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Las diagrammes suivants illustrant la mAthode. 1 2 3 1 2 3 4 5 6 mmmmmm ■E f 1 1 I ■ ' hri.* yf ^ EATON & FRaZEE'S COMMERCIAL ARITHMETIC FOB |h-. ■ ., ■ SCHOOLS, ACADEMIES, COLLEGES, BAHKERS, MERCKIIITS imD MECHimiCS, KMBBACINO . , J ^ V AN EXTbNSIYE COURSE OF THEORY AND PRACTICE. > COMPILED B\ T. A. BRYCE, M. A., LL. D. BEYISED AND COBREOTED BY % . ^K.iv- A. H. EATON, PBINCIPAL OP THE COMMERCIAL COLLEGK, , / ST. JOHN, N. B.» ^ "■ ' -. *"■ ; AND .:•■ f ■•'■'" ^ ■.;'i^/. J. C. P. FRAZEE, PRINCIPAL OF THE COMMERCIAL COLLEGE, HALIFAX, N. S. .."tff..-,K-- V iir .fi -f ArT? • BT. JOHX. N. B.. AKD HALIFAX. K. 8.. PUBLISHED BY EATON & FRAZEK. 1874. REVISOES' PREFACE. «♦> This Work, originally compiled by T. A. Brycf, M. A., LL. D., has undergone extensive alterations in the hands of the proprietors. In order to make it more useful as a Text Book in the Schools and Acadevnies of these Provinces, where it is already extensively used, it has been entirely re-written as far as the Exercises in Simple Proportion. The Explanatioi-S have been shortened, and put in such a shape as to be more easily understood by the unassisted student. The Exercises are also nearly all new, and are of such a nature as, it is hoped, will more speedily conduce to the advancement of those into whose hands the work may be placed. The articles on Equation of Payments and Averaging Accounts have also been re-written, and these subjects are handled in a manner that cannot fail to ,be appreciated by Teachers, Accountants, and Mercantile men. The remaining partF of the work have merely been cor- rected where it was found necessary. 4,? ' •; ;i ,,f- ^ :■ X^i n \\ PREFACE. THOTTOn elementary works on Arithmetic are in abundance, yel it seems desirable that there should bo added to this an extensive treatise on the commercial rales, and commercial laws and usages. It is not enough that the school-boy should be provided with o course suited to his age. There must bo supplied to him something higher as ho advances in age and progress, and ncars the period when ho is to enter on real business life. The Author's aim has, therefore, been to combine these two objects, and to produce a work adequate to carry the learner from the very elements up to the highest rules required by those preparing for business. As the work proceeded, it was found necessary t"* exf'nd the original programme considerably, and, therefore, also the limits of the book, so as to make it useful to all classes in the community. In carrying out this plan, much care has been taken to unfold tho theory of Arithmetic as a science in as concise a manner aa seemed consistent with clearness, and at the same time to show its applications as an Art. Every effort has been made to render tho business part 30 copious and practical as to afford the young student ample information and discipline in all the principles and usages of commercial intercourse. For the 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 Homans, Esq., New York City, Editor and Proprietor of the "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 mere •calculating machine that can work but z certain round, and is almost «are to be at fault when any novel case arises. The ezplanations PREFACE. are, of course, more or less tho result of reading, but, nevertheless, they aro mainly derived from personal study and experience in teaching. The great mass of the exercises arc likewise entirely new. though the Author hus not scrupled to make selections from some el the most approved works on tho subject ; but in doing so, he haa confined himself almost entirely to such questions as are to he fount? in nearly all popular books, and which, therefore, are to bo looked upon as the common property of science. Algebraic forms have been avoided as much as possible, as being unsuited to a largo proportion of those for whom tho hookas in- tended, and to many altogether unintelligible, and besides, those who understand Algebraic modes will have all tho less difficulty in understanding tho 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, but tho intelligent teacher aad learner will often find it necessary to depart from that order. (See suggestions to teachers.) Every one will admit that rules and definitions should be ex- pressed in the smallest possible number of \words, consistent with perspicuity 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 prin- ciple into his mind, as his rule, without the encumbrance of words. Copious exercises are appended to each rule, and especially to- the most important, such as Fractions, Analysis, Fercentage, with its applications, &c. Besides these, there have been introduced extensive collections of mixed exercises throughout the body of th& woirk, besides a lai^e number at the end. The utility of such miscellaneous questions will be readily admitted by all, but the reason why ^ey are of so much importance seems strangely over- looked or misunderstood even by writers on the subject. They are spoken of as review exercises, but their great value depends on some* thing still more important. An illustration will best serve here. A class is working questions on a certain rule, and each member of the class has jvst heard the rule enunciated and explained, and therefore readily applies it. So far one important object is attained, jAm., freedom of operation. But something more is necessary. The fuefage. 't. 'learner nmst be taught to disoem what rule is to be applied for the solution of each question proposed. The pupil, under careful teach' ing, may bo able to understand fully every rule, ard never con- found any one wLu any other, and yet be doubtful what rule is to bo applied to an individual case. Tho iniDcollaneous problems, therefore, are intended not so much as exercises on tho ojyerationt of tho d-lferent rules as on tho mode of applying these rules ; or, in other words, to practice the pupil^ in perceiving of what rule any proposed question is a particular case. Great importance should be attached to this by the practical educator, not only as regards readi^ ness in real business, but also as a mental exercise to the young student. ■i''. » -. v^- ; t ■.■i/,.: . Tho Author is far from supposing, much less asserting, that tho work is complete, especially us the whole has been prepared in less than the short space of six months. It is presented, however, to tho public in the confident expectation that it will meet, in a great degree at least, the necessities of the times. With this view, there are given OAtensivo collections of examples and exercises, involving money in dollars and cents, with, however, a number in pounds, shillings and pence, sufficient for tho purpose of illustration. This seems necessary, as many must have mercantile transactions with Britain and British America. The Rule 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 llule for finding tho Cube B«ot is a modification of that given 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, as in Whole Numbers Addition is the Bule that regulates all others ; 50 in Fractions, which originate from Division, we see, in like manner, ':hat all other operations result from Division, and, in con- nection with it, Multiplication. Several subjects, commonly treated of in works on Arithmetic, have 1 3i;n omitted in order to leave space for more important matter bearing on commercial subjects. Duodecimals, for example, have been omitted, as that mode of calculation i» now virtually superseded ?i. PREFACE. by that of Dooimals. Barter, too, has been passed by, as qnestion? of that olass can easily be solved by the Rule of Proportion, i»I:ich has been fully explained. The sabjoot of Analysis has been gone into at considerable length, and it is hoped that the new manner in which tho oxplana* tions and solutions are presented, and the extensive collection of exercises appended, will contribute to make this a valuable part of the treatise. The view given of Decimal Fractions seems the only true one, and calculated to give the student clear notions regarding 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 beeween Dedrnvah and Decimal Fractions has been ignored as being " A distinction without a difference." D-cimah is merely a short way of writing Decimal Fractions ; thus, .7 is merely a convenient mode of writing /g. These differ in form only, but otiherwise are as perfectly identical as f and g. The contracted methods of Multiplication and Division will be found, after soiiio practice, extremely useful and expeditious in Decimals expressed by long lines of figures. The averaging of Accounts and Equations of Payments, Gash Balance and Partnership Settlements, have been introduced as essential parts of a oommeroiai eduo&tion, and, it is hoped, will form a most important and usefUl study for those preparing for business^, and probably a safe guide to many in business who have not sys- tematically studied the subject ■'X. ';''*.'»^ ■Ji ifi" ■^:rk.SJ- '!;:v.»'.1' <:■;»>■-•- -t •• >1^>4 '4; 4> :;.» »-i\';' M , '■. ./»CJ!: / » SUGGESTIONS TO TEACHERS. '..i .')•.:(• Ths author would first refer to the remark made in the Prsfaoe that he does not expect that the Teacher will follow the logioal order adopted '.q the ]ix)ok, and oven advises that ho should not do tui in many cases. He knows by ozperionco that the same order does not suit all students any more than the samo medical treat- ment suits all patients. The ooure requires to bo varied according to age, ability and acquirements. The greatest difficulties generally present themselves at the earliest stages. What more serious diffi- culty, for example, has a child to encounter than the learning of the alphabet ? Though this is perhaps the extreme case, yet others will be found to be in proportion. For b^inners, therefore, wo recom- mend the following course. Let the elementary rules be carefully explained and illustrated by simple examples, and the pupil shown how to work easy exercises ; this done, let the whole be reviewed, rud exercises of a more difficult kind pro^K)8ed. The decimal coinage should then be taken up. In ex- plaining this part of the subject the teacher ought to notice carefully that the operations in this case di£fer in no way from ^hose already j(one through in reference to whole numbers, except in the preserv^'ng of the mark that separates the cents from the dollars, usually called the deoinal point. The next step ought to be the whole subject of denominate numbers, and in illustration and application, the rule of practice. After a thorough review of all the ground now gone over, Simple Proportion may be entered upon, using such questions as do not involve Fractions. Then, after a course of Fractions has been gone through, Proportion should be reviewed, and questions which involve Fractions proposed. After this it will generally be found de- arable to study Percentage, with 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 the strongest manner possible, impress on thv minds of teachers the great ut"'.ty of frequent reviews, and especiallv of constant ezeroise in the addition of money oolomns T viii. RUOOE8TION8 TO TEACnERS. To make the oxoroiscs under each rule of progressive diffioulty, as far 08 poflsiblo, has boon an object kopt oonstantlj in view, as also to give caeh oxoreisc the semblance of a real question, for all persons, ospceially the young, take greater interest in exercises that assume the form of reality than in such as are merely abstruot ; and, besides, this is a preparatory exercise to the application of the rules afterwards. At every stage the greatest care should be taken that the leamor thoroughly understands the meaning of each rulq, and the conditions of each question and the terms in which it is expressed, before ho attempts to solve it. The Teacher should not always be talking or working on tho black-board ; ho should require the pupils to speak a good deal in answer to questions, and also work much on their slates, and each in his turn on tho board for illustration to the rest. '.^t :^i\ ><> Finally, it is suggested to every Teacher to keep constantly be- fore his mind both of tho two chief works he has to accoiuplish. Fir it, tho dovolopomont of tho mental powers of his pupil ; and, secondly, imparting to him such knowledge ad he will require to use when he enters upon life, either as a professional man, or a mer- chant or clerk. Some seem 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 errror, for the more the mental powers are cultivated, the more readily and rapidly will any species of knowledge bo apprehended, and the more surely, too, will it be retained when it has been mastered. Mental culture is at once the foundation and the means ; the other is the super- structure raised on that foundation and by that means ; or it may be compared to a great capital judiciously embarked in trade, and often turned, and therefore yielding good profits. It frequently happens, however, from the peculiar circumstances of individuals and families, and even communities, that young men require to be hurried into business, so as to be able to support themselves ; but even in such cases the desired object will be much more readily and securely attained by such a course than by what is usually and not inappro- priately called " Cramming." Every effort has been made to give to this book the character here recoiunended, especially in the explana* torypartfl. ^vw^ a •"Ml- i '- >i , i'lAi. *%•* **iv^J'' SUGGESTIONS TO COMMERCIAL STUDENTS. ii , Tbk foregoing suggestions are addressed dircetly to the Teacher, but a careful eonsidoration of thoui by the Student will, it is hoped, be found highly profitable. A few additional hints are Hubjoined for the benetit of those seeking a liberal and praetical oommeroial education. As in all branches, so in Arithmetic, it is of the utmost conse- quence to digest the rules of the art thoroughly, and storo them in the memory, to be reproduced when required, and applied with acouroey. But this is not enough ; something more is needed by the Student. To be an eminent accountant he roust acquire rapidity of operation. Accuracy, it is true, should be attained first, especially as it is the direct means of arriving at readiness and rapidity. Acou<- racy may be called the foundation, readiness and rapidity the two wings of the superstructure. Either of these acquirements is indeed valuable in itself, but it is the combination of them that constitutes real efieotive skill, and makes the possessor relied upon, and looked up to in mercantile circles. Some one may ask, " How are these to be acquired? " The answer is as simple as it is undeniably true; only by extentive practice, not in the counting-house or warehouse, indeed, though these will improve and mature them, but in the school and college, so that you may take them with you to the busi- ness office when you go to your first day's duty. Oo prepared is a maxim that all intelligent business men will affirm. Be so prepared thaC you will not keep your customers waiting restlessly in your office or warehouse while you are 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* tieed in the note at foot of page 18, is not to use the tongue in oalcu- lating but the eye and the mind. Nor should the course of self-disoipline end here. ?^o be an ex- pert aocountant even, is but or' part, though an important one of a qualification for business. Study Oommeroial geography— oommer^ oial and international relations— politioal eoonomy — iadSs, &o., &o. mm Z. SUGGESTIONS TO OOMMEBCIAL STUDETS. Study even politics, not for their own sake but on account of the manner in which they affect trade and commerce. Do not, except in the case, of some aerious di£Boulty, indulge in the indolent habit of asking your teacher or fellow student to work the question for you ; work it out yourself ^rely upon your self, and aim at the freedom and correctness which 'U give you confidence in yourself, or rather in your powers and acqairements. Another can* tion will not be out of place. Many students follow the practice of keeping the text book beside them to see what the answo^ is; this has th'^ same effect as a leading question in an examination, being a ffsxde to tho mode by seeing the result. Study and use the mode to come at tho result ; gain that knowled^i,^ of principles and correotnese of operation that will inspire the confidence that your answer is cor- rect without knowing what answer the text bouk or the teacher may assign to it. Th'iro are t-;70 things of such constant ooonrrenoe and requiring such extreme accuracy that they must be specially mentioned, — they are tho addition of money columns &nd the making of Bills of Par* eels. Too muc^ care and practice can scarcely be bestowed on these. '/, ;,; ik. TABLE OF CONTENTS. ''''' ••• h* bi. Arithmetic, ... ^• Nunieraticn, ... - Notation, Addition, Subtraction, Multiplication, ... Division, ... .. >*•¥' Canada Currency, Properties of Numbers, Greatest Conimon Divisor, Least Comnaon Multiple, Fractions, Beductiou of Fractions, Multiplication of Fractions, Division of Fjactions, Least Common Denominator, Addition of Fractions, Subtraction of Fractions, Decimal Fractions, Addition and Subtraction of Docimnls, Multiplication of Decimals, Division of Decimals, Denominate Numbers, Tables of Weights and Measures. Canada and U.S. Money, Avoirdupois, Produce, Long or Linear, „, Square Measure, " ..; Land Measure, . «.. Cubic or Solid, - .... Measure of Capacity, Liquid and Dry, X^UXci ••• St* ••• Angular and Circular, Miscellaneous, ... ... Metric, ... ..; Lo^ Measure, Surfaces, ... '" :.;<#!. :' Weights, Capacity, Beduction of Denominate Nv js, ... Addition " Subtraction, " " Multiplication " Division " The Ctatal, Longitude and Time, Batio and Proportion, Simple Proportion, Oompoond rroportion, Page. t** 13 • •• 14 1 •• 16 • •• 16 ••• 19 *•• 21 ■ •• .26 ... 31 • •• 34 • •• 36 • •• 37 *4* 39 • •• 40 • •• ■ 42 «•• 46 • ■• 49 • •• bl • •• 63 • •• 64 • • • 59 • •■ 61 • •• 64 ... 66 *•» S6 'm* '. •67 ...'• 67 »•■ 67 68 • •• 68 • •• 68 M a «•• ■ •• 70 • ft •" ■'-" • • ■ 70 • •• • •• 71 • •• • •• 71 • • • • •• 71 • •• • •• 71 • ■ • • •• 72 • •• • •■ 73 • «• ■ •• 79 • •• • •• 81 »•• • •• 88^ • •• • •• 86 • • • • 88 • • • • 90 • •• ■•• 93 • <•• • •• 94 • •• »•• lOL ^■H^pmnmi zu. INDEX. Analysis and SynthesiB, Fntcttce, Accounts and Invoices, Percentage, Interest, Simple ••• Commercial Paper, Partial Payments, C!ompound Interest, Discount and Present Worth, Banks and Banking, Bank Discount, Commission, ... . Brokerage, ... ;- ... ' Insurance, ... ... Life ; '. ... Profit and Loss, " ..♦ Storage, ... ... General Average, Taxes and Customs Duties, Stocks and Bonds, Partnership, • .. Bankruptcy, .t Equation of Payments, Averaging Accounts, Cash,Balance, Account of Solos, Alligation, ... .. Medial Alternate Money-rlts Natui.^ and Value, Paper Currency, Exchange, American Sterling Arbitration Involution, Evolution, Square Boot, .. .. Cube Root, . . . . Progression, . . ^ . by a common difference, by Batio, . . Annuities, Partnership Settlements, Conamercial Questions, Mensuration, of Solids, Piling of Balls and Shells, Mensuration of Timber, Miscellaneous Exercises, Foreign Qold Coins, Silver Coins, ■■''■ Fxes. 1 • •• • • t • ft Ill i •«• • •• • •• 119 ll •t* • •• '•«• 122 \1 • •• • •• • •■ 129 1 • •• •«• ■ •» 1S4 • ^r . • •• • •• • •• 135 ■ ' ! • •• • *• ^;--"»r ••• 149 f • •• • •• • •• 158 • •• • •• 168 "-'/" • •• • •• ''N>^/. ■• ... 171 • •• • •■ K'i' ••• 173 • •• • •• .".' ' ••• 175 rliv.V • •• • •• • •« 180 • •• • •• ••• 184 • ■• • •• • •• 189 • •• • •• • •• 195 • •• • •• / 198 <•• • •• / ••• 204 • •• • •• • •• 206 • • • • • • 209 • • • • , • • 212 • • • • • • 221 • • • • • • 226 • • • • • • 228 • • m m • • 235 • • m • • • 241 • • • • • • 243 • • • • • • 248 • • • • • • 248 • • • • • • 249 • B • • • • 254 • • • • ■ ' . • • 256 • • « • • • 257 • • • • 4 • • 258 • • • • « • 263 • • k • • • 267 • • • • • • 272 • • • • • • 274 • • • • • • 275 1 • • • • • • • 281 • • • • • • 286 • • • • - ■« • * 286 • • • .• • • 295 • • • • « • 302 • • • • • • 306 • • • • • ■ 315 • ■ ■ • • • 320 • • • • • • 328 • • • • • • 329 • • • • • • 331 • • • • • • 334 • • • • t • 342 34C ''fr»t:i'--j^- 34!: •w ■ l' ARITHMETIC. '^ •."V::r:.;.^' Article. 1. — Quantity is anything which can be increased or diminished. Thus, numbers, lines, space, time, motion, and weipfht are quantities. Art. 2. — Mathematics is the science of quantity. Art. 3. — The fundamental branches of Mathematics are Arithmetic, Algebra and Geometry. Art. 4. — Arithmetic is the science of niuubers, and the art of computing by them. Art. 5. — A Number is a unit or a collection of units. Art. 6. — A Unit is a single thing, or one. Art. 7. — A Problem is a question proposed for solution. • Art. 8. — A Theorem is a truth to be proved. f Art. 9. — A Demonstration is a process of reasoning by which a proposition is shown to be true. Art. 10. — An Axiom is a self-evident truth: that is, a proposition so evident that it cannot be made plainer by any demonstration ; as, — the whole is greater than a part. AiiT.^ 11. — Arithmetic is founded on Notation, and its oper- ations are carried on by means of Addition, Subtraction, Mul- tiplication and Division. These are the fundamental rules of Arithmetic. Art. 1 2. — ^Notation is the art of representing numbers, by figures, letters or other characters. Art. 13. — There are two systems of Notation in coTimon use, — the Arabic and the Koman. Art. 14. — The Arabic system (made known through the Arabs) represents numbers by ten characters or symbols, called figures, viz : — 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine are called significant figures, because they indicate some value. The cipher, 0, when alone, indicates no value. Art. 15. — Every number is represented by one of these figures, or a combination of them. AR,t. 16. — ^The Boman system of Notation represents num- bers by letters. The letter I represents one; V, five; X^ ten ; L, fifty ; G, one hundred ; D, five hundred, and M, one thousand. Art. 17. — The other numbers are represented according^ to the following principles : — 1. Every time a letter is re- peated its value is repeated, thus,: — ^11, denotes two; XX» 14 ARITHMETIC. twenty, &c. 2. When a letter of less value is placed before one of greater value, the less is taken from the greater. Thus, IV denotes four, while VI denotes six ; IX denotes ninO; while XI denotes eleven, &c. A bar, — , placed over a letter multiplies its value by one thousand; thus, v denotes five thousand. '';•::> NUMERATION. Art. 18. — Numeration is the art of reading numbers ex- pressed by figures, or letters. Art. 19. — There are two methods of Numeration, the French and the English. Art. 20. — The French method is almost universally used. It separates the figures into groups of three figures each, th Foriod. Trillions. 00 FRKirCH NUMEBATION TABLE. I ^ pq I •73 . ' — r^ 4th Period. BUlioiu. /^-.'-isV^ 3rd Period. Millions. 2nd Period. Thousands. -3 WEiP Ist Period. Units. The periods above Trillions are Quadrillions, Quintillions, Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, ^c. Art. 21. — In the English method, which is seldom used, the figures are separated into periods of six places each. The first period is regarded as units and thousands of units ; the sucond, as millions and thousands of millions ; the third, as billions and thousands of billions, and so on. Art. 22. — Rule fof. Numeration. — Begin at the rightj and point off in periods of three jlgv/rea each; then, begin at the left and read vn aucceaaion eachperiod ivith ita name. NOTATION. 15 EXERCISES IN NUMEBATIOir. Example.— 36827 1 927. Read thus :— ^'2 dp *» S I •^ -S 1 1 • V 3 6 8, 2 125. 58763. 372. 86552. 864. 155731. 1076. 196472. 1884. 251103. 2750. 564989. 5890. 2285432. 9759. 2711511. 10864. 5318754. 17651. 9871832. 42414. 11867438. 'f'S ?0 H 1,9 2 7 25643287. / 87418389. 234656431. 761118445. • 4519876314. 37965432819. 98740811087. 880195038604. 9108630106543. 86419038765789. 386480967318640. ..>■' •v.- - NOTATION. Art. 23. — Rule for Notation. — Write first the figures 'of tJie highest period, then of the other periods in their proper succession, filling vacant places with ciphers. Note.— Eveiy period (except sometimes the highest) must have three figures, and if any period is omitted in the given number, its place must be supplied with three ciphers. BXEBCI8E8 IN NOTATION. ' Write in figures the following ^umbers i— 1. Forty-six thousand, seven hundred and one. i 2. Six thousand, six hundred and sixty. 3. Eight hundred and eighty-eight thousand, eight hun- dred and eighty-nine. 4. Eight hundred and eighty-eight thousand, eight hun- dred and nine. 5. Eight hundred thousand and nine. 6. Ten millions, ten thousand and ten. 7. Ten millions and ten. 8. Ninety millions, nine thousand and ninety. 16 ARITHMETIC. 9. Ninety millions, nine hundred and nine. 10. Seven hundred and seventy billions, tive thousand and seven. 11. Eleven millions and eleven. ~ 12. Eleven billions, eleven millions, one himdred and eleven. , 13. Two trillions, thirty millions and thirty. 14. Nine quadrillions, twenty trillions, five hundred bil- lions, two hundred millions, three thousand and thirty-three. MATHEMATICAL SIGNS. Art. 24. — For the sake of brevity, characters called Signs v^e used in Mathematics. Those most commm in Arith- metic are +j — , X, -T-, =. Art. 25. — The + , called plus,* is the sign of Addition^ and denotes that the numbers between which it is placed are to be added together. Thus 4+5 equals 9 ; 3 + 1+4 equals 8. Art. 26. — The sign — , called minus* or less, is the sigii of Subtraction, and denotes that the number which follow^ it is to be subtracted from that which precedes it. Thus 7 — 3 equals 4. Art. 27. — The sign x , read multiplied by, is the sign of Multiplication, and denotes that the niunbers between which it is placed are to be multiplied together. Thus 5x5 equals 25. Art. 28. — The sign -^, read- divided by, is the sign of Division, and denotes that the number which precedes it is to be divided by that which follows it. Thus 30-^5 equals 6. Division is also frequently represented thus ^ meaning that 30 is to be divided by 5. Art. 29. — ^The sign =, is the sign of equality, and de- notes that the quantities between which it is placed are equal. Thus 3+5=8; 3x5=15. ADDITION. Art. 30. — ^Addition is 'the process of combining two or more niunbers into one, which shall be equal to the whole. The result is called the sum. Thus 13 is the sum of 5 and 8. * Plus and fniii«« are two Latin words signifying more and loss. ADDITION. 17 Art. 31. — Numbers to be added must be of the same kind, or such as may be brought under the same denomination, and the figures of the same order of units. Thus 9 horses and 4 horses can be added because tliey are of the same kind ; but 9 horses and $4 cannot be added, because they are not of the same kind. Again, 3 cows and 4 sheep cannot be added to make 7 cows or 7 'sheep, but they may be brought under the same denomination and added to make seven animals. What is the sum of 82, 543 and 639 ? Solution. — Having arranged the numbers as in 82 the margin, placing units under units, tens under 543 tens, &c., because the figures to be added must be 639 of the same order of units, say 9 and 3 are 12, and 2 are 14 imits ; 1 ten and 4 units, write the 4 units 1264 beneath, and add the 1 ten with the column of tens. ^ Then one and 3 are 4, and 4 are 8, and 8 are 16 t?ns, that is 6 tens to be written beneatli, and 1 hundred to be added to the column of hundreds. Lastly, 1 and 6 are 7 and 5 are 12 hundred, that is 2 hundreds to bo written under the column of hundreds, and 1 thousand to occupy the thousand's place. BULE FOR ADDINO SISIPLE NUMBERS. Write the numbers to he added so tliat figures of the same order may stand in column^— units under units, tens under tens, hundreds under hundreds, &c. Begin at the right, and add each coluimn separately, placing the units of e^^h sum under the column added, and adding the tens with the next column. At the last column, set drn'm the comf plete sum. TsooF. — Begin at the top and add the figures downwards. If the result is the same the work is probably correct. EXERCISES. -J Find the sum of the following quantities:- — . ^^} . (2) (3) (4) 895763 99876 49176 987654321 63879 89765324 283527 123456789 54387 42356798 659845 908760504 789 56798423 7984 890705063 137568 23567989 31659 759086391 278652 79842356 968438 670998767 85945 65324897 2896392 4340661835 721096 357655787 2 1 • \ • - i 18 ARITHMEIIO. • 1 * (6) (') 738 1 78563 659 471 897 47986 12345 • 658 5798 67890 918273 856 19843 98765 651928 789 56479 43219 374859 978 28795 87654 263748 654 Zr. 897 32169 597485 999 : 1984 78912 986879 888 68195 65439 98765 777 3879 98765 9876 666 * 698 43288 987 555 k 5879 77877 456879 897 17985 98989 345678 978 % 336981 805312 4705357 12460 (9) (10) (11) (12) 189 1298 976 98 47 764 85 89 96 5837 73 76 88 6495 338 67 • 59 789 793 281 74 638 49 592 82 546 75 678 97 98 218 58 68 475 1 365 67 75 394 > 113 98 49 89 279 149 76 157 ^•;-v ^^ 67 .54 638 ^ 76 54 78 594 ' 84 72 69 789 1 1379 1 ^1^9 298 2744 87 1044 114 19715 SUDTBACTION. 19 SUBTRACTION. t Art. 32. — SUBTBACTION is the process of finding the dif- ference lietwoen two numbers. The hirj^er number is culled the minuend; the less, the subtrahend; the result, the difference^ or remainder. Subtraction is the converse of Addition, and since none but numbers of the same kind can be added, it follows that unless the minuend and subtrahend be of the b ne kind Subtraction cannot be performed. Subtract 3742 from 8396. SoLUTioN.-^Haviug arranged the numbers as in 8396 the margin, Ixjgin at the right and say 2 from 6 3742 leaves 4 ; 4 from 9 leaves 5 ; 7 from 3 you can not, bon'ow 1 from 8, which, beiAg in the fourth place, 4654 is 1 thousand, and, considered as hundreds, is 10, add the 10 to 3 = 13, then 7 from 13 leaves 6; 3 from 7 and 4, or, add 1 to 3 is 4, 4 from 8 leaves 4. BULB FOR STTBTRAOTIOir. Write the leas number under the greater, placing units t{.nder units, tens under tens, &c. ; begin at tlie right, sub~ troAit each figure from, tlie one above it,placing the remainder beneath. If any figure in tlie subtrahend exceeds the one above it, add ten, to the upper oTie, subtract the lower from, tlte sum, and eWier diminish tJie next figure in the mi/nur' end by one, or increase the next in tlie svijbtrahend by one aa you proceed. ^ ' EXEBCISES. (1) (2) (3) Minuend, 847639021 1010305061 59638743 Subtrahend, 476584359 670685093 18796854 Difference, 371054662 339619968 40841889 4. From 7813257 take 3745679= 4067578. 5. 6. 7. 8. 9. 10. (( (( 111111111 « 8420163 « 910486 " 350000147 " 7130426 « 33333333 " 98657293=12453818. 78590= 8341573. 91364= 819122. 250197319=99802828. 9999= 7120427. 23333334= 9999999. MIXED EXERCISES IN THE APPLICATION OP ADDITION AND SUBTRACTION. 1. I have in my desk $43 ; my neighbor, A, owes me $8 ; so ARITHMETIC. B,owes rae )?;147; C, $409; D, $649; K, $961; F, $91;: liow many doUurs will I have if they all pay me ? An8. $2308. 2. Last night I had $2308 ; to-day I received nothing, but paid away $973 ; how many dollars have I l6ft ? Ann. $1 335. 3. A farmer gathered from his orchard 1C83 bushels of apples, and sold 558 bushels ; how many bushels had ho left ? Ans. 525. 4. A farmer fatted and took to market 7 hogs ; one weighed 163 pounds, another 270 pounds, another 328 pounds, another 197 pounds, another 449 pounds, another 95 pounds, and another 256 pounds; how many pounds of pork did lie have- to sell? ■ Ans. 1758. 5. A wholesale merchant received 8 notes from his custo- mers in one day; the Ist was for $725, the 2nd for $197, the 3rd for $75, the 4th for $19, thd 5th for $473, the 6th for $690, the 7th for $84, and the 8th for $69 ; what was the value of the whole ? Ans. $2332. 6. In 1871, the population of Nova Scotia was 387800, and that of New Brunswick was 285594, how many more people were there in Nova Scotia than in New Brunswick ? Ans. 102206. 7. If a man were worth $3112 on new year's day, and gained during the year $849, how much yroiUd he be worth the next new year's day ? Ans. $3961. 8. If a man were worth $4000 on new year's day, and lost $1943 during the year, how much would he be worth the next new year's day ? Ana. 2057. 9. A man intending to move from the country to the city sold his farm for $1743, his horses for $395, his cows for $98, his sheep for $137, his farming utensils tor $249, his hay for $217, his grain for $75, and his poultry for $29, how many dollars worth did he sell altogether ? Ans. $2943. 10. In the last question, how much more did the man get for his hay than for his grain ? Ans. $142. 11. In 1871 the population of the pounties of New Bruns- wick was as follows : St. John 52120, Charlotte 25888, King's 24593, Queen's 13847, Sunbury 6824, York 27140, Carleton 19938, Victoria 11641, Eestigouche 5575, Gloucester 18810, Northumberland 20116, Kent 19101, Westmorland 29335, Albert 10672, what was the population of the whole Pro- vince ? Ans. 285594. MULTIPLICATION. f 31 MULTIPLICATION. Art. 33. — Is the process of findin.'( the result of repeating a given number as often as there are units in another given number. Tlie number tr» be repeated is the Multiplicand. The number which shows how often the multiplicand is to be repeated is the Multiplier. The result of the operation is called the Product. The multiplicand and the multiplier are called Factors of the Product. AuT. 34. — The product of two numbers is the same, which- ever factor is taken as the multiplier. Art. 35. — The product is of the same name as the mul- tiplicand. Art. 36. — The multiplier must always be considered as an abstract number. It is absurd to talk of multiplying dollars by dollars. It would be as rational to propose to find the product of 5 apples by 4 potatoes. MULTIPLICATION TABLE. Twice 1 are 2 2— 4 3— 6 4— 8 6—10 6—12 7-14 8 — 16 9 — 18 10 — 20 11 — 22 -12 — 24 3 times | li ftreS 2 - 6 8 - 9* 4 -12 6 "15 6 -18 7- -21 8 -24 9 -27 10 -30 11 -33 12 -36 4 times 1 are 4 2— 8 3 — 12 4 — 16 6 — 20 6 — 24 7 — 28 8 — 32 9 — 86 10 — 40 11 —44 12 — 48 6 times 1 are 6 2 — 10 3 — 15 4 — 20 6 — 26 6 — 80 7 — 35 8 — 40 9 — 46 10—60 11 — 66 12 — 60 6 times 1 are 6 2—12 3 — 18 4 — 24 6 — 30 6 — 36 7 — 42 8 — 48 9 — 64 10 — 60 11—66 12-72 7 times 1 are 7 2 — 14 3 — 21 4 — 28 6 — 35 6 — 42 7—49 8 — 66 9 — 63 10 — 70 11-77 12 — 84 8 times 1 are8 2—16 3 — 24 4 — 32 5 — 40 6 — 48 7 — 66 8 — 64 9 — 72 10 — 80 11-88 12 — 96 9 times 10 times 11 times 12 times 1 are 9 larelO lareli 1 are 12 2—18 2 — 20 2 — 22 2 — 24 8 — 27 3-30 8 — 33 3 — 86 4 — 36 4 — 40 4-44 4-48 6 — 46 6 — 60 6 — 66 6 — 60 6 — 64 6 — 60 6 — 66 6-72 7 — 63 7 — 70 7 — 77 7-84 8-72 8 — 80 8 — 88 8 — 96 9 — 81 9 — 90 9 — 99 9-108 10 — 90 10-100 10-110 10-120 11-99 11 -110 11 -121 11 -182 12-108 12-120 12-132 12-144 S2 ARITHMETIC. ' Multiply 9246 by 7. Solution. — For convenience set the Multiplier 924< under the Multiplicand as in the marp;in. Multiply 7 each figure of the multiplicand by the multipher, •■ ■ - ■ thus, 7 times 6 units are 42 units, write 2, and re- 64722 serve the 4 to be added to the product of the tens ; then 7 times 4 tens are 28 tens, and 4 tens are 32 tens, write 2 in the tons place, and reserve the 3 to be added to the pro- duct of tlie hundreds; then 7 times 2 hundred are 14 hundred, add 3 from last product makes 17 hundreds, write 7 in the hundreds place, and reserve the 1 to be added to the product of the thousands; lastly, 7 times thousands are 63 thousands, and 1 are 64 thousands, which you will write in full. Multiply 34.C186 by 268. Solution. — Set the multiplier under the 345186 multiplicand as before, units under imits, t^ns 268 under tens,&c. Begin at the right, and multi- ply each figure of the multiplicand by 8, sett- 2761488 ing down the product as in the preceding 2071116 example ; then by 6, which by netting the pro- 690372 duct one place to the left, gives the product by 60, or 6 tens ; then by 2, which by setting 92509848 the product two places to the left, gives the Eroduct by 200. The three lines are then, 1st, the product y 8, 2nd, the product by 60, 3rd, the product by 200, and their sum, the product by 268 which was required. RULE FOR MULTIPLICATION. Set the multiplier under the multiplicandf units under • units, tens under tens, &c., then, 1. When the multiplier consists of only one figure, mul- tiply each figure of the multiplicand by the multiplier, writing under each tJie right hand figure of its prockbct increased by the rermainviig figure or figures of the product immediately p)xcedlng it, observing to xvrite the last pro- dtLct so increased in full. 2. When the multiplier consists of more than one figure, multiply tlie multiplicand by each figure of the multiplier in succession, beginning each partial pi^od^ct under the figure which produces it. The stim of these partial pro- ducts will be the total product. Proof. — Multiply the multiplier by the multiplicand. If the result is the same the work may be deemed correct. Multiplicand, Multiplier, Product, MULTIPUCATION. EXERCISES. (1) (2) 7896 581067 5 8 (3) 938746 4 38 (4) 193764 7 5. 391876 X 9 6. 987456 X 6 7. 496783x52 8. 719864x43 9. 375967x64 15. Find the 16. Find the 17. Find the ■ 18. Find the 39480 4655736 3754984 1356488 = 3526884 10. 27859 x 29= 807911 = 5924736 11. 679854 x 83= 56427882 =25832716 12. 7596d4 x 187=142060908 =30954152 13. 5372x1634= 8777848 =24061888 14. 7986x3795= 30306870 squaru or second power of 389. Ans. 151321 cube or third power of 538. Ans. 155720872 fourth power of 144. Ans. 429981696 cube 0/991 970299 Art. 37. — Operations in multiplication may sometimes be shortened, as in the following case : — When the multiplier is a composite number, that is a num- ber which is the product of two or more whole numbers, each greater than. one, BuLE. — Separate tJie multiplier into two or jnore factoTB. Multiply first by one of these fadorsj then this prod/act by another, and so on till each factor has been used as a mul- tiplier. The Uzst product is the complete product required. I EXERCISES. 1. Multiply 7325 by 24. 24 is a composite number, and is the product of the factors 4 and 6, therefore if 7326 be multiplied by 4, and that pro- duct by 6, the required product will be obtained. Thus: Multiplicand, - - - - 7325 1st factor of the multiplier, 4 ■■!(>. 2nd « (i 29300 6 i .i Product by 24, 175800 2. Multiply 1728 by 64. 3. Multiply 5673 by 42* 4. Multiply 7916 by 72. 5. Mtdtiply 19743 by 88. 6. Multiply 9173 by 144. kmi -•g.|li;,. Ans. 110592 Ans. 238266 Ans. 569952 Ans. 1737384 Ans. 1320912 i! u ABITHMEriC. I: Ans. 15936 Ans. 674478 Ana. 4462577 Ans. 686208 7. Multiple 498 by 32. 8. Multiply 10706 by 63. 9.. Multiply 91073 by 49. 10. Multiply 7148 by 96. Art. 38. — When the multiplier is 1 with ciphers annexed, as 10, 100, 1000, &c.. Rule. — Annex to the multipliccmd as many cvphera as there are in tJie mvMiplier ; the result will be the required product, EXERCISES. 1. Multiply 7394 by 10. 2. « , 5786 100. a. *< 7120 1000. 4. Multiply 648 by 10000 5. « 7863 100000 6. *• 9104 1000000 Art. 39. — When ciphers ate at the right of one or both factors. Rule. — Multiply as usual, but omitting th^e ciphers r«- f erred to ; then annex to the product as many cyphers aa are to tlib right of both fa>ctors. Find the product of 4300 by 1600. SOLUTION.—Multiply 43 by 16, and to the 4300 product 688 annex four ciphers, viz: the num- 1600 ber to the right of both factors. 258 43 EXERCISES. ^s^ 1. 738000 X 7300. 4. 2. 5900 X 2700. 5. 3, 1070 X 79000. 6. 6880000 270 X 860. 11 1000 X 610. 790000x43000. APPLIC 1. If one yard of cloth cost 25 yards cost ? 2. If 1 pound of cheese cost I give for 9 pounds. 3. If 7 boxes contain 144 all? 4. If a laborer earn $7 a he earn in 35 weeks ? 5. How many bricks would if he took 1625 at a load ? ATIONS. 75 cents, how many cents will Ans 1875 cents. 18 cents, how many cents must Ans. 162 cents. pens each, how many in them Ans. 1008. week, how many dollars would Ans. $24^. a teamster remove at 23 loads, Ans. 37376. DIVISION. 25 * 6. If a wagon wheel make 586 revolutions in a mile, how many revolutions would it make in a journey of 67 miles ? Ans. 39262. V. An ordinary clock strikes 156 strokes in a day, how many strokes does it strike in a year of 365 days ? Ans. 56940. 8. A bushel of potatoes weighs 60 pounds, what is the weight of 350 bushels ? Ans. 21000 pounds. 9. At $15 per acre, what would be the price of a field measuring 29 acres ? Ans. $435»- 10. If an acre of land yield 47 bushels of wheat, how many bushels will 109 acres yield ? Ans. 5123. DIVISION. ' Akt. 40. — Division' is the process of finding how often one number is contained in another ; or it is the process of find- ing one of two factors of a given niunber when the other is known. The number to be divided is the Dividend. ' " The number by which to divide is the Divisor. The result, which shows how often the divisor is contained in the dividend, is the Quotient. The number sometimes left, and which shows the excess -of the dividend over the Divisor repeated as often as there are units in the quotient, is called the Bemainder. Note. — ^Tbe remainder is always of the same kind as the dividend, and must always be less than the divisor. How often is 3 contained in 672 ? , Solution. — For convenience Divisor 3)672 Dividend, write the divisor on the left of ^24 Quotient, the dividend as in the margin ; then say 3 in 6 (hundred) 2 (hundred) times ; write two in the hundreds' place ; then 3 in 7 (tens) 2 (tens) times and 1 • '(ten) over, write 2 in the tens' place, and add 10 to the next figure, 2=12 ; then 3 in 12, 4 times, write 4 in the units' place. The quotient is 224. In practice we. do not name the orders of units, but pro- ceed thus, 3 iu 6, twice, write 2 ; 3 into 7, twice, and 1 over, write 2 and prefix 1 to the next figure in the dividend, making 12 ; 3 in 12, 4 times, write 4. 26 ARrinMKnc. Again, divide 533965 by 7. '—^ Solution. — Arrange the numbers as be- 7)533965 ^; fore ; then 7 in 5, times, but 5 remains 76280-5 undivided ; join it with the next figure ; 7 in 53, 7 times and 4 over, write 7, and mentally prefix the 4 to the next figure 3 ; 7 in 43, 6 times and 1 over, write 6 and prefix 1 to 9, the next figure ; 7 in 19, twice, and 5 over, write 2, and prefix 5 to 6, the next figure; 7 in 56, 8 times even, write 8 ; 7 in 5, times, and 5 over, write 0, and indi- cate the division of 5 thus, | and annex it to the other figures. Wlien division is performed in this way, which is the case when the divisor is not greater than 12, the process is called Short Division. ^ y RULE FOR SHORT DIVISION. 1. Tfriie the divisor on the left of the dividend, with a curved line between them; divide successively each figure of the dividend by the divisor, and set the quotient beneath tlie figure divided. 2. Wien^vei' there is a remainder, prefix it, mentally, to the next figure in the dividend, and divide as before, 3. If any partial dividend does not contai/n the divisor, 'filace a dphsr beneath, and form a new dividend by iwc/Zoj- ^ng the old one to the succeeding figure, and divide asoefore, 4. If there is a remainder after dividing the last figure, indicate its division by placing the divisor under it, and annex it to tJie quotient. ?J1*.» EXERCISES. 1. Divide 7936427 by 3. Ans. 26454751 2. (( 87965328 « 4. Ans. 21991332 3. (( 7963821 « 5. Ans. 1592764^ 4. it 6875324 « 6. Ans. 1145887f 5. (( 3987654 " 7. Ans. 569664| e. (( 19876532 « 8. Ans. 2484566|- 7. C( 2976532 « 9. Ans. 330725J. 8. (( 4967854 « 10. Ans. 496785 jL D. (( 46879352 "11. Ans. 4261759:1 10. (( 18765314 « 12. Ans. 1563776^ Methods of Proof. — 1. Multiply the quotient by the DIVISION^'^ 4 divisor, or the divisor by the quotient, adding the remainder, if any, to the product. Tlie sum should be the dividend. 2. — Subtract the remainder, if any, from the dividend;, the result divided by the quotient should be tlie divisor. EXAMPLE. 9) 77 . i'\ ?■■ .,^■' 8—5. 1st Proof. 9x8=72 and 72 + 5=77. 2nd t»roof. 77—5=72, and 72-^8=9. Divisor. Dividend. Quotient. 15)3465(231 - 30 ,H •.;,,.•.,.. . 46 . ; 45 ;.-: 15 15 00 remainder Art. 41. — ^When the divisor exceeds 12, the process em- ployed is generally such as is shown below, and is called Long. Division. Divide 3465 by 15. Solution. — Write the divisor on the left of the dividend as in the margin. Then as 15 is not contained in 3 (thous- ands), the quotient has no thousands. Next see how often 15 is contained in 34 (hundreds) and set the number 2 (hundreds) in the quotient on the right of the dividend; multiply the divisor ■ by this figure (2) and set the product under 34, and subtract, the remainder is 4 (hundreds). Then bring down the next figure of the dividend which is 6 (tens), and take 46 (tens) for the next partial dividend. 15 is contained in 46 (tens), 3 (tens) times; multiply the divisor by 3, set the pro- duct under 46 and subtract, — the remainder is 1 (ten), to which annex the next figure of the dividend (5), v hicL makes the last partial dividend 15. 15, the divisor, is contained in this once; set one in the quotient, multiply the divisor by it, and subtract the product as in other cases. The remainder is nothing. 15 is therefore contained in 3465, 231 times which is the quotient. RULE FOR LONG DIVISION. 1. — Draiv a curved line on the left, and one on the Hght of tJie dividend, and set the divisor on the left. 2. — See how often the divisor is contained, in the feivest of the left hand figures of the dividend that will contain it, a7id set the number in the quotient on the right of the divir^ dend. ■■i as ARITUMETIC. 3. — Multiply the divisor by this quotient figure, and set the product under the figures ueed hi tlie dividend, cmd subtract it from tliem. 4,— Annex to the remainder tJie next figure of the divi- dend, and divide as before, continuing the process until ea-ch figure of tfie dividend lias been used. 5.—^ at any time, after a figure of the dividend has been brought down, the number thus formed is too small to con- tain the divisor, a cipher must be placed in the quotient, and anotJier figure brougid down, afte/r which divide as before. 6. — If tJiere is a final remainder, write the divisor under it as in Short Division. Proof. — The same as in Short Division. Note 1. — In finding any quotient figure, tria! niay be made of a num- ber which, when the Divisor is multiplied by it, will give a product greater than the partial dividend used. If so, the quotient figure is too large. Try the next smaller figure, and so on till the product obtained is not greater than the partial dividend. Note 2. — ^When the poduct of the divisor by any quotient figure sub- tracted from the partial dividend gives a remrinder greater than the divisor, the quotient figure is too small. Try the next greater figure, and 80 on till a remainder less than the divisor is obtained. EXERCISES. 1. Divide 2. «« 3. « 4. 5. 6. 7. 8. 9. 10. « C( ■it cc (i (C i( 11. 12. 13. 14. 35190 by 81306 « 66683 « 273284 « 666774 « 2317565 « 7910683 «« 9621304 « 10045643 « 57300847 « 41807164- 89709786- 37418007- 45676827- 15. 170064915561-v- 759= 15. 18. 21. 47. 93. 275. 74. 396. 571. 1078. ^3727: ^4509: ^5763: ^7609: 759: Ans. 2346 Ans. 4517 Ana. 3175-8- 91 Ans. 581411 41 Ans. 7169»1 • 8 Ans. 84271*0 Ans. 106901-1- 74 Ans. 24296_»«- 898 Ans. 17593*1 671 Ans. 53154A»»- 1078 112l71*»i. 3787 19895^U1 4609 6492*111 6788 6003 2240644473M 769 if4 i DIVISION. 29 COXTBACTIONS IN DIVISION. Case 1. — When the Divisor is u composite number. Divide 5775 by 25. 25 is the product of two factors, 5 x 5. Rule. — Divide the dividend by one of the 5)5775 factors, and that quotient by another, and 5)1155 80 on till each factor ho a been used as a di- 231 visor. The last quotient luill be the one required. To find the true remainder, multiply each remainder by all the preceding divisors except the one which produced it, and to the sum of the products add the remainder from the first division, if there be one. Thus, divide 247 by 15. 3)247 2x3=6, and 6 + 1=7, true remainder. 1. Divide 2. « 3. 4. 5. 6. 7. (( (( (( (( (t EXERCISES. 90144 by 24. 69090 « 35. 2507639 « 42. 3827864 « 63. 57834722 "81. 55937133 " 56. 497021 " 72. 5) 82-1. 16. 2. Ana. 3756 Ans. 1974 Ans. 59705M 43 Ans. 6075911 63 Ans. 7140081* 81 Ans. 998877»i 58 Ans. 6903-1- 73 From! the above it appears that if the divisor and the div- idend both be divided by the same nimiber, the quotient will not be changed. Case 2. — When the r'livisor is 1 with a cipher or ciphers annexed, as 10, 100, 1000, &c., Rule. — C^U of as many figures from the right of the dividend as there are ciphers in the divisor; the figures out off will be the remavnder, and tlie others %vill be t1\a quotient. EXERCISES. 1. Divide 35243 by 100. 110 0)352143 352-11 100 Or simply from inspeotion, write all the figures except the mmmmm 30 ABITHMEnC. t I I two last for the quotient, and place the two right hand figures above the divisor and annex thereto. 2. Divide 7639 by 10. 3. « 154736 " 100. 4. " 791084 « 1000. 5. « 9018765 " 10000. 6. « 5587391 « lOOOOO. Case 3. — When there are ciphers on the right of the di- visor, and the remaining part greater than 1. KULE. — Gut off the cipJiera from the right of the divi80i\ and as wZbriy figures from, the right of the dividend^ and divide the rem^avning jUgurea of the dividend by the remain- ing figures of the divisor. To the remainder annex tJie Jigurea cut off from the dividend, and you will have th6 true remxiinder. EXEBCISES. 1. Divide 54300 by 130. 52 . 23 13 100 91 90 130 2. D ivide 47689 by 170. 3. (( 916«>345 « 29000. 4. cc 6008074 « 7500. ^ « 7961745 «« 60. 6. (C 15728 « 20. ' 7. i( 9186437 « 70. 8. (( 820186 " 40. 9. (( 7310745 "1290000. lOL u 750900 « 500. •fk- Ans. 280-»-»- 170 Ans. 316 MM 89000 Ans. 801 -»11- 1600 Ans. 132695M • Ans. 786-1- 80 Ans. 13123411 70 Ans. 20504M 40 Ans. 5-MMM. 1S90000 Ans. 1501i CANADA CUBRKNCY. 81 APPLICATI0N3. 1. What is the half of 9786? Ana. 4893. 2. What is the one third of 768594 ? Ana. 256198. 3. Find one eighth of 673915 ? Ans. 84239|. 4. Find the one fortieth of 976183 ? Ans. 24404j§ 5. If 5 barrels of apples cost $20, what is the price of one barrel ? Ans. $4. 6. If a clock strike 1092 strokes in a week, how many strokes does it strike each day ? Ans. 1 56. 7. If 19 yds. of cloth cost 1805 cents, wliat is the price per yard ? Ans. 95 cents. 8. If an orchard of 27 trees produce 5103 apples, how many on each tree, on an average ? Ans. 189. 9. If 56 men earn $30072 in a year, what is the salary of «ach man on an average ? Ans. $537. i» 10. If there he 54432 pens in 378 boxes of equal size, how many in each box ? Ans. 144. 11. How many acres in a field which produces 4277 but^h- els of oats, at the rate of 29 bushels to the acre. Ans. 14711 90 12. If each family in a city consume 72 eggs in a year, and it require 1229688 eggs to supply the town, how many families in the town ? Ans. 17079. 13. If 379 bushels of com cost 13265 cents, how much is that per bushel ? Ans. 35 cents. CANADA CURRENCY. Art. 42. — The currency of Canada is of two denomina- tions, viz : Dollars and Cents. The dollar is the unit of value, of which the cent, from centum — a hundred, is the one hundredth part. When dollars and cents are to be expressed, the cents are written to the right of the dollars, and separated from thepi only by a period (.) called the decimal point. The sign of dollars ($ ) is prefixed to the whole. Thus, two dollars, forty- five cents are written, $2.45. We may estimate the value of the figures in any expres- sion of dollars and cents thus : — Take $436.75 ; the 6, that is the first figure on the left of the point, represents units of dollars, the next, tens of dollars^ • '> »M (i 82 ARITHMEnC. and 80 OD, as explained under the head of Numeration. The 7, that is, the first figure on the right of the point, re- presents tenths of a dollar, and as a dollar is 100 cents, one tenth of a dollar is 10 cents, and 7 tenths are 70 cents, therefore the 7 represents 70 cents. The next figure repre- sents hundredths of a dollar, and as the hundredth part of a dollar is a cent, the 5 represents 5 cents. Or thus : — The right hand figure represents imits of cents, or 5 cents ; the second figure tens of cents, and as 7 tens are 70, the 7 represents 70 cents ; the third represents hundreds of cents, and as a hundred cents are a dollar, the 6 represents 6 dol- lars; the fourth, tens of dollars, and so on as explained under the head of Numeration. Thui^ it will be seen that the figures throughout bear the same relation to one another as in simple numbers, and may therefore be added, subtracted, multiplied and divided by the- rules already given. ADDITION. • Place the quantities to be added so that the decimal points shall be directly under one another, add as in simple num- bers, and cut off two figures from the right of the sum for cents. 1. Add $125.75, $98.50, $25.15, $76.05, $9 l.lH,$43.87i, $84.20, $67.62^, $39.80, and $17.37^. Ans. $669.44. 2. What is the sum of $13.19, $14.16, $85.92, $64.15, $37.25, $91.20, $18.75, $29.10, $47.85, $55.55, and $72.63. Ans. $529.75. 3. Find the sum of $85.50, $49.63, $92.18, $37.09, $8.92, $76.45, $25.75, $64.16, $18.60, $59.11, $148.17, and $265.90. Ans. $931.46. 4. Add together $116.20,$291.45, $89.75, $365.84, $91.50^, $76.15, $485.00, $157.92, $263.75, $188.25, $39.48, and $136.13. Ans. 2301.42. 5. Add together $175.18, $1.74, $2864.91, $3.24, $876.45,, $79.79, $0.85, $278.01, $5371.56, $17.20, and $740.95. Ans. 10409.88. SUBTRACTION. Place the quantities so that the decimal point in the sub- trahend shall be directly under that of the minuend, subtract as in simple nun^bers, and cut o£f from the di£ference the two right hand figures for cents. CANADA CURRENCY. 1. Subtract $278056.80 from $567810.83. Ans. 288862.94 2. From $83756.17 take $76480.71. Ans. 7266!46'. 8 What is the di£ference between $17423.37^ and $9645.634 ? Ans. $7777.74. 4. Find the difference between $100623.40 and $9781.37. Ans. $00842.03. MULTIPLICATION. Multiply as in simple numbers, and cut off two figures from the right of the product for cents. 1. Multiply $365.75 by 7. 2. Multiply $1873.47 by 60. 3. Multiply $865.63 by 03. 4. Multiply $24786.38 by 145. Ans. $2560.25. Ans. $120260.43. Ans. $80503.50. Ans. $3504025.10. DIVISION. Divide as in simple numbers, and point off two figures from the right of the quotient for cents. 1. Divide $60500.68 by 7. 2. Divide $28642.14 by 20. . 3. Divide $37133.34 by 87. " 4. Divide $1043243.55 by 083. Ans. $8644.24. Ans. $087.66; Ans. $426.82. Ans. 1076.85. MIXED EXERCISES IN THE APPLICATION OF MULTIPLICATION AND DIVISION. 1. What is the cost of 17 acres of land at $52.50 per acre? Ans. $802.50. . 2. A hammer factory turns out 37440 hammers in a yfor of 52 weeks, how many is that per week on an average ? Ans. 720. 3. How many yards of calico at 8 cents a yard can I buy for $2.80? Ans. 35. 4. How many yards of ribbon at 25 cents per yard can be purchased for $3 ? Ans. 12. 5. I sold 15 kegs of butter,,each containing 25 pounds, for $60, how much was that a pound? Ans. 16 cents. 6. Bought 21 barrels of apples at $1.05 a barrel, what did they cost ? Ans. $22.05. 7. If 11 tons of hay cost $214.50, what will 1 ton costf What will 27 tons cost ? Ans. I ton, $10.50. 27 tons, $526.50; S4 ARITHMETIC. 8. 1125 bbls. fish were sold for $5906.25, how much per barrel ? Ans. $5,*4S, 0. 269 persons pay a tax of $1312.72, what is the average tax on each ? Ans. $4.88. 10. Huppose a manufacturing company employs 250 men, and pays them on an average $1.75 per day, what is the cost to the company for 1 day ? for 1 week ? for 1 year ? Ans. For a day, $437.50. For a week, $2625.00. For a year, $136500.00. 1 1 . If the liouses in a town are worth on an average $950 each, and their total value is $1168500, how many houses in the town ? Ans. 1230. 12. If the total value of 1230 houses be $1039350, what is the value of each house on an average ? Ans. $845. 13. What would be the value of 1230 houses, if the aver- age value were $845 each ? Ans. $1039350. 14. What is the value of 437 sheep at $4.75 each ? Ans. $2075.75. 15. If a man travel 3 miles an hour every day for 40 days of 12 hours each, how many miles will he travel ? Ans. 1440. 16. If a railway train runs 264 miles in 12 hours, what the average rate per hour ? Ans. 22 miles. 17. At 45 cents per bushel, what must be paid for 1195 bushels potatoes ? Ans. $537.75. 18. A cargo of 4700 bushels oats sold for $1504, how much is that per liushel? Ans. 32 c^nts. 19. What is the weight of a cargo of 5000 bushels of wheat weighing 60 pounds per bushel ? Ans. 300000 pounds. 20. 180 chaldrons of coal were sold for $1035, what was the price for 1 chaldron? Ans. $5.75. PROPERTIES OF NUMBERS. Abt. 43. — 1. An Integer i» a whole number, as 1,2, 3, &c. 2. An Even number is one that can be evenly divided by 2,08 2,4, 6, 8,10, &c. 3. An Odd number is one that cannot be evenly divided by 2, as 1, 3, 5, 7, 9, 11, &c. 4. A Prime number is one that cannot be exactly divided by any whole number except itself and 1, as 1, 2, 3, 5, 7» 11, 13, 17, 19, &c. rUOPEHTIES OF MUMBKIU. 85 5. A Composite number is one that can ))e exactly divided by some wliole number boHideH itHctlf and 1, as 4, 6, 8, 9, 12, 14, 1.5,&c. fi. Every composite num})er in tbe product of two or more prime numbers. 7. Two numbers ure prime to each other when 1 is the only number wliich will exactly divide both, as 5 and 9. 8. A number which will exactly divide two or more num- bers is called a common factor of titem. Thus, 3 is a com- mon factor of 6, 9, 12 and 15. ■ 7 is a common factor of 14 and 35, 9. A Prime Factor of a number is a prime number which will exactly divide it. Thus, the prime factors of 21 are 3 and 7. The prime factors of 24 are 2, 2, 2, and 3. To resolve a composite number into its prime factors, Rule. — Divide me given numher hy any pHme number r/reater than 1 that tvill exactly divide it, repeat the process ^vith the quotient^ and so on till a prime number ia ob- tained ; the divisors and remaining numbers are tlw prime factors required, EXERCISES. 1. What are the prime factors of 90 ? ■■ ' 2 )90 3>45 .„ ;, 3)15 5 2. What are the prime factors of 35 ? 3. What are the prime factors of 75 ? 4. Besolve 651 into its prime factors. 5. Besolve 1764 into its prime factors. Ans. 3, 3, 4, 7, and 7. 6. What are the 'prime factors of 198? Ans. 3, 3, 2, and 11. 7. What are the prime factors of 171 ? Ans. 3, 3, and 19. 8. What are the prime factors of 210 ? X Ans. 2, 3, 5, and 7. 9. What are the prime factors of 2310 ?, i Ans. 2, 3, 5, 7, and 11. 10. What are the prime factors of 362880 ? Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, and 7. 11. Find the prime factors of 180642. Ans. 2, 3, 7, 11, 17, and 23. Ans. 2, 3, 3, and 5. Ans. 5 and 7. Ans. 3, 5, and 5. Ans. .3, 7, and 31. Jo ARITHMBTIC. 12. What uro iho prime factora of 51051 ? Ans. 3, 7, 11, 13, and 17^ ORBATKST COMMON DIVI80R. A (Common Divisor of two or more niunlieni in a number which will divido eiieh of them without a remuider. Thus, 3 in a common divisor of 12', IM, 24 and 30. The Oreateflt Common Divisor of two or more numbers is the ffreatest numlier that >vill divide each of them without a remainder. Thu^ 4 is the greatest common divisor of H, 12, and IH. To find the greatest common divisor of two or more numl)ers, Rule 1. — Resolve the fjiven numhera into tJteir prime factoi'8 ; the product of the factora commaii to all %mll be the greatest cormnon divlaor. EXERCISES. 1. Find tlie greatest common divisor of 18, 27 and 36. The prime factors, common to 18=2x3x3. 27=3x3x3. 36=2x2x3x3. all, are 3 and 3. 3 x 3=9, the greatest common divisor. ; 2. Find the greatest common divisor of 16, 40 and 72. Ans. 8. 3. Find the greatest common div' or of 36, 54, 90 and 72. Ans. 18. 4. Find the grerttest common divisor of 126, 210, 84 and 168. Ans. 42. 5. Find the greatest common divisor of 175 and 245. Ans. 35. Rule 2. — If there be onhj two numhera, divide the greater by tJie leasy that divlaor by the remainder, and ao on till there ia no remainder. The laat diviaor will be tJie greateat com' mon diviaor. When there are more than ttvo nwmbera,find the greateat common diviaor of two, f:hen pf tJiat and another, and so on to t/ie laat. TJie last greoiteat com/mon diviaor wiU be the greateat common diviaor of tht. whole. riujFKKTiKri or numbkus. EXERCISRS. 1. Find the greatest common divisor of 21 and 98. 21)98(4 H4 ' > ; - , ^ ;. . IT •jf,' 14)21(1 14 7)14 ' ■ : .^. '■=: , 2 The last divisor, 7, is the greatest common divisor. 2. Find the greatest common divisor of 52 and 299. Ans. 13. 3. Find the greatest common divisor of 108 and 9342. Ans. 54*. 4. What is the greatest common divisor of 1 638 and 2106 ? Ans. 234. 5. What is the greatest common divisor of 31185 and 50457? Ans. 33. 6. Required the greatest common divisor of 24, 108, and 464. Ans. 4. 7. Required the greatest common divisor of 576, 1344, and 2592. Ans. 96. 8. Required the greatest common divisor of 576, 768, 480 and 1360. Ans. 16. 9. What is the largest price per acre, in whole dollars, at wliich each of three farms, containing respectively 128, 224, and 376 acres, can be bought or sold ? Ans. $8. 10. A gentleman's garden is 162 feet long, and 138 feet wide. He wishes to set posts for fencing at the greatest dis- tance apart that will make equal spaces on all sides. Required the number of feet from centre to centre of posts. Ans. 6 ft. LEAST COMMON MULTIPLE. A Multiple of a number is a number which can be divided by it witho!'t a remainder. Thus, 15 is a multiple of 3 ; so is 12 ; so is 21. A Common Multiple of two or more numbers is a number which can be divided by each of them without a remainder. Thus, 15 is a common multiple of 3 and 5 ; 24 is a common anultiple of 2, 4, and 6. The Least Common Multiple of two or more numbers ia t ^ ARITHMETIC. |i I the least numl)er that can be divided by each of them with- out a remainder. Thus, 45 is the least common multiple of A and 9 ; 12 is the least common multiple of 2, 4, and 6. To find the least commjn multiple of several numbers, — > BuLE. — Arrange the nttmbera in a horizontal line, omit- ting such of t1ie,n as are factm'8 of any of the others; divide by any prime number that will divide two or more of them witJtout a remainder, vn'iting the quotients and undivided numbers in a line below ; contimue tlie 'process till a line is obtained the numbers in which are altprvme to each other. Multiply together the divisors and the numr bers in the last UnCy and the product will be tfie least com- mon multiple required. Note 1. — This process gives the product of all the prime factors in the given numbers. Note 2.— If the given numbers are prime to each other, their product is their least common multiple. '■ EXERCISES. 1. What is the least common multiple of 6, 8, 12, and 15 ? 2)8, 12, 15. Omit 6 because it is a factor of 12. The divisors are all prime numbers, and the num- bers in the last line are prime to each other. Then 2x2x3x2x5=120, which is the least common multiple of 6, 8, 12, and 15. 2. Find the least common multiple of 12, 16, 18, 30, and' 48. Ans. 720. 3. Find the least common multiple of 3, 4, 5, 6, and 7. Ans. 420. 4. What is the least common multiple of 2, 4, 7, 12, 16, 21, and 56 ? Ans. 336. 5. What is the least common multiple of 2, 9, 11, and 33 ? --»i.*^ ........ . • - Ajjg^ i9g^ 6. Find the least coihmon multiple of 2, 3, 4, 5, 6, 7, 8, and 9c Ans. 2520. 7. Find the least common multiple of 8, 12, 16, 24, and 33? Ans. 528. 8. What is the least number into which 2, 4, 8, 16, 32, 64, and 128 will divide without a remainder? Ans. 128. 9. F' d the least common multiple of 3, 9,27, 81, 243, and 729. Ans. 729. 10. Wliat is the least common multiple of 2, S, 5, 7, 11 ?-' Ans. 2310. . 2 )4, 6, 15. 3 )2, 3, 15. 2, 1, 5. FRACTIONS. FRACTIONS. ,. V ■Ui/ Art. 4C — A Fraction is a part of a unit or whole thing,' supposed to be divided into equal parts. Fraction is derived from the Latin, /roc^tts, broken. Fractions are divided into two classes, Common and Becimal. A Common Fraction is expressed by two numbers, one above the other with a horizontal line between them, as ^, read two-thirds. The number below the line is called the Denominator. The number above the line is called the Numerator. The namvj of a fraction and the value of its parts depend on the number of parts into which the imit is divided. When the unit is divided into 2, 3, 4, 5 or 6 equal parts, the frac- tion is named halves, thirds, fourths, fifths or sixths. The Denominator gives a name to the fraction, and shows the number of equal parts into which the unit is divided. The Numerator shows how many of these parts are ex- pressed by the fraction. Thus, in the expression | of a mile, a mile is supposed to be divided into four equal parts, called fourths, three of which are expressed by the fraction, |. A fraction is either Proper or Improper. A Proper Fraction has its numerator less than its denomi- nator, 8 s |. An Improper Fraction has its numerator equal to, or greater than its denominator, as ^, i . A Mixed Number is a whole number with a fraction an- nexed, as 4^ I Again, a Fraction is either Simple, Compound or Complex. ^^j A Simple Fraction has but one numerator and one denom- inator, and may be proper or improper. Thus f and ^ are both simple fractions. A Compound Fraction is a fraction of a fraction, that is, it is two or more fractions connected by the word of, as f of I- of h A Com.plex Fraction has a fraction in its numerator or denominator, or in both. Thus, i,?, f, ^,£,?i are all complex fractions. 4| 1 3 3^ 9^ 40 ABITHMETIC. All common fractions represent division, the numerator being the dividend, and the denominator the divisor. The value of the fraction is the quotient arising from perform- ing the operation of division. Thus the fraction f is 4. When the fraction is proper the division cannot be performed but is merely indicated, and the quotient can only be ex- pressed in the fractional form. REDUCTION OP FRACTIONS. Reduction of Fractions consists in changing their forms without altering their values. A fraction is in its lowest terms when its numerator and denominator are prime to each other, as f , ^, f , but not f . Since the nimierator and denominator of a fraction are a dividend and divisor, they may be divided by the same num- ber without changing the quotient, or value of the fraction. Therefore, To reduce a fraction to its lowest terms, ' ' Rule. — Divide both terms by any common factor, and continue the process till tJiey are prim^ to each other. Or, divide both terms by their greatest common divisor. EXERCISES. ^ .i,..,^::H ^■^■-' 1. Reduce ff to its lowest terms. '-^^ 2|ff=U and 41^=4 Ans. Or, 8l^=f Divide both terms by 2, which is a common factor, then the resulting terms by any common factor of them, say 4, which makes a fraction, ^, the terms of which are prime to each other. | is the fraction ff , in its lowest terms. 2. Reduce -J-| to its lowest terms. 3. Reduce -j^ to its lowest terms. ^ ^ .. /' 4. Reduce -^^^ to its lowest terms. ■'' '■ 6. Reduce |4f to its lowest terms. ■ 6. Reduce -jV^s" ^^ ^^^ lowest teims. v .. 7. Reduce ^j^y to its lowest terms. & ,^; 8. Reduce -^/f^ to its lowest terms. v 9. Reduce ^^\ to its lowest terms. ,.. .\^ 10. Reduce ^o~f4 ^^ ^^ lowest terms. 11. Reduce ^^^ to its lowest terms. 12. Rfeduce flM to its lowest terms. 13. Reduce ^|y ^ to its lowest terms. 14. Reduce xVWA^ *^ ^^ lowest terms. 15. Reduce |4^dF ^ ^^^ lowest terms. 16. Reduce f !^j| to its lowest terms. Ans. ^. Ans. ^, Ans. -^. Ans. ^i Ans. I-. Ans. 1^. REDUCTION OF FRACTIONS. 41 17. Reduce tV^Ws ^ ^^^ lowest terms. 18. Reduce ^VoWo ^^ ^^^ lowest terms. 19. Reduce H^ ^ ^^^ lowest terms. 20. Reduce j||^| to its lowest terms. ) To reduce an improper fraction to a whole or mixed number, Rule. — Divide the numerator by the denominator; the quotient will be the whole or mixed number. Note.— If there be a fraction in the answer, reduce it to its lowest terms. EXERCISES. 1. Reduce f^ to a whole or mixed number. The denominator shows that the unit is divided into 16 equal parts ; hence 16 sixteenths make 1, and there are as many units in || as 16 is contained times in 87, and 16)87(5^ *. ^ i 'i' . ... <■>,;.■. • • 80 ,..■-... ■ ■ •' ■■ /;■ ;..vL. . 2. Reduce *^i to a whole or mixed number. Ans. 49. 3. Reduce "^ to a whole or mixed number. Ans. 71. 4. Reduce ^ to a whole or mixed munber. Ans. 4^. 5. Reduce i-J* to a whole or mixed number. Ans. 19^. 6. Reduce ff to a whole or mixed number. Ans. 7-^. 7. Reduce i-f^* to a whole or mixed number. Ans. 83|-. 8. I?educe \^ to a whole or mixed nuAiber. Ans, 54. 9 }i*iduce ^^ to a whole or mixed number. Ans. 29f. W. liriiice ^^VVu' *® * whole or mixed number. Ans. 5U^. 11. Reauce V ^^^ dollar to dollars. Ans. $3|. 12. In y of a dollar how many dollars? r|^i«. Ans. 9^. 13. How many bushels in if of a bushel? Ans. 17|.' 14. In '^^ of a gallon how many gallons ? Ans. 38f . To reduce a mixed number to an improper fraction, Bulb. — Multiply the whole nurftber by the denominator of the fraction, to the product add the numerator, and write the denominator under the sum. A whole number may be expressed fractionally by writing 1 under it for a denominator. A whole number may be reduced to a fraction having any proposed denominator, by multiplying it by the proposed 42 ARITHMETIC. denominator for a numerator, and writing the denominator under it. EXERCISES. 1. Beduce 4^ to an improper fraction. 4^ The denominator of the fraction is to become the de- } nominator of the answer, therefore, the answer is to be halves, and as two halves make a whole, there will be twice as many halves as whole ones, that is twice 4, and the 1 half expressed ])y the fraction makes 9 halves or f . 2. Reduce 4| to an improper fraction. 3. Reduce 27^ to un improper fraction. i.f 4. Reduce 66^^ to an improper fraction. 5. Reduce 18f to an improper fraction. 6. How many sevenths in 9| ? V 7. In $7^ liow many eighths of a dollar? 8. In 17§ gallons how many thirds of a gallon ? 9. Change 27 to a fraction . 10. Express 9 as a fraction having 7 for its denominator Ans. ^*, 11. Reduce 19 to twelfths. Ans. ^^. 12. Reduce 28 to a fraction having 19 for its denominator. Ans. «^, Ans. y. Ans. Y' Ans. «-|«. Ans. i|'' Ans. ^. Ans. y.. Ans. y* Ans. V-^ MULTIPLICATION OP FRACTIONS. * Multiplication o^ Fractions is the process of multiplica- tion when one or both of the factors are fractional. To multiply a fraction by a whole number, Rule. — Multiply tlie numerator of the fraction by the whole number, and set the product over the denominator. Or, when it can be done without a remainder. Divide the denominator by tJie whole number, and set the quotient under the nuTneratoi'. NoTK. — Resulting fractions are, in all cases, if improper, to be reduced to whole or mixed numbers, if proper, to their lowest terms. EXERCISES. .irj 1. Multiply ^ by 3. 2. Multiply I by 8. 3. Multiply -^ by 9. ^.4. Multiply -iV by 8. Ans. f x3=V=li. Ans. ^=6f.. Ans. 3^. Ans. 3|> MULTIPLICATION OF FRACTIONS. To multiply a whole numheir by a fraction, Rule. — Multiply the whole number by the numerator of the fraction, and set tJie product over tlw denominator. EXERCISES. 1. Multiply 7 by g. 2. Multiply 12 by |. 3. Multiply 18 by ». 4. Multiply 5 by ^j,, ' ' 5. Multiply 4783 by ^g. :«'•'>, Ist. Divide 8 by 2=4; but 2 is three times the given cliviHor, and if 2 is contained 4 times in 8, one-third of 2 is contained 3 times us often. Therefore, 2nd, multiply 4 by 3=12 Ans. The second method is merely reversing the order of the operations for the sake of convenience. 2. Divide 9 by ?. 3. Divide 18 by ^V- • 4. Divide 27 by f . 5. Divide 43 by i. 6. Divide 78 by ||. To divide one fraction by another. Rule. — Invert the divisor and proceed as in multipU' cation. Note 1.— If the divisor or dividend or both be mixed numbers they must be reduced to improper fractions. Note 2,— The rule for this case will solve any case in division of frac- tions ; but t!\e operations are sometimes more tedious than by the niles laid down for other coses. EXERCISES. 1. Divide § by |. ♦ . 7 J m ■■,:■■-. n'r V t Ans. 21. Ans. 39. Ans. 33|. Ans. 49|. Ans. 84. This operation may be explained by reference to the pre- ceding rules, thus : 1st., divide | by 3, that is (case 1) mul- tiply the denominator by 3 ; now we have divided by a num- l)er 4 times the given divisor, hence the quotient -^ is only ^ of what is required, therefore, next, nidtiply by 4 which give^ ||.=-J=l-j^, ans. By cancelling i-s above the work is shortened. ■^.' 2. Divide I by 3^. 3. Divide || by |f. 4. Divide || by ||. 5. Divide iH- ^J M- 6. Divide 4^ by |. 7. Divide 9^ by ||-. 8. Divide ^ by l}. 9. Divide ^V by 4}. 10. Divide 5^ by 4f 11. Divide 4^ by 5|. When it is required to divide the product tions by the product of several others. Ans. fj Ans. Ir^ Ans. i^ . t, Ans. Jf Ans. 5^ Ans. 13^ Ans. 1^ Ans. -^ Ans. l4 Ans. -^ of several frac- 48 IBrrHMETIC. Invert all the faetor$ of the divieor and nvultiply alt together. 12. Divide the product of |, | and \ by the product of ^, I and •^. Ans. 1{. Jg 13. Divide the product of |, 4^ and f by the product of 2, 1^, 3^ and I. Ans. 11. 14. Divide ft of 2f by f of 4||. Ans. f . 15. Divide f of | by J of ||. Ans. 1^^^- A Complex Fraction is an expression of division of Frac- tions, — the denominator being the divisor, and the numera- tor the dividend. Hence, To reduce a complex fraction to a simple fraction, Divide the numerator by the denominator. 16. Find the value of £ 1 4^$-4-"**- 17. What is the value of 4? 18. Reduce | to a simple fraction. 19. Find the value of 7. 20. Seduce -I to a simple fraction. H 21. Find the value of 7^^. 22. Beduce 9| to a simple fraction. 12f 23. Divide ^ of ^ of 5^ by Sf of | of 18. 2i 6i To divide a mixed number by a vrhole number, when the- dividend is greater than the divisor. Rule. — Divide the integral part of the dividend by the dnmaor, Tfie remainder with the fraction^ or the fraction alonej if there be no remainder, will form the nwmerator of a complex fraction of which the divisor is the denominator. Beduee this complex fraction to a simple one, and arme»- it to the qvatient. Ans. 1^. Ans. 2| Ans. ^, Ans. 6\ Ans. f ' Ans. 18 Ans. 14 Ans. T^. DIVISION OF FRACTIONS. 49 KXKKCI8KS. 1. Divide 5876§ by 3. Bemaindcr, 2K_h 3" »•" 2. Divide 7918| })y .1. 3. Divide 491 8 J bv ». 4. Divide 68355| by 7. 5. Divide 19864? by 27. 6. Divide 9131} by 51. To dividu a whole or mixed number by a mixed number, when the dividend is greater than the divisor, BuLE. — Multiply both the dlvinor mid the diindend by the dencyminator of tJm fractioa in the divisor, and prO" oeed by the last rule. EXERCISES. 1. Divide 372 by 4^. 3).'587({§ " 1958| Ans. AnM. 15831 AuH. 546f| Ans. 9705^^ An8. 735| Ans. 17}-^! 1^. 372 2 ,• 9 744 82if Ana. . 4'' ' '■ 2. Divide 5973| bv 8J. Ans. 6941^. 3. Divide 386g by' 5f Ans. 68,»j^, 4. Divide 5987 by ^. Ans. 1651f|. 5. Divide 9176 by 5^. Ans. 1668^ 6. Divide 763^ by 2^. Ans. 286f NoTK.— The probloms in the laHt two cases, as well an all others in divi- ■ion of fractions, may be solved by the f^eneral rule under CuKe 3. The methods here mven are deductions from that rule ; and when the dividend is a large number they are very convenient, and those generally adopted. LEAST COMMON DENOMINATOR. It has already been shown that to divide both terms of a fraction by the same nmnber does not change the value of the fraction. Hence, also. To multiply both terms of a fraction by the same uimiber does not change the value of the fraction. Two or more fractions have a common denominator when their denominators are alike. Thus f , ^ and | have a com- mon denominator, 7. Any two or more fractions may be reduced to equivalent fractions, having a common denominator. A common denominator of two or more fractions must be a common multiple of their denominators, in order that the 6 50 AIlITUMKriO. flquivalcnt fmctioug ))uvin{{ the common ilenominator shall be simple tViictiunB. Thus u coiinnun denominator for the fractions } and | must be u common multiple of 3 and 4, as 12, 24, 36, &o., and ^ and '} muy be reduced to equivalent fractious having 12, 24, 3() oi' any other common multiple of 3 and 4 for their common denominator. The leitHt, common denominator of two or more fractions \H the Least Common Multiple of their denominators. To re(hice two or more fractions to equivalent fractions having u common denominator. lieduce f and j to equivalent fractions having a common denominator. 2x4=8 3x4=12 3x3= n 4x3=12 like r vason Tak(? 1 2 for a common denominator. Then, liintje the numerator and denominator of a frac- tion may lie multiplied by tlie fiame number without alterin^if itn value, multiply both terms of ^ by 4, l)ecau8e it makes the denominator 12; and multiply botli tiTiriH of ^ by 3, for a and we obtain -,^y and ^\j as equivalent fractions havinj^ a common denominator. RuLK. — Multiply both tennn of e,ach fraction by the pro- duct of all the deiiomlnatorti except its own, 2. Reduc(i § , 3 and 5 to equivalent fractions having a com- mon denominator. 5x8x3=120 3x6x3= 54 2x6x«= 96 8 = 111- •i — 00 a— 14 4- M- 6x8x3=144 3. Reduce ■§-, 4^ and g to equivalent fractions having a common denominator. Ans. -f^, ||^, -^j. To reduce two or more fractions to oniTnon denominator. The leaHt common (Umominutor la 30. Then, 8)3 «■' 3)r«o 5)30 thatiH, ^^jj^ The proctwH in eiiuul to tho foHowinjjf : — < 1 X 1 ;>= U 2 X 10=20 3 X = 18 2x15=30' 3x10=30' 5xfi=3d' and the multiplier for each fraction la fomul by dividin{( the leatit common denominator hy the denominator of the fraction. 2. Reduce ^, J, 3, to their least common denominator. 3. Reduce ^, J, -j^y, to their least common den iminator. Alls. -jV fj, tV- 4. Reduce ^, J, ^'»J^, ^J-, to their lejist common denomina- tor. AnU'lhihU^U- 5. Reduce ^j -fV> *^°^ ^o t^* their least common denomi- nator. Ans. II, f|, V4*. 6. Reduce J, 2^ and ^'j- to their least common denomi- nator. Au8.^f, W»li- 7. Reduce } J, 1 -^'g-, 7^ and ^''^ to their least common de- nominator. Ana. «}, ]§, 8,"5/>, f|. 8. Reduce §, ^ of 3^- and § of ^ to their 'least common denominator. Ans. ;f{|, i/^^, 1^. 9. Reduce f of «, f of §, ^ of | of ? of 2g to their least common denominator. Ana. f^g, f^J' HJ* 10. Reduce -} of ^, -^ of 4^ and ^ to their least common denominator. 9 Ans. j%-, |^«, x^. ADDITION OP FRACTIONS. Addition of F'ractions is the process of finding the sum of several fractions. To add fractions, Rule. — If the fractions to he aililed have the 8anie rfe- iiominator, adtl t/telr numerators, and write tl^ sum over fliecoriUium ilenomlnator. If the f radio ns have not the naine il^.noniinator, reduce tJiein to a coiti/niou ilenmnvlncir- tor; add tJie new numerators, and set the sum over tlie common denominator. 52 ARITHMCTIC. >H EXERCISES. 1. Add together S, 5, ^ and |. 6 3 ,7 rr» 1 f > If 2. Add together , , . 3. Add togetlier H, f »-, 4« and 4. Add together -j and -J. ,,, and r^p it* X 4 -4- X ,, 4— « - !» ly + lJ— IT— *Tf 5. Find the Kr.m of i and ^. * , ./'• < H. Find the sum of ,^. and -|^. 7. Find the sum of ^ and }-^. 8. Find tlie sum of \^ J- and }-»] . 9. Find the sum of |, j} and -j^. 10. What is the sum of ^, f , -J^ and I ? Ans. Y=:2.. Ans. 2^^ Ans. l|.. Ans. 1 J.. Ans. -j-j. Ans. Iff-. Ans. 2u\f Ans. l|-»r. Ans. 2|. When there are mixed numbers it is as well to add the fractions and whole numbers separately, and add their sums. 11. Find the sum of 1§ and 2^. X2 = 3 y 3 _ 9 t IT 10 I 9 15 1+2=3 Ans. 5-§. Ans. 10-^^. Ans. 11|-J. Ans. 40^. Ans. 21^'^. 4j^ Ans. 12. What is the sum. of 2^ and 3^ ? 13. Wliat is the sura of 2^^, 3f and 41 ? 14. Add together U, 2^, 3^ and 4j^. 15. Add together 16|, 12^, 8f and 2^. 16. Find the sum of f, 4^, -/^r, 9|^- and o||. 17. Add together ^ of f and | of -f of -/v. Ans. -^. 1 8. What is the value of ^ of 6f + -^ of f of 7^ ? Ans. 4:J^. 1 9. Find the value of f of 96^ + § of -]^ of 5 J- ? Ans. 59 J|. 20. Add together f|-, 7^?'^, «- of 4 J- and o^V ^^g jg »7 8k .' T'*^* The following will be foimd \;seful : — To adf^ two fractions both of which have 1 for a numerator. Rule. — Add the denominators for a numerator, and multipli/ them for a denominator. , - EXERCISES. 1. Add ^ and ^. 7. Add 1 and y. 2. Add ^ and ^. . 5. 8. Add ^t|-and ^V. 3. Add ^ end ^. ' ' 9. Add ^and^. 4. Add ^ and 4. 10. Add 1^ and i. 5. Add ^ and f . ■ \. Add J^ and y*g-. 6. Add ^ and ^V- -2. Add 1 and ^\r. SUmUACTION OF FllACTIONS. 53 I SUBTRACTION OF FRACTIONS. v ., .'.i Subtraction of Fractions is the process of finding the dif- ference between tw Ans. 1|. Ans. 4f. Ans. 2-^-. 54 .'* ARriHMETIC. 18. From IG subtract 3^. ^ ' '' ' ' '^ Ans. I25. Here it is easier to subtract the fraction and whole num- bers separately. Thus, '■' 16 Subtract ^ from 1 (§") and | remain, carry M 3j^ 1 to 3=4 ; 4 from 16=12. ' "'' \2i Ans. 19. From 391§ take 14Vf. 39 1 § Add 1 to f = 1 1 = i[ ; then. \"i*: r>:s^ri'X\i< Ans. 243^. 147$ -i^x 4-^3 n 243 1-^ Ans. f x l| x-^V and f^— 1^=1^- Carry 1 to the whole nuiubors. 20. From 320} take 249f . Ans. 70^» 21. A has $725| and B has $690f how much more has A than B ? Ans. $34|. 22. A man pwned |-§ of a ship, and sold ^ of his share ; how mucli had he left ? Ai " 23. What is the difference between g of 1|- and 4^^ Ans. -f^g. Ans. 7^^« 24. After selliiig 4- of f +i of | of a farpa what part of it remained ? Ans. ^-g. When the numerator of both fractions is 1, The difference of the denoininatcyra will he the numerator and their product the denominator of tJie difference. EXERCISES. • 1. From ^ take 4. 2. From | take -?y. 3. From ^ take i. ' ■ - ' ' 4. From ^ take -J. 5. From ^ take J . ' ' 6. From ^ take -jU-. ^ ' ' DECIMAL FRACTIONS. * A Decimal Fraction is one that has 10, or some power of 10 for its denominator, as -^^, -j-|^, yj-^ir, &c. The word decimal is derived from the Latin, decern, ten. Observe the relation. betwv.en the decimal fractions, -^j Tinr» 1 (1 ' ^^*- The first is six tenths, or -j*^ of units ; the second, six hundredths, or -j'^^ of -^^^ ; the third six thonsandths, or -^ of Hence, it is seen that these fractions bear the same relation 1 )I X'lM AI. FU ACTIONS. m. to one another as exists between the same (lijj;it3 in a whole number, and also tliat the first of the series })ear8 the same relation to 6 units. It' therefore the numerators of these fractions be arranj^ed side by side, thus, 666, tliey form an extension of the Arabic system, and may be used alone, or annexed to whohj numbers without their denominators. Decimals are generally so written, and »re known by being preceded by a period (.) called the decimal point. TABLE OF DECIMAL OKDEBS. s . ■ - ■ i : « J ■'■■'.■- "^ :- • . ■ ' it il 1st place .5 read 5 tenths. 2nd " .04 « 4 hundredths. 3rd " .006 " 6 thousandths. 4th " .0007 " 7 ten-thousandths. 5th " .00003 " 3 hundred thousandths. 6th « .000002 " 2millionths. 7th « .0000009.... " 9 ten-millionths. 8th " .00000001.. " 1 hundredth millionths. 9th " .000000008 " 8 billionths. Sum— .546732918 " five hundred and forty-six million, seven hundred and thirty-two thousand, nine hun- dred and eighteen billionths. And it will be found that if each of these decimals be ex- pressed in the common fractional form, viz.: -j*^, -j-ffs, n/W» &c., and added together by the rule for adding common frao- tions the sum will be JlA«:7^_s_w^_. By examing the above table and what has l)een said it will be seen that the value of a decimal figure depends on the place it occupies, diminishing in a one tenth ratio for every place it is removed farther from the decimal point. Hence to place a ciplier on the right of a decimal does not alter its value. r)ecause tlie cipher is nothing in itself, and, so placed, does not change the place of the figures. But a cipher placed on the left, between the decimal and the point, removes the figures one place to the right, and thus divides the value of the decimal by 10. 56 AIirrUMETIC. ' To read decimals expressed by figures, ^m, n,r BuLE. — Read tlie declmat as a. whole nuihber and give it the name of the right hand figure. 48.7804. , 83.0084. A.J . EXERCISES. Read the following : — .2. . ,.. .8004. , , .04. ., .4010. h , .138. .21042. 121.18006. .4531. .000014. 345.000018. .0098. .1743196. 909.000999. .00006. .0008980. 1203.080764. To write decimals in figures. Rule. — Write the decimal figures as a whole number; then place the point so that the right liand figure shall have its expressed value, x>lacing ciphers to the left of the signif- icant figures if necessary. EXERCISES. Write decimally the following quantities : — J 1. Five tenths. 2. Twenty-two hundredths. '^ .,, ■',. 3. Eight^'-seven thousandths. , - .v * , 4. Fiity-six ten-thousandths. ' ' ^' ,-.• . .'-. 5. Tliree hundred and four ten-thousandths. 6. Five thousand three lumdred and forty-seven ten thousandths. - j - - « 7. Eighty-eight millionth s. 8. Eight lumdred and eight, and eiglit thousand and eijiht millionths. 9. Ten tliousand and fifty-seven hundred-thousandths. 10. One lumdred and twenty-one, and one hundred and twenty-one thousand, one hundred and one millionths. 11. Seven thousand and seven ten-millionths. 12. Twelve thousand, and twelve thousand, one hundred and one ten millionths. 1 3. Six liundred thousand six hundred and seven millionths. 14. Twenty-seven thousand nine hundred and five, and forty thousand and four millionths. 15. Ninety-seven million, four hundred and fifty-three thousand, one hundred and sixty-eiglit billionths. A decimal is deduced from a common fraction by changing DECIMAL FRACTIONS. 57 seven ten the unit of the fraction to tenths, hundreths, &c., and per- forming the division indicated. Thus ^ means ^ of 1, or ^ of 4 imits ; but 4 units equal 40 tenths, hence ^ of 4 units =-J- of 40 tenths, or Y tenths =8 tenths, or .8. Again, | of of a unit=''^^ tentli8=.8, and g- of a tenth over, that is *^ hundredths =.07 and -*• of a hundreth over, that is ^ thous- andths =.005, and adding these three parts together they make .875 as a decimal equivalent to |. Therefore, To deduce a decimal from a common fraction, RuLK. — Annex ciphers to the numerator and divide by the deaaminator, placing the 'point in the quotient so as to make as many decimal fir/urea as ciphers annexed to the 7humerator. ■ o' t EXERCISES. 1. Deduce an equivalent decimal from \. 4 )1.00 .25 2. Deduce an equivalent decimal from ^. 3. What decimal is \ equal to ? 4. Reduce ^ to a decimal. 5. Reduce % to a decimal. 6. Reduce ^ to a decimal. 7. Deduce a decimal from -j\. . Ans. .25. 8. Deduce a decimal from ^ ;^. 9. What decimal is \1^ e([ual to? 10. What decimal is \% equal to ? 11. Reduce \ to a decimal of tux places. 12. Reduce \ of ^^,T- to its equivalent decimal. 13. What decimal is equal to -^^^ • Ans. .6. Ans. .75. Ans. .5. Ans. .375. Ans. .625. Ans. .4375. Ans. .52. Ans. .53125. Ans. .47916 + . Ans. .714285 + . Ans. .3125. US- Ans. 3.6. In deducing decimals from common fractions when any quotient figure or figures are found to continually repeat, as in exercises 9 and 10 above, the decimal is called an Infinite or Circulating decimal. The part of the decimal which repeats is called a Ke- petend. A repetend may be terminated at any point where it begins or ends by taking it as the numerator of a common fraction, and as many 9*3 as there are repeating figures for a denomi- nator and annexing the fraction to the preceding decimal if any. Thus I is equal to .8333, &c., in which the figure three is M ARITHMCTIC. a repetend. This decimal is correctly expressed thus, — .8^> or .83;|, or .833^, &c., that is the ^ is ^ reduced to its lowest terms. Again, | is equal to .714285 repeated ad infinUumy and is correctly expressed -J^^^H = \, or .71 4285 1-, or .714285714285^, &c. A repetend of one figure is distinguished by a point placed above it, thus .83. A repetend of more than one figure is denoted by a point over both the first arid the last figure, thus .712485. 14. Reduce ^ to a decimal. 15. Reduce § to a decimal. 16. Reduce % to a decimal. V 1 17. Reduce f^ to a decimal. 18. Reduce •j'j- to a decimal. 19. Reduce y^ to a decimal. 20. Reduce \^ to a decimal. 21. Reduce ^ to a decimal. 22. Reduce -j^j to a decimal. Ans. .3, or .3^. Ans. .6, or .6f . Ans. .5, or.5-§. / Ans. .8, or .81. Ans. .63, or .63-j^. Ans. .583, or .58^. Ans. .916, or .91|. Ans. .428571, or .42857 If Ans. .461538, or. 46 1538yV To reduce a decimal to a common fraction. Rule. — Write tJie decimal for a numerator, omitti/ng the point and cipJtera on tJie left ; and for a denominator, 1 with 0,8 many cipJisrs annexed as there are figures in the decimal, and reduce the fraction to its lowest terms. EXERCISES. -^ 1. Reduce .5 to a common fraction. 2. Reduce .25 to a common fraction. 3. Reduce .75 to a common fraction. 4. Reduce .875 to a common fraction. 5. Reduce .0625 to a common fraction. 6. Reduce .125 to a common fraction. 7. Reduce .3125 to a common fraction. 8. Reduce 2.125 to a common fraction. 9. Reduce 16.002 to a common fraction. 10. Reduce .0175 to a common fraction. 11. Reduce .390625 to a common fraction. 12. Reduce .003125 to a common fraction. 13. Reduce .15234375 to a common fraction. Ans. ^. Ans. \. '■"- Ans. |. Ans. |. Ans. -^. Ans. \. Ans- j5({. Ans. 2|. Ans. IQ-sl-^, Ans. ^J-^. Ana. f 2. Ans. 3^. Ans. 5^^. DKCIMAL FRACTION'S. m. Wien tfie decimal is a rfipetendf make the decimal with the point omitted t/ie numerator, and as many 9'« aa tliere are repeatiiuj Jirjurea for denominator, and reduce the I fraction as before. Ans. ^. 14. Reduce .3 to a common fraction. 15. Reduce .8 to a common fraction. 16. Reduce .8888 to a common fraction. 17. Reduce .72 to a common fraction. 18. Reduce .307694 to a common fraction. 19. Reduce .857142 to a common fraction. '.'t'' Ans. 5. Ans. -J. Ans. -^3. Ans. 1.- When the decimal is composed of a finite paH and a re~ petend, convert the repetend into a cmnmon fraction, and annex it to the finite part ; under this write the denomi- nator of the decimal, and reduce the complex fraction to a simple one. j; 'V 20. Reduce .83 to a common fraction. .83=.8|=8i, that is 8t^=8^-^10=lg^ x ^\.: 21. Reduce .916 to a common fraction. 22. Find a common fraction equal to .583. 23. Find a common fraction equal to .7083. 24. Reduce .027 to a common fraction. 25. Reduce .78545 to a common fraction. : J. Ans. Ans. \^. Ans. ^. Ans. ^, Ans. ^. The followiuf^ rule deduced from tlie above will be found [Convenient in solving (juestions like the last six. Rule. — Srblract the finite part of the decimal from tJie Iwhole, use the remainder for a numerator, and for a de- \nominator as many 9'« as there arejiijures in the repetend \ and as many ciphers annexed as thsre are fifjures in the initepaH. „ ... '< . ■ -j ■■ '. ADDITION AND SUBTRACTION OF DECIMALS. As decimals are an extension of tlie common Arabic system, they are added and subtracted in the same manner as whole numbers ; and it should be remembered that fif/ures of the same order must be placed under one another, that is tenths under tenths, hundredths under hundredths, &c. In other words, «60 ARITHMETIC. Arrcmge the quantities to be added or ftubtraded ao that the decimal points eJiall stand i^i a vertical column, add or subtract as in whole numbers, and place the decvmal point in the sum or difference directly under those in the numhers added or subtracted. *t EXERCISES. Add together .575, .0456, .73, and .1642i). Observe the decimal points in a column, so that tenths are under tentlis, Imndredths under hundredths^ &c. The column of tenths, with what is carried to it, amounts to 15 tenths, that is, 1 unit and 5 tenths. .575 .0456 .73 .16425 1.51485 2. Addtogether 21.611,6888.32, 3.4167. Ans. 6913.3477. 3. Add 6.6J, 636.1, 6516.14, 67.1234, 1233. Ans. 8458.9734. 4. Add 14.034, .25. .0000625, and .0034. Ans. 39.0374625. 5. Add 16.75, .375, 5, 3.4375 and .000875. * Ans. 25.563375. 6. Add 173, 7000.0005, 1.7, 125.728 and .0005. : , Ans. 7300.429. 7. Add .16, 39.5, .7283. Ans. 40.3949(i. y 8. Add 700.83, 16.765, .72835, 81.9. Ans. 800.227238. 9. Add .142857, .0186, 920, .0139428571. Ans. 920.1754. 10. Reduce to decimals and find the sum of 2^, 4| and 5-^^^. | ( < ■.■'^■■:yr'->:.^i ■■ Ans. 12.775.' >C 11. What is the sum of .76, .416, .46, .648, .23 ? Ans. 2.52087. 1 \¥\i Find the sum of .427, .416, 1.328, 3.029, and 5.476 J Ans. 10.67803712. 13. Find the sum of 35 units, 35 tenths, 35 hundredths! and 35 thousandths. Ans. 38.885. | 14. From 8.53 subtract 3.643. Arrange the numl)er8 so that the points shall be in the same column, and subtract as in whole numbers. Tlie place of thousandths being vacant in the minuend, we borrow one 8.53 3.643 4.887 Ans.! DKCIMAL FIIACTIONS. Gl [from tlu; hundredtliH, which U 10 thouHundthH, subtract 3 [thousandths i.ud carry one as in whole numlxjrs. 15. From 20.03()| subtract 8.77^. 20.0365 20.03(>J 8.7733 or 8.773| ■>.>\' .(V '1 ; 11.2«32 Ans. 11.2631 . \. 16. From 24.0042 take 13.7013. Ans. 10.3029. 17. From 170.0035 take 68.00181. Ans. 102.00169. 18. From .0142 take .005. Ans. .0092. 19* What is the diflferencc }»etween .05 and .0024? Ans. .0476.. 20. What is the diflference between 72.01 and 72.0001 ? Ans. .0099. 21. From 19 take 8.9991. ;: "^ Ans. 10.0008, or 10.00(^5. 22. From .4 take. 04i^. ,' Ans. .356, or .35§. 23. From 2^ take ij. Ans. .95. 24. From 1.169? take .93^V Ans. .238857142, or .238^. 25. Wliat is the ditference between 24^ tenths and 3701 [thousandths? Ans. 1.251. 26. Subtract 1^ liundredtlis from 49§ tenths. Ans. 4.9225, MULTIPLICATION OF DECIMALS, Multiply .375 liy 7. Operation by common fractions. Opprallon, .375 7 "'■ ' ,\r mn v7 — 3636 — onss — 9 fi'j/i To 0' ^ T — J 000 — 'to — — "'i'' Multiply 2.75 by .9 , operation. Operation by common fractions. ^ 2.75 \ .J •^Tfftr— ro (T ^ Iff— to'i) — ^To ——•*'«> Hence, to multiply decimals. Rule. — Multlplt/ as in ■whole nvmhers, and point off in Ithe product as inany decimal places as there are in the Imultiplicand and multiplier tor/ether. If there be not \ enough figures in the product to give the required number lof decimal pluces, supply the deficiency by prefixing ciphers. 62 ARITHMETIC. 1. Multiply 2.54 by .34 yji. Multiply 4.16 by .014. ,3. Multiply 4.5 ])y 4. 4. Multiply .01 by .15. 5. Multiply .08 by 80. «. Multiply 18.46 by 1.007. 7. Multiply .00076 by .0015. . H. Multiply 7.49 by 63.1. 9. Multiply .0021 by 21. 10. Find the continual product of .2, .2, .2, .2, .2, .2. ^ Ans. .000064. 11. Find the continual product of .101, .011, .11, 1.1, and 11. AnB. .001478 741 Ans. .020736. Ans. .86.36. Ans. .05824. Ana. 18. Ana. .0015. Ann. 6.4. AnB. 18.58922. Ana. .00000114. Ans. 472.619. Ans. .0441. 12. Multiply . 144 by .144. 13. Multiply 14.583 by 2.75 «:J 14.583 2.75 72916 1020833 2916666 40.10416 33 &c. 66 &c. 33 &c. G6&C. In this exercise the last figure in the midtiplicand in a repetend, and muat be treated as such. In multiplying by 5 we must carry 1 from the product of 3 understood on the right, and the 6 in the product is a repetend. In a similar manner we carry two when we begin to multiply by 7, and the 3 in the product is also a repetend for which reason we till up the place on the right of the product usually left blank. Also in multiplying by 2, as the 3 in the multiplicand is a repetend so is the 6 in the product, and we must fill up the two places on the right with 6'8. Then in adding the partial products, we must allow for other columns on the right, made up of the repeating figures, and so carry 1 at the beginning. Note. — The above method answers very well when the multiplicand alone contains a repetend of only one figure ; but when the repetend consists of more than one figure, or when there is a repetend in both multiplicand and multiplier, the process becomes complicated, and it is usual to proceed by the following RuLP]. — Reduce the decimals to common fradiona and perform tm multiplication required; then reduce tJie frac- tion, if any. In the product to a decimal, 14. Multiply 7.416 by 8.5. Ans. 63.041*6. 15. MiUtiply .078 by 7. Ans. .552. > " Ml LTI PLICATION OF DECIMAI-S. 68 1(». Multiply ;).<53H ]>y .27.54. » Ana. 1.5">295. 17.Mnltiply.7aby2.fi. ' Ana. 1.95. y,lH. Multiply r>.7.3fi by .41(5. Ana. 2.39015. 19. Multiply 9.4*>7142Hhy ..)384f>i. Ann. 5.0923()7fi. To Multiply by 10 or any power of 10, as 100, 1000, 10000, &e. , . : ; ,, ; - .,; , ., RuLK. — Af. One poiuiil Hterlinj; is worth ^4.H(i§ ; what is the value ofLlOO? Aus. !t^48G.6G§. 7. What will 1000 barrels of Flour cost at $6.75 per barrel ? Ana. ^()750. 8. What is the cost of 100 acres of land at $17.37^ per acre? Ans. $1737.50. 9. What is a million pounds Bterlin^ worth at $4.86f each? . , - Ans. $486(>G()6.fi6§. 10. Multiply 6^ by 100000. Ana. 671428.'571428. To multiply by 15. Move the point one place to the rir/ht, taJce the half and add it. ^ To multiply by 25. Move the point two places to the right and divide by 4. To multiply liy 250. Move the point threp: places to the ri(fht, and divide 4. To multiply by 75. Move the point two places to the right and suhstract a a fourth part. To multiply by 7^. . ,, More the point one pkice to the right and suhtract a Jourth part. lo multiply by 12^. Move the point two places to the right and divide by 8. To Multiply by 2^- ■Move the point one pla^e to tJie right and divide 4. • 64 ABmiNi>7no. 'ii KXKRCI8B8, 1. Multiply 2. Multiply 3. Multiply 4. Multiply 5. Multiply G. Multiply 7. Multiply 8. Multiply 9. Multiply 10. Multiply 2r).7fi4 by 25. .0890 by 1.5. .798.5 by 2.10. 240.8 by 7^. 3.987 by 75. 19.50 by 2^. 100.5 by 1.50. .0(X)32 by 2.50. 73.5 by 750. 99 by 2^. Ana. 644. K Adb. 1.344. An8. 199.625. AuB. 1806. AnH. 299.025 Ans. 48.7.5. An8. 24075. AnH. .08. An8.5512.5. Ana. 247.5. • DIVISION OF DECIMALS. Division is the converse and proof of multiplieution, — the product becoming the dividend, the midtiplier or multipli- cand, the divisor, and the multiplicand or multiplier, the quotient. Hence, since the product contains as many decimal places as the two factors to 875. Divide 8735.724 by .9. Divide 724.573 by .7. Divide 573.183 by .6. Divide 6927.8516 by 78.5 to seven places of deciinalt). Ans. 88.2528874-}-. Divide 9.6 by .55. Ans. 17.57. Divide 12173.9583 ])y .3,1416. 3.1416)12173.958'33(3875 Ans. Ans. 70. Ans. 42. Ans. .3.13. Ans. 7.191. Ans. 1.122. Ant*. 440. Ans. 63.445. Ans. .0084. Ans. 9706.36. Ana. 10,35.104 + . Ans. 955.305. 942490 »«• «?-.,.,': I I V.l ;' 2748958 3 3 2513.333 3 3 2356250 219916«« 1570833 1570838 ',i> ■«» f The above method of dividing, when the divisor contains a repetend, is somewhat tedious and requires great care. The more usual method is to reduce the repetend to a coip- mon fraction, and then divide by the mixed numb«r^ Thus — ''., 3.141§)12173.9583(3875 Ans. •■ «;4f«|!ufM Ji 3 3 . .•J.V^tWJMliH 1^ 9.425 36521.875. n- > 66 ARITHMETIC, 16. Divide .8 by 2.6. 17. Divide 6020.06 by 4.86. 18. Divide 1.77975 by 25425. AnB. .3. Ans. 1237. Ad8. .00007. To divide by 10 or any power of 10, as 100, 1000, 10000, 1000000, &c. Rule. — Move the decimal point as many places to the left aa there are ciphers in the divisor. Ans. 342.55. EXERCISES. 1. Divide 3425.5 by 10. 2. Divide 57.75 by 100. 3. Divide 1444.755 by 1000. 4 Divide 8.J39 by 100. 5. Divide .75 by 10000. 6. Divide 5863.72 by 100000. 7. If it cost $7000 to furnish a meal for an army of 100000 men, what is the cost of each man's meal ? Ans. 7 cents. h S ■' DENOMINATE NUMBERS. An abstract number is simply a number without reference to any object, as, 7, 16, 39, &c. A Concrete number is a number in connection with some object oro})ject8 named, as, 1 h'jrse, 7 men, 39 ships, &c. Denominate numbers are coi.crete numbers applied to the denominations of weights and measures. TABLES OF WEIGHTS AND MEASURES. CANADA C 100 cents (cts.) are 1 dollar BRITISH OR STERLING CURRENCY. 4 farthings aro 1 perny (d.) 12 pence " 1 shilling (s.) 20 shillings " 1 pound (.i\) 5 shillings *' 1 crown. 21 shillings " 1 guinea. URRENCY. (S). UNITED STATES CURRENCY. 10 mills li) cents 10 dimes ^0 dollars are a 1 cent (ct.) 1 dime (d.) 1 dollar ($.) 1 eagle (ea.) Although the abovo is in theory the U. »S. table of currency, in prac- tice it is thtt 8Amc as that of Canada. ■-k ■"I DENOMINATK NUMMUW. 67 AVOIRDUPOIS WEIGHT. The Imperial pound avoirdupois, containing 7000 grains, is the standard for weighing heavy articles, as meat, groceries, vegetables, grain, &c. 16 drams (drs.) are 1 ounce (oz). ■ 16 ounces " 1 pound (flj). 100 pounds " 1 hundred weight (cwt.) or cental. 20 hundredweight " 1 ton (t). In Great Britain 28 lbs. are 1 quarter, and 4 quarters, or 112 lbs. 1 cwt. This manner of reckoning weight of heavy articles has been used in this country, and still is, in some exceptional cases, but it is gradually gro Mng less prevalent. In the Dominion of Canada and the United St;ites a quarter means 25 l!)s. ; but it is not much used. 112 lbs. is the weight of a. quintal of fish ; and a ton of coal is 2240 lbs., or 20 cwt. of 1 1 2 1)8. each . ' The "Weights and Measures Act" of 1873, which fixes the ton at 2000 flis. will probably have the effect of doing away with the " long ton," in weighing coal as well as other articles. In Great Britain, 14 lbs. make 1 stone. TROY WEIGHT. The Troy ounce (480 grains) is the standard for weighing gold, silver, platina and precious stones. 24 grains (gi's.) are 20 pennyweights " 12 ounces a 1 pennyweight (dwt.) 1 ounce (oz.") 1 pound (lb.) APOTHECAllIES WEIGHT. Apothecaries mix their medicines by this weig'it, using the Troy ounce, but they buy and sell by Avoirdupois. 20 grains (grs.) are 1 scruple ( 3). 3 scruples " 1 drum ( 3 ). 8 drams " 1 ounce ( ? )• * 12 ounces " 1 pound (tti.) ■•■;*-'^ ^i^i... . , ■ . - . PUODUCE WEIGHT TABLE. By the "Weights and Measures Act 1873*' the weights of produce are fixed as follows : — 68 ARITHMETIC. To the bU!<}[■ miscellaneous ^jeasures. make 12 articles 20 « 144 « 24 sheets of paper " 20 quires 196 fcs. Flour 1 dozen. (( t( 1 1 1 1 1 score, gross, quire, ream, barrel. 200 " Beefor Pork make 1 barrel. ■ f 31 days. 'l 31 " 1 30 << 1 31 " 1 30 " m 31 <« m February H iar consist h| 3 the time ^ d the sun, 1 r is llm., M 3 days in M m\y every ^ fe the two ^W ear by 4 ; ^ ear whose H ts number H UKNOMIXATH NUMLKUS. OF HOOKS. A sheet folded in 2 leaves is called a folio. A sheet folded in 4 A slieet folded in H A sheet folded in 12 A sheet folded in 18 71 a , a' quarto, or 4 to. a n an octavo or 8 vo. a ^i a duodecimo or 12 mo. 4b a an 18 mo. METRIC WEIGHTS AND MEASURES. The following are the principal tables of tho Metric sys- tem. The use of these weijahts and measures has been made legal by Act of Parliament; but as it will likely be many years before they come into general use in thi» country, the following brief synopsis of them is all that is deemed necessary in this work. 10 millimetres 10 centimetres 10 decimetres 10 metres 10 decametres 1 hectometres 10. kilometres LONG MEASURE. 1 centimetre = 1 decimetre = 1 MKTIIE = 1 decametre = 1 hectometre = 1 kilometre = .010939 yards. .109394 " 1.093944 « 10.939444 " 109.394444 " 1093.944444 « 1 miriametre = 10939.444444 " The metre is the unit for measuring common distances, and the kilometre, the unit for long distances. SURFACE MEASURE. 100 centiares = 1 ARE = 119.6714 sq. yards. 10 ares = 1 decare = 1196.7144 " (C ^ 10 decares = 1 hectare = 11967.1444 « (( The hectare is a little less than 2^ acres. Sr- :; ■:: --.^: : ■ I WEIGHTS. 10 milligrams = 1 centigram = .0000220 fcs. aroir. 10 centigrams = 1 decigram = .0002204 (( u 10 decigrams = 1 GRAM = .002204 U (( 10 grams = 1 decagram = .022046 (( (( 10 decagrams = 1 hectogram = .220462 (( « 10 hectograms = 1 kilogram = 2.204621 U it 10 kilograms = 1 myriagram = 22.046212 (k (i 10 myriagrams = 1 quintal = 220.46212 u (( 10 quintals = 1 millier =2204.62125 or tonneau (( u S'l'i i f% ARITHMETIC. MEASURE OF CAPACITY. 10 centilitres = 1 decilitre 10 decilitres = 1 litre 10 litres = 1 decalitre 10 decalitres = 1 hectolitre 10 liectolitrcs = 1 kilolitre The Litre is used in measuring liquida. is used in measuring grains, &c. = .022 Imp. gal. — .2202 " " = 2.2024 « « = 22.0244 « « = 220.2443 « « The Hectolitre REDUCTION OP DENOMINATE NUMBERS. Reduction is the process of changing the denomination of a quantity without altering its value. Reduction may be considered as of two kinds — REDUCTION Descknding and Reduction Ascending. Reduction Descending consists in reducing a quantity to a lower denomination than that in which it is expressed. Thus, reducing dollars to cents, pounds to shillings, tons to ounces, bushels to quarts, &c., is Reduction Descending. Reduction Ascending consists in reducing a quantity to a higher denomination than that in which it is expressed. Thus, reducing cents to dollars, pence to shillings, ounces to pounds, quarts to gallons, 4&c., is Reduction Ascending. Rule for Reduction Descending. — Multiply by that nuTnber which expresses Jioiv many of tJie lower name make one of the higher. .: Reduce £26 to shillings. £'2Q 20 520 shillings. We multiply by 20 because there are 20 shillings in £1, that is, 20 of the lower name make one of the higher. Or, because, since there are 20 shillings in £1, there are 26 timea 20, or 20 times 26 shillings in £'26. Reduce 18 days, lOh. 23m. 40 sec. to seconds. ^.i fr\ ..■*" d. 18 24 ii. 10 m. 23 40 : ^^ ^ w; 442 60 26543 60 1592620 Multiply by 24, because 24 hours niaku 1 da^. Add in 10 hours. Multiply by yO, because 60ni. make 1 hour. Add in 23 minutes. Multiply by 60, because 60 sec. make 1 minute. Add in 40u8ecoDd«, DKNOMINATE NUMBERS. 73 1. 2. (( (( .' 1 3. u (( •■ 4. (( (( 5. Hectolitre 6. 7. 8. ERS. 9. nination of 10. 11. Reduction 12. juantity to expressed. gs, tons to nding. 13. 14. 15. mtity to a expressed. igs, ounces icending. \y by that icume make 16. 17. 18. 19. 20. i 21. " 22. lillings. ings in £1, gher. Or, re 26 times 23. 24. 25. 26. , ■ ■- :~ ■, :-- -;■- -. iv 21. , .;i j« -^ U^ 28. . ^u 41 ^'t 29. «-4t--?4 -^ m 30. 1 31. ii..,U-M-^-Sfi 1 32. fti^iim-il^: ■ 33. - ' EXERCISES. Reduce £25 12u. to pence. How many pence in £'325 1 9s. 7d. ? Reduce i-l9 to farthings. Keduce £27 17s. l^d. to farthings. Reduce $273 to cents. How many cents in $478.25 ? Reduce $16 to mills. Reduce 17ea. 7dol. 3 dimes to cents. Reduce 37 tons to cwts. How many pounds in 3 tons 17 cwt. ? Ans. 6144 d. Ans. 78235 d. Ans. 18240 far. Ans. 26781 far. Ans. 27300 cts. Ans. 47825 cts. Ans. 16000 mills. Ans. 17730 cts. Ans. 740 cwt. Ans. 7700 fl)s. Reduce 7 cwt. 59 lbs. 7 oz. 12 drs. to drams. Ans. 1 94428 drs. How many ounces in 18 tons? Ans. 576000 oz. Reduce 25Ibs. Troy to grains. Ans. 144000 grs. Reduce 6lbs. 8oz. 15dwt. to pennyweights. Ans. 1615 dwt. How many grains in 3oz. i6dwt. 18grs. ? Ans. 1842grs. Reduce 251bs to grains by Apothecaries' weight. Ans. 144000 grs. Reduce 5fljs. 6oz. 4dr8. Iscr. 8grs. to grs. Ans. 31948 grs. How many scruples in 7oz ? Ans. 1 68 scr. How nrany lbs. in 25 bushels of wheat. Ans. 1500 B)s. What IS the weight of 245 bushels of oats ? Ans. 8330Ibs. Reduce 17 bushels of potatoes to lbs. How many rods in 7 miles ? How many yards in 40 rods ? How many feet in 47 miles ? Reduce 15 miles 5 fur. 35 rods, 3 yds. 1 ft. 7in. to inches. Ans. 997057 in. How many square yards in 3 acres ? Ans. 14520 yds. How many rods in 2 acres, 3 roods, 20 rods ? Ans. 460 rods. Reduce 5a. Ir. 37rds. 20yds. 6ft. 112in. to square inches. Ans. 34408804 sq. in. How many chains in 20 miles ? , , Ans. 1600 chs. How many inches in 4 chains ? ' ' Ans. 3168 in. How many square chains in 27 acres ? Ans. 270 sq. chs. Reduce 12 cubic yards to cubic inches. Ans. 559872 cu. in. How many pints in 75 gallons ? . Ans. 600 pts. Ans. 1020Ibs. Ans. 2240 rods. Ans. 220 yds. Ans. 248160 ft. 74 AmTUMETIC. I! 34. How many ^illsin 9 hlids. ? Ans. 18144 gills. 3o. How inauy <(uarts in 17 busliels? Ans. 544 qrts. 36. Keduce 12bu. Spks. 5qrta. to pints. Ans. 82G pts. 37. How many days in 1873 years. Am. 084113^ days. 38. Reduce 5y. 24Ud. 12h. 42m. 36s. to seconds. Ans. 178569756 sec. 39. How many seconds in 47<^, 50 ', 25 " ? Ans. 1 72225 ' '. 40. How many days in the first six months of tjje year ? Ans. 181. 41. How many days in the last six months of the year? . , .. . . Ans. 184. Rule fou Reduction AHCJENDiNn. — Divide by tJiat num- ber which expresses how many of the loiver name make one of the higher. 2,0)52,08. £26 Ans. Reduce 520 shillings to pounds. We divide by 20 because there are 20 shillings in a pound, that is, 20 of the lower name make one of the higher. Or, since there are 20 shillings in £1^ there are in any number of shillings ^ as many pounds, which is found by dividing by 20. Reduce 1592620 seconds to days. ' , Ist step, from seconds to 6,0)159262,0 s. ' • ' ' 2nd « " minutes to 6,0)2654,3 m. 40s. hours 3rd (( (( to 24)442 h. 23m. 40s. 24 (1 8d. lOh. 23m. 408. Ans. 202 192 lOh. ' ' The remainder after each division is of the same name as the dividend. ^- ' EXERCISES. ^■ 1. Reduce 6144 pence to pounds. 2. Reduce 78235 pence to pounds. 3. How many pounds in 18240 farthings? t ,> . . 4. Reduce 26781 farthings to pounds. > 5. Reduce 27300 cents to dollars. ,,;.(,;' * 6. How many dollars in 47825 cents ? i- i ^ >f ;^ •J: 7. Reduce 16000 mills to dollars. .» 8. Reduce 17730 cents to eagles, &c. ,' > "^ c DICNOMINATK NUMBKRH. 75 9. ITow many pounds Apothecaries' weijjht? 17. In 31948 grains how many poundd, etc., Apotliecarie** weight ? 18. How many onnccw in 108 scruplos? 19. In 1500 lbs. wheat how many bushels ? 20. How many bushels in 8330 il)s. of oats? 21. In 1020 lbs. of potatoes how many bushels? 22. Reduce 2240 rods to miles. 23. Express 220 yards in rods. 24. How many miles are e2i ct'nts. 11. Reduce ^\ of a ton to ounces. Ans. 1400yoz. 1 2. Whut is the value of § of a pound sterling ? 5 :(:^x^0=f=l2is. ^ 'sx \ Ans. 12s. 6d. 13. Reduce j\ of an acre to sq. rods. Aus. 56 rods. 14. Reduce | of a shilling to the fraction of a pound. 6 =^\V- Ans. 15. What is the value of -,7^ of a ton. Ans. llcwt. 66§B)s. 16. What is the value of -^jy of a yard. Ans. 2ft. 8|^in. 17. What is the value of ^ a pound Troy. Ans. 8oz. 18. Find the value of -j^g- of a shilling. Ans. ■'>j'jd. 19. Reduce -| of a dollar to its value in cents. Ans. 88^ts. 20. Reduce 401bs to the fraction of a cwt. Ans. Divide 40 by 100, thus, V'b'V=fcwt. 21. Reduce 12 shillings to the fraction of a pound. Ans. -}. 22. Reduce 9d. to the fraction of u shilling. Ans. |. 23. Reduce 6oz. to the fraction of a pound, Avoirdupois. Ans. §. to the fraction of a pound. Ans. ^j. 25. Reduce 35Ibs. to the fraction of a ton. Ans. ^^-j-. 26. Reduce 5 days to the fraction of a year. Ans. y'g-. 27. What part of a bushel of wheat is 251bs. ? Ans. ^y. 28. Reduce 4235 sq. yards to acres. Aus. |. 29. Reduce 1 28. 6d. to the fraction of a pound. Begin with the lowest denomination, 6d., reduce it to shil- 24. Reduce lOd. sterlin;^ UKNUJllNATJi NLMUKIiS. 77 lin^H a« you would any other numl>or of pence, tliat in, diviflo it by 12. Ni)W ^lic only way you can divide it l»y 12 in by makiiij;; 'J tlu* nuiiuTator and twflvo the denominator of a fniction, thun, /y, \vhi<'h, when rcducoil iw ^. l^s, (Id., there- fore i« 12i^H. liediice this to pound by dividing,' by 20, that is, make 12^ the? luinunator of u fr.ution and 20 the ^denom- inator, thuH, '*' a t'omnU'x fraction ; reduce it, and it becomes 3, which i.s the answer. Hee tlni work ; — i'^-=i;;i^ = f5 = {l Ans. Or, reduce the whoh» (piantity to the lowest denomination, and divide by th(^ number of tliat denomination which makes one of tlie hijjher name to which it is to be reduced. Thus — 12s. (id. 12 L)0 240 = i, Ans. 8' Ans. |. Ans. ^,[.. Ans. jj. Ans. }{. Ans. -[|. Ans. -,V\t- Ans. -j. Ane. f ^J. Ans. ;jy^. .30. Reduce 17s. (Id. to tlie fraction of a poimd. M. Reduce .^s. (Id. to the fracti«m of a pound. .32. Reduce 7s. (Id. to the fraction of a ])ound. 33. Reduce 4^d. to the fra<'tion of a sliillin^jf. 34. Reduce 9^d. to the fraction of a shilling. 3.5. Reduce 1 Is. 7-^d. to the fraction of a pound. 3(1. What part of a dollar is 40 cents ? 37. Reduce 56Ih3. Hoz. to the fr;«5tion of a cwt. 38. Reduce 65 lbs. to the fraction of a ton. 39. Reduce 19cwt. 281bs. 12oz. to the fraction of a ton. 4rtu t S 4 3 40. Reduce Goz. 13 dwt. 8grs. to lbs. Ans. §. 41. Reduce 3fur. 4 rods, 2yd8. 1ft. 4in. to the fraction of a mile. ' ,; 42. Reduce o.'J days to the fraction of a year. 43. What is the value of .875 of a pound sterling ? » Reduce as in the whole; number, observing to point off the decimals properly. Thus, — £.875 ~ , 20 Ans. 17g. 6d. Ans. -j\. Ans. 1^-. '-'•M 17.500 shillings. 12 6.(X)0 pence. n AUITHMETIO. :: \ ; I 44. Reduce £.02/) to its vahiP in MJiillingR JinKN( )M I N ATK NUMBKIW. 70 IS follows — ' >r« 4)2.0 flirt liitigfl. 12)l()..'5(X)d. 2t)).'5.H7.'5H. i,'. 2937/5 AnH. f)2. Reiluco 10^(1. totluMhwimalof apound. A n». £.04375. C3. Reduce l.'Js. })Jd. to ilio ducimul of :i pound. Ans. £.790r>25. 04. Reduce 3 rooda, 1 1 rods to the dccirnul of an acre. Ans. .H1H7.5. fl5. Reduce 3cwt. 32lt)8. to tlie decimal of a ton. Anri. .166. 66. Reduce 37 rods to the decimal of a mile. Ans. .115625. 67. Reduce 7oz. 4dwt;. to the decimal of a pound. Ans. .6. 68. Reduce 5 hours, 48min. 4i).7s«'(^ to th(? decimal of a day. . Ans. .2422419 nearly. ADDITION OF DENOMINATE NUMBERS. Rule. — Write the quantitiea to he added 8o that numbers of the same denomination may stand in cohimn. Begin at the right hand^ or lowest denomination, add each denom- ination separatehjf reducing each sum, to the next higher denomination, the nunther of which cai'ry to the colujnn to which it belongSf arid set tJte remainder, if any, under the column added. EXERCISES. (1) ydR. ft. in. 12.2. 9 16.1.11 27.3. 8 36.3. 6 (2) £76.18. 4 17.11. 4i 99.19. 9 11.11.11 67.15.10i 79.19. 9 28.12. 1 63. 8. 4| (3) lb«. oz. (In. 13.14.10 15.11.10 11. 4. 9 8.12.13 15. 7. 8 10.13.11 8. 9. 6 4.15.15 t. cwt. qn. IbSk 26.17.3.21 18.11.0.19 25.15.1.16 13.17.2.20 39. 4.1.23 28.16.3.14 04. 2. 10 .445.17. 5^ 89.10. 2 153. 3.2.13 80 ARITHMETIC. 5. Add together 31bs. lloz. 16dwt. 21gr8. : 51b8. 8oz. 7dwt^ llgrs; 7fl5s. Goz. 18dwt. 23grH.; lifts. lOoz. 15dwt. ITgrs. ; 12ft8. 7oz. 9dwt., Sgrs. ; IGfts. lOoz. lldwt. 22gr8.; ISfts. 8oz. 19dwt. ISgrs. * Ans. 77fcs. 8oz. Odwt. Ogrs. 6. Find the sum of 5fr)8. lloz. 7drs. 28cr. 19grs. ; 4fts. lOoz. 4drs. Isci'. 7grs. ; Sfts. lloz. 6dr8. 28cr, Idgrs. ; 1ft. 9oz. Sdrs. Iscr. 12gr8. ; 2fts. 4oz. Sdrs. lOgrs. ; Gfts. 7oz. 2dr8. 28cr. 9gr8. ; 2fts. 8oz. Idr. Iscr. 13grs. Ans. 28ft8. 4oz. Odrs. Iscr. 4grs. 7. What is the sum of 176m. 7fur. 39 rods, .'>yd8. ; S.'jm. 4fur. 20 rods, 1yd. ; 79m. 6fur. 29 rods, 3yds. ; 42m. 3fur. 8 rods, 2yds.; 67m. Ifur. 11 rods, 2yds. ; 118m. 3fur. 10 rods, 3yds. ; 81m. 2fur. 31 rods, 1yd. ; 79m. 21 rods, 2yds. ; 18m. 3fur. 33 rods, 3yds ? Ans. 749m. 2fur. 6 rods. 8. Find the sum of 18yds. 2ft. llin. ; 14yd^. 2ft. 7in ; 8yd8. 1ft. lOin. ; 11yds. 7m.; 7yds. 2ft. 8in.; 16yds. 2ft. 9in.; 8yds. 1ft. 7in. Ans. 86yds. 2ft. llin. 9. Add together 29a. 3 roods, 39 rods; 57a. 2 roods, 18 rods; 118a. 26 rods; 7oa., 3 roods, 11 rods; 51a. 1 rood, 8 rods; 94a. 1 rood, 19 rods ; 63a. 2 roods, 21 rods ; 78a. 1 rood, 15 rods ; 19a. 3 roods, 33 rods. Ans. 589a. roods, 30 rods. 10. Add together 39 rods, 30yds. 8ft. 143in.; 18 rods, llyds. 4ft. 68in. ; 24 rods, 4yd8. 7ft. 118in. ; 11 rods, 21yds. 2ft. 96in. : 15 rods, 27yd8. 124in. ; 27 rods, 6yds. 3ft. 87in.; 19 rods, 25yds, 2ft. 38in., — square measure. Ans. 157 rods, 6yds. 3ft. 98in. 11. What is the area of 7 farms, measuring as follows : — the 1st., 79 acres, 9 chains, ^999 links'; the 2nd,, li7a. 4ch. 36501.; the 3rd., 47a. 5ch., 941 1; the 4th., 56a. 2ch. 11821.; the 5th., 27a., 7ch.. 28131.; the 6th., 36a. Ich. 7711. ; and the 7th., 84a. 8ch. 11601. ? Ans. 449a. 8ch. 516 links. 12. Find the sum of 35 bushels, 3 pecks, 1 gallon, 3 quarts, 1 pint; 18b. 2pk8. Iqrt, Ipt. ; 7b. Ipk. Igal 1 pt; 26b. Iqrt. ; 18b. Igal. Ipt. Ans. 106b. 3qtfl. 13. Add together 6 tuns, 1 pipe, Ihhd, 39gal. 3qrts. Ipt. 3 gills ; 4 tuns, 1 hhd. 47gals. 2qrts. 2 gills ; 5 tuns, Ipi. Iqrt. 1 pt. 1 gill; Ihhd. 52 gals. Iqrt. Ans. 17 tuns, 1 hhd. 14 gals. Ipt. 2 gills. 14. Find the sum of 4 tons, 7cwt. 86ft8. ; 2 tons, 9cwt^ DENOMINATE NUMBERS. 81 43lb8. ; 1 ton, 8cwt. OOlbg. ; 1 ton, 16cwt. 33B»s. ; 4 tons, 8cwt. 41fl)8. ; 2 tons, 17cwt. 89D)8. Ans. 17 tons, Scwt. 82ft)8, 15. What is the sum of 359^ 59', 59/'; 153", 40', 45''. 270«',0',0'/; 179°, 45', 30''; 81^30', 10'/; 89^59',59". Aus. 1134^56', 23/'. 16. It is required to find the sum of the following^ pe- riods : — 33 years, S64 days, 23 hours, 59 minutes, 59 seconds ; 28y. 113d. llh. 48m. 488. ; 17y. 97d. 12h.; ly 307d. 23h. 48m. 498.; 12y. 114d. Ans. 93y. 267d. 23h. 37m. 368. To add denominate fractions, Red/uce the fractions to tlieir value in lower d&nomvna^ UonSy and add the results. 17. Add f of a pound to f of a shilling. .8. d. |x*» = Vd.= 0.10 13.4 Ans. 18. Add 4^ of a ton to -^ of a cwt. Ans. 12cwt. Ifl). 3oz. O^fdra. 19. Add together ^ of a mile, ^ of a furlong, and -^^ of a rod. Ans. 4fur. 13 rods, 4yds. 2ft. 9|in. 20. Add together i^^j^ of a cv^t. f of a ton, and f of a pound. Ans. 12cwt. 945)8. 6oz. lOfdra. 21. Add together £-^^ and .875 of a shilling. Ans. 12s. l^d. 22. What is the sum of .79685 of a ton, and .1778125 of a cwt. ? Ans. 16cwt. lllbs.- 7oz. 11.2drs. 23. Find the sum of .675 of a pound Troy, and .97625 of an ounce Troy. Ans. 9oz. Idwt. 12.6gr3. 24. Add together £.790625, .5416 of a shilling, and .75d. Ans. 16s. 5d, SUBTRACTION OF DKNOMINATB NUMBPlllS. Etjle. — Write the smaller quantity under the largely setting numbers of the same denomination under each other. Begin at the right, and take the numbers in ths suifro- hbiidfrom those immediately above them in tfie minu,Gnd. ARITHMETIC. When any number in tJte subtrahend exceeds that of the same denomination in the minuend^ add to the number in the minuend^ as rnaiiy of that denomination as make one of the next hlf/her, subtract the number in the subtrahend from the sum, and carry one to the next denomination as ^ you proceed; or consider the next number in the minuend diminished by \. (1.) £ s. d. From 1573, 11, 4^ Take 976, 15, 10^ EXERCISES. (2.) tons. cwt. fb». 47, 17, 43. 29, 18, 97. £ 596, 15, 6. 17, 18, 46. (3.) miles, fur. rods. 1407, 1, 16. 161, 1, 20. 1245, 7, S6. 4. A farmer possessed 1279 acres, 2 roods, 21 rods of land, and l)y his will left 789 acres, 3 roods, 36 rods to the elder of his two sons ; how much was left for the younger ? Ans, 489 acres, 2 roods, 25 rods, 5. The latitude of London, (England) is 51^^, 30 ', 49 '/ N., and tliat of Gibraltar 3(5*^, (> ', 30^ ' N, ; how many degress is Gibraltar south of London ? Ans. 15*^, 24 ', 19 ". 6. The earth performs a revolution round the sun in about 365 days, 5 hours, 48 minutes, 49 seconds, and the planet Jupiter in about 4332 days, 14 hours, 26 minutes, 55 seconds; how much longer does it take Jupiter to perform a revolu- tion than the earth? Ans. 3967d., 8h., 38m., 6s. 7. What is the difference between 21 hours, 19 min., 24 eec, and 15 hours, 37 min., 45 sec? Ans. 5 hours, 41 min., 39 sec. 8. IIow many months and days from August 29th, 1872, to April 15th, 1873? Ans. 7 mos., 17 davs. 9. How many ihonths and days from December 3rd, 1872, to October 2nd, 1873? ' Ans. 9 mos., 29 days. 10. What is the difference in time between March 3rd, 5 hours, 36 min., 42 sec, and March 2nd, 21 hours, 52 min., 47 sec? Ans. 7 hours, 43 m'n., 55 sec. 11. From 107", 40', 33'', take 69^ 50', 19"? Ans. 37^50^ 14". 12. A man who owes you £19 lis. 5^d. gives you £20; how much have you to give him back ? Ans. 8s. 6^d. i ^ h DENOMINATE NUMBEIIS. 83 To subtract denominate fractions, Rule. — Reduce the fractions to their values in loiver de- nominations, and then subtract. 13. What is the difference between j'j- of a mile and ^- of a furlonj^ ? ' ' , niile. fur. ^"j-x 8=:]f = 6fur. 21 rods, 4yds. 1ft. Gin. far rods. -^X 4o = 3.oo=:Ofur. 28 rods, 3yds. Oft. 5|in. 5fur. 33 rods, 1yd. 1ft. O^in. Ana. 14. P^rom j'^Tj- of a ton take -^ of a cwt. Ans. 7cwt. 44^H)s. 15. AVhat is the difference between | of a 11). Troy, and j^^ of <'m ounce Troy ? Ans. 8oz. 1 Gdwt. Gfj^rs. IG. Find tlio ditterence between ^-of a bushel, and -^ of a peck. Ans. lijrt. O^^T^pt. 17. What is tiie difference between -^^ of a pound and -J of a shilling;? - Ans. 4s. 9^d. 18. Find the difference between i'H and £.4G25. Ans. 3s. lOd. 19. Find the difference between jC.7G825 and .925 of a shillin^^ Ans. 14s. 5.28d. 20. From .G90484375 of a ton, take .87790875 of a cwt. Ans. 12cwt. 931bs. 2oz. 12drs. 21. Find th(} ditTereuce between .875 of a quart and .90625 of a gallon. Ans. 2qrts. Ipt. 2 gills. MULTIPLICATION OF DENOMINATE NUMBERS. Rule. — S)et the niultipUer under the lowest denomination of the multipticand, and muUljjhj each denomination in succession, ohservincf to redtice each -product to the next higher denomination. Write the remainder, if any, from each reduction, and carry the quotient to the next product. 1. Multiply 27 17 5^ by 6 EXERCISES. 167 4 7^ 6 farthinp;8 are Hd., write ^ and carry Id. 3 Id. are 2s. 7d., write 7d. and carry 2s. 104s. are £5 4s., write 4s., and carry £5, &c. I' ): I il 84 • ARITHMETIC. (2) £ 8. d. fi4 11 dh 3" £ 8. (1. 78 5 llf 9 (4) £ n. (1- 147 12 U 12 29 6 45 (8) lbs. oz. drs. 8or. gn. 3 7 6 1 15 11 40 2 1 5 106 6 14 (9) mllra. fur. rods. 5 7 15 8 47 3 193 15 4^ 704 13 9f 1771 5 3 (5) (6) (7) ana. cwt. His. « Ms. oz. dm. n>H. or. dwt. gn 5 17 29 17 11 13 7 4 15 21 5 6 7 51 9 11 » ao) 9 1 » I 10 93 2 K s: I. (11) 5 31 42 4 22 6 48 (12) h. III. 800. 7 12 55 , 7 50 30 25 h. m. Bee, 4 56 28 5 24 42 20 When the multiplier is more than 12 it is usual to multi- ply by factors. Thus, 14. Multiply 24 18 101 by 28. 28=7x4. 7 174 12 3^ 4 698 9 1 product by 28. =# DENOMINATE NUMBERS. 96 acres. ;15. Multiply 15 , 15 roodd 3 roda. 25 by 243 25x3 10 Product by 10, 159 10x4 10 If " 100,1590 2 20 2 j " 200,3181 } « 40, 636 <« « 3, 47 1 1 2 35 ^- 3865 35 Product by 243. 16. Multiply 18 tons, 12 cwt., 61 lbs. by 84. Ans. 1564 tons, 19 cwt., 24 fcs. 17. Multiply 27 acres, 2 roods, 29 rods by 72. Ans. 1993 a., roods, 8 rods. 18. Multiply 11 yds., 2 ft., 7 in. by 150. Ans. 1779 yds., ft., 6 in. 19. Multiply 49 lbs., 11 oz., 12 drs. by 67. Ans. 1 ton, 13 cwt., 32 fl)s., 3 oz,, 4 drs. 20. Bought 7 loads of Hay, each weighing 1 ton, 3 cwt., 87 BJE.; what was the weight of the whole ? Ans. 8 tons, 7 cwt., 9 I^s. 21. If a man can reap 3 acres, 35 rods per day, how much will he reap in 30 days ? Ans. 96 acres, 90 rods. 22. If a man saw a cord of wood in 8 hours, 45 min., 50 sec, how Ion'' will he be sawing 1 1 cords ? Ans. 96 h , 24 m., 10 s. 23. If 12 gals., 3 qrts., 1 pint of molasses be used in a hotel in a week, how much would be used in a year at the same rate? Ans. 669 gals., 2 qrts. 24. If 13 wagons carry 3 tons, 15 cwt., 40 lbs. each, how i much do they all carry ? Ans. 49 tons, cwt., 20 flis. f^. iti'X^^-'.j,,, ARITHMETIC. DIVISION OP DENOMINATE NUMBERS. Rule. — Begin with the highest denomination, and divide each in succession, writing the quotient beneath. When a remainder occurs, reduce it to the next lower denomination, adding in the number of that denomination, and use the sum as tite next dividend. So jproceed to the end. Divide £47 13 8^ by 7. £ a. d. 7 ) 47 13 8^ / , £' 6 16 2f |. Ans. 7 into 47, 6 times and £5 over ; write 6, and reduce £5 to shillings, thus, 5x20=100, add 13=113; 7 into 113, 16 times and 1 shilling over ; reduce the 1 shilliiig to pence, and add 8=20 pence; divide by 7, twice and 6 pence over; re- duce 6 pence to farthings, and add 2=26; divide by 7, =3 times and 5 over, which divided by 7=5. Ans. £6 16 2| |. EXERCISES. 2. Divide £476 19 5 by 5. Ans. £95 7 10^ f 3. What is the ^ of £927 4 IH? Ans. £115 18 \\ f. 4. Find the -^ of £1728 1 3|. Ans. £192 If. 5. Find the ^ of 27 ton% 16 cwt., 56 fcs. Ans. 2 tons, 6 cwt., 38 lbs. 6. Find the ^ of 147 Jbs., 14 oz., 6 drs. Ans. 13 fcs., 7 oz., 2 drs. 7. What is the \ of 62 ibs., 5 oz., 16 dwts., 1 gr.? » Ans. 8 fcs., 11 oz., 2 dwt., 7 grs. 8. Divide 483 acres, 3 roods, 35 rods by 10. Ans. 48a., Ir., 23^ rods. When the divisor is more than 12, ive may either di/vide by factors, or employ the process of long divisign. 9. Divide £7629 14 2 by 28. - ^-. i v---.:^-:- p:rst mktuud. £ 1. d. 4 ) 7629 14 2 7 ) 1907 8 6^. 272 1) 9^. A»3.. PENOMINATK NUMBERS. 87 8KC0ND METHOD. JB s. d. £ s, 28 ) 7629 14 2 (272 9 . . 56 d. Ans. .';r .."sr 202 196 69 56 - 13 20 « 274 252 22 12 ' 266 252 14 4 . . 56 56 - 10. Divide 1564 tons, 19 cwt, 24 lbs. by 84. Ans. 18 tons, 12 cwt., 61 fi)B. 11. Divide 1993 acres, roods, 8 rods by 72. Ans. 27a., 2r., 29 rods. 12. Divide 1 ton, 13 cwt., 32 flm., 3 oz., 4 dis. by ()7. Ans. 49 lbs., 11 oz., 12 drs. 13. 7 loads of hay woi*/f/} be th« price per bushel ? $3.25 60 IU5.00 $1.95 price per bushel. , 2. Wiien oats are $1.30 per cental, what should they be jjet bushel. Ans. 44^ cents. 3. If Timothy seed sells for $10 per cental, what is the price per bushel ? Ans. $4.80. ,' i. When clover seed is $1U per ceulal, what is the price Jiui lii|H he) ? Ans. $7.20. 5. Wneh rye ia $9.y4 per ohiiIuI, what in tl»e price per WM'i Ans. $1.^ 90 ARITHMETIC. LONGITUDE AND TIME. Given the difference of longitude of two places to find the differ* !»ce of time. • Since the earth makes a complete revolution of 3(iO de- grees in 24 liours, the Hun appears to pass over tlie earth at that rate, which is lo dejT^rees per hour. Therefore if the number of def>;rons of longitude between two places bo divided by 15, the (juotient will represent tlio number of hours occu- pied by the isun in passing from the meridian of one of the places to the meridian of the t)ther ; and since the ratio of degn^CH, (") m ill I ites (') and seconds ( '^) to one another is the Eume as that of lio.ns, minutes ;md seconds, if any ditterence of longitude, expressed in degrees, minutes and seconds, be divided by 15, the »[Uotient will express the number hours, minutes and seconds in the diflference of time. What is the difference of time between two places whose difference of longitude is 56^* 28 ^ ? 15)56'' 28 '(3 45 mm. 45 sec. 52. Ans. '1 60 688 60 I 60 ■(' 780 75 30 LON(HT[^DP! ^ND TIME. 9T A,i(.'iin, sinco (50 is PxacHy 4 fiincs 15, if anv .|u;nitity ho multiplied l)y 4, uiid the product l)o divided by (|(), tlio residt will 1)0 the Biinio jis dividiii;^ l)y l^. Ai.d tliis is tho more convenient here, hec:ms(! (10 is the ratio of tlw; table, which rediicea the process to simply niiiltiplyiu;^ 1)y 4. The above ([iiestion will therefore be solved thus : — J Difference of lonijf it I ide, CA't^ 28' 4 Difference of time, 3h. 4")m. 528ec. Ans. Rule. — Multi'pljj tha diferrncc of hmuUvde hy four ; oh' 8c.rvln$i- ^ .^/ Hiotografiiic Sdaices Corporation 23 WBT MAIM STMiT WIBSTIR,N.Y. 14SS0 (71«)t72-4S03 N> ^.\ ,^. :\ 92 ABITHMEIIC. 8. Greenwich, England, is on the first meridian; what time should be shown by a ship's chronometer, showing Greenwich time, when the ship is in longitude 74** W., ar^d her correct time, 9h. 30m. A. M. ? Ans. 4h. 34m. Given the difference of time between two places, to find the difference of longitude. , . , , , , ^ , This is the converse of the last case. Rule. — Multiply the difereTice of time by 15. Or, multi- ply the hours by 60, add in the minutes, and divide the sum and seconds by 4. 1. The difference of time between two places is found to be 3 hoiu-8, 45 min., 52 sec. ; required the difference of longi- tude. ■.,.;■■■ ,;.i ;;:■ ■; -: ;^. SKCONB METHOD, h. m. a. 3 45 52 .60 FIRST METHOD. h. m. - 3 45 52 3 11 17 36 5 4)225 52 56'' 28/ Ans. Se** 23' 00" Ans. 2. The difference of time betweei Halifax and Fredericton, N. B., is 12 minutes, 36 seconds ; required the difference of longitude? Ans. 3«» 9' 3. When it is noon at Yarmouth, N. S., it is llh., 43mi.n., 24sec., A. H., at Portland, Me. ; what is the difference of longitude ? Ans. 4«> 9' 4. When it is noon at Greenwich, England, it is 7 hours, 36 min., A. M., at St. John, N. B. ; what is the longitude of St. John? Ans. eO** 0' W. 5. What is the longitude of Montreal, if when it is noon at Greenwich, it is 7 hours, 5 min., 4 sec. A. M., at Montreal ? Ans. 73^ 44/ W. 6. What is the longitude of a ship whose correct time is found to be 5 hoiurs, 35 min., 40 sec. faster than the time at Greenwich? Ans. 83® 55/ E. 7. Whet is the longitude of a ship whose correct time is 8 hours 43 min., slower than the time at Greenwich ? Ans. 130« 45' W I *f ^.•*- BATIO AND PROPORTION. 93 ■4 -%yt ): K\ RATIO AND PROPORTION. Katio is the relation which one quantity boars to another of the same kind with respect to magnitude. Thus, the ratio of 2 to 6 is tlie relation which 2 bears to 6 in respect to the quantity expressed by «ach ; and since 2 is the \ of 6, tliis ratio is equal to ^. Hence the ratio of one number or quantity to another is measured by the tjuotient obtained by dividing the first by the second. Thus the ratio of 4 to 8 is * ; of 5 to 20, \ ; of 12 to 6, 2 ; of 27 to 3, 9. Ratio is generally expressed by the sign (:) placed between the quantities, thus 3 : 12 expresses the ratio of 3 to 12, and is equal to \. The two numbers or quantities of a ratio are called its terms. The first term is called the antecedent ; the second the consequent. A Simple Katio is an expression of the .relation of two quantities only, as 7 : 21. A Compound Ratio is a combination of two or more simple ratios, as J 3 : 6. 1 \2:3.j A compound ratio is reduced to a simple one by multipli- cation. Thus, 3 : 6| ^6 . 18, or |. X f = A=6 : 18. .. :81 ' i 5 ^ =120 : 120, or, | x^ x-|=H>=120 : 120. :3j Also, 5 : 8" 4 6 A Ratio of Equality is one in which the antecedent is equal to the consequent, as 7:7. ' A Ratio of Majority is one in which the antecedent is greater than the consequent, as 12 : 8. A Ratio of Minority is one in which the antecedent is less than the consequent, as 8 : 24. Note. — The English method makes the antecedent the numerator, and the conse(i[uent the denominator of the frac- tion. The French method makes the consequent the nmne- lator, and the antecedent the denominator ; thus, 3 to 7, by the English, f , by the French, |. 94 ARITHMETIC. Proportion is an expression of two or more ratios equal to one another. A proportion or Analogy is an expression of the equality of two ratios. A Simple Proportion expresses the equality of two simple ratios, usually by means of the sign^ (: :) Thus 2 : 4 : : 7 : 14, which indicates that the ratio of 2 to 4 is equal to the ratio of 7 to 14, and is read, 2 is to 4 as 7 to 14. The four quantities of a simple proportion are called its terms. The first and fourth terms are called the Extremes ; the second and third, the Means. In every proportion the product of the Extremes is equal to the product of the Means. The fourth term is generally known as the Fourth Pro- portional. ^ ; , .: ' To find a fourth proportional, the first three terms being given. r What is the fourth proportional to 3, 21 and 10 ? Multiply the means together, — 21 x 10=210. Now since the product of the means is the same as the product of the extremes, the number, 210, is the product of two factors, one of which is 3. Therefore, if 210 be divided by 3, the quotient will be the other extreme, or fourth proportional. 210^3=70, Ans. Rule. — Multiply the second and third terms together, and divide theproduct by thefirst. { •. >v EXEKCISES. ' ! 1. Find the fourth proportional to 5, 15 and 24. ' '!a. Find the fourth proportional to 17, 34 and 19. ^ 3. What is the fourth proportional to 9, 36 and 48 ? > ■ 4. What is the fourth proportional to 8, 48 and 72 ? * Whenever the first terra, or any factor of it, is a factor of one of the others, the operation may be shortened by can- RATIO AND PROPORTION. <'95 Thus, in the last question, i 0:^0::72 or, $:4S::'^ji 6 6 9 9 ' ^ ^1, ' 4S2, Ans. 432, Ana. 5. Find the fourth proportional to 27, 72 and 31. Jit : :t;i : : 31 8 is a factor of the first 3 8 8 and second terms. 3)248 82f, Ans. ^^ Find the fourth proportional to 16, 27 and 56. i$ : 27 : : ^0 8 is a factor of the first 2 7 7 and third terms. , , , • 2)189 94^, Ans. 7. Find the fourth proportional to 14, 21, 32. Ans. 48. . 8. Find the fourth proportional to 22, 37, 363. '^ Ans. 610^. 9 "What is the fourth proportional to 9, 19, 99 ? Ans. 209. The following principles will be found useful to the learner : In the following or any other proportion : — 8 : 6 : : 12 : 9. By inversion, the second is to the first, as the fourth is to the third, thus, .; ■ ' •' ' ■ , tr ., i^./v.: y;-; 6 : 8 : • 9 : '12. ' 7--; By alternation, the first is to the third, as the second is to the fourth,^ thus, 8 : 12 : : 6 : 9. By composition, the sum of the first and second, is to the second, as the sum of the third and fourth is to the fourth, thus, :- I * ' 14 : 6 : : 21 : 9. 91 96 ARITHMETIC. I'r i By additMn, the first is to the sum of the first and second as the third is to the sum of the third and fourth, thus, 8 : 16 : : 12 : 21. By division, the difference between the first and second is to the second, as the diffe'^ence between the third and fourth is to the fourth, thus, 2 : 6 : : 3 : 9. By conversion, the first is to the difference between the firot and second, as the third is to the difference between the third and fourth, thus, 8 : 2 : : 12 : 3. By mixing, the sum of the first and second is to their difference, as the sum of the third and fourth is to their dif- ference, thus, 14 : 2 : : 21 : 3. f f SOLUTION OF QUESTIONS BY SIMPLE PROPORTION. Questions to be solved by Simple Proportion contain, or indicate three terms, two of which are alike, and are to be taken as the terms of one ratio ; and the third is of the same kind as the required answer, and between which and the answer there exists, by the nature of things, the same ratio as between the first two. If 3 barrels of apples cost $7, what will be the price of 12 barrels. Now in this question, the two terms, 3 barrels and 12 bar- rels, are of the same kind, — let them be taken as the terms of a ratio, thus 3: 12. This ratio is evidently equal to that of the price of 3 barrelb, $7, to the price of 12 barrels, which is the required answer. We may, therefore, state the question in the form of a proportion, the fourth term of which is to be found. Thus, 3 : 12 : : 7: the fourth proportional which is obtained by the rule already given. The completed proportion will be ~ . f r 3 bbla. 12 f 7 28. RATIO AND rROPORTION. &7 * By examining the oreviouB examples of Proportion, it will be seen that whenever the fourth term is greater than the third, the second is greater than the first ; and when- ever the fourth term is less than the third the second is less than the fiist. Therefore, To state questions in Simple Proportion : EuLE. — Place tlie term, or qvantity which is of the same kvnd as the required answer in the third place. Then, when the answer, or fourth term, is to he greater tJtan this third term, make tJie greater of the other two the second term, and the less the first ; hut wlten the answer is to he less than the third term, Tnake tJie less of the other two the second term, and the greater the first If 14 reams of paper cost $44.10, what will 3G reams cost ? 14 : 3G : : $44.10 36 26460 13230 14)158760($113.40, Ana. ^BXSBOIBSS. 1. If 6 barrels of flour cost $32, what will 75 barrels cost ? Ans. $400. 2. If 18 yards of cloth cost $21, what must bo paid for 12 yards ? Aua. $14. 3. How much must be paid for 15 tons of coal, if 2 tons can be purchased for $15 ? 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 can be bought for $270 ? Ans. $1260. 6. What must be paid for a certain piece of cloth, if § 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 2^ days ? Ans. 10. 8. If 16 sheep are f of a flock, how many are there in the same ? Ans. 24. 9. tVliat must be pud for 4^ cords of wood, if the cost of 3 cords iaZlO? Ans. $15. ^8 ABirmCETIO. 10. What is the height of a tree which oasts a shadow of 125 feet, if a stake 6 feet high produees a shadow of 8 feet ? Ans. 93f • 11. ilow lung will it take a train to ran from Syracuse to Os- wego (a distance of 40 miles), at ihe rate of 5 miles in 15/., minutes ? 12. If 15 men can build a bridge in 10 days, how many men will bo repaired to erect iJiroo of the same dimensions in ^ the time ? Ans. 90. Va. If a man receive ^-1.50 for 3 day.s* work, how many days ought lio to remain in his place for $25? Ans. 1G§ days. 14. How much may a porsoti spend in 01 days, if he wishes to save $73, .50 out of a salary of $500 per annum ? Ans. $109. H4. 15. If n cwt., 3 (!;•»., 14 Ibd. of sugar cost $36.50, what will 2 .jrs , 2 lbs. cost ? Ans. $4.879-f-. ll). 5 men are employed to do a piece of work in 5 days, but alter workinH; 4 days tl>oy find it impossible to complete the job in Khs thiiii 3 day.s mojv, 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 ai day, what will be the time by the watch at a quarter past 10 3'cloek, A. M., on the following l^aturday ? Ans. 10 h. 40 m. 36-/g s. 18. A bankrupt owes $972, and his property, amounting to $007.50, is dirttributed among his creditors ; what does one receive whose denuMitl is $11.33^ ? Ans. $7,083-}-. 19. What is the value of .15 of a hhd. of lime, at $2.39 per hhd.? Ans. $.3585. 20. A garrison of 1200 men has provisions for | of a year, at the rate of I of a pound per day ; how long will the provisions last at the same allowance if the garrison be reinforced by 400 men 'f Ans. (jjl mouths. 21. If a piece of land 40 rods in length and 4 in breadth make an acre, how long must it be when it is 5 rods 5^ feet wide ? Ans. 30 rods. 22. A borrowed of B $745, for 90 days, and afterwards would return the favor by lending B $1341 ; for how long should he . lend it? /; r ,^ Ans. 50 days. V. 23. If a man can walk 300 miles in G successive days, how many miles has he to walk at tlio end of 5 days ? Ans. 50c RATIO AND PROPORTION. 99 24. If 495 gallons of ivine cost $394 ; how much will $72 pay .for ? Ads. 90 gal+. 25. If 112 head of oattlo consume a certain quantity of hay in 9 ' days ; how long will the sutue quantity last 84 head? Ans. 12 da^s. 2C. If 171 men can bui' I a house in 168 days; in what time will 108 men build a similar house ? , Ans. 266 days. 27. It has been proved that the diameter of every circle is to the rircuniicrence as 113: IJ55; what then is the circumlcreiico of the moon's orbit, the diameter being, in round numbers, 480,000 miles ? Ans. 1,507,964 .Ya m. 28. A round table is 12 ft. in circuu'ference ; what is its diameter ? Ans. 3 ft. 9!|nin. 29. A was sent with a wanant ; after he had ridden 65 miles, B was sent after hitii to stop the execution, and for every 16 miles that A rode, B rode 21 ; How fur had each ridden when B overtook A? Ans. 273 miles. .30, Find a fourtli proportional to 9, 19 and 99. Ans. 209. " 31. A detective chu.sed 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. 3^ IJ', w much rum may bo bought for §119.50, if 111 gallons cost :j89.625 ? Ans. 148 gallenr-, 33. If 110 yards of cloth cost $18 ; what wUl $63 pay for ? Ans. 385 yards. 34. If a man walk from Rochester to Auburn, a distance < 1 (say) 79 miles in 27 hours, 54 minutes ; in what time will he wulii. at the same rate from Syracuse to Albany, supposing the distance to he 152 miles ? Ans. 53 h. 41 m. nearly. 35. A butcher used a false weigh i ' l.J oz., instead of 16 oz. k/i* a pound, of how many lbs. did he defraud a customer who bought 112 just lbs. horn him? Ans. 9|^ lbs. , 36. If 123 yards of muslin cost $205 ; how much will 51 yards cost? Ans. $85. 37. In a copy of Milton's Paradise Lost, containing 304 p""-"?, the combat c^ Michael and Satan commences at the 139th page ; at what page may it he expected to commence in a copy containing 328 pages? , ' Ans. The "ourth proportional is 149|| ; and hence the papsage will commence at the foot of page 150 38. Suppose a man, by travelling 10 hours a day, performs • 100 AniTHMEno. journey in four weeks without desecrating tho Sabbath ; now many weeks would it take him to perform the some journey, providou ho travels only 8 hours per day, aud pays no regard to tho Sabbath ? Ans. 4 weeks, 2 days. 39. A cubic foot of pure fresh water wciglis lOOO oz., avoirdu- pois ; find the weight of a vessel of water containing 217^ cubic in. Ans. 7 1bH., 131.1;] oz. 40. Suppose a certain pasture, in which arc 20 cows, is sufficient to keep them G 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 48. ; what is tho value of an ounce ? Ans. £,?i. 42. A mason was engaged in building a wall, when another cumo up and asked him how many feet he had laid ; ho repli -l, that tho part ho had finished bore the same proportion to one league which ^y does to 87 ; how many feet had he laid ? Ans. 3^4,%^' 43. A farmer, by his will, divides his farm, consisting of 97 acres, 3 roods, 5 rods, between his two sons so that the share of tho younger shall be J- the share of the elder; required the shares. Here the ratio of the shares is 4 : 3, and we have showiythat if four magnitudes arc proportionals, the first term increased by tho second is to the second as tho third increased by the fourth is to tho fourth. Now, 97 acres, 3 roods, 5 rods, being the sum of tho shares, we must take the sum of 4 and 3 for first term, and either 4 or 3 for the second, and therefore 7 : 4 : : 97 acres, 3 roods, 5 rods : F.P., i. e., the sum of the numbers denoting the ratio of the shares is to one of them as tho sum of the shares is to one of them. This gives for tho elder brother's share, 55 acres, 3 roods, 20 rods, and the younger's share is found either by repeating the operation, or by subtracting the share thus found from the whole, giving 41 acres, 3 roods, 25 rods. , : . . 44. A legacy of $398 is to be divided among three orphans, in parts which shall be as the numbers 5, 7, 11, the eldest receiving tho largest share ; required the parts ? 23 : 5 : : 398 : 86^*, the share of the youngest. ' 23 : 7 : : 398 : 121^, the share of the second. = J .:_ 23 : 11 : : 398 : 190j«3, the share of tho eldest. , , :. " t 45. Three sureties on $5000 are to be given by A, B and C, so that B's share may be one-half greater than A's, and G's one-half greater than B's ; required the amount of , the security of each ? COHPOUSn) PBOPOBTION. 101 Ans. A'8flhare, $106? 63^=^ ; B's, 11578.94}^ ; 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 Boston, to meet him, ut the rate of 3 miles an hour, how far from Washington will they meet, the whole distance hcing 432 miles ? Ana. 246f miles. 47. A certain number of dollars is to be divided between two persons, the less share being § of the greater, and the difference of the Hliares $800 , what are the shares, and what is the whole sura to be divided ? Ans. Less share, $1600 ; greater, $2400 ; total, $4000. 48. A certain number of acres of land are to bo divided into two parts, such that the one shall be i} of the other ; required the parts and the whole, the difference of the parts being 716 acres ? Aus. the less part 537 acres; the greater, 1253 acres ; the whole, 1790. 49. A mixture is made of copper and tin, the tin being ^ of the copper, the difference of the parts being 75 ; required the parts and the whole mixture ? Ans. tin, 37^ ; copper, 112^ ; the whole. 150. 50. Pure water consists of two gasses, oxygen and hydrogen ; the hydrogen is about -f^ of the oxygen; how maiy ounces of water will there be when there are 764 J | oz. of oxygea more than of hydrogen ? I r . , ^^- 1000 oz. COMPOUND PROPORTION. ■1 ■ 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, How many men would be required to reap 65 acres in a given time, if 96 men, working equally, can reap 40 acres in the same time ? Here there lis but one condition, viz., that 96 men can reap 40 acres in the given time, which implies but one ratio, and ^hen the question has been stated 40 : 65 : : 96 : F.P., and the required term is found to be 156, and the proportion 40 : 65 : : 96 : 156, we have the propor> tion, czprc sed 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 102 AltrrRMETIO. I N Mone rate, 10 houra eaoh day, to travel 400 miles ? Hero there arc two conditions, vii. : jirtt^ that, in the one oaso, ho travoli> 12 honra a day, und in the other 10 houra ; and, iecondlj/, that the distanoefl are 250 and 400 miles. The statement, as we shall presently show, would be 10 : 12 ) . . g . 17 7 ^^^^° ^^^^ condition im- 250 : 400 J ■ ' " ^*' pUes one ratio, 10 : 12 and 250 : 400, and when the required term, which in IT^^, is found, there are four rntioH, viz., the two already noted, and 9 : IT-^^, gives two more, one in relation to 10 : 12, and one in relation to 250 : 400. Thiti will bo evident, when wo have shown the method of statement and operation. EXPLANATORY STATEMXMT AND OPERATION. 11 1 33; 3 12 12 F.P. 36 PRACTICAL STATEMENT AND OPERATION. 11:33 18: 5 }■■■ 12 : F. P. 18 , 1 6 5 36 2 F.P. 10. 1 3 3 5 2 : F. P. }: I}:: 2:10. Let the question be, How many men would be required to reap 33 acres in 18 days, if 12 men, working equally, can reap 11 acres in 5 days ? Wo firet proceed, as on the left mai^n, as if there were 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 36, the number of men required, if the time were the same in both cases. The question is now resolved into this : How many men will be required to reap, in 18 days, the same quantity of crop that 36 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 mar^. We firet notice that the last term is to represent » COMTOUND r'^OPORTlON. 108 11X18:33X5 198 eortain number of men, und, tliorcforo, wo place 12 in the third plaoo; next, wo boo that, other thinga buittg equals it will take more mon to roap :i3 than to reap 1 1 acrcH, and that, therefore, oo far as that M concerned, tho fourth term will be greater than the third, and so wo put 1 1 in the fir^t place, and WW in the second. A^ain wo see that, olhQr thimjn bving iqmtl, u le«3 nunibiir of men will be required when 18 days are allowed tor doin*^ the work, than when it is re- quired to be dono iit 5 days, and that therefore the fourth term, oh far mi that in concerned, will be Ichh than tho third, iiiiJ theruforo wu write 18 : 5 below the other ratio as on the margin. Then by cou- traotion wo got .^ ; |, i : : 2 : F. P. Now, uh 3 in the first term is to bo a multiplior, and 3 in the second u divisor, we may omit those also, and we obtain . ' . |- : : 2 : 10, the answer as before. The full unoontractcd operation would bo to multiply 18 by 11, which gives 198, then to multiply 33 by 6, which gives 105, then multiply 165, tho product of tho two second terms, by 12, and divide the result, 1980, by 198, tho product of tho two first terms, which gives 10 as before. Bocause in the analogy 198 : 165 : : 12 : 10, the first two terras aio products, this kind of proportion has been called compound^ and the ratio of 19 to 165 is called a compound ratio. Wo can show the strict and original meaning of tho term compound ratio more easily by an example, than by any explanation in words. Lot us take any scries of numbers, wl Ac, fractional or mixed, say 5, ^, 1, 19, 12, 1, 17, 11, 15, 25, then the ratio of tho first to tho last is said to bo compounded of tho ratio of tho first to the second, the second to the third, the third to the fourth, &c., &o., &c., to the end. Now the ratio of 5 to 25 is ^/^=5, and the several ratios are in this order,ix|xi-Xl'5XT3X-'r''-X!4xifx||^^»o'» ^^'^^^"S fio'^^J ''/=5 OS before. If we took them in reverse order, viz., \=^, it is obvious that all therein could be cancelled, as each would in suo- oossion be a multiplier and a divisor. Wc would also remark ihat compound proportion is nothing else than a number of questions in simple proportion solved by one opentr 165::*12:F. P. 166X12:^10 198 104 ABITHMETEO. tion. This will bo evident from our second example by oomparing the two operations on the opposite margins. Again, we remarked that every condition implies a ratio, and that therefore the third and fourth terms of our first example really involve two ratios, one in relation to each of the preceding. Hence v 'vcrsally the number of ratios, expressed and implied, must always ^o double the number of conditions, and therefore always even. As the third ratio is only written once, the number of ratios appears to be odd, but is in reality even. RULS: Place, as in simple jproportion, in the third place the term that is the same as the required tern . Then conrider each condition separately to see which viust he placed Jirst, and which second^ other things being equal. EXAMPLE . 1. If $35.10 pay 27 men for 24 days; how much will pay 16 men 18 days ? Hero we first observe that the answer will be money, and therefore $35.10 must be in the third place. Again, it will take less money to pay 16 men than 2? men, and therefore, other things being equal, the ant>wer, as far as this is concerned, will be lesa than $35.10, and therefore we put the less quantity, 16, in the second pkce. So also because it will take less to pay any given num- ber of men for 18 days than for 24 days, therefore we put the less quantity in the second place, which the statement shows*in the margin. 27 24 16:: $35.20 18 3: 3: . 1 2 2 ^ 9: 4:: $35.10 4 • 9)140.40 Ans. $15.60 Vf--- BXEBCISES. 1. If 15 men, working 12 hours a day, can reap 60 acres in 16 days ; in what time would 20 boya, working 10 hours a day, reap 98 acres, if 7 men can do as much as 8 boys in the same tkne ? #t: , •^^'*' 26|| days. 2. If 15 men, by working 6f hours a day, l»n dig a trench 48 ftot long, 8 feet broad, and 5 feet deep, in 12 days; how many houn ft day must 25 men work in order to dig a trench 36 feet long, 12 fti|( broad, and 3 feet deep, in 9 days? Ans. 3{. COMPOUND PROPORTION. 105 :t, 3. If 48 incr can build a wall 864 feet long, 6 feet high, and Tl feet wide, in 36 days ; how muuy men will be required io build a wall 36 feet long, 8 feet high, and 4 feet wide, in 4 days? Ana. 32. 4. In what time would 23 men weed a quantity of potato ground which 40 women would weed in 6 days, if 7 men can do as much as 9 women ? Ans. 8g^j, days. 5. Suppose that 50 men can dig in 27 Hays, working 5 hours a day, 18 cellars which are each 48 feet long, 28 feet wide, and 15 feel* de(p; how many days will 50 men require, working 3 hours each day, to dig 24 cellai-s 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 ? . r Ans. 2 cwt., 2 qrs., 8 lbs. 7. If tl2 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 j how far should 35/j cwt. bo carried for $18.75 ? Ans. 9^5 9. If a family of persons can live comfortably in Philadelphia for $2500 a year j what will it cost a family of 8 to live in Chicago, all in the same style, for seven months, prices supposed to bo | of what they would be in Philadelphia ? a -^ . Ans. $1037.04 . ,i 10. If 126 lbs. of tea cost $173.25 ; what will 63 lbs. of a differ- ent quality cost, 9 lbs. of the former being equal in value to 10 lbs. of the latter? Ans 884.15. 11. If 120 yards of carpeting, 5 quarters wide, cost $60; what will bo the price of 36 yards of the same quality, but 7 quarters wide? Ans. $25.20. 12; If 48 men, in 5 days of ^ hours each, can dig a canal 139f yards long, 4| yards wide, and 2 J yards deep ; how many hours per day must 90 men work for 42 days to dig 491 -^ yards long, 4| yards wide, and 3^ yards deep ? Ans. S§^|-. 13. A, standing on the bank of a river, discharges a cannon, and B, on the oppo W^. 23. From f of 2* of 4, take VV of 6| of ^. Ans. 2|. 24. What is the sum of J, J, J, {, J, 4, ^, and ^? Ans. l^U- 25. What is the sum of | ^ of S^-^ J of 85 ? Ans. 22-S ^SJ. 26. How long will it take a person to travel 442 miles, if he travels 3^ miles per hour, and 8^ hours a day ? Ans. 16 days. . 27. Find the sum of 2J of /^, 3J of | of ^% of 4J and J. Ans. 6^^. 28. A has 2] times 8| dollars, and B 6^ times 9f dollars ; how mnoh more has B than A? ^ Ans. $44||. S, 29. If I sell hay at $1.75 per owt. ; what should I give for 9| tons, that I may make $7 on my bargain. Ans. $329. 30. If 7 horses cat 93^ bushels of oats in 60 days ; how many 'bu' hols will one horse eat in 87f days ? Ans. 19|. 31. Bought 14/q yards of broadcloth for $102.90 ; what was tho ^alue of 87| yards of the same cloth ? Ans. $612. 32. How many bushels of wheat, at $2f per bushel, will it re- •quire to purchase 168^^9 bushels of corn worth 75 cents per bushel? Ads. 47y\. • '-' 33. If in 82j^ feet there are 5 rods ; how many rods in one mile ? Ans. 320. 34. Suppose I pay $55 for g of an acre of land ; what is that per «ore ? Ans. $88. •^ 35. If I of a pound of tea cost $1.66^ ; what will ^ of a pound «ost? Ans.$1.55||. 36. Subtract the 2ium of 2} and 1 /j, from the sum of f , 7^ and 3, and multiply the remainder by 3^. 37. If i lb. oost 23^ cents i what wiU 2}^ cost ? Aaa. 71^ oeats* Ans. 24|f MISCELLANEOUS EXEBaSES. 109 38. What is the difference between 2^X3^ and 2JX3,'g ? Ana. ^^. 39. If I lb. cost $3 ; what will | ^ lb. cost ? Ans. 3C^ centa. 40. What is thd difference between J of J+H-^X.', and tH+l ? Ans. |2§. 41. If 4|^,- yards cost Sl^'j , what will 2^ yards cost? ' . - ; ^ Ans. 475 cents. 42. Bought 5 of 2000 yards of ribbon, and sold 3 of it; how muoh remains? Ans. 285!) yards. 43. Divide the sum of ^, f, h h 1111 J. i^»^i»^(l5,n2by«iesumofJ, , and divide the quotient by t»yX«, and multiply the result by f of ^. Ans. g. V 44. I bought g of a lot of wood land, consisting of 47 acres, 3 roods, 20 rods, and have cleared ^ of it ; how much remains to be cleared ? Ans. 20 acres, 3 roods, 31^ rods. 45. What is the differon«e between If^^ and Igg ? Ans. |gj. 46. If $[2 pay for a IJ st. of flour; for how much will $| pay ? Ans. h\ St. 47. Mount 31anc, the highest mountain in Europe, is 15,872 feet above the level of the sea ; how far above the sea level is a clim- ber who is y'j of the whole height from the top, i, c, ,'5 of perpen- dicular bight ? Ans. 12896 feet. ^48. What will 45.94375 tons cost if 12.796875 tons cost $54.64 ? Ans- $196.17. ■ " 49. If I gain 837.515625 by selling goods worth $324.53125 j what shall I gain by selling a similar lot for $520.6635416. ? / . Ans. $60.1884. 50. If 52.815 cwt. cost $22.345 ; what will 192.664 cwt. cost at the same rate ? Ans. $81.512-|- 51. Kcquired, the sum of the surfaces of 5 boxes, each of which is 5^ feet long, 2^ feet high, and 3^ feet wide, and also the number of cubic feet contained in each box. The box /supposed to be made from inch lumber 7 Ans. 369^ Superficial ft. 52. If I pay $j^^ for sawing into three pieces wood that is 4 ft.' long ; how muoh more should I pay, per cord, for sawing into pieces of the same length, wood that is 8 feet long ? Ans. 22^ cents. 53. A sets out from Osw^, un a journey, and travels at the ' rate of 20 miles a day ; 4 days citer, B sets out from the same place, and travels the same road, at the rate of 25 miles per day ; how many 4ays before B will overtake A ? Ans. 16^ 110 ABITHMETIO. 54. A farmer having 56^ tons of hay, gold | of it at $10§ per ton, and the remainder at $9.75 per ton ; how much did he receive for his hay? Ans. S5804g. 55. If the sum of 87}^ and 117^j^ is divided by their difference ; what will be the quotient ? Ans. 6^5 J. 56. If 8f yards of silk make a dress, and 9 dresses be 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- maining only -^l as much money as he had at first ; how much money had he at first ? Ans. $3430. 58. If a person travel a certain distance in 8 days and 9 hours, by travelling 12 hours a day; how long will it take him to perform the same journey, by traveling 8f hours a day ? Ans. 12 days. 59. 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 the same time? Ans. 105. 60. Beduce 1 pound troy, to the fraction of one pound avoirdu- pois. ' Ans I 44 61. Beduce to a Himple fraction. Ans. ^. ' 62. What will be tho cost of 8 cwt., 3 qrs., 12J lbs. of beef, if 4 cwt. cost $34 ? Ans. $75y'g. 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 ■Pj as much as a girl ; what sum will each receive ? Ans. Each girl, $66§f j each boy, $42|f . 65. If A can cut 2 cords of wood in 12^ hours, and B can cut 3 cords iu 17^ hours ; how many cords can they both cut in 24^ hours? ' ■ ^«-".- •--r-^ ■;>-; ^^":r- :■■:■ "-f'-'-^ Ans. 8,^. 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 J yards wide, will be necessary to cover the same flooi ? 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 so as to make $200 by his bargain ? Ans. $1.98 nearly. 68. What must be the breadth of u piece of land whose length is 40^ yards, in order that it may be twice as groat aa another piece of ANALYSIS AND SYNTHESIS. m. 14' land whoso length m 14§ yards, and whose breadth is ISy-, yards? Ans. 9^ yards. 69. If 7 men can reap a rectangular field whose length is 1,800 feet, and breadth 960 feet, in 9 days of 12 hours each ; how long will it take 5 men, working 14 hours a day, to reap a field whose length is 800 foet, 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 number of men in 7 days of 9 hours each ; how many feet of water wiw it capable of holding ? Ans. 2268 cubic feet. 71. If 100 men, by working 6 hours each day, can, in 27 days, dig 18 cellars, each .40 feet long, 36 feet wide, and 12 feet deep ; how many cellars, t^at are each 24 feet long, 27 feet wide, and 18 feet deep, can 240 men dig in 81 days, by working 8 hours a day ? Ans. 256. 72. A gentleman left his son a fortune, ^ of which he spent in 2 months, ^ of the remainder lasted him 3 months longer, and f of what then remained lasted him 5 months longer, when he had only $895.50 left; how much did his father leave him ? 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 each 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 th : 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 weeks, and in similar manner, 5 seizes will keep 35 oxen 2 weeks ? A should turn into the field 18 oxen, and pay $72. B should turn into the field 12 oxen, and pay $48. Ans. -j -It*"' 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. »»i 112 ABirHUxno. The oonvorae of Analysis is Synthesis. The meaning and nsa of these terms will probably be most readily comprehended by referonoe to thoir derivation. Thoy are both pure Greek words. Analysis means looking up. The general reader woald here probably expect looting down, as employed in most popular definitions; but we may illustrate the Greek term, loosing up, by our own everyday phrase, tearing up, which means rending into shreds, the English up 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. We might give many illustrations, but ode will suffice, and we choose the one which will be most generally understood. When we analyse a sentence, we loose it up, or tear it up, into its component parts, and by synthesis v^e write or compose, i. e., put together the parts, which, by analysis, we have found it to consist of. When we commence to analyse a problem we reason from a given quantity to its unit, and then from this unit to the required quan- tity ; hence, all our deductions are self-evident, and we therefore require no rule to solve a problem by analysis. , Although this part of arithmetic is usually called analysis, yet, as it is really both analysis and synthesis, we have given it a title in aocordoiice with the principles now laid down. EXAMPLE. i 1. If 12 pounds of sugar cost $1.80, what will 7 pounds cost ? * ' ■• - t SOLUTION. 12)1.80 .15 $1.05 If 12 lbs. cost $1.80, one pound will cost the j^2 of $1.80=15 cents. Now, if 1 lb. cost 15 cents, 7 lbs. will cost 7 times 15 cents=to $1.05. Therefore, 7 lbs. of sugar will cost $1.05, if 12 lbs. cost $1.80. Note. — ^The work may be somewhat shortened, especially in long ques- tions, by arranging it in tlie following, manner, so as to admit of canceUinn^ if possible : — ■ 15 1 . ,1M. .7 105_ft, ns Ans. M U^ 1 ^1 1 2. If 5 bushels of pease cost $5.50, for what can you purchase 19 bushels? Ans. $20.90. ANALTOIS AND SYNTHESIS. 118 3. If 9 men can perform a certain piece of labor in 17 dojfl, how long ivill it take 3 uen to do it ? Ana. 51 days. 4. How many pigs, at $2 each, must bo given for 7 sheep, worth $4 a bead ? Ans. 14. 5. If $100 gain $G in 12 months, how much would it gain in 40 months ? Ans. $20. 6. If 4j bushels of apples cost $3^, what will be the cost of 7^ bushels ? , SOLUTION. In the first place, 4'1 bu8hela=V bushels, and 83J=$-%9. Now, since -'j'- bushels cost S^, one bushel will cost p^-~-i^^= ^Xi\=9ij and 7i or '/- bushels will cost J^- times $'-=l X-^jt'= $5, the value of 7^ bushels of apples, if ^ bushels arc worth $3^. OPERATION 5 181 9 5 1' 7. Iff of 3f lbs. of tea cost $11 ; what will bo the cost of 5^ pounds ? Ans. $4.12^. 8. 100 is f of what number? ' r - r Ans. 150. 9. If ^ of a mine cost $2800 ; what is the value* of § of it ? . Ans. $4200. 10. ^ of 24 is If times what number ? Ans. 10. 11. I of 40 is y»3 of how many times J of | of 20 ? Ans. 9. 12. A is 16 years old, and his age is § tim6s 'j of his father's ige ; how old is his father? Ans. 36. 13. A and B were playing cards ; A lost $10, which was I times } as much as B then had ; and when they commenced f^ of A'a money was equal to f of B's ; how much had each when he begMi to play? Ans. A $45; B $40. 14. A man willed to his daughter $560, which was I of f of what he bequeathed to his son ; and 4 times the son's portion was | Ihe value of the father's estate ; what was the value of the estate ? Ans. $13,440. 15. A gentleman spent ^ of his life in St. Louis, ^ of it in Bos* Ion, and the remainder of it, which was 25 years, in WashingtSi; what age was he when he died ? \ns. 60; ! 114 ARITHICSTIO. 16. A owt.H ^, and B j^ of a ship ; A's part is worth $650 more than B'h ; what is the value of the ship ? Ans. $16,600. 17. A post stands ^ in the mud, ^ in tho water, and 15 feel above the water ; what is the length of tho post ? Ans. 36 feet. 18. A grocer bought a firkin of butter containing 56 pounds, for $11.20, and sold ^ of it for $8'| ; how much did ho get a pound ? Ans. 20 cents. 10. Tho head of a fish is 4 feet long, tho tail as long ns tho head and ^ the length of ho body, and the body is as long as the head and tail ; what is tho length of tho fish ? Ans. 32 feet. 20. A and B have tho sumo income ; A saves \ of his ; B, by spending $65 a year more than A, finds himself $25 in debt at tho end of 5 yours ; 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 com&icnced and worked alone for 3 days, when B assisted him to complete tho job ; how long did it take them to finish tho work ? SOLUTION. If A can do the work in 8 days, in one day ho can do the ^ of it, and if B can do the work in 6 days, in one day he can do tho ^ of it, and if they work together, they would do ^-\-i=^i of the work in one day. But A works alono for 3 days, and in one day ho can do |f of tho work, in 3 days he would do 3 times |r=f o^ ^^^ work, and as the whole work is equal to g of itself, there would be | — '^=^ of the work yet to be 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 g'^j 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 |X V^^^l ^^J^- "22. A and B can build a boat in 18 days, but if C assists them, they can do it in 8 days ; how long would it take C to do it alone? Ani, 14| days. 23. A certain polo was 25^ feet high, and during a storm it was broken, when ^ of what was broken ofi^, equalled § of what remained ; how much was broken off, and how much remained ? • ■ Ans. 12 feet broken off, and 13^ 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 min., 34| see. ANALYSIS AND SYNTHESIS. 116 A^i 25. A. nnd B. start top;cther by railway train from Buffalo to £rio a distanco of (»uy) 100 inilRs. A •;oc.s by rroii^iit train, at the rate of 12 luilcH per hour, anil B by mixed train, at tliu rato of 1^ milcH |K!r hour, C loaves Erie r«ir BufTulu at the nanio time by ex press train, wliieli runs at (ho rato uf 22 uiiloM per hour, how far from Buffalo will A and B each bo when C moots thorn. A,?).')^^ B, 45. 20. A cistern ban two pipoH, ono will Gil it in 48 minutos, and the other will empty it in 72 minutes ; what time will it rccjuiro to fill the cistern whon both aio running? Ann. 2 hours, 24 min. 27. If u man spends ,'\ of his time in working, J in sloopitiir, /^ in eating, and 1^ hours each day in reading; how much tini ditious plan, and the one generaly adopted, is by the following BULB. Place the antecedents in one column and the oonsequ,cnt$ in another, on the right, with the sign of equality between them. Di- vide the continued product of the terms in the column containing the odd term hg the continued product of the other column, and the guotient wiU he the amwer. ANALYSIS AND 8YNTHESIH. 117 Let us now take oor last oxhoidIo (No. 33), and sohe it by thii G bushels of whoat=9 btubels of barley. 6 biuhelH of barley--? buahela of oats. 12 bushels of oat8=::=10 bosbels of pease. 13 bushels of pease=^ ton of hay. 1 ton of hay:-2 tons of ooal. — tons of ooal:=80 bushels of wheat. Ans. 9. %, ni^ %, ^, %, ?. 34. If 12 bushels of wheat in Boston are equal in value to 12| bushels in Albany, and 14 bushels in Albany are worth 14^ bushels in Syracuse ; and 12 bushels in Syracuse are worth 12| bushels in Oswego ; and 25 bushels in Oswego are worth 28 bushels in CIeye< land ; how many bushels in Cleveland ore worth 60 bushels in Boston ? Ans. 75|g. 35. If 12 shillings in Massachusetts are worth 16 shillings in New York, and 24 shillings in New York are worth 22| shillings in Pennsylvania, and 7^ shillings in Pennsylvania ore worth 6 shillings in Canada ; how many shillings in Canada are worth 50 shillings in Massachusetts ? Ans. 41 f . 36. If 6 men can build 125 rods of fencing in 4 days, how many days would seven men require to build 210 rods ? f SOLUTION. ,j. • If 6 men can build 120 rods oi fencing in 4 days, one man could do I of 120 rods in the same time ; and ^ of 120 rods is 20 «ods. Now, if one man can build 20 rods in 4 days, in one day he would build ^ of 20 rods, and ^ of 20 rods is 5 rods. Now, if one man can build 5 rods in ono day, 7 men would build 7 times 5 rods In one day, and 7 times 5 rods=35 rods. Lastly, if 7 men can l)uild 35 rods in one day, it would take them as many days to build 210 rods as 35 is contained in 210, which is 6 ; therefore, if 6 men ean build 120 rods of fencing in 4 days, 7 men would require 6 days to build 210 rods. 37. 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 daysi of 8 118 ABITBMETIO. ( )■ honrs eaofa, woald the same lot (^ men bnild a wall 20 feet long, 12 feet high, and 2^ fedt thick ? Aiis. 23/b. 38. If 5 men can perform a piece of work in 12 days of 1^ hours each; how many men wiH perform a piece of work four times as large, in a fifth part of tK time, if they work the same number of hours in a day, supposing that 2 of the second set can do as much work in an hour as '6 of the first set ? Ans. 66§ men. No^K. — Such questions ns this, where the nnswer involves • frnction, may frequently occur, and it may be aaked how } of a man can do any work. Th(« answer is simply this, that it requires 60 men to do the work, and one man to continue on workinj[r } of a duy more. 39. Suppose that a wolf was observed to devour a sheep in ^ of an hour, and a bear in f of an hour ; how long would it take them together to eat what remained of a sheep after the wolf had been eating ^ an hour ? Ans. 10, ^^m in. 40. Find the fortunes of A, B, C, D, E, and F, by knowing that A is worth $20, which is ^ as much as B and C are worth, and that is worth ^ as much as A and B, 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, J of E's, and J of F's fortune. f Ans. A, 20 ; 3, 55 ; C, 25 ; and D, E and F. 1200 each. , ' 41. A and B set out from the same place, and in the same direc- tion. A travels uniformly 18 miles per day, and after 9 days turns and goes back as far as B has travelled during those 9 days ; he thea 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 per day. 42. A hare starts 40 yards before a greyhound, and is not per- ceived by him until she has been running 40 seconds, she scuds away at the rate of 10 miles an hour, and the dog pursues her at the irate of 18 miles an hour ; how long will the chase last, and what dis- tance will the hare Jiave run ? Ans. 60^^ sec. ; 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 ia> taken sick and leaves, B continues on working alone, and after 2. days he is joined by 0, and they finish it together in 1^ days^ how long would be doing it alone ? Ans. 12 days. 44. A, in a scufiSe, seized on f of a parcel of sugar plums ; B caught § of it out of his hands, and laid hold on -f^ more ; D ran off with all A had left, exc^t 4 which E afterwards secured dyly for himself; then A and C jointly set upon B, who, in the oonflioti leC I'lULCnCE. m fall i he liud, which were equally picked up by D and E, 3iFl}Q..la]^ perdu. B then kicked down C's hat, and to work they all . w^nt'. anew for what it contained ; of which A got :^, B ^, D ^, und C acfd E equal shares of what was left of that stock. D then struck 'f of what A and B lust acquired, out of their hands ; they, with somo difficulty, recovered f of it in equal shares again, but tho other three carried off f a piece of the same. Upon thia, they called d truce, and agreed that tho ^ of the who\e left by A at -first, sb.ould be equally divided among them ; how many plums, after this distribu' tion, had each of the competitors ? Ans. A had 2863 ; B, 6335 ; C, 2438 ; D, 10294 and E, 4950. .1^ '» • * ' PRACTICE '" The rule which is called Practice is nothing else man a partica« far case of simple proportion, viz., when the first term is unity. Thus : if it is required to find tho price of 28 tons of coal, at $7 a ton — as a question in proportion, it would be, if 1 ton of coal costs $7, what will 28 tons cost? and the statement would be 1 : 28;^;>7 : F. P. Here the first term being 1, the question becomes one of simple multiplication, but the answer, $196, is really the fourth term of an analogy. ^ Again, to find the price of 46 barrels of flour, at $7.62J per barrel, we have only to multiply $7.62j^ by 46. In many cases, however, it is more conveni- ent to multiply the 46 by 7, which will give the price of 46 barrels at $7 each. Now, 50 cents being half a dollar, the pricoi of 46, at 50 cents, will bo $23, aind 12^ cents being \ of, 50 cents, the price at 12^ cents will be the fourth of that at 50 cents, or $5.75, and the whole comes to $350.75. T(r find the price of 36 cwt., 2 qrs., 15 lbs., at $4.87|f. Hero the question stated afc length would be, if 1 cwt. cost $4.87|^, whai^v ^ill 36 cwt., 2 qrs., 15 lbs. cost? Th#< statement would be 1 : 36., 2., 15 : : $4.87^: $350.75 ^' ^' ^^^ becoces a question of multi* $7.62^ 46 23 4572 3048 $350.75 SO I2i \ 46 7 322 23 6.75 120 jaaimcEno. plioation beoanse the first term is unity, and divided 1^ 1 wonld not alter the prodnot of the other two terms. Thus : 2 qrs. I of 1 owt. I 10 lbs. 5 *^ j^ of 2 qrs. ^ of 10 lbs. 4.87^ 36 18 2922 1461 V,-. 'V. ' ■■>. !■< ' ;? ^li 175.50 == pnoe of 3 owt., (^ 94.87} per owt. 2.437= « 2 qrs. « « " .487= « 10 lbs. « « « .243= « 5 " " " ** $178,667= " 36owt., 2qr8., 151b8. We would call the learner's special attention to the following direction, as the neglect of it is a fertile source of error. Whenever yon take any quantity as an aliquot part of a higher to find the price of the former, he sure you divide the line which is the price at the rate of thai higher denomination. To find the rent of 189 acres, 2 roods, 32 rods, at $4.20 per acre. 2 roods=]^ of 1 acre, 20 rods=^ of 2 roods, 10 rod8=^ of 20 rods, 2 rods=^ of 10 rods, Since the rent of 1 acre is $4.20, the half of it, $2.10, will be the rent of 2 roods, the rent of 20 rods will be .525, the ^ of the rent of 2 roods, the half of that, . 2625, will be the rent of 10 rods, and, lastly, .0525 will be the rent of 2 rods, which is the ^ of 10 rods. We then multiply by 189, and set the figures of the product in the usual order, so that the first figure of the product by 9 shall be under the units of cents, &c., and then adding all the partial resi^lts, we fibd the final answer, $796.74, the rent of 189 acres, 2 roods and 32 perches. XXXBOISES. 1. What is the price of 187 owt. at $5.37^ per ewt. ? Ads. $lt)0&.l!2|. 4.20 189 210 525 .y. 2625 525 3780 3360 420 $696.74 SBAOTXCE. 121 2. What is the value of 1857 lbs., at 13.87^ per lb. ? Ana. $7195.871 3. What will 4796 tons amount to at (4.50 per ton ? "'■■''- -■'--[ .^ x-:,'^i-. Ans. $21582. I. What is the price of 29 score of sheep; at $7.62^ each ? Ans. $4422.5&- . 5. Sold to a cattle dealer 196 head of cattle at $18.75 each, fin^ the ar unt. Ans. $3676 u. Sold to a dealer 97 head of cattle, at $1G.12| each, on th« average ; find the price of all. Ans. $ L564.12|. 7. What Is the price of 16 tons, 17 cwt., 2 qrs. of coal, at $8.62^ per ton ? ^ Ans. $145.54. 8. What is the yearly rent of 97 acres, 3 roods, 20 rods, at $4.37i per acre ? Ans. $428.19. 9. If a man has $12.50 per week ; how much has he per year ? Ans. $650. 10. If a clerk has $2.12| salary for every working day in the year ; what is his yearly income ? Ans. $665.12J. II. If a tradesman earn $1.64 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. 6d. 13. What is the price of 479 cwt. of sugar, at $17.90 per cwt. ,.^^., . Ans.$8574.10. i. 14. Find the price of 879 articles, at $1.19 each. 15. Find the cost of 1793 tons of coal, at $7.8T| per ton. 16. What is the value of 2781 tons of hay, at $S.62J per ton ? 17. What is the rent of 189 acres, 2 roods, 32 rods, at $4 20 per acre? Ans. $796.74. 18. What is the price of 879 hogs, at $4.25 each ? :ri^- Ans. $3735.75. 19. What will 366 tons of coal come to at $8.12^ per ton ? I -.; Ans. $2973.75. 20. What ia the price of 118 acresi 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 t24i60 an ai»e; 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 bow much ? Ans. B $989.08. 222 ARITHMETIC. ACCOUNTS AND INVOICES. > AccocNTS are atatemento from merchants to customers that baro pW obaeed 'goods on credit, and ure generally made out. periodically, ualefir specially called for. An mvoice is simplv a statement rendered by the noller to the buyer, « time of purchase, showing the articles bought, and the prices of each. 1. New YoKK, July Ist, 18Ga. Mb. James Axdebson, lo Pbesch. WnrrE & Co., 2)r. . ;^ j i; > , ; 1866. 2.«« !.»» 2.00 Jany. 4, To 2 Ite tea, l.^*^-^ : 8 lbs. coffee, 45c. ; 20 lbs. rice, lOc. . . 4.iT>i 2.»'s " 29, " 2i yds. Amer. tweed, !.•» ; 1 vest Feb. 10, " 14 lbs. Mus. sugar, 12Ac. ; 10 lbs. cms. white sugar, 20c. . . 60c. 2fip. l."* • <' 22, " lib. bk. soda, ; 1 lb. car. soda, ; 4 lbs. coffee, 45c.. . 3.«» 87Jc. Mar. 11, " 10 yds. print, 30c. ; trimming, &c., per bill I."" 85c. 2.«» " 19, " 2 lbs. tobacco, 90c. ; 1 gal coal oil, : 2 gals, syrup, l.»«> . I.''* 1.*^^ Apr! 12, " i yd. blk. silk, 3.«o ; i yd. blk. velvet, Gfi»h ; : , ' 3.«« 40c. 60c. t May 6, " 2 lbs. toa, 1.««'4 ; 1 bottle pickles, ; 1 lb. pepper, .. 35c. l.o» !.«<» « 20, " 1 bag salt, ; 10 lbs. sugar, lOo. ; 3 lbs. raisins, 50c. ... 75c. 2.«o " 31, " 3 lbs. currants, 25o. ; 10 tbs. white sugar, 25c l.»» 12jc. 2.«>» June 10, " 2 lbs. tobacco, 75c. ; i lb. B. soda, 25c. : 20 lbs. rice, 10c. . 40c. 10c. 30c. l.T* « 17, '* 1 lb.cloTes, ; \ lb.Butmegs, ; } cinnamcn, ; 1 Ib.tea, '• . ■ * ,-..' :■'■■'. ■ ,.,:.. ;:W:..: $47.61 t. ^^ • '■-'■ JitsamQiRa, Oct. Ist, 1866. Mb. Wmxui Fattcbsok, To Moffat k Mubrat, Br. 1866. Ai _ July 3, To 14 yds. fancy print, 20c. ; 12 yds. ool'd silk, 2.f^ « 14, " 2 ladies' feli hats, 2.oo; 2 prs. kid gloves, l.«» « 22, " 4 prs. cotton hose, 40c. j 3 yds. red flannel, 8O0 Aug. 19, " 2^ yds. blk. cassimere, S.*"' ; 2( yds. cotton, 20c " 27, " l| yds. white flannel, 75c. ; buttons, 10c. ; twist, 15c.. .. Sept 1, " 2 suits boys' clothes, 9.«o ; 2 felt hats, l.» » •* 8," 2 prs gloves 80c. ; 2 neckties, 62Ao «♦ 22, " I doz. prs. cotton hiose, 7.«« ; \ doa. shirts, 26.««> Ocyidra. Or. 20.00 15.00 Aug. 18, By Cash, ;27,Cash, |35.00 u 25. " firkin butter, 95 lbs., at 22c 20.90 55.90 Balance due |41.71i Aeoeived payment in full, MOFFAT &MUBBAT. AOOOUMTS AND INVOIOES. 128 -!):»1P>9* ' ' RocHESTEB, Jan. 2nd, 186Si lb. John Deans. To Wood & FuoGEU, jDr. ' ' ' '• ' ^^ 1866. -*•;'.' ■--:'• ':'^- ' ' ■■ V." r, , ..'i' ^ .v. l- July 4, To 12 Ibx sugar, 10c. ; 3 lbs. tea, 1*' ; 2 lbs. tobacco, 87Jo. " 11, " 1 bbl. salt, 2»» ; 2 lbs. indigo, 25c. ; 11 lbs. pepper, 30c. " 18, " 2 prs. socks, 45c. ; 1 ncclc-tie, 75c. ; 2 scarfe, 25c " 25, " 10 lbs. sugir, lie; 20 lbs. dr'd apples, 10c. ; 2 Iba. coffee 28o " " " 18 lbs. dried peaches, 12Jc.; 1 bush.. onions, li«^«. Aug. 4, *' 12 lbs. rice, 7c. ; 2 gals, syrup, 75c. ; 14 lbs. BUgar^ 12c... " " " 13 lbs. mackerel, 12c. ; 2 lbs. ginger, 20c. ; 2 lbs. tea, l.»* " 21. " 2 prs. kid gloves, 1«« ; 2 boxes collars, 37^0 Sept. 12, "10 lbs. sugar, 15c. ; 2 lbs. coffee, 35c. ; 1 lb. chocolate, 40c. Oct 4, " 2 felt bats, 1><<; shoe blacking. 25c " 21. " 2 lbs. pepper, 15c. ; soda, 40c. ; salpetre, 30c. ; salt, 75c. Contra. Or. 10.OP 6.0* Sept. 14. By Cash, ; Oct. 4, Cash Oct 17. " 2 bbls. winter apples, 2« Boston, Nov. 1st, 1866, Mb. Wm. To Aug. 4, " 17. Sept 4, " 26. Oct 11, " 22, •« 27, " 30. Reid. Camfbell, Linn & Co., Lr. To 2 prs. kip boots, S** ; 2 prs. cobourgs, 2«* , " 7 yds. fancy tweed, 2*> ; trimmings, loo . buttons 25c.. " 2 prs. gloves, 75u. ; 3 prs. socks, 35c. ; 2 straw hats, 40c, " 10 yds. print 35c. ; trimmings. 1"* j ribbons, 75c. " 3 neck-ties, 62 Jc; 2 prs. boyj' gaiters, if"; shoe tie8,12Jc. " 1 business coat, 14oo ; 2 felt hats, 1^^ ,• 1 umbrella, 2^" " 2 flannel shirts, 4as ; 1 pr. pants, 8»o ; over-coat, 160». " 2 lace scarfs, 2^" ; 3 prs. woollen mits 75c. ; pins, 25o. / i I Contra. Sept 12. Oct 24. a-. 10o» 800 „, ^ .,,^ ^ .. ByCash ; Oct 4, Cash, ,« •• . 300 lbs. cheese, 10c. ; 76 Ibo. butter, 25c Balance due $37.60 *^ " Reeeivetf payment CAMPBELL, LINN ft Oa 124 '■'''-■ ''''"^AtOTBxn^ '^'' AtBUBN, Sept. Ist, 1868. Mb. S. Bhob 3b WiuoN, Rat & Co., J)r. ...^„_ 1866. ■Jan. 15, To 6 yds B. cloth, 4/o; 2 doz. bnttons, 30c.; 9 ozs. thread,lSo. ''-^'^ " 20, « 40 ydfl. &c. cot, ICc. ; 7 spools cot, 4c. ; 12 yds. rib., S5a ^''* " 30. '< 16 yds. B. sUk, 2.>o^ 16yds. lining, ISc. ; Ssilk spools, Ilo. "* Feb 20. " 3yds.drill,Slc. ; 5yds.cob'rg,34o. ; 2papei<)need.l8c. liar. 18, " 9 yds. coating, (i.^o; l^ydiB. vesting, l.*o; 6 pr. hose, 40o. * " 31. « 21 yds. print, 20c.; 19} yds. muslin, 30c.; 2 prs. gloves, lA* Apr.l5, " I prs. gloves, l.^o; 16 yds. ribbon, 18o. ; 6 hand'Ic. 36c.' '^^'^ " 25. " 3prs.blanket8 6.3o. 4 counterpanes, 3.*o; 15 yds. cot.,25o. ' May29, " 2 summer hats, l."; 6 yds. ribbon, 40o. ; 2 feathers, 2** June 5, " 4 prs. slippers, l.*"] 4 prs. hose, 60c. ; 3 [prs. hose, 40c " 15. " 3 wool shawls, 6.3 o; i b. suit, S0.«»; 9 ozb. thread, 18o. July 6. " 40 yds. cotton, 30c. ; 3 spools, 12c. ; 2 spools, 10c Aug. 10. " 13 yds. flannel, 75c. ; 4 hand'ks., 35o. ; 12 yds. tape, 13c. COnira. . Or. 15.««> 10.«« Jan. 15. By Cash, ; 22. Cash, \ {- Feb. 20. " 60 lbs. butter, 40c. ; 6 cwt. pork, 10*« Hay 15. " 6 geese, 80c. ; 14 fowls, 40c Junes. " 60 lbs. wool, 60c. ; 16 lbs. wool, 60o. 30.00 r lo.oo " ^. July 6. "Cash, ; Aug. 10, Cash, Bbooklin, July 15th, 1866ii Ub. R. R. Hilus. 2b J. WiLLUMS, Dr. , -,/. 1866. )^'^' h '^^'' '■■' Jan. 10, To lblb8.H.8ugar,15c.; 16ibB.W.sngar,20c.; 121bs.Cj9ugar,18c. " 30. " 151bs. raisins, 16c.; 13 lbs. raisins,16c.; 10 lbs. raisins,18o. Feb. 12. " 9 lbs. oar'nts, 13c.; 12 lbs. cur'nts, 14c.; 6 lbs. cur'nts,20o. Har. 30. ** 60 lbs. salt, 2 ; 2 lbs. wash, soda, 23c.; 1 lb. bak. Boda,25o. Apr. 6, " 61bs.D.apple8,12c.;101bs.bi8o'ts, 17c.;61bs.bisc'ts,21o. <* 25. " .3 cwt flour, 4.«o;^ 2 cwt C. meal, 2.30; 3 lbs. butter, 26o. r May 1. " 161bs.pork, 20c. ; 19 lb«. cheese, 10c. ; 14 lbs. 8ugar,16o. Jnnel5. " 6 lbs. tea, 1."; 9 gals, molasses, 40c. ; 6 doz. eggs, 12c. Jalyl2, <' 6 lbs. sngar,16c.;9| lbs. raisins, 16c.; 10 lbs.Gur'nt8,12}o. *' 29, <• 14 lbs. bacon, 12c. ; 5 lbs. cheese, 16c. ; 4 lbs. butter,25o. ** 31, '< 4 lbs. tea, 1.* o; 2 lbs. tea, l.a o; 6 lbs. coffee, 35o. ^ , MM « 40 lbs. salt, lie. ; 3 lbs. indigo, 90o. ; If lbs. blue, 30e. **<*** 31be.8altpetre,36o.;4doz.egg8,12|o.;61b8.bntter,15c. liiiiit Xeoeived payment 4^> J. WlLUAMS. iOCOUKTB AND XNYOIG£S. 125 ,^n i AlbaaT, Dec. 1, 1868 li;.*' ■'■ Er. Geo. Smfsoi?, To Tatlor A Grant, i9r. i 1866. T July 'i, To 12 lbs. sugar, 15c. ; 2 lbs. tea, l.«* ; 3 lbs. coffee, 35c. .. . " 12, " 21b8.tobacco,87)c.;31b8.rateins,30c.;121bs.cnnanta,16c. « 2't " 3 lbs. gunpowder, 62)0.; Clbs. shot, 18c. ; 2 lb8.glne,26o. Aug. 4, '* 12 lbs. washing soda, l&c. ; 4 lbs. baking soda, 25c " 12, " Iboxmustard,l.«o;21b8.fllbert8,30c.;21b8.alm'd8,35c. Sept.21, " 8 lbs. sugar, 14o. ; 1 *b. tea, l.^»^ ; 3 lbs. chocolate, 40o. Oct. 12, " 4 lbs. figs, 15c. ;.2 lbs. orange peel, 30c. ; spices 40c ** 20, " 2 lbs. but. blue, i8o.; 2 lbs. sulphur, 20c.; 3 lbs. soda, 35o. 18.«» Not. 4, " 2 lbs. smok. tobacco, 90c.; 2 lbs. snuff, 20o.;l business suit, Contra. x:i't .' Or. 8.00 6.00 Aug. 12, By Cash, ; Sept 21, Cash, ;..... Oct 20, « 100 lbs. dried apples, 15c. ; 60 lbs. peaches, 20c. I ■ ii Balance due. $7.01 ISPi^ ■?v«. ■»■•..« jy ■^' Detroit, bept. 30th, 1866. MR.S.S1IITH, f - - — .■ t- ' ■ 3bBAT,Hiu.&Co.,J9r., -^ •• ,-)■ w j:,,jA /y ' i , v 1866. ,:..-,...:..»-.,..„.*.;.v..^:i;..v.^ ^ ■& ^^ ■ 1, To 6 lbs. tea, l.'o ; 15 lbs. sugar, 15c.; 1} lbs. cinnamon, 2."o. 10, " 18 lbs. rtce, 10c. ; 16 lbs. salt, 4c. ; 34 lbs. oat meal, 6c.. . ; [, 13, " 12 lbs. raisins, 18c. ; 3 lbs. tobacco, 58c. ; } lb. snuff, 34c.. 2, " 10 lbs. cur'nts, 17c.; 10 lbs. ginger, 41c.; 5 lbs. mustard, 42c. 8, " 6 lbs. sugar, 18c.; 13 lbs. rice, 8c.; 21 lbs. dr'd apples, 16c. 13, " 25 lbs. raisins, 18c. ; ^ lb. B. fioda, 30c. ; f lb. nutmegs, 22c. 4, " 12 lbs. coffee, 36c.; 6 lbs. M. r^ngar, 15c.; 4 lbs.W. sugar, 20e. " 15," 41bB.mn8tard,30c.;31bs.tobacoo,30o.;121bs.ginger,27c. A:^7l. 6, " 2 lbs. currants, 20c. ; 14 lbs. rice, 8c. ; 9 lbs. tur. seed, 45c. ** 14, " 1| lbs. cin'mon, 70c.; 12 lbs. sago, 31c.; 14 lbs. sugar, 21c. Hay 10, '< 16 lbs. salt, So. ; 2 lbs. indigo, 90c. ; 61 lbs. com starch. 14c. June 12, » 40 lbs. floor, 4c. ; SO lbs. com meal, 3c. ; 25 lbs. coffee, 88c. Jan. « Feb. « « Mar. $88.41 126 i'M • ABITHIIBIIO. 31. Chioaoo, Jan. 4th, 1866. Mr. EUAS 0. CONKLIN, > '. J* Bought of J. BuNTiN & Co., 12 reams of foolscap 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 doBcn American Commercial Arithmetic @ 18.00 .. ■■- '■■Wv>.' ' $367.90 ' ' Bcoeiyed payment, .,^ ' J. BUNTIN & Co. 32 ToBOKTO, Jan. 12tii, 1866. Mr. James H. BuaniTT, ^ou^A< o/MoBBisoN, Tatlob & Co., 15 owt. of cheese @ $9.00 4cwt. offlour @ 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 owt. maple sugar @ 8.00 25 bags of common salt @ 1.15 57 barrels of mess pork @ 13.00 ^8 " beef @ 9.75 ^ 13 bushels of clover seed @ 7.50* " ? V $2143.80 Beoeived payment by note at 30 days. Fob MORRISON. TATLOR&Cc, I BILLB OF PABOELS. ( 127 33 "*• ' ^ '-'«c; Hamilton, January 2nd, 1866. Mr. M. MoCuLLOcn, 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 mourn in<; envelopes @ 1.06 " 2 reams mourninj; 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 'i « 5 doz. Third Books @ 1.62J - " 10 quires blank books, half bound @ .35 ** 2 packs visiting cards @ .37^ $71.98 NofE.— Bills should not be signed until settled. I t . ' ' 34. Brookvillb, Jan. 5th, 1866. N. D. GALBREAITa, 7b R. FiTZSiHMONS & Co., 2>r. /' For 24 lbs. Mackerel @ (i^o, " 3 gallons Molnsses @ 45 " 13 lbs. Young Hyson Tea @ 87^ " 13 lbs. brown Sugar @ 11 " 15 bushels of Potatoes @ 45 ; vy- ■ V . . _ .:>,..,^ .■ , $22.23 v. "..' "'' Cr. "r-i* ■ :■ -'^ I'or 10 lbs. Butter @ 17o. -. " 5doz.EggB @ 12J " 3 gallons Maple Molasses @ &5 " Note at 20 days, to balance 17.05 $22.23 BFITZSDiMONS&Co. IfoTB.— Such a Bill as this would be termed a Barter Bill. 128 ABIXBMSTIO. ^ 85 > ,^ ,v. . V . . KmoiTON, Jan. 2nd, 1866. James Thompson, Esq., ;. ^4^ To A. Jabdinx & Co., Dr. ;, For 3dos. Buttons @ $0.12 ..^„ " 5^ yards of block Broadoloth @ 5.50 *< 20 yards Sheeting % .15 " 1 chest Y. H. Tea, 83 lbs @ .95 " 18 yards French Print @ .20 " 2 skeins of Silk Thread @ .09 " 6 yards black Silk Velvet % 3.60 *' 20 lbs. Loaf Sugar @ .18 " 2 gallons Molasses @ .40 « 1 bog of common Salt @ 1.15 "25 lbs. Rice @ .09 « 3 saoks Coffee, 70 lbs. oooh % .12 Ob. #166.74 By Cash 50.00 Balance duo -. $116.7«i • c ,*>■; :.■■■.''•'•'•■•.■• ;• 36 Algonqitiit, Jan« 15th| 1865. W. FMMINO' & Co., ' ,^^-.. ^>'.^s'V--VfV'>.i''- ; ./i;v'*, , ^ Bought o/J, & A. Wbight, : » ; ;< > , .jrt ; * u-M^- 1500 lbs. Canadian Cheese (^$.09 300 bushels Fall Wheat... @ 1.25 9 brls Pot Ash, net 7056 lbs @ 5.75 per owt. 150 bushels Spring Wheat @ 1.15 : 200 " Potatoes @ .45 600 " Oats @ .37J ' 150 " Pease @ .65 'J-i 60 " Indian Com @ .50 • 60 " Apples @ .60' * 3 kegs Butter, 110 lbs. each @ .18 . , 60 bushels Eye @ .70 _n: * 40 " Barley. @ .80 ^m^.:/:rr $1688.12: J.&A.WBIGHT. ■fHf,^'* PBBdMTAOS. •••^^ ••* PEROENTAOE. 129 I 1^.— PiBOBNTAOX is an allowance, or redaction, or estimate ol a certain portion of each 100 of the units that enter into any given eakmlation. The term is a contraction of the Latin expression for one hundred, and means literally hy the hundred. In calculating dollars and cents, ti per cent, means 6 dollars for every 100 dollars, or 6 cents for every $1, or 100 cents. If we are estimating the rate of yearly increase of the population of a rising village, and Ond that at the end of a certain year it was 100, and at the ond of the next it was lOG, wo say it has increased 6 per cent. i. "J., G persons have been added to the 100. So, also, if a largo 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 wo 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 tliousnnds 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 6,000, giving a total population of 106,000 as above, or an increase at the rate of 6 per cent. A decrease would be estimated in the same manner. Thus, a falling ofif in the population of 6 persons in tho hundred would be denoted by 100 — 6=94, as an increase of 6 in the hundred would be denoted by 100-j-6=106. So, also, in our first example, a deduetion of $6 in $100 would be $100— 6=:$94, ond a gain would bo $1004-$6=^106. The portion of 100 so allowed or estimated, is colled the rate per cant, as in the examples given, 6 denotes 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, cstimato what is to be allowed on it at the same rate. Thus, if 6 is to be allowed for 100, then 3 must be allowed for 50, and 1^ for 25, &c. The number on which the perc^tage 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 called the amount. If the rate per ecnt. 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 amonnt. This latter result is sometimes called the remainder. i I i .1 180 ABnHMITXO. From what has b«6n laid, it if plain that peroentige ii noiUn^ else than taking 100 as a standard unit of measure — (See Art, 1)— and malting the rate a fraction of that unit, so that 6 per cent, if -)'Jo=(Art. 15, V.) .06. Wo may obtain the same result bj thf rule of proportion. Thus, in our illustrative example of an inoreoso of 6 pcTbons fur every 100 on a population of 100,000, the analogy will bo 100 porsoHH : 100,000 persons : : 6 (the increase on 100) : 6,000, the inorooso on 100,000. It is manliest that the same result will be obtained whether we multiply the third by the second, and divide by the first, or whether we divide the third b> the first, and multiply the result by the second ; or, which is the same thing, mul« (iply tlio second by the result. Now, we already found that 6-^-100— J J{^— .06, the same as before. So also, 7 per cent, of any loss is seven one-hundredths of it, t. e., ^ Jg=.07. It should bo earefully observed that such decimals represent, not the rate per cent., iut the rate per unit. Though this is easily comprehended, yet wo know by experience that learners arc constantly liable to commit errors by neglecting to place the decimal point correctly. We would therefore direct parti- eular attention to the above caution, which, with the rule already laid down, under the head of decimal fractions, should be sufficient to guide any one who takes even moderate pains. EXERCISES ON FINDING THE BATE FEB UNIT. At ^ per cent., what is the rate, per unit ? .,^. At ^ per cent., what is the rate per unit ? , At 1 per cent., what is the rate per unit ? At 2 per cent., what is tlie rate per unit ? At 4 per cent., what is the rate per unit ? At 7^ per cent., what is the rate psr unit ? At 10 per cent., what is the rate );/er unit ? At 12^ per cent., what is the rate per unit ? At 17 per cent., what is the rr^te per unit ? At 25 per cent., what is the rate per unit ? At 33^ per cent., what is the r&te per unit ? At 66f per cent., what is the rate per unit ? At 75 per cc: .t., what is the rate per unit ? At 100 per cent., what is the t«te per unit ? At 112^ per cent., what is the rate per unit ? At 150 per cent., what is the i-ftte per unit? At 200 per cent., what is the rate per unit ? ^ Ans. .00^. Ans. .00|. Ans. .01. Ans. .02. Ans. .04. * Ans. .07J. Ans. .10. Ans. .12^. Ans. .17. Ans. .25. ^ Ans. .33^. Ans. .66f . Ans. .75. Ans. 1.00. Anf. 1.12|. Ans. 1.50. Ans. 2.00» FEBOENTAOB. 181 I. To find the {Mroontage on any given qnantity at a giTcn fate: On the priooiples of proportion, we have as 100 : given qaan^ iity : : rate : porceutago, and as tho third .term, divided by the fint, gives tho ruto per unit, we have tho Himple rulk: Multiply the given quantity by the rate per unitf and theproduet will he the percentaye. EXAHPLB8. To find how much 6 per cent, is on 720 bushels of wheat, wo have 6-;-100— .06, tho rate per unit, and 720X.06=431 bushels, the porcontago. To find 8 per cent, of $7963-75, in like monnor, we have .08, the rate per unit, and $7963.75 X 08 gives $637.10, the percentage. Instead o{per cent the mark ("/„) is now commonly used. XXXaOISKS ON THK RUTiE. I. What does 6 per cent, of 450 tons of hay amount to ? Ans. 27. ' 2. What is 10 per cent, of $879.62^ ? Ans. $87.06. 3. If 12 per cent, of an army of 47,800 men be lost in killed and jg^- -wounded ; how many remain ? Ans. 42,064. 4. What is 5 per cent, of 187 bushels of potatoes ? Ans. 9.35. ''■ 6. What is 2^ per cent, of a note for $870 ? Ans. 21.75. 6. Find 12^ per cent, of 97 hogsheads ? Ans. 12.12^. II. To find what rate per cent, one number is of another given number : — Let us take as an example, to find what per cent. 24 is of 96. Hero the basis is 96, and we take 100 as a standard basis, and these are magnitudes of the same kind, and 24 is a certain rate on 96, and wo wish to find what rate it is on 100, and by the rule of proportion, we have the statement 96 : 100 : : 24 : F. P.=l^git=25. Therefore 24 is 25 per cent, of 96. From this we can deduce the simple *" ' * " " ~ — y- * BULE. Annex two cipher$ to the given percentagef and divide that by the ioiiSf the quotient will be the rate per cent. 7. What per cent, of 150 is 15 7 Ans. 10. 8. What per cent, of 240 is 36 ? ' Ans. 15^ f 5 i! 11 in2 ABTTHMETIO. • 9. What per cent, of 18 is 2 ? ^ Ans. ll^^V 10. V/hp*. per cent, of 72 is 48 ? '' ' ' Ans. 66§. 11. What per cent, of 576 is 18? Aus. 3J. 12. What per cent, is 12 of 480 ? ' ' ' Ans. 2J. j( 13. Bought a block of buildings in King street for $1719, and sold it at a gain of 18 per cent. ; what vras the gain ? '^ ' ' .'* "' y^ -• :'iv,:i^ ■; Ans. $309.42;" 14. Vested $325 in an oil well speculation, and lost 8 per cent. ; what 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 ; A bought 30 per cent, of it ; B, 25 per coot. ; C, 20 per cent. ; and D purchased the remain- •'f der ; what per cent, qf 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 $1370 at 2| per cent. ? Ana. 37.C7J. 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 fi id that number. As 40 : 100 : : 400 : F. P.=: . 4M^M=1,000. Hence we derive the • ' Bn L£ . Annex two ciphers to the given numhery and divide by the rate per cent. E X X R 1 8 E S . 1. A bankrupt can pay $2600, which is 80 per cent of his debts ^ W how much does he owe ? Ans. $8250. 2. A clerk pays $8 a month for rent, which is 16 per cent, of w hb salary ; what is his yearly salary ? Ans. $*)00. 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 ? , j Ans. 160,000. 4. Bought a certain number of bags of flour, and sold 124 of ihcm, which is 12^ per cent, of the whole. Bequired, the number/* of bagf] purch^jcd. Aus. 992. 5. In a shipwreck 480 tons are lost, and this amount is 15 per cent, of the whole cargo. Find the caigo. An!]. 3210 tons* PEBOENTAOE. 138 6. A firm lost $1770 by tho failure of another firm ; the loss was ^ 30 per cent, of their oapitul ; what was their capital ? Ans. $5900. ^> IV. To find the basis when the amount and rate are given : — Suppose a man bays a piece of land for a certain sum, and by selling it for 0300, gains 25 per cent. ; what did he pay for it at first ? — Here it is plain that for every dollar of the co^t, 25 cents are gained by tho sale, i. e., 125 cents for every 100, which gives us tho analo- ' gy, 125 : ICO : : 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, giyes -^f |^^=$240, the original cost. Again, suppose the farm had been sold at a loss of 25 per cent. This being a loss, we subtract 25 from 100, and say, as 75 : 100 : : 300 : P. P.=^^o^=$^0, the prime cost in this case. , . Hence we derive the . RULE. Divide the given amount hy one increcued or diminished hy the given rate per unit^ according as the question implies increase or decrease, gain or loss. ,^;,:.i .,.,- .^ ■M.-i EXERCISES. ; ■.., 1. Given the amount $198, and the rate of increase 20 per cent. to find the number yielding that percentage. Ans. $165. 2. A fiel(^ yields 840 bushels of w)ieat, which is 250 per cent, on the seed ; how many bushels of seed were sown ? Ans. 336 bu. 3. At 5 per cent, gain ; what is the basis if the amount be $126 ? ''':■, ^- ;::■'::;:"■ > Ans. $120. ' 4. At 10 per cent. loss; what is the basis, the 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. "^ 6. A gambler lost 10 per cent, of his money bv a venture, and had $279 left ; how mucu 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 fiour, and having lost 20 per cent, of tile whole, had 160 bags remaining; how many bags did he buy ? Ans. 200. 8. A merchant lost 12 per cent, of his capital by a bankruptcy, «Dd had still $2200 left ; what was his whole capital ? Ans. $2500. Ill y i 184 ABTTHMETIO. . 9. Sold a sheep for |5, and gained 25 per cent. ; vrhat did I pay /^rit? Ans. $4. 10. Lost $12000 on an investment, which was 30 per cent, of the whole ; what was the investment ? Ans. $40000. .b . :,-.::;: ,„,,:. _ INTEREST. ,:;„-: _ \^^^i_ From a transition common in language, the word interest has been inappropriately applied to the mm paid for the use of money, but its original and true meaning is simply the Mse of money. To illustrate this, we will suppose that A borrows of B $100 for one year, and at the end of the year, when A wishes to settle the account, he gives B $107. Wore we to ask tl\p question of almost any per- son except an accountant, whether A or B received the interest, we should undoubtedly receive for an answer that B received it. But such is not the case. A having had the nse of that money for one 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 the same. It is only by considering this subject in its true light that account-^ ants are able to determine upon the proper debits and credit^ that arise from a transaction where interest is involved. If an individual borrows money, he receives the use of that money, and when ho 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 lias parted with the use of that money, and when he re- ceives value for that use or interest, he places ti.e sum so received to that side of his " interest account" which represents interest de> livered. • . -;bi* ■;«.-■ -'-i We think that this explanation is sufficiently clear to illustrate the difference between interest and the value received or paid for it. It will also be noticed that we have given many of the exercises in the usual form, e. g., we say what is the interest on $100 for one year, instead of saying what must be paid /or the interest of $100 for one year, but we have done this more in accordance with custom than from any intention to deviate from the true meaning of the word interest. Interest is reckoned en a scale of so many units on every $100 for one year, and hence it is called so much per cent, per annvrnj from the Latin per centum, by the hundred, and pa^ annum, by the year. Thus, $6 a year for eyery $100, is called six per cmUper IKTEBEST. 135 annum. The term is also extended to desigmite the return aooruini from any investment, such as shares in a joint stock company. To show the object and use of such transactions, we may su'ppose ft case or two. A person feels himself cramped or embarrassed in his circum- stances and operations, and he applies to some friendly party thai lends him $100 for a year, on the condition that at the expiration of the year he is to receive $106, that is, the $100 lent, and ^6 more a» a return for the use of the $100 ; or, if the Borrower gets $G00, he pays at the and of the stipulated time not only the $600, but nls« $36 ($6 for each $100) in return for the use of the $600. By thii means the borrower gets dear of hvn difficulty, and maintains hit credit at a small sacrifice. i:p;\':;--^.^ix. «<--%- -;t :■;/,- -■^)=; -^■i^r ;. :.:k--^j^ The ,5m on which interest is paid is called the principal. "^ f The sum paid for the use of money is called the interest. The sum paid on each $100 is called the rate. The sum of the principal and interest is called the amount. When interest is charged on the principal only, it is called simpk interest. "-■'■' •■'"'■'''■ When interest is charged on the amount, it is called compound interest. When a certain rate per cetti. is established by law, it is oalied legal interest. When a higher rate per cent, is charged than is allowed by law, H is called usury. The legal rsk; per cent, differs in different States and in different oonntries, so r.';' iojs the mode of calculation differ. In some States it is consiocr^d legal, to reckon the month as consisting of 30 daySj in the calculaiii-.g of interest on any sum for a short period, in others it is considered illegal. We have given the different modet of calculation in order to make the work applicable to all the States. For the legal rate per oent. of each State, see ".Laws of the States,'* at the end of this work. ^ ^'-^^^ y--w*.-H^?-M«i^^^..5J.>^v.",, -^^^v;: ^t; fri i m U M ill ,^ u I SIMPLE INTEREST. As sibtple interest, when calculated for one year, differs in no way from a percentage on a given sum, we have only four things to consider, vis., the principal, the Mte (100 being tb«> hms), the inters 136 t ABkTHMETIO. ost, and tho time, any three of which being known, the fourth oan bo found. The finding of the interest indudes by far the greatest number of oases. We shall first show the general principle, and from it deduce an easy practical rule. Let it be required to find the interest on |468 for one year, at € per cent. As 100 is taken as the basis principal in relation to which all calculations are madC) it is plain that 100 will have the same ratio to any given principal that the rate, which is the interest on 100, has to the interest on the ^ven principal. Hence, in the question proposed, we have as $100 : $468: : $6 : interest=$468XTBQ= $468x.06=$28.08. Now .06 is the rate per unitj and from this we can deduce rules for all cases. : : t ' ■ ' .-^■^••i>'---v>k'S.-v).r-:„. CASE l.:>-''&^'i'-^'^''-'^i-^-^- '■'•^'i'^-'^^- To find tho interest of any sum of money fbr one year, at any f'lven rate per cent. ■. "" ' RULE. -A^'-U:- MuUiply the principal by the rate per unit. EXEBOISES. 1. What is the interest on $15, for 1 year, at 3 per cent. ? An«. $0.45. 2. What is the interest on $35, for 1 year, at 5 per cent. ? Ans. $1.75. 3. What is the interest on $100, for 1 year, at 7 per cent. ? Ans. $7.00. 4. What is the interest on $2.25, fer 1 year, at 8 per cent. ? Ans. $0.18. 5. What is the interest on $6.40, for 1 year, at 8]^ per cent. ? Ans. $0.54. 6. What is the interest on $250, for 1 year, at 9^ per cent. ? Ans. $23.75. 7. What is the interest on $760.40, for 1 year, at 7^ per cent.? Ans. $57.03. 8. What is the interest on $964.50, for 1 year, at 6^ per cent. ? Ans. $62.69. 9. What is the interest on t568.75, for 1 year, at 7| per cent. ? Ans. $41.23. an; • v>:j IKTEBESP. OABE II. 187 M^t To find the interest of any sum of moneyi for any number of years, at a given rate per cent. .: BULE. ^ -*Find the intereit for one year, and muUiply by the ntimier oj years. '• '^; 10. What is the interest of $4.60, for 3 years, at 6 per cent ? Ans. $0.83. 11. What is the interest of $570, for 5 years, at 7| per cent. ? An's. $213.75. •^^ 12. What is the interest of $460.50, for 3 years, at 6J per cent, f Ans. $86.34. 13. What is the interest of $17.40, for 3 years, at 8^ per cent. ? ;,.: •.■■:-:;■'. . •"■": Ans. $4.35. 14. What is the interest of $321.05, for 8 years, at 5f per cent. ? Ans. $147.68. 15. What is the interest of $1650.45, for 2 years, at 9 per cent. ? Ans. $297.08. 16. What is the interest of $964.75, for 4 years, at 10 per cent. ? Ans. $385.90. ',:,.. 17. What is the interest of $1674.50, for 3 years, at.lOJ per «tot ? Ans. $527.47. , . 18. What is the interest of $640.80, for 5 years, at 4f per cent. ? . Ans. $152.19. 19. What is the interest of $965.50, for 7 years, at 5^ per cent. ? '''' Ans. $371.72. 20. What is the interest of $2460.20, for 4 years, at 7 per cent. 7 , Ans. $688.86. CASE III. To find the interest on any sum of money foi' any number of months, at a given rate per ce&t. R U^L B . Find the interest fw one year, and take oXiqwit j^rte for thit fMiith* ; or, Find the interest for one yeaTf divide hy 12, oa^ mviUiply by th/e ktmtber of months. .-],., '< iit^ I ; * i m m I I I'll 5! :! 1 'I I I i 188 ABITHMBTIO. V, ^•, RXBB0I8BS. ,..v - . /: -:.!. ..vVy.,... 21. What is the interest on $684.20, for 4 months, at 6 per centt Ans. $13.68. ^ 22. What is the interest on $760.50, for 5 months, at 7 per cent. 1 ^ Ans. $22.18. 23. What is the interest on $899.99, for 2 months, at 8 per cent. ? Ans. $12.00. 24. What is the interest on $964.50, for 4 months, at 9 per cent. ? Ans. $28.94. 25. What is the interest on $1500, for 7 'Jionths, ut 10 per cent. ? Ans. $87.50. 26. What is the fnterest on $1560, for 11 months, at 7^ pet oent.? Ans. $107.25. '"^ 27. What is the interest on $1575.54, for 8 months, at 6^ per cent. ? Ans. $65.65. 28. What is the interest on $1728.28, for 9 months, at Sji per eent. ? Ana. $110.18. 29. What is the interest on $268.25, for 13 months, at 7 per eent. ? Ans. $20.34. 30. What is the interest on $1569.45, for 1 year, 3 months, at ^ 8 per cent. ? Ans. $156.95. 31. What is the interest on $642.9l, for 1 .year, 5 months, at % 10 per cent. ? Ans. $91.09. 32. What is the interest on $560.45, for 1 year, 6 months, at 9^ per cent. ? Ans. $79.86. 33. What is the interest on $48.50, for 3 years, 9 months, ac 10| percent.? : /""' '• Ans. $19.10. 34. What is the interest on $560.80, for 2 years, 8 months, at llf per cent. ? Ans. $175.72. 35. What is the interest on $2360.40, for 19 months, at 12 per cent. 7 Ans. $448.48. CASE IV. To find the interest on any sum of money, for any number of months and days, at a given rate per c«nt. a. v>, ,% . ; ; :- ^ V ,,';.■ V.-;K5>:>?:>^-„.-v.-iI,S'':i?..i;--. RULE. ■?i;;^U.v-i :.::;;;;,;:*;,: y.i- Find the interest /or the fnontha, and take aliquot jpartt for the dayt, reckoning the month at conmting o^SO dag$, EXAMPLE. 36. What is the interest on $875.50, for 8 months, 18 days, tft 11 per oent. ? iT.:«« SDCFIiB INTEBE8T. 139 V • ' SOLUTION. I , ,' }. Principal $876.60 Rate per unit .11 Interest for 1 year 96.3050 Interest for 6 months ; or, ^ of interest for 1 year......!.. 48.1525 Interest for 2 months ; or, ^ of interest for 6 months 16.0508 Interest for 15 days ; or, ^ of interest for 2 months 4.0127 Interest for 3 days ; or, ^ of interest for 15 days 8025 Interest for 8 months, 18 days., $69.0185 We find the intereF:t for 1 year to be $96,305, and as 6 months »re the ^ of 1 year, the interest for 6 months will be the ^ of the interest for 1 year ; likewise the interest for 2 months will be the § of the interest for 6 months, and as 15 days are the |- of 2 months or 60 days, the interest for 15 days will be the ^ of the in- terest for 2 months, and likewise the interest for 3 days, will be the I of the interest for 15 days. Adding the interest for the months and days together, we obtain $69.02, the sum to be paid for the uso of $875.50, for 8 months, 18 days, at 11 per cent. EXERCISES. 37. What is the interest on $468.75, for 4 months, 15 days, at 7 per cent. ? Ans. $12.30. 38. What is the interest on $1654.40, lor 3 months, 8 days, at 5 per cent. ? Ans. $22.52. 39. What is the intercst'on $345.65, for 11 months, 25 days, at 6 per cent. ? Ans. $20.45. 40. What is the interest on $ 14.85, for 5 months, 22 days, at 9 per cent. ? Ans. $3.22. 41. What is the interest on $673.75, for 8 months, 19 days, at 7J percent.? i' • . .^< ^-^ —tv . Ans. $36.35. 42. AVhat is the interest on $57.45, for 1 year, 2 months, 12 days, at 6 per cent. ? Ans. $4.14. 43. What is the interest on $2647, for 1 year, 5 months, 18 days^ at 61 per cent. ? Ans. $242.64. 44. What is the interest on $268.40, for 2 years, 1 month, 1 day, at 8 per cent. ? Ans. $44.79. 45. What is the interest on $2345.50, for 3 years, 7 months, 20 days, at 10 per cent. ? : ,^ r , . Ansi. $853.50. I ;si 140 ABEFBMBnO. 46. What 18 the interest on $4268.45, for 4 yean, 11 months, 11 days, at llf per cent. ? Ana. $2481.24. 47. What is the interest of $642.20, for 2 years, 7 months, 24 HojB, at 12 per cent. ? Ads. $204.22. 48. What is the interest of $64.50, for 2 years, 11 months, 2 days, at 7 per cent. ? Ans. $13.19. 49. What is the amount of $746.25, for 1 year, 10 mooths, 12 days, at 5 per cent. ? Ans $69.65. 50. What is the interest of $680, for 4 years, 1 month, 15 days, at 6 percent.? Ans. $138.30. A 8 E V . To find the interest on any sum of money, for any number of days, at a given rate per cent."' BULE. Find the interest for one year, and iay, as one year (365 days,) u to the given number of days, so is the interest for one year to the interest required ; or, Having found the interest for one year^ muUiply it by the given Humher of days, and divide by 365. EXEBOISES. 51. What is the interest on $464, for 15 days, at 6 per cent. ? •^ Ans. $1.14. 52. What is the interest on $364, for 12 days, at 7 per cent. ? Ans. 84 cents. 53. What is the interest on $56.82, for 14 days, at- 8 per cent. ? Ans. 17 cents. > * To find' how many years elapse between any two dates, wo have only to subtract the earlier from the later date. Thus, the number of years from 1814 to 1865 is 51 years. To find months, we must reckon from the given date in the first named month, to the same date in each successive month. Thus, five months from the 10th of March brings us on to the 10th of August. To find days, we reqidre to count how many days each month contains, for to consider every month as consisting of 80 days, in the calculation of inter- est, is not strictly correct, although for portions of a single month it causes ne serious error. Thus, the correct time from March 2nd to June 14th, would l>e 104 days, viz., 29 for March, 30 for April, 31 for May, and 14 for June. A rery convenient plan for reckoning time between two given dates is to cooot the number of months and odd days that intervene. • Thns, from June 14th to November 20th, woold be 6 moBths and 6 days. SIMPLE VXTSBMffT, 141 64. What is the interest on $75.50, for 18 days, at 8^ per cent. ? Ans. 32 cents. 55. What is the interest on $125.25, for 20 days, at 5 per cent. ? ' Ans. 34 cents. 56. What is the interest on $150.40, for 33 days, nt 6 per cent. ? Ans. 82 cents. 57. What is the interest on $56.48, for 45 days, at 6^ per cent. ? Ans. 45 cents. 58. What is the interest on $75.75, for 65 days, at 7 per cent. ? Ana. 94 cents. 59. What is the interest on $268.40, for 70 days, at 7^ per cent. ? Ans. $3.86. 60. What is the interest on $464.45. for 80 days, at 8 per cent. ? Ans. $8.14. 61. What is the interest on $15.84, for 120 days, at 9 per cent. ? ' '-*'■■ Ans. 47 cents. 62. What is the Interest on $240, for 135 days, at 9^ per cent. ? Ans. $8.43. 63. What is the interest on $2460, for 145 days, at 10 per cent. ? Ans. $97.73. 64. What is the interest on $1568, for 170 days, at 11 per cent. ? Ans. $80.33. f.5. What is the interest oi $2688, for 235 days, at llf per sent. ? Ans. $203.35. , 66. Wliat is the amount of $364.80, for 320 days, at 11^ per ' ^nt? Ans $401.58. 111 If] ik OASB YI. To find the interest on any sum of money, for any time, at 6 per cent. Since .06 would be the rate per unit, or the interest of $1 for 1 year, it follows that the interest for one month would be the j\i of .06, or yij of a cent, equal to ^ cent or .005, and for 2 months it would equal ^ cent, or .005x2=.01. Therefore, when interest is at the rate of 6 per cent., the interest pf $1, for every 2 months, is one cent. Again, if the interest of $1, for ane month, or 30 days, is ^ cent or .005, it fellows that the interest for 6 days will be the | of .005 or .001. Therefore, when interest is at the rate of 6 per oent.^ the interest of $1 for every 6 days is one mil. Hence tho '■I if I 142 • ABXTHXEIXO. BULL Find the irUerett of $1 for the given time hy reckoning 6 cwte for every year, 1 cent for every 2 monthe, and 1 miU/or every 6 dayt; then multiply the given principal by the number denoting tJuit t'n- tereet, and the prodv.ct will be the interest required. Note.— This method can be adopted for any rate per cent by first finding the Interest at G per cent., then adding to, or subtracting from the interest so found, such a part or parts of it, as the given rate exceeds, or is less than 6 per cent. This method, although adopted by some, Is not exactly correct as tho year is considered as consisting of 360 days, Instead of 365 ; so that the in- terest, obtained in this manner, is too large by g{y or i^, which for every $73 interest, is $1 too much, and must therefore be subtracted if the exact amount be required. BXAMPLS. 67. What is the interest of $24, for 4 months, 8 days, at 6 per cent. ? , ;,.? V . • SOLUTION. The interest of |1, for 4 months, is 02 The interest of $1, for 8 days, is ,.001J Hence the interest of $1, for 4 months, 8 days, is 021^ Now, if the interest of $1, for the ^ven time, is .021^, the inter* «st of 124 will be 24 times .021^, which is $.512. ; ,,, ■''•"■' "' '' SXBBOISKS. '"' 68. What is the interest on $171, for 24 days, at 6 per cent. ? Ans. 68 cents. 69. What is the interest on $112, for 118 days, at 6 per cent. ? "'■'•^<"- • .-■ '"■■ - Ans. $2.20. 70. What is the interest on $11, for 112 days, at 6 per cent. ? Ans. 21 cents. 71. What is the interest on 50 cents, for 360 days, at 6 per •ent. ? Ans. 3 cents. 72. What is the interest on $75.00, for 236 days, at 6 per cent. ? '^W -^ . Ans. $2.95. 73. What is the interest on $111.50, for 54 days, at 6 per cent. ? Ans. $1.00. 74. What is the interest on $15.50, for 314 days, at 6 per cent. ? ,„ V ,, . * Ans. 81 cents. ■and SIMPLE INTEBE8T. 148 75. What is the iDterest on $174.25, for 42 days, at 6 per cent. 7 Ads. $1.22. 76. What cent. 77. What 78. What per oont. ? 79. What . 80. What at 7 per cent. ? 81. What 82. What 83. What at 10 per cent 84. What per cent. ? 85. What 7 per cent. ? 86. What per cent. ? 87. What 8 per cent. ? 88. What per cent. ? 89. What cent. ? 90. What per cent. ? s tho interest on $10, for 1 month, 18 days, at 6 per , ' * Ans. 8 cents. 8 tho interest on $154, for 3 months, at 6 per cent. ? Ans. $2.31. s the interest on $172, for 2 months, 15 days, at 6 Ans. $2.15. s the interest on $25, for 4 months, at 6 per cent. ? Ans. 50 coats. H the interest on $36, for 1 year, 3 months, 11 days, Ans. $3.23. 8 tho interest on $500, for 160 days, at G per cent. ? Ans. $13.33. s the interest on $92.30, for 78 days, at 5 per cent. ? Ans. $1.00. s the interest on $125, for 3 years, 5 months, 15 days, Ans. $43.23. s the amount of $200, for 9 months, 27 days, at 6 Ans. $209.90. s the interest on $125.75, for 5 months, 17 days, at Ans. $4.08. a the interest on $84.50, for 1 month, 20 days, at 5 Ans. 59 cents, s the amount of $45, for 1 year, 1 month, 1 day, at Ans. $48.91. s tho interest on $175, for 7 months, 6 days, at 5^ Ans. $5.78. s the interest on $225, for 3 months, 3 days, at 9 per . Ans. $5.23. s the interest on $212.60, for 9 months, 8 days, at 8^ Ajis. $13.95. OASE VII To find the interest on any sum of money, in pounds, shillings, and pence, for any time, at a given fate per cent , BULB. ^j MuUiphf the principal by the rate per cent., and divide hjf 100. ill! m m i ,ii 'ill ■ : < i ' 'II ' i!? '. ! ||: i ;.'1 'h 1 » 111 144 ABXIHICBIZC. IZAMPLI. 91. What is the interest of £47 15s. 9d., for 1 year, 9 monthly 15 days, at 6 per cent. ? . . , . . v SOLUTION. £ 8. D. £ S. D. Interest for 1 year...: 2 17 4 47 15 9 Interest for G mos., or ^ of int. for 1 year, 18 8 6 Interest for 3 mos., or J of int. for mos., 14 4 Interest for 15 days, or i of int. for 3 mo8., 2 ^ 2^86 14 6 20 Interest for 1 yoarj 9 mouths, 15 days.... £5 2 S^ 12 4;i4 92. Wliat is the intcreat of £25, for 1 year, 9 months, at 5 per cent. ? Ans. £2 3s. 9d. 93. What is the interest of £75 12a. 6d., for 7 months, 12 days, at 8 per cent. ? Ans. £3 14s. l^d. 94. What is the amount of £64 lOs. 3d., for 3 months, 3 duys, at 7 per cent. ? Ans. £65 13s. 7d. 95. What is the interest of £35 4s. 8d., for 6 months, at 10 per cent.? Ans. £1 15s. 2|d. 96. What is the junonnt of £18 12s., i'oi 10 months and 3 days, at 6 per cent.? ^/^'^'^^i CASE VIII. To find the pkincipal, the interest, the time, and the rate per cent, being given. . ,. , EXAMPLE. 't. •'..^ 97. What principal will produce $4.50 ii)' .ost in 1 year, 3 months, at 6 per cent. ? If a principal of $1 is put on interest for 1 year, 3 months, at 6 per cent., it will produce .075 interest. Now, if in this example, .075 be the interest on SI, the number of dollars required to produce $4.50, will be represented by the number of times that .075 is con- tained in $4.50, wMoh is 60 times. Therefore, $60 will produce $4.50 interest in 1 year, 3 months, at 6 per cent. Hence the SIMPLE INTEBE8T. RULI. 145 Divide the given interest hy the intereet o/$l for the given (tW, at the given rate per cent. KXER0I8K8. , '■ 98. What principal will produce 77 couts interest in 3 months, 9 j^ days, at 7 per cent. ? . Ana. 940. 99. Whiit principal will produce $10.71 interest in 8 months, 12 dayc, at 7| per cent. ? Ans. $204. 100. What principal will produce $31.60 interest in 4 years, at 3^ per cent. ? Ans. $225. 101. What sum of money will produce $79.30 interest in 2 years, v^ 6 months, 15 dovs, at 6^ per cent. ? Ans. $480. ^ 102. What sum of money is suffioio' .> produce $290 interest in 2 ye» and 6 months, at 7|- per c^ . 't Ans. $1600. ' CASK IX. To find the rate per cent., the principal, the interest, and tho time being given. E X A M P T. E . 103: If $3 be the interest of $60 fo. 1 j car, what is the rate per ■ * . SOLUTION. ' • •'■ If the interest of $60 for 1 year, at 1 per cent, is .60, the re- quired rate per cent, will bo 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 year, the rate per cent, is 5. Hence the RULE. ^••■/v Divide the given interest hy the interest of the given principal at 1 per cent, for tJie given time. ,._.. EXERCISES. . ^ %'%®it'.'' " 104. If the interest of $40, for 2 years, 9 t , aths, 12 days, is* $18.36 ; what is the rate per cent. ? Ans. 12. 105. If I borrow $75 for 2 months, and pay $1 interest ; what is the rate per cent. ? , AniJ;8. .1 In i lO 1 ! I MMA if I \i :i!' ',''.' ;'.! \ i;' 15 !)■ IIS) ,E r 146 ABITHICETIO.. 106. If I give $2.25 for the use of $30 for 9 months ; what rat* per oont. am I paying ? Ana. 10. 107. At what rate per cent, will $150 amount to $165.75, in 1 year, 4 months, 24 days ? . Ans. 7J. lOS. At what rate per cent, must $1, or any sum of money, be on interest to double itself in 12 years ? ^ Ans. Ans. 3J. 109. At what rate per cent, must $425 be lent to gain $11.73 /\ in 3 months, 18 days ? Ans. 9|. S^ 110. At what rate per cent, will any sum of money amount to ^hree times itself in 25 years ? Ans. 8. 111. If I give $14 for the interest of $125 for 1 year, 7 montha, ^K,. 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. EXAMX'LE. 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 $6. 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 represcnte 1 by the number of times that 6 is contained in 12, which is 2. Therefore, $75 will have to be at interest for 2 wjan to gain $12. II once the RULE. Divide the given interest ty the interest of the principal for one year, at the given rate per cent. EXEROIEES. 113. In what time will $12 produce $2.88 interest, at 8 per cent?- ^, - r : . Ans. Syears. •f-, 114. In what time will $25 produce 50 cents interest, at 6 per cent.? '.H< I Ans. 4mopth8. . ^; 115. In wh^ time will %¥^ produce \ ^ cents interest, at 6| per cent ? ^ Alls. 3 montlis, 18 days. BIMFLE INTEREST. 147 116. In viheA time will any sum of money double itself, at 6 per cent. ? Ans. 16 years, 8 months. 117. In what time will any mm of mon^y 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.? Ans. 1 year, 4 months, 15 days. 119. Borrowed, January 1, 1865, $60, at 6 per cent, to bo paid )f as soon as the interest amounted to one-half the principal. When is it due ? Ans. May 1, 1873. 'A 1' \ u b If 120. A^^n^rchant borrowed a certain sum of money on January ^^ 2, 1856, at 9 per cent , agreeing to settle thj account when the in terest equalled the principal. When should he pay the same ? A. y Ans. Feb. 12, 1867. merchants' table VV^ ' 'V . ^ ' •\ K For showing in, what time any sum of money will double itself, at any rate per cent., from one to twenty, simple interest. \' h, Per cent. Years. Per cent. Years. Per cent. Years. Per cent. Years. 1 100 6 16f 11 ?r 16 ^ 2 50 7 ;4| i2i 12 17 5|4 3 m 8 13 7-^H 18 5i 4 25 9 in 14 "?? 19 h% 5 20 10 10 15 4 20 5 A" 7 <^^ '/ f4 A.-"' ^ s , MIXEDEXEECI8E.S. 121. What is the interest on $64.2.- lor 3 years, at 7 per cent.? ' Ans. $13.49. 122. What is the interest on $125.4t; for 6 months, at 6 per cent. ?* Ans. 3.76. 123. What is the amount of S369.29 for 2 years, 3 months, 1 » day, at 9 per cent. ? Ans. $444,16. ^^ 124. What milst be paid for the use of 75 cents for 6 years, 9 \J^ 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. ^C • This and the following exercises (marked with a •) are to be worked by Case VI. i.i,,Vv.:a.(v''*,',' ««t I.' I !i ■t i 148 ARITHMETIC. '■i 126. What must be paid for the interest of $45 for 72 days, at 9 percent.?* ^ Ans. 81 ceiks. v^ 127. Wliat is the interest of $240 from January 1, 1866, to /\ June 4, 1866, at 7 per cent. ? Ans. $7.14. 128. What will $140.40 amount to from August 29, 1865. to y November 29, 1866, at 6J per cent. ? Ans. $151.8 . 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 $44.40, at 8 per cent/;' Ans. 1 yr., 4 raos., 15 days. 131. At what rate per cent, will $40 produce in 1 yr., 4 mos., 15 %^ days, $4.40 interest ? Ans. 8. 132. What must bo paid for the interest of $145.50 for 240 days, at 9A per cent. ?* Ans. $9.22. 133. What will $160 amount to in 175 days, at 6 per cent. ?* Ans. $164.67. 134. At what rate per cent, must any sum of mondy be on ^>ii^^ interest to quadruple itself in 33 years and 4 months ? Ans. 9. 135. In what time will any sum of money double itself, at 10 per cent. ? Ans. 10 years. CASE X I. To find the interest on bonds, notes, or other documents draw- ing 7 ,",j per cent, interest. Since .07-,^g or, .073 would be the rate per unit, or the interest of $1 for I year or 365 days, it follows that the interest for 1 day would be the ^ J^ part of .073 which is .0002, equal to two tenths of a mill, hence the BULL . 3Iultlplii the principal hy the number of days, and the product hy two tenths of a mill the result will be the answer in mills. EXAMPLE. . > ; What must be paid for the use of $75 for 36 days at 7/jj per jeut. ? SOLUTION. , The interest on $75 for 36 days would be the same as the inter- est on $75x36=r$2700 for 1 day, and at ~-q of a mill per day would be $2700 X. 0002=54 cents. "^-^ Vfe:^^' '^^^ %**iW •i 2. What would be the interest on $118.30 for 42 days at 7/^ per cent. ■ ' Ans. 99cts, X COMMEBCIAL PAPER. 140 COMMERCIAL PAPER. OoMMEBOiAL paper is divided into two classes^— NEGOTIABLE and NON-NEGOTIABLE. NEGOTIABLE COMMERCIAL PAPER. Negotiable commercial paper is that which may be 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-ofis, 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 marker intends to give the payee authority to tiansfer it to a third party, Irce from jjl set-offs, or equitable or legal defences existing between himself and the payee. NON-NEGOTIABLE COMMERCIAL PAPER. Nbn-ncgotiahle commercial paper is that wfcich is made pajrable to the pnyee therein named, without authority to transfer it to a third party. It may be passed from one owner to another by assign- ment, 04: by indorsement, but it passes subject to all set-offs, and legal or equitable defences existing between the original parties. now THE TITLE PASSE>i. The title to negotiable paper passes from one owner to another by delivery, if made payable to payee or bearer, or to bearer. It passes by iudorsement 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 promissory note there are two original parties — the maker and the payee. The obligation of the maker 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. When the drawee accepts the bill, he 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 does not come within the Statute. There is no presumption that it Is founded upon a valuable consideration. A consideration must be ^■Q tall r^% Ml !;^>!' 'IS Si ■, 150 ▲BITHIIETIO. alleged in the complaint, nnd proved on the trial. The aoknowledg* mentof a consideration in such promissory note, by inserting the words " value recrived" is sufficient to cast upon tho defendant the burden of proof that there was no consideration. Tho acknowledg- ment of '* value received," raises tho presumption that tho note was given for value ; but this presumption may bo rebutted by the de- fendint. A negotiable instrument is a written promise or request for the payment of a certain sum of 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 whoso order a negotiable instrument is made payable, must lio as^ccrtainable at the time tho instrument is made. A negotiable instrument may give to the payee an option between the payment of the sum specified therein, and the performance of another act. A negotiable instrument may be with or withput date; with or without seal ; and with or without designation of the 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 contjiin any other contract than such as is specified. Two different contracts cannot bo 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 doea ;iot 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 negotifiblc in form, payable to a person named, but adding the words, " or to his order," or " to bearer," or equivalent thereto, is in the former case rayablo to the written order of such person, bnd in the latter case, payable to the bearer. A negotiable instrument, made payable to the order of tho 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 payable to the bearer. A negotiable instrument, mado payable to the order of a person obviously fictitious, is payable to the bearer. The signature of every drawer, acceptor and indorser of a aego- aiM»»MaaaMaitgai»i n COilMEKCIAL r.\rER. 151 tiable instwimcnt, is presumed to. have been niaoc for a valnabla consideration, before the maturity of the instnimcnf, and in thtj ordinary course of business, and the words '• value received," noknowledgo a consideration. Ono who writes his name upon a negotiable instrument, otlierviise than as a, maker or acceptor, and delivers it, with lii.s name thetoon, to another person, is called an itidorser, and his act is culled ab indorsement. One who agrees to indorse a negotiable instrument is bound to write his .signature upon the back of the instrument, if there ia suJEcient space thereon for that [)urpose. When there is not room lor a signature upon thr h;ick of a nego- tiable instrument, a signature equivalent to an indorsement thereof may bo made upon a paper annexed thereto. An indorsement may be general or .'•pecial. A general indorsement is one by which no indorser is named. A special indorsement (jpeciiies the indorsee. A negotiable iiistr'.^ient bearing a general indor.'-ement cannot be afterwards specially indorsed ; but any lawful holder may turn a p;eueral indorsement into a special one, by writing abovo it a direction for payment to a particular per.son. A special indorsement may, by express words Ibr tliat i)urpo.sc, but not olherwine, be so made as to render the instrument not negoti- able. Every indorser of a negotiable instrument warrants to every subs^ quent holder thereof, who is not liable thereon to him : 1. That it is in all respects wrmt it purports to be; 2. That he has a good title to it; 3. That the signatures of all prior parties are binding upon them ; 4. That if the instrument is dishonored, the indorser will, upon notice thereof duly given unto him, or without notice, wlioro it ia excused by law, pay so much of the same as the holder paid therefor, with interest. One who indorses a negotiable instrunjent before it is delivered to the payee, is liable to the payee thereon, as an indovf-cr. An indorser may qualify his indorsement with tho words, " with- out recourse," or e([uivalent words ; and upon such indorsement, he is responsible only to the same extent as in the case of a (ran:- Per without indorsement. Except as otherwise prescribed by the last section, an indorse- ment " without recourse" has the same effect as any other indoise- ment. An indorsee of a negotiable instrument has tlic same right against every prior party thereto, that he would have had if tho contract had been made directly between them in the first instance. An indorser has all the rights of a guarantor, and is exonerated from liability in like manner. ■:| '11 f«ji > ■', i ^11 lir id -I 'i:r; J 'if if II s '111 I 'ill ■m m III; n 152 AlOTHMEKO. One who Indorses a negotiable instrument, at the request, and for the " accommodation" of another party to the instrument, has all the rights of a surety, and is exonerated in like manner, in respect to Kvery one having notice of the facts, except that he is not entitled to contribution from subsequent indorsers. The wiint of consideration for the undertaking of a maker, Acceptor, or indorser of a negotiable instrument, does not exonerate him from liability thereon, to an indorsee in good faith for a consid- eration. An indorsee in due course is one who in good faith, in the ordi- nary course of business, and for value, before its apparent maturity or presumptive dishonor, acquires a negotiable instruuMint 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 generolly void or voidable, and notwithstanding any defect in the title of the person from whom he acquired it. One who makes himself a party to an instrument intended to be negotiable, but whicli is left wholly or partly in blank, for the pur- pose of filling ul'tcrwards, is liable upon the instrument to an indorsee thereof iw du(j course, in whatever manner, and at whatever time it may bo filled, so long ds it remains negotiable in form. It is not necessary to make a demand of payment upon the principal debtor in a negotiable instrument in order to charge him ; but if the instrument is by its terms payable at a specified place, and he is able and willing to pay it there at maturity, such ability and willingness are equivalent to an oiFer of payment upon his part. Presentment of a negotiable instrument for payment, when necessary, nmst be made as follows, as nearly as by reasonable dili- gence it is practicable : . 1. The instrument must be presented by the holder, or his authorized agent. 2. The instrument must be presented to the principal debtor, if he can be found at the place "where presentment should be made, 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 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 place for its payment, must be presented there, and if the 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 whioli does not specify a place for its payment, must be presented at the place of residence or business of the prin- cipal debtor, or wherever he mav be found, at the option of the presenter ; and, COMMERCIAL PAPER. 153 5. Tho instrument must bo presented upon the day of its appar* ent maturity, or, if it is payable on demand, at any time before its apparent maturity, within reasonable hours, and, if it is payable at a baukin<; house, within the usual banking hours of the vicinity ; but, by the consent of the person to whom it should be presented, rt may be presented at any hour of the day. The apparent maturity of a netj^otiable instrument, payable at a particular time, is tho day «n which by its terms it becomes due ; or, when that is a holiday, it should be paid the previous day. A bill of exchancje, payable at a specified time after sight, which is not accepted within ten days after its date, in addition to the time which would suffice, witli ordinary diligence, to forward it for acceptance, is presumed to have been dishonored. The apparent maturity of a bill of exchange, payable at sight or ©n demand, is : 1. If it bears interest, one year after its date ; or, 2. If it docs not bear interest, ten days after its date, in addition to the time which would' suffice, with ordinary diligence, to forward it for acceptance. The apparent maturity of a promissory note, payable at eight or on demanc., is : 1. If it bears interest one year after its date ; or, 2. If it docs not bear interest, six months after its date. When a promissory note is payable at a certain time after sight or demand, such time is to bo added to tho periods mentioned in the last paragraph. A party to a negotiable instrument may require, as a condition concurred to its payment by him : 1. Tluit the instrument be surrendered to him, unless it is lost or destroyed, or the holder has other claims upon it ; or, 2. If the holder has a right to retain the instrument, and does not retain it, then that a receipt for the amount paid, or an exonera- tion of tho party paying, be written thereon ; or, 3. If the instrument is lost, then that the holder give to him a bond, executed by himself and two sufficient sureties, to indemnify him against ajiy lawful claim thereon ; or 4. If the instrument is destroyed, then that proof of its destruc- tion be given to him. A negotiable instrument is dishonored when it is either not paid, or not accepted, according to its tenor, or presentment for tho purpose, or without presentment, where that is excused. Notice of the dishonor or protest of a negotiable instrument may be given : 1. By a holder thereof; or, 2. By a party to the instrument who might be compelled to pay it to the holder, and who would, upon taking it up, have a right to reimbursement from the party to whom the notice is given. ) •■ I* •'A : 1 i; ;^ :\M If 'I ■rii m 4 154 ABn^miETic. A notice of dishonor may bo given in any form which dcscribe» the instrunicnt nvith reasonable certainty, and substaiitiully informs the puity receiving it that the instrument has been dishonored. A notice of dishonor may be given : 1. By delivering it to the party to bo chained, ])cr.sonally, at any place ; or, 2. By delivering it to some person of discretion at tlie placo of residence or business of such party, apparently acting f u' liini ; or, ii. By properly folding the notice, directing it to the party to be charged, at his place of resilience, according to the hos-t information that the person giving the notice can. obtain, depositing it in the post-office most conveniently accessible from the place where the presentment was made, and paying the postage thereon. In case of the death of a party to whonj notice of dishonor should otherwise be given, the notice must be given to one «!' liis personal representatives; or, if there are none, then to any member of hiS family who resided with liim at his death , or, if then; is none, thea it must be mailed to his last place of nsidence, as prescribed by subdivision 3 of the last paragraph. A notice of dishonor sent to a pnrty after his death, but in igno- ranee thereof, and in good faith, is valid. Notice of dishonor, when given by the holder of an instrument, or his agent, otherwise than by mail, must be given on the day of dishonor, or on the next business day thereafter. When notice of dishonor is given by mail, it must bo deposited in the post-office in time for the first mail which closes alter neon of the first business day succeeding this dishonor, and which leaves tho place where tlie instrument was dishonored, for tho place to which the notice should be sent. When the holder of a negotiable instrument, at the time of its dishonor, is a mere agent for the owner, it is sufficient for him to give notice to his principal in the same manner as to an indorser, and his principal may give notice to any other party to be charged, as if he were himself an indorser. And if an agelit of the owner em- ployes a sub-agent, it is sufficient for each successive agent or sub- agent to give notice in like manner to his own principals. Every party to a negotiable instrument, receiving notice of ita dishonor, has the like time thereafter to give similar notice to prior parties, as the original holder had after its dishonor. But this addi- tional time is available only to tho particular party entitled thereto. A notice of the dishonor of a negotiable instrument, if valid in favor of tho party giving it, inures to the benefit of a)l other parties thereto, whose right to give tho like notice hxs cot then been lost. i COMMERCIAL PAPER ir)/> 18 "^ ' FORMS OF FOBEiay BILLS OF EXOHANOE. FiiEXcn. mie, le28 Septemfjre. 1 84 8. Bon pour £ 1 58 9 Sterllnga. Ah vinrjt cinq IXrevihre prochaln, 11 vovn plnira pnytr par ce mamlat d Vordre de iioiisin^ities la sirmrnc de cent einr/uante huil livrea aterlings U ahelUnga valeur en iii,40 -. ■ ^ . Interest on tho name from February 7*H, *i3G:.'-" i < ,..^-,er.a,' •■ ■■ " PARTIAL PAYMENTO. 163^ interests on the annttnl interests to the amovmt of the principal^ ana from this amount take the payments, and the interest on each, from the time they were paid to the time of settlement, the remainder will he the amount due. 6. $500. Prescott, May 1st, 1864, One yewr after date, for value received, I promise to pay Musgrovc do Wright, or order, Five Hundred Dollars, at their office^ in the city of Toronto, with interest at Q per cent., payable annually, James Manning. There was paid on this note : May 4th, 1865 $150 Dec. 18th, " 300 How much was due Juno 1st, 1866 ? SOLUTION. Face 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 562.50 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 Amount of interest upon annual intere. 2.10 Total amount of principal $564.4>0 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 Payments and interest on the same 467.90 Amount due June 1st, 1866 $96.70 ':!i 'ill ■"r''\ ■•u'i B m it Is i if '1 1 1 164 ABTTHMETIC. 7. $700. Cincinnati, January 2nd, 1863. Eighteen months after date, I promise to pay to tht order of J. H. Wilson, Seven Hundred Dollars, for value received, with interest at 6 per cent., payable annually. • Thos. a. Bryoe. There was paid on this note : January 15th, 1864 $350 July 2nd, 1864 300 What amount was duo January 2nd, 1865 ? Ans. $107.22, 8. $950. . Indianapolis, Jan.3rd, 1863. Two years after date, I promise to pay A. R. Tennison, or order, Nine hundred and Fifty Dollars, loith interest at 9 per cent., payable annually, value received. James S. Parmenter. The following payments were receipted on the back of this note : February 1st, 1864, received $500 May 14th, « " 100 • January 12, 1865, " 300 What was due May 6th, 1865 ? Ans. $188.94. 9. $250. Mobile, January 2nd, 1863. J Three years from, date, for value received, I promise to pay Michael Wright, or order, Two Hundred and Fifty Dollars^ vnth interest, payable annually, at Qper cent. Calvin W. Pearsons. At First National Bank here. What was the amount of this note at maturity ? Ans. $297.70. CONNECTICUT RULE. The Supreme Court of the State of Connecticut has adopted the following RULE. Compute the interest on the principal to the time of the first pay- ment ; if that be one year or m^jre from the time the interest com- menced, add it to tJie principal, and deduct the payment from the sum total. If there be afterpayments made, compute the interest on the balarice due to the next payment, and then deduct the payment as dbvve, and in like manner from one payment tf^ apother. t'V oV thr TARTIAL PAYMENTS. -: ''^■'^ payments are absorbed, provided tJui time between one payment and another he one year or more. If any payments be made he/ore one year's interest hasaccruedy 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 the sum paid, from the time it loas paid, tip to the end of the year ; add it to the sum paid, and deduct that mm from the principal and interest, added as above. ^ ' • If any payments be made, of a less sum than the interest arising at the time of such payment, no interest is to be comjmted, but only on the principal sum for any period. Note.— If a year extends beyond the time of settlement, 6nd the amount of tho remaining principal to the time of settlement ; find also the amount of the payment or payments, if any, from tho time they were paid to the time of settlemunt, and subtract their sum from the amount of the principal. K>:A MPLES. 10. $900. Kingston, June 1st, 1862. On demand wc promise to pay J. R. Smith & Co., or order, nine hundred dollars, for value received, with interest from date, at G per cent. Jones & VViiiaHT. On the back of this note were receipted tho following payments : June 16th, 1863, received $200 August 1st, 1864, " 160 Nov. 16th, 1864, " 75 Feby. 1st, 1866, " 220 Wlaat amount was due August 1st, 1866 ? SOLUTION . Face of note or principal $900.00 Interest on the same from June 1st, 1862, to June 16tli, 1863 56.25 Amount of principal and interest, June 16th, 1863 956.25 First payment to be 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 1st, 1864 807.296 " il H M I. 166 ABTTHMETIO. Second payment to be taken from this amount 160.000 Balance due 647.296 Interest on the same for one year 38.837 Amount duo August 1st, 1865 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 due August 1st, 1866 644.422 Amount of 4th payment from February 1st, 1866, to August 1st, 1866 226.600 Balance duo August Ist, 1866 6417.822 MEBOHANTS' RULE. It is customary among merchants and others, when partial pay- ments of notes or other debts are made, when the note or debt is settled within a year after becoming due, to adopt the following RULE. 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 settlementf and subtract their sum from the amount of the principal. EXAMPLE. 11. 6400. Maitland, January 1st, 1865. For value received, I promise to pay J. B. Smith & Co.j or order, on demand, four hundred dollars, with interest at 6 per cent. A. B. Cassels. The following payments were receipted on the back of this note : February 4th, 1865, received $100 May 16th, " " 75 August 28th " « . 100 November 25th, " « 80 .. What was due at time of settlement, which was December 28th, 1865? PARTIAL rAYMENTS. ^g^ SOLUTION. Principal or face of note $400.00 Interest on the same from Jan. 1st, 1865, to Dec. 28th, 1865 23.80 Amount of principal at settlement 423.80 First payment $100.00 Interest on the same from Feb. 4th, 1805, to Dec. 28th, 1865 5.40 Second payment 75.00 Interest on the same from May 10th, 1865, to Dec. 28th, 18G5 2.77J Third payment 100.00 Interest on the same from August 28th, 1865, to Dec. 25th, 1865 2.00 Fourth payment 80.00 Interest on the same from Nov. 25th, 1865, to Dec. 28th, 1865 44 Amount of payments to be taken from amount of principal 365.61^ Balance due, December r8th, 1865 '. $58.18^ 12. $500. Cleveland, January 1st, 1865. Three months after date, I promise to pay James Man- nine/, or order, Jive hundred dollars, /or value received, at the First National Bank of Buffalo. Cyrus King. Mr. King paid on this note, July 1st 1865, $200. What was due April 1st, 1866, the rate of interest being 7 per cent ? Ans. $324.50. 13. $240. Philadelphia, May 4th, 1865. On demand, I promise to pay A. K. Frost & Co., or order, two hundred and forty dollars, for vfllue received, with in- terest at 6 per cent. David Flock. The following payments were receipted on the back of this note : September 10th, 1865, received $60 January 16th, 1866, *' 90 What T?«« ouirt> nrrauiT for akt RmtBiR of tram NOT RxouniNO rirrr. No. T 3 per cent. 3}i per cont. 4 per cent 6 por cont por cent. 7 por cont 1.030 000 1.035 000 1.040 000 1.060 000 l.OoO 000 1.070 000 2 l.OliO 900 1.071 225 1.081 600 1.102 600 1.123 600 1.144 900 3 1.092 727 1.108 718 1.124 864 1.157 625 1.191 016 1.225 043 4 1.125 609 1.147 623 1.169 859 1.215 606 1.262 477 1.310 796 6 1.159 274 1.187 686 1.216 653 1.276 282 1.338 226 1.402 652 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 1.407 100 1.503 630 1.605 781 8 1.2C6 770 1.316 809 1.368 669 1.477 455 1.693 848 1.718 186 9 1.304 773 1.362 897 1.423 312 1.551 328 1.C89 479 1.838 45a 10 1.343 916 1.410 599 1.480 244 1.628 895 1.790 848 1.967 161 11 1.384 234 1.459 970 1.539 454 1.710 339 1.898 299 2.104 852 12 1.425 761 1.511 069 1.601 032 1.795 856 2.012 196 2.252 192 13 1.468 534 1.563 956 1.665 074 1.885 649 2.132 928 2.409 845 14 1.512 590 1.618 694 1.731 676 1.979 932 2.260 904 2.578 634 15 1.557 967 1.675 349 1.800 944 2.078 928 2.396 658 2.759 032 16 1.604 706 1.733 986 1.872 981 2.182 875 2.640 352 2.952 164 17 1.652 848 1.794 675 1.047 901 2.292 018 2.692 773 3.158 815 18 1.702 433 1.857 489 2.025 817 2.406 619 2.8.J4 339 3.379 932 19 1.753 506 1.922 501 2.106 849 2.626 950 3.025 600 3.616 626 20 1.806 111 1.989 789 2.191 123 2.653 298 3.207 135 3.869 684 21 1.860 295 2.059 431 2.278 768 2.785 963 3.399 664 4.140 562 22 1.916 103 2.131 612 2.369 919 2.925 261 3.603 637 4.430 402 23 1.973 587 2.206 114 2.464 716 3.071 524 3.819 750 4.740 630 24 2.032 794 2.283 328 2.563 304 3.225 100 4.048 935 6.072 367 25 2.093 778 2.363 215 2.665 836 3.386 355 4.291 871 5.427 433 26 2.156 591 2.445 959 2.772 470 3.555 673 4.549 383 5.807 353 27 2.221 289 2.631 567 2.883 369 3.733 456 4.822 346 6.213 868 28 2.287 928 2.620 177 2.998 703 3.920 129 5.111 687 6.648 838 29 2.356 566 2.711 878 3.118 651 4.116 136 5.418 388 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 6.088 101 8.145 113 32 2.575 083 3.006 708 3.508 059 4.764 941 6.453 387 8.715 271 33 2.652 335 3.111 942 3.648 381 5.003 189 6.840 690 9.325 340 34 2.731 905 3.220 860 3.794 316 5.253 348 7.251 025 9.978 114 35 2.813 862 3.333 690 3.946 089 5.516 015 7.686 087 10.676 581 36 2.890 278 3.450 266 4.103 933 6.791 816 8.147 252 11.423 942 37 2.985 227 3.671 025 4.268 090 6.081 407 8.636 087 12.223 618 38 3.074 783 3.696 Oil 4.438 813 6.385 477 9.154 252 13.079 271 39 3.167 027 3.825 372 4.616 366 6.704 751 9.703 607 13.994 820 40 3.262 038 3.959 260 4.801 021 7.039 989 10.285 718 14.974 458 41 3.359 899 4.097 834 4.993 061 7.391 988 10.902 861 16.022 670 42 3.460 696 4.241 258 5.192 784 7.761 688 11.557 033 17.144 257 43 3.564 617 4.389 702 6.400 495 8.149 667 12.250 455 18.344 355 44 3.671 452 4.643 342 6.616 616 8.557 150 12.985 482 19.628 460 45 3.781 696 4.702 368 6.841 176 8.985 003 13.764 611 21.002 462 46 3.895 044 4.866 941 6.074 823 9.434 258 14.590 487 22.472 623 47 4.011 895 5.037 284 6.317 816 9.905 971 15.465 917 24.045 707 48 4.132 262 5.213 689 6.670 528 10.401 270 16.393 872 25.728 907 49 4.256 219 6.396 066 6.833 349 10.921 333 17.377 604 27.529 930 60 4.383 906 6.684 927 7.106 683 11.467 400 18.420 154 29.467 025 ;m 'I ii'i m Noa.->lf each of the namberB In the table bo dimiaished by 1, the reouinder will denot* the interoat of $1, inatead of ita amount. 170 ABITHMBTia ISXBROISIB. 2. What is the oompoimd intorost on $75, for 3 yoars, at 7 p«r coDt. ? Ans. $10.87. 3. What will $50 amount to io 3 years, at 6 per cent., compound interest ? Ans. $59.55. 4. What is the compound interest on $600, for 2 years, at G per cent., payable half-yearly ? Ans. $75.31. 6. What will $320 amount to in 2J- years, at 7 per cent, com- pound interest ? Ans. $379.19. C. What is the compound interest of $150, for 3 years, at 9 per cent. ? Ans. $M.25. 7. What is the compound interest on $1,000, for 2 years, at 3|f per cent, payable qunrterly ? Ans. $72.18. 8. What will $460 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 $1860, for 8 yc; at 7 per cent.? Ans. ^1335.83. 10. What will be the compound interest on $75.20, for 20 years^ at 3J j^r 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 ? Ans. $173.03. 13. If a note of $60.60, dated October 25th, 1856, with tho interest payable yearly, at 6 per cent., be paid October 25th, 1860; what will it amount to at compound interest ? Ans. $76.51. 14. What remains due on the following note, 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., or vrder, one thousand dollars on demand, icith interest at 7 per cent, 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 rBESENT WOIITH. 171 Nov. 11, 18(J1, Juno 5, 1862, u «( 30 50 Ans. $1022.34. DISCOUNT AND PRESENT WORTH. Discount boing of the same nature as interest, i.s, strictly speak- ing, the UHc of money bofuro it is due. Tho term is applied, however, to a deduction of .so mucli per cent, from the fucc of a bill, or the dcductin,!^ of interest from ;i note before any interest lias accrued. This i.s tho practice followed in our Bank-s, and is therefore called Bank discount, in order to distinguish it from true discount. The method of computini; bank discount differs in no way from that of computing simple interest, but the method of finding true discount is quite difforeut, e. g., a debt of $107, duo one year hence, is consider'! to bo worth $100 now, for tlie reason that $100 lot out at interest now, at 7 per cent., would amount to 8107 at the end of a year. In calculating interest, the sum on which interest is to be paid is known, hut in computing discount we have to find what sum must be placed at interest so that that sum, toc!;ether with its interest, will amount to tho BANKINO. m except at the Bank and its branches, for all suras above £5, as a great improvement. Savings Banks have been in use in Europe over fifty year?, and in Canntia and the United States, almost as long. They aie established for tlie purpose of receiving from people in moderate ciroumstanccs, small sums of money on interest. In England the deposits are held by the Government, and invested in the three per cent, funds. In Mew England, New York and other States, the deposits are generally loaned on bond and mortgage tii six or seven per cent, interest. Ft'imdbj Societies. — Friendly Societies are a.ssoci!ilions, mostly in England, of persons chiefly in the humblest classes for the pur- pose of making provision by mutual contribution against those '.'jn- tingencies in hum:m life, the occurrence of which can be calculated by way of average. Tlic principal objects contemplated by such societies are the following : The insurance of a sum of money to be paid on the birth of a member's child, or on the death of a member or any of his family; the maintenance of members in old age and widowhood ; the administration of relief to members incapacitated for labor by sickness or accident ; and the endowment of members or their nominees. Friendly Societies are, therefore, associations for mutual assurance, but arc distingushed from assurance societies, properly so called, by the circumstance that the sums of money which they insure are comparatively small. BANK DISCOUNT. The Bank Discount of a note is the simple interest tu the sum for which it is given from the time it is discovntcd to i\\<^ time it becomes due, including three days of grace. Suppose, for example, in getting a note of $200 discounted at a bank I am charged 812 for discount, which being deducted, I 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. As bank discount is the saoae as interest, we derive the following 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 he the BANK UISOOUNT. Subtract the discount from the/ace of the note, and the remain der will he the PROCEEDS OR PRESENT WORTH. Ik' \\ 'in:? ■ 'A "'-til iill d fin I m 1 I 1 176 ARITHMETia EXER OISES 1 . What is the bank discount on a note, given for 60 days, for $350, at per cent.?* Ans. $3,62. 2. What is the bank discount on a note of $495, for 2 months, at 5 per cent. ? Ans. 4.33. 3. What is the present value ci a note of $7840 discountod at a bank for 4 months and 15 days, at o per cent. ? Ans. $7059 68. 4. How much money should bo 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, 1866, 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. $2381.02. 7. A merchant -sold 240 bales of cotton, each weighing 280 pounds, for 12J cents per pound, which cost him, the same day, 10 cents per pound ; he 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 Clemes, Rice & Co., to the amount of $327.40 dated April 11th, 1806, having six months to run after date, and drawing interest at the rate of 6 per cent, per annum. What are the proceeds if discounted at the Girard Bank on the 10th of August, at 7 /„ per cent. ? Ans. $332.86. Note. When a note drawing interest, is discounted 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 face of the note, and this 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. Ans. $4.34. * Throughout all the exercises, unless otherwise specified, the year is to be considered as consisting of 3G5 days. Since it is cuHtomaiy in business when a fraction of a cent occurs in and result to reject it. if less than half a cent, and f not less, to call it a cent, we have adopted this principal through- out thebcoV- RANK DISCOUNT. 177 k34. $527.-,V{ji Oberlin, Oct. 4, .866. Ninety aays after date for value rcceicnl, icc promist to pay to the order of Smith, Warren,ii' Co., Jivr humlrvd twenty- seven and ,"„'^, dollars at the City Bank, Obrrlin, with iiit'test al eight per neiit. Thompson k BuiiNs. 10. AVhrit will be the discount at 7,-'^, per cent, on ii note for $227.41, drawing interest ut 8 per cent., dated iMay l.st, 1SG5, at 1 year after date, if discounted on March 7th, i8(JG? An,«. .$2.romise to pay James Ames, or order, seven hundred and forty-six ^\\^ dollars, at the Commercial Bank, Adrian, with interest at 7 ,-'y per cent, per annum. . "Wilson & Cummixgs. 13. A holds a note against B to the amount of lj.478.02, dated May 10th, 1805 at 1 year after date drawing 7, •'„ per cent, interest. I purchase this note from A. on August 18th, paying ibr it huch a sum that will allow nic 20 per cent, interest on my money. What shall I pay for it? Ans. 8 148.33. 14. 1 got my note for §2000 discounted at a bank. May 20, 1862, for 2 months, and immediately invesled the sum received in flour June 7, 18G2,' I sold half the flour at 10 per cent. lcs.s than cost, and put the money on interest at percent. August 13, 1862, 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 T sold the cloth at $1.1 6 j per yard, receiving half the pay in cash, which 1 lent oa interest at 7^ per cent, and a note for the other half, to be on inter- N' k-- ;' 1 iiir ■ ■ ■ 1m' !;:■■ •kM.; IS ir 178 Ajm^nMETic est frnm October 4, 1802, at J per cent. When my note at tlio biiiilc became duo 1 renewed it ibr 5 months, and when this noto biciuue due I renewed it for 2 months, and when this note became due T V'.newed it ibr such a time that it becunie duo July 20, 18(].'», at wl.K'h tin#r T collected the amount due me, and paid my note at tlic bank, llequlred the loss or gain by the transaction. , , , , Ans. #29!).i,l It i.s !-omotimcs nL(!cssary to know tlie amount for which a noto must be <;iveii, in order that it hliail produce a given sum when dis- counted at a bank. E X A M P L F, . 1. Suppose wc require to obtain lj'2:]6.22 from a bank, and Hint we. are to give our noto, due in two months ; for what amount must wo draw the note, supposing thai, money is worth 9 per cent. ? SOL i: T ION. Vvom tlio nature of tliis example wo can readily poroei'/^ jiut on the. i'ace ol' the note, that wben dis- ciiutited the jivooeeds wiil \h- exactly ^2o(.).22. If we were iii ( ikdtl iii'.c (l(>.22 wv- would have to put as iii().22, which is 240. Therefore, we must put $240 on the lace of a I'.ole due at the end of two montlis to produce $230.22 when discounted at a bank at 9 per cent. From this wo deduce ilio following n i: L E . .Dc(7Hrf ih<: hanJc (JiscoiDit on 'ii<^ . Jo)' the giroi tivic and rate, from ^'i, atal duldr the drsirrd amount Inj the remainder. The quotient v:Ul he the face of the. vote required. 2. For what sum must a note be given, having 4 months to run, that .^' ili ]iroduce $19r)0, if discounted at a bank at 7 per cent. ? A:is. $1997.78. 0. 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? Ans. S'k>. i'l B.VNK DISCOUNT. 179 '(, I ill 4-. Suppose your note for months is discounted at a bank nt 6 per cont., and $48-1.75 jr fi months, ut 6 j)er cent. ? Ans. .^G18.S.S\ p. A fiirmer boui^ht a farm for $5000 cas^h, and having only one- half (if [jlc Slim on haiui. he wishes to obtain the balance from the baii|c. For what sum must "he give his note, to be discounted for 9 mouljiH, iii II put f!L»((f. ? Ans. $2619.17. 7. II' II uicrcliaut wit,hcs (o obtain $550 of a bank, for what sum must he give his note, ptiyablc in GO days, allowing it to be dis- counted at ^ per cent, per month ? Ans. $555.75. 8. 1 sold A. Mills, merchandize valued at $91S.1G, for which he was to pay uic cash, but being dinappuintcd in receiving money ex- pected, he gave me his endorsed uote at 00 days, for such an amount (Jiat when discounted at the bank at 7 per cent, it would produce the price of the raercliandize. AVliat was tiic face of the note ? Ans. $!J.'J4.82. 9. I am owing T?. Harrington on accouut, now due, $1G8.45 ; liC also holds a note against mo for 8210, duo in 34 days, including 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 will be sufficient if disccmntud at G per cent., to produce the amount of account and note. VV hat will be the face ol new note ? Ans. $380. l\'l, 5. What rate of interest is paid, when a note of $200, payable in 70 days, is discounted at f per cent, a month ? Ans, ^f,y^- 6. When a note of $45, payable in G5 days, is discounted at 7 per cent., to what rate of interest does the bank discount correspond? iins. .i)t to purchase ibr him £534 4s. Od. wort]\ of li i H m 1 1 I:,; 1 i 182 ABixmrnTio. '' I: ■ bacon and hams, and offers him ^\ per cent. ; what does the agent get ? Ans. iE38 I4s. 7d. 9. A book apicnt in Cincinatti, sells $487.50 worth of books for Day & Co., of Montreal, and receives $72.05 for his trouble; at what rate per cent, was he paid ? Ans. 15 nearly. 10. An agent sells 84 sewing machines at $25 each, and his commission amounts to $262.G0 ; what is the rate ? Ans. 12^. 1 When a sum has to be sent to a commission agent, such that it will bo equal botli to the sum to bo invested, and the agent's com- mission, it is plain, as already noted, that this is merely a case of percentage. It is the same as the first part of case IV., and wc will have the corresponding RT LE. Divide the given amount hy 1, increased hi/ tJic given rate per unit, and the quotient will be the sum to be invested ; subtract this from the given amount, and the remainder will be the commission. EXAMPLE. If I send $1890 to a commission merchant, and instruct him to buy merchandise with what is left after his commission at 5 per cent, is deducted ; what will be the sum invested, awd the agent's com- niission ? SOLUTION. It is plain that for every dollar of the proposed investment I must remit 105 cents, 100 towards the investment, and 5 towards the commission, and hence the number of dollars which can be In- vested from the sum remitted will be the same as \\\o number oE times that 1.05 is contained in 1800. Mow, $l890-:4.05 gives $1800, the sum to be invested, and tliifl subtraettid from $1890, loaves $90, the commisBlou to which the agent is entitled. KX WHO IH Hh. 1. lUMulltud to A. U., St. I'lmb, ^088 to purchase flour for me with the balanoti llmt ii'.umliis uitcr deduellug his commission at 4 pov cent. J required the purchase raon^y and percentage ? Ans. $950 and $;J8. 2. Heeeived a commission to buy wheat with $779, less by my commission at 2J per cent. ; re (ulred the price of the wheat and my oommission. Ans. $700, and $11). BKOICEliAGi:. 183 8. Ilrmittcd to my cont.^iM/.ulcnt to Au^'usta S2(»r».70, to pay for luinlx!!' which ho purchased for iiii-, and to pay liis own ((ttiiiiiis- tihn at 4 per cent. ; what was tho price of the hiiubir. Jintl what tiio (onuiiissioti ? ' Ans. liJ25l».Jil», and 610.2(1. 4. John -Tones, Ncwniarkot, cop^niis.sions W. Orr, Portland, to procure for liim a (juantlty tif liiu! liour, and remits !i:!»17.00 shares of Krio 11. 11. stock at G4 or SG per cent, dis., " hcllcr 30," now it is not at all probable that A has this stock on hand that he has sold to 13, but expects to bo able to purchase it before thu expiration of the 30 days, at something less than M. This is called celling "short." It not unfrcquently occurs, in transaclions such as just mentioned, that A may not mtih, or B require, the delivery of tho stock when tho time arwvcs; if tliis is the cafe, A simply pays, or receives from, B the differenc* between what the stock was sold at, and what it is worth at tho time of settlement. In purchases liko the above, unless the parties are known to bo reliable men, a certain amount of money must bo put tip, termed u "margin," that may bo considered suflficicnt to cover flueluations in the value of the stock. If either party is unable to meet his part of Hie contract, tlie term " Lame Duck" is applied to him. Any person may buy or sell stock through a broker at " buyer 30" or " seller 30" by putting up what the broker may consider to bo a sufficient " margin." Interest is generally allowed on this margin. Where there are a number of brokers operating in any one stock, they are, according to brokers' phrases, divided into two classes, called " Bulls" and '• Bears." The " Bear" is always tho seller, while the buyer is always a '• Bull." If A sells to B stock at 97, " seller 30," he is evidently a " Bear," as it is to his interest to constantly bear down the price of the stock he has sold, so that he may be able to purchase at a price less than 97, while it is always the interest of the buyer to "Bull" or raise its value. The par value of stock in the following examples is considered to be $100, and tho brokerage ^ per cent., unless otherwise mentioned^ Some of the answers requested may not belong, legitimately, to ques- tions in brokerage ; but the teacher or learner may ask or give but •ill ^i: 1 1.\ IMAGE EVALUATION TEST TARGET (MT-3) & iLo .5?5^ W v.. 1.0 I.I l^|Z8 |2.5 ■50 "^^^ M^^B ■ii Ki 12.2 £ Ki I •9, valued st $2760, the rate being 8 per cent., and the bro- ker'.^ charge \\ per cent, ; how much had he to pay ? The cost of insurance is $2760X •08=$220.80, and the brokerage $41.40, which Added to $220.80, will give $262.20, the answer. EXERCISES. What will be the premium of insurance on goods worth $1280, at 5J per cent. ? Ans. $70.40. 2. A ship and cargo, valued at $85,000, is insured at 2\ 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 2^ 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. T( at 4 $100^ requii This * It is plain that the rate can be found, if the amount and premium are given, and the amount can be found if the rate and preniUiin are given. In the case of insuring property, a pvofessional surveyor is often employed to value it, and likewise in the case of life insurance, a medical certificate i» veqniredj and hveaeh case the fee must be paid by the person insured. A» 100, the basis of percentage, is a constant quantity, when any tvro of thv ether quantities are given, the third can be foond. m INSUBANOB. 198 5. What must I pay to insure a house rained at $898.50, at } per cent. ? ^ Ans. $6.74. 6. A village store was valued at $1130 ; the proprietor insured it for six years ; the rate for the first year was 3^ per cent., with ft reduction of ^ each succeeding year ; the stock maintained an aver- age value of $1568, and was insured each of the six years, at 2^ per eeut. ; how much did the proprietor pay for insurance during the six years ? Ansv $397.53. 7. A store and yard were valued at $1280, and insured at 1^ ^r cent. ; 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 B. Tomlinson, Toronto, to insure for him a building valued at $976 ; R. 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 *ho 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 bo insured for, so that in case 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. ■:-i>=:=$625, the sum required. Taking the rate per unit we find ^^^Q^=:-^^\i=^ . 96. This gives the » RULE. Divide the value of the pfoperhj by 1, diminished hy the rate per unity and the quotient will be tJie sum required. EXAMPLEJ. 1. A foundry is valued at $874 : for what sum at 8 per cent, most it be insured to secure both the value of the property and the prequum ? One mimia the rate or 1.00— .08=.92, and $874-j-.92 =$950, the answer. i: ') ! I Si ; 9 * 194 AiirnDtL.''UT. *• The promises of a gunsmith, who sells gunpowder, arc valued at 112618.85: for how much, at 15 per cent., must they bo 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 remain- dor is .85 and $2618.85-i-.85 gives $3081, the sum required. ,■.•■-••' , ■ EXERCISES , ' 1. A chemist's laboratory and appurtenances are valuea ac $26,250, for what sum should ho insure them at 6^ per cent., to .secure both property and premium ? $28,000. 2. A New York merchan. sent goods worth $1,186 by water conveyance to Chicago ; ho insured them from New York to Buffalo at 1 J per cent., and from Buffalo to Chicago at 2^ per cent., and in both cases so as to secure the premium as well as the cargo; how much did the insurance cost him ? Ans. $45.42. 3. A person owned a flour mill, valued at $1846.05, which he in^ul'ed at 1^ per cent. He also owned a flax mill, valued at $846.30, which ho insured at 2^ per cent., and in both eases at such n ibum as to secure both property and premium. Which cost him most, and how much more ? Ans. The .flour mill cost him $1.67 more than the other. 4. Collins & Co., of Philadelphia, ordered a quantity of pork from G. S. Coates & Son, Cincinnati, which amounts to $2423.10. They insure it to Pittsburg at ^ per cent., and from PittsbuVg to Philadelphia at 3 per cent., and in all cases so as to secure both the price and premium. How much does the whole insurance come to ? .•...,».,.,■-■:.:.. .■ .-u. .T•.^-.v -^iv^. ■■..;>-. ;i AhH. $87.12. 5. In order to secure both the value of goods shipped and the premium, at 1^ per cent., an insurj^ncc is effected on $1526.72. What is the value of the goods ? Ans. $1500.00. 6. The Mechanics' Institute is valued at $18,000 ; it is insured . at IJ per cent., so that in ease of fire, the property and premium may both be recovered. For how much is it insured ? -:■ • / :■:■-."•'- '-./::'-■ ^r--'-' ■■^•-v;-^: -rw k ■ Ans. $18,227.85. 7. How much must be insured on a cargo worth $40,000, at J per cent., to secure both the value of the cargo and the cost of insurance? ' Ans. $40,201.00. LIFE INSURANCE. 195 8. Tho RosBin House, King-street, Toronto, is valued at, say, $160,000, and is insured at l.jj per cent, so that in case of another oonfln'gralion, both tho value of the property and the premium of insurance may be recovered. For how much must it bo insured ? Ans. $152,(571.70, nearly. 9. A jail and court-house, adjoining chemical works, and there- fore deemed hazardous, will not bo insured under 2^^ per cent. How much will secure both property and premium, the valuation being .$17,550,00 ? ^ Ans. $18,000.00. •' 10. A cotton mill is insured for $12,000, at 4 per cent., to secure both premium and property. What is the value of tho property ? Ans. $11538.46. 11. What sum must bo insured on a vessel and cargo valued at , $40,000, at 5^ per cent., in order to secure both the premium and property ? Ans. $42,328.04. 12. How much must be insured on property worth $70,000, at 4^ per cent., to secure both premium and property, a commission of f per cent, having been charged ? Ans. $73,848.17, 1' :M L'U m I' LIFE INSURANCE. A Life Insurance may be eflFected either for a term of years or for the whole period of life. The former is called a Temporary Insurance f and binds tho insurer to pay the amount to the legal heir or legatee or creditor, if the insured should die within the specified time. The latter is called a Life Insurance, because it is demandable at death, no matter how long the insured may live. The rate per annum that the insured is to pay is reckoned from tables constructed on a calcutation of the average duration of life beyond different ages. This calculation is made from statistical re^rns called Bills op Moutality, and the result is called The Expectation op Lipe. The annual preipium is fixed at such a rate as would, at the end of the expectation of li^'e, amount to the sum insured. From tables of the expectation of life other tables are constructed, show- ing the premium on $100 for one year, calculated on tho supposi- tion that it is^to bo paid annually in advance. I ' ■ Hi ii if hi ft HI ' ' ' I lt>'3 AWTHMETia LIFK INBITRANOI T A B L Jl A|{n tt«xt BlrtbilMy. 1 yi>ar. •* 7 yiMii 1 VoT Utt. Ago next nirth(l;iy. 1 yfur. 7 ynoni. For Lir'f 200 ABITHMEIIO. iince the gain ib 8 per cent, of the eost, it follows that oaeh dollar laid out has brought in a return of $1.08, and therefore the eost must have been as many dollars as the number of times that 1.08 is contained in 127.44, which is 118, and therefore the first cost must have been $118. * - , " :^ ; ■ ""^ BXEBCIBES. 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 e&oh on an average ? $6.25. 3. If 13 sheep be 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 16f per cent, be 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, and 10 per cent, gained, for what sum was it bought ? Ans. $3.64, nearly. u"" CASE IV. To find the gain or loss per cent, when the first cost and soiling price are known. .';,:. ■''... (. . ■ ^ BULB. ^ Divide the gain or lou by the fint oott. ' EXAMPLE. '■ i If a web of linen be bought for $20 and sold for $25, what is the gain per cent? * Here $5 are gained on $20, and $20 is I of $100, therefore $25 will be gained on $100, t. e;, 25 per cent. • . iV. vt EXERCISES. .-!-■!' I ■fV-.^- 1. If a quantity of goods be bought for'$318.60, 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 seUs it at 30 cento per lb., what does he gaia per cent ? Ans. 25. m FBOFIT AND LOSS. 201 j'i:? 4. If a oattlfl dealer buys 20 cows, at an average pric* of $20, and pays 50 cents for the freight of each per railroad, what per eent. does he gain by selling them at $25.62} each ? Ans. 25. b. A tobacconist bought a quantity 4)f tobacco for $75, but a part of it being h^st, he sold the remainder for $60 : what per cent, did he lose 7 Ans. 20. A B E V . ♦ Given the gain or loss per cent, resulting from the sale of goods at one price, to find the gain or loss per <^nt. by selling the same a(i anot^ price. v ' « BULK. t Find hy case in. the first cost, and then hy case IV. the gain or lass per cent, on that cost at the second selling price. > ■ . . EXAMPLE. If a farmer sells his hogs 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 gain per cent, would be on the second supposition, that is $3-h4=.75, or 75 per cent. EXERCISES. 1. If a grocer sells rum at 90 cents per bottle, vcui gains 20 pjier eent. ; what per cent, would he gain by selling it at $1.00 per bottle? Ans 33|. 2. If a hatter sells hats at $1.25 each, and loses 25 per cent. ; whal 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 loRt 25 per eent. ; wquld she have gained or lost by selling them at $1.40 ? Ans. She would have lv)st 16 per cent. 5. A merchant sold a lot of goods for $480, and lost 20 per cent. ; would he have gained or lost b]^ 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 woi|ld bave bew the gala per cent, if it had been sold for $160? " * f : £ . ^ Ass. 60. J!? '}''} i|*j II lltr, 202 ABTTHMETIO. 7. A grocer sold tea at 45 cents per pound, and thereby guned 12]^ per cent. ; what would he have gamed per cent, if he had sold the tea at 54 cents per pound ? Ans. 35. 8. A fanner sold com at 65 cents per bushel, and gained 5 per cent. ; what per cent, would he have gained if he had sold the com at 70 cents per bushel ? Ans. 13j^. !i' MISCELLANEOUS EXBBOISBS. 1. If I buy goods amounting to $465, and nell them at a gain of 15 per cent. ; what arc my profits ? Ans. $69.75. 2. Suppose I buy 400^ barrels of flour, at $16.75 a barrel, and Bell it at an advance >f | per cent. ; how much do I gain? Ans. $25.14. 3. If I buy 220 bushels of wheat, at $1.15 per bushel, and wish to gain 15 per cent, in soiling it ; what must I ask a bushel. $1.32^. 4. A grocer bought molasses for 24 cents a gallon, which he sold for 30 cents ; what was his gain per cent. ? Ans. 25. 5. A man bought a horse 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. 12|. 6. A gentleman sold a horse for $180, and thereby gained 20 per cent. ; how much did the horse cost him ? Ans. $150, f 7. In one year the principal and interest* of a certain note amounted to $810, at 8 per cent. ; what was the face of the note ? Ans. $750, 8. A carpenter built a house for $990, which was 10 per cent, less than what it was worth ; how much should he have received for U so as to have made 40 per cent. 7 Ans. $1040. 9. A broker bought stocks at $96 per share, and sold them at $102 per share ; what was his gain per cent. ? Ans. 6^» 10. A merchant sold sugar at 6^ cents a pound, which was 10 per cent, less than it cost him -, what was the cost price ? . Ans. 7| cents per pound. 11. A merchant sold broadcloth at $4.75 per yard, and gained 12^ per cent. ; what would he llave gained per cent, if he had sold it at $5.25 per yard ? Ans. 24||. 12. I sold a horsJB for $75, and by so doing, I lost 25 per cent. ; whereas, I ought to have gained 30 per cent. ; how much was he sold for under his real value? Ans. $55. m PROFIT AND LOSS. ' 208 13. A watch which cost mo $30 I have sold fbr $35, on a credit of 8 months ; what did I gain bj my bargain, allowing money to bo worth per cent.? . , . Akj $3.65. 14. Bought 84 yai-dfl of broadcloth, at $5.06 per yard ; what must be my asking price in order to fall 10 per cent., ^nd still make 10 per cent, on the cost ? Ans. $6.11^. 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^*ccnts per foot ? ' ' " ~" '■ ' " Ans. 12J^ per cent. loss. 16. What must I ask per yard for cloth that cost $3.52, so that I may fall 8 pe** cent., and still make 15 per cent., allowing 12 per cent, of sales to bo in bad debts ? ■ Ans. $5. 17. A merchant sold two bales 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 oncrfourth 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, and the gain per cent. Ans. $1612.80 and 27^ per cent. 19. A flour merchant bought the following lots : — 118 barrels at.... .$9.25 per barreL 212 « 9.50 " 315 " 9.12^ « ^ 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 discounte 1 at the Fourth National Bank, at six per cent., I found I htui gained 20 per cent, by the transaction. What was the Belling price of each aheep? • Ans.. $6.19. '- . - , •. " ., - ' -. '' " . ■ ■ >y ■'■- : -.'. 'i.:.;:. ,,:■ r^mxr':' ''■'^'■^ y .-■•-■■'■'■. I I 204 ABriHMEno. ■r," STORAGE. ; ■-■- -,- ;-v,-^^ Whan ft charge is njtde for the aooommodation of having goodi kept in store, it is called storage. Accounts of storage contain the entries showing when the goods were l^eceivcd 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 for storage, iii;}iioh is generally determined by an averago reckoned for some specified time, usually one month (30 EZAMPLX8. 1. What will be the cost of storing wheat at 3 cents per bushel . per month, which was received and delivered as Mows : — Received, August 3rd> 18G5, 800 bushels ; August 12th, 600 bushels. De- livered, August 9th, 250 bushels ; September 12th, 350 bushels ; September 15tb, 400 buuheb, and Ootolor Ist^ the balance. SOLUTION. . i 18S5. Bush. Days. Bnah. August 3. Beceived 80C ' X 6 = 4800 in store for one day. « 9. Delivered 250 Balance 550 X 3 = 1650 in store for one day. " 12. Beceived 600 Balance 1150 X 31 =35660 b store for one day. Sept. 12. Delivered 350 - , _ /, ^ ■ '^ Balance 800 X 3 = 2400 in ato^ for one day. " 15. Delivered 400 * :^ ? Balance 400 X 16 = 6400 in store for one day. OotI 1. Delivered 400 Total....: :..... 50900 in store for one day. 50,900 bushels ita store for one day would be the same as B0900-^30=1696f busheb in store for one month of 30 dayt, and the storage of J. 697 bushels for one month, at 3 cents per month, would equal 1697X.03=$50.91. It u customary, in business, when the number of .articles upon wbioh storage is to be charged, as found, contains a fraction leu stohaoe. 205 than a half, to reject the fraction ; but it* it is more than a haJ/, to regard it &s an cntiro article. From the solution of the foregoing example, yre deduce the foU lowing Multiply the number of hunJicls, harrch, vr other articles, by the number of dai/n they (ire. in store, and divide the sum of the pro- Juct.t by .*>(>, or the number of days in any term agreed upon. The quotient will give the number of bushels, barrels, or other articles on which storage is to he. charged for that term, «. 2. What will be the cost of storing salt at 3 ,..- . , . --■ i * ' ■ ' ' ' . • " ';■"'-''." V - ; SOLUTION. • , Loss for general henefit. Contributory tnterats. ' ' Cargo thrown overboard,$3498 Value of steamer i $100,000 .Repairs to steamer less J 584 Invoice price of cargo.... 41,050 Expenses in port 253 Fr'ght, less seamen's wages 3,460" Total loss $4,335 Total contrib. int.... $144,500 $4835-v-144,500=:.03 loss per unit, or 3 per cent. ,y j;;^ «,v $100,000 X.03=$3000.00, steamer's share of loss. ;. , 7,480 X -03== 224.40, T.A.Collins' share of loss. ( ;, ; . 5,365 X. 03=:. 160.95, R. Evan & Co.'s share of loss. •; .,. ' 9,218 X. 03= 276.54, H. C. Wright's share of loss. 11,428X.03= 342.84, W. Manning & Co.'s share of loss. 7,559 X. 03= 220.77, E. Carpenter's share of loss. •.',;,. 3,450 X .03= 103,50, Freight's share of loss. $3000.00- 1123.40- 2374.60- $4335.00, Total loss. • ' ' *^ -837.00=$2163.00, balance payable by steamer. ' , -160.95^ $962.45, balance receivable by R. Evans & Co. -226.77= 2147.83, balance rcocivablc by E. Carpenter. , NoTK.— It is evident tfaat since the steamur lost $8:!7 ($581 by repairs, and $253 by uxpenses), — that tliu net amount required I'rom the steamer will be $3000— 837=$2163. R. Evans & Co. having lost by merchandize being thrown overboard $1123.46, a sum greater than their share of tho general loss, 80 that there must be duo them $1123.40 -160.95=$962.45 ; m also the amount of £. Carpenter's share of the general loss must be deducted from bis iadlTidual loss in order to find the balance due him. 4 ■'0 m in 208 ABirmcEno. R n L 1 . Ji» Find the rate per unit of lo»$, hy which multiply the value 0/ saih contributory intereit, and the product will he the §har& of \ou to be iuatained by each. KXBBOI8I8. ( <; 1. Tho steamship Ocean Queen on her trip finom PHiladelphia to Liverpool, wns crippled in a storm, in consequence of which the captain had to throw overboard a portion of the cargo, amounting in vnluo to $4465.50, and tho necessary repairs of tho vessel cost $423. Tho contributory interests were as follows: — Vessel, $30,000; gross freight, $6225 ; oar^o shipped by J. Jones & Co., $3650 ; by Henry Anderson, $0500 ; by George Millan, $2000 ; by J. Foster & Son, $550 ; by Brown Brothers, $5450 ; and by Wilson &, Carter, $8500. Of tho cargo thrown overboard, thcro belonged to Henry Anderson tho value of $3000, and to Brown Brothers tho remainder, $1465.50. The cost of detention in port, in consequence of repairs, was $116.50; seaman's wages, $2075. How ought the loss to bo shared among the contributory interests ? Ans. 8 per cent. 2. .The steamer Persia left Boston for Halifax, June 30th, loaded with 7210 buchels of spring wheat, shipped by J. M. Mus- grove, and invoiced at 95 cents per bushel; 4815 bushels of corn, shipped by Thomas A Bryco & Co., and invoiced at 60 cents per bushel ; 2180 barrels of flour, shipped by A. B. Smith & Co., and invoiced at $5.50 per barrel. When near Halifax, the steamer collided with the Bay State, and the captain found it necessary to throw overboard 1600 bushels of wheat, 1280 bushels of corn, and 720 barrels of flour. On estimating the proportionate loss, it was allowed that the wheat would have sold in Halifax at an advance of 10 per cent., the corn at an advance of 15 per cent., and tho flour for $5 per barrel. The contributory interests were: — Steamer, $95,000; cargo, $ ; gross freight, ^2361. 20. The cost of fcpairs to steamer was $2198.15; cost arising from detention during repairs, $318; seamen's wages, $1252.50. How much of the loss had each contributory interest to bear ? 3. Tho eteamer Edith left Baltimore for New Orleans with Y600 bushels of wheat, valued at $1.25 per bushel, shipped by Dunn, Lloyd & Co., and insured in the Hartford Insurance Company at If per cent., 9200 bushels of com, valued at* 75 cents per bushel, 4 TAXES AND CUSTOM DUTIES. too ihipped by J. W. Roo, and insured in tho JRtna, Insurance Company at 1'^ per cent.; 14,800 busheln of oatii, valued at .'}7^ conts per bnahel, shipped by Morriit, Wright & Co., and insured in the Mutual Insurance Company at 1^ per cent. ; 1,800 barrels o? flour, valued tt $6.25 per barrel, shipped by Smith & Worth, and imiurcd in the Beaver Insurance Company at l^ per cent. In consequence of a violent gale in the Gulf of Mexico, it was found ncoesaary to throw overboard the flour, 4,000 bushels of oats, and 3,100 bushels oi wheat. Tho propeller was valued at $45,000, and insured in the Beaver Insurance Company for $12,000, at 2 per cent., and in the Western for $25,000, at 2^ per cent. The gross freight was $4950 ; seamen's wages, $340, and repairs to tho boat, $3953.75 ; what was tho loss sustained by each of the contributory interests, the propeller being on her first trip ? TAXES AND CUSTOMS DUTIES. A tax is a money payment levied upon tho subjects of a State or tho members of any community, for the support of tho govern- mcnt. : /,v;.; v; > ■• '/^, .^^ ,. A tax is cither levied upon tho property or tho persons of indi- viduals. When levied upon the person, it is called npoll tax. It may be either direct or indirect. When direct, it is levied from the individuals, or the property in the )> mJs of tlio ultimate owners. When indirect, it is in the nature ' .' a cusioms' or excite duty, which is levied upon imports, or i iuiiuiacturci', bd'orc they reach tho consumer, although in tho end thoy aro pal J liy tho latter. Customs^ duties are paid by the iinportcr cl' gooJ- at tho port of entry, wliero a custom-house is stationed, with government employees called custom-Jiouse ojfficers, to collect these dues. Excise duties are those levied upon articles manufactured in the country. An invoice is a complete list of the particulars and prices of goods sent from one place to another. A Specific duty is a certain sum paid on a ton, hundred WAic^ht. yard; gallon, &c., without regard to the cost of the article An ad valorem duty is a percontogo levied on tho actual com,, or fair market value of the coods in the country from which th*^, v« imported. ,>^ , 210 ' ARn'HMETlC. OroM tcelffht is tho vrcight of good;*, upon which a upeoifio duty iit to be loviod, bolbro any allowunoen uru doduotod. Xet weight M tho weight of tho goods after all allowanoos aro doduottid. Among tho ullowancoH made nro tho following: Breakage — iin allowance on fluids contained in bottles or break- able vessels. Draft — thi! allowonco for waBtc. Lotkagr — an allowance for waste by leaking. Tare and (ret are the deductions lundo for tho weight of the ease or barrel which contuinH tho ;;c)ods. When goods, invoiced at gold value, upon which duty is payable, aro imported Mito this country from any foreign country, tho custom houHc duticH are payable in gold, for cIho manifest injustice might be done. Jf tho duty were payable in greenbacks, it would bo nccce sary, in order to obtain uniformity, cither to incroaso or decrease the rato pep cent, of duty, an greenbacks fluctuated in value, compared with gold (tho invoice price ol'tho goods), or cIjo tho goods imported would rciiuire to bo retJucod to their vuluo in greenbacks at time of delivery. To avoid all this troublo and confusion, goods that are invoiced at their gold value, tho duties arc uiidc payable in tho same currency. When goods aro imported from any country which has a doprc- oiated currency, a note is attached to tho invoice, certifying tho amount of depreciation. This is tho duty of tho Consul represent- ing tho country to which tho goods aro exported, and residing at the port /roM which they aro exported. EXAMPLES. ^ To find the specific duty on any quantity of goods. Suppo.so an Albany Provision Merchant imports from Ireland 59 casks of butter, each weighing G8 lbs., and that 12 lbs. tare is allo^ved un each cask, and 2 cents per lb. duty on the net weight. - We find the grcss is 5I]|XCSrr^4012 lbs. " tare is....; 59X12^- 708 ibs. Hence tI/6 net weight is 3304 lbs The duty is 2 cents per lb..... 2 '"'"' The duty, therefore, is 866.08 TAXES AND CU8T0M DUTIES. 211 To find the ad valorem dvtty od any quantity of goods. Suppoio a Troy dry y: odtt niurobant to import from Montreal 430 yards of nilk, at $1.75 per yard, and that Ii5 per oont. dnty ig chur;!;od on them. Hero wo find the wholo price by tho rulo of Practloo to bo $7C3, tlion thu rest of tho operation in a direct case of percentage, and thoroforo wo multiply $703 by .liO, which gives $207.05, the amount of duty on tliu wholo. ;,. " ., Uonoe wo have tho following >rv fill] ■ ii.' I.' RULE FOR 8PK0IFI0 DUTY. Subtract the tare, or other allowance, and multiply r/i5 remain' dtr by the rate of duty per box, yallnn, dkc. nULBFORADVALOREMDUTT. Multiply the amount of the invoice by the rate per unit, « EXEROIBES. 1. Find the specific duty on 5120 lbs. of sugar, the tare being 14 per cent., and the duty 2^ cents per lb. Ans. $121.09. 2. What is tho ad valorem duty on a quantity of silks, the amount of the invoice being $95,800, and the duty 02^ per cent ? ^ ^ ^, , w/ Ans. $59,875. 3. At 30 per cent., what is the ad valorem duty on an importa- tion of china worth $1200. ? 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 tho specific duty on 950 bags of coffee, each weighing 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, each con- taining 75 gallons, at 18 cents a gallon ? Ans. $270. 8. A. B. shipped irom Oswego 24 pipes of molasses, each con- ,* taining 90 gallons; 2 percent, was deducted for leakage, and 12 cents duty per gallon charged on the remainder ; how much was the duty? Ans. $270.95. 212 AI5ITHBIETIC. 9. Peter Smith & Co., Brooklin, import from Cadiz, 80 baskets of port wino, ut 70 francs ror basket ; 42 boskets of sherry wine, at 35 franes pcir basket ; CO casks of champagne, containing 31 gallons each, at 4 I'rancs per gallon. Tho waste of the wine in the casks was reckoned at u gallon each cask, and the allowance for breakage: 'n the baskets was 5 per cent. ; what was the duty at 30 per cent., 18jJ cents being taken as equal to 1 franc? Ans. $776.54. 10. J, John.son & 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 4.'- per yard, duty 19 cwt. ; 100 woollen blankets, at 2a. (id., duty 10 per cent. ; and shoe-lasting to the cost of £60, duty 4 per cent. Kequired the whole amount of duty, allowing the value of the pound sterling to bo 94.84. Ans. $174.24. 11. John McMaster & Co., of CoUingwood, Canada West., bought of A. M. Smith, of Buffalo, N. Y., goods invoiced at $5430.50, which should have passed through tho custom-house dur- ing, the first week in May, when the discount oa American invoices was 43|^ per cent., but they were not passed until the fourth week in May, when the discount was 36f per cent. Tho duty in both cases being 20 per cent. ; what was the loss sustained by McMaster & Co. on account of their goods being delayed ? Ans. $70.72. STOCKS AND BONDS. Capital is a term generally applied to tho property accumulated by incividuals, and invested in trade, manufactures, railroads, build- ings, government securities, banking, &c. The capital of incorpo- rated companies is generally termed its " capital stock," and is divided into shares ; the persons owning ono or more of these shares, being called stockholders. The shares in England, are usually £100. £50, or £10 each. In the United States they are generally $100, 850, or 810 each. The management of incorporated -companies is generally vested in officers and directors, as provided in the law or laws, wh.* are elected by the stockholders or shareholders ; each stockholder, in most cases, being entitled to as many votes as tho number of shares he holds ; but sometimes the holder of a few shares votes in a larger proportion than the holder of many. The a«»pumulating profits which are distributed among the stork- holders, once or twice a year, are called " dividends," and when ** declared," are a certain percentage of the par value of the shares. In mining, and some other companies, where the shares are only a STOCKS AND BOIJDS. 213 few dollars each, the dividend is usually a fixea sum "per share." Certificates of stock are issued by every company, signed by the proper ofiicers, indicating the number of shares each stockholder is entitled to, and us an evidence of ownership ; these are transferable, and may be bought ;^nd sold like any other property. When the market value equals their nomiaal value they are said to be " at par J" 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 are below par, or at a " discount." Quotations of the market value are generally made by a percentage of their par 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 largo 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 " coupons," or certificates of interest, each of which is a due bill for 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 usually cut off, and presented for payment as they become due. These bonds and coupons are signed by the proper officers, and like certificates of capital stock, are nego- tiable by delivery. The loan is obtained by the sale of the bonds, with coupons attached, but they are sometimes negotiated at par. Their market value depends unon 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 are issued by the United States Government, for the purpose of effecting temporary loans, and for the payment of vfontracts and salaries, which resemble bank notes, and are made payable without interest generally. Recently such notes have been issued bearing one year or three years' interest. "Consols" is a term abbreviated from the expression " consoli- dated," the British Government having at various times borrowed money at different rates of interest and payable at different times, " consolidated" the debt or bonds thus issued, by issuing new stock, drawing interest at three percent, per annum, payable semi-f.nnually, and redeemable only at the option of the Government, becoming practically perpetual annuities. With the proceeds of this, the old Btodc was redeemed. The quotations of these three per cent, per- petual annuities, or " consols," iodicate ordinarily the state of the :» .: r^\ ( ?;■• 214 AUITHMETIO. 1 money market, as they form a large portion of the Briti&h public debt. " Mortgage Bonds*' are frequently issued by owners of real property, with coupons attached, which renijl^r the bonds more saleable as well as more conveDient for the collection of interest. " Coupon Bonds," being negotiable by delivery, are payable to the liolder ; and in case of loss or theft, the amoiint cannot be recovered from the government or corporation issu ug them, unless ample notice is given of the loss. " Begistered Bonds" arc those payable only to the " order" of the holder or owner, and are more safe for investment. By law, stockholders are 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 the statute provides for " Limited" liability, by an Act passed in 1862 termed the "Limited Act." CASE I. ' '''''-' -■"*■■'■■• The premium or discount being known, to find the market value of any amount of stock. '••^••'- 'I.:' ■.-•■. .^' /,;■■. BXA-MPLES. ■■l-\ ,/'■ k 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 nULK. ^. ■;>'- Multipltf the par value hy 1 plut or mintu tne rate per wnitp according as the shares are at a premium or a discount STOCKS AKD B02n)S 2lSr KXKB0ISK8. .'. V 1. What is the market value of $450 stock, at 8^ per cofit/'dlfi!- oonnt ? Ans. $411.75. 2. What is the value of 29 shares of $50 each, when the shares are 11 per cent, below par ? Ans. $129p.50. '6. A man purchased 60 shares oi^ $5 each, from an oil \ireii company, when the shares wore at a discount of 8 per cent!,, and sold them when they were at a premium of 10 per cent; how much did ho gain ? Ans. $54 • 4. A man purcliascd $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 ? Ans. $105. ■ -' ) , CASE II. „, ... To find how much stock a given sum will purchase at a given premium or discount. . Let it be required to find how much stock can be purchased for $21,600 when at a 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 $21600 will be represented by the number of times that $1.08 is oontained in 21600, which gives $20000. Again : Let it be required to find how much stock can be pur- chased for $5520, when at a disc( unt of 8 per cent. When stocks are 8 per cent, below par, $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 tisies that .92 is contained in 5520, which gives $6000 stock. Hence we derive the RULE. ■•■■'"■'''' ' ■'■"'"" Divide the given sum hy 1 plu& or minua the rate per unit, accord^ tuff as the shares are at a premium or a discount. - A*' : SXEB0I3ES. 6. When stocks arc at a premium of 12 per cent., how mooh oanr be purchased for $8064 ? Ans. $7200. 216 ABUHXEITOt 7. When stocks are at a disconnt of 9 per cent., how much can he bought for $3640 ? Ans. $4000. 8. When G. T. R. stock is at 18 per cent, below par, how much can be bought for $42,640. Ans. $52000. 9. When Gt. W. B. stock is at a premium of 9 per cent., how much will $4578 purchase ? Ans. $4200. 10. When government stock is selling at 92^, what amount of stock will $28,G75 purchase, and to what will it amount with broker- age at J per cent. ? * r Ans. $31077.50. CASK III. '' ' " ■^--' '■■'*'■'-■" The premium or discount being known, to find the par value. To find tTiA par value of $1,296, when stock is at a premium of 8 per cent. At 8 per cent, premium, each $1 brings $1.08, hence the par value will be represented by the number of times 1.08 is contained in 1296, which ^ives $1200 for the par value. To find the par value of $1104, when stock is at a discount of 8 per cent. Each $1 will bring $0.92, and therefore the par value w:'l be represented by the number of times that .92 is contained in 1104, which gives $1200, the par value. Hence the ., > y .1 3^- ■ 'U..; --^ ■ - •,.- RULE. IQjivide the market value hy 1 plu» or minui the rate per untV, according as the stocks are selling above or below par. S X B R I M E S . 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- oount of 4 per cent. ? Ans. $11250. 13. When government stocks are at 6 per cent, premium ; how much w.U $20246 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 ho purchase, and how much did he gain ? . Ans. 9 shares; £108 gain. STOOES AMD IVOMDS. 217 i ' OASBIV. 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, the corresponding interest may be 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 the per cent, which was invested will be represented by the num- ber of times that 1.20 is contained in .12, which is .10 or 10 percent Hence tho , ^ , v .. Divide the rate per unit of dividend hy 1 phu or minus the rait per cent, premium or discount, according as the stocks are above or helov) par.^ -^' ".•'«•.".■;;''•,;■ ■'■^»";''' ,•',.,..■, '^', ■ EXERCISES. ;>*.•■-■*'''.:'.'', 16. If a dividend of 10 per cent, be declared on stock vested at 25 per cent, advance ; what is the 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 tho corresponding interest ? Ans. 4j^. 18. If money invested at 24 per cent, yields a dividend of 15 per cent., what is the rate of interest ? Ans. 123^. 19. If railroad stock is invested at 18 per cent, above par, and a dividt^nd of 6 per cent, bu declared, what is the rate of interest ? Ans. 5^^. 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. llif. UISCBLIiANEOUS EXEB0I8BS. 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 what price stock paying a given rate per cent dividend can be purchased, so that th<' noney invested shall produce a given rate of interest^ divide fAe rcrfe per uniA of ditulend ^y mium of 15 per cent. ? Ans. $7600. 3. Bailway stock was bought at 15f below par, for $I895.62| ; how many shares were there, each share being $150 ? Ans. 16 shares. 4. If 6 per cent, stock yields 8 per cent, on an investment, at what per "'"t. discount was it bought? Ans. 25. 5. If bank stock which pays 11 per c^int. dividend, is 10 per cent, above par, what is tho corresponding rate of interest on any investment? ■ ' V' - Ans. 10. 6. When 4 per cent, stocks were at 17J discount, A bought $1000 ; how much did he pay, and how much did he gain by selling when stock had risen to 86^ ? 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 be lost by selling out at 10^ per cent. ? Ans. $10.^3. ^^ 9. What income should I get by laying out $1620 in the pur- chase of 3 per cent, stock ut 81 ? Ans. $60. 10. What sum must be invested m 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 4^ per cent, stocks at 90 ? Ans. 5 per ccnt» 12. A person transfers his capital from the 3^ per cent, stocks at 77, to the 4 per cent, at 117^, what is the increase or decrease per cent, in his income ? Ans. Decrease 25. ~ 13. A person sells out of the 3 per cent, stock at 96, and invests his money in railway 5 per cent, stock at par ; how much per cent. is his income increased ? Ans. 60. 14. What must be the market valne of 5]^ per cent, stock, so that after deducting an income tax of 2 cents on the doUar, It may produce 5 per cent, interest ? Ans. 107|. 15. A gentleman invested $7560 in tlie 3} per cent, stocks at 94^, and on their rising to 95 sold out, and purchased G. T. B. 4 per cent, stock at par; what increase did he make in his annual income ? Ans. $24. 16. How much more may a person increase his annual income by lending $3800, at 6 per cent., than by pahi'i9siDg railway 5 per «eiit stock at 95 ? Ans. $28. STOCKS AND BONDS. 219 17. A person sells $4200 railway stock which pays C per cent, at 116, and iavcsts one-third of the proceeds in the 3 per cent. oon. 60I.4 at 80^, and the balance in saviD<;s bank stock, which pays 9 per cent, at par \ what is the decrease or increase of his annual income ? Ans. Increase $97.80. 18. A person having $10,000 consols, sella $5000 at 94|, and on their rising to 98f he sells $5000 nioro ; on their again rising ho buys back the whole at 96 ; how much docs ho 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 rescivod upon it, it was sold out, the whole increase of capita^ being $302.40; at what price was it sold out ? 1 • ■-.■.,),' Ans. 93|f. 20. Suppose a person to have been an Original subscriber for 500 shares of $50 each, in the First National 'Jank, payable by instal- ments, as follows : — J in three months, which he sold for 5J per cent, advance : § in 6 months, which brought him 0^ per cent, ad- vance, and the balance in nine months, which he was compelled to sell at 8f per cent, riisoount ; what did he gain by the whole transac- tion ? Ans. $808.33. ¥ pr f -f 21. A gentleman purchased $5000 of Fivs-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 riLte per cent, of interest on amount invested ? Ans. 7J per cent. 22. From which would be derived the greater income, Seven- thirties purchased at 104, or Five-twenties (C per cent, gold) at 109^, interest on both bonds payable at the same time, and gold, quoted at 140 ? Ans. From the Five-twenties. 23. On Jan. Ist 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 !f en-forties (interest payable in gold at G 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 could I give for the Ten-forties Ans.9&. K if gold wereoalcnlated at 20 per cient. prem. ? U i*''*.-* .'iCfc,, «*««# *%* !!i!:. ( r^ 220 AlUTHMETIO. 25. On May 2l8t, a broker purchased for me » Seven-thirty bond to the amount of $12,000 at 104| ; the interest on this bond is payable on tho Ist Feb. and August ; what does the bond cost me, the brokerage being ^ per cent. ? Ans. $12861.60. 26. After receiving the interest, on Aug. 1st, on tho bond men- tioned in last question, the broker immediately sold it for me at 103f oharging ^ per cent, for selling ; did I gain or loso by the transac- tion, and how much, money being worth 6 per cent. ? , Ans. Lost $157.94: 27. A gentleman subscribed $15,000 in a railroad company, having a paid-up capital of $^50,000 ; 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 invest- ment? Ans. 8f per cent 28. The 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 the paid- up capital is declared, and 10 per cent, on paid-up capital carried to the credit of the stockholders ; how much money does A receive as « dividend, what per cent, on subscribed capital is carried to credit of stockholders, and what has A still to pay on his stock ? Ans. A receives $75 ; to credit of stockholders 2^ per cent. A lias 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 spc oulate in buying and fielling stocks, bonds, and gold, for 60 days, and then return what the money produced, after deducting brokerage of ^ per cent, on the actual sales and purchases. B immediately purchased 200 shares of Erie R. B. stock at 59 " buyer 30," no margin required, and sold 400 shares Reading R. R. stock, at 102| >< buyer 15." Five days after, B called in the Erie R. R. stock and sold it to C at 61|^ ; May 8th he bought $20,000 of Five-twenties at 109|, at the same time the person to whom the Reading R. R. stock was sold, called it in ; B paid the difference, the stock being valued at 102^ ; May 27th, gold Ifaving appreciated in value as compared with Greenbacks, B sold the Five-twenties at 110^ cash, and at the same time made a farther sale of $15,000 in the same kind of bonds at 110^ " seller ed it in ing and •n what on the lares of ind sold ive days at6l|; he same called it i; May snbaoks, made a " seller STOCKtt AND BONDS. 221 30/' which ho was able to purchase and deliver in ton days at 109}. June 20th, B sold $20,000 in gold at 137^ "seller 10," which was not delivered at tho expiration of the ten days, but settled at 131^ ; hoyi much money is A to receive from B ? Ans. $25,475. L PARTNERSHIP, ,V:tel Partnership has been defined to bo tho rcsiilt of a contract, under which two or more persons ngroo to combine property, or labour, for the purpose of a common undertaking, and tho acquisition of a com- mon profit. A dormant, or sleeping partner, is one who shares in tho concern^ but does not appear to tho world as such. A nominal partner is one who lends his name and credit to % firm, without having any real interest in the profits. AH 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 Partner»htj>" 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 grcht deal of litigation has arisen from carelessness in this respect. These Articles of Partnership should particularly specify the amount of investment by each partner, whether the personal atten- tion of thf) partners is required to tho business, duration of partner- ship, and sometimes an agreement with regard to the vrithdrawal of money from the business. A dissolution can take place at any time by mutual consent. ' A partnership at will is one in which there is no limited time- affixed for its oontlnuanoe, and the whole firm may be dissolved M «i SI 222 ABTTHMITIO* by any of its members at a moment's notice. A dooumont is, how* ever, gonorally drawn up and signed upon a dissolution, called a tettlement, which contaius ti statement of the mode of adjustment of the accounts, and the apportionment of profits or losses. -KXAMPLK. Two persons, A. and B., enter into nartnership. A. invests $300 and B. $400. Thoy gain during one year $210; what is each man's share of the profit ? SOLUTION BT P It PC ATI N. ^ ' A.'s Stock, $300 , '■ B.'b " 400 ' ' Entire stock $700 : 300: : $210 : $90 A.'s gain. ) ;' ^ « " 700:400::$210:120 B.'s " '. :;'• ■.■••;'■ « SOLUTION BY PS BCENTAOX. 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 multiplied by .30 it will represent his share of the gain thus : $300x.30=$ 90 A'sgoin. ■ ^j^ /. 400x.30=: 120 B.'s (( v' ^ Entire stock 700 210 Entire gain. Henpe, — To find each partner's share of the profit or loss, when there is no reference to time, we have the following RULE. As the whole stock is to each partner's stock, so is the whole gain or loss to each partner's gain or loss; or, divide the whole gain or loss hy the number denoting the entire stock, and the quotient will he the gain or loss on each dollar of stock ; which multiplied hy the . number denoting each partner's share of the entire stock, will give hii share of the entire gain or loss. EXEBOISES. 1. ThMo persons, A., B., and 0., enter into partnership. A. advances $500, B. $550, and 0. $600 ; they gain by trade $412.50. What is each partner's share of the profit ? Ans. A.'s $125 ; B.'s $137.50 ; O.'s $100. PABTNEBSinP. 228 2. A, B, and D purohoso an oil well. A payg for G shares, B for 6, for 7, and D for 8. Their not profits ut tho end of three months have aniountod to $7800 ; what sum ought each to receive ? Ans. A, 81800 ; B, $1500 ; C, $2100 ; D, $2400. 3. A und B purchased u lot of land for $4500. A paid ^ of tho price, und \i tlio roniuindcr ; they gained by the sub of it 20 per cent. ] what wau each man's aharc of the profit ? Ans. A, $300^ B, $600. 4. A captain, mate, and 12 sailors, won u prize of $2240, of which tho captain took 14 sharcHi, tho mate 0, and tho remainder was cciuully divided among tho sailors ; how much did each receive ? Ans. Tho captain, $080 ; tho mate, $420 ; each sailor, $70. 5. A und B invest equal sums in trade, and clear $220, of which A is to have 8 shares on account of transacting the bu.sinuss, and B only 3 shares ; what is each man's gain, and what allowance is made A for his time ? Ans. Each n)an'8 gain $60 ; A $100 for his time. 6. A, B, C und D enter into partnership with u joint capital of $4000, of which A furnishes $1U00 ; B $800 ; C $1300, and D tho balance ; at tho end of nine months their net profits amount to $1700 ; what is each partner's share of tho gain, supposing B to re- ceive $100 for extra services ? Ans. A, $400 ; B, $320 ; C, $520 ; D, $360. 7. Six persons, A, B, C, D, E and F, enter into partnership, and gun $7000, which is to be divided among them in tho following I'Qanner: — A to have I; B, ^ ; C, J as much as A and B, and tho remainder to bo divided between D, E and F, in tho proportion of 2, 2^ and 3^ ; how much does each partner receive ? Ans. A, $1400 ; B, $1000 ; C, $800 ; D, $950 j E, $1187.50 ; F, ^1662.50. V 8. A, B and C enter into partnership with a joint stock of $30,000, of which A furnished an unknown sum; B furnished IJ, and C 1^ times as much. At the end of six months their profits were 25 per cert, of the investment ; what was eaeh man's share of tho gain ? Ans. A's, $2000 ; B's, $3000 ; and C's, $2500. 9. A, B, C and D trade in company with a joint capital of $3000 ; on dividing the profits, it is found that A's share is $120 ; B's, $255 ; 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 ccrudn field of wheat, for which they agree to take the sum of III i y '» .• 224 ABITHiaEnO. $19.84; A ind B oftlouUte that they oan do | of tho work ; A and f ; B and { of it ; how mnch ahould oooh reoeiye aooording to these ostimAtes ? Ana. A, $8.32 ; B, $7.04 ; and 0, •4.48. I, r To find each partner's share of the gain or loss, when the capital is invested for different periods. IXAMPLE. ■* Two morohants, A and B, enter into partnership. A invests $700 for 15 months, and B $800 for 12 months; they gain 1603 ; what is each man's shore of tlie profits ? ^ SOLUTION. $700X15==$10500 $800X12= 9600 20100 : 10500 : : $603 : $315 A's gain. 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 $10500 would yield, in one month, the same in- terest that 1(700 would in fifteen montha. Likewise $800 invested for 12 mctiths would be the same as $9600 for one pionth; henoo the question becomes one of the previous case, that is, their invest- ments are tho same as if they had invested respectively $10500 and $9600 for equal times ; hence the . ^- BULL " ■"" "■ ■*' '■^■^--'f Multiply each man^$ atock hy the time he continuei it in trade; then say, as the mm of the products it to each particular prodiict, to it the whole gain or lott to each man't thare of the gain or htt, EXEBOISXS. 11. A, B and are associated in trade. A furnished $300 for 6 months ;'B, $350 for 7 months, and C, $400 for 8 mopths. Their profits amouTited to $1490 at the time of dissolution ; what was tho profit belon{i;ing to each partner ? Ans. A, $360 ; B, $490 ; 0, $640. PABTMIBSHIP. 225 12. A, B uifl eontnot to perform a certain pieoo of work ; A cmployi 40 meo for 4j^ months ; B 46 men for 3^ months, 4nd 60 aien for 2^ months. Their profits, after paying all oxponscs,. art $860 ; how mueh of this belongs to each ? Ans. A, $340 ; B, $297.60 ; 0, $212.60. 13. Four men. A, B, and D, hired a pasture for $27.80 ; A . pats in 18 sheep for 4 months ; B, 24 for 3 months ; C, 22 for 3 months ; and D, 30 for 3 months ; how much ought oaoh to pay ? Ans. A and B each, $7.20 ; (.\ 94.40 ; D, $9. 14. On the first day of January A began business with a capital , of $760, and on the first of February following ho took in B, who invested $640 ; and on the first of June following they took in 0, who put into the business $800. At the end of the year they found they had gained $872 ; how much of this was each man entitled to ? Ans. A, $384.93 ; B, $250.71 ; C, $236.36. 16. Three merchants, A, B and 0, entered into partnership with a joint capital of $6876, A investing his stock for 6 months, B his for 8 months, and his for 10 months ; of the profits each partner took an equal sbnro ; how much of the eopital did each invest ? Ans. A, $2600; B, $1875; C, $1600. 16. Two merchants, A and B, entered into partnership ibr two >' years ; A at first furnished $800, and at the end of one year, $600 more ; B furnished at first $1000, at the end of 6 months, $600 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 $2560 ; 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, 1866 ; A puts in $4000 ; B, $3000 ; and 0, $2600; April 1st, A withdraws $500, and B withdraws $600 ; June 1st, puts in $800 more ; September 1st, A furnishes $700 more, and B $400 more. At the end of the year they find they have guned $1600 ; what is each partner's share of it? Ans. A, $608.68; B, $423.31 ; C, $468.01. 18. John Adams commenced business January first, 1866, with . ft capital of $10000, and after some time formed a partnership with William Hickman, who contributed to the joint stock the sum of $2800 cash. In course of time they admitted into the firm Joseph Williams, with a stock worth $3600. On making a settlement January first, 1866, it was found that Adams had gained $2260; . ^ i tarn ■ ■■ ''I S w I i 226 ARirmisTio. Hickman, 8120 ; and Williams, $405 ; how loQg had Hickman's and Williams' money heen employed in the business, and what rate of interest per annum had each of the partners gained on their btook ? Ads. Hickman's 8 months ; Williams' 6 months. Gain, 22^ per cent, interest. BANKRUPTCY. _ « When any person is unaoic to meet his liabilities, he makes an assignment of his property to some other person or persons, called official assignee or assignees, whose office 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 his estate, but only in case of iKunifest /rawe?, which the word hanhmpt originally implied, though now it is used as nearly 'ynonimous with insolvent. The property to bo divided is called the assets. The shares of the pro- perty which are divided among creditors, are called dividends. EXAMPLE. A bankrupt owes A $400 ; B. $350, and C, $600 ; his net assets amount to $810 cash; Jow much is he able to pay on the $1, and how much will each creditor receive ? SOLUTION. $400-[-$3504-$600=$1350, total liabilities. JNow, if he nas $1350 to pay, and only $810 to pay it with, he will only be able to pay $810-:-1350=.60or eOcents on the $1. Therefore, A will receive $400X.60=$240; B,$350X.60=$210,and d, $600 X. 60 =$360. Hence the RULE. Divide the net assets hy the number denoting the total amount of the debts, and the quotient will be the sum to bepaia on each dollary then multiply each m^n's claim by the sum paid on the dollar^ and the product will be the amount he is to receive. f ' f — : BANKBUPTCY. 227 ^ EXERCISES. 1. A becomes bankrupt. He owes B, $800 ; G, $500 ; D, $1100, and E, $600. The assets amount to $1110 ; how much can he pay on the dollar, and how much does each creditor receive ? Ads. He can pay 37 cents on the dollar, and B receives $296 ; 0, $185 ; D, $407 ; and E, $222. 2. A house becomes bankrupt; its liabilities are $17940; its assets arc $8970 ; what is the dividend, and what is t^ac shase of the chief creditor to whom $1282 are due ? Ans. The dividend is 50 cents on the dollar, and the principal ^ , creditor gets $641. 3. A shipbuilder becomes bankrupt, and his liabilities are 1303000 ; the premises, building and stock are worth $220000, and i>8 has in cush hsd notes $12875 ; the creditors allow him $3000 for maintenance of his family; the costs are 3^ per cent, of the amount available for the creditors; what is the dividend, and how much does a creditor get to whom $1360.60 are due ? .,^; Ans. Dividend, 73^ percent. Creditor gets, $995.95. iit' I !i i I' I? iH 4. Foster & Co. fail. They owe in Albany, $22000; in Balti- more, 818000; in Philadelphia, $1/100; in Charleston, $16000; in Bojton, $4400, and in Newark, $4200. Their assets are : house prop rty, $14000 ; farms, $2200 ; cash in bank, $4400 ; railway stock, $4200 ; sundry sums due to them, $20135 ; what is the divi- dend, and how much goes to each city ? Ans. Dividend, 55 cents on the dollar; to be paid in Albany, $12100; in Baltimore, $9900; in Philadelphia, $9405; in rT ; Charleston, $8800 , in Boston, $2420; in Newark, $2310. 5. The firm of Reuben Ring & Nephews becomes bankrupt. It owes to Buchanan & Ramsay, $1080 ; to Kinneburgh & McNabb, $850 ; to Collier Bros., $1720 ; to David Bryce & Son, $1580 ; to Sinclair & Boyd, $970. The assets are : house and store, valued at $848 ; merchandise in stock, $420 ; sundry debts, $220. What can the estate pay, and what is the share of each creditor? Ans. The estate payr. 24 cents on the dollar, and the payments are: to Buchanan & Ramsay, $259.20; to Kinneburgh & McNabb, $204; to Collier Bros , $412.80 ; to David Bryco , & Son, $379.20 ; to Sinclair & Boyd, $231.80. m\ ! !•■( ,, w5,^f ;-' 228 . ABETHXETIO. EQUATION OF PAYMENTS. Equation of Payments is the process of finding the average or mean time at which the payment of several sums, duo at different times, may all be made at one time, so that neither the debtor nor creditor shall be at any loss. The date to be found is called the equated time^ The mode of finding equated time almost universally adopted is very simple, though, as vrc shall show in the sequel, not altogether correct. It is known as tJie mercantile rule. Let us observe, in the first place, that t'no stftPv'-'Td by which men of business re(jcon the advantage that accrues to them from receiving money beibre the time fiked 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 $3", at 6 per cent. ; and, if $50 be paid a year in advance of the time agreed upon, the gain to the payee is $3, at the same rate. In the former case, the person receiving the money charges the payer $3 interest for the inconvenience of lying out of his money, but, in the latter case, ho deducts $3 from the debt, for the advantage of having the money in hand. If, on the 1st May, A gives B two notes, one for $50, at a term of three months, and the other for $80, at a term of seven months, the first will V; hjgally due on the 1st August, and the 2nd on the 1st December; i>;a / is not able to meet the first at August, and it is held over till 'h.> ** t November, when A finds himself in a position to pay both at c^ce. The first is then three months over-due, and accordingly B claims interest for that time, which, at 6 per cent., is 75 cents, but as A tenders payment of the whole debt at once, and the second note will not be 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. ? '^' ;'; t < Let 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 Ist Aiigust, and the other on the 6th 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 thtft a settlement is made on 6th November, we find that the 1st note is EQUATION OF PAYMENTS. 229 3 months and 6 days over due, and the interest on it for that period is 80 cents, while the second will not he due for 2 months, and the interest on it for that period is also 80 cents ; consequently, the interest that A should pay, and that which B should allow being equal, they balance each other, and the principal only has to be paid. There arc, 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, due by persons of con* tracted means. The second is what has been illustrated above by the first exam* plo, 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, so that the debtor has only to give the principal to the creditor. If, in this last case, 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 rule for the Equation of Payments. For this purpose let us suppose a ease. R. Evans owes J. Jones $200, which he undertakes to pay by two instalments of $100 each, (basis of interest 6 per cent.,) the first payment to be 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 B. 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. e., $1 06 in all, and in return for making the 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. in 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 when there are several payments to bo made at difierent dates. ^ ^ ^ l|V- 230 ARITHMETIC. If A owes B $1200, due by instalments as follows : — $300 in 4 montHs, $500 in 6. months, and $400 in 10^ months, what is the equated time for the payment of the whole ? $300 X 4 = 1200 500x 6 = 3000 400x101= 4200 $1200 ) 8400 ( 7 months 8400 (equated time. The interest of $300 for 4 months is the same as the in- terest of $1 for 1200 months ; the interest of $500 for 6 months is the same as the interest of $1 for 3000 months ; and the interest of $400 for 10^ months is the snme as the interest of $1 for 4200 months. The sura of all these is 8400 months ; therefore the interest of the whole is the same as the interest of $1 for 8400 months. Now, if $1 require 8400 months to produce a certain interest, the whole debt, $1200, will require only ^nj^g^ part of that time to produce the same interest ; and 8400 -j- 1200 =7. Hence the equated time is 7 months. ; ; - > ,; ' Rule 1. — Multiply each payment by its time, and divide the 8um of the products by tJie sum of the payments. Another method of producing the same result is the fol" lowing : — Interest of $300 for 4 months= $12.00 @ 12 per cent. Interest of 500 for 6 months = 30.00 '@ « Interest of 400 for 10^ months= 42.00 @ «« Interest of $1200 for 1 month =12)84.00 ( 7 months. Rule 2. — Fimd the interest of each instalment for its time, at any convenient rate, and divide the sum of the interests by the interest of the ivhole debt, at the same rate for one m,onth. Note. — 12 percent, is a vety convewient rate, because the interest is flo easily found, being 1 per cent, a month, and consequently the hundredth part of the principal f err I month, The interest is therefore found by simply multiplying the principal by the number of months, and pointing off two places of decimals. The process by Eule 2 becomes identical with that by Eule 1 by reckon- ing the interest at 1 200 per cent. EXERCISES. i., ■ ' 1. Find the ecpiated time for tlie payment of three debts, tlie first lor $45, due at the end of 6 months ; the second for §300 ntbs, he in- for 6 mths ; IS the lese is 3 same •equire e debt, irocluce quated divide he fol- ceut. (( •I VJ 18. ^or its of the hne rcbt& [rest is so lundredth found by pointing ly reckon- ^e debts, 2ond for EQUATION OP PAYMENTS. 231 $70, due at the end of 1 1 months ; and the third for $75j due at the end of 13 months. . ■ - ' '. • $45 X ()=270 ' < . ' • 70x11=770 ' 75x13=975 . $190 ) 2015 ( 10.61 80 10 mqs. 18'days. Ans. 18.30 "When the division is not exact, continue it to two phices of decimals, and reduce to days. 2. If a person owes $1200, to be paid in four instalments, $100 in 3 months ; $200 in 10 months ; 300 in 15 montlis, and $600 in 18 montivs, in what time should he pay the whole sum at once ? In this and similar questions, the work may be somewhat shortened by counting no time for the first payment, and deducting its time from that of each of the others. Thus : $100 X 0= 200 X 7= 1400 . ' 300x12= 3600 . ^ 600x15= 9000 $1200 ) 14000 ( 11§, to which add 3 months, and we have for the equated time 14§ months. 3. J. Smith owes R. Evans $1300, of which $700 are to be paid at the end of 3 months, $100 at the end of 4 months, and the balance at the end of 8 months. Required the equated time for the payment of the whole ? Ans. 5 mos. 4. T. C. Musgrove owes H. W. Field $900, of which $300 are due in 4 months, $400 in 6 months, and $200 in 9 months ; what is the equated time for the payment of the whole amount ? Ans. months. 5. A. & W. MoKinlay have in their possession five notes drawn by G. W. Armstrong, all dated 1st January, 1873 ; the first is drawn at 4 months, for $45 ; the second at 8 months, for $120 ; the tliird at 10 months for $75 ; the fourth at 11 months, for $60 ; and the fifth at 15 months, for $90 : for what len<»th of time must a sinirle note be drawn, dated 1st May, 1873, so that it may fall due at the properly equa- ted time ? Ans. 6 months. 6. A gentleman left his son $1500, to be paid as follows : I in 3 months, | in 4 months, ^ in G months, and the remain- ll ;;il. Li,':, IliiS ,' hi!;;;:; m W, M 232 ABITHMETIC. der in 8 monthe ; at what time ought the whole to be paid at once? Ans. 4 months 15 days. 7. A merchant bought goods amounting to $6,000. He agrees to pay $500 down, $600 in 6 months, $1500 in 9 months, and the remainder in 10 months; at what time ought he to pay the whole in one payment ? Ans. 8 mos. 16 days. 8. A r^rocer sold 484 bbls. of rosin, as follows : February 6th, 35 bbls. @ $3.12^ on 4 months time ; March 12th, 38 bbls. @ $3.00 on 4 months time; March 12th, 411 bbls. @ 2.62^ on 4 months time. What is the equated time for the payment of the whole ? Febr'y 6, 109 xo' 6= 22 / March 12, 1193x1 12=1193 476 ■• '• •■' :■■;■' 1302 ) 1.691 1302 ( m. 1—3 m ,f 3890 ' 9.0 days. 1 month, 9 days, not reckoning the credit of 4 months on which the whole was bought. Add 4 months to this time, and »the result is & months, 9 days to be counted forward from the beginning of February — making July 9th the date on which the whole should be paid. In the above example we have taken the beginning of Feb- ruary as a convenient point from which to reckon the time on each item. From that point the time on the first item (from which we have omitted the cents .is of no consequence in the calculation) is 6 days, .and on the second 1 month, 12 days. When the time is expressed in months, wa have simply to multiply by the months. When there are days, multiply ^ the principal by ^ the number of days, for -^ the principal will be the product by -j^ of a month, or 3 days, and as 6 days are twice 3 days, the product to be carried out is twice ^ the principal, -^ the principal is obtained by simply cut- ting off the right hand figure, which if 5 or more should be considered as 1, and added to the remaining figures. Thus, in the above example, we have carried out t\«ice 11 for 6 days, and 4 times 119 for 12 days. When the number of days does not contain 3 an exact nimiber of times, the nearest niunber that does may be taken first, and then the odd days over, which must be either ^ or EQUATION OP rAYMENTS. 233 days. 3ruary th, 38 LbblB. no for f of 3 days. Work 5 days as -J- of a month, 10 days as ^ of a month, 15 days as ^ a month, &c., when more convenient. 9. Purchased Goods of J. R. Worthington & Co,, at differ- ent times, and on various terms of credit, as by the following statement : — March Ist, 1872, a bill of $675.25, on 3 months. t( ,y- nths on ^1 time, brward le date ofFeb- le time rst item aence in 12 days, mply to tiply A rincipal nd as 6 is twice ply cut- lould be Thus, 11 for 6 ,n exact betaken her i or July 4th, Sept. 25th, « Oct. 1st, •« Jan'ry Ist, 1873, Feb'y 10th, « Mar. 12th, « April 15tl, « c; (( (i (( it u 376.18 821.75 961.25 144.50 811.30 567.70 369.80 4 2 8 3 6 5 4 (( (( What is the equated time for the payment of the whole ? m. d. 1 = 3, March 1,1872, 675x3 4, July 4, 2, Sept. 25, 376x8 4 822 X 8 25 = 8, Oct 1, 96i X 15 3, Jan'ry, 1,1873, 145x13 1 1 6, Feb'y, 10, 5, March,! 2, 4, April 1.% 811x17 10 568x17 12 370x17 15 4728 ) 2025 for 3 months. 23 « 1 day. 3008 « 8 months. 38 « 3 days. 13 « 1 " 6576 « 8 months. 656 « 24 days. 27 " 1 « 14415 « 15 months. 32 « 1 day. 1885 « 13 months. 5 " 1 day. 13787 « 17 months. 270 « 10 days. 9656 " 17 months. 228 "12 days. 6290 « 17 months. 185 « 15 days. m. 59119 (12.50 4728 30 4- .■-, 11839 9456 ,, 23830 .23640 L 1900 15.00dayB. 'i: Mil., Ik Iff 1 1 lli iWII 234 ABITHMETIC. 12 months, 15 days, from the beginning of March, 1872, gives March 15th, 1873. Ans. 3 months' credit on the first bill, and 1 day in March gives the time on the first bill ; 4 months from March to July and 4 months' credit with 4 days in July gives the time on the second bill ; 6 months from March to September, and 2 months' credit with 25 days in September, gives the time on the third bill, &c. To carry out the products, — Ist, multiply the first bill by the months ; — for the one day, take ^ of 68. 2nd, multiply the second bill by the months, throw off the 6 and take the remaining figures of the principal, plus 1, for 3 days, — take ^ of that for 1 day. 3rd, multiply 822 by 8, for 8 months, — multiply 82 by 8, for 24 days, — ta)ce ^ of 82 for 1 day, &c. 10. Bought of A. & W. Smith, 1650 barrels of flour, at different times and on various terms of credit, as by the fol- lowing statement : — May 6th, 150 barrels @ $4.50, on 3 months' credit. May 20th, 400 " « 4.'75, on 4 « « July 10th, 500 " « 5.00, on 5 " « . August 4th, 600 " " 4.25, on 4 « « What is the equated time for the payment of the whole ? Ans. November 7th. 11. J. B. Smith & Co. bought of A. Hamilton & Son 576 barrels of rosin, as follows : — May 3, 62 bbls. @ $2.50, on 6 months. May 10, 100 « « 2.50, on 6 months. May 18, 10 « " 2.50, as cash. May 26, 50 « " 2.75, on 30 days. - v ' > May 26, 345 « « 2.50, on 6 months. :i? May 26, 9 " " 2.00, on 6 months. ii^U^ What is the equated time for the payment of the whole ? Ans. November 3rd. 12. T. B. Jones & Co. sold goods on 3 months' credit, as follows : — ^ ^ ^ . - May 9, a bill of $435.60. « 30, (( (( 75.30. July 17, (( (( 183#75. Aug. 28, (( (( 239.18. Sep. 21, t( . (( ' 82.10. Oct. 23, (t (C 39.85. Nov. 30, (( (t 390.67 235 AVERAOINO ACCOUNTS. a-r-f. W. When, in equity, ought they to have received the whole in one sum, and, allowinfr money worth 6 per cent, what sum ought they to have received at the date of the last sale ? Ana. Equated time, Nov. 13th ; cadh to settle Nov. 30th, $1450.49. 13. Bought of T. & E. Kenny, on 6 months' credit, goods as follows: » January 3, to the amount of $250.00 February 6, Maich 9, April 12, May 15, June 18, July 21, August 24, Sept'br. 27, October 30, Nov'br. 29, Dec'br. 11. tt (( (( (( 317.40 171.70 88.12 623.50 49.04 73.90 218.75 8.15 55.84 398.00 191.25 What *3 the equated time of settlement, and allowing in- terest at 7 per cent., if payment be delayed till Februaey Ist, 1874, how much will then be due? Ans. Equated time, Dec. 16th, 1873. Du? Feb. 1st, 1874, $2467.69. AVERAGING ACCOUNTS. When one merchan c trades with another, exchanging mer- chandise, or giving aud receiving cash, the memorandum of the transactions is called an Account Current. The fixing on a time when the account may be settled by simply pay- ing the balance without interest against either party, is called Averaging the Account. A merchant sold goods amounting to $4000 on 8 months' credit. The purchaser paid ^ down, and ^ in 3 months; what time should be allowed him for for the payment of the remainder ? $4000 X 8 = 32000 2000 X = 1000 X 3 = 3000 ' 3000 , ,3000 subtract from 320( 1000 :- 4^ w 29000 ( 29 months = 2 years, 5 mouths ''„* ^1 286 ARTrHMETIC. The buyer, by the terms of the purchase, is entitled to the use or interest of $4000 for 8 montha, which is the same as the interest of $1 for 32000 months. He lias received on the first $2000 no credit, and on the $1000 paid, only 3 months, which is equal to the interest of $1 for 3000 months. He has therefore to receive on the remaining $1000 what is equal 'to the use of $1 for 29000 months. But the interest of $1 for 29000 months is the same as the interest of $1000 for the Y^jfj^ part of 29000 months, which is 29 months, or 2 years, 5 months. A merchant sold W. M. Brown, Esq., goods to the amount of $3051, on a credit of 6 months from September 25th, 1873. October 4th, Mr. Brown paid $476 ; Nov. 12th, $375 ; December 5th, $800; and on January Ist, 1874, $200. When, in equity, ought the merchant to receive the balance ? 6, Sept. 25, 3051 x 6-25=18306 • ^ , " -m. . - ;^>*^'v,:- : . 2440 ■;- 102 20848 Oct 4, - - . Nov. 12, Dec. 5, Jan'y 1, 476x1-4 = 476 •' 16 - ; 375x2-12= 750 , 152 800x3- 5= 2400 183 200x4- 1= 800 7 1851 4782 1200 ) 16066(13.39 mo«. 1200 30 " ' 4066 11.70 d ' 3600 ; ^ 4660 - :;: 3600 ^ ':i 10600 13 months, 12 days, after the 31st of August, 1873, which will be October 12th, 1874. v . AVKRAOINO ACCOUNTS. 237 The intcreut on the debtor Hide from Aiip;. 31, 1873, is equal to the interest of ^1 for 20848 months. The iutorest on the credit side from the same date is e({iial to the interest of $1 for 4782, which leaves a difference, in favor of the credit side, of the interest of ^l for KiOGG months, that is the in- terest of the balance, 1^1200 for thn Tnj'jnr P^rt of 1 6066 months, or 13 months, 12 days. Thereforotho merchant should allow Mr. Brown the use of the balance of the account 13 months, 12 days, from August 31, 1873, or till October 12th, 1874. When did the 1>alance of the followinjif account fall due, the merchandize items being on C months credit. Dr. McDonald Bros. 1872. May 15. July 20. Sept. 27. 1872. To Mdse. $350.75 Juno 9. (( (( 185.10 1873. U i( 431.73 Feb. 18. Mar. 8. By Mdse. By cash... By Mdse. 6. May 15, 351 x 6.15 = 2106 175 6, July 20, 185 X 8.20 = 1480 114 6, Sept. 27, 432x10.27 = 4320 ' 387 968 8594 Cr. $200.20 800.00 290.00 I ii r 4.1V 1 •-. i (, which 6, June 9, 200 x 7. 9 = 1400 60 Feb'y 18, 300 x 9.18 = 2700 , 180 6, March 8, 290 x 16.8 = 4640 58 ■■'i:i--y 790 178 ) ■Jy 9057 463 ( 356 1070 1068 20 2.60 mos. 30 18.00 days. W ((1^: 238 ARITHMETIC. 2 montbH, 18 davHi to finning of May, 1872, wh be counted backward from the ])e- l^nninjf of May, 1872, which gives February 11th, 1872, the time from wJiich interest is to be charged on the balance. The interest of the debit side, from April 30, 1872, is enual to the interest of $1 for 8594 months, while the interest on the credit side from the same date is e(iual to the interrst of $1 for 9057 months, which gives a difierenco in favor of tlic debit Hide, of the interest of $1 for 4G3 months, equal to the interest of the balance, $178, for j [^ part of 403 montliH, that is, 2 months, 18 days. From the examples given wo may deduce the following: Rule. — Proceed with each sloe of the account as in Equation of Payments^ counting the time for each aide, from the hefjinning of the month of the earlic8„ date in the account. Take tJce difference between the aumr ' the products of the tivo 8idc8j and divide it by the bal of the account. Count the quotient 'inonths, and carry it to tivo places of decimals. Reduce the decimals to days. WJien the sum of the products of the larger side is greater than the sum of the products of the smaller side, reckon tlie time denoted by the quotient forward, but when the opposite of this is the case^ reckon backward from the date from which all tlie tlTne has been reckoned. EXERCISES. Find the times at which the balances of the following accounts became due, or subject to interest : — 1. Dr. > J. S. Peckham. Cr. May 16, 1872 $724.45 | July 29th, 1872 $486.80. Ans. December 16th, 1871. 2. Dr. T. B. Reagh. Cr. November 19, 1873,... .$635. | December 12, 1873 $950. Ans. January 28th, 1 874. 3. Dr. * Jno. T. Lithgow & Co. Cr. February 24, 1873... .$512.25 | June 10, 18^2 $309.70. Ans. February 24th, 1874. 4. Dr. ' T. J. Golden & Co. Cr. March 17, 1873 $145 | January 15, 1873... .$695.60. ; . ^y^*'. ^ : Ans. December 30th, 1872. AVEMQINQ ACCOUNTS, 239 itcta of :Gouiit. aces of nde is :r aide, d when om the llowing Or. ;486.80. 1871. 6. Dr. S. E. WiiisTON. Cr. August 27, 1873 $341. | November 7, 1873 $247. Ana. Februury 2l8t, 1873. 6. Dr. L. C. Eaton. Or. July 20, 1873,... $711. | April 14, 1873, $1260. Aus. Docoraber Uth, 1872. 1. Dr. Gordon & Keith. Or. June 24, 1872....! $1418. | September 7, 1873 $2346. An8. July 9tli, 1875. 8. Dr, Gi:o. W. Jones. O?'. December 2, 1873!.. $1040.80. | Augunt 13, 1873,...$1112.40. A , Au8. March 0th, 1869. 9. Required the time wlicn the Imlance of the following account becomes subject to interest, allowing the merchan- dise items to havtj been on 8 months' credit ? Dr. S. T. Hall & Co. C^. 1872. 1873. May 1. To Mdse,. $300.00. .Fan. 1. July 7. (( i( 7r)9.06. Feb. 18. Sept. 11. • 417.20. Mar. 19. Nov. 25. (( (( 287.70. April 1. Dec. 20. (( (( 571.10. May 25. By Cash,.... « Mdse,... " Cash,.... « Draft.... " Cash,.... $500.00. 481.75. 750.25. 210.00. 100.00. ;,, • ' #;. Ans. August 7th, 1873. 10. When will the balance of the following account fall due, the merchandise items being on G months' credit ? Dr. ' Barnes & Co. CV. 1873. May 1. May 23. June 12. July 29. Aug. 4. Sept. 18. To Mdse "Cashpd.dft. "Mdse « Cash $312.40 85.70 105.00 243.80 92.10 50.00 Ans. January 11th, 1874. 11. When does the balance of the following account be- come subject to interest ? 1873. June 14, July 30, Aug. 10, Aug. 21, Sept. 28, By Cash,. " Mdse. " Cash... " Mdse, $200.00 185.90 100.00 58.00 45.10 ill !l!*' !-i( 'W!! I 240 ] i Dr. AKITHMETIC. Beard & Venning. Cr. 1873. Aug. 10, Aug. 17, Sept. 21, Oct. 13, Nov. 25, Nov. 30, Dec. 18, 1874. Jan. 31. To Mdse 4 mos. ^' " 60 days « Cashpd. dft. " Mdse 6 mos. « " 90 days " " 2 mos. (( $285.30 192 60 256.80 190.00 432.20 215.25 68.90 1873. Oct. 13 Oct. 26 Dec. 15 Dec. 30 1874. Jan. 4 Jan. 21 By Cash.., $400.00 150.00 «Mse2m 345.80 « « 4m 230.40 « Cash.... 340.30 180.00 Cash 100.00 Ans. August IQtb, 1»74. 12. In the following account, when did the balance be- come dufc, the merchandise articles being on 6 months' credit ? Dr. S. Kerr in account with T. E. Jones & Co. Cr. To Mdse. (( cash paid drft. Mdse cash paid drft. Mdse $240.00 48.88 50.00 1873. Jan. 4, Jan. 18, Feb. 4, Feb. 4, Feb. 9, Mar. 3, Mar. 24, April 9, May 15 May 21, Ans. December 20th, 1873. When, in equity, should the balance of the following account be payable? Dr. Danei & Boyd. Cr. u it (( (( 1873. $ 96.57 Jan. 30 By Cash, 57.37 April 3 (( (( 80.00 May 22 (( i( 3§.96 50.26 15446 42.30 23.60 28.46 177.19 1873. Jan. 3, Jan. 31, Feb. 8, Feb. 21, Mar. 10, Mar. 24, Apr. 12, June 1, Ju 20, Ju 4^ Sep-.. 27, Dec. 9j ?o Cash,. $200 (( u 300 (( (( 75 U li 100 (( (C 350 U (( 25 (( (( 40 (( (( 80 (( (( 125 (( u 268 (C (t 250 (( (( looj 1872. Sep. 20, Oct. 27, Dec. 5, 1873. Jan. 18, Feb. 26, Apr. 15, June 12, Sep. 21, Dec. 29, By Mdse, 6 mos. « 4 « . a a « 60 days... " 6 mos... U A it ^C • • • « 2 « ... « 6 " ... « 6 « ... $583.17 321.00 137.00 98.75 53.98 634.00 97.23 84.00 132.14 Ans. October 16th, 1847. }' CASH BALANCE. CASH BALANCE. 24l JM EXAMPLE, What is the balance of the following account on January 19th, 1873, a credit of three months l)eing allowed on the merchandise, money being worth 6 per cent ? Dr. KisRu & Thorn. CV. 1872. Mar. 12, Apr. 21, May 6, May 27, July 16, Sep. 10, Oct. 19, To Mdse,. (( (t " cash pd. dft. « Mdso « Cash " Miise (( (( $340.00 150.00 165.00 215.00 100.00 310.00 120.00 1872. Apr. 20, May 4, June 15, Aug. 10, Sep. 23, Nov. 12, Dec. 15, By Mdse. " Cash.. " Mdse.. « Cash.. $200.00 110.00 230.00 180.00 50.oO 50.00 100.00 FIRST METHOD. If the above account bo averaged by the method already f;iven, it will be found that the balance fell duo on September 3rd, lb72. If, therefore, iitbe not paid till January 19th, 1873, the amount due will consist of 9480, the balance, and the interest on it from Se))tember 3rd, 1872, to January 19th, 1873. Now the interest of $480 for 130 days, the intorral referred to, i8S10.89, and $480 + 10.89=8490.89, the sum due Jan. 19, 187S. SECOND METHOD. EuLE. — Find the interest on eich item, from the time it falls due to the time of settlement^ and thence the balance of interest. Add the balance of interest to that side of the account which 'produces the larger amount of interest. The balance oftJie account ivill then be the Cash Balance at the date to which the interest of the several items has been reckoned. This method is illustrated in the following ACCOUNT CURRENT AND INTEREST ACCOUNT. MnigroTC * Wrigbt lu aecoxint current and lotcrcat account to Jan. IBth, 1873, with Eaton * PiMee. DATm iTCMf. raixci- PAU WHtK Uva. TlHI. IKT. .DlTB. iTixa I'aiKci- P4U WUJM DlB. Tma Im. I873L 1873. 1872. 1872. Mar. 19 To Mdic. 3 moa. S34O00 7un<-ll 221 d'K ai23i Apr. 30 VCjlt 4 njr Md««,. 1 mod. S'WOO July 20 ind'i flea Apr. 21 Hat a •• " 3 " 15000 JuItZI 18;; •' 449 ■• Cash .1000 May 4 200" "to "C««h, paid draft, 16S00 May 6 2SS " 7 00 Junp i It 4t 23000 JunelS 218" 834 • n " MdM. 3 mo*. 315 00 Auit.W 145 •' 512 KugAO '■ Mdie.,lm(M. ISO 00 Noy.io 70" SOT Juir IS " Cash, 10000 juiy le 187 " 8 07 Si'pt.33 " Caih 50 CO Sept23 Ul»" 007 eeptlO "Mdiir. Smoc. 31000 Dee. 10 40 " 304 NOT.12 ■1 «, 50 ») Not. 12 89 " OM Oct. 10 .. .. 3 .. 13000 1873. Dec. 13 <« 4( 100 no Deals 33 " oaa 187SL Jaal9 "BaLofinterMt, 10 B Jaa.M " Bal. of mtrrett, loss 1873. Jan. U " Balance, iSOM SU10V3 fl340T . ) gUlODQ MOT — The difforenctt observable in the rrault of the tTro metboda is due to the different modm of counting the time. The latter answer i.t tb» more ancurate. The difference, however, is iasi^ idfleuit, amounting to only about bait a day's interest on the baianoo of account. • i ill I 11; I Itlk 142 ABITHXETIO. 2. The following account was settled in full on December Ist, 1865 ; what amount was paid, intor'^t 6 per cent. ? 0. F. Muds in account cnrrent su< aterest account to Dec. Ist, 1865, with T. R BrowB. Dati. ITEMS. rr.ixc:- PAL. wnfM DUK. TWl. INT. DATZ. I1EII8. PBIirci. PAL. WHKN Due. TIMB. Iirr. San. 1 To Ifdsi'., 6 inoJ. " Cash pd. dft. " UdM., 4 mos. .. .. 4 .. " Cash pd. dft. " Udie.. e mm. (15C lo 100 00 31S»u 103 0(1 100 0(. 313 00 F«b. 1 Mar. 30 Mlj 1 July 1 Sep. 10 Br C.vih *' Mdne., 4 inn3. •• « " .. .. 4 .. " •' 4 " •130 i» 430 18 auooo 60 00 »»84 Veb. S Kw.ao * " 90 ■Uyis Au|t.30 Ans. 961.36. t; 3. What will be the cash balance of the following account if settled «* «, (200 00 30U0U 250 00 600 00 400 00 fiOO'K) 100 00 moo 1864. J Illy 20 Aug. 15 Snpt, 1 Nov. 1 Dec. 6 " 20 1865. Feb, 1 " 28 By Cash if By Merchandise.! By CuKh ByUcrebaBdiMii (300 00 450 00 400 00 820 00 600 00 lOOOO 200 00 1W08 Octr .1 Nut. 20 J)cc. 12 1865. Jan. 15 Jfaur.U !■•••• ACOOUHT OF SALES. Aa account of sales is a statement made by the consignee (gen- erally a commission merchant) to the consignor, the person from whom the merchandise was received tp sell, showing the parsons to whom sold, the price, time, charges, and net proceeds. • The net proceeds is the :; mount due the consignor, from proceeds of sales, after all charges are deducted, and are due to the consignor at the avoragc 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 oa 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 sales. 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 be the amount due A. R. E. on May 14, 1867, discount 6 per cent. ? Ans. Net proceeds due May 18, 1867. Due i. R. E.. 82371.36. rltt lii m i I 4 w ri I 244 ABITHMETIC. Account sales of 8745 lbs. bacon, 2976 lbs. choese, and 1245 lbs. batter, for account and risk of A. R. Eastman, Cbioago, 111. 1867. Mar. 10 Apr. 20 ^ay 14 March 1 « 9 May 14 « (( Sold to R. White, at 30 days— 4000 lbs. bacon, at IGc... 500 lbs. butter, at 40c.. Sold to J. B. Harris, for cash — 4745 lbs. bacon, at 15|c.. Sold to J. C. Parsons, at 60 days- 2976 lbs. cheese, at 22c.. 745 lbs. butter, at 41c.. CHABGES.— $640 00 200 00 Paid freight in cash » Paid for labor resalting biieon Storage : Commission on $2547.51 at 2^ per ct. Net proceeds due per average May 21 654 72 305 45 97 40 8 50 5 00 63 69 $840 oa 747 34 960 17 $2547 51 174 59 2372 92 Cleveland, 0., E. and 0. E. May 14, 1867. R. Felton & Co. September 4, 1866, ve received from W. Cummings, Cincinnati, a consignment of 120 brls. of mess 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 average 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 time are the net proceeds due as cash, and what amount in cash will settle our account with W. Cummings on Jan. 1st, 1867, interest 6 per cent ? . Ans. To last question, $6140.67. Account sales of 120 brls. mess pork, and 742 bushels clover seed, on joint account of W. Cuuunings, Cincinnati, and ourselves (each one-half.) Sept. 12 « 30 Sold to C K. Sing, for cash — 250 bshls. clover seed, at $8.95 Sold to M. HoUingsworth, at 60 dys. — 25 brls. mess pork, at $32 80 bshls. clover seed, at $9.25... $2237 50 1540 00 'M.\i CASH BALANCE. 245 15 lbs. m. ?47 34 960 IT 'Oot> 18 Novr. 2 « 15 Sept. 4 <( « Octr. 8 Nov. 15 Sold to T. M. Ames, at 30 days— 200 bshls. clover seed, at $9.25... 10 brls. mess pork, at $32.50... Sold to T. B. Brown, for cash— 200 bshls. clover seed, at $9.30. Sold to A. W. Purdy, at 6 months— 85 brls. mess pork, at $33....... 12 bshls. clover seed, at $9.30 . CHAROES.- Paid frcii;ht and cartage in cash Paid insurance on $9500, at 1^ p. et. Paid for storage, cooperage, and labor Commission on $10,729.10, at2j^p.c. Net proceeds of sales. Your ^ net proceeds, due as per aver. 1850 00 325 00 2805 00 111 60 210 75 118 75 15 00 268 23 2175 00 1860 00 2916 60 10729 10 612 73 10116 37 5058 18 Columbus, Nov. 15, 1866. E. and 0. E. J. O. Denison & Co. January 2, 1867 — Beceived from D. M. Haiman, 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 16c. per lb., to be sold on joint account, of shippers f , and ourselves ^ ; our ^ of invoice due as cash. January 21 — Cashed I). M. Harman's sight draft in favor of First National Bank, Cleveland, for $1264.50. February 14 — Accepted D. M. Harman's one month sight draft in favor of Thos. L. Elliot, Owosso, for mutual accommodation, for $864. 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, 1867, reckoning interest at the rato of 7 per cent, per annum. Ans. Equated time of sales, April 8, 1867. ^ ^ ;; Equated time of Harman's acct., April 7th. Balance of account on May 14, $2458.56. 'in ! . iii>i 246 ABITHMETIO. Account sales of 200 brls. pork, 3750 lbs. clioese, and 100 firkins butter, on joint account of D. M. Harman §, and ourselves J. tfan. 16 Sold A. h5. Morrison, on 90 days — 40 brls. pork, at $18.75* per brl. 50 firkins butter, tiO lbs. each, at 24c per lb $750 00 960 00 1162 50 149 50 $1710 00 Feby. 9 Sold W. E. Glennie, on 2 months— 60 brls. pork, at $19.37^ per brl. 1150 lbs. cheese, at 13o per lb Sold A. n. Peatman, on his note at 2 months — 50 firkins butter, 80^ lbs. each, at 24c oer lb 1312 00 « 27 966 00 265 30 1895 lbs. cheese at 14c. 1231 30 Sold 11. 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.... March? 1443 75 " 24 496 88 105 75 602 63 122 75 6 50 12 46 11 25 157 49 6299 68 Jan'y. 2 Paid for freight and cartage by cash... Paid for cooperage and extra labor by cash " 15 Paid Insurance at IJ per cent Mar. 24 " 24 Charges for storing in storehouse Com. at 2^ per cent, on $6299.68 is. • Net proceeds of sales ' 310 45 5989 23 Your § of N. P. due, as per average, April 12th ■ t. • 3992 83 .E.J md 0. E. E. Geo. ( 1 JONKLIN & Clevel 5 Co. and, 0.. tt ■ Dec: Ist, 1866--We received from Messrs. Gillespie, Moffatt & Co., Boston, 27 cases Mackinaw blankets, 340 prs. at $3.90; 2 cases ? ^) CASH D.ILAKCE. 247 ^rkins 9 J. no 00 312 00 1231 30 1443 75 602 63 6299 68 • 310 45 5989 23 3^2 82 Co. xnd, 0. Moffatt & )0; 2 case* chintz cotton, 987 yds. at 7^ ; 20 pea. table oil cloth, at $3.70 ; 4 pes. do., at $5.62]^ ; 7 pes. West England broad cloth, 126 yds. at $3.70 -, 7 bales cotton batts, at $6.20 ; to be sold on joint account and risk of consignor and consignee, each one-half, our one-half as cash* Dee. 5th— Wo cashed their demand draft for $1200. Dec. 17 — Accepted their draft on us at 30 days' sight, for $984. Jan. 14 — Cashed their draft on demand, for $500. Sales of merchandise as per account sales annexed. At what date arc not proceeds due as cash ? A. Feb. 22, '67. What is cquatea time of 6. & M.'s account ? A. June 10, '64, What is the cash balance March 24, 1867 ? A. $76.54. 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, Mofifatt & Co., Boston, and ourselves (each one-half). i f Deer. 6 Sold John McDonald & Co., ^ cash, ^ on account 30 days — 13 cases blankets, 260 prs. at $4.20 1 case chintz cotton, 425 yds. at 9c ...; 7 pes. table oil cloth, at $4.50 ... <« u 17 9 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 ** 14|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 $4 Sold James A. Dobbie & Co., note at 90 days 2 bales cotton batts, at $7 4 cases Mackinaw blankets, 80 prs., at $6.70 1 pc. W. E. broad, 18 yds., at $5 w ■'i till ii I 948 ♦' 28 Beer. 1 1 1 ABXTHlOEnO. Sold Thiotnu Speneo & Co., | cash, bolanoo on aoot. at 30 dajs — 3 cases M. blankets, 60 pn., at •6.76 6 bales cotton batts, at $7.25. ... OHAROES. Jany. 1 Paid freight and expenses from depot, cash Storage Com. at 2| per cent, on sales Net proceeds ^ Your ^ of net proceeds, due as per average $94 75 34 48 Milwaukee, Wis., January Ist, 1867. £. and 0. E. J. 0. Spencer & 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 kinds, medial ^and alternate, ALLiaATION MEDIAL. Alligation medial relates to the average value of articles com- ponndei, when the actual quantities and rates are given. 7"" " ■ X AMPLE. A miller mixes three kinds of grain: 10 bushels, at 40 cents a bushel ; 15 bushels, at 50 cents a bushel ; and 25 bushels, at 70 cents a bushel; it is required to find the value of the mixture. j^ iUIOiXIOBf. 249 10 bushels, at 40 cents a bushel, will be worth 400 cents., ^■,,, 16 bushels, at 50 cents a bushel, will be worth 750 cents., 25 bushels, at 70 cents a bushel, will be worth 1750 cents., giving a total of 50 bushels and 2900 cents, and hence the mixtoro is 2900-f 50=58 cents, the price of the mixture per bushel. Hence the BULB. Find the value of each of tlie articles, and divide the sum of their values hy the number denoting the sum of the articleSy and tht quotient will he the price of the mixture, EXEBO ISES. 1. A farmer mixes 20 bushels of wheat, worth $2.00 per bushel, with 40 bushels of oats, worth 50 cents per bushel ; what is the price of one bushel of the mixture ? Ans. $1. 2. A grocer mixes 10 pounds of tea, at 40 cents per pound; 20 pounds, at 45 cents per pound, and 30 pounds, at 50 cents per pound ; what is a pound of this mixture worth ? Ans. 46$ cents. 3. A liquor 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 this mixture worth ? Ans. 90 cents. 4. A farmer mixed together 30 bushels of • wheat, worth $1 per bushel; 72 bushels of rye, worth 60 cents per bushel; and 60 bushels of barley, worth 40 cents per bushel ; what was the value of 2^ bushels of the mixture ? Ans. $1.50, 5. A goldsmith mixes together 4 pounds of gold, of 18 carats fine ; 2 pounds, of 20 carats fine ; 5 pounds, of 16 carats fine ; and 3 pounds, of 22 carats fine ; how many carats fine is one pound of the mixture ? Ans. 18f .' ALLIGATION ALTERNATE. ^ Alligation alternate is the method of finding how much of seve* Tal ingredients, the quantity or value (if which is known, must be combined to make a compound of a given value. CASK I. Given, thQ value of several ingredieuts, to make a oompoand of uneven Tahio. ._ I 250 ^srrBMKno. .^ ,Mi: y ty^^'.m ■ XAMPLB ^ ;'; "■'^ •.'■a ''4 How mnoh Bugor that is worth 6 oents, 10 oents, and 13 oenta per pound, must bo mixed together, so that the mixture may bo worth 12 oents per pound ? SOLUTION. 12 ccntfl. ' 1 lb., at G cents, is a gain of 6 cents. ) Gain. 1 lb., at 10 cents, is a gain of 2 cents. J 8 1 lb., at 13 oents, rs a loss of 1 cent. 1 lbs. more, at 13 cents, is a loss of. Loss. 1 7 Gain 8 Loss 8 It is evident, in forming a mixture of sugar worth 6, 10 and 131 oents per pound so as to bo worth 12 cents, that tho gains obtained^ in putting in sugar of has value than tho average price must exactly balance the losses sustained in putting in sugrxr of (^rca^er valuo than the average price. Hence in our example, sugar that is worth 6 cents per pound when put in the mixture will sell for 12, thereby giving a gain of 6 cents on every pound of this sugar put in tho mixture. So also sugar that is worth 10 cents pur pound, when ioi 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 that is worth 13 cents per pound, on being put into the mixture will cell 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, our losses must equal our gains, and therefore we have yet to lose 7 cents* and as there is only one quality of sugar in the mixture by which we can lose, it is plain that we must take as much more sugar at 131 oenta as will make up the loss, and that will, require 7 pounds. Therefore, to form a mixture of sugar worth 6, 10 and 13 cents per 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 13 cents-f-7 pounds of tho same, which must be ^taken to make the loss equal to the gain. .,..:.,; ^ i»^. By making a mixture of any number of times these answers, it will be observed, that the compound will be^oorrectly fortned. Hence we con readily perceive that any number of answers may be obtained ALLIGATION ALTERNATE. 251 to all ezoroises of this kind, tho Mowing < .; From whnt has boea said wo dodttoo RULE. Find how much i$ gained or lost by taking one of each kind of the proposed ingredients. Then take one or more of the ingredient^ or »uch parts of them as will make the gains and losses equal. KXEROISES. 1. A grocer wiahea to mix together tea worth 80 cents, $1.20, $1.80 und $2.40 per pound, so as to make a mixture worth $1.C0 per pound ; how many pou ids of each sort must ho take ? Ans. 1 lb. at 80 cents; 1 lb. at $1.20; 2 lbs. at $1.80, and 1 lb. at $2.40. 2. How muoli corn, at 42 cents, GO cents, 67 cents, and 78 cents per bushel, must bo mixed together that tho compound may bo worth 64 cents per bushel ? . Ads. 1 bush, at 42 cts. ; 1 bush, at CO cts. ; 4 bush, at 67 cts. ; and 1 bush, at 78 cts. 3. It is required to mix wine, worth 60 cents, 80 cents, and $1.20 per gallon, with water, that tho mixture may 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 com- pound may be 62 cents per bushel ? Give, at least three answers, and prove tho work to be correct. 5. A produce dealer mixed together corn, wortfi 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 waB worth 80 cents per bushel ; what quantity of each did he take ? Givo 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 quantity, to find the other ingredients. BXAMFLS. "■'I I!' ! ^ How much barley, at 40 cents ; oats, at 30 cents, and ooni, at 60 jii ,S5S ARrrBMBTIO. «wti per bttiliel, must bo mixed with 20 bnehels of rye. At 65 eflnte per bushel, so that the mixture may be worth 60 ceuts per bmlial tli SOLUTION. Buih. Cents. ^ Gain. Loai. 1 at 40, gives 20 ... 1 at 30, givos 30 1 at 60, gives 00 .00 20 at 85, gives 6.00 .60 6.00 Oat 40, gives 1.80 9 at 30, gives 2.70 ... v $5.00 $5.00 By taking 1 bushel of barley, at 40 cents, 1 bushel of oats at 30 oents, and 1 boabcl of com at 60, in connection with 20 bushels of rye at 85 cents per bushel, wo observe that our gains amount to 60 oents and our losses to $5.00. Now, to make the gains equal the losses, wo have to tako 9 bushels more at 40 cents, and 9 bushels more at 30 oents. This gives us for the answer 1 bushol-f 9=10 bushels of barley, 1 bushel-{-9=10 bushels of oats, and 1 bushel of corn. From tl- we deduce the -^ RULB. JFind how much is gained or loit, hy taking one of each of the proposed ingredients, in connection with the ingredient which is limitedf and if the gain and loss he not equal, take such of the pro- posed ingredients, or such parts of them,, as will make the gain and loss equal. ^ . > i < . EXERCIB18. 6. How much gold, of 16 and 18 carats fine, must be mind with 90 ounces, of 22 carats fine, that the compound may be 20 carats fine ? Ans. 41 ounces of 16 carats fine, and 8 of 18 carats fine. 7. A grocer mixes teas worth $1.20, $1, and 60 cents per pound, wUh 20 pounds, at 40 cents per jraUnd ; how much of each sort must he take to m^ke the composition worth 80 cents per pound? '^ ■■> 8. Bow much barley, at 60 cents per bushel, and at 60«cent8 per bushel, must be mixed with tra bsshels pf itease, worth 80 oenta ALUOATION ALTKBNATB. 268 P«r Imihel, and 6 bushola of ryo, worth 85 cents per bnshel, to make A mizturo worth 75 oentM por bushol ? Am. !t buBholfl, at 50 oontH ; 2^ bushels, at 60 oonts. 9. How many poundH of Hugar, at 8, 14, and 13 oonts pe pound, must bo mixed with 3 poundn, worth 0\ oontH per pound ; 4 pounds, worth 10^ cents per pound; and (i pounds, worth 13j^ cents per pound, so thut tho mixture may bo worth 12^ cents por pound ? Ans. 1 lb., at 8 ots. ; 9 lbs., at 14 ots. ; and 5^ Iks., at 13 eta 0A8E III. To find tho quantity of each ingredient, when tho sum of tho iogrodients and the average price are given. ■'''' KXAMPLE. A grocer has sugar worth 8, 10, 12 and 14 oonts per pound, and ho wishes to make a mixture of 240 pounds, worth 11 oonts ptK pound ; how much 5f each sort mu,st he take ? ^ \ 8 L u T I N . Gala. Loss. 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 41bs, ^ 4 4 240 lb8.-i-4=:60 lbs. of each sort By taking 60 lbs. of each sort we have Che required quantity, and it will be observed that the gains will exactly balance the losses, oonsequently tho work is correct. Hence the RULE. J, Find the least quantity of each ingredient by Case I., Then divide the given amount by the sum of the ingredients already /ound^ and multiply the quotient by (lie quantities found for the propor- tional quantities. ., ; . 10. What quantity of three different kinds of raisins, worth 15 cents, 18 cents, and 25 cents per pound, must be mixed together to tHk a box containing 680 lbs., and to be worth 20 cents per pound ? Ans. 200 lbs., at 15 cents ; 200 lbs., at 18 cents ; and 280 lb0.» at 25cents. , ^ -. ii. 254 ARITHMETIC. 11. How much sugar, at 6 cents, 8 cents, 10 cents, and 12 cents per pound, must be mixed together, 80 as to form a compound of 200 pounds, worth 9 cents per pound ? r. Ans. 50 lbs. of each. 12. How much water must be mixed with wine, worth 80 cents per gallon, so as to fill a vessel of 90 gallons, which may be offered •at 50 cents per gallon ? Ans. 5G^ gals, wine, and o3|; gals, water. 13. A wine merchant has wines worth $1, $1.25, $1.50, $1.75,and $2. per gallon, and ho wishes to form a compound to fill a 150 gallon- cask that will sell at $ I. -10 per gallon; how many gallons of each i^ort must ho take ? Ans. 54 of SI, 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 mixluro may be worth 1 5 cents per polmd ? Ans. 33] lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 cents, 15. A grocer requires to mix 240 pounds fi? different kinds 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 be taken of each kind ? Aus. 192 lbs. of 8 cents, and 16 lbs. of each of the other kinds, MONEY.; ITS NATURE AND VALUE. Money is the medium through which the incomes of the different members of the community arc distributed to them, and the measure by which they estimate their possessions. The precious metals have, amongst almost all nations, been the standard of value I'rom the earliest lime. Except in the very rudest state of society, men have felt the necessity of having some article, of more or less intrinsic value, Jthat can at any time be exchanged for different commodities. No other substances were so suitable for this purpose as gold and silver. They arc 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 required to purchase other articles. If all tho quad islij Willi silvel currj issud will papel pareq MONEY: ITS NATUBE AND VALUE. 255 money in circulation wore doubled, prices would be doubled. The usefulness of money depends a great deal upon the rapidity of its circulation. A ten-dollar bill that changes hands ton times in a month, purchasas, during that time, a hundred dollars' worth of goods. A small amount of money, kept in rapid circulation, does the same work aa a far larger sum used more gradually. Thcrefr .-< , whatever may bo the quantity of money in a country, only that pan of it will ciFect prices which goes int9 circulation, and Is act slly exchanged for goods. Money hoarded, or kept in reserve by individuals, does not act upon prices. An increase in the circulating medium, conforma^^o in duration and extent to a temporary activity in business, does not raise prices, it merely prevents the fall that would otherwise enauo from its temporary scarcity. . PAPER CURRENCY. Paper Currency may be of two kinds — convertible and incon- vertible. When it is issued to represent gold, and can at any timo 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 anytliing as money, than the persuasion that it will be taken from him on the «ame terms by others. That alone would ensure its currency, but would '.lot regulate its value. This evidently cannot depend, as in the cade 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 got 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 metallic Qurrcncy of other countries. It would be ■I 256 ABirSBCETIC. quite impossible for these results to follow the issue of convertible paper for which gold could at any time be obtained. All variations in the value of the ciroulating medium are mis- chievous ; they disturb existing contracts and expectations, and the liability to such disturbing influences renders every pecuniary engagement of long date entirely precarious. A convertible paper currency is, in many respects, beneficial. It is a nA>re convenient medium of circulation. It ib clearly a gain to the issuers, who, until the notes are returned for payihent, obtain the use of them as if they were a real capital, and that, without any loss to the community. i* due is a eannc persoj make I that ehan^ from have very mcrcl bougl BXOBANOI. 26T : f EXCHANaK It often becomes necessary to send money from one town oi country to another for various purposes, generally in payment for goods. The usual mode of making and receiving payments between distant places is by bills of exchange. A merchant in Liverpool, whom we shall call A. B., has recdlved a consignment of flour from C. D., of Chic. go; and another man, E. F., in Liverpool, has Bhipped a quantity of cloth, in value equal to the flour, to 6. H. in Chicago. There arises, in this transaction, an indebtedness to Chi< cage for the flour, as well as an iudebtedness from Chicago for the doth. It is evidently unnecessary that A. B,,-in Liverpool, should Bcnd money to C. D. in Chicago, and that G. II., in Chicago, should send an equal sum to E. F. in Liverpool. The one debt may be applied in paymo'-t of tht other, and by this plan the expense and risk attending the double transmission of the money may bo saved. C. B. draws on A. B. for the amount which he owes to him; and G. H. having an equal amount to pay in Liverpool, buys this bill from C. D., and sends it to E. F., who, at the maturity of the bill, presents it to A. B. for payment. In this way the debt due from Chicago to Liverpool, and the debt due from Liverpool to Chicago are both paid without any coin passing from one place to the other, ,' : • An iurangement of this kind can always be made when the debts due between the different places are 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 againut 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 his own foreign correspondent ; and to place his correspondent in funds to meet it, he will remit to him all the exchange which he has bought and not re-sold. . i. i "ii is? ' %. 258 AiiniQiBTXi When broken find that they are asked for more bilk than are offered to them, they do not absolutely refuse to give them. To enable their oorrcspondents to meet the bills at maturity, as they have no exchange to send, they have to remit funds in gold and silver. There are the expenses of freight and insurance upon the specie, besides the occupation of a certain amount of capital involved in this ; and an increased price, or premium, is charged upon the exchange to cover all. • The reverse of this happens when brokers find that more bills are offered to them than they can sell or find use for. Exchange on the foreign country then falls to a discount, and can be purchased at a lower rate by those who require to make payments. There are other influences that disturb the exchange between different countries. Expectations of receiving large payments from a foreign country will have one effect, and the fear of having to make larger payments will have the opposite effect. AMEBICAN EXCHANQE Exchange between Canada and the United States, especially the northern, is a matter of every day occurrence on account of the 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 the 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 • 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 nom restored, matters will soon find their former level. '^ It has been deemed essential that this should be distinctly ez> plained, as it has brought about a necessity for a constant calculation AMEBIOAN EXCHANGE. 259 of the 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 the following exercises it is understood to bo Greenbacks. A s E I . To find the value of $1, American currency, when gold is at a premium. EXAMPLE. When gold is quoted at 140, or 40 per cent, premium, what is the value of $1, American currency ? SOLUTION. Since gold is at a premium of 40 per cent., it requires 140 cdhts of American funds to equal in value $1, or 100 cents in gold. Hence the value of $1, American money, will be represented by the number of times 140 is contained in 100, which is .71| or 71| cents. Hence to find the value of $1 of any depreciated currency reckoned in dollars and cents, we deduce the following RULE. Divide 100 cents hy 100 ^Zms the rate of premium, on gold, and the quotient will he the value o/ $1. Subtract this from $1, and the remainder will be the rate of discount on the given currency. CASE II. To find the value of any given sum of American currency when gold is at a premium. % EXAMPLES. What is the value of $280, American money, when gold is quoted at 140, or 40 per cent, premium ? SOLUTION. We find by Case I. the value of $1 to be 71 1 cents. Now, it is evident that if 71 f cents be the value of $1, the value of $280 will be 280 times 71 f cents, which is $200, or $280H-l.40=28000-i- 140=$200. Hence we have the following ''^^ '"' "^ '^ "' 4«ii m 260 abuhmeiio. r \ RULE. Multiply the value of $1 2iy the number denoting the given amount of American money, and the product will he the gold value; or, . ' Divide the given sum of American money hy 100 (the number of cents in $1,) plus the premium, and the quotient will he the value in gold. CASE III. To find tbo premium on gold when American money is quoted at a certain rate per cent, discount. EXAStPLE. « When tbe discount on American money is 40 per cent., what is the premium on gold ? *' * SOLUTION. If American money is at a discount of 40 per cent., the discount on $1 would be 40 cents, and consequently the value of $1 would bo equal to $1.00 — 40 cents, equal to 60 cents. Now, if 60 cents in gold be worth $1 in American currency, $1 or 100 cents in gold would bo worth 100 times g'g of $1, which is $1.'^6§, frc aa which if we subtract $1, the remainder will bo the premium. Iherefore, if American currency be at a discount of 40 per cent., the premium on gold would be 66§ per cent. Hence we deduce the following BULE. Divide 100 cents by tJie number denoting the gold value of $1, American currency, and the quotient will he the value, in American currency, of ^1 in gold, from which subtract $1, and the remainder will he the premium. ^ ' > ' CASE IV. • - , 'y-,y '■'■' To find the value in American currency of any given amount of gold. ^ ;. EXAMPLE., What is the value of $200 of gold, in American currency, gold ( being quoted at 150 ? SOLUTION. \ When gold is quoted at 150, it requires 150 cents, in American currency, to equal in value $1 in gold. Now, if $1 in gold bo worth 91.50 in American currency, $200 will be worth 200 times $1.50, which is $300. Hence the < AMERICAN EXCHAKOE 261 < BULE. Multiply ilie value of %\ hy the number denoting the amount of gold to he changed, and theproditct will he the value in American currency; or To the given turn add the premium on itself at tJie given rate, and the result will he the value in American currency. EXERCISES. ■ 1 If American currency ia at a discou^" > of 00 per cent, what ia the value of $450 ? Ans. $225. 2. The quotation of gold is 140, what is the discount on Ameri* can currency? Ans. 28 ij percent. •3. A person exchanged $750, American money, at a discount of 35 per cent, for gold ; how much did he receive ? Ang. $487.50. 4. Purchased a draft on Montreal, Canada East, for $1500 at a premium of 64J^ per cent. ; what did it cost rac ? Ans. $2473.12. 5. If Amfcrican currency is quoted at 33i^ per cent, discount ; what is the premium on gold ? Ans. 50 per cent. 6. Purchased a suit of clothes in Toronto, Canada West, for $35, but on paying for the same in American funds, the tailor charged ^ me 32 per cent, discount ; how much had I to pay him ? Ans. $51.47. 7. What would be the difference between the quotations of gold, if greenbacks were selling at 40 and 60 per cent, discount ? V "^v^/ Ans. 83^ per cent. 8. P. Y. Smith borrowed from C. 11. King, $27 in gold, and wished to repay him in American currency, at a discount of 3S per eent. ; how much did it require ? Ans. $43.55. 9. J. E. Pekham bought of Sidney Leonard a horse and cutter for $315.50, American currency, but only having $200 of this sum, 'V^ be paid the balance in gold, at a premium of C5 per cent. ; how much did it require ? Ansi $70. 10.^ A cattle drover purchased of a farmer a yoke of oxen valued at $135 in gold, but paid him $112 in American currency, at a y^ discount of 27 j per cent. ; how much gold did it require to pay the balance ? Ans. $53.80. 11. W. H. Hounsfield & Co., of Toronto, Canada West, purchased in New York City, merchandise amounting in value to $4798.40, on 3 months' credit, premium on gold being 79f per cent. At tho I-. , m 262 ARITHMETIC. expiration ol' the three months they purchased a draft on Adams, KimbuU nnd Mooro, of Now York, for tho amount duo, at a discount of 57J per cent. ; what was tho gain by exchange ? Ans. $G47.75. 12. A makes an cxchanp;e of a horso for a carriage with B ; tho horse being valued at $127.50, in gold, and the carriage at S210, American currency. Gold being at a premium of 05 per cent.; what was tho diftbrcncc, and by whom payable ? Ans. B pays A 2.^ cents in gold, or 37 cents in greenbacks. 13. A merchant takes 803 in American silver to a broker, and wishes to obtain for tho same greenbacks which arc selling at a dis- count of 30 per cent. The broker takes the silver at 3i per cent, discount ; what amount of American currency does the merchant receive ? Ans. $80.85. 14. I bought tho following goods, as per invoice, from John McDonald & Co., of 3Iontreal, Canada East, on a credit of 3 months : 1120 J yards Canadian Tweed at 95 cents per yard. 2190 " long-wool red flannel at GO *• " « 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 G3£ per cent. ; what shall I mark each piece at per yard to make a net gain of 20 per cent, on first cost ? « Ans. C. tweed, $2.44 ; red flannel, $1.54 ; white flannel, $1.41. 15. A merchant left Toronto, Canada Wegt, 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 tho balance in greenbacks, which were at a discount of 41^ per cent. When in New York he drew from this amount $23.85 to ''square" an old account then past due. On arriving home he found that he still had in greenbacks $16.40, which he disposed of at a discount of 43f per cent., receiving in payment American silver at a discount of 3^ per cent., which he passed ,oflf at 2 J 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 ? 1 Ads. Total expenses in gold, $71.76; expenses in greenbacks, $118.04 ; silver received, $9.53. '(-;r-i EXCHANQE WITU OHEAT BRITAIN. . ■ EXCHANGE "WITH GREAT BRITAIN^; l. ;,.,i. In Britain money is reckoned by pounds, shillings and )^rAh, and fractions of a penny, and is called Sterling mon^-y,. the gold sovereijrn or the pound sterling, consisting of 22 parts gold and 2 alloy, being the standard, and the sliilling, one-twentieth part of tho pound, a silver coin of 37 parts ailvcr and 3 copper, and the penny, one-twelfth part of the shilling, a copper coin, tho ingrcdicuts and size cf which have frequently been altered. Tho comparative value of the gold sovereign in the United States previous to the year 1834 was $4.44 .J, but by Act of Congress passed in tliat year it was made a legal tender at the rate of ^^f^ cents per pennyweight, because the old standard was less than the intriLsic value and also because the commercial value, though 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 86<| cents. The increase in the standard value was, therefore, equal to 9^ per cent, of its nominal value. The real par of exchange between two countries is that by which an ounce of gold in one country can be replaced by an ounce of gold of equal fineness in the oth'^r country. If the course of exchange at New York on London were 108^ per cent. ] and the par of exchange between England and America 109|per cent., it follows that the exchange is 100 percent, against England ; but the quoted exchange at New York being for bills at 60 days sight, the interest must be deducted from the above differ- ence. The general form for the quotation of exchange with England is: 108, 108J, 109, 109i &c., which indicates that it is at 8, 8^, 9, or 9^ per cent, premium on its nominal value. ■I I K. ■'I" It nn ■\iti-i. ,1 'j'l m I S X A M P L K . What amount of decimal money will be required to purchase a draft on London for JE648 17s. 6d. ?— exchange 108. ., . - Tho old par value or itominai value is $4.44J=^^®=^ of $40 H ij 264 RITHMETia by reaucbg to an improper fraction. Now, tio quotation in 108, or 8 per cent, above *ho nonunal vuluc, vc find the premium on $40 at 8 per cent., which itt $3.20, which added to $40 will gwo $43.20, and $43.20+9=$4.80 to be rcniilUid for every pound sterling, and therefore £648 178. 6d. multiplied by 4.80 or 4.8 will be the value in our mooej. ITs. 6d.=:.875 of a pound, and the operation is ai follows : £648,875 4.8 6191000 2595500 $3114.6000 auLB. Po $40 €tdd the premium on tttel/ at the quoted rate, multipijf the sum hy the number repre$eixting the amount of iterling moneys and divide the rctult hy 9, the quotient will be the equivalent of the aterling money in dollars and ceiits. Note.— If there be shillings, pence, &c., in the Btcrlicg money, tbej are to be reduced to the decimal of £1. To find the value of decimal money in sterling money, at any given rate above par. Lot it be required to find the value of $465 in sterling money, at 8 per cent above its nominal value. Here wo have exactly tho converse of tho last problem, and therefore, havinj; found the value of £1 sterling, we divide the given sura instead of multiplying ; thus the premium on $40, at 8 per cent., is $3.20, which added to $40 makes $43.20, and 43.20->-9r^4.80, and $465-r4.80=£96.17.6. ■- - ♦ a u L 1 . . Divide the given sum hy the nvmber denoting the value of on^ pound sterling at the given rate abo, fpar, and if there be a decimal remaining reduce it to shillings and pence. ^. • * IXIRC ISBS. 1. When sterling exchange is quoted at 108. what is the valua of £17 Ai)5.i4.80» ESCHANQE WTTn GREAT DBITAIN. 266 3. If £1 sterling bo worth $4.84^, what is the promium of cx« y change botweeti Londua and America. Ana. 9 per cent. 3. At 10 per cent, above itn nominal value, what is the worth of £60 sterling, in decimal currency ? Ans. $244.44. 4. When sterling ezohangc is quoted at 9\ per cent, premium, V what is the valuo of $1000 ? Ans. £205 18s. llfd. 5. At 12 per cent, above its nominal value, what will a bill for £1800 cost in dollars and cents ? Ans. $8960. 6. A merchant sold u bill of exchange on London for £7000, at an advance of 11 per cent ; what did he rooeive for it more than its real value? Ans. $466.66| . 7. Bought a bill on London for £1266 15s. at 9^ per cent, pre- mium ; what nhall I have to pay fur 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 ho save ? Ans. $122.66j. 9. A merchant in Kingsto i paid $7300 for a draft of £1500 on Liverpool at what per cent, of premium was it purchaaod ? Ans. 9J. 10. Exchange on London can be purchased in Detroit at 108J ; ^ in New York at 108 J. At which place would it be the most advan- tageous to purchase a bill for £;]58 148. Dd., supposing the N.Y. broker charges ^ per cent, commission for investing and gold drafts Dn New York are at a premium of j| per cent. Ans., Detroit by $6.82. 11. A broker sold a bill of exchange for X2000, on commission, it 10 per cent, above its nominal value receiving a commission of ,'0 per cent, on the real value, and 5 per cent, ou what he obtained for the bill above its real value ; what was his commission ? Ans. $11,955. 12. I owe A. N. McDonald & Co., of Liverpool, $7218, net pro- ceeds of sales of merchandise c£fected for them, which I am to remit them in a bill of exchange on London for such amount as will close the 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. Required the amount of the bill, in sterling money, to be remitted. 18 Ans. £1500. .<-m r^n .tail 1 m IS i 26(J . , ABrrnMsno. . > . . « TADLB or rORHION UONBTB. CiTUM AKD Conmiia London, Liverpool, &c DRiraKt. • t fcimi, the operation may be short- ened in the foUowir 5 manner - L jt i" be required to find the six- teenth power of 2. We first £nd tho second power of 2, which is 4, INVOLUTION, 273 then 4X4=16, which is the fourth power, and 16x16=256, th« eighth power, and 256x256= 65536, the sixteenth power. If wo wished to find the nineteenth power, we should only have to multiply the last result by 8, which is thf) third power of 2, for 2' « X23=2» ». £X£BCISES. 1. Find the second power of 697. 2. What is the third power of 854 ? 3. What is the second power of 4.367 ? Find the fourth power of 75. What is the sixth power of 1.12? What is the second power .7, correct to six places ? Ans. .060893+ Ans.'485809. Ans. 622835864. Ans. 19.070689. Ans. 31640625. Ans. 1.9738+. r.sfii OQ4 9 7. What is the fifth power of 4 ? 8. Find the third power of .3 to three places ? 9. What is the third power of I ? 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 5J ? 16. What part of 8^ is 2" ? 17. What is the difference between 5^ and 4° ? Ans, 11529. 18. Expand 3^ X2^ Ans. 3888. 19. Express, with a single index, 47^ X47^ X47<' ? 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. Ans. 1024. Ans. .027. Ans. »^^. Ans. 1.800943. Ans. 4.538039. Ans. Ans Ans. .7776. Ans. -4^=30 J. Ans. |« 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 ? Ans. .000001. Ans. .00000081. Ans. 1.2762815625. Ans. .000000001. Ans. .00001936+. The second power of any number ending with the digit 6 may be readily found by taking all the figures except the 5, and multi- • This exercise will be most readily worked by finding the sixteenth power, and dividing by 1.04. go in the next exercise, find the thirty-second power, and divide by 1.05. A still more easy mode of working such ques* ttons will be found under the head of logarithms. ! t « I I 'I I "i Ml ■■■*• , 2T4' ABIIBMETIO. ;4ying that by itself, increased by a unit, and annexing 25 to tht result^ I 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, 625 3.5 4 1225 6.5 7 4225 10.5 11 11025 21.5 22 46225 57.5 58 330625 EXERCISES ON THIS METHOD. 26., What is the second power of 135 ? 27. What ia 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 ? Ans. 18225. Ans. 42025. Ans. 112225. Ans. 207025. Ans. 342225. Ans. 632025. Note.— The square root of an7 quantity ending in 9, must end in either 3 or 7. No second power can end in 8, 7, 3 or 2. The second root of any quantity ending in (5, must .^nd in 4 or 6. The second root of any quantity ending in 5, nyist end also in 5. The second root of any quantity ending in 4, must end cither in 8 or 3. The second root of any quantity ending in 1, must end either in 1 or 9. The second rout of any quantity ending in 0, must also end in 0. EVOLUTION. The root of any quantity is a number such that when repeated, to a factor, the specified miinbor of times, will prodiioo that quantity. Thu.'5, -5 rcpcutod twice hh u I'aotur gived 5), and Hidrofon^ 3 is oiiliod the second root nf 9, whilo 3 taken three times as a faotor will givo 27, and therefore 3 is called tlio thini root of 27, Utl4 so also it ia oallod the /(n/r<^ roo< of 81, Thcro aro two waya of liulioatlng this. First, by the mark ;/ 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 marlc i; attached, the Hiniplo quantity nr fitnt rort In liullcated. When Il» • «ntn»il ivut la meant, the mark y alone is placed before the quantity, bu'. if the third, fourth, &c., rouia are tu be iudioated. SECOND OB 8QUABE BOOT. 275 the figures 3, 4, &o., are written in the angular space. Thus: 3=|/9=^ 27=^^81 =^243, &c., &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, 27^ 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 64. Now the third root of 64 is 4, and the second power of 4 is 16, 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 the process of finding any required root of a given quantity. SECOND OR SQUARE ROOT. Jf Extracting the square or second root of any number, 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 tabic of second powers, it will bo found that the second power of any whole number less than 10, consists of either one or two digits ; the 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 either twice the number tC Mgits, or one less than twice the number of digits that the root i^^olf consists of Hence, if wo begin lit Uiii nulls' figure, and mark oil" lliu (flvtiii niimbiM' In porloda of two figures each, We "hall find that the number of digits contained in the root will |)o tlio same as tliti number iil' purloilti. H' llio nuni bet of digits is even, each period will consist of two figures, but if the number of digits be o4d, the last period to the lol't will consist of only one figure. Let it now bo rnf|nirod to find tho Huound root of 141. Wo know by the rule of involution that 144 is the second pou'er of 12. Now 12 may bo resolved into one ten and tico units, or 10-|-2, and 10-f 2 multiplied by itself, as in the mi'Tgin, gives 100-|-404-4, ancl uinm 100 is the second pnwer of 10, and \ the mmn] power of 3, and 40 is twleo the product of 10 and 2, wc conclude that the second I' 'V: M •I 276 AIUTHMETIO. I ti I' I' 10+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 twioo the product of the parts. Hence to find the second root of 144, let us resolve it into the three parts 100+40+4, and wo find that the second root of the first part is 10, and since 40 is twice the product of tho parts, 40 divided by twice 10 or 20 will give the other part 2, and 10+2=12, tho second root of 144. We should find tho same result by resolving 12 into 11+1, or 9+3, or 8+4, or 7+5, or G+6, but the most convenient modo is to resolve into the tens and the units. In tho same manner, if it bo required to find the second root of 1369, we have by resolution 900+420+49, of which 900 ia the second power of 30, and 30x2=00, and 420-j-60=7, the second part of the root, and 30+7—37, the whole TOOt. Again, iel U. Idc required to find the second root of 15129. This may be resolved as below : 10000 is the second power of 100. 400 is the second power of 20. 9 is the second power of 3. 4000 is twice the product of 20 and 100. 600 is twice the product of 100 and 3. 120 is twice the product < f 20 and 3. I : 15129 is the sum of all, and hence 1 is the root of the hundreds, 2 the root of the tens, and 3 tho root of the units. Gietleralizing these investigations, we find that the second power of a number consisting of units alone is the product of that number by itself; that the second power of a number conslsliug of (ens and units is the second power of the tens, plus the secon(^ power of the units, plus twice the product of the tens and units ; that the seoond power of a number, consisting of hundreds, tens and units, is thti fcum of the squares of the hundreds, tho tens, and the units, plus twice the product of each pair. Now since the ooinjiioniont of tho full second power, to the sum of tho seoniir] powers of the parts, is twice the product of the parts, it follows that, when the first figure of tlie root has been found, it must be doubled before used as a divi- sor to find the sccuud Lurm, and for tho Hiunu ronson each figure, whnn fbuiid, must |)p !|Qu|j{e(| to gjyo eorreotjy tho next divisor, ileuoe the r ■ SECOIO) OR 8QUABE ROOT. 277 RULE. Beginning at the v,nit»^ figure, mark off the whole line inpmod» 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 as a divisor, and find how often that quantity is contained in the second partial dividend, omitting the last figure ; annex the figure thus found to both divisor and quotient, multiply and subtract as in common division, and to tJie remainder annex the next period; double the last obtained figure of the divisor, and proceed as before till all the periods arc exha'ustcd,r-if there be a remainder^ annex to it two ciphers, and the figure thence obtained will be a decimal, as will every figure thereafter obtained. EXAMPLES. 1. To find the secon(\ root of 7l)744!V First, comiuiiiuilng with tlio units' figure, wo divide tho line into perigilit, viz., 49, 74 and 79,— wo then note that tho Rrcatofit squaro ooulaiued In 79 is 6-4, — this we subtract from 79, and find 15 remaining, to which 803 we annex the next period 74, and placa 8, the second ropt of 1)4, in the quotient, and ita double IG as a divisor, and try how often 16 is contained in 157, which we find to be 9 times ; placing the 9 in both divisor and quotient, we multiply and subtract as in common division, and find a remainder of 53, to which wo annex tho last period 49, and proceeding as before, we find 3, tho last figure of the root, without remainder, and now wo have tho complete root 893. 2. This operation may bo illustrated aa follows : To find the second root of 273529. ir 6 169 1783 797449 64 1574 1521 5349 5349 600 500x2=1000+20, or 1020 ' , 10004-2X20+3=1043 273529 250000 3129 3129 3=523 > i' Hi. Itlj i; ' I 'if 6 i "'■'is I 278 AIOTHMETIO. 3. To find tho second root of 153687. Hero wo obtaia, by tho same process as in the lost example, thu whole nuiubor 392, with a romaiadur of 23, whioh can produoo only a fraction. 69 782 78402 784049 392.029+ 230000 156804 7319600 7056441 263159 Wo now annex two ciphers, placing tho decimal point after the root already found, but oa the divisor is not contained in this Dew 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 the whole root is 392.029+. Ana. 529. Ans. 8642. Ans. 678. Ans. 28.01785+. Ans. 41.569219+. Ans. 25.8069+. EXEROISBB. 1. What is the second root of 279841 ? 2. What is tho second root of 74684164? 3. What is tho second root of 459684 ? 4. What is tho second root of 785 ? 5. What is tho sojond root of 1728 ? 6. What is the second root of 666 ? 7. What is the second root of 123456789 ? Ans. 11111.11106+. 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 tho length of each side of a square field containing 893025 square rods ? Ans. 945 linear rods. The second root of a fraction is found by extracting the roots of its terms, for it=^X| and therefore |/^«=|/|x!=5- So also, i/i^i=l' Again, since i/j%^q=j%=.09, and .3X.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 deoimai or of a whole number and a decimal: SECOND OB SQUARE ROOT. w Point off period* of two figurea tack from the decimal point towards the right and left, adding a cipher, or a repetend, if th§ number of figurea be odd. , From what has boon said, it is plain that ovcry period, except tho first on the left, must, consiat of two dij^its, and every decimal prcauppoaoH soiuethint^ Koi'^f^ before, for .5 indicates tho h.ilf of somo unit under consideration, an 1 .5 is cijuivalont to .50, and not to .05, from which it is cbviour? that the second root of .5 is not tho root of .05, but of .50, and tlioroforo tho secoJid root of .5 is not .2-|-, af the beginner would naturally suppose, but ,7-f-, for .2+ is th« approximate root of .05. ADDITIONAL EX£HCI8XS. 11. What is the second root of .7 to five places of decimals ? , . Ans. .83666. 12. Find the second root of .07 to six plaoes. 13. What is the second root of .05 ? 14. What is tho second root of .7 ? 15. Find the second root of .5. 16. What is the second root of .1 ? 17. What is the second root of .1 ? 18. What is tho second root of 1.375 ? 19. What is the second root of .375 ? 20. What is the second root of 6.4 ? 21. Find to four decimal places v'^^j'j. 22. Find |/2 to four decimal places. 23. Find the value of v/3271.4207. 24. Find the second root of .005 to five places. Ans. 07071. 25. Find the square root of 4.372594. Ans. 2.09107-|-. 26. 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-f . Ans. .8919-f . Ans. .74535+. Ans. .31622774-. • Ans. .3. Ans. 1.1726, &c.* Ans. 61237, &c.* Ans. 2.52982-}-. Ans. 1.774a Ans. 1.4142. Ans. 57.1964-. Ans. .L Ans. 031624-. Ans. .01. Ans. .001. * The young student would naturally expect that the decimal figures of ^1.376 and y/.^JS would be the same, but it is not so. If it were so, 1/14- y.375 would be equal to y/ 1.376. That such ia not the case, may be showo by a very simple example. i/16-{-y'9z=i-^Z—7, but y'Ki-f 9=^/25=5. Let it be carefully observed, therefore, that the sum of the second roots is not the same aa the second root qf the awn. •tl •,n 1 1 K 4 ■•■IJI m IMAGE EVALUATION TEST TARGET (MT-3) // ^&^/ ?, ^ A^ •4 4^0 1.0 I.I S lis 12.0 |25 2.2 im L25 iU ii.6 6" — /2 V ■*^ /A '/ Hiotographic Sciences Corporation 23 WIST MAIN STRffT Wf»STiR,N.Y. U5S0 (716) •73-4503 ? 280 ABUHScEna OPBRATXON 4 i9.0966 16 4.37 trial 83 4.36 troo. Trial 867 309 249 / ?oo great by 1 6068 6069 True 866 6068 5196 ■ ^'' 872 Fore trd find the remtunder, 872, is greater than the diyiaor, 866, which seems inconsistent with ordinary rules ; but it mnst b« observed that we are not seeking an exact root, bat only the closest possible approximation to it. If the given quantity had been 19.0969, we should have found an exact root 4.37. The remainder 872 being greater than the divisor, shows that the last figure of the root is too small by j^^g, whereas 7 would be too great by y^^, and that 866 is not a correct divisor but an approximate one, and that the true root lies between 4.36 and 4.37. i • v . - When the root of any quantity can be found exMStly, it is called % perfect power or rational qtiantiti/, but if the root cannot be found exactly, the quantity is called irrational or iurd. A number may be rational in regard to one root, and irrational in regard to another. Thus, 64 is rational as regards |/64=8, ]^64=4 and |/64=2, but it is irrational regarding any other root expressed by a whole number. But 64, with the fractional index f, i. e., 64^, is rational, because it has an even root as already shown. We may call 64^ either the second power of the third root of 64, or the third root of the second power. In tho former view, the third root of 64 is 4, and the second power of 4 is 16, and according to tha aeoond view, 64^ is 4096, and thd third root of 4096 is 16, the same aa before. |/81=:3 is rational, and |/81=9 is rational, but 81 is oot rational regarding any other root ; while y'25 is rational only regarding the second root, and ^8:=2 only regarding the third root. The second root of an even square may be readily found by r0> •olrii^ the number into its prime faotorsi and taking each of thost THIBD BOOT OB CUBE BOOT. 281 AetoM once,— the product will be the root. Thus, 441 is 3x3X7X7 and oaoh factor taken once is 3x7=21, the second root. Here let it be observed, that if we used eauh factor twice we should obtain the tecond prywer, but if we use each factor half the number of times that it occurs, , we shall have the second root of that power. 64 is 2X2X2X2X2X2=26, t. e., 2 repeated sis timr^a as a factor gives the number 64, and therefore half the number of those factors will give the second root of 64, or 2x2x2=8, and 2X2X2 multiplied by 2X2X2=8X8=64. As this cannot be considered more than a trial method, though often expeditious, we would observe that the smallest possible divisois should be used in every case, and that If the number cannot be thus xesolved into factors, it has no even root, and must be carried out into a line of decimals, or those decimals may be reduced to common firaotions. i *,.. ,i f -.-M.^ .n ' ■' •■' >•■■!« ■■'■'* ■■■, ■ - THIBD ROOT OR CTJBE ROOT. As extracting the second root of any quantity is the finding of irhat two equal factors will produce that quantity, so extracting the (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 the given quantity be marked off in periods of three digits each, there will be one digit in the first power for each 'period in the third power. The left hand period may contain only one digit. From the mode of finding the third power from the first, we esn deduce, by the converse process, a rule for finding the first power II I : ii i; 282 ABxrmana from the third. We know by the mle of involation that the third power of 25 is 15625. If we resolve 25 into 20+5, and perform the multiplication in that fbnn. we have 20-J-5 400+100 ^ ; . 100+26 i; iv : 400+200+25=(20+5)a 20+5 ' ' 8000+4000+500 2000+1000+125 . • ' "^ 8000+6000+1500+125=(20+5) '=15625 Now, 8000 is the third power of 20, and 125 is the third power •f 6 ; also, 6000 is three times the product of 5, and the second power of 20, and 1500 is three times the product of 20, and the second power of 5. Let a represent 20 and b represent 5, then a»=20» = 3000 3aa6=3X20«x5 = 6000 3 a 6a;=3x20x5« = 1500 . . 6»=5» = 125 15625 * By using these symbols we obtain the simplest possible method of extracting the third «oot of any quantity, as exhibited by the fubjoined scheme : ^ Given quantity. 15625 .. . > a»=20»=:20x20x20 = 8000 ^ >'-■ ^'-' ■'-;■ II. - v.;::'r ' . ^. Bemainder 7625 U; , w 3 oa 6=3X202X5 = 6000 * *^* Bemainder 1625 3 o 62=3X20X5* = 1500 Bemainder 125 6»=5»=5X5X'^ = 125 From this and similar examples we see that a number demoted bj more than one digit may be resolved into tens and units. Tluu, 25 is 2 tens and 5 units, 123 is 12 tens and 3 unite, and so of all aniBben. ■ ' of A of A THIBD BOOT OB CUBE BOOT. 283 power seoond md. tha len method 1 by the denoted . Thii8, 80 of all To find the third root of 1860867 : '' As this number consists of three periods, the root will consist of three digits, and the first period from the left will give hundrads, thft second tens, and the third units, and so also in ease of remainder, each period to the right will give one decimal place, the first being tenths, the second hundredths, &o., &c. We may denote the digits by a, b and e. a=100 a»=1003= and 30000x20= 3 a 6»=3X 100X400= 1860867(100+20+3—123 1000000 860867 remainder. 600000 Si60867 remainder. 120000 i3=203= 140867 remainder. 8000 Now (a+6)=120 . • . 3 (a+6)2=132867 remainder. 43200, which is contained 3 times+ „ r in 132867, . • . c=3, and 3 (a+bW =3X120^X3= 129600 And 3 (a+i) c2=3X 120X9= And lastly, c»=39= 8267 remainder. 3240 27 27 no remainder* BULB. Mark off the given nwnher in periods of thra figures each. Find the highest third power contained in the left hand periodf and subtract it from that period. Divide the remainder and next period by three times the second power of the root thus found, and the guotient will be the second term of the root. From the first remainder subtract three times the product of the, second term, and the square of the first, PLUS three times the product of the first term, and the square of the second, plus the thirt vower of the second. Divide the remainder by three times the square of the *um of Out first and uoond terms, and the auotient will be the third term. ^ ■ ! < i li'l 284 xs:m>i ABITHMZnO. From the Icut remainder subtract three timee the product of th^ term kut found, and the square of the SUM of the preceding ternu^ PLUS the product of the square of the last found term hy the snic of the preceding ones, flub the third power of the last found termf ondsoon. BXBR0I8X8. 1. What is the third root of 46656 ? Ana. 36. 2. What is the third root of 250047 ? Ana. 63. 3. What is the third root of 20Q0576 ? Ans. 126. 4. What is the third root of 5545233 ? Ans. 177. 6. What is the third root of 10077696 ? Ans. 216. 6. What is tLe third root of 46268279 ? ' Ans. 359. 7. What is the third root of 85766121 ? Ans. 441. 8. What is the third root of 125751501 ? Ans. 501. 9. What is t'le third root of 153990656 ? Ans. 536. 10. What is the third root of 250047000 ? Ans. 630. 11. What is each side of a square box, the solid content of which Is 69319 ? Ans. 39 inches. 12. Whal is the third root of 926859375 ? Ans. 975. 13. Fmd 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 cubic vessel whose solid DOiitent is 2936.493568 cubic feet 7 Ans. 1432 feet 16. Find the third root of 6. Ans, 1.7099. 17. A store has its length, breadth and height all equal ; it can hold 185193 cubic feet of goods ; what is each dimension ? Ans. 57 feet. 18. How many linear inches must each dimennon 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. 6.241483. The third root of a fraction is found by extracting the third root of the ter.ms. The result may be expressed either as a common fraction, or aa a decimal, or the given fraction may be reduced to ft decimal, and the root extracted under that form. in the are tiou. TBntD BOOT OB CUBE BOOT. 28S S^-,i> V'l J- Aqs. f =.75. 1XBB0I8I8. 1. What is Ihe third root of || ? '^ ,,, V Otherwise: • g3=.421875. To find tho third root of this we have .42i875(.70+.05=u:.76 703=: 3x702x5 =73500 ■) 3X70 X52= 5250 f- 5»= 125) = 343000 78875 remainder. 78876 no remainder. The third root of a mixed quantity Till lo most readily found by reducing the fractional part to the dec imal form, and applying the general rule. It has been already explained that the second root of an even power may be obtained by dividing the given number by the smallest possible divisors in succession, and taking half tho number of thosa divisors as factors. The same principle will apply to any root. If the given quantity is not an even power, it may yet be found approx- imately. If we take the number 46656, we notice that as the last figure is an even number, it is divisible by 2, and by pursuing the same principle of operation we find six twos as factors, and afterwards six threes; and, as in the case of the second root, we take each factor half the number of times it occurs, so in the case of the third root, we take each factor one-third the number of times it occurs. The same principle on which the extraction of the second and third depends may be applied to any root, the line of figures being divided into periods, consisting of as many figures as there are units in the index ; for the fourth root, periods of four figures each ; for the fifth, five, &c., &o. We may remark, however, that these modes are now superseded by the grand discovery of Logarithmic Computa-- ■tiou. ' .'1' •m k •ft ^■)f.l '■.ill M I 'I ■V- V' ■-•jiti ■ -i ' -Ji,'*»%: s 1 283 ABiiBMina , . PROGRESSION. A «ene« is a saocession of quantities increasing or decreasing by a Common Difference, or a Common Ratio. Progression hy a Common Difference forms a series by the addi- tion or subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23 forms a series increasing by the constant quantity 4, and 28, 21, 14, 7, forms a series decreasing by the constant quantity 7. Such a progression is also called an equidifferent series.^ Progression hy a Common Ratio forms a series increasing or decreasing by multiplying or dividing by the same quantity. Thus, 3, 9, 27, 81, 243, is a series increasing by a constant multiplier 3, and 64, 32, 16, 8, 4, 2, is a series decreasing by a constant divisor 2. The quantities forming such a progression are also called Con- tinual Proportionals,^ because the ratio of 3 to 9 is the same as the xatio of !) to 27, &c., &c. From this it is 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 last terms are called the Extremas^ and all betweea them the Means, , PROGRESSION BY A COMMON DIPPERENOE. In a series incKasing 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=11-1-15=26, and in the second 28-1-7=21 -|- 14=35. ^ '' If the number of terms be odd, the sum of the extremes is equal to twice the middle term. Thus in the series 3, 7, 11, ?5, 19, 3-{-19=2X 11=22, and hence the middle term is half the sum of the extremes. " The names ArUhmeiicdl Progression and OtotMtrical ProgrissUm are often applied to quantities so related, but these terms are altogether inapprc- , priate, oa they would indicate that the one kind belonged solely to arithmetic, and the other solely to geometry, whereas, in reali^, »ach belongs to botb these branches of science. ' PBOORESSION B7 A COMMON DIFFEBENOE. 287 In treating of progressions by difference or cquidifferent seriefl, there are five things to be con'iidercd, viz., the first term, the last term, the common difference, the number of terms, and the sum of the series. These are so related to each other that wlion any three of them are known we can find the other two. Given tho first term of a scries, and the common difference, to find any other term. Suppose it is required to find tho seventh term of tho scries 2, 5^ 8, &c. Here, as the first term is given, no addition is required to find it, and therefore six additions of tho common 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 are 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 liad to subtract the 18 from tho first term 29 to find the seventh term 2. The term thus found is usually designated tJie last term, not beoausc the series terminates there, for it does not, but simply because it is the last term considered in each question proposed. From these illustrations we derive the RULE (1.) Svibtract 1 from the number of terms, and multiply the remainder by the common difference; then if the series be an increasing ontf add the result to the first term, and if the series be a decreasing one, tubtract it. EXAMPLES. To find the fifty-fourth term of the increasing series, the first term of which is 33f, and the common difference 1^. Here 54—1=53, and 53X11=66^, and 66J+33f=100, the fifty-fourth term. ": ■^■.> ■'■'" ■'■' "-' '" Given 64 the first term of a decreasing series, and 7 the common difference, to find the eighth term. Here 8— 1=:7, and 7x7=49, and 64— 49=:15, the eighth term. , » EXERCISES. 1. JB'ind the eleventh term uf the decreasing series, the first term of which is 248f, and the common difference 3^. Ans. 216^. 2. The hundredth term of a decreasing series is 392|, and the common difference :1s 3|, what is the bst term ? Ans; 36. I'' 'i II I m s m I !I88 ABmnntria 3. What is the one-thonauidth torm of the ieriea of the odd figures? AnR. 1999. 4. What is the fiye-handredth term of the Miies of even digits ? Ads. 1000. 5. What is the sixteenth term of the deoreasing series, 100, 96, 92, &c. ? * Ans. 40. To find the smn of any eqnidifforent series, when the number of terms, and either the middle term or the extremes, or two tenns equidistant from them, are given. Wo have seen -already that in any such series the sum of the extremes is equal to the sum of any two terms that are 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 24-7+12+17+22-1-27+32, we have ^21=2111=17, the middle term. It is plain, therefore, that if we take the middle term and half the sum of each equi-distant pair, the series will be equivalent to 17+17+17+17+17+17+17, or 7 times 17, which will give 119, the some as would be found by adding together the original quantities. The same result woj^ld be arrived at when the number of terms is even, by taking half the sum of the extremes, or of any two terms that are equi-distant from them. From these explanations we deduce the RULE (2.) Multiply the middle term, or half the «um of the extreme*, or of any two term* that are equidistant from iher.i, hy the number of temu, , . . : ■'■'■•'' ^ NoTK.— If the sum of the two terms be an odd number, it is generally more convenient to multiply b7 the number of terms before dividing by 2. ' EXAMPLES. ■',^h'*>*i^;^ri^y'^^r\f-^ft Given 23, the middle term of a series of 11 numbers, to find the earn. Here we have onlyto multiply 23 by 11, and we find at once the sum of the series to be 253. Qiven 7 and 73, the extremes of an increasing series of 12 nam- ben, to find the sum. The sum of the extremes is 80, the half of wbiol) ia 40.and 40x12=480. the sum required. PBOOBESSION BT A OOIOCON DIFFEBENCE. 289 .;/lT.l?a . 'ira Two equidistant terms of a series, 35 and 70, arc given in a leries of 20 numbers, to find the sum of tho series. In this eaie, W0 have 96+70=105, and 105x20=2100, and 2100-^2=1050, tho ■am required. ' CXER0I8I8. 1. Find the sum of the series, consisting of 200 terms, the first term being 1 and the last 200. Ahh 20100. 2. What is the sum of the scries whoso first term is 2, and twenty-first 62 ? Ans. 672. '' 3. What is the sum of 14 terms of the scries, the first term of which is }t and tho last 7 ? Ans. 52^. 4. Find the sum to 10 terms of tho decreasing series, tho fint term of which is 60 and the ninth 12. Ans. 360. 6. A canvasser was only able to cam $G during the first month ho was in tho business, but at the eud of two ycar.s was ablo to earn $98 a month ; how much did ho earn during the two years, supposing the increase to have been at a constant monthly rate ? An^. $1248. 6. If a man begins on the first of January by saving a cent oo the first, two on the second, three on tho third, four on the fourth, &o., &c., how much will he have saved at the end of the year, not counting the Sabbarhs ? Ans. $490.41. 7. How many strokes docs a clock strike in 13 weeks ? Ans. 14196. 8. If 8£ is the fourth part of tho middle term of a series of 99 numbers, what is the sum? Ans. 3465. 9. In a series of 17 numbers, 53 and 33 are equidistant from the extremes ; what is the sum of the series? Ans. 731. 10. In a series of 13 numbers, 33 is the middle term ; what is tho turn ? Ans. 429. To find tho number of terms when the extremes and common difference are given : As in the rule (1), we found the difference of the cxtrcma bj multiplying by one leas than the numb&r of terms, and added tho fini term to the result, so now we reverse the operation and find the ,-»*■ ■ i . RULE (3.) 1/ivide the difference of the ex*remet bjf the common diffi and add 1 to the retult. *r:5T^^ • Ui ''':.>] n t-M.j'f'it. :'?*T 990 AIUTBXETIO. I BX AM PLit. , Given the extremes 7 and 109, and the oommon diffieronee, 8, to find t!io nuiubor of tcrnia. In this case wo have 109—7:^102, and 102-i-3=:34, and 344-1—3!), thu number of terms. K X R R I B E 8 . I .■ 1. Wliat iH the number of terms wboft the extremes ore 35 and 333, und tlio common difference 2 ? Ans. 150. 2. Two C(|uidi8t:int turniH are 31 and 329, and the common dif- ffercnco 2 ; what is the number of terms ? Ans. 150. 3. Thu fifi^t term of n series is 7, nnd the lost 142, and the com- mon dilTorcuco \ ; what is the number of terms ? Ans, 541. 4. Tiic first and lust terms of u series are 2^ and 35^, and the •oniiuon difference J ; what is the number of terms ? Ans. 100. 5. The first term of u series is ^ and last 12J, and the common diffen.'nce ^ ; what is the number ot terms ? Ans. 25. Given one extreme, the sum of the series and the number of tcrmn, to find the other extreme. This case may be solved by reversing llulo (2), for in it tho data arc tho same, except that there the second extreme was given to findihe sum, and now the sum is given, to find the second extreme. Therefore, as in that rule we multiplied the sum of the extren.es by tho number of terms and halved the product, so how we must double, tiie sum of the series and divide by the number of terms to find the sum of the extremes, and from this subtract the given extreme, and the remainder will bo the required extreme. This will illustrate the r ■»* *■ BULE (4.) Divide twice the turn of the series hy the numher of terms, and from the quotient subtract the given extreme, and the remainder will be the required extreme. EXAMPLE. ^ . ™ Given 5050, the sum of a series, 1 the first term, and 100 th« Onmbcr of terms, to find tho other extreme. Twice the sum is 10100, which, divided by 100, gives 101, and 101 — 1=:^100, the number of terms. PRoamEssioN by a common difference. 291 ■■■< H-./.a \:^ KXRnOIHES. 1. Given SO, tho greater cxtromo of n decreasing icrios, 442, tha aum, and 17 tho number of terms, to Gnd the other extreme. Ans. 2. 2. If 1212G8 bo tho Hura ofu Norics, 8 tho loss cstrcmc, and 142 tho number of terms ; what is tho ^Toutor oxtrcrao ? Ans. 1700. 3. Tho sum of a scries of 7 terms is 105, tho greater extreme is 21, and tho number of tcniis 7 ; what is tho less extreme ? Ans. 9. 4. Tho sum of a scries is 570, tho numlior of terms 24, and the greater extreme is 47 ; what is tho loss extreme ? Ans. 1. 5. Tho sum of a scries is 30204^, tho greater extreme 312, and tho number of terms 193 ; what is tho loss extreme ? Ans. 1. Qiven tho extremes and nuuiber of terms, to find tho common difibrcnce. As explained in tho introduction to Rule (1), tho number of common differeneos must bo one hss than tho number of terms. It if obvious also, that tho sum of these differences constitutes the differ- cnco between tho cxtrotiics, and that therefore the sum of tho differ- ences is tho same as 1 less than tlic numKcr of terms. Therefore the difforcnco of tho extremes, divided by the sum of the differences, will give one difference, i. c, tho common difference. This gives us tho nuLE (5.) Subtract 1 from the number of terms, and divide the difference of the extremes by the remainder. * ;',^). .V'.^,.^w:....,,\.< .,. EXAMPLE. If the extremes of an increasing series be 1 and 47, and the number of terms 24, wo can find the common difference thus: — 47 — 1=46, and 46-7-23^=^2, tho common difference. EXERCISES. 1. If the extremes are 2 and 36, and the number of terms 18; vhat the common difference ? Ans. 2. 2. What is the com non difference if the extremes arc 58 and 3, and the number of terms 12 ? .y:%^'\^ ■« I ■ Ans. 5. 3. In a decreasing series given 1000 the less extreme, and 1793 the greater, and 367 the number of terms, to find the common difference. Ans. 2^. 11 •n H I 292 abuhhetio. 4. If 6 and 60 are the extremes in a series of 10 numbers, what is the common diffurenoe ? Ans. 6. '^' 5. What is the common difference in a decreasing series of 42 tenr i^he extremes of which are 9 and 50 ? Ans. 1. There are fifteen other cases, but they may all be deduoei from the five here given. We subjoin the ^Jgebraio form as it is more satisfaotoiy and complete, and also more easy to persons acquai ited with the symbcla of that science. Let a bo the first term, d the common difference, n the number of terms, 5 the sum of the series ; the series will be represented by a+(a+d)+(o+2d)+(a-}-3. . , ,V «=^(a+0.(3.) This is a convenient form when the last term is given. Using I for the last term, we have five quantities to consider, viz., a, I, d, n, «, and, as already stated, any three of these being ^ven. the other two can be found from (1) and (2.) - "^ f * ' To find (2 when a, Z, n are given : By (2.) I=a-\-(nr-l)d J i- . • . I — a=(« — l)df j _ I— a . \ This finds the common difference, when the extremes and nam- ber of terms are given, and corresponds to Rule (5.") ^ „ . ,.^~ , ... If a, n, s are given, we have J3y(l.) ,=||2a+(n-l)d|. . 2»=2an+n (n— 1)v;:i-r.:^-?...;^'.'V =6+(20~l)4 ■, =82 20 * tnd hence by (3) »=--.(6+82) j. =880. •* k ; X ' V ; . Giyen 0=3, 2=300, n=33, to find d By (4) cfc=i=? 297 >j^-:-; n 32— »U2- MIXED 2XER0ISES. 1. Given 70, the less extreme, 10 the common difference, and 44 the number of terms, to find the sum. Ans. 12540. 2. What is tl^e less extreme when the grealer is 579, the common difference 9, and the sum of the series 18915 ? Ans. 3. 3. What is the series when *=143, d=2, n=ll ? • Ans. 3, 5, 7, 9, 11, 13, 15, &o. 4. Given 4 and 49, the extremeti, and 6 the number of terms, to find the series. Ans. 4, 13, 22, 31, 40, &c „ -li 5. K 120 stones are laid in a straight line, on level ground, at a K^lar distance of a yard and a quarter, how far must a person travel to pick them all up one by one and carry them singly and place them in a heap at the distance of 6 yards from the first, and in the same line with the stones ? Ans. 10 m. 7 fur., 27 rds., 1^ yds. • 6. Insert three means between the extremes 117 and 477. / Ans. 207, 297, and 387. The other variations are left as exercises for the student FBOaBESSIOMS BY RATIO. 595 7. A courier agreed to ride 100 miles on condition of being pai^ 1 cent for the first mile, 5 for the second, 9 for the third, and so oqf; how much did he get per mile on an average, how much foV tjh? la^t mils and liow much altogether? .| Ans. $1.99 per mile, $3.97 for the last, and $199 for all. 8. A man performed a journey in 11 days on horsebackT-t^jB first day ho rode 45 miles, but, his horse getting lame, he was forced to slacken the puce at a certain rate per day, so that ct the last day he made only five miles ; what was the length of the journey, m^d iat what rate did h« slacken his speed ? Ans. The journey was 275 miles, and the slackening of speed 4 m. per day. 9. Find the series of which 72 is the sum, 17 the first term, and number of terms 6. ' Ans. 17, 15, 13, 11, 9, 7. 10. The Venetian clocks strike the hours for the whole day ; how many strokes will one of these strike in a year. Ans. 109500. 11. An Eastern monarch being threatened with invasion, offered his commander-in-chief a reward equivalent to a mill for the first soldier he would enlist within a month, two for the second, three for the third, and so on ; the officer enlisted 999,999 men ; what was his reward equal to in our money. Ans. $499,999,500. 12. One hundred sailors wore drawn up in line at a distance from each other of 2 yards, including the breadth of tho body — the pay- master, seated a distance of two yards from the fir«t, sent a lieutenant to hand to the first a sum of prize money, then back again to the Becondj and so on to each sugly ; how far had the lieutenant to walk? Ans. 11 miles, 3 fur., 32 rods, 4 yds. . 1 ■■:• PROGRESSIONS BY RATIO. i-1 ■ There are in progression by ratio, as in progression by difference, the same five quantities to be 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, we have increase or decrease hy multiplication «r division. If any three of these are known the other two can be found. We have noticed already that if any quantity, 2, bo multiplied by itself, the product, 4, is called the square, or second power of that 206 ABIFHMETIO. quantity ; if this bo again multipUed by 2, tho prodnot, 8, is ealled the cube, or third power of that quantity ; if this again be multi- pUed by 2, tho product is called the fourth power of that quan- tity, and 80 on to the fifth, sixth, &o., powers. To show the short mode of indicating this, let us take 3x3x3x3x3=243. For brevity this is written 3^, which means that there are 5 factors, all 3, to be continually multiplied together, and 5 is called the index, because it indicates the number of equal factors. Given the first term and the common ratio to find the last pro- posed term. Let it be required to find the sixtn term of tho increasing series, of which the first term is 3 and the ratio 4. This may obviously be found by successive multiplications of tho first term, 3, by the ratio, 4, — thus : — . , . ,,^. , , „ ,• , . 3=l8t term. ,. • „ '■-' :.'^-Jv »X4= 12=2ndterm. V ^ . 12X4= 48=3rdterm. ' ^r . 48X4= 192=4th term. ■ .:.■ ''^^■;ir.,':^/^■' ...■'■■':^ 192X4= 768=5th term. ' ' : ' "^^ f V fc* ^ 768X4=3072=6th term. <: - The series, therefore, is 3, 12, 48, 192, 768, 3072. From this, it is plain, that as to find the last of 6 terms, only 5 multiplications bf the first are required, in all cases the number of multiplications will be one less than the number of terms. But to multiply five times by 4 is the same as to multiply by 1024, the fifth power of 4, for 4X4X4X4X4=1024, and 1024X3=3072.* This gives us tho general RULE (1.) Multiply the first term by that power of the given ratio which M a unit less than the number of terms. If the series be a decreasing one, divide instead of multiplying. BX AMP LBS. ^ Given in a sf^ries of 12 cumbers, 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 lltli 'poprer of 4, which is 2048, and this, multiplied by the first term, 4, glres 8192 for the twelfth term. ^ * For the abbreviated mode see Involution. PBOGBES8I0NS BT BATIO. 29T Given the ninth term of a decreasing series, 39366, and the ratio 3, to find the first term. As there are 9 terms, we take . the 8th power of the ratio, 3, which we find to be 6561, and the first term 39366—6561 =6, the first term. -,..,-■-■'- v-v ^>,v^,--,^; .■■.'-.' .c ^ .:.;,-'■.--' EXERCISES. 1. What is the ninth term of the increasing series of which 5 is the first term and 4 the ratio ? Ans. 327680. 2. What is the twelfth term of the increasing series, the first term of which is 1 and the ratio 3 ? Ans. 177147. 3. In a decreasiDg series 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 20tk term of an increasing scries, the first of rhich is 1.06, and also the ratio 1.06 ? Ans. 3.207135. 0. In a decreasing series the first term is 126.2477, the ratio 1.06 ; what is the last of 5 terms ? Ans. 100. Given the extremes and ratio, to find the sum of the series. It is not easy to give a direct proof of this rule without the aid of Algebra, but the following illustration may be found satisfactory, ■and, in some sort, be accounted a proof. Let it be required to find the sum of a series of continual pro* portions, of which the first term is 5, the ratio 3, and the number of terms 4. Since 3 is the common ratio, we can easily find the terms of the series by a succession of multiplications. These are — 5+154-45-f 13j>, and the sum is 200 15-^45-{-135-|.405 400 ' - ' Let us now multiply each term by the ratio, 3, and, for oonye- nience and clearness, place each term of the second line 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 difference of the two extreme quantities, viz., 400. Now, it will bo seen that this remainder is 'jzaotly double of the sum of the aeries, 200, and consequently 400 divided by 2, will give the som 200. Also, 405 is the product of the last term by the ratio, and 400 is the diffei'cnoe between that product and the first term, and t|M divisor, 2, is a unit less than the ratio, 3. Henoe the 20 298 ABITHMETIO. BULK (2.) Multtplj/, the last term hy the ratio, from this product suhtrad the first term, and divide the remainder by the ratio, diminished by unity, > • .^'r ^«!.,,vr EXAMPLE. Given the first tenn of an increasing series, equal 4, the ratio 3, and the number of terms G, to find the sum of the series. By the former rule we find the las* term to bo 972. This, mut tiplied by the ratio, gives 2916, and the first extreme, 4, subtracted from this, leaves 2912, and this divided by 2, which ial less than tho ratio, gives 1456, the sum of the series. ' ''j ' ,, '' "'-'' '■ '' ' ''■■'^■' EXERCISES. ' 1. What is the sum of the series, of which the less extreme is 4, the ratio 3, and the number of terms 10? Ans. 118096. 2. What is the sum of the series, of which 1 is the less extreme, 2 the ratio, and 1 4 the number of terms ? Ans. 16383. .3. What is the sum of the series, of which the greater cxtremo is 18.42015, tho less 1, and the ratio 1.06 ? Ans. 308.755983. 4. A cattle dealer offered a farmer 10 sheep, at the rate of a mill for the first, a cent fok the second, a dime for the third, a dollar for the fourth, &c., &c. ; in what amount was he '* taken in." supposing that each sheep was worth $11,111? Ans. ^11111 00.00, 5. What is the sum of six terms of the series, of which tho ijreater extreme is J and the ratio | ? Ans. 1 735, or IgVVs- To find the ratio when the extremes and number of terms are given : Let it be required to find the ratio when the extremes are 3 and 192, and the number of terms 7. This is effected by simply reversing the first rule, and Iherefore we divide 192 by 3 and finol 64, and take the 6th root of 64, which is 2, the ratio. Hence the RULE (3.) "^^ Divide th^ greater extreme hy the less, and find that root o/thf quotient, the index 0/ which is one less tJum the number of terms, EXAMPLE. If the greater extreme is 1024, and^the less 2, and the numbet of terms 10, we divide 1024 by 2, and find 512, and then by •ztraoting the ninth root of 512, we find the ratio, 2. FBOOBESSIONS BY BATIO. 299 EXEROISEB. 1. If the first yearly dividend of a joint stock company bo $1, and the dividends increase yearly, so as to form a series of continual proportionals, vrhat will all amount to in 12 years, the last dividend being $2048, and what will bo the ratio of the increase ? Ans. ratio, 2 ; sum, $4095. 2. What is the ratio, in the series of which the less extreme is 3 and the greater 98034, and the number of tcriiis 16. Ans. .10GG05. 3. What is the ratio of a series, the extremes of which are 4 iod 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 ?> and 12288 ? Ans. 4. o. In a series of 23 terms the extremes are 2 and 8388G08 ; what is the ratio? Ans. 2. To insert any number.of means between two given extremes : Find the ratio hy Rule (3), and multiply the first extreme by thin ratio', and the second tciU 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 proportional is found by taking the sguare root of the yrih duct of the extremes. \,..U-Vn:'>'-V'' ■-^■~. !:.■::■ \.''-^ EXAMPLE. '.'u'' '■' Let it be required to insert between the extremes 5 and 1280 three terms, so that the numbers constituting the series shall bo oon« tinual proportionals. The number of terms here is 5, and henc^, by Rule (3), we find the ratio to be 4, and 5 multiplied by this will give the second term, 20, and that again multiplied by 4 will give 80, the third, and that again multiplied by 4 will give the fourth term, 320, so that the full series is found to be 5, 20, 80, 320, 1280. The same result would be found by dividing the greater extreme by 4, and so on downwards, thus: 1280, 320, 80, 20, 5. 'iili •«■>• * EXERCISES 1. Between 5 and 405 insert three terms, which shall make the irhole a series of continual proportionals Ans. 5, 15, 45, 135, 405. 2. Insert between ^ and 27 four terms to form a scries, and give Ihe ratio. Ratio, 3 ; series, I, |, 1, 3, 9, 27. 500 ABITHMETIO. 3. What tluroe numbers inserted between 7 and 4376 will form a lerieK of continual proportionals? Ans. 35, 176, 875. 4. What is the mean proportional between 23 and 8464 ? Ans. 441.2164+. 5. Find a mean proportional between |l} and |. Ans. {. ALOKBRAIO FORM. Let a represent the first term, I the last, r the ratio, n the num- lier of terms, and t the sum. Then $=a-{-ar-{-ar"-\-ar^-\'ar*-\-8K oi*-^-!"****"'* Multiplying the whole equation by r, wo obtain r$=ar-\-ar^-^ar^'{-ar*-{-ar'^-{-&o a»*~*+or*. 'Bxkt$=a-\-ar-^ar^-{-ar^-{-ar*-{-ar'^-{-&c ai*~*. Subtracting, we obtain n — 8==*(r — l)=ar"— a, and therefore . But we found the last term of the series to bo at^^, calling this i, we have from (1.) »=^ (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, at* will become unassignably small, compared with any finite quantity, and may be reckoned as nothing. In this case (1) will heaome 9=^z=r~ (3.) By this formula ^e can find the sum of any infinite series so closely as to diffu from the actual sum by an amount less than any assignable quantity. This is called the limitj an expression more strictly correct than the turn. From the formula *==^—{i any three of the quantities a, r, Z, t being ^ven, the fourth can be found. Let it be required to find the sum of the serira l+i+l+i+ isc., to infinity. Here a=l and r=J . • . »=1— J=-|- =1 X2=2. Therefore, 2 is the number to which the sum of the series continually approaches, by the increase of the number of its terms, and is the limit from whibh it may be made to differ by a quantity less than any assignable quantity, and is also the limit beyond which it can never pass. PB00BES8I0NS BT RATIO. 801 Bj adding the first two terms, we find 1-f ^=^=2 — }=1|. By adding the first three terms, we find ^-f-J— ]=2— J=l^. *' By adding the first four terms, we find |-f |=Y— 2 — ^=lf. By adding the first five terms, we find Y-+ A=ii=2 — j'g= By adding the first six terms, we find ?a+a3=Si=2— g'j= It will bo observed hero that the difference from 2 is continually decreasing. The next term would differ from 2 by g'^ , and the next by j^g, &o., &c. Thus, when the series is carried to infinity, 2 may be taken as the sum, because it differs from the actual sum by a quantity less than any assignable quantity. EXAMPLES. To find the sum of the first twelve terms of the series l-f-3-f9-f- 27-1-&0.: Herea=l, r='6, ' And ,=;:t-_3_^^^=::,2m2JjLTz:l_265720. '* To find the sum of the series 1, — 3, 9, — 27, &o., to twelve terms, II ■iX-» -I •lX-177147— I =—132860. 1 -3-1 ~" , r' . . . In the case of infinite series, if a is sought, « and r being given, we have from (3) a— « (1 — r), and if r is sought, a and « being ♦ given, wc have »*=— or 1 — -. EXERCISES. 1. Find the sum of the series 2, 6, 18, 54, &c., to 8 terms. Ans. 6560. 2. Find the sum of the infinite series \ — ^+t3 — 3i> Observe here r= — \. Ans. §. 3. What is the sum of the series 1, ^, |, &o., to infinity ? *■ Ans. -^. - 4. Find the sum of the infinite series 1 — %-\-% — ^^-\-ka. Ans. 3. 5. What is the sum of nine terms of the series 5, 20, 80, &o. ? Ans. 436905. 6. Find the sum of i/^+l+l/i+^o-) to infinity. Ans. i/J-^l. 7. What is the limit to which the sum of the infinite series f, ^, \% \i ^'i continually approaches ? Am. |. if I) 802' ABITHMEnO. 8. What is tho sum often terini of tho series 4, 12, 36, &o. ? Ana. 118096. 9. Insert three terms between 39 and 3159, so that tho wholo shall be a series of continual proportionals. . - , Ans. 117, .351 and 1053. 10. Insert four terms between ^ and 27, so that the wholo shall form n scries of euntinual proportionals. Ans. ^, 1, 3, 9. 11. The sum of a scries of continual proportionals is 10^, the first term 3^ ; whut is the ratio ? Ans. §. 12. Tho limit of an infinite scries is 70, the ratio | ; what is the first term? Ans. 40. ANNUITIES. Ti. The word Annuity originally denoted a sum paid annually^ and though such payments are often made half-yearly, quarterly, &o., still tho term is applied, and quite properly, because the calculations nro made for tho year, at what iimc soever tho disbursements may be made. By the term annuities certain is indicated such as have a fixed time for their commencement and termination. a "' ' By the term annuities contingent is meant annuities, the com- menceiiient or termination of which depends on some contingent event, most commonly the death of some individual or individuals. By tho term annuity in reversion or deferred, is meant that the person entitled to it cannot enter on the enjoyment of it till after the lapse of some specified time, or the occurrence of some event, geher- ally tho death of some person or persons. An annuity in perpetuity is one that ** lasts for ever," and thore> fore is a species of hereditary property. An annuity forborne is one the payments of which have not been made when due, bt:t have been allowed to accumulate. By the amount of an annuity is meant 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 jbr that time. Tables have been constructed showing the present and final values per unit for different periods, by which the value of any annuity may be found according to the following ANNTTTmS. 308 BULKS. To flrd either tiie amottDt or the present value of ao annoity,— Multiply the value of the unit, as fmnd in the tablet, by ih$ number denoting the annuity. If the annuity bo in perpetuity, — Divide the annuity by the number denoting the »'«/««< 0/ th§ unit for ohe year. If the annuity bo in reversion, — Find the value of the unit up to the date of commencement, and alto to the date of termination, and multiply their difference by the number denoting the annuity. * ;,. '. .. , To find the annuity, the time, rate and present worth being given. ; , Divide the present worth by the worth of the unit. Tables are appended varyini^ from 20 to 50 vcars. EXAMPLES. To find what an annuity of 8400 will amount to in 30 years, at 3^ per eent. We find by the tables the amount of $1, for 80 years, to be $51.G22G77, whieh multiplied by 400 gives $20649.07 nearly. To find the present worth of an annuity of $100 for 45 years, at 3 per cent. By the table we find 024.518713, 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 gives $17142.86. To find the present worth of an annuity on a lease in reversion, to oommenoe at the end of three years and to last for 5, at 3^ per cent. By the table we find the rate per unit ibr 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. Qiven $207.90, the present worth of an annuity continued for 4 years, at 3 per cent., to find the annuity. By the tables, the valuo for $1 is $3.717098. and $207.90, divide] by tiilii, [jivcd ?J5.9U. '!■: •1! 804 AMTHlCEnO. TABLB, nowivo m amocmt or ak Amnrrrr or otn doluk rcB amkum, niraom AT COMTOCND LiTnUeflT VOR ANY NVMBBR OF TRAIU NOT KZORRDrNU niTT. 3 per cent. '3) percent. 1 2 3 4 fi 6 7 8 P 10 11 12 13 14 15 16 17 18 10 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 60 1.000 000 2.03U 000 3.090 UOO 4.1 83 627 5.309 136 6.468 410 7.662 462 8.892 336 10.169 106 ll.4fi:JM79 12.807 796 14.192 030 15.617 790 17.086 324 18.598 914 2) 156 881 21.761 688 23.414 435 23.116 868 26.870 374 28.676 486 30.536 780 32.452 884 34.426 470 36.459 264 38.553 042 40.700 634 42.930 923 46.218 850 47.575 416 50.002 678 62.502 759 55.077 841 67.730 177 60.462 082 63.271 944 66.174 223 09.159 449 72.234 233 75.401 260 78.663 298 82.023 196 85.483 892 89.048 409 92.719 861 96.501 457 100.396 501 104.408 896 108JS40 648 112.796 867 4 per cent. 5 per cent. 1.000 000 2.035 000 3.106 J^S 4.214 9 13 6.:i62 466 6.550 162 7.779 408 9.051 687 10.308 496 11.731 393 13.141992 14.601 962 16.113 030 17.676 98(1 19.295 681 20.971 030 22.705 016 24.499 691 26.3.'»7 180 28.279 682 30.269 471 32.328 902 34.460 414 36,666 628 38.949 857 41.313 102 42.759 060 46.290 627 48.910 799 51.622 677 54.429 471 57.334 602 60.341 210 63.453 152 66.674 013 70.007 C03 73.467 869 77.028 895 80.724 906 84.550 278 88.509 637 92.607 371 96.848 629 101.288 831 105.781 673 110.484 031 115.350 973 120.388 297 125.601 846 130.999 910 1.000 000 2.040 000 3.121 600 4.246 464 5.416 323 6.r.:{2 976 7.898 204 9.214 226 10..')82 795 12.006 107 13.486 351 15.02.'^ 805 16.626 838 18.291 911 20.023 588 21.824 531 23.697 512 25.645 413 27.671 229 29.778 079 31.9G9 202 34.247 970 36.617 889 39,082 604 41.615 908 44.311 745 47.084 214 49.967 683 62.966 286 56.084 938 69.328 335 62.701 469 66.209 627 69.857 909 73.652 225 77.698 314 81.702 246 85.970 336 90.409 150 95.026 616 99.826 636 104.819 698 110.012 382 115.412 877 121.029 392 126.870 668 182.946 390 139.263 206 146.83) 734 6 per cent. 1.000 000 2.050 000 3.152 500 4.310 125 6.625 631 6.801 913 8.11.42 008 9.649 109 11.026 564 12.577 893 14.206 787 16.917 127 17.712 983 19.598 6:i2 21.578 664 23.657 492 25.840 366 28.132 385 30.539 004 33.065 954 35.719 252 38.606 214 41.430 475 44.501 999 47.727 099 51.113 454 54.669 126 68.402 583 62.322 712 66.438 848 70.760 790 75.298 829 80.063 771 85.066 959 90.320 307 95.836 323 101.628 139 107.709 546 114.095 023 120.799 774 127.839 763 135.231 751 142.993 339 151.143 006 159.700 156 168.685 164 178.119 422 188.026 393 198.426 663 7 por cent. 162.667 0841209.347 976 1.000 000 2.060 000 3.183 600 4.:.74 616 5.637 093 6.975 319 8.893 838 0.897 468 11.491 316 13.180 795 14.971 64:t 16.869 94) 18.882 138 21,015 060 23.275 970 25.670 528 28.212 880 30.905 653 33.759 992 36.785 691 39.992 727 43.392 290 46.995 828 60.815 577 64.864 51*2 59.156 383 63.705 766 68.528 112 73.639 798 79.058 18(i 84.801 677 90.889 778 97.343 165 104.183 755 111.434 780 119.120 867 127.268 119 135.904 206 145.058 458 154.761 966 165.047 684 175.950 645 187.607 677 199.768 032 212.74.<\ 614 226Ji08 126 241.098 612 266.664 629 272.968 401 290.33fi 905 1.000 000 2.070 000 3.214 900 4.439 943 6.750 739 7.153 291 8.651 021 10.259 803 11.977 C89 13.816 448 1.5.783 599 17.888 451 20.140 643 22.550 488 25.129 022 27.888 054 30.840 217 33.999 033 37.378 965 40.005 492 44.865 177 49.005 739 53.436 141 68.176 671 63.249 030 68.676 470 74.483 823 80.697 691 87.346 629 94.460 786 102.073 041 110.218 164 118.933 426 128.258 766 138.236 878 148.913 460 160.337 400 172.661 020 IFAMO 292 199.635 112 214.609 670 280.632 240 247.776 496 266.120 861 285.749 311 306.751 763 329.224 386 363.270 093 378.999 000 ,406ii28 9 2ft AMJIU1T1JE8. 806 TABLB, nownio nil nmtm woktr or an akktitt of otni dollau per annum, to ooNTtNUi ron > vr nvmbbr or tbarh not exchidino wtnx. 1 1 3 per cent. ■ 3| per cent 4 per cent, 5 pur cent 6 percent. 7 per cent, g 0.970 «74 0.966 184 0.961 538 0.962 381 0.943 396 0.934 579 1 2 1.013 470 1.899 694 1.8.^6 095 1.859 41U 1.833 393 l.h08 017 2 3 2.H28 611 2.801 637 2.775 091 2.723 248 2.673 012 2.624 314 3 4 3.717 (198 3.673 079 3.629 895 3.545 951 3.4(i5 106 3.387 209 4 6 4.579 707 4.515 052 4.451 822 4.329 477 4.212 364 4.100 195 6 6 5.417 101 5.328 553 6.242 137 5.075 692 4.917 321 4.766 537 6 7 6.-2:i() 2H3 6.114 644 6.002 055 6.786 373 5.582 381 5.389 286 7 8 7 019 092 6.873 95(i 6.732 745 6.463 213 6.203 744 6.971 295 8 7.786 109 7.607 687 7.435 33k 7.107 822 6.801 692 6.515 228 9 10 8.530 203 8.3 1<* 605 8.110 89(1 7.721 735 7.360 087 7.023 577 1 11 9.252 62 1 9.001 551 8.760 477 8.306 414 7.886 875 7.498 669 1 1 12 9 954 004 9.663 334 9.385 074 sMy^ 252 8.383 844 7.942 671 12 13 10.634 955 10.302 738 9.985 648 9.393 573 8.852 68H 8.:ij7 635 13 14 11.2!l« 073 10.920 520 10.563 12:{ 9.898 641 9.294 984 8.745 452 14 . 15 11.U..7 935 11.517 411 11.118 387 10.379 658 9.712 249 9.107 898 15 16 12.561 102 12.094 117 11.6.52 29(i 10.837 770 10.105 895 9.446 632 1 6 17 13.166 118 12.651 321 12.165 66*J 11.274 066 10.477 260 9.763 206 1 7 18 13.753 513 13.189 682 12.659 297 11.689 587 10.827 603 10.059 070 1 8 1!) 14.323 799 13.709 837 13.133 939 12.085 321 11.158 11(1 10.335 578 1 9 20 14.S77 475 14.212 403 13.590 326 12.462 21U 11.469 421 10..''.93 997 2 21 15.416 024 14.697 974 14.029 160 12.821 153 11.764 077 10.835 527 2 1 22 15.936 917 15.167 125 14.451 115 13.163 003 12.041 582 11.061 241 2 2 23 16.443 608 15.620 410 I4.8.-.6 842 13.488 574 12.303 379 11.272 187 I 3 24 16.935 542 16.058 368 15.246 963 13.798 642 12.550 358 11.469 334 2 4 25 17.413 148 lo.481 515 15.622 080 14.093 945 12.783 356 11.653 683 2 6 26 17.876 842 10.890 352 15.982 7(>S 14.275 185 13003 166 11.825 779 2 6 27 18.327 031 17.285 365 16.329 58l> 14.643 034 13.210 534 11.986 709 2 7 28 18.764 108 17.067 019 16.663 063 14.898 127 13.406 164 12.137 111 2 8 29 19.188 455 18.035 767 16.983 715 15.141 074 13.590 721 12.277 674 2 9 30 19.600 441 18.392 045 17.292 03:; 15.372 451 1.3.764 831 12.409 041 3 31 20.000 428 18.736 276 17,588 49'J 15.592 811 13.929 086 12.531 814 3 1 32 20.338 766 19.068 865 17.873 552 15.802 677 14.084 043 12.646 655 3 2 33 20.765 792 19.390 208 18.147 646 16.002 549 14.230 230 12.758 790 3 3 34 21.131 837 19.700 684 18.411 198 16.192 204 14.368 141 12.854 009 3 4 35 21.487 220 20.000 661 18.664 613 16.374 194 14.498 246 14.n20 987 12.947 672 3 5 36 21.832 252 20.290 494 18.908 282 16.546 852 13.035 208 3 6 37 22.167 235 20.570 525 19.142 679 16.711 287 14.736 780 13.117 017 3 7 38 22.492 462 20.841 087 19.367 864 16.867 893 14.846 019 13.198 473 3 8 39 22.808 215 21.102 600 19.584 485 17.017 041 14.949 075 13.264 928 3 9 40 23.114 772 21.355 072 19.792 774 17.159 086 15.046 297 13.331 709 4 41 2.3.412 400 21.599 104 19.993 052 17.294 368 15.138 016 13.394 120 4 1 42 23.701 359 21.834 883 20.186 627 17.423 208 16.224 643 13.452 449 4 2 43 23.981 902 22.062 689 20.370 795 17.646 912 16.306 173 13.506 9C2 4 3 44 24.254 274 22.282 791 20.648 841 17.662 773 16.383 182 13J67 908 4 4 45 24.518 713 22.495 460 20.720 040 17.774 070 16.455 832 13.605 522 4 5 46 24.775 449 22.700 918 20.884 654 17.880 067 16.624 370 13.660 020 4 6 47 25.024 708 22.899 438 21.042 936 17.981 016 16.689 028 13.691 608 4 7 48 25.266 707 23.091 244 21.196 181 18.077 168 15.660 027 13.730 474 4 8 49 25.601 667 23.276 664 21.341 472 18.168 722 15.707 672 13.76C 799 4 9 |80 26.729 764 23.465 618 21.482 186 18.2.5 925 15.761 861 13.800 746 6 306 ■;:'«?' PABTNFKSHIP SETTLEMENTS. .The oireumstances under wbbh partnerships are formed, tha oonditions on which they are made, and the causes that lead to their dissolution, arc so varied that it is impossible to do more than give general dircoUons deduced from thd oases of most common occur- rence. In forming a partnership, the great requisite is to have the terms of agreement expressed in the most clear and yet concise lan<- guage possible, setting forth the sum invested by each, the duration of partnership, the share of gains or losses that fall to each, the sum that each may draw from time to time for private purposes, and any other circumstances arising out of the peculiarities of each case. The ease and satisfaction of making an equitable settlement, in case of dissolution, depends mainly on the clearjess of the original' agree- ment, and hence the necessity for its being distinct and explicit. Even when no dissolution is contemplated, settlements should be 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 Houbq. A dissolution may take place from various causes. If the part- nership is formed for a term of years, the expiration of thosb years necessarily involves either a dissolution or a new agreement. The 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, bowever, can an equitable settlement be 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 the firm may be apprized of the change, and hu,ve their accounts arranged. For the same reason, it is necessary that Bome one of the partners, or some trustworthy accountant appointed by them, should be authorized to oolleot all debts due to the firm, and pay all accounts owing by it. » Partnerships are sometimes formed for a specific speoulation, and tj^erefore^of course, oease with the completion of the transaction, and s! settlement must nec^sarily be then made. No matter for what PABTNEBSHIP SITnUSMENTS. SOT by 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 circumstances of any part- ner may so change, from various causes, as to make it undesirable for him to remain in the business. If one partner is deputed to ce'wtle the 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 the net investment on com- mencing business, being given, to find the net gain or loss. 1. W. Smith and B. Evans ire partners intbusiness, and invested when commencing $1000 each. On dissolving the partnership, the assets and liabilities are &a follows : — Merchandise valusd at $1295 ; cash, $344 ; notes against sundry individuals, $790 ; W. H. Monroe owes on account $86.40 ; E. B. Carpenter owes $132.85, and C. F. Musgrove owes $67.50. They owe on sundry notes, as per bill book, 9212,40 ; E. G. Conklin, on account, $29.45, and H. G. Wright, on account, $41.30. What has been the net gain ? SOLUTION. Assets. Merchandise on hand... $1295.00 Cash on hand .1 344.00 Bills Receivable 790.00 Amt. due from W. H. Monroe 86.40 Amt. due from E. B. ; Carpenter 132.85 Amt. due from C. F. Musgrove 67.50 • it LiaMKties, Total amount Assets.... $2715.75 '« '* LiabiUties, 2283.15 Bills Payable $212.40 Amt. due £. G. Conklin. 29.45 Amt. due H. C. Wright. 41.30 W. Smith's investment... 1000.00 B. Evan's investment.... 1000.00 ^:.>i.f,< - '■■^ ■] „ S2283.15 -.:: •,:v.V.i • ': •- ■,J'i' - •':.■' ,.('i>!4 ^I'iV .^j / ■-*> ' 'S :': ,'^>' ,'i ' it '.:':? . -O-. .»■* ilj •;' ,: '-t-^^.H-V H, J';'. •" . •#- f»\ .■-:i:ji i,^iiv.- .i^v.Ai - Find the sum of the assets and UabiUties ;fnm the assets subtract ike liahilities, (including the net amaunt invested) and the differ- ence toill be the net gain ; or, if the tiabiKties be the larger, subtract ih€ assets from the liabilities^ and the difference will be the net loss. I iiii 808 ARITHMBTIC. 2. Harvey Miller and James Oarcy are partners in a dry good» business '/Karvoy Miller investing $1400, and James Carey $1250. When closing the books, they have un hand-'-^ash, $1125.30; mev- chandiso as per inventory book, $1855.75 ; amount deposited in First National Bank, $1200 ; amount invested in oil lands, $963 ; a site of land for building purposes, valued at $1600 ; Adam Dudgeon owes them, on aooount, ^104.92 ; William Fleming '' iA $216.80 , a note against Afl^rod Mills for $69. 43, and a due bill for $33^ drawn by James Laing. They owe W. S. Hope & Co., on account, $849.21 ; B. J. King & Co., $603.12, and on notes, $1326.14. What has been the net gain or loss ? Ans. $1761.73 gain. 3. James Henning and Adam Manning have formed a co-part- nership for the purpose of conducting a general dry goods and grocery business, each to share gains or losses equally. ^ ^,. '^ At the end of one year they close the books, having $1280 wortb. of merchandise on hand ; casli, $714.27 ; Girard Bank stock, $500 ; deposited in Merchants' Bank, $320.60 ; store and fixtures valued at $3100; amount due on notes and book accounts, $3471.49. Tho firm owes on notes $3400, and on open accounts $747.10. James Henning invested $1200, and Adam Manning, $1000 i. what is each partner's interest in the business at closing ? ^ Ans. James Henning's interest, $2719.63. Adam Manning's • \ interest, $2519.63. Note. — Where the interest of each partner at closing is required, tho gam or loss is first found, as in t'ormer examples, then the share of gain or loss ia added to or subtracted from each partner's investment, and the sum, or difference, is the interest of each partner. If a partner has withdrawn any- thing from the business, the amoiint thus withdrawn must bo deducted trom the sum of his investment, p{u. H . T5» and M, /j. Wiien dissolving the* partnership the resources consisted of cash $4700 ; merchandise, $0855 ;■ 'botes on hand $7G80 ; debentures of tlie city of Albany valued at $$780, on which thuro is interest duo, $123 ; horses, waggons, &o., $1280'; Merchant's bank stock, $5000 ; First National bank stock, $5to; mortgages and bonds, $3600; interest due on mortgages, $345.80 -; store and fixtures, $3000; amount due from W. P. Campbell & Co., $2418; due from R. B.Smith, $712.60; due from J. W.Jones, $1000. The liabilities are : — Mortgage on store and fixtures, $5000 ; Interest due on the same, $212.25 ; due the estate of R. M. Evans, $14675 ; notes and acceptances, $11940, on which interest is due, $85 ; sundry other book debts, $7500 ; I invested $7800, interest on his investment to date of dissolution, $702 ; J invested $6400^ interest on investment, $576 ; K invested $6100, interest on invest- ment, $549 ; L invested $5800, interest on investment, $522 ; M invested $5000, interest on investment, $450. I has withdrawn from the firm at different times, $2425, upon which the interest calcu- lated to time of dissolution is $183.40 ; J has drawn $2960, interest, $267.85; K has drawn $1850, interest $87.30; L has drawn $3000, interest, $460 ; M has drawn $895, interest, $63.45. What is the net gain or loss of each partner, and what is the net capital of eaoh partner ? I Anfl. I's net loss, $1233^29 ; I's net capital, $4660.31. J's net ^' loss, $924.97; J's net capital^ $2823.18. K's net loss, > ' $616.65 ; K'l) net capital, $4095.05. L's net loss, $1541.62 ; ' f L's net capiUvl, $1320.38. M's net loss, $308.32 ; M's net '^ ' capital, $4183.23. " 10. A, B, C and D are partners. At the time of dissolution, and after the liabilities are all cancelled, they make a division of the effects, and upon examination of their ledger it shows tho f'^Dcwing result : — A has drawn from the business $3465, and invested on eonimenoement of business, $4240 ; B has drawn $4595, and invested $3800: C has drawn $5000, and invested $3200; D has drawn $^00, «nd iuvested $2800. The profit or loss was to be divided in 312 ABnmano: proportion to the ori^pnal inTestment. What has heen each partner*! gain or loss, and how do the partners settle with each other? Ans. A's.net gain, $368.43; B's net gain, $330.20; G's net gain, $278.06; D's net gaia, $243.31. B has to pay in $464.80 ; has to pay in $1521.94. A receives $1143.43 ; D receives $843.31. '*'* Three mechanics, A. W. Smith, James Walker and P. Banton, are equal partners in their business, with the understanding that each is to be charged $1.25 per day for lost time. At the oloso of their business, in the settlement it was found that A. W. Smith had lost 14 days, James Walker 21 days, ahd P. Banton 30 dayi. How shall the partners properly adjust the matter between them? Ans. P. Banton jpays A. W. Smith, $9.58^, and James Walker, 83^ cents. 12. There are 6 mechanics on a certain piece of work in tho following proportions :— A 'ia^%; B, ^^ ; C, 3^ ; D, /g, and E, ^jp, A is to pay $1.25 per day for all lost time ; B, $1 ; 0, $1.50; D, $1.75, and E> $1.62^. At settlement it is found that A has lost 24 ; B, 19 ; C, 34 ; D, 12 ; and E, 45 days. They receive in pay- ment for their joint work, $250C. What is each partner's share of ^S 9inount according to the above regulations? Ans; A's share, $374.12; B's, $250.41; O's, $487.83; D'a, $787.24; E's, $600.40. ,. , 18. A. B. Smith and T. G. Musgrove commenced business in partjoership January 1st, 1864. A. B. Smith invested, on com- fitaaoeiment, $9000 ; May 1st, $2400; June 1st, he drew out $1800; September 1st, $2000, and October 1st, he invested $800 more. )Pf . C.rMu^oye; invested on commencing, $3000 ; March 1st, he dr^w o^t $jt600; 3j[ay 1st, $1200; June 1st, he invested $1500 morej jEtnd October 1st, $8000 more. At the time of settlement, on the 31st December, 1864, their merchandise account was — Dr. $32000 ; Cr. $27000 ; balance of merchandise on hand, as pef inventory, $10500 ; cash on hand, $4900 ; bills receivable, $12400; E. l)rap&r owes on account, $2450. They owe on their notes, $189Q, and iGi-. Boe on account, $840. Their profit and loss account is^' jbr. $86^ ; Cr. $1520. Expense account is, Dr. $2420. Com- mission account is. Or, $2760. Interest account is Dr. $480 ; Cr, |^5^T T|)ek|.gaijf^, or.loss is to be divided ia proportion to eaoh ^i||bW'8^-«^^^,, and in proportion to the time it was invested. Eeamred each partner's share of the gain or loss, the net balanot FilBTNEBSHIP- SERLEMENTBU 818 ,.?;' due each, and a ledger specificatioa exhibiting th»«lo8in({ of ill fhs accounts, -and the balance sheet. 4. ^ Ans..A.B. S.'s net gain, $6671.73; his net l)alancer|16071.73. « T. G. M.'s net gain, $2748.27 ; his net balance, $1 2448.27. 14. A, B, G, and D commence business together on July Is^ 1865, with the agreement that all gain or loss is to be ehared 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 rata charged on all amounts withdrawn by each. A is to manage tho business, having a salary of $2,000 per year, payable half-yearly. Tho services of B, G, and D are not required in the business. Tho assets are, on commencing, Gash $7440 ; Mdse. $9686 ; Bills Be- ceivablo, estimated value $4976.C ') (face value $5237.89) invested by G. 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 ao counts Cr. $4292.89. Aug. 20th, A drew cash $75, B $90. Sept 4tb> D drew cash $125. Sept. 30th, A drew $200, G $80. Nov. 20th, B ilrcw $100, D, $50. Dec. 24th, A drew $150. Feb. 27th, 18C6, A drew $200, G $150, D $100. May 12th, A drew $200, D $200. Juno 13th, B drew $150, G $100. July 1st, B made a further investment of $1000, G, $1500, and A drew $400, D $100. Sept. 15th, A drew $150, B $500, C, $750. Nov. 1st, A drew $100, D $75. Dec. 31st, 1866, the books are closed and the partnership dissolved, G and D retiring from tho business, being allowed by tho remaining partners $150 each for their good-will in the business. Before calculating the interest on tho partners' investments, and on the amounts withdrawn by them, and allowing A the amount of hiiii fjalary from time of commencement up to date, the assets and liabili- ties are as follows: Gash $9483.50; Mdse. $14675; Bills Eeceiv- able $6219.85, 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. G. B. B. stock, valued at 105; 60 shares Erie B. R stock at 69. They owe in BiUs payable $5657.45. Personal accounts Gr. $3272.94 ; also, Samuel Zimme> man for rent to date $1250. In the settlement a discoxint of 5 per cent, on the B^ls BeceivaUe, 15 per cent, on tho Dr. personal ao> 21 Ji! 814 ABXTBMBTXOi oonnts, and 10 por oont on the Mdso. shipped to New York, is lU lowed for loss in bad debts. agrees to talce the N. Y. 0. R. R, stock at 105, in part payment of the amount due him ; D takes th« Erie R. R. stock at 69 ; receives cash 01200, and D $1800. What has been the whole gain. or loss, the amount still duo and D, the amount of cash on hand when the books are closed up, and what are A and B each worth, the Billa Roc, persona! accounts, and con- signmcnt to N. Y. being valiicd at par? Ans. Gain $3370.45 ; duo C 8959.72; duo D $512.96; cash on hand $6483.50 ; A's capital $16401.47 ; B's net capital $8183.21. 15. A book-keeper applied at our OoUege for counsel, not long since, to settle the following accounts, between two partners, Jan. 1st, 1866. We'll call them Mr. E. and Mr. F., each \ gain or loss. Books were kept by single entry, and tho accounts and inventory were as follows :— Cash in Rank $5,705. Do. in office $6,000. Bills Rcc. 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, Expense account, dr. baL $1,335, Mdse. account, dr. bal. $210. Bills payable, per B. B., $4,564. Rev. O. Burger, cr. bal. $200. D. C. Weed & Co. dr. bal. $2,000. Shepard & Cottier, or. bal. $300. Jn't and Dis't ao- count, cr. bal. $1,524. Joint account with A. L. Griffon & Co. each ^, net gains were $872. P. P. Dobbins & Son, dr. bal. $l,00a What are B. and F. eaeh worth ? ■ ■> ■) .■"3.'5 >«ffw»t- is al« QT7B8TXOKB VOB OOmCBBOXili 8TDDEMTBL 315 QX7ESTI0NS FOR COMMERCIAL STUDENTS. 1. The following questions may be found interesting and inrtruotiye 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., aud consigned to Ross, Winans & Co., commission mer- chants, 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, Winana & 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 du I receive and what are the journal entries? 2. B. Empey, a merchant doing business in Montreal, Canada East, purchased from A. T. Stewart, of New York city, on a orerUt of three months, the following 'nvoice of goods : ■ • Gold. 845 yds. Fancy Tweed, @ $1.90peryd. 81.99 : 1712 " Amer. black broadcloth, @ 3.85 " " 4.02 ■ 423 " Blue pilot, @ 2.75 « "2.87 , 700 " Black Cassimere. @ 2.10 " "2.20 When the nbove goods were passed through the custom-house, a discount of 27} 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 excKange, if at the expiration of the three months B. Empey remitted A. T. Stewart, to balance account, a draft on Adam?; Kimball & Moore, bankers, New York city, purchased at 32^ per cent, discount, and what are the journal entries ? 3. I purcihased for cash, per the order of J. P. Fowler, 70 boxes 0. 0. bacon, containing on an average 400 lbs. each, at ISf cents 816 ABITBICBTXa I: i. ;. per lb., and 140 firkini buttor, 8312 lbs., at 17} cents per id., on a commission of 2} per cent ; paid ibipping and sundry expenses in oash $13.40. For reimbursement I draw on J. P. Powlcr at sight, which I sell to the bank at } per cent, discount ; what is the face of draft, and what are the journal entries ? Ans. Face of draft $5479.05. 4. Sept. 27th, I received from James Watson, Leeds, England* a consignment of 1243 yards black broadcloth, inyoioed at 13s. 6d. per yard, to be sold on joint account of consignor and myself, each one half, my half to be as cosh, invoice dated Sept. 16th. Oct 5th, I sold K. Duncan, for cash, 207 yards, at $6.10 per yard ; Oct 24th, sold 317 yards to James Grant, at $6.25 per yard, on a credit of 90 days ; Nov. 18th, sold E. 0. Oonklin, for his note at 4 months, 400 yards, at $6.30 per yard ; Dec. 12th, sold the remainder to J. A. Musgrove at $6.00 per yard, half cash and a credit of 30 days for balance; charges for storage, advertising, &c., $13.40; my com* mission, with guarantee of soles, 5 per cent, \yhat would be the average time of sales ; the average time of James Watson's account ; and what would bo the face of a sterling bill, dated Dec. 15th, at €0 days after date, remitted James Watson to balance account purchased at $108^^, money being worth 7 percent, and gold being 70 4)er cent, premium ? 6. Buchanan & Harris of Milwaukee, Wis., are owing W. A. Murray & Co. of Washington, $1742.75, being proceeds of consign- ment of tobacco sold for them, and Simpson & Oo. of Washington, are 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., but Washington funds are 2 per cent, premium over those of Chicago. Kequired the face of the draft and the journal entries^ 6. A. Cummings, of London, England, is owing me a certain sum, payable there, and I am owing Charles Massey, of the same place, $1985.42, being proceeds of consignment of broadcloth sold for him. I remit C. Massey in full of account, after allowing him $21,12 for inserest, my bill of exchange on A. Cummings at 60 days' sight ; exchange 109f, gold 42 per cent, premium. What is pcnter, as capital, $8000; S. Northrup, $6000; Levi Williams, $4561.50. The firm assumed the liability of Levi Williams, which was a note to tho amount of $425.80 ; this note was paid on March 10th. The loss or gain is to be shared equally by the partners, but interest at the rate of 7 per cent, per annum is to be allowed on in* vestments, and charged on amounts withdrawn. £. R. Carpenter is to manage the business on a salary of $1000 a year, payable half yearly (the time of the other partners not being required in tho business). March 14th, S. Northrup draws cash, $300; Levi Williams, $200; April 19th* E. R. Carpenter draws cash, $500 ; S. Northrup, $100. On the 1st of May, they admit Qeo. Smith as a partner, under the original agreement, with a cash capital of $4000. The books not being closed, ho pays each partner for a participation in the profits to this time $450, which they invest in the business. May 14th, E. B. Carpenter draws cash, $160 ; May 24th, Levi Williams draws cash, $100 ; Juno 12th, S. Northrup draws cash, 4250, und E. R. Carpenter, $200 ; July Ist, Levi Williams draws cash, $300, and S. Northrup, $450 ; July 21st, Levi Williams draws •cash, $180 ; August 1st, Levi Williams retires from the partnership, the firm allowing him $500 for his profits and good-will in the busi- ness, this amount, together with his capital, has been paid in cash. Oct. 14th, Geo. Smith draws cash, $340; E. R. Carpenter, $725. November 1st, with the consent of the firm, S. Northrup disposes of his right, title and interest in the business to J. K. White, who is admitted as a partner under the original agreement. J. K. White is to allow S. Northrup $600 for his share of the profits to date, aild his good-will in the business. J. K. White not receiving funds an- ticipated, is unable to pay S. Northrup but $1500, the firm therefbre iMumes the balance as a liability. December 10th, received ttdm QUESnONH FOR OOMMIROIAL 8TUDEMT8. 819 J. K. White, and paid oTor to S. Northrup, tho full amount due him (8. N.) to date. Dooomber Slst, the booka are olosod, and tho fol* lowing cffoots are on hand:— Mdae, $11043.75; cash, |eil0.12; bilb reoeivablo, 16400 ; store and flzturos, $3860 ; personal accounts Pr. $14087.50; personal accounts Cr. $10711 ; Bills Payable nnre- deomod, $4000. What has bocn tho not gain or loss, tho net capital of each partner at tho end of the year, and what arc tho double entry journal entries on commencing business, when Levi Williams retires, when Qeo. Smith is admitted, when S. Northrup Mif\a his interest ond right to J. K. White, for E. R. Carpenter's dalary, tnd tho interest due from and to each partner ? The student may also, in the above example, after finding tho interest on tho partners investments, and on tho amounts withdrawn, give a joomal entry that will adjust the matter of interest between the partners without opening any profit and loss account. ; ';V' ,■;■ i ,, ^' /-'"''■•. . '! ■ . \ •4 ■.,■■•••. r-iii'' \ ■,•■■,'. :'■ .■,-.: .- . : ,< iV ) ' ^ .' ■ 1 ,. ...... ..if- ■■■:. .■ . ' . ■'. . ■ UiV : ■(" '.■■ ',.. '.'x'..'' .':.•■-. / . t... ■/ , ■: > . -iY'. *v '' -■. ■■' ;.■■,'•'-. h'.^y:^-'/ ■ .L - .. ■ , . ■ •.'«• • ' . ' 't: -r . i.^. ... .J/,f<,:-'i .:;«''^i h".-. ,v -^■^i'U. ■ »..'.. i . i .- . vfri^ ■" , -H' .:■,','■, ^ V; ';,-'■ , ■' .. • ' • , J ■ . «.,' V^-a. ^'ir^. >'-■'■• f^; ''••'' ' ''■ ' ! •'■^"^'-r.',,^ • , , . '..: ;, '.J . ■ - ■. ■ y ■ ' ■ 5. What does the surface of a plaalc measure, which is 12 fC 6 in. long and 9 in. broad? Ans. 9 sq. ft. 54 sq. in, 6. What is the area of a rhomboidal field, the length of which is 10.52^hains and the perpendicular breath 7.63 chains ? Ans. 8 acres, roods, 4.2816 rods, 7. What is the area of a rhomboidal field, the length of which 19 24 rods and the perpendicular breadth 24 rods ? -> Ans. 3 acres, 2 roods, 16 rods. 8. What is the length of each side of a square field, the area of which is 788544 square yards ? Ans. 888 yards. 9. The area of a rectangular garden is 1848 square yarcls, and| one side is 56 yards ; what is the other ? Ans. 33 yards. 10. The area of a rhomboidal pavement is 205, and the length is 20 feet ; what is the perpendicular breadth ? Ans. 10^^^ feet. PR0BLE2E U. To find the area of a triangle. -^ • 1. If t]ie base and perpendicular, or data to find them, be given, we have tht ; ^;"''*''•V ':'.'■■ !'-'-'*«sji-«v!4 '■ --,' . '' ^ 'H^' BULB. MuUiply the base by the perpendicular^ and take half the pr(h iiuet i or, multiply half ihe one by the other. ' 2. If the three sides are given f ^ ' ;'<■■': BULB. ^,, From half ihe mm of th/e «u2et tvhftmiA each tide tu^xmiv^ly, and the tquare root of the continual product of the half wm» dmd iheee three remainden mil be the area. 822 ABirmoEna Expressed algebraioally this area=|/«(«— <>)(<— &)(i—c). > XXEROISBB. '-'■' 11. What is the area of a triangle, the base of which is 17 inches, and the altitude 12 inches? Ans. 102 square inches. 12. What is the area of a triangular garden, the length of which is 46 rods, and the breadth 19 rods ? Ans. 437 square rods. 13. Find how many acres, &c., are in a triangular field, the length of which is 49.75 rods, and the breadth 34 J rods. Ans. 5 acres, 1 rood, ISj'^ rods. 14. The area of a triangular inclosuro is 150 square rods, and the base is 30 linear rods ; what is the altitude ? Ans. 10 rods. 15. The area of a triangle is 400 roJs, ar.d the altitude 40 rods, what is the base ? Ans. 20 rods. 16. Three trees are so planted that the lines joining them form a right angled triangle ; the two sides containing the right angle are 33 and 56 yards, what is the area in square yards ? Ans. 924. 17. Let the position of the trees, as in the last example, be represented by the tri- angle ABC, and let the distance from A to B be 50 rods, and from B to C 30 rods. Eequired the area. — (See Euclid I. 47.) Ans. 600 square rods. 18. In the fin;ure annexed to 17, suppose A B to represent the pitch of a gallery in a church, inclined to the ground ai an angle of A5' ; how many more persons will the gallery contiu.:: ih^.a if the •eats were made on the flat B C, supposing B C Ij be 'I ' ^' at and Ihe frontage 60 feet in length ? 2a i J. None. We have introduced this question and the next to correct a common misapprehension on this point. Jl|J■^K'lt Because the distance from B f» A is greater than the distance from B to 0, it is commonly supposed that more per- sons can be accommodated on the slant A B, than on the flat B C. By in- specting the annexed diagram it will be seen that the seats are not perpen- dicular to A B, but to B C, and that precisely the same number of seats can be made, and the same number of per* ions accommodated on B C as on A B. v'^.U-Sf 1 . ■ , ..i A A ' v'«; A"- ''■ f / 4 [ / Tte ■ MEKSTTBATION. 323 Hcan per* AB. 19. If B be half the base of a hill, and A B one of its sloping -aides, and B G=30 yards, and A B=50 yards ; how many more rows of trees can be planted on A B, than on B C, at 1 yard apart ? Ans. None, because the trees being all perpendicular to the horizon, are parallel to each other as represented by the vertical lines in the last figure. 20. How many acres, &c., are there in a triangular field of which the perpendicular length and breadth are 12 chains, 76 links and 9 chains, 43 links ? Ans. 6 acres, roods, 2^ rods. 21. A ship was stranded at a distance of 40 yards from the base of a cliff 30 yards high ; what was the length of a cable which reached from the top of the cliff to the ship ? Ans 50 yds« 22. A cable 100 yards long was passed from the bow to the stem of a ship through the cradle of a mast placed in midships at the height of 30 yards ; what was the length of the ship ? Ans. 80 yards. 23. A man attempts to row a boat directly across a riyer 200 yards broad, but is carried 80 yards down the stream by the current; through how maay yards was he carried ? Ans. 215.4-f-yards. 24. Let the three sides of a triangle be 30, 40, 20 ; to find the area 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. 26. What ia the area of a triangular space, of which the base is 56, and the hypotermse 65 yards ? Ans. 924 square yards. 27. What is the area of a triangular clearing, each side of which is 25 chains ? Ans. 27.0632 acres. 28. What is the area of a triangular clearing, of which the three sides arc 380, 420 and 765 ? Ans. 9 acres, 37^ perches. 29. A lot of ground is represented by the three sides of a right angled triangle, of which the hypotenuse i# 100 rods, and the base 60 rods ; what is the area? .„,..».. Ans. 15 acres. '60. What is the area of a triangular field, of which the sides are 49, 34 and 27 rods respectively ? Ans. 2 acres, 3 roods4-. 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 so as to be represented * This quostion, and some others may be solved by either rule, and it will bo found a good exercise to solve by both. 324 ABITHUEnO. M by three sides of a triangle, 12, 18 and 24 ; how many square »iIoB do they guard within their lines ? Ans. Between 104 and 105 square miles. 33. A ladder, 50 feet long, was placed in a street, and reachec| to a parapet 28 feet high, and on being turned over reached a para- pet on the other side 30 feet high ; what was the breadth of the street? Ans. 81.42+feet. PBOBLBM III. To find the area of a regular Polygon. ^ ' . ' ; ^ ; 1. When one of the equal sides, and the perpendicular on it firom the centre, are given. Multiply the perimeter hy the perpendicular on it frcmi its centre, and take half the product ; or, multiply either by half the other. 2. When a side only is given. Multiply the square of the aide hy the number found opposite the numher of sides in the subjoined table. Note.— This tublu shows the area when the Hide is unity ; or, which is the aame thing, the square is the unit. SIDES. ■'* ,! fi 3 4 6 6 7 8 9 10 11 12 REQULAR FIOmiES. Triangle Square Pentagon Pexagon Heptagon Octagon Nonagon.^.^.. Decagon Heredecagon. Dodecagon... 0.4330127. 1.0000000. 1.7204774. 2.5980762. 3.6339125. 4.8284272. 6.1818241. 7.6942088. 9.3656395. 11.1961524. 34. If tlie side of a pentagon is 6 feet and the perpendicular 3 feet, what is the area ? Ans. 45 feet. 35. What is the area of a r^ular polygon, each side of which is ,15 yards? Ans. 387.107325 Sq. yds. |:j^ 36. If each side of a hexagon be 6 feet, and a lino drawn from the centre to any orngle be 5 feet, what is the area ? Ans. 72 8q. iMi MENSUBAHON. 825 37. Tlio side of a decagon is 20.5 rods ; what is the area 7 Ans. 20 acres, roods, 33.5 rods, nearly. 38. A hexa^ional table has each side 60 inches, and a line firqm die centre to any comer is 50 inches ; how raaay square feet in tho fiorface of the table ? 39. What is ihe area of a regular heptagon, the side being 19|{ and the perpendicular 10 ? . Ans. 678.3. 40. An octagonal ennlosure has each side 6 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 lino is 20 mile.i ; how many square miles are guarded ? Ans. 688.2, nearly. 42. Find tho same if there are 6 divisions, and each 4ine extends 5 miles? Ans. 64.95+ miles. . 43. The area of. a hexagonal table is 73^ feet; what is each side? H--_-, ■ , ,. Ans. 6J feet. PBOBLEM IV. To find the area of tvn irregular polygon. .Divide it into triangles hy a perpendicular on each dicyonal from the opposite angle. Find the area of each triangle separately ^ and the sum of these areas will be the area of the trapezium. ., / Note. — Either the diagoniils and perpendicultire must be given, or data from which to find them. , ,,,' 44. The diagonal extent of a four-sided field is 65 rods, and the perpendiculars on it from the opposite comers are 23 and 33.5 rods; what is the area ? Ans. 1 acre, 1 rood, 22.083 rods. 45. A quadrangle having two sides pardlcl, and the one is 20.5 feet long and the other 12.25 feet, and the perpendicular dbtonce 4)etween them is 10.75 feet ; what is the area ? i^U r . ! , . Ans. 176.03125 sq. feet. 46. Required the area of a six-sided 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 intermediate one 14.25 and 9.35 ? Ans. 531.889 yards. 47. If tho two sides of a hexagon bo parallel, and the diagonal panllel to them be 30.15 feet, and the perpendionlars on it from 'yr%;.VT vr . tin^ 4 A. i f:-x v.. 826 ABTTEMETIO. ^ the opposite angles are, on the left, 10.56, and on the right 12.24;. and the part of the diagonal out off to the left hy the first perpendi- cular, 8.26, and to the right by the second, 10.14 ; on the other, nde, the perpendicular and s^;ment of the diagonal to the left are. 8.56 and 4.54, and on the right 9.26 and 3.93 ; what is the area? Ans. 470.4155 sq. feet»^ PBOBLIK V. To find the area of a figure, the boundaries of which are* partly right lines and partly curves or salients. ^ Find the average breadth hy taking several perpendiculars from the nearest and most remote points, from a fixed base, axd dividing the sum of these by their nurnbeTj the quotient, multiplied by the length, will be a close approximation to the area. Let the perpendiculars 9.2, 10.5, 8.3, 9.4, 10.7, their sum is 48.1, then 48.1-4-5=9.62, and if the base is 20, we have 9.62X20= 192.4, the area. . , , .^ When practicable, as laige a portion of the space as possible should b^ laid off, so as to form a regular figure, and the rest found as above. A field is to be measured, and the greater part of it can be laid off in the form of a rectangle, the sides of which are 20.5 and 10.5, and therefore its area is 215.25, and the offsets of the irregular part are 10.2, 8.7, 10.9, and 8.5, the stm of which, divided by Iheir number, is 7.66, and 7.66X20.5=157.03, the area of the irregular part, and this, added so the area of the rectangles, gives 372.23, tho wh6le area. 48. The length of an irregular clearing is 47 rods, and the breadths at 6 equal distances are 5.7, 4.8, 7.5, 5.1, 8.4 and 6.5; what is the area 7 Ans. 1 acre, 1 rood, 29.86 rods. < PBOBLSM VI. I' . ' To find the circumference of a circle when the diameter is known, or the diameter when the circumference u known.* ,' The most accurate rule is the well-known theorem that tho diameter is to the circumference in the ratio of 113 to 355, and * In strictneas the circumiierence and diameter are not like qoantities, bat we may Boppoae that a cord is stretched round the circumference, and then 'drawn «ut into a straight line, and its linear units compared with thos^of tbedianeier. MENSURATION. 827 Miueqnently the eiroumference to the diameter as 355 to 113. Now, 855-1-113=3.1416 nearly, and for general purposes, guffioieDl •oouracj will be attained by this RULB. ' • . • ' f : . • 1 1 » To find the eircumference from a ^ven diameter, multip^^ .t^; diameter by 3.1416 ; and to find the diameter from a given cirqom* ference, divide by 3.1416. , ^' \ ■.'.' 49. What is the length round the equator of a 15-inch globcf? Ana. 47.124 idcheli.' $-•>■■ 50. If a round log has a eiroumference of 6 feet, 10 inches ; whai is its diameter? Ans. 2 feet, 2^*0 inches nearly. 51. If we take the distance from the centre of the earth to tho equator U. be 8979 ; what is the number of miles round the equator? Ans. 25001 nearly* '^ '-> ^ PROBLBK VII. To find the area of a circle. 1. If the circumference and diameter are known, — Multiply tJhe circumference by the diameteTf and take one-fourA of the product. 2. If the diameter alone is given, — " * Multiply the tquare of the diameter by .*l2Xt^. 3. If the circumference alone bo given, — M%dtiply the tqvMre of the nun^er denoting t?te circumferenu by .07958. 52. If the diameter of a circle is 7, and the eircumference 22 { vhat is the area? Ans. 38^ ., , 53. What k the area of a circle, the radius of which is 3§ yds? »' Ans. 3| square yardf. 54. If a semicircular arc be denoted by 10.05 ; what is the are* of the circle ? Ans. 289.36. 55. If the diameter of a grinding stone be 20 inches; what •uperficial area is left when it is ground down to 15 inches diameter, and what superficial area has been worn away ? Ans. 176.715 sqr. inches left, and 137.445 worn away. 56. If the ehord of an arc be 24 inches, and the perpendiculat on it from the centre 11.9 ; what is the area of the circle ? ' Ans. 2.689804. 828 ABrmimaL MENSURATION OF SOLIDS. To find the Bolid contents of a parallelopiped, or any r^larlj box-shaped hody : Lot it bo required to find tho number of cubic feet in a box 8 feet long, 4^ feet broad, and 6^ feet deep. In tho first place, tho length being 8 feet and tho breadth 4}, the area of the base is 8X4^=36 square feet, and therefore every foot of altitude, or depth, or thickness, will give 36 cubic feet, and as there are 6| feet of depth, tho whole solid content will be 36 times 6|, or 243 cubic feet. Uence the .^. ,.,;^'.^u:.;..r,.v,.iv / BULB. '■' ; . -;• Take the continual product o/the lengthy hreadthy and depth. Note.— Let it bo carefully observed that the unit of measure in the case of solids is to bo talten as a cube, the base of which is a superficial unit used in the measurement of surfoces. Tlio solid content is indicated by the repe- tttio 1 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 tho btwe each 6 fleet, then the area of the base is 36 ; but if, at tho height of 1 foot, the dimensions have each diminished by 1 foot, the area is 25 ; at another foot higher it is 16 ; at the next 9 ; at the next 4 ; at tho next 1 ; and at tho 6th 0, t. e., it has come to a point, and the calculation is, how much remains from the solid cube after bo much has been cut off each side as to give it this form. This gives rise to tho following varieties : v.. , ,.^,t I. To find the solid contents of a cone or pyramid : Multiply the area of the bote hy the perpendicular height^ and take one-third of the product. " "*' ' r'f;:> m*^* ! ; II. To find tho solid contents of a cylinder or prism : ^ ' '' Multiply the area of the hate by the perpendicular Ju^ht. III. To find the surface of a sphere : ^^: ^ > Multiply the square of the diameter by 3.1416. >- *' • TV.. To find the solid contents of a globe or sphere : MiUtiplv the third power of the diameter 2y .5236. ''- /^^'^ ■■rs , 'tn' MZNSUIUXION or BOUDB. 329 V. To find the volume of a spheroid, the axes being g*Ten : Mult^jf the tjuare of tht qsm ^r«^k^iof^ bjf the Jixed axitf •wiidthejaroduUbjfJiZiQ. l*f: ' EXER0ISB8. 57. If the diameter of the base of a cylinder be 2 feet, and its height 5 feet, what is tho solid content ? Ans. 25.708 feet: 58. If the diameter of the base of a cone be 1 foot 6 inches, and the altitude 15 feet, what are the solid contents ? Ans. 8 feet, 120 inches. 59. If the diameter of the base of a cylinder be 7 feet, and the height 5 feet, what is the solid content ? ' Ans. 245 cubic feet. 60. What are the solid contents of a hexagonal prism, each side of the base being 16 inches and the height 15 feet? Ans. 69.282 cubic feet: 61. A triangular pyramid is 30 feet high, and each side of the base is 3 feet ; required tho solid contents. Ans. 39.98 cubic feet. 62. What are the solid contents of the earth, the diameter being taken as 7918.7 mUes ? Ans. 2C '>992732079.87. 63. In a spheroid the less axis is 70 and the greater 90 ; what are the solid contents ? Ans. 230907.6. PILING OF BALLS AND SHELLS. Balls are usually piled on a base which is either a triangle, or square, or rectangle, each side of each course containing one ball less than the one below it. If the base is an equilateral figure, the vertex of a complete pile will be a single ball ; but if one side of the base be greater than the contiguous one, the vertex will be a row of balls. Hence, if the base be an equilateral figure, the pile will be a pyramid, and as the side of each layer contains one layer less than the oiie below it, the number of balls in height will be the same as the number of balls ia one side of the lowest layer. If the pile be rectangular, each layer must also be rectangular, and the number of balls in height will bo the same as the number in the less side of the base. If the base be triangular, we have the 880 •*^ ABUBmriCL BULL .til.' :') 1' Multiply ih0 mmher on ofM nd» of the (oMofii row Sy ittelfVLUB one, and the product by the tame ban row FLUl ^ BULB. . From three time$ the number of balls in the length of the hnoett tourte aubtract one leu than Me nun^er in the breadth of the tame eourte ; multiply the remainder by the breadth^ and thit product by ime-tizth the breadth plus one. If the pile be incomplete, find what it to&uld be if complete ; find alto what the incomplete one would be at a teparatt pile^ andnh* tract the latter from the former. >^ii. BXBB0I8B8. 64. In a eomplete triangular pile eaoh side of the base is 40 ; bow many balls are there ? Ans. 11480, 65. In each side of the base of a square pile there are 20 shells ; how many in the whole pile ? Ans. 2870. 66. In a rectangular pile there are 59 balls in the length, and 20 in the breadth of the base ; how many are in all ? Ans. 11060. 67. In an incomplete triangular pile, each side of the lowest layer consists of 40 balls, and the side of the upper course of 20 ; what is the number of balls 7 . „v Ans. 10150. f riv"^.'/-* ■ Nora.— Since the upper ooune b 30. the first row In the wanting part voold be 19. ..I f. •i.Sffi*i^', i-^t; , *Ji;. fi'.S •.$> ,%^A«MM^-.,.i!fS?*v4;afe- .^^iJj: j^;^ ^{-r vM-;-:?iiti'.i.ji-' ■'v-'A. • XZABUBEMZNT 07 TDCBIB. 881 wOU MEASUREMENT OF TIMBER. .i\ Timber is moMured sometimes by the square foot, and sometimei by the oubio foot. Cleared timber, such as planks, beams, &o., are usually measured by the square foot. . „ What is called board measure is a certain length anu 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, &o. RULE. Multiply the length in feet hy the breadth in incites, and divide 6yl2. Note.— The ihickness being taken uniformly as one inch, tlie rule for find- ing the content!! in square feet becomes the same as that for finding surfaoe. If the thickness bo not an inch,— ' Multiply the hoard meature hy the thickness. - If the board be a tapering one, take half the sum of the two * extreme widths for the average width. If a one-inch plank be 24 feet long, and 8 inches thick, then wo have 8 inches equal f of a foot, and f 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, t. «., 1^ feet, and 30xlf=50 ; or, 30x20=600, and 600-j-12='50. To find the solid contents of a round log when the girt is known. BULK. . ■ , ;, .,,...■, ;-.;., '"' Multiply the square of the quarter girt in inches by tfie length in feet, and divide theproduct by 144. • ; * a^ ^^ * i 'v ' 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-f-144=20|. To find the number of square feet in round timber, when Um mean diameter is given. "^ , . . ^ i. ; .iL'.^ '>.il 382 ABUHMma lULB. MuUijpljf tki diamtiet in ineha iy ka^ the diamder in inehm, €»d theprodvct bjf the length in/eet, and divide the retuU 2y 12. If A log is 30 feot long, and 66 inches mean diuMter, the number of squue feet is 66 X 28 X 30-h12=:3920 feet. To find the solid contents of a log when tho length and mean diameter are given. BULB. Multiply the tquare of half the diameter in inchee ly 3.141 6, and thie product hy the length in feet, and divide by 144. 68. How many cubic feet are there in a piece of jnbor 14x18, and 28 feet long ? Ans. 49-f cubic feet. 69. How many cubic feet are there in a * >d log 21 inches in diameter, and 40 feet in length ? 70. What are the solid contents of a log 24 inches in diameter, and 34 feet in length ? Ans. 106.81 -j-cubic feet 71. How many feet, board measure, are there in a log 23 inches in diameter, and 12 feet long ? Ans. 264^. 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 ield ? Ans. 640 solid feet. 76. How many solid feet are there in a board 16 feet long, 6 inches wide, and 3 inches thick ? Ana. 1/^ 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 26 inchcff in diameter, and 32 feet long ? 79. How many feet, |)oard measure, do^ a log 28 inches in diaiaater, and 14 feet in length contain ? Ans. 467|. 80. How many cubic feet are contained in a piece of squared timber that is 12 by 16 inches, and 47 feet in length 7 Ans. 62|. XEABtmZMEin* 07 TIMBER. 888 31. How many foot, board moaauro, nro there in 22 one-inob boarda, eaeh being 13 inchea in width, and 16 feet in length ? Ana. 381|. BALIB, BiNa, ao. Aa bales are UHually of tho aamc form as boxes, tbo aomo rule applies. 82. Honoo, a balo measuring 4^ inohos in length, 33 in width, and 3^ in doptb, is, in Holid content, 37^ fooC. 83. A orate is 5 foot long, 4| broad, and 3yT} doop, what is the solid content ? Ans. 85:^g. To find how many bushels are in a bin of groin : BULB. Find the product of the length, breadth and depth, and divide hjf 61ft0.4. 84. A bin consists of 12 compartments ; each measures 6 feet 3 inches in length, 4 foot 8 inches in width, and 3 foot 9 inches in depth j how many bushels of grain will it hold ? Ans. 1055, nearly. To find how- many bushels of grain are in a oonicol heap in the middle of a floor : '■«■■ ^'' • '• . " ' '-•'■' BULE . ■•''".•■ '. I ■.■■..•■'. , Multiply the area of the bate by one-third the height. The base of such a pile is 8 feet diameter and 4 feet high ; what is tho content ? The area of tho base is 64X. 7854=83.777, and 83.777xi=- 67.02, tho number of bushels. If it be heaped against a wall take half the above result. If it be heaped in a corner, take one-fourth the aboye result. ^f^v^-^. . -taf; ■;! -;-■ ■-■' ._. ., T (i .»i -. /.-.t*^- e'V h:-i:'^:r--f,-' ' ,• •-».l4 •'.:> -•.•!-•» „ • ,1 -.», - ■*! -^^ - — • - ■ --•,,-;;-= ,-7 — - >• ..- 1 ' -'Hfii ' . r '•'!,-._.. .) - ■^. I . ^.i,v v)!' • i-f 1J-! - .;. : ■•»f j .'Jl%i,.- ^>ij..'. ■■^^M ■ ^.5«?.'"eji' Vl^ flS n -ir Krfffxyy, 884 ABUHlCEnO. MISCELLANEOUS EXEBCISEB. \ 1. What number b that f and f of which make 255 ? Ans. 201-/^,. 2. What must be added to 217^, that the sum may be 17^ timea 19J? Ans. 118f 3. What sum of money must be lent, at 7 per cent., to accumu- late to $455 interest in 3 months ? Ans. $26000. 4. Divide $1000 amcng A, B and C, so that A may have $156 more than B, and B $62 less than C. Ans. A. $41 C§; B, $260§; C, $322f. 5. Where shall a pole 60 feet high be broken, that the top may rest on the ground 20 feet from the stump ? Ans. 26§ feet. 6. A man bought a horse for $68, which was | as much again as he sold it for, lacking $1 ; how mu«b did ho gain by the bargain ? • > . Ans. $12.50. 7. A fox is 120 leaps before a hound, and takes 5 leaps to the hound's 2; but 4 of the hound's leaps equal 12 of the fox's ; how many leaps must the hound take to catch the fox? Ans. 240. 8. A, B and C can do a certain piece of work in 10 days ; how long will it take each to do it separately, if A does i^ times as much \^ as B, and B does ^ as much as G ? Ans. A, 30 days ; B, 45 ; C, 22^. 9. At what time between five and six o'clock, are the hour and minute hands of a clock exactly together ? Ans. 27 min., 16y^ sec. past 5. 10. A courier has advanced 35 miles with despatches, when a second starts with additional instructions, and hurries to overtake the first, travelling 25 miles for 18 that the first travels ; how far will both have travelled when the second overtakes the first ? • T" Ans. 125 miles. ; 11. What is the sum of the series | — ts"^^! — iVs+i^s — ^^' ^ Ans. 2*5. 12. If a man earn $2 more each month than he did the month before **od finds at the end of 18 months that the rate of increase will §D te him to earn the same sum in 14 months ; how much did lie earc in the whole time ? . Ans. $4032. 13. How long would it take a body, moving at the rate of 50 mSCELLAMEOFS EZEBCISES. 885 *.■ miles an hour, to pass over a space equal to the distance of the earth firom the sun, t. e., 95 millions of miles, a year being 365 days ? Ans. 21C years, 326 days^ 16 hours. 14. Two soldiers start together for -a certain fort, and one travels 18 miles a day, and after travelling 9 days, turns back as far as the second had travelled during those 9 days, he then turns, and in 22^ days from the time they started, arrives at the fort at the same time as his comrade ; at what rate did the second travel ? Ans. 18 miles a day. 15. What quantity must be subtracted from the square of 48, so ^^ that the remainder may be the product of 54 by 16 ? Ans. 1440. -"•"^^ 16. A father gave J of his farm to his tion, the son sold § of his share for $1260 ; what was the value of the whole farm ? Ans. $5040.' 17. There were | of a flock of sheep stolen, and 672 were left; how many were there in all ? Ans. 1792. 18. A boy gave 2 cents each for a number of pears, and had 42 cents left, but if he had given 5 cents for each, he would have had nothing left. Eequiifed the number of pears. Ans. 14. 19. Simplify p- -■ ■^"^2Ti* Ans. f. 20. A man contracted to perform a piece of work in 60 days, he employed 30 men, and at the end of 48 days it was only half finish- ed; how many additional hands had to be employed to finish it in the stipulated time? : . 21. A gentleman gave his eldest daughter twice as much as his second, and the second three times as much as the third, and the \^ third got $1573 ; how much did he give to all ? Ans. $15730. '\'^ 22. The sum of two numbers is 5643, and their difference 125 ; - what are the numbers ? Ans. 2884 and 2759. 23. How often will all the four wheels of a carriage turn round m going 7 miles, 1 furlong, and 8 rods, the hind wheels being each \. 7 feet 6 inches in circumference, and the fore wheels 5 feet 7^ inches ? Ans. 23716. 24. What is the area of a right angled triangular field, of which •the hypotenuse is 100 rods and the base 60? Ans. 240'^ sq. rds. 25. Simplify 5i=5i„fli±MI of. ?1±11- Ans. l^f. 836 ABXIHMETIO. 26. Find the value of i_|._-L_' Ans. |. 27. If § of A's age is ^ of Bs', and A is 37|, vrhA ago is B ? Am. 40. 28. What is the excess of ■^^- '— above «iT.-i~r«'«-r ? TOT U7Tf~n:uuT Ans. -rISf ,. 29; The sum of two numbers is 5330 and their difference 1999 ; what are the numbers ? Ans. 3664^ and 1665|. 30. A person being asked the hour of the day, replied that the time past noon was equal to one-fifth of the time past midnight ; what was the time? Ans. 3 P,M. 31. A snail, in getting up a pole 20 feet high, climbed up 8 feet every day, but slipped back 4 feet every night; in what time did he reach the top ? Ans 4 days. 32. What number is that whose jj^, ^, and ^ parts make 48? Ans. 44^*5. 33. A merchant sold goods to a certain amount, on a commission of 4 per cent., and, having remitted the net proceeds to the owner, received ^ per cent, for immediate payment, which amounted to $15.60 ; what was the amount of his commission ? Ans. $260. 34. A criminal has 40 miles the start of the detective, but the detective makes 7 miles for 5 that the fugitive makes ; how far will the detective have travelled before he overtakes the criminal ? Ans. 140 miles. 35. A man sold 17 stoves for $153; for the largest size he received $19, for the middle size $7, and for the small size $6 ; how many did he sell of each size ? Ans. 3 of the large size, 12 of the middle, 2 of the small. 36. A merchant bought goods to the amount of $12400 ; $4060 of which was on a credit of 3 months, $4160 on a credit of 8 months and the remainder on a credit of 9 months ; how much ready money would discharge the debt, money being worth 6 per cent. ? \.,,t;,..:..v':>,..i:"r.-- ■,:.:_. ..•.. ........--.,,.....,.-.. ■Ans.$i2000. 37. If a regiment of soldiers, consisting of 1000 men, are to be dothed, each suit to contain 3f yards of cloth that is If yards wide, land to be lined with flannel 1| yards wide ; how many yards will it &ke to line the whole ? Ans. 5625. 38. Taking the moon's diameter at 2180 miles, what are the folid contents? Ans. 5424617475-f sq. miles. MiSGELLAKEOnS EX£BCISES. 887 39. A certain island is 73 miles in circumference, and if two men fitart out from the same point, in the same direction, the one walking at the rate of 5 and the other at the rate of 3 miles an hour; ia whai time will they come together ? Ans. 3U hours, 30 minutes. 40. A circular pond measures half an acre ; what length of cord will be required to reach from the edge of tlie pond to the centre ? Ans. 83263-j- feet 41. A gentleman has deposited $450 for the benefit of his son, in a Savings' Bank, at compound interest at a half-jearly rate of 3^- per cent. He is to receive the amount as soon as it becomes $1781.60^. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in possession of the money ? Ans. 21 years. 42. The select men of a certain town appointed a liquor agent, and furnished him with liquor to the amount of $825.60, and cash, $215. The agent received cash for liquor sold, $1323.40. He paid for liquor bought, $937 ; to the town treasurer, $300 ; sundry ex- penses, $29 ; his own salary, §205 ; }ic delivered to indigent persons, by order of the town, liquor to the amount of $13.50. Upon taking stock at the end of the year, the liquor on hand amounted to $616.50. Did the town gain or lose by the agency, and how much; has the agent any mopey in his hands belonging to the town ; or does the town owe the agent, and how much in either case ? Ans. The town lost $103.20 ; the agent owes the town $7.4U. 43. A holds a note for $575 against B, dated July 13th, paytc ble in 4 months from date. On the 9th August, A received in advance $62 ; and on the rtth September, $45 more. According to the terms of agreement it t;ill be due, adding 3 days of grace, on the 16th November, but o|i ih'i 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 beyontl the IGth of November; how much must B pay on the 3rd of October? Ans. $111.43. 44. A accepted .'>ii agency from B to buy and sell grain for him. A received from B grain in store, valued at f 135.60, and cash, $222.10 ; ]:c beuj^/it grain to the value of $1346.40, and sold grain to the amcant of $1171.97. At the end of four months B wished to close the agency, and A returned him grain unsold, valued at $437.95 ; A was to receive for «er/ices, $48.12. Did A owe B, pr B owe A, and how much ? Ans. B owed A 45 tHtHm 888 ABITHMETIO. 45. A general ranging his men in the form of a sqnaret had 59 men over, but having increased the side of the square by one man, he looked 84 of completing the square ; how many men had he ? Ans. 5100. 46. What portion, expressed as a common fraction, is a ponnd and a half troy weight of three pounds avoirdupois ? Ans. ■^^^, 47. What would the last fraction bo if wo reckoned by the ounces instead of grains according to the standards ? Ans. |. 48. If 4 men can reap 6^ acres of wheat in 2 J days, by working 8J hours per day, how many acres will 15 men, working equally, reap in 3J days, working 9 hours per day ? Ans. 40 j-J days. 49. Out of a certain quantity of wheat, ^ was sold at a certain gain per cent., ^ at twice that gain, and the remainder at three times the gain on the first lot ; what was the gain on each, the ga:': . on the whole being 20 per cent. ? Ans. 9|, 19 J and 28| per cent. 50. If a man by travelling 6 hours a day, and at the rate of 4^ miles an hour, can accomplish a journey of 540 miles in 20 days ; how many days, at the rote of 4f miles an hour, will he require to accomplish a journey of 600 miles ? Ans. 21 1. 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 $8000 after Jones had made purchases to the amount of $13682.24 ; — 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 $2836.24 ; Smith has paid $364.16 and Jones $239.14 for expenses. At the end of the year Jones has on hands goods worth $2327.34 and Smith goods worth $3123.42. The term of the agreement having now expired, a settlement is made, what has been the gain or loss ? What is each partner's share of gain or loss i '. 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 ^Is 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 g^r; the' sum paid each day to all the boys is double the sum paid to all thei girls, and for every five shillings earned by all ^e bdysj ekoh dby, twelvei shillings are earned by all the men; it laSOELLANEOnS ISXEBCISES. 339 is required to find the number of men, the number of boys and the number of girls, the whole number being 59. Ans. 24 men, 20 boys and 15 girls. 53. A holds B's note for $575, jiayable at the end of 4 months from the 13th July ; on the 9th Augmt, *A received $62 in advance, as part payment, and on the 5th September $45 more ; according to agreement the note will not be duo till IGth November, three days of grace being added to tho term ; but on the 3rd October B tenders such a sum as will, together with the payments already mide, ex- tend time of payment forty days forward ; how much must B pay on the 3rd of October ? Ans. $111.43. 54. If a man commence 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 years? - Ans. $27910. 55. A note for $100 was to come due on the 1st October, but on the 11th of August, the acceptor proposes to pay as much in ad- vance as will allow him GO days after the 1st of October to pay tho balance; how much must he pay on the 11th of August ? Ans. $54. 56. A person contributed a eert:?in sum in dollars to four char* ities ; — to one he 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 the remainder and half a dollar, together with one dollar that was left ; how much did he give to each ? Ans. To the first, $16; to the second, $8; to the third, $4; to the fourth, $3. 57. A farmer being asked how many sheep he had, replied that he had them in four different fields, and that two-thirds of the num- ber in the first field was equal to three-fourths of the number in tho second field; and that two-thirds of the number in the second field was equal to three-fourths of the number in the third field ; and that two-thirds of the number in tho 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, 192i fourth field, 160. Total. 811. 340 AETTHMETIC. *.,- 68. How many hours per day must ?A7 men work lor 5 J days to dig a trench 23J- yards long, o^ yarda vido, and 2^ deep, if 24 men working equally can dij; one 3.} J yarda long, 5} vvidj, and iih deep, in 189 days of l-i hours caoli. Ans. IG liwurd. 59. A man bequeathed one-fourth of his property to his oldest son ; — to the second son one-fourth of tiio remainder, and $350 be- sides ; to tho third one-fourth of tlio remainder, togetlicr with $975 ; to tho youngest ono-fourth of tho remainder and $1400 ; ho gives his wife a life interest in tho remainder, and her share is found to be one-fifth of tho whole j what was tho amount of tho property ? ■ • . Ans. $20,000. 60. Five men formed a partnership which was dissolved after four years* continuance ; the first contributed $G0 at lirsfc and $S00 more at tho end of five months, and again $1500 at the end of a year and eight months; tho second contributed $G00 and . Florin before 1858 New florin New Union dollar Maria There.sttdol'r.nsO Five franca 51.14 « 48.63 i< 73.01 II 1.02.12 Belgium 98.04 Bolivia New dollar 79.07 II Half dollar 39.22 Brazil Double Milreis 1.02.53 Canada 20 cents 18.87 Central America. . . Dollar 1.00.19 Cliili Old Dollar 1.06.79 II New Dollar 98.17 Denmark Two rigsdaler 1.10.65 England Shilling, new 22.96 II Shilling, average Five franc, average Thaler, before 1857 Now thaler 22.41 France Germany, North... 11 98.00 72.67 72.89 Germany, South. , . . II Florin, before 1857 New florin fassumedl . . . 41.65 41.65 Greece Hindostan Five drachms Rupee 88.08 46.62 Japan Itzebu 37.63 II New Itzebu 33.80 Mexico Dollar, new 1.06.62 11 Dollar, average Scudo 1.06.20 Naples. ... Netherlands 95.34 2i guild 1.03.31 Norway Specie daler 1.10.65 New Granada Dollar of 1857 97.92 Peru Old dollar 1.06.20 li Dollar of 1858 94.77 14 Half dollar, 1835-38.... Thaler before 1857 New thaler 38.31 Prussia. 72.68 II 72.89 Rome . / . . . Scudo 1.05.84 Russia Rouble Five lire 79.44 Sardinia 98.00 Spain New pistareen 20.31 Sweden Rix dollar 1.11.48 Switzerland Two francs 39.52 Tunis. Five piastres Twenty piastres Florio 62.49 Turkey 86.98 Tuscany 27.60