IMAGE EVALUATION TEST TARGET (MT-3) // 1.0 1.1 11.25 "IB |M 2.0 1.4 11.6 Fhoiographic Sciences Corporation 23 WEST MAIN STRKT WIBSTM.N.Y. 14SM (716) S73-4S03 \ RV 4 o ««^A '^'°^\' ^ ^^.V^ v\ CIHM/ICMH Microfiche Series. CIHM/ICMH Collection de microfiches. Canadian Instituta for Historical Microraproductions / institut Canadian da microraproductiont historiquaa Technical and Bibliographic Notoa/Notas tachniquaa at bibllographiquaa Tha Inatltuta haa attamptad to obtain tha baat original copy avallabia for filming. Faaturaa of thia copy which may ba bibllographleally uniqua. which may altar any of tha Imagaa in tha raproduction, or which may aignlflcantly changa tha uaual mathod of filming, ara chackad balow. 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Las diagrammes ftuivants illustrent ia mAthode. 32X 1 2 3 4 5 6 it THE BRITISH AMERICAN COMMERCIAL ARITHMETIC, FOR THE USE OF SCHOOLS, COllEGES, AND COUNTING-HOUSES, ^ BMBBAOINa AN EXTENSIVE COURSE BOTH IN THEORY AND PBAOTICE. ■T T. A. BRYCE, M.A, TBACBBR OF MATHKMATICS AND ENOUSH, IN TIUC TOSOBITO B. A. 0. 0. \ -,--^'''A^«^t"-v-C?' " i>>=..".~^<4v":" AND ■ T. C. MUSGROVE and H. C. WRIGHT, PRINCIPALS AND PROPRIETORS OK THE • *' ' • . ' _ ?■ .-*■. BBrnSH AMERICAN COMMERCIAL COIliEOES. ^':(;..'^.» . ^'''i:{v ' ", v(;v , ■-':- I -.^i. TORONTO: PUBLISHED BY MUSGROVE ft WRIGHT, CORMSR OF KINO AND TORONTO STREBXa 1866. \ \ 1.) ■■■'i' .':.■■). Entered, according to the Act of the Provincial Parliament, In the year cue tliouEana eight hundred and slzty-slx, by Thomas C. Huiorovi anh Ebxry 0. Wbiqht, In the Office of the Registrar of the Province of Canada. 1^*: > i n mRMTTPID AND PRINTIO At nm ''avovB" stxak job frxb8^ TomuRo, aw. .Jl ' ' ■ ir /^/J. PREFACE. Wk have for some time looked upon it as a kind of reproaoh that Canada has never prodnoed a treatise on Arithmetic adequate to the wantM of a commercial community sach as this countiy has become. It is not enough that the school-boy should be provided wiUi a course suited to his years ; there must be supplied to him something higher as he advances in years and progress, and nears the period rhen he is to enter on real business life. • We have feli this keenly in our ezperienoe in conducting the British American Conhiroial Gollkokb. We have hitherto been obliged to have recourse to United States' publications, which, without any disparagement to their intrinsic merit, we are forced to say, are not suited, in many important respects, to the wants of Canada. The great increase in our Bjfiiere of operations, and the grave responsibilities devolving on us thereby, have made us feel in duty bound to supply to our students a book such as they require. We have therefore undertaken the task of supplying the deficiency. As we proceeded with the work we found it necessary to extend our original programme considerably, and, therefore, also the limits of the book, so as to make it useful, not only to our own Collies, but to the community at laige. In carrying out our plan we have endeavoured to unfold the theory of Arithmetic as a soisnoe in as concise a manner as seemed condstent with clearness, and at the same time to show its applica- tions as an art. We have striven to make the business part so copious and practical as to afford the young student ample informa- tion and discipline in all the principles and usages of commercial intercourse. For the same reason, we have introduced some articles on Commercial Law, written by J. D. Edgar, Esq., Barrister-at-Law, a prominent part of our aim being to produce a work which shall 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. We have taken especial care not to enunciate any rule without explaining the reason, for, without a knowledge of the principle, the operator is a vr. PBKIAOC. mere ffftmlitting midiuie that can work Irat a oertain roond, and ia almoat lare to be at fault when novel oaaee ariae. In giving thoaa explanatioDi we have not followed any predeoeeion, bat have been guided entirely by our own experienoe in teaching. The great maaa of the exercises likewise are entirely new, though we have not aorupled to make selections from some of the most approved worka on the subject ; but in doing so, we have confined ourselves to suoh queetions as are to be found in almost all popular works, and which, therefore, are to be looked upon as the common property of science. We have, as much as possible, avoided Algebraic forms and notation, as being unsuited to a large proportion of those for whom the book is intended, and to many altogether unintelligible. We have been encouraged to follow out this course from the reflection that |hoBo who understand Algebraic modes will have all the less diffi- culty in understanding the common Arithmetical ones. Even in the Mathematical parts, we have endeavoured to popularise the aa1l>- ject as much as jmssible. We were compelled to follow a certain logical order in arranging the subjects treated of, but the teacher and learner will often find it necessary to depart from that order. (See Suggestions to Teachers.) As we consider that rules and definitions should always be expressed in the smallest possible number of words consistent with perspicuity and accuracy, we have taken great pains to carry out this principle in every case. Wo luivo appended copious exercises to each rule, especially to the most important, as Fractions, Proportion, Analysis, and Interest. Besides these, we have introduced extensive collections of mixed exercises throughout the body of the work, besides a laige number at the end. The utility of such miscellaneous questions will be admitted by all, but the rea$on why they are of such importance seems strangely overloaked or misunderstood, even by writers on the subject. They are spoken of as mere review exercises. Their great value depends on something still more important. A class ia work- ing questions on a certain rule, and each member of the class has just heard the rule enunciated, and readily applies it. So far, one important object is attained, viz., freedom of operation. But some- thing more is necessary. The learner must be taught to discern what ruk it to be applied to the solution of each question proposed. The pupil, under careful teaching, may be able to understand fully every rule, and never confound any one with any other, and yei be u nOEFAOE. ▼. doubtfol what rale to apply to an iadividoal oaae. The miaoeUa- neouB probkaua, therefore, are intended, not to nraeh as ezeroiaea on the opereUiotu of the different rales, as on the mode of applying thoae rales ; or, in other words, to practise the pnpil in peroeiving of what rale any proposed question is a particular case. To this we attach great importanoe, not only as regards readiness in real biiiine«i, hut also as a mental exercise to the yonng student. We are far from supposing, much less asserting, that the work is complete, especially as the whole has been prepared in less than the short space of six months. Wo present it, however, to the public, in the oonfideii» expectation that it will meet, to a great extent at least, the necessities of the times. With this view, also, we have given the great mass of the examples and exercises, involving money, in dollars and cents, with, however, a number in pounds, shillings and pence, sufficient for the purpose of illustration. Wo have fol- lowed this course becfiUse we do not see any use in perpetuating the cumbrous system of the old currency, and we even hope to see the day when the decimal system will be universally adopted in British America. For this reason, too, in teaching Book-keeping, we uni- formly employ the decimal notation. Wo b^, also, respectfully to call the attention of the Qovernment to the desirableness of taking steps to introduce the decimal system in weights and measures as well as in money. Its great simplicity must commend it to every intelligent mind. Some difficulty would, no doubt, for a time attend the change, but this would soon pass away, especially as the country is already familiarized with the nota- tion by the use of the decimal coinage. Wo feci confident that a Parliamentary bill for this object, co-ordinate with the same move- ment now going on in Britain, would be most acceptable to the great majority of the people of Canada, if the matter were only taken up by some persons of energy and influence. The rule for finding the Greatest Common Measure, though not new, is given in a new, and, we think, a concise and convenient form. • The rale for finding the Cube Hoot is a modification of that recommended by Dr. Hind, and will be found very ready and short. On the subjects of Logarithms and Mensuration, we have only given the general principles, and a few of the most important cases. To do justice to these subjects would require separate books. In treating of Common FrwtioM we have placed Multiplication and ff. PREFACE. Diyision before Addition and Subtraction, for two reaeons : — FiBST. In Common Fractions, Multiplication and Diviaion proaont much leaa diflkulty than Addition and Subtraction ; and, occondlt, as in Whole Numbera, Addition is the rule that regulates all others ; so in Fractions, which originate from Division, wc see, in like manner, that all other operations result from Division, and, in connection with it, Multiplication . Wo hope this will be accounted reason sufficient for following so unwonted a course. By the ordinary routine the pupil is, in effect, colled upon to apply rules and principles that ho has not learned. Several subjects, commonly treated of in works on Arithmetic, have been omitted, in order to leave space for more important matter bearing on commercial pursuits. Duodecimals, for example, have been omitted, as that mode of calculation is now virtually super- seded by the use of Decimab. Barter, too, has been passed by, as questions of that class can easily be solved by the rule of Proportion, which has been fully explained. The subject of Analysis has been gone into at considerable length, and vre expect that the new manner in which the explanations and solutions are presented, and tho extensive collection of exercises appended, will contribute to moke this a valuable part of the treatise. Wo think that tho view wo have given of Decimal Fractions is the only true one, and is calculated to give to tho student clear notions of tho nature of tho .notation, and to show tho great conve- nience and utility of Decimals. Wo have ignored the distinction sometimes made between Decimals and Decinval Fractions, as being " a distinction without a difference." Decimal is merely a short way of writing Decimal Fraction. Thus : .7 is merely a convenient mode of writing ^^. These differ in form only, but otlnerwise are as perfectly identical as f and g. Tho contracted methods of Multiplication and Division will be found, after somo practice, extremely useful and expeditious in Decimals expressed by long lines of figures. . x We have, after some hesitation, introduced an article on Log- arithms, and also Tables, as the logarithmic mode of computation ■ saves, in many cases, an immense amount of time aud labour, and will be found extremely useful in surveying and mechanical calcula- tions, especially when angular and linear units have to bo compared. As the book has expanded to much greater dimensions than we * PREFACE. fU. anticipated, we have judged it better not to insert a table of 8qaarea and Cabes, as wo had intended. For the same reason, wo found it impossible to introduce tables of Logarithmic Sines, Cosines, &o. Wo hare entrusted a great part of the composition of tho work to the Teacher of Mathematics and English in our Toronto Collie, T. A. Bryce, M.A., Qlosgow University, Scotland. As our own time was so much occupied in teaching, wo were anxious to prccurc the assistance of a Kontleman who was at once a sound Mathemati< cian, an aooomplished English scholar and an aocurato writer, and at the same time an experienced practical teacher, believing that all these qualifications woro needed for tho composition of such a trea- tise. These requisites wo found in Mr. Bryoo. In the explanation of principles and framing of rules wo think he has been peculiarly happy, and we take pleasure in acknowledging the valuable aid he has rendered us in tho preparing of the book. .))<•■' '■s! >'. ;■• ;n's .-,<» i •■A K SDGGESnONS TO TEACHERS AND STDDENTS. Wi would first refer to our remark in the Preface, that we did not expect the teacher to follow oar logioal arrangement, and even advised that he should not. We know by ozporienoe that the same course does not suit all students, any more than the same medical treatment suits all patients. The course requires to bo varied according to age, ability and acquirements. The greatest difficulties generally present themselves at the earliest stages. What more serious difficulty, for example, has u 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 beginners, therefore, we recommend the following coarse : Let the elementary rules be carefully explained and illustrated by nmple exampha, and the pupil shown how to work easy exerciaea. This done, let the whole be reviewed, and exercises of a more difioult kind be proposed. The Decimal Coinage should then be taken up. In explaining this part of the subject the teacher ought to notice carefully that the operations in this case differ in no way from those already gone through, in reference to Abstract Numbers, except in the preserving of the position of the mark that separates the cents from the dollars, usually called the decimal point. The next step should be the whole subject of Denominate Numbers, and, in illus- tration and application, the rule of Practice. After a thorough review of all the ground now gone over. Simple Proportion may be entered upon, using only such questions as do not involve Frac- tions. Then, after a course of Fractions has been gone through, Proportion should bo reviewed, and questions which involve Frac- tions proposed. Afler this, it will generally be found desirable to study Percentage, with all its applications, the most important of which is Interest. The order in which the rest of the course shall be taken is comparatively unimportant, as the student has now realized a capital on which he can draw for any purpose. i BUOOESnONB TO TEACHEBS AND HTITDENTH. IX. We would, in the Htron^^vrit inannrr posHibl*', imprcM nn th« minds of teaohera and HtudenUi the Ki^'*>t^ utility of fnH|uent raviewH, and oapeoialljr of conitant oxorciao in tho addition of money flolumnH. We have endeavoured, as far aa poaaiblo, to make tho oxeroiaeH under each rule of progroMivo difficulty. Wo have alao made it an object to giTO each ezorciitc tho HombLnoo of a real queetion, for nil persons, and especially tho young, take greater interest in ezcrciscH that assume the form of reality than in such ntt are merely abetraot ; and, besides, this is a preparatory exercise to tho application of tho rules afterwards. • In the foregoing suggestions we have had in view tho case of a child beginning from the very elements, but tho judicious teacher will readily modify and apply these principles according to circum- stances. We shall only briefly add the following cautions and hints : At every stogo tho greatest care should be taken that the learner underbuindM the meaning of each rule, and the terms and conditions of each question, before he attempts to solve it. The teacher should never attempt to explain two things at tho same time, and he should be satisfied that the first idea is fully grasped and stored before he approaches tho second. Unruffled temper and untiring patience are essential on the part of the teacher, in order that he may be able to discover the souroe of every difficulty that presents itself to the pupil's mind, and remove it by careful, and, when necessary, repeated explanation. The teacher should not always be talking or working on the black-board ; he should require the pupils to speak a good deal in answer to questions, and also to work much on their slates. The pupils should never be made to work or listen till they arc fatigued, or till their attention flags. Finally, we would suggest to every teacher to keep constantly before his mind both of the two great works he has to accomplish — Firtt, tho development of the mental powers of his pupil ; and, secondly^ imparting to him such knowledge as he will require to use when ho enters upon life, cither as a professional man or a merchant. Some seem to consider these two objeets incompatible, as if taking up time in mental training left insufficient time for the imparting of actual knowledge. We, however, consider this a palpable error, for the more tho mind is cultivated, tho more readily and rapidly will it take in any species of knowledge, and the more surely too will it retain what it has mastered. Mental culture is at once the foundation and \ 1^' z. SUOOESTIONS TO TEACHERS AND STUDENTS. 4 the means ; the other is the superstructure raised on that foundation, and by that means ; or it may be compared to a great capital judi- ciously embarked in trade, and often turned, aLd therefore yielding good profits. It frequently happens, however, from the peculiar circumstances of individuals and families, and even communities, that young men require to bo hurried into business, so as to be able to support themselves ; but even in such cases the desired object will bo much more speedily and securely attained by such a course than by what is usually and not inappropriately called " cramming." Wo have striven to give the character hero recommended to this book, especially in the explanatory portions. We solicit the attention of the student, as well as the teacher, to these suggestions, and in particular of the self-taught student, whose wants also we have kept in view throughout. ''■'■■-"' '"'"'fr'-"'-' ''"' MUSGROVE & WRIGHT, f r ■t >./, v>-r>'^-i^^ TABLE OF CONTENTS. '' ' i Vjm*. Introduction 13 Numeration 14 Notation 15 Axioms 17 AddiUon 17 Subtraction 20 Multiplication 22 DiviBion 26 Tables of Money, Ac 30 Decimal Coinage 32 Reduction 42 Denominate Numbers 45 Addition 46 Subtraction 50 Multiplication 60 Division (Supplement to) 61 Greatest Common Measure 62 Least Common Multiple 64 Exercises on t)ib Preceding Rules 65 Fractions 68 Classification 60 Vulga Fractions 61 Reduction of Vulgar Fractions 61 Multiplication of " 63 Dividonof " 64 Addition of " 65 Subtraotionof " 68 Denominate 69 Decimal Fractions 71 Reduction of Decimal Fractions 71 Addition of « 79 Subtraction of " 79 Multiplication of " 81 Dividon of " 86 Denominate 90 Ratio and Proportion 92 Compoimd Proportion 101 Miscellaneous Exercises on the Preceding Rules 107 Analysis Ill Practice 119 \ ■ w 3di. r INDEX. ' Bills of Parcels 12C Percentage 129 Interest 134 Simple Interest 135 Negotiable Instnunents 148 Partial Payments 152 Compound Interest 162 Discount 165 Banking 167 Commission 172 Brolcerage 175 Insurance 179 Custom House Business 179 Profit and Loss 180 Storage 194 General Average 196 Taxes and Custom House Business 199 Stocks and Bonds 203 Partnersliip 209 Bankniptcy 214 Equation of Payments 216 Averaging Accounts 223 Alligation 232 Medial 232 Alternate 233 Nature and Value of Money 238 Paper Currency 239 Currency of Canada 240 Exchange 241 American Exchange 242 Sterling Exchange 247 Arbitration of Exchange 251 Involution 256 Evolution 258 Progression 270 by Common Differences 270 by Ratio 279 Logarithms 299 Mensuration 304 Miscellaneous Exercises 318 Tables of Logarithms 326 - . .- ....... . ,, , ;■■- >, ;.:■ ■ Ji ■ ARITHMETrC. Artiolk 1. — ^AaiTHBaTio treats of nambera in theoiy and practice. In relation to theory it ia a science, and in relation to practice it is an art. All computations are made by fixing on a certain quantity, called a unit or one, and repeating that unit any required number of times. Various units are selected, according to the nature and extent of the quantity or space to be measured. For example, in measuring length or distance, if the extent is small, such as the length of a pane of glass, we select a small unit, called an inch, and repeating that unit any required number of times, say twelve, we say the pane is twelve inches long, — if a more extended space is to be measured, it is convenient to adopt a larger unit, — ^thus, if we wish to measure the length of a desk, we should probably select a unit called a foot, equal to twelve of the preceding units, — if we wish to measure the length of a room, we should select a still larger unit, called a yard> equal to three of the last, — again, if we wish to measure the length of a field, we should adopt a unit equal to five and a-half of the last, and called a perch or rod, — ^if we wish to note the distance between Toronto and Montreal, we have recourse to a still larger unit, called a mile, and equal to three hundred and twenty of the last, — finally, when astronomers are estimating the distance of any planet, say the earth, from the sun, they generally use a unit equal to a million of the last-mentioned, and they say that the earth is ninety-five millions of miles from the sun, but they simply note the distance as ninety-five; and in the same manner they mark the distance of Venus as sixty-nine, meaning in both cases that each unit is a million of miles. A similar illustration may be applied to every kind of measurement. The symbols or characters now almost universally used to denote quantity or magnitude, are the Arabic figures, or digits 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These, by various combinations, can be made to represent any quantity or magnitude whatsoever. The first nine are called significant figures, because they always denote some real quantity, — the last, called nought (often improperly ought), or cipher, or zero, simply indicates the absence of any significant figure. •14 AIIITHMETIC, NUMERATION. 2. — Numeration is the mode of marking and reading off any line of figures that has been written down, so as to ascertain its value readily and express that value in words. For this purpose every such lino is divided into sets or lots of three figUtes each, counting from right to left, and each set is called a period, — thus, 888888888 forms three periods by marking the figures in threes from right to left by a character of the same form as the comma in composition, — thus, 888,888,888. The first period is called the period of units, the second the period of thousands, the third the period of millions, and so on, — billions, trillions, quadrillions, &c., &c.. to any required extent, which seldom exceeds millions. The first figure of each period denotes units'''^ of that period, the second tens, and the third hundreds of that period. Thu^, in the example given above, the first figure denotes eight units in the period of units, or eight ones, or, as it is usually read, simply eight ; so, also, the fourth denotes eight units in the period of thousands ; or eight times one thousand, or eight thousands ; the seventh figure again denotes eight units in the period of millions, or eight times one million, or eight millions ; again, the second, fifth, and eighth figures denote tens in the period of units, thousands and millions, respectively ; lastly, the third, sixth and ninth figures denote hundreds in the periods of units, thousands and millions, respectively. Such a line, then, as 888,888,888 is read eight hun- dred and eigety-eight millions, eight hundred and eighty-eight thousands, eight hundred and eighty-eight. Every period but the last must have three figures. Thus, in the line 43,279,865 the iBrst and second periods have three digits each, units, tens and hundreds, but the third has only two, units and tens, but uo hundreds, and therefore is read forty-tljree millions, two hundred and seventy-nine thousands, eight hundred and sixty-five. RULE FOR NUMERATION. Beginning at the right, count off periods of three digits each till not more than three are left ; then vead off each period from left to * It is somewhat awkward that the term units is used for two purposes, viz. : as the name of the fii'st period and also us the name uf the first figure of each period. Thougli we cannot well change what usage has so long estab- lished, yet the teacher may obviate (he difiBculty by s'arying the expression occasionally, if not habitually, sayintc, ^ G., units in the unity period, or the place of units in the units period. NOTATTON. 15 right by naming as many hundreds, tens and units as each contains, and adding at the end of each period its proper name. The name of the unity period is usually omitted. When a cipher occurs no mention is made of that place in the period, but the cipher is counted as a digit; thus, in the lino 360,708,091 each cipher is counted a digit, but the reading; is three hundred and sixty millions, seven hundred and eight thousands and ninety-one. EXERCISE. s Divide into periods and read the following lines : 1.— 586729341 2.-976852734 4.-92879357485 5.— 4638709120 7.-2822828228288 8.— 10904870 3.— 2178427385 6.— 11111111111111111 9.— 1010101010101. NOTATION. 3 — Notation is the mode of expressing any quantity or mag- nitude by the combination of conventional symbols or charactxjrs. Thus, by the Roman notation, the letter I. stands for one, IT. for (wo, X. for ten, &c. ; thus, XII. stands for one ten and two units. By the Arabic notation, any digit standing alone, as 5 in the margin, denotes simply five units, but if another digit (5) be placed to the right of it, then the new 5 denotes units and the other 5 becomes tens, so that appending a second digit makes the first one ten times its original value; agr'n, if another digit (5) be subjoined, it takes the place of units, and the 5 next to it becomes tens and the third becomes hundreds, so that eacli of them has ten times the value in the third line that it had in the second ; so also, if another digit (5) be added, each of the three to the left of it will have ten times the value that it had in the third line, and so on. Universally, every digit placed to the right makes everv one to the left ten times its previous value. The use of the tenth of the Arabic characters, the cipher (0) will be made more clear by the rule of notation than by numeration. If I am counting my cash and find that I have eigJU ten-dollar bills, and eight one-dollar bills, it is plain from Art. 2 that it I write 8 alone this must represent the one-dollar bills, and to represent the ten-dollar bills along with the one-dollar bills I must 5 55 555 5555 16 ARITHMETIO. write 88, for the figure to the left being ten times that to the right, will stand correctly for the ten-dollar billR, just as that to the right, being in the units' place, stands for the one-dollar bills. — But if I have no one-dollar bills and write 8, this would stand for only one-dollar bills, and henoc the necessity for introducing a non-significant character and writing 80, for though the cipher represents no quantity, yet by being put in the place of units it throws the 8 to be in the place of tens, and therefore the 8 now stands fitly for the eight ten-dollar bills, and is written $80. — Again, if I find that I have two one-hundred-dollar bills, six one- dollar bills, but no ten-dollar bills, and I write only 26, this would be plainly incorrect, for the 2 would stand for ten-dollar bills only, but by inserting a zero mark between the figures I throw the 2 into the place of hundreds, and $206 represents correctly that I have two one-hundred dollar bills, and six one-dollar bills, but no ten-dollar bills. The superiority of this simple system over the cumbrous Roman one will be manifest from ita simplicity and brevity by writing eightif'eight according to both systems — thus : LXXXVIII. and 88. ; RULE FOB NOTATION. Write tne significant figures of the first period named in their proper places, filling up any places not named with ciphers, just as if you were writing the units period with nothing to follow; then, to indicate that something is to follow, place a comma to the right, and do the same for every period down to units, inclusive. For example, teacher says : " Write down one hundred and six millions ; " pupil writes 106 and pauses; teacher adds, "ninety thousand;' pupil fills up thus : 106,090, and pauses ; teacher concludes : " anc eighteen ; " pupil completes 106,090,018. If the teacher should say sixteen millions and the pupil write 016, the cipher woulc be manifestly superfluous, as it has no effect on figures placed to thir right of it, but only on those placed to the left. EXERCISES. Write in figures and read the following quantities : 1. Ten millions, seven thousand and eleven. 2. Ninety billions, seven thousand and ten. 3. Eighteen millions, sixty thousand and nine hundred. ■ , 4. Forty thousand and nine hundred. f ADDITION. 17 5. Eighty-fleyen millions and one. G. Ninety thousand, seven hundred and eight. 7. Eleven millions, eight hundred thousand and twenty-four. 8. Six hundred and seven thousand and ninety-seven. d. Eight hundred and .seventy billions, sixty thousand and eighteen. 10. Eleven eleven. billioiu), eleven millions, eleven thousand and AXIOMS. 4. — Axioms used in the sequel : I. Things that are equal to the same thing, or to equals, are equal. II. If equals be added to equals, the wholes are equal. (Joivllary. — If equab be multiplied by the same, tho productH are equal. III. If equals be taken from equals, the remainders are equal. Cor. — If equals be divided by the same, the quotients arc equal. IV. The whole is greater than its part Cor. — The whole is equal to all its parts taken together. V,. Magnitudes vrhich coincide, or occupy the same or equal - - spaces, are equal. ' / N. B. — This axiom is modified by, but still is the principle of) all business transactions, purchases, sales, barters, exchanges, &c., &o., where the articles traded in are not equals, but equivalents. ADDITION. 5. — Addition is the mode of combining two or more numbern into one. The operation depends on axiom II. The result is called the sum. Thus : $8+$9+$6=$23. The sign plus (+) indi- cates addition. To illustrate the operation, let it be required to find the sum of the five numbers of dollars noted in the margin. First, the numbers are placed so that those of the same name are in vertical coluums, i. e., units under units, tens under tens, &c. Next, we find that the sum of the units' column is (Ax. IV., Cor.) 27, t. e., two tens and seven units. Next, we find that the sum of the tens' column is 35, but, as it is the tens' 8 $287654 758287 612873 494768 836195 Mit ABITHMBTIO. 27 350 2400 27000 260000 2700000 $2989777 oolamn, we write (Art. 3) 350 ; in the «une man* ner we find the sum of the hundreds' column to be 2400 ; tlie sums of the others will be seen by inspection. Having thus obtained the sum of each ooluinn, each being $ummed as if iinita, but placed in succession towards the left (by Arts. 2 and 5), wo now take the sum of the partial results, which ; (Axiom IV. Cor.) is the sum of the whole, viz.: $2,969,777. In practice the operation is much abbreviated in the following manner : — When the units' column has been added, and we And the sum to be 27, i. c, 7 units and 2 tens, wc write down the 7 units under the units' column, and add up (Art. 3) the 2 tens with the tens' column, and we find the sum is 35 tens, i. e., 5 teng and 3 hundreds, and we place the 5 tens under the tens' column, and add up the 3 hundreds with the hundreds column, and so on. The transferring of the tens, obtained by adding the units' column to the tens' column, and the hundreds obtained by adding the tens' column to the hundreds' column, &c., &c., is called carrying. In all such operations the learner should carefully bear in mind the principle explained in Art. 3., that every figure to the left is ten times the value that it would have if ono place farther to the right* ■-'" ' 'T' EXERCISES. Find the sums of the following quantities : $287654 758287 612873 494768 836195 $2989777 (U (2) 99876 (4) \ 895763 > 49176 987654231 63879 89765324 283527 123456789 54387 42356798 659845 908760504 789 56798423 7984 890705063 137568 23567989 31659 759086391 278652 79842356 968438 670998767 85945 65324897 ; 2896392 4340661745 721096 357655787 * We would strongly recommend every one who wishes to become an expert accountant, to avoid the common practice of drawling up a column of figure» in the manner that may be sufficiently illustrated by the adding of the units' column of the above erample. Never aay 5 and 8 are 13 ; 13 and 3 are IG : 16 and 7 are 23 ; 23 and 4 are 27 ; but run up your column thus : 5, 13, 16, 23, • ADDmON. 1 ^ (5) («) (7) 659 " 471 78563 897 47986 12345 668 vHi 5798 C7890 918273 m 19843 . 98765 651928 789 56479 43219 874859 978 28795 87654 263748 Jfi4 897 32169 597485 999 •■'= . 1984 78912 986879 888 twr-. 68195 65439 1 98765 777 ^^, 3879 , 98r65 9876 666 698 43288 987 6&0 5879 77877 456879 , 897 17985 98989 346678 978 19 ^ 336981 '.>•:. .M.^.> 5159 805312 (10) 98 89 76 ' 67 281 592 678 58 67 98 149 67 54 72 298 2744 4705357 12460 (12) 1298 764 H 5837 £ 6495 789 638 546 ' 98 475 394 89 157 ; 638 594 789 114 1971J 27, fur that is the mode to secure both rnpidity and accuracy. The same remark will apply equally to multiplication, and ilierefore to every arithme- tical operation. To enforce this advice let us add it pimple example fu r»u- tion the student before he approaches multiplication. In multiplyin<| 407 by ft, avoid the tediousness of saying times 7 is 42 — 2 and carry 4— C times !) is 54, and 4 is 58—8 and curry 5— C times 4 is 24, and 5 is 29 ; but practice the eye, aided by the memory, to take in at a glance times 7 is 42. &c.— The quick operator uses the eye, and not the tongue. /aUTHMETIO. ,3V There is no method of proving the oorreotneu of any additioa with positive certainty, but a very convenient mode of chocking is to add each column both upwards and downwards. Another mode is, to add by parts and take the sum of those. This is a very secure method in the case of long columns, but not so ready as the former. If the same result is found by each method, the sum may be accounted correct. 578643957 235412712 343231245 SUBTRACTION. 6. SUBTRAOTION is tlio convcrso of addition, t. e., it is the mode of finding the di£ferencc between two numbers, or, in other words, the excess of one number above another. The number to be subtracted is called tlio subtrahend, and that from which it is to be taken the minuend, and the result is called the re- mainder, difference or excess. The sign used for subtraction is a line ( — ) called minus, or less. Let it bo required to find the difference 1>etween $578643957 and $235412712. Having placed them in vertical columns, as in addition, it is obvious that 2 units taken from 7 units will leave 5 units, and that 1 ten taken from 5 tens will leave 4 tens, and so on. But if it is required to find the excess of $513674208 above $347895319, we find that each figure of the subtrahend, except the last, count- ing from right to left, is greater than the corresponding one of the minuend, and therefore, to find the correct difference, we have recourse to a simple artifice, which is deduced from the principle of the notation, and may be illustrated in the following manner : — Taking the question in the margin, we are first required to subtract 7 units from 3 units. Now, though the algebraic notation furnishes the means of noting the difference directly, the ordinary arithmetical form does not, but still it furnishes the means of doing it indi- rectly. By Art. 3 each figure to the left is ten times the value of the next to its right, therefore we take one of the 3 tens and call it ten units, and add it to the 3 units, and thus we have 13 units, which let us enclose in a parenthesis or bracket, thus : (13), to indicate that the whole quantity, 13, is to occupy the units' place ; when one of the three tens has been thus transferred to the units' 333,333 177,777 155,556 2(12)(12)(12)(12)(13) 17 7 7 7 7 15 5 5 5 6 4> RUBTIUGTION. H 200000 120000 12000 1200 120 13 333333 place, only two tcna romain in the place of tens, and we are now required to toko 7 tens from 2 tens ; to do this wo have reooarse to the same artifice, by calling one of the hundreds tent, which gives 10 tens and 2 tens, and so on to the end, the last 3 ncceaaarily becoming 2. We can now subtract 7 fh)m 13, &o., &c. This mode of resolution depends on the oorol* lary to Axiom IV. The parts into which the whole is virtually resolved are shown in the margin. This artifice is popularly called borrowing. In practice the resolution can be effected mentally as wo proceed, and as each figure from which we borrow is diminiahed by unity, it is usual to count it as it stands, and to compensate for this to increase the one below it by one, for, as in the example, 7 from 12 is the same as 8 from 13, and 2 from 3 is the same as 1 from 2. We are now prepared to answer the proposed question, as annexed, and wo say 9 from 8, we cannot, and there are no tens to borrow from, wo therefore take one of the hundreds and coll it 10 teiu, and one of the tens and call it 10 units, which with 8 units makes 18 units, and we take 9 from 18 and 9 remain. Wo have now only 9 tens left, but we reckon them as ten, and to compensate for the surplua ten, we reckon the 1 below as 2, and say 2 from 10 and 8 remain. We proceed thus to the end, and find the whole remainder to be $165778889. $513674208 $347895319 $165778889 EXERCISES. REMAINDERS. 1.— From 847639021 take 476584359=371054662. 2. « 1010306061 « 670685093=339619968. 3. « 59638743 " 18796854= 40841889. 4. « 7813;:57 '« 3745679= 4067578. 5. " 111111111 " 98657293= 12453818. In Subtraction, as in Addition, we have no method of proof that arrives at positive certainty, but cither of tbo two following methods may be generally relied upon. 1. — Add the reminder and subtrahend, and if the sum is equal to the minuend, it is to be presumed that the work is correct. 2. — Subtract the remainder from the minuend, and if this second remainder is the same as the subtrahend, the work may be accounted correofi. n ARrrmnTio. W^tt >*i I 'i'« MULTIPLICATION. -'" 7. — Multiplication may bo Himply dcfiacd by Faying; tli.it it iH a ohort method of performing nddition, whuii nil tho <|uiintitic8 ti^ bo added iiro tho Hoinc or c<|ual. ThuB : «4-«-f-«-( ti-j-(5-f-(}-f-i}-f 6, ' moMiiH that eight hIzoh aro to '>c nddod together, or that hix Ih to be repeated iih oCtcii oh th '• av - unittt in eight, and wu ttuy thut H timcH (» i.H 4H, p- 1 ••,.., .huB: 8xU=--48. So aI»o 8) H.(-84.8-fH4-8 -iw 4S. So 'hat J{.8.-8.t»:r-48, and thus wo ran construct ;» ..mlti|,'a(.. 1 n tublc. The number to be ro|)cated is called the mn1'i)il!' and, and the one that uhowd how often it iti to bo repeated is culled the multiplier, and tho result is called tho product, or what is produced, and henee tho multiplier and nmlti- pliciknd aro alHo called the factors or makers, or producers, and the operation may bo called linding u product when tho factors arc given, llenco aldo the mode of carrying is tho somo in multiplication as in addition. MULTIPLICATION TABLK. Twice \i times 4 times 5 times 6 timet 1 7 times 1 is 2 1 1. 3 1 is 4 i is 5 1 is 6 1 is 7 2—4 2 — (i 2 -- 8 '» 10 2 — 1 •» 2 — 14 3 — fi 3 — !) 3 — 12 3 — 15 3—18 3 — 21 4 — H 4 — 12 4 — 16 4 — 20 4 — 2 4 4 — 28 5—10 :> — 15 5 — 20 T) - - 25 5 — 30 5 — 35 6 — 12 (J — 18 <; — 24 (» - - 30 (> — 36 6 ~ 43 7 — 14 7 — 21 t — 28 7 — 35 7 — 4 2 7 — 49 H — 1(5 8 — 24 8 — 32 8 — 40 8 ~ 48 8 — 56 •J — 18 !) — 27 y — 36 y — 45 9 — 54 9 — 63 10 — 20 10 — :to 10 — 40 10 — 50 10 — 60 10 — 70 11 — 22 u — ;j;i 11 — 44 11 — ^5 11 — 6«J 11 ~ 77 12 — 24 12 -- ;w 12 — 48 12 — (>0 12 — 7 •I 12 — 84 rt times 1> time!.-' lot mies 1 L times 12 times : is 8 1 is 1) I i s 10 1 is 11 1 is 12 1 — 1« 2 — 18 2 - - 20 •» — 22 2 — 24 3 — 24 3 — 27 3 - - 30 3 - 33 3 — 36 4 — 32 4 — 36 4 - - 40 4 — 44 4 — 48 :^ — 40 5 — 45 5 - - fi-O 5 — 55 5 — 60 (» — 48 (J — 54 6 - - (iO 6 — 66 6 — 72 7 — 5G 7 — 63 7 - - 70 7 - 77 7 — 84 8 — «4 8 — 72 8 - - 80 S — 88 8 — 96 1) — 72 y — 81 y - - 90 y — 99 9 —108 H) — 80 10 — 90 10 - -1JM> 10 -110 10 —120 11 — 88 11 — yy 11 - -110 11 —121 11 —132 12 — yb 12 —108 12 - -120 12 —132 I a —144 MULTirUCATION. 18 Rf^anliii^ the ibUowing part of thii UbJr, see suggMtioni to TesohcrM 13 titnM 14 Umm 'or 2 lit 2>i .! i» 20 2 3 — 39 3 4-i :< ♦ - 52 4 — 56 1 ft^ Aft ft — 70 r> - « - 78 84 b 7 91 7 - 9H r H - 104 H - 112 h - * - 117 » - I'.m; \) timet >^ 30 4.'» 60 I 1 H I ;i.. 10 timwTT^ timM 32 i it 34 48 ;i fti 64 4 68 80 r> - 85 96 6 102 112 7 119 128 H 136 44 :• is;j 2 3 4 f) iu| t» — 7 — 18 tiOMNI 2 in 36 .^4 72 "JO 108 126 144 162 19 tlniM 2 ia 3H :i 57 4 - 7« 5 - 95 6 - 114 7 133 8 -- 162 9 171 W« liuvu in tlui abuvi! «»ble corr'^'tpd grma i;i tuimittlciil blunder w (»mnion ut'miyiog figbi tim ^ two \, hi\ . un. Wfaoa muro ithiin twr> 1lcto^ \ic given, the operation ih oallvd oontinuod luultiplii ion, a «X3> 2x6—180. When the fuci rs cou.s •" re figurcH than one, the most convenient mode of upc ration is ihu ^hown by the annexed example, where the uiultiplicand ih dr ' ' -u^ 8 timet*, then (10 times, or wbieh is the uamo thing (> tii in i tho tifHt figure of the scoond line is placed under tlie sccot tn f the fir»t line, /. r.. (art. 2,) in the plaec nt' s and then the partial produoto 345186 268 2761488 2070916 690:^72 are added, which Hence wo deduct RULB IV. Cor.) givcfi the full product. II MULTIPLICATION. I- under the multiplicand, units r tens, &c., &c., — commencing 92507&18 Place the multr.: under units, tens un at the right, multiply > ••''h figitro of the multiplicand by each figure of the inul iplicr in succession, placing the results in parallel liiioi:*, and unita, tens, &c., in vertical columns, — add all the lines, and the sum of all the partial products will (Ax. IV. Cor.) be the whole product rcrjutred. As i if 08 the learner has committed a multiplication table to me- mory, say to 12 times 12, the work can bo done by u single operation. • When any number is multiplied by itself, the product is called the square or second power of that number, and the product of three equal factors is called a cube or third power, the pro- duct of four equal factors the fourth power, &c., &c. The terms square and cube are derived from superficial and solid measurement. The annexed oquarc has each of its sides divided into 5 equal Iiaiti), and it will be found on inspection that the whole figure contains ;> — — — — — 24 ABITHMETIC. 25 (=5X5) small squares, all equal in arcn, and having all their sides equal. — Hence because 5x5 represents the whole area, 25 is called the square of 5, or the second power of 5, because it in the product of the two equal factors 5 and 5. A cube is u solid body, the length, breadth and thickness of which are all equal, and hence, if these dimensions be each represented by 5, the whole solid will be represented by 5x5x5=^^125, which is therefore called the cube or third power of 5. The terms square and cube arc often used without any reference to superficial and solid measure. For example, in lineal measure an expression for distance in a straight line is often called the square and cube of a certain number, thus : 81 is called the square, and 729 the cube of 9, although these are only used to show that the distance is not 9 in cither case, but in the one 9X9, and in the other 9X9X9. In such cases the terms second and third power arc therefore to be preferred, and since no solid can have more than three dimensions, we have no term corres- ponding to square and cube for the product of four or more equal factors, tind therefore we arc obliged to use the words fourth power, fifth power, &c., &c. CONTRACTIONS AND PROOF. There are many eases in which multiplication may be performed by contracted methods, but the utility of these, for the purposes of accuracy, is, at least, doubtful. The most secure method in the great majority of eases, is to follow the general rule. Multiplication by 10, 100, &c., is efiectcd at once by adding a cipher for ten, two for 100, &c., &c. The following is, next to the above, the most safe and useful contraction that can be adopted. It is exhibited in the subjoined exiunplcs, but purposely without explanation, as an exer- cise for the learner's reflection : ' Ohpinaky Method. CoNTKACTiiD Mbtiiod. Ordinaky Method. Costractso Mrrnob. 35697X17 17 35697X17 249879 35697X71 71 35697X71 249879 249879 35697 606849 35697 249879 2534487 606849 ' ". 2534487 The only practically useful proof of the correctness of the pi(v duct, is the one subjoined, but even it, though it seldom fails, docs not secure positive certainty : MULTIPUCATION. 26 Add together all the figures of each factor separately, rejecting 9 from all sums that oontaiu it, and multiply the remainders together, rejecting every 9 from the result, — add the figures of the product in the same manner, and if the two remainders arc equal, the work majf ho accounted as correct, hut if they are not equal, the work must he wrong. The reason of this proof depends on the property of the numher 9, that if any numher he divided hy 9, the remainder will be the same as if the sum of its digits were divided by 9. — Thus : 7422153-^9=824G83-{ G, and the sum of the digits is 24, and 24-;-9=2-f-6, t. e. 9 is contained in 24 twice with a remainder 6. Every 9 is rejected because 9 is contained in itself once evenly, and therefore cannot affect the remainder. Let it now be required to multiply 122 by 24. Now, 122=9x13+5, and 24=9X2-j-6, and if wc multiply together the two factors thus resolved, wc get 9X13X9X2+9X2X5+9X13X6+6X5, and since 9 is a factor of nil but the last, the last only will give a remainder when divided by 9, and therefore the whole product will give the same remainder when divided by 9, as 6x5-i-9, which gives the remiainder 3, for 6X5=30 and 30-4-9 gives 3 with a remainder 3. To test this by trial, wo find 122^9=13 with a remainder 5, and 24h-9=2 with n remainder 6, and the product of these remainders is 6x5=30, and 30-4-9=3 with a remainder 3. Again, 122x24=2928, and 2928-i-9=325 with a remainder 3, as in the case of the factors. X EXERCISES. 1. 7896X5=39480. 2. 581967X8=4655736. 3. 938746x4=3754984. 4. 193784X7=1356488. 5. 391876X9=3526884. 6. 987456X6=5924736. 7. 496783X52=25832716. 8. 719864X43=30954152. 9. 375967X64=24061888. 10. 27859X29=807911. 11. 679854X83=56427882. 12. 759684X187=142060908. 13. 5372X1634=8777848 14. Find the second power of 389? Ans. 151321. » 15. Find the third power of 538? Ans. 155720872. 16. Find the fourth power of 144 ? Ans. 429981696. 17. Find the cube of 99 ? Ans. 970299. 18. 5796 seamen have to bo paid 169 dollars each ; what is the amount of the treasury order for that purpose ? Ans. $979,524. 19. A block of buildings is 87 feet long; 38 feet deep, and 29 I'cct high ; how many cubic yards docs it contain ? Ans. 3550| cubic yards: ARITHXETIC. •| <.' 20. If 29 oil wclla yield 19 gallonti an hour each; bow much will they all yield iu a year ? Ans. 201115 gal8. 21. If the rate on each of 1597 houses be $19 ; what is the whole aeaetiBmcut ? Ans. $30343. ^ 22. If 1297 persons have paid up 9 shares each in a railway company, and each share is $15 ; what is the working capital of the combauy ? Ans. $172095 :tV''- I'st- .1 DIVISION. 8- — Division is the converse operation to multiplication. It is the mode of finding a required factor when a product and another factor are given. It bears the same relation to subtraction that multiplication does to addition, as will be seen below. By Ax. IV. Cor. we may resolve any complex quantity into its component parts ; so division is resolving a certain quantity called the dividend into the number of parts indicated by another quantity called the divisor, (dividci,) and the result is called the quotient (how often.) Let " ;' it be required to find how often 8 i;^ con- tained ill 279,856. We can resolve 279,- 856 us in the margin ; then dividing the lines separately by 8, we obtain the partial quotients, the sum of which is tlio whole quotient. But this resolution may be done mentally as we proceed. We first see. that 8 is not contained iu 2, therefore we take 27, and find that 8 is contained in it 3 times, with a remainder 3; next combining this 3 with the next figure 9, we get 39, in which 8 is contained 4 times, with a icniaindcr 7 ; combining this 7 with the next figure 8, wo have 78, iu which 8 is contained 9 times, with a remainder 6 ; combining this with the 5 following, we obtain 65, and 8 is contained in it 8 times, with a remainder 1, which combined with the 6 makes 16, and 8 is contained twice in 16. The correctness of the result may be tested by multiplying the quotient by the divisor. When the divisor consists of more than one figure, the learner must have recourse to a trial quotient, but after some prac- tice he will have little difiiculty in finding each figure by inspection. 8 240000 30000 32000 4000 7200 • 900 640 80 16 2 8 279856 34982 DIVISION. 37 Let it be required to 6nd how olton 2f)8 is cn..tained in 4;U7tiH. — The numbers being arrangod in the conveniont order indicated in the margin, we mark oft" to (ho ri;.^ht of tlic dividend blank spaces for the trial and ti*ue rjuotientf^. We readily set; (hat 2 is contained 1192 . r 1456 1192 • 2646 2384 the true (|Uoticnt. We find our next partial divdend by writing 7, the next figure of the dividend after the re- mainder 133. Our experi- ence of the first case aiv'- gests to us that though 2 is contained (> times in 13, yet on multiplying something will have to be carried from the 98 which we expect will make the result too large, and therefore wi; at once '-. " ' ■ •■ -'"'■ , try 5, but we find that -' - , 298x5^^1490, which is larger than 1337, and so wc try 4, and find 298X4=1192, which being less than 1337, we subtract and find a remainder of 145 ; and having placed the 4 in the true quotient, we bring down the next figure of the dividend, giving a partial dividend 145(5. By in- spection, as before, wc see that ti would be too large, owing to the carrying from 98, we try 5 and find 298x5—1490, which is larger than 1456; we try 4, and find 298X4—1192, which is less than 1456, so wc subtract and find a remainder of 264. Having placed this 4 after the other 4 in the true (luoticnt, we bring down 6, the last figure of the dividend, wc try 9, and find 298 X 9— .2682, which is greater than our last partial dividend, 2646 ; wc try 8, and find 298X8=2384, and this being less than 2646, we subtract it from 28 ABITHMETIO. 298 298000 119200 11920 2384 262 that number, and find a final remainder of 262, and close the question by entering 8 in the true quotient. The mode adopted to indicate that the remainder 262 still remains to bo divided, which cannot be actually done, as it is less than the divisor, is to write the 298 below the 262, and draw a line between them, thus r^g, as also is seen in the margin. The resolution into partial dividends is also shown in the maigin, where it will be seen that the partial dividends, includ- ing the remainder, make up the whole original dividend. So also the partial quotients are exhibit- ed, making up the whole true quoti- ent. That the trial quotient is not a single number, like the true quo- tient, but merely a succession of detached numbers, used as separate trials, is indicated by placing a full point between each pair. When we have multiplied the divisor by any figure in the trial quotient, and subtracted the product from the partial dividend, should the remain- der be greater than the divisor, wo perceive that the trial figure is too small, and we must try a larger. . • From these illustrations we can deduce a Remainder Dividend 431766 =1000 = 400 = 40 = 8 1448 RULE FOR DIVISION. (1.) Place the given numbers in the same horizontal line, put- ting the divisor to the left of the dividend, with a vertical line be- tween them, draw another vertical line to the right of the dividend, and enter the quotient, figure by figure as obtained, to the right of that line. (2.) Find by the principles of multiplication, how often the divisor is contained in the same number of figures of the divi- dend ; place the number thus obtained in the quotient, and multiply the divisor by it, and subtract the product from the corresponding partial dividend. (3.) To the remainder annex the next figure of the dividend, and proceed as before, and so on till all the figures of the dividend are exhausted. (4.) Should there be a remainder, write it and the divisor after the quotient, thus : ~?^ The divisor is often written to the right of the dividend, and the quotient written below it, a horizontal line separating the two. BXAVPLS OF rOBM 1 DIVISION. RX AMPLE OP FORU 2 1860904 87 174 21389;^; 120 - g7 ..> V-:- 476 3833 3808 -' t ,' 2650 2380 339 ! ^ 261 1704 1428 696 276 844 783 «l 1. 2. 3. 4. 5. EXRROIBES. 1554768-T-216=7198. ' 31884470h-779=40930. • 57380625^7575=7575. ' 12810098—732=17500;^. ""* 9313702859-v-4687319=19875jjg?^y5. C. 449148410476-^73885246=607973g||2,3. 7. 109588282929H-1386=7902468yVs"g- 8. 35676210832-:-7909445l=7G4095,'y«o'u?4rr- 9. 536818834-^907=591862. 10. 170064915561 --759=2240644479. 11. 554270297961^7584=73084103^7g:?. 12. 60435674634529~-764095=79094451,9gYif_45- 13. How many bags, eacli containing 87 pounds, will 24,853,464 pounds of flour fill ? Ans. 285,672. 14. 857 houses pay annually a tax of $41136 ; what is the aver age on each per quarter ? Ans. $12. 15. $9297175 of prize money are to be divided among 97,865 Bailors ; what is the share of each ? Ans. $95. 16. 120,815,231 pounds of cotton are made up iu 233,879 bales ; how many pounds in each bale ? Ans 89. DIVISION 1. 49687532-4-2=24843766. 2. 67986327H-3=19328776:^ 30 ARITHMETIC. WfV ^42G1759fV. =1503777^. H. 87965328-=-4^21991332. 4. 7963821 -:-5^r.l592764;)!. 5. 6875324h-()^1 145887 j. C. 3987654-H-7=^569GG4«! 7. 19876532-=-8=.2484566A. 8. 297G854--9.=:330761-;. ' 9. 49G7532-^10=49G763J;. 10. 4G879352-T-llr 11. 187G5.314-=-12r 12. 78G5424G--18==43G9680J. 13. 75088—52=1444. 14. lG74918-=-189:=r8S62. 15. 31884470-:-779=:40930. 10. 573S0G28--7575=:7575,vV 17. 5542702921 98-:-7584=:730841G373>g 18. 88789980979-^9584==:9264397g'5V^. 19. 102030429729-:-12345G:r:=826452?§5]f. , - , 20. 2G781794G000-f-36500r=10077204.' 21. 407 men fell 103708 trees; how many does each fell on an average? Ans 329. 22. If 148 houses pay a tax of $7844 ; what is the rate on each on an average? Ans. $53. ^-,. j . .. 23. If $415143030 are levied from 4455 townships ; what is the portion of each on an average ? Ans $93180. 24. How many lots of 0754 each are contained in 3908091 51372 ? Ans. 58703718. 25. What quotient will be obtained by dividing 961504803 twice by 987? Ans. 987. .^ ' ^u;- V :'j^ '7335' 0— TABLES of MONEY, WEIGHTS & MEASURES. DKCIMAf, COrXAOR. lUUTISlI on STERUNG MONIIV. 4 farthings, or 2 half pennu's, :iro 1 pennj- (d.) 12 pence 1 shilliug (s.) 20 shillings 1 pound (£) 10 mills (M) are 1 cent (ct.) 10 ci-nts 1 dime (d.) 10 dimos, or lOU cents.. . 1 dolliir ($)^ ,, AVOIRDUPOIS WEIGHT. TAHI.K. IC drams make 1 ounce, marked oz. 16 ounces . 1 pound, " lb. 25 pounds 1 quarter, '• qr. 4 quarters 1 Imndrodweight, •• cwt. 20 cwt 1 i{ni, '• t. NoTK. This weight is used in weighing heavj' articles, as meat, groceries, vegetaJ»le.s, grain, etc. TABLES OP MONEY, WEIOHTS AND MEASURES. ai TROY WEIGHT. T.VIILK. 24 grains (gre.) make l i>ennywelght, marked dwt. 20 pennyweights i ounce, '• oz. 12 ounces 1 pound, '• ib. Note.— Troy weiglit is used in weighing the precious metals and stones. APOTHECARIES' WEIGHT. * TADI.K. 20 grains (grs.) make 1 scruple, marked scr. 3 Bcmplca 1 dram, " dr. 8 drams 1 ounce, " oz. 12 ounces 1 pound, " ll». Note.— Apothecaries and Physicians mix their medicines by this weight, but tliey buy and sell by Avoirdupois. ,, . t ?»vj; PRODUCE WEIGHT-TABLE FOR CANADA. ^ GRAIN. Wheat 60 pounds to the busliel. Oats .14 " '• '• Corn 5G Com in cob. 80 " '• " Barley ... 48 " '■ . '• Rye 56 •■ Buckwheat.. 48 " •' " Peas 60 " " " Beans 60 " " Tares 60 " " " .SKROS. Clover 60 pounds to the bushel. Flax Timothy .... Hemp Blue gross . Red Top Hongarlan | grass ... I Millet Rape SO 48 54 14 8 48 48 50 VEGETABLES. Potatoes 60 pounds to the bushel Parsnips 60 •' " Carrots 90 Turnips flO " '• Beets 60 " " ' '• Onions 60 '• " " VROBTABLES. Castor Beans 40 pounds to the bushel. Malt lid DriedPenches a;j • " '• Dried Apples 22 " " Salt 56 Bran 20 '• " •' LINEAR (OR LONG) AND SQUARE MEASURE. UNBAR. .SQUARE. 12 inches (in.) make. 1 foot (ft.) 3 leet 1 yard (yd.) 5 J yards 1 rod or perch. 40 rods 1 furlong (fur.) 8 lurlongs 1 mile (m.) 144 inches mal:e 1 foot (ft.) 9 feet 1 yard (yd.) 30J yards 1 rod (id.^ 40 rods 1 rood (r.) 4 roods 1 acre (a.) LAND MEASURE. LENGTH 7Mr inches make 1 link. 25 links Irod. 4 rods or 100 links 1 chain. 80 chains 1 mile. AREA. 10,000 square links make 1 sq. chiiiii 10 square chains .... 1 acre. 'AV 82 ABITHMETIO. In solid meofiure, i. t., the measuremont orsoUds, 1728 (the third power or cube of 12,) inches malte 1 cubic foot, and 27 cubic Toot (i. t. 3X3X3,) malco 1 cubic yard. In measuring timber, 40 cubic feet of round timber malcu what is oallod a ton, and the same name is given to fiO feet of hewn timber. A cord of firewood is 8 feet long. 4 feet wide, and 4 foot high, and therefore its solid content is 8X4X^=128 feet. Dry goods are measured by the yard, and fractions of a yard, the frac- tions used being one-quarter, one-eighth, and one«ixteenth. MEASURES OF CAPACITY. DRV, 2 pints make 1 quart (qt.) 4 quarts I gallon (gal.) 2 gallons 1 peclc (pk.) 4 pecks 1 bushel (bu.) 3U bushels 1 chaldron (ch.) The last is seldom used. LIQUID. 4 gills make lpint(pt.) 2 pints I quart (qt) 4 quarts 1 ftallon (gal.) C3 gallons 1 hogshead (hhd.) 2 hogsheads 1 pipe (pi.) 2 pipes 1 tun (tun.) MEASURK OF TIME. ()0 seconds make 1 minute. CO minutes 1 hour. 24 hours 1 day. 3C5J days 1 year. ANQULAK UB CIRCULAR MEASURE. ' 60 seconds make. 1 minute (1\) CO minutes 1 degree (I ° .) 3G0 degreea 1 complete circle. There are other units applied to certain articles, e. g., 12 articles, one dozen ; 20 articles, one score ; 144 articles, one gross ; 24 sheets of paper, one quire ; 20 quires, one ream, — 141bs., one stone. This last weight is varied in many places, 15lbs. and IClbs., according to the nature of the arti- cle sold, e. g., — potatoes, as an allowance for earth adhering. THE CALENDAR MONTHS OF THE YEAR. January has 31 days. February " 28 " March " 31 " April " 30 " May " 31 " June " 30 " July , has 31 days August " 31 " September " 30 " October " 31 •« November " 30 " December " 31 " Every fourth year is called Leap-year, in which February has 29 days. — If the last two figures denoting the year can be divided evenly by 4, it is Leap-year. DECIMAL COINAGE. 10. The principle of the decimal coinage is generally understood to depend on the rules of decimal fractions ; but as it is merely a separate and co-ordinate result of the common system of notation, we may explain it here, independently of the theory of decimal fractions. DECIMAL COINAQE. 38 Wo DAve already explained, that according to the Arabic notation, each digit has one-tenth the value that it would have if situated one place tarthor to the left. Thus, in the number 88, the digit to the right expresses 8 units, while that to the left expresses 8 tens. Now we cannot have any integer less than unity, but we may have to make calculations respecting quantities less than the unit under considera- tion, e. (/., in calculating by dollars, we may have to take centa into account, and as the cent is a sub-division of the unit, u dollar, some new character must be introduced to indicate this transition from the integral unit to a part of it. This is done very simply by interposing a mark like the |)eriod or full point (.) in printing. — This is usually called the decimal point, though it sometimes gets the vague and awkward name of the separatrix. This simple but admirable contrivance is ascribed to one Stevinus or Stevens, of the Netherlands, who gave his suggestion to the public about the year 1585. Its excellence consists in its being simply an extension of the common notation. The original system marks only the repeti- tion of the unit of measure, — this applies the same principle to the sub division of the unit into parts. To explain this, we have only to carry out the illustration already given regarding integers. We saw that the extreme right hand figure, 8 in our example, stood for 8 units, and was one-tenth of the preceding one ; just in the somo manner another figure, 8, placed to the right of the units' figure, will express one-tenth of those units, and the decimal point is used to mark this descending from integers to parts of the integral unit, and is written thus : 8.8, and means eight units, and eighth-tenths of that unit. If another 8 be added, thus : 8.88, it will express eight-tenths of the preceding unit, i. e., eight>tcnths of one-tenth, which is the same as eight one-hundredths of unity, and thus we have the descending scale by tenths towards the right of the decimal point, in the same manner as we had the ascending scale by tens towards the left. As a farther illustration, we may begin at the extreme right, as in 888.888, and we find throughout that each figure to the left is ten times that immediately to its right. The decimal coinage adopts a certain unit called a dollar — the dollar is then sub-divided into ten equal parts, and each part is called a dime , the dime, in like manner, is divided into ten equal parts, and each part is called a cent ; and the cent is divided into ten equal parts, and each part is called a mill. The mill enters into many calculations, though no coin i' i luvtinn would bo oompreued into " nut-flhell" dimennioDH, ;tiid iliv luomory would not be oyor I 10 rs m >y I re d LSt ry nd dd its ro- )ry lis (1) (5.) (2) (6.) (3.) (7) ' (4.) ■V • : 185.50 $116.20 $13.19 49.63 ■ 291.45 $126.75 14.16 92.18 89.75 98.50 85.92 37.C'9 365.84 25.15 64.15 8.92 91.50 76.05 37.25 76.45 76.15 91.1U 43.87| 91.20 25.75 485.00 18.75 64.16 157.92 84.20 29.10 18.60 ' 263.75 67.62^ 47.85 69.11 188.25 39.80 55.55 - 14817 39.48 17.37i 72.63 265 90 136.13 669.44 529.75 931.46 , 230142 (8.) $11.27 $55.63 45.15 $44.50 $296.75 17.75 54.72 67.23 176.84 84.18 31.30 89.75 518.50 29.88 49.50 27.63 3C;9.63 45.13 16.75 95.13 (JS!7.45 38.81 84.28 ^. 38.88 258.13 67.25 14,85 17.45 591.18 96.20 9.44 56.64 179.25 , ,.,. 77.63 28.09 73.85 567.42 8.75 345.35 511.06 3585.15 1 621.21 86 ARirmono. (9.) Sold to J. JoNis, 20 7ardB cloth •7&.8| 14 mats 21.5^ IC hats 33.50 6 pain of blaaketH 28.7& 15 yard* Malskin 40.25 15 yards of hoi^ 0.(i3 28 yarda fine cloth 112.88 321.82 (10.) $157.99 266.73 985.45 197.00 385.18 876.76 795.85 667.13 659.63 4893.07 (11.) Sold to S. Fulton, Aurora, 12 pairs of worsted sto<:lcings $13.50 18 " " flannel drawers 22.75 24 " " kid gloves 8.63 56 school books 49.72 , 29 yards of satin 83.23 96 school copybooks 1.84 180 yards of ribbon 29.76 84 yards of ticking 22.68 122 yards of sheeting 23.18 • , 255.29 12. The shares in an oil-well speculation are $5 each ; A. takes 15 shares; B.25; C. 20 ; D. 1 ; E. 11 ; F. 37; 0. 16;H. 18; I. 8; K. 21 ; L. 14 ; and 14 other persona take 10 shares each ; what is the capital of the company, and how many sliarcs are there ? Ans. 31,030 and 326 shares. 13. If 17 vessels bring to the port of Boston cargoes of the follow- ing values ; what does the whole amount to ? $2305.75, $1793.87, $3815.25, $2718.03, $4180.50, $3179.13, $lC2:i.88, $4311.75, $1987.38, $2975.75, and the other 7 average $2689.13. Ans. $47781.80. SubtractioH of dollars and cents. (1.) (2.) (3.) $567819.83 $83756.17 $17423.37* 278956.89 76489.71 9654.63| 28881 '2.94 7266.46 7768.74 dcchluu coinage. 87 a 4. What is the differanoe between 2769 doUum and 50 eento, and 987 (lullara 87^ centa ? Ana. $1781 .62^ 5. Tho debit aide of a ledger iH $1770.80, and the credit aide $870.50 , what iH the balance ? Ans. $894.30. 0. Tho credit aide of a cash book ia $8795.88, and tho debit aide Ih $10358.18 ) what ia tho balance ? Ana. $1562.30. 7. A fimi owcH $227968.25, and the oaUto is worth $98764.75 ; what is the hIaU) of the affaira of tho firm ? Antt. — Tho firm ia unable tu pay $129,203.50 over and above the aaaetH. 8. A ship and carg«) were worth $27500.50, — tho ahip wa« lost, and only $6784.60 worth cf tho cargo saved ; what was the loss ? Ans. $20724.90. 9. A cotton mill was totally destroyed by fire ; tho mill and ita eontonta were worth $78610 ; it was insured in one office for $11760; in another, for $9845; in another, for $10800 ; and, in a fourth, for $12685 ; did the owner lose, and if so, how much ? . ' -■-. < Ans.— Ho lost $33526. 10. I have sold for cash, during tho last month, $2786.88 worth of goods, I have received payment of S. Fulton's account, $255.29 received proceeds of J. Jones' note, $302.64; received interest on sundry debentures, $278.50 ; sold my shores in the G. T. R. for $785.75; received in cash, interest from lloyal Canadian Bank, $187.25; sold block of buildings in King street for $1719.00, and shares in Rossin house for $718.50. Paid S. Smellie's account, $261.88; for sundry insurances, $879.60; rent of office, $150.00; for consignment to Liverpool, $2786.50, and charges on same, $175.63 ; what is the balance of receipts above expenditure ? Ans. $2780.20. MuUiplimtion of dollars and ceiUs. '■'•^ (1.) . (2.) (3.) ' (4.) $365.75 $1873.47 $865.63 $24786.38 87 69 93 45 256025 292600 3182025 1686123 1124082 12926943 259689 779067 12393190 9914552 8050359 111538710 * Wc must hero caution tlie tyro against such modes of expression as this, — " multiply $85 by $12." Such an expression is simply absuid, for to say $12 times $85, might as well mean 1200 times $85, or 12000 thnes $85, I y. 38 ARITHMETIO. Sucfi questi0D8 as the following may be worked in three diiferent ways : (5.) $487.63^ or $487. 63^ or $487.63^ add in 14» 28 28 28 390104 97526 390104 97526 J of 28= 14 390118 97526 1365864 J of 28= 14 1365378 1365378 1365378 It is often convenient to make the number expressing the dollars and cents the multiplier, especially when they form the shorter line. Thus to multiply 63 cents by 3587, we make .63 the multiplier, and 3587 the multiplicand, and so in examples 7, 8 and 9. («•) (7.) 3587 2876 ' .63 8.63 10761 21522 225981 8628 17256 23008 2481988 (8.) (9.) 5796 - ^ 14986 4.87i- 9.12i 40572 29972 46368 14986 23184 134874 2898 • 7493 2825550 13674725 10. If 987 houses pay a tax of $3.37^ j what does the whole amount to ? Ans. $3331.12|. which would all give widely different results. We may indeed have to mul- tiply a denominate number representing $85, by another denominate number representing $12, as often happens in questions involving proportion, e. was to be .1 I DEGDCAL. COniAOE. Diyision of dollara and cents. 1. |28642.14-r-29=|987.66. 5. |1943243.55-r-983==$1976.85. 2. $37133.34^87=1426.82. «. $31421.25-^-63=1498.75. 3. $60509.68-j-76=$796.18. 7. $28479.75H-78=$365.12i. 4. $43009.75-=-98=$438.87i. 8. $2595.37f^7G9=$3.37i. 9. $2927.30 a year; how much per day ? Ans. $8.02. 10. $3953.19 a year; how much for every working d&y? Ans. $12.63. 11. 2G9 persons have to pay a tax of $1312.72; what is the average tax on each ? Ans. $4.88. 12. A collection of $544.04 is made by 1876 persons ; how much did each give on an average ? Ans. 29 cents. To reduce currency money to the denominations of the decimal coinage. Since 100 cents make 1 dollar, and 4 dollars make 1 pound, 400 cents make 1 pound currency, and therefore to find the numbei of cents in any given number of pounds, we must multiply the pounds by 400. Again, since 20 cents make 1 shilling, or 12 pence, to find the number of cents in any given number of shillings, wc must multiply tho shillings by 20. Lastly, 5 cents, are equal to H pence, and 12 farthings are also equal to 3 pence, and (Ax. I.) things that are equal to tho same thing, are equal to one another ; therefore, 5 cents are equal to 12 farthings, and 1 farthing is the -,'2 of 5 cents, or j'lj of 1 cent. Hence to find the number of cents in any number of pence and farthings, we multiply the number of farthings in the given ponce and farthings by 5, and divide tho product by 12. Having obtained the three results, we add them all together. Thus to change £48 ISs. 9|d. to dollars and cents, we multiply 48 by 400, 18 by 20, and tak'e /^ of 9^, or 39 farthings, and add the three together, which gives us 19576| cents, 19576] or $195.76|. 48X400=19200 18 X 20= 360 9|=39f.X /2= 16i repeated 2^ Umes, which would make 68. 3d. (3.) The interpretation might be, that as 2s. tid. is 80 pence, that the other 2s. 6d. is to be repeated 30 times, which would give £3 15s. Od. (4.) Tho phrase may also be interpret- ed 08 meaning that 3*)d. was to be repeated 30 times, wliich would also give £3 158. Od. The last two interpretations are the same in two different forms, and give the same result This is the only view in which the expression has any sense, and proves our statement, that whenever a denominate number is used as a mnltipUer, it ceases to be denominate, and becomes abstract Tho same principle will apply to division. w. (1.) £79 X 400=31600 16 X 20= 320 6idX f2= 10« ABITHUETIO. EXERCISES •L/,V. (2.) £117 X 400=46800 " 17 X 20= 340 ., 8|dX VU- 14^ $319.30^ 3. £87.14.10|=$350.97|.,. ■I I 4. je29.19.9=$119.95. 5. JE67.13.4f =$270.67] 6. £279.15.10i=$1119.17|. 7. i2ll8.11.4^=$474.27i/ 8. £79.8.4=$317.66g. 9. £37.18.8=6151.73^. 10. £57.8.1 If =$229.79-j?.. 11. £49.7.6=8197.50. J'-:, '■.-:.: ::-:r '.,.., ^. «471.54^ 12. £137.16.8=$551.33J. 13. iE236.19.2i=$947.84i. 14. £19.16.8=$79.33J. 15. £98.1.1i=»392.22J. 16. i287.11.8=$350.33J. 17. i?457.12.6=$1830.50. 18. £219.4.7f=$876.92f^. 19. £49.9.4|=$197.87|4. 20. £287.18.10^=61151*. 77^. To change dollars and cents to Halifax currency, we must re- verse the above operation. Thus, to reduce $1 95.76 J^ to £. s. d. — First, uduce the dollars and cents to cents, then divide by 400, which gives 48, the even number of pounds, with a remainder of 376J cents ; then divide this remainder by 20, which gives 18, the number of shillings, with a remainder of 16|^ cent^, as in the converse operation, we multiplied by 5, and divided by 12, so now we multiply by 12, and divide by 5; thus, 16^X12=195, and 195-f-5=39, the number of farthings, and this being reduced to pence and farthings, gives 9f, so that $195.76i=£48.18.9|. Or the work may be shortened by the fol- lowing method. As 64 make £1, the number of £'b in 6195.76^, will be the same as the number of times that 4 is contained in the 195 dollars, which gives £48, and 63 remain- 400)19576^(48 400 3576^ 3200 . , 2(0376J(18 . 1 20 176 , . 160 16J 12 5)195 (39 i>t<= / ', \ T)ECIMAL COINAGE ing. Now, these three dollars arc equiva- lent to 300 ccnlM, which added to the rc- • •* maining 76} cents, gives 376} cents ; this divided by 20, will give the shillings, be- cause 20 cents arc equal to one shilling, and it is self-evident that the number of shillings in 376} cents, will be the same as the num- ber of times 20 is contained in that num- ber, which gives 18 shillings, and 16} cents remaining. Lastly, as 5 cents arc equal to 3 pence, >nc cent will be equal to I of 3 ,, pbncc, which is ? of a penny ; therefore, if one cent is equal to ^ of a penny, the re- maining 16} cents will be equal to 16} times ?, of a penny, which is 9^. ; hence wo have $195.76} equal to £48.18.9f. .. $195—76} : - : , ■;. - 4)195 - ' '-*;^ !■ £48- -300 20 376} 8l8- « ,"'. ■ y 5 48 J 93d. ■»J=!> •<^: 2. Reduce 6270.67 1 A 3. Reduce $474.27^' 4. Reduce $197.50 5. Reduce $1119.17^ 6. Reduce $551 .33 J 7. Reduce $1830.50 8. Reduce $1151.77^ >*.^-iV V i ••,..»',?ji (ft" ■ ..■j?e»^- - ■:•*.<•- J. f. EXERCISES. v: '...,v.:' .z'^;: Mt^o"' 5- Halifax currency. Ans. £29.19.9. , . Ans. £67.13.4f. Ans. £118.11.4^ Ans. £49.7.6. Ans. £279.15.10*. Ans. £137.16.8. Ans. £457.12.6. ■ Ans. £287.18.10^. ■m'n'^h •••i f - r 1. '■.:,, 2. ..r' 4. .,p.i% MIXED EXERCISES ■ 7'>>.^^' ?.■' . 6. . 7. 8. - 9. 10. Reduce £436.7.8^ to dollars and cents. Ans. $1745.54^. Reduce $547.87 to Halifax currency. Ans. £136.19.4!. Reduce £783.13.5]- to dollars and cents. Ans. $3134.68|. Reduce $576.85 to Halifax currency. Ans. 144.4.3. Reduce £606 lO.Sf to dollars and cents. Ans. $2427.94t72. Reduce $375.99 to Halifax currency. Ans. £93.19.11f. Reduce 3s. 8}d. to dollars and cents. Ans. 73| cents. Reduce 17 cents to Halifax currency. Ans. lOJ pence. Reduce lOJ pence to dollars and cents. Ans. 17] ;J cents Reduce 23 cents to old Canadian currency. Ans. 13| pence >\ 42 ABirHXEnc. TUB F1B1IEB8' BDLE VOR IlEDCCINO CENTS TO FENCE, AMD VIMd TO CSMIB. QUKSTION. Said farmer A. to grocer B. There's Bomething here that pozzies mo ; I sold some butter here to^ay, I sold by cents, by pence they pay ; How shall I change the cents to pence, And know the trick from this day hence ? A N H W E U . Five cents are three penco you must know, As twenty cents to twelve pence go ; Three times tho cents, the flflb of that , Is just the thing you would bo at ; And if you buy trom groceins hero, That other case is just as clear, Five times the pence, the third of it ;.. ;. Will make you safe and always fit. / ' m.:A' V »fc -n: >l 'V • REDUCTION. 11, — Reduction is the mode of expressing any given qunucity in terms of a higher or lower denomination, c. g., expressing any given number of dollars ns cents, and vice versa, any number of cents as dollars. When a higher denomination is changed to a lower, (as dollars to cents,) tho process is called reduction ^Zescending, and when a lower is changed to a higher, (cents to dollars,) it is called reduction amending. . , ., , , ,, RULE. ' '• •^"■' '^ ■■^''- I To express any given quantity in terms of a lower denomination, multiply it by the number of units which it contains of the next lower denomination, and add in the given units of that denomina- tion, and eo on to the lowest denomination giv^n. Thus, to express 8 dollars and 25 cents as cents, multiply 8 by 100, giving 800, and add the 25 cents, giving, 825 cents. So also, as in the margin, tho pounds are multiplied by 20, for 20s.==£l, and the 11 shillings added in, giving 511 shillings, then these shillings are multi- plied by 12, and the 4 pence added in, giving 6136 pence, and this finally is multiplied by 4, and tho two farthings added in, giving 24546 farthings. So also $98x100=9800 cents. To express a lower in terms of a higher denomination, divide the lower by the number that denotes how many units of the lower are contained in one unit of the higher. Thu» £25.11.44 20 ^ 511 12 6136 4 24546 X ; so in REDUCTION. 43 to reduce 24546 fnrthings to £,. a. d. — &ince 4 farthingtt make 1 penny, we divide by 4, and got 6136 pence, with a remainder of 2 fartliings, or 1 half-penny. Again, since 12 pence make one shil- ling, we divide 613G by 12, and get 511 shillings, with a remainder of 4 pence. Lastly we divide 51 1 by 20, and get 25 pounds, with a remainder of 11 shillings, so that 24546 farthings make £25 lis. 4|d. So also, since 100 cents make one dollar, to reduce 12579 cents to dollars, divide by 100, and 1257l)-:-100-^- $125.79. Wc thus see that cents can be changed to dollars and cents, by simply cutting off two figures from the right. So also dollars can be changed to cents by adding two ciphers, or dollars and cents can bo changed to cents by removing the decimal point two places towards the right. -,,.;■ ';.■-.■-■,-...■ "Jfi- EXERCISES. 1. How many dollars arc there in 47986 cents? Ans. $479.86. 2. How many cents arc there in 187 dollars? Ans. 18700. 3. How many pence are there in £87.12.8 ? Ans. 21032. 4. How many pence are there in £113.18.4 ? Ans. 27340. 5. How many farthings are there in £79.15.10^? Ans. 76602. 6. How many half pence in £97.17.6 ? Ans. 46980. 7. How many pounds, &c., are there in 7983 pence? • Ans. £33.5.3. 8. How many pounds, &c., are there in 156793 farthings? Ans. £163.6.61^. 9. How many pounds are there in 2 tons 16 cwt. 2 qrs. and 21 lbs.? :-'■ '"^''■^- -:*■'"'■- ^-' -■-''' •■ '"^ "• » ■" " ^"'"'' Ana. 5671. 10. How many pounds arc there in 18 cwt. and 22 lbs.? Ans. 1822. 11. llcduce 14796 lbs. to tons, &c.? Ans. 7 tons, 7 cwt., 3 qrs., 21 lbs. 12. Reduce 7643 quarters to tons, &c. ? « 5 * . '-- ' i ' . ^« Ans. 95 tons, 10 cwt., 3 qrs. 13. How many drams arc there in 18 lbs., 13 oz. and 15 drs. ? Ans. 4831. 14. How many pounds are tlieio in 2785 drams? Ans. 10 lbs., 14 oz,, 1 dr. 15. lIo\V many grains are there in 17 lbs., 11 oz., 18 dwt. and 22 grains? ^ .sj, r Ans. 103654. 16. How many lbs, in 46891 grs. ? Ans. 8 lbs., 1 oz., 13 dwt., 19 grs. / 4A ABITHMETIO. 17. Reduce 98 miles, 5 furlongs and 30 rods to rods ? "'^' Ans. 31590 rods. 18. Uow many inches from Toronto to Hamilton, (38 miles) ? Ans. 2407680. 19. How many miles arc there in 527168 feet? Ans. 99 miles, 6 fur., 29 pr., 2 yds,, 3 ft., 6 in. < 20. Reduce 57 acres, 3 roods and 24 rods to rods ? Ans. 9264 rods. :) 21. How many square yards arc there in 17 acres, 2 roods and 1 8 rods ? Ans. 85244| yards. 22. Find the number of acres, &c., in 479685971 square inches ? Ans. a. 76.1.35.19.2.119. 23. How many acres do 176984 square yards make? ^^^^ Ans. a. 36.2.1 0.21^ 24. How many square links are there in 37 acres ? Ans. 3,700,000 links. 25. How many acres, &c., in 479,863,201 square links? Ans. 4798 a., 6 eh., 3201. 26. 7,864,391 cubic inches ; how many cubic yards ? f , Ans. yds. 168.15.263. 27. cubic yards, 7 cubic feet, 821 cubic inches ; how many onbio inches ? Ans. 432821 cubic inches. 28. How many gills does a tun contain ? Ans. 8064 gills. 29. How many gallons, &c., do 479865 gills make ? Ans. gals. 14995.3.0.1. 30. How many pints are there in 28 bu., 3 pecks and 1 gal. ? — Ans. 1848 pints. 31. 27 yards, 3 qrs., 3 nails ; how many nails ? Ans. 447 nails. 32. 286 nails ; how many yards, &c. ? Ans. 17 yards, 3 qrs., 2 nls. 33. 36 ° 40' 25" ; how many seconds ? Ans. 132025". 34. How many degrees, &c., in 4C'786" ? Ans. 13 « .49'.46". 35. The i)opulation of Toronto is 45,288 ; what would a poll tax of 5 cents each amount to ? Ans. $2264.40. 36. How long would it take a railway train to move a distance equal to that of the earth from the sun, (95 millions of miles,) at a speed of 52 miles an hour ? Ans. 208 years, 201 days, lOJ^ hours. 37. The area of Upper Canada is 94,7?0,000 acres ; how many square feet? Ans. 4,126,003,200,000 square feet. r..-. .t DENOMINATE NUMBERS. 38. Sound moves about 11:^0 feet in n Hccond of time; how long would it be in moving from tho earth to the sun ? Ans. 14year8, 27 days, 15 hours, 50 min., ^i^['^(j see. 39. How many seconds of this century had elapsed at the end of 1864, counting tho day at 24 hours ? Aus. 2,019,686,400". 40. The great bell of Moscow weighs 127,836 lbs.; how many tons, &c., does it weigh, the (juarter being 28 lbs. ? Ans. 57t. Ic. Iq. lOlbs. 41. IIow many days from the 11th July, " ^61, to the 1st of April, 1864 ? • Ans. 995 days. 42. A congregation of 569 persons made a collection of X40.6.1 ; . how many pence did each give on an average ? Ans. 17d. 43. The British mint can strike off 20,000 coins in an hour ; what is the value of all the pennies coined in one day of 12 hours' work? Ans. £1,000. 44. 417 tons of fish were caught at Newfoundland in one season, and sold by the stone ol' 14 lbs., at an average price of 42 cents a stone ; what did they bring ? Ans. $25020. 45. How many feet from pole to pole, the earth's diameter Iding 7945 miles ? Ans. 41949600 feet. DENOMINATE NUMBERS. 12. — ^When numbers arc spoken of in general, without reference to any particular articles, such as money or merchandise, they arc called abstract, but when they are applied to such articles they are dometimes called applicate, as being applied to some particular arti- cles to express their quantity ; sometimes they are called concrete, (growing together,) as attached to some particular substances, and sometimes they are called denominate, as denoting quantities that consist of different denominations, as dollars and cents, — pounds, ounces, &c. The elementary rules of addition, subtraction, multi- plication and division, are performed on denominate numbers, exactly in the same way as on abstract numbers, with this single difference, that when a lower denomination is added, and gives a sum equal to one or more units of the next higher denomination, we carry that unit, or those units, to the next higher denomination. Thus: if the sum were 24 inches, we should call that two feet. In abstract and decimal numbers we always reduce, or carry, by tens. m )*■" ABITHMETIC. j B X A M P L K Hero wo find the sum of tlic pcnc; to bo 28, and as 12 penoo Diako 1 shilling, the number of tthilUugs in 28 pence will be the .<.^a as the number of times thnt 12 is contained in 2B, which is twice, with a remainder of 4 ; therefore we write the 4 pence under the pcpce column, and odd up the 2 shillings with tbu shillings' column, and obtain C>4 shillings, and ns 20 shillings make 1 {lound, the number uf pounds in 54 shillings will be the same as the numlxT of times that 20 is contained in 54, which is twice, with a remainder of 14, and therefore we write the 14 shillings under the shillings' column, and add up the 2 pounds with the units' column of pounds, and now since the remaining columns arc all of the same denomina- tion, we proceed as in simple addition, and find the whole to be £236.14.4. The same illustration will apply to the subtraction, multiplication and division of all kinds of denominate numbers. In the exercises on the addition of denominate numbers, one question in abstract numbers is given to contrast with the denominate. EXERCISES. £49. 7. 83.15.10 67.12. 8 36.18. 4 236.14. 4 (1.) . (2.) (3.) (4.) £76.18. 4 $1967.87* • ■*^v'-''' C V''"^" •" $857.63 17.11. 4* 2075.75 7866437 , . 189.50 . 99.19. 9 3194.62* 198675 684.87* ^ X 11.11.11 1:^4 7658.50 8476154 498.75 ;, , 67.15.104 8976.37* 1869538 867.1 2^ ' 79.19. 9 2873.12* 4187643 365.37| ■K ., 28.12. 1 1769.25 5768299 ^^ 917.25 '^ ^ 63. 8. 4J 445.17. 5^ (7.) 2481.92 28365746 4380.50J (6.) 30997.42 (5.) (8.) lbs. oz. drs. t. cwt. qrs. lbs. lbs. oz. dwt. gre. Il>3. '>z. drsL scr. gra. 13.14.10 26.17.3.21 3.11.16.21 5.11.7.2.19 15.11.10 18.11.0.19 5. 8. 7.11 4.10.4.1. 7 11. 4. 9 8.12.13 15. 7. 8 10.13.11 8. 9. 6 4.15.15 25.15.1.16 13.17.2.20 39. 4.1.23 28.16.3.14 7. 9.18.23 11.10.15.17 12. 7. 9. 8 16.10.11.22 18. 8.19.18 3.11.6.2.14 1. 9.3.1.12 2. 4.5.0.10 6. 7.2.2. 9 2. 8.1.1.13 4^ 89.10. 2 153. 3.2.13 77. 8. 0. 28. 4.0.1. 4 cwt 87 49 28 36 88 57 34 DENOMINATE NTMBEBS. 411' (9.) (10.) (11.) (12.) m. tm.nia.riM. yd& n. iu. 1. QC roodi^ nL mia. jrtift ft. lo. 176.7.39.5 18.2. 11.11 29.3.39 39.30.8.143 85.4.20.1 14.2. 7. 9 57.2.18 18.11.4. 68 79. G. 29. 3 8.1. 10. 7 118.0.20 24. 4.7.118 42.3. 8.2 11.0. 7. 6 75.3.11 11.21.2. 96 67.1.11.2 7.2. 8. 5 51.1. 8 15.27.0.124 118.3.10.3 16.2. 9.10 94.1.19 27. G.3. 87 81.2.31.1 8.1. 7. 6 63.2.21 19.25.2. 38 79.0.21.2 78.1.15 18.3.33.3 19.3.33 749.2. 6.0 87.0. 3. 6 ' 589.0.30 157. 6.3.' 98 '■'•H. (13.) ;<*■ (14.) (15.) (16.) a. cb. links. 79.9.9999 117.4.3650 47.5. 941 56.2.1182 27.7.2813 36.1. 771 84.8.1160 ch. b. p. g. qt pt. 5.35.3.1.3.1 7.18.2.0.1.1 '8. 7.1.1.0.1 3.26.0.0.1.0 4.18.0.1.0.1 tiL pL bhd. gaL qt pt gL 6.1.1.1.3.1.3 4.0.1.1.2.0.2 5.1.0.0.1.1.1 ': 1.1.0.1.0 ydftqiuiiu, 36.3.2 19.1.3 87.2.1 63.0.2 74.2.2 93.3.3 449.8. 516 29.34.0.0.3.0 16.1.1.5.0.0.2 375.2.1 (17.) (18.) (19.) (20.) cwt qm. lbs. 87.3.11 49.1.18 28.3.15 36.1. 8 88.1.16 57.3.14 3590. 59'. 59" 153 .40.45 270 . 0. 179 .45.30 81 .30.10 89 .59.59 jn. dajra brs, raliL sec. 33.364.23.59.59 28.113.11.48.48 17. 97.12. 0. 1.307.23.48.49 12.114. 0. 0. C\Tt qrs. Ib& 18.1.18 22.3.11 9.2.18 12.1.15 8.3.24 31.2. 348.3.7 1134 .56.23 93. 267. 2«. 37. 36 103.3.11 48 ARITHMETIC. LBDOBR A000UNT8 The debit and credit sides of four folios of u ledger are as beVvw, what are the balancea ? (21.) Dr. (21.) Cr. (22.) Dr. (22.) Cr. ei214.75 $2763.80 $198.75 $118.50 863.09 471.38 47.63 9.05 291.45 365.50 18.11 16.25 318.25 297.11 97.38 37.08 1789.87 584.88 85.88 19.13 947.63 963.15 76.20 47.75 # 2000.00 1257.76 4.50 65.92 798.38 189.60 181.60 32.40 2018.50 98.13 19.25 76.50 164.30 756.25 76.38 7.75 277.15 87.50 219.50 197.25 1165.20 163.63 48.75 15.75 367.40 1291.00 93.15 8.38 984.70 784.25 25.50 93.16 - 273.00 79.75 81.05 67.46 584.10 81.18 28.30 5.45 1200.00 318.50 69.08 18.09 68.75 1819.20 157.11 4.12 79.15 58.50 278.00 57.60 56.18 176.25 59.50 28.88 ', 2860.14 11.25 .^..:' ,- 941.12 i (23.) Dr. (23.) Cr. (24.) Du. (24.) Cr. 1581.19 $80.10 $177.88 $156.92 17.11 15.65 291.16 285.15 45.38 39.88 356.13 356.12 19.63 10.13 189.38 178.25 187.13 176.15 471.63 469.10 87.63 89.92 785.88 098.80 87.88 77.81 911.50 930.75 \ 111.11 99.88 583.15 496.20 134.56 16.97 432.61 547.60 179.51 87.63 355.55 478.99 ^ 340.25 75.75 638.27 546.54 ' 224.12 56.51 436.15 372.25 156.12 37.23 325.36 252.12 ^ $ e $ \, DEKOMINATf: NUMBERH. {,2b. Db. (25.1 Cr. (26.) Dr. (26.) Cb. $176.9:i $12.37.76 $1087.63 4786.87 27.85 2763.18 467.88 183.05 79.a7 194.25 190.37 97.75 98.11 39.37 87.12 149.16 35.40 8.25 94.25 13.26 83.50 11.87 47.20 41.18 1127.25 29.05 39.15 8.60 48.18 63.20 8.75 9.75 250.00 71.80 367.40 11.12 779.G3 13.10 18.93 183.62 154.20 45.60 67.45 79.10 69.75 25.20 21.03 814.00 G8.87 43.15 298.50 95.60 18.75 7.50 78 60 218.00 28.63 60.00 189.00 69.87 , 71.38 87.75 47.15 18.05 293.63 5.00 68.10 77.40 185.10 31.60 54.30 38.87 9.05 13.40 12.12 15.62 64.20 90.75 89.75 9.87 38.75 15.15 118.00 14.12 45.45 «7.g:j r»9.50 89.60 ^ 215.87 68.r)<) 48.75 4.20 7.75 67.05 30.12 67.37 vi 93.92 49.:i5 91.20 81.09 81.88 21.25 87.63 7.05 68.25 35.15 90.00 57.20 99.99 20.13 100.75 114.25 \ 18.12 92.87 49.15 297.00 27.13 35.28 87.63 78.76 168.00 81.18 43.25 564.87 75.75 10.80 81.37 961.34 738.38 61.25 92.65 268.34 18.24 67.54 37.49 667.84 136.25 91.12 46.87 987.69 126.72 18.35 91.13 366.78 834.15 42.64 54.12 978.65 128.71 16.21 64.64 646.37 . 136.18 25.51 67.62 786.42 178.16 53.99 38.94 428.97 284.77 62.87 61.87 642.85 326.54 91.64 93.89 5C9.64 412.13 32.21 89.78 428.04 391.15 64.12 21.46 106.70 267.18 77.99 64.98 600.00 (' 126.13 42.61 73.75 250.09 ')»■ «Q 0) $147985.871 86907.75 ABrrmono. SUBTRAGTION. (2.) £1573.11. 41 1573.11. 44 976.15.10i (3.) $810731 341876 .37* .62{ 4. I have taken thin month in trarle £1796.18.11, and paid £673.10.10 for fall goods, and expended for private purposes, £26. 8.1, and lodged the rest in the Ontario Bank; how many dollars liave I banked V Ans. $4348. 5. I bought 47 tons, 17 cwt., 1 qr., 18 lbs. of grain, and have ^kM 29 tons, 18 cwt., 3 qrs., 22 lbs. of it; how rouch have I in ^«tore ? Ans. 17 tons, 18 cwt., 1 qr. 21 lbs. 6. If the distance iVom Toronto to Quebec is 503 miles, 1 fur., 20 rods; and the distance from Montreal to Quebec is 180 miles, 2 fur., 35 rods : what is the distance from Toronto to Montreal ? — f Ana. m. 322.6.26. 7. A farmer possessed 1279 acres, 2 roods, 21 rods, and by hia will left 789 aores, 3 roods, 36 rods to his eldest son, and the rest to the second ; how much had the younger ? Ans. 489 acres, 2 roods, 25 rods. 8. The latitude of London (England,) is 51° .30'.49"N., and that of Oibraltar 36''.6'.30" N. ; how many degrees is Gibraltar south of London? Ans. 15^24'.19". 9. The earth performs a revolution round the »* m about 365 days, 5 hours, 48 minutes and 48 seconds, and tho fA&net Jupiter in about 4332 days, 14 hours, 26 minutes and 55 seof»iids ; how much longer does it take Jupiter to perform one revt^lution than the earth ? Ans. 3967 days, 8 h., 38 min., 7 sec. 10. I bought 54 lbs., 10 oz. of tobacco, and 11 oz. of it were lost by drying; and I sold 36 lbs., 12 oz. of it to A. ; and 11 Iba., 9 oz. to B. ; and used 3 lbs., 14 oz. myself; how much have I remaining, and how much did I get for what I sold, at 6 cents an ounce, and how much did my own consumption and drying come to at the cost price, which was 5 cents an ounce V . ... Ans. (1.) 1 lb., 12 oz. (2.) $46.38. (3.) $3.65. MULTIPLICATION. 1. $1796X47=$84412. 2. £2.19.2ixl44=£426.3.0. 3. $168.8"ix64=$10808. 4. £1.2.9 X225==JE265.18.9 MULTTFUOATION. 51 5. Find Ihe daty on 07 oonttgnmcnto of merobandiw at $86.62]^ each? Ana. 18402.62^. It ia often oonvenient Ui multiply denominate numbcra bj tbe /actoTB of the multiplier. Tbua : to multiply by 84 is tbe aune aa to multiply by 7 and 12. Thua, in the annexed examples, ainoo 12x7—84, 18 tons, 12 owt., 2 qra., 11 Ibo.xB-l, is the aouie aa 18 toua, 12 owt., 2 qra., 11 IbB.Xl2x7, Su. (6.) toua. rwt. qrn Ilia. 18.12.2.11X84 12 223.11.1, 7 7 1564.19.0.24 •0. rood*. rdH. 27.2.29X72 8 221.1.32 9 1993.0. 8 cwt qr& Iba, 23.3.22X49 167.3. 4 7 1174.2. 3 (8.) £3.15.0X150 5 18.17.0 5 94. 7.6 6 566. 5.0 ' no.) "*"' llM. iw. dm. • '^ 49.11.12X63 . 7 348. 2. 4 3133. 4 .4 t "■'v,^"" 1* QuestionH, suoh aa No. 10, may also bu worked by multiplying each denomination separately by tbe whole ^iven multiplier at once, reducing to the next higher Ue- nomination, aud adding thin to the nexi< losult. — Thus: £3.15.6, multiplied by 150, will give (^1.) 900 pence=75 shillings, — (2.) 15 shillings multi- plied by 150, will give 2250 shillings, which, add- ed to the 75 shillings already found, will give 2325 shillings, which reduced will give 116 pounds, 5 shillings, and (3.) the 3 pounds multiplied by 150, will give 450 pounds, which added to what has already been found, will give a final result of £566.5.0, as already obtained by the method of factors. £3.15.0 150 900 75.0 2250.0 2325.0 £116.5.0 450.0.0 £566.5.0 52 a ABITHHETIC. owt qm. llMi 9.3.22+86 86 867.1.17 £2.13.14 125 ■, Je331.18.0^ cwtqn. U«. 1.2.17+27 27 45.0. 9 SUPPLEMENT TO THE MULTXPLICATION OF DENOMINATE NUMBERS. 6. How many seconds has a person lived who has completed his twentieth year, the year consisting of 365 days, 5 hours, 48 minutes, and 48 seconds? Ans. 662688000. 7. Bought 7 loads of hay, each weighbg 1 ton, 3 cwt., 3 qrs., 12 lbs ; what did the whole weigh ? 8. If a man can reap 3 acres and 35 rods per day, how much will he reap in 30 days ? Ans. 96 acres, 90 rods. 9. If a staemboat ply across a channel, the breadth of which is equal to 2°, 25', 10", what angular space has she traversed at the end of 20 trips ? Ans. 48", 23'. 10. If a man saves 3s. 9^d. a day, how much will he save in the year, omitting the Sabbaths ? Ans. £59 6s. 9^. 11. If 12 gallons, 3 quarts, 1 pint of molasses bo used in a hotel in a week, how much would be used in a year at that rate ? Ans. 10 hhds., 39 gals., 2 qts. 12. If a man can saw one cord of wood in 8 hours, 45 minutes, 60 seconds, in what time will he saw 11 cords? Ans. 4 days, 24 hours, 10 seconds. 13. If 13 waggons carry 3 tons, 15 cwt., 1 qr., 15 lbs. each, how much do they all carry ? Ans. 49 tons, cwt., qr., 20 lbs. 14. If a man travel 20 miles, 5 furlongs, and 20 rods a day, how would he travel at that rate in a year ? Ans. 755 m., 7 fur., 20 rods. 15. 'i'here are 24 piles of wood, each containing 3 cords, 42 cubic feet; what is the whole quantity? Ans. 79 cords, 120 ft. 16. If 17 hhds. of sugar weigh 12 cwt., 1 qr., 20 lbs. each, how much will the whole weigh ? Ans. 211 cwt., 2 qrs., 15 lbs. 17. Allowing 75 yards, 18 feet, for the surface of 9 rooms, how much paper would be required to coyer the wall ? Ads. 693 sq. yards. DIVISION. 63a 18. If 11 casks contain 54 gals., 3 qts., 1 pt, 2 gills each, how much would they all contain ? Ans. 604 gals., 1 qt., 2 gills. 19. If the care go 21 miles, 2 furlongs, 10 rods per hour, how far will they go in 15 houre ? Ans. 319 miles, 1 fur., 30 rods. 20. If 1 silver cup weigh 3 oi., 15 dwts., 10 grs., how much will 10 such oupa weigh ? Ans. 3 lbs., 1 os., 14 dwts., 4 gn. .if DIVISION. In Division, all remainders are to be redaoed to the next lower denomination, and in that form divided, to get the onita of that denomination. EXSROISEB. 1 . A silversmith made halfa-dosen spoons weighing 2 lbs., 8 ok., 10 dwts. ] what was the weight of each ? Ans. 5 oz., 8 dwts,, 8 gn. 2. If 45 waggons carry 685 bushels, 2 pecks, 4 quarts, how much does each carry on equal distribution ? Ans. 15 bushels, 1% quarts. 3. If a labourer receives 149 lbs., 13 oz. of meat as payment for 26 days' work, how much is that per day, on an average ? - ? Ans. 5 lbs., 123^ oz. 4. If a steamer occupies 48 days, 17 hours, and 40 minutes, in making 121 trips, what is the average time ? Ans. 9 h. 40 min. 5. If 98 bushels, 3 pecks, and 2 quarts of grain can be packed in 37 et^ual-sized barrels, how much will there be in each ? Ans. 2 bush., 2 pecks, 5^^ qts. 6. If a man has an income of £400 a year, how much has he each day? Ans. £1 Is. ll^^d. 7. An English nobleman has £200,000 a year ; how much has he a day ? Ans. £547 18s. 10^., nearly. 8 In .1 coal mine, 97 tons, 13 cwt., 2 qrs. were raised in 97 day^^ ; how much was that per day, on an average ? Ans. 13 cwt., 3 qrs., 22 lbs.+. 9. If 19 canisters of equal size contain 332 lbs., 8 oz., how much is in each ? Ans. 17 lbs., 8 oz. 10. If $15.50 be the value of 1 lb. of silver, what will be the weight of $500000 worth ? Ans. 32258 lbs., 8 oz., 15 dwts., ltj|f grg. / . 54a ABITHMETIG. V 11. If 1246 bushels of wheat are produced in a field of 16 acres what is the yield per acre ? Ans. 77 bush., 3 pecks, 5 qts., If pts. 12. A gardener pulled IH^.OO bushels of apples off 60 trees; how many, on an average, were in oach bushel ? Ans. 230. 13. If 13 hogsheads of sugar weigh 6 tons, 8 cwts., 2 qrs., 7 lbs., what is the weight of each ? Ans. 9 owt., 3 qrs., 14 lbs. 14. What is the twenty-third part of 137 lbs., 9 oz., 18 dwts., 22 grs. ? Ans. 5 lbs., 11 oz., 18 dwts., 6^^ grs. 15. A shipment of sugar consisted of 8003 tons, 17 cwt., 1 qr., 12 lbs., 10 oz., net weight : it was to be shared equally by 451 gro- cers ; how much did each get ? Ans. 17 tons, 14 cwt., 3 qrs., 18 lbs. 14 oz. 16. If a horse runs 174 miles, 26 rods, in 14 hours, what is his speed per hour ? Ans. 12 exiles, 3 fur., 19 rods. 17. A fanner divided his farm, containing 322 acres, 2 roods, 10 rods, equally among his seven sons and 6 sons-in-law ; what was the share of each ? Ans. 24 acres, 3 roods, 10 rods. 18. If 132 bushels, 3 peck; , > ''ts of com be distributed equally among 23 poor persons ; he - !< h does each get? .},^; ; Ans. 5 bushels, 3 pecks, 1 quart. 19. A man having purchased 119 cwt., 3 qrs., 23 lbs of hay, and drew home in 6 wagons ; how much was on each waggon ? Ans. 19 cwt., 3 qrs., 23 lbs. MIXED EXERCISES ON DENOMINATE NUMBERS. 20. A gentleman, by his will, left an estate worth $2490, to be divided among his two sons and 3 daughters in the following propor- tions : — The widow was to receive one-third of the whole, less $346 ; the younger son $212 more than his mother; the older son as much as his mother and brother, lacking $335.50, and the three daughters were to have the remainder, share and share alike ; what was the share of each ? Ans. The widow got $484 ; the older son got $844^ ; the younger son got $696 ; each daughter got $155^. 21. A gentleman left a property in land, consisting of 448 acres, 3 roods, 24 rods, to be divided among his four children in the following proportions : — The youngest was to get 4 acres, 3 roods, 6 rods rr,0TQ than the eighth part ; the second youngest was to get one- fifth of the remainder ; the oldest but one was to get one-third of the i^mainder, and the oldest the residue; what was the @hare of each? DITttlON. 55a Ans. The youngest got 60 acres, 3 roods, 24 rods ; the next got 77 acres, 2 roods, Iti rods ; the ne^i^t got 103 acres, 1 rood, 34§ rods ; the oldest got 206 acres, 3 roods, 29^ rods. 22. A ship made the following headway on six snocessive days : On Monday, 3*>, 8', 45" south, and 1*, 51' cast ; on Tuesday, 2**, 36' south, and 2°, 1', 15" east ; on Wednesday, 4°, 0', 52" south, and 1*> east; on Thursday, 1", 48', 52" south, and 3", 16', 22" cast; on Friday, r, 19' south, and 48', 29" east; and on Saturday, 69', 30" south, and 3°, 52', 11" east; find her distances south and east from the port of departure. Ans. South 13*', 52', 59" ; Bast 12°, 49', 17" 23. A vintner sold in one week, 51 hogsheads, 53 gallond, 1 quart, 1 pint ; in the next week, 27 hogsheads, 39 gallons, 3 quarts ; in the next week, 19 hogsheads, 13 gallons, 3 quarts; how much did he sell in the three weeks ? Ans. 88 hogsheads, 43 gallons, 3 quarts, 1 pint. 24. In a pile of wood there are 37 cords, 119 cubic feet, 76 cubic inches ; in another there are 9 cords, 104 cubic feet ; in a third there are 48 cords, 7 cubic feet, 127 cubic inches, and in a fourth there are 61 cords, 139 cubic inches. Find the whole amount. Ans. 156 cords, 102 feet, 342 inches. 25. The following cargo was landed at Montreal from Liverpool : 78 tons, 3 cwt., 2 qrs., 26 lbs. of Irish pork ; 125 tons, 15 cwt., 1 qr., 9 lbs. of iron ; 90 tons, 12 cwt., 2 qrs., 20 lbs. of West of England cloth goods ; 225 tons, 9 cwt., 12 ibs. of Scotch coal, and 106 tons, 1 qr. of Staffordshire pottery ; what is the whole amount of the consignment ? f Ans. 636 tons, 1 cwt., 16 lbs. 26. If a man can count 100 one-dollar bills in a minute, and keep working 10 hours a day ; how long will it take him to count a million ? Ans. 16f days. 27. The earth's equatorial diameter is 41847426 feet; how many miles ? Ans. 7925 and 3426 feet. 28. The earth's polar diameter is 7899 miles, 900 feet; how many feet ? Ans. 41707620 feet. 29. Sound is calculated to move 1130 feet per sscond ; how far off is a cannon, the report of which is heard in 1' 9"? Ans. 77970 feet. 30. If the circumference of a waggon wheel be 14§ feet ; how often will it turn round in a mile, (5280 feet) ? Ans. 3^Q times. ARimyETrc. GREATEST COMMON MEASURE. 13' — When att^' quantity is contained an even number of times in a greater, the greater is called a multiple of the less, and the less a submultiple, measure or aliquot part of the greater. Thus : 48 is a multiple of 2, 3, 4, 6, 8, 13, 16 and 24, and each of these is a sub- multiple of 48. When one quantity divides two or more ''others evenly it is called a common measure oi those quantities, and the greatest num- ber that will divide them all is called the greatest common measure. Thus: 7 is a common measure of 63 and 49, and it is also the greatest common measure, for no larger number will divide both evenly. When any quantity is measured evenly by two oi more others, it is called a common multiple of them. Thus : 24 is a common mul- tiple of 2, 3, 4, 6, 8 and 12. A number which can be divided into two equal integral parts is called an even number, and one which cannot be so divided is called an odd number. Heuce all numbers of the series 2, 4, 6, 8, 10, 12, &c., are even, 'while those of the series 1, 3, 5, 7, 9, 11, &c., are odd. Hence the sum of any number of even quantities is even ; also, the sum of any even number of odd quantities is even ; but the sum of any odd number of odd quantities is odd. This principle is of great use in checking additions. A prime niunber is one which has no integral factors except itself and unity ; a composite number is one that has integral fac- tors greater than unity, and numbers which have no common factor greater than unity are said to be prim^ to each other. Of the first kind are 1, 2, 3, 5, 7, 11, &c., of the second, 4, 6, 8, 9, 10, 12, &c. ; also, 2 and 7 arc prime to each other, and so are 6 and 7. If one quantity measure another it will measure any multiple of it. Thus : since 3 measures 6, it will also measure 12, 18, 24, &c., because it is a factor of all these. If one quantity measure two or more others, it will also measure their sum and di£ference, and also the sum and difference of any OBEATE8T COMMON MfiASDBE. 53 mnltiples of them, because it moasures them when they lure taken separately. Hcnoc, if one number divide the whole of another number, and also one part of it, it will divide the other part too. Thus : 6 di- vides 24 and 18, and so the other part, 6 ; 9 divides 45 and 27, and also the remainder, 18. Also, if a number be composed of several parts, each of which has a common factor, that factor vrill also measure their sum. Thus : 9 measures 18, 27, and 36, and their sum, 81. From these principles wc can deduce a rule for finding the greatest common measure of two or more quantities. RULE. Divide the greater by the less, and then the less by the re- mainder, until notliing is left, and the last divisor will be the greatest common measure. EXAMPLE. ■■.f]-^':-v •-., . <. . 2145 3471 , ,:,. ., -■ ■ ,■ . -. ,..:.- .^ - .;j. .!. ^;;-».>(i^ 132b 2145 A concise form of the work is exhibited in 819 607 312 195 1326 the margin. The quotients are omitted as 819 unnecessary. The last divisor, 39, is the Q. C. M., as may be proved by trial. If it is re- 2J2 quired to find the G. C. M. of more than two numbers, first find the G. C. M. of two of 117 78 1 95 them, and then the G. C . M. of that and another, 117 and so on. 39 78 •■ ■ 78 * ., • ' • EXERCISES. Fin« 1 the G. 0. M. of the following quantities: 1. 247 and 323. Ans. 19. 2. 532 an(? 1274. Ans. 14. 3. 741 and 1273. Ans. 19. 4. 10416 and 25761 Ans. 93. 5. 468 and 1266. Ans. 6. 6. 285714 and 999999. •- Ans. 142857. 7. 15863 and 21489. Ans. 29. 8. 8280 and 11385. Aas. 1035. 9. 17222 and 32943. Ans. 79. 10. 19752 and 69132. . Ans. 9876. •\. 6i ABrrBMETIC. We mr^ otton find the G. C. M. by inspection. For example, in ozereise 5, wo see that 2 will measure both quantities (Art. 13), for both are even, and also that 3 will measure both, because it mcasurcH the sum of the digits (Art. 16. i ' ,« t** . •'. The least common multiple of two or more numbers is the smallest number that is divisible by all of them. Thus : 48 is a common multiple of 2, 3, 4, 6, 8 and 12, but 24 is the least common multiple of them. It is plain that the least common multiple of quantities that havo no common factor is their product. Thus : the L. C. M. of 5, 7, 6 is 210. But if the quantities have a common factor, that factor is to be taken only once. Thus : 9G, 48, 24, are all common multi- ples of 2, 3, 4, 6, 8, 12, but the least of these, 24, contains only the factors 3 and 8, which are prime to each other, for 2, 3, 4, 6 are all contained in 12, and 8 and 12 have a common factor, 4, which being left out of one of them, 8, gives 2x12=24, or, being left out •of the other, 12, gives 8x3=24. From this we derive the RULS: 0. »•«>•• •4. • .u..."...lo...^7...DU ■,''■■ , h ' :s 4.. .18.. .27.. .30 . . .A. ^-,.. i,-, 2... 9.. .27.. .15 3 2.. .27.. .15 2... 9... 5 9 . *"' *' 45 .;' . P:V 2 , " - .-^ ;'- 90 -t- .mh 8 5^70 2 540 Expunge all common factors and take the continued product of all the results and divisors. Thus, to find the L. C. M. of 2, 3, 4, 6, 9, 18, 27, 30, ar- range them in a horizontal line, and as 2, 3, 6, 9 are all contained in 18, they may be omitted, aa in the second line, then, as 2 is contained in 4, 18 and 30, it may be divided out, and as 9 in the third line is contained in 27, it may be omitted, as in the fourth line; and 27 and 15 be- ing both divisible by 3, we ob- tain in the fifth line 2, 9, 5, all prime to each other, and the products of these and the divis- ors 3 and 2 is the L. C. M., B40. 0REATE8T OOMMON MEASURE. R X BRCI8 BK Find the L. 0. M. of tho following quantitios i 1. 8, 12, 16, 24, 33. 2.36,42,45,81,100. 3. 2, 4, 8, 16, 33, 64, 128 4. 2,3,5,7,11. 5. 3,9,27,81,243,729. *, ; 6. 12, 16, 18, 30, 48. 7. 3, 4, 5, 6, 7. 8. 2, 3, 4, 5, 6, 7, 8, 9. 9. 2, 4, 7, 12, 16, 21, 56. 10. 2, 9, 11, 33. 05 •■•'»■ ^A. An8.628. Ans. 56700. Ads. 128. Ans. 2310. Ani. 729. Ans. 720. Ans. 420. Ans. 2520. Ans. 336. Ans. 198. EXAMPLES FOR PRAOTIOE 1. What will 320 caps cost at $7.50 each ? Anu. $2400. 2. If you can purchase slates at 20 cents each ; how many can you buy for $7.40 ? Ans. 37. 3. If you can walk 4 miles an hour ; how far can you go in 24 hours? Ans. 96. 4. What will be the cost of 216 barrels of pork at $7.50 per barrel ? Ans. $1620. 5. How many sheep can be bought for $560 at $3.50 per head ? Ans. 160. 6. If 825 pounds of beef are consumed by a garrison in one day - what will be the cost for 6 days at 11 cents per pound for beef? Ans. $544.50. 7. A farmer sold 185 acres of land at $25 per acre, and received in payment 1 7 horses at $70 each, and 12 cows at $20 each ; how much remains due ? Ans. $3195. 8. A merchant bought 120 yards of Canadian tweed at $1.15 a yard ; 60 yards of flannel at 95 cents per yard, and 13 dozen pairs of gloves at 35 cents per pair ; what was the amount of his bill? Ans. $249.60. 9. At $2 per gallon ; how much wine can be bought for $84 ? Ans. 42 gals. 10. A boy had $5.50, and he paid one dollar and five cents for a book ; how much had he loft ? Ans. $4.45. 11. What will 18 cords of wood cost it $4.75 per cord ? Ans. $85.50. / 56 ABITHlfETIO. 12. How many poundo of sugar can be bonght for $9.35, at 11 cents per pound ? Ans. 86 Ibe. 13. What will a jury of 12 men receive for coming from Kings- ton to Toronto at 10 cents a mile each ; the distance being lUO mileH ? Ans. $192. 14. A grocer bought a hogshead of molasses at 32 cents per ^on ; but 18 gallons leaked out, and he sold the remainder at 55 cents per gallon ; did he make or lose, and how much ? i\.nu. He gained $4.59. 15. If a clerk's salary is $600 a year, and his personal expenses $320 ; how many years before he will be worth $6600, if he has $1000 at the present time ? Ans. 20 years. 16. A speculator bought 200 bushels of apples for $90, and sold the same for $120 : how much did he make per bushel ? Ans. 15 cents. 17. A person sells 15 tons of hay at $22 per ton, and receives in payment a carriage worth $125, a cow worth $45, a colt worth $40, and the balance in cash \ how much money ought ho to receive ? ; ' Ans. $120. 18. How many pounds of butter, at 20 cents per pound, must be given for 18 pounds of tea worth 75 cents per pound V Ans. 67^ lbs. 19. A grocer bought 7 barrels of fish at $18 per barrel ; but one barrel proved to bo bad, which he sold for $5 less than cost, and the remainder at an advance of $3 per barrel ; did he gain or iose, and how much ? Ans. Lost $13. 20. A man bought a drove of cattle for $18130, and after sel- ling 84 of them at $51 each, the rest stood him in $43 each ; how many did he buy ? Ans. 406. 21. What will 2 cwt. of cheeso cost at 9^ cents per pound ? < Ans. $19 00. 22. A. is worth $960, B. is worth five times as much as A., less $600, and C. is worth three times as much as A. and B. and $300 more ; what are B. and C. worth each, and how much are they all worth ? Ans. B. $4200 ; C. $15780 ; all $20940. 23. A boy bought a dozen knives at 15 cents each, and after selling half of them at the rate of $2.22 per dozen, he lost three, and sold the balance at 25 cents each ; did he make or lose, and how much ? Ans. Gained 6 cents. 24. A labonrer bought a coat worth $16, a vest worth $3, and a t1 / 0BEATE8T OOMIION MEASURE. 57 pair of pants worth 95.50 ; how many days had ho to work to pay for hiB Bait ; his seryicea being worth 50 cents per day ? Ans. 49 days. 25. What will 14 bushels of olovor seed cost at 12^ cents per pound? Ans. 9105. 26. A farmer sold a load of oats weighing 1836 pounds, at 30 cents per bushel ; how much did ho receive for the same ? Ans. 916.20. 27. A produce dealer bought at one time, one load of wheat weighing 3240 pounds, at 91.05 per bushel; one load of barley weighing 2400 pounds, at 85 cents per bushel; one load of rye weighing 2800 pounds, at 65 cents per bushel; two loads of pease, each 2400 pounds, at 68 cents per bushel ; three loads of buckwheat, each weighing 1400, at 55^ cents per bushel ; and a quantity of oats weighing 578 pounds, at 33 cents per bushel ; what had he to pay for the whole ? Ans. 9250.15|. 28. A farmer has 12 sheep worth 93.50 each ; 9 pigs worth 94.65 each ; one cow worth 935, and a fine horse valued at 9150. He exchanges them with his neighbour for a yoke of oxen worth 975 ; two lamba worth 91.925 each ; a carriage worth 9100, and takes the balance in calves at 94.50 ; how many calves does he receive? Ans. 20. 29. A and B sat down to count their money, and found that they had together 9225, but A had 915 more than B ; how much had each ? Ans. A 9120, B 9105. 30. A miller bought 250 bushels of oats for 985 and sold 225 bushels for 970 ; what did the remainder coat him per bushel ? Ans. 60c. 31. A widow lady has a farm valued at 96720; also three houses, worth 912530, 911324, and 99875. She has a daughter and two sons. To the daughter she gives one-fourth the value of the farm, and one-third the value of the houses, and then divides the remainder equally among the boys , how much did each receive ? Aus. daughter 912923, each son, 913763. 32. A man went into business with a capital of 91500 ; the first year he gained 9800, the second year 9950, the third year 9700, and the fourth year 625, when he invested the whole in u cargo of tea and doubled his money ; what was he then worth. Ans. 99150. : 30 cents f boy paid Eipple 3 cents ; how nvfsxy apples did he purchase ? Ans. 60. 68 ABITHMETIO. 34. A Hohoolboy bought 12 oranges at 3 ocnt« each, and sold thorn for 12 oenUi more than ho paid for them ; how much did ho Mil them at each / Ans. 4u. 35. A clerk's inoome ia 82G9S a year, and his expenses $4.5o per day , how much will he save in two years ? Ans 821 11. 3G. A speottlator bought 200 acres of land at 84& per uuic, «"d afterwards sold 100 acres of it for $11550 ; tlio balance ho sold ut u gain of $5 per acre, and received in payment $250 canh, und the balanoe in sheep at $5 each ; how many sheep did ho receive 'i* Ans. 450 sheep. 37. A butcher bought 9 calves for $54, and 9 lambs for $31.50 ; how muoh more did he pay for a calf than a lamb ? Ans. $2.50. 38. A farmer sold to a grocer 380 pounds of pork, at 7 cents per pound ; 150 pounds of butter, at 17 cents per pound, and one cheese weighing 53 pounds, at 9 cents per pound ; and received in payment 22 pounds of sugar, at the rate of 11 pounds fur a dollar, 150 pounds of nails, at 6 cents per pound ; 15 pounds of tea, at 65 oentn per pound ; one half-barrel of fish, at $18 per barrel, and one suit ot clothes worth $27 ; did the farmer owe the grocer, or the grocer the farmer, and how muoh ? Ans. the grocer owed the farmer 12 cents. 30. A milkman sold 120 quarts of milk, at 5 cents per quart, and took in payment, one pig worth $1.50, and the balance in 8huot- ing, at 10 cents per yard ; how many yards did he receive ? Aus. 45 yards. 40. How many pounds of cheese, at 9 cents per pound, must be given for 27 pounds of tea worth 80 cents per pound ? Ans. 240. FRACTIONS. 14. — VuLQAR OR Common Fractions. — When we have di- vided any number by a less, and find no remainder, the quotient is called an integer, or whole number. When we have divided any number by a less as far as possible, and find a remainder still to be divided, but less than the divisor, and therefore not actually divisible by it, we must have recourse to some method of indicating this. We have seen already that the conventional sign of division is this mark (-f-) ; thus, 3-f-4 means that 3 is to be divided by 4, and this being impossible, we indicate the operation either as above or by writing the three in the place of the upper dot, and the 4 in tlic place of the lower, thus, f • FRACTIONS. 09 The nataro of a trution may Im viewed in two ways. Pirtt, wo luay oonsidor that a unit in divided into a certain nvmher of equal parts and a certain number of those parts taken ; or, seeondly, that a number greater than unity m divided into certain equal parts, and one of these purts tulcon ; thus, ^ lueanH either that a unit ia divided into 4 c(|ual partH and thret; of them taken, or that three is divided into 4 equal partN and one of them taken. For example, if a foot be divided into 4 equal parts, eucli of tbene parts will be 3 inches, and three of them will bo nine inches ; and since 3 feet make 3U inohes, if we divide 3 feet into 4 equal parts, each of these parts will be 9 inches, and hence ^ of 1 r-^^ of 3. The lower figure is called the denominator, because it uhows the denomination or number of parts into which the unit is supposed to be divided, and the upper one is colled the numerator, because it shows the number of those parts considered in any given quoF .on. When both are spoken of together they arc called the temu of the fraction. . What may be considered the fundamental principle on which all the operations in fractions depend is this : that the form, but cot the value of a fraction, is altered, if both the terms are cither multiplied or divided by the same iquantity. If we take the fraction f and multiply its terms by 2, we get |. Now, the ^ of a foot is an inch and-a-half, and therefore g is 6 inches and 6 half-inches, or 9 inches ; but we have seen that f of a foot is 9 inches, therefore ^ of a foot is the same as g of a foot. So also f of £1 and | of £1 are both 15s. The same will hold good whatever the unit of measure may be, or whatever the fraction of that unit. Hence, universally the /orm of a fraction is altered if its ten i be either multiplied or divided by the some number, but its value remains the same. '^ Again, if we multiply the numerator 3 by 2, but leave the denominator 4 unchanged, we obtain f , and, keeping to our first illustration, ^ of a foot is G times three inches, or 18 inches, which 13 double of 9 inches, the value of |. We should have obtained the same result by taking | and dividing its denominator by 2, without dividing its numerator. Hence, a fraction is multiplied by either multiplying its numerator or dividing its denominator. In like manner, if we take the fraction ^ and divide its numerator by 2, we obtain f , and if we multiply the denominator of its equal ^ by 2, wc obtain the same rusult, f . Hence, f is ^ of |, and therefore a frac- tion is divided by either dividing its numerator or multiplying its denominator. These principles may also be referred to the obvious 00 ARITHMETIC. faot that in dividing any quantity the groator the divii»nr the Iom the qaotient, and the leas the diviaor the greater the (|Uotient. Am it ih alwayi desirable to have the smallest numberM poMiiblu to handle, let the operator observe this aM a universal ru!c^r/t»t(/^ when you four different oiroumstanocs. I. They are divided into Proper and Irapropor FrootioiiH. A proper fraction is one whose numerator ih less than its denomi- nator. In strictness such alone is a fraction. An improper fraction is one whoso numerator is greater than its denominator. Strictly this is not really a fraction, but only a odrtaia quantity expressed in the fractional form. II. Simple and Compound Fractions. ' The torm simple fraction, as opposed to compound fraction, expresses that the fraction is multiplied by unity alone, as f , which means either f of 1 or | of B, or tXl=j^X5. A compound fraction is one that is multiplied by some other quantity. A fraction is called compound if cither multiplier or mul- tiplicand, or both, be fractional. Thus: ^ of ^ and | of 11 are both compound, and are written f X§ and f Xll* III. Simple and Complex Fractions. The term simple fraction, as opposed to complex fraction, moans that there is only one division. Thus: [§ means that u single number, 15, is divided by a single number, IG. A complex fraction is one of which either the numerator or de- nominator, or both, arc fractional, that is, it indicates a division, when either the given product or given factor, or both, are fractional. ■V- M II Thus : f -r-TT, or i and ^ and fj_ are complex fVactioua and cx- I I hibit the only three posssiblc forms. IV . Vulgar, or Common, and Decimal Fractions. Decimal fractions are those expressed with a denominator, 10, or a power of 10, e. y., ^\, ^Vg, yg^g. Any fraction not so expressed is called vulgar or common. Thus : J would be called a commou fraction, but its equivalent, ,^,3, would be called a decimal fraction, and is written -75, the denomina- tor being omitted, but ity existence being indicated by the mark (), called the decimal coint. , nuonoNs. m A inlxod qiuntity !i ono ez]Nr«M«d partly by • wbol« numbui uDd partly by a fVaotion, an 4|, 12^. Thiv in not anothor kind u! fraction, but aimply another mode of writing an improper Traction when the division indicated has been poriWnicd tut far m posfiiblo ThuH: ^=45. and 7-^12^. It M often Maid that there are six kinds of fVaotions — proper improper, Himplu, compound, complex, and mixed. Thiit ih logi cally incorrect, for u proper fraction iH Himpic, und a mixed ({uantity is an improper fraction in anothor form. 16. — Opirationh in Common Fbaotionh. — From the prin ciplea laid down (Art. 21,) we can deduce ruloa for oil the operations in fractious. I. An improper fraction in reduced to a mixed ((uantity by per form! ^ the division indicated, as '^^^^=24,\. ^»- A mixed quantity in reduced to an improper fraction by muUiplying tl*o integral part by tlio denominator and adding in the numerator, nL -i^^'Ji:*. 8o also uu integer may bo expressed in the fractional form by writing 1 ivs u denominator, and multiplying the terms by whatever number will bring it to any required denomination. Thus: tn reduce 7 to the same denamination as ^, write ] and multiply thi) terms by G, and the result, *^', will bo equivalent to the integer 7, and of the same form an 'l- EXERCISES. . . 1. Express 2. flxpress 3. Express 4. Express 5. Express C. Express 7. Express 8. Express 9. Express 10. Express 11. Express 12. Express 13. Express 14. Express -^^^^ as a whole or mixed number. ^ ^ us u whole or mixed number. -W as a whole or mixed number. W as a whole or mixed number. 'iWir ''^^ a whole or mixed number 'yy' us a whole or mixed number. W' us a whole or mixed number. ^^ as a whole or mixed number. ~\^- as a whole or mixed number. ^- as a whole or mixed number. 'gY as a whole or mixed number. J- 3" as a whole or mixed number. -,y as a whole or mixed number. Y as a whole or mixed number. 1 1 Ans. 49. Ans. b-;!^. Aus. 71, Ans. ')Af Ana fti t'M > Ans. ll^rf' Ans. 7g'n. Ans. 7/3. Ans. 89. Ans. 10/p Ans. 211 Ans. 19"^. Ans. 12/y. Ans. 4|. 62 ABTTHMETIO. 15. Express ^^- as a Tvholc or mixed number. Ans. 24^. 16. Express Jj'/ as a whole or mixed number. Ans. 5^f . 17. Express ^f as a whole or mixed number. Ans. 3^^. 18. Express -'.j'- as a whole or mixed number. Ans. 6^. 19. Express ij-' as a whole or mixed number. Ans. 30J. 20. Express '-'gi as a whole or mixed number. Ans. 83/g. 21. Express J^"/ as a whole or mixed number, Ans. Sj'g. 22. Express 27^ as an improper fraction. Ans. Aj"-. 23. Express 66 J as an improper fraction. Ans. ^'-%^. 24. Express 15|5 as an improper fraction. Ans. ^y"g^-. 25. Express 7f as an improper fraction. Ans. ^. 26. Express 49 as a fraction with the same denominator asi§. Ans. -",33^. 27. Express 19s. as a fraction of £1. Ans. ^g. 28. Express 11 inches as a fraction of a foot. Ans. |A. 29. Bring \, J, ^, 1, ,', to the same denomination. Id* Ans. 30. Express 11 as a fraction having the same denominator as i^^-^. Ans. -75 X* III. To reduce a fraction to its lowest terms or simplest form, divide the terms by their greatest common measure. This is often readily done by inspection, as ^^='^=.^, but in such questions as off i» ^^ °^ost secure and speedy method is to find the G. C. M. of the terms and divide them by it. Thus : the G. C. M. of the frac- tion ^^11 is 1092, and the terms of the fraction divided by this give \y the simplest form. EXERCISES. 1. Reduce yVaVk *° ^^^ lowest terms or simplest form. 2. Reduce %%\ to its lowest terms or simplest form. 3. Reduce §§§§ to its lowest terms or simplest form. 4. Reduce ^'gg"tMj*(> ^ ^^^ lowest terms or simplest form. Ans. J. Ans. j\. Ans. |. Ans. 5*j. 5. Redu;'e |§|g to its lowest terms or simplest form. Ans. %. 6. Reduce tVsWo ^ ^^ lowest terms or simplest form S" 7. Reduce jIIIt ^ ^^ lowest terms or simplest form. 8. Reduce |||| to its lowest terms or simplest form. 9. Reduce |^f ^ to its lowest terms or simplest form. Ans. ^. Ans. -i\. Ans. f . Ans. I . nMLOiioNS. 03 10. Redaoe ^H^i ^^ ^^ lowest terms or Bimplest form. Ads. § 11. Reduce -^t^iiii to *^ lowest terms or ftimplest form. Ans. tV 12. Reduce ^^^^ to its lowest terms or simplest form. Ans. ^^ 13. Reduce js^^iua *<> '^ lowest terms or simplest form. Ans. f 14. Hedttoo lollt'^ to its lowest terms or simplest form. Ans. 15. Reduce ^^g§ to its lowest terms or simplest foim. IG. Reduce 30 js ^ ^*® lowest terms or simplest form. 17. Reduce ^25 to its lowest terms or simplest form. 1 8. Reduce //t]\ to its lowest terms or simplest form. 19. Reduce y^'^sl to its lowest terms or simplest form. 20. Reduce §^|o^ to its lowest terms or simplest form. 045 TS25' Ans. 5' Ans- l Ans. ^§. Ans. -/g. Ans. Ans. 1 1 T-J' 21. Reduce ^f^mf^^Ma to its lowest terms or simplest form. Ans. h. IV. To multiply one fraction by another, multiply numerator by numerator and denominator by denominator. Thus : J X J= 5 . To illustrate that J of J is ^, take a line and let it be divided into 3 parts, and each of those again into 3 parts, as in the margin, wc find that the result is 9 parts, each, of course, being ^ of tlie unit. 1 ... 1 ... 1 111. ..111. ..Ill Wo have seen that a fraction is multiplied by multiplying the numerator or dividing the denominator. Now, if it were required to multiply '^ by ii, we could not divide the denominator, as 5 is not contained in 4, and therefore we multiply^the numerator and obtain Y', but wc have multiplied by a quantity equal to 7 times the given one, and therefore we must divide the product by 7, i. e. (Art. 21,) wo must multiply the denominator 4 by 7, which gives for the correct product. EXERCISES. 1 5 29 J a v It ' 1. Multiply y\ by 2. What i6 :ho product of f by \i ? 3. AVhat is tlie product of Jj by g ? 4. What is the product of ^fby | •' ? 6. What is the product of ^ by ^§ ? Ans. 2«j|\. Ans. |-». Ans. f^. Ans. «5. Aus. Jg. 64 AfilTHMETIO. TOOK Ana. m- Ans. |§. Ana. jVa- 6. What is the product of | by f\ ? Ans. 7. What is the product of ^%% by /g ? Ans. ^^, 8. What is the p:'oduct of ?« by j\ ? 9. What is the pi oduct of § by I 't 10. What is the product of ]% by {j ? When the product has been obtained it should be reduced to its lowest terms. Thus : the product of ^^ by J ^ is jYg, the terms of which are both divisible by 1 1 , and so we get t he equivalent fraction -^^y But we might as well have divided by 11 before multiplying, for by this method we should at once have found the fraction in its simplest form, viz., ,7^. In the same niauner any number or num- bers which are factors of both numerator and denominator, may be omitted in the operation. This we call cancelling in preference to the excessively awkward term " cancellation." This method will be clearly seen in exercise 11. If either the multiplier or multiplicand be a mixed quantity, it must be reduced to an improper fraction before the multiplication is performed. Thus : 8-' x52=-Y-X-^^-=-Ln"^=51.' ■»■ 11. What fraction is equal to ^ of § of J of i of % of ? of 2 of | ? Ans. 12. What quantity is equal to 12|^ multiplied by 7;] ? Ans. 97j^. 13. What quantity is equal to 19^ midtiplied by l}!^ ? Ans. 36. 14. What is the value of % of A of f f of | ? 15. What is the value of \ of ^ of « of \\ ? 16. What is the product of 27f by 3f ? 17. What is the product of i| by ^g ? 18. What is the product of 5| by 5| ? 19. Find the square and cube of ^5 ? 20. What is the cube of fg ? 21. Multiply 27 by a'f? Ans 2' Ano 4 4 Ans. 107|Jr. Ans. /g. Ans. 30J. Anq 2 89 nn(] 4913 •f^na. |g| ana yggjg. Ans 9 3 19 Ans. 1. V.-DIVISION OF FRACTIONS. To divide one fraction by another, multiply by the recipro- cal of the divisor ; or, in other words, invert the divisor and multi- ply. In the language of science, the reciprocal of a fraction is the fraction with its terms inverted. Thus : f is the reciprocal of | ; ^ of f . To find the reciprocal of a whole number, we must first DIV18I0N OP FRACTIONS. 65 represent it as having a denominator 1, — thus 4=J ; 6=^, and therefore the reciprocals arc ^ and ^. The rule for division may be proved in two ways : IhvmT PROOF. — L. t it be required to divide -^^ by jj. If wo had been required to divide by the whole number 5, we shoul'^ cither have divided (Art. 14,) the numerator, or multiplied the denomina- tor, — as the numerator is not divisible by 5, we multiply the I'e- nominator, and obtain X ; but we have divided by a quantity equal to six times the given one, and therefore, to compensate, we must multiply thoen by tlie annexed examples : f X^~';i I • But i^U ^""1 i—Uy 'jotli greater than 'i\. Also, i^i=iXi=li But §=§! and 1=1.?, both less than 2R i4' If two fractions have a common denominator, their quotient is the quotient of their numerators. We have placed multiplication and division of fractions before ad^-ion and subtraction, because, as in whole numbers, multiplication and division are deduced from addition and subtraction, so conversely in fractions, addition and subtraction are to be deduced from multiplication and division, for a fraction is produced by division, and the multiplication of a fraction is merely the r( peating of the divided unit a certain number of times. Thus : ^ is a unit divided into 8 equal parts, and ^ is that fraction repeated 7 times. 66 ABTTHMETIO. •^ T X E 11 C I 8 E B . 1. Divide -i^j by §; y\-§=i\X^ 2. What is the quotient of ij divided by [? ? Ans. 3. What is the quotient of r.7j divided by ,| l^ ? - Ans. 4. What is the quotient of | g divided by ^ » ? Ans. j^. 5. What is the quotient of ^^^ divided by ^ ^ ? Ans. |^. 6. What is the quotient of 36 divided by 19| ? Ans. V, 7. What is the quotient of 3§ divided by 2f ? Ans 8. What is the quotient of 4^ divided by 15 ? Ans. 9. What is the quotient of «« divided by 2|| ? Ans. »gfl. 10. What is the quotient of 75/5 divided by 9 ? Ans. Si)^. 11. Wlmt is the quotient of 6^ ? divided by 9§ ? Ans. ^. 12. AVhat is the quotient of 5| divided by 8/3 ? Ans. 4^;f . Am. Wt; j« — l"i 24HA IT* 1 f.7 3 13. A? Divide the product off, | and ^ by tho product of ^, f and 15. 16. 17. Ans. -V=^l^. 14. What is the quotient of /y of |4-j-| of f^ of j\-^j% of f ? Ans. 4i^>. How many ,',j arc there in -f\ ? Ans. 8 J 1{. What is the value of g of f -4-f of |A ? Ans. 1/3. Divide 27 by J^ ? * Ans. 729. Hence, any quantity divided by its reciprocal gives the square of that number, anu exercise 21, of multiplication, shows that any quantity multiplied by its own reciprocal gives unity. 3 ? 18. Divide ■fi^-^ by ^, and the quotirni. by 19. Divide 4 by /y, and the quotient by -Sf ? 20. Divide ?§ by ^^ ? 21. Divide 1 1 TH ^Uf Aus. 1/j. Ans. {^. Ann ^ 491 Ans. ojgj,. Ans. f . VI.-ADDrnON OF FRACTIONS. We have seen that no quanlities can be added to^ ther except they are in the same denomination. We can add !\, ^, 9 and -','-, a:s they arc all of the same denomination, sevenths, and wc find ■^^^. Wc can easily see that to add f and f, we h;ivo only vo alter the form of f to |, and we have both fractions of the same denomination, and therefore ca*. add them, — 5+3= -U So also ^-J-^4-i^-l-''?4— '^-^=-'■^4— "t4— '^-l-i^-l— ''^ — 4o — JQ But wc cannot always tell thus by inspection, and therefore must be guided by some rule. To find the value of i+s+s+i"!"/^* I ADDITION OP FKACTIONS. 67 By Art. l.J wo find the L. C. M. of 4, 6, 8, 9, 12 to be 72, and the ro^t of the common operation is equivalent to multiplying the terms of each fraction by 72. Thus : if the terms of ^ bc' both multiplied by 72, wc get H^Jirrr^^gj— ^j, but wc might as well have divided 72 by 4 before multiplying, and, to balance that, have multiplied the numerator 3, not by 72, but by the fourth part of 72, viz., 18, giving ij.j, as tac following scheme will show: — ^ X7 i _- ^ X I H >^ 1 ^^ ^ X j. R -_ o j _ Tho other fractions being altered iu the same manner, wc got 7 j-|-7" + 'ri'i"'7*+r'» *°^ "" *^<^''® ^^ now all of the same denomination, the gb not altered in value, wo can add them, and we find M-Hf , lri'2=42+f2+T3+T5+ Hence the 4i -iS! RULE. Find the L. C. M. of all the denominators, which will bc the common denominator ; divide this common multiple by each denomi- nator, and multiply the quotient by each numerator in succession for new numerators ; add all these new numerators together, and place the common rlcnominator below the sum, and the fraction thus ob- tained will k/o the .sum of the given fractions. If the numerator, thus obtained, bc greater than the denominator, the resulting frac- tion may be reduced to a whole or a mixed number by division. EXERCISES. 14 IB- 1. Express ,5 +7% +15 +1-5 ^^ a single fraction ? Ans. 2. Find the sum of ^, f, f and -^ ? Ans. 2|. 3. Add together 4^,^1 {2, ^h ^i? and 5^^, ? Ans. 18y<\fg. 4. What fraction is equal to i+J+^+Tg-f-as+Gi ? -A-ns. g|. 5. Whet fraction is equal to ij + 2§+3|+4§+5|+6§ ? Ans. 2m^. 6. Express ^ of £+§ of |-|-|^ of | as a single fraction ? Ans. 7. Find the sum of 1|^, 8?, 3-A and 4^ ? 8. Find the sum of ^ oi ;l + ,C of ^-\-^ f 9. What single fraction is equi\ _.. it to ^ of i^-fJ^of f-j-J^of ^ ? .0? 4 7 1 I t Ans. 1833^1^. 101 2T3' Ans. Ans. 3 I Cl- io, What Binc;lc fraction is equivalent to f of g of ]--\-^ of f of Hi of i of I? ' Ans.-i3g. 11. Wh.i!, single fraction is equivalent to j) oi*-^ ct'^f of g of 68 ABITHMETIC. 12. Simplify ]V'J;^V ^ Ans. 1^,. 13. Find a single fraction equivalent to ^ of § of §+^ of ^ ? Ans. ^IJ. U. Divide ibi: sum of /, and ^ by the sum of i and f ? An8.-|4|. 1&. PimpHlj f-ULj4i^ T'g+^^+gV+x i^y ^„g ^27 ^-rt+HII+l^4-g|+ii2 16. Simplify l4:|Tirl? ir«3* \na 3 147 VII SUBTRACTION OF FRACTIONS. Wh;;'. '^c have said of addition enables us to give at once the nlTLE FOR SUBTRACTION. Reduce the given fractions, if necessary, to new ones having a common denominator, as in addition, and subtract the numerator of the less from that of the greater, and place the common denominator below the remainder, and the resulting fraction will be the difference between the given fractions. Examples. — (1.) To subtract ,V from ','', . Here the denomi- nations being the same, we can subtract at once, and find the differ- ence to be ,-, . (2.) To find the value of (] — ^. These fractions brought to a common denominator, as in addition, become ^^' and ^ j, aiid therefore the difference is -^\. (3.) To find the excess of 12 J above 7^, we find new fractions with a common denominator, viz., 5", and ^j, and wc write 12^^, — 7A^'. Now we are required first to subtract ^:] from v,",, but as wc cannot do this directly, we take one of the 12 preceding units, and call it 'i*, (for H^^=l,) then 2H~2J=^^^» ^'^ iif — i4=al> ^^^^ ^^ subtract the 7 from the re- maining 11 ; or, as in simple subtraction, 8 from 12, and we find the total excess to be 4!|. In practice it is most convenient to sub- tract 15 from 24, and add 8 ; thus 24—15=9, and 9-|-8=17, and the answer is 4A]. EXSr.OISES. 1. •tf — 3" -'•II 11 '■t I 0_ h^-h' 4.1- 5. What is the difference bci^^een | and ^S ? 6. What is the difference between ^fg and -{'^^ ? 7. Whnt is the u'fferencc between §2 and -f^ ? 8. What is the ex'iess of 20i above 9^| ? _in — ■24 — Ans. 1 11' 17 Ans. J. Ans. ^Vns. lOj^g, DENOMINATE FRACTIONS. 0. From 5-^ take 3 5 ? 10. What in tho difference bi'twccn .5,''- 11. What H the vahio of J-f-;; -^-j /.-^ ? 12. and 6 AVrt ? Ann ''''"' Ans. \JU' Ans. iJ, What is the difference between 100/„ and 60^^ ? Ans. 49441. 13. What is the difference between ^ of J and J of J ? Ans. 14. What is the difference between | of ^\ and ^ of J ? Ans. 15. What is tho value of A-f f|— f— §-f j \ ? 3 »i nan* Aus. ^. VIII.-DENOMINATE FRACTIONS. Hitherto we have treated of fractions abstractly, and we must now apply the principles laid down to denominate numbers, and show how a fraction may be transformed from one denomination to another of the same kind e. g., how a fraction of a shilling umy bo expressed as a fraction of a pound, and vice versa. RULE. (1.) Reduce the given quantity to the lowest denominationwhich it exjirtisscs. ( 2.) Reduce the unit in the terms of which it is to be expressed tit the same denomination, and (3,) niake the former the numerator and the latter the denominator, and the fraction loill be expressed in the required terms. EXAMPLES. 1. To express 16s. 8d. as a fraction of £1 : Kcduoing 16s. 8d. to pence, wc get 200, and reducing £1 to pence, we get 240, and, therefore, 16s. 8d. is £5.^0, which, in its lowest terms, is £g. 2. In like manner, to express 17s. 6d. as a fraction of £1, wo reduce 17s. 6d. to pence, and find 210, which, divided by 240, the number of pence in £1, gives £;]{}]— £^. 3. So also, 12s. 6d., expressed as a fraction of £1, is £|. 4. 18s. 4d., expressed as a fraction of £1, is £\\. 5. 12s. 6d., expressed as a fraction of £1, is £|. 6. 138. 4d., expressed its a fraction of £1, is £§. EXERCISES. 1. Exorcss 3s. 9d. as a fraction of £1. 2. Express 4s. 4d. as a fmction of £1. 3. Express 4^d. as a fraction of Is, 4. Express 1 oz. troy as a fraction of 1 1). Ans. £y='g. Ans. m 8. Ans. ^ Ans. j*^. 70 AfilTHMETIC. AnB. I cwt. Ans. ;,'„ ton. ! S owt. 5. ExprcHH 40 lbs. as a fraction of 1 cwt 6. KxfTCHH 50 lbs. EH a fraction of 1 ton. 7. Express 72 lbs. as a fr^iction of 1 twt. Ans, 8. A day is 23 hours, 56 minutes, 48 seconds, nearly ; what fraction of this will 7 hours bo ? Ans. /'A'*:,. 9. Express 95 stiuarc yards as v. fraction of an acre. Ans. g'^^g. 10. Express 14 yards as a fraction of a mile. Ans. gg^. 11. What fraction of a year (365^^ days) is one month (30 days ?) Ans 12. Express 100 yards as a fraction of a mile. Ans. 13. Express 45 cents as a fraction of a doUar. Ans. r^Q. 14. Express 60 lbs. as a fraction of a cwt. Ans. g. 15. A man has an income of $3610 a year and saves 5 of it ; how much does he spend ? Ans. $2062:^. To find the value of a fraction in the denominations which the integer contains, reduce the numerator to the next lower denomi- nation, and divide the result by the denominator ; if there bo a re- mainder, reduce to the next denomination, and divide again, and continue the same operation till there is either no remainder, or down to the lowest denomination by which the integer is counted. Thus, ^ of £1 is 120 shillings divided by 7, which gives 17 shillings and 1 shi.ling, or 12 pence reuiamder, and 12-r-7=l§, so that £§:r.rl7 shillings and l-^d. EXERCISES. 1. What is the value of £^2 Stg. ? 2. What is the value of /g of a yard ? 3. What is the value of i,J of a mile ? Ans. 4 fur., 13 rods, 1 yd., 2 ft., 6 in. 4. What is the value of Vg of a shilling Stg. ? Ans. llfd. 5. What is the value of j of a ton? Ans. 11 cwt., 1 qr., 1715 lbs. 6. What is the value of ^ lb. troy ? Ans. 8 oz. 7. What is the value of {'.^ of a shilling? Ans. 5Y''3d. 8. What is the value of $?, ? Ans. 8S\j cts. 9. What is the value of J of $G ? Ans. $4.80. 10. What is tlic value of 1 1 of $8 ? -t..ns. $6.80. To change a fraction to one of a lower denomination, reduce the numerator to that denomination and divide by the denominator. Thus, to express £y,f ^ as a fraction o^' shilling, reduce £i ta shil- Ijnrs, which gives 140 shillings, ani< :.i'^ - >§ of a slrllin;;- i' oH- Ans. lis. 8d. Ans. 2 ft., 8| in. I 4. hi DECIMAL FRACnONS, n inn p. X F. R n H B K . of a foot itH a fraction of an inch. £zprcHs , .^ ^ of a cwt. as a fraction of a lb. 91 1. Express 2. 3. Express 7„ of a lb. an n fraction of an or. 4. Express ^ of ,\ of n yard as n fraction of a foot. 5. Express J^^ oi' a roil as a fraction of a yard. (). Express ^ of ^ of an acre as a fraction of a rood. 7. Reduce /,, cwt. to the fraction of a pound. Ans, 8. Reduce .^', of a day to the fraction of a minute. Ads. 68;} min. 9. What part of a tiocond is tlio ont>iniUionth part of a day ? Ans. ,j'j*s sec. 10. Reduce £.,'ij to the fraction of a penny. Ans. H^d. 11. Reduce v,\ of a pound avoirdupois to tho fraction of an oz. Ans. Am. Ans. j(. Ann. {^. Ans. l'^. Ans. \. lb. "/a Ans. oz. The reducing of a donominato fraction from ono of a lotrer to one of a higher denomination being tho converse of tho last rule, \re must perform tho same operation on tho denominator as was there performed on the numerator. Thus, «d. is £3^5, for £t -^11)20 — ^Bl' 'SXiaXJO EXERCISES. 1 . What part of 1 lb. troy is ^ of a grain ? Z. V^hat part of 4 days ip | ol a minute ? 3. What part of 5 bushels is | of | of a pint? 4. What part of a rod is 3j of ~^'■., of an inch? 5. What part of 2 weeks is /^ vf j. day? Ans. 78^75- Ans. gj^. Ans. gg,. Ans. DECIMAL FRACTIONS. 16, — We have seen already (Art. 3,) that every figure to the right is one-tenth the value it would have if removed our place ; *■ left. Thus, resuming our former example, 8 standing alone means 8 units, but if wc place another 8 after it, thus 88, h nov? means 8 tens, so that tho last 8 is one-tenth of tho first. Now, since the 8 to the right expresses units, another 8 placed to tho right will express eight-tenths of the same unit, and another subjoined will express jgg of tho unit. Thus wo sec that the decimal notation ia directly an extension of the Arabic. Hence arose the convenient mode of writing 8/g in the form 8.7, by which is indicated that all r^K5: 72 iVRmiMETIC. the fi^ureH before the decimal jw>int. (.) repiPHont intcpfors, and all allUjr it fractions, each being one-tenth of what it would Ihj if one plaiso further to the left. Therefore 888.888 \n eight hundred*, eight tenSf eight units, — eight-tcnth«, eight, onc-hiindredthit, and eight one-thousandths; or, t\'\' iunt recufH, and hy purHuin}; tho dlviHion. wu Hhould And H45 roourring without ond. When nil tho fifnircs recur, the fraction is called a pure perindio dooi- inul; when only some of thoiu roour, it is called uiixod, and the term rapcator in applied when only one figure recurs, oh J— .1111, &c.r^.i or /.- 58333, &c.=^ . 583. Since tho denominator ia olwayH 10, or a power of 10, and Hinco 10 Iiuh no factors but 2 and 5, and therefore powers of 10 no fuotorn but 2 and 5, or powers of these, it follows that no deci- mal will terminate except tho denomina- tor be expressed by either or both of these, or some power or product of them. Hence all terminating decimals are deri- ved from common fractions having for denominator some figure of tho series 2, 4, 8, 16, 32, &o., or 5, 25, 125, &c., or, 10, 20, 40, 60, 4111 33300)41 110(.12345. 33300 78100 G(>G00 115000 99900 151000 138200 178000 165500 11500 GO, 80, 100, &c. J EXSROISEB. 1. lleduco tho common fraction ^ to a decimal. Ans. .25. 2. Reduce the common fraction ^ to a decimal. Ans. .5. •i- Reduce the eounuon fraction '^ tu a decimal. Ana. .75. 4. Reduce tho common fraction ^ to a decimal. Ans. .3. 5. Reduce tho common fraction ,', to a decimal. Ans. .1. G. Reduce the common fraction ^ to a decimal. Ans. .125. 7. Reduce the common fraction ^ to a decimal. Ans. .10. 8. Reduce the common fraction ;^ to a decimal. Ans. .142857. 9. Reduce the common fraction 1 to a decimal. Ans. .2. 10. Reduce the common fraction ^j, to a decimal. Ans. .1. 11. Reduce the common fraction y,- to a decimal. Ans. .09. 12. Reduce the common fraction -,'^ to a decimal. Ans. .083. 13. Reduce the common fraction § to a decimal. Ans. .G. 14. Reduce the common fraction ^ to a decimal. Ans. .8. 1 PKriilAL FRACTIONS. It 15. Reduce the ooinmon fraction ^ to n Jcoim&l. IG. Kcducu the uoninton fraction ^ to n decimal. 17. lltiducc the comuiou fraction i; to u decimal. IH. Iloduoo the comuiun fruotioa I Ut u decimal, li). Keduco the common fraction }, to ii decimal. 2U. llodueo the eommun fraction <; to a decimal. 21. Ueducu the common fraction j y to » decimal. 22. Keduco the common fraction j \ to ii decimal. 2.'t. Uediicc the common fraction | j to a decimal. 21. llcducc the common fraction { { to a decimal. 25. Keduco the common fraction | ^ to u decimal 20. Keduce the common fraction 0(1 Ana. .7867142. Ans. .6875. to a decimal. Ann. 075. 27. Keduco the common fraction ^! to a decimal. AnH. .34375. 2S. Keduco the common fraction j^'^r, to >^ decimal. Ans. .004^7804. 29. Keduce the coiumou fractioD if^ to a decimal. Ans. .4083544303797. 30. Keduco the common fraction ^Jt; to a decimal. Ans. .0044. 31. Keduco tho common fraction ^^^ to a decimal. Ans. .020408103205300122448979591830734093877551. 32. Express g'g decimally. Ans. .()i. 33. Express ^.^g decimally. • ^^g (jyj 34. Express ^J^ decimally. Ans. .OOOi. 35. Express -.-^Jjup decimally. Ans. .00059994. To reduce a denominate numbet to the form of a decimal frac- tion, reduce it to the lowest denomination xohich it contains ; reduce the integral unit to the same denomivntion, and divide thefomier hy tJte latter. Thus, to express 18s. 4d. as a decimal of £1, we must reduce it to pence, the lowest denomination given, and divide it by 240, tho number of pence in £1, which gives the fraction 5|{{rn:;4j=::;|^, and this reduced to a decimal, gives .710 or £.910. In like maitncr 15s. lO^d. is reduced to half-pence, viz., 381, and the half-p' i.ce ia £1 are 480, and |§o=|igo, which expressed decimally is .79375. 76 ABITHMETIC. » EXKROISEH. 1. What decimal of £1 is lis. 4^d. ? Ana. .56875. 2. Express 158. 9|d. as a decimal of £1. Ans. .790625. 3. What decimal of a square mile is an aero ? Ans. .0015625. 4. Express 1 pound troy as a decimal of 1 pound, avoirdu- pois.* 5. Reduce 17 owt. to the decimal of a ton. 6. Express j| of a cwt. as a decimal of a ton oz. ll-r-16=^.6875 lbs. 22.6875^25=.9075 2.9075-i-4=.726875 cwt 1 l"726875-T-20=,58634375 Ans. .82285714. Ans. .85. Ans. .046875. 16)11 ■'ic«,. £n 5)22.6875 . 4)2.9075 The operation annexed is often convenient in practice. To reduce 11 cvrt., 2 qrs., 22 lbs., 11 oz., to the decimal of a ton. First, we divide the 11 oz. by 10, the num- ber of oz. in 1 lb., and then annex the 22 lbs., and divide by 25, the lbs la a qr., and so on. The first form of the work is best suited for illustration, the second is neater in practice. The principle is the same as that implied in the general rulo given above, e 20)11.726875 / ' is ^ rt ■ .58634375 ADDITIONAL EXERCISES. 7. Reduce 10 drams to the decimal of 1 lb. Ans. .0390625. 8. Reduce 1 1 dwt. to the decimal of 1 lb. Ans. .04583. 9. Express 1 oz., avoirdupois, sis a fraction of 1 oz., troy, (see note.) -;.. .V . .. Ans. .9114583. 10. Reduce 5 hours, 48 minutes, 49.7 seconds to the decimal of a day. Ans. .2422419. * A caution seems necessary here, for since the pound (troy,) contains 12 ounces, and the pound (avoirdupois,) It!, the natural conclusion would be that the pound (troy) is );§ or | of the pound avoir dupois. This is not correct, for the ounce troy ex- ceeds the ounce avoirdupois by 42^ grains, though the pound avoirdupois (7000 grs.) «'*xcef-ds the pound Troy (5740 gia.) by 1240 (|[raius. Tliis will be manifest frcm the operation on the ma^'gin, where the standard weights according to Act of Parliament, dating from A. D. 182G. are given. 5760-4-12=480 7000-~16r:=:4371 dificreace. . 42^ I DKOIMAL FRACTIONS. 77 II. — RlBVCTION OF DkOIMALS TO COMMON FaAOTIONS.— To find the oommon fHotion corresponding to any given decimal. — This involves three oases according an the fraction in a terminattng decimal, a pure circulating decimal, or a mixed ciroulating decimal. The first case scarcely requires proof. We give it, however, in order to assist those unaccustomed to the algebraic notation, to under- stand more dearly the form of illustration used in the other cases. Let us take the fraction .9375, and use d for decimal. We now write d=.9375, and multiplying both terms by 10000, we obtain 10000 d=9376, and therefore d— yVoVo» ^^i^h reduced to its low- est terms is ||, the common fraction required. This is simply put- ting for denominator 1, followed by a cipher for each figure in the decimal. ,> V ■ ftndJ+4+§.». Ans. 2.34613 +. 22. What is the difference, according to the decimal notation, between | and ^^ true to six places of decimals ? Ans. .636363. MULTIPJ.ICATION OP DECIMAIiJ. 81 23. What is the difference between ^ and 4 ezpnwsed deoimallj tnic to six decimal places ? Ans. .071428. 24. What is the difference between the yulgar fractions oorres- ponding to .49 and .5 ? Ans. 0. 25. Find the value of .786426+.975324-i-.176009+.32+ .62519375— 3.282951 75+. 4, Ans.©. 26. What is the difference between 138.6012, and 128.8512 ? ■'•■---■■--^'■=- ■■':-'^' -''■■-:-■ •-■■■ '^'V'.: Ans. 9.75 27. What is the excess of 31.6322 above 5.674-j-1.83+.3125-|- 18.62+4.3+.395— ,5. Ans. 1.0007. 28. What is the excess, expressed decimally, of 5.83 above 4||. Ans. 1.6682. 39. What is the difference between 8.375 and 7^ true to six decimal places ? Ans. .946428. 30. What is the value of 601.050725— 441.001— .0062F 3.818476— 156.1 +.125. Ana. .25. v.- MULTIPLICATION OF DECIMALS. If wc multiply a decimal by a whole number, the process is pre- cisely the same as if the multiplicand were a whole number, but care must be taken to keep the decimal point in the same relative position. Thus, in the annexed example, as there are three decimal places in the multiplicand, we make three also in the product. If we have to multiply a whole number by a decimal, we must mark off a deci- mal in the product for each decimal in the multiplier. — The rca on of this v/ill be manifest from the considera- tion that if wc multiply 8 units by .6. or ,"(,, we get Jg, or 4.8, i. c, 4 units and S-tenths ; and again, when we multiply, 7 tens by .(> or f^,, wc get -^,>y^=r42 units, which with the 4 units already obtained, j^iakc 46 units, and wc now have arrived at whole numbers. The same illustration will apply to multiplying by .66, which laquires two de- eirial places to be laid oft' from the right. Therefore, for e\ery de- ci.nal place in the multiplier one must be cut off in the product, and we saw ahready that for every decimal plac:; in the multiplicand, a dooi* 6.678 6 34.068 5678 .6 3406.8 I oB ABirmCETIG. . mal place must be cut off in the product, and therefore wo conclude that for every decimal place in both factors, a decimal place must bo marked in the product. It may be well to vary the illuHtration by observing thit as the tenth of a tenth is a one-hundredth, tenths mult' • "^ ^\ tenths give hundredths ; so also the product of tenths i:*r..h AS is thousandths, and so on. Thus : .2 or ,",j, multi- ! ,i or /(,, is jgg. Now, ,G would not represent this, for thai M^rould mean j% ; hence, it is necessary to prefix a cipher, and write .06, and this agrees with what has been already noted (Art. 3) regarding whole numbers, viz., that we arc compelled by the nature of the notation to introduce a zero character, and in the present in- stance the cipher means that there are no tenths, just as it indicated in the case referred to that there were no tens. So, also, ^j'ljg would Iw. written decimally .006, which would mean that there arc no tenth:;, no hundredths, but 6 thousandths. From these explanations wo deduce the I ■ K u L E : Multiply, us ill wlwle numbers, and cut off from, the right a deci- mal place for every one in both vmltiplier and multiplicand. EXAMPLES. Multiply .78 by .12. Here we multiply as if the quantities were whole numbers, and in the product point off a decimal figure for each one in both multiplier and multiplicand. In (1.) .78 ^^- 1? tbc number of ligurcs in the product is the same as the number in both factors, and therefore we have no whole number in the result, but four decimal places. In Ex. 2 there arc four decimal places in the factors, and there arc six figures in the product, and consequently two figures represent whole num- bers. In Ex. 3, wh'jn we multiply 6 by 3, we obtaiu 18, but if wc had (2.) .674 (3.) 4.56 carried the repitend out one place far- ther wc should have had 5 to be mul- tiplied by 3, and consequently 1 to carry, so avc add 1 to the 18, and in like manner wc must allow 2 when multiplying by 4, and 1 when multi- plying by 2. .78 .42 156 312 .3276 .674 34.6 4044 2696 2022 23.3204 4.56 2.43 136'J 1826 913 11.0929 :«JLTIPUCATION OP DECUCALS. 88 ^ EXEKOISKS. 1. Multiply 7.49 by 63.1. Ana. 472.619. 2. Multiply .156 by .143. Ans. .022308. 3. Multiply 1.05 by 1.05, and the pioduct by 1.05. AoB. 1.157625. 4. Find the continual prodnot of .2, .2, .2, .2, .2^ .2. Ans. .000064. 5. Multiply .0021 by 21. Am. .0441. 6. Multiply 3.18 by 41.7. Ana. 132.606. 7. Multiply .08 by .036. Ans. .00288. 8. Multiply .13 by .7. Ana. .091. 9. Multiply .31 by .32 Ans. .0992. 10. Find the continual product of 1.2, 3.25, 2.125. Ans. 8.2875. 11. Multiply 11.4 by 1.14. Ans. 12.996. 12. Find the continual product of .1, .1, .1, .1, .1, .1. Ans. .000001. 13. Multiply 1240 by .008. Ans. 9.92. 14. Find the continual product of .101, .011, .1}, 1.1 and 11. Ans. .001478741. 15. Multiply 7.43 by .862 to six places of decimals. Ans. .640839. Hi. Multiply 3.18 by 11.7, and the product by 1000. Ans. 132606. 17. Multiply .144 by .144. Ans. .020736. 18. What is the continual product of 13.825, 5.128 and .001 ? Ans. .0708946. 19. What is the continual product of 4.2, 7.8 and .01 ? Ans. .3276. 20. What is the continual product of .0001, 6.27 and 15.9 ? Ans. .0099693. Contracted Method. — In many instances where long lines of figures arc to be multiplied together, the operation may be very much Hhortencd, and yet sufl&cicnt accuracy attained. Wc may instance what the student will meet with hereafter, calculations in compound interest and annuities, involving sometimes .iiost tedious operations. By the following method the results in such cases may be obtained with great ease, and correct to a very minute fraction: If we are computing dollars and cents, and extend our oaloolation to four M ABirHMETIG. places of decimals, wo arc treating of the one-hundredth part cf a cent, or the ten-thou8andth part of a dollar, a c^uantity bo minute as to become relatively valueless. Hence wc condu'^' that three or four decimal places are sufficient for all ordinary ru/noses. There are cases, indeed, in which it is necessary to carry or' 'he decimals farther, as, for instance, in the case of I jgarithms to be considered hereafter. The principle of the contracted method will be best ex- plained by comparing the two subjoined operations on the some quantities. Let it be required to find the product of 6.35642 and 47.6453, true to four places of decimals: EXTEMDKD OPERATION. 6.35642 47.6453 CONTRACTED OPERATTON. 6.35642 ' 3646.74 19 06926 2542568 317 8210 444949 2542 568 38138 38138 52 2542 444949 4 - 317 2542568 19 2 carried. 'A(\9. aR^R ?i7«'J!« 302.8535 RULE FOR THE CONTRACTED METHOD. Place the units' Jigurc of the lohole number under the last required decimal place of tJie multiplicar d, and the otlicr integral figures to the right of that in an inverted order, and the decimal figures, also in an inverted order, to the left of the integral unit ; multiply by each figure of the inverted multiplier, beginning with the figure of (lie multiplicand immediately above it, omitting all figures to the right, but allowing for ichat would have been carried if the decimal had been carried out one place farther— place the first figure of each partial product in tJie same vertical column, and the others in verti- cal columns to the left ; the sum of these columns will be the required product. • ■ Thus, in the above example, wc are required to find the product correct to four decimal places, therefore we set the units' figure, 7, under the fourth decimal figure, and the tens' figure, 4, to the right, and the decimal figures, 6453, to tho left in reversed order ; then we MULTIFUCATION OF DECIMALB. St multiply the whole lino by 4, anne place each time towards the right in the multiplier, and one place to- wards the Icfl in the multiplicand. Thiu is so different from the ordinary mode of 'ope- ration, as to be excessively awkward -^nd puzzling, and this gave rise to the idea of reversing the order ( the digits. Wc append this remark as most persons cannot at first oight comprehend the reason of the inversion. * Let the learner observe that all the (Igiires of the (irst column are of the name rank, viz.. tcn-lhousandthid, und therefore may he added together, and as the value of er ch figure ]*»> increased or decreased 10 times according to its position to loft or right, it ioUowg that all figures at equal distances from the decimal pout, whether to right or left, are of the same rank, i. e,, unitt^ will be u-.dei .mits, tens under tens, tenths under tenths, hundredths under bundrruths, &c., &c. The contracted method is not of much use in termina- ting decimals which extend to only a iow places, but it saves a vast deal of labour in questions which involve either repetends or terminating decimaln expressed by a long line of decimal figures ,^ 6.35642 47.6453 2542568 . 444949 38138 2542 317 19 2 allowed. 302.8535 86 „f«V!|-i«' ARTrHmnc. I'jv ADDITIONAL KXCRriBBg. 21. Multiply .26736 by .28758 to (our docimol piaccs AoB. .0769. 22. Multiply 7.^85714 by 36.74405 to five decimal placoH. Ans. 267.70665. 23. Multiply 2.656419 by 1.723 to nix decimal placcH. Ans. 4.578?3- 24. What deoimal fraction, true to six places, will exprep^ product of T*^ multiplied by ,\ ? Ans. .11;; 25. What decimal fraction is equivalent to f^XH ? Ans. .4674 26. What is the second power of .841 ? Ans. .707281. 27. What is the product of 1.(35 by 1.48, true to five places ? Ans. 2.45975. 28. Express decimally 2, »,X 5. Ans. 2.393162. 29. What is the product of 73.637i by 8.143? Ans. 599.6272677. 30. .681472 X 01286, true to five places, will give .00876. In the last exercise it must be observed that since there is no whole number, and five decimal places are required, we must place a cipher under the fifth decimal figure, and write .01286 in reversed order. That the result is u sufficiently close approximation will bo evident from the consideration that the last figure 6 is only six one-hundred-thousandths of the unit, and consequently the next figure would bo only one-mil- lionth Tnnri of the unit. .681472 68210.0 681 136 55 I .00876 ;i VI.-DIVISION OF DECIMALS. We have already seen (1) that we cannot perform any operation except the numbers concerned are of the same denomination, or one of them be abstract ; (2) that when a denominate number is used either as a multiplier or a divisor, it ceases to be denominate, and becomes abstract, and (3) that the rules for addition, subtraction, multiplication and division of integers apply equally to decimals, the only additional requirement being the placing and moving of the decimal point. DIVISION Of DECIMALS. ft? Suppoue tncD we arc required to divide 1.2321 by 11-1, we ninst by (1) bring both quaotiticH to the fame denomination. Now the dividend in carried down t<» tvnthousandtt lor 1.2321— 1-f i^V^)^' and therefore wo czprcsH 11.1 in the oorrcspondiog form, ton-thoa- sandths or 11-t I'oVo'o) "^ HI 000, ho that wo ohango the form, bat not the valua of 11.1, the divisor. Again, by (2) the .1. which originally expretiHcd a ' ■ of some unit, and thtrc/bre was in reality denominate, now bet otiK'!' !' j>, tract as one of tho figures of the given factor of 1.2321, / la ;i < cf which we are to find the otiier factor. Heno: by (3) wc can now divide 1.2321 by ll.lOOO as if both were whole numbers, and this is tho reason for omitting mc decimal point when wc have made tho number of decimal i)lace8 c'- 4% \ «> 98 ABiimnio. 3. Diviile 468.7 by 3.365 to six plaees of dedmals. Aim. 139.809889. 4. EzpreiB deoimally l-^iySg. Ans. 233.3. 6. Bipraw in the dedmal fona | of j^-^f of ^ trae to six places ofdeeimab. Ans. 1.054687. 6. Ditide the whole number 9 by the fraction .008. Ans. 1125. 7. What is the quotient of 5.09 by 6.2 ? Ans. .81 nearly. 8. Divide .54439 by 7777. Ans. .00007. 9. What decimal is obtained by dividing 1 by 10.473654 ? Ans. .09547766. 10. What is the differonoe between %-i-^ and {-s-jf in the deci- mal form? Ans. .2458^. OONTBAGTID UXTHODi The work may often be much abbreviated in the manner ox^ubi- ted by the following example : .14736).23748 14736 9012 8841 170 147 23 14 (1.611 14736)23748(1.611 • • • 14736 40 36 040 736 9012 8842 170 147 O Uv 23 15 8 Here it is required to divide .23748 by .14736. Since both divisor and dividend contain the same number of decimal places, no alteration is needed, and so we can at once reject the decimal point, and divide as in whole numbers. The principle of the contraction is simply what has been already explained, viz., that all we look for in such calculations is a sufficiently close approximation, by which we mean an approximation sufficient for all practical purposes. For this reason, when wo have obtained the int^;:! part of tho quotient, we may omit one figure of the divisor in succession after each opera- tion, as the value of each fisrure decreases in a tenfold d^ree as we descend towards the right, and after three decimal figures the error, DIVISION OF fiCCIMAIS. or ddksit ntlier, beoomes only thousandths, iHiidi ure ray nnly worth taloDg into Mooant. For example, if the ealonlation regards dollars and eents, the error at the fourth decimal place would be only the on»4koutcmdth part of a cent. R DLK. Arrange the /raetUmt a$ in theordinarjf mode; find the fint figwrt of tite quotient and the Jirtt remainder ; then, instead of anmexing a periodic figure or a cipher ^ cutoff the right hand figure of the divieory and use the remaining figurea to find the next figure of the quotient, and to on. It is usual to mark the figures as they are sucoessiroly out ollf by placing a point below each. In multiplying by each figure of the quotient, allowance must be made for wliat would have been carried from the figure of the divisor last cut off, had it been used in the division. The vertical line drawn through the ordinary form shows how closely the two modes correspond. As has already been remarked, it is desirable, in order to secure accuiacy, to carry the figures of repetends to one or two places more than are required. XXIROISKS. (1.) 43232323)73640000(170.3355. 43232323 30407677 30262626 145051 129697 (2.) 54637)43682(.7995 • • • • 38246 15354 12970 2384 2162 222 216 6 S436 4917 619 491 27 1 Divide 73.64 by .432, and .43682 by .54637 to 4 decimal places each. To show that theie will be three integral places in the 90 ABTTHMETIO. quotieDt of Ex. 1, we must conuder that them are two pUMwa of whole namben in the dividead and none in the divisor, and, therefore, if we divide 73 and 6, the firat dcoimal place of the dividend by .4, the first figure of the divisor, we get three integral places. Henoe, since we are to have four deoimai places, we shall have seven figures in all. This contraction is extremely useful when there are many decimal pUoes. 3. Find the quotient of 8.6134-»-7.3524 to four decimal places. Ans. 1.1716. 4. Divide M by 13.543516 to five decimal places. Ans. .04549. 5. Divide .58 by 77.482 to five decimal places. Ans. .00756. G. Divide .812.54567 by 7.34 to three decimal places. Ans. 110.649. 7. Divide 1 by 10.473654 to six decimal places. Ans. .09547. 8. Divide 7.126491 by .531 to six decimal places. Ans. 13.420887. 9. Divide 1.77975 by the whole number 25425. Ans. .00007. 10. Divide to eight places .879454 by .897. Ans. .98043924. Vn.-DENOMINAT?J DECIMALS. To express one denominate numher as a frc.ctym of anotlier of the same leind, reduce both to t/ie lowest denomination contained in eitlier, make the former the numerator and the latter the denomi' nator of a common fraction, an luce the fraction so found to a decimal ill th tities be 48, 96, and 132, written thus : 48 : 96 : : 132 : , the required quantity. Now. 132x96=12672, the product of the means are therefore equal to the product of the extremes. We have, therefore, a product, 12672, and one of its factors, 48 , hence, dividing this product by the given factor, we find the other factor to be 264, which is therefore the fourth proportional, or fourth term of the proportion, and we can now write the whole analogy, thus: — 48 : 96 : : 132 : 264. To prove the correctness of the operation, multiply 264 by 48, and 12672 is obtained, the same as before. Hence, 94 AniTHMSTia. T HI RULX. Divide the product of the aecond and third termiby ihejint, and the qnoHent will be the required fowrth term. To show the order in which the three given quantities are to be amai^, let it be required to find how much 730 yards of linen will oott At the rate of $30 for 50 yards. It is plain that the answer, or fotrth term, must be dollars, for it is a price that is required, and in order that the tliird term may have a ratio to the fourth, the $30 must be thfl tbird term. Again, since 730 yds. will cost more than 60 yds., the fourth term will be greater than the third, and therefore the second must be grlsisr than the first, and therefore the statement is So : ^30: :30 : 4th proportional, and by the rule i^i^^^Q^^iJ^ =438, the fourth term, and we can now write the whole analogy, 60 yds :730 yds:: $30: $438. This may be called the ascending scale, for the second is greater than the first, and the fourth greater than the third. If the ques- tion had been to find what 60 yards of linen will cost at the rate of '$438 for 730 yards, we still find that the answer will be dollars, and that therefore, as before, dollars must be in the third place, but we see that the answer will now be less than 438, as 50 yards, of which the price is required, will cost much less than 730 yards, of which the price is given, and that therefore the second term must be less than the first. Hence the statement is 730 yds : 50 yds : : $438 : F. P., and by the rule 4J^^4ii=30, the fourth proportional. We now have the full analogy 730 yds : 50 yds: : $438 : $30. As the second is less than the firdt, and the fourth less than the third, this may be called the descending scale. If the first should turn out to be eqvial to the second, and therefore the third equal to the fourth, we should say that the quantities were to each other in the ratio of eqmlity. RULE FOR THS ORDER OF THK TXRMS. If the gwttion implies that the consequent of the second ratio mmt be greater than the antecedent, make the greater term of the first raiio the consequent, and the less the antecedent, and vice versa. The questions hitherto considered belong to what is called Direct Proportion, to distinguish it from another kind called Inverse Pro- portion; because, in the former, the greater the number given, the less will be the corresponding number required, and vice versa; RATIO AMD PROPORTION. 95 whereas, in the Utter, the greater the niimher ghren, the lew will be the number required, and vice vena. To illustrate thu, let it be required to find how long a stack of hay will feed 12 hones, if it will feed 9 horses for 20 weeks. Here the answer required is time, tt/xA therefore 20 weeks wil* be the antecedent of the second ratio ; but the greater the number of horses, the shorter time will the hay last, and therefore the fourth term will be less than the third, and there- fore the statement will not be 9 : 12, but the reverse, 12 : 9 ; and henoe the name Invirsb, because the term 9, for which the time (20 weeks,) is given, and which therefore we should expect to be in the first place, has to be put in the second ; and the rm 12, for which the time is required, and which therefore we should expect to be in the second place, has to bo put in the first, and thus the whole ana* logy is 12: 9:: 20: 15.* The principal changes that may be made in the order of the terms, will be more readily and clearly understood by thj subjoined scheme, than by any explanation in words : : 12 : 9 for 8 X 9=72=6 X 12. 9 for 8X9=72=6X12. : 12 for 6X12=72=8X9. : 6::12+9: 9 or 14: 6:: 21 : 9 for -6 : : 12 : 12—9 or 8 : 2 : : 12 : 3 for Let us take Original Analogy : 8 : 6 : Alternately: 8: 12:: 6: By Inversion : 6 : 8 : : 9 By Composition : 8+6 14X9=126=6X21. By Division : 8—6 : 6 : : 12—9 : 9 or 2 : 6 : : 3 : 9 for 2X9 18=6X3. By Conversion : 8 : 8 8X3=24=2X12. Simple transposition is otte^^. of the greatest use an easy practical example. In calcula- ting what power will balance a given weight, when the arms of the lever are known, let P be the power, W the weight, A the arm of power, and B the arm of weight. The rule is, that the power and weight are inversely as the arms. This solves all the four possible oases by tran^sition. * InverBe ratio ia aonietuneH spoken of, but in reality there is no suc> iblog. It ia true that Inverse Proportion requires tlie terms of one of Ui« ratios to be inverted, bat that ia a matter of analogy, notofTatio, for we hnvi seen already tliat 7-i-21 expresses tlie verv same relation as 21-!-7. — (See in- 96 ABITBUZnO. A : B : : W : P, giTM the pow«r when tho othen «re known, B : A : : P : W givM the weight when the others are known, W : P : : A : B gives the arm of weight when the others are known, P : W : : B : A gives the arm of power when the others are known. Tho work may often be contracted in the following manner : — Resuming our example 48 : 96 : : 132 : fourth proportional, we see that 96 is double of 48, and therefore the ratio of 48 to 96 is the same as that of any two numbers, tho second of which b double the first, and 48 : 96 is tho same as 1 : 2, and we reduce the analogy to the simple form of I : 2 : : 132 : 4th prop., and we have xif Xl=264, the term required, as before. In the example DO : 730 : : 30 : 4th term, we have iAogafi=:2Ji^=2JUyiXA=73x 6=438. This is equivalent to dividing the first and second by 10, and the first and third by 5. Henco we may divide the first and second, or first and third by any number that will measure both. The same principle will also be illustrated by the consideration that the second and third are multipliers, and the first a divisor ; and if we first multiply, and then divide by the same quantity, the one operation will manifestly neutralise the other. Thus : 48 : 96 : : 132 : F. P. may be written 1X48 : 2X48: : 132 : F. P. ; where it is plain that since by first multiplying 132 by 48, and then dividing by the same, the one operation would neutralize the other, both may be omitted. In proportion, when the means are equal, such as 4 : 12 : : 12 : 36, it is usual to write the analogy thus — 4 : 12 : 36, and 12 is called a mean proportional between 4 and 36. To prove the correctness of this statement, we multiply 36 by 4 and 12 by itself, and as both give 144, the analogy is correct. Now, as 144 is the square or second power of 12, so 12 is called the second root, or square root of 144, or that which produced it, or the root from which it grew ; hence, to find a mean proportional between two ^ven quantities, we have, the following RULE. Muittply them together, and take the tquare root of the product. Thns, in the above example, 4X36=144, the square root of which is 12. Again, to find a mean proportional between 9 and 49, we mul- troductory remarks.) The term Beciprocal Batio i» liable to the Bame objec- tion, for tiiough 3 and ^ are reciprocals, yet they express the same relatiOQ. When the expresdon Inverse Ralio is legithnately used, it does not refer to a dngU ratio, but means that two ratios are so related that one of them must be Inverted. RAnO AND PROPORTION. if Uply 49 by U, whieh is 441, the wioare root of whieh u 21, whieh is a mefto proportional between 9 and 49, i. e., 9 : 21 : 49, or, wrii* ton at full length, 9 : 21 : : 21 : 49. Proof: 49X9=441 and 21x21=441. As the learner is not supposed, at this stage, to know tho method of finding the mots of quantities beyond the limita of the multiplication table, we append a tablo of squares and roota ai the end of the book. When each quantity in a scries is a mean proportional between two adjacent quantities, the quantities arc said to be continued, or continual proportionals. Thus : 2 : 4 : 8 : 16 : 32 : 64 : 128, and 3 : 9 : 27 : 81 : 243, are scries in which each is a mean pro- portional between two adjacent ones. Let us take 16 aiid the two adjacent ones, 8 and 32— tho analogy is 8 : 16 : : 16 : 32. Proof: , 8X32=256, and 16x16=256. So also, 27 and the adjacent terms, 9 and 81. The analogy is 9 : 27 : : 27 : 81, and the proof, 9X81= 729, and 27x27=729. This subject will bo tnwted of at length in a subsequent part of tho work, but this explanation has been introduced here to fill up the outline and let Uie learner understand tho nature of continued proportionals. KXKROIHBB. 1 . If 6 barrels of flour cost $32, what will 75 barrels oost ? •■f Ans. 1400. « 2. If 18 yards of cloth cost $21, what must be paid for 12 yards ? Ans. $14. 3. How much must be paid for 16 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 oost, 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 § of a flock, how many are there in tho same ? Ans. 24. 9. What must be paid for 4^ cords of wood, if the cost of 3 cords i8$10? Ans. $15. 96 ;c. 10. What ii the height of a troe wbioh oaata a ahadow of 121 (Wl,ifaatake6feethighprodao«iaahadowof8fNl? Am. ddf . 11. How long will it take a train to nin from Toronto to Has* ilton (a diatanoe of 39 mika), at tho rate of ft milea in lft/| minataa ? Ana. 2 boon. 12. If 15 nwn can build a bridge in 10 dayi, bow many men will bo required to eroot three of the aame dimenaiona in ^ the timo ? Ana. 90. 13. If a man reoeive 14.50 for 3 daya' work, bow many days ought be to remain in his plaoo for £6 4i. M. ? Ana. 16|. 14. How mnch may u person spend in 94 days if ho wiaboa tu save 18 guineas out of a salary of $600 per annum ? Ana. $109,297+. 15. If 3 owt. 3 qrs. 14 lbs. of sugar cost 186.50, what will 2 qni. 2 lbs. oost ? Ans. $4,879+. 16. 5 men arc oniploycd to do a piece of work in 5 days, but after working 4 dayH they find it impossible to complete the job in less than 3 daya more, how many additional men must be employed to do the work in tho time agreed upon at first ? Ans. 10. 17. A watoh 18 10 minutes too fast at 12 o'clock (noon) on Mon- day, and it gains 3 minutes 10 seconds a day, what will bo the timo by the watch at a quarter past 10 o'clock, A. M., on tho following Saturday ? Ans. 10 h. 40m. 36/^(1. 18. A bankrupt owes $072, und his property, amounting to $607.50, is distributed among his creditors ; what does one receive whose demand is $11.33^ ? Ans. $7,083+. 19. What is the value of .15 of a hhd. of lime, at $2.39 per hbd. ? Ans. $.3585. 20. A garrison of 1200 men has provisions for f of a year, at tho rate of | of a pound per day ; how long will the provisiona last at tho same allowance if tho garrison be reinforced by 400 men ? Ans. 6| months. 21. If a piece of land 40 rods in length and 4 in breadth make an acre, how long must it bo when it is 5 rods 5^ feet wide ? Ans. 30 rods. ' 22. A borrowed of B £175 58. for 102 days, and at^rwards would return the favor by lending B £210 6s. ; for how long should he lend it ? Ans. 85 days. 23. If a man can walk 300 miles in 6 suooessiye days, how many miles has he to walk at tho ond of 5 days ? Ans. 50. RATIO AND PROPOBTION. 24. If 49ft gyioni of wioe owt $394 , bow uuoh will $73 f^j for? Ads. 90 gal. 25. If 1 12 huftd of cftttlo coniume a otrtain qaantity of hay in days ; how long will tho sauio quantity last 84 head ? Ani. 12 days. 2ti. If 171 men can build a hooso in 168 dnyn ; in what tim* will 108 men build u similar house ? Ans. 266 dayi. 27. It has boon proved that tho diamoter of every circle is to the oiroumferenco an 113 : 35ft ; what then is the eircumforenoo of the moon's orbit, tho diameter being, in round numbers, 480,000 miles ? Ans. 1,607,964 ,","4 ra. 28. A round table is 12 ft. in circumference ; what is its diameter ? \wi. '6 (l. 'Jjll in. 29. A was sent with a warrant ; after ho had ridden Oft mUee, B was sent after him to stop the execution, and for every 16 miles that A rode, B rode 21 ; How far had each ridden when B overtook A ? Ans. 273 miles. 30. Find a fourth proportional to 1, 19 and 92. Ans. 209. 31. A detective chased a culprit for 200 miles, travelling at the rate of 8 miles an hour, but the culprit had a start of 7ft miles ; at what rate did the latter travel ? Ans. ft miles an hour. 32. How much rum may be bought for $119.50, if 111 gallons cost $89.62ft ? Ans. 148 gallons. 33. If 110 yards of cloth cost $18 ; what will $63 pay for ? Ans. 38ft yards. 34. If a man walk from Toronto to Hamilton, a distance of (say) 38 miles in 13 hours, 18 minutes ; in what time will ho walk at the same roto from St. Catherines to London, supposing tho distance to be 102 miles ? Ans. 35 hours, 42 min. 35. A butcher used a false weight 14^ oz., instead of 16 oz. for n pound, of how many lbs. did he defraud a customer who bought 1 12 just lbs. from him ? Ans. 9^ lbs. 36. If 123 yards of muslin cost $20ft ; how much will ftl yards cost ? Ans. $8ft. 37. In a copy of Milton's Paradise Lost, containing 304 pages, tho combat of Michael and Satan commences at the 139th page ; at what page may it be expected to commence in a copy containing 328 pages ? Ans. The fourth proportional is'149g^ ; and hence the passage will commence at the foot of page 150. 38. Suppose a man, by travelling 10 hours a day, perfonus a 100 iBirmannc. 1 1 1 1 jounTey in four weeks without desecrating the Sabbath ; now mftnj weelcs would it take him to perform the same journey, provided he travels only 8 honra per day, and pays no regard to the Sabbath ? Ans. 4 weeks, 2 days. 39. A oubio foot of pure fresh water weighs 1000 oz., avoirdu- pois ; find the weight of a vess^'l of water containing 217^ cubic in. Ans. 7 lbs., 13||| oz. 40. Suppose a certain pasture, in v/hich arc 20 cows, is sufficient to keep them 6 weeks ; how many must be turned out, that the same pasture may keep the rest 6 months ? Ans. 15. 41. A wedge of gold weighing 14 lbs., 3 oz., 8 dwt., is valued at £514 4s. ; what is the value of an ounce ? Ans. £3. 42. A mason was engaged in building a wall, when another came up and asked him how many feet he had laid ; he replied, that the part he had finished bore the same proportion to one league which ,\ does to 87 ; how many feet had he laid ? Ans. 32j^:^^\f. 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 the younger shall be '} the share of the elder; required the shares. Here the ratio of the shares is 4 : 3, and wo have shown that if four magnitudes arc proportionals, the first term increased by the second is to the second as the third increased by the fourth is to the fourth. Now, 97 acres, 3 roods, 5 rods, being the sum of the shares, we must take the sum of 4 and 3 for first term, and either 4 or 3 for the second, and therefore 7 : 4 : : 97 acres, 3 roods, 6 rods : F.P., i. «., the sum of the numbers denoting the ratio of the shares is to one of them as the sum of the shares is to one of them. This gives for the elder brother's share, 55 acres, 3 roods, 20 rods, and the yonnger's share is found cither 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 the largest share ; required the parts ? 23 23 23 \ 5 : : 398 : 86^3, the share of the youngest. 7 : : 398 : 121-/g, the share of the second. 11 : : 398 : 190^^5, the share of the eldest. 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 C's one-half greater than B's ; required the amount of the security of each ? COMPOUND TROPORTION. 101 AuB. A'» Bhiro, $1052.63i»5 ; B's, $1678.94 1 * ; C'», •2368.42,',. 46. Suppose thai one man startu from Montreal, and walks 6 miles an hoar, and another at the same time from Newtonvillc, (5 miles west of Port Hope), at the rate of 5 miles an hour, when will they meet, the whole distance being 286 miles? Ans. y'^, of a mile west of Gananoque, which is 166 miles west of Montreal by the Grand Trunk line. 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 shares $800 ; what are the shares, and what is the whole svn to be divided ? Ans. Less share, $1600 ; greater, $2400 ; total, $4000. 48. A certain number of acres of land are to be divided into two parts, such that the one shall be ^ of the other ; required the parts and the whole, the difference of the parts being 716 acres? Ans. the \ess part 637 acres ; the greater, 1263 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, 160. 50. Pure water consists of two gasses, oxygen and hydrogen ; the hydr(^n is about -^^ of the oxygen ; how many ounces of water will there bo when there are 764|i^ oz. of oxygen Liore than of hydrr^n ? Ans. 1000 oz. COMPOUND PROPORTION. 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 m 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 is but one condition, viz., that 96 men can reap 40 acres in the given time, which implies but one ratio, and when the question haa been stated 40 : 66 : : 96 : P.P., and the required term is found to be 166, and the proportion 40 : 66 : : 96 : 166, we have the propor- tion, expressed by two ratios. But, suppose the question were. If a man, walking 12 hours a day, can accomplish a journey of 260 miles in 9 davs. how manv days would ho require walking at the 102 ABirmCEnc. same rate, 10 hours each day, to travel 400 miles ? Here there are two conditions, viz. : jirtt^ that, in the one cose, ho travels 12 hours a day, and in the other 10 hours ; and, secondly, that tho distances are 250 and 400 miles. The statement, as we shall presently show, would be 10:12 \ •• a»<* ^i^i^® '^« quotient by 6^5,, and multiply tho result by f of ^. Ans. §. 44. I bought S 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. 46. What is the difference between If^'g and 1|3 ? Ans. '|^J. 46. If $|A pay fur a 1^ st. of flour ; for how much will $| pay ? Ans. 1;|'^ St. 47. Mount Blanc, 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 j% of the whole height from the top, i. e., ^\ 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 $37.515625 by selling goods worth $324.53125 ; 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-f 51. Required, 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 ? Ans. Surface, 390| ft. ; each box 42|§| C. F. 52. If I pay $^9 for sawing into three pieces wood that is 4 ft. long ; how much 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 Oshawa, un a journey, and travels at the rate of 20 miles a day ; 4 days after, B sets out from the same place, and travels the same road, at the rate of 25 miles per day ; how many days before B will overtake A ? o Ans. 16. 110 ABiTHmno. 54. A fiurmer haying 56^ tons of hay, sold | of it at flOf par ton, and the remoindor at $9.75 per ton ; how much did he reoeire For his hay? .. Ana. |680|g. 55. If the aum of 87 j^ and 117}l| is divided by their diffdrenoe ; what will be the quotient ? Ana. 6}}f . 56. If 8} yards of silk make a dress, and 9 dresses be made from a piece containing 80 yards j what will be the remnant left ? Ans. 1^ yards. 57. A merchant expended $840 for dry goods, and then had re- maining only IS 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 8| hours a day ? Ans. 12 days. 59. If 15 horses, iu 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. Reduce 1 pound troy, to the fraction of one pound avoirdu- pois. Ans. 141- 61. Reduce to a simple fraction. Ans. ^. 62. What will be tho oostof 8 cwt., 3 qrs., 12} lbs. of beef, if 4 owt. cost $34 ? Ans. $75/,. 63. If 4 men, working 8 hours a day, can do a certain piece of work in 15 days ; how long would it take one man, working 10 hours a day, to do the same piece of work ? Ans. 48 days. 64. Divide $1728 among 17 boys and 15 girls, and give each boy /y as much as a girl ; what sum will each receive ? Ans. Each girl, $66f f ; 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? Ans.8Vy^. 66. If it requires 30 yards of carpeting, which is f of a yard wide, to cover a floor ; how many yards, which is 1} yards wide, will be necessary to cover the same floor ? Ans. 18. 67. A person bought 1000 gallons of spirits for $1500 ; but 140 gallons leaked out ; at what rate per gallon must he sell the remain- der so as to make $200 by hia bargain ? Ans. $2 per gallon. 68. What must be the breadth of a piece of land whose length is 40} yards, in order that it may be twice as great as another piece of ANALTSD AND STItTHESIR. Ill L40 of land whose length ia 14| yards, and whone breadth is 13 j^, yards? Ads. 9| yards. 69. If 7 men can reap a rectangular field whoso length is 1,800 feet, and breadth 060 feet, in 9 days of 12 hours each ; how lon^ will it take 5 men, working 14 hours a day, to reap a field whose length is 800 feet, and breadth 700 feet ? Ana. 3^ days. 70. 124 men dug a trench 110 yards long, 3 feet wide, and 4 feet deep, in 5 days of 11 houra each; another trench was dug by one-half the number of men in 7 days of 9 houra each ; how many feet of water was it capable of holding ? Aua. 2208 cubic feet. 71. If loo men, by working 6 hours each day, can, in 27 days, dig 18 cellars, each 40 feet long, 30 feet wide, and 12 feet deep ; how many cellars, that ore 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. 250. 72. A gentleman left his son a fortune, Jt of which ho spent in 2 months, ^ of tho remainder lasted him 3 months longer, and '^ ol what then remained lasted him 5 months longer, when hu had only $895.50 left; how much did his father leave him ? Ana. $4477.50. 73. A farmer having sheep in two dificrent fields, sold \ of the number from each field, and had only 102 yhcup remaining. Now 12 sheep jumped from the first field into the second ; then the num- ber remaining in the first field, was to tho number iu the second field as 8 to 9 ; how luauy sheep were there in each field at firat ? Ans. 80 in first field ; 5G iu second. 74. A and B paid $120 for 12 acres of pasture for 8 weeks, witli an understanding that A should have the grass that was then on the field, and B what grew during the time they wero grazing ; how many oxen, in equity, can each turn into tho 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 G weeks^ and in similar manner, 5 acres 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 $48. Ans. •j ANALYSIS AND SYNTHESIS. Analysis is the act of separating and comparing all the different parts of any compound, and showins^ their connection with each Other, and thereby exhibiting all its elementary principles. llf ABITBMmO. The oottTone of AnaljiiB ii Sjnthetii. The mMDing and ose of th«M tormi will probtblj be mott roMlily oomprehended bj refovnoo to their derivation. They are both pure Greek words. Aoblysia meaDs loo$ing up. The general reader would here probably expect looting down, m employed in most popular definitions; but wo may illustrate the Greek term, looting up, by our own everyday phrase, tearing up, which meanii rending into threds, the English up conveying thr same idea here as the Greek ana in analysis. The Greek nvr. h .lis means literally placing together ; that is, the component ;iat ' mi.> g known, the word synthesis indicates the act of oombiniu^ ihciu into one. We might give many illustrations, but one 'Vill sutiioe, 'J we choose the one which will be most generally un^li > t d. When we analyse a sentence, wo loose it up, or tear it up, mto its component ports, and by synthesis we write or compose, i. e., put togetiier the ports, which, by analysis, we have fouud 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 tho required quan- tity ; henoe, all our deductions are self-evident, and we therefiuw require no rule to solve a problem by analysis. Although this part of arithmetic is usually called analysis, yet, a9 it in r' ally both analysis and synthesis, we have given it o title in aooordanoe with the principles now laid down. EXAMPLK. 1. If 12 poanda of sugar cost $1.80, what will 7 pounds cost ? SOLUTION. : . -J, • 12)1.80 .16 . 7 $1.06 If 12 lbs. cost $1.80, one pound will cost the tV of $1.80=16 cente. Now, if 1 lb. cost 16 cents, 7 lbs. will cost 7 times 16 oents=to $1.06. Therefor ~ M^. of sugar will cost $1.06, if 12 lbs. cost *?'.0 NoTB.— The work may be ^u.iiu\vfaat Huortened, eapeoially in long quM- ttons, by arraiiging it in the following manner, so as to admit of cancelling, Ifpoisible:- , , , 15 1. IW 105 \%^ 1 ^l 1 Ans. 2. If 6 bushels of pease cost $6.60, for what can yon porobase 19biMhels? Ans. $20.90. ANALYBIH AMD HYNTHEfilH. 118 3. If f> men can perform » oerUin pine* of bbor m 17 dajs, how teii^/ will it uk( .1 men to do it ? Km. 51 days. 4 If many pigi, at $2 each, mu>»t be giw for 7 aheap, worth 14 a beMl 7 Ana. 14. 5. Jf $100 gain $6 in 12 Bonths, how nmoh would it gain in 40 monthi V Aaa920. 6. If 4| boaheJi of iK>pl«" oo^i 9H, what will bo ihr u '>f 7^ boiheb? •J ' 80 LOTION In the first place, 4^ bushela^ -^ bnshels, and |3ji |^ Now, Rtnoe Jj^ bnshelB cost $'^^, one bushel will co«. A owns ^, and B ,'.j of a ship ; A'h part is worth $650 more than B's ; what is the value of tho ship ? Ans. $15,600. 17. A post stands | in tho mud, ^ in the water, and 15 feet above the water ; what is the length of tho post ? Ans. 36 feet. 18. A grocer bought a firkin of batter containing 56 pounds, for $11.20, and sold f of it for $8| ; how much did he get a pound ? Ans. 20 cent8. 19. The head of a fish is 4 feet long, the tail as long as tho head and ^ the length of tho body, and the body is as long as the head and tail ; what is tho length of the fish ? Ans. 32 feet. 20. A and B have the same income ; A saves ^ of his ; B, by spending $65 a year more than A, finds himself $25 in debt at the end of 5 years ; what did B spend each year ? Ans. $425. 21. A can do a certain piece of work in 8 days, and B can do the same in 6 days ; A commenced and worked alone for 3 days, when B assisted him to complete the job ; how long did it take them to finish the work ? SOLUTION. If A can do the work in 8 days, in one day he can do the ^ of it, and if B can do the work in 6 days, in one day he can do the ^ of it, and if they work together, they would do ^-\-^=^-^ of the work in one day. But A works alone for 3 days, and in one day he can do ^ of the work, in 3 days he would do 3 times ^=f of the work, and ns the whole work is equal to § of itself, there would bo ^ — ^=§ of the work yet to bo completed by A and B, who, according to the con- ditions of the question, labour together to finish the work. Now A and B working together for one day can do ,7^ of the entire job, and it will take them as many days to do the balance ^ as ^^ is contain- ed in I, which is equal ^X'^r=T^ days. 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 to do it alone ? Ans. 14| days. 23. A certain pole was 25^ feet high, and during a storm it was broken, when f of what was broken off, equalled f of what remained ; how much was broken off, and how much remnined ? Ans. 12 feet broken off, and 13^ remained. 24. There arc 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. 1 1 ANALT8IB AND 8YNTHE8I8. 116 ■I eeo. 26. A and B start together by G. T. R. from Cobonig to Brook- yille, a distance of 138 miles. A goes by freight train, at the rate of 12 miles per hour, and B by mixed trun, at the rate of 18 miles per hour. C leaves BrookviUo for Coboorg at the same time by express train, which runs at the rate of 22 miles per hour ; how far ftom Coboorg will A and B each be when C meets them ? Ans. A 48j| miles ; B 62^^ miles. "'' 26. A cistern has two pipes, one will fill it in 48 minutes, and the other will empty it in 72 minutes ; what time will it require to fill the cistern when both are running? Ans. 2 hours, 24 min. 27. If a man spends /^ of ^ ^^^ i° working, ^ in sleeeping, j'^ in eating, and 1^ hours each day in reading ; how much time will be left ? Ans. 3 hours. 28. A wall, which was to be built 32 feet high, was raised 8 feet by 6 men in 12 days ; how many men must be employed to finish the wall in 6 days ? ^ v, : ~ ..-^^ Ans. 30 men. 29. A and B can perform a piece of work in 5j\ days ; B and C in 6§ days ; and A and C in 6 days ; in what time would each of them perform the work alone, and how long would it take them to do the work together ? Ans. A, 10 days ; B, 12 days; G, 15 days ; and A, B, and C, together, in 4 days. r,, v . / 30. My tailor informs me that it will take 10^ square yards of cloth to make me a full suit of clothes. The cloUi I am about to purchase is 1} yards wide, and on sponging it will shrink ^^f in width and length ; how many yards of thu cloth must I purchase for my "new suit?" Ans. 6 j^§, yards. 31. If A can do f of a oertain piece of work in 4 hours, and B can do f of the remainder in 1 hour, and G can finish it in 20 min. ; in what time will they do it all working t<^ther ? Ans. 1 hour, 30 min. 32. A certain tailor in the City of Hamilton bought 40 yards of broadcloth, 2^ yds wide ; but on sponging, it shrunk in length upon every 2 yards, j'g of a yard, and in width, 1| sixteenths upon every 1^ yards. To line this cloth he bought flannel 1^ yards wide, which, when wet, shrunk ^ the width on every 10 yards in length, and in width it shrunk ^ of a sixteenth of a yard ; how many yards of flannel had the tailor to buy to line his broadcloth ? ^' '^•' Ans. 71j'g yards. 33. If 6 bushels of wheat are equal in value to 9 bushels of bar- ley, and 5 busheb of barley to 7 bushels of oats, and 12 bushels of 116 ABTTBllFnO. oats to 10 bushels of pease, and 13 bushels of pease to } ton of hay, and 1 too of hay to 2 tons of coal, how many tons of ooal are equal ill value to 80 bushsls of wheat ? SOLUTION. If 6 bushels of wheat are equal in value to 9 bushels of barley, or 9 bushels of barley to 6 bushels of wheat, one bushel of barley would be equal to ^ of 6 bushels of wheat, equal to |, or § of a bushel of wheat, and 5 bushels of barley would be equal to 5 times § of a bushel of wheat, equal to §x5=V=^ bushels of wheat. But 5 bushels of barley are equal to seven bushels of oats ; henoe, 7 bushels of oats are equal to 3^ bushels of wheat, and one bushel oi oats would be equal to 3^-v-7=^f bushels of wheat, and 12 bushels of oats would be equal to 12 times ^^=i^^=bf^ bushels of wheat. But 12 bushelft.of oats are equal in value to 10 bushels of pease, hence, 10 bushels of pease are equal to 5f bushels of wheat, and one bushel of pease would equal b^-i-10=i^ of a bushel of wheat, and 13 bushels of pease would equal ^Xl3=*)f'-=7^ bushels of wheat. But 13 bushels of pease equal in value ^ ton of hay, hence, ^ ton of hay equals 7^ bushels of wheat, and one ton would equal 7|X2= 14^ bushels of wheat. But one ton of hay equals 2 tons of coal, hence, 2 tons of coal are equal in value to 14f bushels of wheat, and one ton would equal 14j|-^2=74 bushels of wheat. Lastly, if 7| bushels of wheat be equal in value to one ton of coal, it would take as many tons of coal to equal 80 bushels of wheat, as 7| is contained in 80, which gives 10|§ tons of coal. Note.— This question belongs to that port of arithmetic usually called Coojoined Proportion, or, by some, the " Oiain Rule," which has each ante- cedent of a compound ratio equal in value to its consequent. We have thought it best not to introduce such questions under a head by themselves, on account of their theory being more easily understood when exhibited by Analysis than by Proportion. Questions ihat do occur like this will most probably relate to Arbitration of Exchange. Although they may all be worked by Compound Proportion as well as by Analysis, yet the most expe- ditious plan, and the ono generaly adopted, is by the following RULE. Place the antecedents in one column and the consequents in another, on the right, ioith the sign of equality between them. Di- vide the continued product of the term* in the column containing the odd term by the continue product of the other colwnny and the quotient will be the anawer. r.- > ?.; -^i^^ r; t^ ANALTSn AND SYNTHESIS. 117 L«t us now take our last example (No. 33), and solve it hj this Di- the the nile: 6 bashels of wheat=:9 bushels of barley 5 bushels of barley=7 bushels of oats. 12 bushels of oat8=10 bushels of peaap. 13 bushels of pea8e=^ ton of hay. 1 ton of hay=2 tons of coal. — tons of ooal=80 bushels of wheat. 20 =W=10iS. Ans. ''ijt,, ■n'iM'ifi .iBvij?. J$''V-.. , .- ^,% \%, 13, \, %, ««. %. 34. If 12 bushels of wheat, in Toronto, are equal in value to 12^ bushels in Hamilton, and 14 bushels in Hamilton are worth 14^ bushels in Woodstock, and 12 bushels in Woodstock are worth 12^ bushels in Guelph, and 25 bushels in Guelph are worth 28 bushels in Barrie, how many bushels in Borrie are worth 60 bushels in Toronto? Ans. 75^ |. 35. If 12 shillings in Massachusetts ore worth 16 shillings in New York, and 24 shillings in New York are worth 22| shillings in Pennsylvania, and 7^ shillings in Pennsylvania are worth 5 shillings u Canada, how many shiUings in Canada are worth 50 shillings in Massachusetts ? Ans. 41§. 36. If 6 men can build 120 rods of fencing in 4 days, how many days would seven men require to build 210 rods ? ^-«-.r-^?.-,.^#3,,.,-,:i SOLUTION. /,';__ i If 6 men can build 120 rods of fencing in 4 days, one man could do ^ of 120 rods in the same time ; and ^ of 120 rods is 20 rods. Now, if one man can build 20 rods in 4 days, in one day ho would build ^ of 20 rods, and ^ of 20 rods is 5 rods. Now, if one man can build 5 rods in one day, 7 men would build 7 times 5 rods in one day, and 7 times 5 rods=35 rods. Lastly, if 7 men can build 35 rods in one day, it would take them as many days to build 210 rods as 35 is contained in 210, which ir> 6 ; therefore, if 6 men can 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 eaeh, build a wall 24 feet long, 16 feet high, and 3 feet thick i in how many days, of 8 118 ABrrHmnc. hoora each, would the same lot of men build a wall 20 feet long, 12 feet high, and 2^ foot thick ? Ans. 23/,. 38. If 5 men can perform a piece of work in 12 days of 10 honra each ; how many men will perform a piece of work four times as large, in a fifth part of the time, if they work the same number of hours in a day, supposing that 2 of the second set can do as much work in an hour as 3 of the first set ? Ans. 66$ men. None. — Such questions as this, where the unswer involves a fnwstion, may frequently occur, and it may be asked how § of a man can do any work. The answer is simply this, that it reqiiires 66 men to do the work, and one man to continue on working f of a day more. 39. Suppose that a wolf was observed to devour a sheep in ^ of nn 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/^ min. \- 40. Find the fortunes of A, B, C, D, £, and F, by knowing that A is worth $20, which is ^ as much as B and are worth, and that C 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. Ans. A, 20 ; B, 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 then turns again, and pursuing his journey, overtakes B 22^ days after' the time they first set out. It is required tO'find the rate at which B uniformly travelled. Ans. 10 miles 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 rate of 18 miles an hour ; how long will the chase last, and what dis-^ tance will the hare have run ? Ans. 60j^^ 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 is taken sick and leaves, B continues on working alone, and after 2 days he is joined by C, and they finish it together in 1^ days ; how long would be doing it alone ? Ans. 12 days. ' 44. A, in a soufBe, seized on f of a parcel of sugar plums ; B caught f of it out of his hands, and C laid hold on y^ more ; J> ran off with all A had left, except 4 which £ afterwards secured slyly for himself; then A and C jointly set upon B, who, in the conflict, let PRACTICE. 119 fall ^ he had, which were equally picked ap by D and E, irho lay pcrdn. B then kicked down C'h hat, and to work thoy all went anew for what it contained ; of which A got ^, B j^, D '^, and and B eqoal shares of what was loft of that stock. D then struck f of what A and B last acquired, out of their hands ; they, with some difficulty, recovered f of it in equal shares again, but the other throe carried off | a piece of the same. Upon this, they called a truce, and agreed that the ^ of the whole Icil by A at first, should bo equally divided among them ; how many plums, after this distribu- tion, had each of the competitors ? ' ^ i Ans. A had 2863 ; B, 6335 ; C, 10,294, and E, 4950. '*.7.;' -.. i. -^^^ v.- PRACTICE. The rule which is called Practice is nothing else than a particu- lar case of simple proportion, viz., when the first term is unity. Thus : if it is required to find the price of 28 tons of coal, at $7 a ton — as a question in proportion, it would bo, if 1 ton of coal costs $7, what will 28 tons cost ? and the statement would be 1 : 28 : : 7 : F. P. Hero 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 36 cwt. flour, at £1 lOs. Od. per cwt. — here again, the question stated at length would be, if 1 cwt. of flour cost £1 lOs. Od., what will 36 owt. cost ? and the statement would be 1 : 36 : : £1 10s. Od. : F. P., which thus becomes a question of multiplication, because the first term is unity, and, dividing by 1, would not alter the product of the other two terms. There are various methods of performing the work, •irhich we shidl now iUttStrate by the example given above. ^^ rS - I £ 1 (1.) 10 36 (2.) 36 30 £54. 0. 20)108(0 £54 10s.= (3.) £36 1 :J|of£l)36 18 £54 1st method— mnltiply[£l lOs. Od. by 36, as in compound multi- plication, and the result, £54, in the answer, because it is the prodqpt of the second and third terms of the analogy. In the second method the £1 10s. Od. is reduced to shillings, and 36 mul- tiplied by it, which gives the answer in shillings, and dividing by 20 lao ABITHMETIO. 36 gives 54, the answer in ponnds. By the third method, we note that 36 owi., At £1 per owt., will he £36, and unoe lOs. is the half of a pound, the prioe at 10s. will be the half of the prioe at £1, and we write in the nsnal place for a divisor 10s.=4 of £1, and we take the half of the price at £1, namely, £18, and then by adding together the prioe at £1 and the prioe at lOs., vis., £36 and £18, we get the price at £1 10s. Od. This is in reality nothing else than multiplying by 1^, as in tho margin, which is obviously correct, for £1 10s. Od. is one pound and ti-hal/. We have chosen an exceedingly simple example for the pur- pose ef illustration, and we now remark that the advan- tage of any one of these methods above the others will not be apparent till more difficult questions are pro- posed. The first method is generally the most con- venient in calculating dollars and cents, because, as we have already shown, the multiplication of all quanti- ties expressed decimally can be performed in precisely the same manner as in the case of whole numbers, care being taken to ^ve the decimal point its proper place. The second and third methods, and especially the third, are the best adapted for calculations in pounds, shillings and pence, although the first may often be used with great advantage. The third is called the method of aliquot parts, i. e., a part that is contained in the next higher denomination without remainder ; thus, 10s. above is an aliquot part of £1, vis., 10s. is ^ of £1. TABLE OF ALIQUOT PARTS. I £54 'Parts of $1. Parts of £1. l8.8d. Is. 4d. =j\ Is. 3d. ==T*8 Is. ==,V Ports of Is. 6d. = I 4d. = 3d. = 2d. = lid.= Id. = 9 Parts of a month. 15 day8= 10 = ? : 3 2 1 —IB * In Canada, the hundred weight being 100 lbs., the aliquot parte of the cwt will be the some as tho aliquot parts of the dollar. In Britain, the hun- dred weight is 112 lbs. PRACTICE. IZBRO I8IH. m Find the prioes of the following nomben of artiolm nt tho given prices: (1) (2.) (3.) 187 cwt., at 15.37^: 187 5.37J 1857 lbs., at $3,871 3.87J 1309 661 935 93J 4796 torn, at 14.50 4.50 $1005.121 (4.) What is the price of 29 score of sheep at $7.62^ each. I 680 i 7.62i 290 1160 3480 4060 928J=4ofl857 12999 239800 14856 19184 5571 $21582.00 $7195.87^ (5) Sold to a cattle deal- er 196 head at $18.75 each. ^"''■" 18.76 196 11260 16876 1875 $3675.00 (0.) Sold to a dealer 97 head at $16.12^ on an average. 16.12^ 97 11284 14508 $1564.121 $4422.50 7. To find the price of 347 cwt. of coffee at £7.11.6 per owt. cwt. 347Xl=£347=price at £1. 7=:No. of pounds lOs. =^of£l. Is. 3d. =:^ of 10s. 3d. =§ of la. 3d. £2429=price at £7, obtained by multiply* ing347by 7. 173.10.0=price at lOs. obtained by dividing 347 by 2. 21.13.9=price at Is. 3d. obtained by dividing 173.10.0 by 8. 4. 6.9=price at 3d. obtained by diri- ding 21.13.9 by 6. £2628.10.6=price at £7.11.6, obtained by adding the four parts. ABTTBlIKnO. Here it is evident that if each owt. ooat one pound, the whole must be repeated as often as there are units in 7, t. c, 7 times, and hence we multiply 347 by 7, and obtain 2429, which being iho price of the hundred weights is pounds, i. e., £2429. Again, since 10s. is the I of £1, the price at 10s. will be the ^ of the price at £1, i. e., the ^ of £347, which therefore we divide by 2, and place the quoti- ent £173.10.0, under the £2429, and the divisor 2, by which it was obtained, opposite to it. Again, since Is. 3d. is the ^ of lOs., the price at Is. 3d, will be the ^ of the price at lOs., wo therefore divide £173.10.0 by 8 and place the quotient £21.13.9 below it, and the 8 opposite to it. In the same way we see that as 3d. is the ^ of Is. 3d., that the price at 3d. will be the > of £21.13.9, the price at Is. 3d., we therefore divide £21.13.9 by 5, and obtain £4.6.9, the price at 3d, and place it under the £21.13.9, and the 5 opposite to it. We have now the price at 7. 0.0, all which make the whole price .^ ^ ^^ n i'q ^ ^^^^ indicated. Having thus , , . «* q'o obtained the partial results, we add • them, and the sum is the price of £7.11.6 347 cwt. at £7.11.6. To keep i)efore us the aliquot parts used, we ]p\ace a memorandum of this on the left, as exhibited in the examples. We have written this first example in an expanded form, in order to show every step of the process. The annexed will show in how much smaller space it may be performed : 347 @ £7.11.6 7 otherwise : 10s. Is. 3d. 3d. 2429 •' 173.10.0 21.13.9 4. 6.9 £2628.10.6 :|i^ 10s. h Is. t'o 6d. A 347 @ £7.11.6 7 2429 > 173.10.0 17. 7.0 8.13.6 £2628.10.6 Wo have taken for granted that those who have learned com- pound division, do not need to be told that when there was £1 over in dividing £347 by 2, it was to be reduced to shillings, (20) and then divided by 2, giving 10s., and so on. There are other modes which will be illustrated by other examples. 8. If a man has an income of $12.50 per week ; how much has he per year ? Ans. $650.00. w vnkcncE. 123 58. i 48. il 6d. TO 479 4 1916 119.16.0 95.16.0 11.19.6 ig2143.10.6 9. If a olerk Las $2.12^ salary jr every working day in tbo year ; what is his yearly income ? Ans. $665.12^ 10. If a tradesman earn 8s. 8d. a day ; how much vrWl he earn in the working days of the year ? Ans. £135.12.8. 11. An Ensign's pay in the British army is 5s. 6d. a day ; how mooh is that in Leap-year? Ans. £100.13.0. 12. If an officer's pay is a guinea and a half a day ; how much haa ho in a common year ? Ans. £574.17.0. 13. What will 479 cwt. of sugar come to at £4.9.6 per cwt. ? In this example, as before, wo multiply 479 by 4, to get the price at £4 ; then ns 98. 6d. is 58.-f-4s.-f-6d., we resolve the shillings and pence into this form, and as 58. i8 J of £1, we divide 479, the price at £1, by 4, and obtain £119.15.0, the price at 5s. Again, as 48. in I of £1, wc divide 479 by 5, and get £95.16.0, the price at 48.; lastly, as 6d. is Jg of 5s. we divide £119.15.0 by 10, and get £11.19.6, the ■—'•■• price at 6d. We might here have taken 6d. as the ^ of 4s., but tho division by 10 is easier than the division by 8. We now take the sum of all the partial results. We would call the learner's special attention to the following directions, as the neglect of it is a fertile source of error. When- ever you 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 that higher denomination. Thus, in tlie last example, 4s. is ^ of £1, so we must divide £479, and not the £119,15.0, for that would imply that 4s. was | of 5s. So also, since 6d. h the ^'jj of 5s., we divide the £119 15s. Od. by 10, but, had wc taken 6d. as the ^of 4s., we should then have divided £95 I63. Od., because that is the price of 48 ; the result would have been the same, how- ever, for £119 15s. Od.-=-10=£ll 19s. 6d., and £95 16a. Od.-7-8= £11 198. 6d. also. METHOD OF OOUPLKMBNTS. ' 14.—879;cwt., @ 19s. 15.— 1793 lbs., @ ISs l8.=:2«5)879 43 19 10)1793 179 6 £835 1 £1613 14 124 ▲BITHMETIO. 16.— 2781 toM, Q 17i. 6d. 17.-987 lbs., @ ICs. 8)2781 347 12 6 6)987 197 8 £2433 7 G £789 12 The principle of these operations may be illustrated by Ex. 14. We observe that if the price were £1, the answer would be £879, but the price is Is. less than a £1, and as Is. is ^'^ of £1, we find £43 19s. Od., the price at Is., and subtract it from £879, the price at £1, and we have £835 Is. Od., the price at 19s. In the same manner in 15, we subtract j'q, because ISs. wants 2s. of £1, and 2s. is y'o of £1. So in 16, we subtract ^, and in 17 we subtract ^. This mode is useful in many other operations. It is not of so much service, however, in calculating by dolUis and cents as by pounds, shillings and pence. We shall give one example, worked both ways, and let the learner judge for himself. (18.) 2479 @ 13.90 A of 2479 @ $3.90 3.90 4 2231.10 7437 :i»a:,^')F'-^^?-.^- 9916.00 247.90 $9668.10 $9668.10 Here $4 will be seen to be too largo a multiplier by 10 cents, and therefore we subtract f'^j of $2479, viz., $247.90, and find the same answer as before. 19. To find the rent of 189 acres, 2 roods^ 32 rods at $4.20 per acre. Since the rent of 2 roods=|of 1 acre, 20 rod8=^ of 2 roods, 10rods:^^of20rods, 2 roda=| of 10 rods, 4.20 189 210 525 2625 525 3780 3360 420 $796.74 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 . >25, 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 I of 10 rods. We then multiply by 189, and set the figures of the W PBAOnOE. w prodttot in the unid order, «o that the first figure of tk* i)««duet t>) 9 Bhall bo under tho unita of ccntit, Slc, and then additi all t! o partial results, wo find the final answer, $796.74, the rent of 189 acres, 2 roods and 32 perches. 20. What is tho price of 118 acres, li roods and 20 rods of clear* cd land, at $36.75 per ucru ? Ans. $4368.66. 21. What is thu price of 286 acres, 1 rood, 24 rods of un- cleared land, at $7.25 per acre ? Ans. 2076.40. 22. A has 84 acres, 2 roods, 36 rods of cleared land, worth $24.60 an acre ; B has 248 acres, 3 roods, 24 rods of uncleared land, worth $4.40 an acre — they exchange, the difierenco of value to be paid iu cash ; which has to pay, and how much ? Ans. B $989.08. 23. What is the price of 675^ cwt. of beef, at 10.86§ per owt. ? Ans. 7340.43. 24. What is the value of 483 cwt. of coffee, at $23.33^ per cwt. ? Ans. $11270.00. 25. What are 195 lbs. of raisins \fOTi\f, at 30 ots. per lb. ? Ans. 58.50. 26. What is the value of 514 gallons of oil, at 43^ cts. per gal- lon? ' ' Ans. $223.59. 27. What will be the price of 576 yards of broadcloth, at $3.75 per yard? Ans. $2160.00. 28. Find the price of 1625 yards of muslin, at 54^t8. per yard. Ans. $880.21. 29. Find the price of 4265 owt. of flour, at $6.82^ per cwt. " Ans. $29108.02^. 30. What ia the prico of 7913 owt. of pearl ashes, at 11.375 per owt. ? Ans. $90010.37^. -/ • ad :,L ::M, 196 ABimmio. BILLS OF PAR0EL8. A Bill op Paboblb is limply a ttetoment randend by the leUor to th« parohoMr, ■howiog the artioloi boaght, and the priooi of oaeh. 31. Hamilton, Jan. 4th, 186G. Ma. Eliab 0. Conklin, Bought of J. Bdntin & Co., 12 reams of foolscap paper @ $3.25 15 doien School Books @ 4.50 23 •' Slates 1.30 7 <• Photograph Albums @ 15.00 3 « Bullions Grammar @ 7.00 8 " Fifth Render @ 3.60 5 gallons of Black Ink @ 1.10 . ' $296.90 I ' Received payment, J. BUNTIN & Co. 32 , Toronto, Jan. 12(h, 1866. Mm. Jamis H. Burritt, A>14^A/ - " 13 lbs. Young Ilyson Tea @ 87J ' , *' 13 Iba. brown Sugar @ H " 15 bushels of Potatoes @ 46 •22.23 Ctt. For 10 lbs. Butter @ 17c. ^; " 5doz.Eg^6 @ 12J " 3 gallons Maple Molasses @ 95 " Note at 20 days, to balance 17.06 $22.23 R. FITZSIMMONS & Co. Note.— Such a Bill as this would bo termed a Barter Bill. 128 ABirmiETio. v 35 KiNOSTONy Jan. 2nd, 1866. Jamis Thompson, Esq., ' ' To A. Jardine & Co., Z>r. For 3 doz. Buttons @ $0.12 " 5^ yards of black Broadcloth @ 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 '« 5 yards black Silk Velvet @ 3.50 " 20 lbs. Loaf Sugar @ .18 " 2 gallons Molasses @ .40 " 1 bag of common Salt @ 1.15 "251bs.Rice @ .09 « 3 sacks Coffee, 70 lbs. each @ .12 Ce. $166.74 By Gash 50.00 ^ Balance due , $116.74 V 36 Aloonquin, Jan. 15th, 1865. W. Fi.iMiNa^& Co., Bought of J. & A. Wbioht, 1500 lbs. Canadian Cheese @ $ .09 300 bushels Fall Wheat @ 1.25 ObrlsPot Ash, net 7056 lbs @ 5.75 per cwt. 150 bushels Spring Wheat @ 1.15 200 " Potatoes @ .45 600 " Oats @ .37J 150 " Pease @ .65 60 " Indian Corn @ .50 60 " Apples @ .60 3 kegs Butter, 110 lbs. each. @ .18 SObuflheb Rye @ .70 40 « Barley @ .80 .$1688.12 Received payment, J. & A. WRIGHT. PEBCENTAQE. PERCENTAaE. 18. — PxROKNTAOE is an allowance, or reduction, or estiinato of a certain portion of each 100 of the units that enter into any given calculation. The term is a contraction of the Latin expression for one hundred, and means literally &y the hundred. In calculating dollars and cents, 6 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 find that at the end of a certain year it was 100, and at the end of the next it was 106, we say it has increased 6 per cent. i. c, 6 persons have been added to the 100. So, also, if a large city has a population of 100,000 at the end of a certain year, and it is found that it has 106,000 at the end of the following year, we say it has increased 6 per cent., which means that if we count the population by hundreds wo shall find that for every 100 at the end of the one year, there are 106 at the end of the next; because one hundred thousands is the same as one thousand hundreds, and we have supposed the increase in every 100 to be 6, the total increase will be one thousand aixea 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. Thut*, a falling ofif in the population of 6 persons in the hundred would be denoted by 100 — 6=94, as an increase of 6 in the hundred would be denoted by 100-j-6=106. So, also, in our first example, a deduction of |6 in $100 would be $100— 6=$94, and a gain would be f 1004-$6=$106. The portion of 100 so allowed or estimated, i» called the rate per cent., 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, estimate what is to be allowed on it at the same rate. Thus, if 6 is to be allowed for 100, then 3 must bo allowed for 50, and 1^ for 25, &c. The number on which the percentage is estimated is called the basis. Thus, in the example given regarding the population of a city, 100,000 is the basis. When the basis and percentage are combined into one, the result is 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 decrecue, or loss, it is to be subtracted from the basis to get the amount. This latter result is sometimes called the remainder. 130 ABITHMETIG. From what has been said, it in phiin that porccntage is nothing else than taking 100 as a standard unit of measure — ^Sce Art. 1) — and making the rate a i'raetion of that unit, so that 6 per cent, is i{jjj=(Art. 15, V.) .06. Wc may obtain the same result by the rule of proportion. Thus, in our illustrative example of an increase of 6 persons for every 100 on a population of 100,000, the analogy will be 100 persons : 100,000 persons : : G (the increase on 100) : 6,000, the increase on 100,000. It is manifest that the same result will bo obtained whether we multiply the third by the second, and divide by the first, or whether wo divide the third by the first, and multiply the result by the second ; or, which is the same thing, mul- tiply the second by the result. Now, we akeady found that 6-f-100=,^n=.06, the same as before. So also, 7 per cent, of any loss is seven one-hundredths of it, i. c, ygg=:.07. It should be carefully observed that such decimals represent, not the rale per cent., but the rate per unit. Though this is easily comprehended, yet we know by experience that learners are constantly liable to commit errors by neglecting to place the decimal point correctly. We would therefore direct parti- cular 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 J per cent., what is the rate per unit ? Ans. .00^. At ^ per cent., what is the rate per unit ? Ans. .OOJ. At 1 per cent., what is the rate per unit ? Ans. .01. At 2 per cent., what is the rate per unit ? Ans. .02. At 4 per cent., what is the rate per unit ? , Ans. .04. At 7 J per cent., what is the rate per unit ? Ans. .07|^. At 10 per cent., what is the rate per unit ? Ans. .10. At 12^ per cent., what is the rate per unit ? Ans. .12^. At 17 per cent., what is the rate per unit ? Ans. .17. At 25 per cent., what is the rate per unit ? Ans. .25. At 33 J per cent., what is the rate per unit ? Ans. .33 J. At 66f| per cent., what is the rate per unit ? Ans. .66§. At 75 per cent., what is the rate per unit ? Ans. .75. At 100 per cent., what is the rate per unit? Ans. 1.00. At 112^ per cent., what is the rate per unit ? Ans. 1.1 2|. At 150 per cent., what is the rate per unit? Ans. 1.50. At 200 per cent., what is the rate per unit ? Ans. 2.00. PERCENTAGE. 181 I. To fiad the percentage on any given quantity at a givon rate: On the principles of proportion, wo have as 100 : given quan- tity : : rate : percentage, and as the third term, divided by the first, gives the rate per unit, we have the simple RULE: Multiply the given quantity by the rate per um'f, and the product will he the percentage. EXAMPLES. To find how much 6 per cent, is on 720 bushels of wheat, wc have 6-1-1 00==.06, the rate per unit, and 730x.06=43i bushek, the percentage. To find 8 per cent, of $7963-75, in like manner, we have .08, the rate per unit, and $7963.75 X -08 gives $637.10, the percentage. Instead of ^er cent the mark (%) is now commonly used. EXERCISES ON THE RULE. 1. What does 6 per cent, of 450 tons of hay amount to ? Ans. 27. 2. What is 10 per cent, of $879.62J ? Ans. $87.96. 3. If 12 per cent, of an army of 47,800 men be lost in killed and wounded ; how many remain ? Ans. 42,064. 4. What is 5 per cent, of 187 bushels of potatoes ? Ans. 9.35. 5. What is 2^ per cent, of a note for $870 ? Ans. 21.75. 6. Find 12^ per cent, of 97 hogsheads ? Ans. 12.12|. II. To find what rate per cent, one number is of another given number : — Let us take as an example, to find what per cent. 24 is of 96. Here the 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, wo have the statement 96 : 100 : : 24 : F. P.=l^gil=25. Therefore 24 is 25 per cent, of 96. From this we can deduce the simple RULE. Annex, two ciphers to the given percentage, and divide that by the batiSf the quotient will he the rate per cent. 7. What per cent, of 150 is 15 ? Ans. 10. 8. What per cent, of 240 is 36 ? Am. 15. 182 ABrrHMETIO. & 9. What per cent, of 18 is 2 ? .*, . , . Ans. 11^. 10. What per cent, of 72 is 48 ? * , ^^^^- ^^S- 11. What per cent, of 576 ia 18 ? Ans. 3^. 12. What per cent, is 12 of 480 ? Ans. 2^. 13. Bought a block of buildings in King street for $1719, and sold it at a gain of 18 per cent. ; what was the gain ? 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 Toronto 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 cent. ; C, 20 per cent. ; and D purchased the remain- der ; what per cent, of the whole was D's share ? Ans. 25. 17. If a man walk at the rate of 4 miles an hour ; what per cent, is that of a journey of 32 miles ? Ans^ 12^. 18. What is the percentage on $1370 at 2f per cent. ? Ans. 37.67i. III. Given, a number, and the rate per cent, which it ia of another number, to find that other number, .400 is 40 per cent, of a certain number, to find that number. As 40 : 100 : : 400 : F. P.== ±iii^iM=l,000. Hence we derive the t RULE . Annex two ciphe/n to the given nurnbery and divide by the rate percent. EXERCISES. 1. A bankrupt can pay $2600, wliich is 80 per cent of his debts ; how much does he owe ? Ans. $3250. 2. A clerk pays $8 a month for rent, which is 16 per cent, of his salary ; what is his yearly salary ? Ans. $600. 3. In a manufacturing district in England, 40,000 persons died of cholera in 1832, this was 25 per cent, of the population ; what was the population ? Aus. 160,000. 4. Bought a certain number of bags of flour, and sold 124 of them, which is 12^ per cent, of the whole. Bcquired, the number of bags purchased. Ans. 992. 5. In a shipwreck 480 tons are lost, and this amount is 15 per cent, of the whole cargo. Find the cargo. Ans. 3200 tons. FEBCENTAOE. 133 6. A firm lost $1770 by the failure of another firm ; thu low was 30 per cent, of their capital ; what was their eapital ? Ans. $&900. IV. To find the basis when the amount and rate are given : — Suppose a man buys a piece of land for a certain sum, and by selling it for $300, gains 25 per cent. ; what did he pay for it at first? — Here it is plain that for every dollar of the cost, 25 cents arc gained by the sale, i c, 125 cents for every 100, which gives us the analo- gy, 125 : li)0: : 300 : F. P.; or, dividing the two terms by 100, 1.25 : 1.00 : : 300 : F. P., which by the rules for the multiplication and division of decimals, gives ^2£^^=$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 : F. 'P.=^%%^=HOO, the prime cost in this case. Hence we derive the BULB. Divide the given amount hy one increased or diminished by the given rate per unit, according as the question implies increase or decreoM, gain or loss. 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 field yields 840 bushels of wheat, which is 250 per cent, on the seed ; how many bushels of seed were sown ? Ans. 240 bushels. 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 $12045, 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 ?iis money by a venture, and had $279 left ; how much had ho at first, and how much did he lose ? Ans. He lost $31, and had $310 at first. 7. A grocer bought a lot of flour, and having lost 20 per cent, of the whole, had 160 b:^ remaining ; how many bags did he buy ? Ans. 200. 8. A merchant lost 12 per cent, of his capital by a bankruptcy, and had still $2200 left ; what was his whole capital ? Ans. $2500. lU ARITHMETIC. 9. Sold a sheep for $5, and gained 25 per cent. ; what did I pay for it? Ans. $4. 10. Lost $12000 on an investment, which was 30 per cent, of the whole ; what was the invostment ? Ans. $40000. INTEREST. From a transition common in language, the word interest has been inappropriately applied to the sum paid (or the use of money, but its original and true meaning is simply the nae of money. To illustrate this, wo will suppose that A borrows of B 0100 for one year, and at the end of the year, when A wishes to settle the account, he gives B $107. Were wo to ask the 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 use of that money for one year, paid B $7 for that use or interest ; hence A received the inter- terest or u^e 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 credits that arise from a transaction where interest is involved. If an individual borrows money, ho receives the use of that money, and when he pays for that use or interest, he places the sum so paid to that side of his *' interest account" which rcpresonte interest received, and if he lends money, he has 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. ' 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 he paid for the interest of $100 for one year, but wo have done this more in accordance with custom than from any intention to deviate from the true meaning of tho word interest. Interest is reckoned on a scale of so many units on every $100 for one year, and hence it is called so much per cent, per annum, from tho Latin per centum, by the hundred, and per annum, by the year. Thus, $6 a year for every $100, is called six per cent, per INTEREST. 135 annum. The term i^ also extcndeil to dcsignat«> the return accruing from any investment, such a.s shares in a joint stock company. To show the object and use ol' such transactions, wo may suppose a case or two. ' A person I'ccls himself cramped or embarrassed in his circum- stances and operation^;, and ho applies to some friendly party that lends him $100 for a year, on the condition that at the expiration oi the year ho is to receive 6106, that is, the $100 lent, and $6 more as a return for the use of the $100 ; or, if the borrower gets $600, he pays at the end of the stipulated time not only the $600, but also $36 ($6 for each $100) in return for the use of the $600. By this means the borrower gets clear of his difficulty, and maintains his credit at u small sacrifice. Again, a merchant may find that there is an opportunity for a speculation by which a good sum may be realized, but he has not capital sufficient, and accordingly he borrows a sum sufficient for the purpose, and pays, say 6 per cent, for it. Wo shall now suppose that the speculation yields him 21 per cent., then it is plain that after paying 6 per cent, for the money borrowed, ho is still a gainer of 18 per cent, on that money ; that is, for every $100 that he borrowed ho clears $18 Wc shall cite one case more of common occurrence. A mercantile house fails ; another house is in danger of being involved in the disaster by having extensive transac- tions with the former, but by effecting a loan to meet present emer- gencies, maintains its credit, and goes on with the business. In such a case, a small sacrifice in the shape of interest is of no recount com- pared with the damage of a failure, and so in numberless other cases. The sum on which interest is paid is called the principal. 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 nmple interest. When interest is charged on the amount, it is called cotnpound interest. SIMPLE INTEREST. As simple interest, when calculated for one year, differs in no way from a percentage on a given sum, we have only four things to consider, viz., the princi]jal, the rate, (100 being the basis,) the inter- 136 c. 08t, and the time, any tlMb'Wwfcioh being known, the fourth can bo fouud. The finding of the interest includes by far the greatest number of cases. , rv ^ ^ ' ^ , j. ^^ ,. v . ^ Wo Hhall first show the general principle, and from it deduce an e>i«y practical rule. Let it be required to find the interest on $468 for one year, at 6 per cent. As 100 is taken as the basis principal in relation to which all calculations arc made, 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 given principal. Hence, in the question proposed, wo have oh 0100 : $468 : : $6 : intere8t=$468XTS0— $468x.06=$28.08. Now .06 is the rate per unit, and from thia we can deduce rules for uU cases. CASE I . ' To find the interest of any sum of money for ono year, at any given rate per cent. RULE. MnUiply the principal by the rate per unit. ' EXERCISES. 1. What is the interest on $15, for 1 year, at 3 per cent.? Ans. $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. , 6. What is the interest on $6.40, for 1 year, at 8J 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.? ■ ii , 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 $668.75, for 1 year, at T^ per cent. ? Aos. $41.23. IKTEBE8T. 0A8X II. 187 I'o find the interest of any sum of money, for any number of y< .4W, At a given rate per cent. RULE. Find the interest for one year, and mvUipljf hf the number of years. -.■>'-- BXKR0ISB8. ■ 'r 10. What is the interest of $4.60, for 3 years, at 6 per cent ? Ans. 10.83. 11. What is the interest of f 570, for 5 years, at 7^ per cent. ? Ans. 1213.75. 12. What is the interest of $460.50, for 3 years, at 6^ per cent. ? 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 10| per cent. ? Ans. $527.47. 18. What is the interest of $640.80, for 5 years, at 4| 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. ? Ans. $688.86. CASE III . To find the interest on any sum of money for any number of months, at a given rate per cent. RULE. Find the interest for one year, and take aliquot parts for the months ; or, Find the interest for one year, divide by 12, and multiply by the mm^)er of months. 188 ARrrHMETIC. EXKBOIBBS. o> 21. What is the interest 00 $684.20, for 4 months, ate per cent.? Ans. 813.G8. 22. What is the interest on $760.50, for 5 montha, nt 7 per cunt. ? Ans. $22.18. . ^ 23. What is the interest on $899.99, for 2 montlis, at 8 percent.? ' Ana. $12.00. 24. What is the interest on $964.50, for 4 months, at !) per cunt, ? Ans. $28.94. 25. What is the interest on $1500, for 7 months, ut 10 per cent. ? Ans. $87.50. 26. What is the interest on $1560, for 11 months, at 7^ per cent.?' Ans. $107.25. 27. What is the interest on $1575.54, for 8 months, at G.} per cent. ? Ans. $G5.G5. 28. What is thf interest on $1728.28, for 9 months, at 8^ per cent.? . • Ans. $110.18. 29. What is the interest on $268.25, for 13 months, at 7 per cent. ? Ans. $20.34. 30. What is the interest on $1569.45, for 1 year, 3 months, at 8 per cent. ? Ans. $15G..05. 31. What is the interest on $642.99, for 1 year, 5 months, at 10 per cent.? Ans. $91.09. 32. What is the interest on $560.45, for 1 year, G months, at 9A per cent. ? Ans. $79.86. 33. What is the interest on $48.50, for 3 years, 9 months, at 10^ per cent. ? Ans. $19.10, 34. What is the interest on $560.80, for 2 years, 8 months, at 11| per cent.? ♦ Ans. $175.72. 35. What is the interest on $2360.40, for 19 months, at 12 per cent. ? 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 cent. m-:<^' t\ > .' Find the interest for the months, and take aliquot parts for the days, recJconfng the month as consisting of 30 dai/s. EXAMPLE. 36. What is the interest on $875.50, for 8 mouths, 18 days, at 11 per cent.? *, BDCPLE INTBUIEUT. 13^ SOLUTION. Principal $876.60 Rate per unit .11 IntereRtfor 1 year 96.3060 [nterest for G months ; or, ^ or interest fur 1 year 48.1626 Interest for 2 months ; or, ^ of interest for months 16.0608 [nterest for 16 days ; or, ^ of interest for 2 months 4.0127 Interest for 3 days ; or, Jt of interest for 1 6 days 8026 Interest for 8 months, 18 days $69.0186 We find the interest for 1 year to bo $96,306, and as G months are the ^ of 1 year, the interest for 6 months will bo the ^ ol the interest for 1 year ; likewise the interest for 2 months will be the ^ of the interest for G mouths, and as 16 days are the ^ of 2 months or 60 days, the intcrcst^for 16 days will be the ^ of the in- terest for 2 months, and likewise the interest for 3 days, will be the ^ of the interest for 16 days. Adding the interest for the monthi and days together, wo obtain $69.02, the sum to bo paid for the use of $876.60, for 8 months, 18 days, at 11 per cent. EXBBOISES. 37. What is the interest on $468.76, for 4 months, 16 days, at 7 per cent. ? Ans. $12.30. 38. What is the interest on $1664.40, for 3 months, 8 days, at 6 per cent. ? Ans. $22.62. 39. What is the interest on $346.66, for 11 months, 26 days, at G per cent. ? Ans. $20.46. 40. What is the interest on $74.86, for 5 months, 22 days, at 9 per cent. ? Ans. $3.22. 41. What is the interest on $673.76, for 8 months, 19 days, at 7^ per cent. ? Ans. $36.35. 42. What is the interest on $67.46, for 1 year, 2 months, 12 days, at G per cent. ? , .. Ans. $4.14. 43. What is the interest on $2647, for 1 year, 5 months, 18 days, at 6 J 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. 46. What is the interest on $2346.60, for 3 years, 7 months, 20 days, at 10 per cent. ? Ans. $863.60. AiunmETio. 46. Whubis tho interest ou $4268.45, for 4 years, U Diontb, 11 aayB, at 11} per cent. ? Ana. $2481.24. 47. What u the interest of $642.20, for 2 years, 7 months, 24 doys, at 12 per cent. ? Ans. $201.65. 48. What is tho interest of $64.50, for 2 years, 11 months, 2 days, at 7 per cent. ? Ans. $13.11). 49. What is tho amount of $746.25, for 1 ycur, lU months, 12 days, ut 5 per cent. ? Au8. $815.90. 50. What is tho interest of $680, for 4 years, 1 mouth, 15 days, at 6 per cent. ? Ans. $168.30. To find the interest on any sum uf money, for any number of days, at a given rate per cent.* RULI. ^ Find the tntereat/or one year, and say, a» one year (365 dayt,) it to the given number of dayt, so is the interest for one year to the interest required ; or. Having found the interest for one year, multiply it by the given nun^ter of days, and divide by 365. EXEROISIS. 61. 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 oonts. ^ 53. What is the interest on $56.82, for 14 days, at 8 per cent. ? Ans. 17 cents. * To flad 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 61 years. To And months, we must reckon from the given date in the first named month, to the same date in each successive month. Thus, Are months from the 10th of March brings ua on to the 10th of August. To And days, we require to count how many days each month contains, for to consider every month as consisting ot° 30 days, in the calculation of inter- est, is not strictly correct, although io ortioas of a single month it causes no serious error. Thus, the correct ti: from March 2nd to June 14tb, would be 104 days, viz., 29 for March, 30 for April, 31 for May, and 14 for June. A very convenient plan for reckoning time between two given dates is to count the number of months and odd days that intervene. Thus, from June 14th to November 20tb, would be 5 months and 6 days. oen j«n cen yeai .06, won att one hci .QOt the HlBirtiE INTEREST. Ul W. What is th« intcrmt on $75.SO. for Ift days, at 8^ per cent. ? Ans. 'A2 oeutn. 55. What is the tnt«raat on $125.25, for 20 days, at 5 per cent. ? Ana. 34 cents. 60. What is the interest on $150.40, for :{3 days, nt per vent. ? Ann. 82 cents. 57. What is the interest on $5G.48, for 45 days, at 6^ per cent. V Ans. 45 ocntM. 58. What iH the interest on $75.75, for i)T) days, nt 7 per cent. 7 Ana. 04 cents. 59. What is the interest on $268.40, for 70 days, at 7^ per cent. ? Ans. $3.8G. 60. What is the interest on $464.45. for 80 days, at 8 per cent. ? Ans. $8.14. Gl. 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 i:{5 days, nt 9^ per cent. ? Ans. $8.43. 63. What is the interest on $2460, for 145 days, at 10 percent. V Ans. $97.73. 64. What is the interest on $1568, for 170 duyH, at 1 1 per cent. ? Ans. $80.33. 65. What is the interest on $26^8, for 235 days, at 11 J per oent. ? Ans. $203.35. 66. What is thu amount of $364.80, for 320 days, at 11^ per Mnt. ? Ans. $401.58. . . , CASE VI . To find the interoat on any sum of money, for any time, at 6 per cent ' "• " '-' ■ • '-'> .^\ Since .06 would be th rate per unit, or the interest of $1 for 1 year, it follows that the interest for one month would be the j\j of .06, or j'It of a oent, 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 of $1, for every 2 months, is one cent. Again, if the interest of $1, for one month, or 30 days, is jt cent or .005, it follows that the interest for 6 days will be the I of .005 or .001 . Therefore, when interest is at the rate of 6 per cent., the interest of $1 for every 6 days is one mill. Hence tho 10 . ■»>v- 1^ ABITHMEnO. BULK. Find tht itUereit of $1 /or the given time by reckoning 6 cenU for every year, 1 cent for every 2 months, and 1 miU/or every 6 dc^s; then multiply the given princijial by the number denoting that in- terest, and the product loill be the interest rehired. None.— This method can be adopted for any rate per cent by first flnding the Interest at C per cent, then adding to, or siibtractine from the interest so found, such a pait or parts of it, as the given rate exceeds, or is less than G per cent. This method, although adopted by some, is not exactly correct as the year is considered as consisting of 360 days, instead of 365 ; so that the in- terest, obtained in this manner, is too large by y^-g or i^, which for every $73 interest, is $1 too much, and must therefore be. subtracted if the exact amount be required. EXAMPLE. 67. What is the interest of |24, for 4 mouths, 8 days, at 6 per coat? SOLUTION . The interest of $1, for 4 months, is 02 The interest of $1, for 8 days, ia 001^ Hence the interest of $1 , for 4 months, 8 days, is 021^ Now, if the interest of $1, for the given time, is .021^, the inter- «t of 124 will be 24 times .021^, which is $.512. il^ ' - "^;SL if?" ' SXKBOISKS. ' *''- 6S. 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. ? v ; > Ana. $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 oent. ? Ans. 3 cents. 72. What is tb<^ interest on $75.00, for 236 days, at 6 per cent. ? ,;t. , ' .' V ■A.ns. $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 $16.60, for 314 days, at 6 per ooi^t. ? '■^■'.""^^ >*jwvj-' An8..81.oents. i'> SIMPLE INTEBE8T. 148 75. What is the iaterest on $1T4.26, for 42 days, at 6 per cent. ? Ads. $1.22. 76. What is the interest on $10, for 1 month, 19 days, at 6 per cent. Ans. 8 cents. 77. What is the interest on $154, for 3 months, at G per cent. ? Ans. $2.31. ^' 78. What is the interest on $172, for 2 months, 15 days, at ti per cent. ? Ans. $2.15. 79. What is the interest on $25, for 4 months, at 6 per cent. ? Ans. 50 cents. 80. What is the interest on $36, for 1 year, 3 months, 11 days, at 7 per cent. ? Ans. $3.23. 81. What is the interest on $500, for 160 days, at 6 per cent ? Ans. $13.33. 82. What is the interest on $92.30, for 78 days, at 5 per <^nt. ? Ans. $1.00. 83. What is the interest on $125, for 3 years, 5 months, 15 days, at 10 per cent. Ans. $43.23. 84. What is the amount of |200, for 9 months, 27 days, at percent.? Ans. $209.90. 85. What is the interest on $125.75, for 5 months, 17 days, at 7 per cent. ? Ans. $4.08. 86. What is the interest on $84.50, for 1 month, 20 days, at 5 per cent. ? Ans. 59 cents. 87. What is the amount of $45, for 1 year, 1 month, 1 day, at 8 per cent.? ' Ans. $48.91. 88. What is tlio interest en $175, for 7 months, 6 days, at 5^ per cent. ? Ans. $5.78. 89. What is the interest on $225, for 3 months, 3 days, at 9 per cent. ? Ans. $5.23. 90. What is the interest on $212.60, for 9 months, 8 days, at 8| per cent. ? , ->'« Ans. $13.95. OASK VII. To find the interest on any sum of money, in pounds, shillings, and pence, for any time, at a ^ven rate per cent. RULE. Vse^^^fe.; -4J; i-Vt i* MvUijUjf thi principal by the rate per cent., and divide hy 100. 144 ABITHIIEnO. BXAMPLE. r^y 91. What it the interest of £47 158. 9d., for 1 year, 9 monthf, 15 days, at 6 per oent. ? ''^ SOLUTION. £ 8. D. £ s. D. Interest for 1 year « 2 17 4 47 16 9 Interest for G mos.^ or 4 of int. for 1 year, 18 8 6 Interest for 3 mos., or f of int. for 6 mos., 14 4 Interest for 15 days, or i of int. for 3 mos., 2 4h 2^86 14 6 20 Interoit for 1 year, 9 months, 15 days. ... £5 2 8^ ^HiCS! ({■■■^^^'Wii 12 4;i4 92. What is the interest of £25, for 1 year, 9 months, at 5 per cent. ? Ans. £2 3s. 9d. 93. What is the interest of £75 12s. 6d., for 7 months, 12 days, at 8 per cent. ? Ans. £3 14s. 7|d. 94. What is the amount of £64 10s. 3d., for 3 months, 3 days, 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 amount of £18 12s., for 10 months and 3 days, at 6 per cent. ? Ans. £19 10s. 9|d. CASE VIII :,-)i To find the principal, the interest, the time, and tho rate per cent, being given. ii EXAMPLE. X 97. What principal will produce $4.50 interest in 1 year, 3 tttonthf, at 6 per cent. ? SOLUTiaN. If a principal of $1 is put on interest for 1 year, 3 months, at G per cent., it will produce .075 interest. Now, if in this enimple, .075 be the interest on $1, 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, which is 60 times. Therefore, $60 will produce $4.50 interest in 1 year, 3 months, at 6 per cent. Hence the BIMPLE INTEBEST. 145 R U Ii B . Divide the given interett hy the interest of %\ for the given lime, tU the given rate per cent. EXSB0I8S8. t 98. What principal will prodnoe 77 cents interest in 3 months, 9 days, at 7 per cent. ? Ans. $40. 99. What principal will produce $10.71 interest in 8 months, 12 dayt:. at 7i|^ per cent. ? Ans. $204. 100. What principal will produce $31.50 interest in 4 years, at 3^ per cent. ? Ans. $225. 101 . What sum of money will produce $79.30 interest in 2 years, 6 months, 15 days, at 6^ per cent. ? Ans. $480. 102. What sum of money is sufficient to produce $290 interest in 2 years and 6 months, at 1^ per cent. ? ,,.--, Ans. $1600. '■-'■• ^'- '"' ' ASE I X . ■ ' To find the bate per cent., tho principal, the interest, and-the time being given. EXAMPLE. 103: If $3 be the interest of $60 for 1 year, what is the rate per cent.? .;id! SOLUTION !h If the interest of $60 for 1 year, at 1 per cent, is .60, the re- quired rate per cent, will be represented by the number of times that .60 is contained in 3.00, which is 5 times. Therefore, if $3 is the interest of $60 for 1 year, the rate per cent, is 5. Hence th? ^■?^. RULE. Divide the given, interest by the interest of the given p mm ap al at 1 per cent, for the given time. EXERCISES. -i. • V/" 104. If the interest of $40, for 2 years, 9 months, 12 days, is $13.36 ; what is the rate per cent. ? Ana. 12. 105. If I borrow $75 for 2 months, and pay $1 interest ; what is the rate per cent. ? Ans. 8. I I'i^ f^ 146 ABITHMETIO. ' ■^r ^- 106. If I give $2.25 for the uec of $30 for 9 moDths ; what rate per oeat. am I paying ? Ano. 10. t07. At what ratn per cent, will $150 amount to $165.75, in 1 year, 4 months, 24 dc^ys ? Ans. 7^. 108. At what rate per cent, must $1, or any sum of money, be on interest to doable itself in 12 years ? Ans. Ans. 8^. 109. At what rate per cent, must $425 be lent to gain $11.73 in 3 months, 18 days ? Ans. 9^. 110. At what rate per cent, will any sum of money amount to throe times itself in 25 years ? Ans. 8. 1 1 1 . If I give $14 for the interest of $125 for 1 year, 7 months, 6 days \ v^iiat rate per cent am I paying? Ans. 7. «H:}, CASE X. To find thj TIME, the principal, the interest, and the rate per cent, being given. EXAMPLE. 112. How long must $7&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 ihc number of years required to produce $12 interest, will be lepresente 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 ytan to gain $12. Hence the )i :*'>a RULE. vX UM. Divide the given interest by the interat of the principal for one year, at the given rate j^ 'cent. ' . EXERCISES. 113. In what time will $12 produce $2.88 interest, at 8 per cent ? Ans. 3 years. 114. In what time will $25 produce 50 cente interest, at 6 per cent. ? Ans. 4 months. 115. In what time will $40 produoe 75 oente interest, at 6| per cent. ? Ans, 3 months, 18 days. BQfPLE IMTERE8T. 147 ' 116. In what time will any Bttm of money doable itself, at 6 per cent. ? Ans. 16 yean, 8 montiu. 117. In what time will any Bum of money quadraple itself, at 9 per cent. ? Ans. 33 yean, 4 months. 118. In what time will $125 amount to $138.75, at 8 peroMt.? Ans. 1 year, 4 months, 15 days. 119. Borrowed, January 1, 1865, $60, at 6 per cent, to bo paid as soon as the interest amounted to one-half the principal. When is it due? Ans. May 1, 1873. 120. A merchant borrowed a certain sum of money on January 2, 1856, at 9 per cent., agreeing to settle the account when the in terest equalled the principal. When should he pay the same ? Ans. Feb. 11, 1867. -%,, merohantb' table ,, , ,. For s^uywing in what time any gum of money will double iUdf, at any rate per cent., from one to twenty, simple interest. Per cent. ■ Yenrs. Per cent. Years. Per cent. Years. Per cent. Years. . 1 2 3 4 5 100 50 33J 25 20 6 7 8 9 10 16f Ui l4 in 10 11 12 13 14 15 1' 16 17 18 19 20 1' MIXED EXERCISE8. 121. What is the interest on $64.25 for 3 yean, at 7 per oent.? Ans. $13.49. 122. What is the interest on $125.40 for 6 months, at 6 per oent. ?* Ans. 3.76. 123. What is the amount of $369.29 for 2 yean, 3 months, 1 day, at 9 per cent. ? Ans. $444.16. 124. What must be paid for the ubc of 75 cents for 6 years, 9 months, 3 days, at 10 per cent. ? Ans. 51 cents. 125. What will $54 amount to in 254 days, at 10 per cent. ?* Ans. $57.81. * This and the following exercises (marked with a *) are to be worked by ■Case VI. -: f,: • , . - ;. . ' .- i 148 ABrrHMETIC. 126. What must be paid for the interest of ^46 for 72 days, at 9 per cent. ?* Ann. 81 cent*. 127. What is the interest of $240 from January 1, 1866, to Juno 4, 1866, at 7 per cont. ? Ans. $7.14. 128. What will $140.40 amount to from August 29, 1865, Ui November 29, 1866, at 6| per cent. ? Ans. $151.83. 129. What principal will give $4.40 interest in 1 year, 4 months, 15 days, at 8 per cent. ? A 's. $40. 130. In what time will $40 amount to $44.. 40, at 8 per cent. ? Ans., 1 yr., 4 mos., 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 be paid for the interest of $145.50 for 240 days, at 9^ per cent. T^ 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 money be on interest to quadruple itseli' 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. 136. What is the interest of $30 for 30 days, at 6 per cent. ?* Ans. 15 cents. 137. What is the interest on $460 from January 2, 1866, to November 15, 1866, at 7^ per cent. ?* Ans. $30. 138. What will $25 amount to from December 24, 1865, to January 1, 1867, at 6 per cent. ? Ans. $26.53. 139. What sum of money will amount to $6400 in 4 years and 8 months, at 6 per cent. ? Ans. $5000. 140. In what time will $480, at 4^ per cent., produce $81.60 interest ? Ans. 3 years, 9 months, 10 days. NEGOTIABLE INSTRUMENTS. Bills of Exchange a'id Promissory Notes constitute, in their dif- ferent shapes, (he commercial medium for the payment of money. A Bill of Exchange is a written order for the payment of a certain sum of money unconditionally. A Promissory Note is a written promise to pay a certain sum of money unconditionally. CJiequcs are only a description of Bilb of Exchange. A Cheque is a written order addressed to a bank, or banker, and directing them I a o I NEGOTIABLE INBTBUMEXTB. 149 MfT to pay on presentment, to a person named in the cheque (or bearer, or order), a certain sum of money. Bank Notes are simple promissory notes, payable on demand. FORM AND REQUI8ITE8. A bill or note is called inland, when both made and payable in one country ; and foreign, when made or payable abroad. The following is a form of An inland bill : $500. Toronto, January 1, 1866. At sight, (or on demand, oral days after sight, or at days after date), pay to Messrs. A. B. Smith & Co., or order, (or bearer), Five Hundred Dollars. MusoRovx & Wright. To Messrs. Jones & Brown, Montreal. The following may be taken as an example of a promissory note : $500. Toronto, January 1, 1866. Three months after date we promise to pay to Messrs. A. B. Smith & Co., or order. Five Hundred Dollars, value received. MusoROVE & Wright. A bill of exchange amounts to an undertaking on the part of the drawer to the payee, and every subsequent holder, that the drawee will accept the bill when requested, and pay it when it becomes due. The maker of a promissory note undertakes to the payee, and every subsequent holder, that ho will pay the note when due. All bills and notes made, drawn or accepted in Canada, are sub- ject to a duty, collected by means of staiups. On every such instru- ment, if the amount do not exceed $25, a duty of one cent is imposed. From $25 up to $50, the duty is two cents ; from $50 to $100 it is three cents; and after that, it is three cents for every hun- dred dollars, or fraction of a hundred dollars. In the case of foreign bills, in sets of two, the duty is two cents for each of the set, on the $100 ; in sets of three or more, the duty is only one cent for each of the set on the $100. ' Foreign bills are usually drawn in sets ; that is, copies of the bills arc made on separate pieces of paper, each part containing a con- dition that it shall continue payable only so long as the others re- main unpaid. The object of this is to diminish the chances of losing a bill ; for if one part should fail to reach its destination, one of the others would be likely to do so. - / .w * ,'i , 160 ABixmosno. '«-.' ' PARTTI8 TO BILLS AND N0TI8. The person who draws a bill of exchange is called the drateer ; ho, to whom it is uddressed, the drawee, and, when he aooepts it, the accqator ; he in whose favor it m made, the payee. The person who signs a promissory note is called the maker ; the person to whom the promise is made, the payee. The moment a promissory note is indoned by the payee, by his writing his name on the back of it, he is called the indorser. The person to Whom it is indorsed is the indorsee. ^ .S • When a bill is accepted, or a note is made by several persons who are not in Partnership, the question whether they are bound jointly, or jointly and bvparatcly, depends upon the wording of the document. If a note begin thus: — " I promise," and bo signed by several per- sons, it is several, as well as joint. The name of the maker or drawer must be inserted or subscribed by himself, or his agent. There must be no uncertainty about the maker or drawer. For example, a note may not be signed '' John Smith, or eZse Robert Jones." ' TRANSFER OP BILLS AND NOTES. A bill or note may be payable to a particular individv ^ or to a particular individual or his order, or generally to bearer. When a bill or note is made payable simply to an individual, it is not negoti- able. If it be payable to an individual or order, he may transfer his right to another, by endorsing his name upon it. If it bo payable to an individual or bearer, it may be transferred by mere delivery, without any indorsement. If a blank be left for the payee's name, any borMfde holder may insert his own. An indorsement is said to be in blank when it does not mention the name of the party in whose favor it is made. A bill or note, when indorsed in blank, is transferable by delivery. A full or special indorsement is one which mentions the name of the party in whose favor it is made ; and it has to be endorsed in blank by the latter before it can be rendered negotiable. A restrictive indorsemont puts an end entirely to any furth'r nf^tiability of a note or bill. For instance, if a note be indorsed thus: — "Pay the contents to John Smith only," — it would bo no longer transferable. N^tiable paper in the hahds of an innocent holder, without notice of anything wrong, is good, although the person from whom y '.i NEGOTIABLE INSTRUMENTS 151 he obtained it may havo cotnc hy it nn a thief or tinder. There iH an exception to this when ii person takeu a bill or note a/ter it is due. He then \n in no better position than the person irom whom it was received, and could not recover if it bad been Araodulenily obtained by the latter. :?,f:; PaSSENTMENT AND ACCEPTANCE. .'sti ■ A bill should be always prctscnted for aoceptauce ; and bills and notes for payment, when payable at or after uight, or at some parti- cular place. A note payable on demand need not bo presented in order to charge the maker. Presentment for payment niuHt be made at a reasonable hour of the day upon which un instrument be- comes payable. An acceptance of a bill may be made without any particular form of words, and even without the signature of the acceptor. An acceptance may be conditional, and in that case is not of any force until the condition is performed. When a bill or note specifies a particular place for payment, it should be presented there ; although the maker or acceptor would not be discharged by a failure to present there, unless the words "and not otherwise or elsewhere" had been added. Presentment for acceptance must be made within a reasonable time. No delay upon the part of the holder should extend over any period not rendered clearly necessary by the circumstanoeB of the case. PROTEST AND NOTICE. When a bill or note is dishonoured, either by non-acceptance or non-payment, it should be protested by a notary public. Notice of such protest should be sent to every party on the instrument to whom the holder desires to have recourse. Notice means more than mere knowledge, and must be a formal communication of the dis- honour of the bill. This notice of protest may be given by being deposited in the post office nearest to the place of presentment during the day the protest is made, or the next lawful day. From the date of protest interest commences to run upon the instrument, although there are no words in it about interest. Besides interest, and the the expenses of protest and notice, the holder is entitled, when foreign bills are protested, to damages varying Trom four to ten per cent, of the principal amount specified in the bill. M 162 ■ r ▲BTTRIOBTIC. DAYB OF OnAOC. The time limited by the wordti of a bill or note for \\» payment is extended in this country by day* of grace ; so that a bill is not really payable till three days after tho day tipon which it purports to fall due. When a bill w made payable ho many days after tho the happening of a particular event, for instance, after sight, tho day on which thai event happens is excluded. Accordingly, a bill paya- ble ten days after sight, and presented on the tst August, would purport to be due on tho 11th, but adding tho days of grace, would in reality not bo payable till tho 14th. PARTIAL PAYMENTS. Partial payments, as the term indicates, arc the part payments of promissory notes, bonds, or other obligationu. When these payments arc made the creditor specifies in writing, on the h(ick of the note, or other instrument, the sum paid, and the time when it is paid, and acknowledges it by signing his name. The method wc generally adopt in Canada for casting interest upon bonds, notes, or other obligations, upon which partial payments have been made, is to apply the payment, in the first place, to the discharge of the interest then due. If the payment exceeds the interest, tho surplus goes towards discharging the principal, and the subsequent interest is to be computed on the balance of the princi- pal remaining due. If the payment be less than the interest, tho surplus of interest must not be taken to augment the principal, but interest continues on the former principal until the time when the payments, taken together, exceed the interest duo, and then the surplus is to be applied towards discharging the principal. ;y*?. bulb: Find the amount of the principal to the time of the first pay- ment ; siAtra4:t the payment from the amount, and then find the amount of the remainder to tlie tim^ of the second payment ; deduct the payment as before ; atui so on to the time of settlement. * ,'^, But if any paymfCnt is less tJtan the interest then due, find the* amount of the sum due to the time when the payments, added to- gether, shall he equal, at least, to the interest already due ; thei^find the balance, and proceed as before. A PABTIAL PATMSMTS. 163 [, B X A M P L B . 1. On the 4th of January, 1865, u note wof given for $800, payable on demand, with iatorest at 6 per cent. The following pay* nienta were receipted on the back of the note : * February 7th, 1865, rooeived $160 . ., . April 16th, " " 100 » ,, _ Sept.,30th, " " 180 . ^V ■ L' January 4th, 1866. " 170 f." March 24th, " " 100 June 12th, '* " 60 Settled July Ist, 1867. How much was due ? ' ^ SOLUTION: Face of the note, or principal $800.00 Interest on the same to February 7th, 1865 (1 month, 3 days) 4.40 Amount due at time of 1 at payment 804.40 First payment to be taken from this amount 150.00 Balance remaining due February 7th, 1865 654.40 Interest on the same from February 7th, 1865, to April 16th, 1865 7.525 Amount due at time of 2Dd payment 661.925 Second payment to be token from this amount 100.000 Balance remaining due April 16th, 1865 561.9LJ Interest on the same from April 16th, 1865, to September 30th, 1865 15.359 Amount d'xe at time of 3rd payment. 577.284 Third payment to be taken from this amount 180.000 Balance remaining due Sept. 30th, 1865 397.284 Interest on the same from Sept. 30th, 1865, to January 4th, 1866 6.290 Amount due at time of 4th payment 403.574 Fourth payment to be taken from this amount 170.000 Balance remaining due January 4th, 1866 233.574 t 164 ARITHXETIC. InterMt on the name fVom Jan. 4th, 1866, to March 24th, 18CG 3.1 H Amoant doe at time of 5th payment 236.688 Fifth payment to bo taken fVom thin amount 100.000 Balance remaining due, March 24th, 1866 136.688 Interest on the same fVom March 24th, 1866, to June 12th, 1866 1.799 Amount due at time of 6th payment 138.487 Sixth pajrment to be taken from this amount &0.000 Balance remaining due June 12th, 1860 88.487 Interest on the same fVom June 12th, 1866, to July 1st, 1^67 6.689 Amount due on settlement 94.076 2. $1600. PiHTH, C. W., February 16th, 1866. On demand, I promiae to pay Jacob Anderson, or order, one tlunuand aix hundred dollar$, wUh ifUere$t, at 7 ]}er cent. John Fortune Jr. There \ras paid on this note, April 19th, 1866 |460 ' July 22nd " 160 August 25th, 1866 60 Sept. 12th, " 100 Deo.24th. " 700 How much was due December 31st, 1866 ? SOLUTION. Face of the note or principal $1600.00 Interest on the same from Feb. 16th, 1865, to April 19th, 1865 19.60 Amount due at time of Ist payment 1619.60 First payment to be taken from this amount 460.00 Balance remaining due, April 19th. 1866 1159.60 PAimAT- PATlfEOTS. 166 loterost on tho same from April 19th, lH(]r>, to Jaly 28od, 1866 20.969 Amount duo at tituo of 2nd payment 1180.&69 Second payment to be taken from thin amount IftO.OOO Balance nmuininK' due, July 22nd, 18G5 1030.5C9 lotoroat un tho name from .luly 22nd, 18(>&, to Aug. 2&th, 18GG, less than 3rd |)ayment,* InteroHt on tho mimo from July 22nd, 1865, to Sept. 12th, 18GG 82.369 Amount duo at time n 4th payment 1112.928 Third and fourth paymentn to bo taken from this amount, 160.000 Balance remaining '*.uc Sept. l2Ui, 18GG 962.928 Interest on the same from Sept. i Jth, 18G' , to Deo. 24th, 1866 19.098 Amount due at time of last pay. nt 982.026 Lost payment to be taken front this vmount 700.000 Balance remaining due Deo. 24th, 1866 282.026 Interest on the same from Dec. 24th, 1866, to Dec. 3l8t, 1866 382 Amount due at time of settlement, Dec. :UHt, 1866 $282,408 3. $350. Ottawa, May 1st, 1864. On demand I promise to pay William Broum^ or order, three hundred and ^fty dollars, with interest, at 6 jyer cent. James Weston. There was paid ou this note, December 25th, 1864 $50 lune30th, 1865 5 • The interest on $1030.509, from July 22nd, 1805, to August 25th, 1800, is $78,752, and the payment made at this date, i-^ only $50. not enough to pay the interest, ao if ne proceeded, as in the former case, to add the interest to the principal, and subtract the payment from the amount obtained, we would be taking interest, until the next payment, on the excess of the inter- est, $78,752, over the payment, $50, which would bo in effect interest upon interest, or compound interest which the law doe^^ not allow. 156 ABITHMETIO. August 22nd, 1866 16 ...,,,;• June4th, 1867 100 .' How much was due April 5th, 1868 ? Ana. $251.67. 4. $609.65. ^ Brantvord, June 8th, 1861. ; Six months after date, tee jointly and severally promiat to pay John Anderson, or order, six hundred and nine j'\f^ dollars, at the Royal Canadian Bank in Toronto, with interest at 6 per cent, after maturity. •^' • '•"'■■-■ Samubl Graham. T. B. Beariian. There was paid on this note, October 4th, 1862 $25.00 -' March 15th, 1863 16.25 ■^ August 24th, 1864 36.56 What was due December 19th, 1865 ? Ans. 679.27. 5. $874.95. Kingston, May 9th, 1863. Three months ifter date, I promise to pay Harmon Cum/minys, or order, eight hundred and seventy-four j^jj dollars, with interest after maturity at 6 per cent. ^ ■ ; V _. .J, Thomas Goodpat. There was paid on this note, April 12th, 1864 $56.30 July 14th, 1865 24.80 Sept. 18th, 1866 240.60 What was due February 9th, 1868? Ans. $773.07. When the interest accruing on a note is to be paid annually adopt the following *s ^'^ RULE.* Compute the interest on the principal to the time of settlement, and on each year's interest after it is due, then add the sum of the * When notes, bonds, or other obligations, are %i\eti, " with interest payable annually," the interest is due at the end of each year, and may be collected, but if not collected at that time, the interest due draws only simple inieresi, and the original principal must not be increased by any addition of yearly interest. If nothing has been paid until maturity on a note drawing annual interest, the amount due consists of the principal, the total annual interest, or the simple interest, and the simple interest on each item of annual interest from the time it became due until paid. ■*# PABTIAL PAIMEMTB. }57 interats on the annual xnterettt to the amount of the principal, and from thit amount take the payment*, and the interest on each, from the time they were paid to^ the time of settlement, the remainder will be the amount due. 6. $500. Prkscott, May Ist, 1864. One yeur after date, for value recHved, I promise to pay Musgrove & Wright, or order, Five Hundred Dollars, at their office^ in the city of Toronto, toith interest at 6 per cent., payable annually. Jamks Manning. There waa paid on this note : May4th, 1865 $150 Deo. 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 June 1st, 1866 15 Amount of interest upon annual interest 2.10 Total amount of principal $564^0 First payment. May 4th, 1865 $150.00 Interest on the same from May 4th, 1865, to June 1st, 1866 • 9.70 Second payment, December 18th, 1865 300.00 , ', In;«rest on the same from December 18th, 1865, to June Ist, 1866 8.20 . . I Payments and interest on the same 467.90 Amount due June 1st, 1866 $96.70 U 108 ARITUMEHb. 7. $700. Bkllkvills, January 2nd, 1863. Eighteen months after date, I promiee to pay to the order of H. C. Wright, Seven Hundred Dollars, for valut received^ with interest at 6 per cent., payable annually. Thos. a. Bbtcb. There was paid on this note : January 15th, 1864 .' $350 . July 2nd, 1864 300 What amount was due January 2nd, 1865? Ans. $107.22. 8. $950. DuNDAS, C. W., Jan. 3rd, 1863. Two years after date, I promise to pay T. C. Musgrove, or order. Nine Hundred and Fifty Dollars, with interest at 9 per cent., payable annually, value received. James S. Parmenteb. The following payments were receipted on the back of this note : February 1st, 1864, rojeived $500 May 14th, « " 100 January 12th, 1865, " 300 What was due May 6th, 1865 ? Ans. 188.94. 9. $250. OsHAWA, January 2nd, 1863. Three years ft )'iv date, for value received, I promise to pay Michael Wright, or order. Two Hundred and Fifty Dollars, with interest, payable annually, at 6 per cent. Calvin W. Psabsons, At Bank of Montreal, Brockville. What was the amount of this note at maturity ? Ans. $297.70. OONNEOTIOUTRULE. The Suprame Court of the State of Connecticut has adopted the following RULE ,•■;-'. Compute the interest on the principal to the time of the first pay- TMnt; if that be one year or more from the time the interest com- menced, add it to the principal, and deduct the payment from the sum total. If there be after payments made, compute the interest on the balance due to the next payment, and then deduct tJie payment as altove, and in like manner from one payment to another, till all the I>AinlAL PATMtNTS. 169 payments are ahorbed, provided the time bettoeenone payment and another be one year or more. If any payments be made before one year's interest has iccrued, then compute Jhe 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 tim,e it was paid, tip to the end of the year ; add it to the sum paid, and deduct that sum from the principal and interest, added as above. If any payments be made, of a less sum than the interest arisen at the time of such payment, no interest is to be computed, but only on the principal sum for any penod. N()TK. — If n year extends beyond the time of settlement, find the ntnonnt ol the remaining principal to tlie time of settlement ; find also the amount of the payment or payments, if any, ftom the time they were paid to the time of settlement, and subtract their sum from the amoimt of the principal. EXAMPLE. 10. $900. Toronto, June Ist, 1862. On demand we promise to pay J. R. Smith & Co., or order, nine hundred dollars, for value received, with interest frmn date, at C per cent. MtrsGRovE & Wright. On the back of this note were receipted the followinf; payments : Juno IGth, 1863, received $200 August Ist, 1864, '' 160 Nov. 16th, 1864, " 75 Feby. 1st, 1866 " 220 What amount was due August 1st, 1866 ? SOLUTION. Face of note or principal $900.00 Interest on the same from June 1st, 1862, to Juno 16th, 1863 56.25 Amount of principal and interest, 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 duo August Ist, 1864 807.296 . ^-.y m ABTTHMETIO. Second payment to be taken from this amount 160.(H Balance due 647.296 Interest on the same for one year 38.837 A- uount due August 1st, 1865 686.133 Amoant of 3rd payment from Nov. 16th, 1S64, to August ist, 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 Ist, 1866 226.600 Balance due August 1st, 1866 $417,822 MSROHANTS' 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. FtTid the am/yunt of the principal from the time it became due until the time of settlement. Then find the amount of each payment from the time it was paid until settlement, and subtract their sum from the amount of the principal. EXAMPLE. 11. $400. Maitland, January 1st, 1865. For value received, I promise to pay J. B. Smith dk Cc, ' or order, on demand, four hundred dillars, with interest at 6 per cent. A. B. Gassels. 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 fiiettlement, which was December 28th, 1865? H^' FAirriAL PAYMENTS. t 161 ; > SOLUTION. /•;.>, -.). > Principal or face of note $400.00 Interest on the same from Jan. Ist, 1865, ^o Dec. 28tb, 1865 23.80 Amount of principal at settlement $423.80 First payment $100.00 Interest on the same from Feb. 4th, 1865, to Dec. 28th, 1865 5.40 Second payment 75.00 Interest on the same from May 16th, 1865, to Dec. 28th, 1865 2.77^ . Third payment 100.00 Interest on the same from A j.?> t 28th, 1865, to Dec. 25th, 1865 2.00 Fourth payment 80.00 Interest on the some from Nov. 25th, 1865, to Dec. 28th, 1865 44 Amount of payments to be taken from amount of principal 365.61^ Balance due, December 28th, 1865 $58.18J 12. $500. Hamilton, January Ist, 1865. Three months after date, Iprofiiise to pay James Man- ning, or order, five hundred dollars, for value received, at the Royal Canadian Bank in Toronto. Cybus Kino. Mr. King paid on this note, J\i1y Ist, 1865, ^^200. What was due April Ist, 1866, the rate of interest being 7 per cent.? Ans. $324.50. 13. $240. Smith's Falls, May 4th, 1865. On demand, I promise to pay A. K. Frost & Co., or order, two hundred and forty dollars, for value received, with interest at 6 per cent. Davit Flook. The following payments were receipted on the back c: this note : September 10th, 1865, received $60 January 16th, 1866, " 10 What was due at the time of settlement, which was May 4th, 1866? Ans, $100,44. ! _m ^ 162 ARITHMETIC. 14. $340. Nkwmarket, June Ifith, 1864. Three months after date, I promise to pay Thomas Cvlverwell, or order, three hundred and for*.>j dollars, with interest, at i^ per cent. Wu.uxi'i MusoROVE. On this note were receipted thu folto.ving [nyment« : October 14tli. 1864, nwivoi! ... ... «P J Frbmary, llith, 18t 5, <' ...... io What was duo .it time of settlement, Aug. 10, 1865 ? Ans. $232.06. COMPOUND INTEREST. When interest in unps;'! at the end of a year, xfc may, by special agreement, be added to ihe principal, and in lUi. mru bear interest, and so on from year to y>. r. When adde\.; it the principal in this way, it i!» said to be compounded. Li is not against the law in Canada to take compound interest; but it can never be collected unless it has been specially agreed upon br^forehand, or unless it is the custom of a house, and known to tho customer to that effect. EXAMPLE. 1. What is the compound interest of $60, for 4 years, at 7 per cent.? SOLUTION. Principal $60.00 Interest on tho same for one year 4.20 New principal for 2nd year 64.20 Interest on the same for one year 4.494 Now principal for 3rd year 68.694 Interest on the same for one year 4.808 New principal for 4th year 73.502 Interest on the same for one year 5.145 Araount for 4 years 78.647 Principal to be taken from same 60.000 Compound interest for 4 yearf , $18.6±7 The method of finding compound interest is usually muoh shor' cued by the following table, w ;-^ s shows the amount of $1 or £1 for any number of yera cot jxcseding 50, at 3, 3^, 4, 5, 6 and 7 per cent. Tho amount ^f ^^ £1 thus obtained, being multiplied by the j^'ivcn principal, w " the required amount, from which, if the principal bo taken. <;ho remainder will be the compound interest : COMPOUND INTEBE8T. 168 TABLE, nowiNo ma amovht or ohb dollar at ooiiroinn> hubmmi ron art mmnw or tubs xoT ucBsniiia virrr. N,„ff t lOr DISCOUNT AND PRESENT WORTH. Nov. 11, 18G1, Juno 5, 1862, 165 30 50 Ans. 11022.34. DISCOUNT AND PRESENT WORTH. Discount being of the same nature ati interest, is, strictly speak- ing, the U9C of money before it ia due. The term is applied, however, to a deduction of so much per cent, from the face of a bill, or the deducting of interest from a note before any interest has accrued. This is the practice followed in our Banks, and is therefore OiEilled Bank discount, in order to distinguish it from true discount. The method of computing bank discount differs in no way from tliat of computing simple interest, but the method of finding true discount is quite different, e. g., a debt of $107, due one year hence, is considered to be worth $100 now, for the reason that $100 let out at interest now, at 7 per cent., would amount to $107 at the end of a year. In calculating interest, the sum on which interest is to be paid is known, but in computing discount we have to find lohat sum must be placed 'l interest so that that sut.?. together with its interest, will amount tr the given principal. The su.^ .'bus found is called the " Present Worth." We ha^e already seen that $1.00 is the present worth of $1.07 due one yoar hence, at 7 per cent., therefore, to get the present worth of any 3um due oiic year hence, at 7 per cent., it is only necessary to find how many tiroes $1.07 is contained in the given sum, auJ wc have the present worth ; hence To find th J present worth of any sum, and the discount for any time, at any rate per cent., we have the following R u L c . Divide the given sum by the amount of $1 for the given time and rate, and the quotient will be the present worth. From the given sum subtract the present worth, and the remainder will be the discount. EXBROISES. i. What is the present worth of $224, due 3 years hence, at 6 per cent. ? Ans. $200. 166 ARITHMETIO. J> i 2. What is the diMOunt on $G70, due 1 yoar and 8 montlui honoe, at 7 per coot. ? Ana. $70. 3. What La tho discount on $501, duo 1 year and 5 months hence, at S \M5r cent. ? Ans. $51 . • What is tho present value of a debt of $678.75, due 3 years and 7 months hence, at 7^ per cent. ? Ans. $534.07^. 6. What is the discount on $88. lU, due 1 year, K months, and 12 days hence, at (i per cent. ? Ans. $8.16. 6. If tho du';ount on $1060, for 1 year, at 6 per cent., is $60; what is tho disori-... . on thn same sum for one-half the time ? Ans. $30.87. 7. How much cash will dischaigc a debt of $145.50, due 2 years, 6 months and 12 days hence, at 6 per cent. ? Ans. $126.30. 8. If I am offered a certain quantity of goods for $2500 cash, or for $2821.50, on 1) months credit; which is tho best offer, and by how much ? Ans. Oash by $200. 9. What is the difference between the interest and discount of $46.16, duo at tho end of 2 years, 6 months, and 24 days, at 6 per cent. ? Ans. 95 cents. 10. A merchant sold goods to tho amount of $1500, one-half to bo paid in 6 months, and tho balance k 9 months; huw :.ntich oash ought ho to receive for them after deducting 1^ per cent. > month ? Ans. s. 1331.25. It. Suppose a merchant contracts a debt of $24000, to lo paid in four instalments, as follows: one-fifth in 4 months; one-fouilh in 9 months ; one-sixth in 1 year and 2 months, and tho rest in 1 year ana 7 months ; how much cash must ho give at once to disohai'^c the debt, money being worth 6 per cent. ? Ans. 22587.65. 12. Bought goods to tho amount of $840, on 9 months credit; how much money would discharge the debt at the time of purchasing the goods, interest being 8 per cent. ? Ans. $792.45. 13. A I jksclier marks two prices in a book, one for ready money, and th-.^ c*\^r for one year's credit, allowing discount at 5 per cent. Tf the jrodiL price be marked $9.80 ; what ought to bo tho price m ked /or cash ? Ans. $9.33. 14. IX iPM having a horse for sale, offered it for $225, cash ; or, $230 at months credit ; the buyer chose the latter ; did the seller losv or make by his bargain, and how much, supposing money to bo woriH 7 per cent. ? Ans. He lost $6.47. lt\ A. B. Smith owes John Manning as follows : — $365.87, to BANKIXn. 167 bt^ paid December 19th, 1863 ; f ir>l.ir», to bo paid July ICth, 1H64 , $112.5(), to l)o p^id .lane ZM, IHtiL' , 8«)«;,S1, to Im? paid April IDth, IHOtI, ftllowinp diHcount :it t'» per cent.; how much cash Bhould Manning rccoivo an an c >/.•■* i^ W*'-" • ' 1. What is the bank discount ou a note, given for 60 days, for $350, at 6 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 of a note of $7840 discounted at a bank, for 4 months and 15 days, at per cent. ? Ans, $7659.68. 4. How much iJioney should be I'cceived 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 G 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 $240, at 60 d&ys, and on the 12th of the same month had it discounted at a bank at / per cent ; how much did I realize upon it. Ans. $237.61. 7. A merchant sold 240 bales of cotton, each wei^Mng 280 pounds, for 12^ 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. ; what will be his profits ? Ans. $1479.10. It is sometimes necessary to know the amount for which a note must be given, in order that it shall produce a given sum when dis- counted at a bank. ...? ■ EXAMPLE. 1. Suppose we require to obtain $236.22 from a bank, and that we are to give our note, due in two months; for what amount must we draw the note, supposing that money is worth 9 ^cr cent. ? " ■ • - . • '■■■i.i.:\ ■ ■ . ■. -^: : ■<■ i i >j V/';;.. SOLUTION. From the nature of this example, wo can readily perceive that such a sum must be put on tho face of the note, that when dis- * Throughout aP the exercises, unless otherwise specified, the year is to be considered as consisting of 3C5 days. Since it is customary iu IjusiueBti wlien a fraction of a cent occurs iu any result to reject it, if le.n of receiving a certain sum, to indemnify another for Homething in case it should in any way be lost. The party under- taking the risk is seldom, if ever, an individual, but a joint stock company, represented by an agent or agents, and doing business under the title of an " Insurance Company,^' or " Assurance Com- pany," such as the " Royal Insurance Company," the " Mutual Insurance Cimpany." Some con^panies are formed on the principle that each individual Bhareholder is insured, and shares in the profits, and bears his portion of ir losses. Such a company is usually called a Mutual Insurance Compapy. The ijum paid to the party taking the risk is called the Premium, of Insurance, or simply the Premium. The document binding the parties to the contract, is called the Policy of Insurance, or simply the Policy. The party that undertakes to indemnify is called the Insurer, or underwriter after he has written his name at the foot of the policy. Th*i person or party guaranteed is called the Insured. As there are many different kinds of things that may be at stake or risked, so there arc different kinds of insurance which may bo classified under three heads. Fir^, Insurance., including all cases on land where property is ex- posed to *\ie risk of being destroyed by fire, such as dwelling houses, stores and "actories. Ma.'iae Insurance. — This includes all insurances on ships and cargdes. Such an insurance may be made on the ship alone, and in that caec it is sometimes called hull insurance, and sometimes bot- tomry, the ship's bottom representing the whole ship, just as we say fifty sail for fifty ships. The insurance may be made on the cargo alone, « \ is theu n«ually called Cargo Insurance. It may be made on both ship and cargo, in which case the general t'"Tm Marine In- surance will be applicable. This kind, as the nan'j implies, insures against u) accidents by sea. Life ':.surance. — This k- an agreement between two pju-ties, that in case the one insured shoulJ die within a certain stated time, the other ahall, in consideration ii having received a stipulated sum annually, pay to the lawful heir of the deceased, or some one men- 180 ABIXBUXnC. tioncd in his will, or ttoiuo other party entitled therok >, tlie amount recorded in the policy. For inBtanoe, a man may, on tV^ oocasion of his marriage, iniuro his life for a certain sum, so that u hJd he die within a certain time, his widow or children shall be paid that sum by the other party. Again, a father may insure the life of his child, so that in case of the child's death within a specified time, he shall bo paid the sum agreed upon, or that the child, if it lives to a certain age, shall bo ontitltsd to that sum. One person may insure the life of another. Supposing that A owes B a certain sum, there is the risk that A may die before he is able to pay B ; another party engages, for a certain yearly sum, to pay B in ease A should fail to do so during his life time. In some instcr.ces, insurances are effected to gain a support in case of siokuess. Such a contract is called a Health Jnturance. In- surances are now also effected for compensation in case of railway Bcoidonts. These wc may call Railwai/ Accident Inmrances. A policy is often transferred from one party to another, especi- ally 08 collateral security for debt or somo analogous obligation. If the payments agreed upon are not regularly kept up, the policy lapses, that is, becomes null and void, so that the holder of it forfeits not only his claim to tho sum insured, but also the inHtnlnieDts pre- viously paid. In many companies a person can insure in such a way as to bo entitled to have a share of tho profits. Tho date at which the system of insurance began cannot be clearly ascertained ; but, whatever its date, its origin seems to have been protection against the perils of tho sea. Wo know that it was practised, in a certain way, by the ancient Greeks and Romans. If a Boman merchant sent a cargo to a distant port, ho made a contract with somo one engaged in such business, that he would advance a certain sum, to be repaid with interest, if tho vessel reached her destination in safety, but should the vessel or cargo, or both bo lost, the lender was to bear the loss. This was termed respondentia, (a rcspondence) a term corresponding pretty nearly to tho English word repayment. It was lawful to charge interest in such cases, above tho legal interest in ordinary eases, on account of the great- ness of the risk. Jhe lender of the money usually sent an agent of his own on board tl o vessel to look after the eai^o, and receive the repayment on tho safe delivery of tho goods. This agent corres ponded preUy nearly to our more modern supercargo. As the art of navigation advanced, and the seoorities afforded by law became INBUIUNCE. IBl more ntringent, and also facilitioM of coiuinunication iiioreaMd, thin sytitoni gradually gavo way, and hoH eventually been supplanted by comnnniioationH by post, and telegraphic lucssogc.^ to agents at the porta uf destination. With regard to the cquitab^oness of inBorances, and their utility in promoting commercial cxtcrprisc, wo may remark that they make tho interest of (^vcry merchant, the interest of every other. To show this, w< inpare an tuBiirance office to a club. Suppo-" tho mcrch < anada to form a club, and establish a fiiPL om of- which ibcr, if a loser, was to be indomni6ed > .n ihiih that I ill on tho individual, except his share as a u^im- bcr of i vcn so tho insurance system causes that each spcculatoi iring his own stake, contributes so much to tho funds of n ooinpai y, which is bound to indemnify each loser. On the '^thcr hand, thf insurer or insuring company, gains in this way, that the profits accruing from cases where no loss is sustained, far exceed the cases where loss is sustained, and tho trifling expense of insuring is of no moment to tho insured, in comparison with tho damage of a disastrous voyage, or consuming conflagration . By the insurance system, loss is virtually distributed over a large commu- nity, and therefore falls heavily on no individual, i'rom which we draw our conclusion, that it is equivalent to a mutual mercantile indemnification club. We must now show tho rules of tho club, and principles on which its calculations are made. The principal thing to be taken into account, in all insurances, is tho amount of risk. For example, a store, where nothing but iron is kept, would be considered safe; a factory, where fire is used, would bo accounted hazardous, and one whero inflammable sub- stances arc used would bo designated extra hazardous, and the rates would be higher in proportion to the increased risks. As, however, the degrees of risk arc so very varied, only a rough scale can bo made, and hence the estimate is nothing more than a calculation of probabilities. In life insurances, the rates arc regulated chiefly by the age, and general health of the individual, and also by tho gen- eral health of the family relations. Connected with this is tho cal- culation of the average length of human life. Almost all the calculations in insurance come under two heads. First, to find the premium of insurance on ;i t^ivcn amount, and at a given rate ; and, secondly, to find how much must bo insured at a IMAGE EVALUATION TEST TARGET (MT-3) ^I0 1.0 I.I Iti^jS |25 ■tt Iii2 122 i« 12.0 us lit 11.25 III 1.4 ■ 1.6 Photographic Sciences Corporation 23 WfST MAIN STRIET WnSTH,N.Y. 14SS0 (716) 872-4503 .*%^ 182 ABrrBXEno. giTen rate, so that in oaae of loss, both the principal and premitun may be recovered. Ah the premium is reckoned as so much by the hundred, insur- ance is merely a particular case of percentage. Hence to find the premium of insurance on any given amount, at a ^ven rate per oent, ^rG deduce the following BULB. MuUipiy the given amount hy the rate per unit.* EXAMPLES. 1. To find the cost of insuring a block of buildings valued at $2688, at 6 per cent. ? Here we have .06 for the rate per unit, and $2688X.06=$161.28, the answer. 2. What will be the cost of insuring a cargo worth $3679, at 3 per cent.? The rate per unit is .03, and $3679 X- 03=1110.37, the answer. 3. A gentleman employed a broker to insure his residence and outhouses, valued at $2760, the rate being 8 per cent., and the bro- ker's charge 1^ per cent, ; how much had he to f <3,y ? The cost of insurance is $2760X08=$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 5^ per cent. ? . Ans. $70.40. 2. A ship and cargo, valued ' $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 oent. ) what is the whole cost ? •" '7 ; T Ans. $1900.00. 4. What will t)e the cost of insuring a building valued at $58,000, at 2| per cent. ? Ans. $1450.00. * It is plain that tbe rate can be found, if the amount and premium are given, and tlie amoimt can be found if the rate and premium are given. In the case of insuring property, a pmfeBBional surveyor \a often employed to value it, and likewise in tbe case of life insurance, a medical certificate is required, and in each case the fee must be paid by the person insured. As 100, the basis of percentage, is a constant quantity, when any two of the other quantities are given, the third can be found. i I INSURANCE. m 5. What must I pay to insure a hoaw valued at $898.50, at j peroent.? Ana. $673.88. 6. A village store was valued at $1180 ; the proprietor iurared it for six years ; the rate for the first year was 3^ per cent., with a reduction of |^ each succeeding year ; the stock maintained an aver- age value of $1«>68, and was insured eaeh of the six years, at 2^ per cent. ; how much did the proprietor pay for insurance during tho s'x years ? Ana. $397.53. 7. A store and yard were valued at $1280, and insured at 1§ per 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 Hope, requests K. Tomlinson, Toronto, to insure for him a building valued at $976 ; R. Tomlinson effects the insurance at 4^ per cent., and ohaj^es | per ct^nt commission ; how much has W. Smith to remit to R. Tomlinson, the latter having paid *he premium ? Aus. $46.36 9. The cost of injuring a factory, valued at $25,000, is $125 what is the rate per cent. ? Ans. ^ 10. A 1|^ per cent, insuring my dwelling house cost me $50 : what is the value of the house ? Ans. $4000.00, To find how much must be insured for, so that in case of loss, both principal and premium may be recovered. Bere it is obvious that the sum insured for must exceed the value of the property in the same ratio that 100 exceeds tho rate. EXAMPLE. To find what sum must be insured for on property worth $600, at 4 per cent., to secure both property And premium, we have as $100— 4=$96 : $100 : : $600 : F. P.=*i>-L ^ • 4. Tidman & Co., Montreal, order a quantity nf pork from T. 8. Goates & Son, Belfast, which amounts to $242.^ They insure it to Liverpool at ^ per cent., and from Liverpool u) Portland at 3 per cent., and in all cases so as to secure the price and the pre- mium both. How much does the whole insurance come to ? Ans. $87.12. 6. In order to secure both the v^Jue of goods shipped and the premium, at If per cent., an insurance is effiected on $1526.72. What is the value of the goods ? Ans. $1500.00. 6. The Mechanics' Institutojs valued at $18,000 : it is insured at 1^ per cent., so that in case of fire, the property and premium may both be recovered. For how much is it insured ? Ans. $18,227.85. 7. How much must be insured on a oai^ worth $40,000, at ^ ^ per oent., to secure both the value of the caigo and the cost of insoranoe? Ans. $40,201.00. '( LIFE mSITBANOE. 186 8. Tlio Ronb House, KingHstrect, Toronto, is vdned at, say, $150,000, and is insured at 1| per cent, so that in case of another conflagration, both the value of the property and the prciniuui of insurance may be recovered. For how much must it be insured ? Ana. $152,671.76, nearly. 9. A jail and court-house, adjoining chemical works, and there- fore deemed hasardous, will not be 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^ p noth premium and property. What is the value of the property ? % Ans. $12,500.00. 1\. What sum must be 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. J LIFE INSURANCE. A LiTK Insuranok may be effected either for a term of years or for the whole period of life. The former is called a Temporary Insurance, and binds the 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 demandablo 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 calculation of the average duration of life beyond different ages. This calculation is made from statistical returns called Bills of Mortality, and the result is called Thb Expectation of Life. The annual premium is fixed at such a rate as would, at the end of the expectation of life, 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 the supposi- tion that it is to be paid annually in advance. I . 186 abuhmetic. Liri INBUBANOK TABLB. A|«next Age next ljr«w. IjMn. ForUb. lyeor. 7 yeam ForUr* Birthday. Birthday. 15 .83 .85 1.44 38 1.19 1.28 2.76 16 .84 .86 1.47 39 1.22 1..31 2.85 17 .85 .87 1.51 40 1.24 1.36 2.95 18 .86 .88 1.54 41 1.27 1.41 3.07 19 .87 .90 1.58 42 1.81 1.47 3.19 20 .88 ' .91 1.62 43 1.35 1.54 3.32 21 .89 .92 1.66 44^ 1.40 1.62 3.45 22 .90 .93 1.70 40# 1.47 1.71 3.60 23 .91 .95 1.74 46 1.54 1.80 3.75 24 .92 .96 1.79 47 1.62 1.90 3.92 25 .93 .98 1.84 48 1.71 2.02 4.09 26 .95 .99 1.89 49 1.81 2.14 4.27 27 .96 1.01 1.94 50 1.91 2.28 4.46 28 .98 1.03 2.00 51 2.03 2.42 4.67 29 .99 1.05 2.06 52 2.15 2.59 4.89 SO 1.01 1.07 2.12 53 2.29 2.76 6.12 81 1.03 1.09 2.18 54 2.44 2.95 5.36 32 1.05 1.11 2.25 55 2.60 3.15 5.62 33 1.07 1.14 2.32 66 2.78 3.38 5.89 34 1.09 1.16 2.40 57 2.96 ,3.62 0.19 35 1.11 1.19 2.48 58 3.17 3.87 0.50 36 1.14 1.21 2.56 59 3.39 4.17 6.83 37 1.16 1.24 2.65 60 3.64 4.50 7.18 EXABIPLES. Supposing a young man, on coming of age, wishes to effect an insurance for $3000 for the whole period of his life. To find the annual premium which ho must pay, we look for 21 in the left hand oolumn, and opposite that, in the column headed for life, we find the number 1.66, which is the premium for one year on $100, and |g§=.01 66 is the premium on $1 for 1 year, and hence $3000 x .0166=$49.80, is the whole annual premium. If the insurance is to last /or seven years only, we find under that heading .92, and ^^g=.092, and $3000x.092=$27.60, the annual premium. If the insurance is to be for one year only, we find .89 under that head, and $3000x.089=$26.70, the premium. >-4' LIFE TN8UBANCE. 187 From these eiplanatbns we can now derive » rale for fiadiog tho anniul premium, when the ago of the indiTidoal and the sum to he insured for are known. » BULK. Find the age in the left hand column of the table, and oppotite this in the vertical eolnmn for the given period will be found the premium on $100 for one year, and thin divided by 100 will give the premium on %\ for one. year, and the given ium multiplied by thi$ toitt be the whole annual premium. XXEROISES. 1. What will be the^Rtaual premium for insuring a person's life, who is 18 years old, for $1000 for 7 years ? Ans. $8.80. 2. What amount of annual premium must be paid by A. B. Smith, who wishes to insure his life for 7 years for $2000, his a^^e being 25 years ? Ans. $19.G0. 3. John Jones, 35 years of age, wishes to oflfeet an insurance for life for $1500. What amount of annual premium must he pay ? - Ans. $37.20. 4. A gentleman in Toronto, 32 years of age, being about to start for Australia, and wishing to provide for his family in case of hifl death, obtains an insurance for seven years for $3000. What amount of annual premium must he pay ? Ans. $33.30. 5. Amos Fairplay, 48 years of age, being bound on a dangerous voyage, and wishing to provide for the support of his widowed mother, in case of accident to himself, insures his life for 1 year for $2500. What amount of premium must he pay ? Ans. $42.75, G. A gentleman, 50 years of age, gets his life insured for $3000, by paying an annual premium of $4.46 on eadi $100 insured; if he should die at the age of 75 years, how much less will be the amount of insurance than the payments, allowing the latter to be without interest ? Ans. $345 7. A gentleman, 45 years of age, gets his life insured for $5000, for which he pays an annual premium of $180, and dies at the age of 50 years. Suppose we reckon simple interest at 7 per cent, on his payments, what is gained by the insurance ? Ans. $3911. .■^■?': «. ?-A--,. •■:■--'■ ■■' '' it.v4 *■• 188 ARITHMETIO. PROFIT AND LOSS. In the langoage of arithmetio, (ho ezprossion Profit and Lo$$ is oraally appliod t(> something gained or something lost in mercantile transactions, and Jie most important rule relating to it directs how to find at what increasod rate above the cost price goods must be sold to produce a fhir remuneration for time, labour and expendi- ture ; or, in case of loss by unforeseen oiroumstanoes, to estimate the amount of that loss as a {i^iide in future transactions. There are other oases, however, which we shall consider in detail. OABB i.^ When the prime cost and selling pnifare known, to find (he gain or loss. BULK. Find, hy the rule of practice, the price at the difference between the prime cost and telling price, which will be the gain or loss ac- cording €u the selling price is greater or less than the prime cost; or, Find the price at each rate, and take the difference. KXAMPLXS. To find what is gained by selling 4 owt. of sugar, which oost 12^ cents per lb., at 15 cents per lb. Here the difference between the two prices is 2^ cents per lb., and 400 lbs., at 2^ cents per lb., will give $10. Also, 400 lbs. at 15 cents per Ib.=$60, and at 12| conts=$50, and $60— $50=$10. Again, if 120 lbs. of tobacco be bought at 92 ots. per lb., and, being damaged, is sold at 75 cents per lb., the loss will be a loss of 17 cents in the pound, and 120 lbs., at 17 cents per lb., is $20.40 ; or, 120 lbs., at 92 cents, will come to $110.40, and at 75 cents, to $90, and $110.40— $90=:$20.40. BXBE0ISB8. 1. If 224 lbs. of tea be bought at GO cents per lb., and sold at 95 cents per lb. ; how much is gained ? Ans. $78.40. 2. A grocer bought 24 barrels of flour, at $5.80 per barrel, and sold 12 barrels of it at $6.10 per barrel, 9 barrels at $6.20 per bar- rel, and the rest at $6.25; how much did he gain ? Ans. $8.55. 3. If a person is obliged to sell 216 yards of flannel, which cost him $86.40, at 37^ cents per yard ; how much does he lose ? Ans. $5.40. w PBOnr AHD LOBS. 189 4. If a datler buyi 78 bosheLi of potatoes, at 62^ oenU per buhel, and retail* them at 87^ cents per bnshel ; how much does he gain? Ans. $19.50. 5. A winemerohant bought 374 gallons of wine, at $3.20 per gallon, and sold it at $3.35 per gallon ; how mooh did he gain ? Ans. $50.10. A SI II. To find at what price any article must be sold, to gain a oertain rate per cent., the cost price, and the gain or loss per cent, being known. ' ' . BULK. '"■ MuUipljf the coit price iy 1 plus the gain, or 1 mimu the lots, XXAMPLK. If a qnantity of linen be bought for 75 cents a yard ; at what price most it be sold to gain 16 per cont. ? Since 16 per cent, is 16 cents for every dollar, each dollar in the cost price would bring $1.16 in the selling price, so that we have $1.16X.76==.86, or 86 cents. * ' =^ ' 7 li 190 ABl ' lHMlim Since the g which was 10 per cent, less than the cost ; what would have been the gain per cent, if it had been sold for $150 ? . run- 'ie- ^ Ans. 50. 192 ABITRICITIO. 7. A grooer lold tot »t 4& oento per pound, and thereby g»iaed 12| per oent. ; what would he have gained per cent, if ho had aold the tea at 54 oente per pound ? Ans. Sft. 8. A farmer auld oorn at 65 oenti per buahel, and gained 5 per oent. ; what per oent. would he have gained if he had told the oom at 70 oante per bwihel ? Ana. 18'|^. MIBOILLANKODS IXIB0ISI8. .^. 1. If I buy goods amounting to |465, and sell them at a guu of 15 per cent. ; what are my profits? 2. Suppose I buy 400J barrels of flour, at $16.75 a barrel, and ■eU it at an advance of | per cen(. ; how much do I gain ? Ans. 126.14. 3. If I buy 220 bushels of wheat, at 91.15 per bushel, and wish to gain 15 per oent. in selling it ; what must I ask a bushel ? 4. A grooer bought molasses for 24 cents a gallon, which he aold for 30 cents ; what was his gain per cent. ? Ans. 26. 5. A man bought a horse for $150, and a chaise for $250, and sold the ohaise 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. 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 it so as to have made 40 per oent. ? Ans. $1540. 9. A broker bought stocks at $96 per share, and sold them at $102 per share ; what was his gain per oent. ? Ans. 6^. 10. A merchant sold sugar at 6^ cents a pound, which was 10 per oent. 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 have gained per cent, if he had sold it at $5.25 per yard ? Ans. 24j|. 12. I sold a horse 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. $56. » . rnorrr and lohh. 198 13. A watch which cost me |30 I have mid Ibr $35, od a cradil of 8 moDthi , what did I gain bj my bargain, allowing money to bo worth H per cent.? Ani. $3.65. 14. Bought M yards of Uhadcloth, at $6.00 per yard; what must be my asking price in order to fall 10 per cent., and still owka lU per cent, on the ooet ? Ana. $6.1 1 1. 1 5. A farmer sold land at ft oonta per foot, and gained 25 per cent, more than it coat him ; what would have been hia gain or low per cent, if he had sold it at 3^ cents per foot ? Ans. 12^ per cent. loss. 16. What must I auk per yard for cloth that cost $3.52, so that I may fall 8 per cent., and still make 15 per cent., allowing 12 per cent, of solos to bo in bad debts ? Ans. $5. 1 7. 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 ho gain or lose by the oprrntion, and how much ? Ans. loss $270. 18. Bought 2688 yards of cloth at $2.16 per yard, and sold one-fourth of it at $2.54 per yard ; one-third of it at $2.75 per yard, and the remainder at $2,90 per yard. Find the whole gun, and the gain per cent. Ans. $1612.80 and 27^*^^ per oent. 19. A flour merchant bought the following lots : — 118 barrels at $9.25 per barrel 212 " 9.50 '« 315 " 9.12J « 400 " 10.00 " The expenses amounted to $29.50, besides insurance at tj'^ per cent. At what price must ho sell it per barrel to gain 15 per cent ? Ans., $11.05. 20. Bought 100 sheep at $5 each ; having resold them at once and received a note at six months for the amount ; having got the note discounted at the Royal Canadian Bank, at six per cent., I found I had gained 20 per oent. by the traDsaction. What was the selling price of each sheep? Ans.. $6.19t 'II s 194 AWTBMETIC. -;-• ^ -Z' • STORAGE. ''-> '--■ When a charge is made for the accommodation of having goods kept in store, it is called ttorage. * Acooants of storage contain the entries showing when the goods were received and when delivered, with the nun^^jcr, the description of the articles, the sum charged on each for a certain time, and the total amount charged, for storage, which is generally determined hy an average reckoned for some specified time, usually one month (30 days). EXAMPLES. 1. What will be the cost of storing wheat at 3 cents per bushel per month, which was received and delivered as follows : — Received, August 3rd, 1865, 800 bushels ; August 12th, 600 bushels. De- livered, August 3th, 250 bushels ; September 12th, 350 bushels ; September 15th, 400 bushels, and October 1st, the balance. SOLUTION. 1885. Bush. Days. Baeh. August 3. Received 800 x 6 = 4800 in store for one day. « 9. Delivered 250 Balance 550 X 3 =: 1650 in store for one day. " 12. Received 600 Balance 1150 X 31 =35650 in store for one day. Sept. 12. Delivered 350 Balance 800 X 3 = 2400 in store for one day. " 15. Delivered 400 Balance 400 X 16 = 6400 in store for one day. Oct. 1. Delivered 400 Total 50900 in store for one day. 50,900 bushels in store for one day would be the same as 60900^-30=1 696§ bushels in store for one month of 30 days, and the storage of 1697 bushels for one month, at 3 cents per month, would equal 1697X.03=$50.91. It is customary, in business, when the number of articles upon which storage is to bo charged, as found, contains a fraction lest STOBAOE. 195 than a half, to reject the fraction ; bat if it is ntore than a half, to regard it m an entire article. Fiom the solution of the forcing example, we dednoe the fol- lowing Multiply the number of huaJiels, barreh, or other articles, by the number of days they are in store, and divide the sum of the pro- ducts by 30, 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 be charged for that term. 2. What will be the cost of storing salt at 3 cents a barrel per month, which was put in store and taken out as follows : Put in, January 2, 1866, 450 barrels; January 3, 75 barrels; January 18, 300 barrels; January 27, 200 barrels; February 2,75 barrels. Taken out, January 10, 60 barrels; January 30, 150 barrels; February 10, 190 barrels; February 20, 300 barrels; March 1, 250 barrels ; and on March 12, the balance, 150 barrels. Ans., $39.44. 3. Received and delivered, on account of T. C. Musgrove, sundry bales of cotton, as follows : Received January 1, 1866, 2310 bales; January 16, 120 bales ; February 1, 300 bales. Delivered February 12, 1000 bales; March 1, 600 bales ; April 3, 400 bales; April 10, 312 bales ; May 10, 200 bales Required the number of bales remaining in store on June 1, and the cost of storage ^p to that date, at tho rate of 5 cents a bale per month. Ans., 218 bales in store; $321.18 cost of storage. 4. W. T. Leeming & Co., Commission Merchants, Montreal, in account with A. B. Smith & Co., Toronto, for stor^ of salt and gunpowder, which was received and delivered as follows : Received January 18, 1866, 400 kegs of gunpowder and 50 barreb of salt; January 25, 250 barrels of salt; February 4, 150 barrels of salt and 50 k^ of gunpowder ; February 15, 100 kegs of gunpowder ; March 5, 64 kegs of gunpowder ; April 15, 50 kegs of gunpowder and 75 barrels of salt. Delivered, February 25, 15 kegs of gunpowder, and 40 barrels of salt; March 10, 150 kegs of gunpowder and 285 barrels of salt; April 20, 200 k^ of gunpow- der ; April 25, 150 barrels of salt and 200 k^ of gunpowder. Required, the number of barrels of salt and k^ of gunpowder in I 196 ARITHMETIC. store May 1, and the bill of storage up to that date. The rate of ntorage for lalt being 3 cent« a barrel per month, and for gunpowder Is cents a kc^ per month. Ans. In store, 50 barrels of salt and 99 kegs of gunpowder ; bill of storage, $206.01. GENERAL AVERAGE. This is the term used to denote the contribution of all persons interested in a ship, freight, or cargo, towards the loss or damage incurred by any particular part of a ship, or cargo, for the preserva- tion of the rest. This sacrifice of property is called jettison, from the goods being cast into the sea to save the vessel ; although not only property destroyed in that way is the subject of general average, but also any damages or expenses voluntarily incurred for the good of all. For example, the expense of unloading the cargo that the ship may be repaired ; masts or sails cut away and abandoned to save the ship. The only articles exempt from contribution are provisions, wear- ing apparel of passengers, and wages of the seamen. The owners contribute according to the clear value of the ship and freight at the end of the voyage, after deducting the wages of the crew and other expenses. Goods that have been subject to jettison, and are lost, are valued, when the average is calculated at the place of the ship's destination, at the price they could have sold for there ; but when the average is to be ascertained at the port of lading, the invoice price is the stand- ard of value. In making an account of the articles which are to contribute, the property lost or sacrificed must bo included, and its owners must suffer the same proportionate loss as the rest. The losses to the dif- ferent parties interested in the vessel, freight, and cargo, are paid by their insurers. When repairs have to be made to a ship — new sails, masts, or rigging, for example, — one-third of the expense is deducted on ac- count of melioration, or the improved condition of the ship by these repairs. When the ship is new, and on her first voyage, the full amount of the expense of repairs is allowed in computation of the loss. , * OEIIERAL AVERAGE. 197 EXAMPLE. On the 26th June, 1865, the steamer Canada left Qnebeo for Liverpool, with a cargo as rollows: — tihippcd by T. A. Collins, $7480 ; K. Evans & Co., $5365 ; II. C. Wright, $1)218 ; W. Man- ning & Co., $11,428; E. Carpenter, $755l>. In the Gulf of SI. Lawrence a heavy gale was experienced, during which cargo to the value of $3498 was thrown overboard ; of this $1123.40 belonged to R. Evans & Co., and the balance to E. Carpenter. The necessary repairs of the steamer cost $876, and the expenses in port, while getting repaired, were $253. The steamer was valued at $100,000 ; gross freight, $4310. The seamen's wages were $860. What was the loss per cent., and what was the loss of each contributory in- terest ? SOLUTION . Loss /or general benefit. Contributory interests. Cargo thrown overboard,$3498 Value of steamer $100,000 Repairs to steamer less § 584 Invoice price of cargo.... 41,050 Expenses in port 253 Fr'ght less seamen's wages 3,450 Total loss $4335 Total contrib. int. .$144,500 $4335-^144,500=.03 loss per unit, or 3 per cent. $100,000 X03=$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. Evans & 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= 226.77, E. Carpenter's share of loss. 3.450 X .03= 103.50, Freight's share of loss. $4335.00, Total loss. $3000.00~837.00=$2163.00, balance payable by steamer. 1123.40—160.95= 962.45, balance receivable by R. Evans & Co. 2374.60—226.77= 2147.83, balance receivable by E. Carpenter. NoTK.— It 18 evident that since the steamer lost $837, ($584 by repairs, and $253 by flxp8n8e8,)-that the net amount rc(iuired teom the steamer will bo $30G0-S.?7=$2163. R. Evans & Co. having lost by merchandise being thrown overboard $1123.46, a sum greater than their share of the general loss, so that there must be due them $1123.40— !60.05=$9G2.45 ; so also the amount of K Carpenter's share of the general loss must be deducted from hfs individual loss in order to And the balance due bini. f 198 ABITHIISTIC. RULE. Find the rate per unit of loss, by which multiply the value of each contributory interest, and the product wiU be the share of loss to be sustained by each. i I > E X E R G 1 b E 8 . 1. The steamship Nova Scotian, on her trip from Qaebeo to Liverpool, was crippled in a storm, in consequence of which the cuptain had to throw overboard a portion of the cargo, amoanting in value to $4465.50, acd the necessary repairs of the vessel cost $423. The contributory interests were as follows : — Vessel, $30,000 ; gross freight, $6225 ; cargo shipped by J. Jones & Co., $3650 ; by Henry Anderson, $6500 ; by George Millan, $2000; by J. Foster & Son, $550 ; by Brown Brothers, $5450 ; and by Wilson & Carter, $8500. Of the cargo thrown overboard, there belonged to Henry Anderson the value of $3000, and to Brown Brothers the remainder, $1465.50. The cost of detention in port in consequence of repairs, was $116.50; seamen's wages, $2075. How ought the loss to be shared among the contributory interests ? Ans. 8 per cent. 2. The steamer Spartan left Toronto for Montreal, June 30th, loaded with 7210 bushels of spring wheat, shipped by J. 31. Mus- grovc. and invoiced at 95 cents per bushel; 4815 bushels of corn, shipped by Thomas A. Bryce & 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 Montreal, the steamer ran upon a rook, and the captain found it necessary to throw over- board 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 Montreal at an advance of 10 percent., the corn at an advance of 15 per cent., and the flour for $5 per barrel. The contributory interests were : — Steamer, $95,000 ; cargo, $ ; gross freight, $2361.20. The cost of repairs to steamer was $2198.15 ; cost arising from detention during repairs, $318 ; seamen's wages, $252.50. How much of the loss had each contribu- tory interest to bear ? Ans. 7^ per cent. 3. The propeller Edith left Hamilton for Kingston with 7600 bushels of wheat, valued at $1.25 per bushel, shipped by Dunn, Lloyd & Co., and insured in the Queen Insurance Company at 1| per cent. ; 9300 busheb of com, valued at 75 cents per bushel, TAXES AND TUBTOM DFTIES. 199 shipped by J. W. Roc, and ins.red in the i^tna Insnrancc Company at If per cent. ; 14,800 bushels of oats, valued at 37|^ cents per bushel, shipped by Morris, Wright & Co., and insured in the Provin- cial Insurance Company at 1 1 per cent. ; 1 ,800 barrels of flour, valued at $5.25 per barrel, shipped by Smith & Worth, and insured in tho Beaver Insurance Company at 1| per cent. When near Kingston, a collision with the steamer Spartan occurred, and it was found necessary to throw overboard the flour, 4600 bushels of oats, and 3150 bushels of 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 Royal for $25,000, at 2J per cent. The gross freight was $4950 ; seamen's wages, $340, and repairs to tho boat, $3953.75 ; what was the loss sustained by each of the contributory interests, the propeller bebg on her first trip ? TAXES AND CUSTOMS DUTIES. A tax is a money payment levied upon the subjects of a State, or the members of any community, for the support of the govern- ment. A tax is either levied upon the property or the persons of indi- viduals. When levied upon the person, it is called Vkjaoll tax. It may be either direct or indirect. When direct, it is levied from the individuals, or the property in the hands of tho ultimate owners. When indirect, it is in the nature of a customs or excise duty, which is levied upon imports, or manufactures, before they reach the consumer, although in the end they are paid by the latter. Customs duties are paid by the importer of goods at the port of entry, where a customrhouse is stationed, with government employees called custom-house officers, 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 i'rom one place to another. A specific duty is a certain sum paid on a ton, hundred weight* yard, gallon, &c., without regard to the cost of the article. An ad valorem duty is a percentage levied on the actual cost, or fair market value of tho goods in the country from which they are imported. 200 ABrrmiETic. Grou weight is the weight of p^ds, upon which a specifio daty is to be levied, before any allowances arc deducted. Net weight is the weight of the goods after all allowMoea lire deducted. . Among the allowances made are the following : — Breakage — an allowance on flnids contained in bottles or break*' ble vessels. Draft — the allowance for waste. Leakage — an allowance for waste by leaking. Tare and tret are the deductions made for the weight of the case or barrel which contains tho goods. When goods, upon which duty is payable, are exported to Canada from any foreign country, through the United States, under bonds, they are only valued for duty as if they were imported direct into Canada from the place of export. An Order in Council has extended this provision to free m well as dutiable goods. As customs duties are payable in gold, or its equivalent, upon goods imported from the United States, it is evident that an injus- tice would be done if the amount mentioned in the American invoice were taken to be the dutiable value of the goods. Yet, as the value of American currency, when compared with tho gold standard, varies almost every day, it would be a source of great confusion and in 8- gularity to allow each custom-house officer to affix his own value to the currency. To avoid this, notices are published every week in the Canada Gazette as to the rate of discount to be allowed on American invoices, which is fixed in accordance with the price of gold, as represented by exchange. This method is at once fair and productive of uniformity. EXAMPLES. To find the specific duty on any quantity of goods. Suppose a Montreal Provision Merchant imports from Ireland 59 casks of butter, each weighing 68 lbs., and that 12 lbs. tare is allowed on each cask, and 2 cents per lb. duty on the net weight. We find the gross is 59X68=4012 lbs. « tare is 59X12= 708 lbs. Hence the net weight is 3304 lbs. The duty is 2 cents per lb 2 The duty, therefore, is t66.08 TAXES AND OTSTOM DUTIES. 201 ,To find the ad valorem duty on any quantity of goods. Suppose a Hamilton dry goods merchant to import from Franco 436 yards of silk, at $1.75 per yard, and that 35 per cent, duty is charged on thorn. Hero we have first the whole price by the rule of Practice to be $763, then the rest of the operation is a direct case of percentage, and therefore we multiply $763, by .35, which gives $267.05, the amount of duty on the whole. Hence we have the following . ' "I " RULE FOR SPEOIFIO DUTY. Subtract the tare, or other allowance, and multiply the remainr der hy the rate of duty per box, gallon, &c. RULE FOR AD VALOREM DUTY. Multiply the amount of the invoice by the rate per unit. EXERCISES. 1. Find the specific duty on 5120 lbs. of sugar, the tare being 14 per cent., and the duty 2f cents per lb. Ans. $121.09. 2. What is the ad valorem duty on a quantity of silks, the amount of the invoice being $95,800, and the duty 62^ per cent ? Ans. $59,875. 3. At 30 per cent., what is the ad valorem duty on an importa- tion of china worth $1260 ? Ans. $378. 4. 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, invoic- ed at $4560, the duty being 40 per cent ? Ans. $1824. 6. What is the 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 contain- ing 75 gallons, at 18 cents a gallon ? Ans. $270. 8. A. B. shipped from Montreal 24 pipes of molasses, each containing 96 gallons ; 2 per cent, was deducted for leakrge, and 12 cents duty per gallon charged on the remainder ; how much was the duty? Ans. $270.95. m ABITHMETIC. 9. What is the ad valorem duty ou a shipment of cutlery from Sheffield, Englaud, to Montreal, the invoice amountiug to $840, aiid the duty charged 25 per cent. ? Ans. $210. 10. What is the duty on 11,900 lbs. of pepper at CJ cents per lb., the duty being charged at 3^ per cent., and the tare being 5 per fcent.? Ans. $25.72. 11. Peter Smith & Co., London, import from Cadiz, 80 baskets of port vine, at 70 francs per basket; 42 baskets of sherry wine, at 35 francs per basket ; 60 casks of champagne, containing 31 gallons each, at 4 francs per gallon. The waste of the wine in the casks was reckoned at a gallon each cask, and the allowance for breakage in the baskets was 5 per cent. ; what was the duty at 30 per cent., 18§ cents being taken as equal to 1 franc ? Ans. $776.54. 12. Mitchell and Graham, Glasgow, import from Quebec, per ship Walter Scott, 24 boxes of sugar, 400 lbs. each, at 5 cents ; 40 hogsheads of molasses, containing 63 gallons each, at 30 cents per gallon ; 260 boxes of oranges, at $2 per box, and 410 boxes of cigars, at $7 a box ; the tare on the sugar was 10 per cent., and tlie leakage on the molasses 2 per cent. ; the duty on the sugar and molasses was 24 per cent. ; on the oranges 8 per cent., and on the cigars 30 per cent. ; what was the whole duty ? Ans. $1184.09. 13. What duty would a merchant in Toronto have to pay on merchandise purchased in New York city to the amount of $1834.60, American currency, the rate of dut^ being 25 per cent., and a discount of 31^ per cent, being allowed at the custom-house on the invoice price ? Ans. $314.18. 14. John MoMaster & Co., of CoUingwood, bought of A. Smith, of Buffalo, N. Y., goods invoiced at $5430.50, which should have passed through the custom-house during the first week in May, when the discount on American invoices was 43^ per cent., but they were not passed until the fourth week in May, when the discount was 36f per cent. The duty in both cases being 20 per cent. ; what was the loss sustained by McMaster & Co. on account of their goods being delayed? Ans. $70.60. In the above case, at what per cent, higher would McMaster & Co. require to mark their goods, above cost, in order to make a clear gain of 25 per cent. ? STOCKS AND BONDS. 203 * .■ , ■ . ^, . :. . . _ , STOCKS AND BONDS. The capital of a company is called ita stock, and is UBually di- vided into portions or shares, which are subscribed for by those who intend becoming members, or stockholders in the company. The whole sum is very seldom paid in at once on each share, but is only paid in by instalments or calU, according as the money is required for the purposes of the undertaking. It very often happens that the whole amount of the stock is never called in ; fur example, if the shares are $100, the first call may be for $20, then the next for $10, and so on, as the necessities of the company demand. It is quite possible that more money may be required after the original shares have all been paid up. To raise this sum the company frequently is compelled to dispose of preference shares, upon which n certain rate of interest is guaranteed out of the first profita. The interest or profits paid upon stock are called dividend*^ because they are declared by the board to be the amount that the shareholders are entitled to have divided amongst them. W^en $100 stock sells for $100 cash, it is said to be vXpar ; when it can be placed on the market for more than $100 cash, it is at vk premium; and when it will only sell for less than $100 cash, it is at a discount. At the meetings of shareholders the election of officers and other questions are usually decided by vote. It is often provided that the right to vote shall not be increased in exact proportion to the num- ber of shares held by any one person. For example, all persons holding, say four shares and under, may cast a vote for each share ; those holding ten shares may give seven votes ; those holding twenty may give ten votes, and so on, with a provision that no shareholder shall have more than thirty votes. The vote may generally be given by a proxy, who must be a qualified stockholder. Governments, municipal corporations, and incorporated compa- nies, contract loans at some fixed rate of interest, and give their bonds to the lenders as security for the repayment. The interest is usually payable half-yearly, and the time for payment of the princi- pal is, in most cases, specified in the bond, altliough in the case of the National debt of England, the time for the payment of the principal is in the option of the government. The rate of interest upon this debt is only three per cent., and is equivalent to a perpe- 204 ARITHMETIO. tual annuity. These stocks are called the Funds, or ConnoU, a contraction for " consolidated annuitien." An the interest on ihom la fixed, the value of the principal will rise and fall according to tho abundance or scarcity of money, and itfl vnluo, from tinio to time. When capital is abundant and there are few channels for profitable investment, the rate of interest will be low, and the fundti will accordingly rise. On the other hand, when there is a demand for capital, and bterest is high, the funds will fall. When the rate of interest is fixed, the value of the principal must vary, and when the amount of the principal is fixed, the rate of interest upon it must vary, in order to make the stock marketable at all times. The interest upon bonds, or debentures, as they are sometimes called, is often expressed to be payable upon the presentment of coupons attached to the bonds originally, and cut off as they are required for use. Certificates of stock are given to shareholders by oompanios, to show what stock they are entitled to upon the books. CASE i'. The premium or diaoount being knowD> to find the muket value of any amount of stock. EXAMPLES. If G. W. B. shares are at 7 per cent, premiunif 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 are 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 depre- ciated value will be 3000 times .93, which gives $2790, and there- fore the loss would be $3000—2790=210. From this we derive the RULE. Multipltf tJie par value hy 1 plus or minus the rate per unit, according as the shares are at a premium or a discount. 8T00I8 AMD B0KD8 206 ^ BZBftOI>B8. 1. What is the market value of |450 atock, at S^ per cent, dit- coant? Ans. •411.75. 2. What is the value of 29 shares of $50 each, when the shares are 11 per cent, below par ? Ans. $1290.50. 3. A man purchased CO shares of $5 each, from an oil well company, when the shares were at a discount of 8 per cent., and sold them when they were at a premium of 10 per cent; how much did ho gain ? Ans. $54. 4. A man purchased $10,000 stock when it was at an advance of 8 per cent., and sold when it was at a discount of 8 per cent. ; how much did he lose ? Ans. $1600. 5. If a man buys 15 shares of $100 each, when the shares are at a premium of 5 per cent., and sells when they liave advanced to 12 per cent., how much does 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 vt premium of 8 per cent. In thb case it will require $108 to purchase $100 stock, and therefore $1.08 to purchase $1 stock, and ticncc the amount that can be purchased for $21600 will be represented by the number of times that $1.08 is contained in 21600, which gives $20000. Again : Let it be required to find how much stock can be pur- chased for $5520, when at u discount 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 times that .92 is contained in 5520, which gives $6000 stock. Hence we derive the ' • RULE. Divide the given turn by 1 plus or minus the rate per unit, accord- in tho par value will bo represented by the number of times 1.08 is contained in 1296, which gives $1200 for tho 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 will be represented by the number of times that .92 is contained in 1104, which gives $1200, the par value. Hence tho RULE. Divide tlie market value by 1 plu» or minu» the rate per «mV, according as the stocks are selling above or behw par. EXERCISES. 11. What is the par value of $24420, when stock is 11 per cent, above par ? Ans. $22000. 12. What is tho par value of $10800, when stocks are at a dis- count of 4 per cent. ? Ans. $11250. 13. When government stocks are at 6 pe*^ eont. prefiium ; how \.'3. $1910f^ i' cciit. discount ; Ans. 134. much will $20246 purchtu^o at par value ? 14. The shares in a canal company arc o ! i"' ^ how many shares of $100 will $11390 purchase ? 15. The shares of a British gas company were selling in 1848, at a discount of 12 per cent. ; a speculator purchased a certain num- l • of shares for £792 ; tho value of the shares suddenly rose to par ; now m%uy share > did he purchase, and how much did he gain ? Ans. 9 shares; £108 gain. 8TO0IB AMD DOKDB. 207 OASI IV . To find to wliAt ntte of iiit«rnst a given dividond oorroMpmidM. If A por8(»n rooeivcH fi 'dividend ul' 12 per cent, on nn iiiv«'MtiBC«t mado at 20 per cutit. abovu pur, (hr<'Nt, or .12, it FoIIuwh that the per cent, which was invested will hv rcprcxoittod hy tliu num- ber of times that 1.20 is contained in .12, which u .10 or 10 per cent. Hodoo tho RULE. Divide the rate per unit of dividend hy 1 ;)^im or minm the rate per cent, premium or discount, according a$ tk^ ttockt arc above or below par. *^ KXBROISKS. IG. If a dividend of 10 per cent, bo declared m stock vested at 25 per cent, advance ; what is tho corresponding it torcst ? Ans. 8 pur cent. 17. If a dividend of 4 per cent, bo declared on stock invested at 12 per cent, below par, what is the corresponding interest ? Ans. 4,«. 18. If money invested at 24 per cent, yields n lividend of 15 per cent., what is tho rate of interest ? Ans. 123^. 19. If railroad stock is invested at 18 per cent, above par, and a dividend of 6 per cent, bo declared, what is the rato ol interest ? Ans. 5.;'3. 20. If bank stock be invested at 15 per cent, below par, and a dividend of 10 per cent, declared, what is tho rate of interest ? Ans. 11}^. MIBOSLLANBOUS EXER0I8KS. 1. What must be paid for 20 shares of railway stock, at 5 per cent, premium, tho shares being $100 each? Ans. $2100. * To find at what price stock paying n given rate per cent divi(]«od can be purchased, so that the money invested shall produce a given rote of interest, divide thv rate per unii of dividend by the rate ver unit of interest. 208 ABFTHMETIC. 2. What is the par value of bank stock worth $8740, at a pre- mium of 15 per cent. ? Ans. $7600. 3. Railway stock was bought at 15f below par, for $1895.62^ ; how many shares were there, each share being $150 ? Ans. 15 shares. 4. If G per cent, stock yields 8 per cent, on an investment, at what per cent, discount was it bought ? Ans. 25. 5. If bank stock which pays 11 per cent, dividend, is 10 per cent, above par, what ia the corresponding rate of interest on any investment? Ans. 10. G. When 4 per cent, stocks were at 17^ discount, A bought $1000 ; how much did he pay, and how much did he gain by selling when stock had risen to 80^ ? Ans. $821.25, and $41.25. 7. What will $850 bank stock cost at a discount of 9f per oent., I 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. $10.03. 9. What income should I get by laying out $1G20 in the pur- chase of 3 per cent, stock at 81 ? Ans. $60. 10. What sum must be invested in the 4 per cent, stocks at 84, to yield an income of $280 ? Ans. $5880. 11. What rate of interest will a person receive by investing in the 4^ per cent, stocks at 90 ? Ans. 5 per cent. 12. A person transfers his capital from the 3^ per cent, stocks at 77, to the 4 per cent, at 89 ; what is the increase 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 value of 5J per cent, stock, so that after deducting an income tax of 2 cents on the dollar, it may produce 5 per cent, interest? Ans. 107|. 15. A gentleman invested $7560 in the 3^ per cent, stocks at 94J, and on their rising to 95 sold out, and purchased G. T. R. 4 per cent, stock at par; what increase did he 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 purchasing railway 5 per cent, stock at 95 ? Ana. $28. ft PARTNERSHIP. 209 17. A person sella $4200 railway stock which pays 6 per cent, at 115, and invests one-third of the proceeds in the 3 per cent, con- sols at 80^, and the balance in savings' 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, sells $5000 at 94f , and on their rising to 98f he sells $5000 more ; on their again rising he buys back the whole at 96 ; how much does he gain ? Ans. $75. 19. The sum of $4004 was laid out in purchasing 3 per cent, stocks at 89|, and a whole year's dividend having been received upon it, it was sold out, the whole increase of capital being $302.40 ; at what price was it sold out ? Ans. 93§. 20. Suppose a person to have been an original subscriber for 500 shares of $50 each, in the Royal Canadian Bank, payable by instal- ments, as follows : — J in three months, which he sold for 5.^ per cent, advance; § in 6 months, which brought him 6^ per cent, ad- vance, and the balance in nine months, which he was compelled to sell at 8| per cent, discount j what did he gain by the whole trans; c- tion? ' Ans. $808.33. PARTNERSHIP. Partnership has been defined to be the result of a contract, under which two or more persons agree to combine property, or labour, for the purpose of a common undertaking, and the acquisition of a com- mon profit. A dormant, or sleeping partner, is one who shares in the concern, but docs not appear to the world as such. A nominal partner is one who lends his name and credit to a firm, without having any real interest in the profits. All the partners may contribute equally to the business ; or the capital may be contributed by some or one, and the skill and labour by tho other. Or, unequal proportions may be furnished by each. The contract need not be in writing, but all parties to be bound must assent to it, and it is usually contained in an instrument called " Articles of Partnership.^' A dissolution can take place at any time by mutual consent. A partnership at will is one to which there is no limited time affixed for its continuance, and the whole firm may be dissolved 210 ABTTHMEnO. Ii by any of its members at a moment's notice. A doctunent is, how- ever, (Tcnerally drawn up and signed upon a dissolution, called » settlement, which contains a statement of the mode of adjustment of the accounts, and the apportionment of profits or losses. " EXAMPLE. Two persons, A. and B., enter into partnership. A. invests $300 and B. $400. They gain during one year $210^ what is each man's share of the profit? SOLUTION BT PROPORTION. A.'s stock, $300 B.'s " 400 Entire stock $700 : 300 : : $210 : $90 A.'s gain. " 700: 400:: $210: 120 B.'s " (( SOLUTION BY PERCENTAGE. Since the entire amount invested is $700, and the gain $210, the gain on every $1 of investment will bo 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 : $300 X. 30=$ 90 A.'s gain. 400X.30= 120 B.'s " Entire stock 700 210 Entire gain. Hence, — 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, &o is the whale gain or loss to each partner's gain or loss ; or, divide the whole gain or loss hy tlie number denoting the entire stock, and the quotient will he the gain or loss on each dollar of stock ; which multiplied by the number denoting each partner's share of the entire ttock^ will give his share of the entire gain or loss. EXERCISES. 1. Three persons, A., B., and C, enter into partnership. A. advances $500, B. $550, and C. $600 ; they gain by trade $412.50. What is each partner's share of the profit ? Ans. A.'s$125; B,'s $137.50; G.'s $150. PAItTNERSHIP. 211 2. A, B, C ODd D purchase an oil well. A pays for 6 shares, B for 5, C for 7, and D for 8. Their net profits at the end of throe months have amounted to $7800 ; what sum ought each to receive ? Ans. A, $1800 ; B, $1500 ; C, $2100 ; D, $2400. 3. A and B purchased a lot of land for $4500. A paid ^ of the price, and B the remainder ; they gained by the sale of it 20 per cent. ; what was each man's share of the profit ? Ans. A, $300 ; B, $600. 4. A captain, mate, and 12 sailors, won a prize of $2240, of which the captain took 14 shares, the mate 6, and the remainder was equally divided among the sailors ; how much did each receive ? Ans. The captain, $980 ; the mate, $420 ; each sailor, $70. 5. A and B invest equal sums in trade, and clear $220, of which A is to have 8 shares on account of transacting the business, and B only 3 shares ; what is each man's gain, and what allowance is mad» A for his time ? Ans. Each man's gain $60 ; A $100 for his time. 6. A, B, C and D enter into partnership with a joint capital of $4000, of which A furnishes $1000 ; B $800 ; C $1300, and D the balance ; at the end of nine months their net profits amount to $1700 ; what is each partner's share of the gain, supposing B to re- ceive $100 for extra services ? Ans. A, $400 ; B, $320 ; C, $520 ; D, $360. 7. Six persons. A, B, C, D, E and F, enter into partnership, and gain $7000, which is to be divided among them in the following manner : — A to fiave I; B, | ; C, ^ as much as A and B, and the remainder to be divided between D, E and F, in the proportion of 2, 2j^ and 3^ ; how much does each partner receive ? Ans. A, $1400 ; B, $1000 ; C, $800 j D, $950 ; E, $1187.50 ; F, $1662.50. 8. A, B and C enter into partnership with a joint stock of $30,000, of which A furnished an unknown sum ; B furnished 1^, and C 1^ times as much. At the end of six months their profits were 25 per cent, of the investment ; what was each man's share of the gain ? Ans. As, $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 certain field of wheat, for which they agree to take thQ 6\ua\ of 212 ARITHMETIC. $19.84 ; A and B calculate that they can do ^ of the work ; A and C ^ ; B and C ^ of it j how much should each receive according to thcso estimates ? Ans. A, $8.32 ; B, $7.04 ; and C, $4.48. |!i ii' To find each partner's share of the gain or loss, when the oapital is invested for different periods. EXAM PLE Two merchants, A and B, enter into partnership. A invests $700 for 15 months, and B $800 for 12 months; they gain $603 ; what is each man's sliaro of the profits? SOLUTION. $700xl5=$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 $1 0500 would yield, in one month, the same in- terest that $700 would in fifteen months. Likewise $800 invested for 12 months would be the same as $9600 for one month; hence the question becomes one of the previous case, that is, their invest- ments are the same as if they had invested respectively $10500 and $9600 for equal times ; hence the RULE. Multiply each man's stock by the time he continues it in trade ; then say, as the sum of the prodiicts is to each particular product, so is the whole gain or loss to each man's share of the gain or loss. EXERCISES. 11. A, B and are associated in trade. A furnished $300 for 6 months ; B, $350 for 7 months, and C, $400 for 3 months. Their profits amounted to $1490 at the time of dissolution ; what was the profit belonging to each partner ? Ans. A, $360 ; B, $490 ; C, $640. PABTNEBSHir. 213 12. A, B and C contract to perform a certain piece of work ; A employs 40 men for 4^ months ; B 45 men for 3^ months, and C 50 men for 2;^ months. Their profits, after paying all expenses, are $850 ; how much of this belongs to each ? Ans. A, 8340 ; B, $297.50 ; C, $212.50. 13. Four men, A, B, G and D, hired a pasture for $27.80 ; A puts in 18 sheep for 4 months; B, 24 for 3 months; C, 22 for 2 months ; and D, 30 for 3 months ; how much ought each to pay ? Ans. A and B each, $7.20 ; C, $4.40 ; P, $9. 14. On the first day of January A began business with a capital of $760, and on the first of February following he took in B, who invested $540 ; 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 tliey had gained $872 ; how much of this was each man entitled to ? Ans. A, $384.93 ; B, $250.71 ; C, $236.36. 15. Three merchants, A, B and C, entered into partnership with a joint capital of $5875, A investing his stock for 6 months, B his for 8 months, and C his for 10 months; of the profits each partner took an equal share ; how much of the capital did each invest ? Ans. A, $2500 ; B, $1875; C, $1500. 16. Two merchants, A and B, entered into partnership for two years ; A at first furnished $800, and at the end of one year, $500 more ; B furnished at first $1000, at the end of 6 months, $500 more, and after they had been in business one year, he was compelled to withdraw $600 At the expiration of the partnership their net profits were $2550 ; how much must A pay B who wishes to retire from the business ? Ans. $2190. 17. Three persons, A, B and C, form a partnership for one year, commencing January 1st, 1865 ; A puts in $4000 ; B, $3000 ; and C, $2500 ; April Ist, A withdraws $500, and B withdraws $600 ; June 1st, G puts in $800 more ; September 1st, A furnishes $700 more, and B $400 more. At the end of the year they find they have gained $1500 ; what is each partner's share of it ? Ans. A, $608.68; B, $423.31 ; G, $468.01. 18. John Adams commenced business January first, 1865, with a 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 fir-st, 1866, it was found that Adams had gained $2250 ; 214 ABITHMETIO. Hickman, $420 ; and WilliamH, $405 ; how long had Hickman's and Williams' money been employed in the buBiness, and what rate of interest per annum had each of the partners gained on their stock ? Ans. Hickman's, 8 months ; Williams', 6 months. Qoin, 22^ per cent, interest. BANKRUPTCY. When a trader in Lower Oanadi, and any person in Upper Canada, is unable to meet hb liabilities, he may make an assignment of all hidi property to an Official Assignee, to be by him distributed for the benefit of the creditors generally. Official Assignees aro ap- pointed by the different Boards of Trade for that purpose. Creditors may also, under certain circumstances, compel a dis- honest debtor to part with all his property for their benefit, and place his estate in bankruptcy. The shares of the property, which are divided among the credi- tors, are called dividends. The property to be divided is called asteU. EXAMPLE. A bankrupt owes A $400 ; B, $350, and C, $600; his net assets amount to $810 cash ; how much is he able to pay on the $1, and how much will each creditor receive ? solution: $400-{-$3504-$6P0=$1350, total liabilities. Now, if ho has $1350 to pay, and only $810 to pay it with, he will only be able to pay $810-=-1350=.60 or 60 cents on the $1. Therefore, A will receive |400x.60=$240 ; B, $350X.60=$210, and C, $600X.60 ==$360. Hence the RULE. Divide the net assets by the nuniber denoting the total amount of the debts, and the quotient will be the sum to be paid on each dollar, then mvMiply each man's claim by the sum paid on the dollar, and the product wiU be the amount he is to reeeive. BANKRUPTCY. 216 f EXKROIHEH. 1. A becomes bankrupt. IIo owes B, $800; €, $500; D, $1100, and C, $GGO. The us^ts amount to $1 110 ; how mach can he pay on the dollar, and how much doea each creditor reociTo ? Ana. IIo can pay 37 cents on tho dollar, and B rcccivcH $296 ; C, $185 ; D, $407, and E, $222. 2. A house becomes bankiiipt ; its liabilities are $17940 ; it^ assets arc $8970 ; what is tho dividend, and what is the share of tho chief creditor to whom $1282 are duo ? Ans. Tho dividend is 60 cents on the dollar, and tho principal creditor gets $641. 3. A shipbuilder becomes bankrupt, and his liabilities arc $303000 ; the premises, buildings and stock are worth $220000, and he has in cash and notes $12875 ; tho creditors allow him $3000 for maintenance of his family ; the costs are 3^ per cent, of the amount available for the creditors ; what is tho dividend, and how much does a creditor get to whom $1360.60 are due ? Ans. The dividend is 75 cents on the dollar, and the creditor specified gets $1020. 4. Foster & Co. fail. They owe in Toronto, $22000 ; in Mon- treal, $18000; in Hamilton, $17100; in Kingston, $16000; in London, $4400, and in Quebec, $4200. Their assets are: house property, $14000 ; farms, $2200 ; Cash in bank, $4400 ; railway stock, $4200; sundry sums due to them, $20135. What is the dividend, and how much goes to each city ? Ans. The dividend is 55 cents in the dollar ; $12100 are paid in Toronto ; $9900 are sent to Montreal ; $9405 to Hamilton ; $8800 to Kingston ; $2420 to London, and $2310 to Quebec. 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 Brycc & 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 pays 24 cents un the dollar, and the payments are : to Buchanan & Ramsay, $259.20 ; to Kinneburgh & MoNabb, $204 ; to Collier, Bros., $412.80 ; to David Brycc & Sou, $379.20 ; to Sinclair & Boyd, $232.80. 216 arubmetio. JSQUATION OF PAYMENTS. Equation of Payments is the process of finding tho average or mean time at which the payment of several sums, duo at different times, may all bo made at ono titno, no that neither the debtor nor creditor shall bo at any loss. The date to bo found is called tho cqmtted time. The mode of finding equated time almost universally adopted is very simple, though, as wc shall show in the sequel, not altogether correct. It is known as the mercantile rule. Let us observe, in the first place, that tho standard by which men of business reckon tho advantage that accrues to them from receiving money before tho time fixed for its payment, and the loss they sustain by tho payment being deferred beyond tho appointed time, is the interest of money for each such period. Thus, if $50 be a year overdue, tho loss is $3, at 6 per cent. ; and, if $50 bo paid a year in advance of tho time agreed upon, tho gain to the payee i& $3, at the same rate. In the former case, tho person receiving the money charges the payer $3 interest for tho inconvenience of lying out of his money, but, in the latter case, ho deducts $3 from the debt, for the advantage of having tho money in hand. If, on the 1st May, A gives B two notes, one for $50, at a term of three months, and tho other for $80, at a term of seven months, the first will be legally duo on the 1st August, and the 2nd on the 1st December; but A is not able to meet tho first at August, and it is held over till tho 1st November, when A finds himself in a position to pay both at once. The first is then three months ovor-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 tho same rate, is 40 cents, and accordingly A, in addition to the debt, pays B 35 cents. 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 1st August, and the other on tho 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 that a settlement is made on 6th November, we find that the 1st note is EQUATION OF PAYMENTS. 217 3 months aod 6 days over due, and the interest on it ' ** that period ia 80 cents, while the second will not bo due for 2 m^ .tis, and the interest on it for that period Ih 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 bo puid. 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 ns it becomes due. This needs no elucidation, nor is it often practised, except in the case of small debts, duo by persons of con- tracted means. The second is what has been illustrated above by the first exam- ple, viz., that interest is added for overdue money, and deducted for sums paid in advance of the stipulated time. The third has been illustrated by the seoond 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 conic 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 case. K. Evans owes J. Jones $200, which he undertakes to pay by two instalments of $100 each, with interest at s''' 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 K. E. tenders payment of the whole amount. For the accommodation of having the first payment deferred for one year he is to pay $0, /. e., $10C 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 $0. He has therefore, by the mercantile rule, to pay $106-|-94=:^$200, so that the $0 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 different dates. /•' 218 .- AUTHMma If A owes B $300, payablo at tho end of 4 months ; 1500, paya- ble at the cod of 6 months, and $400, payable at tho end of 10| months, to find at what time tho whole may bo paid, so that interest sliall bo ohaigeablo to neither party. Tho interest of $300 for 4 months is the same as the interest of $1 for 1200 months; the interest of $600 for 6 months is the same as the interest of $1 for 3000 months, and the interest of $400 for 10^ months is the samo as the interest of $1 for 4200 months. The sum of all these is 8400 months, and tho interest of the whole is tho same as the interest of $1 for 8400 months, and if $1 requires 8400 months to produce a certain interest, tho sum of all tho principals will require only the j^'qq part of 8400 months lo produce the some interest, and 8400-^1200=7, and hence the equated time is 7 months. BULK. Multiply each payment by the time that mtut ehjpte before it becomes due, and divide the zum of these products by the sum of the payments. BXAMPLE. To find tho equated time for the payment of three debts, tho first for $45, duo at the end of 6 months ; the second for $70, due at tho end of 11 months, and the third for $75, due at the end of 13 months. $45 X 6=$270 70X11= 770 75X13= 975 190 2015 and 2015-r-19=10f I, so that the equated time will be 10 months and 18 days, the small remaining fraction being rejected. Let us suppose that nothing is paid until the end of the 13 months, and all paid at once, then the amount to be paid will be, at G per cent.. For first debt overdue 7 months, $45+1. 57|, interest for 7 months $46.57^ For second debt overdue 2 months, $70-4-.70, interest for 2 months 70.70 For third debt just due, $75, no interest 75.00 $192.27^ EQUATION 07 PATMEMTS. Jl% The work mty often be mraowhat nhoKoned by ooonting th<- differeooes of time from the date at which the first pajment beeomcH dno, the mean time between the datefl when the first and last beoome due being alone required. If a person owes $1200 to bo paid in four instalment*, $100 in 3 months ; $200 in 10 months; $300 in 15 months, and $600 in 18 months, then the oxcessen of time of the last three above the first are 7, 12 and 1& months, and the work will stand aa below. $100 (no time.) 200X 7.^1400 300X12=3600 600X15=9000 •jr 1200) 14000(118 and ll|^3=14f months. This gives the BULK. Multipljf each debt, except the one firtt due, by the difference be- tween ita term and the term of the firat ; divide the sum of the pro- ducte hy the tum of the debts, the quotient with tlie term of the first added to it will be the equated time. Another method, which is often convenient, may be illustrated by the example already given, as the two operations will give the same result. Interest on $300 for 4 months=$ 6.00 Interest on 500 for 6 " = 15.00 Interest on 400 for lOJ " = 21.00 Interest on 1200 for 1 month=:6)42.00(7 months as before. BULK. Find the interest on each instalment for the given time, and divide the sum of these by the interest of the whole debt for one month, and the quotient will be the equated time. As the sum of the instalments is equal to the debt, the result will be the some for any rate of interest. For the first instalment, $300, overdue 3 months, A has to pay.. $4 50 For the second instalment, $500, overdue 1 month, A has to pay 2 50 $7 00 220 ARTTHICETIC. For tho third instuimont, |40U, not duo lor '.l^ inonthi, A hoM toj^ot $7 00 fo that tliu iuuount« of interest exactly bulancf, mid tho paying of tho whole, at tho end of 7 nionthH, in prccincly e(|uivalcnt to tho pay- ing ol' each instalment as it falls duo. The only diffurenoo that could arise is, that it might bo inoonvonieut for the creditor to lio out of the tirst instalment for tho thruo mouths. In all other respects tho Hcttlcniont is ntrictly equitahh, avcordiug to tiio uiuhnUinding that exists among busincsH men. In tho first place, the difference between thix and what is called " the accurate rule," is insiguifi- cantly small ; and, in tho second place, tho ' morcantilu rule" saves much time, and time is equivalent to so much capital in mercantile transactions. Independently, however, of any other consideration, wo may remark that when tho mode of reckoning is conventumally understood, it becomes perfectly equitable, becauso every merchant knows the terms on which he can do business with any other, just as bank discount becomes perfectly equitable, because every man, before going to a bank for tho discounting of a note, knows perfectly well on what terms ho can have it. Much warm discussion has been indulged in on this subject ; but, as we consider the discussion more subtle than profitable, we shall dis- miss the subject in a few words. We shall adopt tho usual case, thai A owes B 8200, one-half to bo paid at the present time, and the remainder at the end of two years. It is perfectly obvious that, at the end of the first year, A should pay 810G, that is, the principal, 'plus the interest agreed upon. Regarding the settlement of the second instalment, if A proffers payment of the whole at once, he is clearly entitled to claim a reduction for the unexpired term. Now, the question is, what ought tho reduction to be. liy the mercantile rule ho should pay $94, but the true present worth of $100, due ut the end of the year, would be 94.33^^, so that he would have to pay $10C on the instalment over due, and $94,330^ on the one not due, making $200.33^ J, whereas the object is to find at what time inter- est should be chargeable to neither party. As a further illustration of the general rule, let us suppose that J. Smith owes 11. 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, to find the equated time. We shall suppose that J. Smith agrees to pay R. Evans the whole amount at the time the debt was contracted ; then J. Smith would EQOATION or PATVENTH. 221 0W« K. Etms $1300, minua tlio diMOUut for tho length ol' timo i\w ninouot WM ptid before it becamo duo, vii., throo moDths, equalling th«- diiooant on $210 for 1 month ; $100, 1cm the diaoount for 4 months, equalling tho discount on $400 for 1 month ; $r>00, lomt thu diaoount for H months, equalling the diaoount on $4000 for I month. Thia gives a total of $2100-f$400-f-$4000:: $G500, for 1 month. Now, it is evident tliat if .f. Smith wished to pay tho whole amount at suoh a time that there should bo no loss to either party, hu must retain this amount for such a length of time as it will take this amount to equal the discount on $0500 fur 1 mouth, which will be jj'gn of $6600, that is, for b months. To prove that 5 months must bo tho equated time, wu have recourse to the principles laid down under the head of IntereHt. li a settlement is not made until the expiration of 5 months from the time the debt was contracted, then J. Smith would owe R. Evans $700, plua the interest of that principal during tho timo it remained unpaid after becoming due, viz., two months, which would give nn amount of $707. So also, $100, j92u« tho interest for 1 month, would bc> $100.50, and $500, mintu its discount for 3 months (thu length ul time paid before due), would give $7.50, leaving $492.50, and |707+$100.50+$492.&0=$1300. E XEROISES. 1. T. C. Musgrove owes H. W. Field $900, of which $300 arc due in 4 months ; $400 in months, $200 in 9 months ; what is the equated time for the payment of the whole amount ? Ans. months. 2. E. P. Hall & Co. have in their possession 5 notes drawn by G. W. Armstrong, all dated 1st January, 1865 ; the first is drawn at 4 months, for $45 ; the second at 8 months, for $120 ; the third at 10 months, for $75 ; the fourth at 11 months, for $60 ; and the fifth at 15 months, for $90; for what length of time must a single sete be drawn, dated 1st May, 1865, so that it may fall due at the properly equated time ? Ans. 6 months. 3. A merchant sold goods as follows, on a credit of 6 months : — May 10, a bill of $600 ; June 12, a bill of $450; September 20, a bill of $900 ; at what time will the whole become due ? Ans. January 16. ' 4. A merchant proposed to sell goods amounting to $4000 on 8 months' credit, but the purchaser preferred to pay h in cash and ^ in 3 months ; what time should be allowed him for the payment uf the remainder? , Ans. 2 years, 5 months. 222 ABrrHMETIO. 5. A gcntlomon left his son fl500, to be paid as follows: ^ in 3 months, j^ in 4 months, ^ in 6 months, and the remainder in 8 months ; at what time ought the whole to be paid at once ? Ans. 4 mos., 15 dap. 6. A merchant bought goods amounting to $6000. He agrees to pay $500 in cash, $600 in six months, $1500 in 9 months, and the remainder in 10 months ; at what time ought lie to pay the whole in one payment? Ans. 8|^ months. 7. There is due to a merchant $800, one-sixth of which is to be paid in 2 months, one-third in 3 months, and the remainder in 6 months ; but the debtor agrees to pay one-half in cash ; how long may he retain the other half, so that neither party may sustain loss? Ans. 8^ months. 8.' A merchant sold to W. L. Brown, Esq., goods to the amount of $3051, on a credit of 6 months, from September 25th, 1864. October 4th Brown paid $476 ; November 12th, $375 ; December 6th, $800 ; January Ist, 1865, $200. When, in equity, ought the merchant to receive the balance ? Ans. Oct. 8th, 1867. 9. A having sold B goods to the amount of $1200, left it optional with him cither to take them on 8 month's credit, or to pay one-half in cash, one-fifth in two months, one-sixth in four months, and the remainder at an equated time, to correspond with the terms first named ; what was the time ? Ans. 4 years, 4 mos; 10. A grocer sold 484 barrels of rosin, as follows : February 6tb, 35 barrels @ $3.12^^, on 4 months' time. March 12th, 38 barrels @ 3.00, on 4 months' time. March 12th, 411 barrels @ 2.62^, on 4 months' time. What is tho equated time for the payment of the whole ? Ans. July 8th. 11. Bought of A. B. Smith & Co. 1650 barrels of flour, at dif- ferent times, and on various terms of credit, as by the following statement ; what is the equated time for the payment of the whole ? May 6th, 150 barrels, at $4.50, on 3 months' credit. May 20th, 400 barrels, at 4.75, on 4 months' credit. July 10th, 500 barrels, at 5.00, on 5 months' credit. August 4th, 600 barrels, at 4.25, on 4 months' credit. Ans. November 6th. 12. J. B. Smith & Co. bought of A. Hamilton & Son 576 bar- rels of rosin, as follows : May 3rd, 62 barrels @ $2.50, on 6 months' credit. AVERAGING ACCOUNTS. 223 May lOih, 100 barrels (S, 2.50, od G months' credit. May 18tii, 10 barrels @ 2.50, &a cash. May 26th, 50 barrels @ 2.75, on 31) days' credit. May 26th, 345 barrels @ 2.50, on six months' credit. May 26tli, 9 barrels @ 2.00, on six months' credit. What is the equated time for the payment of the whole ? Ans. November 2nd. 13. Purchased goods of J. R. Worthington & Co., at different times, and on various terms of credit, as by the following statement : March 1st, 18C3. a bill of $675.25, on 3 months' credit. July 4th, 1863, a bill of 370.18, on 4 months' credit. September 25th, 1863, a bill of 821.75, on 2 months' credit. October 1st, 1863, a bill of 961.25, on 8 months' credit. January 1st, 1864, a bill of 144.50, on 3 months' credit. February 10th, 1864, a bill of 811.30, on G months' credit. March 12th, 1864, a bill of 567.70, on 5 months' credit. April 15th, 1864, a bill of 369.80, on 4 months' credit. What is the equated time for the payment of the whole ? Ans. March IGth, 1864. AVERAaiNa ACCOUNTS. When one merchant trades with another, exchanging merchan- dise, or giving and receiving cash, the memorandum of the transac- tions is called an Account Current. If the goods be purchased at different dates, or for different terms of credit, and some arc not duo while others are overdue, the fixing on a time when all may be set- tled, so that no interest shall be chargeable to cither party, is called Averaging the Account. Since interest is the standard to which is referred the benefit of receivbg money before it is due, so that in the meantime it can be used in trade, and also the damage of not getting it when due, it is fair and proper that interest should be charged on all sums overdue, and deducted from all not due. In illustration, let us suppose that A sells goods to B, March 2, on 4 months' credit, and again an equal amount on March 20, on G months' credit ; the first will be due on July 2, and the second on September 20. Should B tender payment of the whole on June 2, he would be entitled to claim interest for '.'•■ '■ .' "'' "' 1""' 224 ABITHMEnO. one month on the fint purchase, and for throe months and ei^tcen days on the second. But if payment be delayed till August 2, A would be entitled to one month's interest on the first purchase, and B to the interest on the second for one month and eighteen days, so that there would be in favour of B, on the whole, a balance of inter- est for eighteen days. Again, supposing the settlement is not made till September 20, when all is due, no interest can be either charged or claimed on the second purchase, the term of credit having just then expired ; but as the first debt iB two months and eighteen days overdue, A is entitled to interest on it for that period. If neither is paid till after September 20, A has a right to claim interest on each for the period it has been overdue. But this r^ulates only one side of the account. In order to settle the other, let us suppose that B has, in the meantime, sold goods to A, it is obvious that B's claims on A must be settled on the very same principle, and that therefore the final result must be simply the finding of the balance. It is more usual, however, in accounts current, to fix on a time such that the interest duo by A shall exactly balance that due by B. To illustrate this, let us suppose a case corresponding to a ledger aooount : ^^^/^/f. R. EVANS. - ^..,-^My -..■ - 1865. Bb. July 21, To Merchandise on 2 months' credit... $200 July 25, To Cash 160 Aug. 24, To Merchandise on 4 months' credit... 100 Sept. 21, To Merchandise on 3 months' credit... 250 i" "' "V ^ •- V.-V ;■ * $700 1865. '^■'^'" ■ ' '"■'^"^"'"^' >• '-"''''J'' Ob. August 1, By Cash $100 /. August 20, By Merchandise at 22 days 110 ' Sept'r 30, By Cash 180 ■B- Balance 310 To find in this case at what time the account may be settled so that interest shall be chargeable to neither party. Equating the time, as in equation of payments, we have the following operation : \\ f AYEBAOINCi ACOOUIfTS. 226 \\ Db. 1865 July 26 160X Sept. 21 200X 68=11600 Deer. 24 260 X 162=38000 Deer. 21 100x149=14900 Ob. 1866. August 1 lOOX Sept. 12 110X22= 2420 Sept. 30 180X71=12780 390 15200 16200—390=39 days. Due 39 days from August 1, viz., on September 9. 700 64600 64500-^700=92 days. Due 9'i days from July 25, viz., on October 25. Time from September 9, to October 26=46 days. Excess of debit above credit 700—390=310. 390X46=17940, and 17940-r-3l0=68 days, nearly. Counting 68 days forward, from October 25, will bring us to December 22, the time required for a settlement, with interest chargeable to neither party. Here the time is counted forward from the average date of the lai^r side which becomes due last, but had it become due first, we should have counted backward. The first transaction on the debit side being two months' credit from July 21, is not to be taken into consideration till September 21. The second transaction, being a cash one, and therefore consid- ered as so much due, will therefore mark the date from which all others shall be reckoned ; and, since there is no interval of time, we write it without a multiplier. The next transaction has a term of credit extending to 152 days, and therefore wo write 260x162= 11600. The term of the next extends from September 21 to December 21, a period of 149 days, and we write 100x149=14900. The sum of the debits is $700, and the sum of the results obtained by multiplying each item by the number of days it has to run from July 25 is $64500. Then 64500-^700=92, the equated time in days for the debit side. Now, as already explained, the interest for $700 for 92 days will be the same as the interest of $64500 for 1 day. Hence, the debits are due 92 days from July 25, viz., on October 25. In like manner, on the credit side, the first transaction being a cash one, we start from its date, August 1, and, as there is no inter- val, we have no multiplier. The second being merchandise, on 22 226 ARITHMETIC. dayH* credit, wo write 11 0x22=^2420. The third is cash paid 71 days after August 1, and we write 180x71=12780. Had the account been settled on September 9, the debits would have been paid 46 days before coining due, and the credit side would have gained and the debit side lost the interest for that time. Again, wo must consider how long it would take the balance, $310, to produce the same interest that $390 would produce in 46 days. It is obvious that whatever interest $390 gives in 46 days will require 46 times $390 for $1 to produce the same interest, tliat is, 390x46rr=17940 days, and it will require 17940^310=58 days, for $310 to produce the same interest. If the settlement is made on October 25, the latest date, then the credit has been due 46 days, and therefore bearing interest ; and in order that the debit side may be increased by an equal amount, the time must be ex- tended beyond October 25, that is, it must be counted /ortrarc^. For the same reason, if the greater side had become due first, then the balance must be considered as due at a previom date, and therefore Mre must count backward. An account may be averaged from any date, but either the first or the last will be found the most oonvonient. The first due is generally used. On the principles now explained may be founded the following BULB. Find the equated time token each side becomes due. Multiply the amount of the smaller side by the nunAer of days between the two average dates, and divide the product by the balance of the account. The quotient thus obtained will be the time that the balance becomes due, counted from the average date of the larger side, FOR- WARD when the amount of that side becomes due LAST, but baok- WARD when it becomes due FIRST. ' " The cash value of a balance depends on the time of settlement. If the settlement be made before the balance is due, the interest for the unexpired time is to be deducted ; but if the settlement is not made till after the balnnco is due, interest is to be added for the time it is overdue. EXEROISES. In J. H. Marsden's Ledger, we find the following aoooants, which, AVEUAOINO ACCOUNTS. 227 on being equated, stand as follows ; bt what time should the respeo- tivQ balances commence to draw interest : 1. Dr. J. S. PzcKHAU. Cr. May IGth, 1865 $724.45. | July 29th, 1865 $486.80. Ans. December 15th, 1864. 2. Dr. Nblsont Bostpord, Cr. November 19th, 1865 $635. | December 12th, 1865 $950. Ans. January 27th, 1866. 3. Dr. James Crow & Co. Cr. February 24th, 1866.... $512.25. | June 10th, 1865 $309,70. Ans. March 27th, 1867. 4. Dr. J. H. BuREiTT & Co. Cr. March 17ih, 1866 $145. | January 15th, 1866 $695.60. Ans. December 30th, 1865. b. Dr. M. McDonald. Cr. August 27th, 1865 $341. | November 7th, 1865 $247. 6. Dr. James I. Musorove. Cr. July 20th, 1866 $711. | April 14th, 1866 $1260. Ans. December 9th, 1865. 7. Dr. Thos. a. Bryce & Co. Cr. June 24th, 1864 $1418. | September 7th, 1865 $2346. 8. Dr. B. R. Carpenter. Cr. December 2nd, 1865... $1040.80. | August 13th, 1865.... $1112.40. 9. Required the time when the balance of the following account becomes subject to interest, allowing the merchandise to have been on 8 months' credit ? Dr. A. B. Smith & Co. ' Cr. 1864. May 1, July 7, Sep. 11, Nov. 25, Dec. 20, To Mdse. $300.00 759.96 417.20 287.70 671.10 1865. Jan. 1, Feb. 18, Mar. 19, April 1, May 25, By Cash.. " Mdse. '< Cash.. " Draft., « Cash.. $500.00 481.75 750.25 210.00 100.00 ! Ans. August 5, 1865. 228 ARITHHETIC. 10. When will the balance of the following Moount fall due, the merchandise items being on 6 months' credit ? Dr. J. K. WniTB. Cf. 1865. May 1, May 23, June 12, July 29, Aug. 4, Sept. 18, To Mdse. Cash paid dft. . Mdse Cash.....!!!!!!.. $312.40 85.70 105.00 243.80 92.10 50.00 lg65. June 14, July 30, Aug. 10, Aug. 21, Sept. 28, By Cash... >i Mdse... <( Cash... (( Mdse... <( <( * • • $200.00 185.90 100.00 58.00 45.10 *'■' Ans. January 12,1866. 11. When does the balance of the following aooount become subject to interest ? Dr. ' W. H. MusoROVB. ^ Cr. 1864. Aug. 10, Aug. 17, Sept. 21, Oct. 13, Nov. 25, Nov. 30, Dec. 18, 1865. Jan. 31, 1864. To Mdse 4 mos. $285.30 Oct. 13, " " 60 days 192.60 Oct. 26, u u 30 « 256.80 Dec. 15, " Cash p'd dft. 190.00 Dec. 30, " Mdse 6 mos. 432.20 1865. " " 90 days 215.25 Jan. 4, " " 2 mos. 68.90 Jan. 21, " Cash 100.00 By Cash. " Mdse2moB « u ^ « '' Cash. $400.00 150.00 345.80 230.40 340.30 180.00 12. In the following account, when did the balance become duo, the merchandise articles being on 6 months' credit ? Dr. B. J. Bbyoe in account with D. IIiCKS & Co. Cr. 186 Jan. 4. 4, Jan. 18, Feb. 4, Feb. 4, Feb. 9, Mar. 3, Mar. 24. April 9, May 15, May 21, To Mdse $ 96.57 57.67 80.00 38.96 50.26 154.46 42.30 23.60 28.46 177.19 1864. Jan. 30, April 3, May 22, By Cash.. . (t « • • • • • • i $240.00 a t( 48.88 *' Cash paid draft. " Mdse 50.00 " Cash paid draft. " Mdse « « \- . (( i( r- t • ■ Ans. December 22nd, 1864. 13. When, in be |«yable ? Dr. V,v..; AVERAOINO ACCOUNTS. 229 equity, should the baluoe of the following account J. McDonald k Co. Or. 1865. 1864. Jan. 3. To Catjh.... $200 Sept. 20, By Mdse, 6 mos... $583.17 Jan. 31, 300 Oct. 27, " 4 " .. 321.00 Feb. 8, 75 Deo. 5, " 6 " .. 137.00 Feb. 21, 100 1865. Mar. 10, 350 Jan. 18, •• 60 days.. 98.76 Mar. 24, 25 Feb. 26, " 6moe... 53.98 Apr. 12, 40 Apr. 15, " 4 " . . 634.00 Juno 1, 80 June 12, " 2 " .. 97.23 June 20, 125 Sept. 21, " 6 " .. 84.00 July 4, 268 Deo. 29, " 6 " .. 132.14 Sept. 27, 250 n , k. r Deo. 9, • ••• 100 .^ ' W . i f^ - ? Aub. October 10, 1866. ' t.f'^-; *■»■ To find the true cash balance of an account current when each item draws interest. EXAMPLE. .;.-■ What is the balance of the following account on January 19th, 1866, a credit of three months being allowed on the merchandise, money being worth 6 per cent. ? Dr. \x>: \'\' MusoBOVE & Wbioht. Cn 1865. Mar. 12, Apr. 21, May 6, May 27, July 16, Sept. 10, Oct. 19, To Merchandise... " Cash paid draft " Mdse " Cash " Mdse 1865. $340.00 Apr. 20, By Mdse ... 1 150.00 May 4, " Cash ... 165.00 June 15, « « 215.00 Aug. 10, " Mdse... 100.00 Sept. 23, " Cash ... 310.00 Nov. 12, " '* 120.00 Dec. 15, (( <( $200.00 110.00 230.00 180.00 50.00 50.00 100.00 280 ABTTHIOETIG. SOLUTION Diif. June 12, July 21, May 6, Aug. 27, July 16, Dec. 10, Jan. 19, Debits. $340X221= 160X182. 165X268= 216X146= 100X187= 310:< 40= 120X 0= 76140 27300 42670 31176 18700 12400 $1400 6)207286 $34,647 Du«. July 20, May 4, Juno 16, Nov. 10, Sept. 23, Nov. 12, Peo. 16, Credits. $200x183= 110X260= 230X218= 180X 70= 60x118= 60 X 68= 100 X 36= 36600 28600 60140 12600 6900 3400 3600 $920 6)140740 $23,466 The diflfcrent items on the dohit and credit sides of the account being on interest from the date on which it becomes due until the timo of settlement, the total interest of all the debit items will be the same as the interest of $207286 for one day, or the interest of $1 for 207286 days, which is $34,647. So also, the total interest of all the credit items will bo the same as the interest of $140740 for one day, or the interest of $1 for 140740 dayn, which is $23,466. Now, since each side of the account is to bo increased by its interest, the cash balance will bo represented by the number denoting the differ- ence between the two sides of the account, after the interest is added ; thus, $1400+ $34.647=:$1434.647, amount of debit side, and $920 -f $23.456=$943.466, amount of credit side, then $1434.647— $943.456=$49.109, cash balance. SEOOND METHOD. Debits. Credits. Days. Int. Days. Int. Int. on $340 for 221=$12.623 Int. on $200 for 183= $6,100 f( 150 " 182= 4.650 110 " 260= 4.766 u 166 « 258= 7.096 230 " 218= 8.356 u ■ 216 " 145= 5.196 180 " 70= 2.100 a 100 " 187= 3.116 60 " 118= .983 ■ t*, 310 « 40= 2.066 50 « 68= .666 ■ jr 120 " 100 " 36= .583 $1400 $34,545 $920 $23,454 Now, $34,545 debit interest— $23,454 credit interest=$11.09. CASH BALANCE. 291 the balance of interont, and $1400, amount of ilebit items-}- til. 09 =r|1411.09, aud 1141 1.09-1920 amount of credit itcmR=$491 .09 the cash balance, v^ ioh is the Hamo oa obtained by the firat eolation. Hence from the lioregoing we deduce the following BULL Multiply each item of debit and credit by the nnmber of days interveniny between its becoming due and the time of settlement. Then consider the sums of the products of the debit and credit items as so many dollars, and find the interest on each /or one day, which will be the interest, respectively, of the debit and credit items. Place the balance of interest on its own tide of the account, and the difference then between the two sides toill be the true balance ; or, Find, the interest on each item from the date on which it becomes due to the time of settlement. The difference of the sums of interests, on the debit and credit sides of the account will represent the balance of interest, which is placed on its own side of the account, and the difference then between the two sides will be the true balance. Note.— If any item should not come due until after the time of settlement, the Bide upon which it is, should be diminished, or (he opposite side in- creased by the interest of stich item from the time of sMtlement until due. EXERCISE8. 1. What will bo the cash balance of the following account if settled on January 1, 1865, allowing interest at 8 per cent, on each item after it is due ? Dr. R. Evans in account with John Jones. dr. 1864. June 11, June 29, July 18, Aug. 25, Aug. 31, Sept. 3, Sept. 20, Oct. 14, Oct. 19. To (( (( « « « Mdse, 4 mos. u (3 « Gashp'ddft. Cash Mdse,2 mos. " 1 " Cash Mdse,a8 cash $315.00 180.00 200.00 75.00 50.00 100.00 80.00 150.00 300.00 1864. Apr. 15, May 10, June 12, June 30, July 15, July 27, Aug. 6, Aug. 20, Aug. 30, By Mdse, 3 mos. " 6 " Cash it K $350.00 120.00 240.00 100.00 90.00 80.00 100.00 175.00 75.00 2. A.B. as follows :- Mdse,ascash Cash. Mdse, 3 mos. Ans. $110.86. Smith is ir account and interest with J. K. Amos & Co., -Debtor, January 1, 1865, to merchandise, on 6 months, ▲Rirmano. $166.10 i February 3, to oMh paid draft, $100 ; March 20, to mer- chandiM, on 4 months, $316.90 ; March 30, to merohandiae, un 4 months, $162; May 15, to oash paid draft, $100; Aognst 20, to mcrcbandiBo, on 6 months, $213. Creditor, February 1, by cash, $120; March 20, by morohandise, on 4 months, $420.16; May 1, by merchandise, on 6 months, $300 : July 1 , by merchandise, on 4 months, $50 ; September 10, by merchandise, on 4 months, $99.84- Required, the true balance, if settled on December 1, 1866, interest being at 6 per cent. ? Ans. $61.36. 3. Required the true balance, March 25, 1865, on the following account, each item drawing 7 per cent, interest from its date. A. B. Lyman in account and interest with John Russell & Co. : — Debtor, July 4, 1864, to merchandise, $200 ; September 8, to mer- chandise, $300 ; September 25, to merchandise, $250 ; October 1, to merchandise, $600 ; November 20, to merchandise, $400 ; Decem- ber 12, to merchandise, $500; January 15, 1866, to merchandise, $100; March 11, to merohandidc, $120. Creditor, July 20, 1864, by oash, $300; August 15, by cash, $350 ; September 1, by oash, $400; November 1, by cash, $320; December 6, by merchandise, $600 ; December 20, by cash, $100 ; February 1, 1865, by cash, $200;Febmary 28, by merchandise, $150. Ans. $60.64. ALLIGATION. Alligation is the method of making oaloolations r^ardiog 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, mcdicU aaddltemate. ^ ALLIGATION MEDIAL. ' Alligation medial relates to the average value of articles oom- pounded, when the actual quantities and rates are given. EXAMPLE. A miller mixes three kinds of grain : 10 busheb, 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. ALUOATTON. 983 10 bnaheli, at 40 oenU a bubhel, will be worth 400 oenta., 1& boshels, at 60 oentji a buahot, will he worth 750 oenta., 2& boahela, at 70 oenta a buahel, will be worth 17&0 oenta., giving a total of 50 bushels and 2900 conta, and henoe the miztare is 2900-^50=58 cents, the price of tho mixture per bushel. Henoe tho BULL Find the value of each of the articles, ami divide the turn o/ their values 6y the number denoting the turn of the artidest and the quotient will be the price of the mixture. BXBRC I8B8. 1. A fanner 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 $1.60 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 GO bushels of barley, worth 40 cents per bushel ; what was the value ot 2| bushels of the mixture ? Ans. $1.50. 5. A goldsmith mixes together 4 pounds of gold, of 18 carats fine ; 2 pounds, of 20 carats fine; 5 pounds, of 16 carats fine; and 3 pounds, of 22 carats fine ; how many carats fine is one pound of the mixture? Ans. 18|. Ml ALLIGATION ALTERNATE. Alligation alternate is the method of finding how much of seve- ral ingredients, the quantity or value (^f which is known, must be combined to make a compound of a given value. ' * CASE I. Given, the value of several ingredients, to make a compound of a given value. <' ' '."■•' 284 ABITHMETIO. K X A M IM. K Flow mnoh sugtr ihtt m worth (> oenUi, 10 oonU, and 13 codU per pound, most bo mixed togotber, so thtl IIm mixture may be worth 12 oenta per pound ? SOLUTION. 12oentii. • 1 lb., ftt 6 cents, is a (^ain of 6 cents. ) (hin. I lb., at 10 oonts, is a gain of 2 cents. 8 1 lb., at 13 cents, is a loss of 1 cent. 7 lbs. more, at 13 oonts, is a loss of. LoM. 1 - € Gain 8 Loss 8 It is evident, in forming a mixture of sugar worth G, 10 and 13 cents per pound so as to bo worth 12 cents, that the gains obtained in putting in sugar of le$» value than the average price must exactly balance the losses sustained in putting in sugar of ^rca^er value 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 fhis sugar put in the mixture. So also sugar that is worth 10 cents per pound, when in the mixture will bring 12, so that a gain of 2 cents is obtained on every pound of this sugar used in the compound. Again, sugar that is worth 13 cents per pound, on being put into the mixture will roll 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 H cents, and a loss of only 1 cent. Now, our losses must equal our gains, and therefore wc have yet to lose 7 centsi and as there is only one quality of sugar in the mixture by which we can lose, it is plain tliat we must take as much more sugar at 13 cents as will make up the loss, aiid that will require 7 pounds. Therefore, to form a mixture of sugar worth G, 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-f7 pounds of the same, which must be taken to make the loss equal to the gain. By utakiug u mixture of any number of ti ^es these answers, it will be observed, that the compound will be correctly formed. Hence we can readily perceive that any number of answers mav be obtained ALUOATION ALTERNATE. »6 to all exareiiM of this kind. From what hu been Mid we dtduoc the following RULl. JPind how tmich it gained or tott hy taking one of each kind oj the propoted ingredient*. Then take one or more of the ingredient$, or auch parti of theni aa will make the gains and loua equal. K X K R c I H I H . 1. A grooer wishes to mix togethor tea worth 80 cents, $1.20, $1.80 and $2.40 per pound, so us to make a mixture worth $1.G0 per pound ; how many pounds 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 much com, at 42 cents, 60 cents, 67 cents, and 78 cents per bushel, must be mixed together that the compound may be worth 64 cents per bushel ? Ans. 1 bush, at 42 cts. ; 1 bush, at 60 cts. ; 4 bush, at 67 cts. ; and 1 bush, at 78 cts. 3. It is required to mix wine, worth 60 cents, 80 cents, and $1.20 per gallon, with water, that the 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 ota. ; 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 way be 62 cents per bushel ? Gix%, at least, three answers, and prove the work to be correct. 5. A produce dealer mixed together corn, worth 75 cents per bushel ; oats, worth 40 cents per bushel ; rye, worth 65 cents per bushel, and wheat, worth $1 per bushel, so that the mixture was worth 80 cents per bushel ; what quantity of each did he take ? Give four answers, and prove tht 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. EXAMPLE. How much barley, at 40 cents ; oats, at 30 cents, and conr, at 60 236 ABTTHMETIO. oents per baahel, most be mixed with 20 bushels of rye. at 86 cents per bushel, so that the mixture may bo worth 60 cents per bushel ? SOLUTION. Bush. Centa. Gain. Loss. 1 at 40, gives 20 1 at 30, gives 30 1 at 60, gives 00 .00 20 at 86, gives ... 5.00 .60 6.00 9 at 40, gives 1.80 9 at 30, gives 2.70 $5.00 $6.00 By taking 1 bushel of barley, at 40 cents, 1 bushel of oato at 30 cents, and 1 bushel of com at 60, in connection with 20 busheb of rye at 85 cents per bushel, we observe that our gains amount to 50 cents and our losses to $5.00. Now, to make the gains equal the losses, we have to take 9 bushels more at 40 cents, and 9 bushels more at 30 oents. This gives us for the answer 1 bu8hel-f-9=10 bushels of barley, 1 bushel-j-9=10 bushels of oats, and 1 bushel of corn. From this we deduce the RULE. Find lum much is gained or lost, hy taking one of each of the proposed ingrtdients, in connection with the ingredient which is limited, and if the gain and loss he not equal, take such of the jjro- posed ingredients, or such parts of them, as will make the gain and loss equal. EXERCISES. 6. How much gold, of 16 and 18 carats fine, must be mixed 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, vath 20 pounds, at 40 cents per pound ; how much of each sort must he take to make the composition worth 80 cents per pound ? Ans. 20 at $1.20 ; 10 lbs. at $1 ; 10 lbs. at 60 cents. 8. How much barley, at 50 cents per bushel, and at 60 cents per bushel, must be mixed with ten bushels of pease, worth 80 cents ALUGATION ALTEBNATE. 237 per Inuliel, and 6 bushels of rye, worth 85 cents per bushel, to make a mixture worth 75 cents per bushel ? Ans. 3 bushels, at 50 cents ; 2^ bushels, it 60 cents. 9. How many pounds of sugar, at 8, 14, and 13 cents per pound, must be mixed with 3 pounds, worth 9^ cents per pound ; 4 pounds, worth 10} cents per pound ; and G pounds, worth 13} cents per pound, so that the mixture may be worth 12} cents per pound ? Ans. 1 lb., at 8 cts. ; 9 lbs., at 14 cts. ; and 5} lbs., at 13 ots CASE III. To find the quantity of each ingredient, when the sum of the ingredients and the average price are given. EXAM PL E. A grocer has sugar worth 8, 10, 12 and 14 cents per pound, and he wuhes to make a mixture of 240 pounds, worth 11 cents pM pound ; how much of each sort must he take ? SOLUTION. Gain. 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 ' 4 lbs. 4 4 240 lbs.-H4=60 lbs. of each sort. By taking 60 lbs. of each sort we have the required quantity, and it will be observed that the gains will exactly balance the losses, consequently the work in correct. Hence the RULE. Find the least quantity of each ingredient iy Casx L, Then divide the given amount by the sum of the ingredients already found, and multiply the quotient by the 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 fill 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 lbs., at 25 cents. 16 238 ABITHMETIC. 11. Howmnoh sugar, at 6 cents, 8 cents, 10 oents, and 12 oenta per pound, must be mixed together, so as to form a compound of 200 pounds, worth 9 cents per pound? Ans. 50 lbs. of each. 12. How much water must 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. 56| gals, wine, and 33| gals, water. 13. A wine merchant has wines worth $1, $1.25, $1.50, $1.75,and $2 per gallon, and he wishes to foinn a compound to fill a 150 gallon cask that will sell at $1.40 per gallon ; how many gallons of each sort must he take ? Ans. 54 of $1, and 24 of each of the others. 14. A grocer has sugars worth 8 cents, 10 cents, 12 cents, and 20 cents per pound ; with these ho wishes to fill a hogshead that would contain 200 pounds ; how much of each kind must he take, so that the mixture may be worth 15 cents per pound ? Ans. 33^ lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 cents. 15. A grocer requires to mix 240 pounds of 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 ? Ans. 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 are distributed to them, and the measure by which they estimate their possessions. The precious metals have, among almost all nations, been the standard of value from the earliest time. Except in the very rudest state of society, men have felt the necessity of having some article, of more or less intrinsic value, that can at any time be exchanged for different commodities. No other substances were so suitable lor this purpose as gold and silver. They are easily divisible, portable, and among the least imperishable of all substances. The work of divid- ing the precious metals, and marking or coining them, is generally undertaken by the Government of each State. Money is a commodity, and its value is determined, like that oi 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 the MONEY: ITS NATUBE AND VALUE. 239 money in circolation were doubled, prices would be doubled. The UBefulness of money depends a great deal upon the rapidity of iu circulation. A ten-dollar bill that changes hands ten times iu a month, purchases, during that time, a hundred doUarn' worth of goods. A small amount of money, kept in rapid circulation, does the same work as a far larger sum used more gradually. Therefore, whatever may be the quantity of money in a country, only that part of it will effect prices which goes into circulation, and is actually exchanged for goods. Money hoarded, or kept in reserve by individuals, does not act upon prices. An increase in the circulating medium, conformable in duration and extent to a temporary activity in business, does not raise prices, it merely prevents the fall that would otherwise emiw from its temporary scarcity. PAPER CURRENCY. Paper Curiisnct may be of two kinds-— convertible and incon- vertible. When it is issued to represent gold, and can at any time be exchanged for gold, it is called convertible. When it is issued by the sovereign power in a State, and is made to pass for money, by merely calling it money, and from the fact that it is received in pay- ment of taxes, and made a legal tender, it i^ known as an inconver- tible currency. Nothing more is needful to make a person accept anything as money, than the persuasion that it will be taken from him on the same terms by others. That alone would ensure it.s currency, but would not regulate its value. This evidently cannot depend, as in the case of gold and silver, upon the cost of production, for that is very trifling. It depends, then, upon the supply or the quantity in circulation. While the issue of inconvertible currency is limited to somethinf^ under the amount of bullion iu circulation, it will on the whole maintain a par value. But as soon as gold and silver are driven out of circulation by the flood of inconvertible currency, prices begin to rise, and get higher with every additional issue. Among other commodities the price of gold and silver articles will rise, and the coinage will rise in value as mere bullion. The paper currency will then become proportion ably depreciated, as com- pared with the metallic currency of other countries. It would bo 240 AfilTHlfETIG. quite impossible for theoe result' m follow the issue of convertiblo paper for which gold could at aaj time be obtained. All variations in the value of the circulating medium are nutf* chievous; they disturb existing contracts and expectations, and the liability to »uch disturbing influences renders every pecuniary a ^agement of long date entirely precarious. A convertible paper currency is, in many respects, beneficial. It is a moro convenient medium of circulation. It is clearly a gain to the issuers, who, until the notes are returned for payment, obtain the use of them as if they wcie a real capital, und that, without any loss to the community. THE CURRENCY OF CANADA. In Canada there are two kinds of currency ; the on*^ is called the old or Halifax currency, reckoned in pounds, shillings, pence and fractions of a penny ; the other is reckoned by dollars and cents as already explained under the head of Decimal Coinage. The equivalent in gold of the pound currency is 101.321 grains Troy weight of the standard of fineness prescribed by law for the gold coins of the united kingdom of Great Britain and Ireland. The only gold coins now in circulation in Britain are the sovereign, value one pound, or twenty shillings sterling ; and the half sovereign, ten shillings. The dollar is one-fourth of the pound currency, and the pound sterling is equal to $4.86§. In the year 1786, the congress of the United States adopted the decimal currency, the dollar being the unit, and the system was introduced into Canada in 1858. By the term ieffal tender is meant the profier of payment of an account in the currency of any country as established by law. Copper is a legal tender in Canada to the amount of one shilling or twenty cents, and silver to the amoun t of ten dollars. The British sovereign of lawful weight passes current, and is a legal tender to any amount paid in that coin. There is a silver currency proper to Canada, though United States' coins are most in circulation. The gold eagle of the United States, coined before July 1, 1834, is a legal tender for $10.66§ of the coin current in this province. The ^ame coin issued after that is a legal tender for $19. EXCHANGE. w. SZCHANaE It often becomes necessary to send money' from cue town or oonntry to another for various purposes, generally in payment for l^oods. 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 received a consignment of flour from C. D., of Montreal ; and another man, E. F., in Liverpool, has shipped a quantity of cloth, in value equal to the flour, to G. H. in Montreal. There arises, in this transaction, an indebtness to Mon- treal for the flour, as well as an indebtedness from Montreal for the cloth. It is evidently unnecessary that A. 6., in Liverpool, should send money to C. D. in Montreal, and that G. H., in Montreal, should send an equal sum to E. F. in Liverpool. The one debt may be applied in payment of the other, and by this plan the expense and risk attending the double transmission of the money may bo saved. C. D. 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 Montreal to Liverpool, and the debt due from Liverpool to Montreal are both paid without any coin passing from one place to the other. An arrangement of this kind can always be made when the debts due between the different places are equal in amount. But if there is a greater sum due from one place than from the other, ..hc debts cannot be simply written oflF against one another. Indeed, when a person desires to make a remittance to a foreign country, he does not make a personal search for some one who has money to receive fronx that country, and ask him for a bill of exchange. There are ex- change brokers and bankers whose business this is. They buy oills 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, ho will remit to him all the exchange which he has bought and not re-sold. 242 ABITHXETIO. When brokers find that they arc aflkerl for more bills than are offered to them, they do not absolutely refuse to give thorn. To (enable their corroepondents to meet the bills at maturity, a» they have no exchange to Hcnd, they have to remit funds in gold and nilvcr. There are the expenses of freight and insurance upon the Hpcoie, besides the oceupation of a certain amount of capital involved in this ; and an increased price, or premium, is charged upon tho exchange to cover all. The reverse of this happens when brokers find that more bills iirc offered to them than they can sell or find use for. Exchange on the foreign country then falls to a discount, and can bo purchased at a lower rate by those who require to make payments. There are other influences that disturb the exchange bctwoen different countries. Expectations of receiving large payments from a foreign country will have one effect, and the fear of having to make large payments will have the opposite effect. AMERICAN EXCHANGE. ExoHANO£ between United States and Canada is a matter of everf day importance on account of the proximity of the two countries, and the inceesant intercourse between them both of a social and commer- cial character. Much inconvenience has been felt latterly in Canada on account of the depreciation of American currency. The imme- diate cause of this was the late war. The exigencies of the Northern States compelled them to issue, to enormous amounts, an inconvertible paper currency known by the name of " Greenbacks.' ' As the value of these depbuded mainly on the stability of the government, and the success of the war, public confidence wavered, and in consequence the value of this issue sunk materially. Much damage was accord- ingly suffered by Canada, as all her commercial intercourse with the States was cramped by this depreciation of the currency, as well as by the fluctuation of the money market generally, and the doubtful issue of the struggle. From these causes the value of gold rose gradually until it reached the enormous premium of nearly two hundred per cent., or quotation of nearly three hundred. But though gold was quoted at nearly two hundred per cent, premium, it must not be understood that AMERICAN EXCHANGE. 243 American inoncj, or greenbacks, was at that rate per cent, discount. For example, when gold in quoted at 150, or 50 per cent, premium, greenbacks are not at a disoount of 50 per cent., an many might be led to euppoBC, but only at 33^. The error will be more apparent from tho consideration that when gold is quoted at 100 per cent, premium, the discount would not be 100, for if it were, money would be worth absolutely nothing. CASK I . To find the value of SI, American onrreney, 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 cents 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 71iJ cents. Hence to find the value of $1 of any depreciated currency reckoned in dollars and cents, we deduce tho follow' n'; RULE. Divide 100 cents h\j 100 jo?«s the rate of premium on gold, and the quotient will he the value of $1. Subtract this from $1, and the remainder vrill be therateof dis- (mtnt on the given currency. CASE II. To find the value of any given sum of American currency when gold is at a premium. EXAMPLE. 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^ cents. Now it is evident that if 71 1 cents be the value of $1, the value of $280 will be 280 times 71f cents, which is $200, or $280—1. 40==28000-t- 140=:$200. Hence we have the following .41 244 ABITBHSnO. B U L E . Multiply the value of $1 by the number denoting the yiveH amount of American money, and the product will be the gold value; or, Divide the given »um of American money by 100 {the number of rente in $1,) plut the premium^ and the quotient will be the value in gold. AHE III. To find tho premium on gold when American money is quoted at a certain rate per cent, discount. EXAMPLE. When the discount on Anierioan money is 40 per cent., what is the premium on gold ? ' , BOLUTION. 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 be 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 be worth 100 times g'^ of $1, which is $1.66§, from which if we subtract $1, tho remainder will bo the premium. Therefore, if American currency bo at a discount of 40 per cent., the premium on gold would be 66§ per cent. Hence we deduce the following KULE. Divide 100 cents by the number detioting the gold value of $1, American currency, and the quotient will be the value, in American currency, of$l in gold, from which wbtmct $1, and the remainder will be the premium. CASE I V . ' ' To find the value in American currency of any given amount of gold. EXAMPLE. ^ What is the value of $200 of gold, in Amerioan enrrency, 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 $1.50 in American currency, $200 will be worth 200 times $1.50, which is $300. Hence the AMIBIOAN nCHANOE. 346 RULE. MiUHply the value of $1 &y the number dmniing the amount of yold to be changed, and the product will be the value in American currency ; or, To the given $um add the premium on itulf at the given rate, and the retuit wiB be the value in Atnerican cwrreneg. EXKR0I8K8. 1 . If American ourrency is at a dieoount of 50 per cent., what is the value of $450 ? Ans. $225. 2. The quotation of gold is 140, what is the discount on Ameri- can currency ? Am. 28;) per cent. 3. A person exchanged $750, American money, at a discount of 35 per cent, for gold ; how much did he receive ? Ans. $487.50. 4. Purchased a draft on New York for $1500, at a discount of 31^ per cent. ; what did it cost mo ? Ans. $1027.50. 5. American currency is quoted at 33^ per cent discount ; what is the premium on gold ? Ans. 50 per cent. 6. Purchased a snit of clothes in Toronto for $35, but on paying for the same in American funds, the tulor charged me 32 per cent, discount ; how much had I to pay him ? Ans. $51.47. 7. What would be the di£ference between the quotations of gold, if greenbacks were selling at 40 and 60 per cent, discount. ? Ans. 83^ per cent. 8. P. Y. Smith borrowed from G. R. King, $27 in gold, and wished to repay him in American ourrency, at a discount of 38 per oent. ; how much did it require ? Ans. $43.55. 9. J. £. Peckham bought of Sidney Leonard a horse and cutter for $315.50, American currency, but only having $200 of this sum, he paid the balance in gold, at a premium of 65 per oent. ; how much did it require ? Ans.^$70. 10. An American drover purchased of a farmer a yoke of oxen valued at $135 in gold, but paid him $112 in American currency, at a discount of 27^ per cent. ; how much gold did it require to pay the balance ? Ans. $53.80. - 11. W. H. Hounsfield & Co., purchased in New York City, merchandise amounting in value to $4798.40, on 3 months' credit, premium on gold being 79| per cent. At the expiration of the three months they purchased u draft on Adams, Kimball and Moore of 940 [0. New York, for the amount due, at a diHOount of 57} per cent. ; what waH the gain by exchange ? Aos. $647.75. 1 2. A makofl an exchange of a horao for a carriage with B ; the horHu bcin^ valued at f 127.50 in gold, and the carriage at $210, American currency. Gold being at a premium of ()5 pt^t cent ; what wa8 the difference, und by whom payable ? Anb. B pays A 23 cents. 13. A merchant takcH $63 in American silver to a broker, and wishcH to obtain for the Hamc grecnbackH which ure selling at a dis- count of 30 per cent. The broker takes tho silver at 3] per cent, discount ; what amount of American currency does the merchant reoeive? Ans. $86.85. 14. I purchased of B. W. Smith a farm in the township of Essa containing 100 acres, at $15 per acre. Sold 50 acres to an Ameri- can speculator, at $23 per acre, American money ; exchanged ^ of the remainder with Isaiah Wright, Esq., for a town lot in Brantford, allowing him for tho difference of barter $400, for which ho agreed to take its equivalent in American money, at a discount of 30 per cent. ; tJio balance I sold to J. 11. Forstcr, at a profit of 20 per cent., receiving in payment his note at 30 days, which I immediately sold to an American for $900 greenbacks. The amount of American currency I then had on hand I czchaugod with my broker for Canada money, which was quoted at a premium* of 50 per cent. ; required the amouDt of profit, and the rate per eont. ef gain. 15. A merchant left Toronto for New York City to purchase his stock of spring goods, taking with him to defray expenses $95 in gold. After purchasing his ticket to the Suspension Bridge for $2.40, he expended the balance in greenbacks, which were at a dis- oouBt of 41^ per cent. When in New York ho drew from this amount $23.85 to " square" an old account tJien past due. On aniifing home he found that he still had in greenbacks $16.40, which h«,4>9PO<^<^ ^^ '^^ ^ discount of 43^ per cent., receiving in payment American silver at a discount of '6^ per cent., which he passed off at 2^ per cent, discount for gold. What were his expenses in gold ; the actual amount in greenbacks paid for expenses, and tho amount of silver received ? Adb. Total expenses in gold, $71.76 ; expenses in greenbacks, $118.04; silver received, $9.53. STEBUM ncaiNoc ii7 STIEELINa EZCHANOE. Im Britoin money i» nekoned by pounds, shilUngi and ponoe, »nd frsotioDB of a penny, and is oalled sterling money, the gold bot- ereign, oonnsting of 22 parts gold and 2 alloy, being the standard, and the shilling, one-twentietii part of this, a silver coin of 37 ptrts silver and 3 copper, and the penny, ono-twelfUi part of this, a copper coin, the ingredients and size of which have frequently been altered. This JH called sterling monei/, and the estimating of that currency in terms of the currency of another country, nhioh in the case of Canada is done by dollars and cents, is called sterling exchange. If a house in Canada has trunsactionR with one in Britain, it is neoessary thai each should be able to estimate what given sum in the onrrenoy of the one country ia worth in that of the other. The comparative value of the pound sterling, or gold sovereign and Canadian money, was formerly $4.44J, but as this waa less than the intrinsic value, and besides as the commercial value, though fluctuating, was always considerably higher, it was settled by Act of Provincial Parliament that the standard value of the pound sterling, estimated in dollars and cents, should be raised to $4.86§. The increase in the standard value was therefore equal to 9^ per cent, uf its nominal value. Sterling exchange is usually quoted in the form : 108, lOi), 109^, 110, &c., which indicates that it is at 8, 9, 9^, or 10 per cent, pre- mium. The REAL PAR VALUE of the British sovereign in Canadian money is $4.86§, the exact equivalent of the pound sterling. The COMMERCIAL VALUE is thc amount in dollars and cents required to purchase one pound Bte.^ling at any given rate of the money market. If exchange is quoted at .V09i^, this means that it will require jSl09^ at par value to purchase £100 at commercial value. Thc rate of exchange between Canada and Britain is usually reckoned from the old par value. Though thc commercial value fluctuates, yet 9^ per cent, above par may be taken as its average, so that if we add the premium of $4.44| to itself at 9^ per cent., we shall have $4.86§, which is neajr ly the average of the oommercinl value, and is generally taken as such. ABimiBno. R X A M P L K . A merehtnt in Canada mit^her, Ut remit £648.17.6 to Britain. The old par Taluc in t4.i4l-^%*^~ }, of $40 by rcdncing to an impro- per fraction. Then if the quotation in 108, or K per cent, above the nominal value, we find the pruniiuni on $40 at 8 per cent., which in $3.20, which added to $40 will give $43.20, and $43.20-4-9=^.80 to bo remitted for every pound aterling, and therefore £648.17.6 mul- tiplied by 4.80 or 4.8 will bo the value in Gaoadian money. 17b. 64.85. 8. A merchant sella a bill on London for £4000, at H ] ;. cent, above its nominal value, instead of iniportinc; Npecie at on xponse of 2 per cent. ; what does he save '{ \m. 8122.66^. 9. A merchant in Kingston paid $7300 for u dro;. of £1500 on Liverpool ; at what per cent, of premium was it purchased ? Ans. 9J. 10. A broker sold a bill of exchange for £2000, on commission, at 10 per cent, above its nominal value, receivinf>; a commission of j'q per cent, on the real value, and 5 per cent, on what lio obtained for the bill above its real value ; what was h'n commiasion ? An.s. $11.95^. 11. I owe A. N. McDonald & Co., of Liverpool, $7218, net pro- ceed.s of sales of merchandise effected for *}iem, which I am to remit them in a bill of exchange on Lont..:. 'or such amount u.s will close the transaction, less ]^ per cent, on I'u-j 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, to bo remitted. Ana. £1500 5s. 6d. 360 ABrrHKEnc. TABLE OF PORKIUN MONKTS. I CiTin Am Coramia London, Liverpool, &C' Paris, Havre, &c Amsterdam, Hague, &c, Bremen DBMoaiirATioxs or MolniT. Hambuig, Lubec, &o... Berlin, Dantzic Belgium St. Peteiflburg. Stockholm Copenhagen. Vienna, Trieste, &c.... Naples Venice, Milan, &c Florence, Leghorn, &c. Genoa, Turin, &c Sicily Portugal Spain Constantinople British India.. Canton , Mexico Monte Video. Brazil. Cuba.. Turkey United States. New Brunswick. Nova Scotia Newfoundland... 12 penoe=l shilling ; 20 shillings =1 pound = 100 centimes=l franc = 100 cent8=^l guilder or florin...— 5 swares=l grote; 72 grotes=l rix dollar = 1 12 pfennings=:l schilling; 16s.= 1 markhanco = 12 pfenniiig8=l groschen ; SO gro. =1 thaler = 100 oentimes=l franc = 100 kopecks=rl ruble = 12 rtindstycks^ie skillings; 48s. =1 rix, dollar specie -= 16 8kiliing8=l mark ; 6 m.=:l rtso dollar = 60 kretttzers=l florin = 10 gram=i-l carlino; 10 car.=l ducat =: 100 centesimi=l lira = 100 cent«8imi=l lira = 100 centesimi=l lira = 20 grani=l taro ; 30 tari=l oz.= 1000 reas=l millrea = i34 maravedis=l r^l i}ellon= 68 maravedis=l real plate. . = aspers=l ^iag/er. r=: 12 pice=l anna; 16 annas=l rupee = 100 candarines=l mace ; 10 m.= 1 tael. = 8 rial8=l dollar = 100 centesimas=l rial ; 8 rials=l dollar = 1000 rea8=l milrea — : 8 reals plate or 20 reals vellon=l dollar =- 100 aspers=l piaster... ^^ 10 mills=^l cent ; 10 cents=l dime ; 10 dimes=l dollar.... =^ 4 farthings=l penny ; 12 pence =1 shilling ; 20 shilUngs=l pound.* = l ViJ.DB. $4.86§ .18^ .40 .78} .35 .69 .18! .75 1.06 1.05 .481 .80 .16 .16 -m 2.40 1.12 .05 .10 .05 .44| 1.48 1.00 .83,3, .821 1.00 .05 variable. 4.00 • The Government of New Brunswick now issues postage stamps in the decimal currency, but so for as we have been able to ascertain, the curreaoy of ABBITBATION OF EXCHANGE. 251 ARBITRATION OF EXCHANGE. Arbitration of Exchange is the method of findint; the rate of exchange between two countries through the intervention of one or more other countries. The object of thin U to ascertain what is the mott advantageous channel through which to remit money tc> a fireigo country. Three things have here to be considered. First, what is the most secure channel; secondly, wb.i: is the least expensive, and thirdly, the comparative value of the currencies of the different countries. Regarding the two first considerations no general rule can be given, as there must necessarily be a continual fluctuation arising from political and other causes. We arc therefore compelled to confine our calculatiou to the third, viz., the comparative value of the coin current of different countries. For this purpose we shall investigate a rule, and append tables. Let us suppose an English merchant in London wishes to remit money to Paris, and finds that owing to certain international rela- tions, he can best do it through Hamburg and Amsterdam, and that the exchange of London on Hamburg is 13^ marcs per pound ster- ling; that of Hamburg on Amsterdam, 40 marcs for 36^ florins, and that of Amsterdam on Paris, 56f florins for 120 francs, and thus the question is to find the rate of exchange between London and Paris. SOLUTION: We write down the equivalents in ranks, the equivalent of the first term being placed to the right of it, and the other pairs below them in a similar order. Hence the first term of any pair will be of the same kind as the second term of the preceding pair. As the answer is to be the equivalent of the first term, the first term in the last rank corresponds to the third term of an analogy, and is there- fore a multiplier, it must be placed below the second rank. The these three Provinces is, as usual, in pounds, sliillings and poiicc. It is to be hoped that when thf Confederation of the Britisli Provitici's taltca place, the decimal currency will be speedily adopted in the Lower Provinces, and that the efforts now being made in Britain to adapt the eame currency will prove successful. 252 ABITHlfETIC. terms being thua arranged, we divide the product of the second rank by that of the first, and the quotient will be the equivalent, aa ezhi* bited below : £1 sterlings: 13^ marcs. 40 marcs = 36^ florins. / 56| florins =120 francs. £1 stg. As it is most convenient to express the fractions decimally, wa have ISSX36.26XI80XI o« Q7 !!>_,_« 1X40X5 e.Tft =£0.ai trancs. The foregomg explanations may be condensed into the form of a BULK. Write down the fint ttrm^ and it$ equivalent to the right of it, and the other pain in the tame order, the odd term being placed under the second rank, and then divide the product of the second rank by the product of the first, the quotient will be the required equivalent. Note. — The true principle on which this operation is founded is, that each pair consists of the antecedent and consequent which are to each other in the ratio of equality in point of DmuNSio value, though not in tett*' to the NUMBERS BY WHICH THET ARE EXPRESSED, and therefore the reqnir«u term and its equivalent must have the same relation to eanh other, that is, they will be an antecedent and a consequent in the ratio ot j{iur.lUy as regards their vaiut, but not as regards the numbers by which they art- expressed. EXKROISBS. 1. If the exchange of London on Paris is 28 francs per pound sterling, and that of Canada on Paris 18 cents per franc ; what is the rate of exchange of Canada on London, through Paris ? Ans. $6.04 per £ sterling. 2. If exchange between Montreal and London is at 8 per cent, premiuui, and between London and Paris 251- francs per pound sterling ; what sum in Montreal is equal to 7000 francs in Paris ? Ans. 3. When exchange between Quebec and Hamburg i.s at 34 cents per mark banco, and between Hamburg and St. Petersburg i.s 2 marks, 8 schillings per ruble ; how much must bo paid in St. Peters- burg for a draft on Quebec for $650 ? Ans. 764 rubles, 70 jj^ kopecks, EXCHANGE. 263 4. If a merchant buys a bill in London, drawn on Paris, at the rate of 25.87 francs per pound sterling, and if this bill be sold in Amsterdam at 120 francs for 56 J florins, and the proceeds be invest- ed in a bill on Hamburg, at the rate of 36^ florins for 40 marcs ; what is the rate of exchange between London and Hamburg, or what is £1 sterling worth in Hamburg? Ans. 13.4494-marci). 5. A merchant of St. Louis wishes to pay a debt of $5000 in New York ; the direct exchange is 1^ per cent, in favour of New York, but on New Orleans it is ^ per cent, discount, and between New Orleans and New York at a J^ per cent, premium ; how much would be saved by the circular exchange compared with the direct ? Anri. $87.56. 6. A merchant in Toronto wishes to remit to J. B. Gladstone & Co.; of London, £3600 sterling. Exchange on London, in To ronto, is at a premium of 10 per cent. Exchange on London can be obtained at Halifax, Nova Scotia, for 9 per cent, premium. If Toronto Bills on Halifax are at a discount of ^ per cent., and the merchant remits a draft to Halifax, and pays his agent ^ per cent, for investing it in bills on London ; what will he gain over the direct exchange? Ans. $123.80. 7. A merchant in London remits to Amsterdam £1000, at the rate of 18 pence per guilder, directing his correspondent at Amster- dam to remit the same to Paris at 2 francs, 10 centimes per guilder, less ^ per cent, for his commission ; but the exchange between Amsterdam and Paris happened to be, at the time the order was received, at 2 francs, 20 centimes per guilder. The merchant at London, not apprised of this, drew upon Paris at 25 francs per pound sterling. Did he gain or lose, and how much per cent. ? Ans. 16^1 per cent. gain. MIXED EXERCISES IN EXCUANQE. 1. When gold is quoted at 150 per cent, premium ; what is the reason American money is not at a discount of 50 per cent. ? 2. I wish to invest $3'4 60.80 in a sterling bill of exchange ; for how many pounds must the bill be drawn, exchange being at a pre- mium of 8 per cent. ? Ans. £783 1 Os. 3. What sum in Canada money must I pay for a bill on Loudon of £76 14s. Id., exchange being 9^ per cent, premium, and the broker's commissioa for negociating the bill being ^ per cent. ? Ans. 8375. 17 254 ABITHiCBnO. Xj: 4. A merchant shipped 2560 barrels of floar to hii igtnt ia Liverpool, who sold it at £1 8b. 6d. per barrel, and charged 2 per cent, commission ; what was the net amount of the flour in Canada money, allowing exchange to be at a premium of 8 per cent. ? Ans. $17160.19. 5. What b he cost of a 30 days' bill on Montreal, at ^ per cent, premium, the fao<> of the bill being $1500 ? Ans. $1507.50. <>. What mu!>' be the face of a 60 days' draft on Hi^lifaz, Nova Sootia, to yib.d $1041.75, when sold at a discount of ^ per cent. ? Ans. $1650. 7. What ia the cost of a 30 days' bill on Quebec, at | per cent, premium, and interest off at 6 per cent. ; the face of the bill being $9266.40 ?* Ans. $9240.20. 8. A merchant paid $14400.12 for a bill on Havre for 79000 francs ; how much was exchange below par ? Ans. 2 per cent. 9. I have in possession the net proceeds of a sale of cotton amounting to $3765, which my correspondent desires rae to remit to him in New Orleans ; exchange on New Orleans is at a discount of 2^ per cent., and I invest the whole in a draft at that rate, whioh I remit to him ; what is the face of the draft ? Ans. $3861.54. 10. The proceeds of a sale of goods, consigned to me from Bremen, is $2764.67, on which I am to charge a commission of 10 per cent., and remit the balance to my consignor in such a way ai shall be most advantageous to him. Exchange on Paris can be had at 92 cents per 5 francs, and in Paris exchange on Bremen is 17 francs to 4 thalers. Exchange on Liverpool can be had at 9 per cent, premium, and in Liverpool exchange on Bremen is 6 thalers to the pound sterling. Direct exchange is 80|^ cents per thaler. Which course will be the best, allowing ^ per cent, brokerage to correspon- dents both in Liverpool and Paris ? Aas By way of Paris. 11. A, of Hamilton, sent articles to the World's Pair in London, which were afterwards sold by B, of London, on A's account, net proceeds £1266 15s. sterling. B was instructrges ^ per cent, on the greenback value for investing. If exchan^u on New York is at 47 per cent, discount, at which place would it be the most advanta- geous to purchase, and how much gain, and if the remittance be made by the way of New York, wTiat would be the face of the draft ? Ans. New York by $141.72 ; face of draft, $11161.2^ In the above exercise, suppose that instead of purchasing a dratt on New York, they remit specie at an expense of f per cent., the New York broker's commission being f per cent, on the gold value of the bill ; what would be his gain or loss ? Ans. Loss $13.64. 14. Hughes, Bros. & Co., purchase of E. Choffey & Co., a ster- ling bill at 60 days on Gladstone & Hart, of London, for £3956 10s. They remit this bill to James Aldlcr, in London, where it is accepted by Gladstone & Hart, and falls due on the 20th November, at which time it is protested, causing an expense of £2 19a. Gladstone & Hart having failed, E. Chaffey & Co.'s agent in London pays James Aldler on the 20th November, £2000 on account. How much must E. Chaffoy Sc Co. pay to Hughes, Brothers & Co., on the 24th December, to cover the amount still due in London, allowing interest at the rate of 10 per cent, from November 20th, to the maturity of a 60 days' bill at date oi' 24th December, and |^ of 1 per cent, commis- sion for their trouble in negociating u new bill ? Ans. $0815.91. Iff M ■■«14'»*#** 250 ABUBMSnO. INVOLUTION. Invohitu)n ii the process of finding a gii'en jjnwer of a gi?tft number. We huyc noted already, under the head of rn'iltiplu!:»*.'os) , that th^ product of any number of cmal factors is oalieu th»i seccid, thifo, fourth, &c., power of the numier, according as \lie factor ia taken two, three, four, &c., limes. Tb.(M : 9=3 < 3 ia the aecond power of 3; 27=3X3X3 is tiio third power of three; 81—3x3x3x3 Ip the fourth power of 3. These aro oftC'S wri'>*!n thus : ii*, 'V\ 3*, &c. The small figures, 2, 3, 4, iudiciito the number of fwtoirs, and tlierefore eiich is called the index or eaLpomrtt of tho powtr. Houcd u» find nny required pcv/er of a given quaniity, wc liuve the a II L E . Multiply (he qwmtity continually by itself until it has been wed as a factor as often as there are units in the indea:. Since the first multiplication exhausts two factors, the number of operations will be one less than the number of factors. Involution, then, is nothing more than multiplication, and for any power above the second, it is a case of continual multiplication. For the sake of uniformity the original quantity is called the Jirst power, and also the root in relation to higher powers. Again, if we multiply 3X3 by 3X3X3, we have five factors, or 3x3X3x3x3, but this being an inconvenient form, it is written briefly 3^, the 5 indicating the number of times that 3 is to be repeated as a factor. Hence, since 3x3 is written 3"^, and 3x3x3 is written 3 3, it fol- lows that 3^ xS^^^S**, and therefore we may multiply quantities so expressed by adding their indices, and so also we may divide such quantities* by subtracting the index of the divisor from that of the dividend. For example 3 3 -f-32 =3 or 3 ' . If we di'.'ide 3 > by 3 » by subtracting the index of the divisor from that of the dividend, we obtain 3<>, but 3 or 3' divided by 3 or 3' is equal to 1, and there- fore any quantity with an index zero is equal to unity. When high powers are to be found, the ov ration may be short- ened iii .he following manner : — Let It be ror^uired to find the six- teenth richer of 2. We first find tht ^ecoiA^ pot ~ of 2, which is 4, INVOLUTION. 267 then 4X4=16, which is the fourth power, and 16X16=256, the eighth power, and 256 X 256— 65536, the sixteenth power. If we wished to find the nineteenth power, we should only have to multiply the last result by 8, which is the third power of 2, for 2 '" X 2 =» -~2 • » . EXERCISES. Ans. 485809. An- 622835864. Ans. 19.070689. Ans. 31640625. Ans. 1. 9738+. 1. Find the second power of 697. 2. What is the third power of 854 ? 3. What is the second power of 4.367 ? 4. Find the fourth power of 75. 5. What is the sixth power of 1.12? 6. What is the second power .7, correct to six places ? Ans. .060893-f-. 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 ^ ? 10. What is the fifteenth power of 1.04 ?* 11. Raise 1,05 to the thirty-first power. 12. What is the eighth power of | ? 13. What is the second power of 4^ ? 14. Expand the expression 6''. 15. What is the second power of 5^ ? 16. Whatpartof 83is2«? 17. What is the difference between S** and 4'' ? 18. Expand S'^xa-'. 19. Express, with a single index, 473x47'' )(47'' ? Ans. 47". 20. How many acres are in ii square lot, each side of which is 135 rods ? Ans. 113 acres, 3 roods, 25 rods. Ans. 1024. Ans. .036963. Ans. f j§. Ans. 1.800943. Ans. 4.638039. Anq (>r>6i Ans. 23^|. Ans. .7776. Ans. ±|i=.30J. Ans. ^. Ans. 11529. Ans. 3888. Ans. .000001. Ans. .00000081. Ans. .1.2762815625. Ans, .000000001. Ans. .00001836. 21. What is the sixth power of .1 ? 22. What is the fourth power of .03 ? 23. What is the fifth power of 1.05 ? 24. What is the third power of .001 ? 25. What is the second power of .0044 ? The second power of any ' number ending with the digit 5 may V'e readily found by taking all the figures except tho 5, aii(^. in ilti- o This exercise will he most readily worked by finding the sixteenth power, and dividing by 1.04. So in the next exercise, find the thirty-second power, and divide by 1.05. A still more ea.oy mode of working such ques- tions will be found under the head of logarithms. 258 ARITHMETir. plying thut by itnclf, increased by a unit, and annexing ^5 to the result. Thu:^, to find the necond 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. St / 1 625 / 3.5 4 r 10,5 11 11025 1225 4225 EXER0IHE8 ON THIS 21.5 22 46225 METHOD 57.6 58 330625 / 1^ 0-. ^ 26. What is the second power of 135 ? 27. What is the wccond power of 205? 28. What is the second power of 335 ? 2D. What is the Bccond power of 455 ? 30. What is the second power of 585 ? 31 . What is " -c -d power of 795 ? Ans. 18225. Ans. 42025. Ans. 112225. Ans. 207025. Ans. 342225. Ans. 632025. Note.— The sqirrtrw ; >0,. Oi .'iu quantity ending in 9, must end in either 3 or 7. No second power can t-ud 'u S, 7, '.i or 2. The second root of any ((uantity ending in 6, must end in 4 or 6. Tb' second root of uuy quantity ending in 5, must end also in 5. The second root of any quantity ending in 4, must end cither in 8 or 2. The second root of any quantity ending in 1, must end either in 1 or 9. The second root of any quantity ending in 0, must also end in 0. EVOI^TJTION. TJie root of any quantity is a number such that when repeated, as a factor, the specified number of times, will produce that quantity. Thus, 3 repeated twice as a factor gives 9, and therefore 3 is called the KB! Olid root of 9, while 3 taken three times as a factor will give 27. and therefore 3 is called the third root of 27, and so also it is called tliG fourth root of HJ . There arc two ways of indicating 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 mark is attached, the simple quantity or Jirst root is indicated. When the second root is meant, the mark |/' alone is placed before the quai ♦(ty, bui. if the third, fourth, &c., miif .re to be indicated, SECOND OR HQUARE ROOT. 269 tho %nreti 3, 4, &c , are written in the angular space. Thus: 3=v/9=^27=v'81=:|/243,&c.,&c. The other method is to writ« the indoz as a fraction. Thu», !)^ moans tho second root of the first power of 9, i. e. 3. So also, 27 ' is the third root of tho first power of 27. In tho same manner (54^ means the third root of the secoDd power of 64, or the second power of the third root of ti4. Now the third root of 04 is I, und the second power of 4 is 16, or the second power of 64 is 4096, and tho 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. 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 table of second powers, it will be found that tho second power of any whole number less than 1" consists of cither oru or tvDo digits; tho scjcnd power of any number ^-rea^or than 9, and less than 100, will in like manner be found to consist ot ihree or four digits j and, univcrs;'.lly, tho second power of any number will consist of either tioice the number of digits, or one less than twice the number of digits that the rof t itself consists of. Hence, if wo begin at the units' figufe, and m \rk off the given number in periods of two figures each, we ^hall find that the number of digits contained in the root will be the same r . the number of periods. If the num- ber of digits is even, each period will consist of two figures, but if the number of digits be odd, (he last period to the left will consist of only one figu\ \ Let it now be required to find the second root of 144. We know by Ihe rule of involution that 144 is the second power of 12. • Now 12 may be resolved into one ten and two units, or 10+2, and 10+2 multiplied by itself, as in tho margin, gives 100+40+4, and since 100 5' :iio secoad power of 10, and 4 the second power of 2, and 40 is tvfice t*\o product of 1 and 2, we conclude that the second 'i^r^ 260 ABITHHETIC. lO-f-2 10+2 lOO-l-Ji.i 20+4 100+40-f4 ))owcr ofnny number thuH rcttolvcd in ctiual to the Hum oftbo Moond powers of the part^, />/«» twic«' the product of the piiri.H. Ilcn^'c to firul the mccouJ nK>t of 144, letuH renolvc it into the three partH 100 |-40-( 4, and wc od that the Hceond root of the firnt part is 10, and uneo 40 h twice the product uf the partH, 40 divided by twice 10 or 20 will give the other part 2, and 10+2rr^l2, the nccond root of 144. Wo Hhould find the uame result by resolving 12 into 11-1-1, or i) l-;<. or 8+4, or 7+5, or 6 +-6, but the luoht lunveniont . ode :» to resolve into the t^ns and the unite. In the di.ino manner, if it be required to find the second root of 1369, we have by resolution 900+420+49, of which 900 is the second power of 30, and 30x2=60, and 420-1-60=7, the second part of the root, and 30+7=37, the whole root. Again, let it be 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 of 20 and 3. 15129 is the ium of all, and hence 1 is the root of the hundreds; 2 the root of the tens, and 3 the root of the units. Generalizing 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 consisting of tens and units is the second power of the tens, plus the second power of the units, j)lus twice the product of the tens and units ; that the second power of a number, cons dng of hundreds, tens and units, is the HVLih. of the squares of the b'.adreds, the tens, and the nnits, plus twice the product of i ach ppir. Now since the complement of the full second power, to he smi of the second powers of the parts, is twice the product of >o vir-ts, it follows that, when the first figure of the root i.?s been I'ound, it must be doubled before used as a divi- sor to find the sv cond term, and for the same reason each figure, fhen found, mu,t be doubled to give correctly the- next divisor. Hen?e tho SECOND Oh HQUARE IIOOT. Beginning at Ihe vnit» jigun , murk *>()' thf vhnlf tin*- in prrindi of twojiguru tach ; find thr grvntf^t /nm'ir roHtninrd in t/n It ft hiind period, ami. Hubtntct it from th-it jHri(ul , ^> the rmutinder nnnrx the next fterioil ; for a nrn- iliiidiml, jihni' tlh flgiirr ihtm nhtninrd ita n tjuotitnt, an. First, commenoinj; with the units' figure, wc divide the line into periods, viz., 49, 74 and 79, — wc then note that the greatest square contained in 79 i.s <)4,— this wc subtract from 79, and find 1 5 remaining, to which 89ii wc annex the next period 74, and place 8, the second root of 64, in the quotient, and its double !♦> as a divisor, and try how often 1(> is contained in 157, which we find to be 9 times ; placing the in both divisor and (quotient, we multiply and subtract as in common division, and find a remainder of 53, to which we annex the last period 49, and proceeding as before, wo find 3, the last figure of the root, without remainder, and now we have the complete root 893. 2. This operation may be illustrated as follows : To find the second root of 273529. 500+20-1-3=523 8 169 1783 I 797449 64 1574 1521 5349 5349 500 500x2=1000+20, or 1020 1000+2x20+3=1043 273529 250000 3129 3129 262 ARirmCETIO. 3. To find tbo (tocond root of 153687. Hero WR nbuin, by tho Hamc proooM m id the iMt ozftmplc, tho wholo number 392, with a romAindor o." 23, which oan produce onlj a fraction. 69 782 78402 784049 392.029-f 230000 156804 7319600 7056441 263159 Wo now annex two ciphen, placing tho decimal point after tho root already found, but as tho divisor ia not contained in this new dividend, we place a cipher in both quotient and di- visor, and annex two ciphers more to tho dividend, and by continuing this process wo find the decimal part of tho root, and the whole root is 392.029-f . Ans. 529. Ans. 8642. Ans. 678. Ans. 28.01785+. Ans. 41.569219+. Ans. 25.8069+. EXER0I8EH. 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 the second root of 785 ? 5. What is tho second root of 1728? 6. What is tho second root of 666 ? 7. What is the second root of 123456789 ? Ans. 11111.11106+. 8. What is tho second root of 5 to three places ? Ans. 2.236. 9. What is the side of a square whose area is 19044 square foot ? Ans. 138 feet. 10. What is the length of each side of a square field containing 893025 square rods ? Aus. 945 linear rods. The second root of a fraction is found by extracting the roots of its terms, for .Jfi=:ix^ and therefore |/Ag=i/|X^— a- So also, i/^ ?=5. Again, since j/-,V5=r/g=.09^and .3X ■3=.09, the second root of .09 is .8. This follows from the rules laid down for the multiplication of decimals. To find the second root of a decimal or of a whole number and a decimal : HEfONH OJt W^r.MlK !im)T. 263 Point off prriiiiln "/ '"'< t> jurt:* r'n/i fhuii flu Jtiimul futinl townrris thr rijhf >in)l, Itj't, aiiliiij >i ilplict^ nr n rrjHtiiul, 1/ iJie Hunkhtr n/ fiffiirrn In- odtt. From what \v.in l)ot'n miid, it is plain that every |wri 1 T I O N A I, K X E II C I M E 8 . 1 1 . Whut is the Hccond root of .7 to live places of decimals ? An:^. KUm). 112. Find the second root ol' .07 to six places. 13. What is th«! Hccond root of .05 ? 14. What is the second root of .7 ? 15. Find the second root of .5. 1 H, Whut is the second root of . I '( 17. What is the second root of .1 ? 18. What is the 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 1,^32'^. 22. Find | 2 to lour decimal places. 23. Find the value of i/327l.4207. 24. Find the second root of .005 to live places. Ans. 07071 25. Find the square root of 4.372594. Ans. 2.09107-j- 2H. 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 1 9.0l)G8 ? An.s. -(54575. Ans. .223G-f . Ans. .8819-f . Ans. .74535-f-. Ans. .3162277+. Ans. .3. Ans. 1.1726, &cvJ^ Ans. 61237, kc/'- Ans. 2.529824 . Ans. 1.7748. Ans. 1.4142. Ans. 57.196-}-. Ans. .1. Ans. 03162-f-. Ans. .01. Ans. .001. * The young Htuilenl would naturally expect that Iho dcclinal li|?uro8 of |/ 1.375 and |/.a75 wo\ild be tho same, but it is not so. If it were tso, j/l-r V^ .37.5 would be ocjual to ^-'i.^i'.j- That such is not tho case, may be shown by a very simple example. (/l(j-pj/ !i— l:-}-o--7, but yliJ-{-i)-.=\/'2')-~o. Let it be carefully observed, therefore, that (lie sum of the necond roots is not the same as the second root of the sum. 264 ABITHMETIO. OPERATION 4 19.0968 16 4.37 trial. 83 4.36 true. Trial 867 309 249 '■».__-■: ' - Too great by 1 True 866 6068 6069 6068 5196 872 Here wo find the remainder, 872, is greater than the divisor, 866, which seems inconsistent with ordinary rules ; but it must be observed thut wc aro not seeking an exact root, but only the closest possible approxim ition 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\j, whereas 7 would be too great by -, ^q, and that 866 is not a correct divisor but an approximate one, and that the true root lies between 4.36 and 4.37. "When the root of any quantity can be found exactly, it is called d perfect jwwer or rational quantity, but if the root cannot be found exactly, the quantity is called irrational or surd. A number may be rational in regard to one root, and irrational in regard to another. Thus, 64 is rational as regards y^64=:8, 6 ]^64=4 and ]/()4i=:2, but it is irrational regarding any other root expressed by a whole number. But 64, with the fractional index ^, 64^, is rational, because it has an even root as already shown. I. c, if We may call 64" either the second power of the third root of 64, or the third root of the second power. In the former view, the third root of 64 is 4, and the second power of 4 is 16, and according to the second view, 64'^ is 4096, and the third root of 409G is 16, the same 4 as before. |/81=3 is rational, and |/81=9 is rational, but 81 is not rational regarding any other root ; while |/25 is rational only regarding the second root, and f/8^:^2 only regarding the third root. The second root of an even square may be -oadily found by re- solving the number into its prime factors, and taking each of these THIRD ROOT OR CITBE ROOT. 265 faoton onoe,— -the product will be the root. Thus, 141 is 3x3x7x7 and each factor taken onoe is 3x7-21, the second root. Here let it be obaervod, that if we used each factor twice we should obtain tho second power, but if we use each factor half the number of times that it occurs, ,wo shall have the second root of that power. 04 is 2X2X2X2x2x2=2fi, /. c, 2 repeated six times an a factor ^ivo.s the number 64, and therefore half the number of these factora will give the second root of G4, or 2x2x2—8, and 2X2X2 multiplied by 2X2X2=8X8=04. As this cannot be considered more than n trial method, though often expeditious, we would observe that the smallest possible divisors should be used in every case, and that if the number cannot bo thus resolved 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 fractions. . THIRD ROOT OR CUBE ROOT. As extracting the second root of any quantity is the finding of what two equal factors will produce that quantity, so extracting tlu^ third root is the finding of what three equal factorH will produce the quantity. By inspecting the table of third powers, it will be seen that no third power has more thtn three digits for each digit of the first power, nor fewer than two less than three times the numb<;r of digits. Hence, if the given quantity be marked off in periods df 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 0U3 digit. From the mode of finding the third power from the first, we can deduce, by the converse process, a rule for finding the first power j^, 266 > ARITHMETIC. i ,, from the third. We know by the rule uf involution that the tUird power of 25 is 15G25, If we resolve 25 into 20-1-5, and perform the multipUcatioa ia that form, we have 20-f-5 fr ■ '/?•■' v :; riv.-. . ..ru-^u ., , 400-j-lOO 100+25 ' , 400-f 200+25=:(20+5)-* 20+5 8000+4000+500 2000+1000+125 ' • 8000+ 6000+1500+125=(20+5) 3=15625 Now, 8000 is the third power of 20, and 125 is the third power of 5 ; also, 6000 is three times the product of 5, and the second iwwcr of 20, and 1500 is three times the product of 20, and th« second power of 5. Let a represent 20 and b represent 5, then a 3 =20 3 =:. 8000 3a3 6 r=3x202x5 ^^ 6000 3 a 62.-^3^20x5- =^ 1500 &3=53 = 125 15625 By using these symbols we obtain the simplest possible method of extracting the third root of any quantity, as exhibited by the subjoined scheme : Given quantity 15625 a^ =203:^=20X20X20 = 8000 .Remainder .' 7625 3a'< 6r=3x 202x5 = 6000 Remainder 1625 3ab'i=3X20x^' -- 1500 . ' Remainder 125 1*3^53^,5X5X5 = 125 From this and similar examples we see that a number denoted by more than one digit may be resolved into tens and units. Thus, 25 is 2 tens and 5 units, 123 is 12 tens and 3 units, and so of all numbers. THIBD BOOT OR CUBE ROOT. 2«7 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 hundreds, the second tens, and the third units, und so also in care of remainder, each period to the right will give one decimal place, the first being tenths, the second hundredths. Sic, &c. We may denote the digits by a, b and c. ' a=100 a»=1003: 1860867(100+20+3 ^123 1000000 and 30000X20= =20+, 860867 remainder. 600000 3 a 62=3X100X400— 260867 remainder. 120000 63=203= 140867 remainder. 8000 Now (a+6)=120 .-. 3 (a+6)2 =132867 remainder. 43200, which is contained 3 times+ in 132867, . • . c=3, and 3 (a+6)2c* =3X120^X3= 129600 And 3 (a+6) c2=3X 120X9= And lastly, 0^=3^= 3267 remainder. 3240 27 27 no remainder. RULE. Mark off the given nm.ih^ in periods of three figures each. Find the higheM third power contained in the left hand period, and svhtract it from that period. Divide the remainder and next period by three times the stcondpower of the root thus found, and the quotient loill be the second term of the root. Prom 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 third power of the second. Divide the remainder by three times the square of the sum of the first and second terms, and the auotient will he the third term. ;*Ki[| ;-^i 268 ARirnMETIO. from thii hit remainder subtract three times the jtroduct of the term last found, and the square of the SUM of the preceding terms, HLU8 t/ie product of the square of the last found term by the SUM of the preceding ones, plus the third power of the last found term, and so on. EXERC I8E8 1. 2. a. ■i:. 5. (). 7. 8. 9. What is the third root of 46656 ? ^^ns. 36. What is the third root of 250047 ? Ans. 63. W^at is the third root of 2000576 ? Ans. 126. Ky.: -8 the third root of 5545233 ? Ans. 177. What is the third root of 10077696 ? Ans. 216. What is the third root of 46268279 ? Ans. 359. What is the third root of 85766121 ? Ans. 441 . What is the third root of : 5751501 ? Ans. 501 . What is the third root of 153990656 ? Ans. 536. 10. What is the thiiJ root of 250047000 ? Ans. 630. 1 1 . What is each side of a square box, the solid content of which is 69319 ? Ans. 39 inches. 12. What is the third root of 926859375 ? Ans. 975. 13. Find the third root of 44.6. Ans. 3.416-|-. 14. What is the third root of 9 ? Ans. 2.080O";-j-. 15. What is the length of each side of a cubic vessel whose solid content is 2936.493568 cubic feet ? Ajis. 1432 feet. 16. Find the third root of 5, Ans. 1.7099. 17. A store has its length, breadth and height all equal ; it can hold 185193 cubic feet of goods; what is each dimension? Ans. 57 feet. 18. How many linear inches must each dimension of a cubic vessel be which can hold 997002999 cubic inches of water ? Ans. 999 inches. 19. What is the third root of 1 ? Ans. 1. 20. What is the third root of 14'< ? Ana. 5.241483. The third root of a fraction is found by extracting the third root of the terms. The result may bo expressed either us a comtnou fraction, or as a decimal, or the giveu fraction may be reduced to n decimal, and the root extracted under that form. tl 3 3 r P I c t i THmD BOOT OB GOBE BOOT. 269 An8. |=.7D. K X E R I B E 8 . 1. MTIwt ia the third root of jj| ? 4 Otherwise : f^ .-^iin: V. ||=:.42l876. To find the third root of this we have .42i875(.70f. 05^.75 703=: 3X70^X6 =73600) 3X70 X52= 5250 [ 53= 125) 343000 78875 remainder^ 78875 no remainder. The third root of a mixed quantity will be most readily found by reducing the fractional part to the decimal 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 the number of those divisors as factors. The same principle will apply to any root. If the given quantity is not an even power, it maj yet be found approx- imately. If wc 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 secoud and third depends may be applied to any root, the line of figures beinj^ divided intx) 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., &c. We may remark, hovrever, that these mode.s are now superceded by the grand .diaco^ ^^ry of Logarithmic Computa- tion. (See Logarithms. ..asi ■'Ml 18 '.rl, .|i?>r' 270 [0. PROGRESSION. A »erie» is a raoo«wion of qaantities increaaing or dmreasiog by n Common Differenct, or a Common Ratio. Progression by a Common Difference forms a series by i''? .if - tion or subtraction of the same quantity. Tlius 3, 7, 1!, ' ' '/ , 23 forma a series increasing by the constant quantity 4, and 26, 21, 14, 7, forms a series decreasing by the constant quantity 7. Sttofa a progression is also called an equidifferent series.* Progression by 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 ratio of 9 to 27, &c., &c. From this it is [lain 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 Extreme$, and all betweea them the Means. PROaRESSION BY A COIOION DIFFERENCE. In a series increasing or decreasing by a common di£ferenoe, the sum of the extremes is always equal to the sum of any tw6 that are equally distant from them. Thus, in the first example 3-|-23=74- 19=11+15=26, and in the second 28-{-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, 15, 19, 3+19=2x11^ the extremes. =22, and hence the middle term is half the sum of * The names Arithmetical Progression and 04ometrical Progression are often applied to quantities so related, but these terms are oltogeUier inappro- priate, as they would indicate that the one kind belonged solely to arithmetic, and the other solely to g'eometry, whereas, iu reality, each belongs to both these branches of science. PROOBESSION m A COMMON DIFFERENCE. 271 In treating of progressions Ly difference or oqaidiffcrcot scries, there are five thing?< to h« conrfidcrcd, viz., the first tcnn, the lost term, the common difference, the number of terms, and the sum of the series. These arc so related to each other that when any three of them are known we can £nd the other two. Given the first term of a scries, and the common difference, to find any other term. Suppose it is required to find the seventh term of the .scries 2, 5), 8, &c. Hero, as the first term is given, no addition is rcciuircd to find it, and therefore six additions of the common difference will complete the series on to seven terms. In other words, the common difiorencc is to be added to the first term a.s often as there are units in th3 number of terms diminished by 1. This gives 7 — 1 - C, and 6x3:=18, which added to the first term 2 gives 20 for the seventh term. If we had taken the series on the descending scale, 20, 17, 14, &c., we should have had to subtract the IS from the first tenii 20 to find the seventh term 2. The term thus found is usually designated the last term, not because the series terminates there, for it does not, but simply because it is the last term considered in each question proposed. From these illustrations we derive the RULE (1.) Svhtract 1 from the mimher of terms, and multiply the remainder by the common difference; then if the series he an increasing our, add tike result to the first term, and' if tlie series be a decreasing one, subtract it. EXAMPLES. To find the fifty-fourth term of the increasing series, the first term of which is 33|, and the common difference IJ. Here 54—1=53, and 53x1^=66^, and G6^+33|^a00, 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. Find the eleventh term of the decreasing series, the first term of which is 248|, and the common difference 3 J. Ans. 21G^. 2. The hundredth term of a decreasing series is 392;, and the common difference is 3|, what is the last term ? Ans. 30. ""•HI 'mil 'vr 272 ABIKHMBTIC. 3. What is the one-thoanndth term of the series of the odd figares? Ans. 1999. 4. What is the five-hundredth term of the series of even digits ? Am. 1000. fi. What is the sizteeath tuim of the deoreasiog series, 100, 96, 92, &o.? Ads. 40. To find the snm of - j equidifferent series, when the number of terms, and either the middle term or the extremes, or two terms equidistant from them, are given. We have seen already that in any such series the sum of the extremes is equal to the sum of any two terms that are equidiatan from them, and when the number of terms is odd, to twice the mid- dle term. Hence the middle term, or half the sum of any two terms equi-distant from the extremes, will be equal to half the sum of those extremes. Thus, in the series 2+7 {-12+17+22+274-32, we have 2i32=mi— 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 same as would be found by adding together the original quantities. The same result would 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 mm of the extremes, or of any two terms that are equidistant from them, by the number of terms, NoTii!.— If the sum of the two terms be an odd number, it is generallf more convenient to multiply by the number of terms before dividing by 2. EXAMPLES. Given 23, the middle term of a series of 11 numbers, to find the sum. Here we have onlyto multiply 23 by 11, and we find at onoe the sum of the series to be 253. Qiven 7 and 73, the extremes of ar increasing series of 12 num> bers, to find the sum. The sum of the extremes is 80, the half of which is 40, and 40x 12=480, the sum required. PBOOBE8HI0N BY A COMVON DUTESEMCE. 273 Two eqaidiatuDt '<;ri.i8 of a nctiw, 35 and 70, arc given in a aeries of 20 numbers, to Had the sum of the scriea. In this cue, we have 354-70:=105, and 10^X 20=2100, and 2100-i-2=10&0, the suiu re^^ired. rXEROIBEH. 1. the I of the series, consisting of 200 terms, the first term being 1 and the last 200. Ans. 20100. 2. What is the stun of the series whose first term is 2, and twenty.first 62 ? Ans. 672. 3. What is the sum of 14 terms of the series, the first term of which is ^ and the laat 7 '. Ans. 62^. 4. Find the sum to 10 terms of the decreasing series, the first term of which is 60 and the ninth 12. Ans. 360. 5. A canvasser w, the number of terms 24, and the greater eztremo is 47 ; v ^ loss extreme ? Ans. 1. 6. The sum of a ser '^^, the greater eztrcme 312, and the number of terms I'Ji. .Ji. less eztremo ? Ans. 1. Given the oztremes anu .tu>ab( difference. of terms, to find the oommon As czplained in the introduction to Rule (1), the number of oommon differenccH must bo one less than the number of terms. It is obvious also, that the sum of these differences constitutes the differ- ence between the extremes, and that therefore the sum of the differ- ences is the same as 1 less than the number of terms. Therefore the difference of the eztremes, divided by the sum of the differences, will give one difference, t. e., the oommon difiierenco. This gives us the BULB (5.) Subtract 1 from the number of termty and divide the difference qf the extremes by the remainder. EXAMPLE . If the eztremes of an increasing series be 1 and 47, and the number of terms 24, we can find the common difference thus : — 47 — 1=46, and 46-f-23=2, the common difference. EXERCISES. 1. If the eztremes are 2 and 36, and the number of terms 18; what the common difference ? Ans. 2. 2. What is the common difference if the eztremes arc 58 and 3, and the number of terms 12 ? Ans. 5. 3. In a decreasing series given 1000 the less cztreme, and 1793 the greater, and 367 the nuipber of terms, to find the common difference. Ans. 2^. Hlfl IMAGE EVALUATION TEST TARGET (MT-3) ^ ^^^ %i,M^ Adding the two members of the second to those of the first, ^ we obtain 2»== | 2o+(»— l)d I + | 2a-\-(n-'l)d I + 1 2a-\- (/^— l)d j- -f I 2a-|-(n— l)rf i -f &c., to n tenns. v^,,, // PROORESSION BY A COMMON DIFFERENCE. 277 In the last expression all the termfl are the same, but there are n temu, and therefore the whole will be 2i=» I 2a-\-{n—l)d I and therefore «=*^{2«+(»-l)y(l.) M-=^{2a^(n--l)d] ' . ■ . 2«=x2a?»=i2A2 — c6» * ' ^ . • . dn* -|-n(2a-^l)=2» And t^ BOlving this qnadratio equation, we find ^_.d--2a=bi/|8-6&61=G, the first term. IX1B0I81B. t 1. What is the ninth term of the increasing series of whioh 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 whioh is 1 and the ratio 3 ? Ans. 177147. 3. In a decreasing 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 20th term of an increasing series, the first of which is 1.06, and also the ratio 1.06 ? Ans. 3.207135. 5. In a decreasing series the first term is 126.2477, the ratio 1.06; what is the lost of 5 terms? i 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 whioh the first term is 5, the ratio 3, and the ntunber 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+15+45+135, and the sum is 200 15+45+135+405 . 400 Let us now multiply each term by the ratio, 3, and, for oonre- nience and clearness, place each term of the second lino below that one of the first to which it is equal. Let us now subtract the upper from the lower line, and we find that there is no remainder, except the difibrence of the two extreme quantities, viz., 400. Now, it will bo seen that this remainder is exactly double of the sum of the series, 200, and consequently 400 divided by 2, will give the sum 200. Also, 405 is the product of the last term by the ratio, and 400 is the difference between that product and the first term, and the divisor, 2, is a unit less than the ratio, 3. Hence the ABITHMBTIO. BUL« (2.) Multiply the hit term by the rath, from this product mibtraet thefint term, and< divide the remainder by the ratio, diminiehed by imity. XXAHPLl. Given the first tern- of an increasing series, equal 4, the ratio 3, and the number of termb 6, to find the sum of the series. By the former rule we find the last term to be 972. This, mul- tiplied by the ratio, gi^es :sd1.6, and the first extreme, 4, subtracted from this, leaves 2912, and this divided by 2, which is less than the ratio, gives 1466, the sum of the series. ^ EXBB0I8ES. 1. What is the sum of the series, of which the less extreme is 4, the ratio 3, and the number of terms 10? 3,, •-,,, Ans. 118096. 2. What is the sum of the series, of which 1 is the less extreme, 2 the ratio, and 14 the number of terms ? Ans. 16383. 3. What is the sum of the series, of which the greater extreme is 18.42015, the 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 for 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 011.111 ? Ans. $1111100.00. 5. What is the sum of six terms of the series, of which the greater extreme is ^ and the ratio | ? Ans. ||§c, ^^ ^/iVs- 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 therefore we divide 192 by 3 and find 64, and take the 6th root of 64, which is 2, the ratio. Hence the RULE (3.) Divide tJ^e greater extreme by the less, and find that root o/the qwtient, the index 0/ which is one less than the number of terms, EXAMPLE. If the greater extreme is 1024, and the less 2, and the number of terms 10, we divide 1024 by 2, and find 512, and then by extracting the ninth root of 512, we find the ratio, 2. PBOOBIBBIOim BY BATIO. ■ XlBOItlt. 1. If th« fiftt yearly dividend of a joint stoek oompany be $1, and the diTidends inoreaae yearly, ao as to form a seriea of oontinaal proportionala, what will all amount to in 12 ye^-ra, the last diTidend being $2048, and what will be the ratio of the increase ? Ana. ratio, 2 ; sum, $4095. 2. What is the ratio, in the aeries of which the leas extreme is 3 and the greater 98034, and the number of terms IG. Ans. .196605. 3. What is the ratio of a series, the extremes of which are 4 and 324, and the number of terms 5 ? Ans. 3. 4. What is the ratio of a series, the number of tenm being 7 and the extremes 3 and 12288 ? ' Ans. 4. 5. In a series of 23 terms the extremes are 2 and 8388608 ; what is the ratio? Ans. 2. To insert any number of means between two given extremes : Find the ratio Ity Rule (3), and multiply the fint extreme by (Aw ratiOf and the aecond wilt be obtained, and divide the latt by the ratio, and the last but one will be obtained; continue thit operation until the required term or term* be procured. Note.— A mean proportional is found by taking the aguart root of the pro- dudcflheettrtmea. m Br ay .,,,_ ,, BXAMPLS. Let it be required to insert between the extremes 5 and 1280 three terms, so that the numbers constituting the series shall be con- tinual proportionals. The number of terms here is 5, and hence, by Ru>4. < 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 ao on downwards, thus : 1280, 320, 80, 20, 5. XXKB0I818. 1. Between 5 and 405 insert three terms, which shall make the whole s series of continual proportionals. Ans. 5, 15, 45, 135, 405. 2. Insert between ^ and 27 four terms to form a series, and give the ratio. Batio, 3 ; series, ^, |, 1, 3, 9, 27. 284 ABITHMinO. 3. What thrM nnmbers inierted between 7 and 4375 will form » Mries of continual proportionab? Ans. 35, 176, 875. 4. What ifl the mean proportional between 23 and 8464 ? Am. 441.2164+. 5. Find a mean proportional between ^f and ]. Ans. |. I ALOXBBAIO rOBM. Let a represent the first term, I the last, r the ratio, n the nom- her of terms, and $ the sum. Then $=a-\-ar-\-ar^ -far • -far* -f &o oi*-' -f «»*" ' • Multiplying the whole equation by r, we obtain rt=zar-{-ar^ -far* -far* -far* -f &o m'*''* -f oi* . ' ' S" But i=a-^ar-\-ar'^ -far" -^ar* -f «»•" +&« ai*— ' . Subtracting, we obtain ' ' r» — i=$(r — l)=ar'» — a, and therefore 4 • ' ^ •.•...(1.1 ;■■'?'?- ■ -:^^ ' r'?- ' -i^''* ^t r — 1 ^ But we found the last term of the series to be ar^^, calling this /, we have from (1.) »=^ (2.) If r is a fraction, r* and at^ decrease as n increases, as already shown under the head of fractions, so that if n become indefinitely great, ar' will become unassignably small, compared with any finite quantity, and may be reckoned as nothing. In this case ( 1 ) will become *=;^=-j±;^ (3.) By this formula we can find the sum of any infinite series so closely as to differ from the actual sum by an amount less tlian any assignable quuntity. This is called the limit, an expression more strictly correct than . Let it be required to fiud the sum of the series l+i+i+i4- &c., 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 which 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. raoomsfiioiii bt batio. 286 By tdding the firit two terms, we and l+^=j==r2-4=lf Bj adding the fint three terms, wo find ^-f^=| =2—^—1}. By adding the fint four termi, we find }+i=V=«-~i--U- By adding the fint five terms, we find Y4-i^«=f |=2~i^«=' By adding the fint six terms, wo find ^-f-,^=||=:2-~]'j= It will be observed here that the difference from 2 is oontinoally decreasing. The next term would differ from 2 by ^'4, and the next by Y Jb, &o., &o. Thus, when the scries is carried to infinity, 2 may be taken as the sum, because it diffen from the actual sum by a quantity leas than any assignable quantity. IX AMPLIS. To find the sum of the fint twelve terms of tho series l4-3-f9-|- 27-I-&C.: Here a=l, r=3, To find the sum of the series 1,-3, 9, — 27, &o., to twelve terms, ax— 177M7- 1 '.•"? _ 11 -SX-S -1 =—132860. Ans. ^. ■ — —8-1 — -4 .,, In the case of infinite series, if a is sought, « and r being given, we have from (3) a— a (1 — r), and if r is sought, a and s being given, wo have i'==il? or 1 — -. ,,. ^ - KXKBOISIS. . ^ 1. Find the sum of the series 2, 6, 18, 54, &o., to 8 tenns. Ans. 6560. 2. Find the sum of the infinite series ^ — |+t^ — 3V ■ Observe herer= — |. 3. What is the sum of the series 1, ^, ^, &o., to infinity ? Ans. |. 4. Find the sum of the infinite series 1 — 1-|-^ — 3^+^* 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+J+|/i4-&o., to infinity. Ans. |/i — 1. 7. What is the limit to which the sum of the infinite series ^, ^, ^, I, &c., continually approaches ? Ans. -^. \\ 286 ABiimino* 8. What if tlM rom of tm Unu of Um MiiM 4, 12, SC» 4t.r ▲m. 118096. 9. InMrt three tenni betwoen 39 and 3159, m that the wbolo ■hall be a Mrioa of oontinual proportiooala. Am. 117, .351 and 1058. 10. Inaeii four terma between ^ and 27, w that the whole ihall form a aeries of oontinual proportionals. Ans. ^, 1,3, 9. 11. The sum of a series of oontinual proportionals is 10^, the first term 8| ; what is the ratio ? Ans. |. 12. The limit of an infinita series is 70, the ratio 4 ; what b tha first term? Ana. 40. ANNUITIES. The word Annuxtjf originally denoted a sum paid annutUfy, and though such payments are often made half-yearly, quarterly, &c., still the term ia applied, and quite properly, because the oaloulationa are made for the year, at what time soever the disbursements may be made. By the term annuities certain is indicated such as have a fixed time for their commencement and termination. By the term annuitiee contingent is meant annuities, the com- menoement or termination of which depends on some contingent event, most commonly the death of some individual or individuals. By the term -annuity in revenion 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, gener- ally the death of some person or persons. An annuity in perpetuity is one that " lasts for ever," and there- fore is a species of hereditary property. An annuity forborne is one the payments of which have not been made when due, but 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 for that time. Tables have been constructed showing the present and final values per unit for dififcrent periods, by which the value of any annuity may be found accordiog to the following AMHUrriES. fgf RULIl. To find either the tmoant or the praecnt vahio of ad innnftj, — MuUipljf the value of the unit, aa found in the tabtes, bg the nynmher denoting the annuitjf. ' If the aoQuity bo in perpetuity, — , Divide the annuitjf by the number denoting the intemt of the unit for one year. If the annaity bo in roTenion, — Find the value of the unit up to the date 0/ commencement, and alto to the date 0/ termination, and multiply their difference by the number denoting the annuity. To find the annuity, the time, rate and present worth being giTen. Divide thepretent worth by the toorth of the unit. ' Tables are appended varying from 20 to 60 years. EXAM PLE8. To find what an annuity of $400 will amount to in 30 years, at 3^ per cent. We find by the tables the amount of $1, for 30 years, to be $51.622677, which multiplied by 400 (rives $20649.07 nearly. To find the present worth of an annuity of $100 for 45 years, at 3 per cent. By the table wo find $24.518713, and this multiplied by 100 gives $2451.88. To find the present worth of a property on lease for ever, yielding $600, at 3^ per cent. The rate per unit for one year is .035, and GOO divided by this gives $17142.86. To find the present worth of an annuity on a lease in reversion, to commence at the end of three years and to last for 5, at 3^ per cent. By the table we find the rate per unit for 3 years to be $2.801637, and for 8 years, the time the lease expires, $6.873956 ; the differ- ence is $4.072319, which, multiplied by 300, gives $1221.70. Given $207.90, the present worth of an annuity continued for 4 years, at 3 per cent., to find the annuity. By the tables, the value for $1 in $3.717098. and $207.90, divided by this, gives $55.93. \ TABLE, mowan m amooiit op av uaamr or om waiuM m uamif mnavn AT ooMncMD nrmnr poe axt mnon or tiabs wn izonoiKO nrrr. 1 2 3 4 6 6 7 8 9 10 11 12 IS U 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 [60 3 per cent's) percent 1.000 000 2.080 000 3.090 900 4.183 627 5.309 136 6.468 410 7.662 462 8.892 SS6 10.169 106 11.463 879 12.807 796 14.192 031) 16.617 790 17.086 324 18J»98 914 2). 156 881 21.761 688 23.414 435 25.116 868 26.870 374 28.676 486 30.636 780 32.462 884 34426 470 36.450 264 38JS63 042 40.709 634 42.930 923 46.218 850 47.576 416 60.002 678 52.602 759 55.077 841 57.780 177 60.462 082 63.271 944 66.174 22S 69.169 /i49 72.234 233 76.401 260 78.663 298 82.028 196 85.488 892 89.048 409 92.719 861 96JS01 457 100.896 601 104.408 896 108.540648 112.796 867 4 per cent 1.000 000 2.036 000 3.106 225 4.214 943 6.362 466 6.650 152 7.779 408 9.051 687 10.368 496 11.731 393 13.141 992 14.601 962 16.113 080 17.676 98C 19.295 681 20.971 030 22.706 016 24.499 691 26.357 180 28.279 682 30.269 471 32.328 902 34.460 414 36,666 628 38.94^ 857 41.313 102 42.759 060 46.290 627 48.910 799 61.622 677 64.429 471 67.334 502 60.341 210 63.463 152 66.674 013 70.007 603 73.457 869 77.028 895 80.724 906 84.560 278 88.609 637 92.607 371 96.848 629 101.288 381 105.781 673 110.484 031 116.350 973 120.888 297 126.601 846 6 per oent 1.000 000 2.040 000 3.121 600 4.246 464 6.416 828 6.632 976 7.898 294 9.214 226 10.582 796 12.006 107 18.486 351 15.025 805 16.626 888 18.291 911 20.028 688 21.824 681 2a697 612 25.646 418 27.671 229 29.778 079 31.969 202 34.247 970 36.617 889 39,082 604 41.646 908 44.311 745 47.084 214 49.967 583 62.966 286 66.084 938 69.328 885 62.701 469 66.209 627 69.857 909 73.652 225 77.598 814 81.702 246 85.970 836 90.409 160 95.025 516 99.826 636 104.819 598 110.012 382 115.412 877 121.029 892 126.870 668 182.946 390 139.268 206 146.833 734 6 per oent 180.999 9101162.667 084| 1.000 000 2.050 000 3.152 500 4.810 125 6.526 681 6.801 913 8.142 008 9.549 109 11.026 664 12Ui77 893 14.206 787 15.917 127 17.712 988 19.598 638 21J»78 664 28.657 492 25.840 866 28.182 885 30.539 004 33.066 954 35.719 252 38J>05 214 41.430 476 440)01 999 47.727 099 61.118 454 54.669 126 68.402 683 62.322 712 66.438 848 70.760 790 76.298 829 80.063 771 86.066 959 90.320 807 95.836 323 101.628 189 107.709 646 114.095 023 120.799 774 127.889 763 135.281 761 142.993 389 161.148 006 159.700 166 168.685 164 178.119 422 188.026 393 198.426 663 209.847 976 7 per cent 1.000 000 2.060 000 8.183 600 4.374 616 6.687 098 6.976 819 8.398 838 9.897 468 11.491 816 13.180 795 14.971 643 16.869 941 18.882 138 21,016 066 23.276 970 26.670 628 28.212 880 30.906 668 83.759 992 36.785 591 39.992 727 48.392 290 46.996 828 60.815 577 54.864 612 69.156 883 68.706 766 68.528 112 78.689 798 79.058 186 84.801 677 90.889 778 97.343 165 104.183 765 111.434 780 119.120 867 127.268 119 135.904 206 145.058 458 154.761 966 166.047 684 175.950 645 187.607 677 199.758 032 212.743 614 226JS08 126 241.098 612 266.664 629 272.958 401 290.886 906 1.000 000 2.070 000 3.214 900 4.489 943 6.760 789 7.163 291 8.664 021 10.269 803 11.977 989 18.816 448 16.788 599 17.888 461 20.140 648 22.660 488 26.129 022 27.888 054 80.840 817 88.999 083 37.378 965 40.996 492 44.866 177 49.005 739 63.486 141 6&176 671 63.248 080 68.676 470 74.488 823 80.697 691 87.846 629 94.460 786 102.078 041 110.218 154 118.938 425 128.268 765 138.236 878 148.913 460 160.387 400 nuei 020 185.640 292 199.685 112 214.609 670 280.682 240 247.776 496 266.120 851 286.749 311 306.761 768 329.224 386 363.270 098 378.999 000 406.628 929 Vv •'/ AllHllli'JIB. siowiM m TABLE, WOMB or AM AXKVITT OP OKI OOUiAB MB AXmU, TO OOMTDIOB lOB Altr XCIIBIB OP TBAM NOT EXCnODtO PIPTT. i 3 4 6 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2S 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 3peroeni 0.970 874 1.913 470 2^28 3.717 4Ji79 5.417 6.230 7.019 7.786 8JiS0 9.252 9.954 10.634 11.296 11.937 12.561 13.166 13.753 14.323 14.877 15.415 15.936 16.443 16.935 17.413 17.876 ia327 18.764 19.188 19.600 20.000 20.338 20.765 21.131 21.487 21.832 22.167 22.492 22.808 23.114 23.412 23.701 23.981 24.254 24.518 24.775 25.024 25.266 25.501 25.729 611 098 707 191 288 109 203 624 004 955 073 935 102 118 513 799 475 024 917 608 542 148 842 031 108 455 441 428 766 792 837 220 252 235 462 215 772 400 359 902 274 713 449 708 707 657 764 S^peieent 0.966 184 1.899 694 2.801 3.673 4J»15 5.328 6.114 6.873 7.607 asi6 9.001 9.663 10.302 ia920 11.517 12.094 12.651 13.189 13.709 14.212 14.697 15.167 15.620 1G.058 16.481 16.890 17.285 17.667 18.035 18.392 18.736 19.068 19.390 19.700 20.000 20.290 aoj»7o 20.841 21.102 21.355 21.599 21.834 22.062 22.282 22.495 22.700 22.899 23.091 23.276 23.455 6.*)7 079 052 553 544 956 687 605 551 334 738 520 411 117 321 682 837 403 974 125 410 368 516 352 365 0i9 767 045 276 865 208 684 661 494 525 087 .«nhip has been made, any partner is al ISbntj, al anj time, to withdraw, on showing snffieient canse and giring proper noCiee. This is a jnst prorision, for the eiroomstanoes of any part- ner may so ohange, from varions cianses, as to make it nndesirable for him to remain in the bnsiness. If one partner is deputed to settle the aooonnts 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 R. Evans are partners in business, and invested whm commencing $1000 each. On dissolving the partnership, the assets and liabilities are as follows: — Merchandise valued at $1295 ; cash, $344 ; notes against sundry individuals, $790 ; W. H. Monroe owes on account $86.40 ; E. R. Carpenter owes $132.85, and C. F. Mugrove owes $67.50. They owe on sundry notes, as per bill book, $212.40 ; £. O. Conklin, on account, $29.45, and H. C. Wright, on aoooant, $41.30. What has been the net gain ? SOLUTION. A$tett. Merohandise on hand... $1295.00 Gash on hand 344.00 Bills Receivable 790.00 Amt due from W. H. Monroe 86.40 Amt. due from S. R. Carpenter 132.85 Amt. due from C. F. Musgrove 67.50 Total amount Assets.... $2715.75 " " LiabUities, 2283.15 Net gain $432.60 LiainKliei. Bills Payable $212.40 Amt. due E. Q. Conklin . 29.45 Amt. due H. 0. Wright. 41.30 W. Smith's investment... 1000.00 R. Evan's investment.... 1000.00 $2283.15 ^t..- •\ "-■ ■., ■-' BOLK. ,;. __^„ :, Find the sum of the aueU and liahilities , from the aweu whtract the Uchilitie$y (induding the net amount inveated) and the differ- ence wUl be the net gain ; or, if the liabilities be the larger, subtract the assets from the luUnlities, arid the difference will be the net loss. 292 ABnmcEnc. 2. Harrcy Miller and Junes Carey are partners in a dry goodi business^, Harvey Miller investing $1400, and James Carey $1250. Wben dosing the books, they have on hand — cash, $1125.30 ; mer- chandise, as per inventory book, $1856.75 ; amount deposited in Bank of Toronto, $1200 ; amount invested in oil lands, $963 ; a site of land for building purposes, valued at $1600 ; Adam Dudgeon owes them, on account, $104.92 ; William Fleming owes $246.80 ; a note against Alfred Mills for $69.43, and a due bill for $30, drawn by James Laing. They owe W. 8. Hope & Co., on account, $849.21 ; R. J. King & Co., $608.12, and on notes, $1326.14. What has been the net gain or loss ? Ans. $1759.73 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 worth of merchandise on hand ; cash, $714.27 ; Royal Bank stock, $500 ; deposited in Royal Bank, $320.60; store and fixtures valued at $3100; amount duo on notes and book accounts, $3471.49. The firm owes on notes $3400, and on open accounts $747.10. James Henning invested $1200, and Adam Manning, $1000 ; what is each partner's interest in the business at dosing ? Ans. James Henning's interest, $2719.63; Adam Manning's . interest, $2519.63. Note.— Where the interest of each partner at closing is required, the gain or loss is first found, as in former examples, then the share of gain or loss is added to or subtracted from each partner's investment, and the sum, or diflference, is the interest of "ach partner. If a partner has withdrawn any- thing from the business, the amoimt thus withdrawn must be deducted from the sum of liis investment, plus his share of the gain, or minus his sluure of the loss, and the remainder will be his net capital or interest. 4. F. A. Clarke, W. H. Marsden, and J. M. Muf^ve, are con- ducting business in partnership ; F. A. Clarke is to be ^ gain or loss, W. H. Marsden and J. M. Musgrove, each }. On dissolving the partnership, they have cash on hand $712.90 ; merchandise, as per Inventory Book, $4360 ; bills receivable, as per Bill Book, $1450.75; amount deposited in Bank of Montreal, $3475 ; merchandise shipped to Montreal, to be sold on own account and risk, valued at $995 ; debts due from individuals on book account, $2644.67. They owe on notes $3760, and to Manning & Munson, $1312.60. ■/ PABTNEBSHIP SETTLEMENTS. F. A. Clarke invested $5750, and has drawu out 1875 ; W. H. Manden invested $2500, and has drawn out $500 ; J. M. Mnegmve invested $3000, and has drawn out $750. AVhat has been the net gain or loss, and what is each partner's interest in the business ? Ans. Net loss, $559.28 ; F. A. Clarke's interest, $4595.36 ; W. H. Marsden's interest, $1860.18 ; J. M. Mnsgrovo's inter- • est, $2110.18. NoTR.— In this and succeeding examples, net Interest is to be allowed on investment, or charged on amonats withdrawn, unlets so specified. 5. A, B, and C are partners. The gains and losses are to be shared as follows : A, ,'*, ; B, /j ; and C, /j. A invested $3000, and has withdrawn $2,500, with the consent of B and C, upon which no interest is to bo charged ; B invested $2700, and has withdrawn $1150; invested $2500, and has withdrawn $420. Afler doing business 14 months, C retires. Their assets consist of bills receivable, $2937.20 ; merchandise, $1970 ; cash, $1243.80 ; 50 shares of the Canada Permanent Building and Savings' Society Stock, the par value of which is $50 per share ; cash deposited in the Ontario Bank, $1850 ; store and furniture, $3130 ; amount due from W. Smith, $360.80; G. S. Brown, $246.40; and E. R. Car- penter, $97.12. Their liabilities are as follows : amount due Samuel Harris, $1675; unpaid on store and furniture, $935; and notes unredeemed, $3388.76. C, in retiring, agrees to allow the firm 10 per cent, advance on the Savings' Bank stock. What is the amount due C, and what is A's, and what is B's interest in the business ? Ans. Due C, $815.52; A's interest, $2356.90; B's interest, $2664.14. 6. E, F, 6 and H are partners in business, each to share ^ of profits and losses. The business is carried on for one year, when E and F purchase from G and H their interest in the business, allow- ing each $100 for his good will. Upon examination, their resources are found to be as follows : cash deposited in Quebec Bank, $3645 ; cash on hand, $1422 ; bills receivable, $1685 ; bonds and mortgages, $2746, upon which there is interest due $106; Royal Canadian Bank stock, $1000 ; Quebec Bank stock, $500 ; ntore and fixtures, $3500 ; house and lot, $1800 ; span of horses, carriages, harneps, &c., $495; outstanding book debts duo the firm, $4780. Their liabilities are: notes payable, $2J45, upon which thete is interest due $57 ; due on book debts, $1560. £ invested $5000 ; F. $4500 ; SM AKi'ivumo O, $4000; Md H, $3000. E has dnwn ftom the banneM $1200, uftok which he owm interest $82 ; F has drawn $1000— owes interest $24.50; has drawn $950— owes interest $12; and H has drawn nothing. In the settlement a disoonnt of 10 per cent., for bad debts, is allowed, on the book debts doe the firm and on the bills reoetvsble. G takes the Royal Bank stock, allowing on thasame a premium of 6 per cent. ; and H takes the Quebec Bank stock, at a premium of 8 per cent ; £ and F take the assets and assumes the liabilities, as above stated. What has been the net gain or loss, the balances due and U, and what are E and F each worth after the settlement? Ans. Due G, $3057.75; due H, $3529.75; E's net capital, $4637.75 ; F's net capital, $4345.25. 7. H. C. Wright, W. S. Samuels, and E. P.* Hall, are doing business togeiiher — H. C. W. to have ^ gain or loss; W. S. S. and E. P. H. each |. After doicg business one year, W. S. S. and E. P. H. retire from the firm. On dosing the books and taking stock, the following is found to be the result : merchandise on hand, $3216.50; cash deposited in Quebec Bank, $1627.35; cash in till, $134.16; bills receivable, $940.60; G. Brown owes, on account, $112.40 ; Thos. A. Bryco owes $94.12; W. McKee owes $143.95 ; J. Anderson owes $54.20; K. H. Hill owes $43.60 ; and S. Graham owes $260.13. They owe on notes not redeemed $1864 ; H. T. Collins, on account, $124.45 ; and W. F. Curtis, $79.40. H. C. Wright invested $3200, and has drawn from the business $350. W. S. Samuels invested $2455, and has drawn $140 ; E. P. Hall invested $2100, and has drawn $2000. A discount of 10 per cent, is to be allowed on tho bills receivable and book accounts due the firm, for bad debts. H. C. Wright takes the assets and as- sumes the liabilities, as above stated. What has been the net gain or loss, and what does H. C. Wright pay W. S. Samuels and E. P. Hall on retiring ? Alls. Net loss, $970.74; H. C. W. pays W. S. S., $2072.31^; and E. P. HaU pays H. C. W. $142.68J. 8. T. P. Wolfe, J. P. Towler and E. R. Carpenter have been doing business in partnership, sharing the gains and losses equally. After dissolution and settlement of all their liabilities they make a division * >^ A, B, C and D arc partners. At the time of dissolution, and after the liabilities arc all cancelled, they make a division of the effects, and upon examination of their ledger it shows the following result: — A has drawn from the business $3465, and invested on commencement of business, $4240 ; B has drawn $4595, and invested $3800; C has drawn $5000, and invested $3200; D has drawn $2200, and invested $2800. The profit or loss wag to be divided in ' 10. 296 ABTTBlBnO. proportion to the original investment. What hu been eaoh partner*! gain or loas, and how do the partnera settle with eaoh other ? Ans. A'e net gain, $368.43 ; B'e net gain, 1330.20 ; C'a net gain, $278.06 ; D's net gain, $243.31. B has to pay in $464.80 ; has to pay in $1521.94. A reoeives $1143.43 ; D reoeives $843.31. 11. Three meohanios, A. W. Smith, James Walker and P. Banton, are equal partners in their business, with the understanding that eaoh is to be charged $1.25 per day for lost time. At the close of their business, in the settlement it was found that A. W. Smith had lost 14 days, James Walker 21 days, and P. Ranton 30 days. How shall the partnera properly adjust the matter between them ? Ans. P. Rimton pays A. W. Smith, $9.58|, and James Walker, 83| cents. 12. There ore 5 mechanics on a certain piece of work in tho following proportions :— A is ^% ; B, ^g ; C, 3^ ; D, j",, and E, ^. A is to pay $1.25 per day for all lost time ; B, $1 ; C, $1.50; D, $1.75, and £, $1.62|. At settlement it is found that A has lost 24 ; B, 19 ; C, 34 ; D, 12 ; and £, 45 days. They receive in pay- ment for their joint work, $2500. What is each partner's share of this amount according to the above regulations ? Ans. A's share, $374.12 ; B's, $250.41 ; C's, $487.83 ; D's, $787.24; E's, $600.40. 13. A. B. Smith and T. C. Musgrovo commenced business in partnership January Ist, 1864. A. B. Smith invested, on com- mencement, $9000; May 1st, $2400; June 1st, he drew out $1800; September Ist, $2000, and October 1st, he invested $800 more. T. C. Musgrove invested on commencing, $3000 ; March 1st, he drew out $1600; May Ist, $1200; June 1st, he invested $1500 more, and 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 per inventory, $10500 ; cash on hand, $4900 ; bills receivable, $12400 ; R. Draper owes on account, $2450. They owe on their notes, $1890, and G. Roe on account, $840. Their profit and loss account is, Dr. $866 ; Cr. $1520. Expense account is. Dr. $2420. Com- mission account is, Cr. $2760. Interest account is Dr. $480 ; Cr. $950. The gain or loss is to be divided in proportion to eaoh partner's capital, and in proportion to the time it was invested. Beooired each partner's share of the gain or loss, the net balanoe PB0PEBTIE8 OP KUMBEBS. 297 due eteh, and a ledger speoifioation exhibiting the oloiing of all the aoooants, and the balance sheet. Ana. A. B. S.'s net gain, $6671.73; hia net balance, $16071.73. T. 0. M.'b net gain, $2748.27 ; his net balance, $12448.27. PROPERTIES OF NUMBERS. The tenn Integer, or Whoh Number, is used in contradistinction to the term Fraction. All numbers expressed by the natural series 1, 2, 3.. .10.. .20.. .100, &o., are colled integers, so that 3 and 4 are integers, but ^ is a fraction. All numbers in the natural series 1, 2 3, &o., that can be resolved into factors, are called Compoeite, while those that cannot bu so resolved are called Prime. Since 4=2 X 2, it is called composite, and so 6, 8, 9, 10, &c., but 1, 2, 3, 5, 7, 11, &c., are called prime because they cannot be resolved into factors. Thus, 11 can only be resolved into llXl>orlXll> and these aru not t'actortt in the strict meaning of the word. A Prime Factor is a prime number, which is a factor of a com- posite number. The factors of 10 are 2 and 5, both prime numbers. A composite number may have composite factors, as 36, which has 4 and 9 as factors, and both of these are composite. When any number will divide two or more others, it is called a Common. Factor. Thus, 3 is called a common factor of 6, 9, 12, 15, &c. Numbers that have no common factor, as 4, 5, 9, are said to be prime to each other. "i To resolve a composite number into its prime factors, divide it by the leaet jpomble factor that it contains, and repeat the proccaa tiU a prime number it obtained. EXAMPLES. -^v.;:i,.'rA- .^ ■'• . 2)96 '' ^' -- '■-''' -■■•-■- :^:^ ,,,;s^.., ■- 2)48 - ^..rrJ,... . 2)24 ^^ ^;.^^ .._^ 2)12 . - . 2) 6 10 that the prime factors of 96 are 2X2X2X2X^X3. ■"■■> .1^ ■■■■•*. 298 ABrrHMBno. Alio, beoMue 5X7X11=386, wo mm that 6, 7 ud 11 fti« Um prime iaotora of 386. ■ XB^OISBB. 1. What are the prime faoton of 2310? Ans. 2, 3, 6. 7, 11. 2. What are the prime fkotors of 1764 ? Ans. 2, 2, 3, 3, 7, 7. 3. What are the prime factors of 180642 ? Ana. 2, 3, 7, 11, 17, 23. 4. What are the prime factors of 95 ? Ans. 5, 19. 6. What are the prime factors of 51 ? Ans. 3, 17. 6. What are the prime factors of 99 ? Ans. 3, 3, 1 1 . 7. What are the prime factors of 651 ? Ans. 3, 7, 31. 8. What are the prime factors of 362880 ? Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 6, 7. 9. What factors are common to 84, 105, and 147 ? Ans. 3, 7. 10. What are the prime factors of 308 ? Ans. 4, 7, 11. Whether a number ia prime or composite can only be found by trial. The only even prime number is 2 ; for 4, 6, 8, 10, &c., are all. multiples of 2. > The only prime number ending in the digit 5 is 5 units, and all other numbers ending in either 5 or are multiples of 5. ADDITIONAL EXKRCISKS. 4 . .>. , Ans. Prime. 11. Is 101 prime or composite ? 12. Is 198 prime or composite ? Ans. It has the factors 2, 3, 3, 11. 13. Is 171 prime or composite ? .-• ;j ,V Ans. It has the factors 3, 3, 19. 14. Is 473 prime or composite ? 15. Is 477 prime or composite ? 16. Is 549353259 prime or composite ? 17. I» 674041 prime or composite ? 18. Is 199 prime or composite ? 19. What are the prime factors of 210 ? 20. What are the prime factors of 51051 ? Ans. Prime. Ans. Composite. Ans. Composite. Ans. Composite. Ans. Prime. Ans. 5, 6, 7. Ans. 3, 7, 11, 13, 17. NoTR. — We have thought it suiBciont under this head to give only the leading and most uaefal principles. .^I'i:- LooABfrmn. LOGARITHMS, A logiriUun u tlio index of the power of way ntunber wrkkh that index denotes. Thus 9=3^ where the index 2 is the loKuithn of 9. Taking 3 ' •• ft ooounon root, we hare 3=3' 1=3« 9=3a *=3- 27=3» i=3-^ 8l»-3< ,',=3-» These indioes may, however, bo the logarithms of any root. Thu: 4=4 « 16=4* 64=4« l=a40 ^.=4-« Also, 8=4^, which means that 8 is the second root of the third power of 4. The third power of 4, or 4X^X4=64, and 8 is the second root of 64, for 8x8=64. Again, since 8=2*, and 16=2 «, it follows that the index of any number between 8 and IG, when 2 ia the root, will be 3, plu$ some fraction, that ia, the index will be some number greater than 3, and less than 4. If 4 be the root, then, as above, 8=4^, where the index is 3, or 1.5, greater than 1, but less than 2. In this case, the index expressed decimally is a terminator, but oven when the decimal does not terminate, it may be found approximately with sufficient accu- racy. From this it is obvious that logarithms may be calculated for all numbers. The root from which logarithms are calculated is called tho Base. The bases we have taken for illustration are 2, 3, and 4, but any number may be assumed as base. The base of the original system, invented by Baron Napier, and hence called Napierian logarithms, was 2.718-{-, but the base that has been found most convenient is 10, as it falls in so exactly with the decimal notation. These, by way of distinction, are called Common Logarithms. The common logarithm, then, of a number, ia the index of the potoer of 10, uihich it equeU to that number. 300 AJUTHMmO. Henoe BUrting from the base 10, wo have the following uoondii^ and deMendin); aoalos : l\>wtr. Nit. l4>g. l00=l 10'=10 1 10»=100 2 io«=iooo....a 10«=:.10000...4 Power. No. Uki. lO-'-T^, 1 10-»=tAo 01 10» =To'oo 001 10-4=„4„ 0001 J' Now wo must obieiTO that, aince it is proved by Algebra that any quantity with an index zero is always equal to unity, the Logarithm ofl isO, forx»-^l, a"=:l, m'>==l, and 10«»=1, as exhibited iu the above nchemc. Again, since log. 1=0, and log. 10=1, it follows that the logarithmH of all the numbers between 1 and 10 will be lestt thau 1, but above zero, and also that the log. of any base will be 1. Also, since log. 10=1, and log. 100=2, it follows that the logarithms of all numbers between 10 and 100 will be greater than 1, but leas than 2, and that the logarithms of all numbers between 100 and 1000 will be between 2 and 3, and so on. Logarithms are always expressed in the decimal fonu. The integral part is called the Characteristic. ' ' The decimal part is called th< .Uantiisa. * ' It is not usual in constructiug tables to notice the characteristic, because, aa the above explanations will show, it will always be denoted by as many units as there arc digits in the given numbtr, lest one. Thus, the charaotcristio of 2479G will be 4, and of 879 it will be 2, and therefore it can always bo found by counting the digits. It is also obvious that the characteristics of numbers less than unity must ,e negative, for log. io=log. 10-'== — 1, and log. roo'=^*'8* ^0 '^ =log. .01=: — 2. Since the characteristic is always 1 less than the number of digits, we have the same characteristic for 100 ^ fcir 199, but for 100 we have simply 2, whilo for 199 we have u phi* aome fraction, and since for 4.297, where we have only uno <^i;^U.. ihe characteristic is zero, one less than the number of digits, it follows that for .297, where there is no whole number, the characteristic mu..t l>o — 1 plus some fraction. Hence the mantissa is always positiv.\ and is the same for decimals as for whole numbers. This fracno.i " /7275o, and therefore we have — 1+.472756, which, for coiwcuicr'C& iS written 1.472756, the negative sign being written LooABnmn. aoi a&0M Um eharifSlcruitte, &n %tiiihr> which i» extremely eoii?enieiit in the arrtogiog of tables, u ' !>l»u iu upc 'thm of tho dividend. Also, that a quantity can bo raised to any power by multiplying its logarithm by the index of tho power, and that any root may bo extracted by dividing tho logarithm of tho number by the index. # EXPLANATION OV THK TABLB*. In the first column are marked the natural numbem for tliroe di^ts, and along the top of the page the ten digits. 0, posite the throo digits, on the margin and under tho zero column, wil; be found the mantissa for the number represented by those digits, >r those digits with a cipher annexed, for the annexing of a cipher us akes no difierence in the mantissa, but adds a unit to the index. If tli« given number consists of four digits, the required logarithm will be found under the fourth figure on the top o^ tho page, and opposiu; the other three on the margin. The numbers in the column markod D denote the closely approximate difierence between any two eontiguou:i numbers, or the increment of the logarithms as the natural numtMrs increase by the successivo additions of a unit. To find by the tables the logarithm of any number. If the number docs not exceed 100 look for it at the begianiog of the r.4ible in tho vertical column marked N. or No., and both characteristic and mantissa will be found opposite to it in the column usually marked log. or L. Thus the logarithm of 98 will be found by simple inspection to be 1.991126. If the number consist of four digits, find tl«l- by the Hquarc foot. What in colled board measure in a certain length and breadth, and a uniform thickncHH of one inch. Large <|uantiticH of round timber are often entimatod by the ton. To 6nd either the Nuperficial extent or board moaauro of a plank, &0. R U L K . Multiply the length in feet by the breadth in inchei, and diviif^ by 12. NfyfK. — The thicknofw being taken uniformly as one innh, the rule for flntl- ing the contents in Hquure ftu't becomeH tliu hudih us that for finding Hiirfuce. If the tbicknesH bu not an inch,— Multiply the board meawre by the thicknest. If the board be a tapering one, take half the 8um of the two extreme widths for the average width. If a one-inoh plank be 24 feet long, and 8 inches thick, then we have 8 inches equal § of a foot, and f of 24 feet=16 feet. A board 30 feet long is 2G inches wide at the one end, and 14 inches at the other, honco 20 is the mean width, i. e., 1§ feet, and 30X1^^50 ; or, 30x20=600, and COO-f-12^^50. To find the solid contents of a round log when the girt is known. RULE. Multiply the square of the quarter girt in inches by the length in feet, and divide the product by 144. If a log is 40 inches in girt, and 30 feet long, tho solid contents will bo found by taking the square of 10, the quarter girt in inchcH, which is 100, and 100x30=3000, and 3000-^144=20^. To find the number of square feet in round timber, when the mean diameter is given. 1 1 ■'1 I ir I f • 316 lUiiU ARITHMETIC. BULK . u Multiply the diameter in inche* hy half the diameter in inches, and the product hy the length infect, and divide the result by 12. If a log is 30 feet long, and 56 inches mean diameter, the number of square feet is 50x28x30-5-12=1960 feet. To find the solid contents of a log when the length and mean diameter are given. / RULE Multiply the square of half the diameter in inches by 3.141 6, and this product by the length in feet, and divide by 144. 68. How many cubic feet are there in a piece of timber 14x18, and 28 feet long ? Ans. 49-f cubic feet. 69. How many cubic feet are there in a round log 21 inches in diameter, and 40 feet in length ? Ans. 96^ J. 70. What are the solid contents of u log 24 inches in diameter, and 34 feet in length ? .i: Ans. 106.81 -f-cubic feet. 71. How many feet, board measure, are there in a log 23 inches in diameter, and 12 feet long ? Ans. 2641-. 72. How many feet, board nteasure, are there in a log, the diameter of which is 27 inches, and the length 10 feet. Ans. 486. 73. What are the .solid contents of a round Ic^ 30 feet long, 18 inches diameter at one end, und 9 at the other ? V Ans. 30.e3-|-cubic feet. 74. How many feet of .square timber will a round log 36 inches iu diameter and 10 feet long yield ? • Ans. 540 solid feet. 75. How many solid feet are there in a board 15 feet long, 5 inches wide, and 3 inches thick ? ' '^ Ans. 1/^ cubic feet. 70. 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 a4)iece of timber 12 feet long, 10 inches broad, and 12 inches thick ? Ans. 16 feet. 78. How many cubic feet are there in a log that is 25 inches in diameter, and 32 feet long ? ,. / 70. How many feet, board measure, does a log 28 inches in diameter, and 14 ieet iu length contain ? Ans. 457^. 80. How many cubic feet are contained in a piece of squared timber that is 12 by 16 inches, and 47 feet in length ? Ans. 62§. MEASTTBEMENT OF TIMBER. 317 81. How many feet, board measure, arc there in 22 one-inch boards, each being 13 inches in width, and 16 feet in length ? Ans. 381f BALKS, BINS, *0. hn bnles are usoally of the same form as boxes, the same rule, applies. . •: ■■-.y.. : ■*■ , ,v:.'?-iv,*f'i^' -yr 82. Hence, a bale measuring 4^ inches in length, 33 in width, and 3^ in depth, is, in solid content, 37^ feet. 83. A crate is 5 feet long, 4^ broad, and 3yT. deep, what is the solid content ? Ans. 85^^. To find how many bushels are in a 6m of grain : >■ BULK. Find the product of the length, breadth and depth, and divide by 6150.4. 84. A bin consists of 12 compartments ; each measures 6 feet 3 inches in length, 4 feet 8 inches in width, and 3 feet 9 inches in depth ; how many bushels of grain will it hold ? Ans. 1055, nearly. To find how many bushels of grain are in a conical heap in the middle of a floor : RULE . Multiply the area of the base by one-third the height. The base of such a pile is 8 feet diameter and 4 feet high ; what is the content ? The area of the base is 64 X. 7854=83.777, and 83.777x3=- 67.02, the 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 above result. '. I ' 91 318 abetbheho. MISCELLANEOUS EXERCISES 1. What nnmber is that f and jf of which make 255 ? Adb. 20iy^. 2. What must be added to 217^, that the sum may be 17^ times 19|? Ans. 118^. 3. What sum of money muat be lent, at 7 per oent., to accumu- late to $455 interest in 3 months ? Ans. $26000. 4. Divide $1000 among A, B and C, so that A may have $156 more than B, and B $62 less than C. Ans. A. $416S ; B, $260§ ; C, $322§. 6. Where shall a pole 60 feet high be broken, that the top may fest 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 much did he 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 1^ times as much as B, and B does ^ as much as C ? 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 ? An:^. 27 min., 16^*1- 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 ? Ans. 125 miles. 11. What is the sum of the series |— rV+:}*s— t'/k+^Vs— &c. ? Ans. ^^, 12. If a man earn $2 more each month than he did the month before, and finds at the end of 1 8 months that the rate of increase will enable him to earn the same sum in 14 months ; how much did he earn in the whole time ? Ans. $4032. 13. How long would it take a body, moving at ^ho rate of 50 MISCELLANEOUS EXEBGIBES. 319 miles Ml hour, to pass over a spaoe equal to the distance of the earth fVom the sun, t. e., 95 millions of miles, a year being 365 days ? Ans. 216 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 iVom 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 bo subtracted from the square of 48, so that the remainder may be the product of 54 by 16 ? Ans. 1440. 16. A father gave | of his farm to his son, the ion 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. Required the number of pears. Ans. 14. 1 19. Simplify j— 1+^. • Ans. ^. 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 in 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 th« hypotenuse ia 100 rods and the base 60? Ans. 2400 sq. rds. 26. Simplify ^4=^ of *i±Mlof,li±M- Ans. Iff. 3i+lft 'in n^-H 890 ABUBMEna. \\-. 26. Find the value of 14- m kra, {. 27. If f of A's age is f of Bs', aad A is 37|, what ago is B ? A08.4O. 28. What is the excess of ^9-4-7^7 above vW "thAjt^ Ans. jgl 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 ^, ^, and ^ parts make 48? Ans, 44/3. 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 beibre 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 mouths and the remainder on a credit of 9 months ; how much ready money would discharge the debt, money being worth 6 per cent. ? Ans. $12000. 37. If a regiment of soldiers, consisting of 1000 men, are to be clothed, each suit to contain 8f yards of cloth that is 1^ yards wide, and to be lined with flannel 1^ yards wide ; how many yards will it take to line the whole ? Ans. 5625. 38. Taking the moon's dia^Qeter at 2180 miles, what are the solid contents? Ans. 5424617475-f- sq. miles. MIBCELLANEOUR EXERaSES. 821 39. A certain island is 73 miles in oircamferenoe, and if two men start out from the same point, in the same direction, tho one walking at the rate of 5 and the other at the rate of 3 miles an hoar; in what time will they come together ? Ans. 36 hours, 30 minutes. 40. A cironlar pond measures half an acre ; what length of cord will be required to reach from the edge of the pond to the centre ? . Ans. 83263-f feet. 41. A gentleman has deposited $450 for the benefit of his son, in a Sayings' Bank, at compound interest at a half-yearly rate of 3^ per cent. He is to receive the amount as soon as it becomes $1781.66^. Allowing that the deposit was made when the son was 1 year old, what will be his age when he can come in pofisession of the money ? Ans. 21 years. 42. Tho 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 ; bis own salary, $265 ; he 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 money 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 $7.40. 43. A holds a note for $575 against B, dated July 13th, paya- ble in 4 months from date. On the 9th August, A received in advance $62 ; aiid on the 5th September, $45 more. According to the terms of agreement it will be due, adding 3 days of grace, on the 16th November, but on the 3rd of October B proposes to pay a sum which, in addition to the sums previously paid, shall extend the pay day to forty days beyond the 16th of November; how much must B pay on the 3rd of October ? Ans. $111.43. 44. A accepted an agency from B to buy and sell grain for him. A received from B grain in store, valued at $135.60, and cash, $222.10 ; he bought grain to the value of $1346.40, and sold grain to the amount of $1171.97. At the end of four months B wished to clobe the agency, and A returned him grain unsold, valued at $437.95 ; A was to receive for services, $48.12. Did A owe B, or B owe A, and how much ? Ans. B owed A 45 cents. ABrrHMEinG. 45. A general nnging hb men in the fonn of a aqnare, had 59 men over, bat having increased the aide of the aquare by one num, he lacked 84 of completing the square ; how many men had he ? Ans. 5100. ^ 4t>. What portion, expressed as a common fraction, is a ponnd and a half troy weight of three pounds avoirdnpois ? Ans. ^^. ■' ' 47. What would the last fraction be if wo reckoned by the ounces instead of grains according to the standards ? , Ans. |. 48. If 4 men can reap 6f acres of wheat in 2^ days, by woi^ing 8^ hours per day, how many acres will 15 men, working equally, reap in 3f days, working 9 hours per day ? Ans. 40|f 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 gain on the whole being 20 per cent. ? Ans. 9|, 19i 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 rate of 4§ miles an hour, will he require to accomplish a journey of 600 miles ? Ans. 21^. 51. Smith in Montreal, and Jones in Toronto, agree to exchange operations, Jones chiefly making the purchases, and Smith the sales, the profits to be equally divided ; Smith remitted to Jones a draft for $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 tho 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 ? What is the cash balance, and in favor of which partner ? 52. In a certain factory a number of men, boys and girls are employed, the men work 12 hours a day, the boys 9 hours and the girls 8 hours ; for the same number of hours each man receives a half more than each boy, an;?, each boy a third more than each girl ; the sum paid each day to all the boys is double tho sum paid to all the girls, and for every five shillings earned by all the boys each day, twelve shillings are earned by all the men -, it MISCELLANEOUS EXEBCI8ES. 828 is reqaired to find the number of men, the number of boyt and the number of girls, the whole number being &9. Ana. 24 men, 20 bojs an^ 15 girls. 53. A holds B's note for $575, payable at the end of 4 months from the 13th July ; on the 9th August, A received $62 in advance, as part payment, and on the 5th September $45 more ; according to agreement the note will not be due till 16th November, three days of grace being added to the term ; but on the 3rd October B tenders such a sum as will, together with the payments already made, ex- tend time of payment forty days forward ; how much must B pay on the 3rd of October ? Ans. $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 60 days after the 1st of October to pay the balance ; how much must be pay on the 11th of August ? Ans. $54. 56. A person contributed a certain 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 vhe 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 the 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 the third field was equal to four- fifths of the number in the fourth field ; also that there were thirty- two sheep more in the third field than in the fourth ; how many sheep were in each field and how many altogether ? Ans. First field, 243; second field, 216; third field, 192 > fourth field, 160. Total. 811. 324 ARITHHKnO. 58. How many hoars per day mast 217 men work for &^ days to dig a trench 23^ yards long, 3§ yards wide, and 2^ deep, if 24 men working equally can dig one 33 J yards long, 5^ wide, and 3^ deep, in 189 days of 14 hours each. Ans. 16 hours. 59. A man bcqiieathcd one-fourth of his property to his eldest son ; — to the second son one-fourth of the remainder, and $350 be- sides; to the third one-fourth of the remainder, together with $975 ; to the youngest one-fourth of the remainder, and $1400 ; he gives his wife a life interest in the remainder, and her share is found to be one-fifth of tho whole ; what was the amount of the property ? Ans. $20000. 60. Five men formed a partnership which was dissolved after four years' continuance ; the first contributed $60 at first and $800 more at the end of five months, and again $1500 at the end of a year and eight months ; the second contributed $600 and $1800 more at the end of Hix months ; tho third gave at first $400 and $500 every six months ; the fourth did not contribute till tho end of eight months, ho then gave $900, and the same sum every six months ; the fifth having no capital, contributed by his labor in keep- ing the books at a salary of $1.25 per day ; at the expiration of the partnership what was the share of each, the whole gain having been $20000? Ans. 1st, $2019.65 nearly; 2nd, $4871.81 nearly; 3rd, $4815.81 nearly ; 4th, $6467.74 nearly ; 5tb, $1825.00. lOOAUTHMH. LOQABITHMS OF NUMBERS 89t^ Namtan tnm 1 to loo. Wo. I^. 21 1^. 1-822210 H*. 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