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Tous les autres exemplaires originaux sont film^s en commenpant par la premidre page qui comporte une empreinte d'impression ou d'illustration et en terminant par la dernidre page qui comporte une telle empreinte. Un des symboles suivants apparaitra sur la dernidre image de cheque microfiche, selon le cas: le symbole — »► signifie "A SUIVRE ", le symbole V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent etre filmds & des taux de reduction diff6rents. Lorsque le document est trop grand pour etre reproduit en un seul clichd, il est film6 d partir de Tangle sup6rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d'images n^cessaire. Les diagrammes suivants illustrent la mdthode. 1 2 3 1 2 3 4 5 6 PB AN THE ^'i THE AMERICAN COMMERCIAL ARITHMETIC, I'cll: THK rsK liF TDK Ottawa Business College, of Ottawa, Out. .1. M. Ml SCtH()\ E, Princii'al. AM) Inl; PRIVATE STUDENTS, BUSINESS MEN, SCHOOLS. Universities and Counting Houses, KMMR.UING AN EXTENSIVE COURSE. BOTH IN THEORY AND PRACTICE, •i'n(n.'rnr-.i; wicn THE LAWS Ol' rilK UNITED STATES HJILAl'lNG TO INTEKEST, DAM.MiES ()\ lULF-S. AND THE COLKEOTION OF DEBl'S. « »■•.«! 'M ' IIY r. A. liiLVd:. M. A., ij. I)., «-.,•, Vl IIIDK «»K I'liKATISl^ ON M.(,i:itl{A AMt (JKi).MKTKV. OTTA WA. ONTAK I O: I'LllMSirKK IIV THi; ori'AWV HISINKHH Cnl,LK(tK. IfctOS. 'l ' 264-6^/ \€:>G8> Entered aoeordhif? to Act of Congresa, in the vearlHCr,. l,y GILBERT Y. BURNS, in the Clerk's Office of the District Court of tlie Northern District of New York. '' .'*.vH #*:, ' PREFACE. TnoUGn elementary works on Arithmetic are in abundance, yel it seems desirable that there should be added to this an extensive treatise on the commercial rules, and commercial laws and usages. It is not enough that the school-boy should be provided with a course suited to his age. There must be supplied to him something higher as he advances in age and progress, and nears the period when he is to enter on real business life. The Author's aim has, therefore, been to combine these two objects, and to produce a work adequate to carry the learner from the very elements up to the highest rules required by tho.se preparing for business. As the work proceeded, it was found necessary to extend the original programme considerably, and, therefore, also the limits of the book, so as to make it useful to all classes in the community. In carrying out this plan, much care has been taken to unfold the theory of Arithmetic as a science in as concise a manner as seemed consistent with clearness, and at the same time to show its applications as an Art. Every cfi'ort has been made to render the business part so copious and practical as to afford the young student ample information and discipline in all the principles and usages of commercial intercourse. . For the same reason some articles oa Commercial Law have been introduced, as it was a prominent part of the Author's aim to produce a work which should be found useful, Dot only in the class*room, and the learner's study, but also on the merchant's table, and the accountant's desk. The Author begs to ' tender his best thanks to J. Smith Homans, Esq., New York City, Editor and Proprietor of the '.'Banker's Magazine and Statistical Kegister," for the able manner in which he supplied this part of the work. Throughout the work particular care has been taken not to enunciate any rule without explaining the reuKn of the operation, for, without a knowledge of the principle, the operator ia a merk calculating machine that can work but a certain round, and is almost sure to be at fault when any novel case arises. The explanations %: ■J' IV. PREFACE. are, of course, more or loss the result of reading, but, ncvcrtbclcss, tlicy arc mainly derived from personal study and experience in teaching. The great mass of the exercises are likewise entirely new, though the Author has not scrupled to make selections from some ol t'le most approved works on the subject; but in doing so, he has confined himself almost entirely to such questions as are lo be ibund in nearly all popular books, and which, therefore, are to be looked upon as the common property of science. Algebraic forms have been avoided as much as possible, as being unsuited to a largo proportion of those for whom the book is in- tended, and to many altogether unintelligible, and besides, those who understand Algebraic modes will have all the less difficulty in understanding the Arithmetical ones. Even in the more purely mathematical parts the subject has been popularized as much as possible. In arranging the subjects it was necessary to follow a certain logical order, but the intelligent teacher and learner will often find it necessary to depart from that order. (Sec suggestions to teachers.) Every one will admit that rules and definitions should be ex- pressed in the smallest possible number of words, consistent with perspicuity and accuracy. Great pains have been taken to carry out this principle in every case. Indeed, it might be desirable, if practicable, not to enunciate any rules, but simply to illustrate each case by a few examples, and leave the learner to take the prln- cijjle into his mind, as his rule, without the encumbrance of words. Copious exercises are appended to each rule, and especially to the most important, such as Fractions, Analysis, Percentage, with its applications, &3. Besides these, there have been introduced extensive collections of mixed exercises throughout the body of the •work, besides a large number at the end. The utility of such miscellaneous questions will be readily admitted by all, but the reason why they are of so much importance seems strangely over- looked or misunderstood even by writers on the subject. They are epokea of as review exercises, but their great value depends on some- thing still more important. An illustration will best serve here. A class is working questions on a certain rule, and each member of the class has just heard the rule enunciated and explained, and therefore readily applies it. So far one important object is attained, riz., freedom of operation. But something more is necessary. The rHEFACE. V. learner must be taught to discern wJiat rule is to he applied for the solution of each question proposed. The pupil, under careful teach ing, may bo able to understand fully every rule, and never con found any one with any other, and yet be doubtful •what rule is to be applied to an individual case. The miscellaneous problems, therefore, are intended not .=0 much as exercises on the npcratinnf of the d'.ITerent rules as on the mode of applying those rules ; or, in Other words, to practice the pupil in perceiving of what rule any proposed question is a particular case. Great importance should be attached to this by the practical educator, not only as regards readi ness in real business, but also as a mental exercise to the young student. The Author is far from supposing, much less asserting, that the work is complete, especially as the whole has been prepared in less than the short space of six months. It is presented, however, to the public in the confident expectation that it will meet, in a great degree at least, the necessities of the times. With this view, there are given extensive collections of examples and exercises, involving money in dollars and cents, with, however, a number in pounds, shillings and pence, sufficient for the purpose of illustration. This seems necessary, as many must have mercantile transactions with Britain and British America. The Rule for finding the Greatest Common Measure, though not new, is given in a new, and it is hoped, a concise and convenient form of operation. The llulo for finding the Cube Root is a modification of that given by Dr. Hinds, aiyl will be found ready and short. In treating of Common Fractions, Multiplication and Division have been placed before Addition and Subtraction, for two reasons. First, — In Common Fractions, Multiplication and Division present much less difficulty than Addition and Subtraction ; and, secondly, as in Whole Numbers Addition is the Rule that regulates all others; '50 in Fractions, which originate from Division, we see, in like manner, that all other operations result from Division, and, in con- nection with it, Multiplication. Several subjects, commonly treated of in works on Arithmetic, have been omitted in order to leave space for more important matter bearing on commercial subjects. Duodecimals, for example, have tccn omitted, as that mode of calculation is now virtually supcrscdet'. '^< f •« VI. rnEFACE, by that of Decimals. Barter, ton, hus been passed by, as questions of that class can easily be solvcJ by the Ilulc of Proportion, which has been fully explained. Tlio subject of Analysis has been gone into at considerable length, and it is hoped that the new manner in which the explana- tions and solutions are presented, and the extensive collection of exorcises appended, will contribute to make this a valuable part of the treatise. The view given of Decimal Fractions seems the only rue one, and calculated to give the student clear notfons regarding tlie nature of the notation, as a simple extension of the common Ar.ibie system, and also appropriate to show the convenience and utility of Decimals. The distinction becwetn Dccunah and Decimal Fractions has been ignored as being " A distinctioi* without a dilTcrcnce." Decimals is merely a short way of writing Decimal Fractious ; thus, .7 is merely a convenient mode of writing ,',). These differ in form only, but otherwise are as perfectly identical as J and y. The contracted methods of Multiplication and Division will be found, after some practice, extremely useful and expeditious iu Decimals expressed by long lines of figures. The averaging of Accounts and Equations of Payments, Cash Balance and Partnership Settlements, have been introduced as essential parts of a commercial education, and, it is hoped, will form a most important and useful study for those preparing for business, and probably a safe guide to many in business who have not sys- tematically studied the subject. SUGGESTIONS TO TEACHERS. TuE author would first refer to the remark made in the Preface that he docs not expect that the Teacher will follow the logical order adopted in the book, and even advises that ho should not do so in many cases. lie knows by experience that the same order does not suit all students any more than the same medical treat- ment suits all patients. The courc roi^uircs to be varied accordin-^ to age, ability and acriuirements. The greatest difficulties generally present themselves at the earliest stages. What more serious diffi- culty, for example, lias a child to encounter than the learning of the alphabet ? Though this is perhaps the extreme case, yet others will be found to be in proportion. For beginners, therefore, we recom- mend the following course. Let the elementary rules bo carefully explained and illustrated by simple examples, and the pupil shown how to work easy exercises ; this done, let the whole be reviewed, and exercises of a more difficult kind proposed. The decimal coinage should then be taken up. In ex- plaining this part of the subject the teacher ought to notice carefully that the operations in thfs case differ in no way from those already gone through in reference to whole numbers, except in the preserving of the mark that separates the cents from the dollars, usually called the decinal point. The next step ought to bo the whole subject of denominate numbers, and in illustration and application, the rule of practice. After a thorough review of all the ground now gone over, Simple Proportion may be entered upon, using such questions as do not in^'olve Fractions. Then, after a course of Fractions has been gone through, Proportion should be reviewed, and questions which involve Fractions proposed. After this it will generally be found de- sirable to study Percentage, with its applications. The order in which the rest of the course shall be taken is com- paratively unimportant, as the student has now realized a capital on which he can draw upon for any purpose. The author would, in the strongest manner possible, impress on the minds of teachers the great utility of frequent reviews, and especially of constant exercise in the addition of money columns • * « Till. SUGGESTIONS TO TEACHERS. To make the exercises under each rule of progressive difficulty, as fur as possible, has been an objct kept constantly in view, as also to give each exorcise the semblance of a real question, for all persons, especially the youn;:;, take greater interest in exercises that assume the form of reality than in such as are merely abstract ; and, besides, this is a preparatory exercise to the application of the rules afterwards. At every stage the greatest care should be taken that the learner thoroughly understands the meaning of each rule, and the conditions of each question and the terms ia which it is expressed, before he attempts to solve it. 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 work much on their elates, and each in his turn on the board for illustration to the rest. Finally, it is suggested to every Teacher to keep constantly be- fore his mind both of the two chief works he has to accomplish. Fir it, the developcment of the mental powers of his pupil ; and, secondly, imparting to him such knowledge as he will require to use when he enters upon life, either as a professional man, or a mer- chant or clerk. Some seem to consider these two objects incompati- ble, as if taking up time in mental training left insufficient time for the imparting of actual knowledge. This is a palpable errror, for the more the mental powers are cultivated, the more readily and rapidly will any species of knowledge be apprehended, and the more surely, too., will it be retained when it has been mastered. Mental culture is at once the foundation and the means ; the other is the super- structure raised on that foundation and by that means; or it may be compared to a great capital judiciously embarked in trade, and often turned, and therefore yielding good profits. It frequently happens, however, from the peculiar circumstances of individuals and families, and even communities, that young men require to be hurried into business, so as *o be able to support themselves ; but even in such cases the desired object will be much more readily and securely attained by such a course than by what is usually and not inappro- priately calleJ " Cramming." Every effort has been made to give to this book the character here recommended, especially in the explana- tory parta. SUGGESTIONS TO COMMERCIAL STDDE.MS. ■AS ■• The foregoing suggestions arc addressed directly to tlic Tcaclicr, but a careful consideration of them by the Student •will, it is lw>pcd, be found highly profitable. A few additional hints arc subjoined for the benefit of those seeking a liberal and practical commercial education. As in air branches, so in Arithmetic, it is of the utmost conse- quence to digest the rules of the art thoroughly, and store them iu the memory, to be reproduced when required, and applied with accuracy. But this is not enough ; something more is needed Y.y the Student. To be an eminent accountant he must acquire rapidity of operation. Accuracy, it is true, should be attained first, especially as it is the direct means of arriving at readiness and rapidity. Accu- racy may be called the foundation, readiness and rapidity the two wings of the superstructure. Either of these acquirements is indeed valuable in itself, but it is the on'jjjination of them that constitutes real effective skill, and makes the possessor relied upon, and looked up to in mercantile ciril j. Some one may ask, " How are these to be acquired? " The answer is as simple as it is undeniably true; onlt/ hi/ extensive p/acticc, not in tlie counting-house or warehouse, indeed, though these will improve and mature them, but in the school and college, so that you may take them with you to the busi- ness (tT' • > when you go to your first day's duty. Go prepared is a maxim luat all intelligent business men will affirm. Be so prepared that you will not keep your customers waiting restlessly in your office or warehouse while you are puzzling through the account you are to render to him, but strive rather to surprise him by having your bill ready so soon. Another important help to the attaining of this rapidity, as no- ticed in the note at foot of page 18, is not to use the tongue in calcu- lating but the eye and the mind. Nor should the course of self discipline end here. To be an ex- pert accountant even, is but one part, though an important one of a qualification for business. Study Commercial geography — commer- cial and international relations — political economy — tariffs, &c , &c. Ji X. SUGGESTIONS TO COMMERCIAL STUDENTS. Study cvai poUtics, not for tlioir own sake but on account of the manner in which they aflfect trade and commerce. Do not, except in the case of some serious difficulty, indulge in the indolent iiabit of a.skin- your touclici or fellow Htudent to work the (Question for you ; work it out yoursu.f -rely upon your .self, and aim at the freedom and correctness whicli will give you confidence in yourself, or rather in your powers and acriuircmenta. Another cau- tion will not be out of place. 3Iariy students follow the practice of keeping' the text book beside them to .see what the an.swer is; this has the sumo cIToct as a h-udliuj qHrntum in an examination, bein"-a ^uide to the moth by secin^' the nsult. Study and us? the mmh ic come at the result ; jj;ain that knowledge of principles and correctness of operation that will inspire the confidence that your answer is cor- rect without knowing what answer the text book or the teacher may assign to it. There are two things of such constant occurrence and requiring such extreme accuracy fcliat they must be specially mentioned,— they arc the addition of money columns and the making of Bills of Par- cels. Too much care and practice can scarcely bo bestowed on these, It*. TABLE OF CONTENTS. PACK. Aritlunotio 13 A(Miti()ii 17 Alli;4ati()ti 2.1(5 Aiialysi,') •. 117 AiiiniiUcs 310 A v.Tiii^c, < IciK'nil 212 7\(';oiuil.s uiiil Invoii'i'.s 128 Avi'iii^iii^:^ 1)1' A('i!ouiits 241 A.Nioms 17 A ci.'oii 11 1 of .Siik'.s 251 U:tiil('iinal ( '()iiiii;^(! '. 32 DitiiDuiinalis M uiabcr.s 45 Sulitrnction of 50 JlulUiiliciiliou of 50 ■ Division of 53 Discount 177 .___-__ IJaiik 181 Divi.iou, Sim]ilc 2(5 Ki |uat 1(111 of I'avnii'nt.s : '. . . 234 Kxcivisc.i, Set r V... (il Set II 113 Set III I!t3 — Set IV 342 E?;elmugi) 205 ■ Aihitratiou of 275 • Anieiii'iiu 20fJ Sterling 271 Kvolutiou 232 Fiju^l ioiiM- 04 Fractions, ( 'onimon 01 Division of 70 • Mult i jilicat ion of .' 69 AiMilion of 72 Subtraction (jf 74 Doiiomiualu 75 f ' Xii. nnoEX. Fractions, Docimtil 77 llwluotiou of 83 Addition luid Subtraction of 85 Multiplication of 87 ; Contracted iMethod 89 ' Division of 92 • Contractwl Method 94 Denominate 96 Foreign Jloncys 274 Greatest (Jonmion Measure 68 Interest 140 — Simple 141 ■ — • Compound 174 1 nsurunee 195 — — — Life 201 Involution 280 Interest Laws, United States 352 Canada 364 Least Common Multiple CO Mensuration 328 Measun-ment of Timber : , ; 339 Money, its Nature and Value 262 Multiplication,- Simple 22 Notation 15 Numeration 14 Numbers, Properties of 56 Preflice 3 Partial Payments 164 Partnership 227 Settlements 314 Percentage 135 Practice 125 Profit and Loss 204 Progression ^ 204 — liy a Connnon Dilferenco 294 by Piatio 303 Proportion 98 Sim])lo 98 Compound 101 Patio 98 licdwct ion 42 Suggestions to Teachers 7 to Commercial i?tudents ; 9 Stocks and Bonds 218 Square IJoot 283 Storage 210 Subtraction, Simple 20 Tables of Money, Weights und Measures 80 Taxes 215 ARITl IMETIC ARTICLE 1.— Arithmetio treats of numbers in theory and practice. In rclatioti to tlieory it is a science, and in relation to practice it is an art. All computations arc 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 mca.suring 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 s.iy 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, wo should adopt a unit equal to five and a-half of the last, and called a perch or rod, — if we wish to note tlic distance between New York and IJulTalo, 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 tho last-mentioned, and they say that the earth is ninety-five millions of miles from tho 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 oases that each unit is a million of miles. A similar illustration may be applied to every kind or 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, (i, 7, 8, 0, 0. These, by various combinations, can be made to represent any quantity or magnitude whatsoever. The first nine uro called significant figures, because they always denote some real (luanlity, — the last, called nought (often improperly ought,) or cipher, or zero, simply indicates the absence of any significant figure. ''s^^^^^- .1 ,.»> 14 AMTHJIETIC. NUMERATION. 2. — Numehation is the mode of marking and reading off any line of fip^urcs that lias been written down, so as to ascertain ita value readily and express that value in words. For this purpose every such line is divided into sets or lots of three figures each, counting from right to left, and each sot is called a period, — thus, 8SSS88SSS forms tliree 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, «S:c., &c,. to any required extent, which seldom exceeds millions. The first figure of each period denotes units* of tliat period, the second tens, and the third hundreds of that period. Thus, 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 j 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 cigety-cight millions, eight hundred and cighty-uight thoupunds, eight hundred and eighty-eight. Every period but the last must have three figures. Thus, in the line 43,279,805 the first and second periods have three digits each, units, tens and hundreds, but the third has only two, units and tet>s, but no hundreds, and therefore is read forty-three 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 read off each period from left to * It is Bomewhat awkward that the term units is used for two purposes, viz. : as tbo name of tlio first period and also as the namo of the first figuro of each period. Thougii wo cannot well change what uaoge has so long estab- lished, jct the teacher may obviate the difficulty by varying the oxpreesion occasionally, if not habitually, fiayiuo;, E. G., units in the unity period, or the place of unlt8 in the units period. col scl 1. 4.- 7.- •■•>«* NOTATTOir. 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 omitt«d. 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 line 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. EXERCISES Divide into periods and read the following lines : 1.— 586729341 4.-92879357485 7.-2822828228288 2.-976852734 5.— 4638709120 8.-10904870 3.— 2178427385 6.— 11111111111111111 9.— 1010lt)lrtlOl01 ' NOTATION. 3 — XoTATioN is the mode of expressing any quantity or mag- nitude by the combination of conventional symbols or characters. Thus, by the lloiuan notation, the letter I. stands for one, IT. for two, X. for tai, &c. ; thus, XII. stands for orte ten and twt) 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 ; again, 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 each 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 BO on. Universally, every digit placed to the right makes every 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 eight ten-dollar bills, and eight one-dollar bills, it is plain from Art. 2 that if I write 8 alono this must represent the ono-doUar bills, and to represent the teu-dollor bills along with the one-dollar bills I must 5 55 555 5555 "» . t IG AMTHMETIC. write 88, for the figure to the left being ten times that to the right, will stand correctly for the ten-dollar bills, 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 hence 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 tea-dollar bills, and is written $80. — Again, if I find that I have two onc-hundrcd-doUar bills, six one- dollar bills, but no ten-dollar bills, and I write only 2G, 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 $200 represents correctly .that I have two one-hundred dollar bills, and six one-dollar bills, but no ten-dollar bills. The superiority of this simple system over the cumbrous Roman one will bo manifest from its simplicity and brevity by writing eighty-eight according to both systems — thus ; LXXXVIII. and 88. RULE FOR NOTATION. "Write tne significant figures of the first period named in their proper places, filling up any places not named with ciphers, just as if you were writing the units period with nothing to follow; then, to indicate that something is to follow, place a comma to the right, and do the same for every period down to units, inclusive. For example, teacher says : " Write down one hundred and six millions ; " pupil writes 106 and pauses; teacher adds, "ninety thousand;'' pupil fills up thus : 106,090, and pauses ; teacher concludes : " and eighteen ; " pupil completes 106,090,018. If the teacher should say sixteen millions and the pupil write 016, the cipher woulc bo manifestly superfluous, as it has no cfiect on figures placed to th' right of it, but only on those placed to the left. EXERCISES. Write in figures and read the following quantities : 1. Ten millions, seven thousand and eleven. 2. Ninety billions, seven thousand and ten. 3. Eighteen millions, sixty thousand and nine hundred. 4. Forty thousand and nine hundred. ADDITION. 17 5. Eif^hty-scvcn millions and one. 6. Ninety thousand, -Bcven hundred and eight. 7. Pjlovcn millions, eight hundred thousand and twenty-four. 8. Si$ hundred and seven thousand and niaety-scvcn. 9. Eight hundred and seventy billions, sixty thousand and eighteen. 10. Eleven billions, olcvcn millions, clovon tliouaand and cloven. AXIOMS. 4. — Axioms used in the sequel : I. Things that are c({Ual to the same thing, or to equals, arc equal. II. If equals be added to equals, the wholes are equal. Curolliiri/, — It' ociuaia be multiplied by the same, the products arc equal. III. If equals bo taken from equals, the remainders arc equal. Cor. — If equals bo divided by the same, the quotients are equal. IV. The whole is greater than its part Cor.— The whole is equal to all its parts taken together. V. Magnitudes which coincide, or occupy the same or equal spaces, arc equal. N. B. — This axiom is modified by, but still is the principle of, all business transactions, purchases, sales, barters, .exchanges, &c., &c., where the articles traded in arc not equals, but equivalents. ADDITION. 5. — Additiox is the mode of combining two or more numbers into one. The operation depends on axiom II. The result is called the sum. Thus: $8+ $9+$6=$23. The sign plus (+) indi- CDtos addition. To illustralo 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 columns, i. c, units under units, tons 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 ten£ $237051 758287 6I2S73 4947G8 836195 f 18 AiirrmrETTc. 27 350 2400 ^i 000 2G0000 2700000 $2089777 $287054 758287 G12873 494768 83G195 $2989777 column, vre write (Art. 3) 350 ; in the same man* ncr we find the sum of the hundreds' column to be 2400 ; the sums of the others will be seen by inspection. Having thus obtained the sum of each column, each being summed as if itntls, but placed in succession towards (he left (by Arts. 2 and 5), we 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 find the sum to be 27, i. c, 7 units and 2 tens, we write down the 7 units under the units' column, and add up (Art. 3) the 2 tens with the tens' column, and we find the sum ia 35 tens, i. c, 5 tens 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 wHi's' column to the tens' column, and the hundred* obtained by adding the tens^ column to the hundreds' column, &c., &o., is called carrying. In all such operations the learner should carefully bear in mind tlie principle explained in Art. 3., that every figure to the left is ten times the value that it wQuld have if one place farther to the light.* EXERCISES. Find the sums of the following quantities : (1) (2) 895763 4917G 987654231 283527 123450789 G59845 908760504 7984 890705063 31659 759086391 968438 G70998767 2896392 4340661745 (3) (0 S9876 63879 89705324 54387 42356798 789 56798423 137568 23567989 278652 79842350 85945 G5324897 721096 357655787 • We would strongly recommend every one who wisht's to beer aio an expert accountant, to avoid the common practice of driiwling up a c^ luun o( figures in the manner that may be sufficiently illustrated by the adding of the units' column of the above example. Never say 5 and 8 are 13 ; 13 and 3 are IC : 16 and 7 are 23 ; 23 and 4 are 27 ; but ran up your column thus : 5, I^, 16, 23y ADDITION. . ] (B) (6) (7) (8) 738 G59 471 78563 897 47986 12345 658 5798 C7890 918273 856 19843 987G5 651928 789 5G479 43219 374859 978 28795 87654 263748 654 897 32169 597485 999 1984 78912 086879 888 68195 65439 98765 777 3879 98,-65 987G ■ 666 G98 43288 987 555 5879 77877 456879 897 17985 98989 345678 978 19 336981 805312 4705357 12460 '\^\ brt pa (9) (10) • (11) (12) 189 1298 976 98 47 764 85 89 96 5837 73 76 83 6495 338 67 59 789 793 281 74 638 49 592 82 546 75 678 97 98 218 58 es 475 365 67 75 394 113 98 49 89 279 149 76 ■' 157 67 67 64 638 76 54 78 594 84 72 69 789 1379 298 2744 87 1044 114 5159 19715 27, for tliat is the mode to secure both rapidity and accuracy. The same remark will apply equally to multijplication, and therefore to every arithme- tical operation. To enforce this advice let us add a simple example to cau- tion the student before he approaches multiplication. In multiplying 497 by 6, avoid the tediousness of saying G times 7 is 42 — 2 and carry 4 — C times 9 is 54, and 4 is 58 — 8 and carry 5 — 6 times 4 is 24, and 5 is 29 ; but practice the eye, aided by the memory, to. take in at a glance 6 times 7 is 42, &c.— The quick operator uses the eye, and not the tongue. 20 ARITHMETIC. There in no method of proving the correctness of any addition with positive certainty, but a very convenient mode of checking 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 metliod in the case of long columns, but not so ready aa the former. If the same result is found by each method, the »nm may bo accounted correct. 578G430.J7 235412712 343231215 SUBTRACTION. 6. Subtraction is tlic converse of addition, t. c, it is the mode of finding the difference between two numbers, or, in other •words, the excess of one number above another, The number to be subtracted is called the subtrahend, and that from which it is to bi) taken the minuend, and the result is called the re- mainder, dllTcrence or excess. The sign used for subtraction is a lino ( — ) called ininm, or loss. Let it be required to find the difference between §578043957 and $235412712. Having placed tliem in vertical columns, as in addition, it is obvior.a 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 $513074208 above S347S95319, 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 siujplo artifice, which is deduced from the principle of the notation, and may be illustrated in the following manner : — Taking the question in the margin, wo are first required to subtract 7 units from 3 units. Now, though the alirebraic notation furnishes tlie means of notini' the difference directly, the ordinary arithmetical form doea not, but still it furnishes the means of doing it indi- rectly. By Art. 3 each figure to the left is ten times tlio value of the nest to its right, therefore we take one of the 3 tens and call '\t ten units, and add it to the 3 units, and thna wc have 13 unitg, which let ttg 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 haa been thus transferred to tho units' 333,oo.j 177, 4 t 155,550 2(12)(12)(12)(12)(13) 17 7 7 7 7 1 5 5 G SUBTRACTION. 21 200000 120000 12000 1200 120 13 333333 place, only two tens remain in tlic place of tens, and wc arc now required to take 7 tens from 2 tens ; to do this wo have rc^coiirsc lo the same artifice, by calling one of the hundreds tms, which fives 10 teas and 2 tens, and so on to the end, the last 3 necessarily becoming 2. AVe can now subtract 7 fiom 13, &c.,' &c. This mode of resolution depends on the corol- lary to Axiom IV. The parts into which the whole is virtually resolved arc shown in the margin. This artifice is popularly called borrowing. In practice the resolution can be effected mentally as we proceed, and as each figure frqm which we harrow is diminished by unity, it is usual to count it as it staiKJs, and to compensate for this to increase the one below it by one, for, as in the example, 7 from 12 is the same as 8 from 13, and 2 from 3 is the same as 1 from 2. We arc now prepared to answer the proposed question, as annexed, and we say 9 from 8, we cannot, and there are no tens to Iwrrow from, we therefore take one of the hundreds and call it 10 tens, and one of the tens and call it 10 units, which with 8 units makes 18 units, and we take 9 from 18 and 9 remain. We have now only 9 tens left, but wc reckon tliem as ten, and to compensate for the surplus ten, we reckon the 1 below as 2, and say 2 from 10 and 8 remain. A\'o proceed thus to the end, and find the whole remainder to be $165778889. $313074208 S347895319 $165778889 ,. V* ♦ ^1 • k ■ EXERCISES REM^VINDEUS. 1.— From 847G39021 take 476584359=371054602. 2. " 1010305061 '•' 670685093=^339019968. 3. " 59638743 " 18796854= 40841 889. 4. " 7S13J57 " 3745079= 400757S. 6. " 111111111 " 98657293= 12453818. In Subtraction, as in Addition, we have no method of proof that arrives at p6sitivc certainty, but cither of the two following methods may be generally relied upon. 1. — Add the remainder and subtrahend, and if the sura 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 correct. •4..,- ■ ' .'5* ' ■J' 23 ARITHMETIC. MULTIPLICATION. 7. — Mui.Tll'MOATioN may bo .simply «li'!inoil by nnyiii}^ (hat it is !i f*liort motliul of jHTronnitip; lulditiDii, when all {\w (|uaiiti(i(>s Id bo adtUul aio tlio saiiuror iMpial. Tims : C. | (1 | ('» f (J-| (1 | |-(j-|-(), moans that oij;ht sixos aro to bo adilod tojijothor, or that six is to bo ropoatoil as orton as tlioro aro units in oi;^hl, and wo say (hat S tiinos (» is IS, and writo it (1ms: 8xi3~t8. So also 8 |- S 1-vS ]-8 |-S 1-8 givos -iS. So that U.S S.d IS, and thus wo can oonstruot a multiplioation lahh'. Tho imnibor (o bo rcpoatod is falKnl tho mnltiplioand, and tho ono (hat shows how ol'toii it is to bo ropoatod is oallod tho nmltii)lior, antl tho rosnlt is oallod (ho product, or what is produced, ami honoo tho nmlliplior and multi- plicand aro also oallod tho factors or makers, or producers, and tho operation may bo called I'liuiufj a product when tho factors aro j^ivon. Hence also tho mode of carrying is tho stuno in multiplication hh in addition. INmLTIPLlCATTON TABLK. twice ( times 4 times ;") limes timer 7 times 1 is 'J I is 3 1 is 4 1 is f) 1 is 1 is 7 •J ■ 1 2 — (1 2 -- 8 '2 — 10 2 — 1 2 2 — 14 ;{ — t; { - '.) 3 — 12 3 — 1.") 3 — 18 3 — 21 •t — s 1 -- 12 4 -- lU 4 - — 20 4 — 2 t 4 — 28 f) — 10 ^ -- 1.-) T) — 20 2.") f) — 30 .0 — 35 () — VI (•) — IS (1 — 21 — 30 — 3 -- 42 7 — n 7 — 21 7 — 28 7 — • ;!;■) 7 — 4 2 7 — 49 y ~ 1(1 8 — 21 8 — 32 8 — 40 ■8 — 48 8 — 5() '.) — IS .) - 27 <) — 30 9 ~ 4.") 9 — f) 1 9 — 03 10 — *J0 10 — - 30 10 — 40 10 — oO 10 — 00 10 — 70 1 1 — 'I'l LI — 33 11 — 44 11 — f).-) IL — 00 IL — 77 VI - -21 1 2 — 3« 12 — 48 12 T- 00 12 — 7 2 12 — 84 tS tunes '.) times lot imes 1 I times 12 times 1 i 8 1 is y 1 1 s 10 1 is 11 1 is 12 •2 — IC) 2 — 18 2 - - 20 •) ')•_) 2 — 24 ;5 — '2i 3 -- 27 3 - - 30 f) — 33 3 — 3(J 4 — 32 4 — 30 4 - - 40 4 — 44 4 — 48 f) — 40 5 — 4;") 5 - - r.o 5 — f)-) 5 — 00 G — 48 (1 — 54 - - 00 (> — 00 — 72 7 — 50 7 — (13 7 - - 70 7 — 77 7 — 84 8 — 04 8-72 8 - - 80 8 — 88 8 — 9(> 9 — 72 1) — 81 9 - - 90 9 — 99 9 —108 10 — 80 10 — 90 10 - -100 10 —110 10 —120 11 — 88 11 — 9!) 11 - -110 11 —121 11 —132 12 — 0(i 12 —108 12 - -120 12 —132 12 —144 > ':■**• m}. MULTTPLTCATION. Uf^'jiirtliiij; tlio following juirt of this tulilu, boo HuggcKtionH t( ToacluTH :i tinnvH II litiii<.> 1.) Illiir- 1(1 lillU'.-l 17 liiiii',-1 IM tiliii'.-i It) tiiiio i i.< 2u :: \a x'h •J is .'((1 'J i.H ;(:.' •J lA :ii 2 u :i() 2 iri .'{ ] — :»!) ;{ - -iv ;j - - 4:. ;j — 4H ;j - - r.i ;i - r.i ;j — .0" I -- f)!: I • - rx; •1 (Id 1 - CI ■I - - OH -I - - 7li -i ■ - 7( t • r... r, — 7(' r» — 7. ;. ■ - H(( f) - - h:> r. -. 'Ml 5 -- 0. ; — 7H — HI (1 !)() C — DC C, ]U1 r, - lOH ._.. 11, r — j)i 7 - - !),^ 7 - lit:. 7—1! '.' 7 11 :» 7 - r-'(i 7 -■ i;j. i - i(ii M - .- 11: M I'Jll H -.. 12)- :< i:;(; M 111 H - ir,: » - - 1 1 i :» lL'(i 1) - i.j:. !) - Hi :» 1 ;.:i :» \i;> !» 17 ll U'c have in lln' iilmvr l.ilili' fiiiriMtlcl (In- ;;rii.H ),'r;iiiiiiiiilit;iil hliititliT mi O'liiiniiiii olHiiyiiiK •■'K'''' liiiK'H Iwo.AiU'; hixli-mi. WIkmi iiioro tliati two I'actors am pivcii, tlio'oponifion is called coiitiiiiiiul uiulliplicitinii, as (lX.'5X-/'.f> l''^". Wlicii till) I'.ictorH consist ol" iiioio figures tliaii ono, the most CDiivcniciit nio(|(! of operation is that shown by tliu anncxud example, whori! this multiplicand is (iist rcji: 'cd H times, thon GO times, or wliich is (III! samo tiling (> times w . i» tiie lirst li^u^o of the second lino is placed under the second li^iiro oi' the. first line, -/. r. (art. -,; in the place of (ens, and then the partial products «ro added, which (Ax. IV. Cor.) ;^ivoh the lull jjroduct. llenco wo diuluce tho SdniRfi L7»;MS8 2070.) 10 (;:)o:}72 RirLK roll Miri.TII'MCATION. 02507848 Place tho multiplier under tho multiplicand, units tmdor units, tetis imder (ens, &.C., &c., — commoncin;^ at the rij;ht, niulliply each li;^ure of the multiplicand Jjy each (iL;ure of (ho multiplier in sucocssion, placing; tho results in parallel lines, and units, tons, &c., in vertical columns, — add all the lines, and the sum of all tho partial products will (Ax. 1\^ Cor.) he the whole product rcijuired. As far as tlio learner has oommit(cd a multiplication table to me- mory, aay to 12 times 12, the work can be done by a sinj^jc opcradori. Wiien any number is multiplied by itself, tho product is called the square or second power of that nuudjcr, and the product of tlirec equal factors is called a cube or third power, the pro- duct of four equal factors the fourth power, &c., &c. Tho terms square and cube are derived from superlicial and solid measurement. The annexed square has each of its sides divided into 5 equal parts, and it will be found on inspection that the whole figure contains 5 *^ /^ 24 ARITHMETIC. 25 (=5X5) small squares, all equal in area, and having all their sides equal. — Hence because 5X5 I'^^prcscnts the whole area, 25 is called the square of 5,' or the second power of 5, because it ib the product of the two equal factors 5 and 5. A cube is a 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^=1-5, which is therefore called tho cube or third power of 5. The tcruis square and cube are often used without any reference to superficial and solid measure. For example, in lineal measure an cxpressioa for distance in a straight line is often culled 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 iu 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 ha.vc no term corres- ponding to square and cube for tho product of four or more equal factors, and therefore wc are obliged toniso tho words fourth pow?v, fifth power, &c., &c. CONTRACTIONS AND rROOF. There are many cases in which multiplication may bo performed by contracted method':., but tho utility of these, for the purposes of accuracy, is, at least, doubtful. The most secure method in tlio groat majority of cases, is to follow the general rule. Multiplication by 10, 100, &c., is cfToctcd at once by adding a cipher for ten, two for 100, &c., &c. The following is, next to tho above, the most aaio and useful contraction that can be adopted. It is exhibited in tho subjoined examples, but purposely without explanation, as an cxcr cisc for the learner's reflection : ORDINAKY S[ETII()I). 35097X17 17 CO.NTRACTKn JIlCTIlOD. 35697X17 249879 Okdinakt XIktiiod, 35097X71 71 Co.NTnArTKI) IirKTHOt) 35097x71 249879 249879 35G97 G00S49 35697 249879 2I344S7 G0G849 2534487 The only practically useful proof of tho correctness of tho pro- duct, if the one subjoined, but even it, though it seldom fails, duos not secure positive certainty : MULTEPUCATION. •?5 Add together all tbe figures of each factor separately, rejecting 9 from all sums that contain 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 mai/ be accounted as correct, but if they are not erjual, the work rmist be wrong. The reason of this proof depends on the property of the number 9, that if any number be divided by i), the remainder will be the same as if the sum of its digits were divided by 9. — Thus: 7422153-:-9^824G83-j-(), and the sum of the digits is 24, and 24-f-9=2-|-G, t. e. 9 is contained in 24 twice with a remainder G. 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 24^9 X 2+0, and if we multiply together the two factors thus resolved, wc get 9 Xl3X0X2-f9X 2X5+9X13X6+0X5, and since 9 is a factor of all but the liist, 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 Gx5-:-9> which gives the remiaindcr 3, for Cx5=:;30 and 3(l-:-9 gives 3 with a remainder 3. To test this by trial, wc find 122-f-9=13 with a remainder 5, and 24-:-9=r2 with a remainder G, and the product of these remainders is G X 5=30, and 30-=-9=3 with a remainder 3. Again, 122x24=2928, and , 2928-7-9=^325 with a remsuudcr 3, as in the case of the factors. •^*. EXERCISES. 1. 789GX 5=39480. 2. 5819G7X8=4G5573G. 3. 938740X4=3754984. 4. 193784X7=1350488. 5. 391870X0=3520884. G. 987450X0=5924730. 7. 490783x52=25832710. 8. 719804X43=30954152. 9. 375907X04=24001888. 10. 27859X29=807911. 11. 079854X83=50427882. 12. 750084X1 87=142000908. 13. 5372X1034=8777848 14. Find the second power of 389 ? Ans. 151321. 15. Find the third power of 538 ? Ans. 155720872. 10. Find the fourth power of 144 ? Ans. 4299S1090. 17.- Find the cube of 99 ? Ans. 970299. 18. 5790 seamen have to be paid 109 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 feet high ; how' many cubic yards doca it contain ? Ans. 3550f cubio yards: iillll 26 ARITHMETIC. 20. If 29 oil wells yield 19 gallona an hour cachj how much will they all yield iu a year ? Ans. 201115 gals. 21. If the rate on each of 1597 houses bo $19 ; what is the whole assessment ? Ans. $30343. 22. If 1297 persons- have paid up 9 shares each in a railway company, and each share is $15 ; what is the working capital of the comnany ? Aus. $172095 DIVISION. 8. — Division is the converse operation to multiplication. It is the mode of finding a rociuired factor when a product and another factor arc given. It boars the same relation to subtraction that multiplication docs 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, (divider.) and the result is called tlie quotient (how often.) Let it bo required to find how often 8 is con- tained in 279,850. We can resolve 279,- 85G as iu the margin ; then dividing the lines separately by 8, we obtain the partial quotients, the sum of which is the whole quotient. But this resolution may bo done mentally as wc proceed. We first «ci; that 8 is not contained in 2, therefore wo take 27, and find that 8 is contained in it 3 times, with a remainder 3 ; next combining tlus 3 with the next figure 9, wo get 39, in which 8 is contained 4 times, with a remainder 7 ; combining this 7 with the next figure 8, wo have 78, in which 8 is contained 9 times, with a remainder G ; combining this with the 5 following, we obtain 65, and 8 is contained in it 8 times, with a remainder 1, which combined with the G makes IG, and 8 is contained twice in IG. The correctness of tho result may bo tested by multiplying tho quotient by tho divisor. When tho divisor consists of more than ouo figure, the learner must have recourse to a trial quotient, but after some prac- tice he will have little difficulty in finding each figure by iu.spcction. 8 240000 30000 32000 4000 7200 900 C40 80 IG •J 8 li79S5G 349S2 '■^*k' DIVISION. 27 «.«, Let it be required to find how often 298 is coi.taincd in 4317GG. — Tiic numbers being arranged in the convenient order indicated in the margin, we mark off to the right of the dividend blank spaces for the trial and true quotients. We readily see that 2 is contained twice in 4, but cdnnot so easily sec •whether the whole divisor 293 is contained twice in the same number of figures of the diviilciid, (viz. 431,) wo therefore make trial, and place tho 2 in the trial quotient, and multiply the divisor by 2 to find liow much we shall have to subtract from 431. We find 298x2-^501), larger than 431, and therefore we reject 2 and try 1. Now 29Svl: -298, less than 431, so we subtract and find a remainder of 133, and as this proves correct, we place the 1 obtained in 298)4317GG(2.1.5.4.5.4.9.8 trials. 298 14.4a true quotient. 1337 ito6 1192 • • . 2G46 2384 262 298 the true quotient. AVc find our nciit, partial divdend by writing 7, tho next figure of the dividend after the re- mainder 133. Our experi- ence of the first case sug- gests to us that though 2 ia contained G times in 13, yet on multiplying something will have to be carried from the 98 which we expect will make tho result too large, and therefore we at once try 5, but we find that 298x5=--d490, which is larger than 1337, and so wo try 4, and find 298X4=1192, which being less than 1337, we subtract and find a remainder of 145 ; and having placed tho 4 in the true ([uotient, we bring down tho next figure of the dividend, giving a partial dividend 145(1. By in- spection, as before, we sec that G would be too large, owing to the carrying from 98, we try 5 and find 293x5—1490, which is larger than 145G; we try 4, and find 298x4—1192, which is less than 145G, so we subtract and find a rf^ni'dnder of 2G4. Having placed this 4 after the other 4 in the true quotient, we bring down G, the last figure of the dividend, wo try 9, and find 298x9^- 2G82, which is greater than our last partial dividend, 2G4G ; wo try 8, and find 298x8=2384, and this being less than 2G4G, wo subtract it from ■•.*-% ,- * 1 , '6 ' ,f^ K, 28 ARirmiETic. 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 202 still remains to bo divided, which cannot bo actually done, aa it is less than the divisor, is to write the 298 below the 262, and draw a, line between them, thus H^b) "s ^Iso is seen in the margin. The resolution into partial dividends is also shown in the margin, where it will be sect that the partial dividends, includ ing the remainder, make up tht whole original dividend. So nlse 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. AVhen we have multiplied tho divisor by any figure in the trial quotient, and subtracted the product from the partial dividend, should the remain- der bo greater than the divisor, we perceive that the trial figure is too small, and wo must try a larger. From these illustrations we can deduce a 298 I 29SrOO 119200 11920 2384 Remainder 202 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 tho left of the dividend, with a vertical lino be- tween them, draw another vertical lino to tho right of tho dividend, and enter the quotient, figure by figure as obtained, to tho right of that lino. (2.) Find by tho principles of multiplication, how often the divisor is contained in tho same number of figures of the divi- dend ; place tho number thus obtained in the quotient, and multiply the divisor by it, and subtract tho product fl-om tho corresponding partial dividend. (3.) To tho remainder annex the next figure of tho dividend, and proceed as before, and so on till all tho figures of the dividend are exhausted. (4.) Should there be a reinainder, write it and ihe divisor after the quotient, thus : "i^^^^^ I Tho divisor is often written to the right of the dividend, and tho quotient written below it, a horizontal line separating the two. DRISION. 29 EXASirLE OP rOKM 1 KXAMPLE OF FORK 2. 47G)8593504(18053§:j8. 47G 3833 3808 18G0904 174 120 87 87 21389g} 2550 2380 339 261 780 €90 1704 112. .■ 270 1 • E X K n C 1 S K S . 844 783 CI ^1. 15547GS-:-21G=7193. ^ 2. 318S-1470-:-779--=: 40030. /3. 573S0G25--7575^7575. ^4. I2810098--732^.17500,n3fl3. > '' 5. 9313702850-^4087319== 1987:ff;j5f3Tl5' ^G. 4-19148il0476-f-738S524G=:G079,3^«370. - 7. lG95Sa282929-:-138G=:^79024G8-,V^"^. T -, iy 8. 35G7G210832-:-79094451--rr7G4095^>i;'Jy^40j8.\.: ^ fW^^ *^ 0. 63G818834-;-907=501SG2. VIO. 1700G49155Gl-:-7G9=r2240G44479. /[I. 5542702979Gl-:-7584n:=730S41G3^^«.?. .12. C0435G74G34529-^-7G4095r=:7D094451,Oi7i'5„"J'5. 13. How many bags, each containing 87 pountla, will 24,853,404 pounds of flour fill ? Ans. 285,072. 14. 857 houses pay annually a tax of $41130 ; what ia the aver- age on each por quarter ? Ans. $12. 15. ^9297175 of prize money aro to bo (lividcd among 97,8G3 sailors ; what is the share of each ? Ans. $95. 10. 120,815,231 pounds of cotton arc made up iu ^33,879 bales ; how many pounds in each bulo ? Ans 89. » ••%. ft ^'^% 'r'^!^^,i>^ ' . > * 1 DI V IBION. 1. 49087532-^2=24843700. 2. 57986327-h3=19328775'!. 7 / ^JLh^-' i h ^M 30 AKirmiEnc. 3. 4. 5. 0. 7. 8. 9. 10. 11. 159/^. :15G3777J. 87965328-r-4:r^21991332. 7963821h-5==15927G4!. G875324-f-G=1145887i. 3987G54-r-7z=5G9GG4;'. 1987G532h-8=24845GG1. 297G854^9=3307G1-S. ^ 49G7532-^10=49C753i. 4G879352-r-ll=42C17i 187G5314-^12. 12. 78G5424a-:-18=43G9G80]. 13. 75088-:-52=144^. >^-"'"^ • 1G74918-^189=.8SG2. 31884470-:-779=40930. 57380G28-:-7575=7575y.3,^. 55427029219S-^7584=73084163-r2'gg. 88789930979--9r)84=9264397yV«V lG2030429729-:-123456r=82G452f§of|. 2G781794GOOO--3G500=:10077204. 21. 497 men fell 1G3798 trees; how many does each fell on an average ? 22. If 148 houses pay a tax of $784-1 ; what is the rate on each on nn average ? Ans. $Q3. 23. If $415143G30 arc levied from 4455 townships ; what is the portion of each on an average ? Ans $9318G. 24. IIow many lots of G754 each are contained in 3968091 51372 ? Ans. 587G3718. 25. What quotient "will he obtained by dividing 961504803 twice by 987 ? Ans. 987. . . 14. 15. IG. 17. 18. 19. 20. 0.— TABLES of MONEY, WEIGHTS & MEASURES. nECIMAL COINAQS. 10 mills (M) are 1 cent (ct.) 10 cents 1 dime (d.) 10 dimes, or 100 cents.. . 1 dollar ($) nitlTISU OB STEUUXO MONEY. 4 farthings, or 2 half penny's, arc 1 penny (d.) 12 pence 1 shilling (s.) 20 shlUiugs 1 pound (£) AvoiRDuroia weight. TABLE. 10 drams make 1 otmce, 16 ounces 1 pound,' '/l^ • V • r ' 1 quarter, ^ • . 1 liuudredwelgbt, •-fO^ < '"v 1 ton, ■ ;o!"i .« • ... ■'.^ -Ight ia UBcd in weighing i.eavy articles, as meat, grocOTles, Y'.'v,,,^;!^" - ■•■ ■. «tC. marked 02. lb. " qr. " cwt. " t. 24 20 12 nJ 20 gJ 3 8enn3n>-eiglit, markea dwt. 29 pennyweights 1 ounce, '• oz. ' 12 ounces 1 pound, " lb. Note.— Troy weight is used in weighing the precious metals and atones. .A I k APOTUECARIES' WEIGOT. TABLE. 20 grains (grs.) make 1 scruple, marked scr. 3 scruples ; 1 dram, '• dr. 8 drams 1 ounce, " oZ. 12 ounces 1 pound, " lb. Note. — Apothecaries and Physicians mix their medicines by this weiglit, but they buy and sell by Avoirdupois. PRODUCE WEIGHT-TABLE. GRAIN. Wheat CO pounds to the bushel Oats M " " (Jorn 5li " " " Corn in cob. 80 " " " Earloy 48 " " " Rye 56 " " " Bucliwheat . . 48 Peas (ji) Deans CO Tares GO « SEEDS. Clover CO pounds to the bushel. Flax 50 •' •' Timothy 48 " '• " Hemp 54 " " •' Blue grass .14 " " " Red Top 8 •' <• " Hungarian I ,g „ „ „ grass . . . j Millet 48 " " " Rape fiJ » " . « ■■.*'.^ A'M VEGETABI,E.S. , Potatoes .... CO pounds to (ho bushel. Parsnips CO " •' . " Carrots... . . €0 •< " Turnips (JO " " ' Beets CO " " " Onions CO " " " VEOETABLEg. Castor Beans 40 pounds to the bushel. Malt ac " " •' DriedFeaches 33 " " " Dried Apples 22 " " " Salt 5G " •' '• Dran 20 " " •« LINEAR (OR LONG) AND SQUARE MEASURE. LINEAR. WJUAKE. 12 inches (in.) make. 1 foot (ft.) 3 feet 1 yard (yd.) 6 J yards 1 rod or perch. 40 rods 1 furlong (fur.) 8 furlongs 1 mile (m.) 114 inches make 1 foot (ft.) 9 feet 1 yard (yd.) 30J yards 1 rod (rd. ) 40 rods 1 rood (r.) 4 roods 1 acre (a.) L.iND MEASURE. LENQTB. 7^A inches make 1 link. 25'links Irod. 4 rods or 100 links 1 chain. 60 chains 1 mile. • AKiiA. 10,000 square links make 1 sq. chalii 10 square chains .... 1 ncre. 82 AEITHMETIC. In RolM measure, t. c, the measurement of solids, 1728 (the third powet or cube of 1 2,) inches make 1 cubic foot, and 27 cubic feet (i. c. SX^X^O make 1 cubic yard. lu measuring timber, 40 cubic feet of round timber make what is called a ton, and the same name is given to 50 feet of hcwu timber. A cord of firewood is 8 feet long. 4 feet wide, and 4 feet high, and therefore its solid content is 8X1X'*=12S feet. Dry goods are measured by the yard, and fractions of a yard, the frac- tions used being one-quarter, one-eighth, and one-sixteenth. "MEASURES OF CAPACITY. DKV. 2 pints make 1 quart (qt.) 4q\iart3 1 gallon (gal.) 2 gallons 1 peck (pk.) 4 pecks 1 bushel (bu.) 3U bushels 1 chaldron (ch.) The last is seldom used. LIQUID. 4 gills make 1 pint (pt.) 2 pints I quart ( trader, llierefire, of eitlier re. (juires to l>e |)erleelly familiar not ..vly with tlie eoniparalive valiu' of tlio eurreiieies ol'liotli emiiitries, hut also with theeoins an Hritish 1'rovii\ees, exeept to Canada, where the deeimal system has been adeptitl, thon:;h, nnfortiinately, not universally followed ; Imt it is highly jirohahle that, if tlu! proposed eonfedi'ralion of the IJrilish Provinees should lie earriod, the deoinud coina;:;e will hi; uni- versally adopted, ami universally mlhered to hy the next f^ononit ion at least, if not hy the present. h'or the same reasons the mode of ehani^ini:; l''ederal into Sterliti;^ lueney. and vice rrrsit, has been I'xplained under the head of Sler- liniij J'lxehange. This khmus (piito as necessary as (he preceding-, beeauso the tralVu' between the States and Britain is on an extensive, senle, and the eomiuL; and piiiij; of passengers may now be reekoned by thousaiuls, all of whom require to understand tlior(Mi_t^hly liotli ourreneies and the eireidatinj; miHli.i of both eountries. T1h> in- ereasinp; faeilities of eominunieation are prou;ro.ssivoly and rapidly oxtendin;.:; the trade, ineludini; the passen;_'or trallie, between the twc countries, and bonce the {greater necessity that all ])ersoni. en;j;aire(l in business, or in any way exehan^inf; operations, should Intimately luulerstand hew to chaUL^e the money of each country into that ol the other. for the hviriier is l»it too apt to hiok at tlieqiiestion just us it stands, willmu, ever thiaixinjri'.t (lieiiriiu'ipleoa whieli it isialeiuleil to try liim. Tiioabsintlity ot'tlieexpres.-ioainay he shown l)y lliedilVerenl li;.,'hlsia whieli thelonjj;(liseiis:<- «"(l iiueslion. to umltiiily lis. tiil. by 2s. (id. may lie viewed. ( 1.) As 2s. (id, is J ot a poimd. (he (inestiou may lie taken as nieuiiin^i; tliat 2s. (id. is to lie dividrd into S I'mial pans, and 1 el'ihenj taken, wliicli would lie ;Vjd. (2.) As 2s. Cd. is 2 J shillings, the (luesliou might bo lalvou us meaning that 2a. (id. wuus to lac DRCIMAL COfNAOi: 30 Tim (irii^in of llio mark (8) I'nr (loll.irH is hoiiu'wIiiiL iiiincrtniii. SoiiKi MiiipiiHo it, to 1)0 ;i cfiiitraclidii Hir I'. S., Ilii! iriitinl.s ol" tlio riiilc(I Sliitrs, liiit, it, wuMiis to liiivo Ix'Oii ill UMfi iiicnntiiiciit.'il MiirnfH! Iiclurc llic (IJMtovciy of America, ami tliiTd'orc, imint lie :iii iinpdrliitiou. Till' I'lillnwiii;.^ (!.\|»l!iiiatii>ii (if it^ ()ii;^Mii Hci'iiiH mori! i.rohalili', I'^ir if tim oilier were cmrci!!, \\r hIkiuM Hunly liavc souk; record ol'it. Ac- cordiir; In an aiiciiMit lalilo or I'.iiicy, tin; piliarM ol" llcpdiiles inarlieil llu! liiiiils of lli(« world lowards (lie wo.st. and \v(!r(! waid lo Mipjiort tlu) world. l'"rom llieir |)o.-^itioii at. tlie eiitraiKM! to llie ,^^•dilerra- iii'im Sra. tliey Wert! olijectn of iiilire.st. In llie Sjiaiiiaid.-i and wern relire;;eiiteil oil one Kifl(! of tliuir eniii eailcd lli(! rr.n/^ and in t.li(! cuin lor S veai.s tli(! H was warped around tliciii, tiin.s formiiij^ tliu iiiarL, To rednci! (uirnjiiiiy money to the denomination.H oi' llie fleciinal coliiau't'. Sinoo lot) cunts make I dollar, and I dollarnuake I |ioiiml, •100 cents mako ! pound curnjncy, and thcrelore to lind llie nniidx:p oi' ci'iils in any ^\\v\\ number of pounds, wo must, mnlliply tli'' pounds liy 100. A^^ain, sinco 20 iionts inako 1 .Hliillin^i; or I 2 p(jice, to lind till! nuinher of (!on(s in any ^iven iinmlier ol" ,'~Iiillin;.'s, wn rmst multiply llio shilliii;^s l)y UO, Lastly, 5 cents are ((pial l.o .'I pence, and lli larthinjjs are also (-(lual to ',\ pencej and (Ax. I.) tliinj:;s lliat. iiro (Kjual to the .'■ame lliin^r, are cfjual to on(! anotiier ; llierorore, 5 i!ents arc ('(jual lo lli lartliin;,';!, and I lartlun;^' is tin A' f) cents, or I i A' i C(!nt. Hence to lind the nnnilK!! f cents in any numlier of pcnc(! and fartliin^rs, we multiply tlie riuml)' )• Ol' i':irtiiini:,s in tin; f^ivcn pence and farthin^^s hy 5, and divid"tlo product by !-■ Having obtained the threi! results, we aiM them .ill t()j,'etlier. Thus to chariw .CIS li^ls. O'l'd. to dollars and cents, W(! multiply .fSy<.'i00--rzl0200 4S by dOO, IS by '2U, :m<\ take ■', of ._'>( 1S< 20: )fX li" f)i?, or .'}!) fartliin''s, and a'ld tho thrco •I' 'n'^t 1 to^'cther, which f^ives us lir-Tli]- cents, l!i:i7(>^ or8lJ)5.7(]|. lepeiueil 'J.',- tinier, wiiieli would inal^e On. ;iii. ('.',.) Tlit; interinotiitioii mi^'lit lie. tliat. as 'Js. (id. is liO iienee, tli:it tlie other 'Js. Oil. is to Ix! ri'jM'ated ;;() tiines, wliicii would f^ivo .C\ l.'ts, Od. (I.) The pliruHi! may iil;-io be iaterprel.- I'll as meaning' that IJDd. wiw lo bo repeated 'M times, which would hIhO give x;> l.'is. Oil. Tlie last two hilerpi'etatlons are the same in two different fonii.-J. and pive the name result. This is the only view in whic:li tlii! e.vpreMnion has any neiise, and proves our stateineiit, that whenever a deiioiiiiiiiile niimlier is used as ii iiiulliplier. it ceases (o bi' deaomiualo, imd becoiue.i uh.-;truct. The baiuc prluciplo will apply lo division, 40 AniTHJIETIC. EXEECISES (1) £79 X400=31G00 IG X 20r=. 320 6MX j%^ in (2.) £117 X 400=40800 17 X 20--:. 340 8^dX V^- 14t^ $31 9.30 il 3. £87.14.10,f=S350.97J.,. 4. £29.19.9rr.$l 19.95. 5. £G7.13.4|- :$270.G7!,',. £279.15.10J,=$1119.17^. 7. £118.1 l.'UL$474.27A-.'' 8. £79.8.i^i317.GG§. 9. i;37.18.8-.rei51.73j. 10. £57.8.1 1J^$229.79/:,. 11. £19.7.G=:-8197,50. 6471.54/^ 12. £137.1G.8^e551.3;}J. 13. £23G.19.2i:---,«;947.sit. 14. £19.1G.8^$79.33^. 15. £98.1.U=.$392.22^. IG. £87.11.8:^.6350.331. 17. £457.12.15 ::=$1830.50. 18. £219.4.7J:::=8S7G.92|!. 19. £49.9.4f^::::.§197.87|lf 20. £2S7.18.10i:---$115L77-^. To cliango tlollara and cents to Halifax currency, wo must re- verse tho above operation. Tims, to reduce $195.7G} to £. s. d.— First, reduce the dollars and cents to cents, then divide by 400, which gives 48, the even number of pounds, with a remainder of 370] cents ; then divide this remainder by 20, wliich gives 18, the number of shillings, with a remainder of l(]\ cents, as in the converse operation, we multiplied by 5, and divided by 12, so now wc multiply by 12, and divide by 5; thus, lG;tX 12—195, and 195-^;-5^ 39, tho number of farthings, and this being reduced to pence and farthings, gives 9|, so that $195.7(;.t=£48.18.9.iv. Or tho work may bo shortened by tho fol- lowing method. As §4 make £1, the number of £'s in $105.70}, will be the same as tho number of times that 4 is contained in tlm 195 dollars, which gives £48, uiid $3 rcniui* 400)1957G}(48 400. 3570 1 3200 21)3704(18 20 17G IGO 12 5)195 (3a ■V ^ECIMLVL COINAGE 41 4)195 £-18—300 $195 — 7GJ ing. Now, tlicsc three dollara are equiva- lent to 300 cents, •which added to the re- maining 70.}^ cents, gives 310^ cents ; this divided by 20, will give the shillings, be- cause 20 cents are equal to one shilling, and it is self-evident that the number of shillings in 370.^ cents, will bo the same as the num- ber of times 20 is contained in that num- ber, which gives 18 shillings, and 10,^ cents remaining. Lastly, as 5 cents are equal to 9.id. 3 pence, one cent will be equal to J of 3 pence, which is ? of a penny ; therefore, if one cent is ecjual to jj of a penny, the re- maining 10 J- cents will be e(jual to IG.^ times ?, of a penny, which i3 9|d. ; hence we have $195.70.^ e(|ual to £48.18.9£. 20,370 J sl8— 16^ 3 5 48| 2. lleduce $270.07" 1 3. Reduce 647-1.27^' 4. Keduce $197.50 5. lleduco $ni9.17?r (5. Keduce $551.3;j.-^ 7. llcduce $1830.50 8. Reduce §1151.77^- EXERCISES. n Halifax currency. Ans. £29.19.9. i( Ans. £G7.13.4J. i( Ans. £118.11.4^ (( Ans. £49.7.0". (( Ans. £279.15.10-1. (( An.s. £137.10.8. (( Ans. £457.12.0. ^l Ans. £287.18.10i. MIXED EXERCISES. 1. Reduce «... Reduce 3. Reduce 4. Reduce 5. Roduco 0. Reduce 7. Reduce 8. Reduce 9. Reduce 10. Reduce £43li.7.8^ to dollars and cents. Ans. §1745.54J. $547.87 to Ilalifa.^c currency. Ans. £130 19.4,',. £783,13.01 to dollars and cents. An». e3134.08|. $570.85 to Halifax currency. Ans. 144.4.3. £000 19.8;] to dollars and cents. Ans. $2427,94,'',. $375.99 to Halifax currency. Ans. £9:5.19.11^, 3s. 8}d. to dollars and cents. Ans. 73J cents, 17 cents to Halifax currency. Ans. 10! pence. 10| pence to dollars and cents. Ans. 17] J, cents. 23 cents to old Canadian currency. Ans. 13J pence 42 AKITHMETIC. EEDUCTION. 1 1 . — Heduction is the mode of expressing any given quantity in terms of a higher or lower denomination, c. ^r., expressing any given number of dollars as cents, and vice versa, any number of cents as dollars. When a higher denomination is changed to a lower (as dollars to cents), the process is called reduction tfesccnding, and when a lower is cUangod to a higher (cents to dollars), it is called reduction ascending. Beginners arc generally puzzled by the word reduction, which in its ordinary acceptation means making less, whereas the learner finds that when dollars arc changed to cents, the number denoting the amount is increased a hundred fold. The explanation lies in tlio original use of the word reduo, to hring hack, which would sug- gest that the dollars were originally cents and are hrougid back to cents, or that the cents were originally dollars and are hwitght hack to dollars. Thus, by a transition common in all languai^es, the idea 0^ bringing back was gradually lost, a' id the; idea of changing from one denomination to another alone retained. Again, since one dol- lar is equal to one hun^dred cents, it is plain that the number repre- senting any amount in cents will be one hundred times greater, taken abstracth/, than that representing the same in dollars, and so in all denoniinaie numbers. Some explain the term reduction as taken originally from the changing of a higher to a lower denomination, and afterwards applied to the converse operation. This seems satis- factory enough as regards the present meaning of the word but does not accord with its derivation. Either explanation will clear up the young learner's conception of the term. If we wish to express 17 cwt. 3 qrs. 20 lbs., in c'.vt. nr?. lbs. terms of the lowest denomination,viz. lbs.,wc must first 17.3.20 find how many quarters are equivalent to 17 cwt. 4 3 qrs. which we find by multiplying the 17 cwt. by — -1 and adding in the 3 qrs. for 4 qrs. make 1 cwt, — 71 and then since 25 lbs. make 1 qr. we multiply the 25 71 qrs. by 25 to find the number of lbs. wliich, with the 20 odd lbs. added in, is 1795 lbs., and thus we see that 1795 lbs. are equivalent to 17 cwt. 3 qrs. 20 lbs. The proof depends on the converse operation, as in the margin, for, since the number denoting the pounds is, abstractly, 25 times the num- ber denoting the quarters, wo must divide the number denoting the pounds by 25 to obtain that denoting the 3uarters, anj, in like manner, we must ivide the number representing the 17 cwt. 3 qrs. 20 lbs. quarters bv 4 to find that denoting the 375 142 1795 25)1795 i) 71 qrs. 20. EEDUCTIOK. 4a 2G acres, 2 roods, 3G rods. 4 106 40 427G rods. — Ans. hundreds weight, the remainders in each case being ^vriltcn as sub- dcnomiQations. In the same manner 20 acres, 2 roods, 36 rods will bo reduce 1 tc rods by multiplying the acres by by 4 and adding the odd roods, which gives 106 roods, and this multiplied by 40 with the odd rods added in gives the rods, for 4 roods make one acre and 40 rods 1 rood. Conversely the rods di- v'lclcd by 40 will give 100 roods and 36 rods over, and 106 roods divided by 4 will give 26 acres and 2 roods over, the same as the original question — 26 acres, 2 rods, 36 roods. EXERCISES . 1 . How many dollars arc there in 47986 cents ? Ans. $479.80. 2. How many cents are there in 187 dollars? Ans. 18700, 3. How many pounds are there in 2 tons 10 cwt. 2 qrs. and 21 lbs ? Ans. 5071. 4. How many pounds are there in 18 cwt. and 22 lbs. ? Ans. 1822. 5. Reduce 14790 lbs. to tons, &c. ? _ Ans. 7 tons, 7 cwt., 3 qrs., 21 lbs. 6. Reduce 7043 quarters to tons, &c. ? Ans. 95 tons, 10 cwt., 3 qrs. 7. How many drams are there in 18 lbs. 13 oz. and 15 drs. ? Ans. 4831. 8. How many pounds arc there in 2785 drams ? Ans. 10 lbs., 14 oz., 1 dr. 9. How many grains are there in 17 lbs., 11 oz., 18 dwt. and 2? grains? ' Ans. 103654. 10. How many lbs. in 40891 grs. ? Ans. 8 lbs., 1 oz., 13 dwt., 19 grs. 11. How many gills in 4 tuns, 1 pipe, 1 hdd, and 52 gals. ? 12. How many tuns, &c. in 198402 drams? 13. How many bushels in 8904 lbs. of wheat ? 14. How many bushels in 14382 lbs. of barley ? 15. How many bushels in 48028 lbs. of peas ? 10. How many bushels in 4083 lbs. of timothy seed ? V*-, ">. <• ■}. u ARITHMETIC. 17. Reduce 98 miles, 5 furlongs and 30 rods to rods ? Ans. 31590 rods. 18. How many inches from Albany to New York (_150 miles). 19. How many miles are there in .5271 G8 feet ? Ans. 99 miles, G fur., 29 pr., 3 yds., ft., 6 in. 20. Reduce 57 acres, 3 roods and 24 rods to rods ? Ans. 9264 rods. 21. How many square yards are there in 17 acres, 2 roods and IB rods ? Ans. 85244J 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. 36.2.10.21J. 24. How many square links are there in 37 acres ? Ans. 3,700,000 links. 25. How many acres, -tc, Ir 1*79,863,201 square links? Ans. 4798 a., 6 ch., 3201. 26. 7,864,391 cubic inches; how many cubic yards ? Ans. yds. 168.15.263. 27. 9 cubic yards, 7 cubic leet, S21 cubic inches ; how many jubic inches ? . Ans. 432821 cubic inches. 28. How many gills does a tun contain ? Ans. 8064 gills. 29. IIow 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. 28G nails ; how many yards, &c. ? Ans. 17 yards, 3 qrs., 2 nls. 33. 36° 40' 25" ; how mJny seconds ? Ana. 132025". 34. How many degrees, &c., in 4978G" ? Ans. 13° 49'.46". 35. The area of New York State is 29.440,000 acres ; how many square miles ? 36. lIow long would it take a railway train to move a distance ?qual to that of the earth from the sun (95 millions of miles), at a jpccd of 52 miles an liour ? Ans. 208 years, 201 days, lO^'j hours. 37. The area of Pennsylvania is 47000 square miles ; how many square feet ? ^^ DEKOMIXATE NUMBERS. 45 38. Sound moves about 11.30 feet in a second of time; how long would it bo in moving from the earth to the sun ? Ans. 1 lycars, 27 day.s, 15 Hours, 50 niin., B,Y'j*o ^*^*"'' 39. IIoW many seconds of this century had elapsed at the end of 18G4, counting the day at 2-i hours ? Ans. 2,019,G8G,400". 40. The great bell of Moscow weighs 127,830 lbs.; how many tons, &c., does it weigh, the quarter being 28 lbs. ? Ans. 57t. Ic. Iq. IGlbs. 41. How many day^ from the 11th July, 18G1, to the 1st of April, 18G4 ? Ans. 995 days. 42. A congregation of 5G9 persons made a collection of X40.G.1 ; how many pence did each give on an average ? Ans. 17d. 43. The British inint c:in strike olf 20,000 coins in aji hour ; what is the value of all the pennies coined iu one day of 12 hours' work? Ans. XI, 000. 44. 417 tons of fish were caught at Newfoundland in one season, and sold by the stone of 14 lbs., at an average price of 42 cents a. Btono ; what did they bring ? Ans. $25020. 45. How niany feet from pole to pole, the earth's diameter being , 0*' TO 945 miles ? Ans. 41049G00 feet. DENOMINATE NUMBERS. 12. — When numbers are spoken of in general, without reference to any particular articles, such as money or merchandise, they are called abstract, but when they arc applied to such articles they are Bometimes called applicatc^ as being aj^jiUcd to some particular arti- cles to express their quantity ; eometimcs 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 us on aVstraot numbers, with this single difference, that when a lower dououiinatiou is added, and gives a sum equal to ono or more units of the next higher denomination, wo 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 wo always icduco, or carry, l)y tons. ^ I 46 ARITHMETIC. Here we find the sum of the inches to bo 34, and as 12 inches make one foot, the number of feet in 34 inches will be the same aa the number of times that 12 is contained in 34, which is twice, with a remainder of 10, therefore wc write the 10 under the column of inches, and add up the 2 feet with the column of feet, and obtain 11 feet, and as 3 feet make 1 yard, the number of yards will bo the same as the number of times that 3 is contained in 11, which is 3 tipies with a remainder of 2 ; we therefore write the 2 odd feet under tho column of feet, and add up the 3 yards with tho column of yards, and the whole amounts to 94 yds., 2 ft., 10 in. The same operation would be carried out if we had rods, &;c., given, and is applicable to all operations in denominate numbers of any kind. In the exercises on the addition of denominate numbers, ono ques- tion in abstract numbers is given to contrast with the denominate. Vila ft. In. 12 •> 9 10 1. 11 27 •> 8 30 3. 4 94.2.10 Jn. EXERCISES. (1.) (2.) (3.) £70.18. 4 (4.) $1907.874 $857.03 17.11. 4J 2075.75 7805437 189.50 99.19. 9 3194.02^ 198075 084.87^ 11.11.11 7058.50 8470154 • 498.75 G7.15.10i 8970.374 2873.124 1809538 807.12-i" 79.19. 9 4187043 305. 37^ 28.12. 1 1709.25 5708299 917.25 03. 8. 4i 445.17. 5^ (7.) 2481.92 28305740 4380. 50i 30997.42 (5.) (6.) (8.) lbs. oz. lira t. cwt. qrs. Ibg. lbs. oz. (Iwt. grs. lbs. oz. drs. scr. prs. 13.14.10 20.17.3.21 3.11.10.21 5 11.7.2.19 15.11.10 18 11.0.19 5. 8. 7.11 4 10.4.1. 7 11. 4. 9 25.15.1.1G 7. 9.18.23 3. 11.0.2.14 8.12.13 13.17.2.20 11.10.15.17 1 9.3.1.12 15. 7. 8 39. 4.1.23 12. 7. 9. 8 2 4.6.0.10 10.13.11 28.10.3.14 10.10.11.22 6 7.2.2. 9 8. 9. 6 18. 8.19.18 2. 8.1.1.13 4.15.15 89.10. 2 153. 3.2.13 77. 8. 0. 28 4.0.1. 4 y ^> (9.) DENOMINATE NUIIEEES. (10.) (11.) 47 (12.) tn. fur. rods. j-ds. yds. ft. in. I. oc. ro)da, rd. rijJ.i yds. ft. iu. 17G.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 IH.11.4. GS 79.6.29.3 8.1.10. 7 :iS.0.2G 24. 4.7.118 42.3. 8.2 11.0. 7. G 75.3.11 11.21.2. !)G 07. 1.11. 2 7.2. 8. 5 51.1. 8 15.27.0.124 118.3.10.3 1G.2. 9.10 94.1.19 27. G.3. 87 81.2.31.1 8.1. 7. G 63.2.21 19.25.2. 38 79.0.21.2 78.1.15 18.3.33.3 19.3.33 749.2. CO 87.0. 3. G 589.0.30 157. 6.3. 98 ' ?:- (13.) (14.) (15.) (IG.) n. ch. Imkp, ch. b. p. g. qt pt. t«. pl.hhd. B'L qt.pt. gl. y(!s,qrs.iilg. 79.9.9990 5.35.3.1.3.1 6.1.1.1.3.1.3 36.3.2 117.4.3G50 7.18.2.0.1.1 4.0.1.1.2.0.2 19.1.3 47.5. 941 8. 7.1.1.0.1 5.1.0.0.1.1.1 87.2.1 56. 2. 1182 3.26.0.0.1.0 1.1.0.1.0 63. 0.2 27.7.2813 4.18.0.1.0.1 74.2.2 36.1. 771 03.3.3 84.8.11G0 449.8. 51G 29.34.0.0.3.0 16.1.1.5.0.0.2 375.2.1 i I (17.) (18.) (19.) (20.) cwt. qra. lbs, 87.3.11 49.1.18 28.3.15 36. 1. 8 88.1.16 57.3.14 359". 50'. 50" 153 270 179 81 89 .40 .45 . 0. .45 .30 .30.10 .5D.59 yrs. days. hrs. min. 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\Vt. qm. 11)8. 18.1.18 22.3.11 0.2.18 12.1.15 8.3.24 31.2. 348.3.7 1134 .56.23 03.2G7.23.37.3G 103.3.11 M 48 AEITHSIETIC. LEDGER ACCOUNTS. The debit and credit sidca of four folios of a ledger arc as bel ;w, what are the balances ? (21.) Du. 21.) Cr. 22.) Dk. 22.) Cr. §1214.75 62703.80 $198.75 $118.50 803.09 471.38 47.63 9.05 291.45 305.50 18.11 16.25 • 318.25 297.11 97.38 37.08 1789.87 584.88 85.88 19.13 947.03 903.15 76,20 47.75 2000.00 1257.75 4.50 65.92 798.38 189.00 181.00 82.40 2018.50 98.13 10.25 76.50 104.30 756.25 70.38 7.75 277.15 87.50 219.50 197.25 1105.20 163.03 48.75 15.75 307.40 1291.00 93.15 8.38 984.70 784.25 25.50 93.15 273.00 79.75 81.05 67.45 584.10 81.18 28.30 5.45 1200.00 318.50 09.08 18.09 68.75 1819.20 157.11 4.12 79.15 58.50 278.00 67.00 60.18 176.25 59.50 28.83 2800.14 11.25 \ 941.12 (23.) Dr: (23.) Cr. (24.) Dr. (24.) Cr. 381.19 $80.10 $177.88 $156.92 17.11 15.05 291.16 285.15 45.38 39.88 356.13 360.12 19.63 10.13 189.38 178.25 187.13 176.15 471.03 409.10 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 87.03 365.55 478.99 . 340.25 • 76.75 638.27 546.54 224.12 56.51 436.15 372.25 156.12 37.23 326.36 252.12 1 $ « $ DEKOMINATE KUiTBEKS. 19 25.) Dr. 25.) Cr. (26.) Dr. (26.) Cr. $17G.93 $1237.75 $1037.63 4780.87 5iV.85 2703.18 457.88 183.05 79.37 194.25 190.37 97.75 98.11 39.37 87.12 149.15 35.40 8.25 94.25 13.25 83.50 11.87 47.20 41.18 1127.25 29.05 39.15 8.50 48.18 63.20 8.75 9.75 250.00 71.80 3G7.40 11.12 779.03 13.10 18.93 183.62 154 20 45.50 67..15 79.10 59.75 25.20 21.03 814.00 G8.87 43 15 208..50 95.50 18.75 7.50 78 GO 218.00 .28.03 50.00 189.00 59.87 71.38 87.75 47.15 18.05 293.03 5.00 68.10 77.40 185.10 31.G0 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 67.03 69.50 . 89.50 215.87 58.50 48.75 4.20 7.75 67.05 30.12 67.37 93.92 49.35 91.20 81.09 81.88 21.25 87.63 7.05 68.25 3r).i5 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 78.75 168.00 81.18 43.25 664.87 75.75 10.80 81.37 901.34 73838 51.25 92.65 268.34 18.24 67.54 37.49 567.84 136.25 91.12 46.87 987.69 126.72 18.35 91.13 356.78 834.15 42.54 54.12 978.65 128.71 16.21 64 54 546.37 130.18 ■ 25.51 57.62 786.42 178.16 53.99 38.94 428.97 284.77 62.87 61.87 642.85 326.54 91.54 93.89 529.64 412.13 32.21 89.78 428.04 391.15 54.12 21.46 106.70 267.18 77.99 64.98 600.00 125.13 42.51 73.75 250.09 T^. 50 ARITHMEMIO SUBTRACTION. (1.) • (2.) (3.) ei47985.87J £1573.11. 4^^ $810731.37^ 8G997.75 97().15,10j 341876.62^ 4. I have taken this month in trade $179G. 18, and paid $073.10 for Fall goods, and expended for private purposes $30.80 and lodged the rest in the Bank, how much have I hanked ? Ans. $980.28 5. I bought 47 tons, 17 cwt,, 1 qr., 18 lbs. of grain, and have sold 29 tons. 18 cwt., 3 qrs., 22 lbs. of it ; how much have I in store ? Ans. 17 tons, 18 cwt, 1 qr, 21 lbs. G. If the distance from Washington to Dover be 101 miles, 1 furlong and 20 rods, and that of Baton llouge 1407 miles, 1 fur- long, 30 rods, liow much farther is Baton llouge from Washington than Dover? Ans. 1245 m. 7 f. 30r. 7. A farmer possessed 1279 acres, 2 roods, 21 rods, and by his will left 789 acres, 3 roods, 30 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 Gibraltar 30''.G\30" N. ; how many degrees is Gibraltar south of London ? Ans. 15».24M9'' 9. The earth performs a revolution round the sun in about 305 days, 5 hours, 48 minutes and 48 seconds, and the planet Jupiter in about 4332 days, 14 hours, 20 minutes and 55 seconds ; how much longer decs it take Jupiter to perform one revolution than the earth ? Ans. 3907 days, 8 h., 38 min,, 7 sec. 10. I bought 54 tbs„ 10 oz. of tobacco, and 11 oz. of it were lost by drying ; and I sold' 30 lbs., 12 oz. of it to A. ; and 11 lbs., 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 G cents an ounce, and how much did my own consumption and loss by drying come to at the cost price, which was 5 cents an ounce ? Ans. (1.) 1 lb. 12 oz. ^2.) $46.38. (3.) $3. C5. 1 1 MULTIPLICATION. 1. $r;96X47^$84412. 2. £2.19.3ixl44=£426.3.0. 3. $168.87*X64=$10808. 4. £1.2.9 X*'225=£255.18.9. MULTIPLICATION. 51 5. Find the duty on 97 consignments of merchandise at $80.02^- each ? Ans. $8402.62^ It is often convenient to multiply denominate numbers by the factors of the multiplier. Thus : to multiply by 84 is the same as to multiply by 7 and 12. Thus, in the annexed examples, since 12X7=84, 18 tons, 12 ewt., 2 qrs., U lb3.xS4, is the same as ■•» tons, 12 cwt., 2 qrs., 11 lbs.Xl2X7, &c. (6.) tona cwt. qrs lbs. 18.12.2.11X84 12 223.11.1. 7 7 1564.19.0.24 (7.) -j/^-.*^ ac. roods, rda 27.2.29. X72 8 221.1.32 9 & (8.) 1993.0. 8 (9.) cwt. qrs. lbs. 23.3.22X49 7 167.3. 4 7 ^— — m^ 11.^1— —» 1174.2. 3 yds. ft. iiu 11.3. 7X150 5 60.2.11 5 304.2.76 1829.0. (! [10.) lbs. oz. drs. 49.11.12X63 7 348. 2. 4 9 3133. 4. 4 yds. It in. 11 .. 3 .. 7 150 1050 87 .. 6 450 .. 537 .. 179 .. .. 6 1650 .. .. 1829 .. .. 6 Thus: 11 yds., 3 ft., 7 in., multiplied by 150, will give (1) 150 times 7, which is 1050 in., and divided by 12, is 87 ft., 6 in.,— ^2) 150 times 3, which is 450 ft., and added to the 87 already found, gives 537 ft., and divi- ded by 3, gives 179 ft. without remainder, — (3) 150 times 11 is 1650 yards, which, added to the 179 already found, gives 1829 fb., so that the final result is 1829 yds., ft., 6 in., as already obtained by the method of factors. f".. #• C9 AlllTHMETIO. JC'J r*l, qlK III*. I.U.I7 I 27 27 «r)7.i.i7 £:jai.iH.o| an.o. t> (>. How tunny pooiuuIh hns (i iHM-aim llvod who niifl ooiDitlotod Mi twoiitiolh v<':>t', 11\o j'oai' «on«isting ofljOB ilnyH, H lumr.i, <\^ tninuloM, nmi IS Hoi-iMiiis? AiiH. (;:nr!^Hr»(;o. 7. U»\ii;li( 7 loads of liny, ciicli woifrliiuji; 1 ton, .'{ owl., It (jrFi., I'J lbs; \vl\!»( did tl\o wliolo wcij^h ? H. \t IV loan o!ii\ reap ',\ lUM'os mid 115 rodn pof diiy, how moch will Iu> roup in lU) days? Aiih, '.)(» norow, DO ioiIh. !>. ir !i stoaioltout '(dy noross no1\nnnol, tho hroiidtli of which in p.\U!\l to 'J*\ L'.V, 10", (vimt luij^ulni" ppaoo has hIio (raviTMnd nl, the end of 'JO trips? 10. Iluuiiltou, lloss & (>o., of lloidoij, havo clmr^^od mo on nii invoice ol" (>0 tons, 17 owt., I qr., and 20 ll>s. of Icon, at $55 por ton, and 1 pipo, I hhd., 34 p;als. and 'A t|ls. of wino, at. $.'1.00 prr ^!vl. if 121 1157, how much is this amount astrny 'i* 11. If a man huvcs d5 cents a day, how nmoh will ho buvo in tho year, onntting the Sahbaths 'i' 12. If 1'2 gallons, W (]»iarts, 1 pint of nudassos bo used in n hotel in a week, how nuioh would ho used ii\ a year at that ralo ? Au8. 10 Uhds., :i9 gala., 2 qts. 1[\. If ft man can saw one cord of wood in 8 hours, 45 niiuutes, 50 seconds, in what timo will ho saw 1 1 cords ? Ans. 4 days, 24 minutes, 10 sooonds. 14. If iJi waugons carry II tons, 15 cwt., 1 :|r., 16 Ib.s. each Low much do they all carry '{ Aus. 49 tons, cwt,, qr., 20 lbs. 15. If a man travel 20 miles, 5 furlongs, and 20 rodt u day, how much would ho travel at that rato iu a year 'I Ans. 7650 m., 7 fur., 20 reels. ll>. Thoro nro 24 piles of wood, «ach containing 3 cords, 42 cubic feet; what is the whole quantity? Ans. 79 cords, 120 ft. 17. If 17 hhds. of sugar weigh 12 owt., 1 qr., 20 lbs. each, how much will tho wholo weigh? Ans. 211 owt., 2 qrs., 16 lbs. 18. Allowing 75 yards, 18 feet, for tho surface of 9 rooms, how muoh paper would be required to ooyer the wall ? Ad0. C93 '>q. yard& MVTfK cvrL, :\ qrn., 21 Iha. of iron ft( ? rcnfM p(ir II). ; wlmt. dooH il ntnoiint. to ? 20. Wlmt, must T rp(5olvo lor '£ !!)«., 5 oz«., 1 i dwtfl., 21 i^rw. o( {jmIiI, a», H^IHM pordz. 7 21. DolivtTcil .Iii!ii(<:4 (Urnnt 7 liifiH, I pipe, 4!) p^alH. of I'ort Wiru p* 82.75 i^ei' gill, I whul iH tijo Uiiiuutit of lliu invoiuo ? DIVISION. In Pivimon, nil rnmamdorrt nro U) hn nvliioo*] f.o tlio next lowfii (Iniioiiiiiiiilioii, iiii'l in that form divulcl, to {^'it tlio unit« of that (Iciiutniiiatioii. E X B II O I H K H . 1. A 8lIv(!irHiiiilli niado hair !h1ozom spootis wci;j;liiri;^ 2 llw., Hoz<<.. 10 dwtH. ; what waH the woi;^ht of each 't Aiim. 5 oz«., H (lwt«., H ^rH, 2. If '15 wa;.';jj;nnH carry i)Hl> hiiHhclM, 2 pf.rkH, 4 (|iinrt«, liovr much docH each carry on cjual diHtrihiition '{ Aoh, I.'j hushclH, 7{) quarts, 11. If a lahounir rowlv(^H 14!) Ih.n., l.'S f)ZH. rif mnut fw payment foi 20 days' work, how much in that p(;r day, on rui uvcrn'^c^i'.'f At IS. :i lliH, , i2;',oz3. 4. If a ptoamnr ocniipiofl 48 d.iyH, 17 liour.M, and 40 minuf,*!H, in making; 121 trips; what is thn avornj^o timo? Ans. '.) h. 40 rnin. 5. If J)8 IttiHhols, 3 potiks, and 2 rpiarts of f^rain, can ho. packed in 5i7 cfjual-slzcd harrol.' ; liow much will thoro ho in each ? . Ans. 2 hush., 2 pcck.i, 5)^^ f|tfl 0. If a man has an income of $75000 a year; how much ha.s he an hour, allowinj^ th« year to consist of just .'505 days ? 7. An Kn;^lish nohlciiian h.xs £124,0^5 a year ; how much haH he per minute, the pound bcinj^ worth 84. H4, and the year to consist of 305 days, 5 hours, 4fi miiiutfls, and 48 sccondii? Ans. $1,144- 8. In n coal mine, 07 tons, l'.i owt., 2 qrs. were raLscd in 97 days ; how much was that per day, on an average ? 0. If $15.50 bo the value of 1 lb. of BJlvcr, what will be the weight of $500000 worth ? Ans. 32258 Iba., oi., 15 dwts., 11] J grs. 54 AEITHMETIO. 11. If 1246 bushels of wheat are produced in a field of 16 acres what is the yield per acre ? 12. A gardener pulled 13500 bushels of apples off 00 trees; how many, on an average, were in each bushel ? 13. If 13 hogsheads of sugar weigh tons, 8 cwts., 2 qrs., 7 lbs., what is the weight of each ? Ans. 9 cwt., 3 qrs., 14 lbs. 14. What is the twenty-third part of 137 lbs., 9 oz., 18 dwts., 22 grp. ? Ans. 5 lbs., 11 oz., 18 dwts., 5^\ grs. 15. A shipment of sugar consisted of 8003 tons, 17 cwt., 1 qr., 12 lbs., 10 oz., net weight ; it was to bo shared equally by 451 gro- cers ; how much did each get ? Ans. 17 tons, 14 cwt., 3 qrs., 18 lbs. 14 oz. 16. If a horse runs 174 miles, 20 rods, in 14 hours, what is his speed per hour? - Ans. 12 miles, 3 fur., 19 rods. 17. A farmer divided his farm, containing 322 acres, 2 roods, 10 rods, equally among his seven sons and 6 sons-in-law ; what was the share of each? Ans. 24 acres, 3 roods, 10 rods. 18. If 132 bushels, 3 pecks, 7 quarts of corn bo distributed equally among 23 poor persons ; how much does each get ? Ans. 5 bushels, 3 pccks; 1 quart. 19. A man having purchased 119 cVt., 3 qrs., 23 lbs of hay, and drew liomc in 6 waggons ; how much was on cach^ waggon ? Ans. 19 cwt., 3 qrs., 23 lbs. MIXED EXERCISES ON DENOMINATE NUMBEUS. 20. A gentleman, by his will, left an estate worth $2490, to bo divided among his two sons and 3 daughters in the following propor- tions : — Tho widow was to receive one- third of tho whole, less $346 ; the younger son $212 more than his mother; tho older son as much as his mother and brother, lacking $335.50, and tho three daughters were to havo the remainder, share and share alike ; what was tho share of each ? Ans. Tho widow got $484 ; tho older sou got $844i ; tho younger son got $090 ; each daughter got $155^. 21. A gentleman left a property in land, consisting of 448 acres, 3 roods, 24* rods, to bo divided among his four children in tho following proportions : — Tho youngest Vas to get 4 acres, 3 roods, 6 rods more than tho eighth part ; tho second youngest was to get one- fifth of tho :emainder ; tho oldest but one was- to pet one-third of tho remainder, aud the oldest tho residue ; what wus tho share of each ? i'' *»,^*,,, DI7ISI0N. 55 Ans. The youngest got 60 acres, 3 roods, 24 rods ; the next got V7 acres, 2 roods, IG rods; tho next got 103 acres, 1 rood, 34§ rods ; the oldest got 20G acres, 3 roods, 29 J rods. 22. A ship made the following headway on six successive days : On Monday, 3", 8', 45" south, and 1°, 61' cast ; on Tuesday, 2°, 30' south, and 2", 1', 15" east ; on Wednesday, 4°, 0', 52" south, and 1° cast; on Thursday, 1°, 48', 52" south, and 3°, 10', 22" east; on Friday, 1°, 19' fiouth, and 48', 29" east; and on Saturday, 59', 30" south, and 3°, 52', 11" cast; find her distances south and east from the port of departure. Ans. South 13°, 52', 59" ; East 12", 49', 17" 23. A vintner sold in cue week, 51 hogsheads, 53 gallons, 1 qua.'t, 1 pint ; in tho next week, 27 hogsheads, 39 gallons, 3 quarts ; in the next week, 19 hogsheads, 13 gallons, 3 quarts; how uiucli did ho sell in the three weeks ? Ans. 98 hogsheads, 43 gallons, 3 quarts, 1 pint. 24. In a pile of wood there are 37 cords, 119 cubic feet, 70 cubic inches ; in another there are 9 cords, 104 cubic feet ; in a third there arc 48 cords, 7 cubic feet, 127 cubic -inches, and in a fourth there are 01 cords, 139 cubic inches. Find tho wholo amount. Ans. 150 cords, 102 feet, 342 inches. 25. Tho following cargo was landed at Portland from Liverpool : 78 tons, 3 cwt., 2 qrs., 20 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 lbs. of Scotch coal, and 100 tons, 1 qr. of Staflfordshiro pottery ; what is the whole amount of tho consignment ? 20. If a man can count 100 one-dollar bills in a minute, and keep working 10 hours a day ; how long will it take him to count a million? Ans. 10§ days. 27. Tho earth's equatorial diameter is 41847420 feet; how many miles ? Ans. 7925 and 3420 feet. 28. Tho earth's polar diameter is 7899 miles, 900 feet; how many feet ? Ans. 41707020 feet. 29. Sound is calculated to movo 1130 feet per second ; how far off is a cannon, tho report of which is heard in 1' 9"? Ans. 77970 feet. 30. If tho ciroumferenco of a waggon wheel bo 14§ feet ; how often will it turn round in a mile, (^5280 foot; ? Aus. 300 times. 4- ■ 56 AKITIIMETIC. PROPERTIES OF NUMBERS. The term Integer, or Whole Numher, is used in contradistinction to the term Fraction. All numbers expressed by the natural series 1, 2, 3. ..10. ..20. ..100, &o., are called integers, so that 3 and 4 are integers, but f is a fraction. All numbers in the natural series 1, 2 3, &c., that caa be resolved into factors, arc called Composite, while those that cannot be so resolved are called Prime. Since 4=2X2, it is called composite, and BO 6, 8, 9, 10, &c.,< but 1,2, 3, 5, 7, 11, &c., arc called prime because they cannot be resolved into factors. Thus, 11 can only be resolved into 11x1, ov lXll> and these arc not factors 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 ai.y number will divide two or more others, it is called a Common Factor. Thus, 3 is called a common factor of G, 9, 12, 15, &c. Numbers that have no common factor, as 4, 5, 9, are said to be prime to each other. To resolve oi composite number into its prime factors, divide it by the least possible factor that it contains, and repeat the procesa tiU a prim^ number is obtained. EXAMPLES. 2)96 2)48 2)24 2)13 2) 6 3 00 thai Jie prime factors of 96 are 2X2X2X2X2X3. PROPERTIES OP NUMBERS. 67 Also, becauso 5x7X11=385, wo see that 5, 7 and 11 are the prime factors of 385. EXERCISES . 1. What are the prime factors of 2310? Ans. 2, 3, 5, 7, 11. 2. What are the prime factors of 1764 ? Ans. 2, 2, 3, 3, 7, 7, 3. What are the prime factors of 180642 ? Ans. 2, 3, 7, 11, 17, 23. Ans. 5, 19. Ans. 3, 17. Ans. 3, 3, 11. Ans. 3, 7, 31. 4. What are the prime factors of 95 ? • 5. What are the prime factors of 51 ? 6. What arc the prime flxctors of 99 ? 7. What are the prime factors of 651 ? 8. What are the prime factors of 362880 ? Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 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 is 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 endinj; in the digit 5 is 5 units, and all other numbers ending in cither 5 or are multiples of 5. ^^ •:l '^e ADDITIONAL EXERCISES. 11. Is 101 prime or composite ? Ans. Prime. 12. Is 198 prime or composite ? Ans. It has the factors 2, 3, 3, 11. 13. Is 171 prime or composite ? 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. Is 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. NoTK.— Wo have thought it Bufllclcnt under this head to give only the leading and most useful principles. 5S ARrrmiETTw. GREATEST COMMON MEASURE. 13- — When any quantity is contained an even number of times in a greater, the greater is called a multiple of the less, and the less a suhmultiple, measure or aliquot part of the greater. Thus : 48 is n multiple of 2, 3, 4, G, 8, 12, 16 and 24, and each of thc3C is a sub- multiple of 48. V^-2n one quantity divides two or more 'others evenly it is called a common measure of those quantities, and the greatest num- ber that will divide them all is called the greatest cor"~\on measure. Thus; 7 is a common measure of G3 and 49, and it is also tho greatest common measure, for no larger number will divide both evenly. When any quantity is measured evenly by two or more others, it is called a common multiple of them. Thus : 24 ia n common mul- tiple of 2, 3, 4, G, 8 and 12. A number which can be divided into two «qual integral parts is called an even nuinhcr, and one which canno'i. bo so divided is called an odd number. Ilcnceall numbers of the oeries 2, 4, G, 8, 10, 12, &c., are even, while those of the series 1, 3, fi, 7, 9, 11, &c., are odd. Hence the sum of any number of even quantities ia 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 groat uso in checking additions. A prime number is one which haa 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 ho prime to each other. Of tho first kind arc 1, 2, 3, 5, 7, 11, &c., of the second, 4, 6, 8, 9, 10, 12, &o. ; also, 2 and 7 are 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 G, it will also measure 12, 18, 24, &o., because it is a factor of all these. If one quantity measure two or more others, it will also measure their sum and difference, and also the sum and difference of any 819 507 312 195 117 78 39 ■'..'■■ .» • i GREATEST COMMON MEASUKE. 69 multiples of them, bccauso it measures them when they arc taken separately. Ilcncc, if one number divide the whole of another number, and also one part of it, it will divide the other part too. Thus : G di- vides 24 and 18, and so the other part, G ; 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 will also measure their sum. Thus : 9 measures 18, 27, and 30, and their sum, 81. From these principles we can deduce a rule for finding the greatest common measure of two or moro 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 bo tho greatest common measure. E X A BI P L E . 2145 3471 ■ 132G ^1 4t A concise form of the work is exhibited in 819 507 312 195 132G the margin. The quotients are omitted as 819 unnecessary. The last divisor, 39, is the G. C. M., as may be proved by trial. If it is re- 0,9 quired to find the G. C. M. of more than two numbers, first find the G. C. M. of two of 117 78 195 them, and then tho G. C. I!^^. of that and another, 117 and so on. 39 78 78 EXEROISKS. . Fin< 1 the G. 0. M. of the following quantities : 1. 247 and 323. Ans. 19. 2. 632 and 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. 8. 15863 and 21489. ^280 and 11385. Ans. 29. Ans. 1035. 9. 17222 and 32943. Ans. 79. 10. 19752 and 69132. Ans. 9876. M^ '#f .-Sf' ' L 60 AEITHMETIC. We may ottcn find the G. C. M. by inspection. For example, in exercise 5, we sec that 2 will measure both quantities (Art. 13), for both arc even, and also that 3 will measure both, because it measures the sum of thcdigits (Art. IG.) 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 have 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 : 96, 48, 24, are all common multi- ples of 2, 3, 4, C, 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 RULE; 2...3...4...0...9...18...27...3Q 214.. .18.. .27.. .30 2... 9*. .27.. .15 3I2...27...15 2... 9... 6 9 45 2 90 8 270 2 640 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, 1*8, 27, 30, ar- range them in a horizontal line, and as 2, 3, C, 9 are all contained in 18, they may be omitted, as 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 lino 2, 9. 6, all prime to each othnr, ,and the products of these and the divis- ors 3 and 2 is the L. C. M., 540. GREATEST COMMON MEilSUKE. EXERCISES Find tho L. C. M. of the following quantities: 1. 8, 12, 16, 24, 33. •2. 35, 42, 45, 81, 100. 3. 2, 4, 8, 10, 32, G4, 128 4. 2, 3, 5, 7, 11. 6. 3, 9, 27, €1, 243, 720. C. 12, IG, 18, 30, 48. 7. 3, 4, '5, G, 7. 8. 2, 3, 4, 5, G, 7, 8, 9. 9. 2, 4, 7, 12, IG, 21, 5b'. 10. 2, 9, 11, 33. EXAMPLES FOE PRACITOE. Gl Ans. 528. An3. 56700. Ans. 128. Ans. 2310. Ans. 729. Ans. 720. Ans. 420. Ans. 2520. Ans. 33G. Ans. 193. I 1. What will 320 caps coat at $7.50 each ? Ans. $2400. I 2. If you can purchase slates at 20 cents each; how many can you buy for 67.40 ? Ans. 37. 3. If you can walk 4 miles au hour ; how far can you go in 24 hours ? Ans. 9G. 4. What will be tho cost of 216 barrels of pork at $7.50 per barrel? Ans. 61620. 5. How many ehccp can be bought for $560 at 63.50 per head ? Ans. IGO, G. If 825 pounds of beef are consumed by a garrison in one day ; what will be the cost for G 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 17 horses at 670 each, and 12 cows at 620 each ; how much remains due ? Ans. $3195. 8. A merchant bought 120 yards of American tweed at 61-15 a yard ; GO yards of flannel at 95 cents per yard, and 13 dozen pairs of gloves at 35 cents per pair ; what was the amount of his bill ? Ans. 6249.G0. 9. At $2 per gallon ; how much wine can bo bought for 684 ? Ans. 42 gals. 10. A boy had $5.50, and he paid one dollar and llvo cents for a book ; how much had he loft ? Ans. $4.45. 11. What will 18 oo'rds of wood cost it $4.75 per cord ? Ans. $85.50. )l»; K 1* . 1' : I ' ^ c C2 ahithmetio. 12. IIow many pounds of sugar can be bought for $9,35, at 11 cents per pound ? Ana. 85 lbs, 13. What ■will a jury of 12 men receive for coming from ICings- fon to Albany at 10 cents a mile each; the distance being GO miles ? 14. A grocer bought a hogshead of molasses at 32 cents per gallon ; but 18 gallons leaked out, and he sold the remainder at 55 cents per gallon ; did he make or lose, and how much ? Ans. He gained $4,59. 15. If a clerk's salary is $G00 a year, and his personal expenses 8320 ; how many years before he will bo worth $6000, if he has SIOOO at the present time ? Ans. 20 years. 10. A speculator bought 200 bushels of apples for $90, and sold the same for 8120 ; how much did ho make per bushel ? Ans. 15 cents. 17. A person sells 15 tons of hay at $22 per ton, and receives in payment a carriage worth $125, a cow worth $45, a colt worth $40, and the balance in cash ; how much money ought lie to receive ? Ans. $120. 18. How many pounds of butter, at 20 cents per pound, must be given for 18 pounds of tea worth 75 cents per pound ? Ans. GU lbs. 19. A grocer bought 7 barrels offish at $18 per barrel ; but one barrel proved to be bad, which he sold for $5 less than cost, and tho remainder at an advance of $3 per barrel ; did he gain or lose, and how much ? Ans. Lost $13. 20. A man bought a drovo of cattle for $18130, and after seU ling 84 of them at $51 each, the rest stood him in $43 eacb ; how many did he buy ? Ans 406. 21. What will 2 cwt. of cheese cosl at 9^ Cents per pound ? Ans. $19 00. 22. A. is worth $9G0, B. is worth five times as much as A., less $G0O, and C. is worth threo times as much at A. and B. and $300 more ; what are B. and C worth each, and how much are they all worth ? Ans. B. $4200 ; 0. $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 dolea, 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 labourer bought a coat worth $16, a vest worth $3, and a GBEATEST COMMON MEASURE. C3 pair of pants worth 65.50 ; how many days had he to work to pay for his suit j his services being worth 50 cents per day ? Ans. 49 days. 25. What will 14 bushels of clover seed cost nt 12 J cents per pound? Ans. $105. 26. A farmer sold a load of oats weighings 183G pounds, at 30 cents per bushel ; how much did he receive for the same ? Ans. SI 0.20. 27. A produce dealer bought at one time, one load of wheat weighing 3240 pounds, at $1.05 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 ? 28. A farmer has 12 sheep worth $3.50 each ; 9 pigs worth $4.65 each ; one cow worth $35, and a fine horse valued at $150. Ho exchanges them with his neighbour for a yoke of oxen worth $75 ; two lambs worth $1,925 each ; a carriage worth $100, and takes the balance in calves at $4.50 ; how many calves does lie receive ? j Ans. 20. 29. A and B sat down to count their money, and found that they had together $225, but A had $15 more than B ; how much had each ? Ans. A $120, B $105. 30. A miller bought 250 bushels of oats for $85 and sold 225 bushels for $70 ; what did the remainder cost him per bushel ? Ans. 60c. 31. A widow lady has a farm valued at $0720; also three houses, worth $12530, $11324, and $9875. She has a daughter and two sons. To the daughter she gives one-fourth the value of the farm, and one-third the value of the houses, and tbcn divides the remainder equally among the boys, how much did each receive? Ans. daughter $12923, each son, $13763. 32. A man went into business with a capital of $1500 ; the first year he gained v^OO, the second year $950, the third year $700, and the fourth year 625, when he invested the whole in a cargo of tea and doubled his money ; what was he then worth. Ans. $9150. 33. A boy paid out 30 cents for apples, at the rate of 6 for 3 cents ; how many apples did ho purchase ? Ans. 60. >^*». fr^ C4 AKITHMETIC. 34. A schoolboy bought 12 oranges at 3 conta each, and sold thcni for 12 cents more than ho paid for them ; how much did he sell them at each ? Ans. 4c. 35. A clerk's income ia $2098 a year, and his expenses $4.59 per day ; how much will lie save in two years ? Ans. $2111. 30. A speculator bought 200 acres of land at $45 per acre, and afterwards sold 150 acres of it for $11550 ; the balance he sold at a gain of §5 per acre, and received in payment S-50 cash, and the balance in sheep at $5 each ; how many sheep did he receive ? Ans. 450 sheep. 37. A butcher bought 9 calves for $54, and 9 lambs for $131.50 ; how much 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 1.7 cents per pound, and one checiso weighing 53 pounds, at oonts per pound ; and received in payment 22 pounds of sugar, at the rate of 11 pounds for a dollar; 150 pounds of nails, at G cents per pound ; 15 pounds of tea, at G5 cents per pound ; one half-barrol offish, at $18 per barrel, and one suit of clothes worth $27 ; did the farmer owe the grocer, or the grocer the fiirmer, and how much ? Ans. the grocer owed the farmer 12 cents. 30. A milkman sold 120 quarts of milk, at 5 cents per quart, and took in payment, one pig worth $1.50, and the balance in sheet- ing, at 10 cents per yard ; how many yards did he receive ? Ans. 45 yards. 40. How many pounds of cheese, at 9 cents per pound, must be givcu for 27 pounds of tea worth 80 cents per pound ? Ans. 240. FRACTIONS. 14. — ^VtTLaAR OR Common Fractions. — ^Wheu 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 (^-) ; thus, 3-^4 means that 3 is to be divided by 4, and this being impossible, we indicate the operation either as above or by writing the three in the place of the upper dot, and the 4 in the place of the lower, thus, f . 'h FK.VCTTONS. 65 Tlic nature of ji Iractioa may be viuwcd in two ways. First, wo may consider that a unit i.s dividod into a certain number of equal parts and a certain number of these parts taken ; or, secondly, that a number greater than unity is dividod into certain equal parts, and one of tliesc parta taiten ; thus, '^ means eitlier that a unit is divided into 4 equal parts and three of thorn taken, or that three is divided into 4 equal parts and one. of them taken. For example, if a foot bo divided into 4 equal parts, each of these parts will bo o inches, and three of them will be nine inches ; and cineo 3 feet make 3l! inches, if wc divide 3 foot into 4 equal parts, each of these parts will bo 9 inches, and hcnco f of 1 -=^\ of 3. The lower figure is called the denominator, because it shows the denomination or number of parts into whicli the unit is supposed to be divided, and the upper one is called the numerator, because it shows the number of those parts considered in any given question. When both are spoken of together they arc called the terms of the fraction. What may be considered the fundamental principle on which all the operations in fractions depend is this : that the form, but not the value of a fraction, is altered, if both the terras are cither multiplied or divided by the same quantity. If we take the fraction ^ and multiply its terms by 2, wo get l- Now, the ^ of a foot is an inch and-a-half, and therefore g is G inches and G half-inches, or 9 inches ; but we have seen that J of a foot is 9 inches, ttiereforc J of a foot is the same as | of a foot. So also f of £1 and g of £1 are both lys. The same will hold good whatever the unit of measure may be, or whatever the fraction of that unit. Hence, universally thc/onn of a fraction is altered if its terms be cither multiplied or divided by the same number, but its vqlm remains the same. Again, if we multiply the numerator 3 by 2, but leave the denominator 4 unchanged, we obtain |, and, keeping to our first illustration, | of a foot is G times three inches, or 18 inches, which is double of 9 inches, the value of J. We should have obtained the 5 ao ro y taking g and dividing its denominator by 2, without .viding its numerator. Ileuce, a frwtion is multiplied by cither multiplyin;; cs numerator or dividing its denominator. In like manner, if wo take tho fraction | and divide its numerator by 2, we oh iin ^, and if we multiply the denominator of its equal | by 2, wc obtain the same rusult, ^. Hence, f is J of £, and therefore a frac- tion is divided by either lividing its numerator or multiplying its denominator. These principles may also bo referred to tho olivioua ■"•■fi '**-*''•: ' I c.i; Ai;rriiMi",TTa I'u'l (li.il. ill (liviilii);:; any i|H:iiiliiy llio fi;roiitt>r l\w diviMitr llio Ichm ili« <|\ii)(it'nl, uu'l llu< li"^< »r (In' (:;n>iif(>r the (|ii()lir.i(()r oliMiTvi' (his as ii uiiivorMal nilo — diiHtli'. in/n n i/mi «vrn, iMii'liiiiH ar.> oliiHMil'unl in lour c^illVroiil. wayw, adcortUn^^ x Fractions. Thi> term simpK> iVaetion, a,s opposod to complex fraction, nicann that there is only one division. Thu.s: -j j) niuaiiH that a nin;^lo number, IT), is dividi'd by a hiiii^lo number, 1(5. .\ ei)inplex IVaetion is one r.f which cither tlio niimcrntor or de- nominator, or both, are fractional, that ih, it indiwitcs a diviKion, when cither the given product or given factor, oy both, arc fractional. Tiius; ;J :-,",, or land ,, and x.i arc complex fractions and cx- hibit the only three posssiblc fonua. iV, Vulgar, or (/Oinnion, and Dociinal FraotioiiH. Decimal fractions arc thoso expressed with a denominator, 10, or apowerof liy.j?., ^, .V,,, .Th',.)- Any fraction not so expressed is called vulgar or cominon. Thus : ^ would be c.Ulcd a common fraction, but its equivalent, ,',i\^, would be called a decimal fraction, and is written -75, the donomina* tor being omitted, but its existence Voing indicated by the mark (•), called the decimal ooint. rnAfTioNH, CJ A talxoil f(imn}ity !« (inu rxpronwul pnrtly l»y n wholr iiiiml»fr nnil I'lirtly by a fViirtion, an ^J^, \2^. Tliin in not uiintlKT kind n* fraclifin, bi;! "itiiply iuhiUkt inrMJo of writ.iri}^ an iinjjrojwr Craclioii wln'ti Ui(! (liviHioii iiii|i<;aU!(l IriH btM^ii jiorforiticil .i« far nn poHKihlc TIiiih: 'V i/„ and 7 .11^^. It i'h nlYcMi Haiti tliaf. lli'ini am nix kiiidn of fracti'iriH — projur iiiipr()|i<^'i (•ally iiic'irncf., for a prfipor rraclion Ih Kiiiiplc!, and a mixed (|uatitily is uti iiiiprfipor fraclinii in nnoMior i'onn. 15. — OPKttATioNH IN (I()ii.\ff)N l»'iiAf;TroNH. — F« a wbok or mixed nunibor. ■^j" ' iiH a wholo or mixed number. 'JY "" >* wbolo or mixod number. V'dVt" •"* '^ wliol»; or mixod number '^''fj" aH a wliolo or mixod numbor. ',"/• 'w " wbolo or mixod numbor. ^!J UH a wb.'do or mixed number. I j !]'' as a wholo or ^lixed number. Ant, 49. J,'/- a8 a wholo or mixod numbor. '^Y as a wholo or mixed number. J-Jfi as a wholo or mixed number, ^f'f^ as a wliolo or mixed numbor. Y lui • whole or mixed numbor. Ahh. ii,-'^. Ann. 71. Ann. r>yi. AnH.r,'i;^i. Ans. 11,',. AnH. 7.' . Ans. 7,V. An.H. ^'J. An«. 10/,. Ans. 19». An.. 12,',. Ans. 4|. 68 AMTHMETIC. 15. IG. 17. 18. 19. 20. 21. 22. 23. 24. 25. 20. 2S. 29. 30. Express -1/- as a whole or mixed number. Express -y^ as a whole or mixed number. Express ^ ,- as a whole or mixed number. Express -V- as a whole or mixed number. Express J|J- as a whole or mixed number. Express -f ^J- as a whole or mixed number, Express J^^'- aa a whole or mixed number. Express 27^ as an improper fraction. Express 6GJ as an improper fraction. Express 15] g as an improper fraction. Express 7'^ as an improper fraction. Express 49 as a fraction with the same denominator asjj. Ans. -"i-V- Express lOs. as a fraction of £1. Ans Express 11 inches as a fraction of a foot. Ans, Bring -\, J, ^, J, jV to the same denomination. Anq " 4 3 _2 1 ■"■""• T^» 13> i2» T^» T3' Express 11 as a fraction having the same denominator as ^y,^. An* ''"T ' 1 Ans. 24J. Ans. 5Af . Ans. S/y. Ans. 5A. Ans. 30|. Ana. S3j\. Ans. 9^',. Ans. ■^, Ans. ^^. An3.Ao^. 3.1 I 9 1 t Ans. 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 iospection, as |.5^=^-=:j^, but in such questions as hVih ^^^ ™os*^ 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- *'0" hU'i 's 1092, and the terms of the fraction divided by this njivc I, the simplest form. EXERCISES. 1. Reduce ^^\^^^^g to its lowest terms or simplest form. 2. Reduce HH to its lowest terms or simplest form. 3. Reduce 'jlj^g to its lowest terms or simplest form. 4. Reduce tjVo'}j1,*j to its lowest terms or simplest form, 5. Reduce ^sso ^q \ig lowest terms or simplest form. 0. Reduce yVsV/a ^ '^^ loweit terms or simplest form, 7. Reduce xi^Jr to it» lowest terms or simplest form. 8. Reduce |||| to its lowest terms or simplest form. 9. Reduce §]^| j to its lowest terms or siiuplest form. Ans. J. Ans. ~^j, Ans. g, Ans. ;j5j. Ana. g. Ans. -^ a* Ans. 1^*3. Ans. £. Am. f . FRACTIONS. 10. Reduce sJ'^^i *^ '^^^ lowest terir\ or simplest form. 11. Reduce y^'^j^/j^ to it, lowcoc terms or simplest form. 12. Reduce r;!]J]fj to its lowest ti;rms or simplest form 13, Reduce li !'» v.. 69 Ans. §. - Ans. Z'^. Ans. JJ. f JJu'y^Jb ^'^ ^^ lowest terms or simplest form. Ans. 14. Reduce IJldf *° ^^ lowest terms or simplest form. 1' Ans. 15. Reduce w^Jj to its lowest terms or simplest form 10. Reduce 17. Reduce 18. Reduce 19. Reduce 20. Reduce jt t r, 1 5 25' Ans. |. ao 1 1 ^^ ^^^ lowest tonus or simplest form. '• I i to its lowest terms or simplest form. ^"{Vs *^ i'^ lowest terms or simplest form. uiiTUM ^^ ^'^ lowest terms or simplest form. fin a 'J Ans' Ans. Ans. I r> t) Ans. 3* 1^0^ to its lowest terms or simplest form. Ans. 1 1 21. Reduce -if^iriio^oooo to its lowest terms or simplest form. Ans. h. =* 111. ..111. ..Ill IV. To multiply one fraction by anotlier, multiply numerator by numerator and denominator by denominator. [ Thus:i^Xj=i. To illustrate that A^ of ^ is J, take a line and lot it be divided into 3 parts, '^ and '^ach of those again into .3 parts, !is in the margi, we find that the result is 9 parts, each, of the unit. of course, being We have seen that a fraction is multiplied by multiplying tho numerator or dividing the denominator. Now, if it were ro(iuired to multiply J by ^, wo could not divide the denominator, as 5 is not contained in 4, and therefore we multiply the numr tor and obtain -Y, but we have multiplied by a fjuantity equal . , V times tho given one, and therclbrc wo niu.st divide tho product by 7, i. e. (Art. 21,) we must multiply tho denominator 4 by 7, which gives ^^ for tbu correct product. EXEUCISE8. 1. Multiply {', by | ,^ ? Aug. ^Vi- 2. What i.^ Jio product of J by ] ] ? Ans. ^ . 3. Wliat is tho product of j^ by § ? Ans. ^']. 4. What is the product of {hy \ J ? Ana 6., "What is tho product of \ by iJ} ? Aus. r,n h\' ' P 70 APJTIDIETIC. G. What is the product of I by /g ? Ans. |g 7. What is the product of -j^g by /^^ ? Ans. f^^^. 8. What is the product of 3 5 by /, ? Ans. g-^|. 9. What is the product of % by .^ ? Ans. I5. 10. What is the pr'^'luct of ] 5 by /y ? Ans. -^,^r^. When tlie product has been obtained it should be reduced to its lowest terms. Thus : the product of ,'',- by J .', is -jY^, the terms of whicli arc both divisible by 11, and so >vc j^et the equivalent fraction ■j^^. 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 ibrm, viz., /.. In the same manner iiny number or num- bers which arc f\ictors of both numerator and denominator, may bo omitted in the operation. This we call cancelling in prcferenco to the excessively awkward term "cancellation." This method will bo clearly seen In exorcise 11. If cither the multiplier or multiplicand be a mixed quantify, it must be reduced to an improi)cr fraction before the multiplication is performed. Thus: 8^X5;;=^/^X^=iHp=^5U,' Lo.,. 11. What fraction is equal to i of § of ^ of I of ^ of « of I of % ? Ans. ^. 12. What quantity is equal to 12^ multiplied by 7|} ? An^. 97{A. lo.-What quantity is equal to 19 J multii)lied by Ij-^ ? Ans. 30. 14. What is the value of % of I of ^8 of j'. ? 15. What is the value of J of ij of « of J^ ? 10. What is the product of 27^ by 3« ? 17. Vv'hat is the product of j'^ by J » ? 18. What is the product of 5^ by 5^ ? \ 19. Find the square and cube of AJ ? 20. What is the cube of .■' j; ? 21. Multiply 27 by .V ? AnrsV?,. Ans. 107i;,'.. Ans. -i. Ans. 30^. ivns. 5^4 nnu 7uu,„. ■'»-y=' 1000* 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 tho language of science, the reciprocal of a fraction is tho fraction with its terms inverted. Thus : ^ is tho reciprocal of 1 ; :| of ^. To find tho reciprocal of a whole number, we must first DIVISION OF FRACTIONS. 71 represent it as having a denominator 1, — thus 4=1 ; 6=^, and therefore the reciprocals are J and ^. The rule for division may bo proved in two ways : First ruooF. — Let it be required to divide /,- by ^<. If wc liad been required to divide by tlio whole number 5, wc should cither have divided (Art. 11,) the numerator, or multiplied tlip denomina- tor, — as the numerator is not divisible by 5, wc multiply the de- nominator, and obtain -.'''-- ; but wc have divided by a (quantity equal to six times the given one, and thercrorc, to compensate, wc must multiply the result by G, which gives ^j. Second puoof. — Write the question in the complex form — -T. 7 •!^i, then (Art. 14,) multiply both terms by 11, and ', {" is obtained ; « """ and again multiply the terms by G, and H is the result as before. — The two operations are virtually the same, though exhibited in dif- ferent forms, and both are equivalent to the technical rule, " Invert the divisor and jnultiply," Mixed quantities must be reduced to improper fractions as in multiplication. The expressions mult! plication and division, as ap- plied to i'ractions, are extensions of the ordinary meanings of those terms, for in their original meaning, the former implies increase, and the latter decrease ; but when two proper fractions are multiplied together, the product is less than cither of the factors, and when one proper fraction is divided by another, the quotient is greater tlian ' cither the divisor or dividend. This will be seen by the annexed examples : iX5=^i. But ^^ll and 1=^1^, both greater than ? J. Also, t-^.i^5Xa^;:.?. But l^.-^\ and i^\°^, both less than as If two fractions have a common denominator, their quotient is the quotient of tiieir numerators. Wo have i)laced multiplication and division of fractions before addition and Bubtraction, because, as in whole numbers, multiplication and division are dwduced from addition antl subtraction, so conversely in fractions, addition and Bubtraction a'o to bo deduced from multiplioation and division, for a fraction is produced by division, and the multiplication of a IVaction is merely the repeating of the divided unit a certain number of times. Thus : ^ is a unit divided into 8 equal parts, aud § is that fraction repeated 7 times. AnrniMETic. r-XEUCISES. 1. Divide w l>y § ; V\-§-Ax^. 2. What is the quotient of ; .? divided by i| ? 3. What is the quotient of ,?r, divided by ]*! ^ ? 4. What is the quotient of ;i? divided by 'j!* ? Ans. -?r Ans. If: =1t't Ans. |l. Ans. -^:]. Ans. IJiJ. Ans 1 "'^- .fills. 1 , y ,^. Ans. v';. 1 1 ' 5. What is the quotient of l \ j} divided by ^ 7 ? 6. What is the quotient of 31j divided by 19^ ? 7. What is the quotient of ofj divided by 2§? 8. What is tl^e quotient of 4V divided by 15 ? 9. What is the quotient of ';'^ divided by 2\l? 10. What is the quotient of TOj'^g divided by 9 ? 11. What is the (|uotient of G^» divided by 9j ? 12. What is the f,uoticnt of 5:} divided by 8 ^^^ ? 13. Divide the product off, ^ and ^ by the product of -^. f r, d Ans. Jg'-:^1^ 14. What is the quotient of fy of J J-v-? of f J of V't-^A- o^ # Ans. 45 «. Ans. 8]:;. What is tlie value of ^ of ^— f of {^ ? Ans. !)[;?. A«s.8;]j. Ans. 5. Ans. |««. 15. IG. 17. How many ^/f arc there in y^j ? Divide 27 by ^\ ? Ans. l.,-.j. Aus. 729. Hence, any quantity divided by its reciprocal gives the square of that number, and exercise 21, of multiplication, shows that any quantity multiplied ))y its own reciprocal- gives unity. 18 19 Divide -rX"i by 4, and the quotient by -^^ ? Divide }, by -,'^j-, and the quotient by §j ? 20. Divide ^'i by .{^? 21. Divide Ans. 1/y. Ans.iie- Ans. 3-,V2',. ic-liby^^? Ans. -i. •VI.-ADDITION OF FRACTIONS. Wc have seen that no quantities can be added together except they arc in the sanrc denomination. Wo can add ^, ^, ^ and J,'-, ns they are all of the same denomination, sevenths, and wo find -Y- ^^^ ^''^" easily sec that to add f and ^, wo havo only to alter the form of J to ^', and wc havo both fractions of the same denomination, and therefore can add them, — (j-j-^^^^ But wc cannot always tell thus bv inspection, and therefore must be guided by some rule. To find he value of i+a+b+'a+ia* ADDITION OF FRACTIONS. 73 By Art. 13 we find the L. C. 51. of 4, C, 8, 9, 12 to be 72, and the rest of the common operation is equivalent to multiplying the terms of oucli fraction by 72. Thus: if the terms off be both multiplied by 72, wo get g^"— ^^JJ:}— '^j, but wo might as v;cll have divided 72 by 4 before multiplyinfr, and, to balance that, have muhiplied the numerator 15, not by 72, but by the fourth part of 18. 'vivino: 5:1, as the following scheme will show: — The other fractious being altered iu the same manner, we get ra + ?" + 7 2 + 7 5 + r;> '^^^ ^^ *^^^^° ^^^ now all ot the same denomination, though not altered in value, wo can add thorn, and we find £-l-;j-|-|-|-^-{-^::=|i«+«^4-^^ + t4|-|- 72, viz., r> I 3X7'2 3X1 8X» 3X1 fl T. I 4 X7 vt — I X I 8X4 — 1 X IH — 1;i -•.16 "73 • Ilcncc the ■v% *. / RULE. Find the L. C. M. of all the denominators, which will be the common denominator ; divide this coii.mon multiple by each denomi- nator, and multiply the quotient by each numerator in fiuecessioa for new numerators ; add all these new numerators together, and place the common denominator below the sum, and the fraction thus ob- tained will be the sum of the given fractions. If the numerator, thus obtained, bo greater than the denominator, the resulting frac- tion may be reduced to a whole cr a mixed number by division. ■ EXEUCISES. 1. Express i^ + i'.;+A+i\- ^^ ^ single fraction ? 2. Find the sum of h f, ^ and g ? 3. Add together 4 J, 1 17 >}i n 3;j?and5,'.j? 4. 5. What fraction is equal to ^+^+8+ I'a+a'a+c't ^ ''^"^• What fraction is equal to li-|-2j4-3.J-f-4^+5;|+G« ? Ans. {i. Ans. 2|. Ans. 18;>jt. )i.3 Ans. 25|3i. G. Express ^ of f -{-§ of jj-|-J pf ^ as a single fraction ? 7. Find the sum of 1 1 ^, 8^, 3 A and 4^ ? 8. Find the sum of ^ of H-i'sOf ;}■ Av,q 4 7 111 iVnS. .,,5 — l3j{. Ans. 18^V,. Ans. 1 I ii 10* 9. AVhat single fraction is equivalent to ^ of J-f J of ^-{-^ of ^ ? Ans. A- 10. What single fraction is equivalent to f of ^- of ^-f ^ of § of JHof^of^? Ans. /g. 11. AVhat single fraction is equivalent to g- of § of f-f § of g of i;v Ans. -fjHj. u AIHTHMETIC. 12. Simplify ,-V^? Ans. 1,^5. 13. Find a single fraction equivalent to ^ of § of §-f -| of J? Ans. ?8!' 14. Divide the sum of -{'j aud ^ by the sum of j and ^ ? 3 80' Ann 340 IG. Simplify l^y^^? Ans. ^"i^ 3 3 r YII.-SUBTRACTION OF FRACTIONS. What wo have said of addition enables us to give at once the 11 U L E r n, SUBTRACTION. ricduce the given fractions, if necessary, to new ones having a ccmnion denominator, as in addition, and subtract the numerator of the less from that of the greater, and place the connnon denominator below the remainder, and the resulting fraction will be the difference between the given fractions. I'^x.vMPLKS. — (1. ) To subtract /',- from -^^. Here the denomi- nations being the same, we can subtract at once, and find the differ- oncfl to bo ,'-,. (2.) To find the value of f, — IJ. These fractions brought to a common denominator, as in addition,, become {]§ and f j, ard therefore the difference is ,;-,. (3.) To find the excess of 12 J above 7^, we find new fractions with a common denominator, viz., ,,", and ^:j, and we write 12./, — 7!;['. Now we arc required first to subtract \'l from 0*!,, but as we cannot do this directly, wo take one of the 12 preceding units, and call it H.J, (for 2]=1,) then .j^ ij-/^.,— s;j, and ij— ^:; = n> ^'^^ ^^^ subtract the 7 from the re- maining 11 ; or, as in simple subtraction, 8 from 12, and we find the total excess to be 4J]. In practice it is most convenient to sub- tract 15 from 24, and add 8 ; thus 24— 15===9, and 9-1-8=17, and the answer is 4!]. EXERCISES. — C — S- "• Tj II — 11- ^ 1. .J - I 3" 1 n I ;i What is the difference between '^ and I']? What is the difference between ^i^ and -f'J'g What is the difference between g '^ and j\ ? 4. 1-41=7^ Ans. i? ? Ans. I Ans. /g Ans. 10[i5 . .' -1*. DENOMINATE FriACTIONo. to I 9 o •» ■> :: I On' Ans. 1. 0. From 5J] take 3^ ? Ans. 2..^",. 10. What ia the diffcrcnco between 5, Tj- and C^'.V^ ? Aus. .'.]'5,',. 1 1 . What u tlic valuo of 'i^ 5__^^ .7^^_^ y * " ~ ^Yns" 12. What i.s the difference between 100 /'y and 50,";' ? Ans, .19^j-. l.'l. What is the difference between ^ of J and -J of {r ? Ans. 11. What is the difference between -J of -,\j and (; of j^ ? Ads. 15. What is the value of ^-f ^— J— j-]- 1 ! ? VIII.-DENOMINATE FRACTIONS. Hitherto wc have treated of fractions abstractly, nnd we must now api'ly the principles laid down to donouiinato numbers, and ?liow liuw a fraction may bo transformed from one denomination to another of the suino kind, c. y., how a fraction of a shilling may be expressed as a fraction of a pound, and vice versa. RULE. ( 1.) Reduce the given quantitj/ to the lowest denomination lohieli, it expresses, i 2.) Reduce the unit in the terms of ichich it is to he ircprcsscit to the same denomination, and (3,) mn/ce the former the iiuni/rafor (iiid the latter the denominator, and the fraction will he tcpressed in the required terms, EXAMPLES. 1. To express 2 ft. 9 in. as a fraction of a yard. Rcduciiu; 2 ft. 9 in. to inches, wo get 33 inches, and one yard is 3G inches, — the fraction therefore is iji? or ' ,',. 2. In like manner to express 2 qrs., 24 lbs. as the decimal of a cwt. wc have 2 qrs., 21 lbs^=71 lbs., and 1 cwt., is 100 lbs., so that the fraction is jVo^^u- 3. So also 3 roods, 32 rods, expressed as a fraction of an aero is 10 8 "^" ill of an acre. 4. To express 17 cwt., 2 qrs., 10 lbs. as a fraction of a ton vrc \\nvp :"", n — :ifi — i« — !)_ 5. Exjyess 48 minutes, 48 seconds as a fraction of an hour. C. Express 13s. 4d. as a fraction of £1. 7. Express 36 rods as a fraction of a mile. 8. Express 43. 4d. as a fraction of £1. 9. Express 4Jd. as a traction of Is. 10. Express 1 oz. troy as a fraction of 1 lb. Ans. Ans. £j. ,1<'. !» 3 iO 80* Ans. £iij. Ans. j]s. Ans. ,'^. ^:^ .' •« / * AKITIDIETIC. 4 ft B6- •> 3 11. Express 40 lbs. as a fraction of 1 cwt. Ans. ? cwt. 12. Express 50 lbs. as a fraction of 1 ton. Ans. .,'y ton. 13. pjxprcss 72 lbs. as a fraction of 1 cwt. Ans. }. |J cwt. 14. A day is 2:j hours, 5G minutes, 43 seconds, nearly; what fraction of this will 7 hours be ? Ans. ,V'J'ii' 15. Express 95 Sfjuarc yards as a fraction of an acre. Ans. -J^*^. IG. E.xprcss 14 yards us a fraction of a mile. Ans. j,^^. 17. What fraction of a year (oG5]- day.s) is one month (30 days) ? Ans 18. Express 100 yards as a fraction of a mile, Ans. 19. Express 45 cents as a fraction of a dollar. Ans. 20. E.'jprcss GO lbs. as a fraction of a cwt. Ans 21. A man has an income of S3G10 a year, and saves ^ of it; how much does he spend ? Ans. S20G2I5. To find the value of a fraction in the denominations which the integer contains, reduce the numerator to the next lower denomina- tion, and divide the result by the denominator; if there be a re- mainder, reduce to the next denomination, and divide again, and continue the same operation till there is cither no remainder, or down to the lowest denomination by which the integer is counted. Thus, since a ton is 20 cwt., ^ of G tons is 120 tons divided by 7, which gives 17 cwt., with a remainder of 1, which, reduced to qrs., will give 4, in which 7 is not contained, and the 4 qrs. reduced to lbs., will give 100, and# this divided by 7 produces 14;j ; so that 5 of a ton is 17 cwt., qrs., 14| lbs. EXEllCISES. What is the value of ^\ cf a ton ? Ans. 11 cwt., 2 qrs., 1G§ lbs. AVhat is the value of j'^ of a yard ? Ans. 2 feet., 85 in. What is the value of 1| of a mile ? What is the value of \ ■• of a shilling Stg. ? Ans. 11 ?d. 5. What is the value of ;} of a ton ? Ans. 11 cwt., 1 qr., 17^ lbs. 6. What is the value of ^ lb. troy ? Ans. 8 oz. 1. o 3. 4. 7. What is the value of {\j of a shilling ? 8. What is the value of $«? 9. What is the value of I of $G ? 10. What is the value of H of 88 ? Ans. 5,Tjd. Am. 88g cts. Ans. 84.80. Ans. $G.80. To change a fraction to one of a lower denomination, reduce tho numerator to that dgnomination, and divide by the denominator. Thus : jj^ of a dollar is 700 cts. divided by 145, which gives 4^;]. ciniul pnint, (.) roprcsent intcc;pr!», ninl nil iiftor it frnctions, onoli hdw^ onc-tinfh of what "t would bo if one pliice further to tlio left. Therefore S88.88S is dijht hundrals, right tens, rujht vtiits, — cirf/it-fcntfis, viglit our-hii7i(hcdt!is, and rii/ht vnc-thnumndlhs ; or, ,''0 ' i oivr i fMur These added will j;ivi3 ,"ro-|-,:!.1o-!-,.;;M,' ">• ,^i;i,. whieh, n.r brevity, is written .888, and ni;iy b;i road ei;;ht hnndred iind <^i_L;ht y-ei^jht one-thousandths ; or, iis is usual, jvnnt 888, nr ihrlmil 888, but never {iroporly eight hundred and eiLrlity-eJL'Iit. In the same manner as (-(> means 8 tens and no units, so .08 means no tenths, but S hnndn Iths, and .008 means no tenths, n ) bnndredlhs, but ei,i;ht one-thousandths, tVc. — iront'O wo SCO that for every cipher in the denominator, which i.i always 10 of a power of 10, there nuist be a fissure in the numerator when expressed decimally. Thus : , (,"„,( must bo written decimally .008. From tins we see that removing the decimal point one place to the riij;ht is the same as multiplying by 10, and removin.g it ono place to the loft is t'.o .same as dividing by 10 ; so, also, removing the point two places to the right is tho same as multijilying by 100, and rcuiovir.g it two places to the left is the same aa dividing by 100. This is the principle alre;idy laid down for tho reduction of dollars to cents, and cents to dollars. I. — Reduction of Co.mmon Filvctions to Decimals. — Let it now bo required to express the common fraction ^ a.s a decimal. AVo have seen (Art. 14,) that wo may multiply tho terms of any fr.iction by the s:nnc nundicr without changing the value of the frac- tion. Let us then multiply the terma of jj by 1000, and wo get hllolr ^^" *'*° .same ju'inciplc wo can divide the terms by the same number without altering the value. Let us then divide by 8, and wo got 1%'^'^, where the denominator is a power of 10, and therefore the fraction is in tho decimal form, and may be written .025, the denominator being omitted. But as it is not always apparent by what power of 10 wo nuLst multiply, bo tliat when the terms are divi- ded by the given denominutor, that denominator may bo transformed into 10 or a power of 10, i, c, into 1 followed by a certain number of ciphers, we may as well add ciphers, one by one, as we prooccd. This is exhibited in the annexed exam* plo. From these principles we can deduce a rule for reducing a common fraction to a decimal. F)50(0.G25 48 20 16 40 40 DECIMAL niACTIOXa. 7J lt»>110(.0875 140 120 112 HO 80 n U L E . Diiudf. tJic unmrrafoi; vulh (iripfirr nr rlpJirrit (nin>.T((l, hi/ f/ir. dtnwiinutor. Thuf [,1 inlll rjivr, an ill thr iifinjin, .(i.S?.'). Ii! tlio oxiiinplcH ^Mvcn wo. liml tliat, tlic inMition ol" tlino ciplicrs to llio first, aiid four ti» (Ik; n'cnnil, iiinkcs tlip niiiiicratfir divisiliK^ liy IIk; (IciKiiiiiiiii- t(ir witlioiit nMiiiiiiidcr. SikIi IVactioti.s an: c .1- loil tcnuiriatinij; dcciuuilH. From this \v(! ico tliut tlicn' ail! cniiiiiion fractions wli(ts(> tiTiiis can bo iiiultiplii'd liy Midi |K)wcrs (if 10 an will iiiak(! th(> iiimior.-itur divi^ibli! l)y tlio drtioinitia- tor witlioiit rciiiaiii(K'r, l)iit it ol'tcii hapiictis lliat no power of tt.ii will oH'ci't this, and that niuaiti- dorn occur which cannot be niiidc divisible even- ly by the denominator, by tlio addition of any nutnbtf? of ciphorK. Sneh fracttlons will never teriijiiiate, and there- fore arc called interniinate, and IIk? eoniinon l'ra« tion can n<'ver bo expressed exactly in the decimal I'oriii, and all we can do is to niako an approximation more or less close, accordinj^ to the nuniljer is ecn- taincd ones in 10, with a remainder 1, — annexin;^ nnothcr ciplicr, wo apiln obtain 1 in the quotient, nnd this will obviously continue a terminating decimal, a pure circulating decimal, or a mixed circulating 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 clearly the form of illustration used in the other cases. Let us take the fraction .9375, and use d for decimal. Wo now write d=.9375, and multiplying both terms by 10000, we obtain 10000 d=9375, and therefore dr=-,y(TSg, ^hidi reduced to its low- est terms is -\l, the common fraction required. This is simply put- ting for (Icnomiuator 1, followed by a cipher for each figure in the decimal. To find the value of a pure circulator, suppose .6 J Put d— .(), or d— .666, and multiply by 10, which gives 10 d=6.0G, and writing the former expres- sion beneath, and subtBacting, we get 9 d=:6, and ccT'sequently d=§ or §, the common fraction d=.666-f 9d=6 w .zht. d=.V2 100 d=72.72 99 d=72 Let us now seek the vulgar fraction correspond- ing to .72. Put d=^.72, multiply by 100, and subtract as before, and there results a remain- mainder of 99 d=72, or d=p= . 8 -Tf d=.B68i 10000 d-::BG81.8i 100 d= 66.81 9900 d=5625 Again, to find the vulgar fraction cores- ponding to .5081. Multiply first by 10000, and then by 100, and subtract the latter from the former, and you obtain 9900 d-= 5625, and houce d-^^g'^J^, which reduced to its lowcK^ terms is j^. From these investigations the three following ruloa for the three cases mentionod are derived : I. If the fraction he a terminating decimal make it the numera- tor , and for denominator write 1, followed hy aa many cipliers as there arcfgurcs in the decimal, II. If the decimal he a pure circulator, make the digits of the decimal the num,erator, and for dinominator write a$ many nines (U there are/igurct in the period 84 ABITHMETIC. III. If the decimal he a mixed circulator, subtract the non-cir- culating 2>art from the whole decimal to the end of thefirtt period, both being treated as whole numbers ; make the remainder the nume- rator, and for denominator write as many nines as there are circula- ting figures, and after them a$ many ciphers as there are noi^rcu- lating figures. /» all cases reduce to the lowest terms. EXERCISES. AnS. ■g'y. Ans. ^. Ans. ^. Ans. I. Aqs. f . 1. Find the vulgar fraction corresponding to .04. 2. Find the vulgar fraction corresponding to .54. Ans. -Pj, 3. Find the vulgat fraction corresponding to .245^. 'Ans. ^jVo'u* 4. Find the vulgar fraction corresponding to .1. 5. Find the vulgar fraction corresponding to .3. 6. Find the vulgar fraction corresponding to .7. 7. Find the vulgar fraction corresponding to .75. 8. Find the vulgar fraction corresponding to .47543. Ans. ^1^1 9. Find the vulgar fi action coiTesponding to .4683544303797. Ans. ^. 10. Find the vulgar fraeticn corresponding to .4^. Ans. ^. 11. Find the vulgar fraotioa corresponding to .162. Ans. j^'^j. 12. Find the vulgar fraction corresponding to .14. Ans. ^J. 13. Find the vulgar fraction corresponding to .0138. Ans. ,'j. 14. Find the vulgar fraction corresponding to .56Si. Ans. ||. 15. Find the vulgar fraction oo*responding to .592. Ans. ^!^. The last rule may be deduced from the other two in the follow- ing manner : — Let us tak^ the mixed circulator .418, and thiu being multiplied by 10, the four becomes a whole number, and to preserve the same value, 10 is put as a divisor, which gives ^^\^ or ■^♦, j^*, but by rule II, we have .i§=J|, and hence the whole may bo writ- ten 4-1- ^5=-^^"^^^=^ J 2=11, and this result corresponds to rule III. 5. •- ,1. I ADDITION AND SUBTEACTION OP DECIMALS. 85' IV-ADDITION & SUBTRACTION OF DECIMALS. From what has been said, it is plain that decimals can be added and subtracted just as whole numbers, 'care being taken to keep tho decimal points in tho same vertical line. In all operations into which rcpetcnds enter, it slionld be observed that in order to have a result true to any given number of places, it is generally desirable to carry out the rcpctcnd to one or two places more than the required number. It is often sufficient, however, to allow for what would bo carried, which can usually be done by inspection. In all cases, res- pect should be had to the degree of exactness whioh the nature of the calculation requires. The figures beyond those required can bo estimated and added in. Thus, if only five places are required, and the calculation bo carried to six places, and tho seventh figure is a large one, it should be added to the sixth figure. This may bo stated in tho form of a , RULE. Add and suhtract cw-iu wlioh numbers, keeping the decimal points in the same vertical line. (1) 1.78645 3.978G3 7.84390 4.32782 9.54179 ll.G98fi7 5.48491 44.GG213 EXERCISES. (2.) (3.) 8.58333333+ 51.250000000 17.74747474+ 3.444444.144^ 112.08080808+ 7.637373737+ G.12500000 .885555555+ 15.GGGGGGG7 11.87500U00O .rG9G9G97 7.875875875+ 11.00000000 • 7.111111111+ 171.1*7297979 90.0793G0724 In exercise 2, the ci-.'hth figure of each of the fifth and sixth lines is made 7 instead of G, which renders it unnecessary to make any allowance for the rcpetends that would follow, but this change k not mado on any of the lust figures of exercise 3, and therefore wc add 2 for what would bo carried from tho tenth decimal place to tlio ninth. 4. Find in the decimal form the sum of J, §, J, ^. Ans. 2.31G. 5. Find in tho decimal form the sum of *, |, J, }§, ^i, gjf, |H2- Aus. G.0078126. V"«. ^* .■y 86 AEITHMETICJ. % G. Find ia the decimal form the sum of |§, |, /j, {|. Ads. 2.345. 7. Find ia the decimal form the sum of 25, 4^, D/^. Aus. 12.775. 8. What is the sum of .7S6425, .975324, .176009, .32, .C2519375, .4? Ans. 3.28295175. 9. Add to G places 18.1276, 11.349, 12.145, 8.G48, 15.23. Ans. G5.504414. 10. Find to 6 places the sum of 15.7, 12.4, 18.387, .416, .74687, .9, .45, 10.45, .12345. • Ans. 59.351152. 11. What is the sum of .76, .416, .45, .648, .23 to five places of decimals ? Ans. 2.52087. 12. Ileduco to decimals, and find the sum of ^, j'j, £.i, H- Ans. 1.410. 13. Fiud the sum of .427, .416, 1.328, 3.029, 5.476 to ^ix places of decimals." " Ans. 10.678037. 14. Required the sum of 1.25, 1.4, 1.6:^7, 1.885, 1.684, 1.937, 1.148 and 1.764085. Ans. 12.750458. 15. Find the sum of .40321, .81532, .154926, .7532 to true to four places. Ans. 2.18G7. 16. From 3.4G8 substract 1.2591, and you find the excess 2.2089. 17. What is the excess of 10.008576 above 5.789 ? Ans. 4.219576. 18. From 11.4 take 1.48, and there remains to six places 9.959596. 19. What is the excess of 7.8 above 1.3754658 ? Ans. G.424534i. 20. What is the difference between 9.46574, and 4.18345 ? Ans. 5.28229. 21. Express, decimally, the difference between f+i+s+o+i* andi+H'ai- . Ans. 2.34613+. 22. What is the difference, nccordinj; to the decimal notation, between ^ and J,l true to six places of decimals? Ans. .636363. 5.' fc-'- ^ MXJLTIPLICATION OP DECIMALS. 87 23. "What is tae difference between ^ and ^ expressed decimally true to six decimal places ? Ans. .071428. 24. What is the difference between the vulgar fractions corros- ponding to .49 and .5 ? Ans. 0. 25. Find the value of .786425+.975324-I-.17G009-1-.32+ .625193 75~3.28295175+.4. Ans. 0. 26. What is tho difference between 138.6012, and 128.8512 ? Ans. 9.75 27. What is iho excess of 31. 6322 above 5.G74-f-1.83+.S125-|- 18.62+4.3+.395— .5. Ans. 1.0007. 28. What is the excess, expressed decimally, of 5.83 above 4J^. Ans. 1.6582. 29. 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— .00625— 3.818475— 156.1+.125. Ans. .25. X ^^ » <« V.-MULTIPUCATION OF DECIMALS. If wo multiply a decimal by a whole number, the process is pre- cisely the same as if the multiplicand were a whole n amber, but care must bo 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 wo have to multiply a whole number by a decimal, wo must mark off a deci- mal in the product for each decimal in the multiplier. — The reason of this will be manifest from the considcru' tion that if we multiply 8 units by .6, or -,"„, we get *{], or 4.8, i, c, 4 units and 8-tenths ; and again, when wo multiply 7 tens by .6 or {\^, we get -',-„" -^42 units, which with the 4 units already obtained, make 4G units, and wo now have arrived at whole numbers. The same illustration will apply tu multiplying by .06, whieli reijuires two de- cimal places to bo laid off from the right. Therefore, for every de- cimal place in the multiplier one must'bc cut off in the product, and wo saw already that for every decimal place in the multiplicand, a dcci- 5.d78 6 34.068 5678 .6 3406.'8 88 ABITHaCETIC. mal place mnsi 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 illustration by observing that as the tcnih of a tenth is a one^hundrcdtb, tenths multiplied by tenths give hundredths ; so also the product of tenths and hundredths is thousandths, and so on. Thus : .2 or ,'5, multi- plied by .3 or Jg, is ygg. Now, .G would not represent this, for that would mean /^ ; 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 are 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^''^^ would be written decimally .006, which would mean that there are no tenths, no hundredtlis, but 6 thousandths. From these ezolanations we deduce the rule; Multiply, as in tohole numhers, and cut off from the right a dec^ mal place /or every one in both multiplier and multiplicand. Multiply were whole (1.) .78 .42 156 312 .3276 (2.) .674 34.6 4044 2696 2022 23.3204 EXAMPLES. .78 by .42. Here we multiply as if the quantities numbers, and in the product point off a decimal figure for each one in both multiplier and multiplicand. In Ex: 1, the number of figures 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 are six figures in the product, and consequently two figures represent whole num- bers. In Ex. 3, when wo multiply 6 by 3, we obtain 18, but if wo had (3.) 4.66 earned the n pitend out one place far- ther wo should have had 5 to be mul- tiplied by 3, and consequently 1 to carry, so we^add 1 to the 18,' and in like manner wo must allow 2 when multiplying by 4, and 1 when multi- plying by 2. 4.56 2.43 1369 1826 913 11.0929 Ai A^ MULTIPLICATION OF DECIMALS. 89 EXERCISES. 1. Multiply 7.49 by 63.1. Ans. 472.619; 2. Multiply .166 by .143. Ans. .022308. 3. Multiply 1.05 by 1.05, and the prodnct by 1.05. Ans. 1.157625. 4. Find the continual prodnct of .2, .2, .2, .2, .2, .2. Ans. .000004. 5. Multiply .0021 by 21. Ans. .0441. 6. Multiply 3.18 by 41.7. i^s. 132.006. 7. Multiply .08 by .036. Ans. .00288. 8. Multiply .13 by .7. Ans. .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, .11, 1.1 and 11. Ans. .001478741. 15. Multiply 7.43 by .862 to six places of decimals. Ans. .640839. 16. 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. .0099093. Contracted Method. — In many instances where long lines of figures are to be multiplied together, the opdV-ation may bo very much shortened, and yet sufficient accuracy attained. We may instance what the student will meet with hereafter, calculations in compound interest and annuities, involving sometimes most tedious operations. By the following method the results in such cases may bo obtained with great ease, and correct to a very minute fraction. If we are computing dollars and ccnts^ and extend our caloulation to four ■^ "^ .r.« -' . f 90 AIUTHMETIC. places of decimals, we are treating of the one-hundredth part of a cent, or the ten-thousandth part of a dollar, a quantity so minute as to become relatively valueless. Ilenec we conclude that three or four decimal places are sufl&cient for all ordinary purposes. There are cases, indeed, in which it is necessary to carry out the decimals farther, as, for instance, in the case of Logarithms to be considered hereafter. The principle of the contracted method will be best ex- plained by comparing the two subjoined operations on the same quantities. Let it bo required to find the product of 6.35G42 and 47.G453, true to four places of decimals : EXTENDED OPERATION. G.35G42 47.G453 CONTRACTED OPERATION. 6.35642 3546.74 19 317 2542 38138 444949 2542568 06926 8210 568 52 4 302.8535 37826 2542568 44494» 38138 2542 317 19 2 carried. 302.8535 RULE FOR THE CONTRACTED METHOD. Place the imits' Jlgure of the whole numher under the last required decimal place of the multiplicand, and the other integral Jigures to the right of that in an inverted order, and the decimal fgitrcs, also in an inverted order, to the left of the integral unit ; multiply hy each figure of the inverted multiplier, beginning tcith the figure of the multiplicand, immediately above it, omitting all figures to the right, but allowing for what icould have been carried if the decimal had. been carried out oiie place farther — place the first figure of each partial product in thescime vertical column, and the others in verti- cal columns to the left ; the sum of these columns will be the required product. Thus, in the above example, we are required to find the product correct to four decimal places, therefore we set the units' fijiure, 7, under the fourth decimal figure, and the tens' figure, 4, to the right, and the decimal figures, 6453, to the left in reversed order ; theu we digits. '-■*? n YULTIPLIOATION OF DECI3LVLS. 91 multiply the whole lino by 4, and then we multiply by 7, omitting the 2 which stands to the right, but allowing 1 for what would have been carried, that is, we say 7 times 4 ia 28, and 1 is 29, and wo write the nine under the 8, the first figure of the first partial product. By comparing the contracted method with the figures of the extend- ed form, wliich are to the left of the vertical line drawn after the fourth decimal figures, it will be seen that the figures of each column are the same but .placed in reversed order, which makes no difTerenco in the sum, as 5-|-3=:3+5=^8. This is the same principle as the contracted method of multiplying by 17, 71, &c., suggested in tho article on simple multiplication '■'■• The object of writing the multiplier ia a reversed order is simply to make the work come in the Uotxal form, as otherwise we should bo crossing and recrossing, so to speak,^ as will be seen by the operation in the margin. — Beginning with the left hand figure of the multiplier, and the right hand figure of tho multiplicand, we find the first partial pro- duct ; then taking the second figure of tho multiplier from the left, (7) and the second figure of the multiplicand from the right, we get the second partial product, and so on, moving one place each time towards tho right in tho multiplier, and one place to- wards tho left in tho multiplicand. This is so dificrent from the ordinary mode of ope- ration, as to be excessively awkward and puzzling, and this gave rise to the idea of reversing the order of tho digits. We append this remark as most persons caimot at first sight comprehend the reason of the inversion. * Let the learner observe tlmt till the flgtnvs of the fust column arc of tlio same rank, viz., ten-thousandths, and therefore may be added together, anil as tho value of each figure i? increased or decreased 10 times according to its position to left or right, it follows that all figures at eriuul distances from the decimal point, whether to right or left, are of the same rank, i. c, units will be under units, tens under tens, tenths under tenths, hundredths under hundredths, &e., &c. Tho contracted method is not of much use in termina- ting decimals which extend to only a lew places, but it saves a Ta.st deal of labour in questions which involve either repetenda or terminating deciiuaLs txpressed by a long line of decimal figures 6.35642 47.6453 2542568 444949 38138 2542 317 19 2 allowed. 302.8535 •» IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 144 IM 112 .^ m 1^ |Z2 M 1.8 1-4 ill 1.6 Photographic Sciences Corporation ,\ d V 2b \ \ ^. \ % V '<^ 9) 33 WEST MAIN STREET WEBSTER, NY. MS80 (716) 873-4503 <>'t^: ,y "# Q< Q- i/j. 92 ARITHMETIC. ADDITIONAL EXERCISES: 21. Multiply .26736 by .28758 to four decimal piaccs Ans. .0769. 22. Multiply 7.285714 by 36.74405 to five decimal places. Ans. 267.70665. 23. Multiply 2.056419 by 1.723 to six decimal places. Ans. 4.578932. 24. What decimal fraction, true tc six places, •will express the product of -pf multiplied by ^^^ ? Ans. .113445. 25» What decimal fraction is equivalent to IfXHi ? Ans. .40748. 26. What is the second power of .841 ? Ans. .707281. 27. What is the product of l.o5 by 1.48, true to five places ? Ans. 2.45975. 28. Express decimally 2,33X1. . Ans. 2.393162. 29. What is the product of 73.637i by 8.143 ? Ans. 599.6272077. 30. .681472 X -01286, true to five places, will give .00876. In the last exercise it must be observed that sinco 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 ia a 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 be only one-mil- lionth Ti)»rt of the unit. .681472 68210.0 681 136 55 I .00876 VI.-DIVISION OF DECIMALS. Wo 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 npply equally to decimals, the only additional requirement being the placing and moving of the decimal point. ■^ DIVISION OP DECIMALS. 93 Suppose tncn we are required to divide 1.2321 by 11.1, we must y (1) briug both quantities to the same denomination. Now the dividend is carried down to ten-thousands for 1.2321=:l+/jf„y5, and therefore we express 11.1 in the corresponding form, ten-thou- sandths or ll+iood"o> "^^ 11.1000, so that wc change the form, but not the valus of 11.1, the divisor. Again, by (2) the .1, which originally expressed a tenth of some unit, and therefore was in reality denominate, now becomes abstract as one of the figures of the given factor of 1.2321, by means of which we are to find the other factor. Hence by (3) we can now divide 1.2321 by 11.1000, qs if both wero whole numbers, and this is the reason for omitting the decimal point when wc have made the number of decimal places equal. Beginners generally feel a difficulty in conceiving how a fraction divided by a fraction can give a whole number. The difficulty may be easily re- moved by noticing that ^ Is contained ticice in ^ for 5=;^, c. g., a half dollar contains, or is equivalent to, two quarter dollars. Thus the fraction J divided by the fraction J, gives the whole number 2. So, also, J is contained 4 times in J, and therefore ^-^-^=4, a whole number. Hence, when we have reduced the divi«or and dividend to the same denomination, we may omit the decimal point, as wc have only to find how often the one is contained in th§ other. Hence the AULE. ^ the number of decimal places in the divisor and dividend he not equal, make them equal hi/ supplying ciphers or repctends, and then divide as in whole numhtrs, and the* quotient to far will he a whole number, but if there is a remainder, annex ciphers or rrpetends, and the part of the quotient thus obtained will be a decimal. The decimal places may be supplied as the work proceeds, as it is easy to see liow many ciphers or repctends must be supplied ; for wo have seen in mtiltiplication that the number of decimal places in any product must be equal to all the decimal places in the factors, and, since a dividend must always be viewed as a product, it follows that the difference between the number of decimal places in dividend and divisor will indicate how many ciphers or rcpotends mtwt i»e Bvpplied. £XEROI«E£. 1. Divide 47.58 by 26.176 to six decimal places. 2. Divide .70.8940 by 13.825 to three places. 4ns. 1.817705. Am. {^.128. ''^^r 9t APJrHMETIC. 3. Divide 4G8.7 by 3,3G5 to six places of decimals. Ans. 139.309889, 4. Express decimally l-j-,--]^. Ans. 233.3. 5. Express in tho decimal form ^ of f -i-| of § true to six places of decimals. Ans. 1.054687. G. Divide the whole number 9 by the fraction .008. Ans. 1125. 7. What ia the quotient of 5.09 by 6.2 ? Ans. .81 nearly. 8. Divide .54439 by 7777. Ans. .€0007. 9. What decimal is obtained by dividing 1 by 10,473654 ? ^ Ans. .09547766. 10. What ia the difference between f-T-| and g-H|J in the deci- mal form? CONTRACTED METHOD. The work may often be nrach abbreviated in tho manner exhibi- ted by the following example : .14736).23748 (1.611 14736 9012 8841 170 147 23 14 14736)23748(1.611 • • • • 14736 0. 6 40 36 040 736 9012 8842 170 147 8 304 23 15 Hero it is required to divide .23748 by .14736. Since both divisor and dividend contain the same number of decimal plaocs, no alteration is needed, and ao we can at once reject tho decimal po>at, and divide as in whole numbers. The principle of the contract'on 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 appvo.\imation sufficient for all practical purposes. XTor this reason, when we liave obtained the integral part of tho quotient, we may omit one figure of the divisor in succecffiion after each opcm/* tion, as the value of each figure decreases in a tenfold degree as 'we descend towards the right, and after three decimal figurea the error, DIVISION OF DECDLVLS. 95 or deficit rather, becomes only tliousandths, wliich are very rarely worth taking into account. For example, if the calculation regards dollars and cents, the error at the fourth decimal place would be only the cme-thousandth part of a cent. RULE. Arrange the fractions as in the ordinary mode ; find the first figure of the quotient and the first remainder; then, instead of annexing a periodic figure or a cipher, cut off the right hand figure of the divisor, and use the remaining figures to find the next figure vf the quotient, and so on. It is usual to mark the figures as they are successively cut off by placing a point below each. In multiplying by each figure of tho quotient, allowance, must be made for what would have been carried from the figure of the divisor last cut off, had it been used in the division. Tho vertical line drawn through the -ordinary form shows hov closely the two modes correspond. As has already been rcmar!:cd, it is desirable, in order to secure accuracy, to carry the figures of repetends to one or two places more than arc required. EXERCISES. ! (1.) 43232323)73040000(170.3355. 43232323 30407677 302G2626 145051 129697 (2.) 54637 43682 ('.7995 • • • • 38246 15354 12970 2384 2162 343G 4917 519 491 28 27 M •*■'■ 0^ 222 216 1 6 Divide 73.64 by .432, and .43682 by .54637 to 4 decimal placos each. To ehow that thcr« will be three integral places iu the 96 ARITHMETIC. quotient of Ex. 1, we must consider that there arc two places of whole numbers in the dividend and none in the divisor, and, therefore, if we divide 73 and 6, the first decimal place of the dividend by .4, the first figure of the divisor, we get three integral places. Hence, since we are to have four decimal places, we shall have seven figures in all. This contraction is extremely useful when there arc many decimal places. 3. Find the quotient of 8.613'4-*-7.3524 to four decimal places. Ans. 1.1715. 4. Divide .61 by 13.543516 to five decimal places. Ans. .04549. 5. Divide .58 by 77.482 to five decimal places. " Ans. .00756. 6. Divide .812.54567 by 7.34 to three decimal places. Ans. 110.649. 7. Divide 1 by 10.473654 to six decimal places. Ana. .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.-DENOMINATE DECIMALS. To express one denominate number as a fraction of another of the same kind, reduce both to the lowest denomination contained in either, make thcfoiincr the numerator and the latter the dcnomi' nator of (i common fraction, and reduce the fraction so found to a decimal in the manner already jtointed out. XXAMPLES. To express 16 cents as a fraction of a dollar : Hero the lowest denomination mentioned is cento, and we reduce a dollar to cents and write ,'o'(5=i*B» '"^*^> . .81875. DEDUCTION OF DENOMINATIONS. C7 5. Express li cwt., 1 qr., 7 lbs., as a decimal of a ton. Ans. . 166. G. Reduce 37 rods to the decimal of a mile, Ans. .115625. 7. Reduce 7 ozs'. 4 dwts., to the decimal of a pound. Ans. .6. 8. Reduce a pound troy to the decimal 6f a pound avoirdupois ; correct to six decimal places-* Ans. .822857-}-. 9. Reduce 5 hours, 48 minutes, 49.7 seconds, to the decimal of a day, taken as 24 hours. Ans. .2422419. 10. Express an ounce avoirdupois as a decimal of a pound troy. Ans. .9114583. viii:-r5;duction to denominations. To find the value of a fraction in the lower denominations ^ expressed as a decimal of any given denomination, multiply in suc- cession hyfhe numbers tchich express the given and lower denomina- tions, and after each multiplication cut off from the right as many decimal figures as are contained in the given decimal, and the figures to the left of the decimal point will give the required value. EXAMPLES. I. To find the value of .64379 of a pound (apothecary). "We 12 multiply by 12, by 8, by 3 7.72548 8 5.80384 3 2.41152 • 20 and by 20, which gives 7 ozs., 5 drs., 2 scrs., and a little over 8 grs. Repetends must bo reduced to common frac- tions, or found approximately. 8.23040 -' To find the value of .7 : of a day, which is 18 hours, S9 min. and nearly 59 J sees. .77777 77 ) 24 j carry 1. 311109 155555 18.66659 60 39.99540 60 59.72400 *The standard pounds are meant here, viz. : troy, 5760 grains, aud avoirdupois 7(K)0 graioa. Taking tlie ounces would give ^=^=.75 ■";■, f r' f. -M 98 AEITHMETIO. Ans. 9s. Gd. EXERCISES. 1. What is the vaiue of £.475 ? 2. What is the value of .7 of a cwt. ? Ans. 3 qrs., 3 lbs., 1 oz., 12| drs. 3. What is the value of .541 G of a shilling sterling ? Ans. 6|d. 4. What is the value of .6845 of s cwt.? Ans. 2 qrs., 20 lbs., 10 oz., 9l||. 5. What is the value of .4 of Os. 4|d ? We liavo .4=^ and 9s. 4|d., multiplied by 4, and the product, divided bjr 9, gives 4s. 2d., the exact value. 6. What is the value of .026 of 1° 15' ? Keducing .026 to a vulgar fraction, we get ^2jia=7*5> and multiplying 1° 1^' by 2, and dividing by 75, we find 2'. RATIO AND PROPOR^nON. 17. — Ratio is the relation which one quantity bears to another of the same kind with respect to magnitude, or the number of times that the less is contained in the greater. TJius, tlie ratio 7 to 21 is 3, because 7 is contained 3 times in 21, or 21 is 3 times 7. The Bame result is obtained if we divide 7 by 21, for we then find 2^=^^, which means that 7 is J of 21, and this expresses the very same relation as before ; for, to say that 7 is J of 21 is precisely the same as to say that 21 is 3 times 7. (See note under Inverse Pro- portion.) And, therefore, 3 is called the measure of the ratio. The numbers thus compared are called the terms of the ratio — the first the antecedent and the second the consequent, and the relation is •written 7 : 21. The sign ( : ) originally indicated division. That the magnitudes must be of the same kind will bo obvious from the consideration tliat 7 bags of flour could have no ratio to 21 dollars, for multiplying 7 bags of flour by 3 would not make them 21 dollars, but 21 bags of flour, and multiplying 7 dollars by 3 would not make them 21 bags of flour, but 21 dollars. Hence, the less could not be increased to make the greater, except they are homogeneous, or of the same kind. Proportion is the equality of ratios. The ratio of 9 to 27 is 3, but wo have seen that the ratio of 7 to 21 ifi also 3, therefore the iitioa of 7 to 21 and of 9 to 27 are the BATio Airo phopoetion. 99 same, or 7-j-21=9-i-27, and these quantities arc, therefore, called proportionals. The sign ( : : ) was formerly used for equality, and is still retained for equality of ratios, and the sign (=) is used for the actual equality of quantities, though occasionally used for equality of ratios. Hence, the usual mode of writing the equality of two ratios is 7 : 21 : : 9 : 27. Such a statement is called a pro- portion, or an analogy, and is read — 7 is to 21 as 9 to 27, i. e., 27 exceeds 9 as many times as 21 exceeds 7, and this is expressed by saying 27 is the same multiple of 9 that 21 is of 7, or that 9 is the same sub-multiple, measure, or aliquot part of 27 that 7 is of 21. The four quantities are called the terms of the proportion ; the first and last are called the extremes, and second and third the means ; also, the first and third are called homologous, or of the same name, l. €., both are antecedents, and so the second and fourth are homo- logous, for they are both consequents. The last term is called a fourth proportional to the other three, and we shall denote it by F. P. There are two simple ways of testing the correctness of an analogy. The first is to divide the second term by the first, and the fourth by the third, and if the quotients are equal, the analogy is correct. This is manifest from what has been already said. The second principle is, that, if the analogy be correct, the product of the extremes is equal to the product of the means. To prove this, let us resume the analogy, 7 : 21 : : 9 : 27. W« have seen that 21-^7=27-^9, or 3=3. Now, if each be multiplied by 63, wo have (by Ax. II., Cor.,) 189=189. But 189 is the product of 27 by 7, the extremes, and also of 21 by 9, the means — these products then are always equal. From this simple principle we readily deduce a rule for find- ing a fourth proportional to three given quantities. Let the quan- tities be 48, 96, and 132, written thus : 48 : 96 : : 132 : , the required quantity. Now, 132x96=12672, the product of the means are therefore equal to the product of the extremes. We have, tlierefore, 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 wo can now write the whole analogy, thus : — 48 : 96 : : 132 : 264. To prove the correctness of the operatipn, multiply 264 by 48, and 12672 is obtained, the same as be^jrc. Heaoe, ■^^ , M. ■**^-^ im ■, ■ Mi 100 AMTflilETlC* THE RULE. Divide the product of the second and thud te'rmahy ihejlrst, and tlte quotient will he the required fourth term. To show the order in which the three given quantities are to be arranged, let it be required to find bow much 730 yarda of linen will cost at the rate of $30 for 50 yards. It is pit in that the answer, or fourth term, must be dollars, for it is a price that is required, and in order that the third term may hare a ratio to the fourth, the $30 must be the third term. Again, since 730 yds. will cost morethan 50 yds,, the fourth term will be greater than the third, and therefore the second must be greater than the first, and therefore the statement is yds. yrti. t ^ 50 : 730: :30 : 4th proportional, and by the rule 5Jioxjo^.^).rHV) =438, the fourth term, and we can now write the whole analoizy, 50 yds: 730 yds:: $30: $438. This may be called the ascending scale, for the second is greater than the first, and the fourih greater than the third. If the ques- tion had been to find what 50 yards of linen will cost at the rate of 8438 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 sec that the answer will now be less than 438, as 50 yards, of which the price is required, will cost much less thiin 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 A-3,.«3XJi^=30, the fourth proportional. We now have the full analogy 730 yds . 50 yds : : $438 : $30. As the second is less than the first, and the fourth less than the third, this may be called the descending scale. If the first should turn out to be equal to the second, and therefore the third equal to the fourth, we should say that the quantities were to each other in the ratio of equalitj/. RULE FOR THE ORDER OF THE TERMS. If the question implies that the consequent of the second ratio must he greater tJuin the antecedent, make the greater term of tk first ratio the conseqitent, 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, thv less will bo the corresponding number required, and vice vcn^j r... IIATIO AND rnOPOETION. 10' whereas, in the latter, the greater the number given, the less will ])o the number required, and vice versa. To illustrate this, lot it be required to find how long a stack of hay will feed 12 horses, if it will feed 9 horses for 20 weeks. Here the answer required is lime, and therefore 20 weeks will 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, 12:0; and hence the name Inverse, 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 term 12, for which the time is required, and which therefore we should expect to be in the second place, has to be put in the first, and thus the whole anii- logyis 12: 9::20 : 15.* The principal changes that may be made in the order of tho terms, will be more readily and clearly understood by the subjoined scheme, than by any explanation in words : G; G : 12 : 9 for 8x9=72=6x12. 9for8x9=72=Gxl2. : 12 for GX 12=72=8X9. : 6::12+9: 9' or 14: G::21 9 for 12—9 : 9 or 2 : 6 : : 3 : 9 for 2X9= 12 : 12—9 or 8 : 2 : : 12 : 5 for Let us take Original Analogy : 8 Alternately: 8 : 12: By Inversion : 6 : 8 : : 9 By Composition : 8-J-6 14X9=126=6X21. By Division : 8 — 6 : 6 : : 18=6X3. By Conversion : 8 : 8 — 6 8 X 3=24=2 X^'". Simple transposition is often of tho greatest use an "risy practical example. In calcula- ting wiiat power will balance a given weight, Wi:cn 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 aa the arms. This solvos all the four possible cases by transposition. * Inverse ratio is sometimes spolicn of, but in reality there is no sucl thing. It is true that Invcrs^i Proportion requires the terms of one of tht ratios to be inverted, but that is a matter of analogy, not of ratio, for we havt seen already that 7-<-21 expresses the verv same relation as 21-i-7.— (See in- #,. 102 AKITHMETIC. A : B : : W : P, gives the power wlicn the others are known, B : A : : P : W gives the weight when the others are known, W : P : : A : B gives the ann of weight when tlie others are known, P : W : : B : A gives the arm of power when the others are known. The work may often be contracted in the following manner : — Resuming our example, 48 : 9G: : 132 : fourth proportional, we see that 90 is double of 48, and therefore the ratio of 48 to 90 is the same as that of any two numbers, the second of which is double the first, and 48 : 90 is the same as 1 : 2, and we reduce the analogy to the simple form of 1 : 2 : : 132 : 4th prop., and we have J-^ j^^=264, the term required, as before. In the example 50 : 730 : : 30 : 4th term, wc have 7.ao)<_3n^iL3X3o=l3_x/LXii=^73xG^438. This is equivalent to dividing the first and second by 10, and the first and third by 5. Hence we may divide the first and second, or first and third by any number that will measure both. Thd 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 neutralize the other. Thus r 48 : 96 : : 132 : F. P. may be written 1X48 : 2X48:4 132 : 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 : 3G, it is usual to write the analogy thus — 4 : 12 : 30, 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 propcjrtional between two given quantities, we have the following RULE Multiply them together, and take the square root of the produa. Thus, in the above exampla, 4X36=144, the square root of which is 12. Again, to find a ii,3an proportional between 9 and 49, we mul- troductory remarks.) The term Reciprocal Ratio is liable to Uie same objec- tion, for though 3 and J are reciprocals, yet they express the same relation. When the expression Inverse Ratio is legitimately used, it does not refer to a single ratio, ))ut mcaos th^it two ratios are so related that one of them must bu inverted. EATIO AND PROPORTION. 103 tiply 49 by 9, which is 441, the square root of which is 21, whicli is a mean proportional between 9 and 49, i. e., 9 : 21 : 49, or, writ- ten at full length, 9 : 21 : : 21 : 49. Proof: 49X9=441 and 21X21=^441. As the learner is not supposed, at this stage, to know the method of finding the roots of quantities beyond the limits of the multiplication table, we append a table of squares and roots at the end of the book. When each quantity in a series is a mean proportional between two adjacent quantities, the quantities are said to bo continued, or continual proportionals. Thus : 2 : 4 : 8 : 16 : 32 : G4 : 128, and 3 : 9 : 27 : 81 : 243, are series in which cacli is a mean pro- jwrtional between two adjacent ones. Let us take IG and the two adjacent ones, 8 and 32 — the analogy is 8 : IG: : 16 : 32. P oof: 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, 9X31= 729, and 27x27=729. This subject will be treated of at length in a subsequent part of the work, but this explanation has been introduced lu^ye to fill up the outline and let the learner understand the nature of continued proportionais. EXERCISES. 1. If 6 barrels of flour cost $32, what will 75 barrels cost ? Ans. $400. 2. If 18 yards of cloth cost $21, what must be paid for 12 yards? Ans. $14. 3. How much must be paid for 15 tons of coal, if 2 tons can be iiurchased for $15 ? Ans. $112.50. 4. If you can walk 84 miles in 28 hours, how many minutes will you require to walk 1 mile ? Ans. 20. 5. What will 14 horses cost, if 3 of the average value can be bought for $270? Ans. $1260. 6. What must be paid for a certain piece of cloth, if | of it cost 89. Ans. $13.50. 7. If 5 men are required to build a wall in 5 days, how many men will do the same in 2^ days ? , Ans. 10. 8. If 16 sheep are f of a flock, how many are there in the same ? Ans. 24. 9. What must be paid for 4^ cords of wood, if the cost of 3 cords is $10 7 Ans. $15. 104 AIUTHMETIC. 10. Whau is the height ot a tree wlxich casts a shadow of 125 feet, if a stake 6 feet high prc.uuoes a shadow of 8 feet ? Ans. 93|. 11. How long will it take a train to run from Syracuse to Os- wego (a distance of 40 miles), at the rate of 5 miles in 15 /g minutes ? 12. If 15 men can build a bridge in 10 days, how many men will be required to erect three of the same dimensions in ^ the time ? Ans. 90. 13. If a man receive $4.50 for 3 days' work, how many days ought he to remain in his place for $25 ? 14. How much may a person spend in 94 days, if ho wishes to save $73.50 dut of a salary of S500 per annum ? 15. If 3 cwt., 3 qrs., 14 lbs. of sugar cost $36,50, what will 2 qrs., 2 Iba. cost ? ' Ans. $4,879+. 16. 5 men are employed to do a piece of work in 5 days, but after working 4 days they find it impossible to complete the job in less than 3 days moje, how many additional men must be employed to do the work in the time agreed upon at first ? Ans. 10. 17. A watch is 10 minutes too fast at 12 o'clock (noon) on Mon- day, and it gains 3 minutes 10 seconds a day, what will be the time by the watch at a quarter past ^0 o'clock, A. M., on the following Saturday ? . Ans. 10 h. 40 m. SG/g s. 18. A bankrupt owes $972, and his property, amounting to $007.50, is distributed among his creditors ; what does one receive whose demand is $11.33J ? Ans. $7.083-f-. 19. What is the value of .15 of a hhd. of lime, at $2.39 per hhd. ? ' Ans. $.3585. 20. A garrison of 1200 men has provisions for | of a year, at the rate of | of a pound per day ; how long will the provisions last at the same allowance if the garrison be reinforced by 400 men ? Ans. G^ months. 21. If a piece of land 40 rods in length and 4 in breadth make an acre, how long must it be when it is 5 rods 5^ feet wide ? Ans. 30 rods. 22. A borrowed of B $745, for 90 days, and afterwards would return the favor by lending B $1341 ; for how long should he lend it? 23. If a man can walk 300 miles in 6 suooessive days, how many miles has he to walk at the end of 6 days ? Ans. 50. BATIO AND PROPORTION. 105 0. 24. If 495 gallons of wine cost $394 ; how much will $72 pay for? * • Ans 90 gal. 25. If 112 head of cattle consume a certai* quantity of hay in 9 days ; how long will the same quantity last 84 head? Ans. 12 days. 26. If 171 men can build a house in 168 days ; in what time will 108 men build a similar house ? Ans. 266 dajs. 27. It has been proved that the diameter of every circle is to the circumference as 113: 355; what then is the circumference of the moon's orbit, the diameter being, in round numbers, 480,000 miles ? Ans. 1,507,964/vSj m. 28. A round table is 12 ft. in circumference ; what is its diameter ? Ans. 3 ft. 9^1 in. J 29. A was sent with a wan-ant ; after he had ridden 05 miles, B was sent after him to stop the execution, and for every 10 miles that A rode, B rode 21 ; How fiw had each ridden when B Overtook A? Ans. 273 miles. 30. Find a fourth proportional to 9, 19 and 99. Ans. 209. 31. A detective chased a culprit for 200 miles, travelling at the rate of 8 miles an hour, but the culprit had a start of 75 miles ; at what rate did the latter travel ? Ans. 5 miles an hour. 32. IIcw much rum may be bought for S119.50, if 111 gallons cost 189.025 ? Abs. 148 gallon?. 33. If 110 yards of cloth cost $18 ; what will $63 pay for ? Ans. 385 yards. 34. If a man walk from Rochester to Auburn, a distance o\ (say) 79 miles in 27 hours, 54 minutes ; in what time will he waii; at the same rate from Syracuse to Albany, supposing the distance to be 152 miles ? 35. A butcher used a false weight It J oz., instead of 16 oz. for a pound, of how many lbs. did he defraud a customer who bought •112 just lbs. from him ? Ans. 9?f lbs. 36. If 123 yards of muslin cost $205 ; how much will 51 yards cost? Ans. $85. 37. In a copy of Milton's Paradise Lost, containing 304 pr"»fs, the <;ombat of Michael and Satan comra&nr>es at the 139th page ; at what page may it be expected to commence in a copy containing 328 .pages? Ans. The fourth proportional is 149^|; and hence the passage will commence at the>foot of page 150 38. Suppose a man, by travelling 10 hours a day, performs s ««.• 106 ARITTrvrF.TIC. journey in four weeks without desecrating the Sabbath ; now many weeks would it take him tc perform the same journey, provided he travels only 8 hours per^day, and pays no regard to the Sabbath ? Ajis. 4 weeks, 2 days. 39. A cubic foot of pure fresh water weighs 1000 oz., avoirdu- pois ; find the weight of a vessel of water containing 217^ cubic in. Ana. 7 lbs., 13] ff oz. 40. Suppose a certain pasture, in which are 20 cows, is sufficient to keep them G 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 j^^ does to 87 ; how many feet had he laid ? 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 f the share of , the elder; required the shares. Here the ratio of the shares is '1 : S, i»nd we have shown that if four magnitudes are, 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 sura of 4 and 3 for first term, and either 4 or 3 for the second, and therefore 7 : 4 : : 97 acres, 3 roods, 5 rods : F.P., i. e., the sum of the numbers denoting the ratio of the shares is to one of them as 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 younger'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 45. 86Jj^, the share of the youngest. 121: 5:: 398 7 : : 398 11 : : 398 : IWi, the share of the eldest. Three sureties on. $5000 are to be given by A, B and C, so pg, the share of the second. that B's share may bo one-half 'greate* than A's, and C's one-half greater than B's ; required the amount of the security of each ? «« COMPOUND PROPOnnON. 107 Ans. A's share, $1052.63^5 ; B's,$1578.94|^; C's, 02368.42/^. 46. Suppose that A starts from Washington and walks 4 miles an hour, and B at the same time starts from Boston, to meet him, at the rate of 3 miles an hour, how far from Washington will they meet, the whole distance being 432 miles ? 47. A certain number of dollars is to be divided 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 sum to be divided ? Ans. Less share, $1600 ; greater, $2400 ; total, $4000. 48. A certain number of acres of land are to bo divided into two parts, such that the one shall bo ^ of the other ; required Ihc parts and the whole, the difference of the parts being 710 acres ? Ans. tho less part 537 acres; the greater, 1253 acres ; the whole, 1790. 49. A mixture is made of copper and tin, the tin being J of the copper, the difference of the parts being 75 ; required the parts and the whole mixture ? Ans. tin, 37^; copper, 112i; the whole, 150. 50. Pure water consists of two gasses, oxygen and hydrogen ; t'lo liydrogcn is about -f^ of the ox}gcn ; how many ounces of water will there be when there are 764J-? oz. of oxygen more than of hydrogen ? Ans. 1000 oz. !«K^i 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 is found, by two ratios, and com- pound, by more than two. Thus, if the question be, How many men would be required to reap 05 acres in a given time, if 90 men, working equally, can reap 40 acres in the same time? Hero there is but one condition, viz., that 96 men can reap 40 acres in the given time, which implies but one ratio, and when the question has been stated 40 : 65 : : 96 : F.P., and the required term is found to bo 150, and the proportion 40 ; 65 : : 96 : 156, we have the propor- tion, expressed by two ratios. But, suppose the (Question were, If a man walking 12 hours a day, can accomplish a journey of 250 miles in 9 days, how many days would he require walking at the las AEITHMETIO. same rate, 10 hours eacli day, to travel 400 miles ? Here there are two conditions, viz. : first, that, in the one case, ho travels 12 hours a day, and in the other 10 hours ; and, secondly, that the distances are 250 and 400 miles. The statement, as wq shall presently show, would be' 10:12 ).. 0.177 TS.qtq each condition im- 250 : 400 f ■ ' * ^^' plies one ratio, 10 : 12 and 250 : 400, and when the required term, which is 17/^, is found, there are four ratios, viz., the two already noted, and 9 : Vl-^^, gives two more, one in relation to 10 : 12, and one in relation to 250 : 400. This \friU be evident, when we have shown the method of statement and operation. EXPLANATORY STATEMENT AND OPERATION. 11 1 33; 3 12:F.P. 12:36 PRACTICAL STATEMENT AND OPERATION. 11:33 18: 5 ]■■ 12 : F. P. 18 1 5 36 : 2 F.P. 10. 1 3 3 2 : F. P. }:^}::.:,0. Let the question be, How many men would be required to reap 33 acres in 18 days, if 12 men, working equally, can reap 11 acres in 5 days ? Wc first proceed, as on the left margin, as if there were only one condition in the question ; or, in other words, as if the number of days were the same in both cases, and tlie question were — If 12 men can reap 11 acres in a given time, how many men will be re- quired to reap 33 acres in the same time. This, then, is a question in simple, proportion, and by that rule we have the statement — 11 : 33 : : 12 : F. P., which, by contraction, becomes 1 : 3 : : 12 : F. P. ; and thus, we find F. P. to be 36, the number of men required, if the time were the same in both cases. The question is now resolved into thia: How many men will be required to reap, in 18 days, the same quantity of crop that 36 men can reap in 5 dq,ys ? This is obviously a case of inverse proportion, for the longer tho time allowed tho less will be the number of men required, and hence tho statement, 18 : 5 : 36 : F. P., which, by contraction, becomes 1 ; 5: : 2 : F. P.," which gives 10 for the number of nu5n. The work may be shortened by making tho two statements at once, as ou the right margin. We first notice that tho last term is to represent a COMPOUND PROPOETION. 109 certain number of men, and, therefore, we place 12 in the third place; next, we see that, other things Icing equal, it will take more men to reap 33 than to reap 11 acres, and that, therefore, as far as that is' concerned, the fourth term will be greater than the third, and BO we put 11 in the first place, and 33 in the second. Again we see that, other things being equal, a Ipss number of men will be required when 18 days are allowed for doing the work, than when it is re- quired tojbo done in 5 days, and that therefore tho fourth term, as far as that is concerned,- will be less than the third, and therefore we write 18 : 5 below the other ratio as on the margin. Then by con* traction we get ^ ; g j- : : 2 : F. P. Now, as 3 in the first term is to be a multiplier, and' 3 in the second a divisor, we may omit these oisO) and we obtain i ". 5 f : : 2 : 10, the answer as before. The full uncontractcd operation would be to multiply IS by 11, which gives 198, then to multiply 33 by 5, which gives 1G5, then multiply 1G5, tho product of the two second terms, by 12, and divide the result, 1980, by 198, the product of the two first terms, which gives 10 as before. Because in the analogy 198 : 165: : 12 : 10, the first two terms arc products, this kind of proportion has been called compound, and the ratio of 19 to 165 is called a compound ratio. Wo can show the strict and original meaning of the term compound ratio more easily by an example, than by any explanation in words. Let us take any series of numbers, whole, fractional or mixed, say 5, f , I, 19, 12, 1, 17, 11, 11, 25, then the ratio of the first to the last is said to be compounded of tho ratio of tho first to the second, the second to the third, tlie third to the fourth, &c., &c., &c., to tho end. Now the ratio of 5 to 25 is ^^ 11X18:33X5 198 165::12:F. P. 165X12=10 198 =5, and the several ratios are in this 1.5 order,|x|X-^'X-l3XT3X-V-Xi|Xi|X}|which leaving finally ¥ If wo took them in reverse order, viz., ^^- ^,it 5 as before. is obvious that all therein could be cancelled, as each would in sue* cession be a mtlltiplier and a divisor. We would also remark ihat compound proportion is nothing else than a number of questiona in simple proportion solvod by ono opera* ■A*;.*^^ 'f *" *H,: ¥». W 110 ARITH5IETI0. tion. This will be evident from our second example by comparing the two operations on the opposite margins. Again, we remarked that every condition implies a ratio, and that therefore the third and fourth terms of our first example really involve two ratios, "one in relation to each of the preceding. Hence universally the number of ratios, expressed and implied, must always be double the number of conditions, and therefore always even. As the third ratio is only written once, the number of ratios appears to be odd, but is in reality even. BULE Flace, as in simple proportion, in the third place the term that is the same as the required tr m. Then consider each condition scparateli/ to sec which must oe placed first ^ and Mchich second^ other things being equal. EXAMPLE. 1. If $35,10 pay 27 men for 24 days; how much will pay 16 men 18 days ? Here we first observe that the answer will be money, and therefore $35.10 must be in the third place. Again, it will take less money to pay 16 men than 27 men, and therefore, other things being equal, the answer, as far as this is concerned, will be less than $35.10, and therefore we put the I^s quantity, 16, in the second, place. So also because it will take less to pay any given num- ber of men for 18 days than for 24 days, therefore we put the less quantity in the second iVns. $15.60 place, which the statement shows in the margin. 27:16 24: 18 : $35.10 3: 3: 2 o 9: 4:: $35.10 4 9)140.40 EXERblBES. 1. If 15 men, working 12 hours a day, can reap 60 acres in 16 days ; in what time would 20 boys, working 10 hours a day, reap 98 acres, if 7 men can do as much as 8 boys in the same time ? Ans. 26|| days. 2. If 15 men, by working 6f hours a day, can dig a trench 48 feet long, 8 feet broad, and 5 feet deep, in 12 days;- how many hours a day must 25 men work in order to dig a trench 86 feet long, 12 feot broad, and 3 feet deep, in 9 days ? Ans. 3|. ? COMPOUND PROPORTION. Ill 3. If 48 men can build a wall 864 feet long, 6 feet high, and 3 feet wide, in 36 days ; how many men will be required to build a wall 36 feet long, 8 feet high, and 4 feet wide, in 4 days? Ans. 32. 4. In what timo would 23 men weed a quantity of potato ground which 40 women would weed in 6 days, if 7 men can do as much as 9 women? Ans. 8jAj days. 6. Suppose that 50 men can dig in 27 days, working 5 hours a day, 18 cellars which are each 48 feet long, 28 feet wide, and 15 feet deep; how many days will 50 men require, working 3 hours each day, to dig 24 cellars which are each 36 feet long, 21 feet wide, and 20 feet deep ? Ans. 45 days. 6. If 15 bars of iron, each 6 ft. 6 in. long, 4 in. broad, and 3 in. thick weigh 20 cwt., 3 qrs., (28 lbs.) 16 lbs. ; how much will G bars 4 ft. long, 3 in. broad, and 2 in. thick, weigh ? Ans. 2 cwt., 2 qrs., 8 lbs. 7. If 112 men can seed 460 acres, 3 roods, 8 rods, in 6 days ; how many paen will be required to seed 72 acres in 5 days ? " • Ans. 21. 8. If the freight by railway of 3 cwt. for 65 miles be $11.25; how far should 35 o\ cwt. be carried for $18.75 ? ' 9. If a family of 9 persons can live comfortably in Philadelphia for $2500 a year ;• what will it cost a family of 8 to live in Chicago, all in the same style, for seven months, prices supposed to bo ^ of what they would bo in Philadelphia ? 10. If 126 lbs. of tea cost $173.25; what will 68 lbs. of a differ- ent quality cost, 9 lbs. of the former being equal in value *to 10 lbs. of the latter ?. 11. If 120 yards of carpeting, 5 quarters wide, cost $60; what will be the price of 36 yards of the same quality, but 7 quarters wide ? Ans. $25.20. 12. If 48 men, in 5 days of 12J hours each, can dig a canal 139^ yards long, 4^ yards wide, and 2 J yards deep ; how many hours per day must 90 men work for 42 days to dig 491 j'g yards long, 4^ yards wide, and 3 J yards deep ? Ans. 4. 13. A, standing on the bank of a river, discharges a cannon, and B, on the opposite bank, counts six pulsations at his wrist between the flash and the report ; now, if sound travels 1142 feet per secood, ■••**, » M ■m 112 AEITHMETIO. and the pulse of a person !n health beats 75 stroKes in a minute, what is the breadth of the river? Ans. 1 mile, 201 1 feet. 14. If 264 men, working 12 hours a day, can make 240 yards of a canal, 3 yards wide, and 12 yards deep, in 5 days j how long will it take 24 men, working 9 hours a day, to make another portion 420 yards long, 5 yards wide, and 3 yards deep ? 15. If the charge per freight train for 10800 lbs. of flour be $16 for 29 miles; how much will it be for 12500 lbs. for 100 miles? Ans. $92^5. 16. If $42 keep a family of 8 persons for 16 days; how long, at that rate, will $100 keep a family of 6 persons ? Ans. 502g days. 17. If a mixture of wine and water, measuring 63 gallons, con- sist of four parts wine, and one of water, and bo wcrth §138.60 ; what would 85 gallons of the same wine in its purity be worth ? Ans. $233.75. 18. If I pay 16 men $02.40 for 18 days work ; how much must I pay 27 men at the same rate ? 19. If 00 men can build a wall 300 feet long, 8 feet high, and 6 feet thick, in 120 days, when the days are 8 hours long ; in what time would 12 men build a wall 30 feet long, 6 feet high, and 3 feet thick, when the i^iys are 12 hours long ? Ans. 15 days. 20. If 24 men, in 132 days, of 9 hours each, dig a trench of four degrees of hardness, 337 J feet long, 5| feet wide, and 3 J feet deep ; in how many days, of 11 hours each, will 496 men dig a trench of 7 degrees of hardness, 465 feet long, 3^ feet wide, and 2^ feet deep ? Ans. 5|: 21. If 50 men, by working 3 hours each day, can dig, in 45 days, 24 cellars, which arc each 36 feet long, 21 feet wide, aiid 20 feet deep ; how many men would be required to dig, in 27 days, working 5 hours each day, 18 cellars, which are each 48 feet long, 28 feet wide, and 15 feet deep ? Ans. 50. 22. If 15 men, 12 women, and 9 boys, can complete a certain piece of work in 50 days ; what time would 9 men, 15 women, and 18 boys, require to do twioA as much, the pa^ts performed by each, in the same time, being as the numbers 3, ? i^nd 1 ? Ans. 104 days. 23. If 12 oxen and 35 sheep eat 12 tons, 12 cwt. of hay, in 8 days ; how much will it cost per month (of 28 days,) to feed 9 oxen and 12 sheep, the price of hay being $40 per ton, and 3 oxen being supposed to oat as much as 7 sheep ? Ans. $924. .'•hN» <^' ViW MISCELLANEOUS EXERCISES. 113 24. A vessel, whose speed was 9J miles per hour, left Bellevillo it 8 o'clock, a. m., for Gananoquc, a distance of 74 miles. A second t'essel, whose speed was to that of the first as 8 is to f), starting from the same place, arrived 5 minutes before the first ; what time did the second vessel leave Belleville ? Ans. 55 min. past 10 o'clock, a. m. 25. If 9 compositors, in 12 days, working 10 hours each day, cau compoce 36 sheets of IG pages to a sheet, 50 lines to a page, and 45 letters in a line ; in how many days, each 1 1 hours long, can 5 com. positors compose a volume, consisting of 25 sheets, of 24 pages in a sheet, 44 lines in a page, and 40 letters in a line ? Ans. IG days. ^^ki'^ MISCELLANEOUS EXEECISES ON THE PRECEDING RULES. 1. What is the value of .7525 of a mile ? Ans. 6 fur., rd, 4 yds, 1 ft., 2| in. 2. What is the value of .25 of a score ? Ans. 5. 3. Reduce 1 ft. 6 in. to the decimal of a yard. Ans. .5. 4. What istho value of 14 yards of cloth, at $3,375 per yard ? Ans. §47.15. 5. What part of 2 weeks is j\ of a day ? Ans. j'^-jr. 6. What part of £1 is IBs. 4d ? . Ans. §. 7. Reduce jjPg of a day to hours, minutes and seconds. Ans. 2 hours, 52 min., 48 sec. 8. Add I of a furlong to ^ of a mile. Ans. 7 fur., 31 rds, yd,, 1 ft., 10 in. 9. What is the value of .857^ of a bushel of rye ? Ans. 48 pounds. 10. Reduce 47 pounds of wheat to the decimal of a bushel. Ans. .783J. 11. Redube 9 dozen to the decimal of a gross. Ans. .75. 12. Add /jj of a owt. to |- of a quarter. Ans. 3 qrs., 10 lbs. 13. Subtract | of a day from -^ of a week. Ans. 4 days, 3 hrs. 14. From jg of 5 tons take ^ of 9 cwt. Ans. 2 tons, 17 cwt., 1 qr., |§ lbs. 15. How many Jrards of cloth, at $3^ a yard, can be bought for USi,? Ans. 13| 5 yards. 16. A man bought g of a yard of cloth for $2.80 ; what was the rate per yard ? Ans. $3.20. 17. How mony tons of hay, at $16J per ton, can be bought for $196|? _ » Ans. llf^ tons. c, ,,,J_L 114 ABITHMETIO. 18. At $17| por week, how many weeks can a family board for $7G5| ? Ads. 43^- weeks. 19. What number must be added to 26|, and the sum multipli* ed by 7f , that the product may be 496 ? Ans. 37|. 20. A man owns f of an oil well. He sells § of his share for $3500 ; what part of his share in the well has he still, and what is it worth at the same rate ? 21. How long will 119^ hhds. of water last a company of 30 men, allowing each man | of a gallon a day ? Ans. 627 days. 22. Reduce f of 2|, f^ of 1|, and 3 J of 2|, to equivalent frac- tions having the least common denominator. Ans. ||, |§, \%^. 23. From f of 2§ of 4, take yHy of G| of ^. Ans. 2|. 24. What is the sum of J,*^, I, |, J, ^, J, and i? Ans. IsgSg. 25. What is the sum of f^ of 3|4-H' o^ ^^^ ^o"*- ^^Uh• 26. How long will it take a person to travel 442 miles, if he travels 3|- miles per hour, and 8^ hours a day? Ans. 16 days. 27. Find the sum of 2^ of j\, 3J of I of /, of 4J and i. Ans. .63*3. 28. A has 2\ times 8| dollars, and B 6^ times 9| dollars ; how much more has B than A? Ans. $44||. 29. If I sell hay at $1.75 per cwt. ; what should I give for 9| tons, that I may make $7 on my bargain. Ans. $329. 30. If 7 horses cat 93^ boshels of oats in 60 days ; how many bushels will one horse eat in 87| days ? Ans. 19|. 31. Bought 14^5 yards of broadcloth for $102.90 ; what was tho value of 87 1 yards of the same cloth ? Ans. $&12. 32. How many bushels of wheat, at $2^ per bushel, will it re- quire to purchase IQS^y bushels of com worth 75 c«nt» per bushel ? Ans. 47y\. 33. If in 82^ feet there are 5 rods ; how many rods in one mUe ? Ans. 320. 34. Suppose I pay $55 for | of an acre of land ; what is that per acre ? * Ans. |«8. 35. If § of a pound of tea cost $1 .66^ ; what will ^ of a pound cost? ADS.|1.65|f 36. Subtract the sum of 2^ and 1,\;, from the sum of f, 7^ and 3, and multiply the remarndor by 3j\. An». 24|f . 37. If I lb. cost 23s\ eonts ', what will 2}^ cost ? An», 77j2\ eeat»r from MISCELLANEOUS EXEHCISES. 115 knd 111' ite. 1 Ans. 39f cents. 38. What ia the difference between 2^X3^ and 2lXhh ? Ans 39. If I lb. cost II ; what will J } lb. cost ? 40. What is the difference between f of J+i+4Xi, and 41. If 4/j yards cost ^l^'^ , what will 2^ yards cost ? Ans. 47| cents. 42. Bought I of 2000 yards of ribbon, and sold f of it ; how much remains? Ans. 285;^ yards. 43. Divide the snm of J, f, |, ig, U, |f, JH by the sum of J, h h T0. 52» bV> T3B. and w many times J of i of 20 ? Ans. 0. 12. A is 16 years old, aud his ago is j times § of his ftithcr's age ; how old is his father ? Ans. 3G. 13. A and B were playing cards; A lost 810, which was ^ times g as much as B then had ; and when they commenced I of A's money was equal to f of B's ; how much had each when they began to play? Ans. A $45; B §40. 14. A man willed to his daughter $5G0, which was I of f of what he bequeathed to his son ; and 4 times the son's portion was '] the value of the father's estate ; what was the value of the est;ite ? Ans. $13,440. 15. A gentleman spent •J^ of his life in St. Louis, I of it in Bos- ton, and the remainder of it, which was 25 years, in Washington; irhat ago was ho when ho died ? \ns. GO years. 120 ABITHWETIO. 16. A owns ^, and B j'j of a ship ; A's part is worth $650 more than B's ; what is the value of the ship ? Ans. $15,G0O. 17. A post stands ^ in the mud, ^ in the water, and 15 feet above the water ; what is the length of the post ? Ans. 3G feet. 18. A grocer bought a firkin of butter containing 56 pounds, for $11.20, and sold f of it for $8| ; how much did he get a pound ? Ans. 20 cents. 19. The head Of a fish is 4 feet long, the tail as long as the head and ^ the length of the body, and the body is as long as tlio head and tail j what is the length of the fish ? Ans. 32 feet. 20. A and B have the same income ; A saves | of his ; B, by spending $G5 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 G days, in one day he can do the ■}^ of it, and if they work together, they would do ^-f ^==37 of the work in one day. But A works alone for 3 days, and in one day he cnn 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 be ^— rj^=::=§- of the work yet to l)o completed by A and B, who, according to Il:e con- ditions of the question, labour together to finish the work. Now A and B working together for one day can do -^^f of the entire job, and it will take them as many days to do the balance ^ as -^^ is contain- ed in ^, whicli is equal ^X V^^^'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 C to do it alone ? Ana. 14 1 days. 23. A certain pole was 25^ feet high, and during a storm it was broken, when J of what was broken off, equalled § of what remained J how much was broken oif, and how much remained ? 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, tho second in 30 minutes, and the third in one hour ; in what time will they all fill it together ? Ans. 8 min., 34| soo. •■jll^ ANALYSIS AND SYNTHESIS. 121 25. A. and B. start together by railway train from Buffalo to Erio a distance of (say) 100 miles. A goes by freight 'train, at the rate of 12 miles per hour, and B by mixed train, at the rate of IS miles per hour, C leaves Erie for Buffalo at the same time by ex- press train, which runs at the rate of 22 miles per hour, how far from Buffalo will A and B each be when C meets them. 2G. A cistern has two pipes, one will fill it in 48 minutos, and tho other will empty it in 72 minutes ; what time will it require to fill the cistern when both arc running? Ans. 2 heurs, 2-1 min. 27. If a uian .«;pend3 f.^ of his time in working, J in .sleeping, -,'^ in eating, and 1^ hours each day in reading ; how much time will be left? Ans. 3 hours. 28.' A wall, which was to be built 32 feet high, was raised 8 feet by G men in 12 days; IioAV many men must be employed to finish the wall in G days ? ' Ans. 30 men. 20. A and B can perform a piece of work in 5,'^,- dnys; B and in Gg days; and A iind C in days; in what time would each cf sm perform the work alone, and how long would it take them to do ...: work together? Ans. A, 10 days; B, 12 days; C, 15 days;^ and A. B, and C, together, in 4 days. 30. IMy tailor informs me that it will take' 10}- square yards of cloth to make mo a full suit of clothes. Tho eloth . I am about to purchase is 1^ yards wide, and on sponging it will shrink .}^j in width and length ; how many yards of this cloth must I purchase for my " new suit ?" ^ Ans. G j ;; ;' .j yards. 31. If A can do § of a certain piece of work in 4 hours, and B can do £ of the remainder in 1 hour, and C can finish it in 20 min. ; in what time will tliey do it all working together ? Ans. 1 hour, 30 min. 32. A certain tailor in tho City of Brooklyn bought 40 yards of broadcloth, 2^ yds wide ; but on sponging, it shrunk in len-th upon every 2 yards, -j'jv of a yard, and in width, 1^ sixteenths upon every 1^ yards. To lino this cloth, he bought flannel !]■ yjirds wido which, when wet, shrunk h tho width' on every 10 yards in length, and in width it shrunk ^ f"f p sixteenth of a yard ; how many yards of Hannel had tho tailor to buy to lino his broadcloth ? Ans, 71 1'.j yards, 33. If G bushels of wheat are equal in value to 9 bushels of bar- Icy, and 5 bushels of barley to 7 bushels of oats, and 12 bushels of V 122 ABITHMETIO. oats to 10 bushels of peoso, and 13 bushels of pease to } ton of hay, uud 1 tea of hay to 2 tons of coal, how many tons of coal are equal in 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=-'3"-;=3J^ bushels of wheat. But 5 bushels of barley are equal to seven bushels of oats ; -hence, 7 bushels of oats are equal to 3J bushels of wheat, and one bushel of oats would be equal to 3j-f-7=:^y bushels of wheat, and 12 bushels of oats would bo equal to 12 times {(S==;-y_a=:55 bushels of wheat. But 12 bushels of oats are equal in value to 10 bushels of pease, hence, 10 bushels of pease are equal to 5^ bushels of wheat, and one bushel of pease would equal 5|h-10=:^ of a bushel of wheat, and 13 bushels of pease would equal ^Xl3=-''7*-=7^ bushels of wheat. But 13 bushels of pease equal in value ^ ton of hay, hence, i ton of hay equals 7^ bushels of wheat, and one ton would equal 7^X2= 14^ bushels of whoat. But one ton of hay equals 2 tons of coal, hence, 2 tons of coal arc equal in value to 14 1 bushels of wheat, and one- ton would equal_ 14|-4-2=7| bushels of wheat. Lastly, if 7^ bushels of wheat be equal in value to one ton of coal, it would take as many tons of coal to equal 80 bushels of wheat, as 7| is contained in 80, which gives 10}^ tons of coal. NoTK. — This qiu'stioa belongs to that part of arithmetic usually called Conjoined Proportion, or, by some, the "Chain Rule," which has each ante- cedent ot a compound ratio equal in value to its consequent. We have thought it best not to iiitroducc such questions under a head by themselves, on account of their theory being more easily understood when exhibited by Analysis than by Proportion. Questions that do occur like this will most probably relate to Arbitration of E.xchange. Although they may all be worked by Compound Proportion as well as by Analysis, yet the most expe- ditloua plan, and tho ono geueraly adopted, is by the foUowiug RULE. Place the antecedents in one column and the consequents in another, on the right, with the sign of equality between them. Di- vide the continued j)roduct of the terms in the column containing the odd term hy the continued product of the other column, and tha quotient will be the answer. ANALYSIS AND SYNTHESIS. 123 »\le: Let us now take our last examole (No. 33), and solve it by tUIs 6 bushels of wheat=9 bushels of barley. 5 bushels of barley:;rr7 bushels of oats. 12 bushels of oats=10 bushels of pease. 13 bushels of pease=:^ ton of hay. 1 ton of hay;=2 tons of coal. — tons of coal=::80 bushels of wheat. 20 !<, 7, iq. \X %. m ^, ^, n, 13, 1, zsMo^sioy. Ans. 34. If 12 bushels of wheat in Boston are equal in value to 12J bushels in Albany, and 14 bushels in Albany are worth I'ih bushels in Syracuse ; and 12 bushels in Syracuse are worth 12^ bushels in Oswego ; and 25 bushels in Oswego are worth 28 bushels in Cleve- land ; how many bushels in Cleveland are worth GO bushels in Boston? Ans. 75|^. 35. If 12 shillings in Massachusetts are worth 16 shillings in New York, and 24 shillings in New York are worth 22^ shillings in Pennsylvania, and 7J shillings in Pennsylvania are worth 5 shillings in Caaada ; how many shillings in Canada are worth 50 shillings in Massachusetts? Ans. 41 j. 30. If G men can build. 125 rods of fencing in 4 days, how many days would seven men require to build 210 rods ? SOLUTION. If G men can build 120 rods of fencing in 4 days, one man could do J 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 he would build J of 20 rods, and J of 20 rods is 5 rods. Now, if one man can build 5 rods in one day, 7 men would build 7 times 5 rods in one day, and 7 times 5 rod8=35 rods. Lastly, if 7 men can build 35 rods in one day, it would take them as many days to build 210 rods as 35 is contained in 210, which is 6 ; therefore, if G men can build 120 rods of fencing in 4 days, 7 men would require G days to build 210 rods'. 37. If 12 men, in 36 days, of 10 hours each, build a wall 24 feet long, 16 foct high, and 3 feet thick j in how many days, of 8 ■H*^^ *,«' 1C4 AIUTHMETIC. hours cnch, would the same lot of men build a wall 20 feet loujr, 12 feet hidi, and 2-\ feet thick ? Ans. 23 '. 33. If 5 men can perform a piece of work in 12 days of 10 hours 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 uunibcr of hours in a day, supposing that 2 of the second set can do as mucli work in an hour as 3 of the first set ? Ans. GG* men. NoTK.— Such questions ns (his, vliero the answer involves a fnicliun. niny frequently occur, and it may be aslvcd bow g of u man can do any worU. The •answer is simply thin, that it requires 6(5 men to do tho worli, and one muii to continue on working S of a day more. 39. Suppose that a wolf was observed to devour a sheep in § of imhour, and a bear in ;|- of an hour ; how lon<^ would it take th(^m together to eat what remained of a slicep after the wolf had been eating ^ an hour? Ans. 10, \ min. 40. Find the fortunes of A, B, C, D, E, and F, by knowing t!iat A is worth :;;2 J, which is ^ as much as li and are worth, and that C is worth i^ as much as A and B, and also that if 19 tinios tlie sum of A, B and C's fortune wa.s divided in the proportion of J, h and J, it would respectively give ^ of D's, ^ of E's, and -J of F's "ibrtunc. ' 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 alter the time they first set out. It is required to find the rate at which B uniforndy ti'a veiled. Ans. 10 miles per day. 42. A base 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 13 miles an hour ; how long will the chase last, and what dis- tance will the hare have run ? Ans. 60 A 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 foj 3 days, when A is taken sick and leaves, B continues on working alone, and after 2 days ho is joined by 0, and they finish it together in 1^ days ; how long would C be doing it alone ? Ans. 12 days. 14. A, in a scuffle, seized on § of a parcel of sugar plums ; B caught -^ of it out of his haiids, and C laid hold on -,>'y- more ; D ran ■iff with all A had left, except | which E afterwards secured slyly for ivmscU'j then A and C jointly set upon B, who, in tho conflict, lei ..'^ > PRACTICE. 125 full i lie liailj wliich were equally picked up by D and E, who lay perdu. B then kicked down C's hat, and to work they all went anc for what it contained ; of which A got }, B J^, D ?, and C and E equal shares of what was left of that stock. D then struck J of what A nnd B last acquired, out of their hands; they, with Bomo difficulty, recovered ^ of it in equal shares again, but the other three carried off ^ a piece of the same. Upon this, they called a. truce, and agreed that the J of the whole left by A at first, should bo equally divided among them ; how many plums, after this distribu- tion, had each of the competitors ? Ans. A had 2863 ; B, 6335 ; C, 2438 ; D, 10294 and E, 4050. '%. ■- ' ■ P ■J-^\ m ' ' ♦r -^.^' PRACTICE The nilc whicli is called Practice is nothing else man a particu- lar case of siuiplo 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 be, 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, $190, is really the fourth term of an analogy. Again, to find the price of 4G barrels of flour, at $7. 62 J per barrel, we have only to multiply 67.62 J by 46. In many cases, liowever, it is tooro conveni- ent to multiply the 46 by 7, which will give the price of 46 barrels at $7 each. Now, 50 cents being half a dolljir,- the price of 46, at 50 cents, will be $23, and 12J cents being J of 50 cents, the price at 12 J cents will be the fourth of that at 50 cents, or $5.75, and the whole comes to $350.75. To find the price of 36 cwt., 2 qrs., 15 lbs., at $4.87^, Hero the question stated at length would be, if 1 cwt. cost $4.87^, what will 36 cwt., 2 qrs., 15 lbs. cost? The statement would be 1 : 36., 2., 15: : $4.87^: F. P. This becomeB a question of multi- $7.62^ 46 23 4572 3048 $350.75 50 12J ^ 46 7 i 322 23 5.75 $350.75 12 C ABITHMETIO. plication because the first term is unity, and divided by.l would not alter the product of the other two terms. Thus : 2 qrs. 10 lbs. 5 " J of 1 cwt. 1^ of 2 qrs. i of 10 lbs. 36 18 2923 1461 175.50 = pnce of 3 cwt., @ 34.87J per cwt. 2.437= " 2 qrs. " " " .487= " 10 lbs. « " " .243= '« 5 " " " «' $178,667= " 36cwt., 2qrs., 151bs. " We would call the learner's special attention to the following direction, as the neglect of it is a fertile source of error. Whenever 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 thej^rice at the rate of that higher denomination. To find the rent of 189 acres, 2 roods, 32 rods, at $4.20 per acre. 2 roods=: J of 1 acre, 20 rod8=J of 2 roods, 10 rod8=J of 20 rods, 2 rods=^ of 10 rods, 4.20 189 210 525 2625 525 3780 3360 420 $696.74 Since the rent of 1 acre is $4.20, the half of it, 02.10, will be the rent of 2 roods, the rent of 20 rods will be . 525, the J 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 4 of 10 rods. We then multiply by 189, and set the figures of the product in the usual order, so that the first figure of the product by 9 shall be under the units of cents, &c., and then adding all the partial results, we find the final answer, $796.74, the rent of 189 acres, 2 roods and 32 perches. XXEROISSS. 1. What is the price of 187 cwt. at $5.37^ per cwt. ? Adb. 11006.12^. PRACTICE. 127 2. What is the vako of 1857 lbs., at $3.87| per lb. ? Ans. $7195.87J 3. What will 4796 tons amotint to at 814.50 per ton ? Ans. $21582. 4. What is the price of 29 score of sheep, at $7.62 J each ? Ans. .c. ; 1 lb.chocolaU','10c. Oct. 4, " 2 felt hats, 1««; shoo blacking, 25c *' 21. " 2 lbs. pepper, 15c. ; soda, 40c. ; salpctre, 30c. ; salt, 75c. Contra, ' Or. lO.oo 5.00 Sept, 14. By Cash, ; Oct. 4, Cash ,... Oct. 17. " 2 bblb. winter apples, 2»« $ 18.42 *;*■;«' j«i- ■=^. *r fJI.- Mb. Wm. 2b Aug 4, (1 17. Sept 4, II 26. Oct. 11, 11 22, .< 27, K 30. Contra. Sept. 12. Oct. 24. BasTON', Nov. Ist, 18CC. Reid. Campbeix, LuiN & Co., Dr. To 2 prs. kip boots, 3»« ; 2 prs. cobourgs, 2^^ " 7 yds. fancy tweed, 29* 5 trimmings, po ; buttons 25c.. " 2 prs. gloves, 75c. ; 3 prs. socks, 35c. ; 2 straw hats, 40c, " 10 yds. print, 35c. ; trimmings, 1*5 ; ribbons, 75c " , 3 neck-tics, 62 Jc; 2 prs. boys' gaiters, 2^^] 8hoeties,12Jc. " 1 business coat, 14oo ; 2 felt hats, l^s ; i umbrella, 2«'» " 2 flannel shirts, 49S ; 1 pr. pant«, 8«o ; over-coat, 16o<». " 2 lace scarfs, -S** ; 3 pre. woollen raita 75c. ; pins, 25c. (V. 100* 8«» ByCash ; Oct 4, Cash, " SQO lbs. cheese, 10c. ; 75 IInu bntter, 25c Balance due $37.60 Received payment. 9 CAMPBELL, LINN & CO. I'-' I ' "..A ...iUi ICO ARITHMETIC. Auburn^ Sept. let, 18G6. Mn. S. S>nTn To Wilson, Rat & Co., Dr. 18CG. Jan. 15, To C yda B. clotli, 4.'50; 2 doz. buttons, 30c.; 9 ozs. thread,15c. ' 20, " 40 yds. fac. cot., Ific. ; 7 spools cot, 4c. ; 12 yds. rib., 35c. " 30. " 15 yds. Ji. silk, 2.0 0; IC yds. lining, 15c. f 3 Bilk spools, lie. Feb. 20. " 3 yds. drill, 31c. ; 5 yds. cob'rg; 34c. ; 2 papers need. 18c. Mar, 18, " 9 yds. coating, .''>.*'*; 1 J yds. vesting, l.»<»; 5 pr. hose, 40c. " 31. " 21 yds. print, 20c.; 19J yds. muslin, 30c.; 2"prs. glovea, l.*" Apr.l5, " 4 prs. gloves, l.i"; ItJ yds. ribbon, 18c. ; C hand'k. 3Gc. " 25. " 3prs.blankctsC.3''; 4 counterpanes, 3.«o; 15 yds.cot.,25c. May 29, " 2 sammcr hats, l.os; 6 yds. ribbon, 40c. ; 2 feathers, 23 o Juno 5, " 4 prs. slippers, l."*"; 4 prs. hose, COc. ; 3 prs. hose, 40c. '• 15. " 3 wool shawls, 5.»o; IB. suit, 30.50; 9 ozs. thread, 18c. July (). '•' 40 yds. cotton, 30c. ; 3 spools, 12c. ; 2 spools, 10c Aug. 10. " 13 yds. flannel, 75c. ; 4 hand'ks., 35c. ; 12 yds. tape, 13c. Conira, • Or. IS.oo 10.00 Jan. 15. By Cash, ; 22. Cash, Feb. 20. " oO lbs b\itter,'40c. ; C cwt. pork, I04o May 15. " 6 geese, 80c. ; 14 fowls, 40c June 5. " CO lbs. wool, 50c. ; IC lbs. wool, COc 30.00 10.00 July C. "Cash, •; Aug. 10, Cash Balancedue :. . $82.73 Brooklix, July 15th, 18CG. Mb. R. R. IIilms. To J. WiLUAMS, Br. i8ce. Jan. 10, To 101bB.M.sugar,15c.; lClbs.W.sngar,20c.; 121b8.C.sugar,18o. " 30. " , 15 lbs. raisins, IGc; 13lbs. raisin9,I5c.; 10 lbs. raisins, 18c. 9 lbs. cur'nts, 13c.; 12 lbs. cur'nts, 14c.; 6 lbs. cur'nt8,20c. CO lbs. salt, 2c.; 2 lbs. wash, soda, 23c.;;i lb. bak. 8oda,25c. G lbs.D. apples,12c.; 10 lbs. bisc'ts, 17o.; 5 lbs. bi8c'ts,21c. 3 cwt. flour, 4.30; 2 cwt. C. vxQ^\i, 4» <•; 3 lbs. butter, 25c. IG lbs. pork, 20c. ; 191b8. cbprse I.e. ; 14 lbs. sugar,15c. 5 lbs. tea, l.^oj 9 gals, mola.oses. iOc. ; 6 doz. eggs, 12c. 5 lbs.8Ugar,16c.; *i\ lbs.n>.\;^i!i:', IGc; 10 lbs.cur'nts,12}c. 14 lbs. bacon, 12c. ; 5 lbs. cheese, 16c. ; 4 lbs. bntter,25c. 4 lbs. tea, l.«o; 2 lbs. tea, I."*; 6 lbs. coffee, 35c 40 lbs. salt, lie. ; 3 lbs. indigo, 90c. ; If lbs. blue, 30c. 3 lb8. salt petre, 35c.; 4 doz. eggs,12^c.; 6 lbs.butter,15c. I"eb. 12. J^Iar. 30. Apr. 5, " 25. May 1. Junel5. July 12, " .29, " 31, .« .4( U .U leceiyed payment (83.16 J. Ti .XLIAM& ACCOUNTS AKD INYOICES. 181 AtBAXY, Dec. 1, 1866 Hr. Geo. SniPS0!7, To TATLon Sc Graxt, Dr. 1866. July 7, " 12, " 24, Aug. 4, " 12, Sept 21, Oct. 12, " 20, Nov. 4, To 12 Ibi. Bugar, 15c. ; 2 lbs. tea, l.«» ; 3 lbs. coffee, 35c. ... " 2 lbs.tobacco,87 Jo . ; 3 lb8.rniHin8,30c. ; 12 lbs.currRnte, 15c. " 3 lbs. gunpowder, 62}c.; lb ^ shot, 18c. ; 2 lbs. glue, 25c. " 12 lbs. washing soda, 15c. ; 4 lbs. baking soda, 25c " 1 box mustard, 1.'^ o ; 2 lbs. filberts, 30c. ; 2 lbs. alm'ds, 35c. " 8 lbs. sugar, 14c. ; 1 lb. tea, l.ia>4 ; 3 lbs. chocolate, 40c. " 4 lbs. figs, 15c. ;,2 lbs. orange peel, 30c. ; spices 40c " 2 lbs. but. blue, 180.; 2 lbs. sulphur, 20c.; 3 lbs. soda, 35c. 18.00 " 2 lbs. smok. tobacco, 90o.; 2 lbs. snuff, 20c.; 1 business suit, CorUra. Or. • 8.00 6.00 Aug.l2, ByCash, ; Sept 21, Cash, ;...... Oct. 20, " 100 lbs. dried apples, 15c. ; 60 lbs. peaches, 20c .. . Balance due. $7.01 Detboit, Sept. SOtb, 1866. Mb. S. Surra, To Rat, Hnx & Co., Dr., * 1866. Jan. 1, To 6 lbs. tea, l.'^'o ; 15 lbs. sugar, 15c.; 1} lbs. cinnamon, 2.r fool iM^M-v ('1^ #•' \ i\ i1o»iM\ s«'1K« ('♦) ' Ua Mntos on • T *' l^^.^tl'.^Trt^>^\ nll^untN. i.i»»»». »»...»*. ('ft \^- ',\ '' MMUion'n r.vtnniunv ("^ '* H " liAl* iV!\.W , (;rt M 5 Urtlionf of M;\oK i\nt, riOT.Oi! .1. miNTIN *('i.. ^'2 Mil 1 CO t n I.N M TouoNVo. .Inn. I 'Jill, IHOO, owt. oroh.vso (""\ ^^.^\\l\^^S Ol' l>rt»V>n 0'^ '* ' '^• bushol;* oOvWU woiil 0^ j^^ft tirkins ot'bnH.M- C.^ l'*>fil> hushols oi' »U »o.l nj^j^W i. «.•••• (i|l S^^'-^ft " ^H^nohos ('ft "l'^'' o\M. of ln»A \\1wv\f iKnu" ("^ t^f^'* »Mv<. 1ua^^lo snafvv • C'.') '^■'•'^ IvVsT* ot\>M«H10U SIvU. (.'j^ '''"^ bAvn^s »^f <«css |><>rk 0'^ 13.00 Wr (i^ l}.'?^ Inis^hrls of clover !«oo*l (^^ *f'** For MOUlUi?ON TAYLOll & Co., A. C. Uknuv. iiii,!,H or v^\l(^\^^M. l,i,i '/■(» .liiHl'.l'fl lilMltt, MlilHniif'C, th. Kiir nilO l''ii'Mcli ('iivf|M|it.>t fit, !|t:t Oil pr^r MiotiHfiri'l " ll! i\nr. MiitiHli Aiiiciii'HM ciiji^ li(inlt(«..,((^ I 15 " (J " n. M. Ii-ii-! ,..>ti.'il« {,/♦ r.o " r» I'CHMM inmitllitlir CMVI'lllllCH (f/j 1.05 " '1 I'Miiii'MiiMdiiiiiiL.' riulo jMijicr i((/; .'115 " I " liiitril nolo |m|ti'r ^ {0} )II5 " '.'h " l''HI''IUIt tl'ltl- |l!l(lf'r...i. ..,(>!) it 00 " I till?!. I''I|-h1, lllltlfcM (,(j .15 " fi iH.xr.HJiiiMirM Nn ;io:i |.,.tiH (;/i !»o " (• iln^. Tl.inl M.inh. (,/, \ t]2^ " lo (jiiiiin |i!)imI( liiHilfM, lifilC lininnl (//( .;i5 " 2 |iiM'l hI)(Ii"i| titiUI n<'ll|M|. .11. H«o('KVM-r,n, Jfifi. r»»li, i't:(',(', N. P. (jAitmr.Arrft, To II. I'^lt/RIMMfiMfl k (!o., ffr. I'nr '2[ lie*. IMnrk.'i.'l C^ Or^f!, " ,'l iiilllnim INIulimMnit (tt^ 4r» " i;i llin. Vuniii', llyn'iii 'I'm ('/> >V^J^ " I ;i lltH. lutiwii Miii'iir (if^ 1 I " 15 iMiBlM'lw 111' \'i,\uUm p> 45 Oil. I'.ir lOllm. lliiMor ('f^ 17c. " 5 0 bushels Bye "© .70 40 " Barley © .80 $1688.12 Received payment, J. & A. WRIGHT. PERCENTAOE. ir> PERCENTAaE. 18. — Percentaok is an allowance, or reduction, or estimate of tt certain portion of each 100 of tlio units that enter into any ^'ivca calculation. Tlio term i.i a contraction of the Latin exprcssiorr for one huiulrod, and means literally hi/ t/ic hundred. In (!!ilculatin;^ dollars and cents, fi per cent, means G dollars I'or every 100 dollans, or cents for every $1, or 100 cents. If wo arc estimatinj^ l\iut> only tlwwi, fw wnn»i(K» ftvsw i>> \\^\\ .vV Vi <\< o lotit ol .VuuihI. T'yN'i>^>\<\v ^^ ^^^«»^ ^0* ««*«■> >lrt\« «»rtv>h WomMi »>t>)\(nll(»i, ftii ■<»j!k to »»W »'(;viVj .NMiNVt. !M<'« (>r n t'.ui'lo imtidli II i>i»im>« ifs"- wr*o«» owvvr, ^'^rtv t>H> o^^nxs't Um«> l\\nw M«>vU Snd U> Jum> 1 4ili, «f tiiiM >«♦ tA* Ai\>"Si >iV, tf iM^ ^*)vV. .ix^ Km VjmU. ;U «\m' Muy, n\u\ I i tor Jomv A ^Wt ^vr.Tv»»>V»ts» |rv1»« tiu txvKxM^lnjE liwo tv«\MV»\ Iwo nlvvH tInliNi In (o oouitt MIMt'f >1 f*t'f'f«MW>V iil |j4, \\VIi)(I If) »l(«« liit..H-.i ♦•»( f/h m, f)i^ In ,|/.yrt. ,,( r!|^ |,^f (.Af,» / hrt. WIttif Im lltn lMl^'^^Nl ».»i |(Slf» «f>< fn^ i!0 /|rty«, «! h |;/tf ^'l^f f fill Wllt^^ i« iliH lH^h^hH^ KH »imi^o, tut nn rt«^«, m /) f,hf t-Hif 1 B'r. Wiml Im IIih lH(»'fK«» ^M ftill 4«, Dif 'Ih 'lfiy«, ni «;| |,^r ^^f,» / hM WIml Ifl llih l»(l»<»-»»«» (It. fiU 7h, 0«f fift *1rty«, «f 7 j^h*- ^^fff, / Aft" ^i hMii«. M» Wind h Mih liilhH.M /In^N, ftf, M f,^f A/.fit / Afr« ♦« N. (11, WIimI. U III*' MhtKni. hh lift Ml, fr.r iVi^i thyfi, «♦ f,hf *^f, / An* -(7 ''^rt♦«, Hi! WIimI I« I ho IttlfH.wl f ^^f,» / on. U'liiil U lli« IkIoh'bI, cm $Um, rm VUt >!«/«, *f 10 (K«crtMrf / Am r>'/7.'j (14, Whttl. Irt IliM liilt.ri'wl iitt liriCM, f',t 170 ')i/«, fi> 1 1 (.^r /•/* .>. A OOHIll 14U) Th II»mI IIi*« hilorodl, tin «ny itiitrt »»(' mimhy^ fur huj HrrtA, tiC ti \,t^f tH»M(. Hliii«» ,0)1 wiMilil lit' IImi t(il<< |>"r iitii', lit \\>i\ SuU'tt'n^, f,f fl f'lf i your, II. (itlliiwn llml. Ilm liiMirtiwI, fur otm iMmfh mit\M \t*\ Uift ,'^ '.f .(Ml, or ^";^ itriMmnl, nipiiil In |^ (KWit, ut ,00.%, nri/t fof 2 tni>n^hn it Woulil iM|iiul ^ tipMl,, iir ,(»0ft/2^ .01. Tliflf »for«, whftfi itr^i-rtK^, k* III Uhi iiil4» mI' II \mr mid., Iliti Itifiirt'tif, trf |l, fur m'stj 2 tufrrtih*, i* t)»i»t rrnl> A^hIii, If Uta UtUirntl '»(' |l, f';r ^m* m^mlh, at <'{0 'layi, i* 4 (Wilt iir .005, it, f'oll'iWN lliiil. tlitt iiiUirt^tit, C'lr <; rhij^* «ill ^,a Ui': i f/f .005 or .001. Thflnilbro, wiiun iiil• »i Ibis rtt^ < f ^# per Mat., Um InivrMt (H* II for uyory dajra in &n« rtnU. WtntM th« 14 ji AnrrHMETic. RULE. Find the interest of $1 for the given tinnc hy rcclconing 6 cents /'or every year, 1 cent for every 2 months, and 1 mill for evcrv G days; then midtiply the given 2)ri'icij)al by the number denoting that in- ierest, and the product will be the interest required. Note. — This method can be adopted for any rate per cent, by first flndlng Ibo interest at per cent., then adding to, or subtracting from the" interest so found, such a part or parts of it, as the given rate exceeds, or is less than G per cent. This method, although adopted by some, is not exactly correct as the year is considered as consisting of 3G0 days, instead of 3G5 ; so that the in- terest, obtained in this manner, is too largo by ^ (jj or ij^j, which for every $73 interest, is $1 too much, and must therefore be subtracted if the oxac^ amount bo required. EXAMPLE. G7. "What is the interest of $24, for 4 mouths, 8 days, at 6 per cent. ? . . S L U T I-O N . The interest of 81, for 4 months, is 02 Tho interest of $1, for 8 days, is OOIJ Hence the interest of $1, for 4 months, 8 days, is 021J Now, if the interest of $1, for tho given time, is .021 J, the inter- est of 024 will bo 24 times .021 J, which is 8-512. EXERCISES . 68. What is the interest on $171, for 24 days, at 6 per cent. ? Ans. 08 cents. 09. What is the interest on $112, for 118 days, at per cent. ? Ans. $2.20. 70. What is the interest on $11, for 112 days, at 6 per cent. ? Ans. 21 cents. 71. What is the interest on 50 cents, for 360 days, at 6 per oont. ? An<3. 3 cents. ' 72. What is tho interest on $75.00, for 236 days, at G per cent. ? Ans. $2.05. 73. What is the intercut on $111.50, for 54 days, at 6 per cent. ? Ans. $1.00. 74. What is the interest on $15.50, for 314 days, at 6 per cent. ? , Ans. 81 cents. m SIMTLE INTEEEST. 149 76. What 76. What cent. 77. What 78. What per cent. ? 79. What 80. What at 7 per cent. ? 81. What 82. Who'- 83. .What at 10 per cent 84. What per cent. ? 85. What 7 per cent. ? 86. What per cent. ? 87. What 8 per cent. ? 88. What per cent. ? 89. What cent. ? 90. What per cent. ? 3 the interest on $174.25, for 42 days, at 6 per cent. ? Ans. $1,22. s the interest on $10, for 1 month. 18 days, at 6 per Ans. 8 cents, s the interest on $154, for 3 months, at G per cent. ? • Ans. $2.31. s the interest on $172, for 2 months, 15 days, at 6 Ans. $2.15. g the interest on $25, for 4 months, at G per cent. ? Ans. 50 cents, s the interest on $36, for 1 year, 3 months, 11 days, Ans. $3.23. 3 the interest on $500, for ICO days, at G per cent. ? At", $13.33. s the interest on $92.30, for 78 days, at 5 per cent. ? Ans. $1.00. 8 the interest on $125, for 3 years, 5 months, 15 days, Ans. $43.23. 3 the amount of $200, for 9 months, 27 days, at G Ans. $209.90. 3 the interest on $125.75, for 5 months, 17 days, at • Ans. $4.08. 3 the interest on $84.50, for 1 month, 20 days,' at 5 Ans. 69 cents. 3 the amount, of $45, for 1 year, 1 month, 1 day, at Ans. $48.91. 8 tho interest on $175, for 7 months, G days, at 5^ Ans. $5.78. a the interest on $225, for 3 months, 3 days, at 9 per Ans. $5.23. s the interest on $212.60, for 9 months, 8 days, at 8^- Ans. $13.95. %• ** , I m CASE VII. To find tho interest on any sum of money, in pounds, shillings, and ponce, for any time, at u given rato per cent. RULE. Multiply tlie principal by the rate per cent., and divide by 100. irviWur 150 AEITHMETIO. EXAMPLE. 91. What 19 tho interest of £47 ISs. 9d., for 1 year, 9 months, 15 days, at G per cent. ? ' ' SOLUTION, • £ 8. D. £ 8. D. Interest for 1 year 2 17 4 47 15 9 Interest for I) mos., or ^ of int. for 1 year, 18 8 6 Interest for 3 mos., or l~ of int. for G uios., 14 4 Interest for 15 days, or^i of int. for 3 mos., 2 4J 2;86 14 G ■'■■ '■■ iiU latcrest for 1 year, 9 mouths, 16 days... .£5 2 8^ 17^34 • 12 4;i4 92. What is the interest of £25, for 1 year, 9 months, at 5 per cent, ? Ans. £2 3s. 9J. 93. What is the interest of £75 123. 6d., for 7 months, 12 days, at 8 per cent. ? Ans. £3 14s. 7U. 94. What is the amount of £G4 10s. 3d., for 3 months, 3 days, at 7 per cci.t. ? Ans. £G5 13s. 7d. 95. What is the interest of £35 4s. 8d., for 6 months, at 10 per cent.? • Ans. £1 15s. 2|d. 9G. What is tho amount of £18 12s., for 10 months and 3 days, at G per cent. ? CASE VIII. To find tho rRiNCiPAL, the interest, tho time, and the rate per cent, being given. EXAMPLE. 97. What principal will produce $4.50 interest in 1 year, 3 months, at G per cent. ? SOLUTION. •> 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 example, .075 bo tho interest on 61, the number of dollars required to produce $4.50, will bo represented by tho number of times that .075 is con- tained in 84.50, which is CO times. Therefore, $60 will produce $4.50 intorost in 1 year, 3 months, at G per cent. Hence the 8IMFLE INTEnEST. RULE. 151 Divide the given interest hy tJie interest of%\for the given time, at the given rate per cent. EXEROIBXS. jy ) ■ >' ^ 98. What principal will produce 77 cents interest in 3 months, 9 ^ jja^s, at 7 per cent. ? Ans. S40. 99. "NVlmt principal will produce $10.71 interest in 8 months, 12 day;:, at 7| per cent. ? Ans. $204. 100. What principal will produce $31.50 interest in 4 ycaia, at 3^ per cent. ? Ans. $225. 101. What sum of money will produce $79.30 interest in 2 years, G months, 15 days, at GJ per cent. ? • Ans. $480. 102. What sum of money is sufficient to produce $290 interest in 2 years and G months, at 7 J per cent. ? Ans. $1G00. OASE IX. To find the BATE PER CENT., the principal, the interest, and the time being given. EXAMPLE. 103: If $3 bo the interest of $G0 for 1 year, what is the rate per cent. ? SO'LUTION. . If the interest of $G0 for 1 year, at 1 per cent, is .GO, the re- quired rate per cent, will lie represented by the number of times that .GO is contained in 3.00, which is 5 times. Therefore, if $3 is the interest of $G0 for 1 year, the rate per cent, is 5. Hence the RULE. Divide th". given interest hy the interest of the given principal at \pcr cent, for ttie given time. EXERCISES. 104. if the interest of $40, for 2 years, 9 months, 12 days, is $13.3G ; what is the rate per cent. ? Ans. 12. 105. If I borrow $75 for 2 months, and pay $1 interest j what is the rate per cent. ? • Ana. 8. ■ At * r mm 152 ahithmetic. lOG. If I give $2.25 for the use of $30 for 9 months ; what rate per cent, am I paying ? Ans. 10. 107. At what rate per cent, will $150 amount to $165.75, in 1 year, 4 months, 24 days ? Ans. 7^. 108. At what rate per cent, must $1, or any sum of money, be on interest to double itself in 12 years ? Ans. Ans. 8J. 109. At what rate per cent, must $425 be lent to gain $11.73 in 3 months, 18 days? Ans. 9i. 110. At what rate per cent, will any sum of money amount to three times itself in 25 years ? Ans. 8. 111. If I give $14 for the interest of $125 for 1 year, 7 months, 6 days ; what rate per cent am I paying ? , Ans. 7. Case x. To find the time, the principal, the interest, and the rate per cent, being given. EXAMPLE. 112. How long must $75 be at interest, at 8 per cent., to gain $12? SOLUTION. The interest for $75, for 1 year, at 8 per cent., is $6. Now, if $75 require to be on interest for 1 year to pioduce $6, it is evident that the number of years required tg produce $12 intcv-sfc, will bo Tcpreseuteu by the number of times that 6 is contained in 12, which is 2. Therefore, $75 will have to be at interest for 2 years to gain $12. Hence the RULE. Divide the given interest by the interest of the principal for one ijear, at the given rate per cent. EXERCISES. • 113. In what time will $12 produce $3.88 interest, at 8 per cent ? Ans. 3 years. 114. In what time will $25 produce 50 cents interest, at 6 per cent. ? Ans. 4 months. 115. In what time will $40 produce 75 cents interest, at G^ per cent. ? . Ans. 3 months, 18 days. EMPIRE INTEREST. 153 116. In wbat time will any sum of money double itself, at 6 per cent. ? • Ans. 10 years, 8 months. 117. In what time will any sum of money quadruple itself, at 9 per cent. ? Ans. 33 years, 4 months. 118. In what time will $125 amount to $138.75, at 8 per cent. ? Ans. 1 year, 4 months, 15 days. 119. Borrowed, January 1, 18G5, $60, at 6 per cent, to be paid as soon as the interest amounted to one-half the principal. When is it duo? ' ' 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. 12, 18G7. merchants' table For showing in what time any sum of moner/ will double itself, at any rate per cent., from one to twenty, simple interest. Per cent. Years. Per cent. Years. Per cent. Years. Per cent. Years. 1 2 3 4 5 100 50 33^ 25 20 6 7 8 9 10 16§ 14| iH 10 11 12 13 14 15 • ^ 16 17 18 19 20 6i MIXED EXERCISES. 121. What is the interest on $64.25 for 3 y^ars, at 7 per cent. ? Ans. $13.49. 122. What is the interest on $125.40 for 6 moliths, at 6 per cent. ?* ■ Ans. 3.76. 123. What is the amount of $369.29 for 2 years, 3 months, 1 day, at 9 per cent. ? . Ans. $444.16. 124. What must be paid for the uso of 75 cents for 6 years, 9 months, 3 d»js, at 10 per cent. ? Ans. 51 cents. 125. What will $54 amount to in 254 days, at 10 per cent. ?* Ana. $57.81. '* Ibis and the following exercises (marked with a *) are to be worked by Case VI. •«^ 151 ARITHMETIC. 12G. "What must be paid for the interest of $45 for 72 days, at 9 per cent. ?* . Ans. 81 cents. 127. What is the interest of $240 from January 1, 18G6, to Juno 4, ISGG, at 7 per cent. ? Ans. $7.14. 128. V/hat will $140.40 amount to from August 29, 1SG5, to November 29, 18G6, at GJ per cent. ? ^ Ans. $151.83. 129. What principal will give $4.40 interest in 1 year, 4 months, 15 days, at 8 per cent. ? Ans. $40. 130. In what time will $40 amount to •$l4.i0, at 8 per cent.? Ans. 1 yr., 4 nios., 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 cf $145.50 for 240 days, at 9i per cent. ?=<= Ans. $9.22. 133. What will $100 amount to in 175 days, at G per cent. ?-:= Ans. $1G4'.G7. 134. At what rate per cent, must any sum of money be on interest to quadruple itself in 33 years and 4 months ? Ans. 9. 135. In what time will any sum of money double itself, at 10 per cent. ? Ans.- 10 years. CASE X I. To find the iaterest on bonds, notes, or other documents draw- ing 7 ^"'^y percent, interest. Since .07-,^jj or, .073 would be the rate per unit, or the interest of $1 for I year or 365 days, it follows that the interest for 1 dtiy would be the ^jj^ part of .073 which is .0002, equal to two tenths of a mill, hence the BULB . MultlpJy the'pidncipalhy the number of days, and the product hy two tenths of a mill the result icill he the answer in mills. EXAMPLE. What must be paid for the use of $75 for 3G days at 7j^j per jent. ? SOLUTION. The interest on $75 for 36 days would be the same as the inter- 3st on $75X3G=$2700 for 1 day, and at -f^ of n Aill per day would bo $2700 X -0002=54 cents. 2. What would bo the interest on $} 18.30 for 42 days at per cent. 7 3 'to Ans. 99cts. COMaiERCIAL P.VPER. 155 COMMERCIAL PAPER. Commercial paper is divided into two classes — neootiable and NON-NEQOTIABLE. .NEQOTIABLE 0OM3IERCIAL PAPER. Negotiahle commercial paper is that which may be freely trans- ferred from one owner to another, so as to pass the right of action to the holder, without being subject to any set-offs, or legal or equitable defences existing between the original parties, if transferred for a valuable consideration before maturity, and received without any defect therein. Negotiable paper is made payable to the payee therein named, or to his order, or to the payee or bearer, or to bearer; or sonic similar term is used ; showing that the maker intends to give the payee authority to transfer it to a third party, I'rco i'rom all set-offs, or equitable or legal defences existing between himself and tho payee. NON-NEaOTIAJJLE COMMERCIAL PAPER. Non-negotiable commercial jiaper is that which is made prrablo to tho payeo therein named, without authority to transfer it to a third party. It may be passed from one owner to another by assign- ment, or by indorsement, but it passes subject to all set-offs, and legal or equitable defences existing between the original parties. now THE TITLE PASSES. The title to negotiable paper passes from one owner to another by delivery, if made payable to payea or bearer, or to bearer. It passes by indorsement and delivery, if made payable to payee or order. Tho title to non-negotiable paper passes by a mere verbal assignment and delivery, or by indorsement and delivery. PRIMARY DEBTOR. In a promissory note there arc two original parties — the maker and the payeo. The obligation of tho maker is absolute, and con- tinues until the note is presumed to have been paid under the Statute of Limitations. The maker is the primary debtor. In a bill of exchange there are three parties. When the drawee accepts tho bill, he becomes the primary debtor upon the bill of exchange. j PROMISSORY NOTE NOT PAYABLE IN MONEY. When a promissory note Is payable in anything but money, it docs not come within the Statute. There is no presumption that it is founded upon a valuable consideration. A consideration must bo 150 AEITHMETIO. alleged in the complaint, nnd proved on the trial. The aclcnowlcdsr- rncntof a coasidoration in such promissory note, by inserting tlio words " value recrircJ," is sufficient to cast upon the dcJ'endant tho burden of proof that there was no consideration. Tho aolcuovvlodg- ment of '• value received," raises tlio presumption that the note wua given for value ; but this presumption may be rebutted by the dc- fendunt. A Dogotiablc instrument is a written promise or requcstr for the payment of a certain sum of money to order or bearer. A negotiable instrument must be made payable in money only, and without any condition not certain of fulfillment. Tho person, to whose order a negotiable instrument is made payable, must bo ascertainable at the time the instrument is made. A negotiable instrument may jrive to the payee an option between tho payment of tho sum specified therein, and the perlbrmauce of another act. A negotiable instrument may be with or without date; with or without seal ; aud with or without designation of tho time or place of payment. A negotiable instrument may contain a pledge of collateral secu- rity, with autliority to dispose- thereof. A negotiable instrument must not contain any other contract than .sucli as is specified. Two different contracts cannot be ad- mitted. Any date may be inserted by. tho maker ef a negotiable instru- ment, whotluT past, present, or future, and tho instrument is not invalidated by his death or incapacity at the time of the nominal date. There arc several classes of negotiable instruments, namely : — 1. Bills of Exchange; 2. Promis.sory JNotes ; H. Bank Notes; 4. Cheques on Banks and Bankers; 5. Coupon Bonds; 0. Certifi- cates of Deposit ; 7. Letters of Credit., A negotialjle instrument that doea uut wpecify tho time of pay- ment, is payable immediately. A negotiable instrument which docs not specify a place of pay- ment, is payable wherever it is held at its maturity. An instrument, "Othci-wiso negoti :bk in form, payable to a person named, but addipg the words, " or to his order," or " to bearer," or equivalent thereto, is in the former c;ise r«yablc to the written order of such person, and in the latter caisu, payable to the bearer. A negotiable instrument, made payable to the order of the maker, or of a fictitious person, if issued by the maker for a valid considera- tion, without indorsement, has the same effect against him and alL other per,-. 'US having notice of the facts, its if payable to tho be.'ircr.' A negotiable instrument, made payable to tho order of a person obviously fictitious, is payable to the bea/er. Tho signature of every drawer, acceptor and indorser of a nego- COMMERCIAL r^VTER. 1.7 tiablo instrument, is prcsumetl to have been maao for a valuublc* considcralion, before the maturity of the instrument, and in the* ordinary course of buHincss, and the words ''value received," acknowledge a consideration. Olio who writes bis name upon a negotiable instrument, otherwise than as a maker or acceptor, and delivers it, with his name thereon, to another person, is called au indorser, and hia act is called ua indorsement. One who ngrccs to indorse a negotiable instrument is bound to write his signature upon the back of the instrument, if there is sufficient space thereon for that purpose. When there is not room for a signature upon the back of a nego- tiable instrument, a signature equivalent to au indorsement thereof may be made upon a paper annexed thereto. An indorsement may be general or special. A general indorsement is one by which no indorser is named. A special indorsement Bpeciiies the indorsee. A negotiable instrument bearing a general indorsement cannot bo afterwards specially indorsed ; but any lawful holder may turn a general indorsement into a special one^by writing above it a direction lor payment to a particular person. A special indorsement may, by express words for that purpose, but not otherwise, be so made as to reuder the instrument not negoti- able.^ ... ]wery indorser of a negotiable instrument wariiants to every subse- quent holder thereof, who is not liable thereon to him : 1. That it is in all" respects what it purports to bo ; 2. That he has a good titlo to it; ii. That the signatures of all prior parties are binding upon tbera ; 4. Tliat if the instrument is dishonored, the indorser will, upon notice, thereof duly given unto him, or without notice, where it is excused by law, pay so much of the same as the holder paid therefor, with interest. One who indorses a negotiable instrument before it is delivered to the payee, is liable to the payee thereon, as an indorser. An indorser may qualify his indorsement with the words, " with- out recourse," or equivalent words; and upon such indorsement, he is responsible only to tho same extent as in the case of a transfer without indorsement. Except as otherwise prescribed by the last eection, an indorse- ment " without recourse" has the same effect as any other indorse- ment. An indorsee of a negotiable instrunftnt has the same right ' jainst every prior party thereto, that he would liave had if the contract had been made directly between them in the first instance. An indorser has all the rishts of a guarantor, and ia exonerated from liability in like manner. 153 ARITHMETIC. c'nVa One who Indorses a negotiable instrument, at the rcqucfit, and for t'.io " iiccomniodation" of another party to tho instrument, has all the ri_:;ht3 of a snvoty, and is exonerated in like manner, in respect to every one having notice of tho facts, except that he is not entitled to contribution from subsequent indorsers. Tho Avaiit of consideration for the undertaking of a maker, acceptor, or iiidor.scr of a nr;.^otiablo instrument, docs not exonerate liiin from llubility thereon, to an indorsee in good faith for a consid- eration. An indorsee in due course is ono who in good faith, in the ordi- nary cmirso of bushiGi-S; and for value, before its apparent maturity or presumptive dishonor, acijuircs a negotiable instrument duly indorsed to him, or indorsed generally, or payable to the bearer. An indorscr of 'a negotiable instrument, in due course, acquires an absolute title thereto, so that it is valid in his iiands, notwith- standing any provision of law making it generally void or voidable, and notwithstanding any defect in the title of the person from whom he acquired it, One who makes himself a party to an instrument intended to bo negotiable, but which is left wholly or partly in blank, for tho pur- pose of lillitig afterwards, is liable upon tho instrument to an indorsee thereof in duo course, in whatever manner, and at whatever time it may be filled, so long as it remains negotiable in form. It is not necessary to make a demand of payment i^sn tho principal debtor in a negotiable instrument in order to clnrgo him ; but if the instrument is by its terms payable at a spccilied place, and lie is able and willing to pay it there at maturity, such ability aud willingness are equivalent to an offer of payment upon his part. Presentment of a negotiable instrument for payment, when necessary, must be made ua follows, as nearly as by reasonable diii- gencc it is practicable : 1. The instrument must be presented by the lioldcr, or his authorized agent. 2. The instrument must be presented to the principal debtor, if he can be found at tho place where presentment should bo made, and if not, then it must be presented to some other persop of discretion, if one can be found there, and if not, then it must bo presented to .•^omc other person of discretion, if one can be found there, and if not, then it must be presented to a notary public within the State ; 3. An instrument which specifics a place for its payment, must be presented there, and if the place specified includes more than one house, then at the place of residence or business of the principal debtor, if it can be found* therein ; 4. An instrument which does not specify a place for it must bo presented at the place of residence or business oi cipal debtor, or wherever he mav bo found, st llie optit proscntor ; and, ji'in* if tho COMJIEr^-UAL TXPUi. loO ■ tho 5. The instruincnt must Ic presented upon I'lic dny of its appar- ent mnturity, or, if it is payable on dctiiiind, at nny lime before i(i npp:irent maturity, witliin reasonable liours, and, if it in payable at a baiikin;^ liou:^e, williiu the iiisual bankin;» liours of the viciuity; but, by tl»o consent of the person to whom it should bo presented, it luay be presented at any hour of tho day. The appiirent maturity of a neirotiablc instrutiicnt, payable at :i particuLir time, i.i tho day on which by its terms it becomes due ; or, when that is a holiday, it should bo paid the previous day. A bill of ixclnn'je, payable at a ppocified time after sii^dit, which is not accepted within ten days after it.s date, in addition to the time which would sulRcc, with ordinary diligence, to forward it fur acceptance, is presumed to have been dii^honored. The apparent maturity of a bill of exchange, payable at sight or on demand, is : 1. If it bears interest, one year after its date ; or, 2. If it docs not boor interest, ten days after its date, in addition to the timo which would suffice, with ordinary diligcace, to. forward it for acceptance. The apparent maturity of a promissory note, payable at sight or on demand, is : 1. If it bears interest one year after its date; or, 2. If it docs not bear interest, eix months after its date. When a promissory note is payable at a cerXain time after sight or demand, such time is to bo added to tho periods mentioned in the last paragraph. ' A party to a negotiable instrument may require, as a conditioa concurred to its payment by him : 1. That the instrument be surrendered to him, unless it is lost or destroyed, or the hoklcr has other claims upon it; ox, 2. If the holder has :i right to retnin tho instrument, and does not retrain it, th.cn that a receipt for the amount paid, or an exonera- tion of the party paying, be written thereon ; or, 3. If the instrument is lost, then that the holder give to him a bond, executed by himself and two sufficient sureties, to indemnify him against any lawful claini thereon ; or 4. If the instrument is destroyed, then that proof of its destruc- tion be given to him. * A negotiable instrument is dishonored when it is cither not paid, or not accepted, according to its tenor, or presentment for the purpose, or without presentment,, where that is excused. Notice of tho dishonor op protest of a negotiable instrument may he given : 1. By a holder thereof; or, • 2. By a party to the instrument who might bo compelled to pay it to the holder, and who would, upon taking it up, have a right to reimbursement from the party to whom tho notice is given. .,..i.vv,\ w ' .- ^< M^N v^ \\\\K V\\\\ A* m \\v'\\\\ , k^K II lm»»u h \\i>\i»v \\i •» \->n\^ ♦V\'vs->^t *w ^s\ v\\»v^\ ys>TK \ u\\,l T X' ■ V \*AS M\^HVv\\\\\ji^ ^\^yJi v^\>»l\ny\\v\ »\\\il w\\|»'l» li''U('-\ (!'■' ^V *»S>s\ \\'»-v'A tV V\V\ VI A ^>^svty>\\^V \y\«t^y\\\\«>yyK «* ♦l^'* t<*^*'" >** "- V Wv V^^^v^ HA >V\^N\H«^> \yv\^ \\ i\\v A>)>>yyi \^ tly»* \''\\\w t^u ^ASWV » -^'* ^ s .», i .. «i>\W\Miv\\t 1\V\ \'Avl\ «»\\\'»vv»\\»* rtnt>»> i*< "HI> ^v>«^»^v% »\ tV \M'^\>^\ Vv^^i^'-x ^\\ *>>-,^\ yt* \\y«h>M»\*y \h\^ 0^l«* -ul.li tv*w»: tr^- IS n\-*t)»(\xV x'-^In t«^ V^\^ ^^r.y'♦^»W^y ^vrty»,v \'\\\\\\>'\\ Ou'»\««« ♦»y<>>%\ %4^>N■ vA^^^i ♦v'^^y *!»' \«V* ^x»'/l)/ f i iiiiUlillH )>itifii'4 I'llllH fM,>\ l^liMi /,>4Wf«, ItHH " '!>■ hufHlilnn Ih f * f J M'««tillfi» (il'»« (Id Ji|«/(( |(IC(;/*/t» <((»(<(((// / //^^ i'i('» tKIMlJ l|0(l«i» l'l(( //». /i(|.» ilfijjllil^ h fJ'IHil, Ulili-lh II illlllt^ill II' ii:,l /li h*f ' IV'idK f (MHjl WVfcl /((|.U (•/(//(( ///. (//(• tlihllilllii Mli hlftfimit, .'f -ti fiim/iMil inn "/ >« lillft«<« "III UN flM /MtiiiMphUf, Of'* 't',U,m^ \\\f »ijmH im||', ()ii(> ^iiMi iiff'i'.iiii'fiiiif III Ilif itfihi lit rrifrj'iffit * Mn-i.h, 'ff,-> //iiMilifiJ mill MiMU •(■''"fi f'i'Oiiih Niiit'li'H'i Hliill4f)if» Mii'l Y**» /»*iw* Mt>fHt,ff^ tiiltiK ) Hi*t, hiinhiii *»i\i'\l\, /*/a/w//v «y i^», )*fM ''( Ul» Irt'lAf Iff ^4>lfr>'l/if '»/// m^Pffil ft t,0 '/imii/(h) ittiil Mi-iian^ ttii* iHx/ ^^ IhilUi**, Wj/nM tut^^tttl, iffO */,/«■-/•• '*« •>vri«# ti. Ill iiHiMiiiNf. Unlihwn*. Nit AdftH ♦. rVAVH 10-2 AEITHMETIC. Bills of Exchange are the highest c»ass of commercial paper known to the law, and it has never been the cherished object of ilie law merchant, — which has been permitted by the English courts to insinuate itself into the common law, till it now forms a part of that code, — to uphold them inviolate, as far as possible. While the hx' mcrattoria (or mercantile law) is deeply impregnated with the principles of equity, those principles have been chiefly marked to enable courts of law to enlbrce equitable rights, and upon this principle was the negotiability of bills of exchange insisted upon and llnally maintained at the common law ; but when equitable principles liave been invoked for the purpose of destroying the validity and security of bills of exchange, they have been listeucd to with great disfavor and only admitted us exceptional casea CHECKS. 1. A check is substantially the same aa an inland bill of exchange ; it passes by delivery, when payable to bearer, and tlic rules as to presentment, diligence of the holder, &c., which are applicable to the one, are generally applicable to the other. 2. A check is an appropriation of the drawer's funds, in the liands of the banker, to the amount thereof, and, conse(iuciitly, the drawee has no right to withdraw them bcfbre the check is paid. H. The characteristics which distinguish checks from bills of exchange arc, that checks are always drawn on a bank or bunker ; that they are payable immediately upon presentment, and without days of grace;' and that they are not presentable for acceptance, but only for payment. The want of duo presentment of a check, and notice of the non-payment thereof only exonerates the drawer in £0 far as actual damages have thereby resulted to him. LETTERS OP CREDIT. In addition to the conniiercial paper before mentioned, there is an extensive business done by the issuo of " Letters of Credit.^' These are issued by prominent bankers in London, Paris, New York and other cities, to travellers who are about to visit foreign countries, and who are thus saved the risk and expense of carrying any large amount of cash al)out them. These Letteiw op Credit are addressed by the banker to his correspondents abroad, authorizing any ono or more of them to pay to i\vi person named, any portion of the sum mentioned in thu letter. Thus a person leaving New York for the Paoifio Ports, South America or Arctic Ports, or any city or place in Europe or other portions of the world, need carry very little cash. At the first port of arrival ho is able to realize such funds as may be Doocssary to pay C03OIERCI.iL rAPEE. 1G3 his expenses to . further port by using his Letters of Credit. A traveller may go round *Im world, with the aid of such a Credit, and never havu mcie than ono hundred dollars in his pocket. No loss from exchingc need occur, in such ca.scs : bills on London being in demand throughout ihc civilized world. The usual char'TO by the bankers for such " Letters of Credit," is one per cent, vhere the trader doA not pay the amount of the Letter in advance. Where he pays in advance, no charge is made ; the use of the money 4u the banker's hands being an ecjuivalcnt for the cost of the credit. Letters of Credit are also extensively used by importers when travelling abroad for the purchase of goods ; also by supercargoes and captains of vessels for the purchase of cargoes in foreign ports ; also as remittances to distant ports in Asia, Australia, kc, for the purchase of cargoes of foreign goods. Before Letters of Credit were adopted or in circulation, it was the practice among American and other merchants to remit specie txi remote parts for iavcstmcnt in foreign merchandize DAYS OF GRACE. iicrc IS iredit.^' York Entries, large [to his Ito pay 1 letter. 1 South other It port Ito pay 1. In most countries, when a bill or noto ia payable ar a certain time after date, or after sight, or after demand, it is not payable tho precise time mentioned in the bill or note, but days of grace arc allowed. 2. The dajs of grace are so called, because they wt3re formerly gratuitous, and i ">t to be claimed as a right by tho person on whom it was incumbent i pay the bill, and were dependant on the inclina- tion of tiie holder ; they still retain the name of drfys of grace, though the custom of merchants, recognized by law, has long reduced thenj to a certainty, and established them as a right. 3. In England, Scotland, Wales, and Ireland, three davs grace are allowed ; in other countries they vary from three to twelve days. 4. The days of gnico as allowed in England, are generally allowed in the United States, at least no 'ncs can be found of a contrary decision, except in tho State of Mucsochusetts, where it has been held that no days of grace are allowable, unless stipulated in the contract itself. It is probable that a bill of exchange was, in its original, nothing more than a letter of credit from a merchant in one country, to his debtor, a morchan* in another, requesting him to pajr the debt to a third person, who carried the letter, and happened to be travelling to the place where tho debtor resided. It was discovered, by experi- ence, that this mode of making payments was extremely convenient to all parties : — to the creditor, for ho oould thus receive his debt without trouble, risk or expense — to tho debtor, for tho facility of 1G4 ARITHMETIC. payment was an equal accommodation to hira, and perhaps drc^^^ after it facility of credit to Iho bearer of the letter, who found himself in funds in a Ibrnign country, witlioi^t the danger and incumbrance of carrying specie. At fir-it, perhaps, the letter contained many other thinjis benides the order to give credit. But it waa found that the ori^iniJ bearer might often, with advantage, transfer it to another, '"he letter was therf disencumbered of all other matter ; it was opened and not sealed, and the page on which it waa written, gradually shrunk to the slip now in use. The ;issigncc was, perhaps, desirous to know beforehand whether the party to whom it was addressed would pay, and sometimes showed it to him lor that pur- {)oso; his promise to pay was the origin of acceptances. These otters or bills, the representatives of debts due in a foreign country, were sometimes more, sometimes less, in demand ; they became, by degrees, articles of traffic; and the present complicated and abstruse practice and thedry of exchange was gradually formed. PARTIAL PAYMENTS Partial payments, as the term indicates, are the part payments of promissory notes, bonds, or other obligations. When these payments are made the creditor specifics in writing, on the Inick of the note, or other instrument, the sum paid, and the time when it is paid, and acknowledges it by signing his name. The method approved of by the Supreme Court of the United States, for easting interest upon bonds, notes, or other obligations, upon which partial payments have been made, is to apply the pay- ment, in the first place, to the discharge of the interest then duo. If the payment exceccfe the interest, the surplus goes towards discharg- ing the principal, and the subsequent interest is to be computed on the balance of the principal remaining due. If the payment bo less than the interest, the surplus of interest must not be taken to aug- ment the principal, but interest continues on the former principal until the time when the payments, taken together, exceed the in- terest due, and then the surplus is to be applied towards discharging the principal. RULE. Find the amount of the principal to the time of the first pay- ment ; subtract the payment from the amount, and then find the amount of the remainder to tite time of the second payment ; deduct the payment as before ; and so on to the time of settlement. But if anif jHjymmt is less than the interest then due, find the amount of tlie sum due to the titnc. tchen the payments, added to- gether, shall be equal, at hast, to the interest already due; then find the balance, and proceed as before. |/iC lit PARTLYL PAYMENTS. 1C5 EXAMPLE. 1. On tliO 4th of January, 18G5, a note was given for $800, payable on demand, with interest at G per cent. The following pay- ments were rectipted on the back of the note : February 7th, 18G5, received $150 April IGth, " " 100 Sept., 30th, " " ■ 180 January 4th, 186G. " 170 3Iarch24th, « " 100 Juno 12th, " " 50 Settled July Ist, 18G7. How much was due ? solution: Face of the note, or principal $800.00 Interest on the same to February 7th, 18G5 (1 month, 3 days) 4.10 Amount duo at time of 1st payment 804.40 First payment to bo taken from this amount 150.00 Balance remaining due February 7th, 1805 G54.40 Interest on the same from February 7th, 18G5, to April »'' IGth, 18G5 7.525 Amount due at time of 2nd payment GG1.925 Second payment to be taken from this amount 100.000 Balance remaining due April IGth, 18G5 5G1.1)25 Interest on the same from April IGth, 18G5, to September 30th, 18G.5 1.5.359 Amount due at time >f 3rd payment. 577.284 Third payment to bo taken from this amount 180.000 Balance remaining d*ie Sept. 30th, 18G5 31(7.284 Interest on the same from Sept. 30th, 1865, to January 4th, 1866 0.290 Amount due at time of 4th payment 403.574 Fourth payment to bo taken from this amount 170.000 Balance remaining duo January 4th, 186G 233.574 • . ( 1G6 AnrrnMETic. Interest on the same from Jan. 4tli, 18G6, to Marcli 24th, 18GG 3.114 Amount duo at time of 5th payment 236.688 Fifth payment to be taken from this amount .' 100.000 Bahmcc remaining due, March 21th, 18GG 13G.G88 Interest on the same from March 24th, 18GG, to June 12th, ISGG 1.799 Amount due at time of Gth payment: 138.487 Sixth payment to be taken from this amount 50.000 Balance remaining due Juno 12th, 18GG 88.437 Interest on the same from Juno 12th, 186G, to July 1st, 18Gi 5.589 Amount duo on settlement 94.076 2. $1600. Charleston, February 16th, 1865. On demand, I jyromise to pay Jacob Anderson, or order, one thousand six hundred dollars, with interest, at 7 per cent. John Fortune Jr. There was paid on this note, April 19th, 1865 ^460 July 22nd " 150 August 25th, 1866 50 Sept. 12th, " 100 Dec. 24th. " 700 How mucli was due December 31st, 1866 ? SOLUTION. Face of the note or principal $1600.00 Interest on tho same from Feb. 16th, 1865, to April 19th, , 1865 19.60 Amount duo at tirao of 1st payment 1619.60 First payment to be taken from this amount 460.00 Balance rcmainiiV'' duo, April 19th. 18G5 1159.60 TATiTIAL TAYJUENTS. 1G7 Interest ou the parac from April 19th, 18G5, to July 22ntl, 18C5 20.9G9 Amount duo at time of 2n(l payment 1180.509 Bccond payment to be taken from this amount 150.000 Bal;;iico rnr.aining duo, July 22nd, 1805 1030.509 Interest on the same from July 22nd, 18G5, to Aug. 25th, 1800, {U'cater than 3rd payment,* I:itcrc,st on the same fro n July 22nd, 1805, to Sept. 12th, 1800 82.359 ^ ■ Amount duo at time of 4th payment 1112.928 Third and fourth payments to be taken from this amount, 150.000 Balance remaining due S.-pt. 12th, 18GG 902.923 Interest on the same from Sept. 12th, 18GG, to Dec. 24th, 180G 19.093 Amour. t due at time of last payment ..i 982.02G Last payment to bo taken from this amount 700.000 Balance remaining due Dec. 21th, 18G6 282.02G Interest on the same from Dec. 24th, 1806, to Dec. 31st, 1800 382 Amount duo at time of settlement, Doc. 31st, 18GG §282.408 3. $350.. BosTONjMay 1st, 18G4. On demand I jyromhc to paj/ ]VUUam Brown^ or order, three hundred and fifty dollars, with interest, at G per eent. James Weston. There was paid on this note, December 25th, 18G4 $50 June 30th, 1805 5 • Tho interest ou $1030.5(59, from July 22nd, 18U5, to August 25th, 18(JG. is $78,752, and the payment madf at thi.s dntt>, is only $50, not enough to pay tlio interest, so if we proceeded, as in the former case, to add the interest to the principal, and subtract the payment from tlio amount obtained, wo would bo taking interest, until tho next payment, on the excess of the inter- est, $78,752, over tho payment, $50, which would bo in effect iutere&t upon interest, or compound interest which the law does not allow. ••r. in-} AinTiTMF/nr. AuRtist 2-2nd, IROC* 15 Junoith, IHCT 100 How njuoh w!iH lino April fiili, ISOS ? Aiw. $25l.C7. i. ^GOD.C.r). HiiANTKOKi), Juno Hth, IHCI. .SV.r DtniitLt tij'tir ilntr, irr jitlnt/i/ prr mit. after maturity, SaMUKI. TiUAlIAM, T. U. llKAIlMAN. Thoro WHS piiiil on this nolo, Octobor4th, IHOU $2r>.00 Mimh ir)lli, ISf,:? 1(5.25 Au^'ust2tth, ISJII \\{\M What was duo DcaMubor llHh, ISC..") ? Ana. 079.27. 5. $871.1)5. KiNdSTON, IMiiy Dth, IStUJ. Three months after ilite, I promise to pni/ llnniwn Cnmmimjs, or order, lijhf humhed and gcvcnti/four ^■'^^^j dollars, with interest after matnrlti/ at (\ ])er cent. Thomas Goodpay. Thoro was paid on this nolo, April 12th, isr.l $50.30 July l|th, 1805 21.80 Sept. 18th, 1800 210.00 W]vAt wa,s duo February Dth, 1808? An.«. $773.07. Whou tho interest aceruin;^ on a note is to bo paid annually adopt tho following 11 u L E . =1* Compute the interest on the principal to the time of settlement, and on each years interest after it is due, then add the sum of the ' When noti>8, bonds, or otbor obli}j:iitions, iiro Rivon, " with intoroHt payublo iiunually,'' tho intorost is duo at tho oiul of ouch your, and may bo collocted, but If not collucted at Ihut tinio, tho intorest duo draws only .shiip^e vitncst, and the original principal must not bo incroosod by any addition of yearly intorest. If nothing has boon paid until maturity on u note drawing annual interest, tho amount duo consists of tho principal, tho total annual interest, or the simple interest, and tho simple interest on each item of annual iatori^t from tho time it became duo until paid. TAIITTAL TAYMl'VlTl. Jfjg tHtcrrxfx on the (innwif. iiitrrrsts tn th,: (naniiiit of llir prinn'/nil, and, /nun t/uH iimoitiif take thr. piii/mnifn, nncipal to the time of the first pay- ment ; if that he one year or more from the time the interest com- menced, add it to the principal, and deduct tlic jmy men t from the sum total. If there be after payments made, compute the interest on the balance due to the next 2)ayment, and then deduct the payment as above, and in like manner from one pjymevt to nvofhcr. f'," u]> the PARTIAL PAYMENTS. 171 mymenU tire ahsorhcd, 2>i'<'ivi(lcd the time between one j>a^»ic;if and another he via' y,ar or more. If any jxiymcntH he iiiade hv/orr. one ijca/g interest hax accrued, then eoiiipitfe the iufrrmt on the principal Kiim due on the ohligadon for one i/ritr, mid it to the princijinl, timl compute the interest on t/ir. sum])'! id, from the time it irns jxtid, up to the end of the year ; add it t" the Kinn ptiid, and deduct that mm from the pi'iucipnl and intercut, added (is above. If inn/ jmym cuts he made, of a less sum thin the interest arisinj at the time iil or |i:iyiniuits, it' iitiy, from Iho tiiuu llit-y woru jiuiii to tliu tiuiu of gt'Ulumuut, uiid uubtracl their sum from Uk; luiioiuit uf Ibu |)riacipul. 1. -N A M p L E S . 10. 8900. Kingston, Juik; Ist, 18G2. On demand r-e prnmin*' to pay J. II. Smith L£. 11. $400. Maitland, January 1st, 1865. For value received, I promise to2)ay J. B. Smith & Co., or order, on demand, four hundred dollars, with interest at G per cent. A. R. Cassels. The following payments were receipted on the back of this note : February 4th, 18G5, received $100 May 16th, " " 75 August 28th " " ■ 100 November 25th, " " 80 What was due at time of scttlemcDt, which was December 28th, 1865 ? 100.000 G47.29G 38.837 G8G.133 78.187 607.94G 8G.47G 644.422 22G.G00 PARTIAL PAYMEXTS. SOLUTION. 173 Principal or face of note. $400.00 Interest on the samo from Jan. 1st, 1865, to Deo. 28th, 1865 23.80 Amount of principal at settlement 423.80 First payment $100.00 Interest on the same from Feb. 4th, 1805, to Dec. 28th, 18G5 5.40 Second payment 75.00 Interest on the samo from May IGth, 1865, to Dec. 28th, 18G5 2.77J Third payment 100.00 Interest on the samo from August 28th, 1865, to Deo. 25th, 1865 2.00 Fourth payment 80.00 Interest on the same from Nov. 25th, ICC, to Dec. 28th, 1865 .. .44 Amount of payments to be taken from amount of principal 365.6U Balance duo, December 28th, 1865 3B^..18J 12. 8500. Cleveland, January Ist, 1865. Three montJia after date, J promise to pay James Man- ning, or order, five hundred dollars, /or value received, at the First National Bank of Bxiffalo. CybUS Kino. Mr. King paid on this note, July let 1865, $200. What was duo April let, 1866, the rate of interest being 7 per cent ? Ans. 8324.50. 13. $240. Philadelphia, May- 4th, 1865. On demand, I promise to pay A. K, Frost & Co., or order, two hundred and forty dollars, for value received, with in- terest at 6 per cent. David Flook. The following payments were receipted on the back of this note : September 10th, 1865, received $60 January 16th, 1866, *« 90 What ^nn Ann at the time of settlement, which was 3Iay 4th, 1866? Ans. $100.44. 'im 174 AEITHMETIC. 14. $340. Lowell, June IGtli, ISCi. Three months after (late, I promise to pni/ Thomai Culvrncdl, or order, three hundred andfortif dollars, leilh interest, at G per cent. Wiluaai JIanninu. Ob this note were receipted the following payments : October 14th, 18G4, rcceiyed $86 Fobruary 12th, 1805, " 40 Wliat was due at time of PCttlcraQpt, Aug. 10, 18G5? Aus. $232.0.G COMPOUNP INTEREST.. When interost is unpid at the end of the year, it may, by spoeiil agreement, bo added to the principal, and in its turn bear intercut, and fto on from year to year. When added to the principal in tlua way, it is said to be compound. A person may take compound interest and not be liable to the charge of usury, provided the person to whom ho lends nioncj chooses to pay conipounl interest, but ho cannot legally collect it unless there has been o previous agreement to that effect. £ X A :kf P L E . 1 . What is the compound inteicst of $G0, for 4 years, at 7 per cent. ? SOLUTION. Principal $GO.(IO intercut ou tho same for one year.... 4.2(1 New principal for 2nd year G4.20 Interest on tho same fur one year 4.404 New principal for 3rd year CS.GJ't Interest on the (nmo for ono year 4.808 New principal for 4th year 73.502 Interest on the same ror one year 5.1 15 Amount for 4 ycaw '. , 78,r)t7 Principal to be taken from same GO. 000 Compound intei^t for 4 j'cars $1SG47 Tho method of finding comnound interest is usually much Bhort- cncd by tho following thblc, which shows the amount of $1 or .i'l for any numbor of years not exceeding 60, at '6\ 3^,4, fi, and 7 poreciit. The amoantof $1 ov £i 4hu8 obtained, ijoing mullipliiil by tho given principal, will give the required omount, from which, if tho principal bo taken, the remainder will bo compound interest : %(»' 5' o3 /f Vi COMPOUND INTEREST, 175 TABLE, gnowjso THB AMOiwr Of oxK uotLAn AT coMPoMD rfTcnasT ron axt airMDEii or nuaa xoT BzcKGni.va FirrT. No. T 3 per cent. 8cr cent per cent. 7 iM!r cout. I.UaO \i\i\S 1.033 OOU I.UIO UUU 1.050 000 l.UuU UUO I.UJO 00,t 2 1.060 000 1.071 225 1.081 600 1.103 600 1.123 600 1.111 900 3 1.092 727 1.108 718 1.124 861 1.1.37 625 1.191 016 1.223 013 4 1.125 509 1.147 523 1.169 839 1.213 506 1.262 477 1.310 796 5 1.159 274 1.187 686 1.216 653 1.276 282 1.338 226 1.402 .332 r> 1.194 052 1.229 235 1.263 319 1.340 096 1.418 319 1.501) 730 7 1.229 874 1-.272 279 1.313 932 1.407 UK) i.5o.i (;:;o 1.603 7,^1 8 1.266 770 1.316 809 1.368 569 1.477 4,35 1.593 HJS 1.7 1,'^ 1m; t) 1.31)1 773 1.862 897 1.423 312 1.551 328 1.<;,S9 479 I.S38 43.) 10 1.343 916 1.410 599 1.180 244 1.628 893 1.790 818 l.!)67 131 11 1.384 231 1.439 970 1.539 434 1.710 339 1.898 299 2.104 8 -.2 12 1.425 761 1.311 069 1.601 032 1.795 836 2.012 196 2.232 l'.)2 13 1.468 534 1 363 936 1.663 074 1.883 610 2.132 92,"^ 2.409 >;|3 14 1.512 590 1.618 694 1.731 67(! 1.979 932 2.260 901 2.,378 534 13 1.657 967 1.675 .349 1.800 911 2.078 928 2.396 r38 2.739 032 16 1.604 706 1.733 980 1.872 981 2.182 875 2.540 332 2.932 164 17 1.652 848 1.794 G75 1.947 901 2.202 018 2.692 773 3.158 813 18 1.702 433 1.857 489 2.025 817 2.406 619 2.834 .339 3.379 932 19 1.733 506 1.922 501 2.106 819 2.326 930 3.023 600 :;.6i(; 326 20 1.806 111 1.989 789 2.191 123 2.653 298 3.207 133 3,S(;9 6,<1 21 1.860 2!lo" 2.039 431 2.278 768 2.785 963 3.39.) 561 •1,110 562 22 1.916 103 2.131 512 2.369 919 2.923 261 3.603 337 4.430 402 23 1.973 587 2.206 114 2.464 716 3.071 524 3.819 730 4.7 10 B30 24 2.032 794 2.283 328 2.663 31)4 3.2:'5 100 4.0(8 933 3.072 367 25 2.093 778 2.363 245 2.663 836 15.386 335 4.291 871 3.127 433 26 2.1fi« 591 2.445 959 2.772 470 3.553 673 4.549 383 3.8')7 ,".33 27 2.221 289 2.331 667 2.883 369 3.733 456 4.822 316 6.213 868 28 2.287 928 2.620 177 2,993 703 3.920 129 .'5.111 6S7 6,618 83.S 29 2.356 066 2.711 878 3.118 6.31 4.116 136 6.418 38,S 7,114 237 ao 2.427 262 2.806 794 3.243 398 4.321 912 5.743 4!tl 7.612 233 31 2.500 080 2.903 031 3,373 133 4.338 039 6.088 101 S.1I3 li;! 32 2.575 083 3.006 708 3./y08 059 4.764 941 6.433 387 ,S.715 271 33 2.632 3;'5 3.111 942 3.648 381 5.003 189 6.810 390 9.323 3 to 84 2.731 905 3.220 860 3.794 316 5.263 318 7.231 023 9.978 111 35 2.813 8(-2 3.333 590 3.946 089 6.516 015 7.686 087 10.676 581 36 2.890 278 3.450 266 4.103 933 6.791 816 8.147 232 11.423 912 87 2.9:f5 227 .1.571 023 4.268 090 6.081 407 8,636 0.S7 12.223 61S 38 3,074 783 .3.696 Oil 4,138 813 0.385 477 0.164 252 13.079 271 39 3.167 027 3.825 372 4.616 366 6.704 731 0.703 507 I3.:i:i» .'^JO 40 3.262 038 3.939 260 4.801 021 7.039 9S9 10.283 7 IS 11,971 43S 41 3.339 899 4.097 834 4.993 (161 7.391 988 10.902 861 l(i,()22 670 U 3.460 096 4.241 2.38 3.192 784 7.761 588 11.557 033 17.114 237 43 3.564 517 4.389 702 5.400 493 8.149 667 12,230 4,33 l.S,3ll 333 44 3.671 4,Vi 4.543 342 5.616 515 8 ,337 150 12.ft86 482 19.628 460 15 3.781 596 4.7ii2 338 5.8 H 176 8.G83 OOil 13.761 611 21,002 132 46 3.895 044 4.866 911 6.074 823 9.434 238 14.590 4,S7 22,172 6.3 47 4.011 895 5.037 284 6.317 816 9.Si03 971 15.465 917 .'t.OI3 707 48 4.1.33 2:>2 5.213 689 6.570 528 10.401 270 16,393 872 i3.728 907 49 4.236 219 6,396 065 6.833 349 10.921 333 17.377 .301 27,329 930 50 4.383 906 5,584 927 7.106 683 11.467 400 18.420 134 29.137 023 Nora.— Ifoach of Itio numbers in tho tablo bo tlimluUUvd by 1, Iho romolnUcr will doaut« ths iatoroil of |t, l:istoa'l of Us amount. 176 AMTHMETIO. EXERCISES* 2. What is the compound intoroat on $75, for 2 years, at 7 per cent. ? Anfc. $10.87. 3. "What will $50 amount to in 3 years, at G per cent , eompound interest V Ans. $59.55. 4. Wiiat is the compound interest on $G00, for 2 years, at G per cent., payable half-yearly ? Ans. $75.31. 5. What will $320 amount to in 2^ years, at 7 per cent,, com- pound interest ? Ans. $370.10. G. What is the compound interest of $150, for 3 years, at 9 per cent. ? Ans. $-14.25. 7. What is the compound intei-est on $1,000, for 2 years, at 3J per cent, payable quarterly ? Ans. $72.18. 8. What will $4G0 amour' *,o in 3 years, 4 months, 10 days, at G per cent., compound interest .' Ans. $55*9.74. 9. What is the compound interest on $18G0, for 8 years, at 7 per cent. ? Ans. $1335.83. 10. What will be the compound interest oa $75.20, for 20 years, at 3^ per cent. ? Ans. $74.43. 11. How much more will $500 amount to at compound than simple intci-est, for 20 years, 3 months, 15 days, at 7 per cent. ? Ai.". $7G4.14. 12. What sum will $50, deposited in a savings bank, amount to at compound interest, for 21 years, at 3 per cent, payable half-yearly ? Ans. $173.03. 13. If a note of $G0.GO, dated October 25th, 185G, with the interest payable yearly, at 6 per cent., be paid October 25th, 18G0 ; what will it amount to at compound interest ? Ans. $7G.51. 14. What remains due on the following note, April 1st, 1863, at 7 per cent, compound interest ? $1,000. Cleveland, January 1, 1858. For value received^ I promise to pay A. B. Smith do Co., or order y one thousand dollars on demand, with interest at 7 per cent. J. J). Foster. On the bock of this note wore receipted the following payments: June 10, 1858, received $70 Sept. 25, 1859, " 80 July 4, 1860, '« 100 DISCOUNT XSH PEESENT WOrwTH. 177 Nov. 11, 18G1, Juno 5, 1862, 30 50 Ans. $1022.34. DISCOUNT AND PRESENT WORTH. Discount being of the sarao nature as interest, is, strictly speak- ing, the u?c of money before it is due. The terra 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 called Bank discount, in order to distinguish it from true discount. The method of computing bank discount differs in no way from that of computing simple interest, but the method of finding true discount is quite different, c. g., a debt orf $107, duo one year hence, is considered to bo worth $100 now, for the reason that $100 let out at interest now, at 7 per cent., vrauld amouut to $107 at the end of a year. In calculating interest, the sum on which interest is to be paid is k'totv!!, vut in computing discount we have to find what sum must be ^) i at interest so that that sum, toj;ether with its interest, will amount to the given principal. The sum thus found is called the " Present Worth." Wo have already seen that 81.00 is the present worth of $1.07 duo one year hence, at 7 per cent., therefore, to got the present worth of any sum due ono year hence, at 7 p<*r cent., it is only necessary to find how many times $1.07 is contained in tho given eum, and wo have tho present worth ; hence To find the pvesont worth uf any sum, and tho discount for any time, at any rate per cent., we have tho following RULB . Divide the given sum hif the amount of$\/or the given time and rate, and the quotient will be the present worth. From the given sum subtract the present worth, and the remainder Kill be the discount. EXEROISI?. 1. What ia tho present worth of $224, duo 2 years hcnoc, a( 6 per cent. ? j^ Ans. $200. 178 ahithmetic. 2. Wliat is the discount on $670, duo 1 year and 8 months hence, at 7 per cent. ? Ans. $70. 3. What is the discount on $501, due 1 year and 5 months hence, at 8 per cent. ? Ans. $51. 4. What is the present value of a debt of $678.75, due 3 years and 7 months hence, at 7 J per cent. ? Ans. $534.97^. 5. What is the discount on $88.16, due 1 year, 8 months, and 12 days hence, at 6 per cent. ? Ans. $8.16. 6. If the discount on $1060, for 1 year, at 6 per cent., is $00 ; \rhat is the discount on the same sum for one-half the time ? Ans. $30.87. 7. How much cash will discharge a debt of $145.50, duo 2 years, 6 months and 12 days hence, at 6 per cent. ? Ans. $126.30. 8. If I am oflFcrcd a certain quantity of goods for $2^00 cash, or for $2821.50, on 9 months credit ; which is the best offer, and by how much ? ' Ans. Cash by $200. 9. What is the difference between the interest and discount of $46.16, due at the end of 2 years, 6 months, and 24 days, at 6 per cent. ? Ans. 95 cents. 10. A merchant sold goods to the amount of $1500, one-half to bo paid in 6 months, and the balance in 9 months ; how much cash ought ho to receive for them after deducting 1^ per cent, a month ? Ans. $1331.25. 11. Suppose a merchant contracts a debt of $24000, to bo paid in four instalments, as follows: one-fifth in 4 months; one-fourth in 9 months ; one-sixth in 1 year and 2 months, and the rest in 1 year and 7 months ; how much cash must he give at once to disohargo the debt, money being worth 6 per cent. ? Ans. 22587.65. 12. Bought goods to the amount of $840, on 9 months credit; how much money would discharge the debt at the time of purchasing the goods, interest being 8 per cent. ? Ans. $702.45. 13. A bouksellor marks two prices in a book, one for ready money, and the other for one year's credit, allowing discount at 5 per cent. If the credit price be marked $9.80 ; what ought to be the price marked for cash ? Ans. $9.33. 14. A man having a horse for sale, offered it for $225, cash ; or, $230 at 9 months credit ; the buyer chose the latter ; did the seller lose or make by his bargain, and how much, .supposing money to bo worth 7 per cent. ? Ans. Ho lost $6.47. 15. A. B. Smith owes John Manning as follows : — $305.87, to B.VNKS AM) KANKIXO. 179 be paid December 19th, 1863; S1G1.15, to bo paid July 16th, 1864 ; $112.50, to be paid June 23rd, 1862 ; $96.81, to be paid April 19th, 1866, allowing discount at 6 per cent. ; how much cash should M^nnUig receive an an equivalent, January 1st, 18u2 ? Aus. $653.40. 16. I buy a bill of goods amounting to $2500 on six months' •credit, and can get 5 per cent, off by paying cash ; how much would 1 pain by paying the bill now, provided I have to borrow the money, and pay 6 per cent, a year for it ? Add. $53.75. uld otherwise havo to bo kept, at a considerable risk, in private houses. They also prevent, in a great measure, tho necessity of carrying money from place to place to make payments, and enable thorn to bo made in tho most convenient and least ozpen- biv 1. What is tho bank discount on u note, given for CO days, fur $350, at 6 per cent. ?* Ans. $3,67. 2. What is the bonk 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-aad 15 duys, ut G per cent. ? Ans. $7(559.68. 4. How much money should be received on a note for $125, payable at the end of 1 year, 3 months, and 15 dayij, if discounted at a bank at 8 per cent. ? Ans. 8112. 5. A note, dated December 3rd, 1800, for $100.40, and having' G months to run, was discounted at a b.ink, April 3rd, 1801, at 6 per cent.; how long had it to run, and what were tho proceeds ? Ans. 04 days ; proceeds $158.71. 6. On tho first day of January, 1800, 1 received a note for 82405 ot 00 "days, and on tho 12th of the simic month had it discounted at a bank at 7 per cent. ; how much did 1 realize upon it. Ans. $237.01. 7. A merchant sold 240 bales of cotton, each weighing 280 pounds, for \1\ cents per pound, which cost him, the same day, 10 cents per pound ; ho received in payment u good note, for 4 mouths' time, which he discounted immediately ut a bank at 7 per cent. ; what will bo his profits ? Ans. $1479 10. 8. I hold a note against Clcmes, llieo & Co., to the amount of $327.40 dated April 11th, 1800, having si-K months to run after date, and drawing interest at tho rate of per cent, jwr annum. What are the proceeds if discounted at tho GirardUank on tho 10th of August, at 7 1\ per cent. ? Ans. $332.90. NoTK. AVhon 11 note drawing intorost, w discounted at a bank, the interest is calculated on tho fiico of the note from its date to the time of maturitj-, and added to the face ot the note', and this amount discounted ior the length of time the note has still to nin. 9. What will bo the discount on the following note if discounted at the City Bank on tho 17th of JNouembcr, at per cent. (300 days to a year), • Throughout all the exercises unless olherwiBO specilled, the year Ih lo 1)0 considered as connisting of Gli;') days. Hiaco it ia customary in business when a fraction of a cent occura in and result to reject it, if less than halt a cent, and if not less, to call it a cent, we Lave adopted this principal tlu'uui;h- out the bool^ LANK DISCOUNT. 183 'tor riK. ■uglU rsGo is to n«s3 lit n $527.-P(;^ Oberlin, Oct. 4, .866. Ninety o 'ys after date for value received, wc promist to pay to the order of Smith, Warren & Co., five hundred twenty- seven and -j^^'g dollart at the City Bank, Oherlin, with interest a\ eight per cent. Thompson & Burns. 10. What will be the discount at 7,^jj per cent, on u note for $227.41, dravring interest at 8 per cent., dated May 1st, 1865, at 1 year after date, if discounted on March 7th, 1866? 11. What amount of money will I receive on the following note, if discounted at the First ^'atioQal Bank of Detroit on June 21st, at 9 per cent. ? $473.80. Detroit, May 17, 1866. TTxrec months afterdate I promise to pay to the order oj J. R. Siny , at which time I collected the amount due me, nnd paid my note ut the bank. Beqiiired the loss or gain by the transaction. It is sometimes necessary to know tho amount for which -a note must be given, in order that it shall produce a given sum when dh- 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 mu.st we draw the note, supposing that money is worth 9 per cent. ? SOLUTION. From tho nature of this example we can readily perceive that such a sum must bo put on the face of the pote, that when dis- counted the proceeds will be exactly 8236.22. If we were to take a one dollar note and discount it at a bank for the given time, and at the given rate, tho proceeds would bo .98425. llencc, for every dol- lar wo put upon tho face of tho note wo receive .98425, and to re- ceive $236.22 we would have to put as many dollars on the fiice of the note as are represented by the number of times that .98425 is contained in $230.22, which is 240. Therefore, we must put §240 on the face of a note due at the end of two months to produce $236.22 when discounted at a bank at 9 per cent. From this \yc deduce the following RULK. Deduct the hanJi discount on ^1, for the given time and rate, from $1, and divide the desired amount hi/ the remainder. The quotient will he the face of tJic note required, 2. For wljat sum must a note bo given, having 4 months to run, that shall produce $1950, if discounted at a bank at 7 per cent. ? Aus. $1997.78. 3. What must be the face of a note, so that when discounted lor 5 months and 21 days, at 7 per cent., it will produce $57.97, cash ? Ana. $60. BANK DISCOUNT. 185 4. Suppose your note for 6 months is discounted at a bank at 6 per cent., and $484.75 placed to your credit, what must liave been the face of the note ? Ans. $500. 5. A merchant bought a quantity of goods for $600. For xviii't sum must he write his note, to be diecounted at a bank for C montlis, at 6 per cent. ? Ans. 5GI S.SvS. 0. A farmer bought a farm for $5000 cash, and having only one- half of tho sum on haou, be wishes to obtain the balance from the bank. For what sum must ho give Lis note, to be discounted for months, at per cent. ? Ans. $2019.17. 7. If a merchant wiihos to obtain $550 of a bank, for what sum mu?t he give his note, payable in GO days, allowing it to be dis- counted at ^ per cent, per month ? Ans. $555.75. 8. I sold A. Mills, niorchandizu valued at $D1S.1C, for which he was to pay mc cash, but being disai'puintcJ in receiving money ex- pected,'ho gave me his endorsed note at DO lUiys, for such an amount that when discounted at the bank at 7 por cent, it would produce the price of the merchandize. What was the face of iho note ? 9. I am owing K.. Harrington on account, now due, $108.45 ; l.o also holds a note against mc for $210, due in 34 days, indudinir days of grace ; he allows a discount of 8 per cent, on the note, and if I give him'my note at CO days for an amount that will be sufficient if discounted at 6 per cent., to produce the amount of account and note. What will be the face of new note ? 10. Samuel Johnsop has been owing mc $274.48 for 84 days. I charge him interest at G per cent, per annum for this time, and ho gives me his note at 90 days for such an amount that when dis- counted at the Girurd Bank, at 8 per cent., the proceeds will equal the amount now due. What is the face of the note ? f}' From the many dealings business men have, in regard to dis- count and interest, it is frequently required to know what rate of interest corresponds to a given rate of bank discount. EXAMPLE. 1 , Wh»t rate of interest is paid when a note, payable iu 3C2 days, is discounted at 10 per cent. ?. IMAGE EVALUATION TEST TARGET (MT-3) 7 // {/ :/ ;w i^. «^: ^. 5r /^./i 1.0 I.I 1.25 iia iiiM IIM IIIII2.2 iiii 1.4 1.6 Photographic Sciences Corporation ^ iV \\ ^9) V ^\/#^\ ^^.* % n^ #1 :^^ 23 WEST MAIN STREET WEBSTER, NY. I4S80 (716) 872-4503 ^MP., Q. (/x I 186 ahithmetic. SOLUTION. If we discount $1 for the given time, and at the given rate, the proceeds will bo .90, or 90 cents. Hence, the discaunt being 10 cents, we are paying 10 cents for the use of 90 cents. Now, if we pay 10 cents for the use of 90, for the use of 1 cent we must pay 7j',j of 10 cents, or ^ of a cent, and for $1, or 100 cents, wa must pay 100 times ^ of n cent, or -igi^^.llj, and for $100, $11^, or 11 » per cent. Therefore, to find the rate of interest corresponding to a given rate of bank discount, we deduce the following RULE. Divide the given rate per cent., expressed decimally, or tlu rate per unit, by the number denoting the proceeds of $1 for the given time and rate. The quotient vaill he the rate of interest required. EXERCISES. 2. What rate of interest is paid when a note, payable in 60 days, is discounted at 7 per cent. ? Ans. l-i^ij. 3. AVhat rate of interest is paid when a note, payable in 3 months, is discounted at G per cent; ? Ans. C j^g^g. 4. A note, payable in G months, is discounted at 1 per cent, a month ; what rate of interest is paid ? Ans. 12^|J. 5. What rate of interest is paid, when a note of $200, payable in 70 days, is ■''iscounted at f per cent, a month ? Ans. O^^^'j. 6. When a note of $45, payable in 05 days, is discounted at 7 per cent., to what rato of interest does the bank discount correspond ? 'Ann 7 833 7. A bank, by discounting a note at 6 per cent., receives for its money a discount equivalent to G^ per cent, interest j how long must tho note have been discounted before it was duo ? Ans. 1 yr.j 3 mos., 12d. COMMISSION. GoM^iissiON is the term applied to money paid to an agent to remunerate him for his trouble in buying, selling, valuing, or for forwarding; merchandise or other property. The goods sent to a commission merchant or agent, .to be sold ou account and ri^^k of another, are termed a consignment. COMMISSION. 187 The person to whom these goods are consigned is called the con- si^*^e or correspondent. Tb* term shipment is sometimes used instead of consignment. K X A M r L E , A •ommission merchant sells fur me goods worth §1200, and . c^iargcff 1 per cent. ; what have I to pay him ? SOLUTION. 4 per cent, of 61200 is equal to $1200X-04=:848. Ilcncc I would have to pay $48, and from this wc deduce the following RULE. Find the j^crcentaf} :. on the given sum at the given rate, which icill he the commission. EXEKCISES. 1. Consigned to A.K.Boomer, Esq., Syracuse, by the Troy, N.l ., foundry, agricultural implements which arc sold for $1875.75 , what is the agent's commission at 2J per cent. ? Ans. §40.80. 2. Bought in Boston 12 chests of tea, containing 64 lbs. each, at 61.12^ per lb., on a commission of 1^ per cent. ; what was my com- mission ? . . Ans. $15.12. 3. My Toledo correspondent has bought for mc 2768 lbs. of bacon, at 12^ cts. a pound ; what is his commission at 3,} per cent.? $11.25. 4. Bought a carriage and pair of horses, per Iho order of S. Williams, Portland ; paid for the horses $240, and charged 4^ per cent., and paid for the carriage $160, and charged 1^ per cent. ; how much did I earn ? Ans. $13.20. 5. A commission agent in a Southern State bought cotton worth 82284 for an English manufacturer, and charged 5^ per cent.; what is his commission ? Ans. $125.62. 0. On another occasion the manufacturer gave the conimissiou merchant $165.78, for purchasing for him cotton worth $3684 ; what was the rate per cent ? Ans. 4^ 7. An English.commission merchant buys for a Portland house, X576" 10s. Od. worth of provisions, and charges 4^ per cent. ; wliat is his commission? Ans. £25 ISs. 10],d. 8. A Now York provision merchant instructs a Belfast (Ireland) commission merchant to purchase for him £534 4s. Od. worth of I, 188 ARITHMETIO. bacon and hams, and offers him 7 J per cent. ; what does the agent get? Ans. £38 14s. 7d. 9. A book agent in Cincinatti, sells $487.50 worth of books for Day & Co., of Montreal, and receives $72.05 for his trouble; at what rate per cent, was he paid ? Ans. 15 nearly. 10. An agent sells 84 sewing machines at $25 each, and his 'commission amounts to $262.60 ; what is the rate ? Ans. 12i. When a sum has to be sent to a commission agent, such that it will be equal both to the sum to be invested, and the agent's com- mission, it is plain, as already noted, that this is merely a case of percentage. It is the same as the first part of case IV., and we will have the corresponding RULE. Divide the given amount hy 1, increased hy the given rate per unit, and the quotient will he the sum to be invested ; subtract this from the given o mount, and the remainder will be the commission. EXAMPLE, If I send $1890 to a commission merchant, and instruct him to buy merchandise with what is loft after his commission at 5 per cent. is deducted ; what will be the sum invested, and the agent's com- niissiou . SOLUTION. It is plain that for every dollar of the proposed investment I must remit 105 cents, 100 towards the investment, and 5 towards the commission, and hence the number of dollars which can be in- vested from the sum remitted will bo the same as the number of times that 1.05 is contained in 1890. Now, $1890-1-1.05 gives $1800, the sum to be invested, and this subtracted from $1890, leaves $90, the commission to which the agent is entitled. EXERCISES. * 1. Remitted to A. B., St, Paula, $988 to purchase flour for me with the balance that remains after deducting his commission at 4 per cent. ; required the purchase momy and percentage ? Ans. $950 and $38. 2. Received a commission to bay wheat with $779, less by my commission at 2J per cent. ; required the price of the wheat and my commission. Ans. $7G0, and $19. buokepu^ge. 189 3, Remitted to my co^rt^pcndent to Augusta $266.76, to pay for lumber which ho purchased for pie, and to pay his own commis- siou at 4 per cent. ; what was the price of the lumber, and what tho commission?. Ans. $256.50, and $10,26. 4. John Jones, Newmarket, commissions W. Orr, Portland, to procure for him a quantity of fine flour, and remits $917.01 ; how much flour can he have, after allowing 4^ per cent., and what will the commission amount to ? Ans. $870, and $41.01. 5. John Stalker, London, commissions J. Fleming New York, to purchase for him as much butter as he can procure for the balance between $779.52, and his own commission at 1 J per cent. ; how many pounds butter did he get at 25 cents per lb. ; what the whole price, aud what was the commission? Ans. 3072 lbs., $708, and $11.52. 0. Dr. Gallipot is about to remove to England, and sends to a London cabinet maker $4005.45 towards getting his house furnished, he is charged 3| per cent, over and above the price of tho furniture, lor time and labour, what docs tho furniture cost ? Ans. $3870. 7. Graham Bros., of Nowbury, spnd to R. White, Charleston, bacon and hams worth $1560, they charge 5^ per cent, commission, and tho charge for lading is $75.15; how much does R. White owo them ? . Ans. $1720.95. 8. P. Robson, commission merchant, Albany, buys for T. Black & Co., Baltimore, groceries, the price of which, together with their commission at 4 per cent, comes to $475.02 ; what was tho price of tho goods, and what was the amount of the commission ? Ans. $456.75, and $13.27. 190 ARITHMETIC. BROKERAaE BnoKERAaE is a per centage paid to an agent for negociating bills, exchanging money, buying and selling railroad, bank, and building society stocks, Government bonds and gold. Such an agent is called a broker. A smaller per centage is usually allowed to a broker than to a commission merchant, because the work he has to do requires less time and labor. Brokers charge, generally, one-eighth of one per cent, for buying or selling stocks, bonds, gold, &c., and it is always reckoned on the par, or fiico value. For instance, if a broker were to purchase for you a share of N. Y". C. R. R. stock at 112^, or 12^ per cent, premium, the brokerage would be ^ per cent, on $100, and not on the $112.50. The charge would still be the same if purchased at 85, or 15 per cent, discount. In gold operations,- the brokerage is calculated on the gold, although the brokerage itself is taken in currency. For in- stance, if a broker purchases $10,000 in gold for a customer, the charge would be ^ per cent, on $10,000, viz. : $12.50 in currency. A great many of the transactions made by brokers consist in the buying and selling of Government Bonds, called " Five-Twenties," " Ten-Forties," and " Seven-Thirties." The " Five-Twenties" arc so called because they are payable, at the option of the Governiaent, at five years atYer their date, or at the end of twenty years. The " Ten-Forties" are payable, at the option of the Government, ten years after their date, or forty. The " Seven-Thirties" are so called because they bear interest at the rate of seven and three-tenths per cent, per annum — (7/o"o)' In buying or selling those bonds, the seller of a " Seven-Thirty" always receives the interest that has accrued on it, from the time of last payment of interest by Government, until the time of sale ; but in all other bonds the buyer has the bcUcfit of the interest. The only reason that can be assigned why the " Seven-Thirties" should bo an exception to the general rule, is, that the interest is so easily calculated, being just one cent per day on every fifty dollars. In reality Ihe result is the same, because, if in the " Seven-Thir- ties" the buyer received the interest, the quotation or market value of them would be greater. BEOKERAGE. 191 Brokers frequently, among each other, buy and sell bonds, ^old, &c., at .'{0 days, using the terms " seller 30" or " buyer 30," which signifies, if the term "seller 30" is used, that the seller of the stock can, any time during the thirty days, deliver the stock to the buyer, and deceive his money ; if " buyer 30" is used, the buyer has the privilege of calling in the stock bought, any time during the 30 days. , This practice, an will bo seen at n glance, gives a great range for speculation. To illustrate : Suppose that A sells to B 500 shares of Erie II. R. stock at 64 or 36 per cent, dis., *' seller 30," now it is not at all probable that A has this stock on hand that he has sold to B, but expects to be able to purchase it before the expiration of the 30 ^ays, at something less than 64. This is called selling "short." It not unfrequently occurs, in transactions such as just mentioned, that A may not wish, or B require, the delivery of tl»e stock when the time arrives ; if this is the case, A simply pays, or receives from, B the difierence between what the stock was "sold at, and what- it is worth at the time of settlement. In purchases like the above, unless the parties are known to be reliable men, a certain amount of money must be put up, termed a "margin," that may be considered sufficient to cover fluctuations in the value of the stock. If either party is unable to meet his part of the contract, the term " Lame Duck" is applied to him. Any person may buy or sell stock thrcugh a broker at " buyer 30" or " seller 30" by putting up what the broker may consider to be a sufficient Interest is generally allowed on this margin. Where there arc a number of brokers operating in any one stock, they are, according to brokers' phrases, divided into two classes, called " Bulls" and " Bears." The " Bear" is always the seller, while the buyer is always a " Bull." If A sells to B stock at 97, " seller 30," he is evidently a " Bear," as it is to his interest to constantly boar down the price of the stock he has sold, so that he may be able to purchase at a price less than 97, while it is always the interest of the buyer to " Bull" or raise its value. The par vAlue of stock in the following examples is considered to bo $100, and the brokerage ^ per cent., unless otherwise mentioned* Some of the answers reqaested may not belong, legitimately, to ques* tions in brokerage ; but the teacher or learner may ask or give but .^. 192 ARITflMETIO. one answer. Like commission, brokerage is merely a partictdar case of per ccntage, and hence the RULE. To find the brokerage on any sum, find the per centaje on the given sum at the given rate, which will he the brokerage. 1. I purchased for C. R. Sinp; 10 shares of Iludson River R, R. stock at 103 ; what is the brokerage ? Ans. $1,25, 2. Sold the above stock for the same person at 103| ; what ia th« brokerage ? Ans. $1 .25. 3. Bought for J, C, Baylies 50 shares of N. Y. C. R. R. stock at 107i^ i what is my brokerage ? Ans. $G,25. 4. Sold for Kimball & Co. $5000 in gold, at 137^ ; what is my brokerage at § per cent. ? , . Ans. ftl 8.75. 5. Purchased through my broker 100 shares Harlem R, R, stock at 109|-, " buyer 30 ;" at the expiration of the 30 days he sold the same, per my order, at 110^ j what was my gain, and what the bro- kerage ? Gain, $100 ; brokerage, $25. G. Paid a broker f per cent, for exchanging $245 fractional cur- rency for bills of a larger denomination ; what is the brokerage ? 7. My broker has purchased for mo a " Five-Twenty Bond" for $4500 at 108^ ; what ia the brokerage at f per cent., and what docs it cost me ? 8. Instructed a broker to purchase for me Seven-Thirties to the amount of $7500, which he did on March 8th, 1867, at 107|; in- terest on this bond is payable on the Ist of January and July ; what is the brokerage, and its cost to me ? Ans. Brokerage $9.37^; Cost $8181.75. 9. I purchased through a broker $15,000 gold, at 134 J, he sells it out for mo in a few days at 131 J; brokerage on purchase ^ per cent., on selling -f\ ; what is the brokerage, and what my loss ? 10. A broker purbhascd for mo 150 shares of Michigan Central Btock, at 87^ — brokerage f per cent. ; 80 shares Reading R. R. stock, at 102^ — brokerage ^ per cent., and $3000 of Ten-forties, at lOlf ; what is the brokerage and fall cost ? MISCELLANEOUS EXAMPLES. 103 MISCELLANEOUS EXAMPLES. to the 1. A commission merchant has purchased for me 5G48 lbs. long cut liamp, at 14J cts. per lb. j what is his commission at 1^ per cent. ? Ans. $10.06. 2. My agent in Richmond has purchased cotton for me to the amount of $1785.80 and charges me a commission of ^ per cent.; how much have I to remit him to pay for the cotton and commis- sion ? Ans. $1801.42i. 3. I remit J. Purdy, com. mer., New Orleans, $1142.40, in- structing him to invest it in cotton at 32 cts. per lb. ; after deduct- ing his commission at 2 per cent., how many lbs. of cotton do I re- ceive ? Ans. 3500 lbs. 4. Morrison & Thompson have sold for me 112 bbls. white fish, ftt $9.50 per bbl., and 85 bbls. flour at $12.40 — commission 2^ per cent. I have instructed them to invest the proceeds in bacon, at 13J cts. per lb., after deducting their commission at 1^ per cent. ; how much is the commission, and how many lbs. of bacon do I re. ceive ? 5. A purchased, per the order of Andrew Campbell & Co., Nash' ville, Tenn., 14872 lbs. C. C. bacon at 13J- cts. per lb., charging a commission of 1^ per cent. A wishes to draw on them for reim- bursement ; what must be the face of the draft if it cost ^ per cent, to get it cashed, and what is the commission on purchase ? Ans. Face of dft. $2010.15; Commission $29.56. 6. I have received, from a correspondent in Troy, $4781.25, vrith instructions to invest the same in Five-twenties, at 105j^, first deducting my commission at f per cent. ; what is the brokerage, and (rhat amount of Five-twenties can I purchase ? Ans. Brokerage $33.75 ; invested in Five-twenties $4500. 7. An accountant is entrusted to make schedules of the debts ind- assets of a bankrupt ; he charges only 2^ per cent, on the debts, jn the prinoiple that he will have little trouble in gett'jg the accounts duo by the bankrupt sent in ; but aa he knows very well that he will 13 194 ABITH3IEnC. have trouble in getting correct statements sent in of accounts due to the bankrupt, he stipulates for 5 J per cent, on those ; how much does he get altogether, the debts being $2786, and the assets SGI 8 ? Ans. $103.64. 8. I have sold for Walker & Smith, Cincinnatti, a consignment of 100 bbls. of pork, at 27Y'g"g per bbl. I have paid out for charges $31.40 — my commission is 2 J per cent. I remit them their net proceeds by draft on Cincinnati, purchased at f per cent, discount, charging ^ per cent. com. on face of draft ; what commission do I deceive, and what is the face of dft. that I remit them ? Ans. Commission $72.08 ; Face of dft. $2666.52. 9. On the 14th of March, 1867, a broker purchased for B, 100 shares of Erie R. R. stock, at 71 ; 50 shares C. and R. I. R. R. stock, at 95|; 200 shares N. Y. C. R. R. stock, at 103J, and a Seven- thirty bond for $6000, interest payable Dec. and June, at lOGi. They were sold on April 12th at 68 J, 97f, 103 J, and 106^, respec- tively ; what is the brokerage at J per cent, for buying, and ^ for selling, and B's gain or loss on the transaction ? 10. I sent to Taylor & Morrison, com. merchants. New York, 250 firkins butter, containing on an average 56 lbs. each, at 15 cts. per IJ). They sold at an advance of 10 per cent. ; freight, &c., de- ducted $10.45, commission 2 J per cent. They have remitted me a sight draft for net proceeds, which they purchased at f per cent, premium, charging J per cent, commission on face of draft. What amount of draft did I receive, and what amount of commission charged? aW' INSURANCE. 195 INSURANCE. Insurance is an engagement by which one party is bound, ii> consideration of receiving a certain sura, to indemnify another for something in case it should in any -Why 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 Compavy," or " Assurance Com- pani/," such as the " Globo Insurance Company," the " Mutual Insurance Compojy." Some companies arc formed on the principle that each individual shareholder is insured, and shares in the profits, and bears his portion of the losses. Such a company is usually called a Mutual Insurance Company, The sum paid to the party taking the risk is called the Premium of Insurance, or simply the Premium. The document binding the parties to the contract, is called the 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. The 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 are different kinds of insurance which may be classified under three heads. Fire Insurance, including all cases on land where property is ex- posed to the risk of being destroyed by fire, such as dwelling houses, stores and factories. Marine Insurance, — This includes all insurances on ships and cargoes. Such an insurance may be made on the ship alone, and in that case 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 bo made on the cargo alone, and is then usually called Cargo Insurance, It may be made on both ship and cargo, in which case the general term Marine In- surance will be applicable. This kind, as the name implies, insures, against all accidents by sea. Life Insurance, — This is an agreement between two parties, that in case the one insured should diQ within a certain stated time, the other shall, in consideration of having received a stipulated sum annually, pay to the lawful heir of the deceased, or some one men- 19G ARITHMETIC. tioned in. his will, or soue other party entitled tJicrcto, the amount recorded in the policy. For instance, a man may, on the occasion of his marriage, insure liis life for a certain sum, so that should he die ^vithin a certain time, his -widow or children shall be* paid that sum hy 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 be paid the sum agreed upon, or that the child, if it lives to a certain age, shall be entitled 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 ».'ase A should fail to do so during his life time. In some instances, insurances are cfiFected to gain a support in case of sickness. Such a contract is called a Ilealth Insurance. In- surances arc now also eflfected for compensation in case of railway accidents. These we may call Railway Accident Insurances, A policy is often transferred from one party to another, especi- ally as collateral security for debt or some 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 the sum insured, but also the instalments pre- viously paid. In many companies a person can insure in such a way as to be entitled to have a share of the profits. The 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 the sea. Wc know that it was practised, in a certain way, by the ancient Greeks and Romans. If a Koman merchant sent a cargo to a distant port, he made a contract with some one engaged in such business, that he would advance a certain sum, to be repaid with interest, if the vessel reached her destination in safety, but should th& vessel or cargo, or both be lost, the lender was to bear the loss. This was termed respondentia, (a respondence) a term corresponding pretty nearly to the English word repayment. It was lawful to charge interest in such cases, above the legal interest in ordinary cases, on account of the great- ness of the risk. The lender of the money usually sent an agent of his own on board the vessel to look after the cargo, and receive the repayment on the safe delivery of the goods. This agent corros* ponded pretty nearly to our more modern supercargo. As the art of navigation advanced, and the securities afforded by law beouae INSURANCE. 197 reat- 3nt of the VQ lonos' U art Loune more stringent, and also facilities of communication increased, this eystcm gradually gave way, and has eventually been supplanted by communications by post, and telegraphic messages to agents at the ports of destination. With regard to the cquitableness of insurances, and their utility in promoting commercial cxterprise, wc may remark that they make the interest of every merchant, the interest of every other. To show this, wc may c jniparc an insurance office to a cluh. Suppose tho merchants of a town to form a club, and establish a fund, out of which every member, if a loser, was to be indemnified, it is plain that no loss would fall on the individual, except his share as a mem- ber of the club. Even so the insurance sy?*.. the insurance system, loss is. virtually distributed over a large commu- niiy, and therefore falls heavily on no individual, from wlrch we draw our conclusion, that it is equivalent to a mutual mercantile indemnification cluh. We must now show the rules of the club, and principles on which its calculations are made. The principal thing to be taken into account, in all insurances, h the amount of risk. For f xample, a store, where nothing but iron is kept, would be considered safe; a factory, where fire is used, would bo accounted hazardous, and one where inflammable sub- stances are used would be 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 be 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 the gen- eral health of the family relations. Connected with this is the cal- culation of the average length of human life. Almost all the calculations in iasurance come under two heads. First, to find the premium of insurance on a given amount, and at a given rate \ and, secondly, to find how much must be insured at a l^v*^ AIUTIIMETIC. ffivou rnlo, so o rooovoivtl. As Iho invniimn in .coknnod iia po iinu'li liy llu' liimilrod, iiisur- niioo i« iin'ivly « piWiiodlur Oiiso ol" jM'ifi'ntiijfo. lIcinHi (o lltitl llin jtroiuimn of insmunoo on i\ny givon lunouivl, at u f^ivon I'ulo iior cent., ^•0 ilodvioo tUo Ibllowhi}; u r I, F, . » Mnltiiiiif the yivcit amount hy the. nUc jm- uin't." F, X A M I' I, KH . 1. To find tlio -.••of>t of insiiriiij;- a Mook of Ittuldinps vnlnctl nt $20S"8, at it )or oont. ? Horo wo liavo .('(> for (ho nito jht unit, inxl $::G8vS \ .(H) ,T i (• 1 .2S. Uio a\i.s\vor. 2. \Vh!)t will lio (ho cost of iofivuinj;- a oinivo woidi ?.'1(>7I>, at '.\ ivr oont.? The riito \wr unit iB .(Kl, and §;:!(;T!»X.03- $no.:;7, tho answoi". 3. A pMitlonian oniplovoil a hrokor (o ii\Hino his n^sidonro and outhousos, v:i'iU<'vl «( $L\Tl)(). (ho j«a(o hoiic; S por oon(., and (ho hro- kor's* ohai'ir'^ \h ivr wnt, ; how innoh had ho to pay ? Tho oom( of insurance is 8'J!7(iOX-08~-$220.SO, and (h«< hrokorayo ItjlLlO, whicli {\ddcd to $1220.80, will }\ivc §2i)2.20, tl»o answor. E X K U C 1 S K S . What will 1h3 the prouiiuni ^A' insuranoo on goods wor(h $12S0, at r»^ jwr wnt. ? An.-*. $70.|(). 2. A ship aitd oarp^, valuod at $8.'), 000, is insuivd at 2} per cent, ; what is tho pivniiuni ? Ans. $l!)l2.riO. 3. A ship worth 6;)r).000, is i\i,snrod at 1 \ jvr oont., and hor Ci\rgi\ worth $55,000, at 2| per cent. ; what is thp wliolo oost ? Ans. $1000.00. •1. What will bo tho cost of insuring a Imildiivj; valued at §58,000, at 2\ iH^r cent. ? Ans. $1450.00. • It is |>lain that tlio ratooau lio (oaud, if tlio amount and proinhna uro given, and tho nmouat can bo found if tho rato and pivniiiim aro givon. In tho oa^o of insuring proporty, a prol'i'ssional survo^or U oUcu ouiployod to value it. and likowiso in tho oaso of lilo iiusuraiicf, a inodical corlillcato is reqniroti. and in oach caso tho foo must bo paid l»y the poi*soa iusnrod. As 100. tlio biuns ofporoontajro, is a constant quantity, wlion any two of tho otiior quantities arv given, the third can be louud. Jf iNHimANcn. 100 5. WImt must I pay to iiiHum a liouso valued at 8BO.^.r>0, nt, '^ jwr oont. '{ (i. A villfljro pinro vrfiH valnml nt 611*^'^; tli" propriolor iriflnrorl il I'or h'ix yt'iirs ; Mionitd lor <|i(> liiHt ^cnr wiw 'J\ pir nut., with a rcdiirlion ol'^ cnoli fliiciuv-diii^ ycmr ; Mio Hf^x^k iiiMinlaiiKMl jm uvcr- uyo viiliio til'SUWiH, itml wiiR itiHiirntl ofidi (if tlio hIx yonrK, iit 2| por emit.. ; Iiuw iiiuoli tlid llio pnipiioUtr piiy Tor iiiRurutioo during; flio six yours/ AiiH. $U!>7.r>;{. 7. \ ptoro nnd yard w^ro valund at, 8ll!H(), niid itiRiirod at I jf [ct Ponf. ; llio polioy and survoyor'M Ibo oaiiio to . Tho coHt ol' limurin;:; a factory, valued at 825,000, in $lL'r»; wlint JH Iho rato por cont. ? Arift. ^. 10, A ]\ jjor oont. itiHurinpj my dwelling liouso cost mo $50; wlmt i.4 tho valuo of tho Iiouho ? An.s. $4000.00. To find how mtioh inust bo iuHurcd for, ro that in oaHO of losfi, both principal and promiuiu may bo rocovorcd. Iloro it is obviouH that tho Bum itiHurfid for mu«t exceed tho valuo of tho property in tho samo ratio that 100 cxccodr* tho rate. K X A M 1' U K . To lind what sum must bo iiiHurcd for on property worth 8000, . at 4 per v!ont., t^) Hccuro both jiroporty and pnimium, wo liavc a.H ^l()0_t, $!)(}: $100:: $000 : K. J'. "'"'jY"" $025, tho Bum. ro(piircd. Taking tho rato per unit wo lind -?oo-'"-^/o"o=^ • ''^^' This gives tho n V L K . Divide the value nf the proper ttf hi/ 1, diminished by the rate per unit) and the qxtoticiit will he the sum required. K X A M f l< E H . 1. A foundry is valuod at $874 : for what Bum at 8 per cent, must it bo insured to sccuro both the value -of tho property and tho premium? One minut tho rate or 1.00— .08— .92, and $874-7-.02 ==8950, tho answer. 200 ARITHMLTIC. The premises of a gunsmith, "who sells gunpowder, arc valued at 82618.85: for how much, at 15 per cent., must they be insured in order to recover the value of the property and also the premium of insurance ? Subtract .15, the rate per unit, from 1, and the remain- dcr is .85 and $2G18.85-7-.85 gives $3081, the sum required. EXERCISES 1. *A chemist's laboratory and appurtenances arc valuea ac $20,250, for what sum' should he insure them at 6J per cent., to secure both property and premium ? $28,000. 2. A New York merchant sent goods worth $1,186 by water conveyance to Chicago ; he insured them from New York to Buflfalo at 1^- per cent., and from Buffalo to Chicago at 2^ per cent., and in both cases so as to secure the premium as well as the cargo ; how much did the insurance cost him ? Ans. 645.42. 3. A' person owned a flour mill, valued at $1846.05, which ho insured at 1^ per cent. He also owned a flax mill, valued at ■ $846.30, which he insured at 2^ per cent., and in both cases at such a sum as to secure both property and premium. Which cost him most, and how much more ? Ans. The flour mill cost him $1.07 more than the other. ' 4. Collins & Co., of Philadelphia, ordered a quantity of pork from G. S. Coates & Son, Cincinnati, which amounts to $2423.10. They insure it to Pittsburg at ^ per cent., and from Pittsburg to Philadelphia at 3 per cent., and in all cases so as to secure both the price and premium. IIow much does the whole insurance come to ? Ans. $87.12. 5, In order to secure both the value of goods shipped and the premium, at 1;J per cent., nn insurance is effected on $1526.72. What is the value of the goods ? Ans. $1500.00. 0. The Mechanics' Institute is valued at §18,000 ; it is insured at 1 J per cent., so that in ease 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 cargo worth $40,000, at J per cent., to secure both the value of the cargo and the cost of insurance ? Ans. $40,201.00. LIE*E INSURANCE. 201 8. The Rossin House, King-street, Toronto, is valued at, say, $150,000, and is insured at 1| per cent, so that in case of another conflagration, botli the value of the property and the premium of insurance may bo recovered. For how much must it be insured ? Ans. $152,G71.7G, nearly. 9. A jail and court-house, adjoining chemical works, and there- fore deemed liazardous, will not be insured under 2^ per cent. IIow much will secure both property and premium, the valuation being $17,550.00? ■ Ans. $18,000.00. 10. A cotton mill is insured for Sf2,000, at 4 per cent., to secure both premium and property. What is the value of the property ? 11. "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. IIow 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. LIFE INSURANCE. A Life Insuranck may bo effected cither 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 tlie insured should die within tho specified time. The latter is called a Life Insurance, because it is dcmandablo at death, no matter how long the insured may live. Tho rate per annum that tho insured is to pay is reckoned from tables constructed on a calculation of the average duration of lifo beyond different ages. This calculation is made from statistical returns called Bills op Moutalitv, and the result is called TiiE Expectation op Life. Tho annual premium is fixed at such a rate as would, at tho end of tho expectation of lifo, amount to tho sum insured. From tables of the expectation of lifo other tables are constructed, show- ing tho premium on $100 for ono year, calculated on tho eupposi- tion that it is to bo paid annually in advance. 202 ABmrMTlTIO. LIFE INSURANCE TABLE. Ago next BirlUday. 1 year. 7 years. For Lifo. Ago next Birthday. lycar. 7 years. For Life. 15 .83 .85 1.44 38 1.19 1.28 2.75 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 1.31 1.47 3.19 20 .88 .91 1.G2 43 1.35 1.54 3.32 21 .89 .92 1.G6 44 1.40 1.62 3.45 22 .90 .93 1.70 45 1.47 1.71 3.G0 23 .91 .95 1.74 46 1.54 1.80 3.75 24 .92 .96 1.79 47 1.62 1.90 3.92 26 .93 .98 1.84 48 1.71 2.02 4.09 2G .95 .99 1.89 49 1.81 2.14 4.27 27 .90 1.01 1.94 50 1.91 2.28 4.46 23 .98 1.03 2.00 51 2.03 2.42 4.67 29 .99 1.05 2.06 62 2.15 2.59 4.89 SO 1.01 1.07 2.12 63 2.29 2.76 5.12 31 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 56 2.78 3.38 5.89 34 1.09 , 1.16 2.40 57 2.96 3.G2 G.19 35 1.11 ' 1.19 2.48 58 3.17 3.87 6.50 36 1.14 1.21 2.50 59 3.39 4.17 6.83 37 I.IG 1.24 2.65 GO 3.64 4.50 7.18 EXAMPLES. Supposing a young man, on coming of ago, wishes to effect an insurance for $3000 for the whole period of his life. To find tho annual premium wliich ho must pay, we look for 21 in tho left hand column, and opposite that, in the Column headed for life, we find tho numher 1.66, which is the premium for one year on $100, and |-gg=ir.0106 is the premium on $1 for 1 year, and hence $3000x .0166=849.80, is tho whole annual premium. If the insurance is to last /or seven years only, wc find under that heading .92, and ^,!'jJJg=.092, and $3000X.092=$27.60, tho annual premium. If the insurance is to be for one year only, we find .89 under that bead, and $3000X.089=$26.70, tho premium. LIFE TKSURA!.'CE. 203 From these explanations we can now derive a rule for finding the annual premium, when the age of the individual and the sum to he insured for are known. RULE Find the age in the left hand column of the table, and opposite this in the verttail column for the given j)criod will he found the premium on $100 for one year, and this divided by 100 will give the p'emium on $1 for one year, and the given sum multiplied by this will be the whole annual j)rcmium. '^ EXERCISES. 1. What will bo the annual premium for insuring a person's life, who is 18 years old, for 61000 for 7 years ? Ans. $8.80. 2. What amount of annual premium must be paid by A. B. Smith, who wishes to insuie his life for 7 years for $2000, his age being 25 years ? Ans. $19.G0. 3. John Jones, 35 years of ago, wishes to effect an insurance for life for $1500. What amount of annual premium must he pay ? Ans. $37.20. 4. A gentleman in Chicago, 32 years of age, being about to start for Australia, and wishing to provide for his family in case of his death, obtains an insurance for seven years for $8000. 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 acdident 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 ago, gets his life insured for $3000, by paying an annual premium of $4.4iJ on each $100 insured ; if lie should die at the age of 75 years, liaw 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 wo reckon simple interest at 7 per cent, on his payments, what is gained by the insurance ? Ans. $3911. JUJ ±, »"%ij(i 204 AEITHMETIO. Profit and loss. In the language of arithmetic, the expression Profit and Loss ia usually applied to something gained or something lost in mercantile transactions, and the most impprtant rule relating to it directs how to find at what increased rate above the cost price goods must bo sold to produce a fair remuneration for time, labour and expendi- ture ; or, in case of loss by unforeseen circumstances, to estimate the amount of that loss as a guide in future transactions. There are other cases, however, which we shall consider in detail. CASE I. When the prime cost and selling price are kno\Tn, to find the gain or loss. RULE . Find, hy the rule of practice, the price at the difference between the prime cost and selling price, which will he the gain or loss ac- cording as the selling price is greater or less than the prime cost; or, Find the price at each rate, and take the difference. . EXA31PLES. To find what is gained by selling 4 cwt. of sugar, which cost 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 lb.=i60, and at 12^ cents=$50, and §G0— $50=r$10. Again, if 120 lbs. of tobacco be bought at92,cts. per lb., and, being damaged, is sold at 75 cents per lb., the loss will be a loss of 17 cents in the pound, and 120 lbs., at 17 cents per lb., is $20.40 ; or, 120 lbs., at 92 cents, will come to $110.40, and at 75 cents, to $90, and $110.40— $90=$20.40. EXEROISES. 1. If 224 lbs. of tea be bought at 60 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 $0.10 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 \ liow much does he lose ? Ans. $5.40. PEOnr AND LOSS. 205 4. If a dealer buys V8 bushels of potatoes, at G2^ cents per bushel, and retails them at 87^ cents per bushel ; how much docs ho gain? ' Ans. $19.50. 5. A wine merchunt bought 374 gallons of wine, at §3.20 per gallon, and sold it at $3.35 per gallon ; how much did he gain ? Ans, §5G.10. CASE II. To find at what price any article must be sold, to gain a certain rate per cent.^ the cost price, and the gain or loss per cent, being known. RULE. Multiply the cost price hi/ 1 phis the gain, or 1 mimis the loss, E X A 31 P L £ . If a quantity of linen be bought for 75 cents a yard ; at what price must it be sold to gain 16 per cent. ? Since IG per cent, is IG cents for every dollar, each dollar in the cost price would bring $1.1G in the selling price, so that we have $1.1G X .75=.8 7, or 8 7 cents. EXERCISES. 1. Bailroad shares being purchased for $2500, and sold at a gain of 20 per cent. ; for what amount were they sold ? Ans. $3000. 2. A property having been bought for $2000 was sold at a gain of 10 per cent. For what was it sold ? Ans. $2200. 3. A horse was bought for $50, but, proving lame, was sold at a loss of 15 per cent. At what price was he sold ? Ans. $42.50. 4. Bought a horse for $897 and sold it at a loss of 11 per cent ; for what sum was it sold ? Ans. $798.33. 5. A merchant buys dry goods for $15G2 and sells them at a profit of 22 per cent. For what were they sold ? Ans. $1905.64. CASE III. To find the cost when the selling price and tho gain per cent, aro known. RULE. Divide the selling price hy 1 plus the gain, or 1 minu» the loss. To find what was the first cost of a quantity of flour which produced 8 per cent, profit by being sold for $127.44. 206 ARITHMETIC. vince the gain is 8 per cent, of the cost, it follows that each dollar laid out has brought in a return of $1 .08, and therefore the cost must have been as many dollars as the number of times that 1.08 is contained in 127.44, which is 118, and therefore the first cost must have been $118. EXERCISES. 1. if flaxseed is sold at $17.40 per bushel, and 13 per cent, lost, what was the first cost ? Ans. $20.00. 2. A dealer bought 116 hogs for $580, and sold them at a gain of 25 per cent. ; at what price did ho sell each on an average ? 3. If 13 sheep be sold for $52.05, and 25 per cent, gained on the first cost, how much was paid for each at first ? Ans. $3.24. 4. If 16f per cent, bo lost on the sale of linen at $1.25, what was the first cost ? Ans. $1.50. 5. If a quantity of glass be sold for $4, and 10 per cent, gained, for what sum was it bought ? Ans. $3.64, nearly. CASE IV. To find the gain or loss per cent, when tho first cost and selling price are known. SUXiE. Divide tJie gain or lots hy the first coat. EXAMPLE. If a web of linen bo bought for $20 and sold for $25, what is tho gain per cent ? Here $5 are gained on $20, and $20 is I of $100, therefore $25 will be gained on $100, i. e., 25 per cent, EXERCISES. 1. If a quantity of goods be bought for $318.50, and sold for $299.39, how much per cent, is lost ? Ans. 6 per cent. 2. If two houses are bough , the one for $150 and the other for $250; and the first sold again for $100 and the latter for $350, what per cent, is gained on the whole ? Ans. 12^. A grocer bays batter at 24 cents per lb. and Mils it at 30 cents per lb., what doea he gain p«r cent 7 Ana. 25. PROFIT AND LOSS. 207 4. If a cattlo dealer buys 20 cows, at an average price of $20, and pays 50 cents for the freight of each per railroad, what per cent, does he gain by selling them at $25.62^ each ? Ans. 25. 5. A tobacconist bought a quantity of tobacco for $75, but a part of it being lost, he sold the remainder for $60 : what per cent. did he lose ? Ans. 20. CASE V. Given the gain or loss per cent, resulting from the sale of goods at one price, to find the gain or loss per cent, by selling the same at another price. aULE. Find by lose III. the first cost, and then by case IV. the gain or loss per cent, on that cost at the second selling price. EXAMPLE. If a farmer sells his hogs at $5 each, and realizes 25 per cent. ] what per cert, would he realize by selling them at $7 each. We find by case III., that the cost was #4, and then by case IV. what the gain per cent, would be on the second supposition, that is $3-7-4=.75, or 75 per cent. EXERCISES. 1. If a grocer sells rum at 90 cents per bottle, and gains 20 per cent. J what per cent, would he gain by selling it at $1.00 per bottle ? Ans. 33J. 2. If a hatter sells hats at $1.25 each, and loses 25 per cent. ;* what per cent, would he lose by selling them at $1.60 each ? Ans. 4. 3. If a storekeeper sells cloth at $1.25, and loses 15 per cent. ; would he gain or lose, add how much, by selling at $1.65 ? Ans. He would gain 12 per crat. nearly. 4. A milliner sold bonnets at $1.25, and thereby lost 25 per cent. ; would she have gained or lost by selling them at $1.40 ? Ans. She would have lost 16 per cent. 6. A merchant sold a lot of goods for $480, and lost 20 per cent. ; would he have gained or lost by selling them for $720, and how much ? Ans. Ho would have gained 20 per cent. 6. A quantity of grain was sold for $90, which was 10 per cent, teas than the oost ; what would have been the gain per cent, if it had been sold for $150 7 Ans. 50. ■»«! 203 ABITHMETIC. 7. A grocer sold tea at 45 cents per pound, and thereby gained 1 2i per cent. ; what would ho have gained per cent, if ho had sold tho tea at 54 cents per pound ? Ans. 35. 8. A farmer sold corn at G5 cents per bushel, and gained 5 per cent. ; what per cent, would ho have gained if ho had sold the corn at 70 cents per bushel ? Ans. ISj'j. MISCELLANEOUS EXERCISES. 1. If I buy goods amounting to $465, and sell them at a gain of 15 per cent. ; what arc my profits ? 2. Suppose I buy 400i barrels of flour, at $16.75 a barrel, and Bell it at an advance of | per cent. ; how much do I gain ? Ans. $25.14. 3. If I buy 220 bushels of wheat, at $1.15 per bushel, and wish to gain 15 per cent, in selling it ; what must I ask a bushel ? 4. A grocer bought molasses for 24 cents a gallon, which he sold for 30 cents ; what was his gain per cent. ? Ans. 25. 5. A man bought a horse for $1 50, and a chaise for $250, and sold the chaise for $350, and the horse for 100 ; what was his gain per cent. ? Ans. 12 J. 6. A gentleman sold a horso for $180, and thereby gained 20 per cent. ; how much did tho 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 ? Aris. $750. 8. A carpenter built a house for $990, which was 10 per eent. loss than what it was worth ; liow much should -he have received for it so as to have made 40 per cent. ? Ans. $1540. 9. A broker bought stocks at $96 per share, and sold them at $102 per share ; what was his gain per cent. ? Ans. 6J. 10. A merchant sold sugar at G^ cents a pound, which was 10 per cont. 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 J per cent. ; what would he have gained per cent, if he had sold it at $5.25 per yard ? Ans. 24 Jf. 12. I sold a horse for $75, and by eo doing, I lost 25 per cent.; whereas, I ought to have gained 30 per cent. ; }iow much was he sold for under his real value ? Ans. $55. 9- ' » PROFIT AND LOSS. 209 13. A watch which cost mc $30 I have sold for $35, on a credit of 8 months ; what did I gain by my bargain, allowing money to ho worth 6 per cent. ? Ans. $3.65. 14. Bought 81 yards of broadcloth, at $5.00 per yard ; what must be my asking price in order. ta full 10 per cent., and still make 10 per cent on the cost ? Ans. $6.11 J. 15. A farmer sold land at 5 cents per foot, and gained 25 per cent, more than it cost him ; what would have been his gain or loss per cent, if he had sold it at 3^ cents per foot ? Ans. 12^ per cent. loss. 16. What must I ask per yard for cloth tliat cost $3.52, bo that I may fall 8 per cent., and still make 15 per cent., allowing 12 per cent, of sales to be in bad debts ? Ans. $5. 17. A merchant sold two bales of cotton at $240 each ; for one he received 60 per cent, more than its cost, and for the other 60 per cent, less than its cost. Did ho gain or lose by the operation, and how much ? Ans. loss $270. 18. Bought 2688 yards of cloth at $2.16 per yard, and sold one-fourth of it at $2.54 per yard ; one-third of it at $2.75 per yard, and the remainder at $2,90 per yard. Find the whole gain, and the gain per cent. Ans. $1612.80 and 27j\^*y per cent. 19. A flour merchant bought the following lots : — 118 barrels at $9.25 per barreL 212 « .*..... 9.50 « 315 " 9.12J « 400 " 10.00 « The expenses amounted to $29.50, besides insurance at ^ per cent. At what [Mrioe must he sell it per barrel to gain 15 per cent ? Ans., $11.05. 20. Bought 100 sheep at $5 each ; haying resold them at once and received a note at six months for the amount ; having got the note discounted at the Fourth National Bank, at six per cent., I found I had gained 20 per cent, by the transaction. What was the soiling prico q$ each sheep? Ans.. $6.19. 1« 210 AEITHMETia STORAaE. 'Wlien a charge is made for the accommodation of having goods kept in store, it is called storage. Accounts of storage contain the entries showing when the goods were received and when delivered, with the number, the description of the artiglcfi, the sum charged on each for a certain time, and the total amount charged for storage, which is generally determined by an average reckoned for some specified time, usually one month (30 days). • EZAMPLXS. 1. What will be the cost of storing wheat at 3 cents per bushel per month, which was received and delivered as follows :; — Received, August 3rd, 1865, 800 bushels ; August 12th, 600 bushels. De- livered, August 9th, 250 bushels; September 12th, 350 bushels j September 15th, 400 bushels, and October 1st, the balance. SOLUTION. 1865. Bush. Dayp. Bash. 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 . ■ !■■■■■ ■ t) Balance 1150 X 31 =35650 u 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 = 6^00 in store for one day, Oct. 1. DeUvered 400 Total « 50900 in store for one day. 60,900 bushels in store for one day would be Ae same as 50900-^30=1 696§ bushels in store for one month of 30 dai/s, and the storage of 1697 bushels for one month, at 3 centa per month, would equal 1697X.03=$50.91. It is customary, in business, when the number of articles upon which storage is to be chai^d, as found, contains a fraction let$ STORAGE. 211 than a Iwlf, to reject the fraction ; but if it is more than a half, to regard it as an entire article. From the solution of the foregoing example, we deduce the fol- lowing BULE. Multiply the numhcr of hishcls, hnrrels, or other articles, ly 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 en ivhich storage is to he 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 barrelb January 27, 200 barrels; February 2, 75 barrels. Taken out, , anuary 10, (jO barrels; January 30, 150 barrels; February 10, 190 barrels ; February 20, 300 barrels; March J, 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, 186G, 2310 bales ; January 16, 120 bales; February 1, 800 bales. Deli- vered February 12, 1000 bales ; March 1, 600 bales ; April 3, 400 bales ; April 10, 312 bales ; May 10, 200 bales. Required the num- ber of bales remaining in store on June 1, and the eost of storage up to that date, at the rate of 5 cents a bale per month. Ans. 218 bales in store ; $321.18 cost of storage. 4. W. T. Leeming & Col, Commission Merchants, Albany, in account with A. B. Smith & Co., Oswego, for storage of salt and gunpowder, received and delivered as fol'ov 3 : Received, January 18, 1866, 400 kegs of gunpowder and 50 barrels of salt; January 25, 250 barrels of salt; February 4, 15p barrels of salt, and 50 kegs of gunpowder; February 15, 100 kegs of gunpowder ; March 5, 64 kegs of gunpowder ; April 15, 50 kegs of gunpowder, and 75 barrels of salt. Delivered, February 25, 15 kegs of gunpowder, and 40 barrels of salt ; March 10, 150 kegs of gunpowder, and 286 barrels of salt; April 20, •200 kegs of gunpow- der ; April 125, 50 barrels of salt, and 200 k^s of gunpowder. Required the number of barrels of salt" and kegs of gunpowder in store May 1, and the bill of storage up to that date. The rate of V4-. .ft'i ^n ^v KmWiWS^^. \\\\\>i\\^^'\v\\ \\\ A *^\iv \S\AA\, iv\ v^\«,'», \ t\\Vi »i*Nt\ \\^ *^SV^^ H\«* \1WH\ l t\llHl»(ij4li UtH- V>\^ w *>v\ nVm\\^^'.^\A v^y^>^\w4 \*^\\^\\u\\Vv U\»'»i>\t'ii H li\('H('i'il nt *^ ln^tk^V !^VV\H^V t» \^V^^Wvv^ «t yV^v^ \\\m»»N \Ai\\\\ n\\\\\*s (l»Htl»\(a|i*«, ^■Viv «f«tt«» Vfcxx *\'' >y ^vvK*! <\^ (^ i>)vv^v- ^\»** •nil", w«»if /.(((*! ifi'fft, «/(// M//' h§\^im^ Ifi phit^ mtivi |«l'lllll|| Jiflllli I m-H. |"hf| tl//* (>\hHUii4 4IIIH ^nUtMi Hi 9if(f(f/0i , iliii liiM |iHi» «(tH( i ««/j nifftf WfN /i/« l//iwf'i> w^/ I'mi^lhHiHti iu loiMdl j» III |Im(|iI Id HJMlMltil' !/•«« Ijl HMI ilHuU'h fifll-'^- ttt Mif^J. ,i 4f/fVf )i|(»|iiiiiHiiM 1(1 |i(iM '^M ih'ftiiifimitt>nitiftf n4t^ I I'dlh ! 1 I (.fr(/// ^/'l 1/^*^ |,/.^ /^/(/>^. f/t Hjft^ H.Hi t(ll»Mi»IM m m 1 1, MM 'j^'H.f'/, t ^ ^V/f/?/^«' **/(yM Ilium U]h (Ih, li ifiiitu k fill '0 t4im I A Um * I ^\')\.W, ihiriit,H'ii fliifitii iif Um. $imm/fnhHMi0 I IIKIII III) «('|7 U(t $Jl\milU, hfnuM, ;/fry,t^,l^ f^ i HKil I:"!'! Iff »' l(>. ^f/'- >lf''m |M'/'< 1'V ^^ 'H |»M')r -1'. /.«, ix^r^s flirt ll(i|l»litiiHl I'ld*! Id im\*'t \ft \\U<\ \Stti hroportionato loss, it was allowed that the wheat would have sold in Halifax at an advance of 10 per cent., the corn at an advanco of taper cent., and the flour for $5 per barrel. The contributory iutcro.'its were: — Steamer, $95,000; cargo, $ ; gross freight, i23G1.20. The cost of repairs to steamer was $2198.15; cost arising from detention during repairs, $318; seamen's wages, $1252.50. How much of the loss had each contributory interest to bear ? 3. The steamer Edith left Baltimore for New Orleans with 7600 bushels of wheat, valued at $r.25 per bushel, shipped by Dunn, Lloyd & Co., and insured in the Hartford Insurance Company at If per cent., 9200 bushels of corn, valued at 75 cents per busliel, TAXES AND CUSTOM DUTIES. 215 shipped by J. "W. Roe, and insured in the -ffitna Insurance Company at 1 J per cent. ; 14,800 bushels of oats, valued at 37^ cents per bushel, shipped by Morris, Wright & Co., and insured in the Mutual Insurance Company at 1^ per cent. ; 1,800 barrels of flour, valued at $5.25 per barrel, shipped by Smith & Worth, and insured in the Beaver Insurance Company at IJ per cent. In consequence of a violent gale in the Gulf of Mexico, it was. found necessary to throve overboard the flour, 4,600 bushels jof oats, and 3,150 bushels of wheat. The propeller vras valued at $45,000, and insured in the Beaver Insurance Company for $12,000, at 2 per cent., and in the Western for $25,000, at 2J per cent. The gross freight was $4950 ; seamen's wages, $340, and repairs to the boat, $3953.75 ; what was the loss sustained by each of the contributory interests, the propeller being on her first trip ? TAXES AND CUSTOMS DUTIES. A tax is Vk money payment levied upon the subjects of a State or the members of any comirunity, for the support of the govern- ment. A tax is either levied upon the property or the persons of indi- viduals. When levied upon jthe person, it is called a. poll tax. It may be' either direct, or indirect. When direct, it is levied from the individuals, or the property in the hands of tho ultimato owners. When indirect, it is in the nature of a customs' or' excise duty, which is levied upon imports, or manuiacturcs, before they reach the consumer, although in tho end they are paid by the latter. Ctistoms' duties are paid by tho importer cf goods at tho port of entry, where a custom-house is stationfed, with government employees called custom-lwuse officers, to collect these dues. Excise duties oro those levied upon articles manufactured in, tho country. An invoice is a complete list of tho particulars . and prices of goods sent from 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 tho article. An ad valorem duty is a percentage levied on the actual cost, or fair market value of the eoods in the country from which they are imported. 216 ARITHMETIC. Gross weight Is the weight of goods, upon which a epecific duty is to be levied, before any allowances arc deducted. Net weight is the weight of the goods after all allowances arc deducted. Among the allowances made are the following: Breakage — an allowance on fluids contained in bottles or break* able 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 the goods. When goods, invoiced at gold value, upon which duty is payable, are imported into this country from any foreign country, the custom house duties are payable in gold, for else manifest injustice might be done. If the duty were payable in greenbacks, it would be neces- sary, in order to obtain uniformity, cither to increase or decrease the rate per cent, of duty, as greenbacks fluctuated in value, compared with gold (the invoice price of the goods), or else the goods imported would require to be reduced to their value in greenbacks at time of delivery. To avoid all this trouble and confusion; goods that are invoiced at their gold valne, the duties are made payable in the same currency. When goods arc imported from any country which has a depre- vsiated currency, a note is attached to the invoice, certifying the amount of depreciation. This is the duty of the Consul represent- ing the country to which the goods are exported, and resifling at the yixtfrom which they are exported. £ X A M P L X S . To find the specific duty on any quantity of goods. Suppose an Albany Provision Merchant imports from Ireland 59 casks of butter, each weighing G8 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 69x12= 708 lbs. Hence the net weight is 3304 lbs ' The duty is 2 cents per lb 2 The duty, therefore, is .....»66.08 TAXES AND CUSTOM DUTIES. 21T To find the ad valorem duty on any quantity of goods. Suppose a Troy dry goods merchant to import from Montreal 436 yards of silk, at $1.75 per yard, and that 35 ^/^r cent, dnty is charged on them. Here wo. find tho whole price by the rule of Practice to be $7G3, then the rest of the operation is a direct case of percentage, and therefore we multiply $7G3 by .35, which gives $267.05, the amount of duty on the whole. Ilence we have the following RULE FOR SPECIFIC DUTY. Subtract the tare, or other allowance, and multiply tne remain* der 1)1/ the rate of duty per. box, gallon, &c. RULE FOR AD VALOREM DUTY. Multipli/ the amount of the invoice by the rate per unit. at tho |bs. kbs. EXERCISES. 1. Find the specific duty on 5120 Ibs.-of sugar, the tare being 14 per cent., and tho duty 2;J cents per lb. Ans. $121.09. 2. What ia the ad valorem duty on a quantity of silks, tho amount of the invoice being $95,800, and the duty G2|^ per cent ? Ans. 859,875. 3. At 30 per cent., what is the ad valorem duty on an importa- tion of china worth $1200. ? Ans. $378. 3. What is the specific duty, at 10 cents per lb., on 45 chests of tea, each weighing 120 lbs., the tare being 10 per cent. ? Ans. $480. 5. What is the ad valorem duty on a shipment of fruit invoiced at $4500, tho duty being 40 per cent.? Ans. $1824. 6. What is the specific duty on 950 bags of cofiee,*each weighing 200 lbs., the duty being 2 cents pel: lb., and tho tare 2 per cent? Ans. $3724. 7. What is the ad valorem duty on 20 casks of wine, each con- taining 75 gallons, at 18 cents a gallon ? Ans. $270. 8. A. B. shipped from Oswego 24 pipes of molasses, each con- taining 90 gallons; 2 percent, was deducted for leakage, and 12 cents duty per gallon charged on tho remainder ; how much was tho duty? Ans. $270.95. ■ I "■■ i. 218 ARITHMETIC 9. Peter Smith & Co., I^rooklin, import from Cadiz, 80 baskets of port wine, at 70 francs per basket ; 42 baskets of sherry wine, at 35 francs per basket ; GO casks of champagne, containing 31 gallons each, at 4 francs per gallon. Tho 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. 10. J. Johnson & Co., of Boston, import from Liverpool 10 pieces of Brussels carpeting, 40 yards each, purchased at 5s. per yard, duty 24 per cent. ; 200 yards of hair cloth, at 4s. per yard, duty 19 cwt. ; 100 woollen blankets, at 2s. Ud., duty 16 per cent. ; and shoe-lasting to the cost of £60, duty 4 per cent, llequircd tho whole amount of duty, allowing the value of the pound sterling to be 84.84. Ans. $173.64. 11. John McMaster & Co., of Collingwood, Canada West., bought of A. M. Smith, of I3uffalo, N. Y., goods invoiced at 65440.50, which should have passed thrcfugh the custom-house dur- ing the first week in May, when the discount on American invoices was 43J per cent., but they were not passed until the fourth week in May, when the discount was 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. V^il;'-. ■■■-' If".:-. .:.:■■':•■■ ■■<■!*■ ■ STOCKS AND BONDS. Capital is a term generally applied to the property accumulated by individuals, and invested in trade, manufactures, railroads, build- ings, government securities, banking, &c. The capital of incorpo- rated companies is generally termed its " capital stock," and is divided into shares ; the persons owning one or more of these shares, being called stockholders. . The shares in England, aro usually £100, £50, or £10 each. In the United States they are generally $100, 650, or $10 each. The management of incorporated companies is generally vested in officers and directors, as provided in the law or laws, who are elected by the stockholders or shareholders ; each stockholder, in most cases, being entitled to as fuany votes as tho number of shares he holds; but sometimes the holder of a few shares votes in a larger proportion than tho holder of many. Tho accumulating profits which are distributed among tho stork- holders, onco or twice a year, are called " dividends," and when " declared," are a certain percentage of tho par value of tho shares. In mining, nud sonic other companies, where the shares are only a •<• • STOCKS AND BONDS. 219 few dollars each, the dividend is usually a fixoa sum "per share." Certificates of stock are iSsued by every company, signed by the proper officers, indicating tJio number of sliarea each stockholder is entitled to, and as an evidence of ownership ; these are transferable, and may be bought and sold like any other property. AVhen the market value equals their nominal value they are said to be " at par." When they sell for more than their nominal value, or face, they arc said to be above par, or at a " premium" ; when for less, thoy are below par, or at a " discount." Quotations of the market value are generally made by a percentage of their par value. Thus, a share which is $25 at par, and sells at $28, is quoted at twelve per cent, premium, or 112 per cent. When states, cities, counties, railroad companies, and other corporations, borrow large amounts of money, for the prosecution of their objects, instead of giving common promissory notes, as with the mercantile community, they issue bond's, in denominations of convenient size, payable at a specified number of years, the interest usually payable semi-annually at some well known place. These are usually payable to "bearer," and sometimes to the "order" of the owner or holder. When issued by Government.'! or States, these bonds are frequently called Government Stocks or State stocks, under authority of law. To these bonds are attached, what aro called " coupons," or certificates of interest, each of which is a duo bill for the annual or semi:annual interest on the bond to which it is attached, rcp'-esenting the amount of the periodical dividend or interest; whiijh coupons -vfrere usually cut off, and presented for payment as they become due. These bonds and coupons are signed by the proper officers, and like certificates of capital stock, are nego- tiable by delivery. The loan is obtained by the sale of the bonds, with coupons attached, but they are sometimes negotiated at par. Their market value depepds upon the degree of confidence felt by capitalists of their being paid at maturity, and the rate of interest compared with the rate in the market. Treasury notes are issued by the United States Government, for the purpose of effi^cting temporary loans, and for the payment of contracts and salaries, which I'csemblo bank notes, and are mado payable without interest generally. Recently such notes have beea issued bearing one year or three years' interest. "Consols" is a term abbreviated from the expression "consoli- dated," the British Government having at various times borrowed money at diffi}rent rates of interest and payable at different times, " consqlidated" the debt or bonds thus issued, by issuing new stocky drawing interest at three percent, per annum, payable semi-annually, and redeemable only at the option of the Government, becoming practically perpetual annuities. With the proceeds of this, the old stock was redeemed. The quotations of these three per cent, per- petual annuities, or " consols," indicate ordinarily the state of tho «^* 1* . ■■•■. ', "*.■;•• 220 AKirmiETic. ■m- money market, as they form a large portion of the British public debt. " Mortgage Bonds" are frequently issued by owners of real property, with coupons attached, which render the bonds more saleable as well as more convenient for the collection of interest. "Coupon Bonds," being negotiable by delivery, are payable to the holder ; and in case of loss or theft, the amount cannot be recovered from the government or corporation issuing them, unless ample notice is given of the loss. " Registered Bonds" are those payable only to ihe " order" of the holder or owner, and are more safe for investment. By law, stockholders are liable for the whole debts of the corpo- ration, in case of failure. In some States the law provides that they are liable only to an amount equal to their stock. In England tlu) statute provides for " Limited" liability, by an Act passed in 1862 termed the " Limited" Act." ASE I. The premium or discount being known, to find the market value of any amount of stock. EXAMPLES. If G. "W. R. shares are at 7 per cent, premium, to find the value of 30 shares of $100. Here it is plain that each $100 will bring $107, and that each $1 will bring $1.07, and as the par value is $3000, the advanced value will be 3000 times 1.07^ which gives $3210, the market value, and $3210— $3000=$210, the gain. Again, if the same arc sold at a discount of 7 per cent., it is plain that each $100 would bring only $93, and therefore each $1 vrould bring only $0.93, and therefore as the par value is $3000, the de- preciated value will be 3000 times .93, which gives $2790, and therefore the loss would be $3000—2790=210. From this we derive the RULE. Multiply the par value hy 1 plus or minut tne rate per unit, according as the shares are at a premium or a discount. »■■ STOCKS Ain> BONDS 221 XXXiROISKS. 1. What is the market value of $450 stock, at 8| per cent, dis* count ? ■ Ans. $411.75. 2. What is the value of 29 sharq^ of $50 each, when the shares are 11 per cent, below par ? Ana. $1290.50. 3. A man purchased 60 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 he gain ? • Ans. $54. 4. A man purchased $10,000 stoek 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 arc at a premium of 5 per cent., and sells when they have advanced to 12 per cent., how much does he gain? Ans. $105. CASE II. . To find how mncB 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 njnremiuin of 8 per cent. . In this case it will require $108 to purchase $100 stock, and therefore $1.08 to purchase $1 stock, and hence the amount that can he ptfTchased 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 a 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 sum hy 1 plus or mimu the rate per unit, accord- ing a* the thara are at a premium or a discount. BXBR0I81«. 6. When stooks are at a premium of 12 per cent., how much can be purohaned for $8064 ? Ans. $7200. «itf' 222 ABITHMETIO. m 7. "When stocks arc at a discount of 9 per cent., how much can be bought for $3G40 ? Ans. $4000. 8. "When G. T. R. stock is at 18 per cent, below par, how much can be bought for $42,640. ^ Ans. $52000. 9. When G. W. R. stock is at a premium of 9 per cent., how much will $4578 purchase ? Ans. $4200. 40. When government atook is selling at 92 J, what amount of stock will $28,675 purchase, and to what will it amount with broker- age at^ per cent. ? Ans. $31077.50. CA8X III. The premium or discount being known, to find the par value. To find the par value of $1,296, when stock is at a premium of 8 per cent. • « At 8 par cent, preftxium, each $1 brinpp $1.08, hence tlio par value will be represented by the number of times 1.08 is contained in«1296, which gives $1200 for the par value. To find the par value of $1104, when stodc 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 the RULE. Divide the marlfet value hy 1 plu» or minus (hf rate per unit^ according as the stocks are selling above or hdoxe par, EXERCISES. 11. What is the par value of $24420, when stock is 11 per cent. above par ? Ans. $22000. 12. What is the par value of $10800, when stocks ore at a dis- count of 4 per cent. ? Ans. $11250. 13. When government stocks are at 6 per cent, premium ; how much will $20246 purchase at par value ? Ans. $19100. 14. The shares in a canal company are at 15 per cent, discount ; how many shares of $100 will $11390 purchase ? Ans. 134. 15. The shares of a British gas company were selling in 1848, at a discount of 12 per ccnt^ j % speuulator purchased a certain num- ber of shares for £792 ; the value of the shares suddenly rose to par ; how many shares did he purchase, and how much did he gain ? Ans. 9 shares; £108 gain. .^•' STOCKS AND BONDS. 223 iTicli can . $4000. ow much $52000. int., how \. $4200. mount of h broker- H077.50. value. •emium of • « e the par contained loount of 8 UQ vrill he id iu 1104, per unity 1 per cent. . $22000. re at a dis- B. $11250. ium; how 8. $19100. , discount ; Aus. 134. in 1848, tain num- se tlo par ; in? 108 gain. CASE IV. To find to what rate of interest a given dividend corresponds. If a persor receives a dividend of 12 per cent, on an investment made at 20 per cent, above parj the corresponding interest may bo calculated thus : As the stock was bought at 20 per cent., or .20 above par, $1.20 of market value corresponds to $1 of par value, and as every $1 of par value corresponds to 12 per cent, interest, or .12, it follows that the per cent, which was invested will be represented by the num- ber of times that 1.20 is contained in .12, which is .10 or 10 per cent. Hence the RULE. Divide tJie rate per unit of dividend by 1 plus or minus the rate per cent, premium or discount, according as the stocks are above or below par. ^ _ XXEKOISES. 16. If a dividend of 10 per cent, be declared on stock vested at 25 per cent, advance ; what is the corresponding interest ? . Ans. 8 per cent. 17. If a dividend of 4 per cent, bo declared on stock invested at 12 per cent, below par, what is the corresponding interest ? Ans. 4/t. 18. If money invested at 24 per Jcnt. yields a dividend of 15 per cent., what is the rate of interest ? * Ans. 12j\. 19. If railroad stock is invested at 18 per cent, above par, and a dividend of 6 per cent, be declared, what is the rate of interest ? Ans. 5/g. 20. If bank stock be invested at 15 per cent, below par, and a dividend of 10 per cent, declared, what is the r^te of interest ? Ans. lljf. MIH0XLI.ANX0U8 XX1B0IIX8. . 1. What must be paid for 20 shares of railway stock, at 5 per cent, premium, the shares being $100 each? Ans. $2100. * To find at what pritie itoek paying n giren rote per cent dividend can be purchased, bo that the money invested shall produce a given rate of interest, divide tAe raf e per unt( <4 c^ividend h^ the rate per unit of interest. c? 221 ABITEMETIO. 2. What is the par value of bank stock worth $8740, at a pre- mium of 15 per cent. ? Ans. $7600. 3. Kailway stock was bought at 15f below par, for $1895.62J ; how many shares were there, each share being $150 ? Ans. 15 shares. 4. If C 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 is the corresponding rate of interest on any investment? Ans. 10. 6. When 4 per cent, stocks were at 17f discount, A bought $1000; how much did he pay, and how much did he gain by selling when stock had risen to 8G^ ? Ans. $821.25, and $41.25. 7. What will $850 bank stock co^ at a discount of 9f per cent., ^ per cent, being charged for brokerage ? Ans. $771.38. 8. On the data of the last example, how much would' be lost by belling out at 10^ per cent. ? Ans. $10.03. 9. What income should I get by laying out $1620 in the pur- chase of 3 per cent, stock at 81 ? Ans. $60. 10. 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 tiu of 2 cents on the dollar, it may produce 5 per cent, interest? Ans. 107|. 15. A gentleman invei!itn any DS. 10. bouglit Belling K1.25. H per 771.38. lost by $10.03. the pur- 03. $60. :8 at 84, , $5880. sstiog in »er cent. itocks at jaao per ease 25. invests sr cent. B. 60. itock^rSO it may sks at It. r. 4 annual 8. (24. income ly 5per IS.I28. 17. A person sells $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 94^, and on their rising to 98^ he sells $5000 more ; on their again rising ho 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 89f , 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 First National Bank, payable by instal- ments, as follows : — ^ in threo months, which ho sold for 5J per cent, advance: f in G months, which brought him 63 per cent, ad- vance, and the balance in nine months, which he was compelled to sell at S^ per cent discount j what did he gain by the whole transac- tion ? • Ans. $808.33. 21. A gentleman purchased $5000 of FiYC-twenties (gold 6 per cents) at 108 ; gold at time of purchase was at 85 per^cent. pre- mium ; if it remained so when the interest was payable, what waa the rate per cent, of interest on amount invested ? Ans. 7^ per cent. 22. From which would be derived the greater income, Seven- thirties purchased at 104, or Five-twenties (6 per cent, gold) at lOO^-, interest on both bonds payable at the same time, and gold quotdd at 140 ? Ans. From the Five-twenties. 23. On Jan. 1st I wish to make an investment of money that will allow me 7| per cent, interest on the investment ; what can I afford to pay for Ten-ibrties (interest payable in gold at 6 per cent.) and what for Seven-thirties, calculating gold at 35 per cent. prem. ? Ans. For Ten-forties 108 ; Seven-thirties 97^. 24. In the above example, what could I give for the Ten-forties if gold were oalculated at 20 per cent. prem. ? Ans, 96^ 226 ARITHMETIC. 25. On May 21st, a broker purchased for me a Seven-thitty bond to the amount of $12,000 at 104f ; the interest on this bond is payable on the 1st Feb. and August ; what does the bond cost me, the brokerage being ^ per cent. ? Ans. $12861.00. 26. After receiving the interest, on Aug. 1st, on the bond men- tioned in last question, the broker immediately sold it for me at 103^ charging J- per cent, for selling ; did I gain or lose by the transac- tion, and how much^ money being worth 6 per cent. ? Ans. Lost $157.94: 27. A gentleman subscribed $15,000 in a railroad company, having a paid-up capital of $750,000 ; but only 40 per cent, of sub- scribed capital paid in. A cash dividend of 3^ per cent, on the par value is declared ; what rate per cent, does he receive on hh invest- ment ? Ans. 8f per cent. 28. The capital of the " First National Bank," of Cincinnati, is $1,500,000, of which A has subscribed $7,500. There has been 25 per cent, called in. A cash dividend of 4 per cent, on the paid- up capital is declared, and 1 per cent, on paid-up capital carried to tlic credit of the stockholders ; how much money does A receive as a dividend, what per cent, on subscribed Capital is carried to credit of stockholders, and what has A still to pay on his stock ? Ans. A receives $75 ; to credit of stockholders 2^ per cent. A has still to pay' $5437.50. 29. A having $25,000 for "investment on May 1st, placed it in the hands of B, a broker, advising him to speculate in buying and selling stocks, bonds, and gold, for 60 days, and then return what the money proJuced, after deducting brokerage of ^ per cent, on the actual sales and purchases. B immediately purchased 200 shares of Erie R. R. stock at 59 " buyer 30," no margin required, and sold 400 shares Reading R. R. stock, at 1Q2J " buyer 15." Five days after, B called in the Erie R. R. stock and sold it to C at 6l| ; May 8t!i he bought $20,000 of Five-twenties at 109J, at the same time the person to whom the Reading R. R. stock was sold, called it in ; B paid the difference, the stock being valued at 102^ ; May 27th, gold having appreciated in value as compared with Greenbacks, B sold the Five-twenties at 110^ cash, and at the same time made a further sale of $16,000 in the same kind of bonds at 110| <' seller STOCKS AND BONDS. 227 30," which he was able to purchase and deliver in ten days at 109|. June 20th, B sold $20,000 in gold at 137^ "seller 10," which was not delivered at the expiration of the ten days, but settled at 137^ ; how much money is A to receive from B ? Ans. $25,475. PARTNERSHIP. Partnership has been defined to bo 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 does not appear to the world as such. A nominal partner is one who lends his name and credit to a firm, without having any real interest in the profits. All the partners may contribute equally to the business ; or the capital may be contributed by some or one, and the skill and labour by the other ; or, unequal proportions may be furnished by each. The contract need not be in writing, but all parties to be bound must assent to it, and it is usually contained in az instrument called " Articles of Partnership." Too much pains cannot be taken to have this agreement so plain and explicit in regard to particulars, that it cannot possibly be mis. understood. A great deal of litigation has urisen from carelessness in this respect. These Articles of Partnership should particularly specify the amount of investment by ej»ch partner, whether the personal atten- tion of the partners is required to the business, duration of partner- ship, and sometimes an agreement with regard to the withdrawal of money from the business. A dissolution can take place at any time by mutual consent. A partnership at mil in one in which there is qo limited lime affixed for its contiauance, and the whole firm may be dissolved 228 AKITHMETIC. by any of its members at a moment's notice. A document is, how- ever, generally drawn up and signed upon a dissolution, called a settlement, which contains a statemept 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 ehare of the profit ? SOLUTION BY PBOPOBTION, A.'s Stock, $300 B.'b " 400 Entire stock $700 : 300 : : $21 : $90 A.'s gain. " « 700: 400:: $210: 120 B.'s ." SOLUTION BY PERCENTAGE. Since the entire amount invested is $700, and the gain $210, the gain on every $1 of investment will be represented by the num- ber of times that 700 is contained in $a>10, which is .30 or 30 cents oa the dollar. Now if each man's stock be Aulliplied by .30 it will represout his share of the gain thus : $300 X. 30=$ 90 A.'s gain. . 400 X. 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, wc have the following RULE. As the whole stock is to each partner' s stock, &o is the whole gain or loss to each partner's gain or loss ; or, divide the whole gain or loss by the number denoting the entire stock, and the quotient will be the gain or loss on each dollar of stock ; which multiplied by the number denoting each partner's sluire of the entire stocky will give Am share of the entire gain or loss. EXERCISES. 1. Throe persons, A., B., and 0., enter into partnership. A. advances $500, B. $550, and 0. $600 ; they gain by trade $412.00. What is \\U pen JitnOO. (o Im' \y>\\\\ nn lMln\t«"; ,^ In '.\ wy^wSh-i, \ \\\ \ \\\\''\\\\\'^, \ \» « n»i»(tth3. nni< triiinlmli'i in H tw^M^h^i nt \v1\i\t i4m«* o»}il\» tl»o wholi' to 1h> |»nli! «<• tinwf l'^ ^ny l^i^OO in M'sIv ^«00 in pit nyoMili-". (>in00 in (» nncilliM, mul <1\o »vnn>\n\U>v in \\\ \\\\\\\\\\n\ nt nln»t time tniulil ln> In jmy «l\i> >vMo \\\ onn ^mv»«vntV A»i«. Hj|,\ ntnnlliP. t. TlnMV is \\\\\\ tn n miMt^hrtMt fsOO, nm» uixll* «'l wliii'h Im Ik Im f«i\l i« '.i n\onll\'', onoti(i(»l in ;i n»on(lm, rtnd ll\t' vi'n\n!tiilin' In t^ WiMMI^o , ^nt U>«> ilvUtov ny»A>o'« !l^ ^vvv o\n' hull In cr^sh ; Innt litnn \n^v ^»^ \>'*inn t1\o i^tlnn- ^n\\f, mi ilnU ncitlvov \\\\\{\ mm pnsliiin lunfi .' Ans MJj ntmtHtii Fv V nnnv^rtnt pi,>\,on rt xnv«\it «^i (< n\nn Mopii-nil'tn' SiHtlt, lMll4. tN«h> fiS«^»V .1nnn;n\ ist. IStU'N, jl'>0*V WIn'n. \\\ I'unily. onMltt ilio tn\Hvhftn* t>^ \vt>>vvv ^^^<<'' <<* On' tnninint ol' #1'2()(». loll il «NjMi trtk«» <'o\»t'S|>in\il willt lln> >«,<» U\o tin\o ? \\(s, 4 yoius, 4 luoe. tlV V (ji\>M«in, ns rnll\»\Vf» ; VVtMUMV iHJ\, '^^ brtvvvls (.a lfM.1'.(«t |vtvyn»ont ol' iho wholo ? Auf. July !'lh. U lV>\\jiM of A U, Snnfh \ (V. UU^«» UmvU ol" Honi. «l (Uf fi«Nmt tiv«o>s, «4«tl on v«v\ov\!» knms oC o»Y«li». i»n bf loil-ntinn ft6*(vrvnt , >shAt \» t)>o »s\nrt(«sl Tnno for il»o |>,\ynn>nt, nl' (ho wKolo If M\v 0«1\. 1^0 K'UwK B( $\ ^0. on ;i n»onll»«' orxMlit. "M)^y 20th. 400 KtiwK. «t AM\, on 4 tnotdlm' mviUt. ,1«(t I0<>>. »N00 IvonvU. »t ^,00, on ^ »non(l»»' ojvdit. A«(g;nst 4th, t>(>t^ iNArjvU. M 4.i2^, o.i 4 n^ontlw' oivillt. Vns. Novomlior 7 th. 15. .V U .^with ,^ Civ K^ujiht of A. lUmiltou »^ Sm D70 im- toh of rv\«9n, *.« f\%lK'>'(rs ; Maj Srd, 62 bwrolt ^ 12.50, on C mouUu' oioait. AttiHAMlMM ^^uu^\fHrti. n\ Mmv I nil), mo lmrH.|« VH '^ riJl, MM (! (rintiMiN* f-rft'llf. Wity iMMi, HI littf^J'f'lrt (.ft ',< fill, nn i hO, *)() p|f Mi'i))l((t' ».h.»1ll Mi)jr '.^llll). liDfM.ln (.71 « (H», >iti mU »f,'.»)(l(..' ^ffwlif,. Wimli \n Ihn wjuM»'tl ll»t)« Dti- llio [myiiwul (.(' Ili»> w!ifilfl| IJOKtIfl ««(■ .r II WMfl|)i»)(elf((| II), (Ut, ltil\\ffhft*ui i\\m<*, M)nl titi vti»lii)iM li'»it)« (if ».ff'(||l, itri liy llif. Iiillmtlfit* plrtl''ff««fif, ; M))rt'l) Imi, iM(i!i. t) MM <.itii7ri >r». /.fi 'I »f,it\hn' 6MII.. f^.tlt'HtlH'f ?irillt. I MIDI, ,) l.lll ..f MM! 7r.. (.ft M »r((rtifl)B' owlH. nH.)l)t»f l«l. |M(1!I, (t I.(H Ml' IMJl '/n, f.ft M »f)'»f(fl.«' <'w]|f,, .Imiuny ^f^, I'^-'J, •; Ml! m" III tM*, fir. :i mnnlM' r.rr.f1If K..|if)M)»-y Hill), iwil!. l.lll ..r Ml I !i(», „u n ff('.»if».«' crNli*,. IMtiifl) |y||), I Kill. » 1)111 mC h07 70, f.ti rr rn/.oni^' ^f^flif. A)i»II imi), I Mil I, fl Mil r.f JIflllMO, MM ^ U,nui\,n' t^tt^MK VVIinli Is i!»»» iM()t(ilt'M«>iiil of lit** wltol^ 7 A V HI M A t* I N A a (J V N T « , ..ly !M1i. I . ))( .ur 'ullowl«»n wholo ? ii. It. tt. t. WllBN mm iiifintlMiMl lrn months' credit ? Dr. A. B. Smith & Co. Cr. 1864. May 1, July 7, Sep. 11, Nov. 25, Dec. 20, Jan. 1, Feb. 18, Mar. 19, April J, May 25, To Mdso. $300.00 759.90 417.20 287.70! 571.10 By Cash.. " Md.se. '< Cash.. " Draft. « Cash.. $500.00 481.75 750.25 210.00 100.00 Ans. August 5, 1805. Ili' '■ ■^ A A:. 246 ABITHMETIC. 10. When xvill the balance of the following account fall due, the merchandise items being on 6 months' credit ? * Br. J K. White. Cr. 1866. May 1, May 23, June 12, July 29, Aug. 4, Sept. 18, To Mdse. Cash paid dft. . Mdse Cash 11 1866. I $312.40i!june 14,|By Cash. ... 86.70 105.00 243.80 92.10 50.00 July 30, Aug. 10, Aug. 21, Sept. 28, " 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 account become subject to interest ? Br. W. H. MusOROVB. Cr. 1864. Aug. 10, Aug. 17, Sept. 21, Oct. 13, Nov. 25, Nov. 30, Dec. 18, 1865. Jan. 31, To Mdse 4 mos. " " 60 days « 30 " Cash p'd dft. Mdse 6 mos. " 90 days " 2 mos Cash. $285.30 192.60 256.80 190.00 432.20 215.26 68.90 100.00 1864. Oct. 13, By Cash. Oct. 26, Deo. 15, Dec. 30, 1865. Jan. 4, Jan. 21, Mdse 2 mos Cash. $400.00 150.00 345.80 230.40 340.30 180.00 12. In the following account, when did the balance ])ecome duo, the merchandise articles being on 6 months' credit ? Br, R. J. Bryce in account with D. IliCKS & Co. Cr. 1864. Jan. 4, Jan. Feb. Feb. Feb. Mar. Mar. 24. April 9, May 11, May 21, 18, 4, 4, % 3. To Mdse 1 $ 96.67 57.67 80.00 38.96 50.26 154.46 42.30 23.60 28.46 177.19 1864. Jan. 30. By Cash... • • • t • • $240.00 <( ti April 3, May 22, 48.88 " Cash paid draft. " Mdso 50.00 " Cash paid draft. " Mdse '< «« f( (( • a (( " << An6. December 22nd, 1864. AVERAGING ACCOUNTS. 247 13. When, in equity, should the balance of the following accouni be payable? Jh. J. McDonald & Co. Cr. 1865. * 1864. ' ^ ..t*^1 Jan. 3, To Cash.... $200 Sept. 20, By Mdse, 6 mos. . $583.17 ^4 Jan. 31, 300 Oct. 27, " 4 « .. 321.00 Feb. 8, 75 Dec. 5, « 6 " .. 137.00 Feb. 21, 100 1865. Mar. 10, 350 Jan. 18, " 60 days. 98.75 Mar. 24, .25 Feb. 26, " 6 mos. . 53.98 Apr. 12, 40 Apr. 16, « 4 " .. 634.00 June 1, 80 June 12, " 2 « .. 97.23 June 20, 125 Sept. 21, " 6 " .. 84.00 July 4, 268 Dec. 29, " 6 « .. 132.14 Sept. 27, 250 •t Dec. 9, 100 Ans. October 10, 1866. CASH BALANCE. To find the true cash balance of an account, 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. MusGROVE & Wright. Cr. 1865. Mar. 12, To Merchandise Apr. 21, May 6 May 27, July 16, Sept. 10, Oct. 19, (( Cash paid riraft Mdse Cash Mdse; $340.00 150.00 165.00 215.00 100.00 310.00 120.00 1865. Apr. 20, By Mdse... $200.00 May 4, " Cash.... 110.00 Juno 15, (( It 230.00 Aug. 10, " Mdse... 180.00 Sept. 23, " Cash.... 50.00 Nov. 12, « « 50.00 Dec. 15, U ii 100.00 248 AIIITHMETIO. SOLUTION. Debits. ' Creditt. Due. Dno. Juno 12, $340X221= 75140 July 20, $200X183= .30000 July 21, 150X182..: 27300 May 4, 110X200= 28000 May 0, 105X258= 42570 Juno 15, 230X218= 50140 Aug. 27, 215X145= 31175 Nov. 10, 180 X 70= 12G0O July 1(), 100X187= 18700 Sopt. 23, 50x118= 5900 Deo. 10, 310 X 40= 12400 Nov. 12; 50 X 08= 3400 Jan. lU, 120x 0= Doo. 15, lOOX 35= . 3500 $1400 6)207285 $34,547 $920 6)140740 $23,450 Tho different items on tlio debit and credit eidoB of the account being on interest from tlie date on which it becomes duo until tho time of settlement, tho total interest of all tho debit items will bo tho same as tho interest of $207285 for one day, or tho inforest of $1 for 207285 djiys, which is $34,547. So also, tho total interest of all tho credit items will bo tho same as tho interest of $140740 for one day, or tho interest of $1 for 140740 days, which is $23,450. Now, since each side of tho account is to be increased by its interest, tho cash biilanco will bo represented by tho number denoting tho differ- ence between the two sides of tho account, after the interest is added ; thus, $1400+S34.547=$14.34.547, amount of debit side, and $920 -f$23.450----:.$943.450, amount of credit side, then $1434.547- $943.45G=$491.09, cash balance. BEOOND METHOD. 1 Debits. Credits. Pnvf. Int. Pays. Int.. Int. on $.310for22i=$12.523 Int. on $200 for 183= $0,100 i( 150 " 182= 4.550 « 110 " 20Q= 4.706 « 105 " 258= 7.095 It 230 " 218= 8.350 (( 215 " 145= 5.195 u 180 " 70= 2.100 IC 100 " 187=F= 3.110 (( 60 " 118= .983 Ci 310 " 40= 2.000 (( 50 " 08= .500 i( 120 " i( 100 " 35= .583 $1400 $34,545 $920 $23,454 Now, $34,545 debit intorest— $23,454 credit intcrcst=$11.09, CASH BALANCE. 249 tlio balance of interest, and $1400, amount of debit it«msf 811.09 =$1411.09, and $1411.09—8920 amount of credit itcin8::-$49 1. 09, tho cash balance, which ia the Baino au obtained by tho fintt solution. HcDOO from tho foregoing we doduco tho following^ RULE. Multiply each item of debit and credit hj the numljcr of day» intervening hetwcai its hccominff due and the time of settlement. Then consider the sums nf the products of the debit and credit itcm» as so many dollars, and find the interest on each for one day, which will be the interest, respectively, of the debit and credit items. Place the balance nf interest on its own side of the account, and the difference then between the two sides will be the true balance ; or. Find the interest on each item from the date on which it becomes due to the time of settlement. The difference of the sums of interest, on the debit and credit sides of the account, will represent the balance of interest, which is 2>laced on its own side of the account, and the difference then between the two sides will be the true balance. The following is a form of statement or account current, generally made out by merchants in determining the cash balance. They are usually rendered quarterly or half-yearly, unless called for by tho payer, in which case they aro balanced by cash if paid at once. 1. — What will bo tho cash balance of the following account, on Sept. 1st, 18GG, interest calculated at G p6r cent.? II. C. Wright ill account current and Intercut account t.o Sept. l.st, 1800, with C. O'Dca. Date. ITEMS. PllTNX'l- I'AL. WHRN Due, TIME. I.NT. DATE. ITE.MS. PUTNCI- I'AI.. DUE. ^'*"'- Jiiiio 1 a2dy9 July2i :;;» '• '■ :«i .TI " AuK.'iM 3 " ■■ ];i;i3 " Ual. of Inter. IKT. »Ii\r. 4 Aiir. 2;: '• 3(1 Juno V " m July 17 " ID To Mdnc. (2 41 M 3 (17 Jiinu 1 '■ 21 July 311 AiiK.m .S.;iit. 1 Ily Cimli on iwct. ■' l)niftf.«3i)dyH " ('luh (III iu:(:t. " I)nift(.c27dyn " t'wh nil »ci;t. " UiilJUlco • ■ • ♦ 170 l!i 41 18 .10 31 4H I'l 401 71 IKJl IH ♦2«l U 2» 21 02 !l 87 6 13 ♦ I(HI4 82 ♦10 13 tWA 82 ♦10 1.1 Sept. 1 t'.HII 18 NoTK. — If any item should not come diK? until after tho time of settlement, the side upon wliicli it i.s should be dimiuisbed, or tho opposite side increased, by the interest of such item from tlie time of settlement until due. 737 ^ . I. 250 ABTTHMETIO. 2. The Mowing account was settled in foil on December 1st, 1865; what amount was paid, interest 6 per cent. ? C. p. Meads in account current and interest account to Dec. Ist, 1865, with T. R. Brown. - — ^ Sate. ITEMS. PniNCi- PAL. When TIME, INT. Date. ITEMS. PBINCL PAL. WHEN DUE. TIME. INT. Jan. 1 To MdHO., (i inos. " Caih pd. clft. " Mdue., 4 nios. •' " 4 ■• " Cash p-l. dft " MdsC. fi IJKM. $IM 111 100 0(1 310 Ull 162 00 .100 00 213 0(j KBb. 1 Mar. 20 May 1 July 1 Sep. 10 By Cash •' Udse., 4 mo8. " " a " " ■■ 4 ■' " " 4 " 1120 00 420 le 300 00 60 00 99 84 Feb. 3 JIar.20 " 30 Maris AUK.2U Ans. $61.36. 3. "What will be the cash balance of the following account if settled on January I, 1865, allowing interest at 8 per cent, on each item after it is due ? Ans., $110 86. J. Smith Homans In acct. current and interest account to Jan. Ist, 1865, witli T. C. Musgrove. Date. ITEMS. To MJso., 4 moa. 8 " Ca.d interest account to July 1st, 1807, with J. B. Harris. i ......,, Date. Items. Princi- pal WHEN DUE. Time. INT. Date. Items. Princi- pal. WHEN Due. TME. 1ST. 1860. July 15 Aiig.SO Bop. 1!) Nov. 25 h ToMdse.QSOdys. •' 3 raos. COUya. " 4 moa. flOdya. " 4 mca. " 2 nins. ;ii)dya. »,'i04 50 2! 8 12 116 27 1218 50 110 8-: 428 30 100 00 2,-.7 75 1 IHOIi. Aug. 14 •• 31 Ont. 18 Nov. 1 Dec. 20 Jan. 30 Mar. 4 Junc25 By Cash " Md.ae.V3b'dys " Cash " Drart@60dy8 " Cash " Ord.onJ Jones " Mdse., CO dye $400 00 104 50 200 00 185 14 300 00 250 00 BOO 00 475 00 312 90 • Viv. 20 1 1807. 1 Jan. 12 Mar. 19 Mar 20 5. What will be the balance of the following account on March 25, 1865, each item drawing 7 per cent, interest from its date ? Ans. $50.64. CASH BALANCE. 251 ;, 1865; Brown. TIME. 1ST. J. C Baylies tn account current and interest account to Marcli 25tli, 1SC5, with E. K. Brj-an. DATE. ITEMS. PRINCI- PLE. WHEN DUE. TIME. Int. j DATE. ITEMS. I'ltlNCl- PAL. WHEN Due. Time. INT. 1804. July 4 Bcpt.8 '■ 25 ToMcrcImdisa.. •• ii #200 00 300 00 250 00 600 00 400 00 600 00 100 00 1 18ir». 1 July 211 |Aiib.15 Nov. 1 Dec C •• 20 1865. Feb, 1 " 28 By Cash ByMiTcliauiUiio'.. By OiMli ByMercliauiliJe.. #300 00 450 UU 400 00 320 00 Clio oil 100 00 200 00 130 0« Octr .1 Nov. 20 Dec- 12 i8i>a. Ji>n. 15 Uu.ll if settled tern after 3. Mussrovc. ^ TIME. INT. .. :: :;:::: July Ist B. Harris. EN TIME. INT. ...... [arch 25, ^50.64. ACCOUNT OF SALES. An account of sales is a statement made by the consignee (gen- erally a commission merchant) to the consignor, the person from whom the merchandise was received to sell, showing the persons to whom sold, the price, time, charges, and net proceeds. The net proceeds is the amount duo the consignor, from proceeds of sales, after all charges are deducted, and are due to the consignor at the average time of sales. Commission merchants often become interested in the merchan- dise consigned to them for sale, by accepting a certain share and sell- ing 6n joint account of themselves and the consignor ; when this is the case, the gain or loss is shared according to the way in which the merchandise was originally divided. When the commission inerchant accepts the merchandise to sell on joint account, the terms upon which he becomes responsible for his share should be known, whether payable as cash, some definite term of credit, or at average time of tales. In the following account sales of merchandise sold for A. II. Eastman, at what time will his net proceeds be due, as cash, and what will be the amount duo A. R. E. on May 14, 18G7, discount 6 per cent. ? Ans. Net proceeds due May 21, 1867. Due A. R. E., $2370.94. £52 AEITHMETIC. Account sales of 8745 lbs. bacon, 2970 lbs. cheese, und 1245 lbs. butter, for account and risk of A. 11. Eastman, Chicago, 111. 18ti7. Mar. 10 Apr. 20 May 14 March 1 « 9 May 14 Sold to R. White, at 30 days— 4000 lbs. bacon, at 1 Go. 500 lbs. butter, at 40c. Sold to J. B. Harris, for cash — 4745 lbs. bacon, at 15Jc.. Sold to J. C. Parsons, at 60 days- 2970 lbs. cheese, at 22c.. 745 lbs. butter, at 41 c. CnARGES.— Paid freight in cash Paid for labor resalting bacon Storage Commission on 82547.51 at 2^ per ct Net proceeds due per average May 21 $640 00 200 00 654 72 305 45 97 40 8 50 5 00 63 69 $840 00 747 34 9G0 17 82547 51 174 59 2372 92 Cleveland, 0., • E. and 0. E. May 14, 18G7. R. Felton & Co. September 4, 18G0, we received from W. Cummings, Cincinnati, a consignment of 120 brls. of mess pork at $25.00 per bbl., and 742 bushels clover seed at $8.30 per bushel, to be sold on joint account of consignor and consignees, each one-half; consignees' half subject to average sales. The same date we cashed his demand draft in favor of Third National Bank, for $3450. The following is the account sales. At what time arc the net proceeds due as cash, and what amount in cash will settle our account with W. Cummings on Jan. 1st, 1867, interest per cent? Ans. To last question, 81552.56. Account sales of 120 brls. mess pork, and 742 bushels clover seed, on joint account of W. Cumftiings, Cincinnati, and ourselves (each one-half.) Sept. 12 " 30 Sold to C. K. Sing, for cash — I 250 bshls. clover seed, at 88.95' Sold to M. Hollingsworth, at 60 dys 25 brls. mess pork, at 832 80 bshls. clover seed, at 89.25. ^SOO 00 740 00 82237 50 1540 00 CASH BALANCE. 253 Oct. 18 Novr. 2 <« 15 Sept. 4 (( <' Octr. 8 Nov. 15 Sold to T. M. Ames, at 30 days— 200 bshls. clover seed, at 89.25... 10 brls. mess pork, at $32.50... Sold to T. R. Brown, for cash — 200 bshls. clover seed, at $9.30. Sold to A. W. Purdy, at 6 months— 85 brls. mess pork, at $33 12 bshls. clover seed, at $9.30 . CHARQES.- Paid freight and cartage in cash.... Paid insurance on $9500, at 1^ p. ct. Paid for storage, cooperage, and labor Commission on $10,729.10, at 2^ p. c Net proceeds of sales. Your ^ ne t proceeds, due as per aver, 1850 00 325 00 2805 00 111 60 210 75 118 75 15 00 208 23 2176 00 1860 00 2916 60 10729 10 612 73 10116 37 5058 18 Columbus, ^ Nov. 15. 1866. E. and 0. E. J. 0. Denison & Co. January 2, 1867 — Received from D. M. Harman, Owosso, Mich. 200 brls. pork, invoiced at $18 per brl. ; 3750 lbs. cheese, at 10c. per lb., and 100 firkins of butter, each 80 lbs., at 16c. per lb., to be sold on joint account, of shippers f , and ourselves ^ ; our ^ of invoice due as cash. January 21 — Cashed D. M. Harman's sight draft in favor of First National Bank, Cleveland, for $1264.50. February 14 — Accepted D. M. Harman's one month sight draft in favor of Thos. L. Elliot, Owosso, for mutual accommodation, for $864. February 28 — Cashed D. M. Harman's demand draft in favor of Third National Bank, Detroit, for $1174.75. Find equated time of sales in the following account ; equated time of D. M. Harman's account ; also, give the amount that would bal- ance his account on May 14th, 1867, reckoning interest at the rate of 7 per cent, per annum. Ans. Equated time of sales, April 9, 1867. Equated time of Harman's acct., April 23rd. Bidauce of account on May 14, $2456.38. ■-i:^ ♦ . . kit 254 ABTTHMETIO. Acwunt sales of 200 brla. pork, 3750 lbs. cheese, and 100 tirkins butier, on joint account of D. M. Harman §, and ourselves |. Jan. lb Feby. 9 « 27 March? " 24 Jan'y. 2 " :a " 15 Mar. 24 « 24 tiold A. S. Morrison, on 90 days — 40 brls. pork, at $1 75 per brl. 50 firkins butter, 8^ xus. each, at 24c per lb Sold W. E. Qlennie, on 2 months — 60 brls. pork, at $li).37i per brl 1150 lbs. cheese, at 13c per lb Sold A. n. Peatman, on his note at 2 months — 50 firkins butter, 80^ lbs. each, at 24c per lb 1895 lbs. cheese, at 14c Sold H. D. Wright, for cash— 75 bfls. pork, at $19.25 per brl Sold A. B. May, on 30 days — 25 brls. pork, at $19.87^ per brl. 705 lbs. cheese, at 15o per lb.... ■ CHARGI:8. Paid for freight and cartage by cash... Paid for cooperage and extra labor by cash Paid Insurance at li; per cent Charges for storing in storehouse Com. at 2^ per cent, on $62^9.68 is. Net proceeds of sales. Your § of N. P. due, as per average, April 12th $750 00 960 00 1162 60 149 50 966 00 265 30 49o 88 105 75 122 75 6 50 12 46 11 25 157 49 $1711 00 1312 00 1231 30 1443 75 602 63 6299 68 310 45 5989 23 3992 82 £. and 0. E. E. OSO. CONKLIN & Co. Cleveland, 0. Dec. let, in67--Wc received ftom Messrs. Gillespie, Moffatt & Co., BofttOD, 27 casei Mackinaw blanketsi 340 pra. at $3.90; 2 case* ^^i CAQH BALANCE. 255 )0 tirkins ikes J. n7U 00 1312 00 l| 1231 30 1443 75 5l 602 63 6299 68 310 45 5989 23 3992 82 \k Co. pland, 0. Moffatt & |.90', 2Ga««i chintz cotton, 987 yds. at 7fo ; 20 pes. table oil cloth, at $3.70 ; 4 pes. do., at $5.62| ; 7 pes. West England broad cloth, 126 yds. at $3.70 ] 7 bales cotton batts, at $6.20 ; to be sold on joint acccunt and risk of consignor and consignee, each one-half, our one-half a^j cash. Dec. 5th— We cashed their demand draft for $1200. Dec. 17 — Accepted their draft on us at 30 days' sight, for S984. Jan. 14 — Cashed their draft on demand, for $500!. Sales of merchandise as per account sales annexed. At what date are the net proceeds duo as cash ? What is the equated time of G. & M.'s account ? What is the cash balance on March 24, 1867 ? Account tales of 27 cases Mackinaw blankets, 2 cases chintz cotton, 24 pes. table oil cloth, 7 pes. West England broad cloth, and- 7 bales cotton batts, on joiut account of Gillespie, Moffatt & Co., J^oston, and ourselves (each one-half). Deer. 5 i« 9 ijold John McDonald & Co., ^ cash, ^ on account 30 days — 13 cases blankets, 260 prs. at $4.20 1 case chintz cotton, 425 yds. at 9o 7 pes. tnble oil cloth, at $4.50 ... Sold K. Chisholm & Co., on note at 6 months — 7 cases blankets, 140 prs. at $4.50 8 pes. W. E. broad, 54 yds., at $4.20 4 pes. table oil, at $6 " 14 tt 17 Sold Thomas & Arthurs, for cash — 1 case chintz cotton, 562 yds. at 13 pes. table oil, at $4.40 3 pes. W. E. broad, 54 yds., at $4 Sold James A. Dobbie k Co., note at SO days 2 balea cotton baits, at $7 4 oaaes Mackinaw blankets, 80 prs., at $6.70 1 po. W. £. broad, 18 yds., •* $5 256 « 28 Deer. 1 Jany. 1 " 1 ABTTHMETIO. Sold Thomas Spence & Co., joash, balance on acct. at 30 days — 3 cases M. blahkets, 60 prs., at $6.75 5 bales cotton batts, at $7.25. ... OHARQES. Paid freight and expenses from depot, cash Storage '. Com. at 2^ per cent, on sales Net proceeds »... Your ^ of net proceeds, due as per average $94 75 34 48 Milwaukee, Wis., January 1st, 1867. E. and 0. E. J. 0. SPENOsa & Co. ALLIGATION. Alligation is the method of making oalonlations regarding the compounding of articles of different kinds or different values. It is a Latin word, which means binding to, or binding together. It is usual to distinguish alligation as being of two kinds, medial and altenuUe. ALLIQ-ATION MEDIAL. Alligation medial relates to the averago Value of articles com- poanded, when the actual quantities and rates ars given. BXAUPLB. A miller mixes three kinds of grain: 10 bushels, at 40 cents a boshel ; 15 bushels, at 50 cents a bushel ;' and 25 bushels, at 70 cents a buflheli it in required to find the vtlue of the mixture. ALLIGATION. 257 10 bushels, at 40 cents a bushel, will be worth 400 cents., 15 bushels, at 50 cen*? a bushel, will be worth 750 cents., 25 bushelp, at 70 c-nts a bushel, will be worth 1750 cents., giving a total of 5U bushel^ and 2900 cents, and hence the mixture is 2900-f-50-.=58 cento, the price of the mixture per bushel. Hence the RULE. I^ind the value of each of the articles, and divide the sum of their values hy the number denoting the sum of the articles, and the quotient toill be the price of the mixture. EXERCISES. 1. A fanner mixes 20 bushels of wheat, worth $2.00 per bushel, with 40 bushels of oatSj worth 50 cents per bushel ; what is the price of one bushel of the mixture ? Ans. $1. 2. A grocer mixes 10 pounds of tea, at 40 cents per pound; 20 pounds, at 45 cents per pound, and 30 pounds, at 50 cents per pound ; what is a pound of this mixture worth ? Ans. 46§ cents. 3. A liquor merchant mixed together 40 gallons of wine, worth 80 cents a gallon ; 25 gallons of brandy, worth 70 cents a gallon ; and 15 gallons of wine, worth $1.50 a gallon ; what was a gallon of this mixture worth ? Ans. 90 cents. 4. A farmer mixed together 30 bushels of wheat, worth $1 per bushel ; 72 bushels of rye, worth 60 cents pen bushel ; and 60 bushels of bar ^y, worth 40 cents per bushel ; what was the value of 2^ bushels of tL ■ 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 a pounds, of 22 carats fine ; how many carats fine is one pound of the mixture ? Ans. 1 8 i^ . ALLIO-ATION AL SENATE. Alligation alternate is the method of finding how much of seve- ral ingredients, the quantity or value (if which is known, must be •ombined to make a compound of a given value. CASK I. Given, the value of several ingredients, to make a oompoand of a given i^alao. 258 ASITHMEtTO. EXAMPLE How much sugar that is worth 6 cents, 10 cents, and 13 cents per pound, must be mixed together, so that the mixture may be worth 12 cents per pound ? SOLUTION. 12 cents. 1 lb., at 6 cents, is a gain of 6 cents. ) Gain. 1 lb., at 10 cents, is a gain of 2 cents, t 8 1 lb., at 13 cents, is a loss of 1 cent. 7 lbs. more, at 13 cents, is a loss of... Loss. 1 7 Qain 8 Loss 8 It is evident, in forming a mixture of sugar worth 6, 10 and 13 cents per pound so as to bo worth 12 cents, that the gains obtained in putting in sugar of less value than the average price must exactly balance the losses sustained in putting in sugar of ^reafer 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 ^his sugar put in the mixture. So also sugar that is worth 10 cents per ppund, when in the mixture will brln'j; 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 ecU 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 diiferent qualities of sugar there is a gain of 8 cents, and a loss of only 1 cent. Now, our losses must equal our gains, and therefore we have yet to lose 7 centsi and as there is only one quality of sugar in the mixture by which we can loso^ it is plain that we must take as much more sugar at 13 cents as will make up the loss, and that will require 7 pounds. Therefore, to form a mixture of sugar worth 6, 10 and 13 cents per pound, so as to bo worth 12 cents per pound, we will require 1 pound at G cents, 1 pound at 10 cents, and 1 pound at the 13 ccnts-f 7 pounds of the same, which must be taken to make the loss equal to the gain. By making a mixture of any number of times these answers, it will be observed, that the compound will be correctly formed. Hence we can readily perceive that any number of answers may be obtained ALLiaATION ALTERNATE. 259 to all exercises of this kind, the Mowing From what has been said we deduce BULE. JFind how much is gained or lost by taking one of each kind of the proposed ingredients. Then take one or more of the ingredients, or yuch parts of them as will make the gains and losses equal. EXEfi'oiSES. 1. A grocer wishes to mix together tea worth 80 cents, $1.20, $1.80 and $2.40 per pound, so as to make a mixture worth $1.60 per pound ; how many pounds of each sort must he take ? Ans. 1 lb. at 80 cents; 1 lb. at $1.20; 2 lbs. at $1.80, and 1 lb. at $2.40. 2. How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents per bushel, must be mixed together that the compound may be worth 64 cents per bushel ? Ans. 1 bush, at 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 cts. ; 9 gal. at 80 cts. ; and 1 gal at $1.20. 4. In what proportion must grain, valued at 50 cents, 56 cents, 62 cents, and 75 cents per bushel, be mixed together, that the com- pound may be 62 cents per bushel ? Give, at least, three answers, and prove the work to be correct. 5. A produce dealer mixed together com, 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 thb mixture was worth 80 cents per bushel; what quantity of each did he take? Give four answers, and prove the work to be correctly done in each ease. 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; oata, at 30 cents, and ooro, at 60 t ^ 260 AEITHMETIC. WDta per bushel, must be mixed with 20 bushels of rye. at 85 cents per bushel, so that the mixture may be worth 60 cents per bushel ? SOLUTION. Bush. Cents. Gain. Loss. 1 at 40, gives 20 1 at 30, gives 30 1 at 60, gives 00 .00 20 at 85, gives 6.00 ~!60 5.00 9 at 40, gives 1.80 9 at 30, gives 2.70 $5.00 $5.00 By taJcing 1 bushel of barley, at 40 cents, 1 bushel of oats at 30 cents, and 1 bushel of corn at 60, in connc. 'ion with 20 bushels of rye at 85 cents per bushel, we observe that o r gains amount to 50 cents and our losses to $5.00. Now, to maL the gains equal the losses, we have to take 9 bushels more at 40 cents, and 9 bushels more at 30 cents. This gives us for the answer 1 bu8hcl4-9=10 bushels of barley, 1 bushel-|-9=10 bushels of oats, and 1 bushel of corn. From this we deduce the RULE. Find how much is gained or lost,, hy taking one of each of the proposed ingredients, in connection with the ingredient which is limited, and if the gain and loss he not equal, take such of the jifo- jmsed ingredients, or such parts of them, as will make the gain and loss equal. EXERCISES. 6. How Oauoh 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, with 20 pounds, at 40 cents per pound ; how much of each sort must he take to make the composition worth 80 cents per pound? 8. Bow much barley, at 50 cents per bushel, and at 60 cents per boshel, vsM%i be mixed with ten bushels of pease, worth 80 cents •ii fc ALUGATION ALTERNATE. 261 per bushel, 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 ; 2J- bushels, at GO 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 lOJ cents per pound ; and 6 pounds, worth 13J cents per pound, so that the mixture may be worth 12 J cents per pound ? Ans. 1 lb., at 8 cts. ; 9 lbs., at 14 cts. ; and 5^ lbs., at 13 cts c Asrf III. To find the quantity of each ingredient, when the sum of the ingredients and the average price are given. EXAMPLE. A grocer has sugar worth 8, 10, 12 and 14 cents per pound, and he wishes to make a mixture of 240 pounds, worth 11 cents pec pound ; how much of each sort must he take ? SOLUTION. Gain. Losa. • 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.H-4==60 lbs. of each sort. By taking GO lbs. of each sort we have the required quantity, and it will be observed that the gains will exactly balance the losses, consequently the work is correct. Hence the RULE. Find the least quantity of each ingredient hy Case I., Then divide the given amount hg the sum of the ingredients already found, and multiply the quotient by the quantities found for the jiropor- tional quantities. 10. What quantity of three diflferent kinds of raisins, worth 15 cents, 18 cents, and 25 cents per pound, must be mixed together to fin a box containing 680 lbs., and to be worth 20 cents per pound ? Adb. 200 lbs., at 15 cents ; 200 lbs., at 18 cents ; and 280 lbs., at 25 cents. ■h 262 AEITHMETIC. 11. How much sugar, at 6 cents, 8 cents, 10 cents, and 12 cents per pound, must bo mixed together, so us 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. 56g gals, wine, and 33g gals, water. 13. A wine merchant has wines worth $1, $1.25, SI. 50, $1.75,and $2. per gallon, and he wishes to form 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 he wishes to fill a hogshead that would contain 200 pounds ; how much of each kind must he take, so that the mixture may be worth 15 cents per pound ? Ans. 33J lbs. of 8, 10, and 1 J 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, amongst almost all nations, been the standard of value from the earliest time. Except in tho 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 for this purpose as gold and silver. They arc easily divisible, portable, and among the least imperishable of all substances. The work of dividing tho precious metals, and marking or coining them, is generally undertaken by the Government of tho country. Money is a commodity, and its valuo is determined, like that of other commodities, by demand and supply, and cost of production. When there is a large supply of money it becomes cheap ; in other words, more of it is required to purchase other articles. If all the MONET : ITS NATUKE AND VALUE. 263 money in circulation wore doubled, prices would bo doubled. The usefulness of money depends a great deal upon the rapidity of its circulation. A ten-dollar bill that changes hands ten times in a month, purchases, during that time, a hundred dollars' 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 circu.lating medium, conformable in duration and extent to a temporary activity in business, does not raise prices, it merely prevents the fall that would otherwise ensue from its temporary scarcity. PAPER CURRENCY. 13 Paper CirBaENCT 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 cdlled convertible. When it is issued by the sovereign power in a State, and is made to pass for money, by merely calling it money, and from the fact that it is received in pay- ment of taxes, and made a legal tender, it is known as an inconver- tible currency. Nothing more is needful to njake 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 its currency, but would not regulate its value. This evidently cannot depend, as in the case of gold and silver, upon the cost of production, for that is very trifling. It depends, then, upon the supply or the quantity in circulation. While the issue of inconvertible currency is limited to something under the amount of bullion in circulation, it will on the whole maintain a pat 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 proportionably depreciated, as com- pared with the metallic currency of other countries. It would bo 2C4 ARirHMETIO. quite impossible for iheso results to follow the issue of convertible paper for which gold could at any time be obtained. All variations in the value of the circulating medium are mis- chievous ; they disturb existing contracts and expectations, and the liability to such disturbing influences renders every pecuniary engagement of long date entirely precarious. A convertible paper currency is, in many respects, beneficial. It is a more convenient medium of circulation. It in clearly a gain to the issuers, who, until the notes are returned for payment, obtain the use of them as if they wove a real capital, and that, without any los3 to the community. THE CURRENCY OF CANADA. . In Canada there arc two kinds of currency ; the one is called the old or Halifax currency, reckoned in pounds, shillings, pence and fractions of a penny ; the other is reckoned by dollars and cents • as already explained under the licad of Decimal Coinage. The equivalent in gold of the pound currency is 101.'}21 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 arc 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 64.8G§. In the year 1780, 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 Icyal tender is meant the proffer of payment of an account in the currency of any country as establibhcd by law. Copper is a legal tender in Canada to the amount of one shilling or twenty cents, and silver to the amount of ten dollars. The British sovereign of lawful weight passes current, and is a legal tender to any amount paid in that coin. There is a silver currency proper to Canada, though United States' coing are most in circulation. The gold eagle of the United States, coined before July 1, 1834, is a legal tender for $10.66J of the coin current in this province. The same coin issued after that is a legal tender for $10. .#■ EXCHANGE. 265 EXCHANGE. It often becomes necessary to send money from one town or country to anotlicr for various pufposes, generally in payment for goods. The usual mode of making and receiving payments between distant places is by bills of exchange. A merchant in Liverpool. whom we shall call A. B., has received a consignment of flour from C. D., of Chicago; and another man, K. F., in Liverpool, has Hbipped a quantity of cloth,. in value equal to the flour, to G. IL in . Chicago. There arises, in this transaction, an indebtedness to Clii- cago for the flour, as well as an indebtedness from Chicago for the cloth. It is evidently unnecessary that A. IJ., in Liverpool, should Bond money to 0. D. in Chicago, and that G. II., in Chicago, bhould 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 tho money may be saved. C. D. draws on A. B. for tho amount which he owes to him; and G. II. 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 duo from Chicago to Liverpool, aid the debt due from Liverpool to Chicago aro botli 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, the debts cannot be simply written off against one another. Indeed, when a person desires to make a remittance to a foreign country, he does not mako a personal search for some one who has money to receive from that country, and ask him for a bill of exchange. There arc ex- change brokers and ))ankers whose business this is. They buy bills from those who have money to receive, and sell bills to those who have money to pay. A person going to a broker to buy a bill n'.:iy very likely receive one that has been bought the same day from a merchant. If the broker has not on hand any exchange that he has bought, he will often give a bill on his own foreign correspondent ; and to place his correspondent in funds to meet it, he will remit to him all the exchange which he has bought and not re-sold. 266 AIHTHMETia When brokers find that they are asked for more bills than are offered to them, they do not absolutely refuse to give them. To enable their correspondents to meet the bills at maturity, as they have no exehango to send, they have to remit funds in gold and silver. . There are the expenses of freight and insurance upon the spocic, besides the occupation of a certain amount of capital involved in this ; and an increased price, or premium, is charged upon the exchange to cover all. The reverse of this "happens when brokers find that more bills arc 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 infiuenccs that disturb the exchange between different countries. Expectations of receiving large payments from a foreign country will have one effect, and the fear of having to make larger payments will have the opposite effect. AMERICAN EXCHANGE Exchange between Canada and the United States, especially the northern, is a matter of every day occurrence on account of the proximity of the two countries, and the incessant intercourse between them, both of a social and commercial character. The exigencies of the Northern States arising from the late war, compelled them to issue, to an enormous extent, an inconvertible paper currency, known by the name of " Greenbacks." As the value of these depended mainly on the stability of the government and the issue of the war, public con- fidence wavered, and in consequence, the value of this issue sunk materially. This caused a gradual rise in the value of gold until it reached the enormous premium of nearly two hundred per cent., or a quotation of nearly three hundred per cent., that is, it took nearly three hundred dollars in Greenbacks to purchase one hundred dollars in gold. It is to be hoped and expected, however, that as peace is now restored, matters will soon find their former level. It has been deemed essential that this should be distinctly ex- plained, as it has brought about a ueccssit; for a oum>taat calculatloii i AMERICAN EXCHANGI 237 of the relative values of gold and greenbacks, and haa generated an extensive business in that species of exchange. When the term " American currcn(^" 'is used in the following exercises it is understood to be Greenbacks. CASE I. * . To find the ViUue of $1, American currency, when gold is at a premium. EXAMPLE. When gold is quoted at 140, or 40 per cent, premium, what is the value of $1, American currency ? solution; Since gold is at a premium of 40 per cent., it requires 140 cent.s of American funds to equal in value $1, or 100 cents in gold. Hence the value of $1, American money, will bo represented by the number of times 140 is contained in 100, which is .71 1 or Tr|">Beatau_ Hence to find the value of $1 of any depreciated currency reckoned in dollars and cents, we deduce the followin<. BULK. Divide 100 cents hy lOOpZws the rate of premium on gold, and the quotient will be the value of $1. Subtract this from $1. and the remainder will be the rate of discount on the given currency. CASB II. To find the value of any given sum of American currency whan gold is at a premium. EXAMPLES. "What is the value of $280, American money, when gold is quoted at 140, or 40 per cent, premium ? , SOLUTION. We find by Case I. the value of $1 tolae 71 1 cents. Now, it is evident that if 71 1 cents be the value of $1, the value of $280 will be 280 times 71;^ cents, which is $200, or $280-j-l.40=28000-i- 140=$200. Hence wo have the following 268 ABITHMETIO. RULE. Multiply the value of $1 ly the number denoting the given amount of American mon^y, remium o« itself at the givm rate, and tlit result will he the value in American currency. EXKRCISES. 1 If American currency ia at a discount 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 ? Ans. 28:} per cent. 3. A person exchanged ^TfiO, American money, at a discount of 35 per cent, for gold ; how much did ho receive ?" Ans. $ 4Q7 i& . // ^ y, ^ ^ 4. Purcliased a draft on Montreal, Canada Eabt, for $1500 at a premium of G42 per cent. ; what did it cost me ? Ans. ' ' ^ j 5. If American currency is quoted at 33J per cent, discount ; what is the premium on gold ? Ans. 50 per cent. 6. Purchased a suit of clothes in Toronto, Canada West, for $35, but on paying for the same in American funds, the tailor charged me 32 per cent, discount^ how mucli had I to pay him ? Ans. $51.47. 7. What would bo tho difference 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 C. R. King, $27 in gold, and jvished i" repay him in American currency, at a discount of 38 per eent. , how much did it require ? Ans. $43.55. P. J. E. Pekham bought of Sidney Leonard a horse and cutter fov $315.50, American currency, but only having $200 of this sum, bo paid tho balance in gold, at a premium of 65 per cent. ; how much did it require ? Ans. $70. 10. A cattle drover purchased of a farmer a yoke of oxen valued at $135 in gold, but paid him $112 in American currency, at a discount of 27^ per cent. ; Iiow much gold did it require to pay the balance ? Ans. $53.80. 11. W. n. Ilounsficld Jt Co., of Toronto, Canada West, purchased in New York City, meiohandiso amounting in value to $4798.40, on 3 months' credit, premium on gold being 79§ per coot. At the K 270 AMTHMETIC. expiration of the three months they purchased a tlraft on Adams, Kimball and Mccrc, of New York, for thq amount due, at a discount of 57f per cent. ; what was the gain by exchange ? Ans. $647.75. 12. A makes an exchange of a horse for a carriage with B ; the horse being valued at $127.50, in gold, and the carriage at $210, American currency. Gold being at a premium of 65 per cent. ; what was the difference, and by whom payable ? Ans. B pays A 23 cents in gold, or 37 cents in greenbacks. 13. A merchant takes $63 in American silver to a broker, and wishes to obtain for the same greenbacks which are selling at a dis- count of 30 per cent. The broker takes the silver at 3^ per cent. discount ; what amount of American currency does the merchant receive? Ans. $86.85. 14. I bought the loilowing goods, as per invoice, from John McDonald & Co., of Montreal, Canada East, on a credit of 3 months : 1120^ yards Canadian Tweed at 95 cents per yard. A 2190. " long-wool red flannel at 60 '• " " 3400 " « white flannel at 55 " «« « Paid custom house duties, 30 per cent. ; also paid for freight, $3V.40. Qold at time of purchase was at a prrmiam Of 63f per cent. ; what shall I mark each piece at per yard to make a net gain of 20 per cent, on full cost ? Ans. C. tweed, $2.44 ; red flannel, $1.54 ; white flannel, $1.41. 15. A merchant left Toronto, Canada West, for New York City to purchase his stock of spring goods, taking with him to defray expenses $95 in gold. AfUnt purohastng his ticket to the Suspension Bridge for $2.40, he expended the balance in greenbacks, which were at a discount of 41 J per cent. When in New York he drew from this amount $23.85 to " square" an old account then past duo. On arriving homo he found that ho still had in greenbacks $16.40, which he disposed of at a discount of 43| per cent., receiving in payment American silver at a discount of 3^ per cent., which he passed off at 2^ per cent, discount for gold. What were his expenses in gold ; the actual amount in greenbacks paid for expenses, and the amount of silver received ? Ads. Total expenses in gold, $71.76 ; expenaes in greeobaeks, $118,04 ; silver received, $9.63. -^ EXCHANGE WITH OBEAT BBITADT. 271 1 Aclans, & discount . 0647.75. th B ; the » at 0210, per cent.; TTccnbacks. jrokcr, and ng at a dis- 5^ per cent. e merchant ,ns. $86.85. from John f 3 months ; per yard. for freight, of 63J per ie a net gun mnel, $1.41. |w York City im to defray e SuBpension icks, which ork he drew icn past duo. [aoka $16.40, receiving in ioh he passed expenses in tscs, and the EXCHANaE WITH GREAT BRITAIN. In Britain money is reckoned by pounds, shillings and pence, and fractions of a penny, and ia called Sterling money, the gold soverei[:n or the pound sterling, consisting of 22 parts gold and 2 alloy, being the standard, and tho shilling, one-twentieth part of the pound, u silver coin of 37 parts silver and 3 copper, and the penny, one- twelfth part of the shilling, a copper coin, the ingredients and size (if which have frequently been altered. The comparative value of the gold sovereign in the United States previous to the year 1834 was 04.44^, but by Act of Congress passed in that year it was made a legal tender at the rat« of 94j% cents ^cr pennyweight, because the old standard was less than the intni.sio value and also because the commercial value, though fluc- tuating, was always considerably higher. Hence, the full weight of the sovereign being 5 dwts. 3.274 grs., it was made equivalent to ii i!' Uars and 86f} cents. The increase in the standard value was, thcefore, equal to 2^ per cent, of its nominal value. The real par of exchange between two countries is that by which an ounce of gold in one country can be replaced by an ounce of gold of equal fineness in tho other country. If tho course of exchange at New York on London were 108J per cent. ; and tho par of exchange between England and America 109^ per cent., it follows that the exchange is 100 per cent, against England ; but the quoted exchange at New York being for bills at 60 days sight, tho interest must bo deducted from th>3 above differ- ence. The general form for the quotation of exchange with England is: 108, 108^, 109, 109^ &c., which indicates that it is ut 8j 8^, 9, or 9^ per cent, premium on its nominal value. EXAMPLE. What amount of decimal money will be required to purchase a draft on London for £648 178. 6d. ?— ezohango 108. The old par value or nominal value is $4.44^=Y=9 of $40 TTT- 272 BTTHMETIG. by reaucing to an improper fraction. >iow, the quotation is 108, or 8 per cent, above the nominal value, we find the premium on $40 at 8 per cent., which is $3.20, which added to $40 will give $43.20, and $43.20-v-9=$4.80 to be remitted for every pound sterling, and therefore JE648 17s. 6d. multiplied by 4.80 or 4.8 will be the value in our money. 17s. 6d.=.875 of a pound, and the operation is aa follows : £648,875 4.8 5191000 2595500 $3114.6000 . B U L E . Vo $40 add the premium on itself at the quoted rate, multiply the mm hy the number representing the amount of sterling money, and divide the result by 9, the qu/jtient will be the equivalent of the sterling money in dollars and cents. NoTX.~If there bo Bhillings, pence, &e., in the sterling money, they arc to be reduced to the decimal of £1. To find the value of decimal money in sterling money, at any given rate above par. Let it be required to find the value of $465 in sterling money, at 8 per cent abovo its nominal value. Hero we liavo exactly the converse of tho last problem, and therefore, having found the value of £1 sterling, we divide tho given sum instead of multiplying ; thus tho premium on $40, at 8 per cent., is $3.20, which added to $40 makos S43.20, and 43.20-^-9=4.80, and $465-^.80=:£g6.17.6. BULK.^ Divide the given sum hy the number denoting the value of one l>ound sterling at the given rate above par, and if there be a decimal remaining reduce it to shillings ahd pence. EXEBOIBSS. 1. When sterling exchange is quoted at 108, what is the value of £1 ? Ads. »4.80i is 108, on $40 $43.20, ng, and le value ion is as EXCHANGE WITH GlffiAT BRITAIN. 273 , multiply ng money, hnt of tU fcy, they are aoy, at any money, at exactly the i the value ultiplying ; Ided to $40 ;96.17.6. \Ue of one I a decimal Is the value LBS. »4.80. 2. If £1 eterling be worth $4. 84 ;J,* what is the premium of ex* change between London and America. Ans. 9 per cent. 3. At lO'per cent, above its nominal value, what is the worth of £50 sterling, in decimal currency ? Ans. $244.44. 4. When sterling exchange is quoted at 9J per cent, premium, what is the value of $1000 ? Ans. £205 18s. llfd. 5. At 12 per cent, above its nominal value, what will a bill for .£1800 cost in dollars and cents ? Ans. $8960. 6. A merchant sold a bill of exchange on London for £7000, at an advance of 11 per cent ; wHat did ho receive for it more than its real value ? Ans. $466.66|. 7. Bought a bill on London for £1266 15s. at 9^ per cent, pre- mium ; what shall I have to pay for it ? Ans. $6164.85. 8. A merchant sells a bill on London for £4000, at 8 per cent, above its nominal value, instead of importing specie at an expense of 2 per cent. ; what docs ho save ? Ansi $122.66§. 0. A merchant in Kingston paid $7300 for a draft of £1500 on Liverpool ; at what per cent, of premium was it purchased ? Ans. 9J. 10. Exchange on London can be purchased in Detroit at 108J ; iu New York at 108}. At which place would it be the most advan- tageous to purchase a bill for £358. 14s. 9d., supposing the N.Y. broker charges } per cent, commission for investing and gold drafts }Q New York are at a prerixium of g per cent. Ans., Detroit by $6.82. 11. A broker sold i\ bill of exchange ibr £2000, on commission. It 10 per cent, above its norainal value receiving a commission of y'jj per cent, ou the roal value, and 5 per cent, oh what he obtained for the bill.above its real value ; what was his commission ? Ans. $11 12. I owe A. N. McDonald & Co., of Liverp6bl, $7218, net pro- ;x}cds of sales of merchandise cflfcctcd for them, which I am to remit them in a bill of exchange on London for suoh amount as wiTl close the transaction, lees } per cent, on the face of the bill for my oom- Qiission fdr investing. Bills on London are at 8 per cent, premium. Required the amount of the bill, in sterling money, to be remitted. 18 Ans. £1600. 274 AIOTHMETIC. TABLE OF FOBEIQN MONEYS. Cities and Coc!(trii& London y Liverpool, &c Paris, Ha\TC, &c Amsterdam, JFIagae, &c. Bremen Hamburg, Luboo, &o. Berlin, Dantzic Belgium ■ St. Petersburg. 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..., Denommations of Uonbt. shilling ; 20 shillings 12 pence=:l =1 pound = 100 centiraes=l franc = 100 cents=:l guilder or florin. ..= 5 swaren=:l grote ; 72 grotes=l rix dollar =: 12 pfennings=l schilling ; 16s.=t 1 markhanco ;= 12 pfennings=::=l groschen ; 30 gro. =1 thaler: = 100 centimes=l franc = 100 kopccks=l ruble = 12 rund8tycks=16 skillings; 48s. =1 rix dollar specie .^ =^ 16 8killings=l mark ; 6 m.=l rix dollar : = 60 kreutzor8=l florin = 10 grani=l carlino ; 10 car.=l ducat = 100 ccntesirai:=l lira = 100 centesimi=l lira = 100 ccnte8imi=l lira %.= 20 grani=l taro ; 30 tari=l oz.= 1000 reas=:l millrca = (•34 maravedis=l real vellon= \ 68 maravedis=l real plate. . = 100 a8pers=l /Jtuster ....== 12 pice=l anna; 16 annas=l rupee = 100 candarines=::l mace ; 10 m.= 1 tael = 8 rial»=l dollar == 100 centesimas=l rial ; 8 rials=l dollar = 1000 rcas=rl milrea =i= 8 reals plate or 20 reals vellon=l dollar = 100 asper8=l piaster ..=^ 10 mills=l cent ; 10 cent8=-l dime ; 10 dimes=l dollar....-— 4 farthings=l penny ; 12 pence =1 shilling; 20 shillings^: 1 pound.* = Vaiub. ! $4.86f .40 .781 .35 .69 .18' .75 1.06 1.05 .48J .80 .16 .16 .183- 2.40 1.12 .05 .10 .05 .44^ 1.48 1.00 ■83/0 .82^ 1.00 .05 variable. I 4.00 • The Grovernmcnt of New Brunswick now Issues postage stamps in the decimal currency, but bo for as wo have been able to ascertain, the currency of ARBITIIAIION OF EXCHANGE. 275 Valov. ARBITRATION OF EXCHANGE. $4.86§ .18? .40 .781 .35 .69 .18^ .75' 1.06 1.05 .48J .80 .16 .16 .181 2.40 1.12 .05 .10 .05 "L ' .441 = 1-48 = 1.00 ^1\ .83^ n= 1.00 .05 5=-l liable. vai encel 1=1 .=1 4.00 stamps in tho currency ol the Arbitration of Exchange is tho method of finding the rate of exchange between two countries through the intervention of one or more other countries. The oVjcct of this is to ascertain what is the most advantageous channel through which to remit money to a foreigo country. Three things have here to be considered. First, what is the liQOst secure channel ; seeondli/, what 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 bo given, as there must necessarily be a continual fluctuation arising from political and other causes. We are therefore compelled to confine our calculation to the third, viz., the comparative value of the coin current of different countries. For this purpose wo shall investigate a rule, and appen^Haj ^les. 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 tho exchange of London on Hamburg is \Z\ marcs per pound stetr 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 tho rate of exchange betweea London and Paris. SOLUTION: We write down tho 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. H^nce 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 tho equivalent of the first term, the first term in the last rank corresponds to the third term of an analogy, and is there* fore a multiplier, it must be placed below the second rank.' The these three Provincea is, as usual, in pounds, shillings nud pence. It is to be hoped that when the Confederation of the British Provinces takes place, the decimal currency will be speedily adopted in tho Lower Provinces, and that the efforts now being made in Britain to adapt the fMme currency will prove sacccssiUl. t 276 ABTTHMETICL terms being thus arranged, \ire divide the product of the second rank by that of the first, and the quotient will be the equivalent, as exhi- bited below: £1 sterlings 13^ marcs. 40 marcs = 3Q^ florins. 66| florins =120 francs JEl stg. 9 As it 18 most convenient to express the fractions decunally, we have I3.fiXS O / Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. MS80 (716) 872-4503 ,* €P.. s ^^ t^ i/i k •'•'< ■^r- 280 ABTTHMETIO. INVOLUTION. Involution is the process of finding a given power of a given number. We have noted already, under the head of multiplication, that the product of any number of equal factors is called the second, third, fourth, &o., power of the number, according as the factor is taken two, three, four, &c., times. Thus: 9=3X3 is the second power of 3; 27=3X3X3 is the third power of three; 81=3x3x3x3 is the fourth power of 3. These are often written thus : 3-, 3 3, 3'», &c. The small figures, 2, 3, 4, indicate the number of factors, and therefore each is called the index or exponent of the power. Hence to find any required power of a given quantity, wo have the BULK. Multiply the quantity continually hy itself until it Jmabeen used as a factor as often as there are units in the index. Since the first multiplication exhausts two factors, the number of operations will be one less than the number of factors. Involution, then, is nothing more than multiplication, and for any power above the second, it is a case of continual multiplication. For the sake of uniformity the original quantity is called the Jirst power, and also the root in relation to higher powens. 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^, it fol- lows that 3- X3^=3''j 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^-33=3 or 3 ' . If we di' ido 3 > by 3 » by subtracting the index of the divisor from that of the dividend, we obtain S**, 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 arc to bo found, the operation may be short- ened in, the following manner :— Let it be required to find the six- tcentli power of 2. Wd first find the second power of 2, which is 4, INVOLUTION. 281 tlien 4X4=16, which is the fourth power, and 10X16=256, the eighth power, and 256x256= 65536, the sixteenth power. If we wished to find the nineteenth power, wo should only have to multiply the last result by 8, which is the third power of 2, for 2» ^ x23=:2» ». Aas. 485809. Ans. 622835864. Ans. 19.070689. Aos. 31040625. Ans. 1.9738+. EXERCISES. 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. .000893+. 7. What is the fifth power of 4 ? Ans. 1024. 8. Find the third power of .3 to three places ? Ans. .036963. Ana 343 Ans, ^vfTj* Ans. 1.800943. Ans. 4.538039. Ana (lAfll Ans. 23^|. Ans. .7776. Ans. -Lp=30J. 1 B Ans. 1 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 tho second power of 5J ^ 16. What part of 83 is 2«? 17. Wliat is the diff"erence between 5" and 4" ? Ans. 11529. 18. Expand 35X2'. Ans. 3888. 19. Express, with a single index, 47^X475 X47C ? Ans. 47> 4. 20. How many acres are in a square lot, each side of which is 135 rods ? ♦ Ans. 113 acres, 3 roods, 25 rods. 21. What is the sixth power of .1 ? Ans. .000001. 22. What is the fourth power of .03 ? Ans. .00000081. 23. What is the fifth power of 1.05 ? Ans. .1.2702815625. 24. What is tho third power of .001 ? Ans. .000000001. 25. What is the second power of .0044 ? Ans. .00001836. The second power of any number ending with tho digit 5 may he readily found by taking all tho figures except tho 5, and multi- o This exerciBo will bo most readily worked by finding the 8i.K^eenth power, and dividing by l.Oi. So in tlio next excrclsof find tho thirty-Becond power, and divide by 1.05. A still more cosy mode of working such ques- tions will bo found under tho head of logarithms. '■■• A V 282 ABnTTMKTIC. plying that by itself, increased by a unit, and annexing 25 to the result. Thus, to find the second power of 15, cut off the 5, and 1 remains, and this increased by 1 gives 2, and 2X1=2, and 25 annexed will give 225, the b<;coad power of 15. So also, 625 3.5 4 1225 I' 10.5 11 11025 21.5 22 46225 57,5 58 3.30625 EXERCISES ON THIS METHOP. 26. What is the second power of 135 ? 27. What is the second power of 205 ? 28. What is the second power of 335 ? 29. What is the second power of 455 ? 30. What is the second power of 585 ? 31. What is the secoud power of 795 ? Ans. 18225. Ans. 42025. Ans. 112225. Ans. 207025. Ans. 342225. Ans. 632025. Note. — The square root of any quantity ending in d, must end in either 3 or 7. No second power can end in 8, 7, 3 or 2. The second root of any quantity ending in 6, must i The second root of any quantity ending In 5, must > The second root of any quan^ty ending in 4, must i The second root of any quantity ending in 1, must i The second root of any quantity ending in 0, must i end in 4 or C. end also in 5. ; end cither in 8 or 2. end either in 1 or 9. : also end In 0. EVOLUTION. The root of any quantity is a number such that when repeated, as a factor, tho specified number of times, will produce that quantity. Thus, 3 repeated twice as a factor gives 9, and therefore 3 is called tho second root of 9, while 3 taken three times as a factor will give 27, and therefore 3 is called tho third root of 27, and' so also it is called iliG fourth root of 81. There are two ways of indicating this. First, by the mark -j/ which is merely a modified form of tho letter r, tho initial letter of the English word root, and the Latin word radix (root). When no ;nark' is attached, the simple quantity or jflrst root is indicated. When the second root is meant, tho mark -y/ alone is placed before tho quantity, h\x\ if tho third, fourth, &c., roots arc to be indicated, SECOND OR SQUAEE EOOT. 233 tho figures 3, 4, &c., are written in the angular space. Thus: 3=1/9=^27=^81=^243, &c., &c. Tho other method is to write the index as a fraetion. Thus, 9* means the second root of the first power of 9, i. e. 3. So also, 27 » is the third root of the first power of 27. In the aarlte manner G4^ means the third root of the second power of C4, or the second power of the third root of 64. Now the third root of 04 is 4, and the second power of 4 is IG, or the second power of G4 is 4096, and the third root of 4096 is 16, so that both views give the same result. Evolution is the process of finding any required root of a given quantity. SECOND OR SQUARE ROOT. 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 the second power of any whole number less than 10, consists of cither one or two digits ; the second power of any number greater than 9, and Jpss than 100, will in like manner be found to consist of three or four digits ; and, universally, the second power of any number will consist of either twice the number of digits, or one less than twice the number of digits that the root itself consists of Hence, if wo begin at the units' figure, and mark oiF the given number in periods of two ^gures each, we "hall find that the number of digits contained in the root will be the same as the number of periods. If the num- ber of digits is even, each period will consist of two figures, but if the number of digits be odd, the last period to the left will consist of only one figure. Let it now bo required to find tho second root of 144. We know by the rule of involution that 144 is the second power of 12. Now 12 may bo resolved into one ten and two tmits, or 10-(-2, and 10+2 multiplied by itself, as in tho margin, gives 100-(-40-|-4, and since 100 is tho second power of 10, and 4 tho second power of 2, and 40 is twice the product of 10 and 2, we conclude that the second 284 AEITHMETIC. 10+2 10+2 100+20 20+4 100+40+4 power of any number thus resolved is equal to the sum of the second powers of the parts, plus twice the product of the parts. Ilcncc to find the second root of 144, let us resolve it into the three parts 100+40+4, and we find that the second root of the first part is 10, and since 40 ' is twice ^lie product of tha parts, 40 divided by twice 10 or 20 will give the other part 2, and 10+2.^12, the second root of 144. We should find the same result by resolving 12 into 11+1, or 9+3, or 8+4, or 7+5, or 6+6, but the most convenient mode is to resolve into the tens and the units. In the same 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=00, and 420-^00=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. GOO is twice the product of 100 and 3. 120 is twice the product of 20 and 3. 15129 is the sum of all, and hence 1 is the root of the hunareds^ 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; tliat the second power of a number consisting of ^ns and units is the second power of the tens, jilustYiQ second power of the units, plus twice the product c£ the tens and units ; that the second power of a number, consisting of hundreds, tens and units, is the sum of the squares of the hundreds, the tens, and the units, plus twice the product of each pair. Now since the complement of tho full second power, to tho sum of tho second powers of the parts, is twice tho product of the parts, it follows that, when the first figure of the root has been found, it must bo doubled before used as a divi- sor to find the second term, and for the same reason each figure, when found, must bo doubled to give correctly tho next divisor. Hence the SECOND OR SQUAEE ROOT. 285 )129. This RULE. Beginning at the units' figure, mark off the tshole line in periods of two figures each ; find the greatest power contained in the left hand period, and subtract it from that period ; to the remainder annex the next period ; for a new dividend, place the figure thus obtained as a quotient, and its double as a,divisor, and find how often that quantity is contained in the second partial dividend, omitting the last figure ; annex the figure thus found to both divisor and quotient, multiply and subtract as in common division, and to the remainder annex the next period; double the last obtained figure of the divisor, and proceed as before till all the periods are exhausted, — if there be a remainder , annex to it two cipher^ and the figura thence obtained \DJ,ll ho a decimal, as will every figure thereafter obtained. EXAMPLES. 1. To find the second root of 707449. First, commencing with the units' figure, we divide the line into periods, viz., 49, 74 and 79, — we then note that the greatest square contained in 79 is 64, — this we subtract from 79, and find 15 remaining, to which 893 we annex the next period 74, and place 8, the second root of G4, in the quotient, and its double 16 as a divjsor, and try how often 16 is contained in 157, which we find t<) be 9 times ; placing the 9 in both divisor and quotient, we multiply and subtract as in common division, and find a remainder of 53, to which wc 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 bo illustrated as follows : To find the second root of 273529. 8 109 1783 797449 64 1574 1521 5349 5349 600 500x2=1000-1-20, or 1020 1000+2X20+3=1043 273529- 250000 3129 3129 500+20+3=523 ■•♦•t 286 AurrmrETic. 3. To find the second root of 153687. Here we obtain, by the same process as in the last example, the whole number 392, with a remainder of 23, which can produce only a fraction. fi9 782 78402 784049 392.029+ 230000 156804 7319600 7050441 263159 We now annex two ciphers, placing the decimal point after the root already found, but as the divisor is not contained in this new dividend, we place a cipher in both quotient and di- visor, alid annex two ciphers more to the dividend^ and by continuing this process we find the decimal part of the root, and the whole root is 392.029+. Ans. 529. Ans. 8042. Ans. 678. Ans. 28.01785+. Ans. 41.569219+. Ans. 25.8069+. EXERCISES. 1. What is the second root of 279841 ? 2. What is the second root of 74684164? 3. What is the second root of 459684 ? 4. What IS the second root of 785 ? 6. What is the second root of 1728? 6. What is the second root of 666 ? 7. What is the second root of 123456789 ? Ans. 11111.11106+. 8. What is the second root of 5 to three places ? Ans. 2.236. 9. What is the side of a square whose area is 19044 square feet ? Ans. 138 feet. 10. What is the length of each side of a square field containing 893025 square rods ? Ans. 945 linear rods. The second root of a fraction is found by extracting the roots of its terms, for ^^=1X5 and therefore i/^«=|/4xi=|- So also, i/^ 9=5. Again, since •^j%\z2z-»j^=.09, and .3 X .3=.09, the second root of .09 is .3. This follows from the rules laid down for the multiplication of decimals. ' To find the ieoond root of a decimal or of a whole number and a decimal: SECOND OR SQtJ.UlE ROOT. 287 Point off periods of two figures each from the decimal point towards the rigjit and left, adding a cipher, or a rcpdcnd, if the number of figures be odd. From what has been saii, it is plain that every period, except the first on the loft, must consist of two digits, and every decimal presupposes something going before, for .5 indicates the half of some unit under consideration, and J) is equivalent to .50, and not to .05, from which it is obvious that the second root of .5 is not the rcot of .05, but of .50, and therefore the second root of .5 is not .2-f , as the beginner would naturally suppose, but .7-}-, for .2-j- is the approximate root of .05. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. ADDITIONAL EXERCISES. What is tho second root of .7 to five places of decimals ? _^^^^ Ans. .83G66. Find tho second rootoTlWr to six places. Ans. 264575. What is the second root of .05 ? What is the second root of .7 ? Find the second root of .5. What is the second root of .1 ? What is tho second root of .1 ? What is the second root of 1.375 ? What is tho second root of .375 ? What is tho second root of 6.4 ? Find to four decimal places \^^^xi' Find \/2 to four decimal places. Ans. .2236-}-. ns. .8819-f-. Ansr"rr4635-f-. A^s. .3102277-1-. Ans. .3. Ans. 1.172G, &c.* Ans. 61237, &c.=i« Ans. 2.52982-1-. Ans. 1.7748. Ans. 1.4142. Ans. 57.196-I-. Find tho value of i/3271.4207. Find the second root of .005 to five places. Ans. 07071. Find the square root of 4.372594. . Ans. 2.09107-f-. What is the second root of .01 ? What is the second root of .001 ? What is the square root of .0001 ? What is the second root of .OOOOOl ? What is the second root of 19.0968 ? Ans. .1. Ans. 031G2-I-. Ans. .01. Ans. .001. • The young etudeat would naturally expect that the decima'. figures of ■j/1.376 end |/.a75 would be the same, but it is not bo. If it were so, |/1-}- l/.37d would be equal to y'l^tTd. That such is not the cose, may be abowa by a very simple exftn?.ple. 1^10-1-/9=44-3—7, but |/'H)-f9=|/25=5. Let it bo carefully observed, therefore, that thi awn of the second roots is not the same as tlu second root qf the svm. 288 AEITHMETTO. OPERATION 4 19.0968 16 • 4.37 triaf. 83 4.36 true. Trial 867 . 309 249 Too greiyjb by 1 True 866 G068 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 that we are not seeking an exact root, bui only the closest possible approximation to it. If the given quantity had becu 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 /g^g, whereas 7 would bo too great by yj^, 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 % perfect power 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 |/64=8, 6 ^Ci:=4 and ]/64=2, but it is irrational regarding any other root expressed by a whole number. But 64, with the fractional index %, i. e., 64^, is rational, because it has an even root as already shown. We may call 64* either the second power of the third root of 64, or the third root of the second power. In 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 4096 is 16, the same 4 as before. i/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 ^8=2 only regarding the third root. The second root of an even square may be readily found by ro> flolying the number into its prime factors, and taking each of these THIRD ROOT OR CUBE ROOT. 289 factors once,— the product will be the root. Thus, 441 is 3x3X^X7 and each factor takca once is 3x7=:21, the second root. Here let it be observed, that if we used each factor twice we should obtain the 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. 64 is 2X2X2X2X2X2=2", i. e., 2 repeated six times as a factor gives the number 64, and therefore half the number of these factors will give the second root of 64, or 2x2X2=8, and 2X2X2 multiplied by 2X2X2=8X8=64. As this cannot be considered more than a trial method, though often expeditious, wo would observe that the smallest possible divisors should be used, in every case, and that if the number cannot be 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 fraotioQS. 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 the third root is the finding of what three equal factors will' produce the quantity. By inspecting the table of third powers, it will be seen that no third power has more than three digits for each digit of the first power, nor fewer than two less than three times the number of digits. Hence, if the given quantity be marked off in periods of three digits each, there will be one digit in the first power for each period in the third power. The left hand period may contain only one digit. From the mode of finding tho third power from the first, we can deduce, by the converse process, a rule for finding the first power 18 •-"<•«> 290 ARITHMETIO. from the third. We know by the rule of involution that the third power of 25 is 15625. If we resolve 25 into 20-}-5, and perform the multiplication in that form, tee have 20-J-5 400+100 ' 100+25 400+200+25=(20+5)3 20+5 8000+4000+500 2000+1000+125 . 8000+6000+1500+125=(20+5) 8=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 power of 20, and 1500 is three times the product of 20, and the second power of 6. Let a represent 20 and b represent 5, then a3=203 = 8000 3 a2& =3X202X5 = 6000 3 a 62=3X20X5* = 1500 63=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^=208=20x20X20 ,.=• 8000 Bemainder 76425 3 a» 6=3X20* XS* = 6000 Hemunder 1625 3 a 53=3X20X5* ....= 1600 Bemainder 125 63=53=5X5X5 = 125 From this and (Similar examples we see that a number denoted by more than one digit may bo resolved into tens and units. Thus, 25 is 2 tens uid 6 units, 123 is 12 tens and 3 units, and so of all nnmbem. •^"' it the third that fona, 25 third power : the second 20, and the 5, then iihle method >ited by the tor denoted its. Thus, id 80 of all THIRD ROOT OB CUBE ROOT. 291 To find the third root of 1860867 : As this number consists of three periods, the root vriW consist of three digits, and the first period from the left will give hundreds, the second tens, and the third units, and so also in case of remainder, each period to the right will give one decimal place, the first being tenths, the second hundredths, &c., &c. We may denote the digits by a, 6 and c. a=100 a8=1003= and 30000X20= 3 a 62=3X100X400= i3=203 = 1860867(100-1-20+3=123 1000000 860867 remainder. 600000 260867 remainder. 120000 140867 remainder. 8000 Now (a-f-6)=120 . • . 3 (a+ J)2=132867 remainder. 43200, whic < is contained 3 times -j- in 132867, . • . c=3, and 3 (a-^-b^^c^ =3x120^X3= 129600 And 3 (a-\-b) c8=3Xl20x9= And lastly, c3=33= *3267 remainder. 3240 27 27 no remainder. SULB. Mark off the given number in periods of three Jl^ures each. Find the highest third power contained in the left hand period, and subtract it from that period. Divide the remainder and next period hy three times the second power of the root thus found, and t^ quotient will be the second term of the root. From the first remainder subtract three times t'he product of the second term, and the square of the first, PLCS three times the product of the first term, 4md the square of the second, PLUS the third pover of the second. Divide the remainder by three times the square of the sum oftJtC first and second terms, and the auotient will be the third temb 202 ARITHMETIC. From the last rcmainddr subtract three times the product of the tertn last found, and the i^u^ire of the sum of the j)i'eceding terms, PLUS the product of the square of the last found term by the SUM of the preceding ones, PLUS tJie third power of the last found term, and so on. EXERCISES . 1. Whati stho third root of 4GG5G? Ans. 36. o What i s the third root of 250047 ? Ans. 63. 3. What 1 s the third root of 2000576 ? Ans. 126. 4. What 1 a the third root of 5545233 ? Ans. 177. 5. What ] 3 the third root of 10077696 ? Ans. 216. 6. What 13 the third root of 46208279? Ans. 359. 7. What IS the third root of 85766121 ? Ans. 441. 8. What is tho'^third root of 125751501 ? Ans. 601. 9. What IS the third root of 153990G56 ? Ans. 536. 10. What! 8 the third root of 250047000 ? Ans. 630. 11. What is each side of a square box, the solid content of which is 59319 ? Ans . 39 inches. 12. What is the third root of 926859375 ? Ans. 975. 13. Find the third root of 44.6. Ans. 3.456+. 14. What is the third root of 9 ? Ans. 2.08008+. 15. What is the length of each side of a cubic vessel whose solid content is 2936.493568 cubic feet ? Ans. 1432 feet. 16. Find the third root of 5. Ans. 1.7099. 17. A store has its length, breadth and height all equal ; it can hold 185193 cubic feet of goods; 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 144 ? Ana. 5.241483. The third root of a fraction is found by extracting the third root of the terms. The result may be expressed either as a common fraction, or as a decimal, or the given fraction may be reduced to a decimal, and the root extracted under that form. THmD ROOT OR CUBE ROOT. 293 EXERCISES. 1. What ia Iho third root of || ? • Ans. |=.7D. Otherwise ; §|=.421875. To find the third root of this wo have 703== 3X702X5 =73500") 3X70 X52= 5250 V 53= 125 \ = .421875(.70+.05=.75 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 bo obtained by Jividing tho given number by the smallest possible divisors in succcssioii, 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 may yet be found approx- imately. If we take tho number 4G65G, wc 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 tico» as factors, and afterwards six tlirccs ; and, as in the case of tho second root, we take each factor Tialf tho number of times it occurs, so in the case of the third root, we take each factor one-third tho number of times it occurs. The same principle on which the extraction of the second and third depends may be applied to any root, the line of figures being divided into periods, consisting of as many figures as there are units in tho index ; for the fourth root, periods of four figures each ; for the fifth, five, &c., &e. Wc may remark, however, that these modes are now superseded by the grand discovery of Logarithmic Computa- tion. ^ 294 ABUHUETIO. PROaRESSIpN. A series is a succession of quantities increasing or decreasing by a Common Difference, or a Common Eatio. Progression hy a Common Difference forms a scries by the addi- tion or subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23 forms a series increasing by the constant quantity 4, and 28, 21, 14, 7, forms a series decreasing by the constant quantity 7. Such a progression is also called an equidifiFerent series.* Progression hy a Common Ratio forms a series increasing or decreasing by multiplying or dividing by the same quantity. Thus, 3, 9, 27, 81, 243, is a series increasing by a constant multiplier 3, and 64, 32, 16, 8, 4, 2, is a series decreasing by a constant divisor 2. The quantities forming such a progression are also called Con- tinual Proportionals,^ because the ratio of 3 to 9 is the same as the ratio of 9 to 27, &c., &c. From this it is plain that in a progression by ratio, each term is a mean proportional between the two adjacent ones, and also between any two that are equally distant from it. The first and last terms are called the Extremes^ and all between them the Means. PROaRESSION BY A COMMON DIFFERENCE. In a series increasing or decreasing by a common difference, the sum of the extremes is always equal to the sum of any two that are equally distant from them. Thus, in the first example 3-j-23=74- 19:==114-15=:26, and in the second 28+7=21+14=35. If the number of terma be odd, the sum of the extremes is equal to twice the middle term. Thus in the berics 3, 7, 11, 15, 19, 3+19=2x11=22, and hDuca the middle term is half the sum of the extremes. * The uamea Arithmetical Progression and Geometrical Progression aro often applied to quantities so related, but tlioso terms aro altogether inappro- priate, as tliey would indicate that the one kind belonged solely to arithmetic, and the other solely to ^^eometry, whereas, in reality, each belongs to both these branches of scieuce. PE0GRES8I0N BY A COMMON DUTEBENCE. 295 In treating of progressions by difference or cqnidifferent series, there are five things to be considered, viz., the first term, the last term, the common difference, the number of terms, and the sum of the series. These are so related to each other that when any three of them are known we can find the other two. Given the first term of a series, and the common difference, to find any other term. Suppose it is required to find the seventh term of the series 2, 5^ 8, &c. Here, as the first term is given, no addition is required to find it, and therefore six additions of the common difference will complete the series on to seven terms. In other words, the common difference is to be added to the first term as often as there are units in the number of terms diminished by 1. This gives 7 — 1=6, and 6X3=18, which added to the first term 2 gives 20 for the seventh term. If we had taken the series on the descending scale, 20, 17, 14, &c., we should have had to subtract the 18 from the first term 20 to find the seventh term 2. The term thus found is usually designated the last term, not because the scries 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.) Subtract 1 from the number of terms, and multiply the remainder by the common difference ; then if the series be an increasing one, add the result to the first term, and if the scries 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 53Xli=6GJ, and 66^+331=100, the fifty-fourth term. Given 64 the first term of a decreasing series, and 7 the common difference, to find the eighth term. Here 8 — 1=:7, and 7x7=49, and 64 — 49=15, the eighth term. EXEllOISES. 1. Find the eleventh terra of the decreasing series, the first term of which is 248|, and the common difference 3 J. Ans. 216J. 2. The hundredth ierm of a decreasing scries is 392|, and the common difference is 3f , what ia tho last term ? . Ans. 36. 'tv 296 ABTTHMETIO. 3. What is the one-thousandth term of the series of the odd figures? Ans. 1999. 4. What is the fite-hundredth term of the series of even digits ? Ans. 1000. 5. What is the sixteenth term of the decreasing series, 100, 96, 92, &c. ? Ans. 40. To find the sum of any 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 tetms that are equidistant from them, and when the number of terms is odd, to twice the mid- dle term. Hence the middle term,^r 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+27+32, we have 2^=1^=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 arc equi-distant from them. From these explanations we dednce the RULE (2.) Multiply the middle term, or half the sum of the extremes, or of any two terms that are equidistant from them, by the number oj terms. Note.— If the sum of the two terms be an odd number, it is generally more convenient to multiply by the number of terms before dividing by 2. EXAMPLES. Given 23, the middle term of a series of 11 numbers, to find the sum. Here we have only to multiply 23 by 11, and we find at once the sum of the series to be 253. Given 7 and 73, the extremes of an increasing series of 12 num- bers, to find the sum. . The sum of the extremes is 80, the half of which is 40, and 40x12=480, the sum required. PROGEESSION BY A COMMON DIFFERENCE. 297 Two equidistant terms of a series, 35 and 70, arc given in a series of 20 numbers, to find the sum of the series. In this case, wo have 35+70=105, and 105x20=2100, and ^100^-2=1050, the sum required. EXERCISES. 1. Find the sum of the series, consisting of 200 terms, the first term being 1 and the last 200. Ans. 20100. 2. What is the sum of the series whose first term is 2, and twenty-first G2 ? Ans. G72. 3. What is the sum of 14 terms of the scries, the first term of which is J and the last 7 ? Ans. 52J. 4. Find the sum to 10 terms of the decreasing series, the first term of which is 60 and the ninth 12. Ans. 360. 5. A canvasser was only able to earn 66 during the first month he was in the business, but at the end of two years was able to earn $98 a month ; how much did he cam during the two years, supposing the increase to have been at a constant monthly rate ? Ans. $1248. 6. If air- n begins on the first of January by saving a cent on the first, two on the second, three on the third, four on the fourth, &c., &c., how much will he have saved at the end' of the year, not counting the Sabbaths ? * Ans. $490.41. 7. How many strokes docs a clock strike in. 13 weeks ? Ans. 14196. 8. If 8f is the fourth part of the fniddle terra of a series of 99 numbers, what is the sum? Ans. 3465. 9. In a scries of 17 numbers, 53 and 33 are equidistant from the extremes; what is the sum of the scries? Ans. 731. 10. In a scries of 13 numbers, 33 is the middle term; what is tho sum ? Ans. 429. To find the number of terms when the extremes and common dificrence are given : As in the rule (1), we fo'ind the differcno'. of the extremes by multiplying by one less than the number of terms, and added the first term to the result, so now wc reverse the operation and find the RULE (3.) hivida the difference of the extremes by the common diffcrcncG and ftdd 1 to the result. 298 ahithmetic. EXAMPLE. Given the extremes 7 and 109, and the common difference, 3, to find the number of terms. In this case we have 109—7=102, and 102—3=34, and 34-|-l^=35, the number of terms. EXEBOISES. 1. Whht is the number of terms when the extremes are 35 and 333, and the common difference 2 ? Ans. 150. 2. Two equidistant terms are 31 and 329, and the common dif- ference 2 ; what is the number of terms ? Ans. 150. 3. The first term of a series is 7, and the last 142, and the com- mon difference ^ ; what is the number of terms ? Ans. 541. 4. The first and last terms of a scries arc 2^ and 35J, and the common difference J ; what is the number of terms ? Ans. 100. 5. The first term of a series is J and last 12J, and the common difference J ; what is the number of terms ? Ans. 25. Given one extreme, the sum of the series and the number of terms, to find the other extreme. This case may be solved by reversing Rule (2), for in it the data arc the same, except that there the second extreme was given to find the sum, and now the sum is given, to find the second extreme. Therefore, as in that rule we multiplied the sum of the extremes by the number of terms and halved the product, so now we must double the sum of the series and divide by the number of terms to find the sum of the extremes, and from this subtract the given extreme, and the remainder will be the required extreme. This will illustrate the RULE (4.) Divide twice the sum of the series hy the number of terms, and from the quotient subtract the given extreme, and the remainder will be the required extreme. EXAMPLE. Given 5050, the sum of a series, 1 the first term, and 100 the number of terms, to find the other extreme. Twice the sum is 10100, which, divided by 100, gives 101, and 101 — 1=100, the number of terms. PEOGEESaiON BY A COMMON DDTEEEXCE. 293 HXERCISES. 1. Given J50, the greater extrwne of a decreasing series, 442, the sum, and 17 the number of terms, to find the other extreme. Ans. 2. 2. If 121268 be the sum of a series, 8 the less extreme, and 142 the number of terms ; what is the greater extreme ? Ans. 1700. 3. The sum of a series of 7 terms is 105, the greater extreme is 21, and the number of terms 7 ; what is the less extreme ? Ans. 9. 4. The sum of a series is 576, the number of terms 24, and the greater extreme is 47 ; what is the less extreme ? Ans. 1 . 5. The sum of a series is 30204^^, the greater extreme 312, and . the number of terms 193 ; what is the less extreme ? Ans. 1. Given the extremes and number of terms, to find the comnwn difFerencG. As explained in the introduction to Rule (1), the number of common differences must be 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 extremes, divided by the sum of the differences, will give one difference, i. e., the common difference. This gives us the KULE (5.) Subtract 1 from the number of terms, and divide the difference of the extremes by the remainder. EXAMPLE. If the extremes of an increasing scries bo 1 and 47, and the number of terms 24, we can find the common difference thus : — 47 — 1=46, and 46-h-23:=2, the common difference. EXERCISES. 1. If the extremes are 2 and 36, and the number of terms 18; what the common difference ? Ans. 2. 2. What is the common difference if the extremes arc 58 and 3, and the number of terms 12 ? Ans. 5. 3. In a decreasing series given 1000 the loss extreme, and 1793 the greater, and 367 the number of terms, to find the common difference. Ans. 2^. •i '■ t ' 300 ABITHMETIO. 4. If 6 and 60 are the extremes in a series of 10 numbers, what is the common diflference ? . ' ^ Ans. 6. 5. What is the common difference in a decreasing series of 42 terms, the extremes of which are 9 and 50 ? Ans. 1. There are fifteen other cases, but they may all be deduced from the five here given. . "We subjoin the Algebraic form as it is more satisfactory and complete, and also more easy to persona acquainted with the symbols of that science. Let a bo the first term, d the common difference, n the number of terms, s the sum of the series ; the series will bo represented by a4(a+tZ)+(a+2(Z)+(a-}-3cZ)+&c., to | a+(n— 1)(?. | By in- specting this series it will bo seen that the co-efficient of d is always 1 less than the number of terms, for in the second term where d first appears, its co-efficient is 1, in the third it is 2, and therefore since n represents the number of terms, the co-efficient of d in the last term is n — 1, and that term therefore is a-\-(n — l)d. If the series were a decreasing one, that is, one formed by a succession of sub- tractions, the last term would be a — (n — l)d. To find the sum of an cquidifferent series. We have here s=a+(a-i-cZ)+(a+2d)4-(a-f 3(Z)+ &c -j- \ a-\-(n — l)d. [• Since a-{-(ii — l)d is tlio last term, the last but one will be a-{-(n — 2)d, and the last but two will be a-}-(n — 3)c?, &c., &c. But the sum of any number of quantities is the same in whatever order they may be written. Let us therefore write this Bevies both as above, and also in reversed order : s=:a4-(a+(Z)+(a+2i)-f (a-f3(Z)+(a+4rf)+&c -fa-f-(n — 3)d-\-a-\-{n — 2)d-\-a-\-(n — l)c?. 8=a-\-(n—l)d-\-a-{-{7i—2)d-{-a-\-(n—S)d-\-&0 (a+4i)+(a+3(Z)+(a+2cr)+(a+cZ)-}-a. Adding the two members of the second to those of the first, wo obtain 2s=i2a-{-(n—l)d\-\-^za-^(n—l)dl-\-i2a-\- (ii—l)d J + j 2a+(n— 1)(Z j +&o., to n terms. PROGEESSION BY A COMMON DIFEEEENCE. 301 In the last expression all the terms are the same, but there are n terms, and therefore the whole will be 2s=n I 2a+(n—l')d | and therefore *=^{2a+(n-l)ci| (1.) As we have used no single symbol to represont the last term, wo must now show how it may be obtained from the other data. Wo have seen that the last term is «+(«— l).i, which we may denote by I, which will give us the formula l=q-^(n—l)d. This formula, in the case of a decreasing series, will become '=« — (n — !)(/, and generally l=a± (n—l)d. (2.) This formula is the same as Rule (1.) We may modify (1) by (2) by substituting I for a+(vr-l)d. Thus: *=i(«+0. (3.) This is a convenient form when the last term is given. Using I for the last term, we have five quantities to consider, viz., a, I, d, n, 8, and, as already stated, any three of these being given, the other two can be found from (1) and (2.) To find d when a, ?, » are given : By (2.) l=a-{-(n.—l)d I — a=ln — l)d d. ^-^ . «-l- (4.) This finds the common difference, when the extremes and num- ber of terms are given, and corresponds to Rule (5.) If a, n, 8 are given, we have By(l.) s=^2ai.(n^l)dl . • . 2s=2an+n (n—l)d .' .dn (n — 1)=2 (s — an) /7~2 (»— an) i! n {n-1) t> , *.. 302 ARITHMETIO. If n ia to be found from a, d, s, wc have . • . 2s=2an=dn^ — d7i . • . dn^-\-n(2a—d)=28 And by solving this quadratic equation, we find ^_d—2a±y/i8ds-\-{2ar-d)^ | 2d ~ EXAMPLES. Given a=6, .;rcd a farmer 10 sheep, at tho rate of a mill for the first, a cent for the second, a dime for the third, a dollar for the *1« «,■ > I- 308 ARITHMETIC. 3. What three numbers inserted between 7 and 4375 will form a series of continual proportionals? Ans. 35, 175, 875. 4. What is the mean'proportional between 23 and 8464 ? • Ans. 441.2164+. 5. Find a mean proportional between ^^ and |. Ans. |. AJjaEBBAIC FORM. Let a represent the first term, I the last, r the ratio, n the num- ber of terms, and « the sum. Then 8=a-{-ar-{-ar" -\-ar^ -{-ar'^ -f &c ar»-~ -^ai"^ ' . Multiplying the whole equation by r, we obtain rs=ar-\-ar"-\-ar\-\'ar'*-\-ar^-{-&o ar^~^ •{-ar'^ . 'B\its^=a-\-ar-\-ar'~-^ar^-{-ar'^-{-ar^^&a or^T'. Subtracting, we obtain ra — s=s(r — l)=ar"— ra, and therefore But we found the last term of the series to be ar""*', calling this I, wo have from ( 1 .) «=^ (2.) . If r is a fraction, r" and ar' decrease as n increases, as already shown under the head of fractions, so that if n become indefinitely great, ar" will become unassignably small, compared with any finite quantity, and may be reckoned as nothing. In this, case ^.1; will become 8=-^=v^ (3.) r— 1 1— r V / By this formula we can find the sum of any infinite series so closely as to differ from the actual sum by an amount less than any assignable quantity. This is called the limit, an expression more strictly correct than the sum. ' • From the formula 8==~j, any three of the quantities a, r, 1, s being given, the fourth can be found. Let it be required to find the sum of the series 1+^+4+^+ &c., to infinity. Hero a^l and r=J . • . ar=l_^=-l =1 x2=2. Therefore, 2 is the number to which the sum of the series coatinually approachc?, by the iacreoso 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. PROGRESSIONS BY RATIO. 309 will form 175, 875. 4? .1.2164+. Ane. |. n the num- -[-ar^' , calling thia 3, ag already indefinitely |th any finite asc [Ij will lite series so BS3 than any Iression more ff ties a, r, ?, « -Hi-+H- I Thereferc, 2 ajiproachc?, |e limit from ly assignable br pasa. By adding the first two terms, we find 1-f J=|=2 — i=l|. By adding the first throe terms, we find |-(-J^=|=2 — J=l|. By addi»g the first four terms, we find |-[-^=Jg''-^2 — ^=1^. By adding the first five terms, we find ^-+10=15=2 — j\= 116 By adding the first six terms, we find ^^l-\-^^^=^^=.2 — 2^= 131 ■^32- It will be observed here that the difference from 2 is. continually decreasing. The next term would differ from 2 by g^, and the next by j^g, &c., &c. Thus, when the scries is carried to infinity, 2 may be taken as the sum, because it differs from the actual sum by a quantity less than any assignable quantity. EXAMPLES. To find the sum of the first twelve terms of Mie series l-}-3-(-9+ 27+&C. : Here a=l, r=3. u rl—a 3 3 -1 3X177 147-1 =265720. And s='-±:?_lji_^^ r— 1 — 3—1 To find the sum of the series 1, — 3, 9, — 27, &c., to twelve terms, ,^^3^^!^^ - ■' X- 1^ 7 1 4 7- 1 ^_i328C0. In the case of infinite scries, if a is sought, s and r being given, we have from '(3) a=^5 (1 — r), and if r is sought, a and s being given, wc have r= — or 1 . EXERCISES.* 1. Find the sum of the scries 2, G, l8, 54, &c., to 8 terms. Ans. 6560. 2. Find the sum of the infinite series J — i+12 — -ii- Observe here r=^ — J. . Ans. ?,. 3. What is the sum of the scries 1, J, ^, &c., to infinity ? Ans. i]. 4. Find the sum of the infinite scries 1 — §-h.J — j/^+ic. Ans. 3. 6. What i.s the sum of nine terms of the scries 5, 20, 80, &c. ? Ans. 436905. 6. Find the sum of |/A+J-|-|/^-|-&c., to infinity. Ans. i/i — 1. 7. What is the limit to which tli^ sum of the infinite series f , h, J, I, &c., continually approaches ? Ans. |. 310 ARITHMETIC. 8. What ia the sum often terms of the series 4, 12, 3G, &c. ? Ans. 11809G. 9. Insert three terms between 39 and 3159, so that the whole shall be a scries of continual proportionals. Ans. 117, .351 and 1053. 10. Insert four terms between J and 27, so that the whole shall fDrni a scries of continual proportionals. Ans. ^, 1,3, 9. 11. The sum of a scrioa of continual proportionals is 10^, the first term 3| ; what is the ratio ? Ans. ^. 12. The limit of an iuiinito scries is 70, the ratio ^ ; what is the first term ? • . Ans. 40. ANNUITIES. The word Annuity originally denoted a sum paid annnaUi/, and thoupjh such payments are often made half-yearly, quarterly, &c., still the term is applied, and quite properly, bceauso the calculations are made for tho year, at what time soever the disbursements may be made. IJy tho terra annuities certain is indicated such as have a fixed time for their commencement and termination. Uy tho term annuities contingent is meant annuities, the com- lucnccment or termination of which depends on some contingent event, most commonly the death of some individual or individuals. • By the term annuityui reversion or deferred, is meant that the person entitled to it cannot "enter on the enjoyment of it till after the lapse of some spcc'.ficd time, or the occurrence of some event, gener- ally the death of some person or persons. An annuity in 2^crj)ctuity is one that " lasts lor ever," and there- fore is a siweics 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 anionnt 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 dilferent periods, by which tho value of any annuity may be found according to the following 3G, &c. ? •ns. 11809G. at the whole 1 and 1053. I whole shall ^.J, 1,3, 9. 3 is 101-, the Ans. ^. ; what is the Ans. 40. muaUj/, and i-ly, kc, still iulations are cuts may be have a fixed }?, the com- j contingent dividuals. • ant that the till after the 2vcnt, gencr- " and therc- h have not ;ite. :ho principal ich it would , if forborne t and final due of any ANNUITIES. 311 RULES. To ^nd either tna amount or the present value of an anuuity,- Multip!,/ the value of the unit, as found in the talks, ly \he number denoting tire annuity. ^ If the annuity b'e in perpetuity, • Divide the annuity hy tlie number denoting the interest of the una fur one year. ' "^ If the annuity be in reversion, — Find the vahtc of the unit np to the date of commencement, and also o the date of termination, and multij>ly their difference hy the number denoting th(f annuity. Divide the present worth hy the worth of the unit. Tables are appended varyin- from 20 to 50 years. EXA3IPLE8. To find what an annuity of 8400 will amount to in 30 years at 3J per cent. ' Wo find by the tables the amount of $1, for -30 years, to bo $01.022077, which multiplied by 400 j/ivcs $20049.07 nearly. To find the present Worth of an annuity of $100 for 45 years at 3 per cent. ' By the table we find $24.518713, and this multiplied by 100 gives $2451.88. ^ To find the present worth of a property on lease for ever, yiuldin- $000, at 3\ per cent. The^rato per unit for one year is .035, and GOO divided by this gives $17142.80. 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 U per cent. By the table we find the rate per unit ibr 3 years to be $"2.801G;J7, and for 8 years, the time the lease expires, $0.873950 ; the difier- cnee is- 84.072319, which, umltiplied by 300, gives $1221.70. Given 6207.90, the present Worth of un annuity continued for 4 years, at :j per cent., to find the annuity. By the tables, the value for $1 is $3.717093, and $207.90 divided by this, gives $55.93. ' 312 ARITirSIETIO. TABLE, 8H0WIN0 Tin! AMOtTOT OF AN ANNTITT OF ONE DOLLAR PER ANNTM, IMPIIOTED AT COMPOUND INTEREST FOR ANY NUMBER OF YEAHa NOT EXCEEDINU FIFTT. h 3 per cent. 3J per cent. 4 per cent. 5 per cent. 6 per cent. J per cent. 1 1.000 000 1.000 000 1.000 000 1.000 000 1.000 000 1.000 00( 2 2.030 000 2.035 000 2.040 000 2.050 000 2.000 000 2.070 000 3 . 3.090 900 3.100 225 3,121 600 3.152 500 3.183 600 3.214 900 4 4.183 627 4.214 913 4 240 464 4.310 125 4.:,74 616 4.439 943 6 6.309 136 5.362 406 5.416 323 5.525 631 6.637 093 5.750 739 6 0.468 410 6..5.-)0 152 6.032 975 6.801 913 . 6.975 319 7.153 291 7 7.662 462 7.779 408 7.898 294 8.142 008 8.393 838 8.654 021 8 8.892 38(i 9.051 687 9.214 226 9.549 109 9.897 468 10.259 803 9 10.159 106 10.308 496 10.582 795 11.026 564 11.491 310 11.977 989 10 11.463 879 11.731 393 12.006 107 12.577 893 13.180 795 13.816 448 11 12.807 790 13.141 992 13.486 351 14.200 787 14.971 043 15.783 599 12 14.192 030 14.601 902 15.025 805 15.917 127 16.809 911 17.888 45! 13 15.617 790 16.113 03(1 10.020 838 17.712 983 18.882 \2b 20.140 043 U 17.086 324 17.670 981 18.291 911 19.598 632 21.015 t)60 22.550 488 15 18.598 914 19.295 081 20.023 588 21.578 564 23.275 970 2.5.129 022 16 2(1.156 881 20.971 030 21.824 531 23.657 492 25.070 528 27.888 051 17 21.761 588 22.705 010 23.097 512 25.840 360 28.212 880 30.840 217 18 23.414 43.'i 24.499 691 25.045 413 28.132 385 30.905 653 33.999 033 19 2.).116 868 26.357 180 27.071 229 30.539 004 33.759 992 37.378 905 20 26.870 374 28.279 682 29.778 07;i 33,005 954 36.785 591 40.99.'i 492 21 28.676 486 30.269 471 31.969 202 35.719 252 39.992 727 44.8()5 177 22 30.536 780 32.328 902 34.247 97(1 38.505 214 43.392 290 49.005 739 23 32.452 884 34.460 414 3(!.617 88<.) 41.430 475 40.995 828 53.436 141 21 34.426 470 36,066 528 39,082 001 44.501 999 50.815 577 58.176 071 25 36.459 264 38.949 857 41.645 908 47.727 099 54.8()4 512 03.249 030 20 . 38.553 042 41.313 102 44.311 745 51.113 454 59.156 383 68.076 470 27 40.709 634 42.759 060 47.084 211 54.009 126 63.705 76(1 74.483 823 28 42.930 923 46.290 627 49.967 583 58.402 583 68.528 112 80.697 691 29 45.218 850 48.910 79» 52.906 280 02.322 712 73.639 798 87.346 529 30 47.575 410 51.622 C77 56.084 938 00.438 848 79.058 18(1 94.400 780 31 50.002 678 51.429 471 59.328 335 70.760 790 84.801 (i77 102.073 041 32 52J502 759 57.334 502 02.701 469 75.298 829 90.889 778 110.218 154 33 55.07r 841 00.341 210 00.209 527 8Q.003 771 97.343 165 118.933 425 34 .57.730 177 63.453 152 09.857 9011 85.006 95!! 104.183 755 128.258 705 35 60.462 082 66.674 013 73.052 225 90.320 307 111.434 780 138.236 878 36 03.271 944 70.007 603 77.598 314 95.830 32:S 119.120 867 121268 119 148.913 460 37 66.174 223 73.457 869 81.702 210 101.628 13!) 1 00.337 400 38 69.159 44!) 77.028 895 85.970 330 107.709 54( 135.904 206 172.561 020 39 72.234 233 80.724 906 90.409 150 114.095 02^ 145.058 458 185.640 292 40 75.401 200 84.550 27.^ 95.025 516 120.799 77 1 154.761 960 199.(135 112 41 78.(503 298 88.509 537 99.820'53(. 127.839 763 105.047 684 214.009 570 42 82.023 190 92.607 371 104.819 598 135.231 751 175.950 045 230.032 240 43 85.483 892 90.818 029 110.012 382 142.9!)3 33fl 187.507 577 247.776 490 44 89.048 409 101.238 331 115.412 877 151.143 00( 199.758 032 200.120 851 45 92.719 861 I05.7SI 073 121.029 392 159.700 15( 212.743 514 285.749 311 46 90.501 457 110.484 031 126.S70 508 168.685 161 220.508 1^5 300.751 703 47 100.390 501 115..350 97:) 132.915 .390 178.H9 421 241.098 61;; 329.224 386 48 104.408 39r 120.388 297 139.263 20( 188.025 39L 256.564 i>2! 353.270 093 49 108.540 048 125.001 84f 145.833 731 198.426 061] 272.958 401 378.999 000 '^0 ■12.r9a 867 130.999 010 1152.667 084 209.347 97( 290.335 905 400.528 929 mm] M, IMPROVED 5DINO FIFTY. 7 por cent. i.ooa 00( 2.070 OOO 3.214 900 4.-i:59 94C .'j.7i)0 731) 7.15:i 291 S.G.Vt 021 10.259 803 11.977 989 13.810 44S- 15.783 59!l 17.888 45! 20.140 G43 22.550 4Hb 2.5.129 022 27.888 05-1 30.810 217 33.999 033 37.378 9C5 40.99.'i 492 44.8()5 177 49.005 739 53.43(1 141 58.17() G71 03.249 030 08.(i76 470 74.-183 823 80.097 G91 87..34G 529 P1.4G0 78G 102.073 041 110.218 154 118.933 425 128.258 7G5 13S.23G878 |l48.913 400 G0.337 400 1172.501 020 1S,).G4()292 199.(i3J 112 214.01)9 570 |2;50.G32 240 !47.77G 49G >G(;.120 851 !85.749 311 tWG.751 7G3 ;29.224 386 553.270 093 178.999 000 I0(i..^»28 929 ANNUITIES. TABLE, 313 SUOWIXO TUK niE^ENT WORTO OP AN AXNXITT OP ON'K I^OI.TjlU PER AXNTM, TO CONTINUE FOR ANY NUMBER OF YEAR.S NOT KXCEEDIXO FIFTY. 1 3 per cent. 3 J per cent 4 per cent. 5 per cent. percent. • 0.943 390 7 por cent, i 1 0.970 874 0.9GC 184 0.9G1 538 0.952 881 0.9.!4 579 1 2 1.913 470 1.899 091 1.880 095 1.8.>9 410 1.833 393 1.808 017 2 3 2.S.!8 Gil 2.801 037 2.775 091 2.723 248 2.G73 012 2.G24 314 3 4 3.717 098 3.C73 079 3.G29 895 3.515 951 3.465 100 3.387 209 4 5 4.579 707 4.515 052 4.451 82': 4.329 477 4.212 364 4.100 195 5 6 5.417 191 5.328 553 5.242 137 5.075 G92 4.917 324 4.7G6 637 6 7 0.230 283 G.114 544 Gr002 05." 5. 780 373 5.582 381 5.389 280 7 8 7.019 092 C.«73 950 0.732 745 0.403 213 G.209 744 5JD71 295 8 9 7.780 109 7.()07 087 7.435 332 7.107 822 0.801 092 0.515 228 9 10 8.530 203 8.310 C05 8.110 89(i 7.721 735 7.300 087 7.023 577 10 11 9.252 021 9.001 551 8.7G0 J77 8.30G 414 7.880 875 7.498 C69 11 12 9 954 001 9.GG;3 331 9.385 074 8.803 252 8.383 844 7.942 C71 12 13 10.G34 955 10.302 738 9.985 048 9.393 573 8.852 683 8.357 035 13 14 11.290 073 10.920 520 10.503 12:; 9.898 041 9.294 984 8.745 452 14 15 11.9J7 935 11.517 411 11.118 387 10.379 G5K 9.712-249 9.107 898 15 1(! 12.5G1 102 12.094 117 11.052 29(i 10.837 770 10.105 895 9.440 632 16 17 13.1 GO 118 12.G51 321 12.105 009 1 1.274 000 10.477 200 9.763 20G 17 18 13.753 513 13.189 082 12.059 297 11.089 587 10.827 003 10.059 070 18 1!) 14.323 799 13.709 837 13.133 939 12.085 321 11.158 llfl 10.335 578 19 20 14..S77 475 14.212 403 13.590 320 12.402 210 11.469 421 10.593 997[20| 21 15.415 024 14.G97 974 14.029 100 12.821 153 11.764 077 10.835 527 21 22 15.930 917 15.IC7 125 14.451 115 13.103 003 12.041 582 11.001 241 22 23 10.443 008 15.020 410 14.85ft 842 13.488 574 12.303 379 11.272 187 23 24 10.935 542 10.058 30S 15.246 903 13.798 642 12.550 358 11.469 334 24 25 17.413 148 18.481 515 15.022 080 14.093 945 12.783 350 11.653 583 25 20 17.870 842 10.890 352 15.982 7Gli 14.275 185 13 003 IGO 11.825 '779 26 27 18.327 031 17.285 3G5 10.329 58(i 14.043 034 13.210 534 11.986 709|27| 28 18.704 108 17.007 019 IG.GG3 oo;; 14.898 127 13.400 164 12.137 111 28 29 19.1«8 455 18.035 7G7 10.983 71.0 15.141 074 13.590 721 12.277 074 29 30 19.000 441 18.392 015 17.292 03;; 1,5.372 451 13.704 831 12.409 041 130! 31 20.000 428 18.730 270 17.588 494 1,5.,592 811 13.929 080 12.531 814 31 32 20.338 700 19.008 805 17.873 552 15.802 677 14.084 043 12.646 555 32 33 20.705 792 19.390 208 18.147 040 10.002 549 14.230 230 12.753 790|33| 34 21.131' 837 19.700 G81 18.411 198 10.192 204 14.308 141 12.854 009 34 35 21.487 220 20.000 GGl 18.004 G13 10.374 194 14.498 246 12.947 672 35 3G 21.832 252 20.290 491 18.908 282 10.540 852 14,620 987 13.035 208 30 37 22.107 235 20.570 525 19.142 579 10.711 287 14.730 780 13.117 017 37 38 22.492 402 20.841 087 19.307 804 10.807 893 14.840 019 13.193 473 38 39 22.808 215 21.102 aoi) 19.584 485 17.01-7 041 14.949 075 13.204 928 39 40 23.114 772 21.355 072 19.792 774 17.159 080 15.040 297 13.331 709 40 41 23.412 400 21.599 104 19.993 052 17.294 308 1.5.138 010 13.394 120 41 42 23.701 359 21.834 883 20.185 627 17.423 208 15.224 643 13.452 449 42 43 23.981 902 32.002 089 20.370 795 17.545 912 15.306 173 13.500 902 43 44 21.254 274 22.282 791 20.548 841 17.002 773 15.383 182 13.557 908 44 45 24.518 713 22.495 450 20.720 010 17.774 070 15.455 832 1.3.605 622 45 40 24.775 449 22.700 918 20.884 654 17.880 0G7 15.524 370 1.3.650 020 40 47 25.024 708 22.899 438 21.042 93G 17.981 016 15.589 028 13.691 608 47 48 25.2GG 707 23.091 244 21.195 131 18.077 1C8 15.050 027 13.730 4741481 49 25.501 657 23.276 504 21.341 472 18.108 722 15.707 572 13.706 .799149 50 25.729 704 23.455 018 21.482 185 18.255 925 15.7G1 801 13.800 746|60 '<.. 314 ARITHMETIC. PARTNERSHIP SETTLEMENTS. The circumstances under which partnerships arc formed, tlio conditions on which they are made, and tlu3 causes that lead to their dissolution, are so varied that it is impossible to do more than give general directions deduced from the cases of most common occur- rence. In forming a partnership, tlic great requisite is to have the terms of agreement expressed in the most el'jar and yet concise lan- guage possible, setting forth the sum invested by each, the duration of partnership, the share of gains or lasses that fall to each, the sum that c'ach may ^» h » ^"^ ^> Ti* -^ investod $3000, and has withdrawn $2500, with tho consent of B. and C, upon which no interest is to be charged ; B invested $2700, and has withdrawn $1150; C invested $2500, and has withdrawn $120. After doing business 14 months, retires. Their assets consist of bills receivable, $2037.20 , merchandise, $1970 ; cash, $1240.80 ;.50 shares of tho Chicago Permanent Building and Savings' Society Stock, tlio par valuo of which is $50 per share; cash deposited in tho Third National Bank, $1850'; store and furniture, $3130 ; amount due from W. Smith, $300.80 ; G. S. Brown, $246.40; and E. R. Carpenter, $97.12. Their liabilities are as follows : Amoiyit duo Samuel Harris, $1675 ; unpaid on store and furniture, $933; and notes unredeemed, $3388.76. The Savings' Society stock is valued at 10 per cent, premium, and C in retiring takes it. as part payment* What is tho amount due C, and what is A's, and what is B'a interest in the business ? Ans. Due C, $815.52; A's interest, $2356.90; B's interest, $2664.14. 0, E, F, G and H are partners in business, each to share \ of profit and losses. Tho business is carried on for one year, when E and F purchase from G and II their interest in the businees, allow- ing each $100 for his good will. Upon examination, their resources are found to be as follows : .Cash deposited in Girard Bank, $364o ; cash on liand, $1422 ; bills receivable, $1685 ; bonds and mortgages, $2746, upon which there is interest duo $106 , Metropolian Bank stock, $1000; Girard Bank stock, $500; store and fixtures, $3500; house and lot, $1800; spaa of horses, carriages, harness, &c., $495 ,> outstanding book debts duo the firm. $4780. Their liabilities me : Notes payoble, $2345 ; upon which there is interest due. $5? ; duo on book debts. $1560. E invested $5000 ; F $4500 ; 313 ARITHMEIIC. h- O, 81000 ; and IT, $3000. E has drawn from tho business 1J1200, upon which he owes interest $32 ; F has drawn 61000— owes interest 834.50 ; G has drawn $950 — owes interest 812 ; and H has drawn nothing. In the scttlemeiit a discount of 10 per cent., for bad debts, is allowed, on the book debts due the firm and on tho bills receivable. O takes the Jletropolitan Bank stock, allowin;; on tho same a pre- mium of 5 per cgnt. ; and II takes the Girard Bank stock, at a premium of 8 per cent'. ; E and F take the assets and as^iuc tho liabilities, as above stated. What has been the net gain or loss, tho balances due G and II, and what are E and F each worth after tho Bcttlement ? Ans. Dae G, $3057.75 ; due II, $3529.75 ; E's net capital, 64037.75; F's net capital, 64345.25. 7. II. C. Wright, W. S.' Samuels, and E. P. Hall, are doing business together — II. C. W. to have J gain or loss ; W. S. S. and E. P. II. each J. After doing business one year, W. S. S. and E. P. II. retire from the firm. On closing the books and taking stock, the following is found to be the result : merchandise on hand, $3216.50; cash deposited in Sixth National Bank, $1627.35 ; cash in till $134.16; bills receivable, $940.60 ; G. Brown owes, on ac- • count, 6112.40; Thos.. A. Bryce owes 894.12'; W. McKee owes $143.95; J. 'Anderson owes $54,20 ; II. II. Hill owes $43.60 ; and S. Graliam owes 6260.13. They owe on notes not redeemed §1804 ; H. T. Collins, on account, $124 45; and W. F. Curtis, $79.40. II. C. Wright invested $3200, and has drawn* from the business $850. W. S. Samuels invested $2455, and has drawn $140 ; E. P. Hall invested 62100, and has drawn 62000. A discount of 10 per cent, is to be allowed on tho bills receivable and book accounts duo the firm for bad debts. H. C. Wright takes tho assets and assumes 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 ? ' . * 8. T; ^. Wolfe, J. P. Towlcr atid E. V 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 of the remaining effects without regard to the proper pro- portion each should take. The following is the result according to their ledger :— T. P. Wolfe invested $3495, and has drawn $2941 ; J, P. Towler invested 12900, and has drawn $2200; E. R. Carpenter r-VLTiNsnsnip settlexient& 319 icss 81200, vts interest has drawn bad debts, I receivable, iainc a prc- stock, at a asv>nic tho or loss, tlio ■th after tho net capital, 11, arc. doing ; W. S. S. ir, W. S. S. s and taking iise on hand, 127.35; cash on ac- • cKee owes $43.60 ; and med^lSG-i; tis, $79.40. isiness $850. P. Hall .0 per cent. Mounts duo and assumes net gain s and E. P. have been ses equally, hey mako a proper pro- .ccording to iwn $2941 ; I. Carpcatcr r invested $3150, and lias drawn $3000. How will tnc partners settle with each other ? Ans. E. R. Carpcnter-pays T. P. Wolfe $86, and J.P. Towler8232. 9. I, J, K, L and M have entered into co-partnership, agrccinc; to share tho gains and losses in the following proportion : — I, j'^ ; J, ■f*^ ; K, y-. ; L, /j- ; and M, ^y When dissolving tho partnership tho resources consisted of cash $ 1700 ; morchandiso, $9855 ; notes on hand $7G30 ; dcbantures of tho city of Albany valued at $0780, on whicli there i.i interest due, $123 ; horses, waggons, &c., $1280 ; Merchant's bank stock, $5000 ; First National bank stock, $5000 ; mortgages and bonds, $3600 ; interest duo on mortgages, $345.80 ; Qtorc and fixtures, $3000 ; amount due from W. P. Campbell & Co., $2418 i duo "from R. B.Smith, $712.60; due from J. W. Jones, $1000. The liabilities arc : — Mortgage on store and fixtures, $5000 ; interest due on the same, $212.25 ; duo the estate of R. M. Evans, $14675 ; notes and acceptances, $11940, on which interest is due, $35 ; sundry ot^er book debts, $7500 ; I investod $7800, interest on his invostraenfr to'^te of dissolution, $T63sj.J^iavcsted $6400, interest on investment, $576 ; K investod $6100, inteffcst on invest- ment, $549 ; L invested $5800, intefest on investment, $522 ; M invested $5000, interest on investment, $450. I has withdrawn from the firm at different times, $2425, upon wWch the interest calcu- lated to time of dissolution is $183.40 ; J has drawn $2960, interest, 6267.85; K has drawn $1850, interest $37.30 j L has. drawn $3000, interest, $460 ; M has- drawn $895,* interest, $63.45. What is the net gain or loss of each partner, and what is -the net capital of each partner ? Ans. I's net loss, $1233.29 ; I's net capital, $4660.31. J'h net loss, $924.97 ; J's «et capital, •$2823.18. K's net loss, $616.65 ; K's net capital, $4095.05. L's net loss, $1541.62 ; L's net capital, $1320.38. M's net loss, $308.32 ; M's net capital, $4183.23. 10. A, B, C and D are partners. At the time of dissolution, and after the liabilities are all cancelled, they make a division of the effects, and upon examination of their ledger it shows 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 was to be divided in wmmim \m \\\\\'\\m':\'\\\. Hf>i«\ >M !\^«i'^ n»ul !<•*« il* tl»i< jvMlniM^ 'tt'llli' wl(l( t'!n>l( ulliet ? imUMO. OhnM.>|vvv liilln-llO-l Am't'lvt^ V \V M((\i ^||tl|^M■M^l(»^l^ln^ \\\M x'fiV'h U to W \A\i^\-ixv\\ |l 9\^ \\v\' (Irtv \h\: \m{ Htntv Af llio mIiwb (M th«MV 1»«kI<^om, I« \\w nt>Ml»'>»><»Mf !l »«« IUhh»I lli-tf A. W Pitillll \ ** to |n\ *r'>A V''v »Uv <«•,»■ (.11 l>*'«Mi<(>..i "u. (HI i r, PI mi; i», |H.i»\ n(\ii K, ji\ (\0n (miiu tiio«\v>M>\» \\«, ^iMOO ; M(\) <*l, ♦'ilOO. ,\\\w IhMu» «lh'\r onl ♦UMIO; ^\>^M«''»lWv \«!, jt'JOOO, M\y\ {\<\\\W\ U(, 1\«» inVt'Mlt!l <»MI)I> ItUHO. T. Vi. Mu^-ivw inw-^UM^ oi\ v'vMV»»\o»\>M(\;i, |;iOtlO , Mnn'lt IhI, Iih »Uvw .M\t jilOOO, \Uv \»t. fl'-!00. .\\\w \A, lio iii\i'Ml.«tl »l(Mi(l WAr>\ *wvl iVttvlvv IM^ :^S00<^ (\\i>»v At tl\o {\\\\o (>rH('Uli>iM»M»l, nu tW AIM IKsVmlhM", \S(^, (hv'U- ««Mvl\tUulll»0 «W(Mn\t WM lh\ (^V2\V0 , (\\ ^'KOOO, K\iiU<»v oO u5on>hiu\(lis«» on lii)»»tl. i»fl jh«I' imvntovx, #TO>0v>> »\«nt. flMi'NO. Tlun- »nv»« on ll.cli- iioW*, $l;^V *w»l iv. U*v v^u i^\\v»\\\(, $840 TUoij' j»rolU aiuI Iobh (u'couut ;?, Av t>Y^6, *v\uit (•, ^)»'. •a4'J0. Onm- wi»»i«w .vvvttut is (^■. $"i7(U>. t\»ton>Mt t\iH>tunt U !)>'. •iMI) j (>• $'A5(^. TW gftiw or Kw i? to Iv UiviJovl in |m>poi'tlon to o»oli p»rtift«"'r's «\itAl. i>«vl in jm^j>orti>n» to tl\«> timo il wn» invi'stiMl. K«idaiA\l oAch )^artttv^r'« »haiv of tl\o g^iu or Iv>kk, (ho not buluiioo I'AIIINKillf^Mfr Mf^'^ fl.rMrNfM. m% tH lit ji'^V 1" rt'»- Mttit I'. llt't-MhtlllllttM ,. W Mmllli iitit ;ui ilit^^si. tl lltt-lttf VtoVk III il»"' ,. iiinl 1*1. ,/\,. ik ^ i»iH tt'Kt' «,>f'|« h\\MV of |,..l, on v\^\\\- V .«u( #IMI>0; H'Ul»>l»H>«t, »*» n( wiirt /v. liSUtl. UK l"'«' llu'li' r.nU'S, . *l><0; (V. itlon to o«i'l» iVUK invoNlod. Dot bulunoo ililh Ortcli, HMil It li'i|||fh n|ih('|(lct)(|()H o«!(!ltl(ift« Mt« tAnn\tift, (if fill fh« nt'(«tHH(«, «Mtl HH' l(lll(Hl«'C Kllff-f. Amh a II fi'n fiKf H". I'Hi/l 7;li Id-M'f f.f.l^flf.^. linnvi,?}!. 14, A, M, (', lUJtt f I'Mtftnif'iiW'tl l((f<»I«r-M iiiu,i'\hfr im Ju\if (*♦, I Hurt, will) fith H(t»'t'f'JMf'(if lltrtf nllnrtl«»ir IfiM Im ♦♦» lif «ltfif«>»1 ^'jMrtlljr hy mhmIi |iM»fi(t'r, Iml llttit Inf^rnt* id lli*i fdfnfff ji^f ccftf J>^f «fi- MHMt 1m fit IlK mIIhWmI litt »'flr>l( ((«»«'«» !t(V»'»«f»fl''flf, flM'l Mti) Wiffift ffifrt •'|t(iifi;»'il titt nil fitiMHtiiN «Uli4l'nt»»i liy ^fl»•ll A U In rft»i«/»(»'' l^f? IhimImkmh, liBvliiy h Ndliiry tit ii!«( Tlio n«Mo»« HtM, HH miffMiiMiflittt, Hh«Ii I'MiO, Mdrtf.. lOfiMn , HilU M^ l'i4<»; if^v^«♦^'l liy (', I'l-r, flfi'tiitiiM \h |l!^M7I.K» (»««ififfifif»>(l vfilrf, M» jixr ^'>fi> tllnoMiHit ) liivi'Mlotl liy A. Of (li« n(w>M, flcf" l''-l'iri(<,<< >'» A ♦ UUIH^'JU, II, lOIMMI, I), IMHO, (Mwl «». (» $Uliith ViTm,tmi nf> ««M»niN (!i |1',^l»iJM|». AtiK, «(Mli, A »l«.w fnM. «7r., M J»<»0 R^j/f 'nil. i» .how f..«ii (iiiir. M..|,f ;iO(i,, A 'if».w |'/o(», ri I'iO. n*,*. HOMt. II .lifw •1110, l», inO l»H. ;-!l»l,, A flOO Mf.y li!»l., A 'IrfW »200, it $'.mi ,\um \',Uh, 11 'Ircw ()ir»0, (I »IO(l. .Inly l*», fl m*^]f« « riwllM'r litvi'Htiiiont ••r|l(JOI», (I, ^\Mli, nr*fl A 'Ir^^w tiOO, r> ||00. H«ji« irKli. \ -Irow »ir.O. II liMMi, (!,«7r.O. Moy l«f, A f|r"w|IOO, P |7n hf>i> nisi, iMCid, (||«< |iivik>« nrr. f(|»in»w| nmi tli/t jm, rf fr'tfultJ^ iIIbhiiIvimI, Mtid l> rflifiit^ (r'nn Mic IjiisitM-Mx, li»'if(i/ n)|/,w I Iry fhA roiiinlitlit^ |inrliif)rn #150 oni'lt T'lr lM«'ir /;/»o»l will if» >li'i htiain/'M. Iloloin (mliMilMliii^r llin iiitornnf, (in M»o |»rtr»,n^rrfi, (wmI Mllfiwini;/ A lli/t nncnirif fVf lii^ Pitlnry IVnin llnMM)rtirirnni«itr«>(nt. i»|» lo fInJo, »li«» M»t^N wfi'l li«l/ji»- lli'M HIV iin r.illowB (Inqli IIUKjlOO, M-hc. $1 tOY,', , fJilU |f/./Ai7 nltlo |(l«!n» Hft, l'flrw»n«l nwoiinM i;r |7ri?»l .'50 lfi7»f(t/^7 '.f Mdnn. oot>Nl|^iio(| t^) W, HnillJi, Now Vorfc, i«» hn ntt\i\ on (rrir nw/nnt iiimI link. II'^SIIA.IL'; 50 nUtfsn N. V. , HsirfMi'l /iifftrr»»if iiinn Top rnril l^i dnlo flliKO, In tifi MoiUnnifint n rJi/Ky^nnt of 5 p«r oont. on the IllJIii lUKKiivablo, 15 r> . f> per lb., and 140 firkins butter, 8312 lbs., at 17| cents per ID., on a commissiun of 2J per oont ; paid shipping and sundry cxpensea in cash $13.40. For reimbursement I draw on J. P. Fowler at sight, which I sell to the bank at ^ per cent, didoount ; what is the facq of draft, and what are tho journal entries ? Ana. Face of draft $5479.05. 4. Sept. 27th, I received from Jnmcs Watson, Lcedi, England' a consignment oi 1243 yards black broadcloth,. invoiced at 13s. Cd. per yard, to be sold on joint account of consignor and myself, each one half, my half to be as cash, invoice dated Sept.. IGth. Oct 5th, I sold 11. Duncan, for cash, 207 yards, at $0.10 per yardi Oct 24th, sold 317 yards to James Grant, at $0.25 per yard, on a credit of "00 days ; Nov. 18th, sold E. Cr. Conklin, for his note at 4 months, 400 yards, at $0.30 per yard ; Dec. 12th, jold tho remainder to J. A. Miisgrovo at $6.00 per yard^ half cash* and a credit of 30 days for balance ; charges for storage, advertising, &c., $13.40 ; my com- mission, with guarantee of sales, 5 per cent. What would bo tho avcri^o time of sales ; the average time of James Watson's account; and what would be tho face of a sterling bill, dated Doc. 15th, at GO days after date, remitted James Watson to balance account purchased at $108^^, money being worth 7 percent, and gold being 70 per cent, premium ? » 5. Buchanan & Harris of ^Ijlwaukec, Wis., are owin^ W. A. Murray & Co. of W.-isliington, $1742.75, being proceeds of consign- ment of tobacco sold for thcni, and Simpson & Co. of Washington, arc owing liuchanaii & Harris $2000 payable in Washington. Buchanan & Harris wish to remit W. A. 3Iurray & C». the proceeds of their consignment and they do so by draft on Simpson & Co., but Washington funds arc 2 per cent, premium over tho.sc of Chicago. Hequircd tho face of the draft and the journal entries. G. A. Cummings, of London, England, is owing mo a.certain sum, payable there, and I am owing Charles IMassey, of the same place, $1985.42, being proceeds of consignment of broadcloth sold for liim. I remit C. Massey in lull of account, after allowing him $21,12 for inscrcst, my bill of exchange on A. Cummings at 60 days' sight; exchange 109^, gold 42 per cent, premium. What is the face of tho drai't, and what are tho journal entries ? QUESTIONS FOR COMMERCIAL STUDENTS. 325 D., on a nscs in it sight, I facQ of 479.05. Sngland ' 13s. Gd. 3lf, cuch Oct 5th, let 24th, lit of "OO iths, 400 to J. A. days for my com- Id bo the account; , '15th, at I account bcin^70 W. A. oonsign- Ishipgton, phington. proceeds Co., but I Chicago. I sum, ic place, sold for Iring him |gs at GO What ia 7 . March 10, I shipped per steamer Vandcrhilt ana consigned to'Sarauc! Vestry, Liverpool, England, to be sold on joint account of consignee and consignor, each one half^ (consignee's half as cash), 2'?894 lbs. prime American Cheese at 15|^ cents per pound. Paid Bhipping expenses $18,30. " Insuran-o IJ per cent, and irsurcd for such an amount that, in caso the cheese was lost, the total cost would be rccovcrablo. May 19, I received from Samuel Vestry an account sales, showing my net proceeds to be £298 14 9f, due as per average August 21. Juno 1, I drew on Samuel Vestry at the num- ber of days after date that i^ 'would take to make the bill fall due at the properly equated time of his account. Sold the above bill to K. llamdsey, broker, at 108 J; Required the number of days I drew the bill at, its face, gold being at a premium of 43f per cent., the amount of money_ in greenbacks I received, and the journal entries. 8. T am doing a coiftmission business in New York, and on Sept. 14, I received from A. J. Rice, of Hudson, to be sold on joint ac- count of himself §, A. H. t*catman, of Newburg, ^, and myself ^ ; merchandise invoiced at $1262.40, Jjaid freight $14.20. The same day, I received from A. I|. Peatman to be sold on joint account of himself 'j, A. J. Rico J, ivnd myself ^, merchandise invoiced at $1102.12 ; paid freight $10.00. I also invest to bo sold on joint account of A. J. Rice ^, A. II. Peatman ^, and myself §, merchan- dise valued at $745.35. The chares of each arc subject to average sales. October 29th, I sold J- of the merchandise received from A. J. Rice to S. King at an advance of 20 per cent., on a credit of 90 days. November 9th, sold for cash ono half of tho remainder at 15 per cent, advance, closed tho company, and rendered account sales ; storage $3.50, commission 2J per cent. November 12, sold to A. M. Spafford, on a credit of 30 days ono half of goods received from A. H. Peatman, at an advance of 25 per cent. November 23, sold for cash tho merchandise that I invested at an advance of 15 per cent.; closed the company and rendered account sales ; storage $2.75, commission 2^ per cent. December 4th, sold J of the remainder of merchandize received from A. H. Peatman to G. W. Wright, on a credit of 60 days, at 3^ per cent, advance. December 12th, sold the balance of Peatman's merchandise for cash at 25 per cent, ad- vance ; closed the company and rendered account sales, storage $5.00 commission 2^ per cent. December 23fd, I wish to settle with A. J. Rice, and A. H. Peatman, iu full ; I take to my own account, . f^^dmnm-L. i ■ 826 AWTHJttETIG as cash, the balance of morohandiso unsold at an advance of 8 per cent.. What, is the average tiuio of sales of each Mdso. Co., the average time of A. •). -R. and A. II. P's. accounts, the amount of money I Bhall have to pay them on Dcoembcr 20, how do A, J. R. and A. II. P, stand with each .yiner, and what are the journal entries ? 9. E. R. Carpenter, S. Northrup and Levi Williams, comuicnced business together as partners under the name and style of \]. R. Cai"penter & Co., on January Ist, 18G5, with the following ejects : merchandise, $7844 ; cash, $5000 ; store and fixtures, $3984 ; bills receivable, $1732.50; of this anjount there belonged to E. R. Car- penter, as capital, $8000 ; S. Northrup, $G000 ; Levi Williams, $4j3|i)1.50. The firm assumed the liability of Levi Williams, which was a note to tho amount of $425.80 ; this note was paid on March 10th. The loss or gain is to bo shared equally by tho partners, but interest at tho r.".to of 7 per cent, per annum is to be allowed on in- vestments, and charged c "mounts withdrawn. E. 11. Carpenter is to manage the bi diac..> -,. a salary of $1000 a year, payable half yearly (the time of tho othc?> partners not being required in the business). Marcii 14ti' S. '"^'^i thrup draws cash, $300 ; Levi Williams, $200; April iSthj Fj. K. Carpenter draivs cash, $500 ; S. Northrup, $100. On the 1st of May, they admit Geo. Smith as a partner, under the original agreement, with a cash capital of $4000. The books not being closed, he pays each partner for a participation in the profits to this time $450, which they invest in the business. May 14th, E. R. Carpenter draws c*h, $1G0 ; May 24th, Levi Williams draws cash, $100 ; June 12th, S. Northrup draws cash, $250, and E. R. Carpenter, $200 ; July 1st, Levi Williams draws cash, $300, and S. Northrup, $450; July 21st, Levi Williams draws cash, $180 ; August Ist, Levi Williams retires from tho partnership, the firm allowing him $500 for his profits and good-will in tho busi- ness, this amount, together with his capital, has been paid in cash. Oct. 14th, Geo. Smith draws cash, $340; E. R. Carpenter, $725. November 1st, with the consent of the firm, S. Northrup disposes of liis right, title and interest in the business to J. K. White, who is admitted as a partner under tho original agreement. J. K. White is to allow S. Northrup $G00 for his share, of the profits to date, and his good-will in tho business. J. K. White not receiving funds an- ticipated, is unable to pay S. Northrup but $1500, the firm therefore assumes the balance as a liability. December 10th, received from QUESTIONS FOR COMilERCLiL STUDENTS. 327 J. K. White, and paid over to S. Northrup, the full amount duo him (S. N.) to date. December Slat, the books are closed, and the fol- lowing effects arc on hand:— Mdse, 011943.75; cash, $2110.12; bills receivable, $0400 ; store and fixtures, 83850 ; personal account^ Dr. 614987.50; personal accounts Cr. $10711 ; Bills Payable unre- deemed, $4000. What has been the net gain or loss, the net capital of each partbcr at the end of the year, and what are the double entry journal entries on commencing business, when Levi Williams retires, when Geo. Smith is admitted, when S. Northrup scHs hia interest and right to J. K. White, for E. R. Carpenter's salary, and the interest duo from and to each partner ? The student anay also, ih the above example, after finding tl.-j interest on tlio partners investments, and on the amounts withdrawri* give a journal entry that will adjust the matter of interest between the partners without opening any profit and loss account. K79.ri^ 328 ARITintETIC. 3 a b c MENSURATION." We have already observed that no Bolid body can have mora thaa three dimensions, viz. : length, breadth, and thickness, or depth, and that a line is length, or breadth, or depth, or it is a line or unit repeated a certain number of times. A foot in length is a line mea- sured by repeating the linear unit called an inch 12 times, and a yard is the linear unit called a foot, repeated 3 times, and so on. Thus, 1 ft. 1 ft. 1 ft. - „ ^ _ _, , . ... ' . , =3 feet. But there may be two such hues drawn at righif angles to each other, and each three feet long, and if ^ the figure be completed it is a square. Also, if lines be drawn, each an inch apart from the other, and parallel to the two first-mentioned lines, we shall find that there are three small figures, each an inch square', between the two upper hori- zontal lines, and 3 of the same extent between the two intermediate lines, and 3 between the two lower lines, making 9 in all, or 3 times 3. This is the origin of the expression that 9 is the square of 3. Let the learner mark the difference between 3 square feet and 3 feet square, a, b and c are 3 square/eet, but the whole figure is 3 feet square, and therefore three feet square must be equal to 9 square feet. Three feet square, then, is a square, each of whose sides measures 3 linear feet ; but 3 square feet would denote 3 squares, each side of each measuring one linear foot. The spaco thus inclosed is called the area. , This is the principle on which surfaces are measured. PROBLEM I. To find the area of a paralellogram : RULE. Multiply the length by the perpendicular breadth. If thefigurt be rectangular, one of the sides loiU be the perpendicular breadth. * We have taken for granted that those studying menBuration have learned, at least, the elementary principles of geometry. Wo have, there- fore, only given the rules, as our space would not admit of our giving demonstrations as this would require a separate treatis(> MENSURATION. 320 e more thaa r depth, and line or unit I a line mea- imes, and a , and so on. 70 such lines ; long, and if is a square, ich an inch arallel to tho re shall find ures, each an upper hori- same extent ite lines, and lines, making lion that 9 is se between 3 feet, but the luare must be [uare, each of 'ould denote The space * I If ihefigurt dar breadth. suratioa have lo have, there- lof our giving Tf tJte fijura he not rectanjahr, elfJier the perpendicular breadth Mitst he given or data from which to find it. EXERCISES. 1. How many acres arc there in a square, each side of which is 24 rodi? Ans. 3 acres, 2 roods, 16 rods. 2. What is tho area of a square picture frame, each side of which is B £1, 9 in. ? Ans. 33 ft. 9 in. 3. How many acres are there in a rectangular field, the length of which is .13J chains, and the breadth 9^ ? Ans. 130.625 square chains, or 13 acres, roods, 10 rods. 4. What is the area of a rectangle, whose sides are 14 ft. 6 in. and 4 ft. 9 in. ? •- Ans. 68 ft., 126 sq. in. 5. What does tho surface of a plank measure, which is 12 ft. 6 in. long and 9 in. broad? Ans. 9 sq. ft. 54 sq. in. 6. What'is tho area of a rhomboidal field, the length of which is 10.52 chains and the perpendicular breath 7.63 chains ? Ans. 8 acres, roods, 4.2816 rods. 7. What is tte area of a rhomboidal field, tlie length of which is 24 rods and the perpendicular breadth 24 rods ? Ans. 3 acres, 2 roods, 16 rods. 8. What is the length of each side of a square field, the area of which is 788544 square yards ? Ans. 888 yards. 9. The area of a rectangular garden is 1848 square yards, and one side is 56 yards ; wha't is the other ? Ans. 33 yards. 10. The area of a rhomboidal pavement is 205, and the length is 20 feet; what is the perpendicular breadth? Ads. 1G^ feet. PROBLEM II. To find the area of a triangle. 1. Kthe base and perpendicular, or data to find them, b€ given, we have the RULE. Multiply the base by tJie perpendicular, and take half the pro- dtKt ; or, multiply half the one by the other. 2. If the three sides are given BULK. From half the sum of the sides subtract each side successively, and the square root of the continual product of the half fum, and these three remainders will be the area. SCO ABITHMETIO. Expressed algebraically this arca=r]/«(s — a)(s — 6)(< — c). EXERCISES. 11. What is the area of a triangle, the base of which is 17 inches, and the altitude 12 inches ? Ans. 102 square inches. 12. What is the area of a triangular garden, the length of which ia 4G rods, and the breadth 19 rods? Ans. 437 square rods. 13. Find how many acres, &c., are in a triangular field, the length of which is 49.75 rods, and the breadth 34^ rods. Ans. 5 acres, 1 rood, 1^,'s rods. 14. The area of a triangular inelosure is 150 square rods, and the base is 30 linear rods ; what is the altitude ? Ans. 10 rods. 15. The area of a triangle is 400 rods, and the altitude 40 rods, what is the base ? Ans. 20 rods. IG. Three trees are so planted that the lines joining them form a right angled triangle ; the two sides containing the right angle are 33 and 50 yards j what ia the area in square yards ? Ans. 924. 17. Let the position of the trees, as in the last example, be represented by thf> tri- angle ABC, and let the distance from A to B be "50 rods, and from B to C 30 rods. Required the area. — (See Euclid I. 47.) i Ans. COO square rods. 18. In the figure annexed to 17, suppose A B to represent the pitch of u gallery iu a church, inclined to the ground at an angle of 45 ' ; how many more persons will the gallery contain than if the seats wcro made on the flat B C, supposing B C to be 20 feet and the frontage GO feet in length ? Ans. None. Wo have introduced this question and the next to correct a common niisapprehensioa on this point. Because the distance from B to A is greater than the distance from B to C, it is commonly supposed that more per- sons can be accommodated on the slant A B, than on the flat B C. By in- specting the annexed diagram it will be seen that the seats are not perpen- dicular to A B, but to B C, and that precisely the same number of seats can be made, and the same number of per- sons accommodated on B C as on A B. MENSURATION. 831 19. If B C bo lialf the base of a hill, and A B one of its slopin;^ siJcs, anil B C— 30 yards, and A Ii-"50 yards; liow many ni9ru rows of trees can bo planted on A li, than on B C, at 1 yard apart ? .'vns. None, because the trees buin;^ all perpendicular to the horizon, arc parallel to each other as reprosontcd by the vertical lines in the la«t Hi^ure. 20. IIow many acres, &c., are tliore in a triangular field of which the perpendicular length Aid breadth are 12 cliaias, 7() links and chains, 43 links? Ans. G acres, roods, 2^ rods. 21. A ship was stranded at a distance of 40 yards from the base of a cliff oO yards high ; what was the length of a cable which reached from the top of the cliff to the ship ? Ans 50 yds. 22. A cable 100 yards long wa.s passed from the bow to the stern of a ship through the cradle of a mast placed in midships at the height of 30 yards ; what was the length of the ship ? Ans. 80 yards. 23. A man attempts to row a boat directly across a river 200 yards broad, but is carried 80 yards down the stream by the current ; through how many yards was ho carried ? » Ans. 21 5.4 -[-yards. 24. Let the three sides of a triangle bo 30, 40, 20; to find the area in square feet. Ans. 290.4737 square feet. 25. What is the area of an isosceles triangle, each of the equal sides being 15 feet, and the base 20 feet ?* Ans. 111.803 sq. feet. 20. What is the area of a triangular space, of which the base is 5G, and Uie hypotcrmse 05 yards ? Ans. 924 square yards. 27. What is the area of a triangular clearing, each side of which is 25 chains ? Ans. 27.0032 acres. 28. What is the area of a triangular clearing, of which the three sides arc 380, 420 and 705 ? ' Ans. 9 acres, 37J perches. 29. A lot of ground is represented by the three sides of a right angled triangle, of which the hypoteuuse is 100 rods, and the base GO rods ; what is the area? . Ans. 15 acres. 30. What is the area of a triangular field, of which the sides are 49, 34 and 27 rods respectively ? . Ans. 2 acres, 3 roods-}-. 31. What is the area of a triangular orchard, the sides of which are 13, X4 and 15 yards ? Ans. 84 square yards. 32. Three divisions of an army are jJlaccd so as to be represented • This question, anil some others may be solved hj either rule, and it will be found a good exercise to solve by both. 332 ARITHMETIC. I ' by three sides of a triangle, 12, 18 and 24 j how many square miles do they guard within their lines ? Ans. Between 104 and 105 square miles. 33. A ladder, 50 feet long, was placed in a street, and reached to a parapet 28 feet high, and on being turned over reached a para- pet on the other sido 30 feet high ; what was the breadth of the street ? Ans. 76.123+feet. FROBLCM III. To find the area of a regular Polygon. 1. When one of the equal sides, and the perpendicular on it from the centre, are given. Multiply the perimeter by the perpendicular on it from its centre, and take half the product ; or, multiply either hp Imlf the other. 2. When a side only is given. Multiply tJte square of the side by the number found opposite the number of sides in the subjoined table. Note. — This tiiblo show* the area when the Hide ia unity ; or, which is the aame tiling, the square ia the unit. SIDES. 3 4 5 6 7 8 9 10 11 12 RKUUIAR FIQCUES. Triangle Square Pentagon Pcxagon .Heptagon Octagon Nonagon Decagon Ilcredecagon. Dodecagon... 0.4330127. 1.0000000. 1.7204774. 2.5980762. 3.6339125. 4.8284272. 6.1818241. 7.6942088. 9.3656395. 11.1961524. 34. If the side of a pentagon is 6 feet and the perpendicular 3 feet, what is the area ? Ans. 45 feet. 35. What is the area of a regular polygon, each side of which is 15 yards ? * Ans. 387.107325 sq. yds. . 36. If each side of a hexagon be 6 feet, and a lino drawn from the centre to any angle be 5 feet, what is the area ? Ans. 72 sq. feet. MENSURATION. 333 ly sqtiaro miles 5 squaro miles, t, and reached reached a para- breadth of the I. 76.123+fcet. endicolar on it from its centre, ilf the other. md oj>posUe the ' ; or, which is tho 0.4330127. 1.0000000. 1.7204774. 2.5980762. 3.6339125. 4.8284272. 6.1818241. 7.6942088. 9.3656395. 11.1961.524. perpendicular 3 Ans. 45 feet, jide of which is 107325 sq. yds. ine drawn from Ina. 72 sq. feet. 37. The side of a decagon h 20.5 rods ; what is the area ? Ans. 20 acres, roods, 33.5 rods, nearly. 38. A hexagonal tabic haa each side CO^inchcs, and a lino from the centre to any comer is 50 inches ; how many square feet in the surface of tlie table ? 39. What is tho area of a regular heptagon, tho side being 19 J J and the perpendicular 10 ? Ans. 078.3. 40. An octagonal enclosure has each side yards, what is its ? Ans. 3 acres, 2 roods, 14 rodi. 19 yards. 41. Five divisions of an army guard a certain tract of country- each lino id 20 miles ; how many square miles are guarded ? , ' Ans. 088.2, nearly. 42. Find the same if there are G divisions, and each line extends 5 milca V Ans. 64.954- ™iic8. 43. Tho area of a hexagonal tablo is 73^ foot; what is each side ? Ana. 6J feet. PnOBLEMIV. To find the aroa of an irregular polygon. Divide it into triangles by a perpendicular on each diagonal f*om the opposite angle. Find the area of each triangle separately ^ and the sum of these will he the area of the trapezium. NoTK. — Either tbe diagonals and porpendiculars must be given, or data from which to find them. 44. The diagonal extent of a four-sided field is 65 rod."?, and the perpendiculars on it from tl. ' opposite corners are 28 and 33.5 rods ; what is the area ? Ans. 1 acre, 1 rood, 22.083 rods. 45. A quadrangle having two sides-parallel, and the one is 20.5 feet long and the other 12.25 feet, and the perpendicular distance between them is 10.75 feet ; what ia the area ? i\ns. 176.03126 sq. feet. 46. Required tbp area of a si::-sided figure, the diagonals of which are as follows : the two extreme ones, 20.75 yards and 18.5, and the intermediate 27.48 ; the perpendicular on the first is 8.6, on the eeoond 12.8, and those on the intermediate one 14.25 and 9.35 ? Ana. 531.889 yards. 47. If tho two sides of a hexagon be parallel, and the diagonal parallel to them be 30.15 feet, and the perpendiculars on it froitai ■^T" nn4 AnTTTiMrrtr. . ft llio op|>(wi(o iii\;;loF» iiro, (in ilio IcO, 10,5(1, Mid on Oio li(j,lit 12.24, mill tl>i< piut ol' llio »liii|4;iinfil out dlV In (lie IcH by (lr«l. jirrjuMuli. odiiir, M.LNi, ami to llii> ri^lit by (lii> nct'oinl, lO.II; (in tlin tillior Hide, (lie |u'r|H'nili(MilfU ntitl ncirinoitl of (lie (liiifroiiiil to llio 1«<(Y nro >^.50 nml '1.51, mul on t!io right '.•.LM» nml .T'.Kl ; wliiit. is (lio uiou? Atis. 470.1156 wj. loci. f 11(111 1, r, M V . To llnd i/ hihiiuj sritgitil prrpnuh'ruhirit /intn the uriiri\<>( niut tuni>t rim((' (» tfitxf (r/i/ii"ri,ii)/i(i/i"(i)i ^» thr nirit. luM. (ho jH-rpciulitMilurn O.L', 10.5, H.;i, 0.4, 10.7, their Kum ih 4S.l,th,'n4S.l-,- ft=rlM')'J, luul ii' iho hitMo '\h 20. wohuvo0.02x'J0 lO'J.l, the nren. WIu'H prnotii'.'ihh', nn larf^o n jKn'tiou of the fi|mee hh |)(»HBil»lo slionlil he liiitl oil", HO hm to loriii a regular ll|;ure, atui tho rent (buiui «s ftbow. ,\ liehl is to be nu'iL^ured, and ('>\20.5 157,0M, the urea of the irregular pnrt. nud thi.s, luldod m tho urea ol' tho roctaugloH, given o72.2(^, tho whole nr\>a. 4v'^. The length of an irregular clearing ih 47 rods, and tlio breadths at 0(]ual distwucen aro 5.7, 4.S, 7.5, 5.1, H.4 and 0.5; what is tho urea'/ An». 1 acre, 1 rood, 29.80 rodw. r K o H I. K M VI. To find thr circuinfenMice of a eirdo when the dinmotor is known, or tho diameter when tlw circunifcronco is knoWB.* Tlio most aceurato rule is tho well-known theoroni that tho dinmctcr is to th(j circumforctioo in the ratio of 113 to 355, and * la slrictnoha \h6 oircumfercnco and dtamoler arc not liko quantiUoa, but wo miiy B\ippoM that u cord ia stretched round tho oiruumforonoo, and then drawn out into a straight line, and its linear unit« compared with tlioie of th« diameter. Mr.NfUTUA'nOTI. :m 5tcr is known, (Mmsfvjnnntly llin droitmrcrcnf-o fd lljn rlininotcr nn .'ir»5 In \\'.\. Now, !ir>rt ; 11:1 :I.14I('. nrmrly, nntl for goriorftl purposes, BufTicuHt nocurnoy will "bo nttuiiioJ Ity Wm II II I. K . To (hid Mm riroiim(i. 40. Wind, ir llm loiif(;lli round tlio ivjiuifor of » IT) incli ^^^loho? Aii«. 47,Hi4 ifK^licM. r»0. If 11 jroiii'd ioj,; liiiH ivcirciiiMrcroiiro of <» \'m\^ 10 incli'-H; wind JH il« diutiK'lor? Atip. li ff'fif., 'J,',; iti(di('« nearly. r>l. If wn tiikn flio diHtiiiico fnnri ilio ('("ntrn of flio rnrfli Ut llif oqUHtor to Ik) IJUTU; wind, h (lio !iuiiil»ttrof iiiilon ntiiiid llio wjiiiilorV Ann. 25001 nonrly. r II II I, KM VII. To find llio nron of a rlrolo. 1. If tlio riiTiiui((>rnico ntid dirini<;U«r nro known, — MnUlphi (hr, rinum/ct'cncr. by fhr. (liamr.Ur, avd Vtk" nruyfourfh i)f dtp proiluft. 2. If tl.d dlftinotor fdotm if f!;ivcTi, — Miildphi fhr mjiiiirt; of ihr t/intnifi r In/ .TBI)!. .'I. If tlio (linMiiiiforc.iico iiloiio bo given, — Multiply the, m/uarc of thr nuinlirr iknutintj the circumference hy .07058. 52. If tlin diuniclcr of a circle iH 7, and tlio circumference 22; what iH tlio area? . Ann. .'{Hj^. 5I{. Wliiit .8 ll'O urea of a circle, the rndiuH of Mfliich in .'Ji'j ydw? Ann. .''.[j H(|iiare yardw. B4. If a Rcnilcirculur arc bo denoted by 10.05 ; what !■» the area of tho ciroU) ? Ann. 2Hf).:U{. BB. If the diameter «)f a f^rindin^ Btone be 20 inehex; what Bupcrficial area w hd't when it in ground down to 15 inchcf diameter, and what Huperlieial area has been worn awny ? AnH. 170.715 H(|r. inches hift, and 137.445 worn away. B6. If tho chord of an are be 24 inches, and the j^er^wndicalar on it from tho contro 11.9 ; what is tho area of the circl t ? Ana. 2.689804. 336 AEITHMETia ,^^»^ MENSURATION OF SOLIDS. To find the solid contents of n parallclopipcd, or any regularly box-shaped body : Let it be required to find the number of cubic feet in a box 8 feet Ion};;, 4J feet broad, and 6 J feet deep. In the first place, the length being 8 feet and the breadth 4J, the area of the base is 8X'tJ=oG square feet, and therefore every foot of altitude, or depth, or thickness, will give 3G cubic feet, and as there are G,J feet of depth, the whole Bolid coutout "will bo 3G times Gj, or 243 cubic feet. Hence tho nuLE. Take the continual j>roduct of the lengthy breadth, and depth. Note. — Lot it be carefully observed that the unit of measure in tho case of solids is to bo taken as a cube, tho base of which is a superficial unit used in t'uo nieasuremeat of surfaces. Tlio solid content is indicated by the repe- tition of this unit n certain number of times. If tho body is of uniform breadth the rule needs no modidcatiou, but it it is rounded or tapering, as a globe, cone, or pyramid, tho calculation becomes virtually to find how mnch the rounded or tapering body differs from tho one of nniform breadth. Sup- pose, for example, wo take a piece of wood 6 feet high, in tho form of a pyramid, and having the length and breadth of tho base each G feet, then the area of tho base is 'iti ; but if, at tho height of 1 foot, the dimensions havo each diminished by 1 foot, tho area is 25 ; at another foot higher it is IG ; at the next 9 ; at the next 4 ; at the next 1 ; and at the Gth 0, t. e., it has come to a point, and tho calculation is,-bow much remains from the ftolid cube after so much hoc been cut off each side as to give it this form. This gives rise to the following varieties : I. To find tho solid contents of a cono or pyramid : Multiply tJte area of the base by the perpendicular height^ and take one-third of the product. II. To find tho solid contents of a cylinder or prism : Multiply the area of the bate by the perpendicular height. III. To find the surface of a sphere : , Multiply the squitre of the diameter by 3.1416. rV. To find the pjlid contents of a globe or sphere : Mubiplif the third power of the diameter by .5236. MENSUnATION OP SOLIDS. ?37 V. To find the vc;umo of a spheroid, the axeS being given : Multiply the square of the axis of revolution by the fixed axis, and the product by .523G. EXERCISES. 57. If the diameter of tlie base of a cylinder be 2 feet, and its height 6 feet, what is the solid content ? Ans. 25.708 feet: 58. If the diameter of the base of a cone bo 1 foot 6 inches, and the altitude 15 feet, what are the solid contents ? Ans. 8 feet, 120 inches. 59. If the diameter of the base of a cylinder be 7 feet, and the height 5 feet, what is tha solid content ? An«. 245 cubic feet. 60. What arc tho solid contents of a hexagonal prism, each s"de of the base being lo inches and tho height 15 feet? Ans. 09.282 cubic feet. Gl. A triangular pyramid is 30 feet iiigh, and each side of the base is 3 feet ; required tho solid contents. Ans. 39.98 cubic feet. 02. What are tho solid contents of the earth, the diameter being taken as 7918.7 miles ? Ans. 259992732079.87. 63. In a aphcroid the less axis is 70 and the greater OO ; what are tho solid contents ? Ans. 230907.a PILING OF BALLS AND SHELLS. Balls are usually piled on a base which is either a triangle, or square, or rectangle, each side of each cour.se containing one ball less than the ono below it. If tho base is an equilateral figure, the vertJjx of a complete pile will bo a single ball ; but if one side of the base b« greater than the contiguous ono, the vertex will bo a roto of balls. Hence, if the base bo an equilateral figure, the pile will be a pyramid, and a» tho side of each layer contains ono layer less than the one below it, the number of balls in height will be the same as the number of balls in one side of tho lowest layer. If tho pile be rectangular, each layer must also be rectangular, and the number of balls in lieight will be the samo as tho number in tho less side of tho base. If tho bas« bo triangular, wo have tho n , ► ■ ■ V. 338 ABITHHETIC. ''.■(* '*".;»'K-*^ t \ RULE. Multiply the nMmher on one aide of the bottom row hy itself TLVS one, and the product hy the same base row PLUS tu)0, and divide the . result by six. For a complete squaro pilo we have the RULE. Multiply tJie number of balls in one side of the lowest course by itself VLVSa one, and this product by double the first multiplier PLUS one, and take one-sixth of the result. » . If the pile be rectangular, we have tho .^,. I ' RULE. From three times the number of balls in the length of the lowest course subtract one less than the number in the breadth of the same course,' multiply the remainder by the breadth, and this product by one-sixth the breadth plus one. Jf the pile be incomplete, find what it would be if complete ; find also wluxt the incomplete one would be as or separate pile, and sub- tract the latter from the former. EXERCISES. 64. In a complete triangular pile each side of the base is 40 ; how many balls arc there ? Ans. 11480. 65. In each side of the base of a square pile there are 20 shells ; how many in the whole pile ? Ans. 2870. 66. In a rectangular pile there are 59 balls in the length, and 20 in the breadth of the base ; how many are in all ? Ans. 11060. 67. In an incomplete triangular pile, each side of the IowSst4aycr consists of 40 balls, and the side of the upper course of 20 ; what is the number of balls ? lifiH. 10150. » NoTS.— Since the npper course la 20, the flnst row in the wantinK part would be 19. MEASmiEMENT OF TQCBER. 339 Sy itself VLva id divide the vest course hy lUiplier PLUS of the lowest h of the same is product hy omplete ; Jind oile, and sub' base is 40 ; Ans. 11480. tre 20 shells ; Ans. 2870. jDgth, and 20 Ans. 11060. elowSsHayer of 20 ; what Ana. 10150. > wanting part MEASUREMENT OF TIMBEH. by tti"^wo'f "r"' "°°""°'" '^ "■" •'""'= <■-'■ "^ •»"'-^«-"« measured To and ether «.„ «„p,rfici.l e.tcat or board „ca.„re „L pC,' tK RULE. Multtptg the hoard meamre ly the tUdcnm. To find the solid contont, of a round log whe» the girt i, to™. RULE. Mukiply the square of {he guartstJlJ^t in inches by th. 1^ ,i. - feet, and divide the product by U4 ^ '""^'^ '" will'l!etnl\'?k^^^^^ "' ;?/r ^°"^' the solid contents Which is ^ooX^mTslSZ:^^^^^^^^^ ^"^^es, To find the number of square feet in r^»^A *• u mean diameter is given. ^ *'"^'' ''^^^ ^^^ .'•><* 'iT 1 % 1 „•■ •t. 1* •>f. *■ w 340 ABTTHMETIC. RULE . Multiplg the diameter in inches lif half the diameter in inches, and the product by 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 56 X 28 x30-j-l 2=3920 feet. To find the solid* contents of a log when the length and mean diameter are given. , BULK. Multiply the sqtutre of half the diameter in incites by 3.1416, and this product by the levgth in feet, and divide by 144. G8. How many cubic feet arc 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 "'"and log 21 inches in diameter, and 40 feet in length ? 70. What are the solid contents of a log 24 inches in diameter, and 34 feet in length ? Ans. 1 06.81 +cubio feet. 71. How many feet, board measure, are there in a log 23 inches in diameter, and 12 feet long ? Ans. 264 J. 72. How many feet, board measure, are there in a log, the diameter of which is 27 inches, and the length lb feet. Ans. 486. 73. What are the solid contents of a round log 36 feet loug, 18 inches diameter at one end, and 9 at the other ? 74. How many feet of square timber will a round log 36 inches in diameter and 10 feet long yield ? Ans. 540 solid feet. 75. How many solid feet are there in a board 15 feet long, 5 inches wide, and 3 inches thick ? Ans. ly^g cubic feet. 76. What are the solid contents of a board 20 feet long, 20 inches broad, and 10 inches thick ? Ans. 27| feet. 77. What is the solid content of a piece of timber 12 feet long, 16 inches broad, and 12 inches thick ? Ans. 16 feet. 78. How many cubic feet are there in a log that is 25 inches in diameter, and 3? feet long ? 79. How ma.:y feet, board measure, does a log 28 inches ia diameter, and 14 feet in length contain ? Ans. 457^. 80. How many cubic feet are contained in a piece of squared timber that is 12 by 16 ioches, and 47 feet in length ? Ans. 62f . JtEASUEEMENT OF TIMBER. 341 81. How many feet, board measure, arc there in 23 one-inch boards, each being 13 inches in width, and 16 feet in length ? Ans. 381'. BALES, BINS, iC. As bales are usually of the same form as boxes, the same rule applies. 82. Hence, a bale measuring 4i inches in length, 33 in width, and 3J in depth, is, in solid content, 37^ feet. 83. A crate is 5 feet long, 4| broad,\nd 3/. deep, what is the solid content ? ^^^^ g5_,^ To find how many bushels are in a bin of grain : RULE. Find the jyroduct of the length, breadth and depth, and divide hy 5150.4. 84. A bin consists of 12 compartments ; each measures G 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 conicalteap 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 G4X. 7854=83.777, and 83.777 x:*= 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. 342 ABTTHMETIC. MISCELLANEOUS EXERCISES. 1. What number is that f and f of which make 255 ? Ans. 201-/»g. 2. What must be added to 217^, that the sum may be 17^ times 19|? . Ans. 118^. 3. What sum of money must be lent, at 7 per cent., to accumu- late to $455 interest in 3 months ? Ans. $26000. 4. Divide $1000 among A, B and C, so that A may have $156 more than B, and B $62 less than C. Ans. A. $4165 ; B, $260§ ; C, $3225. 5. Where shall a pole 60 feet high be broken, that the top may rest on the ground 20 feet from the stump ? Ans. 26§ feet. 6. A man bought a liorse 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 leap^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 docs 1 J times as much as B, and B does ^ as much as C ? Ans. A, 30 days ; B, 45 ; C, 22i. 9. At what time between five and six o'clock, are the hour and minute hands of a clock exactly together ? Ans. 27 min., 16j''j- sec. past 5. 10. A courier has advanced 35 miles with despatches, when a second starts with additional instruotions, 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 |— r\+4\— t/s+-4¥s— &c. ? Ans. 365. 12. If a man earn $2 more each month than he did the month before, and finds at the end of 18 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 tho rate of 50 MISCELLANEOUS EXERCISES. 343 miles an hour, to pass over a space equal to the distance of the earth from 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 from the time they started, arrives at the fort at the same time as his comrade ; at what rate did the second travel ? Ans. 18 miles a day. 15. "What quantity must be subtracted from the square of 48, so that the remainder may be the product of 54 by 16 ? Ans. 1440. 16. A father gave | of his farm to his son, the son sold ^ of his share for §1260 ; what was the value of the whole farm ? Ans. $5040. 17. There were § of a flock of sheep stolen, and 672 were left ; how many were there in all ? Ans. 1792. 18. A boy gave 2 cents each for a number of pears, and had 42 cents left, but if he had given 5 ces^sfor each, he would have had nothing left. Required the number ofpears»»« Ans. 14. 1 19. Sim^/lify - 1 Ans. 5 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 7J inches ? Ans. 23716. 24. What is the area of a right angled triangular field, of which the hypotenuse is 100 rods and the base 60? Ans. 2400 sq. rds. OK Q- ye 54— 2i ^ 4i+5Aa n2|+l^. 25. amphfy i^Jof J±^« «f^^^ Ans. lf|. I* Ij 344 AiurmiETic. 26. Find the value of i_|_ 1+i A.DS. 27. If f of A's age ia ^ of Ba', aud A is 37^, what age ia B ? Ans. 40. 28. What is the excess o^ gg-v-757 above ^7jTj-f-Tc3T ^ Ana 200 29. The sum of two numbers is 5330 and their diflFcrencc 1999 ; what are the numbers ? Ans. 36G4^ and 1G65J. 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 snfiil, in getting up a pole 20 feet hbh, 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 J, \, and \ parts make 48 ? Ans. 44 A. I it 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. $2G0. 34. A criminal has 40. miles the start of the detective, but the detective makes 7 miles for 5 that the fugitive makes ; how fur will the detective have travelled before he overtakes the criminal ? Ans. 140 miles. 35. A man sold 17 stoves for $153; for the largest size he received $19, for the middle size $7, and for the smaJl size §6 ; how many did he sell of each size ? Ans. 3 of the large size, 12 of the middle, 2 of the small. 36. A merchant bought goods to the amount of $12400 ; $4060 of which was on a credit of 3 months, $4160 on a credit of 8 months and the I'emainder on a credit of 9 months ; how inuch 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 8.| yards of cloth that is \\ yards wide, and to be lined with flannel \\ yards wide ; how many yards will it take to line the whole? Ans. 5625. 38. Taking the inoou's diameter at 2180 miles, what are the solid contents? Ans. 5424617475+ sq. miles. "*/ MISCELLANEOUS EXERCISES. ?45 39. A certain island is 73 inilcs in circumference, and if two nie:i si^rt out from the same point, in the same direction, the one walking at the rate of 5 and the other at the rate of 3 miles an hour; in what time will they come together ? Ana. 3G hours, 30 minutes. 40. A circular pond measures half an acre ; what length of cord will be required to reach from the edge of the pond to the centre ? Ans. 832G3+ feet. 41. A gentleman has deposited $450 for the benefit of hia son, in a Savings' 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. GG^. Allowing that the deposit was made when the son was 1 year old, what will be his ago when ho can come in possession of the money ? Ans. 21 years. 42. The select men of a certain town appointed a liquor agent, and furnished him with liquor to the amoutit of $825.G0, and ca^, $215. The agent received cash for liquor sold, $1323.40. lie paid for liquor bought, $937 ; to the town treasurer, $300 ; sundry ex- penses, §29 ; his own salary, $2G5 ; 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 661G.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 docs 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 froni date. On the 9tli August, A received in advance $G2j and on the 5th September, $45 more. According to tho terms of agreement it will bo due, adding 3 days of grace, on the IGth 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 IGth of November; how much must B pay ou 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. GO, and cash, $2" 2.10 ; he bought grain to the value of $134G.40, and sold grain to the amount of $1171.97. At the end of four months B wished to close the agency, and A returned him grain unsold, valued at $437.95 ; A was to receive for services, $48.12. Did A owe B, or B owe A, and how much ? Ans. B owed A 45 cents. mmm 'r \i m ARITHMETIC. 45. A general ranging his men in the form of a square, had 59 men over, but having increased the side of the square by one man, he lacked 84 of completing the square ; how many nien had ho ? Ans. 5100. 46. What portion, expressed as a ounimon fraction, is a ponnd and a half troy vireight of three pounds fivoirdupois ? Ans. -jT^-g. 47. What would the last fraction bo if we reckoned by the ounces instead of grains according to the standards? Ans. 48. If 4 men can reap 6] acres of wheat in 2^ days, by working 8.^ hours per day, how many acres will 15 men, working, equally, reap in 3 J days, working 9 hours per day ? Ans. 40] J days. 49. Out of a certain quantity of wheat, t^ 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 way the gain on each, the gain on the whole being 20 per cent.? Ans. 9?, 19 J and 28^ per cent. 50. If a man by travelling 6 hours a day, and at the rate of 4J miles an hour, can accomplish a journey of 540 miles in 20 days ; liow many days, at tlio rate of 4| miles an hour, will he require to accomplish a journey of 600 miles ? \ Ans. 21 i?. 51. Smith in Montreal, and Jones in Toronto, agree to exchange operations, Jones chiefly making the purchases, and Sn^ith the sales, the profits to be equally divided ; Smith remitted to Jones a draft for $8000 after Jones hid maJe purchases to the amount of $13082.24; — 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 j'oar Jones has on hands goods worth $2327.34 and Smith goods worth $3123.42. The term of the agreement having now expired, a settlement is made, what has been the gain or loss ? What is edch partner's share of gain or loss ? What is the cash balance, and in favor of which partner ? 52. In a certain factory a number of men, boys and girls are employed, the men work 12 hours a day, the boys 9 hours and the girls 8 hours ; for the same number of hours each man receives a half more than each boy, and each boy a third more than each girl ; the sum paid each day to all the boys is do.uble the sura 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 K MISCELLANEOUS EXERCISES. 847 is required to find the number of men, the number of boys and tho number of girls, tho whole number being 59. Ana. 24 men, 20 boys and 15 girls. 53. A holds B's note for $575, payable at the end of 4 months from the 13th July ; on tho 9th August, A received 662 in advance, as part payment, and on the 5th September $45 more ; according to agreement the note will not bo duo till IGth November, three days of grace being added to the term ; but on tho 3rd October B tenderji 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 duo 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 tho balance ; how much must he pay on the 11th of August ? Ans. $54. 56. A person contributed a certain sum in dollars to four char- ities ; — to ono he gave one half of the whole and half a dollar ; to a second half the remainder and half a dollar ; to a third half the re- mainder and half a dollar; and also to the fourth half the remainder and half a dollar, together with ono dollar that was left ; how much did ho give to each ? Ans. To tho first, $16 ; to the second, $8 ; to tho third, $4 ; to the fourth, $3. 57. A farmer being asked how many sheep ho had, replied that he had them in four difiiercnt fields, and that two-thirds of the num- ber in tho first field was equal to three-fourths of the number in the second field; and that two-thirds of the number in the second finld was equal to three-fourths of the number in the third field ; and oi (I twi "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- tw sheep more in tho 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 i fourth field, 160. Total. 811. 348 AEITHMETIO. 58. How man J hours per day must 217 men work lor 6 J days to dig a trench 23|- yards long, 3| yards wide, and 2J- deep, if 2-t men working equally can dig one 33| yards long, 5f wido, and 3^ deep, in 189 days of 14 hours each. Ans. 16 hours. 59. A man bequeathed 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 remain'er, together with $975 ; to the youngest one-fourth of the remainder and $1400 ; he gives his wife a life interest in the remainder, and her share is found to be one-fifth of the whole ; what was the amount of the property ? Ans. $20,000. GO. Five men formed a partnership which was dissolved after four years' continuance ; the first contributed 6G0 at first and $800 more at the end of five mocths, and again $1500 at the cud of a year and eight months; the second contributed $G00 and $1800 more at the end of six months ; the third gave at first $ 100 and $500 every six months ; the fourth did not contribute till the end of eight months ; he then gave $900, and the same sum every six mouths ; tha fifth, having no capital, contributed by his labor in keep- ing the books at a salary of $1.25 par day; at the expiration of the partnership what was the share of each, the whole gain having been $20000 ? 61. Four men. A, B, C, and D, bought a stack of hay containing 8 tons, for $100. A is to have 12 per cent, more of the hay than B, B is to have 10 pur oont. more than C, and C is to have 8 per cent, more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 foot high, and 12 feet square at its base, it being an exact pyramid ; and it in agreed that A shall take his share first from the top of the stacks B is to take his share the next, and then C and D. IIow many feet of the perpendicular height of tho stack shall each take, and what sum shall each pay ? Ans. A. takes 13.22+ft., and pays $23.93 J; B takes 3.14+ft., and pays $2r>.83i; takes 2.0G-|-ft., and pays $23.18^ ; D takes 1.58+ft., and pays $21.74^. 62. A merchant bought 500 bushels of wheat and sold one half of it at 80 cents per bushel which was 10 per cent more than it MISCELIAKE0U3 EX2RCISE8. 349 lor 5J daya deep, if 24 do, and 3^ (.16 hours. > his eldest d$350 bc- with $;975 ; ) ; he gives ia found to •opcrty ? [IS. $20,000. isolvcd after St and §800 the cud of a and $1800 ;t 6100 and 1 the end of m every six bor in keep- ation of the laving been y containing hay than B, 8 per cent, quantity ho at its base, lall take his re the next, ar height of 58 n.l4+ft., ^3«23.48i; cost him, aui 5 p3r cent, lesi than ho asked for it. He sold tlie remainder at 12 J par cent, more than it cost him. What was his askiii;^ price for both lata? What did ho rcocivo for the last lot, and how much did he gain on the wliolc ? C.]. May 1st, 18G2, I got ray note for 82000 payable in 4 months discounted at a buik, and immediately invested the money yeceived in woodland. November 9th, I sold Jhc land at an advance of 15 per cent., receiving | of the price in cash, and a note for tho remainder, payable August 10, 1864, without grace, and to be on interest after January 1, ISiil, at 7 por cent. I lent tho cash re- ceived at 6 pjr cant. Whan my note at tho bank became duo I renewed it for tho same time as before, and at the proper time renew J it again, and finally pcnowod it for such a time that the note would become duo August 10, 18G4. Now, if I paid 6 per cent, on tho money borrowed at the bank, and made a complete settlement A.ugust 10, 18t)4, what was the amount of my gains ? 04. My ajront at Mobile buys for mo 500 bales of cotton, avei aging 500 lbs. per bale, at 1.0 cents per pound. I pay hlra 1 J per cjut. on tho amount paid for the cotton, and shipping charges at 60 cents from January 1 for an amount sufficient to pay for tho cotton, charges and commission including also 2 per cent, discount on the draft. On tho rooeipt of the invoice, I insure fo^the amount of tho draft plus 10 per cent. ; I pay l.|- por cent, premium on the amount insured, and from tho amount of tho premium is discounted IJ per cent, for cash. On tho arrival of t'ln «ir>ttnn I pay J of a cent per pound for freight, and 5 per cent, primage to tho ^apUiin on the freight money, ai.d al.so 4 cents per bale for wharfage. I sell it on tho wharf, Januuiy 20, at $1 per bale profit,' and agreed to take in payment the note of tho parohasor for 6 months from January 20. What amount would bo roooivod on tho note wlicii diacountod at a "bank at 7 per cent. ? , • . old cue half lore than it ;'4; '*. 850 AILITHUETIC. % FOREIGN GOLD COINS. JUKT TALUE. COUyTKT, SBXOliniATIOXg. ., «■ Australia.. . . *( Austria * n Belgium .... Uoliria Brazil Ccntr'l America Chill Denmaric Equador Eogland 11 France << Germany, North « « Grermany, South Greece Hindostan Italy Japan Mexico Naples Netherlands. . . . New Granada.. i' ^ra Portugal Prussia Rome Russia Spain II Sweden Tunis Turkey ToAoany Pound of 1852 Sovereign 1 855-60 Uucat SoHverain NewUnion Crown (ossuc^od) Twenty-five francs Doubloon 20 Milreia Two cscudos Old doubloon Ten Pesos Ten thaler Four eseudos Pound or Sovereign, new. . Pound or So verign, average Twenty francs, new Twenty francs, average . . . Ten thaler Ten thaler, Pnissian Krone [crown] Ducat Twenty drachms Mohui 20 lire Old Cobang NewCobailg Dob!)loon, average " new Six duoati, new. Ten guilders Old Doubloon, Bogota Old Doubloon, Popayan. . Ten pesos, new Old doubloon Gk>ld crown NewUnion Crown [aaiumed] 2^scudi,new Five roubles 100 reals 80 reals Ducat 26 piastres 100 piastres Seqoln wnonr. riNB- MB88. VALC*. Oz. Dkc. Thocs. Q.2gl 916.5 15.32.37 0.256.6 916 4.85.58 0.112 986 2.28.28 •0.363 .900 6.75.36 0.357 900 6.64.19 0.254 899 4.72.03 0.867 870 15.59.26 0.675 917.5 10.90.57 0.209 853.5 3.68.75 0.867 870 15.69.26 0.492 900 9.15.35 0.427 895 7.90.01 0.433 844 7.55.46 0.256.7 916.5 4.86.34 0.256.2 916 4.84.92 0.207.6 899.5 3.86.83 0.207 899 3.84.69 0.427 895 7.90.01 0.427 903 7.97.07 0.357 900 6.64.20 0.112 986 2.28.28 0.185 900 3.44.19 0.374 916 7.08.18 0.207 898 3.84.20 0.362 568 4.44.0 0.289 572 3.67.6 0.867.6 866 15.62.98 0.867.5 870.6 16.61.05 0.245 996 6.04.43 o.fts 899 3.99.66 0.868 870 16.61.06 D.867 858 16.37.76 0.525 891.6 9.67.61 0.867 868 15.65.67 0.308 912 6.80.66 0.357 900 6.64.19 0.140 900 2.60.47 0.210 916 3.97.64 0.268 896 4.9«.39 0.216 869.6 3.86.44 0.111 976* 2.23.72 0461 900 2.99.64 0.231 916 4.36.93 0.112 999 2.31.29 Vnluo after Deduction. 115.29.71 4.83.16 2.27.04 6.71.98 • 6.60.87 4.69.67 15.51.46 10.85.12 3.66.91 15.61.47 9.10.78 7.80.06 7.51.69 4.83.91 4.82.50 3.83.91 3.82.77 7.86.06 7.93.09 6.60.88 2.27.14 3.42.47 7.04.64 3.82.34 4.41.8 3.55.8 16.45.22 16.63.26 5.01.91 3.97.67 15.53.26 15.30.07 9.62.68 15.47.90 6.77.7tt 6.60.87 2.59.17 3.96.68 4.93.91 8.84.51 2.22.61 2.98.05 4.34.75 2.80.14 FOREIGN SILVER COINS. 351 FOREIGN SILVER COINS, lOMT TALCK. DKKOVXNATIOSla. wnaBT. 1 VJLLCK. Austria Old rix dollar Oz. Dec. 0.902 0.836 0.451 0.397 0.596 0.895 0.803 0.643 0.432 0.820 0.150 0.866 0.864 0.801 0.9"7 0.182.5 0.178 U.800 0.712 0.595 0.340 C.340 0.719 OJJ74 ^^79 0.279 0.867.6 0.866 0.844 ry.804 0.927 0.803 ^.866' 0.766 0.433 0.712 0.595 0.864 0.667 0.800 0.166 1.092 0.323 0.611 0.770 0.220 THOC& 833 902 833 900 900 838 897 903.5 667 918.5 925 a50 908 900.5 • 877 924.5 925 900 750 900 900 903 900 916 991 890 903 901 830 944 877 896 901 909 650 750 9«0 900 875 900 899 760 899 898.6 830 926 $1.02.27 « Old scudo ,... 1.02.G4 « Florin before 1858 New fl9rin New Union dollar '. Maria Theresa dol'r,1780 Five francs 51.14 i( 48.63 « 73.01 i< 1.02.12 Belgtnm 98.04 Bolivia New dollar 79.07 i; Half dollar 39.22 Brazil Double Milrei^ 1.02.53 Canada 20 cents '.... 18.87 Cental America. . . Dollar. 1.00.19 Chili Old Dollar 1.06.79 « New Dollar 98.17 Denmark Two rigsdalcr 1.10.65 England • . . . . Shilling, new 22.96 « Fraace Germany, North... 4( Shilling, average Five franc, average Thaler, before 1857 New thaler 22.41 98.00 72.67 J2.89 Germany, South. . . . Greece nindostan Florin, before 1857 New florin [assumed], . . Five drachms 41.65 41.65 88.08 46.62 Japan itzebu 37.63 it New Itzebu 33.80 Mexico Dollar, new Dollar, average Scudo 1.06.62 « 1.06.20 Naples 95.34 Nell rlands 2i guild 1.03.31 Norway Specie daler 1.10.65 New Granada Peru .• Dollar of 1857 Old dollar 97.92 1.06.20 er cent, may bo recovered back, provided tho action for recovery shall be brought within one year after pay- ment, (lb. sees. 0& 7.) Damages on Dills —Tho damages on Bills of Exchange drawn or negotiated In Arkansai>, expresses to l>o for value received, and protested for uon-acceptance, or fur non-payment after non-acceptance, are ns follows. — (lb. chap. 2d.) 1. If payable within the State, 2 per cent. 2. If payable In Alabama, Louisiana, Mississippi, Tennessee, Kentucky, Ohio, Indiana, Illinois or Missouri, or at an^ point on the Ohio Kiver, 4 per cent. 3. If payable in anjr otiior t-tate ot' territory, 6. per cent. 4. Jf payable within either of tho United States, and protested for non-payment, after acceptance, 6 (lercenU /brnon /vo any of his property out of tho State, with intent to defraud his creditora 3. When tho defendant fraudulently con- tracted tho debt or Ineurrod tho obligation, respecting to which tho fuit is brought. 4. When the dcfendaut is a non-resident. 5, When ho has Iniudulently convoyed, disposed of or con- coaled his property, or a part of it, or iiite;i(ls to convoy the samo to dolVaud hi,-) creditors. In California tho real estate sliall bo bound, and tho attachment shall bo a lien thereon, olthougli the debt or demand duo tho plaintllf bo not due— lu case tho defendant is about to remove himself or his property from tho Sliito. The law of otlachment applies In California to/ien the contract hot been nuide in that StalCy or when made payable xn that State. t In writing 1 contmcia bd upon.— Iter rate of Icent. may 1 after i»y- |Arkansai>, aont after Indiana, Lent, after Id payable llgtkt bills. lUio law- LrosldoDt, I State; or niry pro- CONNECTICUT. Interest.— The legal rate of Interest In Connecticut Is six percent, and no higher rate Is* ollowed on special contracts. Banks uro Ibrbiddcn, under penalty of $500, from taking directly or indirectly over 6 per cent Law piisscd May, 1851. Penalty for VioUUion of the Usury iaio*.— Forfeiture of all the Interest received. In Eults on u&urious contracts. Judgment Is to bo rendered for tho amount lent, without interest. Damages on Bills.— Tao damages on bills of cxcliango negotiated in Connecticut, pay- able in other states, and returned under protest, are as follows; 1. Maine, New Hampshire, Vermont, Massachusetts, Rhwlo Island, New Yori (interior). Now Jersey, Pennsylvania, Uelawaro, Maryland, Virginia, District of Columbia, 3 per cent 3. New York City, 2 per cent 8. North Carolina, South Carolina, Georgia antl Ohio, 6 per cent 4. All tho other States and Territories, 8 per cent Foreign Bills.— Then Is no statute lu force Id Connecticut in rofcrcncoto damages on foreign bills of exchange. Sighi Bills,— Gmao is not allowed by statute or usago on checks, bills, cct, payable cti tight. Collection <(f/>eb(A— Attachment may bo granted ngalnst tho gomls and chattels nnd land of the defendant ; end likewise against his person when not exempted trom imprisonment on tho execution in tho suit The plalnUlfto give bonds to prosecute bis claim to effect. DELAWABX. Interest— Thv legal rate of interest la irix per cent , and no more Is allowed on direct or indirect contracta Penaty for Violation of the Uutury i««u.— Forfeiture of the money tr:^ other things lent, ono liolf to tho Governor for the support of government, the other half payable to tho person sut Ing for tho same. Damage i on £»//«.— -There Is no statute In force in Delaware is reforDnoe to damages on domestib tr Inland bills of eztSbange. Foreign M/b.— The damages upon bills of exchange drawn upon any person in Eng- land, or ulbor parts of Europe, or beyond tho seas, nnd returned under protest, are 20 per cent, Sit^t £iIb.->Thero is no statute Vith referebco to bllla^ drafu, etc., oi tight. They ore not, by usage, entitled to gr»ca. CoUeetion i^f DthU—K writ of domestic attachment Issues against an Inhabltaut of Dcla> ware when the defendant cannot be found, or has abscondud >Tlth *ntent to defraud his credi- tors; and a writ of foreign attachment when tho defendant Is not ai. Inhabitant of this State. This attachment la dissured by the defeudanl'i appewing uid puttiiiH la apeoiiU boil at mf time before Judgment. 23 354 FLORIDA. hl^retl.— The legal rate of interest is six per cent On special contracts eight per cent, may ixs thar-uii. PenaJlij for Tiolaiion of the Vtury iu bills cif exclian);c, negotiated in Florida, payable i3 other l-utos, uud reaiiiied under (irotest for non-payment, ore uniformly 5 per cent. y^reiya Jiiils. — Duuiag s n writing, iu-o entitled to iho Fame remedy us provided (or Bscuritiea. In all cases tli < attachment li^l^l Hcrrcd shall Ijc llr^tsatislied. No lieu shall bu created by tho levying of an attachment, to ilio cisclusiun I r any judgment obtained by any crcdiUir, before judgiueul is obiaiued by the bttachiug creditor. » ILLINOIS. 7n/oration. (Act of 1S57). Damanet on BilU —Tim d.-im-iges on bills of cxcbango Degotlatcd in Illino's payable in o^her^utes or Territori -8, and returned under protest for non-iiayment, are uuiformily {by act of March 3, 1845) 5 jicr cent in mldition tj tho interest Poreign'BilU.—Thm damages pnya'>!e on foreign bills of oxchaORS, iftumod nndr>r protest, arc [by act of Jilarck U, 1845] 10 |M.>r cent, in addition to Ibe Interest. /tight JKilt.^—nenioTiiTO tber* lies been no statute In force regarding bi11> or dtnds at sight, but by an net of ih' leglsliituro, approved Icbruary 'Wd,4OTl, it is enacted that "no note, chock, draft, bill of exchange, ord'-r or other negnliable or conunercijl Investment payable at sight or ( n demand, < r in prcscntntion, tliall bo entitled to days of grace, but shall bo absolutely iwyabloou pres<. nent All other notes, draitsnr bills ol( exchange, shall be en- titled to tbo usual days grace. This act is in force frtm its i>a$sage. Collection, of /V6t{b.— Tlio damages on bills of exchange, negotiated in LouiBiaua, paya'dc iu other States, are uniformly 6 per cent. Foreign Bi7/s.— Tho damages on foreign bills of exchange, returned under protest, arc uni- formly (Statute of 1838) .....•• lOixircent. Sight BillL—TheTo Is no statute upon this Subject in Louisiana, A dcoisinn ha.1 licen made in one of the inferior couris allowing three days' grac; ou sight bills, but the usngo Is to pay on presentation. ' CoUtction. of Debts— \ creditor may obtain an attachment against the properly ofhis debtor upon ofBdavit; 1, when tho latter is about leaving permanently the State before oblaiuiug or executing Judgment against him; 3. when tho debtor resides out of tho Slate; 3 when ho conceals himself to avoid belBg cited to answer to a suit, and provided tho term of paymcut lias arrived. In tho absence oftho creditor, the oath (nay bo made by his ugcnt or uttome>', to the best ofhis knowledge and belleC MAINE. /nt (liana, »owa, isoD, Wlaouu* at a future ich tho Roods pluittela may btor. Such a |t, In hlB own attuu may Ua MAKYLAND. Intfrnt—Tii!arylaiiil proTlilcs tliat th> rate of interest in tho sUitoKJiiill noL circcU UK pur tent, prr iinniiiii. iiikI iid liitihor rate Hliall lie taken ordt'inindcd. And tho Ipyi.sluluri! hlull jiroviUo liy l.i'v idl uici.so;iry furlolturc:! and iKjiialtics against usury. Penalties. — .tny ixivon (;iiill>i()f umry hIkiII rirlcit all tho oxocsj above tlia real sum or value I'f liio dtrtdils or rlmticm iu tuully Ui.t or ailvaiicrd and the I gal interest on Ruch Kutu or value, wiiicli iDrR'iliire kIiliI! < uuro tw tlio b. jn^Jit o. any dLdondant whu shall plead u-^ury, u:ul pi-ovo the same. Tho plea niii^t, hovvev r, Klato the Kura or amount ((fth.'Uelit.nnd the plain- lill' shall have judg:ncut I'ur that aniDiiut un I W^wX intore^il only. JId. Code, vol. 1, p. 697. Damngp.) on Hills.— 1'ho dania^'e.'* on bills of eyehanin ne;;otlatod in Maryland, payable in other .SiatoJ, and ri'lurned nniler priitesl, are unilbnnly 8 per rent. Tho claimant iixntillcd to receive a sum (■ullleienl to liuy another bill or iiio same tenor, and eight per cent, damage!) on tho value tjitlie iirinelpal ."uni nienlioiu'd in llie bill, and inlerost from the lime of protest, a:cdeoKt.'<. The pmtet cjI an inland bill nm>l bo madu iicconlinj? to tho law or u.=.ago of tho t'tate where it is iiayablo. J'riiclici; incbidos the flistriet td' Cjlumbla 1 1 this law of damages ;.\et of As.s mbly. Its.), ih. .';8); but it is iiucstiuuablo wl.eiucr tho iJistricl be within tho law, which provides only for Slalis. Furiign Jlill.t. — The diinia;:o.^ on forelsn bills of exchango returned under protest arc 15 per rent. TliO claimant j.j to leeelvo a suiii fUllleii'iil to buy another bill of tho .=amo tenor, and 15 per cent, damnnos on the value of tlie {.riucipal 'ium lucutiuncd In the bill, and interest Iroin time of protest, and cost.s. Sight iiiWj.— Grace is nut allowed by the UauUs on b.lls, drafli^, checka, etc., payable at fifeht. Callfxl'on of Di-hls. — K ere litor may obtain an attachment, whether ho be a citizen of Maryland or not, n.?aliin his debtor, who Is not a citizen of ibis Slate, and not residing therein. If any c.li.cen of the State. liein;{ indelitcd to another citizen thereof, shall actuully run away or absKind. or secretly remove himself from hi.s place of alMwIe, with intent to evade the jviy- nient nf his jnst debts, an attaehnient may be obtained n^'ulnst lilm. An attB'diment may be laid upon d bts duo tho 'lelendant upon judiiinents or deerees rend rofl or ]vi-sed by any court of this i;taic, audjudgmcutof cond'eniuaiiun thereof may bu hud, as upon other dcbtd due tiiu dvfenduut. MASSACHUSETTS. Inlrre.tt. — Tho le^al rate of interest iu Mussachusett.s 1;; si.t ]x>r cent, and no higher rate is aiiowcd on special contracts. renalli/ /iir Vi)latiiin of titn I'mr;/ Lav:?.. — N'o rnnlra't for tlie pnym^nt of money with in;ere.-l (;ie.i;er than si.\ jier cent, shall bo void ; but in an action on such contract the defend- i:.t shall reo jver b .s full co.-,t.-', and the i.laintill shall forfeit thrce-fuld the amount of the w holu in'ero.?t roA'rvcd or taken. Dnniagc.^ nn Hills r,/ j-Szc/iangt. — T!io dauinses^on bills of ex'h.mgo negotiated in l.'nssacl.;:- setlg, payable in othei- Stales, and retunie 1 under protest, are as lullowg; 1. Bills payable in Maine. New Hjunjislilre, Vermont, I.hudo Island, Connecticut, or New York, U iicr cent 2. Bil s payable in New Jerse;.-, Pennsylvaiiia, Maryland, or Delaware. 3 per cent 3. Bills iiayable in Virginiii, District of (.'uluinbia, .N'oilh Carolin.i, .-'uuth Carolina, or Georgia 4 p r cent 4. Bills payable elsewhere within tho Ui.ited States or the Territories, 5 per cent 0. Bills for one hundred dollars or more, iiayaMo at any place In Massachuse.ts, not within seventylivo miles of the placo where drawn, 1 i)CT cent Fvrr^gn ij'iz/i.— Tho damages on foreign biJls of cxcUaugo, roturno 1 under pretest, ore na follows: 1. Bills payable beyond the limits of the Tnited Stales (exceplln'? plaees In Africa, beyond till' Cape ofiiood lloiie, and places in Asia and tlic r-lands thereof) shall pay the cur- rent raio of e:;chan','e when diU', ond fl\e p' r cent additional. 2. Dills payable at any place in Afri'-a, beyoivl the Cipo f)f Good Hope, or any placo in Asia or the itlands tliereof, shall pay damages, 21 pej' cent. Sight liill.i.—\ii\li of exchange, dmfls, etc., payable al tigltl, or at a future day certain, within this btate, uro entitled to ilu-ee days' grace. But not bills, uotcii, drafld, etc., payabbi b,i deniaiuL Nota on nemand.—\n order to e'.iarpn an indorser, payment iuust bo demanded Willi. n si.\ty clays Irom its dale, without grac". on luiy note payable on iknuind. Collection of Z>tl/ta.— iiriginal writs mav be framed, either to attach the goods or estate of tho defemUni, or for want Uieieot (o take his body; or there nwy bo an original summons, cither Willi or without an onlcr to attach tho goods or estate. All real estalo, or goods and chatties thill .uo liable lo be taken in execution, may be ottached upon \ho original writ, in any action in which any debt or damages arc lecovcrable, anil may bo held as security tu taiisVy Euch Judguieut as the pliiiiitiif may recover. I :• " . :■/ Ui 358 MicmaAN. Tnltr(it.—Tho legal rate of intorcst In Michigan ia seven per ccnl. But it 13 lawfal tor parties to Btlpulato m writing for any sum not cxcvodlng ton per cent. I'enaltyfor ViohHim 0/ the Utury Lawt. — Purlica suing upon contracts reserving over ten percent interest, may recover judgment lor tho principal unil legal rate of Interest. There is •10 provision for recovering back 'liegal interest |>aid, au4 uu |)enalty for receiving it. Bema fide holders of usuiious negotiable paper taken before maturity, witiiout notice of usury, may recover thn full amount of lis lace. Vamaget on £iMf.— Damages on bills drawn or negotiated In Michigan and payable else o fraudulently contracted tho debt, or Incurred the obligation about wluoli tho suit is brou^iht ; 4, that ho U net n resident of the Stale, or has not resided thero three months immediately prticeding tiio suit ; fi, that the defendant is a foreign corporation. MINNESOTA. IvtereH. — Interest for any Icjsal indobtodnoss slmll be at tho rato of $7 for |tOO (br a year aoless a diifereut rate bo contnicted fur hi writing, but no agreement or contract f t a greatc rate of interest than $12 for every $100 for n year shall bo vaiid for tho ctccsa of inter s over twelve per cent. ; and all ogreemcnts and contracts .sliall bear tho samo rato of interes* after they become due as before, if tho rate be clearly expressed therein. Providtd, tho samo shall not exceed Iwelvo per cent, per annum. All judgments or decrees, made by any court In this State, shall draw interest at the rate «f six (0) percent, per annum. [Laws of 18B0, p. 226.] Penalty for Violation 0/ Interest Law. — Kxcess ol interest over 12 per cent fnrMted. Days of Grace. — On nil bilMcT exchange payable atBlghl, oral a future day certain within chls S^tate, and on ail negotlablo promissory notes, orders and drafts, payable at a future day certain within this State, in which there is not an express stipulation to tho contrary. When Grace not allowed.— On bills of exchange, nolo orUralt, payable on demand. JVhenpresented for Payment, dc. — Bills of exchange, b;ink chocks and promissory notes falling due, or tho presonimont for acceptance or payment whereof should bo mado on "the 1st day of January, tho 4th day of July, tho 25th day ofDccembcr, tho 22d day of Icbrnary, and every day appointed by tho President of tho United States or tho Governor of tho State as n day of fasting or thanksgiving, shall bo presented loracceptaiico or payment on tho Aaypreced. ing. Such days [al>ovo enumei^ed] shall be treated and co;isiderod as tho Urst day uf tho week, commonly called Sunday. "[Col. Laws, 376.] Acceptance of Bills of Exchange — No penon within this State shall bo charged as an nr- (■eptor on a bill of exchange, unless his acceptance shall bo in writing, signed by himself or l.i^ lawlul agent. Damages on Bills of Exchange.— On any bill of cxchinRO drawn or endorsed within this State, and payable without tho limits of tho t?nltod States, v/liich shall bo duly protested for non-acceptance or non-payment, tho party tlitblo for tlio contents at such bill shall, on duo notico ond demand thereof, pay the same at th; current rate of exchange, at tho time of tho demand, and damages at tho rate of ten per cent, upon tho contents thereof, together with interest on said contents to bo computed from tho* dato of tho protest ; and said amount of contents, damages and interest shall bo in full of all damages, charges and expenses. On all bills drawn on any person, botly politic or eorporation oitl of this State, but within somo State or Territory of the United Mates, and protested for non-acceptance or non-payment five per cent, damages and interest, and costand charges of protect. Collection of Debts.— A warrant of attachment may bo Issued agilnst tho property of a defendant when a foreign corporation; or, when not a resident of this Territory; or, has left tho Territory with intent to defraud his creditors. Thus it will bo seen that in all tho States tho property of non-residents and foreign corpora- tion.s is liable to attachments at tho suit of creditors beford judgment in rendered; likcwi.so against domestic debtors when thoy have absconded from tho State, or have fraudulently con- veyed, oraro about to convey, sell, ossign or socroto tlieir cflccta In some few States, how- ever, even this condition is not essential before a writ oi' attachment will Issue In tho states of Alalyima, Massachusetts, Connecticut, Maine, Now Hampshire, Vermont and Rhode Island, tho creditor may have a writ of attachment against tho property of the debtor at tho lirst institution of a suit— and witho it any ground of fraud or fraudulo;,l Intent — such properly being held by the attachment until tho termination of tho suit, or untl Jiidg- inenl; thoplaintilf insuch coses giving bond or security to indemnify tho defendant for any loss or damage sustained, should the case bo decided In favor of tho tatter, (jcneraily, t lie property is liable only when actually levied upon; but In tho i:tato of Kentucky lUo pro- perty \i liable from the timo of delivery of tho order to tho shcriir. 359 MISSISSIPPI. Tntfral.—lm legal rate of lntercat in MigsisBippl is six per cent, by the act passed in March, 1856. Da"\agfs .m liUls. — No damages ore allowed for default In tlio payment of any Mil of ex- rliango drawn liy nny person or persons williin llio btato on any person or |)crson8 in any otliur : tnlo. O.i ull domostioor iiilund billif [drawn on persons wilUin llio eituie], uud pruiestod for non-iayment, live per cen' [.See act of May 11, 18o7. | Foreign liillt. — Tlio dumiiges oii bills of cxclmngo drawn on persons wlltiout tho United Statie, ruiunicd under prolett. are 10 percent., wiili nil incidculal cburgis and lawltil lotcrett. Sight Hills.— -Omco Is not allowed on billij of exchange, dral'.s. etc , puyublo at tight. C'ollecllon of Debts, — An attaclimcnt ogaln.si itio estate. Including real estate, goods, chnltelri, AiC, ot'a delitir, when it lasliown that U.t has remo/.d, or ii uboiit reraiiving or ab^co^din){ I'rotn tho Slate, or privately conceals himself Attew8 in churches, and tlio liauchi.se of any corporation authorized to receive lolls, until the period of thirty days from tho time of rendering the judgment. NEW JERSEY. Interest. — Tho legal rate of interest In New .Jersey is six per cent, and no higher rale ot Inter, stis nllowablo on special cimtracts, except as provided In the following acts : Tho Icgislaturoof New Jersey piuc., That upon all contracts hereafter made in tho city of Jersey City, and Iti tho township of Uoboken, in tho county of Hud.son, i 1 this Stale, for the loan of or forbearance, or gl.iiig day of payment, for any money, wares, nierchandi.se. Riiodsor clialtc .>, it shall Ixf lawful for any person lo take the value of seven dollars for the I'orbeurauco of ouo hundred duUara im-rt. ^votum* 360 for a year, oml nftcr that rolo for n (frcator or lo=8 sum, or for n longer or fhortcr period, nny thing contuliiud ill tlio net, to whicli tUia la ii Bupnlumcnt, to tho contrary notwiili^uiiiilint;. I'roviiUd, Kiich coutruvt be marlo liy und iHitwocn porMOa actually located iu oitber suiU city or townHlii)), or liy pcrsonH not residing In this Hlalo. April 6, ISf).'). Tho latter provlHo was amended, " I'rovldcd tlio cnntractlnR parties, or Cither ur thtiin. reside in oilhoruf Huid places, or out of tho State." The following cbaogca bave Rinco been made so as to make It legal to charge 7 per cent interest : Act, February 21, 1800, Ar/n of Debts.— An attachment may issue at the InRfanco of a creditor (or, In his absence, of his agent or attorneyj, against the property ofa dbbtiir when tho latter is about to , atiscuud from '.he State, or is not a resident of the State, or Is a lurcigu corporation. It i NEW YORK. Interest— The legal rate of Interest in Now York is seven per cent., and no higher rate is allowed on special contracts. Penalty for VioUitinn of the Usury Laws. — Forfeiture of the contract In civil Bcttons. in criminal actions, n lino niit exceeding one thousand dollars; or imprisonment not cxi'eeding six months; or belli. Alf bonds, bills, notes, assurances, conveyances, all other controcts or securities ivhutsoovcr (except bottomry oud respondentia bomfs.and contracts), and all ileposits of goods, or othcr.tliings whaWoever, whereupon or whereby thcro shall bo reservcil o" taken, or secured, or a^roed to bo resen'ed or taken, any greater sum, or greater value for the loan or forbearance of any money, vpoin or other things In actipn than seven per cent, shall bo void. (Uev. Stat. Vel. II., p. I>i21. For the purpose of calculating Interest, a month shall bo confidereuwiic twelfth part of a year, and as consisting of thirty days; and intcre.''t for any number of diys less than a montli shall bo estimated IJy tho proportion which such numbcrof days shall boar to thirty. Datnageson JlilU. — The damages en bills of exchange, negotiated In New York and payable In other State ■, and returuncd uiulor protest for non-iicceptanco or non-payment, are oa follows : 1. Moine, Now iiampshiro, Vermont, JIa.'..=acliUKotts, Rhode Island, Connecticut, Now Jersey, rennsylvania, Delaware, Maryland, Virginia, District of Columlfla, and Ohio, 3 per cent. 2. North Carolina, l-oiith Carolin.i, Cieorgia, Kentucky, and TenneS:oe, 6 per cent. 3. If drawn upon parties in any other State, 10 p^r cent. Tho following day.s, namely, the llrst day of .(anuary, commonly called New Year's day; tho fourth day of July; tho twcnty-lifth day of December, commonly called Chri.stmas day ; nud any day appointed or recommended by tho Governor of tho State, *or tho President of the United States, as a day of lastorthauksxiviug, sha'l, for all puri'oses whatsoever, as regards tho presenting for payiii'iil or acceptance, and of tho protesting and givir^g notice of the dis- honour of bills of cxchau.;!-, bank chocks and promissory notes, made after the passage >.f this act, bo treated and considered as is tho Urst day of the week, commonly called Sunday. [1849, ch. 281.) •Foreign Dills.— Tho. &.im:\^ on foreign bills of exchange, returned undei' protest, are 10 per cent. I Sight Dills. — Grace is not allowed by tho banks ofthe city of Now York and of the interior, upon bills, drafts, checks, kc. , payable at sight. Coltcctinn of Debts. — Any creditor to tho amount of $25 may compel tl\p assignment of their estates by iL'btors lmiiris(med^)n csecutiou in civil causes for more than 60 years. If tho debtor refuses to bo ex;iinlned, and to discloso his affairs, ho Is llaljlo to bo committed to close confhiomeut. If ho rel'iises to render an account Inventory, and make an assignment, lie will not bo entitled to his dischargo : though the officer 'having jurisdiction in tho cai-e is authorized to mal;n tho assignment for him. Tho proceedings and the cflicct of the discharge, when duly obtained, and tho duties of tho debtor, and tho rights of tho creditors, aio essentially tho sameas tn tho caso of proceediU'-'s with tho as-ent of two-thirds of tho .creditors. Kveiy Insolvent debtor may also petition tho proper olllceis tor leave voluntarily to assign his csiiilo lor tho benefit of his creditors ; and tho samo proceedings and che.ks^re substantially jire- sciibed as In other cases of insolvency. His discharge, obtained in such a case, exempts hi» from imprisonment, as to debts duo at the lime of tho assignment, or provjoiisly coniracied, and as to lial>ilities incurred by making or indorsing any promissory^to or bill of exchaii;.'e. But tho dischargo does not afl'ect or impair any debt, demand, payment, or decree against tho Insolvent ; and they remain good against his property acquired after the execution ofthe as- signment ; and tho Hen of Judgment and decree Is not all'ected by the discharge. i:l:^ period, ftny • BuiJ city or ! partlco, or iiig clmngea ,858, Bcrpcn Ijway. Act, ont. lutcre. I I sum Is for- lagca on bills on protcstcil plnco within neJ due iiD'orth America, except tho Northwest Coast end tho West Indies, 10 per cent, i. Eills payaUic! in Maileria, tho Canaries, the Azores, Cape do Verdo Islands, Euruiic and South Aiiienea, 1.') percent. C. HilLs payablo eisewlieie, 20 jier cent. Sig'il Ilillx.—Xty virtue of an net of tho Legislature. pa.=.«ed In January, 1849, ,;racc if nliowcd on Idlls ri( siV/Zi^, unless there is a stipulation to tho contrary. Prior to thatdate the usage wa.", not to nlinw fraco on such biiljj. Colli clwn of J)chls. — An attachment may Issue en tho complaint of a creditor, his agent, nltorney or factor, against the property of a debtor when hi- ha;irenioveil or is about to remove, privately from tho biaio, so that tha ordinary process of lawwill not reach him. • OHIO. Tnlere^t.~The law allows interest nt six per cent, per annum on nil money due, and no more. (The law allowing.lu per cent, on special coitnicis wasiejiealed April Isl, 1850, liut the repeal docs imt allect coptiacts entered into prior to lids date.) Kaiiroad Coiupanics arc authorized to borrow at thc«Mo of 7 jier cent. refirrttrr*. — Tlioro are ifo pena lties ordinarily for I'.snry. Contract."? for greater rates arc void a.i (0 the excess v itli/ ' f MW I I lllLlRJIi lii i juiid H i . rTigr<»T tlion lognl interest, ho may re- cove^ back til) fXCOKS irBiicii foi within nix nicnilis. Act May i!!Ub, 1868. Damage! vn liilU.—'iha daniDKcs on bHIs ol exchniiKo nc'x •tiutod in i'ccnsylvauia, payable in oihor Mates, and rcliirnod under prolosl, un' ii:^ lullnwd [May 13, 1850] : 1. iJppcriind ix)vver( alifornla, New Mexico, and ^^lugon, 10 per cent. 2. All oilier Males, b per cent. I'arf iffn Hills. — 1 bo dumagos on foreign bills of exchange, returned under protest, arc as follows [Muy la, 18J0| : 1. I'nyaO.o in China, India, or other parts of Ai^la, Africa, or Islands in the raciflo Ocean, 20 p re 'Ml 2. Mexico, iSpanish Main, West Indies, or other Atlantic islands, East Coast of South America, (ireul llrilain, or other parts of Kuropo, 1.0 per cent. 3. West (Joasl of >Soutb America, 13 per cent. 4. All other parta of Iho world', 10 per c^'Pt. night liUti.—yy a law passed Wa'- 21, 1857, all drafts and bills c' oxchanRO, payable iil Bight, " shall lio and become duo op prr.senlaliun, wiilwxit gniro ; and shull and may, if dis- biinorcd, bo pmtcsled on and lini-iodiatoly ul'ier sueh prest'nialiuii." Collection <^ Debts.— In tl;i.s titaie the writ > i dctmi'.sljc attadiment Issues against any debtor, being un inhabitant of llio Stale, if ho bifs ab.scondeU from his usual placo of abod', or shall liavo remained alisent from tho Siaie, or Fliall have conflned himself in his own house, or Ciiiiccalod himscll clHOwhoie, lo dcfrdu:! his eredilor." No second atlueliinent wiil bo issued against Iho t-umo property, iinle.s.-< thi< Dist l/e u-.: executed or Imi dissolVL^d bythocouil. A writ of att^ielimont may bo also ls.siie(l aKalnsl tho property of a foreign coriioratloii or n ii«u- residoul. In tho I iltcr cas', tho altarliuienl inures Id tlio lienr lit of tho altacUitij; creditor only. Il\ tho former case, it id for the beucllt of creditors ut lar^ie. h\ RHODE If LAND. Intfnst.—lho legal rate of interest in Itho o Island Issls per cent., and no higher rate is allowed o.i special contracts. I'eiiaUy for Violalion of the Usury Law: . — Forfeiture of tho excess taken nbovo six per cent. /)nmrt(;« on /?i7/».— Tho damages on*l)ills of exchange, iiayiiblo [in other Stales, and re tumeil under protest, nro uniform y f> pi-r cent. Foreign DUU.—lho dainagos on furelgn biils of exchange, returned under protest, aro 10 pet cent. Sight Hills. — By statute it is provided that "all bills of exdiange drawn at sight, which slmli bo duovnd iiayabiu In this .Stale, (Uliodo Island), shall bo deemed to bo duo and puyablo on tho day of presentation, wiiliout graco. " Colltclion of Debts. — In this .Slito a writ of attachment Is flrst levied against tho Inxly of tho defendant, and if ho cannot bo ruind, then against his goods and chatties. Tho property of I'oroigu corporations and dobtois is also liable to attachuiout at Ihu suit of a creditor. SOUTH CAROLINA. Inlerei.t. — ^Tho legal rate of Intero-t in South Carolina is Bovoa per cent., and no higbei rate is aliviwed on special iontract.>». I'enaUy for Violation of the l/'sury Laws. — I/ijs of all tho Iniercst taken. Damages on Hills.— T\w damages on bills <.f exchange negotiated lif ..South Carolina, pay- able in otiicr Slates, and protected lor noii-payinent, aro uniformly 10 per <:cnl. togkliier with co.tIs of protest, A bill drawn in .South Carolina, payable in another State, is deemed a foreign jIU, and d.-xmages may be claimed, alihougli mich bill be not actually returned after protest. Foreign y^Ut. -'Iho dumuges on furcign bids of exchange, negotiated in iroutb Carolina, !;ro as follows; 1. 'Ou Mils on any part of North America other than tho United States and on tho West la vidies, 12 'i per cent. 2. On inils drawn on any other part of tho world, LI per cent. Sight Hills. — The slatuto of 1843 enacLs that " bill.s of exchange, foreign or domestic, pay able at sight, shall bo entitled to the Eanio days of graco as now allowed by law ou bllln ol exchange payable on k nie.'' Hy n statute pa.ssod in 1831, it is enacted that if money or other commodity bo lent or mlNanced upon unlawful interest, tlie |ilaiatilf shall bo allowed to recover tho amouut or value actually lent, but without interest or cost. By iin act passed in 1S39, ii is ciiiictcil that a debtor by bond, note, or otherwise, about to leave tho State, tho debt not be ii;f yet due, may bo sued and held to ball. Tho plaiiilillmust swear to tho debt, and that ho d d not know tlie delitor meant to remove ut the liino llio con- tract was made, liui tho writ mu.^t bo mado returnable to tho term next bucceoaing tho roatiirity of tlyj mto, etc. Cnltedion of Debts. — A writ of attachment will ifi«ue at the Instance of the creditor wherever residing, against a debtor when ho is a nnii-rcsident — or against u citizen who 1ms been absent more limn a y< ar and aday ; or when he ab.scoiul.'i or is removing out of the county; or con- ci*als hlinijelf no that tho orUinary procegs of law cuuuoi roach him. 3G3 TENNESSEE. Interest— The lepnl mlo of Inlcrc8t fn Toniiceaoo Is nix per rent., and uo higher rate can b» recovcroil 111 law. Conlractsat agieattT rato gf Inlcrcst arc vo J as l> the excess, and lliu IodUit i.s liablo to a Uiio of $10 to (1.000. Jtnalttjfor Violation of the. Uaury haws.— I. \a\>\r> to nn Indli'tment for mlsdemcnnnr. II convicteil lo bo lineil :i mim not less than iho wliolo iiHiirinu.s inloroBl tal^n ami ivceivod, anil no lliio iu he I0-8 than ten dollara The borrower uiiil his Judgment rr diturs inuy oltio, ul any timo williiu Kj.x yenr.s bfli'r Ui, .lanii.ny 1, or on any day appointed by the (Jovernur i.s a' day of Thauksgiviiig, or a.sa public holidaj, shall ho payable the lUy preccdinR either of tlio.'.o ideiit debtor.-i, liavi.g any real or personal property iu lUo btalo, it Is required, lu order to ubluiu un utlacU- tuent, lo hie a bill iu chancery . TEXAS. Interests — On nil written contracts opccrtaining the sums due, when no rats of IntercHt I1 oxpretecd, interest may bo recovered at thoTato of eight per cent, per aiiuuin. Tho parties to any written coutrucl may xtipulato f.iruiiy rule of lut rest, not cxcecdlns twclvo per cent, per annum. Judgincul!) bcarcighl per cent. Intere.'-t, except wheroJhey are recovered 011 a contract iu writing which stipulated for mure, uol exceeding t«velv^ iu which cuiw they bear tho mlo eoiitrucled for No Intercal on acccunls, unless there bo an express contract ; but only eight per cent, can be rccovereil on a verbal contract. .• Contracts to i>ay Intire.sl on account will nol li' presumed from previous course of dealln.':. I'enalty/or i'wlation of the Vtunj /,au)j.— lorieiiuio of all llii- Intere.-^t paid or cliargecl. Damages on Uills. — Au act giving damages upon protested ilrafis and bills of exchunjjo drawn upon persous living out of the limiisof the Male, pa.ssed December, IS.'il. ^KCTlo.N 1. Jie it enacted hy the Lrgitlaluie 0/ tlif. Slate of Texas, That the hoMcr ofany protested diaft or bill of exchange, drawn within iho limiisof this JSiaie, upon any per.'ioii or persons living beyond tho llnilia of this Mat', Khali, after having fixed llin liabiliiy of th<> drawer or endorser ofany such draft or bill of exchange, as provideil for in tho act of March 20, 1848, bo entitled to recover and receive 10 per cent, on the amouiit of kucIi dralt or bill, tts dumaijeM, together with liitereHt and co.st of ttuit Ibereon accruing. J'lueidrd, that iho pro- vii^ioiis of this ucl shall nol be bo coniilrueil as to embiuce drafts drawn by persons other than mcrehunts U|it exceed $100, exiliisivuoflnlere.sl, by iii.«liliiliiig sua before Justice of tho P.'nco, within sixty dayH) afler iho right of luiioii acerues ; or by lusliluting suit bcfuro tho sec >nd term of said couil, and sliowiuggood causu why the sulPwas nol instituted before tho first term. Thodruwor of any bill of exchange tMilcIi shall not he accepted when presented forac«rpl- auco, thall bo iinmeUiately liablo lo- tlu' paxnient tliereol. Collection of Debts. — Original altachmouls aio i.ssued ajain.''t Iho prolior'y ofa debtor when ho is nol to bo found In the county; and tho properly allaclied hliaH rci.iaiu in ciiMoilv ni..il ilnal Judgment. Atlacliineiil will also I4U wlieii llio deleiidaiit i.s a mm resident ; or when a rceldcut is about lo rumovo out of tho btalo, and whuthor the Uobl bo matured or not. VERMONT. Interest.— XUo IcRal rnto of Interest In Vermont Is six per d nt , and no higher rate cf In- terest Is allowed on Hyocial contracts, except upon railroad notes or boud.s, wuich may be:ir seven per cent I'enally Jor Violation of the Usury Iauvs.—'IUo excels ol Intvmtl rucolvod beyond six per cent, may Imj recovered by action of aB.sumpslt. Damages on Hills of Kxchnnge.—'lUQto Is 110 Statute In force lu Venuonl In rorercnco Ic ilaiuuiitis ou protested billi ot uxclmuiic. ^%. 364 Foreign I}ilU.~T\\o commenced by trusleo process. VIRGINIA. Interest.— ViiCi legal rate of interest in Virginia is six per cint, and no higher rate Is allowed cu bpoclul contracts. renalltjfor Violation of the Umry Lawt. — All contracts for a greater rale of Interest than six per cent, per annum are void. Damagct on Dills. — The damages on bills of exchange negotiated in Virginia, payable ia other States, and returned under protest, arc uniformly 3 per cent. Foreign liillt.—Wxa damages on foreign bills of exchange, rotumQd under protest, are unil'i -mly, 10 per cent. Sight Billi.—Gnco is not allowed by statute or by usage on bills, etc., payable at sight. Collection of DMs. — The jiropcrty v( the def-.^dant, if a non-resident, or a resident who la abcut to romoVo himself or eilccts from the Hir.' ' j liable to attachment. Au alUichmcnt in £uch au»a will bold bol'ore the claim is due RUd . .. able. WISCONSIN. After Januar>', 1863, lY ■ ' gal rate of interest, by an act of the legislature, is seven per cent. An usurious coutrui.. is void, and the party loanmg the money ia liable to a penalty of Ibrco times the usury in addition. Venallfifor Violation of the Usury /y(ii».».— Whenever any person shall apply to any court In this ij.tato to bo relieved in ca.so of a usurious contract or serurity, or when any person nhnll set up tho plea* of usury in any action or suit iii-stltuled agaiust him, such person, to be en- tiilcdto such relief or tho bencllt (if puch plea, i-hall prove a tender of tho principal sum of money or tiling loaned, to the party eutiilud to recuivo the fcanie. Act March 29, 18i)6. Damages on Bills of Kxchapge. — The ('aninfrcs on bills ot exchange, drawn <>? .:^dor."ed in Wisconsin, payable in nthi^r of the tftate.i adjolui.ig that Stale, and protested for noi, iccept- nuco or non-payment, are 5 | cr cent. Ifdrawnupona j)cison,or body politic or cnri)ornte, within either of the United irtates, and not adjoining to that Stale, the damages arc 10 jier cent. Foreign Bills. — The damages en bills of exchange, dmwn or endorsed in Wisconiin, payable beyond tlio limits of the Vniled .Stale.i, and prnteslod lor non-acoeptanco or non-p.-iyment, are [R. S. 1^40, p. 'IJi], 6 per cent, together with tho current lato of exchange at the time of demand. fiigld Bills.— On nil bills of exchanfo, payable at siRht, or at a future day certain, grace shall be allowed [it. S. Ib49, p. 26U], but not tjii bills of exchange or notes payable on demand. Cftll'-clion of Di'i'ts. — An alticliment will hold against the property of a debtor when ho has al'sconded, or is about lo al)i-cond 1mm tho Stale; or has fraudiilcnllv assigned, disposed of or ro.icoaled his ed'ects; or remnved his property from the Slate; or when the defendant is a n in- resident or a foreign corporatiou. UPPER AND LOWER CANADA. Interest. — .«^ix per cent, is tho legal rate of interest, but any rite agreed upon can be recov- ered. .Iiidginenls boar kIx per centum per annum interest from llio date ort'iiliy. Hanltsaro not allowe I a lii(;lier rate than sevrn per cent. Corporations and associations authorized by liiw lo borrow and lend iiioiicy, unless specially allowed l)y wmio Act of I'arliamoiit, arc prohi- Llied IVom taking a lilglier lal ' ol inlorest than six per cent. In.siirauco Companies, however, life authorized lo take eiglit per cent. 7/1.7.? of Kxthan'je and Vroniiunnj Notts. — Three days of grace are allowed on nil bills and notes i);iyal)le vvuiiin I ppor or Lower Caimdii, except wlien drawn mi ileinand. When tli(> lust day of grace fills on Sunday, or a legal liollday, it Is payable Uio fnllowliig day. Ai'coptnnces imifl bo ill wriilng. No person or corpomllon in l"ppi>r Canada can Issno noios lor le.« than one dollar. Protest may bo made, mid the partloH lo the bill i.r nolo notilled on llie sjinio day the bill or nolo is dislionourod ; but, in ciso ol' nonpayment in I'lipor Caimda, not Ix-foro llireo o'clock, p.m., and in Uiwer Canada any time alter tho Ibrenonn of tho la.- 1 •lay (d (.mice. Diiilionourcd inland liills or notes in Cpper l^anada, when protested, and in Lower Canada willioiit prolest, bear Interest at the rate of .six per cent, from date of i.rotesi, or In Lower Canada torn maturity to time of | uymenl ; but 11' Interest is expressed lo lie payable from a particuar period, then from the time of micIi period to the time of payment, flio daniagea uliowwl upon protested foieigu billB drawn, sold or negotlatei'. within I'pper or Lower Canada, % 865 9 flamagps on pro- sight, or on bills attics or ^stato of iturity (,f a claim, no State, may l» cr rate la allowed of interest than 5inia, payable !a dcr protest, are bio nt sight, rosiilont who fa u atuichinwit in iro, is sown prr to a penally y to nnv roiirt »"y iiorson fIihII ''"■soil, to bo 011- rincliml gum of 20, 18,)6. f'-' .ndorfcil in for nou iccopt- > United States, soniin, payable iiwyiiiunt, nro il tho tlnio of CPrtain, pnco lo OH (loniaiul. r whon lio has illsposotl of or Juiit laamu- rnn bo rorov- '■ HiuikBiiro uthoiizL'd liy It, arc piohi- ios, huwevor, nil 1)111,0 nnd 1 lii'ii ilm last Arcppiaiirrs I'lr IcsH tlimi ic wiino (lay not l)(-fi)rn ".V (if pure, rtfr CaniKlft )r 111 l.iiHor iibli) li-om a 10 In Lower Caiiad i on persons in Upper Caniidii, or If i^ iwn in cither Upper or LowcrCanadn on nny person in iiiiy (iilicr of iho Uriiish Nnrth Aniei .n colonics or Iniled .f lntore.st on tho nnmnnt for which tho bill wa-s drawn, to boreokoncd from tho date of ^Mle.-^t to day of repayment, touethorwith tho current rate of exchange of tho day wlioii lopayniciit is demanded, and tho e.tpcuses of noting and proti-sting Iho liill, I'louii-^si'iy iiotefl marto in I'piier Cuiiadu, payable In tho I nited States of America or British Nortli .American Coloiiie.-i, not being Canada, and not otherwise or elsetuhere, and pro- tested, in iuldition ti) tho principal sum, aro liable to damages at tho rate of four per cent, on Biich priiic i)alFiim, and interest at tho rate of six per centum jier annum, to bo reckoned from tho day of protest to tho day of repayment, together with Iho current rate of oxchango of the day when lepiynicnt i.s demanded and tho expenses of protesting tho note. Tho Statute of Limitaliniis liarH the right of action on bill.s of exchange and promissory notes, in Upper Canada lu si.\- ycar.^, ami In Lower Cauada In llvo year.-'. CclUclinn fi/Pf.bl.i. — f)ebL'J may bo recovered In Lower Canada by actions at law, and in Upper Canada by ndiiiti-int law or suits in equity. Debtors may bo arrested and held to bull in I pper Canaila upon a i nllldavlt of tlie creditor, or of some other indiuidual, showing that ho ha.-< a causo cfai'ti'in to tliciiinounlof $100, orui)wards.nnd has sull'ered damages to that amount, and shows fai't.i aid clrciim-taiH'os tosatisfy tho .judge that th' ro i.s good imdjmjbable cause for believing thai sui h pirsdii nnlefsbc is forlhwilh apprehended, i.Habout to rpiitCanada with Intent to de- frau I bis ( rodinir.i gcnenilly or tho deponent iu iiarlicu'ar. In Lower Canada debturs may be ar- I-i'Stodaiid held to bail iipo|i'allldavil of tho riaintilT, his bookkcoper, clerk, or legal attorney, that llio Defendant l.-i persoiiiilly Indebted to the rialntillfor a sum amoimting to or exceeding S-l'J, uiid tl'at depooi nl bclb-ves, upon gniinds set Ibrth in allldavit, that Doleadant is imiU' di.itely nbo It to leave Iho Province with Intent to defrand his creditors generally, or tho riainlilf iu jiarticular, a' d tliat such doiiarturo would deprivo t!io FlnintiH' ol his remedy ngaiiiM tho l)cl'endaut, (i !iat tho Defondapt has secreted or is about to seorcto his properly with .'^urb Intent. A icMdent of Upper Caliaila, cannot In Lower Canada, bo arrested at Iho suit t f anv person resi ling In Upper Camd.i, unle.po and Great Uritain Irnvj adopted it. But tlia rental system is much preferable to the bushel ir -thod and, no doubt, will soon become universal. .Until it does, a little dilllculty will !«> cxiJeriencetl Jn buyinj? and sclliu;,', as ivery variation in the price iared and given, ss below. • To find the value of grain, vegetables, *c., per cental, when the valu(» jxjr bushel is given- ifvUipfy the given price ptr tnuhel hy 100, and divide bfi the numbti- of Ibt. i» the bushel. ExAXPu:. —For any article weiRliing 00 lbs. to the bushel. Say wheat is quoted^t 32. 10 per bushel, what is the value per cental? Ka of Ibo. in bush. . . 00^#210.00— prod, after multiplying by IOC $3.50 — value per cental. ExAHFtt— For any article weighing 66 lbs. to the bushel. Com is quoted at 60c pet l>|l>a]iel, what is the value per cental? No. of lbs. in bush. . . M,' ?60.00— prod, after multiplying by 100. $1.0714— value per cental. To flnd the vohie per btuhelwhcn tlie price per cental is given, we urerse tlie abov nie, and MtUtiplfi by the number of pounds in the hushtl, and dir>ide fni 100. If wheat is worth $3.S5 i' mlnent articles of merchandise. If tlic value i^t ccntiil shmild l>e re<>r bushel not given in the tables, the de»iroU result moy Iw found by nddiiiR the two nearest quoted rates together, and dividing by 'J. For instawe — the price of wheat j>er cental is re- quired, when the quoted price per bushel is ♦l.ilS. Now, we see by the table t'"it nt $1 (H per bushel it would be 1*2. 73J per cental, and nt i»1.06 it would be$'2.70i Then 8i.73i added to $3.76}, equal to $5.50 divided by 2, equal to $-.'.75 per cental at $1.05 per bushel. POn GRAIM, &c.. WKIOniNG M I.tti. TO THE nU.SHEL. It Per Per Per Per Per Per Per Per Per , Per bdd. cental. bthl. cental. Mhl. cental. ^la. 7.1 cental. . $3 38 1-8 bshl. 9U SO cental. 35 $0 78 l-H 41 $1 38 1-8 67 $1 78 1-8 $2 78 1-8 'M 81 1-4 43 1 31 1-4 68 1 81 1-4 74 3 31 1-4 00 2 PI 1-4 37 84 ,1-8 43 1 34 SS 69 1 84 3-8 <*F 3 94 ?■■$ 91 . 2 84 3..S as 8T 1-2 44 1 37 1-2 00 1 87 1-2 7<5 2 37 1-2 W ' 2 87 1-3 39 »0 6-8 45 1 40 &-« «l 1 90 6-8 77 2 40 1. « 93 3 90 6-3 80 !>S 8-4 40 1 43 3-4 «ii 1 03 3-4 78 2 43 :i-4 94 2 ffii 3-4 Si t>0 90 .■' 00 S3 1 03 1-8 40 1 «3 1-8 *5 •J 03 1-8 81 2 5.1 1-8 97 3 03 1-8 84 1 00 1-4 50 1 50 1-4 60 2 Oti 1-4 63 2 50 1-4 98 3 00 1 4 U 1 09 3-3 M 1 6« 3-8 67 2 00 3-8 83 i 59 3-8 »9 3 09 3-8 M 1 IS 1-3 63 1 03 1-3 08 2 13 1-3 S4 2 02 1-8 1 00 3 IS 1-3 07 1 IS 64 63 1 05 5-8 09 2 15 5-8 85 8 05 5.8 3$ 1 18 8-4 64 1 08 8-4 70 2 18 3-4 80 3 08 ^M S» 1 SI 7-8 66 I 71 7-8 71 2 I'l 7r» 87 3 71 7-8 4« 1 S5 60 1 75 72 2 25 ' 88 2 76 !1M. >te rcer ceutnl is re- 3 table t'"»t at $1 fA k. Then 82. 73i added per buHhct. SHEL. ! Pit Per . , bshl. cental. 8 10 80 «2 78 1-8 * 00 2 PI 1-4 8 91 . 2 84 :5.,-) 2 W ' 2 87 1-3 ■< ftl ^ 2 90 6-S • W , 2 \v.i 3.4 i vr- 2 OU T-8 00 3 00 1 »T :i 11.1 1-8 08 3 00 1 4 00 3 00 3-8 1 00 8 13 1.3