cose is payable, the custom •e might be 1 be neces- iecrease the I, compared Is imported 9 at time of ids that are in the same IS a depre- iifying the represent- iing at the treland 59 is allovicd lbs. lbs. libs To find the ad valorem duty on any quantity of goods. Suppose a Troy dry goods merchant to import from Montreal 43G yards of silk, at $1.75 per yard, and that 35 per cent, duty is charged on them. Hero wo find tho wholo price by the rule of Practice to be $7C3, then tho rest of the operation is a direct case of percentage, aud therefore we multiply $763 by .35, which gives $267.05, tho amount of duty on tho whole. Honoc wo havo tho following EULE FOR SPEOIFIO DUTY. Subtract the tare, or other allowance, and multiply tne remain* der hy the rate of duty per box, gallon, &c. RULE FOR AD VALOREM DUTT» Multiply the amount of the invoice by the rate per unit, EXEROISES. 1. Find the specific duty on 5120 lbs. of sugar, the tare being 14 jwr cent., and the duty 2| cents per lb. Ans. $121.09. 2. What is the ad valorem duty on a quantity of silks, the amount of tho invoice being $95,800, and the duty 62^ per cent ? Ans. $59,875. 3. At 30 per cent., what is the ad valorem duty on an importa- tion of china worth $1260. ? Ans. $378. 3. What is the specific duty, at 10 cents per lb., on 45 chests of tea, each weighing 120 lbs., the tare being 10 per cent. ? Ans. $486. 5. What is the ad valorem duty on a shipment of fruit invoiced at $4560, the duty being 40 per cent.? Ans. $1824. 6. What is the specific duty on 950 bags of coflfee, each weighinj^ 200 lbs., the duty being 2 cents per lb., and the tare 2 per cent ? Ans. $3724. 7. What is the ad valorem duty on 20 casks of wine, eacli con- taining 75 gallons, at 18 cents a gallon ? Ans. $270. 8. A. B. shipped from Oswego 24 pipes of molasf s, each con- taining 96 gallons; 2 percent, was deducted for leuicagc, and 12 cents duty per gallon charged on the remainder : how much was the duty? Ans. $270.95. 218 ABITHMETIC. }:■ !,'! 9. Peter Smith & Co., Brooklin, import from Cadiz, 80 baskets of port wine, at 70 francs per basket ; 42 baskets of sherry wine, at 35 franco per basket ; GO casks of champagne, contain in,^ ol gallons each, at '1 IVancs per gallon. The waste of the wine in the casks wa3 reckoned at a gallon each cask, und the allowance for breakage in the baskets was 5 per cent. ; what was the duty at 30 per cent., 18-J cents being taken as equal to 1 franc? Ans. $77G.54. 10. J, Johnson & Co., of Boston, import from Liverpool 10 pieces of Brussels carpeting, 40 yards each, purchased at 5s. per yard, duty 24 per cent. ; 200 yards of hair cloth, at 4s. per yard, duty 19 cwt, ; 100 woollen blankets, at 2s. Gd., duty IG per cent. ; and shoe-lasting to the cost of £G0, duty 4 per cent. Required the whole amount of duty, allowing the value of the pound sterling to be 64.84. Ans. $173.G4. 11. John McMaster & Co., of Colliugwood, Canada Wcstv, bought of A. M. Smith, of BuiFalo, N. Y., goodi i ivoiccd at 65440.50, which should have passed through the custom-house dur- ing the lirst week in May, when the discount on American invoices was 43J per cent., but they were not passed until the fourth week in May, when the discount was 36j per cent. The duty in both cases being 20 per cent. ; what was the loss sustained by McMaster & Co. on account of their goods being delayed ? Ans. $70.60. ■'Ill STOCKS AND BONDS. Capital is a term generally applied to the property accumtdated by individuals, and invested in trade, manufactures, railroads, build- ings, government securities, banking, &c. The capital of incorpo- rated companies is gsucrally termed its " capital stock," and is divided into shares ; the persons owning one or more of these shares, being called stockholders. The shares in England, are usually £100, £50, or £10 each. In the United States they arc generally $100, 850, or $10 each. The management of incorporated companies is generally vested in officers and directors, as provided in the law or laws, who aro elected by the stockholders ov shareholders ; each stockholder, in most cases, being entitled to us many votes as the number of shares he holds ; but sometimes the holder of a few shares votes in a larger proportion than the holder of many. The accumulating profits which are distributed .imong the stork- bolders, once or twice a year, arc called " dividends," and when "declared," arc a certain percentage of the par value of the shares. In milling, and some other companies, where the shares are only a baskets r wine, at !l gallons the cuska brcukago per cent., 677G.54. irpool 10 t 53. per per yard, )er cent, j [uircd the itcrling to $173.64. a Westv, voiced at ouse dur- 1 invoices irth week ;y in both VIcMaster 1. $70.60. STOCKS AND BONDS. 219 umulated ds, build- incorpo- and is e sliares, usually generally y vested who arc older, in )f shares a larger he stork- id when shares. 1 only a few dollars each, the dividend is usually a fixca sum "per share." Ccrtilicates of stock arc issued by every company, signed by the proper officers, indicating tlio number of sharcrf each- stockholder is entitled to, and as an evidence of ownership ; these arc transferable, and may be bought and sold like any other property. When the market value equals their nominal value they arc said to be " at par." When they sell for more than their nominal value, or face, they are said to be above par, or at a " premium"; when for less, they arc below par, or at a " discount." Quotations of the market value are generally made by a percentage of their p;:r value. Thus, a share which is $25 at par, and sells at $28, is quoted at twelve per cent, premium, or 112 per cent. When states, cities, counties, railroad companies, and other corporations, borrow large amounts of money, for the prosecution of their objects, instead of giving common promissory notes, as with the mercantile community, they issue bonds, in denominations of convenient size, payable at a specified number of years, the interest usually payable semi-annually at some well known place. These are usually payable to "bearer," and sometimes to the "order" of the owner or holder. When issued by Governments or States, these bonds are frequently called Government stocks or State stocks, under authority of law. To these bonds are attached, what are called " coiqwns," or certificates of interest, each of wliicli is a duo biu Lor the annual or semi-annual interest on the bond to which it is attached, representing the amount of the periodical dividend or interest; which coupons were rHualiy cut off, and presented for p:'.yment as they become due. These bonds and coupons are signed by the proper officers, and like certificates of capital stock, are nego- ti ible by delivery. The loan is obtained by the sale of the bonds, witli coupons attached, but they are sometimes negotiated at par. Their market value depends upon the degree of confidence felt by capitalists of their being paid at maturity, and the rate of interest compared with the rate in the market. Treasury notes arc issued by the United States Government, for the purpose of effectinp; temporary loans, and for tho payment of contracts and salaries, which resemble bunk notes, and arc made payable without interest generally. Recently such notes have been issued bearing one year or three years' interest. "Consols" is a terra abbreviated from the expression ''consoli- dated," the British Government having at various times borrowed money at difi'crent rates of interest and payable at diOereiit times, '• consolidated" the debt or bonds thus issued, by issuing new block, drawing interest at three percent, por annum, payable seii!i--innually, and redeemable only at the ojition of the Government, lieeoming practically perpetual annuiticf.. With the proceeds of thi.:', tlio old Biock was redeemed. The quotations of these three per cent, per- petual annuities, or " confsols," indicate ordinarily the state of the 220 ABITHMETIC. ■lii I i ! m\ money market, as they form a large portion of the Briti.sh public debt. " Mortgage Bonds" are frequently issued by owners of real property, with coupons attached, which render the bonds niore saleable as well as more convenient for the collection of interest. "Coupon Bonds," being negotiable by delivery, are payable to the holder ; and in case of loss or theft, the amount cannot be recovered from the government or corporation issuing them, unless ample notice is given of the loss. "llegistercd Bonds" arc those payable only to the "order" of the holder or owner, and arc more safe for investment. By law, stockholders arc liable for the whole debts of the corpo. ration, in case of failure. In some States the law provides that they are liable only to an amount equal to their stock. In England tlic statute provides for " Limited" liability, by an Act passed in 18G2 termed the " Limited Act." ■ -J CASE r. The premium or discount being known, to find the market value of any amount of stock. EXA.MPLES. If G. W. R. shares are at 7 per cent, premium, to find the value of 30 shares of $100. Here it is plain that each $100 will bring $107, and that each $1 will bring $1.07, and as the par value is $3000, the advanced value will be 3000 times 1.07, which gives $3210, the market value, and $3210— $3000=$210, the gain. Again, if the same arc sold at a discount of 7 per cent., it is plain that each $100 would bring only $93, and therefore each $1 would bring only $0.93, and therefore as the par value is $3000, the de- preciated value will be 3000 times .93, which gives $2790, and therefore the loss would be $3000—2790=210, From this we derive the ncLE. Multiply the par value hy 1 plus or minus tne rate per unit, according as the shares are at a premium or a discount. STOCKS AND BONDS 221 I is plain would the de- loo, and zf w»mV, EX EROISES. 1. What is the market value of $450 stock, at 8|- per cent, dis- count? Ans. $411.75. 2. What is the value of 29 sharca of $50 each, when the shares are 11 per cent, below par ? Ans. $1290.50. 3. A man purchased GO shares of $5 each, from an oil well company, when the shares were at a discount of 8 per cent., and sold them when they wore at a premium of 10 per cent; how much did ho gain ? Ans. $54. 4. A man purchased $10,000 stock when it was at an advance of 8 per cent., and sold when it was at a discount of 8 per cent. ; how much did he lose ? Ans. $1600. 5. If a man buys 15 shares of $100 each, when the shares are at a premium of 5 per cent., and sells when they have advanced to 12 per cent., how much docs he gain ? Ana. $105. ' CASE II. To find how much stock a given sum 'rill purchase at a given premium or discount. Let it be required to find how much stock can bo purchased for $21,6oO when at ^premium of 8 per cent. In this case it will require $108 to purchase $100 stock, and therefore $1.08 to purchase $1 stock, and hence the amount that can be purchased for $21000 will be represented by the number of times that $1.08 is contained in 21600, which gives $20000. Again : Let it be required to find how much stock can be pur- chased for $5520, when at a discount of 8 per cent When stocks arc 8 per cent, below pur, $92 will purchase $100 stock, and there- fore $0.92 will purchase $1, and hence the amount that can be pur- chased for $5520 will be represented by the number of times that .92 is contained in 5520, which gives $6000 3tock. Hence we derive the RULE. Divide the given sum hi/ 1 plus or minus the rate per unitf accord- ing as the sJiares are at a premium or a discount. EXERCISES . 6. When stocks arc at a premium of 12 per cent,, liow much can be purchased for $8064 ? Ans. $7200. 222 ARTTHMETIO. 7. When Btocka arc at a discount of 9 per cent., how much con be bought for $3G40 ? Ans. $4000. 8. Wlion G. T. R. stock is at 18 per cent, below par, how much can })e bought for $42,040. Aus. $52000. 9. When G. W. R. stock is at a premium of 9 per cent., how much will i:4r)78 purchase ? Ans. $4200. 10. When government stock is selling at 92^, what amount of Btock will $28,075 purchase, and to what will it amount with broker- age at J per cent, ? Ans. $31077.50. CASE III. The premium or discount being known, to find the par value. To find the par value of $1,296, when stock is at a premium of 8 per cent. • At 8 per cent, premium, cjxch $1 brings $1.08, hence tho par value will bo represented by tho number of times 1.08 is contained in 1290, which gives $1200 for tho par value. To find tho par value of $1104, when stock is at a discount of 8 per cent. Eucii $1 will bring $0.92, and therefore the par value will be represented by the number of times that .92 ia contained iu 1104, wliich gives $1200, the par value. Ilenco the RULE. Divide the marJcet value hy 1 plus or minus the rate per -uniV, according; as the stocks are selling above or below par. EXERCISES. 11. What is the par value of $24420, when stock is 11 per cent above par ? Ans. §22000. 12. What is the par value of $10800, when stocks are at a dis- count of 4 per cent. ? Ans. S11250. 13. When government stocks are at 6 per cent, premium ; how much will $2;)24G purchase at par value ? Ans. $19100. 14. The shares in a canal company are at 15 per cent, discount ; how many shares of $100 will $11390 purchase ? Ans. 134. 15. The shares of a British gas company were selling in 1848, at a discount of 12 per cent. ; a speculator purchased a certain num- ber of shares for £792 ; the value of tho shares suddenly rose to par ; how many shares did he purchase, and how much did ho gain ? Ans. 9 shares; £108 gain. STOCKS AND 7J0NDS. 223 nuch con 3. $4000. low much $52000. ont., how I. $4200. mount of b broker- 1077.50. value, jmium of tho par :ontained ount of 8 3 will be iu]104, )cr unit, )er cent §22000. at a dis- 811250. m ; how $19100. iscount ; ns. 134. n 1848, in nura- ! to par ; n? )8 gain. CASE IV. To find to what rate of interest a given dividend corresponds. If a person receives a dividend of 12 per cent, on an investment made at 20 per cent, above par, tho corresponding interest may bo calculated thus : As the stock was bought at 20 per cent., or .20 above par, $1.20 of market value corresponds to $1 of par value, and as every $1 of par value corresponds to 12 per cent, interest, or .12, it follows that tho per cent, which was invested will bo represented by tho num- ber of times that 1.20 is contained in .12, which is .10 or 10 per cent. Ilenoo tho RULE. Divide the rate per unit of dividend hy 1 j^lus or minus the rate per cent, premium or discount, according aa the stocks arc above or below par.'^ EXERCISES . 10. If a dividend of 10 per cent, bo declared on stock vested at 25 per cent, advance ; what is tho corresponding interest ? Ans. 8 per cent. 17. If a dividend of 4 per cent, be declared on stock invested at 12 per cent, below par, what is the corresponding interest ? Ans. 4/y. 18. If money invested at 24 per cent, yields a dividend of 15 per cent., what is the rate of interest ? Ans. 12^^,-. 19. If railroad stock is invested at 18 per cent, above par, and a dividend of per cent, be declared, what is the rate of interest ? Ans. 5-Aj. 20. If bank stock be invested at 15 per cent, below par, and a dividend of 10 per cent, declared, what is the rate of interest ? Ans. 11 YY* MISOELLANEOtrs EXERCISES. 1. What must be paid for 20 shares of railway stock, at 5 per cent, premium, the shares being §100 each? Ans. $2100. * To find at wliat prico slock payiii"; iv given rate per C( nt. dividend can be purchased, so that the money invested shall produce a givt a rate of interest, divide the rate per unit of dividend hy the rate per unit of interest. •22i ABITBMETIC. 2. What is tho par value of bank stock worth $8740, at a pre- mium of 1 5 per cent. ? Ans. $7600. 3. Railway stock was bought at 15f below par, for $1895.62 J j liow many shares were there, cuch share being $150 ? Ans. 15 shares. 4. If per cent, stock yields 8 per cent, on an investment, at what per cent, discount was it bought ? Ans. 25. 5. If bank stock which pays 11 } r cent, dividend, is 10 per cent, above par, what is the corresponding rate of interest on any investment? Ans. 10. 6. When 4 per cent, stocks were at 17|- discount, A bought $1000 ; how much did ho pay, and how much did ho gain by selling when stock had risen to 8G^ ? Ans. $821.25, and $41.25. 7. What will $850 bank stock cost at a discount of 9f per cent., ^ per cent, being charged for brokerage ? Ans. $771.38. 8. On the data of the last example, how much would bo lost bj selling out at 10| per cent. ? Ans. $10.03. 9. What income should I get by laying out $1620 in the pur- chase of 3 per cent, stock at 81 ? Ans. $60. 10. Wiiat sum must be invested in the 4 per cent, stocks at 84, to yield an income of $280 ? Ans. $5880. 11. What rate of interest will a person receive by investing in the 4J per cent, stocks at 90 ? Ans. 5 per cent. 12. A person transfers his capital from the 3^ per cent, stocks at 77, to the 4 per cent, at 89 ; what is the increase o" decrease per cent, in his income ? Ans. Decrease 25. 13. A person sells out of the 3 per cent, stock at 96, and invests bis money in railway 5 per cent, stock at par ; how much per cent. is his income increased ? A ns. 60. 14. What must be the market value of 5J- per cent, stock, sc that after deducting an income tax of 2 cents on the dollar, it may produce 5 per cent, interest? Ans. 107|. 15. A gentleman invested $7560 in the 3^ per cent, stocks at 94J, and on their rising to 95 sold out, and purchased G. T. R. 4 per cent, stock at par; what increase did he mako in his annual income ? Ans. $24. 16. How much more may a person increase his annual income by lending $3800, at 6 pcr ceRt.^ than by purchottog railway 5 per cent. Btook at 95 ? Ans. $28. STOCKS AND BOKDS. 225 at a pre- I. $7600. I95.62J ; ) bharcs. mcnt, at Ads. 25. 8 10 per t on any Ads. 10. L bought >y selling 1 $41.25. H per $771.38. )o lost by . $10.03. 1 the pur- lins. $60. iks at 84, s. $G880. resting in )cr cent. stocks at rcase per Ircase 25. d invests per cent. A ns. 60. stock, sc it may s. 107|. stocks at T. R.4 s annual .ns. $24. [ income ray 6 per m $28. 17. A person sells $4200 railway stock vthkh pays 6 per cent. at 115, and invests one-third of the proceeds in tho 3 per cent, con- boIm at 80^, and tho balance in savings bank stock, which pays 9 per coat, at par ; what is tho decrease or inoreaso of his annual income ? Ads. Increase $97.80. 18. A person having $10,000 consols, sells $5000 at 94 J, and on their rising to 98f he sells $5000 more ; on their again rising ho buys back tho whole at 96 ; how much does he gain ? Ans. $75. 19. The sum of $4004 was laid out in purchasing 3 per cent, stocks at 89^, and a whole year's dividend having been received upon it, it vrs& sold out, tho whole inoreaso of capital being $302.40 ; at what price was it sold out ? Ans. 93^. SO. Suppose a person to have been an original subscriber for 500 shares of $50 each, in the First National Bank, payable by instal- ments, as follows : — | in throe months, which ho sold for 5^ per cent, advance: f in 6 months, which brought hira G^ per cent, ad- vance, and the balance in nine months, which he was compelled to sell at 8| per cent, disoount; what did he gain by the whole transac- tion ? Ans. $808.33. 21. A gentleman purchased $5000 of Five-twenties (gold 6 per cents) at 108 ; gold at time of purchase was at 35 per cent, pre- mium ; if it remained so when the interest was payable, what was the rate per cent, of interest on amount invested ? Ana. 7J per cent. 22. From which would be derived the greater income, Seven- thirties purchased at 104, or Five-twenties (6 per cent, gold) at 109^, interest on both bonds payable at tho same time, and gold quoted at 140 ? Ans. From the Five-twenties. 23. On Jan. 1st I wish to make an investment of money that will allow me 7^ per cent, interest on the investment ; what can I afford to pay for Ten-forties (interest ray able in gold at 6 per cent.) and what for Seven-thirties, calculating gold at 35 per cent. prem. ? Ans. For Ten-forties 108 ; Seven-thirties 97|^. 24. In the above example, what ooold I give fox tho Ton-forties if gold were oaloulated at 20 per oent. Qtem. ? Ans. 96 226 ABirHUETIO. ii '! !! " 11! Hill! i f 25. On May Slst, a broker purchased for mo a Seven-thirty bond to the amount of $12,000 at 1Q^ ; the interest on this bond is payable on the 1st Feb. and August ; what docs the bond cost mo, the brokerage being ^ per cent. ? Ans, $12801. GO. 26. After receiving the interest, on Aug. 1st, on the bond men- tioned in lust question, the broker immediately sold it for mc at 103^ charging ^ per cent, for selling ; did I gain or lose by the transac- tion, and how much, money being worth 6 per cent. ? Ans. Lost $157.91. 27. A gentleman subscribed $15,000 in a railroad company, having ft paid-up capital of $750,000 ; but only 40 per cent, of* sub- scribed capital paid in. A cash dividend of 3^ per cent, on the par value is declared ; what rate per cent, does he receive on his iuvest- ment ? Ans. S'^ per cent- 28. Tho capital of the " First National Bank," of Cincinnati, is $1,500,000, of which A has subscribed $7,500. There has been 25 per cent, called in. A cash dividend of 4 per cent, on tho paid- up capital is declared, and 10 per cent, on paid-up capital carried to tho credit of the stockholders ; how much money does A receive as u dividend, what per cent, on subscribed capital is carried to credit of stockholders, and what lias A still to pay on his stock ? Ans. A receives $75 ; to credit of stockholders 2J per cent. A has still to pay $5437.50. 29. A having $25,000 for investment on May 1st, placed it in the hands of B, a broker, advising him to speculate in buying and Belling stocks, bonds, and gold, for GO days, and then return what the money produced, after deducting brokerage of -J per cent, on tho actual sales and purchases. B immediately purchased 200 shares of Erie R. R. stock at 59 " buyer 30," no margin required, and sold 400 shares Reading R. 11. stock, at 102^ " buyer 15." Five days after, B called in the Erie R. R. stock and sold it to C at Gl J ; May 8th he bought $20,000 of Five-twcuties at 100^, at the same time tho person to whom the Reading R. R. stock was sold, called it in; B paid the difference, the stock being valued atl025; May 27th, gold having appreciated in value as compared with Greenbacks, B sold the Five-twenties at 110^ cash, and at the same time made a further sole of (15,000 in the some kind ol bonds <\t| U0| "solUr 1" 6T0CES AND BONDS. 227 30," which he was able to purchase and deliver in ten days at 109 J. Juno 20th, B sold $20,000 in gold at 137J "seller 10," which was not delivered ut the expiration of the ten days, but settled at 137J ; how much money is A to rocoivo from B ? Ana. $25,475. PARTNERSHIP Partnership has been defined to be the result ol a contract, under which two or more persons agree to combine property, or labour, for the purpose of a common undertaking, and the acquisition of a com- mon profit. A dormant, or sleeping partner, is one who shares in the concern, but does not appear to the world as such. A nominal partner is one who lends his name and credit to a firm, without having any real interest in the profits. All the partners may contribute equally to the business ; or the capital may be contributed by some or one, and tho skill and labour by the other ; or, unequal proportions may be furnished by each. The contract need not be in writing, but all parties to be bound must assent to it, and it is usually contained in an instrument called " Articles of Partnership*^ Too much pains cannot be taken to have this agreement so plain and explicit in regard to particulars, that it cannot possibly be mis. understood. A great deal of litigation has arisen from carelessness in this respect. These Articles of Partnership should particularly specify tho amount of investment by each partner, whether tho personal atten- tion of the partners is required to tho business, duration of partner- ship, and sometimes an agreement with regard to tho withdrawal of money from the business. A dissolution can take place at any time by mutual consenfw A partnership at vnll is one in which there is no limited time affixed for its oontiauoncc, and tl wholo firm may bo dissolved 928 ABITHMETIC. by any of its members at a moment's notice. A document is, how* ever, generally drawn up and signed upon a dissolution, called a settlement, which contains a statement of the mode of adjuBtment of the accounts, und the apportionment of profits or losses. EXAMPLE. Two persons, A. and B., enter into partnership. A. invests $300 and B. $400. They gain during one year $210; what is oocb man's share of the profit ? SOLUTION BT PJSOFORTION. A.'s stock, $300 B.'8 " 400 :l ii at '> I' I Entire stock $700 : 300 : : $210 : $90 A.'s gain. « " 700:400::$210:120 B.'s " SOLUTION BY PERCENTAGE. Since the entire amount invested is $700, and the gain $210, the gain on every $1 of investment will be represented by the num- ber of times that 700 is contained in $210, which is .30 or 30 cents on the dollar. Now if each man's stock be mullipUod by .SO it will represent his share of the gain thus : $300 X. 30=:$ 90 A.'s gain. 400 X "0= 120 B.'s " Entire stock 700 210 Entire gain. Hence, — To find each partner's share of the profit or loss, uhCD there is no reference to time, we have the following RULE. As the whole stoch is to each partner^ s stock, io is the vhole gain or loss to each partner's gain or loss ; or, divide the whole gain or loss by the number denoting the entire stock, and the quotient will be the gain or loss on ench dollar of stock ; which multiplied by the number denoting each partner' s share o/ tJi^ entire stock, will give hia share of the entire gain or loss. EXERCISES. 1. Three persons, A., B., and C, enter into partnership. A. advances $500, B. $550, and C. $600 ; they gain by trade $412.00. What is each partner's share of the profit ? \ns. A.'s$126i B.'fl 1137.60: C.'s $100. PARTNERSHIP. 229 2. A, B, C and D purchase an oil well. A pays for 6 shares, B for 5, C for 7, and J) for 8. Their net profits at the end of three months have amounted to $7800 ; wha* sum ouglit each to receive ? Ans. A, $1800 ; B, 81500 ; C, C2100 ; D, 02400. 3. A and B purchased a lot of land for i;4500. A paid ^ of the price, and B the remainder • they gained hy the sale o/ it 20 per cent. ; what was each man's share of the profit ? Ans. A, $300 ; B, 8600. 4. A captain, mate, and 12 sailors, won a prize of §2240, of which the captain took 14 shares, the mate 6, and the remainder was equally divided among the sailors ; how much did each receive ? Ans. The captain, $980 ; the mate, $420 ; each sailor, $70. 5. A and B invest equal sums in trade, and clear $220, of whicli A is to have 8 shares on account of transacting the business, and B only 3 shares ; what is each man's gain, and what allowaiico is mado A for his time ? Ans. Each man's gain $60 ; A $100 for his time. 6. A, B, and D enter into partnership with a joint capital ( f $4000, of which A furnishes $1000 ; B $800 ; C $1:J00, and D tho balance ; at the end of nine months their net profits amount to $1700 ; what is each partner's share of the gain, supposing B to re- ceive $100 for extra services ? Ans. A, $400 ; B, $320 ; C, $520 ; i>, $3.>.. 7. Six persons. A, B, C, D, E and F, enter into partnership, aii'i gain $7000, which is to be divided among them in the following manner : A to have I ; B, }. ; C, J^ as nmch as A and B, and the remainder to be divided between I), E and F, in the proportion of 2 2^ and 3^ ; how much does each partner receive ? Ans. A,*'$1400 J B, $1000; C, $800; D, $950; F., $1187.50; F, $1662.50. 8. A, B and C enter into partnership witJi a joint stock of $30,000, of which A furnished an unknown sum; B furnished 1^, and C 1|- times as much. At the end of six months thel- profits were 25 per cent, of the investment ; what was each man's share of the gain ? Ans. A's, $2000 ; B's, $3000 ; and C's, $2500. 9. A, B, C an 1 D trade in company with a joint capital of $3000 ; on dividing the profits, it ia found that A's share is $120 ; B's, $255 i C's, §225 ; and D's, $300 ; what was each partner's stock ? Ans. A's, $400 ; B's, $850; C's, $750 ; and D's, $1000. 10. Three labouring men. A, B and C, join together to reap a certain field of wheat, for which they agree to take the sura of I 230 ABTTHMETIC. $19.84 ; A and B calculate that they can do | of the work ; A and G f ; B and C f of it ; how much should each receive according to these estimates ? Ans. A, $8.32 ; B, $7.04 ; and C, $4.48. ill ' II '< m 1:1 I To find each partner's share of the gain or lose, when the capital is invested for di£fercnt periods. EXAMPLE. Two merchants, A and B, enter into partnership. A invests $700 for 15 months, and B $800 for 12 months; they gain $603 ; what is each man's share of the profits ? SOLUTION. $700xl5=$10500 $800X12= 9G0O 20100 : 10500 : : $603 : $315 A's gaia 20100 : 9600 : : $603 : $288 B's gain. The reason for multiplying each partner's stock by the time it was in trade, is evident from the consideration that $700 invested for 15 months would be equivalent to $700x15 equal to $10500 for one month, that is $1 0500 would yield, in one month, the same in- terest that $700 would in fifteen months. Likewise $800 invested for 12 months would be the same as $9600 for one month; hence the question becomes one of the previous case, that is, their invest- ments are the same as if they had invested respectively $10500 and $9600 for equal times ', hence the RULE . Multiply each man^s stock by the time he continues it in trade ; then say, 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. EXERCISES. 11. A, B and C are associated in trade. A furnished $300 for 6 months ; B, $350 for 7 months, and C, $400 for 8 months. Their profits amounted to $1490 at the time of dissolution ; what was the profit belonging to each partner ? Ans. A, $360 ; B, $490; C, $640. 11(1. i; FABTNEBSHtP. <>?1 12. A, B and contract to perform a certain piece of work ; A employs 40 men for 4^ months ; B 45 men for 3^ months, and 50 men for 2J- months. Their profits, after paying all expenses, are $850 ; how much of this belongis to each ? Ans. A, $340 ; B, $297.50 ; C, 8212.50. 13. Four men, A, B, C and B, hired a pasture for $27.80 ; A puts in 18 sheep for 4 months ; B, 24 for 3 months ; C, 22 for 2 months ; and D, 30 for 3 months ; how much ought each to pay ? Ans. A and B each, $7.20 ; C, $4.40 ; D, $9. 14. On the first day of January A began business with u capital of $7G0, and on the first of February following ho took in B, who invested $540 j and on the first of June following they took in 0, who put into the busine&s $800. At the end of the year they found they bad gained $872 ; how much of this was each man entitled to ? Ans. A, $384.93 ; B, $250.71 ; C, $236.36. 15. Three merchants. A, B and C, entered into partnership with a joint capital of $5875, A investing his stock for 6 months, B hia for 8 months, and C his for 10 months; of the profits cauh partner took an equal share ; how much of the capital did each invest ? Ans. A, $2500; B, $1875; C, $1500. 16. Two merchants, A and B, entered into partnership ior two years ; A at first furnished $800, and at the end of one year, $500 more; B furnished at first $1000, at the end of 6 months, $500 more, and after they had been in business one year, he was compelled to withdraw $600 At the expiration of the partnership their net profits were $2550 ; how much must A pay B who wishes to retire from the business ? Ans. $2190. 17. Three persons. A, B and C, form a partnership for one year, commencing January 1st, 1865 ; A puts in $4000 ; B, $3000 ; and C, $2500 ; April 1st, A withdraws $500, and B withdraws $600 ; Juno 1st, C puts in $800 more ; September 1st, A furnishes $700 more, and B $400 more. At the end of the year they find they hava gained $1500 ; what is each partner's share of it ? Ans. A, $608.68 ; B, $423.31 ; C, $468.01. -18. Jonn Arlams commenced business January first, 1865, with, a capital of $10000, and after some time forinod a partnership with William Hickman, who contributed to the joint stock the sum of $2800 cash. I:i course of time they admitted into tlie firm Joseph WiUiams, with a stock worth $3600. On making ti settlement January first, 1866, it was found tliut Adams had Kuined S2250 ; 1 ■ Ml 1 232 ^BITHIICETIO. m\\ ! P' i il ji ' I I"! I 'if: i^ Hickman, f 420 ; and Williams, $405 ; how long had Hickman's and Williams' money been employed in the business, and what rate of interest per annum had each of the partners gained on their stock ? Ads. Hickman's 8 months ; Williams' 6 months. Gain, 22^ per cent, interest. BANKRUPTCY. | vVlien any person is unanie to meet hk liabilitied, he makes ixn ass^^imeniof his properly to some other person or persons, called official assignee or assignees, whose ofice it is to distribute the avail- able property, after paying expenses, rateably among the creditors. An allowance for maintenance is generally made to the insolvent, but sometimes lie is compelled to surrender all hia estate, but only iu case of manifest ykiMc?, which the word hankmpt originally implied, though now it is used as nearly synonimoua with insolvent. The property to bo divided is called the assets. The bhares of the pro- perty which are divided among creditors, are called dividends. EXAMPLE. AT3aaknipt owes A $400 ; B. 6350, and C, $600; his net assets amount to $810 cash ; how much is lie able to pay oq t^ |J , aiid Low Biutvh TviU each creditor receive V BOLUTION. §400-l-$350-f $600=61350, total liabilities, ^jow, if he nas 61350 to pay, and only $810 to pay it with, he will only be able to pay $810-^1350=.60 or 60 cents on the 61. Therefore, A will receive 6400X.60=|240; B, 6350X.60=$210, and C, efiOOX-OO =$360. Hence the BTTLE. Divide the net assets hi/ the number denoting the total amount oj the debts, and the quotient will be tJie sum to be paid on each dollar, then multiply each man's claim hy the sum paid on the dollar^ and the product will be the amount he is to receive. BA2JKRUPTCY. 2^3 :man's and hat rate of leir stock ? Gaio, 22| I makes an ons, called I tlao avail - creditors, olvent, but ut only iu ly implied, ent. Tho of the pro- nds. net assets S^,aiid ■ lie naa e able to A will inox-on mount oj h dollar, lor^ mid EXERCISES. 1- A becomes bankrupt. lie owes B, $S%; C, $500; D, ^1100, and E, $G00. Tho assets amount to 01110 ; how much cun he pay on the dollar, and liow much does each creditor receive ? Ans. He can pay C7 cents on the dollar, and B receives $296 ; C, $183 ; D, ^i07 ■ and E, $222. 2. A house becomes bankrupt ; its liabilities arc $17940 ; its assets arc $8970 ; what is the dividend, and what is the shase of the chief creditor to whom $1282 arc duo ? Ans. TIk) dividend is 50 cents on the dollar, and tliO principal creditor gets $641. 3. A .shipbuilder becomes bankrupt, and his liabilities -arc 5303000 ; tho premises, building and stock arc worth $220000, and ho has in cash and notes $12875 ; the creditors allow him $3000 for maintenance of his family; the costs are 3-\ per cent, of the amount available for tho creditors ; what is the dividend, and hoxv much docs a creditor get to vrhora $1360,00 arc duo ? 4. Foster & Co. faU- They owe in Albany, $22000 ; in Biilti- more, $18000 ; in Philadelphia, $17100 ; in Charleston, 81G000 : in Boston, $4400, and in Newark, $4200. Their assets arc : hout^t proporty, $14000 ; farms, $2200 ; cash in bank, $4400 ; railway stock, $4200; sundry sums duo to them, $20135; what is the divi- dend, and bow much goes to each city ? Ans. Dividend, 55 cents on the dollar; to be paid iu Albany, .^12100; in Baltimore, $9900; in Philadelphia, $9105; in 'Charleston, $8800 , in Boston, $2420 ; in Newark, $2310. 6. The firm of Reuben Ring & Nephews becomes bankrupt, h owes to Buchanan & Ramsay, $1080 ; to Kinneburgh & McNahl $850 ; to Collier Bros., $1720 ; to David Bryce & Son, $1580 ; i Sinclair & Boyd, $970. The assets are : house and store, valucil : $848; merchandise in stock, $420; sundry debts, $220. What e;.ii the estate pay, and what is the share of each creditor ? Ans. The estate pays 24 cents on the dollar, and the payment," arc: to Buchanan & Ramsay, $259.20; to Kinneburgh & McNabb, $204 ; to Collier Bros., $412.80 ; to David Brycc & Son, $379.20 ; to Sinclair & Boyd, $231 80. 234 AETTHMETIO. EQUATION OF PAYMENTS. i i ,1!' !i M ilni j'll EqvMion of Payments is the process of finding the average or mean time at vrbich the payment of several sums, duo at different times, may all bo made at one time, so that neither the debtor nor creditor shall bo at any loss. The date to be found is called the equated time. The mode of finding equated time almost universally adopted i3 very simple, though, as no shall show in the sequel, not altogether correct. J*^' 's known as the mercantile rule. Let us observe, in the first place, that the standard by which men of business reokon the advantage that accrues to them from receiving money before the time fixed for its payment, and the loss they sustain by the payment being deferred beyond the appointed time, is the interest of money for each such period. Thus, if $50 be a year overdue, the loss is $^, at G per cent, ; and, if $50 be paid a year in advance of the time agreed upon, the gain to the payee is 83, at the same rate. In the former case, tho person receiving the money charges the payer 63 interest for the inconvenience of lying out of his money, but, in the latter case, he deducts $3 from the debt, for the advantage of having the money in hand. If, on the 1st 3Iay, A gives B two notes, one for $50, at a term of three months, and tho other for $80, at a term of seven months, the first will be legally due on the 1st August, and the 2nd on the 1st December; but A is not able to meet the first at August, and it is held over till the 1st November, when A finds himself iu a position to pay both at once. The first is then three months over-duo, and accordingly B claims interest for that time, which, at G per cent., is 75 cents, but as A tenders payment of the whole debt at once, and tho second note will not bij due for another month, A claims a deduction of one month's interest, which, at the same rate, is 40 cents, and accordingly A, in addition to the debt, pays B 35 cents. ict us now suppose another case. A owes B $130, as before, and he gives B two notes — one for $50, on 1st May, at 3 months, and another, on the 6th May, for $80, at 8 months. The first falls due on 1st August, and the other on the Gth January, but A and B agree to settle at such a time that neither shall have interest to pay, but that A shall simply have to pay the principal. Supposing that a setllemout is madti on (ith November, we find that tho lt>t note is verage or diflferent 3btor nor .dopted i3 iltogether by which bem from 1 the loss appointed IS, if $50 be paid 3 payee is iving the 1 of lying from the in the 1st nths, and )G legally 3ut A is the 1st at once. claims but as A note -will month's y A, in 'ore, and ths, and alls due and B to pay, ng that ; uote is EQUATION OP PAYMENTS. 235 3 montha and 6 days over due, and the interest on it for that period is 80 cents, while the second will not be duo for 2 months, and tho interest on it for that period is also 80 cents ; consequently, tho interest that A should pay, and that which B slrould allow being equal, they balance each other, and tho principal only has to be paid. There are, then, three methods for the payment of several debts, or a debt to be paid by instalments. The first is to pay each instal- ment as it becomes due. This needs no elucidation, nor is it often practised, except in the case of small debts, duo by persons of con- tracted means. The second is what has been illustrated above by the first exam- ple, viz., that interest is added for overdue money, and deducted for sums paid in advance of the stipulated time. The third has been illustrated by the second example, viz., to fix on such a time that the interests on the overdue and underdue sums shall be equal, m that the debtor has only to give the principal in the creditor. If, in this last ease, the time should come out as a mixed number, the fraction must be taken as another day, or thrown off, making the payment fall due a day earlier. The principle on which all such settlements are made is, that the interest of any sum paid in advance of a stipulated time is equivalent to the interest of the same sum overdue for a like time. With these explanations we are now ready to investigate a rulo for the Equation of Payments. For this purpose let us suppose a case. R. Evans owes J. Jones $200, which he undertakes to pay by two instalments of $100 each, (basis of interest per cent.,) the first payment to bo made at once, and the second at the expiration of two years. But the first payment is not made till the end of the first year, at which time R. E. tenders payment of the whole amount. For the accommodation of having the first payment deferred for one year he is to pay $6, i. c, $10G in all, and in return for making tho second payment a year before it is due, he claims a discount at the same rate, which gives $6. He has therefore, by the mercantile rule, to pay $106-|-94=:$200, so that the $6 ia the latter case balances the $6 in the former. This takes one year as the equated time, and is the mode usually adopted on account of its simplicity, though not strictly accurate. To find the equated time whcu there nie several payments to bo made at dif erent dates. Sli 1'i ]i!ll I! 236 AnrrnMETio. If A owcg B $300, payable at the end of 4 months ; 8500, paya- ble at the end of G months, and $400, payable at the end of 10^ months, to find at what time the whole may bu paid, so that interest ehall bo chargeable to neither party. The interest of $300 for 4 months is the same us the interest of $1 for 1200 months; the interest of $500 for G months is the same as the interest of $1 for 3000 months, and the interest of $400 for 10^ months is the same na the interest of $1 ibr 4200 months. The sum of all these ia 8400 months, and the interest of the whole ia the same ua the interest of $1 for 8400 months, and if $1 requires 8400 months to produce a certain interest, the sum of all the principals will require only the j25n part of 8400 montha to produce the same interest, nml 8400-5-1200=7, and hence the equated timo is 7 months. BU L £ . Multiple/ each payment hj the time that must clapsa before it becomes duCf and diaido the sum of these products hj the sum of the payments. EXAMPLE. To find tho equated time for the payment of three debts, the first for $45, duo at the end of G months ; the second for $70, due at tho end of XI months, and the third for $75, due at tho end of 1.^ months. $45X 6=$270 70X11= 770 75X13= 975 190 2015 and 2015-5-19=10^^^, so that the cquatca time will be 10 months and 18 days, the small i«emaining fraction being rejected. Let us suppose that notliing is paid until the end of the 13 months, and all paid at once, then tho amount to be paid will be, at G per cent., For first debt overdue 7 months, $45-|-1.57^, interest for 7 months $4G.57J For second debt overdue 2 months, $70-j-.70, interest for 2 months 70.70 For third debt just due, $75. no interest 75.00 |192.27i )0, paya- l of 10^ b interest 00 for 4 interest for 3000 no as tho ia 8400 itereat of )ro(luco II iirc only ireat, »nd before it urn of the tho first ,ue at tho hd of la EQUATION OP PAYMENTS. 237 ) months f the 13 ill be, at 70.70 75.00 fl92.27i The work may often bo soraowlmt shortened by counting tho differences of time from tho date at which tho first payment becomes due, the mean time between tho dates when tho first and last become due being alone required. If a person owes $1200 to bo paid in four instalments, §10C i 3 months; $200 in 10 months; $300 in 15 months, and $600 in 18 months, then the excesses of time of tho last three above tho first ore 7, 12 and 15 months, and the work will stand aa below* 0100 (no time.) 200X 7^1400 300X12=3000 600X15=0000 1200) 14000 (11§ and llf X3=14f months. This j^dves the n u I- E . Multiply each deht, except the one first due, 6y the difference be- tween its term and the term of the first ; dinidc the sum of the pro- ducts hi/ the sum of the debts, the quotient with the term, of the first added to it loill he the equated time. Another method, which is often convenient, may bo illustrated by the example already given, aa the two operationa will give tho same result. Interest on §300 for 4 months=$ G.OO Interest on 500 for G " = 15.00 = 21.00 Interest ou 400 for 10^ " Interest on 1200 for 1 month=6)42.00(7 months as before. KULE . Find the interest on each instalment for the given time, and divide the sum of these by the interest of the whole debt for one month, and the quotient will be the equated time. As the sum of the instalments is equal to tho debt, the result will be the same for any rate of interest. For the first instalment, $300, overdue 3 months, A has to pay U 50 For the second instalment, $500, overdue 1 month, A has to pay 2 50 $7 00 238 AiirnoiEno. For tho third instalment, $400, not duo for 3^ mouths, A has to get $7 00 so that tho uniouuts of interest exactly halancc, and tho paying of tho wliolo, nt tho end of 7 months, is precisely equivalent to tho pay- ing of each instalment as it falls duo. Tho only diflFercnoo that could arise in, that it might bo inconvenient for tho creditor to Ho out of the first instalment for tho three months. In all other respects tho settlement is strictly equitable, according to tho understanding that exists among business men. In the first place, the diflFcrcnco between this and what is called " tho accurate rule," is insignifi- cantly small; and, in the second place, tho *' mercantile rulo" saves mucli time, and time is equivalent to so much capital in mercantile transactions. Independently, however, of any other consideration, ^0 may remark that when tho mode of reckoning is conventionally understood, it ])ccomes perfectly equitable, because every merchant knows the terms on which ho can do business with any other, just as bank discount becomes perfectly equitable, because every man, before going to a bank for tho discounting of a note, knorws perfectly well on what terms ho can have it. Much warm discussion has been indulged in on this subject ; but, as we consider tho discussion more subtle than profitable, we shall dis- miss tho subject in a few words. We shall adopt tho usual case, that A owes B 8200, one-half to bo paid at tho present time, and tho remainder at the end of two years. It is perfectly obvious that, a6 tho end of tho first year, A should pay '$100, that is, tho prinoipaF, plus the interest agreed upon. Regarding the settlement of tho second instalment, if A proffers payment of tho whole at onco, ho is clearly entitled to claim a reduction for tho unexpired term. Now, tho question is, what ought the reduction to be. By tho mercantilo rule lie sliould pay $94, but the true present worth of $100, due at the end of the year, would be 94.33§J , so that he would have to pay $10o on the instalment over due, and $94.33|-] on tho one not due, making $200.33:;J, whereas the object is to find at what time inter- est should be chargeable to neither party. As a furthci- illustration of the general rule, let us suppose that J. Smith owes R. Evans $1300, of which $700 are to be paid at tho end of 3 months, $100 at the end of 4 months, and the balance at *h.c end of 8 months, to find the equated time. \Vc shall suppose that J. Smith agrees to pay R. Evans the whole amount at the time the debt was contracted : then J. 8mith would EQUATION OF PAYMEOT9. 239 owe 11. Evfliis $1300, mintia tho discount for the length ol'tirao the lunount was \\\id bcibro it became duo, viz., three months, equalling the discount on $L'U) for 1 month; $100, less tho discount for 4 months, equalling the discount on $400 for 1 month ; $500, less tho discount for 8 months, equalling tho disc(mnt on $1000 f )r 1 month. This pivc3 a total of 62100+$400-|-§4000:3^8G500, for 1 month. Now, it is evident that if J. Smith wished to pay tho wholo amount at such a time that there should be no loss to citlicr party, ho must retain this amount for such a length of time aa it will toko this amount to equal tho discount on $G0QO for 1 nionth, which will bo ja'oj of 80500, that is, for 5 months; To frove that 5 mouths must bo tho equated time, we h:\te rccouri-o to Die principles laid down under tho head of Interest. If a settlement is not mado until the expiration of 5 months from the time the debt was contracted, then J. Smith would owe R. Evans !$700, 2iilus tho interest of that principal during the time it remained unpaid after becoming due, viz., two months, which would givo an amount of $707. So also, $100, j)lu3 the interest for 1 month, would be $100.50, and $500, minus its discount for 3 montha (the length of time paid before duo), would givo 87.50, leaving 8402.50, and e707-|-$100.50-f-$492.50^81300. ^ EXERCISES. ^ 1. T. C. Musgrovo owes H. "W. Field $900, of which $300 arc due in 4 months ; $400 in G months, $200 in 9 months j what is the equated time for the payment of tho whole amount ? Ans. G months. 2. E. P. Hall & Co. havo in their possession 5 notes drawn by G. W. Armstrong, all dated 1st January, 18G5 ; the first is drawn at 4 months, for $45; the second at 8 months, for $120; the third at 10 months, for §75 ; tho fourth at 1 1 montli.':;, for $G0 ; and tho fifth at 15 months, for 890; for what length cf time must a single note be drawn, dated 1st May, 1865, so that it may fi 11 duo at liio properly equated time ? Ans. G months. 3. A merchant sold goods as follows, on a credit of G mcntlis: — May 10, a bill of $G00 ; Juno 12, a bill of $450; September 20, a bill of $900; at what time will tho whole become due? Ans. January L6. 4. A luevchant proposed to sell goods amounting to $4000 on 8 months' credit, but the purchaser preferred to pay h in cash and J in 3 months ; what time shoiild be allowed liim for the payment of the remainder ? Ana. 2 years. 5 months. 240 AEITHMETIO. 5. A gcntlomon left his son $1500, to bo paid as follows: J in 3 months, j in 4 months, | in G months, and the remainder in 8 months ; at what timo ought the wholo to bo paid at once ? An;-:. 4 mos., 15 days. G. A merchant bought goods amounting to $G000. Ho agrees to pay $500 in cash, $G00 in six months, $1500 in 9 months, and tho remainder in 10 months; ut what timo ought ho to pay tho wholo in one payment? Ans. 8j]^ months. 7. There is duo to a merchant $800, one-sixth of which is to bo paid in 2 months, one-third in 3 mouths, and tho remainder in 6 months ; but tho debtor agrees to pay one-half in cash ; how long may ho retain tho other half, so that neither party may sustain loss? Ans. 85| months. 6. A merchant sold to "W. L. Brown, Esq., goods to tho amount of $3051, on a credit of G months, from September 25th, 1864. October 4th Brown paid $47G; November 12th, $375; December 5th, $800 ; January 1st, 1865, $200. When, in equity, ought tho m.'u-chunt to receive tho balance ? 9. A having sold B goods to tho amount of $1200, left it optional with him cither to tako them on 3 month's credit, or to pay one-half in cash, one-fifth in two months, one-sixth in four months, and the remai idor at an equated timo, to correspond with tho terms first named ; what was tho timo ? Ans. 4 years, 4 mos. 10. A grocer sold 484 barrels of rosin, as follows : February 6th, 35 barrels © $3.12^, or. 4 months' timo. TMarch 12th, 38 barrels @ 3.00, on 4 months' timo. "March 12th, 411 barrels @ 2.62^, on 4 months' time. VTliat is tho equated timo for the payment of the whole ? Ans. July 9 th. 11. Bought of A. B. Smith & Co. 1650 barrels of flour, at dif- ferent times, and on various terras of credit, as by tho following tatomont; what is tho equated time for the payment of Iho whole? May Gth, 150 barrels, at $4.50, on 3 months' credit. May 20th, 400 barrclb, at 4.75, on 4 months' credit. July 10th, 600 barrels, at 5.00, on 5 months' credit. August 4th, GOO barrels, at 4.25, on 4 montlis' credit. Ans. November Tth. 12. J. B. Smith & Co. bought of A, Hamilton & Son 576 bar- rolfl of rosin, as follows : May 3rd, 62 barrolfl ^ 82.50, on 6 months' credit. ! i AVERAGINQ ACCOUNTS. 241 rs: ^ in 3 ndor in 8 > ,15 days. Ho agrees onth8, and to pay tho !,', months, ch is to bo naindcr iu , how long stain loss? 3j| months, tho amount !5th, 1864. December , ought tho 200, left it t, or to pay ur months, ;i tho terms ars, 4 mos. Itimo. Itimo. I time. »le? July 9 th. |our, at dif- foUowing ho whole ? ledit. edit, edit. Icdit. imbertth. in 576 bar- May 10th, 100 barrels @ 2.50, on 6 months' credit. May 18th, 10 barrels © 2.50, as cash. May 26th, 60 barrels @ 2.75; on 30 days* credit. May 26th, 345 barrels © 2.50, on six months' credit. May 26th, 9 barrels @ 2.00, on six months' credit. What is tho equated time for the payment of tho whole ? Ans. November 3rd. 13. Purchased goods of J. R. Worthington & Co., at different times, and on various terms of credit, as by tho following statement : March Ist, 1863, a bill of $675.25, on 3 months' credit. July 4ch, 1863, a bill of 376.18, on 4 months' credit. September 25th, 1863, a bill of 821.75, on 2 months' credit. October 1st, 1863, a I'm of 961.25, on 8 months' credit. January 1st, 1864, a bill of 144.50, on 3 months' credit. February 10th, 1864, a bill of 811.30, on 6 months' credit. Marrih 12th, 1864, a bill of 567.70, on 5 months' credit. April 15th, 1864, a bill of 369.80, on 4 months' credit. What is tho equated time for tho payment of tho wholo ? Aus. March 16th, 1864* AVERAO-lNa ACCOTJNTP "When one merchant trades with ftnother, exchanging merchan- dise, or giving and receiving cash, tho memorandum of tho transac- tions is called an Account Current. If the goods bo purchased at different dates, or for different terms of credit, and some are not duo while others arc overdue, tho fixing on a timo when all may bo set- tled, BO that no interest shall bo chargeable to either party, is called Averaging the Account. Since interest is tho standard to which is referred tho Lenefit of receiving money before it is due, so that in tho meantime it can be used in trade, and also the damogo of not getting it when duo, it is fair and proper that interest should bo charged on all sums overdue, and deducted from all not duo. In illustration, let us ^)U])poso that A sells goods to B, March 2, on 4 mouths' credit, and ji>:;iii. i.u equal amount on March 20, on 6 mouths' credit j tlio CrMt w'>\\ hv 'luo on July 2, and the second on September 20. Should B teuJfi |.aymon< of the whole on June 2, he would be entitled to claim i^Merc^t fox 1« /Ill ■;iii I l^iiii^ i.t i '\ / Iri- 1 jj hi 1 ii H '1 il 1 1 I! 242 ABITHUETIO. one month on the first pnrchaso, and for three months and eighteen days on the second. But if payment be delayed till August 2, A would be entitled to one month's interest on the first purchase, and B to the interest on the second for one month and eighteen days, so that there would be in favour of B, on the whole, a balance of inter- est for eighteen days. Again, supposing the settlement is not made till September 20, when all is due, no interest can be either chained or claimed on the second purchase, the term of credit having just then expired ; but as the first debt is two months and eighteen days overdue, A is entitled to interest on it for that period. If neither is paid till after September 20, A has a right to claim interest on each for the period it has been overdue. But this regulates only one side of the account. In order to settle the other, let us suppose that B has, in the meantime, sold goods to A, it is obvious that B's claims on A must be settled on the very same principle, and that therefore the final result must be simply the finding of the balance. It is more usual, however, in accounts current, to fix on a time such that the inteicbt duo by A shall exactly balance that due by B. To illustrate this, let ua suppose a case corresponding to a lc(?«^ Docount: B. EVANS. 1865. Dr. July 21, To Merchandise on 2 months* credit... $200 July 25, To Cash 150 Aug. 24, To Merchandise on 4 months' credit... 100 Sept. 21, To Merchandise on 3 months' credit... 250 $700 1865. Gb. August 1, By Cash ..$100 August 20, By Merchandise at 22 days 110 Sept'r 30, By Cash 180 Balance 310 §700 To find in this case at what time the account may bo settled so that interest shall be chargeable to neither party. Equating tlie ttmOi aa in equation of payment?, we have the following operation ; eighteen rust 2, A base, and I days, BO ! of inter- not made p chained wing just teen days [f neither iterest on ates only IS suppose 3 that B's and that e balance. time such lyB. To a le(}*r R. ,00 50 loo 150 m )0 I settled 60 lating the ration ; AVERAGma ACCOUNTS. 243 1865 July 25 150x Sept. 2\ 200 X 58=11600 Deer. 24 250X152=38000 Deer. 21 100 X 149=14900 64500 Gb* 1865. August 1 lOOX Sept. 12 110X22=2420 Sept. 30 180X71=12780 390 15200 15200-f-390=39 days. Due 39 days from August 1, viz., on September 9. 700 64500-^700=92 days. • Duo 9C days from July 25, viz., on October 25. Time from September 9, to October 25=46 days. :Exces3 of debit above credit 700—390=310. 390X46=17940, and 17940-;-310=58 days, nearly. Counting 58 days forward, from October 25, will bring ns to December 22, the time required for a settlement, with interest chargeable to neither party. Here the time is counted forward from the average date of the larger side which becomes due last, but had it become due first, we should have counted backward. The first transaction on the debit side being two months' credit from July 21, is not to bo taken into consideration till September 21. The second transaction, being a cash one, and therefore consid- ered as so much due, will therefore mark the date from which all others shall be reckoned ; and, since there is no interval of time, wo write it without a multiplier. The next transaction has a term of credit extending to 152 days, and therefore we write 250x152=: 11600. The term of the next extends from September 21 to December 21, a period of 149 days, and wo write 100x149=14900. The Bum of the debits is $700, and the sum of the results obtained by multiplying each item by the number of days it has to run from July 25 is $64500. Then 64500-^-700=92, the equated time in days for the debit side. Now, as already explained, the interest for 6700 for 92 days will be the same as the interest of $64500 for 1 day. Hence, the debits are due 92 days from July 25, viz., on October 25. In like manner, on the credit side, the first transaction being a cash one, we start irom its date, August 1, and, as there is no inter- val, we have no multiplier. The second being merchandise, on 22 V L: 'm if"' *'ii ill! ; , ,i ill iis: ARITHMETIC. daya' credit, we write 110x22=2420. The third is cash paid 71 days after August 1, and we write 180X71=12780, Had ti.G nccount been settled on September 9, the debits would haTO beei) p -i 1 4G days before coming due, and the credit side would have gained and the debit side lost the interest for that time. Again, wc must consider how long it would take the balance, $310, to produce the rsame interest that $390 would produce in 4G days. It is obvious that whatever interest $390 gives in 46 daya will require 46 times $390 for $1 to produce the same interest, that is, 390X46=17940 days, and it will require 17940—310=58 days, for $310 to produce the same interest. If the settlement ia made on October 25, the latest date, then the credit has been due 46 days, and therefore bearing interest; and in order that the debit side may be increased by an equal amount, the time must I'e ex- tended beyond October 25, that is, it must be counted /orioartZ. Foe ihe same reason, if the greater side had become due first, then the balance must be considered as due at a previous date, and therefore wo must count backward. An account may be averaged from any date, but either the first or the last will be found the most convenient. The first due is generally used. On the principles now explained may be founded the followiug RULE. Find the equated time when each side becomes due. Multiply the amount of the smaller side by the number of days helvoeen the two average dates, and divide the product by the balance of the account. The quotient thus obtained will be the time that the balance becomes due, counted from the average date of the larger side, for- WAED lo/ien the amount of that side becomes due last, but BACK- WARD when it becomes due FIRST. The cash value of a balance depends on the time of scttlcuiont. If the settlement be made before the balance is due, the interest for the unexpired time is to be deducted; but if the settlement is not made till after the balance is duo, interest is to be added for the time it is overdue. EXERCISES. In J. H. Matsdon'a Ledger, we find the following accounts, which, 11 ii i.LL, , N. AVERAGING ACCOUNTS. 245 ih paid 71. bits would side would me. 10 balance, []uce in 4G in 46 days ;erest, that ;tleraent id een duo 4G the debit lUst be cx- mird. Foe ;, then the I therefore er the first irst due is J following r of days he balance le balance side, FOR- but DACK- cttleuicnt. tcrest for nt is not d for the ts, which, on being equated, stand as follows ; at what time should tho respec- tive balances commence to draw interest : 1. Dr. J. S. Peckham. Cr. May 16th, 1865 $724.45. | July 29th, 1865 $486.80.' Ans. December 15th, 1864. 2. Dr. Nelson Bostford. Cr. November 19th, 1865 $635. | December 12th, 1865 0950. i Ans. January 27th, 1866. 3. Dr. James Crow & Co. Cr. February 24tli, 1866....$512.25. | June 10th, 1865 $309,70. Ans. March 27th, 1867. 4. Dr. J. H. BuRRiTT & Co. Cr. March 17th, 1866 $145. | January 15th, 18G0 $695.60. Ans. December 30tli, 1865. 5. Dr. M. McDonald. Cr. August 27th, 1865 $341. | November 7th, 1865 ..$247. 6. Dr. James I. Musorove. Cr. July 20tk 1866 $711. | April 14th, 1866 ^12G0. Ans. December 9th, 18G5. 7. Dr. Thos. a. Bryce & Co. Cr. June 24th, 1864 $1418. | September 7th, 1865...,., $2346. 8. Dr. E. R. Carpenter. Cr. December 2nd, 1865... $1040.80. | August 13th, 1865.... 81112.40. 9. Required the time when the balance of the followbg account becomes subject to interest, allowincc the merchandise to have beea on 8 months' credit ? Dr. A. B. Smith & Co. Cr. 1864. May 1, July 7, Sep. 11. Nov. 25' Dec. 20. ' ] " "\\ IMo. ' To Mdae $300,001 .Tan. 1, " '• I 759.90 Feb. 18, " " 417.20 Mar. 19, 28Y.70 April 1, 571.10iiMay 25. (( (( By Cash..., " Mdse. , " Cash.,., " Draft.., $500.00 481.75 750.25 210.00 100.00 Ans. August 5, 1865. §!■ 'iJ ll • 246 ARITHMETIO. 10. When will the balance of the following account fall due, the merchandise items bcinp: on 6 months' credit ? Dr. J. K. White. Cr. 1865. May 1, May 23, June 12, Ju.y 29, Aug. 4, Sept. 18, To Mdse. Cash paid dft. . Mdse Cash ,,,,.• || 1805, $312.40 June 14 By Cash 85.70; July 30, 105.00 I Aug. 10, 243.80 Aug. 21, 92.10 50.00 Sept. 28, Mdse. Cash. Mdse. u $200.00 185.90 100.00 58.00 45.10 Ans. January 12, 1866. 11. When docs the balance of the following account become subject to interest ? Dr. W. H. MUSGROVE. Cr. Aug 1864. 10, Aug. 17, Sept. 21, Oct. 13, Nov. 25, Nov. 30; Dec. 18, 1865. Jan. 31, To Mdse 4 mos. " " 60day.s " " 30 " Cash p'd dft. Mdse 6 mos. " 90 days " 2 mos. Cash. $285.30 192.60 256^80 I Dec. 15, 190.00 'Dec. 30 432.20, 1865. 215.25 Jan. 4, 68.90 Jan. 21, 100.00 1864. I Oct 13, By Cash. Oct. 26,1 " Mdse 2 mos Cash. $400.00 150.00 345.80 230.40 340.30 180.00 12. In the following account, when did the balance become due, the merchandise articles being on 6 months' credit ? Dr. R. J. Bryce in account with D. Hicks & Co. Cr. 1864. Jan. 4, Jan, Feb. Feb. Feb. Mar, 18, 4, 4, 9, 3, Mar. 24. April y' May 15, May 21, To (( It a u (( (I u u a Mdse $ 96.57 57.67 80.00 38.96 50.26 154.40 42.30 23.60 28.46 177.19 1864. Jan. 30 By Cash . . . U It • • • • • • $240.00 u April 3, May 22, 48.88 Cash paid draft. Mdse 50.00 Cash paid draft. Mdse << (( K «« „..„...... Ans. Dccembe? 22nd, 1864. (Im ill due, the 200.00 185.90 100.00 58.00 45.10 r 12, 1866. mt become CV. • • • $400.00 • • « 150.00 lOS 345.80 (( 230.40 340.30 « • • 180.00 ecorne due, Cr, $240.00 48.88 50.00 2nd, 1864. AVERAGING ACCOUNTS. 247 13. When, in equity, bhould the balance of the following accouni be payable? Br. J. McDonald & Co. Cr. \m% Jan. 3, To Jan. 31, '( Feb. 8, ( Feb. 21, (( .Mar. 10, 1. Mar. 24, .( Apr. 12, June 1, June 20, ^23.454 credit interest=$11.09, CASH BALANCE. 249 the balance of interest, and $1400, amount of debit items-{-$11.09 =81411.00, and $1411.00— $920 amount of credit itcuis^S4a 1.09, the cash balance, which is the same as obtained by the first solution. Hence from the foregoing vo deduce the following RULE. Multiply each item of dclit and cndit hy ihc nnmter of days intervening hetwecn its becoming due and the time of settlement. Then consider the sums of the products of the debit and credit items as so many dollars, andjind the interest on each for one day^ ichtch will be the interest, respectively, of the debit and credit iVcnis-- Place the balance of interest on its own side of the acconnf^ and the difference thc7i beticcen the two sides will be the true balance ; or, Find the interest on each item from the date on which it becomes due to the time of settlement. The difference of the. sums of interest, on the debit and credit sides of the account, will represent the balance of interest, which is placed on its own side of the account ^ aiul lEc difference then between the two sides will be the true balance. The following is a form of statement or account current, geaorally made out by merchants in determining the cash balance. They are usually rendered quarterly or half-yearly, unless called for by the payer, in which case they are balanced by cash if paid at once. 1. — ^What will be the cash balance of the following account, on Sept. 1st, 1366, interest calculated at 6 per cent. 2 n. C. Wright in account current and intnrcst account t^) Sept. 1st, ISuiJ, Willi C. O'Dea. Date. ITEM3. PniN-ci- I'AL. When- DUE. Time. I.NT. Date. ITEM3. Pr.ixci- V\L. ^v^r.^•' DUE. ^^'^• IXT. Mar. 4 To Mdso. & t imn. tm 11 July 4 r.at:y,i fi 37 1 Juno 1 Ey C.isli im ari't. §i;j !'■ I'-lIli) 1 92(.';y3 $2 CI ' - 2:: " «.):» •■ ;!uu o:i ■• 2;i ■ID ■' •i (Id' " 21 •• Dr.irn-: JJilv.t 4t 18 Jr.lyJl :VJ - 29 Ul) ecoiiv?. -ii 07 Junc'J;. (it " 0S1| Julys J " Ca^n o.i a':.:t. ;!,) ;■ ■ •• XU.Zi •• 21 JuiiO 2 " 03 Ca^li. ;u 18 • ( o ai " OVl " " ■■ Driiftf')' ;7.i.vs 48 V \yyS-~\ ^ " 02 •• 18 •■ @'.'>aoi. 171 21 .V.i;.l« 14 " 41, Aii-.IO " Cash o;i actt.l 401 7. " m'ls ■• : «7 July 17 " (!J)1W(1V^ ■M) UJ "It; 1« " Ky ^Cl)t. 1 " Ualauco • - • 1 flul It Ull. of lutCT 13 " 111 a.t C.iih. SvO 0;i July I'J 41 " 3 07 To balance of lut. To Balance CIClil 8-.> $M 13 ♦ii;,ii K ♦1013 Sept. 1 «ni:i l.s Note. — If any itera should not come duo until after the time of settlement, the side upon which it i.s should ha diminished, or the opposite side increascdi by the interest of such item from the time of settlement until due. Tf -'i f 'II 250 AIUTHMETIO. 2. The following account was settled in fall ou December 1st, 1865 ; what amount was paid, interest G per cent. ? C. p. Meads in account current ami interest account to Dec. 1st, 18C5, with T. R. Brown. Date. Jan. Fill. 8 Mar. 2j " 3J May ir> Aug.'.'i) Items. 1 fo MU.\, luo.j. MUdo., 4 r,ia.i. .. 4 .. Caih p(l. est. M(l;p., nio). PRrxci- I'.VL. ^'•M 1 lOO :uo V 1 10 2i;i 01 When- DUK. Time, i.vT. 1 Date. iTtUS. .4 Kib. .Mur. May 1 July 1 Hup. 10 111. Jy C'a>h '' Mdiio,, 4 mo». " .. ^ .. .. 4 ,. Puisc:. PAL. ti'jo 00 42U 10 3o0 00 C» IH) SIU 81 Whbn DUE. TlUt. Int. Ans, $61.36. 3. What Will bo the cash balance of the following account if settled on January I, 18G5, allowing interest at 8 per cent, on each item after it is duo? Ans., $110 86. •T. Smitb Ilomans in acct. current and intircst account to Jan. 1st, 1865, with T. C. Musgrova. Date. Items. PHINCI- r.vL. WHEN DbT. Tlmo I.NT. Date. IKOl. Apr. 15 May 10 JllUClL' •' 30 Jajyl.-, A'l-.'O •' 20 " ;ii Items. Pr.iwi- I'AL. When- Due. TIME. Int. 18 J I. Juucl! To MJ30., 4 mo3. " " " " C;V,U pd. lift. " CuU " MJm., 2 mo.i. " " 1 mo. " Cajh " M'hc. ai Cus'i «."1j 01) 18J0J 2111 0;) 7.j 00 CJ03 103 00 f))03 15DO0 300 00 By Mdso., 3 mos. ,. .. 4 .. J350 00 120 00 *' 2: July IP All" '>0 '.'.'.'.'.'J 1 1 Ca'jii ..." »'^* '^ "31 " M(l<(>. ar) Cash •• C;ish " Mtlso., 3 mo.i. SO 00 80 00 100 00 17.j 00 ir, 00 Sept. ; 1 '.'.'.'.'.'J Oct. 14 " IC '.'.'.'.'.'} 4. What will be the amount due on the following account, July 1st 18G7, interest calculated at 7-^q per cent. ? E. G. Conldiii in account current and interest account to July 1st, 1SC7, with J. B. Harris. DATE. Jaly 1 j Aui'.SO ticp. 1!) N0V.2.J Vkc. 2) 38j7. .T.m. 12 -Mir. 10 41^/20 Items. To:ja-?,e3Dii"s. 3 mo I. OJily.i. 4 mo.i. OOiIyj. 4 r.ioi. 2moi. 30(ly.^. PniNci- TAI.. When Du:-.. TI.ME. IXT. Date. »;oi c:. 2: 8 r: 1'.! 17 1 ' iiij.i. Au3.11 1 " 31 Oi-t. 18 Nov. 1 110 8. 4:8 3 : 100 0- 2.->7 7.- Dec. 21 1 Jan. 3.) Mar. 4 Jimc2.'j 1 II 1. Items. r.v Ca
  • ) 00 4'IJ Oil aj.) 00 CJOOO lUUOO S-IOOO 130 OQ Hoot. « Octr .1 Nov. '.10 Dm- i\i • 1 II II 13^.. .Tun. 16 l.tur.a 36. i if settled item after C. Musgrove, HEN UE. TIME. INT. 1 f t, July 1st r. B. Harris. HEX UE. TIME. I.^^. :;:: March 25, $50.G4. ACCOUNT OP SALES. An account of 6alcs is a Btatcmcnt made by the consignee (^gen- erally a commission merchant) to tho consignor, the person from whom the merchandise was received to sell, showing the pcrsonH to whom sold, the price, time, charges, and net proceeds. The net proceeds is the amount due the consignor, from proceeds of Bales, after all charges are deducted, and arc duo to the consignor at the average time of sales. Commission merchants often become interested in the merchan- dise consigned to them for sale, by accepting a certain share and sell- ing on joint account of themselves and the consignor ; when this is the case, the gain or loss is shared according to the way in which the merchandise was originally divided. "When the commission merchant accepts the merchandise to sell on joint account, the terms upon which he becomes responsible for his share should be known, whether payable as cash, some definite term of credit, or at average time of salc^ In the following account sales of merchandise sold for A. R. Eastman, at what time will his net proceeds be due, as cash, and what will bo the amount due A. R. E. on 3Iay 14, 18G7, discount 6 per cent. ? Ans, Net proceeds due May 21, 1867. Due A. R. E.. S2370.94. Tfi s:*' £52 ARITHMETIC. Account sales of 8745 lbs. bacon, 207(5 lbs. chocsc, and 1245 lbs. butter, for account and risk of A. II. Ktvstuian, Cliicago, 111. Mar. 10 Apr. 20 May 14 Mardi 1 " 9 May 14 Sold ) 11. White, at 30 day.s— 4000 lb.s. bacon, at IGc... DOO Ibij. butter, at 40o.. Sold to J. B. Harris, for cash — 4745 lbs. bacon, at 15jC Sold to J. C. Parsons, at GO days — 2970 lbs. cheese, at 22o 745 lbs. butter, at 41c....... cnAiiaES.— Paid freight in cash Paid for labor rcsaltini; bacon Storaa;o Commission on $2547.51 at 2^perct. Net proceeds duo per average May 21 $640 00 200 00 054 72 305 45 97 40 8 50 5 00 03 09 $840 00 747 34 9G0 17 82547 61 174 59 2372 92 Cleveland, 0., E. and 0. E. May 14, 1867. E. Felton & Co. September 4, 1806, wo received from W. Cummings, Cincinnati, n consignment of 120 brls. of moss pork at $25.00 per bbl., and 742 bushels clover seed at $8.30 per bushel, to be Sold on joint account of consignor and consignees, each one-half; consignees' half subject to uvera2cc sales. The same date we cashed his demand draft in favor of Third National Bank, for $3450. The following is the account sales. At what timo aro the net proceeds due as cash, and what amount in cash will settle our account with W. Cummings on Jan. let, 1SG7, interest G per cent ? AnS. T(j last queation, 81552.56. Account sales of 120 hrh. mess pork, and 742 bushels clover seed, oa joint account of W. Cummings, Cincinnati, and. ourselves (each one-luilf.) Sept. 12 Sold to C. 11. Sing, for cash — I 250 bsh's. clover seed, at $8.95 30 Sold to M. Ilollingsworth, at 60 dys. — 25 brls. mess pork, at $32 80 bshls. clover seed, at $9,25... (( $2237 50 1540 00 Oct. 18 Novr. 2 « 15 CASH BALANCE. Sold to T. M. Ames, at 30 days— 200 bshls. clover eccd, at 89.23... 10 brls. mcBS pork, at iij;[]2.D0... Sold to T. R. Brown, for ca«li — 200 bshls. clover seed, at ^O.^O. Sold to A. W. Purdy, at (5 months— 85 brls. mess pork, at 83!] 12 bshls. clover seed, at S0.30 ... 251 Sept. 4 (( it Octr. 8 Nov. 15 1850 00 :J2'5 00 2805 OC 111 00 CnARQES. Paid frei2;ht and cartago in c:jsh I'aid insurance on §9500, at 1.^ p. ct. Paid for storage, cooperajj;c, and labor Commission on $10,729.10, at 2^ p. c Net proceeds of sales. I Your ^ net proceeds, duo as per aver. 210 118 15 2G8 2175 00 18C0 00 291 G CO 10729 10 75 75 00 23 G12 73 lOllG 37 5058 18 Columbus, Nov. 15. 18G6. E. and 0. B. J. G. D^INISON & Co. January 2, 18G7 — Received from D. M. Ilarmau, Owosso, Mich. 200 brls. pork, invoiced at §18 per brl. ; 3750 lbs. cheese, at 10c. per lb., and 100 firkins of butter, each 80 lbs., at IGc. per lb., to bo sold on joint account, of Bhippera §, and ourselves J ; our ^ of invoice due as cash. January 21 — Cashed J). M. Ilarman's sight draft in favor of First National Bank, Cleveland, for $1204.50. February 14 — Accepted D. M. Ilarman's ono month sight draft in favor of Thos. L. Elliot, Owosso, for mutual accommodation, ibr $8fi4. February 28 — Cashed D. M. Ilarman's demand draft in favor of Third National Bank, Detroit, for §1174.75. Find equated time of sales in the following account ; equated time of D. M. Harman's account; also, give the amount that would bal- ance his account on May 14th, 18G7, reckoning interest at the rate of 7 per cent, per annum. Ans. Equated time of sales, April 9, 1807. Equated time of Harman's acct., April 23rd. Balaace of account on May 14, §2456.38. $ \m ! I 1 iM 254 AEITHMETIO. Account sales of 200 brls. pork, 3750 lbs. cheese, and 100 firkins butter, on joint account of D. M. Harman f , and ourselves i^. Jan, lU Feby. 9 « 27 ►Sold A. n. Peatman, on his note at 2 months — 60 firkins butter, 80J lbs. each, at 24c per lb 1895 lbs. cheese, at 14c March 7 " 24 Jarry. 2 " 14 " 15 Mar. 24 " 24 ^old A. 8. Morrison, on 90 days — 40 brls. pork, at $18.75 per brl 50 firkins butter, 80 lbs. each, at 24c per lb Sold "W. E. Glennie, on 2 months — GO brls. pork, at $19.37^ per brl, 1150 lbs. cheese, at 13o per lb Sold II. D. Wright, for cash— 75 brls. pork, at $19.25 per brl Sold A. B. May, on 30 days— 25 brls. pork, at $19.87^ per brl 705 lbs. cheese, at 15c per lb.... — CHARGES. 6750 00 960 00 1162 50 149 50 966 GO 265 30 490 88 105 75 Paid for freight and cartage by cash. . . Paid for cooperage and extra labor by cash Paid Insurance at IJ per cent Charges for storing in storehouse Com. at 2i per cent, on $6299.68 is. Net proceeds of saloa. Your § of N. P. duo, as per average, April 12th $1710 00 1312 00 1231 30 1443 75 602 63 6299 68 122 7ff 50 12 46 11 25 157 49 310 45 «••••» c-*^ • • • 5989 23 3992 82 E. and 0. E. E. Geo. Conklin & Co. Cleveland, 0. Dec. 1st, 1867— We received from Messrs. Gillespie, Moffatfc & Co., Boston. 27 cases Mackinaw blankets, 340 prs. at $3.90; 2 cases CASH BALANCE. 255 chintz cotton, 987 yds. at 7gc ; 20 pes. tablo oil cloth, at $3.70; 4 pes. do., at $5.G2i ; 7 pes. West England broad cloth, 126 yds. at $3.70 ; 7 bales cotton batts, at $G.20 ; to be sold on joint account and risk of consignor and consignee, each one-half, our one-half as cash. Pec, 5th — Wc cashed their demand draft for $1200. Dec. 17 — Accepted their draft on us at 30 days' sight, for S984, Jan. 14 — Cashed their draft on demand, for $500. Sales of merchandise as per account sales annexed. At what date arc the net proceeds due as cash ?' What is the equated time of G. & M.'s account ? What is the cash balance on Slarch 2-1, 18G7 ? Account sales of 27 cases Mackinaw blankets, 2 cases chintz cotton, 24 pes. table oil cloth, 7 pes. West England broad cloth, and 7 bales cotton batts, on joint account of Gillespie, Moflfatt & Co., Boston, and ourselves (each one-half). Deer. 5 (( 9 " 14 u 17 Sold John McDonald k Co., A cash, -V on account 30 days — 13 cases blankets, 260 prs. at C4.20 1 case chintz cotton, 425 yds. at 9o 7 pes. tablo oil cloth, at $4.50 . . . Sold K. Chisholm & Co., on note at 6 months — 7 cases blankets, 140 prs. at $4.50 3 pes. W. E. broad, 54 yds., at $4.20 4 pes. table oil, at $6 Sold Thomas & Arthurs, for cash — 1 case chintz cotton, 562 yds. at 9o 13 pes. table oil, at $4.40 3 pes. W. E. broad, 54 yds., at Sold James A. Dobbie & Co., note at 90 days 2 bales cotton batts, at $7 4 cases Mackinaw blankets, 80 prs., ai $6.70 1 po. W. iJ. broad, 18 yda., at $5 256 AMTHMETIO. (( 28 Sold Thomas Spenco & Co., J cash, balance on acct. at 30 days — 3 cases M. blankets, GO prs., at $0.75 5 bales cotton batts, at $7.25.... Deer. 1 J any, 1 " 1 -CHARGES. Paid freight and expenses from depot, cash Storage Com. at 2^ per cent, on sales ,... Net proceeds r Your J of net proceeds, due as per average 894 75 34 48 Milwaukee, Wis., January 1st, 1867, E. and 0. E. J. 0. SP£NC£a & Co. ALLIGATION. Alligation is the method of making calculations regarding the compounding of articles of different kinds or different values. It is a Latin word, which means binding to, or binding together. It is usual to distinguish alligation as being of two kiuds, medial and alternate. ALLIGATION MEDIAL. Alligation medial relates to the average value of aif tides com- pounded, when the actual quantities and rates are given. t: X A M r L E . A millernnxes three kinds of grain: 10 bushels, at 40 cents a bushel ; 15 bushels, at 50 cents a bushel ; and 25 bushels, at 70 wnts a bushel; it is required to fiud tho value of the mature. AULIGATION. 257 0. E. OEK & Co. yarding tlie ues. It is ids, medial 10 bushels, at 40 cents a bushel, will be worth 400 cents., 15 bushels, at 50 cents a bushel, will be worth 750 cents., 25 bufihck-i, at 70 cents n bushel, will be worth 1750 cents., giving a total of 50 bushels (ind 2000 cents, ind hence thq mixture is 2900-7-50—58 cents, the price of the mixture pQP buahel. Hence the , a u L E . F'nd the value of each of the articles, and divide the sum oj their values ly the number denotinr) the sutri of th& articles, and the quotient will he the j/rin ■ f the vi'xture, E XEUCISES. 1. A farmer mi'""3 20 bushels of wheat, worth $2.00 per bushel, wi'"' 40 bushels o* .,ats, worth 50 cents per bushel; what is the |. -ce of one bushel of the mixture? Ans. $1. a. A grocer mixc ; 10 pounds of tea, at 40 cents per pound; 20 pounds, at 45 cents per pound, and 30 pounds, at 50 cents' per poun 1 ; what is a pound of this mixture worth ? Ans. 46§ cents. 3. A luj[Uor merchant mixed together 40 gallons of wine, worth 80 cents a gallon ; 25 gallons of brandy, worth 70 cents a gallon ; and 15 gallons of wine, worth $1.50 a gallon ; what was a gallon of tills mixture worth ? Ans. 90 cents. 4. A {\irmer mixed togctlirr 30 bushels of wheat, worth $1 per busliel; 72 bushels of rye, worth "0 cents per bushel; and 60' bushels of barley, worth 40 cents per bushel ; what was the value of 2^ bushels pf the mixture ? Ans. $1.50, 5. A goldsmith mixes together 4 pounds ol gold, of 18 caratt* fine; 2 pounds, of 20 carats fine; 5 pounds, of 10 carats fine; and 3 ; oundd, of 2t'^ carats fine; how many carats fine is one pound of tlio mixture? Ans. 18|. icles oom- cents a it 70 oents ALLIG-ATION ALTERNATE. Alligation alternate is the method of finding how much of seve- ral ingredients, the quantity or value cf which is known, must be combined to make a compound of a given value. CASE I. Given, the value of several iugi'cdients, to make a compound oi a given value. 17 1 ■("■ 258 ABITHMETIC. EXAMPLE How much sugar that is worth 6 centd, 10 cents, and 13 cents per pound, must be mixed together, so that the mixture may bo worth 12 cents v"- pound ? eoLUTJoir, 12 centos. ' 1 lb., at 6 cents, is a gain of S cents. ) (rain. 1 lb., at 10 cents, is a gain of 2 cents. \ 8 1 lb., at 13 cents, is a loss of 1 cent. 7 lbs. more, at 13 cents, is a luns of. , 1 Gain 8 Loss u It is evident, in forming a mbt+nre of sugar worth 0, 10 and 13 cents per pound so as to be worth 12 conts, that the gains obtained in pu'/'-^g in sugar of less value than the average price must exactly balance the losses sustained in putting in sn^ar of greater value tlir.a the average price. Hence in our example, su'^'.r that is worth G cents pc.' J ■ and when put in the mixture will sell for 12, thereby giving a gain of 6 cents on every pound of this sugar put iu the mixture. So also sugar that is worth 10 cents per pound, when in the mixture will bring 12, so that a gain of 2 cents is obtained on every pound of this sugar used in the compound. Again, sugar thnt is worth 13 cents per pound, on being put into the mixture will Ecll for only 12 cents, consequently a loss of 1 cent is sustained on every pound of this sugar used in forming the mixture. In this manner we find that in taking one pound of each of the different qualities of sugar there is a gain of 8 cents, and a loss of only 1 cent. Now, cur losses must equal our gains, and therefore we have yet to lose 7 centsi and as there is only one quality of sugar in the mixture by w! :eh we can lose, it is plain that we must take as much more sugar at 13 cents as will make up the loss, and that will require 7 pounds- Therefore, to form a mixture of sugar worth G, 10 and 13 cents pci pound, so as to be worth 12 cents per pound, we will require 1 pound at 6 cents, 1 pound at 10 cents, and 1 pound at the 1.^ cents+7 pounds of the same, which must be taken to make the loss equal to the gain. By making a mixture of any number of times these answers, it will be observed, that the compound will be correctly formed. Hence we can readily perceive that an^ number of answers mav be obtained 13 cents ro may be 1 Loss i.J 10 and 13 13 obtained ust exactly • value thr.a is worth 12, thereby : put iu the id, when in sbtained on sugar thnt are ■will sell cd on evi-vy his manner qualities of Now, cur ose 7 cents» by wl 'eh Isugar at 13 7 pounds- 3 cents pd require 1 at the 13 tkc the loss answers, it led. Ilenco Ibe obtained ALIJGATION ALTEENATE. 2£9 to all exercises of this kind, the fallowing From what has been said we deduoe RULE. Find how much is gained or lost hy talcing one of each kind nj the i^roposed ingredients. Ti. :: taJce one or more of the ingredients, or such parts of them as will make the gains and losses equal. EXEUOISES. 1. A grocer wishes to mix together toi worth 80 cents, $1.20, $1.80 and $2.40 per pound, so as to make a mixture worth $1.G0 per j5ound ; how many pounds of each sort must he take ? Ans. 1 lb. at 80 cents; 1 lb. at $1.20; 2 lbs. at $1.80, and 1 lb. at S2.40. 2. How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents per bushel, must be mixed together that the compound may be worth 64 cents per bushel ? Ans. 1 bush, at 43 cts. ; 1 bush, at 60 cts. ; 4 bush, at 67 cts. ; and 1 bush, at 78 cts. 3. It is required to mix wine, worth 60 cents, 80 cent.s, and $1.20 per gallon, with water, that the mixture m- ;■ be worth 75 cts. per gallon ; how much of each sort must be taken :' Ans. 1 gal. of water ; 1 gal. of wine at 60 cts. ; 9 gal. at 80 cts.; and 1 gal at $1.20. 4. In what proportion must grain, valued at 50 cents, 56 cents, 62 cents, and 75 cents per bushel, be mixed together, that the c m- pound may be 02 cents per bushel ? (Give, at least, three answers, and prove the work to be correct. ^. A produce dealer mixed together corn, worth 75 cents per bushel ; oats, worth 40 cents per bushel ; rye, worth 65 cents per bushel, and wheat, worth $1 per bushel, so that the mixture was worth 80 cents per bushel ; what quantity of each did he take ? Give four answers, and prove the work to be correctly done in each case. CASE II. When one or more of the ingredients are limited In quantitj, to find the other ingredients. EXAMFLJB. How much barley, at 40 ceats; oats, at 30 cents, and com, at 60 260 AEITHMETIO. cents i)er bushel, must bo mixed with 20 bushels of rye. ut 85 cents per bushel, so that the mixture may bo woi^^ 60 cents per bushel ? SOLUTION. Ltvdi. '"onls. Gain. Loss. 1 ut 40, gives 20 1 iit ;]0, p;ives 30 1 at GO, gives 00 .00 20 at 85, gives 5.00 .50 5.00 9 ut 40, gives 1.80 9 at 30, gives 2.70 S5.00 85.00 By taking 1 bushel of barley, at 40 cents, 1 bushel of oats at 30 cents, and 1 bushel of corn at GO, in connection with 20 bushels of rye at 85 cents per bushel, we observe that our p,uin!s amount to 50 cents and our losses to $5.00. Now, to make the i),aitiw ctiual the losses, wo have to take 9 bushels more at 40 cents, and 9 busliols more at 30 cents. This gives us for the answer 1 buHhel-|-!).— 10 bushels of barley, 1 bushel-|-9=10 bushels of oats, and 1 b\Jshol of corn. From this wo deduce the RULE. Find how much is gained or lost, hi/ taking one of each of the proposed ingredients, in connection with the ingredient which in limited, and if the gain and loss he not equal, take snch of the 'pro- posed ingredients, or such parts of them, as will make the gain and loss equal. EXERO IS E3. 6. How much gold, of IG and 18 carats fine, must bo mixed with 90 ounces, of 22 carats fine, that tho compound may be 20 carats fine ? Ans. 41 ounces of 16 carats fine, and 8 of 18 carats fine. 7. A grocer mixes tons worth $1.20, $1, and GO cents per pound, with 20 pounds, at 40 c^nts per pound ; how much of each sort must ho take to make the composition worth 80 cents per pound? 8. 13 ow much barley, at 50 cento per bushel, and at GO cents per bushel, must be mixed with ten bushels of pe- worUi 80 cents at 85 cents )cr bushel ? S3. 10 )0 )0 >f oats at 30 buslicls of uount to 50 js ciiuul thu ,d 9 busliols sliol-l-0.^10 1 1 bushel of each of the mi lohich is of the pro- fhe gain and st bo mixed may be 20 carats fine. s per pound, Lch sort niv.st md? at CO cents 3rUi 80 cents ALLIGATION ALTEENATE. 261 ' per bushel, and 6 bushels of rye, worth 85 cents per bushel, to mako a mixture worth 75 cents per bushel ? Ans. 3 bushels, at 50 cents; 2J bushels, at CO cents. 9. TIow many pounds of sugar, at 8, 1-1, and 13 cents per pound, muiit bo mixed witli 3 pounds, worth 0.^ cents per pound ; 4 pounds, worth lOiV cents per pound; and U pounds, worth lLI|coiit3 per pound, so that tlie mixturo may bo worth l-^^ cents por pound ? Ans. 1 lb., at 8 cts. ; 9 lbs., at 14 cts. ; and 5J lbs., at 13 eta C A H E III. To find the quantity of each ingredient, when the sum oi' the mgredicnts and the average price are given. E X A JI i> L E . A grocer has sugar worth 8, 10, 12 and 14 cents per pound, and he wishes to mako a mixture of 240 pounds, worth 11 cents per pound ; how much of each sort must he take ? 8 T; u T 1 N . Gain. T.obs. 1 lb., at 8 cents, gives 3 1 lb., at 10 cents, gives 1 1 lb., at 12 cents, gives 1 1 lb,, at 14 cents, gives , 3 4 lbs. 4 4 24^> lb3.~4— 60 lbs. of each sort. By taking GO lbs. of each sort we have the required quantity, and it will be observed that the gains will exactly balance the losses, ronscquently the work is correct. Hence the RULE. Find the least quantity of each ingredient 6_y Case I., Then divide the given amount by the snm of the ingredients already found, and multiply the quotient by the quantities found for the propoT' tional quantities. 10. What quantity of three different kinds of raisins, worth 15 nents, 18 cents, and 25 cents per pound, must be mixed together to fill a box containing 680 lbs., and to bo worth 20 cents per pound ? Ans. 200 lbs., at 15 '-'-'ts ; 200 lbs., at 18 cents j and 280 lbs., at 25 cents. 262 ABITHMETIO. 1 1 How much sugar, at G cents, 8 cents, 10 cents, and 12 cents per pound, must bo mixed together, so as to form a compound of 200 pounds, worth 9 cents per pound ? Ans. 50 lbs. of each. 12. How much water must bo mixed with wine, worth 80 cents per gallon, so as to fill a vessel of 90 gallons, which may be offered at 60 cents per gallon ? Ans. 56g gals, wine, and 33'^ gals, water. 13. A wine merchant has wines worth $1, $1.25, $1.50, $l.'75,and t2.per gallon, and ho wishes to form a compound to fill u 150 gallon cask that will sell at $1.40 per gallon; how many gallons of each sort must he *iko ? Ans. 54 of $1, and 24 of each of the others. 14. A grocer has sugars worth 8 cents, 10 cents 12 cents, and 20 cents per pound ; with these he wishes to fill a hogshead that would contain 200 pounds ; how much of each kind must he take, so that the mixture may be worth 15 cents per pound ? Ans. 33^ lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 centi. 15. A grocer requires to mix 240 pounds of different 'ands of raisins, worth 8 cents, 12 cents, 18 cents, and 24 cents per lb., so that the mixture shall be worth 10 cents per pound ; how much must bo taken of each kind ? Aus. 192 Iba. of 8 cents, and 16 lbs. of each of the other kuids. MONEY; ITS NATURE AND VALUE. Money is the medium through which the incomes of the different members of the community are distributed to them, and the measure by which they estimate their possessions. The precious metals have, amongst almost all nations, been tlie standard of value from the earliest time. Except in the very rudest state of Dociety, men have felt the necessity of liaving some article, of more or less intrinsic value, that can at any time be exchanged for different commodities. No other substances were so suitable for this purpose as gold and silver. They are easily divisible, portable, and among the least imperishable of all substances. The work of dividing the precious metals, and marking or coining them, is generally undertaken by the Government of the country. Money is a commodity, and its value is determined, like that of other commodities, by demand and supply, and cost of production. When there is a large supply of money it becomes cheap ; in other words, more of it is reauircd to purchase other articles. If all the MONEI I ITS NATUEE AND TALTO. 263 d 12 cents and of 200 bs. of each, h 80 cents T be offered ^ala. water. , $l.'75,and 150 gallon )ns of oacli the others, nts, and 20 that would akc, so that of 20 ccnt.<3. )nt kinds of 3 per lb., so ■ much must other kuids. UE. the different ic measure is, been the very rudest lOme article, exchanged suitable for Ic, portable, Che work of them rr a 13 like that of production, i; in other If all the money in circulation were doubled, prices would be doubled. The usefulness oF money depends a great deal upon the rapidity of ita circulation. A ten-dollar bill that changes hands ten times in » month, purchases, during that time, a hundred dollars' worth of goods. A small amount of money, kept in rapid circulation, does- the name work as a far larger sum used more gradually. Therefore, whatever may be the quantity of money in a country, only that part* of it will effect prices which goes into circulation, and ia actually cxclianged for goods. Money hoarded, or kept in reserve by individuals, does not act upon prices. An increase in the circulating medium, conformable in duration and extent to a temporary activity in business, does not raise prices, it merely prevents the fall that would othen/iso ensue from its temporary Bcarcity. PAPER CURRENCY Paper Cctirency may be of two kinds — convertible and incon- vertible. When it is issued to represent gold, and can at any time be exchanged for gold, it is called convertible. When it is issued by the sovereign power in a State, and is made to pass for money, by merely calling it money, and from the fact that it is received in pay- ment of taxes, and made a legal tender, it is known as an inconver- tible currency. Nothing more is needful to make a person accept anything as money, than the persuasion that it will be taken from Jam on the same terms by others. That alone would ensure its currency, but would not regulate its value. This evidently cannot depend, as in the case of gold and silver, upon the cost of production, for that is very trifling. It depends, then, upon the supply or tho quantity in circulation. While the issue of inconvertible currency is limited to something under the amount of bullion in circulation, it will on the whole maintain a par value. But as soon as gold and silver are driven out of circulation by the flood of inconvertible currency, prices begin to rise, and get higher with every additional issue. Among other commodities the price of gold and silver articles will rise, and the coinage will rise in value as mere bullion. Tho paper currency will then become proportion ably depreciated, as com- pared with the metallio currency of other countries. It would bd I ll n.-: 264 AEITHMETIO. quite impoasiblo for tbcso results to follow tho issue of convortiblc paper for which gold could at any time bo obtained. All variations in the value of tho circulating medium are mis- chievous ; they disturb existing contracts and expectations, and tho liability to such disturbing influences renders every pecuniary engagement of long date entirely precarious. A convertible paper currency in, in many I'cspccts, beneficial. It is a more convenient medium of circulation. It is clearly a gain to the issuers, who, until tho notes aro returned Ibr payment, obtain the use of them as if they NVfcie a real capital, and that, without any loss to tho community. THE CURRENCY OF CANADA. In Canada there are two kinds of currency ; tho one is called the old or Halifax currency, reckoned in pounds, shillings, pence and fractions of a penny ; the other is reckoned by dollars and cents as already explained under the head of Decimal Coinage. Tho equivalent in gold of the pound currency is 101.321 grains Troy weight of the standard of fineness prescribed by law for the gold coins of tho united kingdom of CJreat Britain and Ireland. Tho only gold coins now in circulation in Britain are the sovereign, valuo one pound, or twenty shillings sterling ; and the half sovereign, ten shillings. The dollar is one-fourth of the pound currency, and the pound sterling is equal to 84.8G§. In the year 178G, the congress of the United States adopted the decimal currency, the dollar being the unit, and the system was introduced into Canada in 1858. By the term legal tender is meant the proffer of payment of an account in the cuircucy of any country as established by law. Copper is a legal tender in Canada to the amount of one shilling or twenty ceutrt, and silver to the amount of ten dollars. The British sovereign of lawful weight passes current, and is a legal tender to any amount paid in that coin. There is a silver currency proper to Canada, though United States' coins are most in circulation. The gold eagle of the United States, coined before July 1, 1834, is a r ^?,i.\ tender for $10.66f of the coin current in this province. The same coin issued after that is a legal tender for $10 EXOHMGE. 265 EXCHANGE. It often becomes necessary to send money from one town ox country to another for various purposes, generally in payment for (];oocls. The usual niodo of uuikhiL; and receiving payments between distant places is by bills of exchange. A iiiorchaiit ia Liverpool, whom wc shall call A. B., has rcc-ivcd a consignment of flour from C. D., of Chicago; and another man, E. F., in Liverpool, has ahipped a quantity of cloth, in value equal to the flour, to G. IL in Chicago. There arises, in this transaction^ an indebtedness to Chi- oago for the flour, as well as an indebtedness from Chicago for tho cloth. It is evidently unnecessary that A. B., in Liverpool, should send money to C. D. in Chicago, and that G. IL, in Chicago, should send an equal sum to E. F. in Liverpool. Tho one debt may be applied in payment of the other and by this plan the cspenso and risk attending tho double transmission of tho money iiiay bo saved. C. D. draws on A. B. for the amount which ho owes to him: and G. II. having an equal amount to j;a^ in Liverpool, buys this bill from C. D., and sends it to E. F., who, at tho maturity of tho bill, presents it to A. B. for payment. In this way the debt duo from Chicago to Liverpool, and tho debt due from Liverpool to Chicago arc both paid without any coin passing from one place to the other. An arrangement of this kind can always be made when the debts due between the different places arc equal in amount. But if there • is a greater sum due from one place than from the other, the debts cannot be simply written off against one another. Indeed, when a person desires to make a remittance to a foreign country, he does not make a personal search for some one who has money to receive from that country, and ask him for a bill of exchange. There arc ex- change brokers and bankers whose business this is. They buy bills from those who have money to receive, and sell bills to those who have money to pay. A person going to a broker to buy a bill may very likely receive one that has been bought the same day from a merchant. If the broker has not on hand any exchange that he has bought, he will often give a bill on liis own foreign correspondent j and to place his correspondent in funds to meet it, he will remit to him all the cxohango which he has bought and not re sold. 2G6 ARITHMETIC When brokers find that they are nskcd lor moro bills than are oflFcrcd to thcin, they do not absolutely refuse to give them. To enable their correspondents to meet the bills r.t maturity, aa they have no exchange to send, they have to remit funds in gold and silver. There are the expenses of freight and insurance upon tho specie, besides the occupation of a certain amount uf capital involved in this ; and an increased price, or premium, is charged upon the exchange to cover all. Tho reverse of this happens when brokers find that moro bills are oflfcred to them than they can sell or find use for. Exchange on tho foreign country then fulls to a discount, and can bo purchased at a lower rate by those who require to make payments. There are other influences that disturb tho exchange between different countries. Expectations of receiving large payments from a foreign country will have one effect, and the foai of Uaviug to make larger payments will have the oonosite effeob. AMERICAN EXCHANGB 11 il Exchange between Canada and tho United States, especially tho northern, is a matter of every day occurrence on account of tho proximity of the two countries, and the incessant intercourse between them, both of a social and commercial character. The exigencies of the Northern States arising from the late war, compelled them to issue, to an enormous extent, an inconvertible paper currency, known by the name of " Greenbacks." As the value of these depended mainly on the stability of the government and the issue of tho war, public con- fidence wavered, and in consequence, the value of this issue sunk materially. This caused a gradual rise in the value of gold until it reached the enormous premium of nearly two hundred per cent., or u quotation of nearly three hundred per cent., that is, it took nearly/ three hundred dollars in Greenbacks to purchase one hundred dollars in gold. It is to be hoped and expected, however, that as peace is now restored, matters will soon find their former level. It has been deemed essential that this should be distinctly ex- plained, as it has brought about a necessity for a oonAtaut culoulation ^kAi J^^IERICAN EXCHANGE. 267 than are cm. To , aa they ;old aud upon tho involved upon tho noro bills hango on mrohascd I between }nt3 from having to cially tho int of tho c between cncics of 1 them tf) known by cd mainly ublic con- isuo sunk d until it cent., or ok nearly ed dollars 8 pcacG is inctly ex- Iculatioii of tho relative values of gold and greenbacks, and has generated an extensive business in that species of exchange. When the term " American currency" is used in tho foUowlDg exercises it is understood to bo Qrceubacks. CASE I. To find tho viUuo of $1, American currency, when gold is at a premium. EXAMPLE. Wlicn gold is quoted at 140, or 40 per cent, promiain, what is tho value of $1, American currency ? SOLUT ION. Since gold is at a premium of 40 per cent., it requires 140 cents of American funds to equal in value 61, or 100 cents in gold. Henco tho value of 81, American money, will bo represented by tho number of times 140 is contained in 100, which is .71 i} or 71^ cents. Hence to find tho value of $1 of any depreciated currency reckoned in doUojcs and cents, we deduce tho following RULE. I)mde 100 cents by 100 plus tho rate of premium on gold, and the quotient loill be the value of $1. SuhtiHct this from $1, and the remainder imll be the rate oj discount on the given currency. CASE II. To find tho value of any given sum of American currency when gold is at a premium. EXAMPLES . What is tho value of $280, American money, when gold is quoted ut 140, or 40 per cent, premium ? SOLUTION . Wo find by Case I. the value of $1 to bo 71 ^ cents. Now, it is evident that if 71^ cents bo the value of $1, the value of $280 will be 280 times 71 1 cents, which is $200, or $280—1.40=28000-1- 140=3200. Henco we have tho following 268 EU L E . Multijili/ the value of $1 hi/ the numlcr denoting *he given amount of American monci/, and the product will l/c the gold value; or, Dioidc the giocn sum of American monrg hj 100 (the numhcr oj cents in $1,) plus the premium, and the quotient will he tJiAi value in gold. CASE III. To find tho prcniiiun on gold when American money is quoted at .1 certain ruto per cent, discount. E X A JI r L E . Wlien the discount on American money is 40 per cent., what is tho premium on gold ? BOLTITION. If American money is at :i discount of 40 per cent., tho discount on $1 would be 40 cents, and consequently tho vahio of SI would he equal to SI. 00 — 40 cents, equal to 00 cunts. Xow, if (iO cents in gold bo worth $1 in American currency, $1 or 100 cents in gold would 1)0 worth 100 times -J^j of $1, which is 81.G(."|t, from which if we subtract $1, the rcm-indcr will be tho preiniuui. Therefore, if American currency be at a discount of 40 per cent., the premium on gold would be GG!| per cent. Ilencc we deduce tho following K u L E . Divide 100 cents hg the numhcr dcnotiw^ the gold value of $1, American currcnrg, and the quotient will he the value, in American c. :icg, I'f^l in gold, from which auhtract $1. and the renvxindcr wilt he the prcnium. CASE I V . To find the value in American currency of any given amount of gold. • E X A Jl P L E . What is the value of 8200 of gold, in Ai-iovican currency, gold being quoted at 150 ? SOLUTION. When gold is quoted at 150, it requires 150 cjuts, in Americ"n currency, to equal iu value $1 in gold. Now, if %1 in gold bo w:^'*th 81.50 in American currency, $200 will be worth 200 times $lo50, which is $300. Hence tho VI i If I" ill" AMERICAN EXCHANGE 269 ,, Sf)k\ cnc"n w?rth $1,50, K U L E . Multiply the value Oj $1 hy the number denoting the cmount oj gold to he changrxlj and the product loill be the vaf'ue in American currency ; or To the given sum add the premium on itself at the gium rate, and the result will be the value in American currency. EXi;ilCISES. 1 If Americau currency is at a discount of 50 per cent., T.'liat ia tliO value of $-150 ? An«. $225. 2. The quotation of gold is 140, what is the discount on Ameri- can currency ? Ans. 2S:J percent. 3. A person exchanged $750, American money, at a discount ol' 35 per cent, for gold ; how much did lie receive ? ^ns. $427.50. 4. Purchased a draft on Montreal, Canada East, ibr !;;15G0 :it ti premium of 04^ per cent. ; what did it cost me ? AnK'. 5. If American currency is quoted at 33.\- per cent, discount; what is the ^ "cniiuni on gold? Ans. 50 per ceut. G. Purchased a suit of clothes in Toronto, Canada West, for (^35, but on paying for tho same in American funds, llio t.iilor charged mo 32 per cent, discount ; how much had I to pay him ? Ans. s^51.47. 7. What would bo tho diflcrencc between the quotations of gold, if greenbacks were selling a^ 40 and GO per cent, discount ? Ans. 83J> i)cr cent. 8. P. Y. Smith borrowed from C. II. King, S27 in gold, and wished to repay him in American currency, at a discount of 35 percent.; how much did it require? .. > Ans. 5543.55. 9. J. Vj. Pckh im bought of Sidney Leonard a horse and cutter Cor S315.50, American currency, t'ut only having $200 of this sum, 'jg paid the balance in gold, at a premium of G5 per cent. ; liow much did it require ? Ans. $7(1. 10. A cattle drover purcLused of a farmer a yoke of oxen valued -it $135 in gold, but paid him (illl: in American currency, at a discount of 27.V per cent. ; how much gold did it require to pay tlic balance V Ans. $53.80. 11. AY. II. Ilounslield 6^ Co., ol' Toronto, Canada West, purchased in New York City, merchandlso nniounting in value to ^1708.40, on 3 months" credit, proiniuni on gold being 70i| per cent. At the 270 ARITHMETIC. expiration of the three months they purchased a draft on Adams, Kimball and Moore, of New York, for the amount due, at a discount of 57£ per cent. ; -what vras the gain by exchange? Ans. $647.75. 12. A makes an exchange of a horse for a carriage with B ; tho horse being valued at $127.50, in gold, and the carriage at $210, xlmcrican currency. Gold being at a premium of G5 per cent. J \yhat was the difference, and by whom payable ? Ans. B pays A 23 cents in gold, or 37 cents in greenbacks. 13. A merchant takes $G3 in American silver to a broker, and wishes to obtain for the same greenbacks which are gelling at a dis- count of 30 per cent. Tho broker takes the silver at 3J per cent. discount ; what amount of American currency does the merchant recpivc? Ans. $86.85. 14. I bought tho following goods, as per invoice, from John McDonald & Co., of Montreal, Canada East, on a credit of 3 months : 1120^ yards Canadian Tweed at 95 cents per yard. 2190 " long-wool red flannel at 60 '■ " " 3400 " « white flannel at 55 " '' " Paid custom house duties, 30 per cent. ; also paid for freight, $37.40. Gold at time of purchase was at a premium of 63f per cent. ; what shall I mark each piece at per yard to make a net gain of 20 per cent, on full cost ? Ans. C. tweed, $2.44; red flannel, $1.54; white flannel, $1,41. 15. A merchant left Toronto, Canada West, for New York City to purchase his stock of spring goods, taking with him to defray expenses $95 in gold. After purchasing his ticket to the Suspension Bridge for $2.40, he expended the balance in greenbacks, which were at a discount of 41A- per cent. When in New York he drew from this amount $23.85 to "square" an old account then past due. On arriving homo ho found that ho still had in greenbacks $16,40, which he disposed of at a discount of 432- P*-^^" cent., receiving in payment American silver at a discount of 3\ per cent., which he passed off at 2^ per cent, discount for gold. What were his expenses in gold; the actual amount in greenbacks paid for expenses, and the amount of silver received ? Aus. Total expenses in gold, $71.76 ; cupensc^ in greenbacks, $118.04 ; silver received, $9.53t EXCHANGE WITH GEEAT BRITAIN. 271 EXCHANaE WITH GREAT BRITAIN. In Britain money is reckoned by pounds, shillings and pence, and fractions of a penny, and is called Sterling money, the gold 5overei[:n or the pound sterling, consisting of 22 parts gold and 2 alloy, being tho standard, and the sliilling, ono-twcntieth part of the pound, a silver coin of 37 parts silver and 3 copper, and the penny, one-twelfth part of the shilling, a copper coin, the ingredients and size C'f which have frequently been altered, Tho comparative value of the gold sovereign in the United States previous to the year 1834 was $-i.44i, but by Act of Congress passed in that year it was made a legal tender at the rate of 94 /'g cents per pennyweight, because the old standard was less than the intrii.sic value and also because the commercial value, thougli fluc- tuating, was always considerably higher. Hence, the full weight of the sovereign being 5 dwts. 3.274 grs., it was made equivalent to 4 dollars and 8Cf cents. Tho increase in the starvdatd value was, tlicrelbro, equal to 9|- per cent, of its nominal value. 'J'ho real par of exchange between two countries is ihat By which an ounce of gold in one country can be replaced by an ounca of gold of equal fineness in the other country. If th" course of exchange at New York on London were 108J per cent. ; and the par of exchange between England and America 100^ per cent., it follows that the exchange is 100 percent, against England ; but the quoted exchange at New York being for biii^ ^t 00 days sight, the interest must bo deducted from tho above differ- L'uce. The general form for the quotation of exchange with England i.s: 108, 108A, 100, 100^ &c., which indicates that it is at 8, 8J, D, or 9^ per cent, premium on its nominal value. EXAMPLE. What amount of decimal money will bo required to purchase u draft on London for £G48 17s. Gd. ? — exchange 108. The old par value or tominal value ia $4.44;^- ^u*>=i of S40 mm 272 UITHMETia by reaucing to an improper fraction. INow, the quotation is 108, or 8 per cent, above the nominal value, we find the premium on $-10 at 8 per cent., which is 83.20, Vvhich added to $40 will give $43.20, and 8-i3.20--.-9i:i^$-1.80 to bo remitted for cvciy pound sterling, and tliercforc XGiS I'^s. Gd. multiplied by 4.8-9 or 4.8 will be the value in our money. 1 's. Gd,=.875 of a pound, and the oneratiou is as follows : £648,875 4.8 6191000 2595500 ^3114.0000 RULE. fo $40 add tJic premium on itsclj- at the quoted rate, multijni/ the sum hj the number representing the amount of sterling money, and divide the result hi/ 9, the quotient will be the equivalent of the sterling money in dollars and cents. KoTK.— If there bo sbUlhigs, pence, &c., in the sterling money, they are to 1)0 reduced to the decimal of £1. To find the value of decimal money in sterling money, at an/ given rate above par. Let it be required to find the value of $4G5 in sterling money, at 8 per cent above its nominal value. Hero we havo exactly tho converse of tho last problem, and therefore, liaving found the value of £1 sterling, we divide the given .sum instead of multiplying ; thus the premium on $40, at 8 pci cent., is §3.20, which added to $40 makes 843.20, and 43.20-^9^4.80, and $465-H^.80=£96.17.6. IIULE. Divide the given sum by the number denoting the value of one pound sterling at the given rate above par, and if thcrebca decimal remaining reduce it 'o shillings and pence. EXERCISES. 1. When stCTling oxchunge is quoted at 108. what is tho value of£l?* Ads. E4.80. 1 is 108, n on S-iO 2 843.20, ■ling, and the value itiou is as , multiply ng money, lent of the jy, they are icy, at au/ money, at [sactly the the value iltiplyiug ; led to $40 16.17.6. \ue of one- la decimal EXCHANGE WITH GEEAT BRITAIN. 273 the value Ids. S4.80. 2. If £1 sterling be worth $4.84^-, what is the premium of ex- change between London and America. Aos. 9 per cent. 3. At 10 per cent, above its nominal value, what is the worth of £50 sterling, in decimal currency ? Ans. $244.44. 4. When sterling exchange is quoted at 9^ per cent, premium, what is the value of SIOOO ? Ans. £205 18s. lip. 5. At 12 per cent, above its nominal value, what will a bill for £1800 cost in dollars and cents ? Ans. $8960. G. A merchant sold a bill of exchange on London for £7000, at an advance of 11 per cent ; what did he receive for it more than its real value ? Ans. $406.66 j. 7. Bought a bill on London for £1266 15s. at 9J per cent, pre- mium ; what shall I have to pay for it ? Ans. $6164.85. 8. A merchant sells a bill on London for £4000, at 8 per cent, above its nominal value, instead of importing specie at an expense of 2 per cent. ; what does he save ? , Ans. $122.66;!. 9. A merchant in Kingston paid $7300 for a draft of £1500 on Liverpool ; at what per cent, of premium was it purchased ? Ans. I)J. 10. Exchange on London can be purchased in Detroit at 108|- ; ill New York at 108J. At which place would it be the most advan- tageous to purchase a bill for £358 14s. 9d., supposing the N.Y. broker charges J per cent, commission for investing and gold drafts Ml New York are at a premium of j] per cent. Ans., Detroit by $0.82. 11. A broker sold a bill of exchange for £2000, on commission, jt 10 per cent, above its nominal value receiving a commission ot ,'y per cent, on the real value, and 5 per cent, on wliat ho obtained for the bill above its real value ; what was his couiiiiis.sion ? Ans, $11,955. 12. I owe A. N. McDonald & Co., of Liverpool, $7218, net pro- ceeds of sales of merchandise ciFcctcd for them, which I am to remit thoni in a bill of exchanc:o on London lor such amount as will closo ilio transaction, less ^ per cent, on the face of the bill for my com- mission for investing. Bills on London are at 8 per cent, premium. Hccjulred the amount of the bill, in sterling mouoy, to be remitted. 18 Ans. .£1500. 274 ABTTHMETIO. TABLE OP POREION MONEYS. ClTIKa AKD COtTS-iUIBS. London, Liverpool, &c Denomlvatioss oy Money. Paris, Havre, ik.c Amsterdam, Hague, &c, Bremen Hamburg, Lubec, &c. Corlin, Dantzic Dclgium , St. Petersburg. Stockhobn , Copenhagen Vienna, Trieste, &c.... Naples Venice, Milan, &c Florence, Leghorn, &c. Genoa, Turin, &c Sicily... Portugal Spain Constantinople Biitish India.. Can ten. Mexico Monte Video. Brazil. Cuba.. Turkey United States. New Brunswick. Nova Scotia Newfoundland..., 12 pence=;l shilling ; 20 shillings' rrrl poUnd =z 100 ccntimcs:^l franc ==■ 100 cents^l guilder or florin... =^ 5 swarcs^=:l grote ; 72 grote8=l rix dollar r= 12 pfcnnings=l schilling ; 1G3.= 1 mark banco ^= 12 pfcnnings=l groschen ; 30 gro. 1 thaler = 100 centime3=:l franc = 100 kopccks=:l ruble = 12 rundstycks=::::l() skillings; 483. =1 riz dollar specie = 16 skillings=:::l mark ; 6 m.=l rix dollar =: 60 kreutzers=l florin = 10 grani=l carlino ; 10 car.=l ducat =^ 100 centesimi=l lira = 100 centesimi=l lira =^ 100 ccntesimi:==l lira = 20 grani=l taro ; 30 tari=l oz.= 1000 reas=:l millrea := ( 3-i maravcdis=:l real vellon= I 68 morrivedis=l real plate. . ■^=^ 100 aspers=l ^lasfer. = 12 pice=::il anna; 16 annas=l rupee. 100 candarines=l macoj 10 m.=: 1 tael. := 8 rials=l dollar = 100 ccntesimas=l rial ; 8 rials=l dollar = 1000 reas=l milrea = 8 reals plate or 20 reals velion=l dollar =^ 100 aspers==l piaster = 10 mills=il cent ; 10 cent3=l dime ; 10 dimes=r:l dollar.. ..= 4 farthings=l penny ; 12 pence =1 shilling; 20 shillings^:' pound.* = Vawji. $4.8G§ .18^ .40 .78-2 .09 .18^^ .75 1.06 1.05 .48J .80 .16 .16 .18| 2.40 1.12 .05 .10 .05 .4^ 1.48 1.00 •83A .82| 1.00 .05 variable 4.00 • The Government of Now Drtm-swick now issues postage stamps in the decimal cu].Teacy, but bo for &a we bavo bjou ablo to aacertain. the currency of —* VALim. 64.8G§ — _ .181 - .40 :1 = .78-2 I .35 0. .09 =1 a =1 ce =1 .18^ .75 1.06 1.05 .48J .80 .16 .16 .181 2.40 1.12 .05 .10 .05 .44J 1.48 1.00 •83A .82-1 1.00 .05 variable 4.00 tamps in the I carwucy of ABBITBAnON OF SXCHANQE. 275 ARBITRATION OF EXCHANG-E Arhitratlon of Exchange is tho method of finding the rate of exchange between two countries through the intervention of one or more other countries. Tho object of this is to ascertain what is tho most advantageous channel through which to remit money to a foreign countrJ^ Three things have hero to bo considered. First, what is tho most secure channel; secondly, what is the least expensive, and thirdJij, the comparative value of the currencies of the diffcront countries. Regarding tho two first considerations no general rule can bo given, as there must necessarily be a continual fluctuation arising from political and other causes. We are therefore compelled to confine our calculation to the third, vi?:., the comparativo value of tlie coin current of different countries. For this purpose we shall investigate a rule, and append tables. Let us suppose an English merchant in London wishes to remit mo\iey to Paris, and finds that owing to certain international rela- tions, he can best do it through Hamburg and Amsterdam, and that, the exchange of London on Hamburg is 13j marcs per pound ster- ling ; that of Hamburg on Amsterdam, 40 marcs for 30^ florins, and that of Amsterdam on Paris, 56f florins for 120 francs, and thus tho question is to find tho rat« of exchange between London and Paris. solution: We write down the ccjuivalents in ranks, tho equivalent of tho first term being placed to the right of it, and the other pairs below tbcm in a similar order. Hence the first term of any pair will be of the same kind as tho second term of the preceding pair. As the answer is to be tho equivalent of the first term, the first term in the last rank corresponds to the third term of an analogy, and is there- fore a multiplier, it must bo placed below the second rank. The theso threo Pi-ovincos irf, ns usual, in pounds, shillings and pence. It is to bo hoped that, when the Confederation of tho British Provinces takes place, the decira-.il currency will be speoculy adopted iu the Lower Provinces, and that the efforts now boiog (Qodo ia Britain to udaoi the eomo currency will prove Bucceasful. 276 MlITHMETia terms being thus arranged, wo divido the product of the second rank by that of the first, and the quotient will be the equivalent, as exhi- bited below : £1 sterling^:: 13J marca. 40 raarcs = 36|- florins. 56| florins =120 francs £1 stg. As it 13 most convenient to express the fractions decimally, we have 1 n..'>xnr,.2r.xi2 0Xi =25.87 francs. 1X40X5 C. 75 The foregoing explanations may be condensed into the form of a RULE. Write down the first term, and its equivalent to the right of it, ajia the other pairs in the same ordc) , the odd term being j^l'iiced under the second rank, and then divide the product of the second rank hi/ the product of the first, the quotient will he the required equivalent. NoTK.-l he true principle on which this operation if* founded, is that each pail- consists of the antecedent and consequent -which arc to each other in (he ralio of equality in point oii" iNxnixsic value, though not in regard to TiiK NL'MBEits VY wuicu TiiKY Miv: KXPPKssKi), and therelorc the required term tnd its equivalent must have the same relation to each other, that is, they will bo an antecedent and a consequent in the ratio of equality as regurda lieii' value, but not as regards the numbers by which they are expressed. EXERCISES. 1. If the exchange of London on Paris is 28 irancs per pound sterling, and that of America, on Paris 18 cents per franc j what is the rate of exchange of America on London, through Paris? Ans. $5.04 per £ sterling. 2. If exchange between New York and London is at 8 per cent, premium, and between London and Paris 25;J- francs per pound sterling ; what sum in New York is equal to 7000 francs in Paris ? 3. When exchange between Portland and Hamburg is at 34 cents per mark banco, and between Hamburg and St, Petersburg is 2 marks, 8 schillings per ruble ; how much must be paid in St. Peters- burK for a draft on Portland for $660 ? Aus. 764 rubles, 70|;^ kopecks. EXCHANGE, 277 4. Ifu merchant buys ;■ ))iU ia London, drawn on Paris, at the rate of 25.87 franc i \hr pound .stcrlinLT, and if this bill be sold in Anisterdiiiii ;ir 120 iv.xucH I'm- uil^^ llorins, and th(f proceeds bo in- vested ill a bill on llaialiiuj:-, at the rate of oGJ florins for 40 raarcs j ■what is tlu; rato ( i' ( Xuhange between London and Hamburg, or wliat is JLl s;Lriiiiji woith.iii Ilaniburg? Ans. 13.4--9-f-marcs. f). A iiiorcliant oi' St. Louiti wishes to pay a debt of $5000 in New York; the direct exchange is li^ per cent, in favour of New York, but on New Orleans it is ^ per cent, discount, and between New Orleans and New York at a J- per cent, premium ; how much would be saved by the circular exchange compared with the direct ? Ans. §87.56- r». A merchant in Detroit wishes to remit to J. B. Gladstone & Co., of London, X3G0O sterling. Exchange on London, in Detroit, is at a premium of 10 per cent. Exchange on London can bo obtained at New York for 9 per cent, premium. If Detroit bills on New York are at a discount of |- per cent., and the merchant remits a draft to New York, and pays his agent i per cent, for investing it in bills on London ; vdiat will ho gain over the direct exchange ? Ans. 8123.80. 7. A merchant in London remits to Amsterdam £1000, at the rate of 1V> pence per guilder, directing his correspondent at Amster- dam to remit the same to Paris at 2 francs, 10 centimes per guilder, less J per cent, for his commission ; but the exchange between Anis- terdum and Paris happened to be, at the time the order was received, at 2 francs, 20 centimes per guilder. The merchant at London, not apprised of this, drew upon Paris at 25 francs per pound ster- ling. Did he gain or lose, and how much per cent. ? Ans. 1G^§ per cent. gain. MIXED EXERCISES IN EXCHANGE. 1. When gold is quoted at 150 per cent, premium ; what is the reason greenbacks arc not at a discount of 50 per cent. ? 2. Bar gold in London is 77s. 9d. per ounce standard; required, the arbitrated rate of exchange produced by its import to this coun- try for coinage, at the rate of 232^ grains of fine gold for the caglo of 10 dollars. 3. What sum in decimal money mu&t I pay for a bill on London of £76 14s. Id., exchange being 9^ per cent, premium, and the broker's commission for negotiating the bill being i per cent. ? iV*' 278 ASn'HMETIC. 4. A merchant shipped 2560 barrels of flour to his agent in Liverpool, who sold it at £1 8s. 6d. per barrel, and charged 2 per cent, commission ; what was the net amount of tho flour in decimal money, allowing exchange to bo at a premium of 8 per cent. ? Ans. $17100.19. 5. What is tho cost of a 30 days' bill on Montreal, at ^ pjr cent, premium, the face of tho bill being $1500 ? Ans. $1507.50. 6. What must be tho face of a 60 days' draft on New Orleans to yield $1641.75, when sold at a discount of ^ per cent. ? Ans. $1050. 7. What is the cost of a 30 days' bill on Chicago, at § per cent, premium, and interest off at G per cent. ; the face of tho bill being 89256.40?* . Ans. $9240.20. 8. A merchant paid $14400.12 for a bill on Havre for $79000 francs ; how much was exchange below par ? Ans. 2 per cent. 9. I have in possession the net proceeds of a sale of cotton amounting to $3765, which my correspondent desires nic to remit to him in New Orleans; exchange on New Orleans is at a discount of 2J per cent., and I invest the whole in a draft at that rate, which I. remit to him; what is the face of the draft ? Ans. $3801.54. 10. The proceeds of a sale of goods, consigned to mo from Bremen, is $2764.07, on which I am to charge a commission of 10 per cent., and remit the balance to my consignor in such a ^vuy as shall bo most advantageous to him. Exchange on Paris can bo had at 92 cents per 5 francs, and in Paris exchange on Bremen is 17 francs to 4 thalers. Exchange en Liverpool can be had a per cent, premium, and in Liverpool exchange on Bremen is G thalers to the pound sterling. Direct exchange is 80J cents per thaler. Which course will be the best, allowing J per cent, brokerage to correspon- dents both in Liverpool and Paris ? Ans. By way of Paris. 11. A, of Buffalo, sent articles to the World's Fair in London, which were afterwards sold by B, of London, on A's account, not proceeds £1200 ISs. sterling. B was instructed to invest thi.s amount in bills on New York, and remit to A, which was accordingly done. B charged J per cent, brokerage on the face of tho bills for investing, and purchased the bills at 7 per cent, discount. Required *When there is interest to bo computed, it must bo reckoned on tho face of the bill or draft. When other than tho value or cost of the bill is to be found, proceed as in percentage. agent in ^ed 2 ijer n decimal ,t.? L71G0.19. ■ pjr cent. S1507.50. V Orlcana » 13. $1G50. \ per cent, bill being $9240.20. or $79000 2 per cent, of cotton :o remit to iiscoutit of ;c, which I $3801.54. uio from bioti of 10 a wiiy as ian bo hud imen is 17 a 9 per thalcrs to Which correspon- y of Paris. London, icount, not nvest this ccordingly 10 bills for Required lU thfc face )iU ia to be EXCHANQE. 27V it the amount of the bill A must receive in dollars and cents to oloso the transaction. Ans. $6037.53 nearly. 12. A merchant in Boston having to remit £434 ISs. to Liver- pool, wishes to know which is tlio most profitable, to buy a set of exchange on Liverpool at lOJ^ per cent, premium, or send it by way of France ; exchange oh the latter place being lOf cents per franc, and exchange on Liverpool can be bought in France at the rato of 24 J francs per pound sterling, and he has to pay his correspondent in "Franco :^ of 1 per cent, for purchasing the bill on Liverpool. Ans. By way of France, 815.09. 13. John DcDonald & Co., of Toronto, Canada "West, wish to remit to a creditor in London £1241 15s. 9d. Exchange on London can be bought in Toronto at 109f , but Exchange on Lon- don can be purchased in New York for gold at 108L In New York it takes $1.85 greenbacks to equal $1 in gold. The broker in New York charges f per cent, on the greenback value for investing. If Exchange on New York is at 47 per cent, discount, at which place would it be the most advantageous to purchase, and how much gain, and if the remittance be made by way of New York, what would be the face of the draft ? Ans. N::w York by $141.72; face of draft, $111G1.21. 14. Find the arbitrated rate of exchange between London and Amsterdam when the exchange of London on Madrid is 37 pence for one dollar of plate, and that of Amsterdam on Madrid is 100 florins, 75 cents, for 40 ducats of plate. 15. Hughes Bros. & Co., purchase of E. Chaffey & Co., a stcr liug bill at GO days on Gladstone & Hart, of London, for £3950 lOs. They remit this bill to James Alder, in London, where it is accepted by Gladstone & Hart, and falls due on the 20th November, at which time it is protested causing an expense of £2 19s. Gladstone &. Hurt having failed, E. Chaffey & Co.'s agent in London pays James Alac on the 20th November, £2000 on account. How much must,,. Chaffey & Co., pay to Hughes, Brothers & Co., on the 24th Decc.u- bcr, to cover the amount still due in London, allowing i.itorcsiL ..i the rate of 10 per cent, from November 20th, to the maturity < / . 60 days' bill at date of 24th December, and J of 1 per cent, comi;....- sion lor their trouble in negociadng a new bill ? Ans. S9815.9i. IMAGE EVALUATION TEST TARGET (MT-3) V // .// ^J^ :/ 'Q>. 1.0 I.I 1.25 f ilia - IB ■' IM 4b IM 2.2 1.8 U IIIIII.6 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 ^ ,\ ,v \\ '% .V ^\/#^\ fc '^'■y. %^ 'i?," 280 ABirmffETio. INVOLUTION. Tnvdhtion is the process of finding a given power of a given iramber. We have noted already, under the head of multiplication that the product of any number of equal factors is called the second, third, fourth, &c., power of the number, according as the factor is taken two, three, four, &c., times. Thus: 9=3X3 is the second power of 3 J 27=3X3X3 is the third power of three; 81=3x3X3X3 iS the fourth power of 3. These are often written thus : 3^, 3-, 3*, &c. The small figures, 2, 3, 4, indicate the number of factors, and therefore each is called the index or exponent of the power. Hence to find any required power of a given quantity, wo have the RULE. Multiply tJie quantity continually hy itself until it has been used as a/actor as often cw there are units in the index. Since the first multiplication exhausts two factors, the number of operations will be one less than the number of factors. Involution, then, is nothing more than multiplication, and for any power above the second, it is a case of continual multiplication. For the sake of uniformity the original quantity k called the first powei', and also the root in relation to higher powers. Again, if wc multiply 3X3 by 3X3X3, we have five factors, or 3X3X3X3x3, but this being an inconvenient form, it is written briefly 3^, the 5 indicating the number of times that 3 is to be repeated as a factor. Hence, since 3X3 is written 3-, and 3X3X3 is written 3^, it fol- lows that 3-X3''=3°, and therefore we may multiply quantities so expressed by adding their indices, and so also we may divide such quantities by subtracting the index of the divisor from that of the dividend. For example 33-^3-:=3 or 3^ If we divide 3' by 3* by subtracting the index of the divisor from that of the dividend, wc obtain 3°, but 3 or 3' divided by 3 or 3' is equal to 1, and there- fore any quantity with an index zero is equal to unity. When high powers are to be found, the operation may bo short- ened in the following manner : — Let it be required to find the six- teenth power of 2. We first find the second power of 2, which is 4, INVOLUTION. 281 f a given 1 that the ad, third, : is taken 1 power of X3X3iS ^ 3=, 3% otors, and . Hence iO been used lumber of 1, and for iplication. the Jirst ain, if wc 3X3X3, *, the 5 8 a factor. 33, it fol- mtitics so ividc such kat of the 3» by3» idcnd, we md there- bo short- d the sisr hioh is 4, then 4X4=1*6, which is the fourth power, and 10x16=256, the eighth power, and 256x256:= 05536, the sixteenth power. If wo wished to find the nineteenth power, wo should only have to multiply the last result by 8, which is the third power gf 2, for 2» x23=2» ». EXERCISES. Ans. 48580a. Ans. 022835864. Ans. 19.070689. Ans. 31640625. Ans. 1. 9738+. 1. Find the second power of 697. 2. What is the third power of 854 ? 3. What is the second power of 4.367 ? 4. Find the fourth power of 75. 5. What is the sixth power of 1.12? 6. What is the second power .7, correct to six places ? Ans. .060893+. 7. What is the fifth power of 4 ? 8. Find the third power of .3 to three pliices ? 0. What is the third power of ^ ? 10. What is the fifteenth power of 1.04 ?* 11. Raise 1.05 to the thirty-first power. 12. What is the eighth power of | ? 13. What is the second power of 4| ? 14. Expand the expression 6^. 15. What is the second power of 5^ ? 16. What part of 83 is 20? 17. What is the diflcrencn between 5° and 4° ? 18. Expand 3^X2". 19. Express, with a single index, 47 = X47'''X47<' ? Ans. 1024. Ans. .030963. Ans. |;Jt5. Ans. 1.800943. Ans. 4.538039. Ans. 23^^ Ans. .7776. Ans. i^i^soj. Ans. ^. Ans. 11529. Ans. 3888. Ans. 47'-'. 20. How many acres are in a square lot, each side of which is 135 rods ? Ans. 113 acres, 3 roods, 25 rods. 21. What is the sixth power of .1 ? 22. What is the fourth power of .03 ? 23. What is the fifth power of 1.05 ? 24. What is the third power of .001 ? 25. What is the second power of .0044 ? The second power of any number ending with the digit 5 may he readily found by taking all the figures except the 5, and nmlti- * This exercise will bo most readily worked by finding llie .sixteenth power, and dividing by 1.04. So in the next exerciso, find tlio thirty-second power, and divide by 1.05. A still more easy mode of working sucJi quos- tiona win bo i'ound under the head of logarithms. Ans. .000001. Ans. .00000031. Ans. .1.2702815625. Ans. .000000001. Ans. .00001836. 282 ABITHMETIC. plying that by itsolf, increased by a unit, and annexing 25 to the result. Thus, to find the second power of 15, cut off the 5, and 1 remains, and this increased by 1 gives 2, and 2X1=2, and 25 annexed will give 225, the second power of 15. So also, 3,5 65 10,5 21.5 67.5 4 7' 11 22 58 I' 625 1225 4225 11025 46225 330625 EXERorsEs ON THIS jirTnop. 26. What is the second power of 135 ? 27. What is the second power of 205 ? 28. What is the second power of 335 ? 29. What is the second power of 455 ? 30. What is the second power of 585 ? 31. What is the second power of 795 ? Aus. 18225. Ans. 42025. Ans. 112225. Ans. 207025. Ans. 342225. Aus. 632025. Note.— The square root of any quantity ending in 9, must end in oitbor b or 7. No second power can end in 8, 7, 3 or 2. The second root of any quantity ending in C, must end in 4 or 6. The second root of any quantity ending in 5, must end also in 5. The second root of any quantity ending in 4, must end cither in 8 or 2. The second root of any quantity ending in 1, must end either in 1 or 9. The BecoQd root of my quantity ending in 0, must also encl in 0. EVOLUTION. The tf06t of any quantity is a number such that when repeated, as a factor, the specified number of times, will produce that quantity. Thus, 3 repeated twice as a factor gives 9, and therefore 3 is called the second root of 9, while 3 taken three times as a factor will givo 27, and tlierefore 3 is called the third root of 27, and so also it is called the fourth root o£ 81, There arc two ways of indicating this. First, by the mark i which is merely a modified form of the letter r, the initial letter of the English word root, and the Latin word radix (root). When no mark is attached, the simple quantity or first root is indicated. When the second root is meant, the mark |/ alono is placed before the quantity, buj if the third, fourth, &Q,, roots oro to be indicated, remains, SECOND OB SQUABE BOOT. 283 tno figures 3, 4, &c., are \Tritten in the angular space. Thus; 3=|/9=|^27=-i/81=^243, &o., &c. The other method is to write the index as a fraction. Thus, 9^ means the second root of the first power of 9, i. e. 3. So also, 2T^ is the third root of the first power of 27. In the same manner G4^ means the third root of the second power of 64, or the second power of the third root of 04. Now the third root of 64 is 4, and the second power of 4 is IG, or the second power of 64 is 4096, and the third root of 4096 is 16, so that both views give the same result. Evolution is tho proooss qS fiadiog any required root of a given quantity. SECOND OR SQUARE ROOT. Extrading the square or second root of any cumber, is the find- ing of a number which, when multiplied by itself, will produce that number. To find the second root, or square root of any quantity. By inspecting the table of second powers, it will be found that tho second power of any whole number less than 10, consists of cither one or two digits ; tho second power of any number greater than 9, and less than 100, will in like manner be found to consist of three or four digits ; and, universally, the second power of any number will consist of cither twice the number of digits, or one less than twice the number of digits that tho root itself consists of. Henco, if wo begin at the units' figure, and mark off tlic given number in periods of two figures each, we r-hall find that tho number of digits contained in the root will be the same as the number of periods. If the num- ber of digits is even, each period will consist of two figures, but. if the number of digits be odd, tho last period to the left will consists of only ono figure. Xet it now bo required to find the second root of 144. Wo know by the rule of involution that 144 is the second power of 12. Now 12 may be resolved into one ten and two units, or 10 -j- 2, and 10-1-2 multiplied by itself, as in the margin, gives 1004-40+4, and since 100 is the second power of 10, and 4 the second power of 2, and 40 is twioo the product of 10 and 2, wc conclude that the second 284 ARITHMETIC. 104-2 10+2 100+20 20+4 100+40+4 power of any number thus resolved is equal to the sum of the second powers of the parts, plus twice the product of tho parts. Ilcn'.^c to find the second root of 144, let us resolve it into the three parts 100+40+4, and we find that the second root of the first part is 10, and since 40 is twice the product of the parts, 40 divided by twice 10 or 20 will give tho other part 2, and 10+2=12, the second root of 144. Wo should find tho same result by resolving 12 into 11+1, or 9+3, or 8+4, or 7+5, or G+G, but tho most convenient niodo ia to resolve into tho tens and the units. In tho same manner, if it be required to find the second root of 13G0, wo have by resolution 900+420+49, of which 900 is the second power of 30, and 30x2^G0, and 420h-G0=7, the second part of tho root, and 30+7=37, the whole root. Ag..in, let it be required to find thq second root of 1512D. This may be resolved as below : lOOOO is the second power of lOa. 400 is the second power of 20. 9 ia the second power of 3. 4000 ia twice the product of 20 and 100. 600 is twice tho product of 100 and 3. 120 is twice the product of 20 and 3. 15129 is the sura of all, and hence 1 is the root of the hundreds 2 the root of the tens, and 3 tho root of the units. Generalizing these investigations, we find that tho second power of a number consisting of units alone is the product of that number by itself; that the second power of a nuu.ber consisting of tens and units is the second power of the tens, plus the second power of the units, plus twice the product of the tens and units ; that tho second power of a number, consisting of hundreds, tens and units, is the sum of the squares of the hundreds, tlio tens, and the units, plus twice the product of each pair. Now since the complement of tho full second power, to the sum of the second powers of thp parts, is twice the product of tho parts, it follows that, when the first figure of the root has been found, it must be doubled before used as a divi- sor to find the second term, and for tho same reason each figure, when found, must be doubled to give correctly tho next divi3or. Henoo the ■ SECOND OR SQUAEE BOOT. 235 RULE . Beginning at tJie units' figure, mark off the whole line in periods of two figures each; find the greatest power contained in the left hand period, and subtract it from that period; to the remainder annex the, next period; for a new dividend, place the figure thus obtained as a quotient, and its double us a divisor, and find how often that quantity is contained in the second partial dividend, omitting the list figure ; annex the figure thus found to both divisor and quotient, muUiplg and subtract as in, common division, and to the remainder annex the next period; double the last obtained figure of the divisor, andp>roc.ecd as before till all the periods are exhausted, — // there he a remainder^ annex to it two cij^hcrs, and the figure thence obtained will he a, decimal, as will every figure thereafter obta,ined. EXAMPLES. 1. To find tho second root of 797449. First, comniencin;:? with tho units' figure, wo divide tho lino into periods, viz., 49, 74 and 79,— -.vu th(3n noto that tho greatest squaro contained in 79 is G4,— this wo subtract from 79, and find 15 remaining, to which 893 wo annex the next period 74, and place 8, the second root of 04, in tlie quotient, and its double 16 as a divisor, and try how often IG is contained in 157, which we find to be 9 times ; placing tho 9 in both divisor and quotient, we multiply and subtract as in common division, and find a remainder of 53, to which wc annex tho last period 49, and proceeding as before, we find 3, the last figure of the root, without remainder, and now we have *ho complete root 893. 2. This c;»'^ration may bo illustrated as follows : To find tho second root of 273529. 8 109 1783 7"97449 64 1574 1521 5349 5349 500 500X2=1000+20, or 1020 1000-f2x20-f3=1043 500-f20-f3=523 3129 3129 I 286 ABTTHMETIO. 3. To find tho second root of 153687. Hero we obtain, by the same proocsa as in the last example, tho whole number 392, with a remainder of 23. which can produce only u fraction. 69 782 78402 784049 1536871 392.029-1- 9 230000 156804 7319600 7056441 203169 We now annex two ciphers, placing the decimal point after the root already found, but as the divisor is not contained in this new dividend, wo place a cipher in both quotient and di- visor, and annex two ciphers more to the dividend, and by continuing this process wo find the decimal part of tho root, and tho whole root is 392.029-f . Ans. 529. Ans. 8642. Ans. 678. Ans. 28.01785+. Ans. 41.569219+. Ans. 25,8069+. EXEROISEB. 1. What is the second root of 279841 ? 2. What Is the second root of 74684164? 3. What is the second root of 459684? 4. What is tho second root of 785 ? 5. What is tho second root of 1728 ? G. What is the second root of 666 ? 7. What is the second root of 1,23456789 ? Ans. xllll.lll06+. 8. What is the second root of 5 to three places ? Ans. 2.236. 9. What is the side of a square whose area is 19044 square feet ? Ans. 138 feet. 10. What is the length of each side of a square field containing 893025 square rods ? Ans. 945 linear rods. Tho second root of a fraction is found by extracting the roots of its terms, for ^f=|X| and therefore i/^«-=;/|X*^f,. So tOso, i/^a=i. Again, since i/tVo=/o~-^9^ a^d -3 X .3=.09. the second root of .09 is .3. This follows from the rules laid down for the multiplication of decimals. To find the second root of a deoim&l or of a whole number and a decimal: BEOOKD OB SQUABE BOOT. 287 mplo, tho luoo only ) ciphers, oint after d, but as ktaiacd in place a it and di- ciphers I, and by IS wc find 3 root, and 029-I-. Ans. 529. .ns. 8642. Ans. 078. 01785+. 69219+. 5.8069+. .11106+. ns. 2.236. 44 square 138 feet, lontaining near rods, ic roots of ^/.. So =.00, tho down for iber and a Point off periods of two figures each from the decimal' poin, towards the right and left, adding a cipher, or a repetend, if the number of figures be odd. From what has been said, it is plain that every period, except tho first on the left, must consist of two digits, and every decimal presupposes something going before, for .5 indicates the half of some unit under consideration, and .5 is equivalent to .50, and iWt to .05, from which it is obvious that tho second root of .5 is not the rcot of .05, but of .50, and therefore tho second root of .5 is not .2-f-, as tho beginner would naturally suppose, but .7+, for ,2-\- is tli- approximate root of .05. ADDITIONAL EXERCISES. 11. What is tho seooad root of .7 to five places of decimals ? Ans. .83666. 12. Find the second root of .07 to six places. 13. What is the second root of .05 ? 14. What is the second root of .7 ? 15. Find the second root of .5. 10. What i* the second root of .1 ? 17. What is tho second root of .1 ? 18. What is the second root of 1,375 ? 19. What is the second root of .375 ? 20. What is tho second root of 6.u ? 21. Find to fcnr decimal places i,' 3^^. 22. Find |/2 to four 'Iccimal placed. 23. Find the value of t/3271.4207. 24. Find the seoona root of .005 to fivo places. Ans. 07071. 25. Find the square root of 4.372594. Ans. 2.09107+. 20. What is the second root of .01 ? 27. What is the second root of .001 ? 28. What is the square root of .0001 ? 29. What is the second root of .000001 ? 30. What is the second root of 19.0968 ? Ans. 264575. Ans. .2236+. Ans. .8819+. Ans. .74535+. Ans. .3162277H-. Ans. .3. Ans. 1.1726, &cJ^ Ans. 61237, &c.* Ans. 5^.52982+. Ans. 1.7748. Ans. 1.4142. Ans. 57.196+. Ans. .1. Ans. 03162+. Ans. .01. Ans. .001. " Tbo yonn?; student would naturally expect that the decimal figures of j/ 1.375 and |/ ^76 would be the same, but it is not so. If it were so, 1/1+ 1/.375 woul v iiL> equal to -^ 1.67 5- That such is not the case, may bo shown by a very simple example. i/lC+|/9=4+3— 7, but y'lG+9=y'25=5 Let it be carefully obaerved, therefore, tbat the awn of the eecond roots ia not the same aa iJu second root qf the sum. 288 AETTHMETIO. 0P3IIATION 4 19.0968 16 4.37 trial. 83 4*(J() true. Trial 867 309 249 Too great by 1 6068 6069 True 866 6068 5198 873 Here wo find the remainder, 872, ig greater tlian the divisor, 866, wl'.iclrfsocms inconsistent witli ordinary rules j but it must bo obsci*ved that wo arc not socking an exact root, but only tho closest possible approximation to it. If tho given quantity had been 19.0969, wc should have found an exact root 4.37. Tho romaLadcr 872 being greater than tho divisor, shows that the last figure of thn root is too small by j^q, whereas 7 would bo too great by -, Ju' ^"-"^ that 806 is not a correct divisor but an approximato one, and that tho true root lies between 4.36 and 4.37. When the root of any quantity can be found exactly, it is called a 2)crfect jiower or rational quant tti/, but if the root cannot be found exactly, tho quantity is called irrational or surd. A number may be rational in regard to one root, and irrational in regard to another. Thus, 64 is rational as regards |/64=8, c •^64=4 and ]/64=2, but it is irrational regarding any other root expressed by a whole number. But 64, with tho fractional index §, i. €,, 64* is rational, because it has an even root as already shown. Wc may call 64^ either the secon 1 power of tho third root of 64, or the third root of tho second power. In tho former view, tho third root of 64 is 4, and the second power of 4 is 16, and according to the second view, 64^ is 4096, and tho third root of 4096 is 16, the same as before. i/81=:3 is rational, and |/81=9 is rational, but 81 ia not rational regarding any other root ; while |/25 is rational only regarding tho second root, and ]^8;=2 only legarding tho third root. 1h.Q second root of an oven square may be readily found by re- solving the number into its prime factors, and taking each of these i THIRD ROOT Oa CtTBE BOOT. 289 10 dirisor, It muab bo tho closest had been rcmaLadcr ;urc of the. J TOO' aad , and th.-it factors once,— the product will bo tho root. Thus, 441 is 3x3X7x7 and each factor taken once is 3x7=21, tho second root. Horo lot it bo observed, that if wo used each factor twice wo should obtain tho teco7id power, but if wo use each factor half the number of times that it occurs, wo shall have tho second root of that power. G4 is 2X2X2X2X2X2=2«, i. e., 2 repeated six times as a factor gives the number G4, and therefore half tho number of those factors will givo tho second root of 64, or 2x2X2=8, and 2X2X2 multiplied by 2X2X2=8X8=04. As this cannot bo considered more than a trial method, though offcn expeditious, we would observe that the smallest possible divisors should bo used in every oase, and that if the number cannot be thus resolved into factors, it has no even root, and must bo carried out into a lino of deoiiuals. or thooe deoimals ma? be red'ioed to oommon fractiono. i is called t be found irrational l/64=8, other root il index f , dy shown. C4, or the third root ing to the , the same but 81 is tonal only bird root, ind by ro- of these THIRD ROOT OR CUBE ROOT. AS extracting the second root of any quantity is tho finding of what two equal factors will produce that quantity, so extracting tho third root is the finding of what three equal factors will produce tho quantity. By inspecting the table of third powers, it will be seen that no third power has more than three digits for each digit of the first power, nor fewer than two less than three times the number of digits. Hence, if tho given quantity be marked off in periods of three digits each, there will be one digit in the first power for eacb period in tho third power. The left hand period may contain only one digit. From tho mode of finding tho third power from tho first, we can deduce, by the oonverse process, a role for finding the first power 19 290 AMTHMETIO. from the third. Wo know by tho rule of involution that tho third power of 25 is 16C25. If wc resolve 25 into 20-|-5, and perform tho multiplication in that form, we have 20-|-5 400-f-lOO 1004-25 4004-2004.25=:(20+6)=» 80004-4000-1-500 20004-10004-125 8000-]-G000-^15004-126=(20-|-5)3=15625 Now, 5000 is iTio third iwwcr of 20, and 125 is tho third power of 6; also, COOO is three times tho pmJuct of 5, and the second power of 20, and 1500 is three times tho product of 20, and tho sooond power of 6. Let a represent 20 and h represent 5, tlien «3=203 -^ 8000 Sa" I =3X202x6 ^ COOO 8 a 68=3X20X5=* ^ 1500 '»8==53 = 125 15625 By usmg these symbols wo obtain the simplest possible method of extracting tlo third root of any quantity, as exhibited by tho snbjoinod scheme : Given quantity 15625 0^=203=20X20X20 = 8000 Bemaindcr 7625 3 a3 6=3x20^x5 = 6000 Bemainder 1G25 3 a 63=3X20X52 :^ 1500 Brcmainder 125 6»=53=5X5X5 = 125 From this and similar examplod wo see that a nnniber denoted by more than ono digit may bo resolved into tens and units. Thus, 25 is 2 tens and 6 y9its, 123 u 12 t«DS and 3 units, und 80 of ail numbcm. lat tho third 1 that form, 25 third power 1 tho second 20, and (ho 5, then ilblo method 5ited by tho ber denoted lits. Thus, id 80 of ail THIRD BOOT OB OUBE BOOT. 291 To find tho third root of 1860867 : As this number consists of thrco periods, tho root will consiflt of three digits, and the first period from the left will give hundreds, the second tens, and the third units, and so also in ease of remainder, each period to the right will give one decimal plaoe, the £rst being tenths, the second hundredths, &o., &o. We may denote the digits by a, b and c. a=10O a«=100»= ft=3aa 6-1-3 aa=m^'y=20+, and 30000X20= 3 a 6»=3X100X400= 1860867(100+20+3=123 1000000 860867 remainder. 600000 260867 remainder. 120000 ^8—203: 140867 remainder. 8000 Now (a+J)=120 . • . 3 (a+2»)2=132867 remainder. 43200, which is contained 3 times+ in 132867, . • . c=3, and 3 (a4-iyo* =3X1202X3= 129600 And 3 (a-1-6) c3=3Xl20X9= And lastly, c5=3»= 8267 rein&inder. 3240 27 27 no remainder. BULB. Mark off t%e gitm nwaCbif in period of three Jtguret each Find the highett third power contained in the left hand period, and tultract it from that period. Divide the remainder and next period by three timet the second power of the root thue/ound^ and the quotient toill be the tecond term of the root. From thefirtt remainder subtract three times the product of the Mcond term, and the square o/thejirttf PLUS three times the product t)f the first term, and the square of thi KOOnd, plus the third power of the tecond. Divide the remainder by three timet the tquare of tho sum of the first and tecond terms, and the amtient will be the third term. 292 ARTTHMETIC. From the last remainder subtract three times the product of the term last found, and the square of the sum of the preceding terms, PLUS the product of the square of the last found term hi/ the sum of the preceding ones, PLUS tfui third power of the last found term, and so on. £X£ROISEB. 1. Whati s the third root of 46656? Ans. 36. 2. What i s the thi.-d root of 250047 ? Ans. 03. 3. What ] IS the third root of 2000576 ? Ans. 120. 4. What ] 13 the third root of 5545233 ? Ans. 177. 6. What 1 3 the third root of 10077696 ? Ans. 216. 6. What is the third root of 46208279? Ans. 359. 7. What 13 the third root of 8576G121 ? Ans. 441. 8. What is the third root of 125751501 ? Ans. 501. 9. What ] IS the third root of 153990656 ? Ans. 536. 10. Whati 18 the third root of 250047000 ? Ans. 630. 11. What is each side of a square box, the solid content of which is 59319 ? Ans . 39 inches. 12. What is the third root of 926859375 ? Ans. 975. 13. Find the third root of 44.6. Ans. 3.456-f . 14. What is the third root of 9 ? Ans. 2.08008-|-. 15. What is the length of each side of a oabic vessel whose solid content is 2936.493568 cubic feet? Ans. 1432 feet. 16. Find the third root of 5. Ans. 1.7099. 17. A store has its length, breadth and height all equal ; it can hold 185193 cubic feet of goods; yrhat is each dimension? Ans. 57 feet. 18. How many linear inches must each dimension of a cubic vessel be which can hold 997002999 cubic inches of water ? Ans. 999 inches. 19. What is the third root of 1 ? Ans. 1. 20. What is the third root of 144 ? Ans. 5.241483. The third root of a fraction is found by est meting the third root of the terms. The result may be expressed cither as u common fraction, or as a deoimol, or the given fraction may bo reduced to n dooimtJ, and the root extracted onder tliat form. THIRD HOOT OR CGBE BOOT. 293 duct of the '.ding terms, hy the SUM ^QUiid term, Ans. 36. Ans. G3. Ans. 12G. Ans. 177. Ans. 216. 359. 441. 501. 536. Ans. Ans. Ans. Ans. Ans. 630. rt of wliich 1. 39 inches. Ans. 975. is. 3.456+. 2.0800S-I-. whose solid 1432 feet, ns. 1.7099. lal; it can ? ns. 57 feet. of a cubic X? 999 inches. Ans. 1. 5.241483. c tliird root u common iduccd to A Ans. }=i75. UXKROISES. I. Wliat ifl the third root of l\ ? Othcrwiso : «]=.42l875. To find tlio t)iird root of .42i875(.70-f.05=;.76 this w^ have 70 a ~ 3X70- X5 =73500") 3X70 X5-=^ 5250 \ 5^= 125 3== 343000 78875 roniiuniur. 78875 no remair. Icr. Tho third root of a mixed quantity will be most readily fonnd hy reducing the fractional part to tho decimal form, and applying the general rule. It ha.s been already explained that the second root of an even power may be obtained by dividing the given number by tho smallest possihlo divisors in succession, and taking half the number of those divisors as factors. The same principle will apply to any root. If the giver) quantity is not an even power, it may yet bo found approx^ iniately. "f wo take t'lo number 4GC5G, wc laotice that as the last figure is an oven number, it is divisible by 2, and by pursuing the same principle of operation wo find six. twoi as factors, and afterwards BIX threes ; and, as in tho caso of the second rcot, wo take cac/i factor half the number of times it occurs, so in the case of the tliird root, we take each factor one-third tho number of times, it occurfi. Tho same principle on which tho extraction of tho second and third depends may bo applied to any root, tho lino of figures being divided into periods, consisting of as many figures as there arc units in tho index ; for tho fourth root, ueriods of four figures each ; for tiie fifth, five, &o., &o. Wo liiay remark, hciwovor, that thoso modes arc HOW superseded by tho grand disoovory of I^Rorithmio Oomputu- tion. 294 sscBMnno* PROGRESSION". A tenet is a saccc.^sion of quantities increasing or decreasing by a Common Di^erc^ee, or a Common Ratio. .Progression hy a Common Difference forms a series by tbe addi- lion or subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23 forms a series increasing by t!ic constant quantity 4, and 28, 21, 14, 7, forms a scries decreasing by the constant quantity 7. Such a progression is also c )Ilc J an cquidiffercnt series.* Progression by a Common Ratio forms a series increasing or decrciiding by multiplying or dividing by the pame quantity. Thua, 3^ 9, 27, 81,. 243, is a Bcrics increasing by a constant multiplier 3^ nnd 04, 32, IC, 8, 4, 2, is a series decreasing by a constant divisor 2. 'Ihc quantities forming such a pro,:^ression are also called Con- tinual Proportionals/'^ because the ratio of 3 to 9 is the same as the ratio of 9 to 27, &c., &c. From this it i i plain that in a progression by ratio, each term is a mean proportional between the two adjacent ones, and also between any two that are equally distant from it. The first and lost terms are called the JExtrcmetf and all between them the Means. PROGRESSION BY A COMMON DIFFERENCE. In a series increasing or decreasing by a common difference, the sum of the extremes is always equal to the sum of any two that are equally distant from them. Thus, in the first example 3-|-23=:7-|- 19=114-15=26, and in t.-c second 28+7=21+14=35. If the number of terms be old, the sum of the extremes is equal to twice the middle term. Thus in the jieries 3, 7, 11, 15, 19, 3+19=2x11=22, and hence the middle term is half the sum of tuj extremes. • The names Arithindiixd Progression and 0«omf>trical Progrtssion ar« olten applied to quantities bo related, bat these torms arc altogether inappro* priate. as they would indicate that the one kind belonged solely io arithmetic, and the other solely to e^eometry, whereas, in reality, each bolonga to both these branches of science. rROORESsioi: by a cojMon difference. 295 In treating of progressions by difference or cquidifferent senes, there are five things to be considered, vis:., the first term, the last term, the common difference, the numbev of terms, and the sum of tho series. These are so related to each other that when any three of them are known no can find tho other two. Given the first term of a series, and the common difference, to find any other term. Suppose it is required to find the seventh term of tho series 2, b^ 8, &c. Here, as the first term is given, no addition is required to find it, and therefore six additions of the CDmmon difference will complete the series on to seven terms. In other words, the common difference is to be added to the first term as often as there arc units in the number of terms diminished by 1. This gives 7 — 1=::6, and 6X3=18, which added to the first term 2 gives 20 for the seventh term. If we had taken the series on the descending scale, 20, 17, 14, &c., we should have had to subtract the 18 from tho first term 20 to find the seventh term 2. The term thus found is usually designated the last term, not because the series terminates there, for it does not, but simply because it is the last term cansidcred in each question proposed. From these illustrations we derive the RULE (1.) Subtract 1 from the number of terms, and multiply the remainder by the common difference; then if the series be an increasing one, add the result to the first term, and if the series be a decreasing one, subtract it. EXAMFLEB. To find the fifty-fourth term of tho increasing series, the fiist term of which is 33f, and the common difference 1^. Here 54—1=53, and 53x11=66^, and 66i+33|-=100, the fifty-fourth term. Given G4 the first term of a decreasing series, and 7 the common difference, to find the eighth term. Hero 8 — 1=7, and 7x7=^49, and 64— 49=:15, the eighth term. EXERCISES. 1. Find the eleventh term of the decreasing series, the first term of which ia 248f, and the common difference 3\, Ans. 216 J. 2. The hundredth term of a decreasing series is 302|, and the common difference ia 3|, what is the lost term ? Ans. 36. t I ! 1]' I 296 ABTTHMETia 3. What ia the one^thoasandth term of the series of the odd figures? Ans. 1999. 4. What is the five-hundredth term of the series of oven digits ? Ans. 1000. 5. What is the sixteenth term of the decreasing series, 100, 96, il2,(&o.? Ans. 40. To find the snm of any cquidifferent series, when the numher of terms, and either the middle term or the extremes, or two terms equidistant from them, are given. We have seen already that in any such scried the sum of the extremes is equal to the sum of any two terms that arc equidistant from them, and when the number of terms is odd, to twice the mid- dle term. Hence the middle term, or half the sum of any two terms equi-distant from the extremes, will be equal to half the sum of those extremes. Thus, in the series 2+7H-12-|- 17+22+27+32, wo have 2^=2^=17, the middle term. It is plain, therefore, that if we take the mi Jdle term and half the sum of each equi-distant pair, the serieswillbe equivalent to 17+17+17+17+17+17+17, or 7 times 17, which will give 119, the s^. as would be found by adding together the original quantities. The same result would bo arrived at when the number of terms ia even, by taking half tlie sum of the extremes, or of any two terms that arc equi-distant from them. From these explanations we deduce the RULE (2.) Multiply the middle term, or lidlf the sum of the extremeg, or of any two term* that are equidistant from them, hy the number oj terms, ^'OTK,— If the sum of the two terms bo an odd number, it is generally more convenient to multiply by the number of terms before dividing by 2. EXAMPLES. Given 23, the middle term of a series of 11 numbers, to find the sum. Here we have onlyto multiply 23 by 11, and we find at once the sum of the series to be 253. Given 7 and 73, the extremes of an increasing series of 12 num- bers, to find the sum. The sum of the extremes is 80, the half of which is 40, and 40x12^480, the sum required. PBOORESSION BY A COMMON DUTPEBENCE. 2107 of the odd Ana. 1999. ivcn digits ? Ans. 1000. es, 100, 96, Ans. 40. the number t two terms mm of the equidistant CO the mid- y two terms um of those 27+32, wo rcforo, that ;qui-distant '+17+17, Q found by t would bo alf the sum from them. ernes, or of number oj is generally ling by 2. to find the nd at once )f 12 num- ho half of Two equidistant terms of a series, 35 and 70, are given in a series of 20 numbers, to find the sum of the series. In this case, we have 35+70=105, and 105X20=2100, and 2100-7-2=1050, the sum required. EXEBCISES. 1. Find the sum of the aeries, consisting of 200 terms, the first term being 1 and the last 200. Ans. 20100. 2. What is the sum of the series whose first term is 2, and twenty-first G2 ? Ans. G72. 3. What is the sum of 14 terms of the series, the first term of which is \ and the last 7 ? Ans. 52J. 4. Find the sum to 10 terms of the decreasing series, the first term of which is 60 and the ninth 12. Ans. 360. 5. A canvasser was only able to earn $6 during the first month he was in the business, but at the end of two years was able to cam $98 a month ; how much did he earn during the two years, s^apposing the increase to have been at a constant monthly rate ? Ana. $! 248. 6. If a man begins on the first of Jftuuary by saving a cent on the first, two on the second, three on the third, four on the fourth, &c., &c., how much will ho have saved at the end of the year, not counting the Sabbaths ? Ans. $490.41. 7. How many strokes does a clock strike in 13 weeks ? Ans. 14196. 8. If 8| is the fourth part of the middle term of a series of 99 tium])crs, what is the sum? Ans. 3465. 9. In a series of 17 numbers, 53 and 33 are equidistant from ihe extremes ; what is the sum of the series? Ans. 731. 10. In a series of 13 numbers, 33 is the middle term . what is the sum ? Ans. 429. To find the number of terms when the extremes ap5. t^ommon diifcrence are given '. As in tho rule (1), we found the difference of the extremes by nmltiplying by one less than the number of terms, and added tho first term to li.o result, so now we reverse the operation and find the RULE (3.) Ifivide the difference of the extremes hy the common difference and odd 1 to the result. 298 AKITHMETIO. EXAMPLE « 5^9 h\ Given the extremes 7 itnd 109, and the common difference, 3, to find tho number of terms. In this case wo have 109— V--102, and 102—3^^34, and 34+1=35, tho number ti terms. EXEROtSES. 1. What is t'-.c number of terms when the ejctrcmos arc 35 and 333, and the common difference 2 ? Ans. 150. 2. Two equidistant terms are 31 and 329, and the common dif- fercnco 2 ; what is the niunber of terms ? Ans. 150. 3. Tho first term ol u series is 7, and tho last 142, and the com- mon difference |- ; what is the number of terms ? Ans. 541. 4. The first and lust terms of a series are 2J and 35J-, and tho common difference J ; what is the number of terms ? Ans. 100, 5. The first term of a series is J and last 12^ and the common difference J ; what is the number of terms ? Ans. 25, Given one extreme, the sum of the soiies and tho number of terms, to find the other extreme. Tins ease may bo solved by reversing Rule (2), for in it the data are tlie same, except that there the second oxiremc was given to llnd the sum, and now the sum is given, to find tho second extreme. Therefore, as in that rule we multiplied tho sum of tho extremes by the number of terms and halved the product, so now wo must double the sum of tho series and divide by the number of terms to find the sum of the cs.tremes, and from this subtract tho given extreme, and the vcmuindcr will be the required extreme. This will illustrate the RULE (4.) IXvidc twice the sum of the series hy the number of terms, and from the quotient subtract '' ' ^'ven extreme^ and the remainder will hcthcrecj d extreme. EXAMPLE. Given 5060, the sum of a series, 1 the first term, and 100 tho number of terms, to find the other exticnic. Twico the sura is 10100, whichj divided by 100, gives 101, and 101—1=100, the number of tonus. PBOOBESSZON BT A COMMON DIFFEBENCS. 299 SXEBOISia. 1. Given 60, the greater extreme of a decreasing series, 412!, the ;ium| and 17 the number of terms, to find the other extreme. Ads. 2. 2. If 121268 bo the sum of a series, 8 the less extreme, and 142 the uumbcT of tcrm^' ; what is the greater extremo ? Ans. 1700. 3. The sum of a scries of 7 terms is 105, the greater extremo is 21, and the number of terms 7 ; what is the less extreme ? Ans. 9. 4. The sum of a scries is 576, the number of terms 24, and the greater extreme is 47 ; what is the less extreme ? Ans. 1. 5. The sum of a series is 30204^, the greater extreme 312, and the number of terms 193 ; what ia the less extreme ? Ans. 1. Given the extremes and number of terms, to find tho common difference. Aa explained in tho introduction to Rule (1), tho number of common differences must be one less than the number of terms. It ia obvious also, that the sum of these differences constitutes tho differ- ence between the extrenjcsi, and that therefore the sum of the differ- ences is the same as 1 less than the number of terms. Therefore the difference of tho extremes, divided by the sum of the differences, will give one difference, t. e., the common diSercncc. This gives us the BULB (5.) Subtract 1 from the numhcr of ienm, and divide the difference of the extremes by the remainder. BXAUPLB. If the extremes of an increasing sorijs bo 1 and 47, and tho number of terms 21, we can find the common difference thus :— 47—1=46, and 46-7-23=2, tho common difference. XXEBOISBS. 1. If the extremes are 2 and 36, and the number of terms 18 ; what the common difference ? Ans. 2. 2. What is the common difference if the extremes are 58 and 3, nnd the number of terms 12 ? Ans. 5. 3. In a decreasing series given 1000 the less extreme, nnd 1793 the greater, and 367 the number of terms, to find the common difference. Ans. 2^. 300 ARITHMETIC. 4. If G and 60 aro the cxtromcs rn a porica of 1 numbers, what is the common difiFcrcncc ? Ans. 6. 5. What is tho coniiuon diCercnco in a dcoreasing ccrics of 42 terms, the extremes of which nto 9 and 60 ? Ans. 1. There arc fifteen other cases, but thcj may all be deduced from tho five here gi' 'cn. Wo subjoin tho Algebraic form as it is more satisfactory and complete, and also more easy to persona acquainted with tho symbols of that science.. Let a be tho first term, d tho common difference, n the number of terms, 8 the sum of tho seriiss ; the series will bo represented by ;(_}-(a-f-J)-j-(a-l-2cZ)-f (a+37, and 387. PROGRESSIONS BY RATIO. There are in progression by ratio, as in progression by difference, the same five quantities to bo considered, except that in place of a com- mon difference we have a common ratio ; that is, instead of increase or decrease by addition and subtraction, wo have increase or decrease by multiplication or division. If any three of these are known the other two can be found. We have noticed already that if any quantity, 2, be multiplied by itself, tho product, 4, ia called the square, or ;i(coud power of that 804 ABTTHMETIO. quantity ; if this bo again multiplied by 2, tbo prodtioi, 8, is oallecl tho cubo, or third power of that quantity ; if this again bo multi* plied by 2, Lhe produet is called tho fourth power of that quan- tity, and BO on to the fifth, sixth, &o., powers. To shoM' the fihort mode of indieating this, let us talco 3x3X3x3X3—243. For brevity this is written 3^, whieh means that there are 5 faotora, all 3, to bo continually multiplied tojuether, and 5 is oallod tho index, because it indicates the number of equal factors. Given tho first term and tho common ratio to find the lost pro- posed term. Let it bo required to find tho sixth term of tho increasing Borics, of which tho first term is 3 and tho ratio 4. This may obviously bo found by successivo multipUcatiooa of iha first term, 3, by tho ratio, 4, — thus : — 3=lst term. 3X4= 12=2nd term 12X4= 48=3rdterm. 48X4= 192=4th term. 192X4= 768=5th term. 768x4=3072=6th term. The scrieg, therefore, is 3, 12, 48, 192, 768, 3072. From this, it is plain, that as to find tho last of 6 terms, only 5 multiplications oi tho first arc required, in all cases the number of multiplications will be one less than tho number of terms. But to multiply five times y 4 is tho same as to multiply by 1024, tho fifth power of 4, foi .X4X4X4X4=1024, and 1024X3=3072.* . This gives us tho general BULB (1.) Multiply the first term ly that power of the given ratio tohich is a unit less than the number of terms. U the series be a decreasing one, divide instead of multiplying. ZXAMFLES. Given in a series of 12 numbers, the first term 4 and the ratio 2, to find the last term. Since 11 is one less than the number of terms, we find the 11th power of 4, which is 2048, and this, multiplied by the first term, 4, gives 8192 for tho twelfth term. * For tho abbreviated mode see I&7olutlon. 3, is called i bo multl* that qaon* Bbov the X3— 243. 5 footorf, 1 iho index, le lost pro- ising series, tioDS of thd pnoanEssioNS by iiatio. 305 ?rom tlus, it plications ol icatioDS will y five timea VQT of 4, foi ratio which multiplying. the ratio 2, nd tbe 11th irst term, 4, Given the ninth term of a dccroasing scries, 393GG, and the ratio 3, to find the first term. Aa thoro arc 9 terms, wq tako the 8th power of the ratio, 3, which wo find to bo C561, and the first t^rm 30360-^6561^0, the first term. EXEROIHES. 1. What 18 the ninth term of the increasing series of which 6 is the first term and 4 the ratio ? Ans. 327G80. 2. "What is the twelfth term of the increasing series, the first term of which is 1 and the ratio 3 ? Ans, 177147. S. In a decreasing scries the first term is 78732, the ratio 3, and the number of terms, 10 ; what is the last term ? Ans. 4. 4. What is the 20th term of an increasing scries, the first of rhich is LOG, and also the ratio LOG ? Ans. 3.207135. .. ^n a decreasing series the first term is 126.2477j the ratio LOG ; -what is the last of 5 terms ? Ans. 100. Given the extremes and ratio, to find tho sum of the scries. It 18 rol easy to give a direct proof of this rule without the aid of Algebra, out the following illustration may be found sa'-iafactory, and, in some sort, bo accounted a proof. Let it bo required to find tho sum of a series of continual pro- portions, of which the first term is 5, tho ratio 3, and tho number of terms 4. Since 3 is the common xi^tio, wo can easily find tho terms of the scries by a succession of multiplications. Theso arc — 5+15+45+135, and tho sum is 200 15+45+135+405 400 Let us now multiply each term by tliO ratio, 3, and, for oonve- uience and clearness, place each term of the second lino below that one of the first to which it is equal. Let us now subtract the upper from the lower line, and we find that there is no remainder, except the dificrence of the two extreme quantities, viz., 400. Now, it will bo scon that this remainder is exactly double of the sum of the series, 200, and consequently 400 divided by 2, will give tho sum 200. Also, 405 is the product of the last term by the ratio, and 400 is the difference between that product and the first term, and the divisor, 2, is a unit less than the ratio, 3. Hence the 306 ARITHMETIC. RULE ,2.) Multiply the last term by the ratio, from this product subtract the first term, and divide the remainder by the ratio, diminished by unity. EXAMPLE. Given the first term of nn increasing scries, equal 4, the ratio 3, and the number of terms G, to find the sum of the scries. By the former rule we find the last term to .)e 972. This, mul- tiplied by the ratio, i^ives 291G, and the first (stremc, 4, sub acted from this, leaves 21)12, and this divided by 2, v;hiohisl Icsstlian tho ratio, gives 1456, the sum of tho series. EXERCISES. 1. What is the sum of the series, of wliich the less extreme is 4, the ratio 3, and tho number of tonus 10 ? Ans. 118096. 2. What is the sum of the scries, of which 1 is the less extreme, 2 (ho ratio, and 14 the number of terms? Ans. 1G383. ~3. Whut is the sum of the series, of which the greater extreme is 18.42015, the less 1, and the ratio LOG ? Ans. 308.755083. 4. A cattle dealer offered a farmer 10 sheep, at the rate of a mill for the first, a cent for tho second, a dime for the third, a dollar lor tho fourth, &c., &c. ; in what amount was he " taken in," supposing that each sheep was worth $11,111 ? Ans. $1111100.00. 5. What is the sum of six terms of the series, of which the greater extreme is i and the ratio I ? Ans. 3 7 -2 ■» 3 liJO> or 1 I f) i 10, we divide 1024 by 2, and find 512, and then by extracting the ,i>iuth root of 512, wc fi^d the ratio, 2. PROGRESSIONS BY RATIO. 307 ud subtract ninished by , the ratio 3, 5. This, iiiul- 4, Bub acted llcsatlian the I extreme is 4, Ans. 118096. B less extreme^ Ans. 1G3B3. •vcatcr extreme s. 308.755983. ic rule of u mill .rd, a dollar ibr in," supposing . $1111100.00. L of which the sr of terms arc l:cme8 tea 3 and 3ted by simply ty 3 and fiad llcnoc tho that root of the ibir of terms. land tho number |l2, and then by I2. EXERCISES. 1. If the first yearly dividend of a joint stock company be $1, and tho dividends increase yearly, so as to form a series of continual proporti.inals, what will all amount to in 12 years, the last dividend being $2048, and what will be the ratio of the increase ? Ans. ratio, 2 ; sum, $4095. 2. "What is tho ratio, in the series of which tho less extreme is 3 and the greater 98034, and the number of terms 16. Ans. .196605. 3. "What is tho ratio of a series, the extremes of which arc 4 and 324, and the number of terms 5 ? Ans. 3. 4. What is the ratio of a series, the number of terms being 7 and the extremes 3 and 12288 ? Ans. 4. 5. In a series of 23 terms the extremes are 2 and 8388605; what is the ratio ? Ans. 2. To insert any number of means between two given extremes j Find the ratio by Rule (3), and multiply the first extreme by this ratio, and the second icill be obtained, and divide the last by the ratio, and the last but one will be obtained; continue this operation until the required term or terms be procured. Note. —A mean proporlional ia found by taking the sgucre root of the pro- dvxil of the extremes. EXAMPLE. Let it bo required to insert between the extremes 5 and 1280 three terms, so that the numbers constituting the series shall be con- tinual proportionals. The number of terms here is 5, and hence, by Rule (3), we find the ratio to be 4, and 5 multiplied by this will give tho second term, 20, and that again multiplied by 4 will give 80, the third, and that again multiplied by 4 will give tho fourth term, 320, so that the full scries is found to 0. 5, 20, 80, 320, 1280. Tho same result would be found by dividing tho greater extreme by 4, and so on dowuwards, thus : 1280, 320, 80, 20, 5, EXERCISES. 1. Between 5 and 405 insert three terms, which shall make tho ybole a series of continual proportionals. Ans. 5, 15, 45, 135, 405. 2. Insert botweca ^ and 27 four terms to form a series, and give \h'> ratio. Batio, 3 ; series^ ^, J, 1, 3, 9, 27. ^^ ARITHMETIC. 3. What three numbers inserted between 7 and 4375 will form a Berics of continual proportionals? Ans. 35, 175, 875. 4. What is the mean proportional between 23 and 84G4 ? Ans. 441.21G4+. 5. Find u uiean proportional between A 7 and ^. Ans. ^. ALGEBRAIC FORM. Let a represent the first term, I the last, r the ratio, n the num- ber of terms, and s the sum. Then s^:::^a-\-ar-}-ar~-{-ar'^-\-a7''*-\-&c ar'»- - -f-ar**- ' . Multiplying the whole equation by r, wo obtain rs=:ar-{-ar'^ -{-ar''^-\-ar'*-^ar^ -f-^c or**"* -|-M^'* • But S-— a-far+^jy- -^ar'^-\-ar^-\-ur^ -f-^o «r"— ' . Subtracting, we obtain ^ rs — 8=:s(r — l)=ar'^ — a, and therefore '=T=r (!•) But we found the last term of the series to be ar'^*', calling this I, we have from (1.) s=~ (2.) If r is a fraction, r" and ar" decrease as n increases, as already shown under the head of fractions, so that if n become indefinitely great, ar" will become unassignably small, compared with any finite quantity, and may be reckoned as nothing. In this case 1 , will become «^ J=l— -5- (3. > By this formula we can find the sum of any infinite series so closely as to differ from the actual sum by an amount less than any assignable quantity. This is called the limit, an expression more strictly correct than the sum. From the formula 8=^rEj) ^^J three of tho quantities a, r, I, s being given, the fourth can be found. Let it be required to find the sum of the series l+i+i^-r^+ &c., to infinity. Here a^l and r=^ . • . s=i—^=-j^ .--=1 X 2=^2. Thcrefere, 2 is the number to which the sum of the series continually upproachc.«, by the increase of the number of its terms, and is the limit from which it may be made to differ by a quantity lesa than any assignable quantity, and ia ulso the limit })cyrnd which it cau never i>as3. PROGRESSIONS BY RATIO. 309 75 will form 5, 175, 875. 4G4? 441.:nG44-. Adb. 5- n the num- '-\-af -1. .»v— I ' , calling this es, as alrciidy c iadofinitcly ith any finite case ly will nito series so less than any )rcs8ion luoro ities a, r, ?, s Thcrcferc, 2 ly approachc?, ic liniit from iiiy assignable Cr 1)089. '■IB' By adding the first two terms, we find l-j-J=:|=2 — J=1J. By adding the first three terms, we find 4+J=|=2— ^=1|. By adding the first four terms, we find |-f-^=-V-=2 — J==lf' By adding the first five terms, we find •'g'i-+7'g=yg=2 — Jg= By adding the first six terms, we find f J+g'jzs^ =2 — g'3= It will bo observed here that the diflference from 2 is continually decreasing. The next term would differ from 2 by g\, and the next by jAg, &c., &c. Thus, when the series is carried to infinity, 2 may be taken as the sura, because it differs from the actual sum by a quantity less than any assignable quantity. EXAMPLES . To find the sum of the first twelve terma of tlie seiics l*4r3-f 9+ 27+&C. : Here a=l, r=3, And ,='-ir:«_l:l!zl==32«/t«v. To find the sum of the series 1, — 3, 9, — 27, &0., to twelve terms, 11 •8X-a -1 —.IX- 177147—1 -132860. J^— —a— 1 —4 . " In the case of infinite series, if a is sought, s and r being given, wc have from (3) a— 8 (1 — r), and if r ia sought, a and « being given, wc have r=~' or 1 — j. EXERCISES. 1. Find the sum of the series 2, 6, 18, 54, &c., to 8 terms. Ans. 6560. T^2 Observe Ans. §. 2. Find the sum of the infinite series J — J here r=—^. 3. What is the sum of the series 1, ^, ^, &c., to infinity ? Ans. ^. 4. Find the sum of the infinite series 1 — |-|--i— ■jV-h&o. Ans. 3. 5. What is the sum of nine terms of the series 5, 20, 80, &c. ? Ans. 436905. 6. Find the sum of i/i+i-fi/^-f*&C') ^ infinity. Ans. |/^ — 1 . 7. What is the limit to which the sum of the infinite series f , ^, }i |i ^1 continually approaches ? Ans. 4'. 310 AEITHMETTO. 8. What is the sum often terms of the series 4, 12, 36, &c. ? Ans. 118096. 9. Insert three terms between 39 and 31 5. t, so that the whole shall be a series of continual proportionals. Ans. 117, .?/>! and 1053. 10. Insert four terms between ^ and 27, s-o tint the whole shall form a series of eontinual proportionals. Ans. J, 1,3, 9. 11. The sum of a series of continual proportionals is 10^, the first term 3^ ; what is the ratio ? Ans. §. 12. Tho limit of an infinite series is 70, the ratio -^ ; what is the first term ? Ans. 40. ANNUITIES. The v/ord Annuity originally denoted a sum paid annuaUy^ and though such payments are often made half-yearly, quarterly, &c., still the term is applied, and quite properly, because the calculations are made for the year, at what time soever the disbursements may be made. By the term annuities certain is indicated such as have a fixed time for their commencement and termination. By the term annuities contingent is meant annuities, tho com- mencement or termination of which depends on some contingent event, most commonly the death of some individual or individuals. By the term annuity in reversion or deferred, is meant that the person entitled to it cannot enter on the enjoyment of it till after tho lapse of some specified time, or the occurrence of some event, gejiui- cilly the death of some person or persons. An annuity in perpetuity is one that " lasts for ever,'* uud there- fore IS a species of hereditary property. An annuity forborne is one the paymsnts of which have nofc been made "when due, but have been allowed to accumulate. By the amount of an annuity is meaut the sum that the principal and compound interest will amount to in a given time. The present worth of an annuity is the sum to which it would amount, at compound interest, at the end of a given time, if forborne for that time. Tables have been constructed showing tho present and final valaes per unit for different periods, by which tlio value of any annuity may bo found according to the following 36, &c. ? ns. 118096. ;it the wholo 1 and 1053. whole shall 3. J, 1,3,9. 3 is 10^, the Ans. ^. j y?hut is the AnH. 40. nnualli/i and !rly, &c., still culations are ticnta may he ; hoAe a fixed ies, the com- ic contingenli ndividuals. leant that the t till after the event, gcwer- r,'" and therc- ich liavQ nofe ilate. t the principal rhich it would no, if forborno lent and final value of any ANNUnTEg. 311 RULES. To find either the amount or the present value of an annuity,— MuUij)li/ the value of the unit, a« fomd in the tables, by th€ number denoting the annuity. If the annuity be in perpetuity, — Mvide the annuity by the number dmoHny the iitterest o/ (he imityor one year. If the annuity bo in reversion, — Mnd the value of the unit up to the date of commencementj and also to the date of termination, and multiply their difference by the number denoting the annuity. To find the annuity, the time, rate and present -.vorth beJDg given. Divide the present worth by the worth of the unit. Tables are appended varying from 20 to 50 years. EXAMPLES. To find what an annuity of $400 will amount to in 30 years, at 3J per cent. We find by the tables the amount of $1, for 30 years, to be $51,622677, which multiplied by 400 gives $20649.07 nearly. To find the present worth of an annuity of 0100 for 45 years, at 3 per cent. Bj the table wo find $24.518713, and this multiplied by 100 gives $2451.88. To find the present worth of a property on lease for ever, yielding $600, at 3^ per cent. The rate per unit for one year is .035, and 600 divided by this givc3 $17142.86. To find the present worth of an annuity on a lease in leversion, to commence at the end of three years and to last for 5, at 3^ per cent. By the table we find the rate per unit for 3 years to be $2.801637, and for 8 years, the time the lease expires, $6,873956; the differ- ence is $4.072319, which, multiplied by 300, gives $1221.70. Given $207.90, the present worth of an annuity continued for 4 years, at 3 per cent., to find the annuity, By the tables, the value for $1 is $3.717098, and $207.90, divided by this, gives $55.93. w^ 312 ABITIIMETIO. TABLE. SBOwiMO Tna amount of am ^NNMrrrr of oxb voUiKR p£K ji^Vii ntrsovEO AT COMPOUND INTEREST FOR ANY KCMBEB OF TEARa NOT KXCEEDINO FIFTT. I ill 1 1 3 per cent. 3,} percent. 4 per cent. 5 per cent. l.OOJ OON 6 per cent. I per cent. 1 1.000 000 1.000 000 1.000 000 1.000 OOO 1.000 001 2 2.030 000 2.035 000 2.010 000 -J.OJOOOO 2.060 000 2.070 OOC 2 3.090 900 3.1O0 1'2') 3.121 60!;! 3.152 500 3. '83 600 3.214 90C 4 4.183 627 4.214 913 4 246 464 4.310 125 4.o74 616 4.439 942 5 5.309 136 6.:;02 4GC 5.416 323 6.525 G31 5.637 093 6.750 73J 6 6.468 410 6.550 152 6.r,.T2 975 6.801 913 6.975 31p 7.153 291 7 7.662 462 7.779 408 7.81)8 294 8.142 008 8.393 838 8.054 021 8 8.892 330 9.051 687 9.214 226 9.549 109 9.897 468 10.259 803 9 10.159 106 10.368 406 10.^82 795 11.026 564 11.491 316 11.977 98Q 10 11.463 R79 11.731 393 12.006 107 12.577 893 13.180 795 13.816 448 11 12.807 796 13.141992 13.486 351 14.206 78t 14.971 643 15.783 69;i 12 14.192 030 14.601 962 15.025 805 15.917 IZV 16.869 941 17.888 451 13 15.617 790 16.113 030 16.620 838 17.712 983t 18.882 13tJ 20.140 643 U 17.086 324 17.676 98( 18.291 911 19.598 632 21,015 066 22.550 48i: 15 18.598 914 19.295 681 20.023 58.8 21.824 531 21.578 564 23.275 970 25.129 022 16 20.156 881 20.971 030 23.657 492 25.670 628 27.888 054 17 21.761 688 22,705 016 23.697 512 25.840 366 28.212 880 30.840 217 18 23.414 435 24.499 621 ?5.645 413 28.132 385 30.905 653 33.999 033 19 25.116 866 26.357 180 2/.671 229 30.539 004 33.759 992 37.378 965' 20 26.870 374 28.':;G 682 29.778 079 33.065 954 36.785 591 40.995 492 21 28.676 486 30.269 471 31.969 202 35.719 252 39.992 727 44.865 177 22 30.536 780 32.328 1)02 34.247 970 38.505 214 43.392 290 49.005 739 23 82.452 884 34.460 414 36.617 889 41.430 475 46.995 828 53.430 141 24 34.426 470 36,666 628 39,082 604 41.501 999 50.815 577 58.176 071 25 36.459 264 38.949 857 41.645 908 47.727 099 54.804 612 63.249 03IJ 26 38.553 012 41.313 102 44.311 745 61.113 454 69.156 383 68.67C 471) 27 40.709 631 42.759 060 47.084 214 64.669 126 63.705 766 74.483 82^. 28 42.930 923 46.290 627 49.967 583 68.402 583 68.528 112 80.697 69 ;i 29 4.5.218 850 48.910 799 52.966 286 62.322 712 73.639 798 87.340 62L' 30 47.575 41C 51.622 677 56.084 938 66.438 848 70.058 186 94.460 780 31 60.002 678 54.429 471 59.328 335 70.760 790 84.831 677 102.073 041 32 62.502 759 57.334 502 62.701 469 75.298 829 90.8S9 778 110.218 154 33 65.077 841 60.341 210 66.209 527 80.003 77] 97.343 105 118.933 425 34 67.730 177 63.453 15. C9.857 909 85.0G6 95!) 104.183 755 128.258 765 35 60.4G2 082 6G.674 013 73.652 225 90.32.0 307 111.434 780 138.236 87t 36 63.271 914 70.007 603 77.598 314 95.836 323 119.120 867 148.913 400 37 66.174 223 73.457 869 81.702 216 101.628 139 127.268 119 160.337 400 38 69.159 449 77.028 895 85.970 336 107.709 54C 135.904 200 172.561 020 39 72.234 233 80.724 906 90.409 150 114.095 023 145.058 458 185.040 292 40 75.401 260 84.550 27S 95.025 51(; 120.799 771 154.701 906 199.035 112 41 78.663 298 88.509 537 99.826 53C 127.830 76."^ 165.047 681 214.009 570 42 82.023 196 92.607 371 104.819 598 135.231 751 175.950 615 230.632 240 43 85.483 892 96.848 629 110.012 382 142.993 339 187.507 577 247.776 490 44 89.048 409 101.238 331 11.5.412 877 151.143 OOC 199.758 032 26G.120 851 45 92.719 861 105.781 673 121.029 392 159.700 15C 212.743 514 285.749 311 4G 96.501 457 110.484 031 126.870 568 168.685 IG"^ 22r,.508 125 300.751 763 47 100.396 501 115.350 973 132.945 390 178.110 422 241.098 612 329.224 386 48 104.408 396 120.383 297 l.^>9.263 206 188.025 39L i5G.564 629 353.270 093 49 108.540 C4& 125.601 84G 145.833 734 198.426 663 272.958 401 378.999 COO 50 112.796 867 130.999 91C 152.667 084 209.347 97G 290.335 905 406.528 929 ini niTBovEO !ED1N(J FIFTT, / per cent. 1, 2. 3, 4, 6 7 8 10, 11, 13, 15, 17 20 22 25 27 30. 33, 37, 40 44, 49, 53 58 C3, 68, 71, 80, 87, 94, 102 110 118 128, IC8, 148, I GO 172 185 199 214. 200. 217. 2 GO. 285, ;')0G, 329 353 378 406 ,000 001 .070 OOC ,214 90e .439 94c .750 73f .153 291 .054 021 .259 803 .977 98Q ,81G 44fc ,783 59:1 .888 451 .140 043 .550 4Sh .129 022 .888 054 ,840 217 .999 033 ,378 9C5 .995 492 ,865 177 .005 739 .436 141 ,17G 071 .249 030 .67G 471; ,483 82-;. .697 09 ;i .34G 52L' .400 780 .073 041 154 .933 42!; ,258 7C5 .236 87fc ,913 460 .337 400 ,561 020 .640 292 .G35 112 609 570 Joli 240 77G 490 120 851 749 311 ,751 763 ,224 386 ,270 093 .999 000 ,528 929 ANNUITIES. TABLE, . 3ia 311,)WIN0 TIIE PIIE3EKT WOHTH OF AN AXsmTT OV O.VB POLUTl TEK ANNUM, TO CONTINTTE VOH ANT %'UMBER OF YEAB.S NOT ESCEEDIXO FIFTT. 1 3 por coDt. 3J per cent 4 per cent. .) per cent. por cent. 7 per cent. 0.970 H7-\ 0.9CG 184 0.961 538 0.952 P,?>1 0.943 390 0,934 579 1 2 1.913 4V0 1,899 694 1.8S0 095 L859 410 1.833 :;';) 1.808 017 2 3 2.8.8 Oil 2.801 637 2.775 091 2.723 248 2.073 012 2.024 31! y 4 3 717 098 3.073 079 3.029 895 3.545 951 3.405 IOC 3.387 20L' 4 5 4.579 707 4.515 052 4.451 822 4.329 477 4,212 304 4.100 19.5 r, 6 5.417 191 5.328 653 6.242 137 6.075 092 4.917 32 J 4.70G 537 7 0.230 283 0.114 644 0.002 055 5.786 373 5.5S2 381 5.389 280 4 8 7.019 092 0.873 956 0.732 745 0.403 213 0.209 744 5.971 295 8 9 7.786 109 7.G07 087 7.435 332 7.107 822 0.831 €92 0.515 228 U 10 8.530 203 8.316 605 8.110 890 7.721 735 7.300 087 7.023 577 10 11 9.252 024 9.001 551 8.760 477 8.306 414 7.8S0 87:; 7.498 009 tl 12 9.954 004 9.663 334 9.385 074 8.!:03 252 8.383 844 7.942 071 12 13 10.634 955 10.302 738 9.985 048 9.393 673 8.852 OSL 8.357 035 Hi 14 11.20G 073 10.920 520 10.503 123 9.898 041 9.294 9Si 8.745 452 12 15 ll.'Ju7 935 11.517 411 11.118 387 10.379 C58 9.712 24'J 9.107 89 r, 15 16 12.561 102 12.094 117 11.652 290 10.837 770 10.105 89.-^ 9.4 4'J 032 It; 17 13.1G6 Ufi 12.651 321 12.1G5 069 11.274 060 10.477 200 9.703 20C 17 18 13.753 513 13.1^9 682 12.059 297 11.089 687' (.0.827 COu 10.059 070 IC 19 14.3:3 799 13.709 837 13.133 939 12.085 321 11.158 110 10.335 678 19 20 14. .'77 475 14.212 403 13.590 326 12.462 210 11.409 421 10.593 997 20 21 15.415 024 14.097 974 14.029 100 12.821 153 11.704 077 10.835 527 21 22 15.936 917 15.1G7 125 14.451 115 13.103 003 12.041 582 ll.OGl 241 22 23 1G.443 608 15.020 410 14.S5G 842 13.488 674 12.303 379 11.272 187 23 24 10.935 542 10.058 3GS 15.246 963 13.798 042 12.550 358 11.409 331 24 25 17.413 148 16.481 515 15.022 080 14.093 945 12.783 350 11.053 583 25 26 17.876 842 16.890 352 15.982 7G9 14.275 185 13 003 IOC 11.825 779 20 27 1S.327 031 17.285 365 10.329 580 14.043 034 13.210 534 11.986 709 27 28 18.7C4 108 17.0G7 019 1G.G63 0G3 14.898 127 l.T 406 104 i;?.l37 111 28 29 19.188 455 18.035 767 16.983 715 15.141 074 13.590 721 12.277 G74 2 832 13.005 522 45 40 24.775 440 22.700 018 20.884 054 17.880 067 15.524 370 13.050 020 40 47 25.0 2 i 708 22.899 438 21.042 930 17.981 016 15.689 028 13.091 008 47 48 25.266 707 23,091 244 21.195 131 18.077 158 15.050 0271 13.7:50 474 48 49 25.501 657 23.276 564 21,341 472 18.1G8 722 15,707 672 13.706 799 49 50 25.729 764 23.455 618 21.482 185 18.255 925 15.7G1 861 13.800 746 50 314 ARITHMETIO. PARTNERSHIP SETTLEMENTS. The circumstances under which partnerships are formed, the conditions on which they arc made, and the causes that lead to their dissolution, arc so varied that it is impossible to do more than givo general directions deduced from the cases of most common occur- rence, lu forming a partnership, the great requisite is to have the terms of agreement expressed in the most clear and yet concise lan- gua-ic possible, setting forth the sum invested by each, the duration of partnership, the share of gains or losses that full to each, the sum that each may draw from time to time for private purposes, and 4iny other circumstances arising out of the peculiarities of each case^ The ease and satisfaction of making an equitable settlement, in casa -of dissolution, depends mainly on the clearness of the original agree- ment, and hence the necessity for its being distinct and explicit. Even when no dissolution is contemplated, settlements should ha frequently made, in order that the parties may know how they stand to each other, and how the business is succeeding. This is of great importance in preserving unanimity and securing vigour and regu- larity in all the transactions of a mercantile house. A dissolution may take place from various causeg. If the part- nership is formed for a term of years, the expiration of those years necessarily involves either a dissolution or a new agreement. Tho death of one of the partners may or may not cause dissolution, for the deceased partner may have, by his will, left his share in the business to his son^ or some other relative or friend. In no case, however, can an equitable settlement bC made, except by the mutual consent of the parties, or else in exact accordance with the terms of agreement. It is also necessary that when a dissolution takes place public notice should be given thereof, in order that all parties having dealings with tho firm may bo apprized of the change, and have their accounts arranged. For the same reason, it is necessary that some one of the partners, or some trustworthy accountant appointed by them, should be authorized to collect all debts due to the firm, and pay all accounts owing by it. Partnerships are sometimes formed for a specific speculation, and therefore, of course, cease with tho completion of the transaction, and a settlement must necessarily be then made. No matter for what PiiRTNEESHIP SETTLEMENTa, 315 TS. brmccl, the !ad to their than givo mon occur- to have tho concise lan- hc duration each, the jrposes, and )i' cacli case^ icnt, in casa iginal agrce- ,nd explicit. ;s should Bd jf they stand s ia of great ir and regii- If the part- those years iment. Tho isolution, for itirc in the In no case, ^ the mutual the terms of a takes place arties having nd have their ry that some ippointcd by the firm, and culation, and nsactioD, and iter for what time tho partnership has been made, any partner is at liberty, at any time, to withdraw, on showing sufficient cause and giving proper notice. This is a just provision, for the circuiustanccs of any part- ner may so change, from various cauics, as to make it undesirable for him to remain in the business. If one partner is deputed to Bcttlc tho accounts of the house, it would be reckoned fraudulent for any other partner to collect any moneys due, except that on receipt of them he hands them directly over to the person so deputed. The resources and liabilities, with tho net investment on coii]- luencing business, being given, to find the net gain or loss. i. W". Smith and R. Evans are partners in business, and invested when commencing $1000 each. On dissolving the partnership, tho assets and liabilities are as follows : — Merchandise valued at §1295 ; cash, $344 ; notes against sundry individuals, $790 ; W. II. Monroe owes on account $86.40 ; E. R. Carpenter owes $132.85, and C. F. Musgrovc owes $"07.50. They owe on sundry notes, as per bill book, 6212.40 ; E. G. Conklin, on account, $29.45, and H. C. Wright, on account, $41.30, What has been the net gain ? SOLUTION. Assets. Merchandise on hand...S1295.00 Cash on hand ' 344.00 Bills iioccivablc 790.00 Anit. due from W. 11. Monroe 8G.40 Amt. due from E. R. ■Carpenter 132.85 Aint. due from C. F. Musgrovc 07.50 Total amount Assets . . . .$2715.75 " '' Liabilities, 2283.15 Net gain LialUitics, Bills Payable $212.40 Amt. due E. Cr. Conklin. 20.45 Amt. due II. C. WrI-ht. 41.30 W. Smith's investment... 1000.00 ll» Evau'a inveatinent.... 1000.00 S2283.15 ....$432.60 RULE. Find the sum of tJie assets and UahiUties I from the assets sxibtract the liabilities, (including the net amount invested) and the differ- ence loill be the net gain ; or, if the UabiUties be the larger, subtract the assets from the liabilities, and the difference will be the net loss. 316 ARITHMETIC. 2. Harvey 3Iillcr and James Carey arc partners in a dry goods business; Harvey Miller investin;^ $1100, and James Carey $1230. Wliea closing the books, they liavc ou hand — cash, §1125.30; mcr- chandiso us per inventory book, $1855.75; amount deposited in First National Bank, $1200; amount invested in oil lands, $9G3; a site of land lor buildin_L^ purposes, valued at §1000 ; Adam Dudgeon owes them, o:i account, .$101.92 ; William Fleming owes $210.80 , a noto against xVlfred 3Iills for $00. -IJ, and a duo bifl for $3.), drawn by Jatucs Laing. They owe \V. S. Hope & Co., on account, $840.21 ; n. J. King & Co., $003.12, aud on notes, $1320.14. What has been the net gain or loss ? Ans. $1701.73 gam. 3. James iJcnning and Adam Manning have formed a co-part- nership for the purpose of conducting a general dry goo'ls and grocery business, each to share gains or losses equally. At the end of one year they close the books, having $1280 worth of merchandise on hand ; cash, $711. 27 ; Girard Bank stock, $500 ; deposited in Merchants' Bank, $320.00 ; store and lixiurcs valued at $3100; amount due on notes and book accounts, $3171.49. Tho firm owes on notes $3400, and on open accounts §747.10. James Ilcnning invested $1200, and Adam Manning, $1000 ; what is each partner's interest in tho business at closing ? Ans. James Ilcnning's interest, .$2719.03. Adam Manning's interest, $2519.03. Note.— Whcro tho interest of each partner at closing isrotinlrecl, the gam or loss is first found, as in former examples, then tho share of gain or loss is added to or subtracted from each partner",-! investment, and the sum, or differoace, is the interest of each partner. If a partner has withdrawn any- " thinjj from the business, tho amount thus withdrawn must bo deducted trom tho Slim of his investment, pjtis his sharo of tho ff.vin, or minus his share of tho lo.s3, and the remainder will bo his not capital or interest. 4. F. A. Clarke, W. II. Marsden, and J. M. Musgrove, arc con- ilucting business in partnership ; F. A. Clarke in to be ^ gain or l')ss, W. n. Marsden and J. M. Musgrove, each J. On dissolving the partnership, they have cash on hand $712.90 ; merchandise as per Inventory Book, $4300 ; bills receivable, as per Bill Book, $1450.75 ; amount deposited in Third National Bank of Syracuse $3475 ; merchandise shipped to Richmond, to be sold on own account and risk, valued at $095; debts due from individ- uals on book account, $2044.07. They owe on notes $3700, and to Manning and Munson, 81312.G0. PARTNERSHIP SETTLEMENTS. 6n dry goods cy ei230. 1.30 ; mcr- >(! in First )G3 ; a sito Igcon owc» 30 , a noto drawn by , $810.21 ; NVliat has G1.73 gain. id a co-part- coods and J12S0 worth itock, 8500 ; :c6 valued at ri.40. Tho .0. ling, $1000 ; 2? Mauuing's Hired, the gam )f gain or loss d the sum, or ithdrawa any- leducted trom bis share of rove, arc con- be ^ gain or aud $712.90 ; •ivablc, as per ional Bank of to be sold on from individ- $3700, and to V. A. Clarke invested $5750, and has drawn out $875 ; W. H. Marsden invested $2500, and has drawn out $500 ; J. M. Musgrovo invested 83000, and has drawn oiit $750. What has been tho not gain or loss, and what is each partner's interest in tho business ? Ans. Net loss, $559.28; F. A. Clarke's interest, $4595.36 ; W. II. Marsdcn's interest, $1800.18 ; J. M. Musgrovo'a inter- est, $2110.18. Note.— la this ivnd succeeding examples, no iutorest is to bo allowed on iavestmeat, or churgec' on amouats withdrawn, unless so speciflod. 5. A, and C aro partner-). Tho gains and losses are to bo shared as follows : A, {\ ; B, -i\ ; and 0, j% A invested $3000, and has withdrawn $2500, with tho consent of B. and C, upon which no interest is to bo charged ; B invested $2700, and has withdrawn $1150 ; C invested $2500, and has withdrawn $420. After doing business 14 months, C retires. Their assets consist of bills receivable, $2937.20 , merchandise, $1970 ; cash, $1240.80 ; 50 shares of the Chicago Permanent Building and S vings' Society Stock, the par value of v aich is $50 per share ; cash deposited in tho Third National Bank, $1850; store and furniturC; $3130; amount due from W. Smith, $360.80 ; 0. S. Brown, $240.40; and E. R. Carpenter, $97.12. Their liabilities aro as follows : Amount duo Samuel Harris, $1075 ; unpaid on store and furniture, $933 ; and notes unredeemed, $3388.76. Tho Savings' Society stock is valued at 10 per cent, premium, and C in retiring takes it as part payment* What is the amount due C, and what is A's, and what is B's interest in the business ? Ans. Due C, $315.52; A's interest, $2356.90; B's interest, $2664.14. 6. E, F, G and H aro partners iii basiness, each to share J of profit and losses. Tho business is carried on for ono year, when E and F purchase from Gr and H their interest in tho business, allow- ing each $100 for his good will. Upon examination, their resources arc found to bo as follows : Cash deposited in Girard Bank, $3045 ; cash on hand, $1422 ; bills receivable, $1085 ; bonds and mortgages, $2746, upon which there is interest due $106 , Metropolitan Bank stock, $1000; Girard Bank stock, $500; store and fixtures. $3500 ; house and lot, $1800 ; span of horses, carriages, harness, &c., $495; outstanding book debts due the firm. $4780. Their liabilities uic : Notes payable, $2345 ; upon whi.oh there is interest due. $5-2 : due on book debts, $1560, B invested $5000 ; F $4500 ; 318 ARITHMETIC. G, a-lOOO ; mid II, i<3000. E ha.s drawn IVoin the bosi -; 51200, upon which ho owc.-i interest $32 ; F haa drawn ^1000 — owes interest $24.50 ; O has drawn 6950— owc3 interest $12 ; and II has drawn nothing. In the settlement a discount of 10 p-or cent., for bad debts, is allowed, on the book debts duo the firm and on the bills receivable. O takes the Metropolitan Bank stock, allowing on tho same a pre- mium of T) per cent. ; and II takes tho Girard Bank stock, at a ITTcmium of B per cent. ; E and F take tho assets and assume tho liabilities, as abovo stated. What has been tho net gain or loss, tho balances duo G rmd II, and what arc E and F each worth after tho settlement ? Ans. Duo G, $3057.75; duo IT, $3520.75; E'rnct capital, $4037.75; F's not capital, $4345.25^ 7. H. C. Wright, W. S. Samuels, and E. F. HalT, aro doing business together — II. C. W. to have ^ gain or loss ; W. S. S. aD(7 E. P. II. each ^-. After doing business one year, W. S. S. and, E. P. II. retii-c from tlic firm. On closing the books and taking stock, tho following is found to be tho result : merchandise on hand, $3210-50; cash deposited in Sixth National Bunk, 81027.35; cash in till $134.10 ; bills receivable, $940.00 ; G. Brown owes, on ac- o-ount, $112.40 ; Tho3. A. Bryce owes 89412 ; AV. McKeo owes $143.95 ; J. Anderson owes $54,20 ; R. II. Hill owes $43.00 ; and S. Graham owes $250.13. They owe on notes not redeemed $1864 ; H. T. ColUus, on account, $124.45; and W. F. Curtis, $79.40. II. C. Wright invested $3200, and has drawn from the business $350. W. S. Samuels invested $2455, and has drawn $140; E. P. Ilall inTcstcd $2100, and has drawn $2000. A discount of 10 per cent, is to bo allowed on the bills receivable and book accounts duo the firm for bad debts. II. C. Wright takes tho assets and assumes tho liabilities as abovo stated. What has been tho net gain or loss, and what does H. C. Wright pay W. S. Samuels and E. P. Ilall on retiring ? 8. T. P. Wolfe, J. P. Towler and E. K. Carpciifer havo been doing business in partnership, sharing the gains and losses equally. After dissolution and settlement of all their liabilities they make a division of the remaining effects without regard to the proper pro- portion each should take. The following is the result according to their ledger :— T. P. Wolfe invested $3495, and has drawn $2941 ; J, P, Towler invested $2900^ and has drawn $2200 ; E. E. Carpenter PiUlTKLinSHIP BETTLEMENTS. 3rJ -■i .^200, wcs interest has drawn r bad debts, 3 receivable. same a prc- Btock, at a assume tho or loss, tho rtU after tho ' net capital, IT, nxo dain<5 ; W. S. S. xr, W. S. S. :3 and taking Use on hand, ;27.35; cash owes, on ac- McKcc owes $43.G0 J and mcdS1861; rtis, $70.40. iisiucss $350. E. P. Hall 10 per cent, ccounts dao and assumes 10 net gain 3 and E. P. havo been )sse3 equally, they make a 5 proper pro- according to awn $2941 ; &, Carpenter invested $3150, and has drawn $3000. How will tno partncrd Bettlo with each other ? Ana. E. R. Carpenter pays T. P. Wolfe 880, and J.P. Towler$233. 9. I, J, K, L and M Invo entered into co-purtncrrfhip, a^'recing to share tho gains and losses in tho followinj; proportion : — I, ^\ ; J, A I ^) f'i i ^1 t:i i ^"*^ ^^) i\' ^Vhon didsalvin;^ tho partnership tho resourced consisted of c:it!h $1700 ; morciianJiso, $9355 ; notes on hand §7030 ; debentures of tho city of Albany valued at §0780, on which thorj U iutorejt duo, 812.) ; hordes, wa^j;oas, &c., §1230 ; Merchant's bank stock, 85000 ; First National bank stock, $5000; mortgages and bonds, $3000 ; interest duo on raortga-^os, §345.80 ; store and fixtures, §3000; amount duo from W. P. Campbell & Co., §2418 J due from R. B.Smith, §712.00; duo from J. W. Jones, §1000. Tho liabilities are : — Mortgage on store and fixtures, §5000 ; interest duo on the same, §212.25 ; duo tho estate of 11. 31. Evans, $14675 ; notes and aocoptanccs, ;;-ill040, on whicli interest is duo, §85 ; sundry other book debts, $7500 ; I invested §7800, interest on his investment to date of dissolution, §702 j J invested !)G400, interest on investment, §570; K invested §3100, intorost on invest- ment, §549 ; L invested §5300, interest on invostmcnt, §522 ; M invested §5000, interest on investment, $450. I has withdrawn from tho firm ut dilFerent times, $2425, upon which the interest calcu- lated to time of dissolution is §183.40 ; -T has drawn §2900, interest, §267.85; K has drawn §1850, interest §37.30; L has drawn §3000, interest, §460 ; M has drawn $895, interest, §63.45. What is tho not gain or loss of each partner, and what is tho net capital of each partner ? Aus. I's net loss, §1233.29 ; I's net capital, §4660.31. J's net loss, $924.97; J's net capital, §2323.18. K's net loss, §616.65 ; K's net capital, §4095.05. L's net loss, §1541.02 ; L's net capital, §1320.38. M's net loss, §308.32 ; M's net capital, §4183.23. 10. A, B, C and D are partners. At the tfmo of dissolution, and after tho liabilities are all cancelled, they make a division of tho effects, anl upon examination of their ledger it shows tho following result : — A has drawn from tho business §3405, and invested on commencement of business, §1240 ; B has drawn §4595, and invested $3800; C has drawn §5000, and invested §3200; D has drawn §2200. and invested S2800. Tho profit or loss was to bo divided in 320 ARITHMETIC?, proportion to thu oviginal investment. What has been each partner's gain or loss, and liow do the partners settle witli each other ? Ans. A'3 net gain, $3G8.43 ; B's net gain, $330.20 ; C's net gain, 8278.0G ; D's net gain, $2i3.31. B has to pay in «:4C-4.80 ; C has to pay in 81521.94. A receivoa $1143.43; D receives 8843.31. 11. Tlircc mechanics, A. W. Smith, James Walker and P. Ranton, i\rc equal partners in their business, with the understanding that each is to be charged 81.25 per day for lost time. At tho close of* their business, in tho settlement it was found that A. W. Smith hud lost 14 days, James AVaker 21 days, and P. Ranton 30 days. Iloiv shall the partners properly adjust tho matter between them? Ans, P. Ranton pays A. W. Smith, 89.58J, and James Walker, 831- cents. 12. There mo 5 mechanics on a certain pieco of work in the following proportions : —A is -^'^J ; B, rfj ; 0, -^^^ ; D, jjij, and E, ^^q. A is to pay 81.25 per day for all lost time; B, 81 ; C, $1.50; D, $1.75, and E, $1.G2^. At settlement it is found that A has lost 24 , B, 19 ; C, 34 ; i), 12 ; and E, 45 days. They receive in pay- ment for their joint work, 82500. What is each partner's share of this amount according to tho above regulations ? Ans. A's share, 8374.12 ; B's, S250.41 ; C's, 8487.83 ; D's, 8787.24; E't<, 8000.40. 13. A. B. Smith and T. C. Musgrovo commenced business in partnership January 1st, 1864. A. B. Smitl: invested, on com- mencement, 89000; May 1st, $2400; Juno 1st, ho drew out 81800; September 1st; $2000, and October 1st, ho invested $800 more. T. C. Musgrovo invested on commencing, $3000 ; March 1st, ho drew out 81000 ; May 1st, 81200 ; June 1st, ho invested 81500 more, and October 1st, 8SCC0 more. At tho time of settlement, on the 31st December, 18G4, their mcrchandiso account was — Dr. $32000 ; €>'. $27000 ; balance of mcrchandiso on hand, a^ per inventory, 810500 ; cash on hand, 84900 ; bills receivable, $12400; R. Draper owes on account, 82450. They owe on their notes, $1890, and G. Roe on account, 8840. Their proiit and loss account is, Dr. 88GG ; Cr, 81520. Expense account is, Dr. $2420. Com- mission account is, Cr. 827G0. Interest account is Dr. $480 ; Cr. $950. Tho gain or loss is to bo divided in proportion to each parluer'a cupital, and in proportion to tho time it was invested. Heauircd each partner's share of the gain or loss, the uct balanoc PARTNERSHIP SETTLEMENTS. 321 5h partner's icr? ; C's net IS to pay in 161143.43; :er and P. dcrstanding A.t the close .. W. Smith Lon 30 days, en them? nes Walker, work in the , and E, i^. ), $1.50; D, t A has lost leive in pay- ler'a share of 87.83; D's, business in ;ed, on comr r out $1800; $800 more, arch 1st, ho stcd $1500 3ttlcment, on \t was — Dr. land, as per )lc, $12400; their notes, loss account J420. Com- . $480 ; Cr. ■tion to each vas invested, uct balauoc due each, and u ledger specification exhibiting the dosing of all the accounts, and the balance sheet. Ans. A. B. S.'s net gain, $6671.73; his net balance, $15071.73. T. C. M.'s net gain, $2748.27 ; his net balance, $12448.27. 14. A, B, C, and D commenced business together on July Ist, 1865, with the afz;rccnicnt that all gain or loss is to be shared equally by each partner, but that interest at the rate of 6 per cent, per an- num is to be allowed on each one's investment, and the same rate charged on all amounts withdrawn by each. A is to manage the business, having a salary of $2,000 per year, payable half-yearly The services of B, C, and D arc not required in the business. The assets are, on commencing. Cash $7440 ; Mdse. $9586 ; Bills Re- ceivable, estimated value $4976.00 (face value $5237.89) invested by C. Per. accounts Dr. $12271.40 (estimated value, 10 per cent, discount) invested by A. Of the assets, there belongs to A $13492.40, B, $6000, C, $5750, and to D $5000. Personal ac- counts Cr. $4292.89. Aug. 20th, A drew cash $75, B $90. Sept. 4th, D drew cash $125. Sept. 30th, A drew $200, C $80. Nov. 20th, B drew $100, D, $50. Dec. 24th, A drew $150. Feb. 27th, 1800, A drew $200, C $150, D $100. May 12th, A drew $200, D $200. June 13th, B drew $150, C $100. July 1st, B made a further investment of $1C00, C, $1500, and A drew $400, D $100. Sept. 15th, A drew $150, B $500, C,$750. Nov. 1st, A drew $100, J) $75. Dec. 31st, 1866, the books are closed and the partnership dissolved, C and D retiring from the business, being allowed by the remaining partners $150 each for their good-will in the business. Before calculating the interest on the partners' investments; and on the amounts withdrawn by them, and allowing A the amount of his salary from time of commencement up to date, the assets and liabili- ties arc as follows: Cash $9483.50; Mdse. $14675; Bills Receiv- .nblc $0219.85. Personal accounts Dr. $7694.30. Inventory of Mdse. consigned to W. Smith, New York, to be sold on our account and risk, $1265.12 ; 50 shares N. Y. C. R. R, stock, valued at 105; 50 shares Eric R. R. stock at 69. They owe in Bills payable $5057.45. Personal accounts Or. 83272.94 ; also, Samuel Zimmer- man for rent to date $1250, In the settlement a discount of 5 per cent, on the Bills Receivable, 15 per cent, on the Dr. personal no- 822 ARITHMETIC. counts, and 10 per cent on the Mdse. shipped to New York, is al- lowed for loss in bad debts. C agrees to take the N. Y. C. R. R, stock at 105, in part payment of the amount duo him; D takes tho Erio R. R. stock at 69 ; C receives cash $1200, and D $1800. What has been the whole gain or loss, the amount still due C and D, the amoun*^^ of cash on hand when the books arc closed up, and wliat are A and B each worth, tho Bills Rec, personal accounts, and con- signment to N. Y. being valued at par ? Ans. Gain $3370.45; due C $959.72; due D $512.96; caslton hand $6483.50; A's capital $16401.47; B's net capital $8183.21. 15. A book-keeper applied at our College for counsel, not long since, to settle the following accounts, between two partners, Jan. 1st, 1865. We'll call them Mr. E. and Mr. F., each J gain or loss. Books were kept by single entry, and the accounts and inventory were as follows :— Cash in Bank $5,705. Do. in office $6,000. Bills Rec. on file $4,921.33, upon which there was interest duo $78.67. Mdse. unsold $4,000. Propeller Toledo account, (techni-. cal) dr. bal. $6,210. Their shares in the boat valued at $5,000. E drew out $1,010. F drew out $3,339, FiXpense account, dr. bal. $1,335, Mdse. account, dr. bal. $210. Bills payable, per B. B., $4,564. Rev. 0. Burger, cr. bal. $200. D. C. Weed & Co. di bal. $2,000. Shepard & Cottier, cr. bal. $300. In't and Dir't ac- count, cr. bal. $1,524. Joint account vath A. L, Griffen & Co. each J, net gains were $872. D. P. Dobbins & Sou, dr. bal. ,|1,000. What are E. and F. each worth? QUESTIONS FOR COMMERCIAL STUDENTS. 323 York, is al- Y. C. R. R, D takcy tho i D $1800. lue C and D, ap, and wlmt its, and con- .96; caslton il $8183.21. isel, not long artners, Jan. r gain or loss. nd inventory Dffico 8G,000. i interest duo aunt, (teclini- led at $5,000. jount, dr. bal. )lc, per B. B.. eed & Co. dx iand Dir'^' ac- en & Co. each bal. $1,000. QUESTIONS FOR COMMERCIAL STUDENTS. 1. The following questions may be found interesting and instructive to young men preparing for the practical duties of accountants. On the 1st of May I purchased for cash, on a commission of 2^ per cent., and consigned to Ross, Winans & Co.. commission mer- c'.iauts, Baltimore, Md.; 380 bbls. of mess pork, at $27.50 per bbl., to bo sold on joint account of himself and myself, each one half. Paid shipping expenses, $7.40. July 7th, I received from Ross, Winans & Co. an account sales showing my net proceeds to be $5319,79, due as per average, August 12th. August 8th, I draw on them at sight for the full amount of their account, which I sell at ^ per cent, discount for cash ; interest 7 per cent. "What amount of money do I receive and what are tho journal entries ? 2. B. Empey, a merchant doing business in Montreal, Canada East, purchased from A. T. Stewart, of New York city, on a credit of three months, the following invoice of goods : 845 yds. Fancy Tweed, @ $1.90 per yd. 1712 " Amer. black broadcloth, @ 3.85 " " 423 " Blue pilot, @ 2.75 '• " 700 " Black Cassimcre. 2.10 " ' \Vhen the above goods were passed through tho custom-house, iv discount of 27J per cent, was allowed on American invoices ; duty 25 per cent., freight charges paid in gold, $29.35. What must each, piece be marked at, per yd., to sell at a net profit of 15 per cent, on full cost ? What would be the gain or loss by exchange, if at the expiration of the three months B. Empey remitted A. T. Stewart, to balance account, a draft on Adams, Kimball & Moore, bankers. New York city, purchased at 32J per cent, discount, and what arc the journal entries ? 3. I purchased for cash, per the order of J. P. Fowler, 70 boxes 0. C. bacon, containing on an average 400 lbs. each, at 13^ cents 324 ■ ARlTHMEnO. per lb., and 140 firkins butter, 8312 lbs., at 17J cents per id., on a commission of 2^ per cent ; paid shipping and sundry expenses in cash $13.40. For reimbursement I draw on J. P. Fowler at sight, "which I bc'll i.» the bank at ^ per cent, discount j what is the face of draft, ^nd wiMt arc the journal entries ? Ans. Face of draft $5479.05. 4. Sept. 27th, I received from James Watson, Leeds, England' u consignment of 1243 yards black broadcloth, invoiced at los. Gd. per yard, to bo sold on joint account of consignor and myself, oacli one half, my half to be as cash, invoice dated Sept. IGth. Oct 5th, I sold 11. Duncan, for cash, 207 yards, at $G.10 per yard ; Oct 21th, sold 317 yards to James Grant, at $G.25 per yard, on a credit of 90 days; Nov. 18th, sold E. G. Conklin, for his note at 4 montht*, 400 yards, at $6.30 per yard ; Dec. 12th, sold the remainder to J. A. Musgrove at $G.OO per yard, half cash and a credit of 30 days for balance ; cliarges for storage, advertising, &c., $1 ^^40 j my com- mission, with guarantee of sale 6 per cent. What would bo the average time of sales ; the avcnigc time of James Watson's account ; and what would bo tho lace of a sterling bill, dated Dec. 15th, at 60 days after date, remitted James Watson to balance account purchased at §108|, money being worth 7 percent, and gold being 70 per cent, premium ? 5. Buchanan & Harris of Milwaukee, Wis., are owing W. A. Murray & Co. of Washington, $1742.75, being proceeds of consign- ment of tobacco sold for thcu, and Simpson & Co. of Washington, arc owing Buchanan & Harris $2000 payable in Washington. Buchanan & Harris wish to remit W. A. Murray & Co. the proceeds of their consignment and they do so by draft on Simpson & Co., bat Washington funds are 2 per cent, premium over those of Chicngo Uequired the face of the draft and the journal entries. 6. A. Commings, of London, England, is owing mo a certain sum, jjayablo there, and I ani owing Charles JIasscy, of tho same place, $1985.42, being proceeds of consignment of broadclotli sold for lum. I remit C. Massey in full of account, after allowing him 821,12 for inserest, my bill of exchange on A. Cummlngs at 60 days' sight; exchange 109|, gold 42 per cent, premium. What is the face of the draft, and what are the journal entries ? QUEiTIONS FOB COMMERCL\L STUDENTS. 'VlCy per ID., on a expenses in jhv at sight, s the faoe of Ft $5479.05. ds, England' 1 at 13s. Gd. myself, o;icli h. Octoili, 1; Oct2ith, , credit of 90 months, 400 idcr to J. A. 30 days lor :0 ; ray cora- would bo tho en's account ; Dec. 15th, at anco account gold being 70 )wing "NV. A. Isof consij^n- Washington, Washington. the proceeds on & Co., but E5 of Chicago a certain sum, same place, ;loth sold for allowing him nmings at 60 Lim. What is ? 7. March 10, I shipped per steamer F^, 2A- rods. 21. A ship was stranded at a distance of 40 yards from tho baso of a cliff 30 yards high ; what was the length of a cable which reached from tho top of the cliff to the ship ? Ans. 50 yds. 22. A cable 100 yards long was passed from the bow to the stern of a ship through tho cradle of a mast placed in midships at the height of 30 yards j what was tho length of the ship ? Ans. 80 yards. 23. A man attempts ';o row a boat directly across a river 200 yards broad, but is carried 80 yards down the stream by the current ; through how many yards was he carried ? Ans. 21 5.4-f yards. 24. Let the three sides of a triangle be 30, 40, 20 ; to find the aroa in square feet. Ans. 290.4737 square feet. 25. What is the area of an isosceles triangle, each of the equal sides being 15 feet, and the base 20 feet ?='^ Ans. 111.803 sq. feet, 2G. What is the area of a triangular space, of which the baso is 56, and the hypotermse G5 yards ? Ans. 924 square yards. 27. What is the area of a triangular clearing, each side of which is 25 chains ? Ans. 27.0G32 acres. 28. What is the area of a triangular clearing, of which the three side's are 380, 420 and 7G5 ? Ans. 9 acres, 37A^ perches. ^(29. A lot of ground is represented by the three sides of a right angled triangle, of which the hypotenuse is 100 rods, and the base GO rods ; what is the area? Ans. 15 acres. oO. What is the area of a triangular field, of which the sides are t9, 34 and 27 rods respectively ? Ans. 2 acres, 3 roods-j-. 31. What is the area of a triangular orchard, the sides of which are 13, 14 and 15 yards ? Ans. 84 square yards. 32. Three divisions of an army are placed bo as to be represented * This question, ani some others may be solved by either rule, and it will bi' foimd a good exercise to solve by bo*,h. I I Hi 832 A.IHTEMETIO. by throo sides of a triangle, 12, 18 and 24; how many square miles do they guard within their lines ? Ans, Between 104 and 105 square milcs- 33. A ladder, 50 feet long, was placed in a street, and reached to a parapet 28 feet high, and on being turned over reached a para- pet on the other bide 30 feet high ; what was the breadth of the street ? Ans. 70.123+fcet. PROBLEM III. To find the nrca of a regular Polygon. 1. When one of tho equal sides, and the perpendicular on it from tho centre, are given, 3Iu2tip1y the perimeter hy tJic perpendicular on it from its centre^ and take half the product ; or, multiply cither by half the other, 2. "When a side only is given. Multiply the square of the side by the number found opposite the mimber of sides in the subjoined table. Note.— This tablo Bhowa the area when Ibonido Ja unity ; or, which ig tho damo thing, tho square la tho unit. SIDES. BEOULAR FIGURES. 3 Trianirlo 0.4330127. 4 Sauarc 1.0000000. 5 Pentagon 1.7204774. G Pexacon 2.59807G2. 7 IIcDtacron 3.G339125. 8 Octao'on 4.8281272. 9 Nonagon 6.1818241. 10 Decagon 7.0942088. 11 Ileredeca- "^n 9.3G5G395. 12 Dodecajron 11.19G1524. 34. If the side of a pentagon is G feet and the perpendicular 3 feet, what is the area ? Ans. 45 feet. 35. What is the area of a regular polygon, eacli side of which ia 15 yards ? Ans. 387.107325 sq. yds. 3G. If ef^ch side of a hexagon be 6 feet, and a line drawn from the centre to any angle be 5 feet, what is the area ? Alia. 72 sq. feet. square railos iqunrc milcs- and reached ichcd a para- •cadth of tho r0.123+fcet. dicular on it 'om its ccntrCt '■ the other. d opposite the or, wbicb i3 tho MENSURATION. 33M 0.4330127. 1.0000000. 1.7204774. 2.59807G2. 3.0339125. 4.8284272. 0.1818241. 7.G942088. 9.3G5G39J. 11.19G1524. irpcndicular 3 Ans. 45 feet, ie of which is i7325 sq. yds. le drawn from ;i8. 72 sq. feet. 37. The side of a decagon h 20.5 rods ; what is tho area ? Ans. 20 acres, roods, 33.5 rods, nearly. 38. A hexagonal taUo has each side GO inches, and a line from tho centre to any corner is 50 inches ; how many square foot in tho eurfaoe of tho table ? ■39. What is tho area of a regular heptagon, tho sicfd fecfng 19 Jg and tho perpendicular 10 ? Ans. G78.3. 40. An octagonal enclosure has each side G yards, what is its area ? Ans. 3 acres, 2 roods, 14 rods. 19 yards. 41. Five divisions of an army guard a certain tract of country — each line is 20 miles ; how many square miles arc guarded ?" Ads. G88.2, nearly. 42. Find tho same if there aro G divisions, and each line extends 5 miles ? Ans. G1.95-f miles. 43. The area of a hexagonal tabic is 73^ feet; what is each side ? Ans. 5^ feet. PROBLEM IV. To find the area of an irregular polygon. Divide it into triangles hy a perpendicular on each diagonal from the opposite angle. Find the area of each triangle scparatcli/, and the sum of these areas will he the area of the trapezium. IsoTE. — Either tho rlUigonala and perpoudiculars must bo given, or data from which to find them. 44. The diagonal extent of a four-sided field is C5 rods, and the perpendiculars on it from tho opposite corners are 28 and 33.5 rods ; what is the area ? Ans. 1 acre, 1 rood, 22.083 rods. 45. A quadrangle having two sides parallel, and tho one is 20.5 feet long and tho other 12.25 feet, and the perpendicular disfancu between them is 10.75 feet ; what is the area ? Ans. 17G.03125 sq. feet. 46. Required tho area of a six-sulcd figure, the diagonals of which are as follows : the two extreme ones, 20.75 yards and 18.5, and the intermediate 27.48 ; the perpendicular on the first is 8.6, on the second 12,8, and those on the int§yraediate one 14.25 and 9.35 ? Ans. 531.889 yards. 47. If the two sides of a hexagon be parallel, and tho diagonal parallel to tliom be 30.15 feet, and the perpendiculars on it from 1 1 334 ARITHMETIC. the opposite angles arc, on the left, 10.56, and on the right 12.24, and the part of the diagonal cut off to the left by the first perpendi- cular, 8.20, and to the right by the second, 10.14; on the other side, the perpendicular and segment of the diagonal to the left arQ 8.5G and 4.54, and on tho right 9.26 and 3.33 ; what is the area? Ans. 470.4155 sq. feet. PROBLEM V. To find the area of a figure, the boundaries of which arc partly right lines and partly curves or salients. Find the average breadth hi/ taking several perpendiculars from the nearest and most remote points, from a fixed hajp, axd dividing f.he 811)71 of these hi/ their number, the quotient^ multiplied bi/ the length, icill be a close approximation to the area. -Let the jocrpendiculars 9.2, 10.5, 8.3, 9.4, 10.7, their sum is 48.1, then 48.1-:-5=9.G2, and if the base is 20, we have 9.62X20^1. 192.4, the area. When practicable, as lai^o a portion of the space as possible should bo laid off, so as to form a regular figure, and the rest found as above. A field is to bo measured, and the greater part of it can be laid off in the form of a rectangle, the sides of which arc 20.5 and 10.5, and therefore its area is 215.25, and the offsets of the irregular part arc 10.2, 8.7, 10.9, and 8.5, the sum of which, divided by their number, is 7.GG, and 7.66x20.5=157.03, the area of the irregular part, and this, added so the area of the rectangles, gives 372.28, tho whole area. 48. The length of an irregular clearing is 47 rods, and tho breadths at G equal distances are 5.7, 4.8, 7.5, 5.1, 8.4 and 6.5; •AJiat is the area ? Ans. 1 acre, 1 rood, 29.86 rods. PROBLEMVI. To find the circumference of a circle when the diameter is known, ur the diameter when the circumference is known.* The most accuratu rule is tho well-known theorem that the diameter is to the circumference in tho ratio of 113 to 355. and * III strictness the circumference and diameter are not lilio quantitioa, but wc may suppose that a cord la stretched round tho circumference, and then drawn cvut into a straij^ht line, and ltd linear units compared with those of Uic diaiaetor. ight 12.24, 5t pcrpcndi- a the other the left arQ is the area ? .55 sq. feet. :h arc partly iculars/rom xd dividing [plied hij fho heir sum is i 9.62X20^ as possible le rest found it can be laid 5 and 10.5, regular part ed by their ic irregular 372.28, tho )ds, and tho 4 and 6.5; 29.86 rods. tor is known, !m that the to 355. and quantities, but ence, and then with those of MENSUEATION. 335 consequently the circumference to the diameter as 355 to 113. Now, 355-^-H 3=3.141 6 nearly, and for general purposes, sufficient accuracy will be attained by this K u L E . To find the circumference from a given diameter, multiply thi diameter hy 3.1416 ; and to find the diameter from a given circum- ference, divide hy 3.1416. 49. What is the length round the equator of a 15-inch globe? Ans. 47.124 inches. 50. If a round log has a circumference of 6 feet, 10 inches ; what is its diameter^ Ans. 2 feet, 2j'^ inches nearly. 51. If we take the distance from the centre of the earth to the equator to be 3979 ; what is the number of miles round tho equator? Ans. 25001 nearly. PROBLEM VII. To find the aron of a circle. 1. If the circumference and diameter are known,^— Multi]ily the circumference hy the diameteTf and take OiU'fourth of tlic product. 2. If the diameter alone is given, — Multiply the square of the diameter hy .7854. 3. If the circumference alone be given, — Multiply the square of the number denoting the circumference hy .07958. 52. If the diameter of a circle is 7, and tho circumference 22 ; wliat is the area ? Ans. 38J. 53. What is the area of a circle, tho radius of which is 3^. yds? Ans. 3;] square yards. 54. If a semicircular arc be denoted by 10.05 ; what is the area "{ tho circle ? Ans. 289.30. 55. if the diameter of a grinding stone be 20 inches; what superficial area is left when it is ground down to 15 inches diamctei*, and what superficial area has been worn away ? Ans. 176.715 6(ir. inches left, and 137.445 worn away. 56. If tho chord of an arc bo 24 inches, and the perpendicular en it from tho centre 11.0 ; wliat is the area of the circle ? Ans. 2.C89804. 336 AnrrHMETia MENSURATION OP SOLIDS. To find the solid contents of a parallelopiped, or any regularly box-sliapcd body : Let it bo required to find the number of cubic feet in a box 8 feet long, 4| foot broad, and Gf feet deep. "^ the first place, the length being 8 feet and the breadtli 4^, the area of the base is 8X4J=36 square feet, and therefore every foot of altitude, or depth, or thickness, -will give 36 cubic feet, and as there are G| fcct of depth, the whole solid content "will bo 36 times 6f , or 243 cubic feet. Hence the RULE. '"" 'V ' : . ^ ' :'' ' Take the continual product of the length, breadth, and depth. Note. — Lot it be carefully observed that the unit of tncasure in the case of solids is to be taken as a cube, the base of which is a superficial unit iised in the measurement of suifaces. The solid content is indicated by the repe- tition of this unit a certain number of times. If the body is of uniform breadth the rule needs no modification, but if it is rounded or tapering, as a globe, cone, or pyramid, the calculation becomes virtually to find how much the rounded or tapering body differs from the one of uniform breadth. Sup- pose, for example, we take a piece of wood 6 feet high, in the form of a pyramid, and having the length and breadth of the base each 6 fieet, then the area of the base is 36 ; but if, at the height of 1 foot, the dimensions have each dimuiished by 1 foot, the area is 25 ; at another foot higher it is 16 ; at the next 9 ; at the next 4 ; at the next 1 ; and at the 6th 0, i. e., it has come to a point, and the calculation is, how much remains from the solid cube after so much has been cut off each side as to give it thia form. This gives rise to the following varieties : I. To find the solid contents of a cone or pyramid : Multiply the area of the lase hy the perpendicular heights and take one-third of the products II. To find the solid contents of a cylinder or prism : Multiply the area of the hase hy the perpendicular Jieight, III. To find the surface of a sphere : Multiply the square of the diameter hy 3.1416*. IV. To find the solid contents pf a globe or sphere : Multiply the third power of the diameter hy .5236. MENStrRATIOK OP SOLIDS. 3S7 IS. my regularly it in a box 8 brcadt'i 4^, erclbrc every bio feet, and t ^ill bo 36 and dep<^. ore in the case •ficial unit used ed by the repc- j is of uniform r tapering, as a find how much breadth. Sup- I the form of a 6 feet, then the sions have each it is 16 ; at the has come to a d cube ftfter so \T heighU oy itself VLm id divide tha. II it .tu- OT. ocst course lij dlipUcr PLUf> 'i of the loiocst th of (Jig same Ills product hij complete ; ftnd pile, and suh- ■ IP T «- lO baso IS 40 ; Ans. 11480. are 20 shells ; Ans. 2870. length, and 20 Ans. IIOGO. he lowest layer c of 20 ; what Ans. 10150. he wanting part MEAST7BEMENT OF TIMBER. Timber is measured sometimes by the square foot, and sometimes by the cubic foot. Cleared timber, such as planks, beams, &c., are usually measured by the square foot. What is called board measure is a certain length and breadth, and a uniform thickness of one inch. Large quantities of round timber are often estimated by the ton. To find either the superficial extent or board measure of a plank, &c. *•";■' RULE. Multiply the length in feet hy the breadth in inche», and divide hy 12. Note. — Tlio (hickness being taken uniformly as one inch, the rule for find- ing Ihe contents in square feet becomes the eamo as that for finding eurfacc. )f the thickness be not an inch, — Multiply the board measure by the thickness. ' If the board be a tapering one, take half the sum of the two extreme widths for the average width. If ia one-inch plank be 24 feet long, and 8 inches thick, then we have 8 inches equal f of a foot, and § of 24 feet=16 feet. A board 30 feet long is 26 inches wide at the one end, and 14 inches at the other, hence 20 is the mean width, i. c, 1§ feet, and 30xl§^50 ; or, 30x20=600, and 600-5-12=50. To find the solid contents of a round log when the girt is known. Multiply the square of the quarter girt in inches by the length in feef, and divide the product by 144. If a log is 40 inches in girt, and 30 feet long, the solid contents will be found by taking the square of 10, the quarter girt in inches, which is 100, and 100x30^3000, and 3000-7-144=20^;. To find tlie number of square feet in round timW, when the medJX diameter is given. 840 ,ti. ABITHMETIC. Hft RULE. IfulHplg th€ diameter in incites by half the diameter in inches, and the product by the length in feet, and divide the result hi/ 12. If a log is 30 feet long, and 56 inches mean diameter, the uuiubcr of square feet ia 56x28x30-^-12=3020 feet. To find the solid contents of a lo^ Tvhcn the length and moan diameter are given. SirAvj') ^m 1/iuod ImlUiO hi HdH BULK. iJ taioliiiii a lui i Multiply the square of half the diameter in inches by 3.141G, and (his product by the length in feet, and divide by 144. ' ' T" , C8. How many cubic feet are there in a piece of timber 14X18, and 28 feet long ? • Ans. 49-|-cubic feet. ')> 69. Kow many cubic feet are there in a round log 21 inches in diameter, and 40 feet in length ? 70. What arc the solid contents of a log 24 inches in diameter, and 34 i'ect in length? Ans. 10G.81-|-cubic feet. 71. How many feet, board measure, are there in a log 23 inches in diun'.eter, and 12 feet long ? Ans. 264J. 72. How many feet, board measure, are there in a log, the diameter of which is 27 inches, and the length 16 feet. Ans. 486. 73. What are the solid contents of a round log 36 feet long, 18 inches diameter at one end, and 9 at the other ? 74. How many feet of square timber will a round log 36 inches in diameter and 10 feet long yield ? Ans. 540 solid feet. 75. How many solid feet are there in a board 15 feet long, 5 inches wide, and 3 inches thick ? <> mnuiinK, Ans. l^'^jj cubic feet. 76. What are the solid contents of a board 20 feet long, 20 inches broad, and 10 inches thick? Ans. 27^ feet. 77 What is the solid content of a piece of timber 12 feet long, 16 inches broad, and 12 inches thick ? Ans. 16 feet. 78. .How many cubic feet are there in a log that is 25 inches in diameter, and 32 feet long ? 79. How many feet, board measure, does a log 23 inches in diameter, and 14 feet in length contain ? Ans. 457]. 80. lluw many cubic feet are contained in a piece of squared timbei' that is 12 by 10 inches, and 47 f^t in lengtli ? Ans. 62f . ler in inches, mlt hi/ 12. ', the uumbcr ndao jdi V^ ,h and uicim iMl^t ■■•«* ';• d that tho midjii^ht J ins. 3 r.M. id up 8 feet time did he A.ns 4 days. iko 48? Ans. 44 j%. commission ) the owner, mounted to Ans. ^260. ivc, but tho low far will rial? 140 miles, est size he ze $6 ; how f tho small. 00; $4060 f 8 months 3ady money ? ns. $12000. are to bo yards wide, arda will it Ans. 5625. Lat aro the - sq. miles. iiO. A certain island id 73 niilc^ iu clrcumforenco, and if two men start out from the same point, in tho same direction, tho one walking at tlio rate of 5 and tho othor at the rate of 3 miles an hour; in what tiuic will thoy come together ? Ans. 36 hours, 30 minutes. 40. A circular pond measures half an acre ; what length of cord will be required to roach from tho edge of tho pond to the centre ? Ans. 83203 -f- feet. 41. A gentleman has deposited $450 for tho benefit of his aon, in a Savings' Bank, at compound interest at a half-yearly rate of 3 J per cent. IIo is to receive tho amount as soon as it becomes $1781. 66J. Allowing that the deposit was made when tho sou was I year old, what will bo his ago when ho can come iu possession of the money ? Ans. 21 years. 42. The select men of a certain town appointed a liquor figont, and furnished him with liquor to tho amount of $825.00, and cash, $215. The agent received cash for liquor sold, 81323.40. lie paid ibr liquor bought, $937 ; to the town treasurer, §300 : sundry ex- penses, $29 ; his own salary, $205 ; ho delivered to indigent persori|, by order of tho town, liquor to tho amount of 613.50. Upon taking .stock at the end of tho year, the liquor on hand amounted to $610.50. Did tho town gain or lose by the agency, and how much; has the agent any money in his hands belonging to tho town ; or docs tho town owe the agent, and how much in either case ? Ans. The town lost $103.20 ; the agent owes the town $7.40. 43. A holds a note for $575 against B, dated July 13th, paya- ble in 4 months from date. On tho 9th August, A received iu advance $02; and on the 5th September, $45 more According to tho terms of aguoement it will be due, adding 3 days of grace, on tho 16th November, but on the 3rd of October B proposes to pay a sum which, in addition to the sums previously paid, shall extend the pay day to forty days beyond tho 10th of November; ho'v much must B pay on the 3rd of October ? Ans. $111.43. 44. A accepted an agency from B to buy and sell grain for him. A received from B grain in store, valued at $135.60, and cash, $222.10 ; he bought grain to the value of $1346.40, and sold grain to the amount of $1171.97. t tho end of four months B wished to close the agency, and A returned him grain unsold, valued at 1^437.95 ; A was to receive for services, $48.12. Did A owe B, or B owe A, and how much ? Ans. B owed A 45 ccntB. iJ4tJ ARrrHMETIO. 45. A general ranging his men in tlio form of a Bquarc, had l^}*J men over, but having iticrcusoJ tho uido of tho square by one man, ho lacked 84 of complotiu^ thu square ; how umuy mon bad ho ? • Ana. 5100. 4G. What portion, expressed as a common fraction, is u poniid and a half troy weight of three pounds avoirdvpois ? Ans ,"•,'-.,. 47. AVhat would the lust fraction bo if wo reckoned by tho ouiic(!h instead of grains according to tho standards? Ans. -J. 48. If 4 men can reap G^ acres of wheat in 2^ days, by working 8.} hours per day, how "lany acres will 15 men, working equally, reap in 3£- days, working 9 hours per day ? Ans. 40]^ day.^. 49. Out of a certain quantity of wheat, ^ was sold at a certai;i gain per cent., J- at twice that gain, and the remainder at thre ^ tinios the gain on tho first lot ; what was tho gain on each, the gain on the whole being 20 per cent.? Ans. 9?, 191 and 231 per cent. 50. If a man by travelling G hours a day, and at tho rate of 4 J miles an hour, can accomplish a journey of 540 miles in 20 days ; how many days, at the rate of 4§ miles an hour, will he require to accomplish a journey of GOO miles ? Ans. 21^'. 51. Smith in Montreal, and Jones in Toronto, agree to exchange operations, Jones chiefly making the purchases, and Smith the sales, the profits to be equally divided ; Smith remitted to Jones a draft for 88000 after Jones had maJo purchases to the amount of $13G82.24; — Jones had sent merchandise to Smith, of which the latter had made sales to the value of $9241.18 ; Jones had also made sales to the worth of 6283G.24; Smith has paid $3G4.16 and Jones $239.14 for expenses. At the end of the year Jones has on hands goods worth 62327.34 and Smith goods worth 83123.42. The term of tho agreement having now expired, a settlement is made, wha.' has been the gain or lo.ss ? What is each partner's share of gain or loss ? What is the cash balance, and in favor of which partner ? 52. In a certain factory a number of men, boys and girls are employed, the men work 12 hours a day, the boys 9 hours and the girls 8 hours ; for the same number of hours each man receives a half more than each boy, and each boy a third more than each girl ; the sum paid each day to all tho boys is double the sum paid to all the girls, and for every five shillings earned by all the bovs each day, twelve sliillinga we earned by nil the men j it MIHCELL,iNE0U8 EXERCISES. U7 ;, had M ono lunii, 1 ho ? .ns. 5100. s 11 ponml AiiH ;';-,.. ho OUtlClIH AllH. •]. ly vrorkinj» g equally, [0 1 -J (lays. a cortnin hro: tiinc» ;ain on the ^ per cent. rate of 4?f 20 days; require to Ans.21^. exchange 1 the sales, lies a draft amount of which the also made and Jones on hands The term I, wha.' has )f gain or irtaer? id girls are irs and the receives a than each |c the sum icd by all ke men ; it is required to find the number ol" men, the number of boys and the iiuijiber of girls, tbo whole number being 50. Ana. 24 men, 20 boys and 1 5 girls. 63. A holds B's note for $575, payable at the end of 1 months from the 13th July; on tho 9th August, A received 6i>:i in advance, as part payment, and on tho 5th September ^45 more; according to agreement tho note will not ba duo till IGth November, three days of grace being added to the term; but on tho iJrd October 13 tenders Buch a sum as will, together with the payments already made, ex- tend time of payment forty days forward; how much must B pay on the 3rd of October ? Ans. ^ 1 1 1 .13. 54. If a man commcneo business with a capital of $5000 and realises, above expenses, so much as to increase his capital each year by one tenth of itself less $100, what will his capital amount to in twenty yeirs? Ans. $27910. 55. A note for §100 was tj como due on the 1st October, but on tho 11th of August, the acceptor proposes to pay as much in ad- vance as will allow him GO days after the 1st of October io pay the balance; how much must ho pay on tho 11th of August ? Ans. ;;;54. 56. A person contributed a certain sum iu dollars to four char- ities ; — to one ho gave one half of the whole and half a dollar ; to a second half the remainder and half a dollar; to a third half the re- mainder and half a dollar; and also to the fourth half tho remainder and half a dollar, together with ono dollar that was left ; how much did ho give to each ? Ans, To the first, $10; to the second, $8; to the third, ipi; to the fourth, $3. 57. A farmer being asked how many sheep ho had, replied (hat he had them in four different fields, and that two-thirds of the num- ber in tho first field was equal to three-fourths of the number in the second field; and that two-thirds of the number in the second field was equal to three-fourths of the number in tho third field ; and that two-thirds of tho number in the third field was equal to four- fifths of the number in the fourth field; also that there were thirty- two .sheep more in the third field than in the fourth ; how many sheep were in each field and how many altogether ? Ans. First field, 243; second field, 216; third field, 1921 fourth field. 160. Total. 811. 348 ARITHMETIC. 58. How many hours per day must 217 men work lor 5^ aays :o dig :i trench 23.J- yards long, 3| yards wide, and 2J- deep, if 24 men working equally can dig one 33| yards long, 5| wide, and 3 J deep, iu 189 days of 14 hours each. Ans. IG hours. 59. A man bequeathed one-fourth of his property to i?is eldest son J — ^to the second son one-fourth of the remainder, and $350 be- sides ; to the third ono-fourth of the remainder, together with $975 ; to the youngest one-fourth of the remainder and $1400 ; he gives his wife a life interest in the remainder, and her share is found to be one-fifth of the whole ; what was the amount of the property ? Ans. $20,000. GO. Five men formed a partnership which was dissolved after four years' continuance ; the first contributed $60 at first and $800 more at the end of five months, and again $" 300 at the end of a year and eight months; the second contributed $000 and $1800 more at the end of six months ; the third gave at first $400 and $500 every six months ; the fourth did not contribute till the end of eight months ; he then gave $000, and the same sum every six months ; the fifth, having no capital, contributed by his labor in keep- ing the books at a salary of 1^1.25 per day; at the expiration of the partnership what was the share of each, the whole gain having been $20000 ? " Gl. Four men, A, B, C, and D, bought a stack of hay containing 8 tons, for $100. A is to have 12 per cent, more of the hay than B, B is to have 10 per cent, more than C, and C is to have 8 per cent, more than D. Each man is to pay in proportion to the quantity ho receives. The stack is 20 feat high, and 12 feet square at its base, it being an exact pyramid ; and it is agreed that A shall take his share first from tlic top of tlie stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sura shall each pay ? Ans. A. takes 13.22-f ft., and pays $23,93^ ; B takes 3.14-|-ft., and pays $25.83 J; C takes 2.0G+ft., and pays $23.48J ; D takes 1.58-|-ft., and pays $21.74J. 62. A merchant bought 500 bushels of wheat and eoid one half of it at 80 cents per bushel which was 10 per cent more than it jnSCELLAMEOUS EXERCIfiES. 319 5^ aays ep^ if 24 and '^ ,G hours. is eldest ^350 bc- :h $975 ; he gives found to crty? $20,000. Ived after and $800 Q end of a id $1800 $400 and the end of every six )r in keep- ion of the ring been ontaining ,y than B, per cent, iiantity ho t its base, 1 take his the next, height of J.14+ft., |$23.48i ; one half \q than it oost him, and 5 per cent. Ie.s3 than ho asked for it. He sold the reinaindor at 12^ por cent, more than it co.st him. What wad hii- asking price for both lots ? What did ho receive for the last lot. and how much did he gain on the whole ? 63. May Ist, 18G2, I got my note for $2000 payable in 4 months discounted at a bank, and immediately invested the money received in woodland. November 9th, I sold the land at an advance of 15 per cent., receiving | of the price in cash, and a note for the remainder, payable August 10, 1864, without grace, and to bo on interest after January 1, 1864, at 7 por cent. T lent the cash re- ceived at 6 por cent. When my note at the bank became due I renewed it for the same time as before, and at the proper time renewed it again, and finally renewed it for such a time that the note would become due August 10, 1864. Now, if I paid G per cent, on the money borrowed at the bank, and made a complete settlement August 10, 1864, what was the amount of my gains ? 64. My agent at Mobile buys for mo 500 bales of cotton, avei aging 500 lbs. per bale, at 1.0 cents per pound. I pay him 1 J per cent, on the amount paid for the cotton, and shipping charges at GO cents from January 1 for an amount sufficient to pay for the cotton, charges and commission including also 2 per cent, discount on the draft. On the receipt of the invoice, I insure for the amount of the draft plus 10 per cent, j I pay 1;^ per cent, premium on the amount insured, and from the amount of the premium is discounted 1^ per cent, for cash. On the arrival of the cotton I pay f of a cent per pound for freight, and 5 per cent, primage to the captain on the freigbt money, and also 4 cents per bale for wharfage. I sell it on the wharf, January 20, at $1 per bale profit, and agreed to take in payment the note of the purchaser for 6 months from January 20. What amount would bo reoeiveu on the uote when discounted at a hank at 7 ]per Cfiuli- ? 350 ArJTHMETlC. FOREIGN GOLD COINS. MD,T VALCB. COnUTKY. Australia. Austria (I (• Belgium ... Bolivia Brazil Ccntrl America|Two CMli « DenmarK... Equador England . . . France « Germany, North Germany, Sonth Greece Ilindostan. . . Italy Japan « Mexico Naples Netherlands. . . New Granada. Peni . . . . , Portugal Prussia .. Rciiio . . . , Russia . . , Spain. . ., Sweden Tunis Turkey Tuscany .,,,,, DENOMINATIONS. WEIGHT. Pound of 1852 Sovereign 18.'35-G0 . . . Ducat .... Souveraiu NewUnion Crown (assumed; Twenty-flvo irancs Doubloon 20 Milreis oscudos. Old doubloon Ten Pesos Ten thaler Four eseudos Pound or Sovereign, new. . Pound or Soveri,Q;n,avcrago Twenty Irancs, new Twenty francs, average. . . Ten thaler Ten thaler, Prussian Krone [crown] Ducat Twenty drachms Mohur 20 lire Old Cobang New Cobang Doubloon, average. '•■ new ... Six ducati, new Ten guilders Old Doubloon, Bogota Old Doubloon, Popayan. Ten pesos, new Old doubloon Gold crown NewUnion Crown [assumed] ?i scudi,new Five roubles 100 reals 80 reals Ducat 25 piastres . ■ . . 100 piastres Scquio ,.fi..< Oz. Deo. 0.281 0.25G.5 0.112 0.3G3 0.357 0.254 0.8G7 0.575 0.20"J 0.8G7 0.'192 0.427 0.433 0.25 G.7 0.25G.2 0.207.5 0.207 0.427 0.427 0.357 0.112 0.185 0.374 0.207 0.3G2 0.2S9 0.8G7.5 0.8G7.6 0.245 0.215 0.8G8 0.8G7 0.525 0.8G7 0.308 0.357 0.110 0.210 0.268 0.215 0.111 O.lCl 0.231 0.112 riNE- KESS. Tiions. 91G.5 91G 98G 900 900 899 870 917.5 853.5 870 900 895 8U 91G.5 91G 899.5 899 895 903 900 98G 900 91G 898 5G8 572 86G 870.5 99G 899 870 858 891.5 8G8 912 900 900 91G 896 809.5 975 900 915 999 VALCB. $5.32.37 4.85.58 2.28.28 G.75.35 G.Gl.U: 4.72.03 15.59.25 10. 90.57 3.G8.75 15.59.2C 9.15.35 7.90.01 7.55.4 G 4.8G.34 4.84.92 3.85.83 3.84. C9 7.90.01 7.97.07 0.6 J. 20 2.28.28 3.44.19 7.08.18 3.84.2 G 4.44.0 3.57.6 15.52.98 15.61.05 5.04.43 3.99.5G 15.61.0G 15.37.75 9.67.51 15.65.67 6.80.66 6.64.19 2.60.47 3.97.64 4.96.39 3.86.44 2.23.72 2.99.54 4.36.93 2.31.29 Value nftei Dcductioa $5.29.71 4.83.16 2.27.04 G.7 1.9 8 G.C0.87 4.69.67 15.51.46 10.85.12 :}.GC.91 15.51.47 9.10.78 7.8G.0G 7.51.09 4.83.91 4.82.50 3.83.91 3.82.77 7.86.06 7.93.09 6.G0.88 2.27.14 3.42.47 7.04.64 3.62.34 4.41.8 3.55.8 15.45.22 15.53.26 5.01.91 3.97.57 15.53.26 15.30.07 9.62.68 15.47.90 6.77.7G 6.60.87 2.59.17 3.95.66 4.93.91 3.84.51 2.22.61 2.98.05 4.34.75 2^0.14 lvalue aftci Deduction. .37 .58 (.28 ..35 kl !.03 ).25 ).57 ^.75 ).2G i.oo :1.01 3.40 0.34 4.92 5.83 4.69 0.01 7.07 !.20 28 ,19 .18 .20 .0 .G .98 05 ,43 ,50 ,00 .75 .51 .67 .00 .19 .47 .64 .39 .44 1.72 .54 .93 .29 $5.29.71 4.83.1G 2.27.04 0.71.98 C.G0.87 4.09.07 15.51.40 10.65.12 3.GG.91 15.51.47 9.10.78 7.8G.00 7.51.69 4.83.91 4.82.50 3.83.91 3.82.77 7.86.06 7.93.09 6.00.88 2.27.14 3.42.47 7.04.64 3.62.34 4.41.8 3.55.8 15.45.22 15.53.25 5.01.01 3.97.57 15.53.26 15.30.07 9.62.68 15.47.90 5.77.70 6.60.87 2.59.17 3.95.66 4.93,91 3.84.51 22.61 I'OKEiGN' silve:: FOREIGN SILVER COINS rUNT VALUE. 351 2.98.05 4.34.75 2.80.14 — ' • — 1 1ES-0MIN.\TI0X!3. WLIGHT. 1 FI.NEXESS. ViLl-K. Au>'4ria Old rix dollar Oz. Dec. 0.902 0.836 0.451 0.397 0.596 0.895 0.803 0.043 0.432 0.820 0.150 0.8GG 0.SC4 O.SOl 0.927 0.182,5 0.178 O.800 0.712 0.595 0.340 0.340 0.719 0.374 0.279 0.279 0.867.5 0.800 0.844 0.804 0.927 0.803 0.866 0.766 0.433 0.712 0.595 0.864 0.667 0.800 O.IGG 1.092 0.323 0.511 0.770 0.220 Tnors. 833 902 833 900 900 838 897 903.5 CG7 918 5 925 850 90S 000.5 877 921.5 925 030 750 900 900 903 900 91G 991 890 903 901 830 944 877 896 901 909 650 750 900 900 875 900 899 750 899 898.5 830 m $1.02.27 l.O'' 64 A: Old scudo (( Florin before 1858 New florin Now Union dollar Maria Theresa dorr,1780 Five francs 51.14 ti 48.03 73.01 Eelgium .Bolivia 1.02.12 98.0* New dollar 79.07 if Half dollar 39.2/J .TJrazil Canadii Double Milrei? 20 cents 1.02.53 18 87 Ceniral America, . . Dollar. 1,00.19 Chill Old Dollar 1.00.79 Denmark Now Dollar.. Two rigsdalor 9S.17 1,10.05 England Shilling, new 22.91 Prance Germany, North. . . Shilling, average Five franc, average Thaler, before lbj7 New thaler 22.'! 1 !18.C0 72.07 72.89 Germany, South. . . Greece Slindostan Florin, before 1857 New florin [assumed] . . . Five drachms Ihipeo Itzobu 41.05 41.05 88.08 46.02 •f apan 37.03 a New Itzebu 33,80 Mexico . Dollar, new Dollar, average Scudo 2J guild Specie dalcr Dollar of 1857 Old dollar Dollar of 1858 1.06.02 1.00.20 Naples Netherlands Norway. }Sii\v Granada Peru 95.54 1.03.31 1.10.05 97.92 1.06.20 ti 94.77 1. Half dollar, 1835-38.... Thaler before 1857 New thaler 38.31 I'russia 72.08 ti 72.89 Rome Scudo 1.05.84 Russia Sardinia. Rouble. . Five lire 79.44 98.on Spain New pistaroen 20.31 Rwcdeu Ri.v dollar 1.11.48 Switzerland Two francs 39.52 Tunis Turkey Fiv<,» piastres Twenty piastres. Florin..: 62.49 86.98 'JL^OBcanv 27.60 LAWS OF THE UNITED STATES BEIATINO TO INTEREST, DAMAGES ON BILLS, AND THE COLLECTION OF DEBTS Tho following brief skttclica of tho laws of the different States of Uio Union, will be fonud usolul, not only to business inou but also to privato individuals. Tho information on which tlioy nro founded, has been derived from authentic sources and condensed into a convenient cpltomo, which may Ic relied upon as correct as regards tbo present slate cf tho law, which is all that anj' ono can bo nnswcrablo for, as alterations may herepftcr be made on eomo points. ALABAMA. I 'J Interest.— Tho rato of interest in Alabama is cij;ht per cent, per annum. Penallij for Violalion of the Usury Laws. — All contracts mado at a higher rato ol interest than eight per cent, aro usurious, and cannot bo enforced except as to tho principal. Damages en Bills. — Damages on inlan.l bills of ex -lango protested for nonpayment, aro per cent. ; on foreign hilla of exchango 10 per cent, on tlio sum drawn for. All bills drawn and payable within this t talo aro termed inland bills ; those drawn in this Ktato and payablo elsewhere, aro eoniidercd forci^ni billa Sight Hills.— Oraco is allowed ou bills, drafts, etc., payablo at sight. Collection cf Debts. — Original attachments, foreign and domesiie, aro i.?suod by judges of th5 circuit or county courts?, or justices of tho peace. An attachment may Issue, although tho debt or demand of tho plaintiff bo not duo : and shali be a lien ou tho property attuclied until tho debt or demand becomes due, when ju'\;ment shull bo rendered and csocution issued. A nonresident plaintiiTmaybavo an attachmen' against tho property cf a non-resident defend- ant, provided Lo gives good and Bufflcient resident security in tho required bond, making oath that tho defendant has not sufflcient property within tho Btato of defendant's residence to eatisfy tho debt or demand. ARKANSAS. Interest— Ihc legal rate of interest in Arkansas is six per cent. Special contracts in writing will admit an interest not to exceed ten percent. Alljudgraents or decrees upon contracts bearing moro than six per cent, shall bear the same rato of interest originally agreed upon.— (GouW'.s Digest, chap. 02, sec. 1, 2, &c., I8J8.) renaUufor Violaiion(fthe Usury Lav}S.—AX\ contracts for rcscr\'ation of a greater rato of interest th;;n ten per cent aro void. Tho excess taken or charged beyond ten per cent, may bo recovered back, provided the action for recovery shall bo brought within ono year after p.iy- mont. (U). Bcca & 7.) Damages en Dills — Tho damages on Bills of Exchange drawn or negotiated in Arkansas, expresses to bo for valuo received, and protested for non-acceptance, or for non-payment after nou-acccptauco, aro as follows. — (lb. chap, 'lb.) 1. Ifpayaljlo within tho State, 2 per cent^ 2. If payablo in .Alabama, Louisiana, Mississippi, Tennessee, Kentucky, Ohio, India a, Illinois or Missouri, or at any point ou tho Ohio River, 4 per cent. 3. Ifpayablo in any other fc'tato or territory, 5 per cent. 4. If payablo within cither of tho United States, and protested for non-i)ayment, after ucc:ptance, percent. I'oreign ZJi//;;.— Tho damages ou bills cf exchange, expressed for valuo received, undpayablM beyond tho limits of Iho United States (tb. chap. 25), aro 10 per cent Sight Dills.— X\icxo\3 no Btatuloiii fnco in Arkansas in reference to grace or right bills, rcction 15, G.-uld'u Digest, says '• l''oreign nqd inland bills bl-all bo governe I by the law- merchant as to days of grace, protest and notices." CoUeeiioii (f Debts.— \a attachment may bo issued against tho property of ii non-resident, i.nd also against a resident of tho l-^tato when tho latter is about to remove cut of tho State; cr is nl)out to remove his goods or effects, or about to eecrete himself, bo that tho ordiniry pr - cosa of law couuot bo served oa him. 863 BILLS, jn, will bo found nation on wblch ito a convenient f tho law, which I made on eomo CALIFORNL/L /n<«r«*t— Thft legal rato of Lr ^v-e i in California is, oy stutnte, fixed at mi per cent. On Epociol contracts any rato of lnte:eii may bo agreed upon or paid. Penalty for Violation of the Interal Zaw.— Thuro is no law In California fixing any fienalty Cot charging any rato or interest abovo tan per cent. Tho matter is thus left entirely tee between tho contracting parties. Damages on Billg.— Tho damages on billfl of oxchango drawn or negotiated iu California p.nyablo in any State cast of tho Roclcy Mountains, and roiumed under protest for non-occfpt- arix or Tum-paipnent, aro uniformly, 15 per cent. Foreign Bills. — Tho damages on foreign of exchange relumed under protest, ar* '. per cent. £ight j.UlIs.~GTaco is not allowed by tho bankers on bills, chocks, drafts, etc., payable al tight. Tho notarial fees for protesting a bill of cxchango or promissory nolo aro $5 or moro, flccordmg to tho number of notices sent. Act March 13, 1860. Collection of Debts.— 1. Creditors may proceed by attachment when tho defendant haa nbtconded, or is about to abscond from tho Ktato, or is concealed tliorciii to tho injury of his creditors. 2. When tho defendant '^is removed or is about to remove any of his property out of tho State, with intent to dclV^ui his creditors. 3. When tlio defendant fraudulently con- tracted tho debt or inoiirrad tho obliRation, respecting to which tlio suit is brought. 4. When tho defendant is a non-resident. 5, When ho has fraudulently convoyed, ftisposcd or or con- cealed bh property, or n part, of it, or intends to convey tlio same to dofrauu liia creditors. In California tho real cstata ehall bo bound, and tho attachment shall bo a lieu thereon, although tho debt or demand duo tho plaintill" bo not duo— in case tlio defendant is about to remove limself or his property from tho tJtato. Tho law of attachment applies iu CaliUrnia when the cr>riract hat been made in tliai Slate, or when made payable in thai Stale. a higher rato o£ to tho principal. or uon payment, for. ; thoso drawn in pucd by judges of [sue, nlthougU tho ■ty attached until mtion issued. A i-resident defend- ind, making oath it's residence to Lutractsinwriilntf [es upon contracts ly agreed upon. — • a greater rato of len per cent, inay |do year after pay- dtcd la Arkansas, pn-paymcnt after Ohio, India a, [n-imymcnt, afl^r lived, and payablo ace or right bills, [lel by tuo law- Lfiiuonresideul, ft of tho State ;cr Tthe ordinwy pr> CONNECTICUT. /nferi'rt.— Tho legal ratool interest In Connecticut Is six per cent., nnd no higher rato is allowed on fipecial coatracts. Banks aro forbidden, imler penalty of |500, from taking directly or iudirccily over per cent. Law passod May, 1854. Penalty for Violation of the Vsury Laws. — Forfeiture of all the interest received. Iu su'ts oa usurious contracts, judgment is to bo rendered tor tho amount lent, without interest. Vamafjcs on Bills.— Tho damages on bills of cxchango negotiated iu Connecticut, pay- uble in oilier ; tales, and returned under protest, are as follows: 1. lla.ac, New Ilampshirc, Vermont, Massachusetts, Rhodo Island. N'-w York (interior), Now Jersey, Pennsylvania, Delaware, Maiyloud, Virginia, District of Columbia, 3 per ceatk 2. Kcw York City, 2 per cent 3. North Carolina, South Carolina, Georgia and Ohio, 6 per coat il. All tho other t-'tates and Territories, 8 per cent. Foreign BilU.—T'hGTo is no statute in force in Connecticut in reference to damages on foreign bills of exchange. Sight Billi.—GTSiCO is not allowed by statute or usage ou checks, bills, cct., puyabliio^ M^ht. Collection o/Z>e6te.— Attachment may bo granted against the goods and chattels and land of llio dcl'endanl ; and likewise against his person when not exempted from imprisoiuucutou tho execution in Uio suit. Tho plaintifi* to give bonda to prosecute Lis claim to clle^^.t. DELAWARE. jtnlerest.—'Iho legal rate of fntcrost ia sir per cent , and no tnoro iii allowed oa direct or Indirect contracts. JPenally for Violation of the Uumnj X«x«.— Forfeiture of tho money tr .'. other things lent, one half to tho Governor for the support of government, tho other lialf payable to tho person sueing for the same. JDamaget on Bills.— Tbora fa no statute in force in Dolawaro ii reference to damases on domestic or inland bills of exchange. J^oreign Bills.— Tho damages upon bills of exchange drawn upon any person In Eng- Lnnd. or other parts of Europe, or beyond tho was, and vcturncd under protest, aro 20 per cent, .Sight Bills.— ThoTQ is no Btatuto with rcforenco to bills, drafts, etc., al tight. They aro not, 'by usage, entitled to grace. -Collection <^ Debit.— K writ of Uomcsiio attachment issues against nn Inhab taut of Dela- ware when the defendant cannot bo found, or has absconded with intent to defraud his credi- tors; and a writ of foreign attachment when the defendant U not an Inhabitant of this State. This attachment is dissolved by the d«(badaut'a tppooriug and patting in suocial bail at J3V lime Iwfore Judgment. 23 354 FLORIDA. *»<«•«<,— The legal rate of interest l3 six per cent. On epecial contracts eight per com. may bo cLarRcJ. .Penalty for Violation of the Usury Zaw*.— Forfeituro of tho whole interest paid. Damages on Vills.—Tiio damages on bills of exchange, n jctiatcd in Florida, payable in other Btatcs, ami returncil under protest for non-payment, aro iiuilbrmly 5 per cent. Foreign /ji(7i'.— Damag'S cu foreign bill3 of exchange C per cent. Sight Bills.-— GrM'Xi i3 not allowod on bills, drafts, etc., payable at sight. There is no Btatuto in Florida iiron this subject. Collection of Debts An attachment issues when tho amount is actually due, and tho defendant ia actually moving out of tho State, or absconds or couceala.lnmBeli; GEORGIA. Interest— ThQ legal raio of interest in Georgia is ecven per cent., and no higher rate is allowed on special contracla Open accounts, uiilic;::i'j:;tc'd, do not boar interest. Penalty for Violation of the Usury Laws.— FotMiu:o cf only the excess of intorcs' <'M'.' 83ven per cent. Principal and legal interest aro recoverable. (Acts of 1855-6, page 259. ) Damages on Dills. — The damn' i oii bills of exchange, nc.'^o'.iatcd ia Georgia, payable iii other States, and returned under ] ; • .;t, aro uniformly 5 per ce: .,'. Foreign Bills.— Tho damages on ioreign bills of exchange, returned under protest, aro 10 per cent. tiiglit Bills.— ^- Three days, commonly called the thrco days' of grace, shall not bo allowed ■upon any sight drafts or bills of cschaugo drawn payable atsight, after tho passagxjof thi.i Act; but tho samo shall bo payablo on presentation thereof, subject to tho provisions of tho l^rsl eection of this Act. Tho'lirst fcectiondc.-i£:-'n:itestho holidays." Act passed Feb. 8, 1850. (i'ea Cobb's New Tigcst of tho l.iws of Georgia, pp. 519, 522.) ■ £nrfowers.— Eudorser.s aro not entitled to notice of lUflionour, except upon notes and bills payablo nt l)ank, or negotiated in bank, or placed in bank in collection. Collection of Debts. — A judge of tho (Superior Court, or a justice of tho inferior court, or a justico of tho peace, may grant an attachment against a debtor whether tho debt bo matured or not, when tho latter is removing Avithout tho limits cf tho State, or any county, or conceals Iiimseir. Tlio remedy by attachment may bo resorted to by non-resident as well as by resident creditors. Tho necessary afildavit may bo made before any commissioner ni:-^i!ited by tho State to tako affidavits. Indorsers of notes, obligations and all other instruments In writing, aro entitled to tho eamo rer.icdy as provided for securities. In all cases tho attachment (i;tt eerved shall bo lirst Bati;;;f.!. Ko lien shall bo created by tho l;Vv viri.:; of anattachmcatk to tho exclusion ofany jutlj.iu.-.t •. btained by any creditor, before judgment is obtaiuetl by tlio Qltachiiig creditor. ILLINOIS. Jnfercsi.— Tho L'CtSlaturo, in 1857, passe V; > following act: • t-EcnoN 1. That from and after tho pash:a-j of this act, tho rato of interest upon al! con"- tract and agreements, written <,.■ verbal, express or implied, for tho payment of money, shail bo Bis per cent, per annum upon every one hundred dollars, unless otherwise provided by law f.. .;.iN 2. That in all contracts hereafter to be made, whether written or verbal, it shall be lawful lor the parties to stipulato or agree that ten per cent, per annum, or any loss sum of interest, shall bo taken and paid upon every one hundred dollars of money loaned, or in any manner duo and owing from any person or corporation to any person or corporation in this State. Penalty for Violation of the Usury Laws.—li any person or corporation iu this ft no shall contract to receive a greater rato of interest than ten per cent, upon any contract .. rbal or written, such person or coriioration shall ibrfoit tho whole of said interest eo" con- ;. acted to bo received, and £h:Jl bo entitled only to recover the principal sum duo to such person or coiporation. {Act of 185"). Damages on Bills— Tho damages on bills of exchange negotiated in Illlno's, payablo in other states or Territories, and returned under protest for non-payment, aro uniformily (py act of March 3, 1845) C per cent, in addition to tho interest. Foreign Bills.— Tho damages payablo on foreign bills of exchange, returned under protest, aro Iby act of March ?, 1845] 10 per cent, in addition to tho interest. Sight Bills.— TIcTcMoTO there has been no statute iu force regarding bills or drafts al sight, but by an act of tho legislature, approved February 22d, 1861, it is enacted that "no note, check, draft, bill of exchange, order or other negotiable or commercial investments payablo at Bight or on demand, or on presentation, shall bo entitled to days of grace, but shall bo ahsoiutcly payablo on presontmout. All other notes, drafts or billa of cschangOj shall be en- titled to tho iisual days grace. This act is in Ibrce from its passage. Collection cf TJc&is.— Attacliments aro issued by tho clerks of tho Circuit Court, when affidavit is lllod that tho dofondnut has depr.r'.cd, or is about to depart, out off tUe Stftto, or luacoola Iiiiuaelf so that tUo proccsa cauuot bo eervfid upon hir^ 855 INDIANA. I oight per com. ;st paid. Floritln, payable per cent. it. There Is no I? due, and Uio no higher rate is rest. i of intcrcs' <'m-.' 3, pago2J9.) jorgia, payable in >r protest, arc 10 JI not bo allowed issago of tlii.j Act: isiona of tbts Lrsl i'ob. 8, 1850. (iea jn notes ami bills ifcrior court, or a debt bo matured ounty, or conceals well as by resident a^"^inted by tho meuta la writiug. ) attachment liiti an attacbraont» to 3 obtaiued by tho rest upon nil mr^'- t of money, Bball rwiso provided by or verbal, it shall or any less sum Dncy loaned, or in or corporation in poration iu this upon any contract d interest so con- sum duo to such Illjno'3, payable iro uriiformily (py acJ under protest, I or tlrafld at sight, that "no nolo, i-CEtment3 payable jrace, but shall bo tiangOj shall bo cn- rciiil Court, when It of.' tUo State, or IttUretL—Tbo legal iaterest in Indiana is sis percent., wbicb may botalcen la advance, If BO expressly agreed. Penalty for Violation of the Usury Latcs.—lt a greater rate of interest than as above ehall be contracted for, received or reserved, tho coatract shall not therefore, bo void ; but if it ia proved in any action that a greater rate than six per cent, per annum has been contracted for, tho plaintiff sha:i only recover his principal wit'.i six per cent, interest and costs; and if tho defendant ha.s paid thereon over six percent, interest, such excess of interest shall bo deducted trom tho plaintiff's recovery. If any action for a recovery of a debt, it is proved that previous to tho conwncnccment of tho suit tho defendant has tendered tho amount due, with legal interest, tho defendant shall recover costs, and tho plaintiff shall only recover tho amount tendered. Damages on Bills.— Damages, payable on protest for non-payment cr nonaccptanco of a bill of exchange, drawn ornegotiatcd within tho 8tato of Indiana, if drawn upon ony person nt any placo out of this State, aro, ot 5 per cent. Beyond such damages no interest or charges accruing prior to protest shall bo allowed, and tho rate of exchange shall not bo taken Into occount Foreign Bills.— Iho damages payable on protest for non-payment or non-acceptance ofa bill of cxc;:auge, drawn on any placo not in tho United States, are, on tho principal of such bill, 10 per cent. No damages beyond tho cost of protest aro chargeable against tho drawer or tho endorser ofcither species of bill, If upon notice of protest and demand of tho principal sum, Iho &umo id paid Sight Diils.—GTacQ la allowed ou ell bills of exchange payable In Indiana, whether sight or limu bills. tV/trorations, or individuals, aro not prevented, in discounting bills of exchange, from taking a fair rato of exchange between the place whoro it u bought, and tho placj wliero it Is payable, ,n addition to tho discount for intei'ost. But nuc! privilego of buying bills of exchange at less than par value, shall not be uaod tu disguiis a '"> ' of money At a greater rato of discount than the legal interest or discount Damap'* on £tZ2<.—Ko statute is tu force ia Ketttuokynpon tbe euUJoot oi , on Likud bll jxchonEa 366 Foreign Bills.— '\^horcs any bill of cxcnanpo, drawn on nny iiorcnn out of tho Unltdl !'tatc8, Bhall Ijo protested for non-payment or non-iicceptancp, It Pball bear ten per tout, jicr year In- terest from thoUay of protest, for notl n^fT Ihauciglitocnniontlis.tnUcsapayineut bo t=oonci domandiHl from tho party to bo charS''fl. t:^ucli iutcrest Kliall bo recovered up t > ibo limn o) thojudginont. and tho judgment Bliall boar lesal interest tlicivalter. Damages ou all otlioj billa aro disullowvl. [uoviaod Statutes, pages 103 and 194.] Sight BiUs. — Oraco ts allowed, by Rome banks on bills, drafts, etc payablo at tight, but tho point is not y. i fully settled in this Mate. •Collection of Dahli-~\. Tho plaintiff may havo an atiachmont against tho property cpf ihg defendant when tho luttor is n foreign corporation, or a non-rcsidci.t of this Jitato ; or, 2, \\ha has boon absent thorelrom four months ; or, y, lias departed from tho ; tato with luteal lo Ca- fraud Lis creditors ; or, 4 has lelt Iho county of bis rcsiden<'o to avoiil tho Eorvico f f : Earn- mo jti, or conceals hrnitolf that a tummouH cannot reach him ; or, 5, is about tor movohls property, or n raatcrlul part thereof, out or tho,- tato ; tr, has sold or convoyed hist ropcrty with tho luiont to defraud lii.^ creditors, or is ab:ut no to 'clI or convey, f- uch attachment I.* binding upon tho doPndaiit'a property in tho county from Iho tUno of tho delivery (f th« order to tho Shcrifr, LOUISIANA. Interett.—l. All dubtafihall bear interest at tho rate cf fivu per cent, from tho lliiic they bocomoduo, unless otherwise elipulatcd. (Act JIarch 15, 18o5.) 2. Conventional interest not, exceeding eight per cent, per annum may bj lontr.ujt ■ fui. —Ibid. 3. Tho owner of any promLssory note, bond, or written obligation for tho pa.anent o! money to order or to bearer, or transforablo by assignment, shall h.' -o tho right to c'jilccl tho whole amount of such promissory note, bond, or written obligation, notwiths;anding tiich promissory i.Oii>, bond, or written obligat.on may include u greater ralo of interest or discoinit than eight per cent lutcri'St per a:'.nutn. Provided that such obliKa'. ions shall not bear ui ro than eight per cent. Interest per annum after their maturity until paid. (Act of March 2d, 18G0.) Damajcs on i:i'J:. — Tho damages on bills of exchange, negotiated In Louisiana, p.iyu lo la other States, aro uuilbrmly 5 per cent. Forcifin Hill .—Tho damages o'.i foreign bills of oxchange, returned under i rotest, aro uni- formly (Statute of 183S) .....•• 10 per cent. Sight Hills —Thoro is no Etatuto upon this subject in Louisiana. A decision has been m:'.do in ouo of the inferior courts allowing three days' graco on sight blll.s, but the usago is to pay on presentation. CoUcdiox, o/Tkbts—A creditor may obtain an attachment against tho property of lii.s debtor upon aflldavit: 1, when tlio latter Is about Icavii.g permanently tho State before obtaining or executing judgment against him; 2, v,-hcn tho debtor resides cut of tho State; 3, when ho concoalshimself to avoid being cited to answer tj a suit, and provided tho term ol iiaymcnt has arrived. In tho absence cf tho creditor, t^^o oatli may bo made by hia agent or attorney, to tho best of his knowledge and bcllcC MAINE. Inter'est.~'Tho legal rato of interest in Maine is sis per cent., and no higher rate is allowcl on special contracts. [R. S. 322. Cap. 46, sec. 2.] renalhj for Violation of the Usury iaws. —Excosa of interest not recoverable, nor costs wbero cxrcss of interest has been taken ; but tho defendant may recover costs of tho party taking tho excess. Kxcc.'iS of Interest may bo recovered back by the party having paid it. Tho provisions do not extend to bona fide holders of negotiable paper for value without notice. [LI. S 323. Cap. 45, sees. 2 and 3. Laws of 18C2, ch. loO ] Damages on Vii:s.— Tho damages on bills of exchange ne.;;otlated in Slaino, payablo in other States, and returned under protest, aro as follows : [It. ti. 019. Cap. G2, eoo. 35.] 1. Now Hampshire, Vermont, J 'assachusctts, Rhodo Island, Connecticut, Now York, 3percent- 2. Now Jersey, Pennsylvania, Delaware, Maryland, Virginia, District of Columbia, South Caro- lina, Georgia, per cent. 3. All others, namely. North Carolina, Alabama, Arkansas, Florida. Illinois, In.liana, Iowa, _Kcutucky, Louisiana, Michigan, Mississlpp., J.il.ssourl, ■ hlo, Tennessee, Texas, AViscon- .consin, California, 9 per cinit. JSiglit Bills. — Grace is allowed on bills, drafts, checks, &c., payablo in this State at a flitnT» day or at sight, but not on those payablo on demand. [U. f. 264.] ■Collection of Debts. — In this State an original writ may bo framed either to attach tho goods cr estiito of tho defendant, or for w.int thereof to tako his body. All goods and chattels may 1)0 attached by tho creditor and held bs security pending anya nit again.st tho debtor. Such a ■writ will authoriz') an attachment of goods and estate of tho principal defendant, in his own hands, w well as in tho hands of tnistjcs. T.cal cstato. liable to bo token in excoutiou may bo pttaohed 367 MARYLAND. l):iyii lo ia nf a fiitnre' tnterett—Tho rovisofl constltntion of Maryland jirovldcs that tho rato of Intorost In the Statofchall not cxccrd six percent, per annum, nml no hifiher rato shall bo takenordemindoJ. And tho legilbturo ehall proviJo by law all uccestjary forlciturca and ponaltloa against usury. Pemiltki. — Any person gailty of usury Fhall forl'cit all tho excess abovo tho real sura or valuo ( f tl.o goods 01- cliatto .s ttctr.ally lent or advanced and tho Itgal Interest on such sum or va!i!0, which lodoituio .'^hall eniiro to tho boheilt, oi'auy del'ondant who shall plead Uiury, and prove tlio Fame. Tho plea must, howov r, Btato tho sum oramount ol'th !dobt,and tho plain- lillLaball liavo juda Damages on lUlls.— Tho daina;,'es on billa of cxclmngo negotiated in Maryland, paynblo in other btates, and returned under protest, aro uniformly 8 per cent. Tho cluiraanl is ontilloil to receivo a snin sunicicnt to buy another bill of tho sumo tenor, and eight per cent, damages on Iho valuo of iho principal kuui inontroncd la tho bill, and interest from tho timo of protest, a!id costs. 'I'lio protot of an inland bill must bo mado according to tho law or usago of tho Stato whero it ij payable. Praciica includes tho District of Columbia in this law of damages (Act of As3 mbly, 178J, cli, C8j; but it Is questionablo whether tho District bo within tho law, wliich provides only for States. Foreign Hills.— 'i'hu damages on foreign bills of cxchango roturuod under protest aro 15 per cent. Tho claimant is to receive a sum sufflciont to buy another bill of tho eamo tenor, and 15 per cent, damages on the valuo of tho principal aum montionod iu tho bill, and intoreat from timo of prolc:it, and costs. Sight Z!t7/«.— Grace is not allowed by tho Ban'AS on bills, drafts, chocks, olc. , payi»ole at sight. Collection o/ Debts.— X creditor may obtain an attachment, whether ho bo a citizeuof Maryland or not, agai:ist his debtor, who is not a citizen of this Stato, and not residing therein. If any citizen of tho State, being indebted to another citizen thereof, shall actually run away or abscond, or Becrc. remove himself from his placo of abode, with intent to cvado iho pay- ment of his just debt!.-, an attachment may bo obta'ued against him. Au attachment may bo laid upon d jbts duo tho defendant upon judgments or decrees rendered or pa-sod by any court of thistitato, undjudgiuentcfcondomuation thereof may bo Lad, as upon other debts duo tho (Ivifeuduut. MASSACHUSETTS. Inters!. —Tho legal rato of Interest in Massachusetts Is six per cent., and no higher rato Is allowed uu r.pocial contracts, Penally J >r Violation of the Usury Zai««.— No contract for the payment of money svi'.h Interest grca ; :• than six per cent, shall bo void; but in an action on such contract tlio deli :.d lilt shall rec a\ :• h s full costs, and tho plaintill shall forfeit three-fold tho amount of tho \> holj interest reserve i or taken. Damages on IJilts of Excnange. — Tho damagcs^on bills of exchange negotiated lu Massachu- setta, payable in other States, and returned under protest, aro as follows: 1. Bills payablj iu Maino, Now Hampshire, Vermont, Kbodo Island, Couuecticu., or New York, » per cent. 2. Bills payable in Xow Jersey, Pennsylvania, Maryland, or Delaware. per cent. 8. Bills payable in Virginia, District of Columbia, North Carolina, South Carolina, or Ocrgia 4 p T cent. 1 Bills payable elsewhere within tho United States or the Territories, 5 per cent 6. Uilld for one luindrcd dollars or more, payable at anyplace iu Massachuseits, not within fovcuty-flvo miles of iho placo whero drawn, Iper cent. Foreign Bills.— tho damages on foreign bills of exchange, returnoi under protest, areas follows: 1. Bills payable beyond tho limits of the United States (excepting places In Africa, »joyond iho Cape of Good Uope, and places ia Asia and tho islands thereof) shall pay tho cur- Tont rato of cxchango when duo, and five per cent, additional. 2. Bills payable at any placo in Africa, beyond tho Capo of Good Hope, or any placo . .. Asia •or the islands thereof, shall pay damages, 20 per eent. Sight BiVJs.— Bills of cxchango, drafts, etc., payable at sight, or at a future day ct.tain, within this Stat^, aro entitled to three days' grace. But not bills, notes, drafts, etc., payable (n demand. Jiotes on Demand.— lu order lo charge an indorSer, payment must be demanded within sixty days from its date, without grace, ( n any nolo payable on demand. Collection of DeJyts.—OHgmal writs may bo framed, either to attach the goods or estate ot the defcndaai, or for want theieol to take his body; or there may bo au original turamons. either with or without an order to attach tho goods or estate. All real estate, or goods and chatties that aro liable to bo taken iu execution, may be attached upon tho origir.al writ, in any action in which any debt or damages aro recoverable, and tnny bo liold as securtly ♦« sntisfv such judgment as tho ulaintiff may recover. 35fl uicbioan. /wtoTjf.— The lofc«l rnto of InteTest In lIlchiRan Is R«!Tt«n per cent. But it m lawAil f* parties tu Rtipuluio in torUing tor any Bum not cxcaodiDi; tou pt-r cent. Penalty/or Viot'iiionc/ the Utury Laws.— Varxica e\^g upon rontracra reserving over tea per cent, inlcrcst, may recover jmlRiucut lor tho priucipoJ unU legal rato of interest. Thcro ii «jo provision for locrivcriug back 'llcgal interest paiti, uuJ no pcuiJty lor receiviua it. t^ona /«;<: boIUcry of USUI ions ucsotiablo paper taken bol'oro maturity, witlioul notico of usury, may recover thn full amount of its lace. Damages on Hills.— Damapos on bills drawn or negotiated in ilicbigon oud payablo else wbcro ami protested nrj ns follows : 1. If payt bio out of tho United States, 5 per centi. L'. K payablo in Wisconsin, Illluois, Indiana, Ohio, Pennsylvania, or New York, 3 per cent, 8. If payablo in Missouri, Kentucky, New England, New Jersey, Eelawaro, Maryland, Vir ginia, or District cf Columbia, 5 per cent. i. If payiiblo in any other State or Tcnltory, 10 per cent*. Sight Hills.— Oraco ij allowed on all paper not [layablo on demand. Collection rf Debts.— Tho erounds of attachment iu ihii ttatocro: 1, that tho Uefoadant naa nb;;conc'.ed, cri.s about, to iibscond, or has concealed him. elf ; 2, thatlio has assiRuodo- concealed, or i.s about to rcmovo his property with a view to defraud : 3, that ho fraudulently contracted tlio debt, or incurred tho oblisation about which tho Euit is brought ; 4, that liu ia not (I resident cf tho State, or has not rcifidcd thcro thrco monUiaimmodialoly p^'c-coUiug t'le suit ; 0, that tho defcudonl is aforcigu corporation, MINNESOTA. Inte>-esi.—\na:Tcil for UDy legal iudcbtcdness Bb.ill bo at tho roto of $7 for $100 for a year unleiH a diilercat rato bo contracted for in wTitiDR, but no agreement or contract fjr n greato rato of interest than $liJ for every $100 for a year Bhall bo valid for tho csccs3 of intercf ovrv twelve per cent.; and all n^rcemcuts r.nd contracts shall bear tho samo rato of intcrcs* afu r they becorao duo as before, if tho rato bo clearly expressed therein. Provided, tho somo shall not exceed twelve jier cent, per annum. All judgments cr decrees, mado by any court In this State, shall draw interest at tho rata of six (0) per cent, per annum. [I*n\i-3 of ISGO, p. 220.] PcnaUjifor Violation of Interest Zau'.— Kscoes cl interest over 12 per cent, forfeited. Days of Grace.— On all bills cf exchange payablo at sight, or at n future day certain withia ihiat^tate, nndonall negotiablo promissory notes, orders and drafts, payablo at a futuro day c rtain within this State, In v.hicli there i.j not an express stipulation to tho contrary. When Cracenot allowed. — On bills olcxcliange, note ordraVt, payablo on demand, \Vhe7i presented for Payment, rving over tea rest. Thcro ia viui{ H. Bona of usury, may i payable clso 3 per cent. MarylauU, VJr tho dcrcatlant laa ossiRaoil o- 10 fraudulently ; 4, that ho ia rprccoJlDgt'ie 100 Torn year ct f jr u grcato ;cs3 or intcrof ato of inlcrc3» rided, thoBomo rest at tho rata forfeltod. certain withia at a futuio (lay itrary. omand. omissory notei lado ou tho 1st I'cbruary, aud tho Siato OS a ;ho day preced- rst day uf tha 'gcd 03 on oc- blmsolforbls 3cd within this y protested for > Bball, on dua ho tlmooftbo , together with aid amount of msci to, but within r non-payment property of a or, bos left th» sreign corponi- lered;likewi83 udulently con- w States, bow- ihire, Vermont nroperty of tho iulcut intent— , or until judg- mdant fur any or. Gcnerallj, tiickythepro' 359 MISSISSIPPI. InteraU—\ao legal rate of interest in Mississippi \3 six per cent, by tho act passed in March, 1860. Vafutga on 73t7/4.— No damages nro allowed for default in tho payment of any bill of cx- chango drawn by any person or ponsons within tho huito on any iicr.-ion or persona in any oth-or ; lato. O.i all duiucstio i r liiiMld bill.s [drawn on pursons within tiio btale], uud prolMtod for uon-|iaymcnt, live per tent, [.-soo act of ilay 1 1, ISoT. ) Fomga Jiitls.—lhti damat;eH un biiU of e.\chango drawn on persons without tho United StatrK, TLiurncd under prouht, aio 10 percent., with all iuciilental charges aiul lawlul intcrosU. ,%'i{jlit liills. — (Jiaco is ni)l allowed on b.ii.-^ of exciiaugo, dral'is, etc., luiyablo at sir/lit. C'oUedUiii of l)cbU.—\i\ allaclinicnt against tln) estato, including real estate, gooils, chattels^ &c., cf a debti r, when It Us shown that h 1 haa reino>\d, or U about removing or ab.scoudintj fioia tho State, or privately conceals himself. Attachment also lies against tho nroperty ol nonresident descoUcnls. It may bo obtained beforo tho debt is duo lor which it Issues, wLen thocroditur lias ground to hoiiovo that the debtor will romoTO WUli bis cUecUi out of tLc iilato^ 01 bati removed. MISSOURI. /n£erf»<.— Tho legal rate of interest in Missouri Is six per cent, whcc no other rate Id npycvfX upon. I'artiosmay ngreo in writing fur any larger rate, not oxcocdiug ten per cent. I'artiea may so contract as t(< compound tho interest annually. Penalty for Violation i'f the Usury //awi'.— Forfeit uro of tho entire interest; but judgment to bo rendered lor tho principal with ten per cent, interest, tho interest to bo approiiriuted to tho Bchool land. Tho damages allowed on bills of c-.xchango payable in other States or Verrltorics of tho United Slates r^turaed under protest, aro unifoinily 10 per cent. Oa bi!l3 of cxchango payable within tho State, 4 per cent. On ncgoliablo notes, W actually negotiated, 4 per cent. In thcsj last two cases no damages can bo recovered, if payment is made o." tendered within twenty days alter demand or notice of dishonour. Foreign Hilts. — Tho damages allowed on foreign bills of exchange, protested for nou pay- ment, aro -O per cent. Tho, damages allowed In all of tho abovo cases aro in I'eu of interest, char,L;es of protest and other c:;pcuses incurred previous to or at tho timo of giving uotico of dishonour, or maturity of ujtc or bill when uotico is required; but alter protest tho interest will bo allowed nu tho aggregiiio sum of principal and damages. Sight JUtts.—A statute of 1853-4 provides, that on bills of exchange, payable a: .Ught, grace shall not bo allowed. Collcctio.i of Debts.— An attachment maybe issued hero when tho debtor is not a resident ol tho!- tale; or if a resident, when ho absconds, absents or touccals hiiuseU; or is aljout. to remove hid property or fraudulently convey it, with a view to hinder or delay his creditors,, or when tho debt was contracted out of tlio t:I.2t?, and tho debtor baa Bccrclly reniovod bin cU'cc.i: into this State with intent to dcfrand. NEW HAMPSHIRE. Jnfercsi.— Tho legal rato of interest in Kew Hampshire is six p:r coat., and no moro (9 allowed ou contracts, direct or indirect. Penalty for Violation of the Usury Laws. — Tho pcnson rcceivin;:; interest at a higher than tho legal rate, shall forfeit for every such oll'cnco threo times tho sum so received. Damages on Dills. — No statute in forco in New Hampshire. /o)V(/;ii CiWs.—Xo statute In force in Now Hampshire allowing Jamages on foreign bills returned umlcr protest. Sight Dills.— 'So bill of exchange, negotiable promissory note, order or draft, cxcenl such as aro payable 0)1 rfcnianiZ, shall bo payable until days of grace have been allowed liioreon, unless it appear in Hio instrument that it was tho intention of tho parties that days ol graca should not be allowed. [Ueviscd h^t. 3S9, § 10.] Collection of Debts. — in this State a writ of attachment may bo issued upon the institution of any personal action ; au^ will hold real and personal property, shares of stock in corpora- tions, i)cws in churches, and tho franchise of any corporation authorized to rccclvfi tolls, until tlio period of thirty days from the timo of rendering tho judgment. NEW JERSEY. /rtfercs<.— Tho legal rate of mtorest in New Jersey is six per cent., and no higher rate o( interest is allowable (ju special contracts, except as provided in tho following nets : Tho legislature of New Jersey passed i\- -^ilowiug special net in March, 1852, supplementary to an act u'-aiust usury, app'oved April \', 1840, tho provisions of which act hjw ii;)ply nl.-o to tho counilo., ofHudson, B^^rgen and Essex, and to tho town of Paterson, in Passaic Coi:.'ity: Dc it cnacle'l. etc That upcu all contracts hereafter mado in tho city of Jonsey City, and in tho township of Hobokcn, In the county of Hudson, ialliis Stale, for tho loan of or forbearance, or Kiviug day of payment, for a/iy mfraoy, wares, merchandise, goods or chatte s, it shall bo lawful for any pcnion to toko tho vo^uo uf sovcu dollars for tho forbearance of ono hundred dollars 860 fbr year, irnd after that rate tor a grcftter or lest mim, or for a Idngw or tborttr period, uy thing coatulocd 111 tlio act, to which thin la n Bupplsmcnt, to tUocuiitrary uotNTitliiitaiidlDg. I\oviiUd, such contract bo made by and botwocn porsous actually locatoil lu either aaiU city ur township, or by persiuua not rualding lu Ma Stato. April C, 18j5. Tbo luitcr pruvlsu wau ainondod, " Trovldcd tbo CDntraRtinK parties, or either of thoiu, roftldo in olihcrof said placoH, or out of tho Ijtato." The following cbaugca bave sioue Im'Oii iniulo ho as tu iniiko it legal to chargo 7 per cout interest : Act, February 'Zl, 18U0, Acquackauonndo, Powulo County. Act, Fobmary 0. 1858, Bergen County. Act, Fcbruaiy 18, 18j8, Union County. Act, March 18, 1868, OUy onuUway. Act, March 20. 18&7, to ull t^uviiiga Inytitutlona la tho Stato. J}y Act of March 'J8, 18UJ, tho Icgislaturo authorized contracts at seven per cent, luterest by parties residing in Middlesex County. J'enallyfor VMlation of Iha Utury Laws.— Tho contract is void, aad tho wholo sum is for- reltod. Damaget on DilU o/ Exchange.— Tbon is uo Btatuto in force lu refcronco to damages on bllU of exchange. J'oreign nUli.—Thon Is llkowlso no Btatuto in force in reforouce to damagca on protested forcljin bills of exchange. £ight liitls.—Thal all bills of exchange or drafts drawn payable at slKht, nt anyplace within this tituto, other than thosu upon bauka or baul;.iu); association.'), shall bo dooruod duo and payable at tho expiration of thrco days' graco after the sumo shall bo presented for accoptuuco. Collection of Debts.-^Aa attocbmcnt may issue at tho instance of a creditor (or, iu hla nbscnco, of his agent or attorney), agulust tho property of a debtor when tho latter 13 about to abscond l^om tho ijiato, or is not » roeidont of tho tiuto, or ia a foreign coiporalion. NEW YORK. 7n{erU(.:h anil payablo troua fallows; New Jersey, 3 per ccul Year's day; istmas day; idcnt oftbo as regards oof tliodla- issagoof tbia uUay. [1849, roteet, ore 10 the interior, ssignment of 30 years. If lommitted to assignmcDt, n the case is 10 discharge, 'o essentially ,ors. Every _n his cslato intially pro- :empts him contracted, )f exchange. against the in of tbo OS- NORTH CAROLINA. InUrat.—Tho legal rate of interest in North Carolina ii sis per cent., and no higher rata la allowed on special coutnicta Penally fur ViolMion of i\» Viury /,au't.— A forfeiture of tbo principal and Interest ; and If usurloua Interest id collected, a liubilliyto pay doiiblo (bo umoiiutot' principal and interest paid— one half of (bo amount recovered tor tbo uto of tbu tiiate, tbo other half for tbo claimant. DanioDtt or% DUU.—tho damages on bills of exchange negotiated In North Carolina, pay- ablo in otber States, and returned under protest, aro unilbrmiy per cent. Foreign liilU.—Tho damagea on foreign biila of ezcbaugo returned under protest, ore as fullows : 1. Uilla payablo in any part of North America, except tho Northwest Coast and tho West Indies, 10 per cent, t, Oiilu puyublo in Maderia, tho Canaries, tho Azores, Cope do Vordo Islania, Europo and South Americn, 16 percent. & DIUm payable clsowhttre, 20 per cent. Sight ViUt.—Hf Tirtuo of an act of tbo Legislature, passed In January, 1840, grnco ti alloned on Mlla at tight, unless there is n stipulation to the contrary. Prior to that date tho «Mga iraa, not to allow graco on such bllla CoUfctten of VtUU. — An attachment may issue en tho complaint of n creditor, his agent, attorney or factor, against the property of a debtor when ho boa removed or is about to remove^ priTa(«(f ft*m tli« 6l«t«i w that tbo otiiaaiy procaci of htTrwUl not roach him. OHIO. tnUreit.—Vie law allowi Interest at six per cent per annum on nil money due, and no more. o same days of grace as now allowed by law on bills of cxchango i)ayablo on time." l)y ii statute passed in 1831, it is enacted that if money or other commodity bo lent or advanced upon unlawful interest, tho plainlilf shall bo allowed lo recover tho amount or value actually lent, but without ipterest or co.st. Uy an aet passed iu 1SG9, it is enacted that a debtor by bond, note, or otherwise, about to leave tho State, tho di.'bt not beng yet due, may bo rued and held to bail. Tho plaintilf must swear lo tho debt, and that ho d d not know tho debtor meant to remove at tho time tho con- tract was made. But iho writ must bo made returnable to tho term next succeeding tho «nalUii'-y of tho n.ite, etc. Collection of Debts. — A writ of attachment will Issue at the instance of tho creditor wherever residing, against a debtor when ho is a non-rcsidonl — or against a citizen who has been absent more than u year and a day: or when he absconds or is removing out of tho county; or con- c«al8 himself 80 that the oruiaary process of law cannot reach him. 3R3 I legal intercfll St, ho may le- ;anip, iwyable protest, aro as Pacific Ocean, auth America, go, payable at lU may, if dis- s against any laco or aboil.', jia own houso. , will bo ioduca tho cuuit, A tiou or a non- 2hing croUiUir higlicf rato Is I abovo Bix per Itatcs, and ro- protcst, aro 10 ' sight, which o auU payable St tho body of TI'o property reditor. nJ no higboi Carolina, pay- tcgethor with cigu bill, and St. outh Carolina, 1 tho West In lomcstic, pay law on billa oJ lity bo lent or louut or valuo ise, about to plaiatill'must limo tho con- accceding tho litor wherever a boci\ absent unty; or con- TENNESSEE. Interui— tho leRni rato of Interest l:i Tcnnessoo is six per cent., and no higher rat6 ran be recovered nt law. Coiitractaat agrciuc'r ruio of imcrost uro vu J as tj tho excess, and the Uuder in liublo to a lino of 110 to f l.COO. VenaUj/or VMalion vf thif. duij Laws.— l^wMo to au Indictment for misdemeanor. II convicicd, to bo uncd :i mini ijni, lej^d iliaa ilio wholo usurious interest taken and received, and no Hue to 1)0 ics tliaii ten do...-.r.j Tho borrower a:.d his judgment cr dilors may also, at any liuio wiiliiii six years i.ficr usuiy iiaici, recover it bacli Irom tho lender. Damagi'.t on 'ihUa.—Iha I'.aniigesi ii bi.lscreschango ucgoiiatod in Tennessee, payable ia other .' tales, and protested lor iiou-payraeut, aio 3 per cent. I'orcign />'W/#.— Tho damages allowed on Ibr^iga bills of exchange, returned under protest, lire ns follows : 1. 11' upon any person out of tlio Unite! f tate?, and in Korth America, bordouug Uiic. the eity iu tho State, it ia rmuir;;d, iu order to obtain an attach- Hwiil to ai« » bill iu cboocory > TEXAS. JntfeH$ — <)n i\ll WTitt<;n contracts ascertaining tho sums duo, when no rate of interest is •xprusaed, interest may bo recovered at tho rato of eight p"" cent, per annum. Tho parties to any written contract may stipulato fji any rato of interest, not exceeding twelve per cent, per annum. Judgments bear eight per cent. Interest, except where they aro "covered on a contract In writing which stipulated for moro, not exceeding twelve, iu which case they bear tho rale contracted for Ko interest on accounts, unless there bo au cxpresfi contract ; but only eight per cent, can ka recovered on a verbal contract. Contracts to pay interest on account will not bj presumed from previous course of dealing. fenallj/for Violation of the Usury Zau j.— I'orfeituio of all tho Interest paid or cliavgcd. Damages on Dills. — -Vu act giving Uatnagw upon protested drafis nud bills of ciichango drawn upon persons living out of the limits cf ihoh'tato, p-.issed Deccni'ocr, ISOl. tK(rno:s 1. Se it enact'.d hy the Lrgnlatnie (•/ the tSLatc of Texas, That the holder of r.uy protested draft or bill of exchange, draT>-u v.iiLiu tho li-ni'.i cf thij fciiatc, upon r.uy pt iscn or ^rsons living beyond tho limits of this ! tat ■, shall, after Laving Ii.\od «ho liabiliiy ..I'tho drawer or endorser of any such draft or bill of cicliajigc, as provided lor iu tho act cf Murc'.i 20, 1B18, bo entitled to recover and receive 10 per cent, on ihonmouut cf EULlidiaii or bill, OS damages, together with interest and cost of suit thereon accruing. I'rovida'., tUut the pro- visions of this act shall not bo so construed as to embrace drafts drawn by ^erso.u other than merchants upon their agents or factors. Sight Bills. — By usage, grace Is not goncniUy allowed on bil!.^, I'.ral't.i. etc., payable at iigM, but Iho rule is not invariable iu this State. Billt vf Exchange. — The general rule is that the holder of any bill ofcixliange may fix tho liability of tho Urawer (whtio bill has been accepted) or any enUor.ser, wiUiout protest f notice, by instituting suit against the acceptor bel'or j tho lirsl ter.i cf t l;o clisirict court to v. hich Euit can bo brought, {i.tation of the Usury Lawn.—'iha excess jt interest r.ccivod beyond six jiet cent, may bo rficoverod by action of assumpsit. Damaget o» liilU of Exchav ec—Ilaxo id lio 5^^t^to lu force hi Vtraoat iu vofiirnicc ti: damagM on protested bilii ot oxchauge. 364 Foreiftn Bills.— Tharo is no statute in forco in Vermont in referonco to damagfes on iiro- tcstea foit'igu bills of txcliango. Sipt't /?i7;,t.— Graco ia iint allowed on bill", drafts, checks, etc., payable at Fi.'lit. or oii bills »ud iiotea niado und p.iyablo within tho !~tato. [U. S. sxiii. 1 ] 'ctioti cfDehts.—'^rW.n orattachmcDt may issuo nsainst tho poods, chnttlfs or rstrto of tl- ndaiit, or for want (hereof, against hi3 body, bolero or nf'-r tho maturity of.' • laim. A ■sogaiustnon-ret;idcnts, or when tho defendant lias abseortljd Irom tho Ptatr, :iy bo coiu.^onccd by trustco iiroccss. VIRGINIA. Interest.— Tho legal rato of interest in Virginia is six per cent,, and no higher j-nto la rlloced on special contracts. PenaUyfor Violation of II i Vsur'j Laws.— All contracts for agreatvt ra.o U iui.ne.-it tlia.i Fix per cent, per annum aro void. Damages on Bills.— Tho damages on bills of cxchaufco neototiated ia Virginia, piyuLl"'' tthor States, and returned under protest, aro uuilormly a per cout. Foreign Bills.— Thi damages on foreign bills of exchange, returned under prolet:, ur.- uniformly, 10 per cent. Sight Bills. — Grace is not allowed by statuto or by usage on bills, i tc, payable ut liigUt. Collection of Debts. — Tho projierty ( f tho deCcudant, if a non-resident, or a resident who if abcut to removo himself or cQ'ects from the State, is liablo to attadimeat. An uttnchmcnt in eucli c«i£C3 wlU hold bel'oro tbo claim is duo end payable. "VVISCC/NSIN". '' •After Ja'2uary, 18C^ the legal rato of interest, by an itct of tho k-fislaturo, Is ecwu per cent. An usurious contract is void, and tho party loaning tho money U liablo to u penalty i!' '.''.rcc times tho usury in addition. , 'I'naltyfor Violation of the Usury iaws.— Whenever any 'son shall apply to any court l;i tlij.^ Stato to bo relieved in case of a usurious contract or sc ^rity, or when any person shall Ect u;) tho jilta of u^ury in any action or sniit instituted against liim, such person, to bo on- tiJed to suih relief or tho bonelit of such plea, shall prove a tender of tho !^j'_i:i',)al sum of woncy or thing loaned, to the party entitled to rcccivo the same. Act March -9, 1806. Damages on Bills of Exchavge. — Tho dam;i'"-s on bills of cschango, drawn or indorsed In Wisconsin, payable in either of tho States ud.ii,i;;i g that State, and protested for nou-accept- nuco or non-payment, nro 5 per cent. Ifdrawnupona person, or body politic or corporate, within either of tbo United Ptatcsi and not adjoining to that State, th • damages aro 10 per cent. Foreign Bills.— The damages en bills of exchange, drawn or endorsed in Wisconsin, payable beyond tho limits of tho United States, aud protested for ncnacceptauco or non-payment, r.ro [H. S. 18i9, p. 2.."], 5 per cent, together wii'> the current ratu of exchange at tho timo of demand. Sight Bills.— On all bills of cxchan';e, payable at sight, or at a future day certain, graco ehall bo allowed [U. S. 1849, p. 2C3], but not on bills ol exchange cr notes payable oa deme.iid. Coi: ■ '-.n of Dell! ■ —An attachment will hold against tho property of a debtor when ho has absco:.'.. .'.. or is aUn.i to abscond I'rom tho Stato; or has fraudulently assigned, disposed of or concealed his cflects ; or removed his property from tho State ; or whci) tho defendant ia a n in- resident or a foreign corporation. UPP2II AND LOWEIo CANADA. tnicrest.—Six per cent, is tho legal rato of interest, but any r.Uo agreed opon can bo rocov- <.rcd. Judgmcnt.s bear six per centum per annum interest from tho dato of entry, lianksaro not nllov.e I a higher rato than seven per cent. Corporations and associations authorized by law to borrow and lend money, unless specially allowed by some Act of i'arliatncut, aro proh - bitcd IVoin taking a higher rato of interest than six per cent. In^iirinre (.'(^r'n nics, however, aro authorized to take eight per cent. Bills of Kxchange. and Promissory Notrs. — Three days of grace aro idlowed on all ';il!s ;,i.d notes I lyablo within I'pper or Lower Canada, except wlieii drawn on demand . \Vl)>>n tho hist day < i .-I'aco falls on ! r.nday, or a legal holiday, it is piiyr.'.ile tho following day. AcccjUanccs mnti 1.0 in writing. No person or corporation in Ujijiir Canada can issuo notes for less than one dollar. I'rolest may bo made, and the jiarties to tho bill tr noto notiftcil on tho samo day tUo bill or noto is dishonoured ; but, in ciso of non-payment in Upper Canada, not buforo tlirco o'clock, I'.m., and in Lower Canada any timo after tho forenoon of tho la'^t liay of graco. Dishonoured inland bills or notes in Upper Canada, when protested, nnd in Lower Canada without protest, bear interest at tho rato of six per cent, from dato of protest, or in Lower Canada from maturity to time of payment ; but if interest is expressed to bo payalilo frotn a particular period, thcij from tho timo of such perlgd to tho timo of payment. Tho damages allowed upon piotcsted forolgn billa dra. n. Bold or tircotiated within fjppcr or Lower Cauada. 365 iraagts on pro- 'lit. nr on bills Ics or rstfto of rity (if ■•' - Liim. Ptalp. -.ly bo nitoiarllovTcd r iui'.'ivtft tlm.i iia, iviyi-.l.l" '■• fip l'roU'i■^ itr.' lo ut slgbt. resident who 1? lultiiclimciu iu TH, Is EC V. 11 per a to u penalty y to any court uy person sball irson, to bo on- !''_L:i',)al Eum v( •29, 18&6. or- indorsed In or nou-accept- ) United states; conpin, payablo payment, nro at liio 11 mo of certain, graco )lo oa donia'-id. jr when lie has disposed ol" or ndant is a n iii- can bo recov- ry. banlarly ail of Europe and Great Britain liavc adonted it. But tho cental system ia much prefcraTilo ti> tho Inishel mcthoil, and, no doubt, ■» 'U Boon become universal. Until it does, a little dlfflcxilty will bo experienced in buying and selling, as every vai-iation in tlie price per bushel or cental requires new work to determine tho rolativo values. Hogs, cattle, &c., liave been bought and sold by the cental system for years. To save time and labor, and insure correctness, a table has been prepared or.d givtn, as below. To find the value of f^rain, ve^^etables, &c., per cental, when the valuo per bushel is given- MuUiply the given price 2icr bushel by 100, and divide hy tlie number of lbs. in the bushel. Example.— For any articlo weighing 60 lbs. to the bushel. Say wheat is quoted n,t $2.10 per bushel, what is the valuo per cental? No. of lbs. in bush. . . CO^ §210.00— prod, after multiplying by 100. S3.50— %'aluo per cental. Example.— For any articlo weigliing 60 lbs. to the bushel. Com is quoted r.t COc per bushel, what is tho valuo per cental? No. of lbs. in bush. . . 56^ $60.00— prod, after multiplying L>y 100. $1.0714— value per cental. To find the valuo per bushel when tho price per cental is given, wo r6v«a« tlia abov riUe, and Multiply 111 the number of pounds in, the bushel, and divide by 100. If wlieat is wortli §3.25 per cental, what is tJic valuo per bushelf $3.25 — prico per cental. 00— lbs. in bnshel lOO; 195.00 $1.95— per bushel. The folloTvinj? tables show the valuo per cental at any given price ptt bnshel of t!ie r«> mlnent articles of merchandise. If tho value per cental should bo required at a rata pei buslicl not given in tlio tables, the desired result may bo found by adding the two nearcsf quoted rates together, and dividing by 2. For instance — tho price of wheat per cental is re- quired, when the quoted price per bushel is $1.05. Now, wo sec by tho table t>'\t at ?1.7 she! of Vao r^o id at ft rats pei he tTTo nearest per cental is re- lo t:"\t (It $1.64 icn$2 7Di added jughcl Per Per ighl. ) SO cental. 12 78 1-8 90 2 81 1-* 01 a 84 3-S 03 3 87 1-2 03 2 00 5-3 94 2 OS .1-4 05 2 90 7-3 Of. 3 00 07 .1 03 1-3 03 3 OC 1-4 l»9 3 09 3-3 00 3 12 1-2 •'■-■- — Tor Tor Per Per Per Per Per " ■"" " ■ - Per Per Per bshl. cent.il. bshl. SO 72 cental. b.shl. cental. b;ihl. cental. bshl. cental. 40 $0 S3 1-3 SI 50 .?1 04 $2 10 2-3 $1 "o $2 S3 1-3 ■*1 03 ?3 50 12 87 1-2 74 1 54 1-0 1 0(3 2 20 5-0 1 3S 2 87 1-2 1 70 3 54 1-0 il 01 2-3 70 1 5S J -3 1 Oi 2 25 1 40 2 91 2-3 1 72 3 53 1-3 40 05 C-0 7S 1 02 1-2 1 10 2 £9 1-0 1 42 2 03 5-0 1 74 3 02 1-2 2 , 1 OS 1-3 1 fl 1 75 1 IC. 2 41 2-3 1 48 03 1-3 1 60 3 75 ni : 1 12 1-2 m 1 79 1-0 1 13 2 45 5-0 1 50 3 12 1-2 1 £2 3 79 1-0 SQ 1 1 10 2-3 m 1 8:i 1-3 1 20 2 50 1 52 3 10 2-3 1 84 3 63 1-3 r>8 i 1 20 £-0 00 ] S7 1 " 1 22 2 54 1-G 1 54 3 20 5-0 1 80 3 67 1-2 CO ; 1 25 01 T \n ? ■' 1 24 2 53 1-3 1 50 3 25 1 63 3 01 2-3 fi2 1 29 1-0 04 1 'Jj ., 1 20 2 C2 1-2 1 58 3 29 1-0 1 00 3 05 C-0 Ci 1 B3 1-3 !)(5 2 CO ' 1 23 2 CO 2-3 1 CO 3 33 1-3 1 92 4 CO i;c 1 S7 1-2 OS 2 01 :.-o 1 , 1 SO 2 70 5-0 1 C2 3 C7 1-2 1 04 4 04 1-0 R8 1 41 2--! 1 CO 2 03 . -3 1 32 2 75 1 '■rl :. 41 2-3 1 06 4 03 1-2 VO 1 45 c-r. 1 02 2 12 1-2 . 34 2 70 1-0 1 Cn 3 45 5-0 2 00 4 10 2-3 For grain, &o., wc!2hinij50 ILs. to tic bushel, doirjlo tlio prico per bushCi wliicU wil! ^iv.) tho prico pir cental. FOR GRAIN, tc. V-T/iOniXO 55 Las . TO THE BUSHEL. Per Per 1 Per Per ' Pel- ro. [ Pep Per Per Per balil. cental. bK'..!. cental. bshl. cvutal. bshl. cental. bslil. cental. 32 §0 53.1 CO ei 20 SI 00 01 61.3 SI 3t S2 43.0 •?1 CS S3 C5,4 VA CI. 8 C3 1 23.0 1 02 1 65.4 1 30 2 47.2 1 70 3 09 Uj C5.4 70 1 £7.2 1 01 1 69 1 38 2 50.9 1 72 12.7 05 69.0 72 1 30.9 1 CO 1 92 7 1 40 2 54.5 1 74 3 10.3 43 72.7 74 : :.1.5 1 03 1 00.3 1 42 2 53.1 1 70 3 20.9 42 70.3 70 1 8S.1 1 10 2 00 1 44 2 CI. 8 1 78 3 £3.0 '14 80 78 1 41. r, 1 12 2 03.0 1 40 2 C5.4 1 80 3 27.2 40 E3.0 80 1 45.4 1 14 2 07.2 1 43 2 C9 1 82 3 30.9 43 87.2 82 1 49 1 10 2 10.9 1 CO 2 72.7 1 64 3 .34.6 50 90.9 84 . 52.7 1 13 2 14.5 1 52 2 70.3 1 60 3 GS.l 52 04.5 60 1 50.3 1 20 2 18.1 1 54 2 80 1 83 3 41.8 G4 03.1 63 1 CO 1 £2 2 21.3 1 50 2 63.0 1 00 3 45.4 56 1 01.8 00 1 03.0 1 24 2 25.4 1 58 2 87.2 1 02 3 49 58 1 05.4 02 1 07.2 1 20 2 29 1 CO 2 00.9 1 04 3 52.0 (iO 1 CO 94 1 70.9 1 23 2 32.7 1 C2 2 04,5 1 00 3 50.3 -2 1 12.7 DO 1 74.5 1 30 2 30.3 1 04 2 08.1 1 03 3 CO C4 1 10.3 i)S 1 73.1 1 32 2 40 1 CO 3 01.8 2 00 3 03,0 FOR GRAIN, &c., TVEIGmNG 50 Ens. TO THE nUSHED. Tcr Per Tor Per 1 Per Per Per Per Per Per bshl. cental. bshl. cental. bshl. cental. bshl. cental. bshl. cental. 20 J?0 C5 6-7 62 .50 92 C-7 j SO 84 81 50 SI 10 S2 07 1-7 SI 43 ?2 C4 £-7 22 ■89 £-7 64 00 3-7 ' 80 1 53 4-7 1 13 2 10 5-7 1 60 2 07 0-7 24 42 C-7 60 1 00 1 68 1 57 17 1 20 2 14 2-7 1 52 2 71 C-7 20 46 8-7 68 1 O.T 4-7 ' 00 1 CO 5-7 1 22 2 17 C-7 1 54 2 75 28 50 60 1 07 1-7 02 1 04 2-7 1 24 2 21 3-7 1 60 2 73 4-7 80 63 4-7 02 1 10 5-7 04 1 77 0-7 1 £0 2 25 1 63 2 82 1-7 32 57 1-7 C4 1 14 2-7 00 1 71 3-7 1 2S 2 2.8 4-7 1 CO 2 65 6-7 84 CO 5-7 CO 1 17 C-7 03 1 75 1 30 2 32 1-7 1 02 2 89 £-7 8o C4 2-7 C3 1 21 3-7 1 00 1 73 4-7 1 32 2 35 5-7 1 G4 2 02 0-7 S8 07 C-7 70 1 25 1 02 1 62 1-7 1 34 2 39 2-7 1 CO 2 00 3-7 40 71 3-7 72 1 28 4-7 1 04 1 65 6-7 1 30 2 42 C-7 1 C3 3 00 42 75 74 1 32 1-7 1 00 1 80 2-7 1 33 2 40 3-7 1 70 3 03 4-; 44 78 4-7 76 1 35 6-7 1 08 1 02 C-7 1 ■<0 2 60 1 72 3 07 1-7 46 82 1-7 78 1 39 2-7 1 10 1 95 0-7 : 2 63 ■!-7 1 74 3 10 6-7 4fl 85 5-7 80 1 42 C-7 1 12 2 00 2 67 1-7 1 76 3 14 £-7 50 80 2.7 n 1 '0 p , 1 U 2 03 4 y 1 i . Vj 2 C3 0-7 1 78 .3 17 C-7 B6d AETTHMETIC. FOR GRAIN, «e.» WEIGHING 60 Lb8. TO THE BUSHEL Per Per Per Per Per Per Per Per Per 1 Pfir ))m1iI. cental. l)3hl. SO 93 cental. bshl. cental. l.Slll. 1 centiil. bshl. cental. 40 42 $0 CO 2-3 $1 03 1-:J ai 66 $2 CO §2 14 $3 60 2-3 4'^ 72 $4 53 1-S 70 1 00 1 CO 2-3 1 63 2 63 1-3 2 16 3 CO 2 74 4 50 2-3 44 73 1-3 1 02 1 70 1 60 2 CO 2-3 2 18 3 03 1-3 2 70 4 00 40 70 2-3 1 04 1 73 1-3 1 62 2 70 2 20 3 CO 2-3 2 78 4 03 1-S 48 80 1 00 1 70 2-3 1 04 2 73 1-3 2 22 3 70 2 SO 4 CO 2-3 50 83 1-3 1 08 1 SO 1 CO 2 70 2-3 2 24 3 73 1.3 2 82 4 70 52 54 80 2-3 1 10 1 63 1-3 1 OS 2 SO 2 20 ! 3 70 2-3 2 84 4 73 1-3 00 1 12 1 80 2-3 1 70 2 83 1-3 2 23 1 3 80 2 80 4 76 2-3 CO 93 1-3 1 14 1 90 1 72 2 80 2-3 2 30 j 3 63 1-3 2 88 4 80 53 90 2-3 1 10 1 03 1-3 1 74 2 90 2 32 3 60 2-3 2 90 4 83 1-3 CO 1 00 1 18 1 90 2-3 1 70 2 93 1-3 2 34 3 90 2 92 4 80 2-3 C2 1 03 1-3 1 20 2 00 1 78 2 00 2-3 2 30 3 93 1-3 2 94 4 90 64 1 00 2-3 1 22 2 03 1-3 1 80 3 00 2 38 3 90 2-3 2 90 4 93 1-3 60 1 10 1 24 2 00 2-3 1 82 3 03 1-3 2 40 4 00 2 98 4 96 2-3 C3 1 13 1-3 1 20 2 10 1 84 3 00 2-3 2 42 4 03 1-3 3 00 6 00 70 72 1 16 2-3 1 28 2 13 1-3 1 80 3 10 2 44 4 00 £-3 3 92 5 03 1-3 1 20 1 30 2 10 2-3 1 88 3 13 1-3 2 40 4 10 3 04 6 00 2-3 74 1 23 1-3 1 S2 2 20 1 00 3 10 2-3 2 43 4 13 ^-^ 3 00 5 10 7G 1 20 2-3 1 34 2 23 1-3 1 92 3 20 2 60 4 10 2-:; 3 08 5 13 1-3 78 1 30 1 30 2 20 2-3 1 94 3 23 1-3 2 52 4 20 3 10 6 10 2-3 80 1 33 1-3 1 38 2 30 1 90 3 20 2-3 2 64 4 23 1-3 3 12 6 20 82 1 30 2-3 1 40 2 33 1-3 1 93 3 30 2 66 4 20 2-.'> :• 14 5 23 1-3 84 1 40 1 42 2 30 2-3 2 00 3 33 1-3 2 68 4 30 ;; 10 5 20 2-3 80 1 43 1-3 1 41 2 40 2 02 3 30 2-3 2 CO 4 33 1-y 3 IS 6 30 83 1 40 2-3 1 40 2 43 1-3 2 04 3 40 2 02 4 CO 2-.'l 3 20 5 33 1-3 90 1 60 1 48 2 46 2-.; 1 2 00 3 43 1-3 04 4 40 3 22 6 30 2-3 02 94 90 1 63 1-3 1 60 2 60 1 2 OS 3 40 2-3 CO 4 43 1-3 3 24 5 40 1 60 2-3 1 62 2 63 1-.! 2 10 3 60 2 68 4 40 2-3 3 26 5 43 1-3 1 CO 1 54 2 60 2-ci ■-' 12 3 53 1-3 2 70 4 50 3 28 5 40 2-3 Per Per bshl. cental. 2 72 !$4 63 \-% 2 74 4 50 2-3 2 70 4 .00 2 78 4 03 1-3 2 SO 4 CO 2-3 2 82 4 70 2 84 4 73 1-3 2 80 4 76 2-3 2 88 4 SO 2 90 4 83 1-3 ^' 92 4 86 2-3 -• 94 4 90 2 00 4 93 1-3 2 08 4 06 2-3 ^ 00 5 00 3 92 5 03 1-3 ) 04 h 00 2-3 1 00 5 10 ; OS 5 13 1-3 ! 10 5 10 2-3 3 12 6 20 '. 14 5 23 1-3 ; 10 5 20 2-3 1 18 5 :!0 J 20 5 S3 1-3 i 22 6 30 2-3 J 24 5 40 ( 26 r. 43 1-3 5 28 6 46 2-3 the interest of $448.12 for 1 month. 229.70 " 2 mo8. 664.40 " 3 " 7J2.60 " 4 « 964.30 " 5 •' 667.60 " 6 •' 1267.70 " 7 " 1461.12 " 8 " 48.48 " 9 " 194.14 •' 10 " 3344.20 "11 .' 1616.60 •' 12 " ti It (I 1718.80 " 16 •' 2120.40 " 2>oarBand 1 nionll,, 369 RULE TO FIND THE INTEREST FOR DAYS. Por'fi*'!?/'"''''- •:"'"«•' If *'"' ,^'"''"'' ^" '*>" principal, and it becomes the interest d«v ' pf; t; • **'-,'^'^^'^' ''»^. '^''''""t "'I'-ts of the inteict found as above for 6^ day?. For other nunabers. multiply the interest found a. abov. for eduyBbv >/ fh„ «iven number of days. The product is the answer. 'or bauys by y^ the KXAMPLK. !• Requiied 2. 8. •< 4. 6. «♦ Ob •• 7. «• 8. « 9. 10. 11. 12. 13. 14. 15. « 16. 17. 18, 19. 20. •< 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. «« 36. 37. " 38. •• 39. «« 40. «« 41. <« 42. " 43. «« It It It It II ' fSWBR. $2.24 2.297 9.966 14.25 24.107 20.028 44.369 58.444 2.181 9.707 183.931 96.996 137.504 265.05 5.5.636 311.219 407.998 475.653 493.724 808.752 .05 .007 .15 .391 .665 3.041 8.846 6.328 9.603 6.965 14.143 7.323 6 325 8.004 16.4.53 15.399 .204 .123 .112 .848 .448 1.834 , , ^^'f ?'''«• though admitted to be sliRhtly inaccurate, haa been sustained by iudi- o.«l decisions, and it s almost universally used by merchants in this country itt based on the supposition of the year being divided into twelve equal months oTso Ll e.oh-360day8. It therefore gives the interest ^, = ^^ part to^Zlch ' ^' A /^. Y ( r ' <- HiJ iV f^-^l ^^r- / / ''c /'V* / /^ /?^ yv^r^ M^k-'it f-y^ f, /' ^F'^^'i^Htmmmm W .; 4 li O JJ a