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WII.I.IAM STRKhHS. 1869. /, )\\ \ 1/ ' i 11 I i: r:h/ in:i/ .w ^^-^'' ^^ \ ! 1 ^ 4 I ! 1 ! ■ ■ t / 1 t'i i' > y,^\ .UjO') i» ' ! /. i / 1/ ./ I )'l • ' ' ' ./ H I \ 1 PREFACE. Tii()U(Hi elementary works on Arithmetic arc in abundance, yet it seems dcsirablo tliut tlicrc should be added to this an extensive treatise on tlio commercial rules, and commercial laws and usages. It is not enough that the school-boy should bo provided with s course suited to his age. There must be supplied to him sometliinj.' Iiiglier as he advances in ;ige and progress, and ncars 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 elen)ents up to the highest rules required by those 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 AritLmctic 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 cflbrt has been made to render the business part so copious and practical as to afibrd the young student ample information and discipline in all the principles and usages of commercial intercourse. For the same reason some articles on Commercial LiiW have been introduced, as it was a prominent part of the Author's aim to produce a work which should be found useful, not only in the class-room, and the learner's study, but also on the merchant's table, and the accountant's desk. The Author begs to tender his best thanks to J. Smith Ilomans, Esq., New York City, Editor and Proprietor of the '•'Canker's Magazine and Statistical- Register," for the able manner in which he supplied this part of the work. Throughout the work particular care has been taken not to enunciate any rule without explaining the reason of the operation, for, without a knowledge of the principle, the operator is a mere calculating machine that can work but a certain round, and is almost sure to be at fault when any novel case arises. The explanations i'. i IV. ITEFACi:. are, of course, more or less Iho result of roailinj', but, ncvortlieloss. ^licy ar(! iiiaiiily iliTived from j)cr,sonal study aiil oxporiouco in toacliiiiLT. TIio f;rcat mass of the exercises an; likewise entirely new. tliouuh IJK! Author lias not f-cruiik-il Id make selections iVom s^nie ol the inosL apjirovoil works on iho .suKjocl; hut in Join,:,' so, he lias eitnlined liinisell' almost entirely to sueli (lueslions as are to lie i'ound ill nc;irly all popular hooks, and which, therefore, are to he looked upon as I he common )>roperfy of science. Aluehraie lorm.s have been avoided as iimcli as pf)ssihlo, as beini* unsuited to a lari,'e i)roporlion of those for wlioia tlii> hook is in- I ended, and to many altoirctlier nninfelli;j;il)le, and licsiiles, tho.so who understand Ali^'chraic modes will have all tiie le-s diiricrilty in unJersLandiiijj the Arithmetical ones. Even in tin; more purely mathematieal parts tiio subject has been popularized as much a3 possible, 111 arraniring the subjects it was necessary to lollow a certain hii^ical order, but the intellii^ent teacher and learner will often lind it necessary to depart from that order. ''See suir^'cstions to teachers.) l^iVery one will admit that rul I definitions should be ex- pressed in the .smallest po.ssible ..jor of words^ consistent with per.spicuity and accuracy. Great pains have been taken to carry out this principle in every case. Indeed, it might be desirable, if practicable, not to enunciate any rules, ))ut .simply to illustrate each ca.so by a few examples, and leave the learner to take the prin- ciple into his mind, as his rule, without the encumbrance of words. Copious exercises arc appended to each rule, and especially to the most important, such as Fractions, Analysis, l*erccnta'::o, with its applications, kQ. Besides these, there liave been introduced extensive collections of mixed exercises throutrhout the bodv of the work, besides a large number at the end. The utilit}' ol" such auisceUancous questions will be readily admitted l)y all, but the o'cason why they arc of so much importance seems strangely over- looked or misunderstood even by writers on the subject. They arc spoken of as review exercises, but their great value depends on some- thing still more important. An illustration will best serve here. A class is working questions on a certain rule, and each member of the class has just heard the rule enunciated and exidained, and therefore readily applies it. 80 far one important object is attained, Tiz., freedom of operation. But something mor ncccs ry. hav bcai bcei ruKF.vri:. V. hnbcr and Jiincd, The Icarnor nui.st he t;ui;;iit to ilisroni f/iat ndr is to ir. nppllal fur tlic solution of each (jnostiim proposoil. Tho pupil, tiiulor careful teach iiiLi'. may ho ahlr to i!iiilcr>tiiinl I'uUy every rule, ami never con fouml any one with any otlu-r. ami yet ho douhtl'ul \vli:it rule is tc bo iip[ilie')iJf ofnppfi/iny those rules; or, in utiier words, to practice the jiupil in pisrceiviu^' of what rule any proposed (juchiion \^ a particular case. Great importanco should be attached to this hy (he practical educator, not oidy as reirards readi- ness in real business, but. also iis a mental exercise to the )oung (student. The Author is far from supposing', much less assertimr, that the work is complete, especially as tlic whole has be'cn prepared in less than the short space of .six months. It is presented, however, to tho public in tiie conlideiit expectation that it will meet, in a f^roat Jc^•rco at least, the necessities of tho times. With thi^ view, there are given extensive collections of examples and exerci.scs, involving money in dollars and cents, with, however, a number in pounds, (•hillings and pence, sufficient for tho purpose of illustration. This seems necessary, as many must have mercantile transactions with Britain and British America. The Rule for finding tho Greatest Common Measure, though Dot new, is given in a now, and it is hoped, a concise and convenient form of operation, The llulc for finding tho Cube Root is a modification of that given by Dr. Hinds, and will bo 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 Ti'ractions, Multiplication and Division present much less difliculty than Addition and Subtraction ; and, secondly, as in "Whole Numbers Addition is tho 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 been omitted, as that mode of calculation is now virtually superseded VI. Tiir.rArn. ;»y til, it dl' I), rliiiuls. IJartiT, ton. has licon passed liy, as fiiicstiona (if tli;it cliiss can easily )).• si>lvcil l»y tin; lliili.! ol' Proportion, which lias lrablc loniitli. and it is hoped that thoiK'lit (if those ^^ockiii^ !i liljoral and jinctioal couuncrcial tducatidii. As ill all ])ranchcs, so in AriUunotl*', it is of tlio utmost cnnfc- fjuonco to (lij:oHt the rules of the art thoroughly, and store thoni in tlu! memory, to bi; rciroduced \\hvn ro{|uircd, and api)lied with accuraey. lUit this is not cnou;^'h ; soinothiiijj; more is needed by the Student. To bo ar. eminent accountant he nmst acquire rapidity of operation. Accuracy, it is true, .should be attained first, csppcially as it is the direct means of arriving at readiness and rapidity. Accu- racy may be called the foundation, readiness and rapidity the two winL's of the superstructure. Either of these acquirements is indeed .valuable in itself, but it is the condunation of them that constitutes real cffoctivc skill, and makes the possessor relied upon, and looked up to in mercantile circles. Some one may ask, " IIow arc these to be acquired? " The answer i.s as simple as it is undeniably true; on?// hi/ extensive practice, not in the countin!:;-houso or warehouse, indeed, though these will improve and mature them, but in the school and college, so that you may talce them with you to tlic busi- ness office when you go to your first day's duty. Go prcjnircd is a maxim that all intelligent business men will afiirm. Be so prepared that you will not keep your customers waiting restles-sly in your office or warehouse while you arc puzzling through the account you arc 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 rr.pidity, as no- ticed in the note at foot of page 18, is not to use the tongue in calcu- lating but the e)je and the mind. Nor should the course of self-discipline cud 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, &o., &c. X. SUCIGESTION'S TO COM.AIERCIAL STUDENTS. Study rvrn jioJitics, not for tlioir own sake bxit on account of the manner in wliicli ilioy affect tratlo and cnninicree. Do not, except in tlic case of sonic serious ditliculty, indulge in tlic indolent iiabit of askin.i:^ your teacher or fellow student to work the question for you ; Avork i*^ out yourself — rely upon your self, and aim iit the freedom and corrcctnes;'- which will give you confidence in yourself, or rather in your powers and acquirements. Another cau- tion will not be out cf place. IMr.ny students follow the practice of keeping the text book beside them to see what the answer is ; this has the same cffeot as a leading question in an examination, being a g'uido to the vioJc by seeing the irsuU. Study and use tlic mode tc come at the result ; gain that knowledge of principles and correctness of operation that will inspire the confidence that your answer is cor- rect without knowing what answer the text book or the teacher may assign to it. There are two things of such constant occurrence and requiring such extreme accuracy that they must be specially mentioned, — thej are the addition of money columns and the making of Bills of Par- cels. Too much care and practice can scarcely be bestowed on these. i> ■ 'r i I TABLE OF CONTENTS. AcfiMiijts anil onvoices. ....' . 1J2 A(Mi.ion 17 Alligation 232 Analysis Ill Annuitii's 2S() Averaging of Accounts 2'J;> Average, General I'JG Axioms 17 IJanking 10" Bankruptcy 211 Dook-keeping Exercises 299 Brokerage. 17;' Commercial Paper 149 Commission , 172 Coins, Foreign 326 Currency of Canada 240 Custom House liusineaa 199 Decimal Coinage. . 32 Denominate Numbers 45 Diijcount 1C5 Division, Simple 20 Eqiiation of Payments 21G Exercises— Set I 55 Set II 107 Set III :U8 Exchange . . . . , 241 Arbitration of 251 American 242 Britain 247 Evolution 258 I'ofi'ign Moneys, Table 250 Fractions 58 Common C i Decimal 71 Greatest Common Measure 52 Insurance 179 itileresi i;}! Simple 1 ;]5 Compound 102 luiroduclion 13 Xll. IMDEX. I I;iv!. For this purpose every .such line is divided into sets or lots of three ri::ures eneh, counting iVoni ri^ht to left, and each set is called :i j criod. — thus, 8SSSSS8S8 forms three periods by marking,' the figures in threes from right t(i left by a character of the same form as tlie conniia in composition,— tlius, SSS,SS8,S88. The first period is called the periiid of units, the second the period of thousands, tlic third the period of millions, and so on, — billions, trillions, (juadriilions, ka., &c.. to any required extent, which seldom exceeds millions. The lirst figure of each period denotes units"'- of that period, thij second tens, and the third hundreds of that jicricd. Thus, in the example given above, the fir. i figure denotes, eight units in the period of units, or eight ones, or, as it is usually read, simply eight ; so, also, the fourth denotes eight units in t!ic [)eriod of thou.sands ; or eiglit times one thousand, or eight thou-^and^; the seventh figure again denotes eight units in the period of miilious, or eight times one millinn, or eight miliioiis ; again, the second, fifth, and eighth figures denote tens in the period of units, thousands and millions, respectively; lastly', the third, a'lxih and ninth figures denote liundreds in the periods of units, thousands and miliions, respectively, Such a line, then, as 888,888,888 is read eight hun- dred and eigety-eight millions, eight hundred and eigh^y-^ight tliousands, eight hundred and cighty-eiglit. Every period but the last must h;ivo 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 tens, but no hundreds, and therefore is read fort^y-three millions, two hiindred and seventy-nine thousands, eight hundred and sixty-uve. RULE l-'OIl NUMEltATIO-N'. ^- , Beginning at tlie right, count off periods of three digits each till not more than three are left ; then read o'lf each period from left to * It is somewhat awkward that the term ijinits is used for two pin-poses. viz. : as the name of the flrst period and also fs tlie name of the llrst ligiu'c ol each period. Though we cannot Avell changt^; what usage has so long estab- lished, yet the teacher may obviate the diffioilty by varying the expression occasionally, if not habitually, saying, E. G.., units in the unilt/ period, or the place of uuiis in the units neriod. / NOTATION. 15 ■'I right by naminu' as many huiiJrcfls, tons ainl uiiitH as each contains, and adding at the end of each period its iiniprr nai:;c. Tlio name of the unity period is usually omitted. V\'lKn a oiplior occurs no mention is made of lliat place in the i)C'iiod, Imt the cipher is counted as a dipit ; thus, in the Hue .'IiJO.TOS.ODl each cipher is counted a digit, hut the reading is three hundred and si::iy millions, seven hundred and eight thousands and ninet} -one. K X E u c r S E s Divide into periods and read the following lines : 1.— :iSG7293ll ::.— U7(;S52734 H.— 2178427:185 4.— 92S7'j;]."j748.5 .",. — i(.;3:s7oyi20 (;.— Ill 11111 iiniiiu 7.— 2S2282822B2SS 8.— 10U04870 !).— lOlOlOluKdOl lun- iight tlie y.ich, tens, two iive. !U till il to •poses. rm-C ot .'Stiib- rossion or llio N O T A T I O N . 3 — N'oTATlON is the mode of expressing any quantity or mag- nitude h}"^ the combination of conventional syndiols or characters. Thus, by the lloman notation, the letter I. stands for unr, IT. for tiro, X. for ten, &c. ; thus, XII. stands for one ten and tiro units. IJy the Arabic notation, any digit standing alone, as 5 in the margin, denotes simpl}' five units, but if another digit (5) be placed to the rii':t of it, llien the new 5 denotes units and the other 5 becomes tens, so that appending a second digit makes the first one ten times its oriy inserting a zero mark between the figures I throw the 2 into the place of hundreds, and :v'20lJ represents correctly that I have firo one-hundred dollar bills, and .s/,c one-dollar bil's, but no ten-doll;;r bills. The superiority of this sini])le system over the cuniln'ous llonian one will bo manifest from it.s simplicity and brevity by writing cicjht-ij-eijkt according to both system-s — 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 wnned 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 counna to the right, and do the same for every period down to units, inclusive. For example, teacher says : " Write down one hundred and six niillioas; " pupil writes lOG and pauses ; teacher adds, ''ninety thousan,!;' pupil fills up thus : 100,000, and pauses; teacher concludes: " anc eighteen;" pupil completes 100,090,018. If the teacher should say sixteen millions and the })upil write OIG, the cipher woulc bo manifestly superfluous, as it has no etlbct on figures phiccd to thf right of it, but only on those placed to the left. EXERCISES. Write in figures and read the following quantities : 1. ! on millions, seven thousand and eleven. 2. Ninety billions, seven thousand and ten. iJ. Eighteen millions, sixty thousand and nine hundred. 4. Forty thousand and nine hundred. a I al &( in th ca ADDITION. 17 5. Ei^hty-scvcn millions ;mi] one. G. Ninety thousand, sovon linndrod and ciirht. 7. Eleven millions, ei^ht hundred tliousund and twentv-lbur. 8. Six hundred and seven thousand and ninety-seven. 0. Eight hundred and seventy billions, sixty thousand and eighteen. 10. Eleven billions, cloven millions, eleven thousand and eleven. ?t as if then, ri[;;ht, For lions ; ^ana; '' auL should ■\voulc to tlu AXIOMS. 4. — Axioms used in the sequel : I. Things that arc equal to the same thing, or to equals, arc equal. TI. If equals be added to equals, the wholes are equal. Corollary. — It' equals be multiplied by the same', the product'^ arc cquah III. If equals be taken from equals, the remainders are equal. Cor. — If equals be divided by the same, the quotients are equal. IV. The whole is greater than its part Cor. — The whole is equal to all itf. parts takca together. V. Magnitudes which coincide, or occupy the same or equal spaces, ai\: ( '^ual. 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. — Addition is the mode of combining two or more numbers into one. The operation dope ids on axiom II. The result is called the sura. Thus: $8+$94 $6=:$23. The sign plus (+) indi- cates addition. To illustrate the operation, lot it be required to find the sum of the five numbers of dollars jioted in the margin. First, the numbers are placed so that those of the same iiamc arc in vertical columns, i. e., units under units, tens under tens, &c. Next, we find that the sum of the units' column is (Ax. IV., Cor. ) 27, i. e.. two tens and seven units. Next, we find that the sum of the tens' column is 35, but, as it is the tens $2871^54 758287 612873 4947G8 83G195 ^-i 18 ARITHMETIC. ¥ I 27 350 2400 27000 260000 2700000 $2989777 column, we write (Art. ',]) 350 ; in the same man- ner we find the sum of the hundreds' column to be 2400 ; tlie sums of the others will be seen by inspection. Having thus obtained tlic sum of each column, each being summed as if units, but placed in succession towards the left (by Arts. 2 and 5), wc 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 wo find the sura to be 27, L c, 7 units and 2 tens, wc write down the 7 units under the units' column, and add up (Art. 3) the 2 tens with the tens' column, and we find the sum is 35 tens, i. c., 5 tens and 3 hundreds, and we place the 5 tejis under the tens' column, and a<\d up the 3 hundreds with tlie hundreds column, and so on. The transferring of the tens, obtained by adding the units' column to the teiis' column, and the hundreds obtained by adding the tens' column to the hundreds' column, &c., &c., is called carrying. In all such operations the learner should carefully bear in mind the principle explained in Art. 3., that every figure to the left is ten times the value that it would have if one place farther to the right.* EXERCISES. Find the sums of the following quantities : $287054 758287 G12873 494768 836195 $2989777 ^^l (2) • 99876 (4) 895763 49176 987654231 63879 89765324 283527 123456789 54387 42356798 659845 908760504 789 56798423 7984 890705063 137568 23567989 31659 759086391 278652 79842356 968438 670998767 85945 65324897 2896392 4340661745 721096 357655787 * We would strongly recommend every one who wishes to become an expert accountant, to avoid the common practice of drawling up a column of 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 16 : IG and 7 are 23 ; 23 and i are 27 ; but run up your colamn thus : 5, 13, 16, 23, 1 ADDITION. 1 (5) (6) (7) (8) • 738 659 471 78563 897 47986 12345 658 5798 G7890 918273 856 19843 98765 651928 789 56479 43219 ■ 374859 978 28795 87654 263748 654 897 32169 597485 999 1984 78912 986879 888 68195 6543: 98765 777 3879 98765 9876 666 G98 43288 987 555 5879 77877 456879 897 17985 98989 345678 978 19 336981 805312 4705357 12460 (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 63 475 365 67 75 394 113 98 49 89^ 279 149 76 157 67 67 54 638 76 54 78 594 84 72 69 78^ 1379 298 2744 37 1044 114 5159 19715 27, for that is the mode to secure both rapidity and accuracy. The same remark will apply equally to multiplication, and therefore to every arithme- t'Cal 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 tediousaess of saying 6 times 7 is 42 — 2 and carry 4 — C times 'J is 54, and 4 is 58 — 8 and carry 5 — G times 4 is 24, and 5 is 29 ; but practice the eye, aided by the memory, to talvo in at a glance 6 limes 7 is 42, i tons, and we aro iKnv rcfjuircd to take 7 tens from 2 tons ; to ilo tliis we liavc rceoiirse to the hixino, iirtilicc, by calling; one of llie luindrods tcim, wliich j-ivcs 10 tens and 2 tens, and 8«. on to the end, the lust 3 neeossarily becoming; 2. AVo can now subtract 7 froui 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 maru;in. This artifice is popularly called borrowing. In practice the resolution can be cifectcd mentally as wo proceed, and as each figure from which wo borrow is diminiriued by unity, it is usual to count it as it stands, and to compensate for this to increase the one below it by one, for, Jia in the example, 7 from 12 is the same as 8 from 13, and 2 from 3 is the same as 1 from 2. We are now prepared to answer the proposed question, as annexed, and we say 9 from 8, we cannot, and there are no tens to borrow from, we therefore take one of the hundreds and call it 10 tens, and one of the tens and call it 10 units, \\ ich with 8 units makes 18 units, and we take 9 from 18 and 9 remain. We have now only 9 tens left, but we reckon them as ten, and to compensate for the surplus ten, we reckon the 1 below as 2, and say 2 from 10 and 8 remain. We pioceed thus to the end, and find the whole remainder to be $165778889. $513074208 $347895319 18165778889 EXERCISES. REMAINDERS. 1.— From 847639021 take 476584359=371054062. 2. " 1010305061 " 670685093=339619968. 3. " 59638743 " 18796854= 40841889. 4. " 7813J57 " 3745679= 4007578. 5. '• 111111111 " 98657293= 12453818. In Subtraction, as in Addition, we have no method of proof that arrives at positive certainty, but either of the two following methods may be generally relied upon. 1. — Add the remainder and subtrahend, and if the sum is equal to the minuend, it is to be presumed that the work is correct. 2. — Subtract the remainder from the minuend, and if this second remainder is the same as the subtrahend, the work may be accounted correct. AIUTIIMETIC. W M MULTIPLICATION. 7. — MULTIPLICATION Tuiiy 1)0 siiiiply (lefined by sayinj; tliat it is a Khort mcthotl of pcrforiniti;^ mlditioii, when ull tlic quantities to bo adtleil are the same or ecjual. Thu.s : O-l-G j-G-|-0-:-0-;-G|-U |-G, means that eij^ht sixes are to ba added together, or that six ia to bo repeated a.s often as there are units in eight, and wc say that 8 times Ct is 48, and writo it thus: 8x0—48. So also 8-r 8+8 [-8-(-8-l-8 gives 48. So that G.8::^8.G^^48, and hus wc can construct a multiplication table. The number to bo repeated is called the multiplicand, and the one that shows how often it is to be repeated is called the multiplier, and the result is called the product, or what is produced, and hence the multiplier and multi- plicand are also called the factors or makers, or producers, and the operation may be called finding a product when the factors are given. Hence also the mode of carrying is the same in multiplication as in addition. MULTIPLICATION TABLE. Twice 3 times 4 times 5 times 6 timoii 7 times 1 is 2 1 is 3 1 is 4 1 is 5 1 is 6 I is 7 2—4 2—6 2 — 8 2 — 10 2 — 12 2 — 14 3—6 3—9 3 — 12 3 — 15 3 — 18 3 — 21 4—8 4 — 12 4 — 16 4 — 20 4 — 24 4 — 28 6 — 10 5 — 15 5 — 20 5 — 25 5 — 30 5 — 35 6 — 12 6 — 18 6 — 24 6 — 30 6 — 36 6 — 42 7 — 14 7 — 21 < — 28 7 — 35 7 — 42 7 — 49 8 — 16 8 — 24 8 — 32 8 — 40 8 — 48 8 — 56 y — 18 9 — 27 9 — 36 9 — 45 9 — 54 9 — 63 10 — 20 10 — 30 10 — 40 10 - 50 10 — 60 10 — 70 11 — 22 11 — 33 11 — 44 11 — 55 11 — 66 11 — 77 12 — 24 12 — 36 12 — 48 12 — 60 12 — 7 2 12 — 84 8 times 9 times lot imes 1 L times 12 times 1 is 8 1 is 9 1 i 3 10 1 is 11 1 is 12 2 — 16 2 — 18 2 - - 20 2 — 22 2 — 24 3 — 24 3 — 27 3 - - 30 3 — 33 3 — 3G 4 — 32 4 — 36 4 - - 40 4 — 44 4 — 48 f) — 40 5 — 45 5 - - 50 5 — 55 5 — 60 6 — 48 6 — 54 6 - - 60 6 — 66 6 — 72 7 — 56 7 — 63 7 - - 70 7 — 77 7.— 84 8 — 64 8 — 72 8 - - 80 8 — 88 8 — 96 9 — 72 9 — 81 9 - - 90 9 — 99 9 —108 10 — 80 10 — 90 10 - -100 10 —110 10 —120 11 — 88 11 — 99 11 - -110 11 —121 11 —132 12 ~ 96 12 —108 12 - -120 12 —132 12 —144 MULTII'LICATIUN. 23 Rof»ariliii TcachcrH : the following l)art (tf tliia tahle, soo sufrj^cstions to 13 timoH 41 timort 1.) tilllOH l(j tiim'f< 17 tiinort IH tinios I'J tiUKM 2 is '2C> 2 h L'H 2 H 311 2 is 32 2 id 31 2 is 3i; 2 is 3,H 3 — 3!) 3 - 4'2 3 — 4;-) 3 ~ 4y 3 — f)l 3 r.i 3 - - rr, 4 — r)2 4 -- r»c 4 --- «50 4 — CI 4 — C8 4 -- 72 4 — 70 6 — f.:i :> — 7(1 f) — 7:) 5 --- 80 f) — 8.-) 5 — 90 r. — 95 G — 7H r, ■ HI - DO (; — \)(\ « -- 102 (1 — 108 (; - 114 7 — ni 7 — Of 7 — lO". 7 — 112 7 - 110 7 — 12(i 7 — 133 8 -- lOJ H - 112 H — 120 8 — 12^- H - i;{(; K — 114 H -- ir)2 9 — 117 :) - I'JG 9 — 13.") 9 — in !) -- ir)3 !) — 102 9 — 171 Wo lirtvo irj tho above tablo oorruclecl tliu gi-ons graiiimiitlciil bliiailer ho common ol'miying cijilit tiiiu's two auk Hixtoen. When uioro than two factors arc given, tlio operation is called continued multiplication, as 0x3X2 XiJ ISO. When the factors consist of more fij,'uros than one, the most convenient mode of operation is that 8hown by the annexed example, where the multiplicand i? first repeated 8 times, then GO times, or which is the same thing times when the first figure of the second •line is placed under the second figure of the first line, i. c. (art. 2,) in tho place of tens, and then tho partial products are added, which (Ax. IV. Cor.) gives the full product. Hence wo deduce the 345186 268 2761488 2070blG 690372 RULE FOR MULTIPLICATION. 92507848 Place the multiplier under tho multiplicand, units under units, tens under tens, &c., &c., — commencing at the right, multiply each figure of the multiplicand by each figure of the multiplier in succession, placing the results in parallel lines, and units, tens, &c., in vertical columns, — add all the lines, and the sum of all the partial products will (Ax. IV. Cor.) be the whole product required. As far as the learner has committed a multiplication table to me- mory, say to 12 times 12, the work can be done by a single operatic n. Wlicn any number is multiplied by itself, the product is called the square or second power of that number, and the product of three equal factors is called a cube or third power, the pro- duct of four equal factors the fourth power, &c., &c. The terms square and cube are derived from superficial and solid measurement. The annexed oquarc has each of its sides divided into 5 equal parts, and it will be found on inspection that the whole figure contains 5 5 24 AmXIIMETIC. III II ysii 25 (=::=r)X5) small squarea, all equal in area, and having all their sides equal. — Ucncc because 5X5 ''tipresents the whole area, 25 is called the square of 5, or the second power of 5, because it i.s the product of the two equal factors 5 and 5. A cube i^ a solid body^ the length, breadth and thickness of which arc 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 culled the cube or third power of 5. The terms square and cube are often used without any reference to superficial and solid measure. For example, in lineal measure an expression for distance in a straight line is often called the square and cube of a certain number, thus : 81 is called the square, and 729 the cube of 9, although these arc only used to show that the distance is not 9 in cither case, but in the one 9X9, and in the other 9X9X9. In such cases the terms second and third power are therefore to be preferred, and since na solid can have more than three dimensions, we have no term corres- ponding to square and cube for the product of four or more equal factors, and therefore we are obliged to use the words fourth power, fifth power, «Sic., &c. CONTRACTIONS AND PROOF. There are many cases in which multiplication may be performed by contracted methods, but the utility of these, for the purposes of accuracy, is, at least, doubtful. The most secure method in the great majority of cases, is to follow the general rule. Multiplication by 10, 100, &c., is effected at once by adding a cipher for ten, two for 100, &c., &.C. The following is, next to the above, the moot safe and useful contraction that can be adopted. It is exhibited in the subjoined examples, but purposely without explanation, as an cxer cisc for the learner's reflection : Ordinauv JlETnoi). 35397x17 17 Contracted llETiion, 35697X17 2-49879 Okdinauy JlETnOD. 35097X71 71 CONTRACTF.U METH0I» 35697X71 249879 249879 35697 006849 606849 35697 249879 2534487 2534487 The only practically useful proof of the correctness of the pro- duct, is the one subjoined, but even it, though it seldom fails, doen not secure positive certainty . MULlirLICATION. 25 Add together all the 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 are equal, tlic work may be accounted as correct, but if they arc not equal, the work must be wrong. The reason of this proof depends on the property of the number 9, that if any number be divided by 9, the remainder will be the same as if the sum of its digits were divided by 9. — Thus: 74221 53-:- 9.^i:82-iG83-(-6, and the sum of the digits is 24, and 24-:-9:=2-|-6, ^. e. 9 is contained in 24 twice v.'ith a remainder 6. Every 9 is rejected because 9 is contained in itself once evenly, and therefore cannot affect the remainder. Let it now be required to multiply 122 by 24. Now, 122=^9X13+5, and 24=0X2+G, and if we multiply together the two ftxctors thus resolved, we get 9x13X9X2-1-9x2x5+9x13x6+6x5, and since 9 is a factor of all but the last, the last only will give a remainder when divided by 9, and therefore the whole product will give the same remainder when divided by 9, as 6x5^-9, which gives the rcmiainder 3, for 6X5=30 and 3(f~-9 gives 3 with a remainder 3. To test this by .trial, we find 122-f-9=:13 with a remainder 5, and 24-:-9;:^2 with a remainder G, and the product of these remainders is Gx5=30, and 30-1-9=3 with a remainder 3. Again, 122x24=2928, and 2928-7-9=::325 with a remainder 3, as in the case of the factors. EXERCISES. 1. 2. 3. 4. 5. G. 7. 7890X5=39480. 581967X8=4055736. 938740X4=3754984. 193784X7=1350488. 391870X9=3520884. 987450X0=5924730. 490783X52=25832710. 8. 719804X43=30954152. 9. 375907X64=24001888. 10. 27859X29=807911. 11. 679854X83=50427882. 12. 759684X187=142000908. 13. 5372X1634=8777848 14. Find the second power of 389 ? Ans. 151321. 15. Find the third power of 538 ? Ans. 155720872. 16. Find the fourth power of 144 ? Ans. 429981696. 17. Find the cube of 99 ? Ans. 970299. 18. 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 does it contain? Ans. 3550 1 cubic yards: '•!( II 26 ARITHMETIC. 20. If 29 o]l wells yield 19 gallons an hour each; how much will tlicy all yield iu a year ? Ans. 201115 gals. 21. If the rate on each of 1597 liouses be $19; what is the whole assessment ? Ans. lJ30o43. 22. If 1297 persons have paid up 9 sliares each in a railway company, and each share is $15 ; what is the working capital of the comDany ? Ans. $172095 K< 'I '" i I;' -I n DIVISION. 8. — Division is the converse operation to multiplication. It is the mode of finding a required factor when a product and another factor are given. It bears the same relation to subtraction that multiplication docs to addition, as will be seen below. By Ax. IV. Cor. we may resolve any complex quantity into its component parts ; sc division is resolving a certain quantity called the dividend into the number of parts indicated by another quantity called the divisor, (divider,) and the result is called the quotient (how often.) Let it bo required to find how often 8 is con- tained in 279,856. We can resolve 279,- 856 as in 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 be done mentally as wc proceed. We first see that 8 is not contained in 2, therefore we take 27, and find that 8 is contained in it 3 times, with a remainder 3 ; next combining this 3 with the next figure 9, we get 39, in which 8 is contained 4 times, with a remainder 7 ; combining this 7 with the next figure 8, we have 78, in wliich 8 is contained 9 times, with a remainder 6 ; combining this with the 5 following, we obtain 65, and 8 is contained in it 8 times, with a remainder 1, which combined with the 6 makes 16, and 8 is contained twice in 10. The correctness of the result may be tested by multiplying the quotient by the divisor. When the divisor consists of more than one figure, the learner must have recourse to a trial quotient, but after some prac- tice he will have little diiEculty in finding each figure by inspection. 8 240000 300O0 32000 4000 7200 900 640 80 16 2 8 279856 34982 DIVISION. 27 Let it be required to find how often 298 in coi.taincd in 431 76G. — The numbers being arrauj^ed in tlic 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 tioicc in 4, but cannot so easily sec whether the whole divisor 298 is contained twice in the same number of figures of the dividend, (viz. 431,) we therefore make trial, and place the 2 in the trial quotient, and multiply the divisor by 2 to find how much we shall hav(> to subtract from 431. AVo find 298x2-^590, larger than 431, and therefore we reject 2 and try 1. Now 298x1=298, less than 431, so we subtract and find a remainder of 133, and as this proves correct, wc place the 1 obtained in 298)4317G6( 2.1.5.4.5.4.9.8 trial s. 298 144S truo quotient. 1337 1192 1456 1192 2646 2384 262 298 the true quotient. We find our next partial divdcnd by writing 7, the next figure of the dividend after the re- mainder 133. Our experi- ence of the first case sug- gests to u9 that though 2 ia contained 6 times in 13, yet on multiplying something will have to be carried from the 98 which we expect will make the result too large, and therefore wc at onco try 5, but we find that 298X5=-1490, which is larger than 1337, and so we try 4, and find 298X4=1192, which being less than 1337, we subtract and find a remainder of 145 ; and having placed the 4 in the true quotient, we bring down the next figure of the dividend, giving a partial dividend 145G. By in- spection, as before, we .see that 6 would be too largo, owing to the carrying from 98, we try 5 and find 298x5=1490, which is larger than 1456; we try 4, and find 298X'4— 1192, which is less than 1456, so we subtract and find a remainder of 264. Having placed this 4 after the other 4 in the true quotient, w^ bring down 6, the last figure of the dividend, we try 9, and find 298'X 9^^2682, which is greater than our last partial dividend, 2646 ; we try 8, and find 298X8=:2384, and this being less than 2646, wc subtract it from 28 ARITHMETIC. I!i| il-l 111: that number, and find a final remainder of 2G2, and close the question by entering 8 in tho true quotient. The mode adopted to indicate that the remainder 2G2 still remains to be divided, which cannot bo actually done, as it is less than the divisor, is to write the 298 below the 262, and draw a line between them, thus Hy|, as also is seen iu the margin. Tiie resolution into partial dividends is also shown in the margin, whore it will be seer that the partial dividends, includ ing the remainder, make up tht whole original dividend. So alsc • the partial quotients are exhibit- ed, making up the whole true quoti- ent. That the trial quotient is not a single number, like the true quo- tient, but merely a succession of detached numbers, used as separate trials, is indicated by placing a full point between each pair. When we have multiplied the divisor by any figure in the trial quotient, and 3ubtrE,c{ed ihc product from the partial dividend, should the remain- der be greater than the divisor, we perceive that the trial figure is too small, and we must try a larger. From these illustrations we can ('.educe a 298 298 3808 120 87 2550 2380 1704 1428 276 339 261 780 090 844 783 01 E X K R C I S K ' . . 1. 2. 3. 4, 5. C. 15547C8--216=7198. :17500,»,B3. i 31884470-:-779=:40930. 57380025-f-7575=:7575. I2810098"-T-732. 9313702859--4687319=:L987:jjjg».5T^. 44914841047G-;-73885246=0079=y^g||2:rg. 7. 109588282929--138G=:r7902468yVg3g. 8. 3507G210832-r-79094451=3704095,'^fo^g3jWT- 9. 53081 8834--907:i=591862. 10. 1700649155el-^-759=2•240644479. 11. 5542702979ei--7584r^73084ie3^|g;f. 12. 00435074034529-^704095=79094451,9ii^/'jfg'',-. 13. IIow many bags, each containing 87 pounds, will 24,853,404 pounds of flour fill? Ans. 285,072. 14. 857 houses pay annually a tax of $41130 ; what is the aver- age on each per quarter ? Ans. $12. 15. St 9297 175 of prize money are to be divided among 97,805 sailors ; what is the share of each ? Ans. $95. 10. 120.815,231 pounds of cotton are made up in 233,879 bales; how many pounds in each bale ? Ana 89. DIVISION. 1. 49G87532--2:r:::24843700. 2. 57980327—3^19328775^. 30 ARITHMETIC. 3. 87905328-^4=21991332. ill I '! 'I V 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 79G3821- -5-^1592764^. 6875324—6=11458874. =330761 §. =496753^. 3987654~:-7=569664?. 19876532-;-8=2484566A. 2976854-:-9. 4967532—10= 46879352--ll=42617597\. 18765314-r-12=:1563777^. 78654246--18=4369680J. 75088-52=1444. 1G74018--189=8.,32. 31884470--779=:40930. 57380628--7575=7575^^3^^. 554270292198-:-7584=73084163y2'gg. 88789980979-:-9584=9264397^Vg4' 1 02030429729 ---123456=:826452f|S||. 267817946000-:-3G500=10077204.^ 497 men fell 163798 trees ; hov^ many does each fell on an Ans. 329. 22. If 148 houses pay a tax of $7844 ; what is the rate on each on an average ? Ans. $53. 23. If $415143630 are levied from 4455 townships ; what is the portion of each on an average ? Ans. $93186. 24. How many lots of 6754 each are contained in 3968091 51372 ? Ans. 58763718. 25. What quotient will be obtained by dividing 961504803 twice by 987 ? Ans. 987. average ? 9.— TABLES of MONEY, WEIGHTS & MEASURES. DBITISU OB STERUNO MONEY. 4 farthings, or 2 half pennies, are 1 penny (d.) 12 pence 1 shilling (s.) 20 Bhillings 1 pound (£) nECIMAL COINAGE. 10 mills (M) are 1 cent (ct.) 10 cents 1 dime (d.) 10 dimes, or 100 cents... 1 dollar ($) AVOIRDUPOIS WEIGHT. TABLE. 16 drams make 1 ounce, 16 ounces 1 pound, 25 pounds 1 quarter, 4 quarters 1 lumdredweight, 20 cwt 1 ton. marked oz. lb. *' qr. " cwt. " t. Note. — Tliis weiglit is used in weighing heavy articles, as meat, groceries, grain, etc. vegetables TABLES OP MONEY, WEIGHTS AND MEASURES. 31 TROY WEIGHT. TABLE. 24 grains (grs.) make 1 pennpveight, marked dwt. 20 pennyweights 1 ounce, " oz. 12 ounces 1 pound, <' lb. Note.— Troy weight is used in weighing the precious metals and atones. APOTHECARIES' WEIGHT. TABLE. 20 grains (grs.) make 1 scruple, marked scr. 3 Bcniples 1 dram, '• dr. 8 drams 1 ounce, " oz. 12 ounces 1 pound, '• lb. Note. — Apothecaries and Physicians mix their medicines by this weight, but they buy and sell by Avoiruupois. PRODUCE WEIGHT-TABLE. GRAIN'. Wheat GO pounds to the bushel. Oats 34 " " " Corn 56 Com in cob . 80 Barley 48 Rye 56 •Buckwheat. . 48 Peas 60 Beans 60 Tares 60 u It (( 1( SEEDS. Clover 60 pounds to the bushel. Flax 50 " " " Timothy 48 " " " Hemp 64 Blue grass.. 14 Red Top 8 Hungarian I ^g grass . . . f Millet 48 Rape 50 It It It ti tt It It It TEOETABLES. Potatoes .... 60 pounds to the bushel Parsnips 60 " " " Carrots 60 " " " Turnips 60 " " < Beets 60 " " " Onions 60 " " " VEOBTABLES. Castor Beans 40 pounds to the boahel. Malt 36 " " " DriedPeaches 83 " " " Dried Apples 22 " " " Salt 56 " •' " Bran 20 " " •' LINEAR (OR LONG) AND SQUARE MEASURE. LKEAB. SQUARE. 12 inches (in) make. 1 foot (ft.) 3 feet 1 yard (yd.) 6 J yards 1 rod or perch. 40 rods 1 furlong (fur.) 3 furlongs 1 mile (m.) 144 inches make 1 foot (ft) 9 feet 1 yard (yd.) SOJ yards 1 rod (rd.> 40 rods 1 rood (r.) 4 roods 1 acre (a.) LAND MEASURE. LEXOTII. 7^ inches make. 9r^ linta llink. 25Tinks 1 rod. 4 rods or 100 links 1 chain. 80 chains 1 mile. ak£:a. 10,000 square links make 1 sq. chain 10 square cliains .... 1 acre. 32 ARITHJIETIC. Ill solid measure, i. c, the measurement of solids, 1728 (the third powei or cub(! ol" 12,) iuclicH make 1 cubic loot, uiid 27 cubic feet (i. r. SX'^X'^)) make 1 cubic yard. Iix uieaHuriiijj timber, JO cubic feet of round timber malio what ia called a ton, and the same name is given to 50 I'eot of hewn timber. A cord of firewood is 8 feet long. 1 feet wide, and 4 feet high, and thcrcforo its solid content is 8X1X-'=128 feet. Dry goods are measiux'd l)y the yai'd, and fractions of a yard, the frac- tions used being one quarter, one-eighth, and one-sixteeuth. MEASURES OF CAPACITY. DRV. 2 pints make 1 quart (qt.) 4 quarts 1 gallon (giil.) 2 gallons 1 peck (pk.) 4 pecks 1 busliel (bu.) 3G bushels 1 chaldron (ch.) The last is seldom used. LIQUID. 4 gills make 1 pint (pt.) 2 pints 1 quart (qt.) 4 quarts 1 /rallon (gal.) G3 gallons 1 hogshead (bM.) 2 hogsheads 1 pipe (pi.) 2 pipes I tun (tun.) liii'i I.,, I MEASURK Ol'" TIMK. CO seconds make 1 minute. CO minutes 1 hour. 24 hours 1 day. 3tj5i days 1 year. ANGULAR OR CIRCULAR MKASURE. CO seconds make. 1 minute (1\) CO minutes 1 degree ( 1 °.) 3G0 degrees 1 complete circle. There are other units applied to certain articles, e. g., 12 articles, one dozen ; 20 articles, ono score ; 144 articles, one gross ; 24 eheots of paper, one quire ; 20 quires, ono ream. — 141bs., one stone; This last weight is varied in many places, 151bs. and IGlbs., according to the nature of the arti- cle sold, e. g., — potatoes, as an allowance for earth adhering. THE CALENDAR MONTHS OF THE YEAR. January has 31 days. February " 28 " March " ,^1 " April " :jo " May " 31 " June " 30 " July has 31 days August '• 31 " September " 30 October " 31 November " 30 " December " 31 " Every fourth year is called Leap-year, in which February has 29 days. — If the last two figures denoting the year can bo divided evenly by 4, it is Leap-year. iiiii DECIMAL COINAGE. 10. The principle of the decimal coinage is generally understood to depend on the rules of decimal fractions ; but as it is merely a separate and co-ordinate result of the common system of notation, we may explain it here, independently of the theory of decimal fractions. DECIMAL COINAGE. 33 Wc have already explained, that according!; to tlic Arabic notation, each digit has one-tenth the value that it would have if situated one place farther to the left. Thus, iu the number 88, the dii,'it to the right expresses 8 units, while that to the left expresses 8 tens. Now wc cannot liave any inteirer less than unity, but we may liave to make calculations respecting quantities less than the unit under considera- tion, c. g., in calculating by dollars, we may have to take cents into account, and as the cent is a sub-division of the unit, a dollar^ some new character must be introduced to indicate this transition from the integral unit to a port of it. This is done very simply by interposing a mark like the period or full point (.) in printing. — This is usually called the decimal point, though it sometimes gets the vague and awkward name of the separatrix. This simple but admirable contrivance is ascribed to one Stevinus or Stevens, of the Netherlands, who gave his suggestion to the public about the year 1585. Its excellence consists in its being simply an extension of the common notation. The original system marks only the repeti- tion of the unit of measure, — this applies the same principle to the sub division of the unit into parts. To explain this, we have only to carry out the illustration already given regarding integers. We saw that the extreme right hand figure, 8 in our example, stood for 8 units, and was one-tenth of the preceding one ; just in the same manner another figure, 8, placed to the right of the units' figure, will express one-''-I require us to divide a less number by a greater, or in the case of remainders, the division is indicated by writing the dividend above the divisor, and separating them by a line, — thus: 7-:-8 is written 1. So to indicate tiiat 1 dollar is divided into 100 cents, we write $,io, which means the one-hund- redth part of a dollar, and thoreibre dollars and cents arc sometimes written, especially in bills and drafts, in this manner, $12.,Yo, l>ut the form $12.25 is generally preferable. To show the reason of the. form $1.05, for one dollar and five cents, wc have oniy to notice that the form $1.5, would mean one dollar and five dimes, or lifty cents; \,iiereas $1.05 means one dollar, no dimes, and five cents. From the foregoing explanations, it is plain that the rules for the addition, subtraction, multiplication and division of abstract numbers, or applicate numbers of only one denomination, apply also to dollars and cents, because they increase from right to left, and decrease from left to right, according to the same law, that is, in the former case by tens, and in the latter by tenths. It would be of great benefit to the whole commercial community, and perhaps still greater to the farmer, if tlie decimal scale were adopted in weights and measures, as well as in money, as it would materiaMy simplify and expedite all calculations. Every one must feel and admit the very great ease and rapidity with which every operation is effected, accounts made up, and books kept in dollars and cents, in comparison with the sub-division into pounds, shillings and pence, and the difference would be at least as great i-egarding weights and measures. It would also very inuch accelerate the learner's pro- gress, for it would save him the heavy labour of committing to memory the formidable host of tables, through which he has now to cut bis i i i: ; il::i n DECIMAL COINAGE. 35 ^vay — the wlmlo processes of reduction would bo compressed into " nut-shcU" dinu'usions, and the nioniory would not be over-taxed in after years to keep up the recollection of tlio tables conned in youth. Besides, by the i)lan we have sugj,'ested, the pupil could pass at onco from the elementary rules to the hij,'hcr ones, sueli as proportion and interest, and could either get into business iu a mucli shorter timo than is po.->siblo at present, or devote his timo to higher and more important studies. (1.) EXEUCISES. Addition of dollars and cents. (2.) (3.) W JJS5.50 $11G.20 $13.19 49. G3 291.45 $125.75 14.1G 92.18 89.75 08.50 85.92 37.09 3G5.84 25.15 G4.15 8.92 91.50 7G.05 37.25 7G.45 70.15 •Jl.lU 91.20 25.75 485.00 43.87-A- 18.75 ei4.iG 157.92 84.20 29.10 18.G0 263.75 G7.G21 47.85 59.11 183.25 39.80 55.55 148.17 39.48 17.37^ 72.G3 205.90 130.13 G69.44 529.75 931.40 2301 42 3vcry and and lights pro- |mory It his -tt (5.) (G.) (7.) (8.) $11.27 $55.63 45.15 $44.50 $296.75 17.75 54.72 G7.23 176.84 84.18 31.30 89.75 518.50 29.88 49.50 27.63 369.03 45.13 10.75 95.13 627.45 38.81 84.28 38.88 258.13 67.25 14.85 17.45 591.18 96.20 9.44 56.64 179.25 77.63 28.09 73.85 5G7.42 8.75 345.35 511.0G 3585.15 521.21 I' 1 I!ii I 3G AIITTHMETIC. CJ.) Sold (n .1. Jo NFS, 20 yanls clotli Z'^.2') It luiit.^ 2l.r)(J 10 huLs 3:}.r)0 5 pairs of blankets *JH.75 15 yards Ncalskln 40.25 15 yiird.s of scrjro 0.03 28 yards fino cloth 112.88 321.82 1 1 (10.) 1 $157.20 H 208.73 :j 985.45 i 1!)7.00 1 385.18 J 870.75 ,« 795,85 1 507 13 i 059.03 4893.07 (11.) Sold to S. Fulton, Aurora, 12 pair.s of worsted stockings .' $13.50 18 " " flannel drawers 22.75 24 " " kid gloves 8.03 50 sehool books 49.72 29 yards of satin 83.23 90 school copy books 1.84 180 yards of ribbon.. 29.70 84 yards of ticking 22.68 122 yards of sheeting 23.18 255.29 12. The shares in an oil-well speculation arc $5 each; A. takes 15 shares; B. 25; C. 20 ; D. 1 ; E. 11; F. 37 ; G. 16; II. 18; I. 8; K. 21 ; L. 14 ; and 14 other persons take l^ bhares each; what is the capital of the company, and how many shares are there ? Ans. $1,030 and 320 shares. 13. If 17 vessels bring to the port of Boston cargoes of the follow- ing values ; what does the whole amount to ? $2305.75, 81793.87, $3815.25, $2718.63, $4180.50, $3179.13, $1623.88, $4311.75, S1987.38, $2975.75, and the other 7 average $2089.13. Ans. $47781.80. Subtraction of dollars and cents. (1.) (2.) (3.) $567819.83 $83756.17 $17423 37^ 278956.89 76489.71 9654.63t^ i 288862.94 7206.46 7768.74 DECIMAI, COINAOK. 87 4. What is the (lifioroiico between 'JTllO dolhirs. and 50 CMits, luul 987 ilullar.s H7A cents? An.«. i^lTSl.ii'JA. f). The debit side of u ledger in 81770.80, and the eredit side 8S7iJ.5U ; what is tlic bahmcc ? Aus. $89 l.iJO, G. The credit side of a cash book is 68795.88, and the debit side is 810358.18 ; what is the bahiuee? Aus. §15G2.;J0. A firm owes 6li'279G8.25, and the estate is worth $98704.75 ; what is the state of the affairs of tlic liriu ? Ans. — The tirni is vnubio to pay S129,20U.50 over and above the assets. 8. A ship and cargo were worth $27509.50, — the ship was lost. nd only 80781.00 worth of the cargo saved ; what was the loss ? Ans. $20721.90. Multiph'c'ition of dollars and cents. * (1.) (2.) (3.) (4.) 6365.75 $1873.47 S8G5.C3 $24780.38 87 09 93 45 250025 292600 31820.25 lG86i23 1124082 129209.43 259689 7790G7 80503.59 12.393190 9914552 1115387.10 Division of dollars and cents 1. |28G42.14-r-29=-8987.GC. 5. 81 943243.55-^983=r-$l 970.85 2. $37133.34--87=842G.82. 0. 831421. 25-^03==$498.75. 3. 800509.08--7G:^879G.18. 7. 828479.75-:-78=.3G5.12i. 4. 843009.75->98^8438.87A. 8. 82595,37^-r-7G9=$3.37j. 9. $2927.30 a year ; how"much per day ? " Ans. 88.02. 10. $3953.19 a year; how much for every working day ? Ans. 812.03. 11. 209 persons have to pay a tax of $1312.72; what is the average tax on each ? Ans. $4.88. 12. A collection of $544.04 is made by 1870 persons ; liow much did each give on an average ? Ans. 29 cents. * Wo must hero caution the tyro against such modes of expression as this, — " multiply §85 by $12." Such an exprf s.sion is simply absurd, for to say $12 times $85, might as well mean 1200 times $Ho, or 12000 times $85, which would all give widely different results. Wo may indeed have to mul- tiply a denominate number representing §85, by another denominate number representing $12, as often happens in questions involving proportion, e. (j., iu interest ; but so soon as we use the number 12, or any denominate number as a multiplier it eeases to be denominate, and becomes abstract, and no longer re- presents any denomination,but merely the number of times the other is to Ijo re- peated. We object even to the putting of such questions as "catch questions,"' 38 ArJTH^rETIO I" DECIMAL AND DUODECIMAL CURRENCIES. As there is frequent intercourse between the United States and the Lower British American Provinces, it has been thought desira- ble to show the method of changing Decimal or Federal currency into Duodecimal or Halifax currency, and vice versa. The traffic between the coast line of the States and liiat of the Lower British Provinces is very considerable. The trader, therefore, of either re. quires to be perfectly familiar not only with the comparative value of the currencies of both countries, but also with the coins and paper money used by both. Besides there is constant personal intercourse by travelling and migration, and this makes an intimate acquaint- ance with all the details of both currencies a most important acquire- ment. This applies more or less, though in different degrees, to all the British Provinces, except to Canada, where the decimal system has been adopted, though, unfortunately, not universally followed ; but it is highly probable that, if the proposed confederation of the British Provinces should be carried, the decimal coinage will be uni- vcrsally adopted, and universally adhered to by the next generation at least, if not by the present. For the same reasons the mode of changing Federal into Sterlin2 money, and vice versa, has been explained under the head of Ster- ling Exchange. This seems ({uite as necessary as the preceding, because the traffic between the States and Britain is on an extensive •scale, and the coming and going of passengers may now be reckoned by thousands, all of whom require to understand thoroughly both currencies and the circulating media of both countries. The in- creasing facilities of communication are progressively and rapidly extending the trade, including the passenger traffic, between the twc countries, and hence the greater necessity that all persons engaged in business, or in any way exchanging operations, should intimately understand how to change the money of each country into that of the other. for the learner is but too apt to look at the question just as it stands, witliou over tliinking of the principle on which it is intentled to try him. The aljsurdity of the expression may be shown by the different lights in which the long discuss- ed question, to multiply 2s. (!d. by 2s. Gd. may be viewed. (1.) As 2s. lUl, is J of a pound, the (juestion may be taken as moaniug that 2s. (id. is to be divided into 8 ccpial parts, and 1 of them taken, which would bo 3:Jd. (2.) As 2s. Gd. is 2i shillings, the question might be taken as meaning that 2s. Cd. was to be k DECIM.\L COINAGE. 39 rdity iCUSS- is -J s. oa. The origin of the mark (8) for dollars is somewhat uncertain. Some suppose it to be a contraction for U. S., the initials of the United States, but it seems to have been in use in continental Europe before the discovery of America, and therefore must bo an importation. The followiuL;' explanation of its origin seems more probable, for if the other were correct, Ave should surely have some record of it. Ac- cording to an ancient fable or fancy, the pillars of Hercules marked the limits of the world towards the west and were said to support the world. From their position at the entrance to the Mediterra- nean Sea they were objects of interest to the Spaniards and were represented on one side of their coin called the 7'cal, and in the coin for 8 reals the 8 was warped around them, thus forming the mark. To reduce currency money to the denominations of the decimal coinage. Since 100 cents make 1 dollar, and 4 dollars make 1 pound, 400 cents make 1 pound currency, and therefore to find the number of cents in any given number of pounds, we must multiply the pounds by 400. Again, since 20 cents make 1 shilling or 12 pence, to find the number of cents in any given number of shillings, we must multiply the shillings by 20. Lastly, 5 cents arc equal to '6 pence, and 12 farthings are also equal to 3 pence, and (Ax. I.) things that are equal to the same thing, are equal to one another ; therefore, 5 cents are equal to 12 farthings, and 1 farthing is tho -jV of 5 cents, or -[\ of 1 cent. Hence to find the number cf cents in any number of pence and farthings^ we multiply tho number of farthings in the given pence and farthings by 5, and divide the product by 12. Having obtained the three results, we add them all together. Thus to change £48 18s. 9£-d. to dollars and cents, we multiply 48 by 400, 18 by 20, and take y\j of 0£-, or 39 fiirthings, and add the three together, which gives us 1957G^ cents, or $195.76,^. 48X400^=19200 18x 20== 300 9|=39fX /.= IGi 19576;if repeated 2J times, which woukl make Cs. 3d. (o.) Tbo interpretation miglit bo. that as 2s. Cd. is GO pence, that tho other 2s. Cd. is to bo repeated oO times, which would give £3 15s. Od. (1.) Tho pliraso may also bo interpret- ed as meaning that 30d. was to bo repeated 30 times, wliich would ahso giro £3 las. Od. Tho last two interpretations are tho same in two dilTeront forms, and givo tlio same result. This is the only view in which the expression has any sense, and proves our statement, tliat wlienever a denominate number is used as a multiplier, it ceases to l)c (ienorain.ito, and becomes abstract. Tho same principle will apply to division. 40 AHITHMETIC. EXERCISES A (1.) £79 X400=31G00 16 X 20= 320 64dX 1 .; 10^. S319.30^ (2.) £117 X400=46800 17 X 20=:=. 340 83d V -'^-— 14-''- $471.54/5 I ''I IP; :i ilil 3. £87.14.10|=^$350.97J^. 4. £29.19.9=^119.95. £67.13.4f=8270.G7-jA. £279.15.10i=:$1119.i7i. £118.11.4i=S474.27i." 5 6 7 8, 9. £37.18.S=$151.73J-. 10. £57.8.11£^=$229.79/2. 11. £49,7.G=$197.50. £79.8.4=$317.66f. 12. £137.16.8=$551.33J. 13. £236.19.2^=$947.84f 14. £19.1G.8=$79.33J. 15. £98.1.1i=S392.22^. 16. £87.11.8=$350.33J. 17. £457.12.6=$1830.50. 18. £219.4.7f=:l$87G.92i^. 19. £49.9.4f=$197.87|Ar 20. £287.18.10i=:$115f.77^. To change dollars and cents to Halifax currency, we must re- verse the above operation. Thus, to reduce §195. 76|- to £. s. d. — First, reduce the dollars and cents to cents, then divide by 400, which gives 48, the even number of pounds, Avith 11 remainder of 37G| cents ; then divide this remainder by 20, which gives 18, the number of Khilling.s, with a remainder of IG:^- cents, 0"^ in the couvcrsc operation, we multiplied by 5, and divided by 12, so now Ave multiply by 12, and divide by 400)195761(48 400 357Gi- 3200 2( )376:K18 20 176 160 12 5)195 (39 5; thus, 16^X12=195, and 195---5=39, the number of farthings, and this being reduced so that gives n, to pence and farthings, $195.7G.j:=£48.18.9J. Or the work may be shortened by the fol- lowing method. As 84 make £1, the number of £'s in 8195.761 ^viU be the same as the number of times that 4 is contained in the 195 dollars, which gives £48, and $3 remain- & I I I T)ECI>LVL COINAGE 41 $195—70:} mg. Now, tlicsc three dollars are equiva- lent to 300 cents, which added to the re- maining 7G^ cents, gives 37G|- 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 37G|- cents, will be 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 3 pence, one cent will be equal to I of 3 pence, which is | of a penny ; therefore, if one cent is equal to | of a penny, the re- maining 10} cents will be equal to 10} times f of a penny, which is 9fd. ; hence we have $195.70} equal to £48.18.9;^. 4)105 £48- -300 20 370} sl8- -10} 3 5 ,48i 9|d. EXERCISES . 1. 2. 3. 4. Reduce $119.95 to Halifax currency. Reduce $270.07{^ " Reduce 0474.27^ Reduce $197.50 5. Reduce $1119.17^ 0. Reduce $551. 33J^ 7. Reduce $1830.50 8. Reduce 81151.77^ Ans. £29.19.9. Ans. £07.13.4f. Ans. £118.11.4-^ Ans. £49.7.0"! Ans. £279.15.10^. Ans. £137.10.8. Ans. JW57.12.0. Ans. £287.1 8.1 0J-. MIXED EXERCISES. 1. Reduce 2. Reduce 3. Reduce 4. Reduce 5. Reduce 0. Reduce 7. Reduce 8. Reduce 9. Reduce 10. Reduce £430.7. 8A- to dollars and cents. $547.87 to Halifax currency. £783,13.5} to dollars and cents. $570.85 to Halifax currency. £000 19. 8 J to dollars and cents. $375.99 to llalifa.\ currency. 3s. 8}d. to dollars and cents. 17 cents to Halifax currency. 10;2 pence to dollars and cents. 23 cents to old Canadian currency Ans. $1745.541 Ans. £130 19.41. Ans. $3131,08|. Ans. 144.4.3. Ans. 32427.94-,^^-. Ans. £93.19.11 f;. Aus. 73|- cents. Ans. 10' pence. Ans. 17]! cents Ans. 13 ', pence 42 AlUTHMETIC. BEDUCTION. XI. — liEDUCTioN is tlio inodo of expressing any given fjuantity in tonus of a liiglicr or lower dononiination, e. g., expressing any given number of dollars as cents, and vice versa, any number of cents as dollars. ^Vhen a Iiiglier denomination is changed to a lower (as dollars to cents), the process is called reduction ttecending, and when a lower is changed to a higher (cents to dollars), it is called reduction t/scending. Beginners are gcnerall^ 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 sxplanation lies in the original use of the word reduc, to bring hack, which would sug- gest that the dollars were originally cents and are brought back to cents, or that the cents were originally dollars and are brought back to dollars. Thus, by a transition common in all languages, the idea of bringing back was gradually lost, and the idea of changing from one denomination to another alone retained. Again, since one dol- lar is equal to one hundred cents, it is plain that tiie number repre- senting any amount in cents will be one hundred times greater, taken abstractly, than that representing the same in dollars, and so in all denominate numbers. Some explain the term reduction as takca originally from the changing of a liigher to a lower denomination, and afterwards applied to the converse operation. This seems satis- factory enougli 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 terms of the lowest denomination,viz. ibs.,we must first find how 3nany quarters are equivalent to 17 cwt. 3 qrs. which wc find by multiplying the 17 cwt. by 4 and adding in the 3 qrs. for 4 qrs. make 1 cwt, — and then since 25 lbs. rnako 1 qr. we multiply the 71 qrs. by 25 to find the number of lbs. which, with the 20 odd lbs. added in, is 1735 lbs., and thus we see that 1795 lbs. arc 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, we must divide the number denoting the pounds by 25 to obtain tliat denoting the quarters, and, in like manner, wc must cwt. qi-s. lbs. 17.3.20 4 71 25 375 142 1795 25)1795 4) 71 .jrs. 20. divide the number representing the 17 cwt. 3 qrs. 20 lbs. Quarters bv 4 tc find thatdcnotinir the ItEDUC'J'IOX. 43 2G acres, 2 roods, 3G rods. 4 100 40 427G rods. — Ans. Imndrcds wcii:,Iit, the remainders in each case fjeini^ ^\•rittcn as sub- Ucnouiinatious. In tlu! same manner 2G acres, 2 roods, .')G rods will be reduced tc rods by multiplyini^' the acres by by I and adding the odd roods, whicli gives lOG roods, and this multiplied by 40 with tho odd rods added in gives the rods, for 4 roods make one acre and 40 rods 1 rood. Conversely tlic rods di- vided by 40 will give lOG roods and 3G rod^ over, and lOG roods divided by 4 will give 2G acres and 2 roods over, tho same as tlic original question — 2G acres, 2 rods, 3G roods. EXERCISES. 1. How many dollars arc there in 4798G cents ? Ans. $479.80. 2. How many cents are there in 187 dollars? Ans. 18700, 3. How many pounds arc there in 2 tons IG cwt. 2 qrs. and 21 ^bs ? Ans. 5G71. 4. How many pounds are there in 18 cwt. and 22 lbs. ? Ans. 1822. 5. Reduce 1479G lbs. to tons, &c. ? Ans. 7 tons, 7 cwt., 3 qrs., 21 lbs. 6. Reduce 7643 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 22 grains? An.s. 103654. 10. How many lbs. in 46891 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 1984G2 drams ? 13. How many bushels in 8964 lbs. of wheat? 14. How many bushels in 14382 lbs. of barley ? 15. How many bushels in 48028 lbs. of peas ? 16. How many bushels in 4683 lbs. of timothy seed ? Eil'M 1 '!"■ 'vH 44 ARITHMETIC. 17. llcducc 98 iiillcH, 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 5271C8 feet ? Ans. 99 miles, G fur., 29 pr., 2 yds , 3 ft., G in. 20. Reduce 57 acres, 3 roods and 24 rods to rods ? Ans. 92G4 rods. 21. How many square yards arc there in 17 acres, 2 roods and 18 rods ? Ans. 85244^ yards. 22. Find the number of acres, &c., in 479G85971 square inches. Ans. a. 7G.1.35.19.2.119. 23. How many acres do 17G984 square yards make ? Ans. a. 3G.2.10.21J. 24. How many square links are there in 37 acres ? Ans. 3,700,000 links. 25. How many acres, &c., in 479,803,201 square links ? Ans. 4798 a., G ch., 3201. 2G. 7,864,391 cubic inches ; how many cubic yards ? Ans. yds. 1G8.15.263. 27. 9 cubic yards, 7 cubic feet, 821 oubic inches ; how many 3ubic inches ? Ans. 432821 cubic inches. 28. TIow many gills does a tun contain ? Ans. 80G4 gills. 29. How many gallons, &.C., do 479805 gills make ? ^Ans. gals. 14995.3.0.1. 30. How many pints are there in 28 bu., 3 pocks and 1 gal. ? — Ans. 1848 pints. 31. 27 yards, 3 qrs., 3 nails ; how many nails? Ans. 447 nails. 32. 280 nails ; how many yards, &c. ? Ans. 17 yards, 3 qrs., 2 nls. 33. 36° 40^ 25'" • how many seconds? Ans. 132025". 34. IIow many degrees, &c., in 4978G" ? Ans. 13° 49'.4G". 35. The area of New York State is 29.440,000 acres ; how many iquarc 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 «peed of 52 miles an hour ? Ans. 208 years, 201 days, 10j\ hours. 37. The area of Pennsylvania is 47000 square miles ; hew many square feet ? DENOMINATE NUJILEIIS. 45 38. Sound moves about 1 130 feet in a second of time ; how long would it be in moviiij^ from the earth to the srn? Ana. 14years, 27 days, 15 hours, fjO niin., 5,Y';i"<} sec. 39. How many seconda of this century had elapsed at the end of 18G4, counting the day at 24 hours? Ans. 2,010,080,400". 40. The great bell of Moscow weighs 127,830 lbs.; how many tons, &c., do'^s it weigh, the quarter being 28 lbs. ? Ans. 57t. Ic. 1({. IGlbs. 41. IIow many days from the 11th July, 1801, to the 1st of April, 1804 ? Ana. 995 days. 42. A congregation of 509 persons made a collection of £40.0.1 ; how numy pence did each give on an average ? Ans. 17d. 43. The British mint can strike off 20,000 coins in an hour ; what is the value of all the pennies coined in one day of 12 hours' work? Ans. £1,000. 44. 417 tons offish were caught at Newfoundland in one season, and sold by the stone of 14 lbs., at an average price of 42 cents a stone ; what did they bring ? Ans. $25020. 45. How many feet from pole to pole, the earth's diarreter being 7945 miles ? Ans. 41949600 feet. many DENOMINATE NUMBERS. 12. — When numbers arc spoken of in general, without reference to any particular articles, such as money or merchandise, they are called abstract, but when they are applied to such articles they are sometimes called appUcate, as being applied to some particular arti- cles to express their quantity ; sometimes they are called concrete, (growing together,) as attached to some particular substances, and sometimes they arc called denominate, as denoting quantities that consist of different dcnoti^inations, as dollars and cents, — pounds, ounces, &c. The elementary rules of addition, subtraction, multi- plication and division, arc performed on denominate numbers, exactly in the same way as on abstract numbers, with this single difference, that when a lower denomination is added, and gives a sum equal to one or more units of the next higher denomination, we carry that unit, or those units, to the next higher denomination. Thus : if the sum wcc 24 inches, we should call that two feet. In abstract and decim.l numbers we always reduce, or curry, by tens. 46 ARITHMETIC. Here we find tlic sum of the inches to be 34, and as 12 inches make one foot, the number of feot in 34 inches will be the same as the number of times that 12 is contained in 34, which is twice, with a remainder of 10, therefore we write the 10 under the column of inches, and add up the 2 feet with the column of feet, and obtain 11 feet, and as 3 feet make 1 yard, the number of yards will be the same as tlie number of times that 3 is contained in 11, which is 3 times with a remainder of 2 ; we therefore write the 2 odd feet under the column of feet, and add up the 3 yards with the column of yards, and the whole amounts to 94 yds., 2 ft., 10 in. The same operation would be carried out if we had rods, &c., given, and is applicable to all operations in denominate numbers of any kind. In the exercises on the addition of denominate numbers, one ques- tion in abstract numbers is given to contrast with the denominate. yds. ft. iu. 12.2. 9 10.1. 11 27.3. 8 30.3. 4 94.2.10 U'.A I !' !i I EXERCISES. (1-) (2.) (3.) £70.18. 4 $1907. 87i $857.03 17.11. 44 2075.75 78G543T 189.50 99.19. 9 3194.02-i 198675 C84.87i 11.11.11 7058.50 8470154 498.75 07. 15.101 8970. 37iV 1809538 807. 12i 79.19. 9" 2873. 12|- 4187043 3Qb.3n 28.12. 1 1709.25 5708299 917.25 63. 8. 4^ 2481.92 28305746 4380. 50'i- 445.17. 5i 30997.42 (5.) (6.) (7.) (8.) Ihs. oz. drs. t. cwt. qm. lbs. lbs. oz. dwt. grs. lbs. oz. dra. scr. grs. 13.14.10 26.17.3.21 3.11.16.21 5 11.7.2.19 15.11.10 18 n.0.19 5. 8. 7.11 4 10.4.1. 7 11. 4. 9 25.15.1.16 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 16.10.11.22 6. 7.2.2. 9 8. 9. 6 18. 8.19.18 w ■ 8.1.1.13 4.15.15 89.10. 2 153. 3.2.13 77. 8. 0. 28 4.0.1. 4 1 cv 1 8 4 ■■^ o ^s 1 3 o 1 5 ■■''-^k -# o fj 'i' .'P. — "i'r ■■■*»;' •i -■It -^ "'iK'l -^ ■.'Ji DENOMIN^VTE NUMBERS. 47 cig (10.) (11.) (12.) m. fur. rods. yds. y's. n. in. 1. nc. roods, nl. rods. yds;, ft. in. 17G.7.39.5 18.2.11. 11 29.3.39 39.30.8.143 85.4.20.1 14..?. 7. 9 57.2.18 18.11.4. iiS 79.6.29.3 8.1.10. 7 118.0.26 24. 4.7.118 42.3. 8.2 11.0. 7. 6 75.3.11 11.21.2. 96 C7.1.11.2 • 7.2. 8. 5 51.1. 8 15.27.0.124 118.3.10.3 16.2. 9. 10 94.1.19 27. 6.3. 87 81.2.31.1 8.1. 7. 6 63.2.21 19.25.2. 38 79.0.21.2 78.1.15 18.3.33.3 19.3.33 749.2. 6.0 . 87.0. 3. 6 589.0.30 157. 6.3. 98 (13.) (14.) (15.) (16.) a. ch. links. Cll. h. p. g. qt. pt. tiL pi. hhd, gaL qt. pt. gl. yd&qr3.iiis. 79.9.9999 5.35.3.1.3.1 6.1.1.1.3.1.3 36.3.2 117.4.3650 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 93.3.3 84.8.1160 449.8. 516 29.34.0.0.3.0 16.1.1.5.0.0.2 375.2.1 (17.) (18.) (19.) (20.) cwt. qra. lbs. 87.3.11 49.1.18 28.3.15 36.1. 8 88.1.16 57.3.14 359^59'. 59" 153 .40.45 270 . 0. 179 .45.30 81 .30.10 89 .59 .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. cwt. qrs. 11).=. 18 .1 18 22 .3 11 9 2. 18 12 .1 15 8 .3 .24 31 .2 348.3.7 1134 .56.23 93.267.23.37.36 103.3.11 1 48 ARITHMETIC, li E D G E H ACCOUNTS. The debit and credit sides of four folios of a ledger arc as be! /W, what are the balances ? (21.) Dr. 21.) Cr. ,22.) Du. (22.) Cr. $1214.75 $2703.80 $198.75 $118.5(1 863.09 471.38 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 19.13 347.G3 963.15 76.20 47.75 2000.00 1257.75 4.50 65.92 798.38 189.60 181.60 32.40 2018.50 98.13 19.25' 76.50 164.30 756.25 76.38 7.75 277.15 87.50 219.50 197.25 1165.20 103.63 48.75 15.75 367.40 1291.00 93.15 8.38 984.70 784.25 25.50 93.15 273.60 79.75 81.05 67.45 584.10 81.18 28.30 5.45 1200.00 318.50 69.08 18.09 68.75 1819.20 157.11 4.12 79.15 58.50 278.00 1 57.60 56.18 176.25 59.50 28.88 2860.14 11.25 941.12 (23.) Dr. (23.) Cr. (24.) Dr. (24.) Cr. $81.19 $80.10 $177.88 $156.92 17.11 15.65 291.16 285.15 ^5.38 39.88 356.13 356.12 19.63 10.13 189.38 178.25 187.13 176.15 471.63 469.10 87.63 89.92 785.88 698.80 87.88 77.81 911.50 930.75 111.11 99.88 683.15 496.20 134.56 16.97 432.61 547.60 179.51 87.63 355.55 478.99 340.25 75.75 638.27 546.54 224.12 56.51 436.15 372.25 156.12 37.23 325.36 252.12 $ $ $ $ DENOMINATE NUMBEItH, 40 (25.) Du. $17G,0;{ 27.85 79.:{7 08.11 :{5.4U 83.50 1127.25 48.18 250.00 779.03 .IS'i 20 59.75 68.87 :^8.75 28.G3 71. .38 293.63 185.10 9.05 64.20 38.75 45.45 215.87 7.75 93.92 81.88 68.25 99.99 18.12 27.13 168.00 75.75 738.38 18.24 136.25 126.72 834.15 128.71 136.18 178.16 284.77 326.54 412.13 391.15 267.18 125.13 (25. Cii. 6i2;j7.7r) 27(i;j.lS 194.25 39.37 8.25 11.87 29.05 (k:.20 71.80 13.10 45.50 25.20 43.15 7.50 50.00 87.75 5.00 31.60 13.40 90.75 15.15 67.63 58.50 67.05 49.35 21.25 35.15 20.13 92.87 35.28 81.18 10.80 51.25 67.54 91.12 18.35 42.54 16.21 25.51 53.99 62.87 91.54 32.21 54.12 77.99 42.51 (2(5.) ])R. $1087.03 i 457.88 190.37 87.12 94.25 47.20 39.15 8.75 307.40 18.93 67.45 21.63 298.50 78 00 189.00 47.15 68.10 54.30 12.12 89.75 118.00 69.50 48.75 36.12 91.20 87.63 90.00 100.75 49.15 87.63 43.25 81.37 92.65 37.49 46.87 £ 1.13 51.12 64.54 57.62 38.94 61.87 93.89 89.78 21.46 64.98 73.75 (26.; Cr. 4786.87 183.05 97.75 14915 13.25 41.18 8.50 9.75 11.12 183.02 79.10 814.00 95.50 218.00 59.87 18.05 77.40 38.87 15.62 9.87 14.12 89.50 4.20 67.37 81.09 7.05 57.20 114.25 297.00 78.75 564.87 961.34 268.34 567.84 987.69 356.78 978.65 546.37 786.42 428.97 642.85 529.64 428.04 106.70 500.00 250.09 50 AUrrilMEMIO ^ I:' > 11 $147085.87^ 80997.75 SUBTRACTION. (-•) (3.) £ir)7.3.11.'U $81 07.31. .^7X 97G.15.10J"^ :M1870.G25 4. I liavc taken this month iutniilc $1790.18, and imid $071]. Ul for Fall goods, and expended for private purposes 8'}0.S() and lodged the rest in the Bank, liow much have 1 banked ? Ausj. $980.2^ 5. I bought 47 tons, 17 cwt., 1 qr., 18 lbs. of grain, and have sold 29 tons, 18 cwt., H (jrs., 22 lbs. of it; how much have I in store ? Ans. 17 tons, 18 cwt., 1 (jr. 21 lbs. 0. 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, how much farther is Baton Uouge from Washington than Dover? Ans. 1245 m. 7 1". 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 ol Loudon? 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 dees 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.) 846.38. (3.) $3. G5. MULTIPLICATION. 1. S1796X47=S84412. 2. £2.19.2ixl'i4=^£42G.3.0. 3. $108.87i-X64r=$10808. 4. £1.2.9 X*'225=£255.18.9. i i ■1 MULTIPLICATION. CI 5. Fiii'l tho duty ou 97 consignments of merchandise at 880.02^ each ? AnH. §8402.02^ It in often convenient to multiply denominate numbers by tho factors of tlio multiplier. Thus : to multii)ly by 84 is tho same n» to multiply by 7 and 12. Thus, in the annexed examples, sir.co 12x7—84,18 tons, 12 cwt., 2 qrs., 11 Ib3.x84, is tlio same as 18 tons, 12 cwt., 2 qrs., 11 lbs. X 12X7, &c. (G.) (7.) (8.) tona cwt. ({Tfi Ib^. 18.12.2.11X84 12 nc. roods, rd* 27.2.29. s X72 yds. n. III. 11.3. 7X150 5 223.11.1. 7 7 221 1 32 9 GO. 2. 11 5 1564.19.0.24 1093 .0 8 304. 2. 7G (9.) cwt. (ITS. lb.". 23.3.22X19 7 lbs. 49. 1829.0. (• [10.) oz. drf. 11.12X03 7 107. 3. 4 7 348 '!. 4 9 1174.2. 3 3133. 4. 4 j-da ft. in. 11 .. 3 .. 7 150 1050 87 450 .. .. 537 .. 179 1650 .. .. .. G .. 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., G in.,— f2) 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 1050 yards, which, added to the 179 already found, gives 1829 ft., so that the final result is 1829 yds,, ft., G in., a?. already obtained by the method of factors. 1829 .. .. C r^:r# I If* i cwl. qrs. Il'a I). 3. 224-80 AIUTHMETIO. £2.13.1} 125 857.1.17 cvvt. qiu lbs. 1.2.17-1-27 27 45.0. £331.18.0} G. How many seconds has a person livca 'wno nas completed hia twentieth year, the yc: consistinpj of 365 days, 5 hours, 48 minutes, and 48 seconds? ' Ans. G31138560. 7. Bouglit 7 loads of hay. each wcigliini^ 1 ton, 3 cwt., 3 qrs., 12 lbs; what did the whole wei^h ? 8. If a man can roan 3 acres and 35 rods per diiy, how much will he reap in 30 days? Ans. 9G acres, 90 rods. 9. If a steamboat ply across a channel, the breadth of which is equal to 2*^, 25\ 1»/ , what angular space has she traversed at the end of 20 trips ? Ans. 48"", 23\ 10. Hamilton, lloss <& Co., of Boston have charged me on an invoice of GO tons, 17 cwt., 1 qr., and 20 lbs. of iron, at $55 per ton, and 1 pipe, 1 hhd., 34 gals, and 3 qts. of wine, at $3. GO per gal. $4213.57, how much is this amount astray ? 11. If a man saves 45 cents a day, how much will he save in the year, omitting the Sabbaths ? 12. If 12 gallons, 3 quarts, 1 pint of molasses be nsed in a hotel in a week, how much would bo used in a year at that rate ? Ans. 10 hhds., 39 gals., 2 qts. 13. If a man can saw one cord of wood in 8 hours, 45 minutes, 50 seconds, in what time will he saw 1 1 cords ? Ans. 4 days, 24 minutes, 10 seconds. 14. If 13 waggons carry 3 tons, 15 cwt., 1 qr., 15 lbs. each how much do they all carry ? Ans. 49 tons, cwt., qr., 20 lbs. 15. If a man travel 20 miles, 5 furlongs, and 20 rods a day, how much would he travel at that rate in a year ? Ans. 7550 m., 7 fur., 20 rods. 16. There are 24 piles of wood, each containing 3 cords, 42 cubic feet; what is the whole quantity? Ans. 79 cords, 120 ft. 17. If 17 hhds. of sugar weigh 12 cwt., 1 qi'., 20 lbs. each, how much will the whole weigh ? Ans. 211 cwt., 2 qrs., 15 lbs. 18. Allowing 75 yards, 18 feet, for the surface of 9 rooms, how much paper would be required to cover the wall ? Ana. G93 sq. yards. S i)TVISION. 53 c 19. Purclia-scd from 11. Boll 493 cwt., 3 qrs., 21 lbs. of irou a- ? cents per lb. ; what docs it amount to ? 20. What must I receive for 2 lbs., 5 ozs., 14 dwts., 21 grs. o: g^jld, at $18.G0 per oz. ? 21. Delivered James Grant 7 tuns, 1 pipe, 49 gals, of Port Win( ul $2.75 per gal. ; what is the amount of the invoice ? DIVISION In Division, all remainders arc to be reduced to the next lowei denomination, and in that form divided, to get the units of that denomination. EXERCISES. 1. A silversmith made half-a-dozen spoons weighing 2 lbs., 8 ozs.. 10 dwts. ; what was the weight of each ? Ans. 5 ozs., 8 dwts., 8 grs. 2. If 45 waggons carry G85 bushels, 2 pecks, 4 quarts, how much does each carry on equal distribution? Ans. 15 bushels, 7-^ quarts. 3. If a labourer receives 149 lbs., 13 ozs. of meat as payment foi 20 days' work, how mucli is that per day, on an average ? Ans. 5 lbs., 12/'^ ozs, 4. If a steamer occupies 48 days, 17 hours, and 40 minutes, in making 121 trips ; what is the average time ? Ans. 9 h. 40 niin. 5. If 98 bushels, 3 pecks, and 2 quarts of grain, can be packed in 37 equal-sized barrels ; how much will there bo in each ? Ans. 2 bush., 2 pecks, 5!! qt.« G. If a man has an income of $75000 a year ; how much has he an hour, allowing the year to consist of just 3G5 days ? 7. An English nobleman has .£124,GS5 a year ; how much has he per minute, the pound being worth $4.84, and the year to consist of 3G5 days, 5 hours, 48 minutci', and 48 seconds? Ans. $1.14-1- 8. In a coal mine, 97 tons, 13 cwt., 2 qrs. were raised in 97 days ; how much was that per day, on an average ? 9. If 815.50 bo the value of 1 lb. of silver, what will bo the weight of S500Q00 worth ? ** ' ' Ans. 32258 lbs., 8 oz., 15 dwts., 11.^; grs. 1)1. i! i '4^ i m 54: a ARITHMETIC. 11. If 1216 bushels of wheat are produced in a field of 16 acres what is the yield per acre ? Ans. 77 bush., 3 pecks, 5 qts,, If pts. 12. A gardener pulled 13500 bushels of apples off 60 trees; how many, on an average, were in each bushel ? Ans. 230. 13. If 13 hogsheads of sugar weigh 6 tons, 8 cwts., 2 qrs., 7 lbs., what is the weight of each ? Ans. 9 cwt., 3 qrs., 14 lbs. 14. What is the twenty-third part of 137 lbs., 9 oz., 18 dwts., 22 grs. ? Ans. 5 lbs., 11 oz., 18 dwts., b^ 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, 26 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 be distributed equally among 23 poor persons ; how much does each get ? Ans. 5 bushels, 3 pecks, 1 quart. 19. A man having purchased 119 cwt., 3 qrs., 23 lbs of hay, and drew home in 6 waggons ; how much was on each waggon ? Ans. 19 cwt., 3 qrs., 23 lbs. MIXED EXERCISES ON DENOMINATE NUMBERS. 20. A gentleman, by his will, left an estate worth $2490, to be divided among his two sons and 3 daughters in the following propor- tions : — The widow was to receive one-third of the whole, less $346 ; the younger son $212 more than his mother; the older son as much as his mother and brother, lacking $335.50, and the three daughters were to have the remainder, share and share alike ; what was the share of each ? Ans. The widow got $484 ; the older son got $844-^7; the younger son got $696 ; each daughter got $155 J. 21. A gentleman left a property in land, consisting of 448 acres, 3 roods, 24 rods, to be divided among his four children in the following proportions : — The youngest was to get 4 acres, 3 roods, 6 rods more than the eighth part ; the second youngest was to get one- fifth of the remainder ; the oldest but one was to get one-third of the remainder, and the oldest the residue ; what was the share of each ? 1 DIVISION. 55 « 230. I Ans. The youngest got GO acres, 3 roods, 24 rods ; the next gut 77 acres, 2 roods, 1(J rods; the next got 103 acres, 1 rood, 34§ rods ; the oldest got 20G acres, 3 roods, 2*Jj rods. 22. A ship made the following headway on six successive days : On Monday, 3°, 8', 45" south, and 1°, 51' east ; on Tuesday, 2°, 3G' south, and 2°, V, 15" east ; on Wednesday, 4°, 0', 52" south, and 1° east; on Thursday, 1°, 48', 52" south, and 3°, IG', 22" east ; on Friday, 1°, 19' south, and 48', 29" east; and on Saturday, 59', 30" south, and 3°, 52', 11" east; 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 one week, 51 hogsheads, 53 gallons, 1 quart, 1 pint ; in the next week, 27 hogsheads, 39 gallons, 3 quarts ; in the next week, 19 hogsheads, 13 gallons, 3 quarts; liow much did ho sell in the three weeks ? Ans, 88 hogsheads, 43 gallons, 3 quarts, 1 pint. 24. In a pile of wood there are 37 cords, 119 cubic feet, 7G cubic inches ; in another there are 9 cords, 104 cubic feet ; in a third there are 48 cords, 7 cubic feet, 127 cubic inches, and in a fourth there are Gl cords, 139 cubic inches. Find the whole amount. Ans. 15G cords, 102 feet, 342 inches. 25. The 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 ot England cloth goods; 225 tons, 9 cwt., 12 lbs. of Scotch coal, and 106 tons, 1 qr. of Staffordshire pottery ; what is the whole amount of the consignment? Ans. G3G tons, 1 cwt., IG lbs. 26. If a man cai^ 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. 1G§ days. 27. The earth's equatorial diameter is 4184742G feet; how many miles ? Ans. 7925 and 3426 feet. 28. The earth's polar diameter is 7899 miles, 900 feet; how many feet ? Ans. 41707G20 feet. , 29. Sound i.'i calculated to move 1130 feet per second ; how far off is a cannon, the report of which is heard in 1' 9"? Ans. 77970 feet. 30. If the circumference of a waggon wheel be 14| feet ; how often will it turn round in a mile, ^^5280 feet; ? Ans. 3G0 times. Iv'l;. I -I -i 52 ARITHME' XiU- G-REATEST 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 submultiple, measure or aliquot part of the greater. Thus : 48 is a multiple of 2, 3, 4, 6, 8, 12, 16 and 24, and each of these is a sub- multiple of 48. When one quantity divides two or more '^others evenly it is called a common measure of those quantities, and the greatest num- ber that will divide them all is called the greatest common measure. Thub: 7 is a common measure of 63 and 49, and it is also the greatest common measure, for no larger number will divide both evenly. When any quantity is measured evenly by two or more others, it is called a common multiple of them. Thus : 24 is a common mul- tiple of 2, 3, 4, 6, 8 and 12. A number which can be divided into two «qujkl integral parts is called an even number, and one which cannot bo so divided is called an odd number. Hence all numbers of the ceries 2, 4, 6, 8, 10, 12, &c., are even, while tliose of the series 1, -3, 6, 7, 9, 11, &c., are odd. llence the sum of any number of even quantities is even ; also, the sum of any even number of odd quar»<^itics is csn ; but the sum of any oaa number of odd quantities is odd. This principle is of great use in checking additions. A prime nun\bcr is one which has no integral factors except itself and unity ; a. composite number is one that has integral fac- tors greater than urity, and numbers which have no common factor greater than unity are said to ho prime to each other. Of the first kind are 1, 2, 3, 5, 7, 11, &c., of the second, 4, 6, 8, 9, 10, 12, &c. ; also, 2 and 7 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 6, it will also measure 12, 18, 24, &c,, because it is a factor of all these. If one quantity measure two or more others, it will also measure their sum and diffbi'ence, and also the sum and difference of any GREATEST COMMON MEASURE. 53 multiples of them, because it measures them when they are taken separately. Hence, 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 21 and 18, and so the other part, 6 ; 9 divides 45 and 27, and also the remainder, 18. Also, if a number bo composed of several parts, each of which has a common factor, that factor will also measure their sum. Thus : 9 measures IS, 27, and 36, and their sum, 81. From these principles we can deduce a rule for finding tho greatest common measure of two or more quantities. \i r L E . Divide the greater by the less, and then the less by the re- mainder, until nothing is left, and the last divisor will bo tlio greatest common measure. EXAMPLE. 2145 1326 819 507 312 195 117 78 39 3471 2145 1326 819 507 312 195 117 78 78 A concise form of the work is exhibited in the margin. The quotients are omitted as unnecessary. The last divisor, 39, is the G. C. M., as may be proved by trial. If it is re- quired to find the G. C. M. of more than two numbers, first find the G. C. M. of two of them, and then the G. C. M, of that and another, and so on. EXERCISES. Find the G. 0. M. of tho following quantities : 1. 247 and 323. 2. 532 and 1274. 3. 741 and 1273. 4. 10416 and 25761 5. 468 and 1266. 6. 285714 and 999999. 7. 15863 and 21489. 8. 8280 and 11385. 9. 17222 and 32943. 10. 19752 and 69132. Ans. 19. Ans. 14. Ans. 19. Ans. 93. Ans. G. Ans. 142857. Ans. 29. Ans. 1035. Ans. 79. Ans. 9876. 54 ARITHMETIC. i I'i::':*' ,; \- ^■k Wc may oiton lind tlio G. C. 31. by inspection. For example, in exercise '), avc sec that 2 Avill measure bntli quantities (Art. 13), for botli arc even, and also that U will measure both, because it measures tiic sura of the digits (Art. IG. ' The least common multiple of two or more numbers i.s the smallest number that is divisible by all of them. Thus : 48 is a common multiple of 2, 3, 4, C, 8 and 12, but 24 is the least common i.uiltiple of them. It is plain that the least common multiplf^ of quantities that have no common factor is their product. Tlius : the L. C. M. of 5, 7, 6 is 210. But if the quantities liave a common factor, that factor is to be taken only once. Thus : 9G, 48, 24, are- all common multi- ples of 2, 3, 4, G, 8, 12, but the least of these, 24, contains only the factors o and 8, which arc prime to each other, for 2, 3, 4, G 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 tlic otlier, 12, gives 8x3=^24. From this we derive the RULE 2.. . 3.. . 4.. .G... 9. ..18.. .27.. .30 2|4...18...27...30 ^..« br. . vAj I • • alt) 312.. .27.. .15 2... 9... 5 9 45 2 90 3 270 o 540 Expunge all common factors iind take the continued product of all the results and divisors. Thus, to find the L. C. M. of 2, 3, 4, 6, 0, 18, 27, 30, ar- range them in a horizontal line, and as 2, 3, G, 9 are all contained in 18, they may be omitted, as in the second line, then, as 2 is contained in 4, 18 and 30, it .'nay be divided out, and as in the third line is contained in 27, it may be omitted, as in the fourth line; and 27 and 15 be- ing both divisible by 3, we ob- tain in the fifth line '2, 9, 5, all prime to each other, and the products of these and the divis- ors 3 and 2 is the L. C. M.. 540 m GREATEST COMMON ME.VSURE. 55 EXERCISES Find tlio L. (!. M. of tho followiiij^ quantities : 1. 8, 12, 1(3, 24, ^3. Ans. 523. 2. 33, 42, 45, 81, 100. Ans. 5G700. 3. 2, 4, 8, 10, 32, 04, 128 ' Ans. 128. 4. 2, 3, 5, 7, 11. Ans. 2310. 5. 3, 0, 27, 81, 243, 720. Ans. 729. G. 12, 16, IS, 30, 48. Ans. 720. 7. 3, 4, 5, 6, 7. Ans. 420. 8. 2, 3, 4, 5, G, 7, 8, 9. Ans. 2520. 9. 2, 4 7, 12, IG, 21, 5l3. ^Vns. 33G. 10. 2, 9, 11, 33. Ans. 198. EXAMPLES FOR P R A T f K . 1. What will 320 caps cost at $7.50 each ? Ans. $2400. 2. If you can purchase slates at 20 cents each; how many can you buy for 67.40 ? Ans. 37. 3. If you can wnlk 4 miles an hour ; how far can you go in 24 hours ? Ans. 96. 4. What will be the cost of 21 G barrels of pork at $7.50 per barrel ? Ans. $1620. 5. How many sheep can be bought for $560 at $3.50 per liead ? Ans. 160. G. If 825 pounds of beef are consumed by a garrison in one day ; what will be the cost for 6 days at 11 cents per pound for beef? Ans. $544.50. 7. A farmer sold 185 acren of land at $25 per acre, and received in payment 17 horses at $70 each, and 12 cows at $20 each ; how much remains due ? Ans. $3195. 8. A merchant bought 120 yards of American tweed at $1.15 a yard ; GO yards of flannel at 95 cents per yard, and 13 dozen pairs of gloves at 35 cents per pair ; what was the amount of his bill ? Ans. $249.60. 9. At $2 por gallon ; how much wine can be l)ought for $84 ? Ans. 42 gals. 10. A boy had $5.50, and he paid one dollar and five cents for a book ; how much had he left ? Ans. $4.45. 11. What will 18 cords of wood cost it $4.75 per cord ? Ans. $85.50. 56 ARITHMETia. 12. IIow many pounds of sugar can be bought for $9.35, at 11 cents pci pound ? ^ Ans. 85 lbs IIJ. AVhat will a jury of 12 men receive for coming from Kings- ton to Albany at 10 cents a mile each ; the distance being GO miles ? I 14. A grocer bought a hogshead of molasses at 32 cents per gallon ; but IS gallons leaked out, and he sold the remainder at 55 cents per gallon ; did he make or lose, nd liow much ? Ans. He gained $-i.59. 15. If a clerk's salary is $C00 a year, and his personal expenses $320 ; how many years before he will be worth $GGOO, if ho has SI 000 at the present time ? . Ans. 20 years, IG. A speculator bought 200 bushels of apples for $90, and sold the same for $120 ; how much did lie make per bushel? Ans. 15 cents. 17. A person sells 15 tons of hay at $2*^ per ton, and receives in payment a carriage worth $125, a cow worth $45, a colt, worth $40, and the balance in cash ; how much money ought ho to receive ? Ans. $120 18. How many pounds of butter, at 20 cents per pound, must be given for 18 pounds of tea worth 75 cents per pound ? Ans. G7i- lbs. 19. A grocer bought 7 barrels offish at $18 per barrel ; but one barrel proved to be bad, which he sold for $5 less than cost, and the remainder ac an advance of $3 per barrel ; did he gain or lose, aud how much ? Ans. Lost $13. 20. A man bought a drove of cattle for $18130, and after sel- ling 84 of them at $51 each, the rest stood him in $43 each ; how many did he buy ? Ans. 40G, 21. What will 2 cwt. of cheese cost at d^r cents per pound ? Ans. $10 00. 22. A. is worth $9G0, B. is worth five times as much as A., less $600, and C. is worth three times as much as A. and B. and $300 more ; what are B. and C. wortli each, and how much arc they all worth ? Ans. B. $4200 ; C. $15780 ; all $20940. 23. A boy bought a dozen knives at 15 cents each, and after selling half of them at the rate of $2.22 per dozen, he lost three, and sold the balance at 25 cents each ; did he make or lose, and how much ? Ans. Gained G cents. 24. A labourer bought a coat worth $1G, a vest worth $3, and a GREATEST COMMON MEASUKE. 57 pair of i)aiVs worth S5.50 ; how many clays had he to work to pay for hia suit ; his services being worth 50 cents per day ? Ans. 49 days. 25. What will 14 bushels of clover seed cost at 12^ cents per pound? Ans. $105. 26. A farmer sold a load of cats weighing 1836 pounds, at 30 cents por bushel ; how much did he receive for the same ? Ans. SI 6.20. 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 ? Ans. $250.1 5 J. 28. A farmer has 12 slieep worth $3.50 each ; 9 pigs worth $4.65 each ; one cow worth $35, and a fine horse valued at $150. He exchanges them with his neighbour for a yoke of oxen worth $75 ; two lambs worth $1,925 each ; a carriage worth $100, and takes the balance in calves at $4.50; how many calves does lie receive? Ans. 20. 29. A and B sat down to count their money, and found that they had together $225, but A had $15 more than B ; how much had each ? Ans. A $120, B $105. 30. A miller bought 250 bushels of oats for $85 and sold 225 bushels for $70 ; what did the remainder cost him per bushel ? Ans. 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 ftirm, and one-third the value of the houses, and then 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 $800, 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 cf 6 for 3 cents ; how many apples did he purchase ? Ans. 60. >'A ii &\ }■ i ^'^ IjiU: 58 ARITHMETIC. 'IL A schoolboy bouglit 12 oranges at .'} ccnt.s each, and sold thcni lor 12 cents more than he paid tor thcni ; liow much did ho gcll tliem at cucli ? Ans. 4c. 35. A clerk's income is $2098 a year, and his expenses $4.50 per day ; how much will he save in two years ? Ana. S2111. 30. A speculator bought 200 acres of land at $45 per acre, and afterwards sold 150 acres of it for $11550 ; the balance lie sold at a gain of $5 per acre, and received in payment $250 cash, and the balance in sheep at $5 each ; how many sheep did he receive ? Ans. 450 sheep. 37. A butcher bought calves for $54, and 9 lambs for $31.50 ; how much more did he pay for a calf than a lamb ? Ans. $2.50. 38. A fiirmer sold to a grocer 380 pounds of pork, at 7 cents per pound ; 150 pound.9 of butter, at 17 cents per pound, and one cheese weighing 53 pounds, at 9 cents per pound ; and received in payment 22 pounds of sugar, at the rate of 11 pounds for a dollar; 150 pounds of nails, at G cents per pound; 15 pounds of tea, at 05 cents per pound ; one half-barrel offish, at $18 per barrel, and one suit of clothes worth $27 ; did the farmer owe the grocer, or the grocer the farmer, and how much ? Ans. the grocer owed the farmer 12 cents. 39. 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 sliect- ing, at 10 cents per yard ; how many yards did lie receive ? Ans. 45 yards. 40. How many pounds of cheese, at 9 cents per pound, must be given for 27 pounds of tea worth 80 cents per pound ? Ans. 240. FRACTIONS. 14. — Vulgar or Common Fractions. — When we have di- vided any number by a loss, and find no remainder, the quotient is called an integer, or whole number. When we liave 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, i^. rilACTIOXS. 51) Tlio nature of a Iniction may bo viewed in two ways. Flnt, wo may consider that a unit is divided into a certain number of c([ual parts and a certain number of I heso parts taken ; or, secondlt/, that a number greater than unity is divided into certain equal parts, and 07ie of these parts taken ; thus, ^l means either that a unit is divided into 4 equal parts and three of them taken, or that three is divided into 4 equal parts and one of tliem taken. For example, if a foot bo divided into 4 equal parts, each of these parts will be 3 inches, and three of them will be nine inches ; and aince 3 feet make 30 inohcs, if we divide 3 feet into 4 equal parts, each of these parts will be 9 inches, and hence f of l-=\ of 3. The lower figure is called the denominator, because it shows the denomination or number of parts Jito which 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 arc spoken of together they are culled the terms of the fraction. What may bo considered the fundamental i)rinclple 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 terms are eitlier multiplied or divided by the same quantity. If we take the fraction ^ find multiply its terms by 2, wo get |. Now, the Jj- of a foot is an inch and-a-half, and therefore g is 6 inches and G half-inches, or inches ; but we have seen that f of a, foot is 9 inches, therefore ^- of a foot is the same as g of a foot. So also :J Oi' £1 and | of £1 are both 15.« The same will hold good whatever tic unit of measure may be, or whatever the fraction of that unit. Hence, universally the form of a fraction is altered if its terms be eithor multiplied or divided by the same number, but its value 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 6 times three inches, or 18 inches, which is double of 9 inches, the value of £. We should have obtained the same result by taking | and dividing its denominator by 2, without dividing its numerator. Hence, a fraction is multiplied by either multiplying its numerator or dividing its denominator. In like manner, if we take the fraction g and divide its numerator by 2, we obtain §, and if we multiply the denominator of its equal f by 2, we obtain the same rusult, g-. Hence, ^ is ^- of f , and therefore .i frac- tion is divided by either dividing its numerator or multiplying its denominator. These principles may also be referred to the obvious GO ARITHMETIC If 1- itei |i I fact that in dividin;^ any quantity tho greater tho divisor tlio less th« quotient, and tho le33 tho divisor tho greater the quotient. As it la always desirable to have tho smallest numbers possiblo to handle, lei tho operator observe this as u universal r\x\o~divide lohcn you can. Fractions arc classified in four different ways, according to four different circumstances. I. They arc divided into Proper and Improper Fraction.s. A proper fraction is one whose numerator is less than its denomi- nator. In strictness such alone is a fraction. An improper fraction is one whose numerator is greater than its denominator. Strictly this is not really a fraction, but only a certain quantity expressed iu the fractional form. II. Simple and Compound Fractions. Tho term simple fraction, as opposed to compound fraction, expresses that tho fraction is multiplied by unity alone, ns f , which means either §• of 1 or ^ of 5, or f X 1=^X5- A compound fraction is one that is multiplied by some other quantity. A fraction is called compound if cither multiplier or mul- tiplicand, or both, be fractional. Thus: ^ of § and ^ of 11 are both compound, and are written f Xg and f XH- III. Simple and Complex Fractions. The term simplo fraction, as opposed to complex fraction, means that there is only one division. Thus: jg means that a single number, 15, is divided by a single number, 16. A complex fraction is one of which either the numerator or de- nominator, or both, are fractional, that is, it indicates a division, when either the given product or given factor, or both, are fractional. Thus : f -T-j\, or ± and f and jji are complex fractions and cx- 1 1 hibit the only three posssible forms. IV. Vulgar, or Common, and Decimal Fractions. Decimal fractions are those expressed with a denominator, 10, or a power of 10, e. g., j\, j^^%, y^^^. Any fraction not so expressed is called vulgar or common. Thus : -J would be called a common fraction, but its equivalent, /(fg, would be called a decimal fraction, and is written '75, the denomina- tor being omitted, but its existence being indicated by the mark (•), called the decimal Doint. rnACTioNs. 01 A T.iixcd quantity i.s ono cxprcsacd ptirtly by a whole nuniboj tind partly by u fraction, as 'IJ, 12A-. This is not uuother kind o' fraction, but simply another mode of writinj^ an improper IVaction when the division indicated h is boon pori'ornjcd as far as possible Thus: -y^ 4J, and -^-lliL It is often said that there ure six kinds of fractions — proper improper, simple, compound, complex, and mixed. This is lo^i ciliy incorrect, for a proper fraction is simple, and a mixed (juantit} is an improper fraotidu in another form. 15. — OPKaATioNM IN Common Fuactions. — From tho prin cipksf laid down (Art. 21,) wo can doduco rulc;4 for all tlie operations in fnt'tious. I. An improper fraction U reduced to a mixed quantity by per- forrain* tho division indicated, as 3.}^l^:=:24:}y II. A mixed quantity is reduced to an impron ar fraction by multiplying the integral part liy tho denominator and adumg in tho numerator, as 12J=-'2^. So also an integer may bo expressed in tho fractional form by •writing 1 as a denominator, and multiplying tho terms by whatever number will bring it to any requirtd denomination. Thus: to reduce 7 to tho same denamination as §, write } and multiply the terms by 6,. and the result, y, will bo equivalent to the integer 7, and of the same form as ^. RXERCISES. 1. Express A^± as a whole or mixed number. 2. Express ^J as a whole or mixed number. 3. Express ^/'J- as a whole or mixed number. 4. Express -'A^- as a whole or mixed number. 5. Express -'/gVif ^^ ^ whole or mixed number G. Express Yg''- ^ * whole or mixed number. 7. Express ^^ as a whole or mixed number. 8. Express f 3 as a whole or mixed number. 9. Express -['3- as a whole or mixed number. 10. Express -Y,^- as a whole or mixed number. 11. Express '^Y as a whole or mixed number. 12. Express ^7j-- as a whole or mixed number. 13. Express ^,'^'- as a whole or mixed number. 14. Express >" as a whole or mixed number. 5 Ans. 49. Ans. b-^Q. Ans. 71. Ans. 5A^. Ana Jil 9H 1 Ans. 11^^'g. Ans. 7,'3. Ans. 7j\, Ans. 89. Ans. 10/,. Ans. 211 Ans. 19'^. Ajis. 12/^. Ans. 44. C2 Ar.miMETic. 15. IG. 17. 18. 19. 20. 21. 22. 23. 24. 25. 2G. 27. 28. 29. 30. Express J-|J- Express -Jj)- Express Y" ^^ a whole or mixed number. Express -\J/*- as a whole or mixed number. Express ^ 'l as a whole or mixed number. Express -V- as a whole or mixed number, as a whole or mixed number. as a whole or mixed number. Express -\f^^^~ as a whole or mixed number. Express 271 as an improper fraction. Express 66 J as an improper fraction. Pjxpress 15}| as an improper fraction. Express 7f as an improper fraction. Express 49 as a fraction with the same denominator asis. Ans. -«,V- Express 19s. as a fraction of £1. Ans. J^ Express 11 inches as a fraction of a foot. Ans. -j-A i) h' ii o» 12 ^^ ^^^ same denomination. Ana " ■' 3 2 1 X1.US. jTT, ]T7, , rr, J 3, ,o Ans. 2^. Ans. 5^|. Ans. 3/7. Ans. 5A. Ans. 30|. Ans. 83/g. Ans. 9^0. Ans. -Af. Ans. ^K .Vns. iU'^. -V- Ans. Bring 1 7 Express 11 as a fraction having the same denominator as ^ ^ j Ans. -VdV-. III. To reduce a fraction to its lowest terms or simplest form, divide the terms by their greatest common measure. This is often readily done by inspection, as ^^=i='^, but in such questions as h^i% the most secure and speedy method is to find the G. C. M. of the terms and divide them by it. Thus : the G. C. M. of the frac- tion J2§^ is 1092, and the terms of the fraction divided by this give ^, tlie simplest form. EXERCISES. t: :,'!| a . 1. Reduce y's^j^J'g to its lowest terms or simplest form. 2. Reduce 1^^] to its lowest terms or simplest form. 3. Reduce jfrff to its lowest terms or simplest form, 4. Reduce yVuVo^; to its lowest terms or simplest form, 5. Reduce H^^ to its lowest terms or simplest form. 6. Reduce TVybVo ^^ ^^^ lowest terms or simplest form. 7. Reduce j^s It ^^ '^^^ lowest terms or simplest form. 8. Reduce ,J.J^| to its lowest terms or simplest fortu 9. Reduce pf;* to its lowest terms or simplest form. Ans. ^. Ans. -^j. Ans. |. Ans. Jg. Ans. |. Ans. 1, Ans. y'^. Ans. f . Ans. f . FRACTIONS. 63 10. 11. 12. 13. Reduce Reduce Reduce Reduce 3M yT *° ^^^ lowest terms or simplest form. Ans. T^^I'-ih *^ ^^^ lowest terms or simplest form. Ans. ^^^'l to its lowest terms or simplest form. Ans iViiSVifb *^ ^^^ lowest terms or simplest form. 3" 11' 19 14. Reduce |J-|f f to its lowest terms or simplest form. Ans. f . 15. Reduce ^jg| to its lowest terms or simplest form. 16. Reduce ayf | to its lowest terms or simplest form. 17. Reduce j^g to its lowest terms or simplest form. ] 8. Reduce aVg^ *° ^*^ lowest terms or simplest form. 19. Reduce 9 1,4 31 *^ ^^^ lowest terms or simplest form. 20. Reduce {||j§g to its lowest terms or simplest form. Ans. Ans' Ans. Ans. 7 U' 3 \ns. l^. 35* I- Aus. 1 1 T3' 21. Reduce fi^"Hf74'oi?o%"(fd ^o its lowest terms or simplest form. Ans. i. 1 ... 1 ... 1 111. ..111. ..Ill IV. To multiply one fraction by anotlicr, iimitiply numerator by numerator and denominator by denominator. Thus : JXj==i- To illustrate that I of J- is ?„ take a line and let it be divided into 3 parts, and each of those again into 3 parts, as in tlie margin, we find that the result is 9 parts, cncli, of course, being ^ of the unit. We have seen that a fraction is multiplied by multiplying the numerator or dividing the denominator. Now, if it were required tx> multiply f by ij, we could not divide the denominator, as 5 is not contained in 4, and therefore we multiply'the numerator and obtain ^^^-, but we have multiplied by a quantity equal to 7 times the given one, and therefore we must divide the product by 7, i. c. (Art~ 21,) we must multiply the denominator 4 by 7, which gives i| for the correct product. EXEKCISEB. 1. Multiply -j^ by -|^? 2. What i.. :he product of 5- by \l ? 3. What is the product of ,Tj by -^ ? 4. What is the product of '^ by \^ ? 5. What is the product of ^ by Ag ? Aus. /i Ans. Ans. Ans. Ans. 3 3 ■r-i* fif. 1 i) 4 0' 64 ARITHMETIC. m vt§h n.ns. ■Q^y Ans. Ul 3 2 4 5'' ins. Ana -?■'• 6. What is the product of I by -j^^ ? 7. What is the product of -,oj'g by -j^j ? 8. What is the product of j;^ by -j\ ? 9. What is the product of |]- by ^ ? 10. What is the product of ] ~ by /^ ? When tlio product lias been obtained it should be reduced to its lowest terms. Thus : the product of /,- by ] .', is -^^"ri, the terms of which arc both divisible by 11, and so we get the equivalent fraction ■j-'^j. But we might as well liave divided by 11 before multiplying, for by this method we should at once have found the fraction in its simplest form, viz., y^. In the same manner, any number or num- bers which are factors of both numerator and denominator, may be omitted in the operation. This wo call cancelling in preference to the excessively awkward term ''cancellation." This method will bo clearly seen in exercise 11. If either the multiplier or multiplicand be a mixed quantity, ifc must be reduced to an improper fraction before the multiplication is performed. Thus: 8|x55-iy^X¥=iii^=5l2lf. 11. What fraction is equal to J of -| of f of | of | of « of | of f ? Ans. ^. 12. What quantity is equal to 12|- multiplied by 7§ ? Ans. 97j^. 13. Wliat quantity is equal to 19^ multiplied by 1{'^? Ans. 3(5. 14. What is the value of g of § of f « of J- ? 15. What is the value of J of f of § of j-} ? 16. What is the product of 27^ by 3| ? 17. What is the product of -| § by -]» ? 18. What is the product of 5^ by 6h ? 19. Find the square and cube of 1% ? 20. What is the cube of fg ? 21. Multiply 27 by .^ '^ An«! 4 4 Ans. 107f^ Ans. y^jj. Ans. 30|. Ans &-^2 mid _4J)_L3 Ans. 1. V.-DIVISION OF FRACTIONS. To divide one fraction by another, multiply by the recipro- cal of the divisor ; or, in other words, invert the divisor and multi- ply. In the language of science, the reciprocal of a fraction is the fraction with its terms inverted. Thus : ^ is the reciprocal of | ; 4 of f . To find the reciprocal of a whole number, we must first # ..^' #' DinsiON OF rruVCTioxs, 05 represent it as having a denominator 1, — thus 4---1 ; 0:^';, and therefore the reciprocals are J and ^-. The rule for division may bo proved in two ways : First troof.— Let it be required to divide -^^ by •). If we had been required to divide by the wliole number 5, we sliould cither have divided (Art. 14,) tlie numerator, or multiplied the denomina- tor, — as the numerator is not divisible by 5, we multiply the de- nominator, and obtain J^ ; but we have divided by a (juantity equal to six times the j^iven one, and therefore, to conipensate, we must multiply the result by G, which gives |^. Second proof. — Write the question in the complex form — 1 1 ~, then (Art. l-l,) multiply both terms by 11, and Til? is obtained ; and again multiply the terms by G, and ;1 1 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 multiply." Mixed quantities must be reduced to improper fractions as in niultlplication. The expressions multiplication and diuision, as ap- plied to fractions, are extensions of the ordinary meanings of those terms, for in their original meaning, the former implies increase, and the latter decrease ; l)ut 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 than either the divisor or dividend. This will be seen by tlio annexed examples : f X^=y- Cut f:=2^ and ^=3^, both greater than ^},. I3,} and f^^f , both less than Also, ^-r-^^^X 7\/4- -2 8 But -|: 28 54- If two fractions have a common denominator, their quotient is the quotient of their numerators. Wc have placed multiplication and division of fractions before addition and subtraction, because, as in whole numbers, multiplication and division are deduced from addition and subtraction, so conversely in fractions, addition and subtraction arc to be deduced from multiplication and division, for a fraction is produced by division, and the multiplication of a fraction is merely the repeating of the divided unit a certain number of times. Thus : i- is a unit divided into 8 equal parts, and -J is that I'raction repeated 7 times. u (36 iVr.rrrDiETic. ^"0 EXERCISES. Iri ii! m\^ i 1. o 3. 4. AnS. r;^:T Ans. If: Vi Ans 5. 2 4 8;-. Ans. j|. Ans. ^5. -^ Jb'J" _3_ 10* Ans. aofl. Ans. 8i;;^. Ans Ans Ans Divide ^V by S ; TV--5=-AXi;. Wliat is the quoticiit of ] ■^^ divided by -j-l ? AVliat is the quotient of ^'^^ divided by ^1-v ? AVhat is the quotient of ~ -^ divided by 5 '^ ? AVhat is the quotient of ■}. \ ;] divided by ^ ij ? G. AVbat is the quotient of 3G divided by 19^ ? 7. What is the quotient of 3-;; divided by 2§? 8. What is the quotient of 4^ divided by 15 ? 9. What is the quotient of f !,' divided l)y 2}3 ? 10. What is the quotient of 75/q divided by 9 ? 11. What is the quotient of liAf divided by ^ ? 12. What is the quotient of ^ divided by 8^^ ? 13. Divide the product of |-, A and § by the product of A-, f and Ans. -y-=::;l?. 14. What is the quotient of /p of U~| off} of JL-^-Sj off? Aus.4g«. 15. How many -./,- are there in -f^j? Ans. 8|-!). 16. What is the "value of § of -J~-f of |J ? Ans. 1^%. 17. Divide 27 by r,\- ? Ans. 729. Honce, any quantity divided ])y its reciprocal gives the square of that number, and exercise 21, of multiplication, shows that any quantity multiplied l)y its own reciprocal gives unity. Ans. Ans. •10 8 "7M f J 1 • Divide ^^JY by jj, and the quotient by -fy ? Divide 4 by -/p, and the quotient by 3^ ? Divide :5§ by fa *? 18 19 20, 21. Divide 47 1 1 byil^v Ans. 1/t. Ans. ^§. Ans. S-j-VsV- Ans. h. VI.- ADDITION OF FBACTIONS. We have seen that no quantities can be added together except they are in the same denomination. We can add 4, |, | and -y-, as they are all of the same denomination, sevenths, and we find -v'. AVe can easily sec that to add :} and |, wo have only to alter the form of ± to {', and we have both fractions of the same dcnomiuation, and therelbre can add them, — |]-}-|=r: -y . 00, aibO, ;j-r5 I 4 I u I i-^-lul i l;J I Jv; I l;; I 13 — li — 3 ' But we cannot always tell thus by inspection, and therefore must be guided by some rule. To find the value of a-j-;!- |-a + (i+Ti• ^■ ADDITION OF FPiACTIONS. G( 72, By Art. 13 we find the L. C. M. of 4, 0, 8, D, 12 to b and the rest of the common operation is equivalent to multiplying the terms of each fraction by 72. Thus : if tlie terms of '} be both multiplied by 72, we get 5a "=7 4^:1=71, but we might as well have divided 72 by 4 before multiplying, and, to balance that, have multiplied the numerator o, not by 72, but by the fourth part of 72, viz., x8, giving !•;], as the following scheme will show: — The other fractions bein'jr altered iu ■AXTi -:!X1 8X1 iXlbX'l- ..1X1 ft 51 •1X1 8 1\i- the same manner, we get 7iH~?5+72+r"~r^-' '^"^ ^^ ^^*^^*^ ^^^ 7 -J) now all of the same denomination, though not altered in value, wo can add them, and we find f+l+^J- 2.1.^^5 4 _|.««_^«^_|_^^ 4 2 -2 3 6 72 — "73 ' Hence the RULE Find the L. C. M. of all the denominators, which will bo the common denominator ; divide this common multiple by each denomi- nator, and multiply the quotient by each numerator in succession for new numerators ; add all these new numerators together, and place the common denominator below the sum, and the fraction thus ob- tained will be the sum of the given fractions. If the numerator, thus obtained, be greater than the denominator, the resulting frac- tion may be reduced to a whole or a mixed number by division. EXERCISES. 1. Express -/ j+-f5-rTs+i''5 as a single fraction ? Ans. ||. 2. Find the sum of i, f, f and g- ? ' Ans. 2|. 3. Add together 4^,1 jj, 2||, 3|o and 5/3 ? Ans. 18j«(fg. 4. What fraction isequal to ^+i+^-|-T(i-i-3 3+o'i '• ^^^s. 5^. 5. What fraction is equal to UH-2f+3f +4^+5i}+6« ? Ans. 25.1 3 i- 6. Express ^ of f -{-5 of g+f of A- as a single fraction ? Ans. ^^=::l^l. 7. Find the sum of 1 j 4, 8|, ii^ and 4^ ? Ans. IB^V,-. 8. Find the sum of -J of H-73" of 7 -r3 of « ? Ans. 1%. 9. What single fraction is equivalent to jr of ^-i-;\ of '\~\-^ of J ? ■ Ans. Yo- lo. What single fraction is equivalent to f of '^ of l-!-^ off of ■lof^of^? ' Ans. -V 11. AVhat single fraction is equivalent to § of -^ of -^-[-f of ^ of Ans. I 1 ^ !| ^ it 1) Cm i i '^ n: "1 ''!i ■.\ I-' Mi 1'!) &t mil I( ,. — .t'.i'i ii ^;ii 3 Kit C8 MIITHMETIC. Ans. 1..1 nn' 12. Simplify -Vi±-iii? 13. Find a singlo fraction equivalent to -l- of § of §+§ of ^ ? 14. Divide the sum of /j and ^ by the sum of § and ^ ? 16. Simplify |-i|ijl ? Ans 127 \n<5 •'"'1''' VII.-SUBTRACTION OF FRACTIONS. Wnat we have said of addition enables us to give at once the llULE FOR SUBTRACTION. Reduce the given fractions, if necessary, to new ones having a common denominator, as in addition, and subtract the numerator of the less from that of the greater, and place the common denominator . below the remainder, and the resulting fraction will be the difFercnee between the given fractions. Examples. — (1.) To subtract /',- from {'^. Here the denomi- nations being the same, we can subtract at once, and find the differ- ence to be -j\. (2.) To find the value of ^ — ^. These fractions brought to a common denominator, as in addition, become §^ and 2I, ai:.d therefore the difference is ^?^. (3.) To find the excess of 12^ above 7^, we find new fractions with a common denominator, viz., .,^1 and 1;J, and we write 12r,^j — 71;['. Now we are required first to subtract l'] from ■.f,^, but as we cannot do this directly, we take one of the 12 preceding units, and call it ^ |, (for H;i---1, ) then :-;^^»_[-_8j.rr=.'];^, and |'| — 1^:=11, then we subtract the 7 from the re- maining 11 ; or, as in simple subtraction, 8 from 12, and we find the total excess to be 41]. In practice it is most convenient to sub- tract 15 from 24, and add 8 ; tlms 24—15^9, and 9-f 8=17, and the answer i? 4 ■■I 1 ' EXEIICISES, a. e a (j o* -• II 11 II* '^'•13 J 5. What is the difference between 3 and — J i J 3' 2;> .', 10. Whutis the difference between 5, ^j and Gw'Airt "i" Aus. '.-i^,', 1 1 . Wluit is the value of ^J-j-jj—^-f -i— J ? ^ " ' Any" 12. What is the dilfercnec between 100,^,, ajid 50,'<,' ? Ans. 49^^-5. 13. What is the difference between h of -J- and J^ of ^ ? Ans. 14. What is the difference between ^ of /„ and j} of ;] ? Ans. ^,^. 15. What is the value of *+§—$—;• + [A ? Ans. ^. VIII.-DENOMINATE FRACTIONS. Hitherto we have treated of fractions abstractly, and we must now apply the principles laid down to denominate numbers, and show how a fraction may be transformed from one denomination to another of the samo kind, e. g., how a fraction of a shilling may be expressed as a fraction of a pound, and vtce versa. RULE. (1.) Reduce the given quantity to the lowest denominatiomohich it ex2)r€sscs . 2.) Reduce the unit in the terms of which it is to be expressed to the same denomination, and (3,) make the former the numerator and the latter the denominator, and the fraction will he expressed in the required terms. E X A JI r L E S . 1. To express 2 ft. 9 in. as a fi-actionof a yard. Reducing 2 ft. 9 in. to inches, wo get 33 inches, and one yard is 36 inches, — the fraction therefore is •• ■: or ] .', . 2. In like manner to express 2 qrs., 2llbS: as the decimal of a cwt. we have 2 qrs., 2-1 lbs=:=7-l: lbs., and 1 cwt., is 100 lbs., so that the fraction is -^^\y .37 3. So also 3 roods, 32 rods, expressed as a fraction of an acre is 1 a2 or .10 rp V^ or -.v:; of an acre. 4. To express 17 cwt., 2 qrs., 10 lbs. n^ a fraction of a ton wo have -^"" ■.\r, "2 0" " i " _!» .) (3 • Ans 5. Express 4S minutes, 48 seconds as a fraction of an hour. G. Express 13s. 4d. as a fraction of £1. 7. Express 3G rods as a fraction of a mile. 8. Exm'ess 4s. 4d. as a fraction of £1. 9. Express 4M. as a fraction of Is. 10. Express 1 oz. troy as a fraction of 1 lb. Ans. £'2. sr, — <)_ "SiO So* Ans. £11. Ans. -^'s. Ans. -f'.T. 1 >M i' 'i V Hi: V, ftj: 'M I . :\:' 1^^ III i t i!«^^ 70 AltlTHMETIC. 18. Express Iv. ' /a 19. Express 45 coiu-. 20. Express GO ll 11. Express 40 Iha. as n, fraction of 1 cwt. Ans. r cwt. 12. Express 50 lbs. us a fraction of 1 ton. Ans. ,',, ton. 13. Express 72 lbs. r,» a fraction of 1 cwt. Ans. h ^ cwt. 14. A day is 23 hours, 56 minutes, 48 seconds, nearly ; what fraction of this will 7 hours be ? Ans. yV.j^j- 15. Express 95 square yards as a fraction of an acre. Ans. y'^'y. 10. Express 14 yards as a fraction of a mile. Ans. yj,,. 17. AVhat fraction of a year (3G5,^ days) is one month (30 days)? Ans. ^Vi- ai a fraction of a mile. Ans. jfg. fraction of a dollar. Ans. jv'g. X, r,!ic*ion of a cwt. Ans. ?. 21. A man has an income oi ^3010 a year, and saves ^ of it; how much does he spend ? Ans. S20G2IJ. To find the value of a fraction in the denominations whicli 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 either 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| ; so that IJ of a ton is 17 cwt., qrs., 14| lbs. EXERCISES. What is the value of ■{'., cf a ' on ? Ans. 1 1 cwt., 2 qrs., 1G§ lbs. What is the value of /„- of a yard ? Ans. 2 feet., 8 r in. What is the value of ^4' of a mile ? AVhat is the value of Jlj of a shilling Stg. ? Ans. 11 Sd. 5. What is the value of j of a ton ? Ans. 11 cwt., 1 qr., 17^ lbs. G. What is the value of ^ lb. troy ? Ans. 8 oz. 1. 2. 3. 4. 7. What is the value of -f'.^ of a shilling ? Ans. 5/3d. 8. What is the value of $« ? Ans. 88« cts. 9. What is the value of ^ of $6 ? Ans. 84.80. 10. What is the value of A J of 88 ? Ans. $G.80. To change a fraction to one cf a lower denomination, reduce the numerator to that dcuominatiou, and divide by the denominator. Thus : yj- of ^ dollar is 700 cts. divided by 145, which gives 41^. DECIJLVL I'liACTIONS. 71 Ans. §i. Ans. ;]. Ans. ,). Ans. i;;. Ans. ti. exj:ucises. 1. Express j^jj of a foot as a fraction of an inch. 2. Express j!,j of a cwt. as a fraction of a lb. 3. Express v,',j of a lb. as a fraction of an oz. 4. Express ^ of ,•'_, of a yard as a i'raction of a foot. 5. Express ■.^\ of a rod as a fraction of a yard. (]. Express ^j of -J of an acre as a fraction of a rood. Ans. \. 7. llcduco ;j',; cwt. to the fraction of a pound. Ans. 11 ,'^., lb. 8. llcduco v,',- of a day to the fraction of a minute. Ans. 68 :| inin. 9. What part of a second is the one-millionth part of a day ? Ans. jfv,^^ sec. 10. Reduce £^\j to the fraction of a penny. Ans. O'jc'. 11. Reduce r,', of a pound avoirdupois to the fraction of au oi: Ans. ;' oz. The reducing of a denominate fraction from one of a lov .u o one of a higher denomination being the converse of the last r.w, w,; must perform the .same operation on the denominator as wa3 tli^. o performed on the numerator. Thus, ■^d. IS Xjh:}, for ^gxTixTo ^^^ivm^^^til' EXERCISES. 1. What part of 1 lb. troy is ^ of a grain ? 2. What part of 4 days is | of a minute? 3. What part of 5 bushels is ^ of -^ of a pint? 4. What part of a rod is 2 ^ of -^^ of an incli ? 5. What part of 2 weeks is -^^j of a day ? Ans. ^Jgj,. Ans. g^j- Ans. 1 'JG" DECIMAL FRACTIONS. 16, — We have seen already (Art. 3,) that every figure to the right is one-tenth the value it would have if removed one place to the left. Thus, resuming our former example, 8 standing alone means 8 unita, but if we place another 8 after it, thus 88, it now means 8 tens, so that the lust 8 is one-tenth of the first. Now, since the 8 to the right expresses units, another 8 placed to the right will express eight-tenths of the same unit, and another .subjoined will express ^ go ^^ *^^° ^^"^^' Thus wc see that the decimal notation i.s directly au extension of the Arabic. Hence arose the convenient mode of writing 8j"(j in the form 8.7, by which is indicated that all 72 Ar.ITII.'^IETIC. the fij^urcs before tho dociinnl point (.) reprosent intecjcr?, and all after it fractions, each bciiii; onc-fcnth of whiit -t would be W one place further to the left. Therefore 888.888 is cjjht hioidycds, eight tens, eight, nnllx, — elght-tciitha, eight onc-hundrcdths, and eight one-thousandths; or, /()-[• too"!" id'oo- 'fheso added will fiivo 800 I HO i_ . fl _ „.. R8fl which for brevitv is writ ton 888 1000 I 1000 I 10(»0' 1000' *''"^") "" UH-VllJr, in MllllLll .OOO, and may bo read eight hundred and eiglity-eight one-thousandtli.s ; or, as is usual, ^)om< 888, or decimd 888, but never properly eight liundrcd and eighty-eight. In the same manner as 80 means 8 tens and no units, so .08 means no tenths, but 8 hundredths, and .008 means no tenths, no Imndrodths, but eiglit one-thousandths, &c. — llenee we see that for every cipher in the denominator, which is always 10 of a power of 10, there must be a ligure in the numerator when expressed decimally. Thus : , (,"oj; must be written decimally .008. From this we pee that removing the decimal point one place to the right is the same as multiplying by 10, and removing it one place to the left is the same as dividing by 10 ; so, also, removing the point two places to the right is the same as multiplying by lOQ^ and renwving it two places to the left is the same as dividing by l(iQ. This is the principle already laid down for the reduction of dollars to cents, and cents to dollars. I. — Reduction op Common Fractions to Decimals. — Let it now be required to express the common fraction ^ as a decimal. We have seen (Art. 14.) that we may multiply tlie terms of any fraction by the same number without changing the value of the frac- tion. Let us then multiply the terms of g by 1000, and we get •iuoo* ^" ^^^^ same principle we can divide the terms by the same number without altering the vislue. Let us then divide by 8, and we get -/y-„^(j, where the denominator is a power of 10, and therefore the fraction is in the decimal form, and may be written .025, the denominator being omitted. But as it is not always apparent by Avhat power of 10 we must multiply, so that when the terms arc divi- ded by the given denominator, that denominator may be transformed into 10 or a power of 10, i. e., into 1 ibllowed by a certain number of ciphers, we may as well add ciphers, one by one, as we proceed. This is exiiibitcd in the annexed exam- ple. From these pi-inciples we can deduce a rule 8)50(0.025 48 20 16 40 40 for reducing a common fraction to a decimal. DECIMAL FHACTIONa. 78 lt.)110 .0875 90 140 128 120 112 80 80 II \J L E . Divide the numerator, vnth a cipher or ripherit anjicxed, bi/ the dcnomiiuxtor. Thus M will (jive, as in the vitrgin, .0875. In tlic examples given wo find that the addition of three ciphers to the first, and lour to tlie second, makes tlie numerati • divisible by the denomina- tor without remaindt r. Such fractions arc cal- led terminating decimals. From this we see that there arc common fractions who.sc terms can be multiplied by such powers of 10 as will make the numerator divisible by the denomina- tor without remainder, but it often happens that no power of ten will effect this, and that remain- ders occur which cannot be made divisible even- ly by the denominator, by the addition of any number of ciphers. Such fractions will never terminate, and there- fore arc called interminate, and the common fraction can never be expressed exactly in the decimal form, and all we can do is to make an approximation more or less close, according to the number of de- cimal places to which wo carry it. Let us take the fraction ^J. — First, 9 is not contained in 1, and therefore we place the decimal point in the quotient, and add a cipher to the numerator, and we find that 9 is con- tained ones in 10, with a remainder 1, — annexing another cipher, we again obtain 1 in the quotient, and this will obviously continue ad wfinitmn. — This recurrence is marked by a dot or dash over Ike figure, thus: .1 or .1'. If we express ^ as a decimal, we find that after we have got six figures in the quotient, we have a remainder 1, the same as the original rnmcrator, and therefore we f^hould again obtain the same quotient .142857, and hence this is called a circulating or periodic deci- mal, and the first and last of the recurring figures are marked with a point or trait. Thus : .142857 or .1'42857'. Again, it often happens that some figures do not recur whilst others following them do, as in the annexed exainDle, after we have qot 7)10(.142857 7 30 28 20 14 60 56 40 35 50 49 74 ARirnMETir. H'l 4111 a3a()(041110(.12345. 33300 78100 GGGOO 115000 1)9900 151000 133200 178000 165500 11500 five figures tlit; 11500 whiuli ^mvo u.s the third li-uro 3, in tho quoti cnt recurs, and hy pursuiiij; tho division, wu should find 315 rccjurrin;j; without end. When idl the lis^urcs recur, the fraction is called ii pure jHiriodic deci- mal ; when only some of them recur, it is culled mixed, and the term rapeatcr is applied when only one figure recurs, as -J-— .1111, &c.-_--.i or -/.=-.58333, &o.= . 583. Since the denominator is always 10, or a power of 10, and since 10 has no factors but 2 and 5, and therefore powers of 10 no factors but 2 and 5, or powers of these, it follows that no deci- mal will terminate except the denomina- tor be expressed by cither or both of these, or some power or product of them, llcnce all terminatiriL; decimals are deri- ved from common fractions having for denominator Kome figure ot the series 2, 4, 8, KJ, 32, &c., or 5, 25, 125, etc., or, 10, 20, 40, 50. GO, 80, 100, &c. E X E 11 C I S E S . 1. Reduce tho conmion fraction ■}( to a decimal. Ans. .25. 2. Reduce the common fraction ^ to a decimal. 3. Reduce the common fraction £- to a decimal. 4. Reduce the common fraction ^ to a decimal. 5. Reduce the common fraction J to a decimal. 6. Reduce tho common fraction ^ to a decimal. 7. Reduce the common fraction ^ to a decimal. 8. Reduce the common fraction | to a decimal. 9. Reduce the common fraction -l to a decimal. 10. Reduce the common fi-action ^'^ to a decimal. 11. Reduce the common fraction Jy to a decimal. 12. Reduce the common fraction -jV- to a decimal. 13. Reduce the common fraction § to a decimal. 14. Reduce the common fraction 4 to a decimal. ^xns. .0. Ans. . 75. Ans. .3. Ans. .i. Ans. .1 25. Ans. . IG. Asia. .142857. iins. .2. Ans. .1. Ans. . 09. Ans. .083. Ans. .G. Ans. .8. DECIMAL rnACTION'a. 76 15. IG. 17. 18. 19. 20. 21. 22. 23. 24. Reduce llcduco lleduco Reduce Reduce Reduce Reduce Reduce Reduce Reduce the common the common the common tlic common the common the common the common the common the common the common iVaction tVuction fraction fraction fraction fractioA fraction fraction fraction fraction ;'; to u decimal. •} to !i decimal. IJ to a decimal. I to a decimal. ;', to a decimal. !y to u decimal, i V to a decimal, j ^ to a decimal. } 'j to a decimal j J to a decimal. Ans. .8a. Ans. Ml'). Am. .625. Ans. .875. Ans. .4. Ans. .714285. Ans. .66. Ans. .DK;. Ans. .It2307(;. Ans. .0875. 25. Reduce the common fraction j ,'j to a decimal. 20. Reduce the common fraction jf^j to a decimal. 27. Reduce the common fraction I \ to a decimal. Ans. .o4.']75. 28. Reduce the common fraction j^'Vj to a decimal. 29. Reduce the common fraction ^l to a decimal. Ans. .4083544303797. 30. Reduce the common fraction tjotj to a decimal. Ans. .0044. 31. Reduce the common fraction ^j'.j to a decimal. Ans. .020408103205300122448979591830734003877551. 82. Express t^'.j- decimally. Ans. .01. 33. Express Tjig decimally. Ans. .(ioi. 34. Express Tjjg decimally. 35. Express j/^j^ , decimilly. Ans. .00059994. To reduce a denominate numbei to the form of a decimal frac- tion, reduce it tu the lowest denomination ivhich it contains ; reduce the integral unit to the same denomination^ and divide the former by the latter. Thus, to express 18s. 4d. as a decimal of £1, we must reduce it to pence, the lowc&v denomination given, and divide it by 240, the number of pence in £1, -which gives the fraction ^^o^^^f^^Ii. ^"fl this reduced to a decimal, gives .910 or £.910. In like manner 15s. lOi^d. is reduced to half-pence, viz., 381, and the half-pence in £1 are 480, and :ig^=TjJ, which exDressed decimally is .79375. fi'i! 'Iff' l„ m I " ' .11, 76 ARITHJIETIC. EXEllCISEB. 1. What decimal of £1 is lis. Ud. ? " Ans. .50875. 2. Express 15s. 9|d. as a decimal of £1. Ans. .790025. 3. AVhat decimal of a square mile is an acre ? Ans. .0015025. 4. Express 1 pound troy as a decimal of 1 pound, avoirdu- pois.* Ans. .82285714. 5. llcducc 17 cwt. to the decimal of a ton. Ans. .85. 0. Express m of a cwt. as a decimal of a ton. Ans. .040875. 11--10 .0875 22.G875-r-25=.9075 qrs. 2 9075-:-4^.72G875 cwt. 11.7PG875--20=:^58G34375 16)11 ^5)22.0875 4)2.9075 The operation annexed is often convenient in practice. To reduce 11 cwt., 2 qrs., 22 lbs., 11 oz., to the decimal of a ton. First, we divide the 11 oz. by 10, the num- ber of oz, in 1 lb., and then annex the 22 lbs., and divide by 25, the lbs in a qr., and so on. The first form of the work is best suited for illustration, the second is neater in practice. The principle is the same as that implied in the general rulo given above. 20)11.726875 ; 58634375 ADDITIONAL EXERCISES. 7. Reduce 10 drams to the decimal of 1 lb. Ans. .0390625. 8. Reduce 11 dwt. to the decimal of 1 lb. Ans. .04583. 9. Express 1 oz,, avoirdupois, as a fraction of 1 oz., troy, (sec note.) Ans. .9114583. 10. Reduce 5 hours, 48 minutes, 49.7 seconds to the decimal of a day. * A caution seems necessary here, for since tlie pouud (troy,) contains 12 ounces, and the pound (aroirdupois,) 16, llie natural conchi-sion would bo that the pound (troy) is J^ or | of the pound avoir dupois. This is not correct, for the ounce troy ex- ceeds the ounce avoirdupois by 4:2^ grains, though tho pound avoirdupois (7000 grs.) exceeds the pound Troy (.'J710 grs.) by 1240 grains. Tlii.s will be manifest from tho operation on the margin, where the standard weights are given. 5760--12 7000—16 .480 difference . . 42 J DECIMAL TRACTIONS. 77 II. — Reduction oi' Decimals to Common Fractions. — To find the coiiimon fraction corresponding to any given decimal. — This "nvolves three cases according as the fraction is a terminating ) 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, juwjj would be written decimally .006, which would mean that there are no tenths, no hundredths, but G thousandths. From these exDlanations we deduce the rule: MuU!j}h/, as in wJiolc numbers, and cut off from the right a deci- mal place foi every one in hath multiplier and multiplicand. EXAMPLES. Multiply .78 by .42. Here we multiply as if the quantities were whole nuuibors, and in the product point off a decimal figure for cacli one in both multiplier and multiplicand. In (1.) .78 ^^- !• t-bt*' number uf 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 are four decimal places in the factors, and there are six figures in the product, and consequently two figures represent whole num- bers. In Ex. V), when wo multiply G by 3, we obtain 18, but if we had (3.) 4.56 carried tlic repiteud out one place far- ther wc should have had 5 to be mul- tiplied by 3, and consequently 1 to carry, so wc add 1 to the 18, and in Ukc inanuor we must allow 2 when multiplying by 4, and 1 when nmlti- plying by 2. .78 .42 156 312 .3276 (2.) 34.6 4044 2G9G 2U2::. ''].3204 4.56 2.43 13G9 1826 '.n:\ 11.0920 MULTIPLICATION OP DECIMALS. EXERCISES 83 1. Multiply 7.49 by G3.1. Ans. 472.G10. 2. Multiply .15G by .143. Ans. .02230S. 3. Multiply 1.05 by 1.05, and the product by 1.05. AuH. 1.157G25. ■4. Find the continual product of .2, .2, .2, .2, .2, .2. Ans. .0000G4. 5. Multiply .0021 by 21. Ans. .0441. G. Multiply 3.18 by 41.7. Ans. 132.G0G. 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. 12.99G. 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 .8G2 to six places of decimals. Ans. .640839. 16. Multiply 3.18 by 11.7, and the product by 1000. Ans. 132G06. 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. Wha. is the continual product of .0001, G.27 and 15.9 ? Ans. .0099693. Contracted Metuod. — In many instances where long lines of figures are to bo multiplied together, the operation may be very nmch shortened, and ^-et sufficient accuracy attained. AVe may inbtancc "what tlu! student will meet with hereafter, calculations in conipouiid interest and annuities, involving sometimes most tedious operations. By the following method the results in such cases may be obtained with great ease, and correct to a very uiinute fraction. 1 •' we are -computing dollars and cents, and extend our calculation to four .v^- JIM *ii.. Hi'.- 84 AHITHMETIC. places of decimals, wc arc troatiiii^ of the onc-lmnclrcdth ])f rt of a cent, or the ten-thousandth part of a dollar, a quantity .so minute as to become relatively valueless. Hence we conclude that three or four decimal places are sufficient for all ordinary purposes. There arc cases, indeed, in which it is necessary to carry out the decimals farther, as, for instance, in the case of Ijogarithms 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. Lot it be required to lind the product of C.35G42 and 47.0453, : rue to four places of decimals : EXTENDED OPERATION. 6.35642 47.6453 CONTRACTED OPERATION. 6.35642 3546.74 19 06926 317 8210 2542 568 38138 52 444949 4 2542568 302.8535 37826 2542568 44494? 38138 2542 317 19 2 carried. 302.8535 RULE FOR THE CONTRACTED METHOD. Plncc tJiC luii.ts' figure of the whole nianher under the hist required decimal place of the midtipUcand, and the other integral figures to the right of that in. an inverted order, and the decimal figures, also in an inverted order, to the left of the integral unit; multiply hij each fgnre of the inverted multiplier, beginning with the figure of the multijdicand immediateli/ above it, omitting all figures to the right, but allowing for what icould have been carried if the decimal had been carried out one place farther — place the first figure of each partial product in the same vertical column, and the others in vcrii- eal columns to the left ; the sum of these columns will he tite required product. Thus, in tlie above example, we are required to find the product correct to four decimal places, therefore we set the units' figure, 7,. under the fourth decimal figure, and the tens' figure, 4, to the right, and the decimal figures, 6453, to the left in reversed order ; then we. MULTIPLIvJATlON OF DECIMALS. 85 iDultiply tlio wliolo line by 4, ami then wo multiply by 7, oinittinu; tlio 2 wliicli stiinds to tlio riji,lit, but allowing 1 lur what would h'lvo boon cariitid, that is, wo .say 7 times 4 i.s 28, and 1 is 2I>, and wo write the nino under the 8, the lirst tij^ure of the first partial product. By comparinjf the contracted method with the li;j;urc,s of the extend- ed form, which are to the lol't of the vertici' line drawn after the fourth decimal figures, it will bo seen that the figures of each column are the same but placed in reversed order, which makes no differenco in the sum, as 5-|-l)^ ^o-| 5- 8. This is the same principle as the contracted method of multiplying by 17, 71, &c., suggested iu the article ou simple multiplication '■''' Tho object of writing the multiplier in a reversed order is simply to make the work come in the uaual form, as othci wise we should be crossing and rccrossing, so to speak, as will be seen by the operation in the margin. — Bccinning with the left hand figure of the "'ultiplicr, and the right lumd figure of the multiplicand, we find the first partial pro- duct ; then taking the second figure of the 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 the right in the multiplier, and one place to- wards the left in the inultiplicand. This is so different from the ordinary mode of ope- ration, as to be excessively awkward and puzzling, and this gave rise to the idea of icvcrsing tlio order of the digits. We append this remark as most persons cainiot at lirst sight comprehend the reason of the inversion. * Let the learner observe that all the figiu'os of the first column aro of thi; same rank, viz., ton-thonsaiulths, and therefore may bo added together, and as tho value of ouch figure i.'r inereasod or docioasod 10 times according to its position to loft or right, it follows that all figures at equal distances from the decimal point, whether to right or left, aro of the same rank, i. e., units will bo under units, tons under tons, tontlis imdcr tenths, himdrodths nndor hundredths, &c., Sec. The contracted method is not of much use in tormiua- ting decimals which extend to only a lew places, but it saves a vast deal of labour in questions which involve either rcpetends or terminating (' limals expressed by a long line of decimal figures 6.35G42 47.G453 2542568 444949 38138 2542 317 19 2 allowed. 302.8535 8G .\EITHMETIC. ADDITIONAL KXERCISES 21. Multiply .2G73G l^y .28758 to four decimal piaccs Ans. .0709. 22. Multiply 7.285714 by 3G.74405 to five decimal places. Ans. 2G7.70GG5. 23. Multiply 2.G5G419 by 1.723 to six decimal places. Ans. 4.578932. 24. What decimal fraction, true to six places, will express the product of ,"f multiplied by v\ ? Ans. .113445. 25. What decimal fraction is equivalent to IfX^f' ? ' Ans. .4G748. 26. What is the second power of .841 ? Ans. .707281. 27. What is the product of 1.G5 by 1.48, true to five places ? Ans. 2.45075. 28. Express decimally 2^%X^g- Ans. 2.393162. 29. What is the product of 73.6371 by 8.143 ? Ans. 599.6272077. 30. .081472 X -01286, true to five places, will give .00876. lii the last exercise it must be observed that since there is no whole number, and five decimal places are rc([uired, we must place a cipher under the fifth decimal figure, and write .01286 in reversed order. That the result is a sufficiently close approximation will be evident from the consideration that the last figure 6 is only six one h'li'dred-thousandths of the unit, and consequently the next figure would be only one-mil- lionth rinrt of the unit. .681472 68210.0 GSl 136 55 t .00876 VI.-DIVISION OF DECIMALS. We have already seen (1) that we cannot perform any operation except the numbers concerned arc of the same denomination, or one of them be abstract ; (2) that when a denominate number is used either as a multiplier or a divisor, it ceases to be denominate, and becomes abstract, and (3) that the rules for addition, subtraction, multiplication and division of integers apply qually to decimals, the only additional requirement being the placing and moving of the decimal jwint. DIVISION OF DECIMALM. 87 Suppose tncn wc aro requir(Ml fo divido l.li.'J21 by U-l, we must y (1) bring both lividcnd is carried down to ten-tliousands for \.'2'.)21:=r:l-\--^^•^^fJ^\^, and tliorefore wo express 11.1 in the correspondii)g form, ten-thou- sandths or 11- i ,'„"."(>' nj "^' n.lOOO, so that wo change tlie form, but not the valu« of 11.1, tlie divisor. Again, by (2) the .1, which originally expressed ~T^o-Tu '^^^ ^y ■^^' which uives 7 oi.?., 7.«2o4o _ , ,j ■ ' ' ,. , c. o drs., 2 scrs., and a little over 8 gns. Rcix;tcnds must ri.803S4 be reduced to common frac- tions, or found approximately. 2.41152 20 8.23040 2- -I-o find the value of 7 : of a day, which Is 18 hours, 39 min. and nearly 59 J sees. 77777 ) , 24 ]• carry I. 311109 155555 18.GG(359 00 39.99540 00 59.72400 *Tho standard pounds are meant hero, viz.: troy. ;')7(i() j,'iains, mid avoirdupois 7000 Rraina. TakiDR the ounces wouM giv- \'f. —■[ -.7 j « 92 AWTIIMETIO. E X E 11 C I S E S . 1. \>hat is the vamo of £.475 ? ' Ans. Os. Gd. 2. What is the vahio of ,7 of a cwt. ? Ans. 3 qrs., 3 lbs., 1 oz., I'l] dn;. 3. Wiiat is tljc vaUio of ,541 (5 of a shilling sterling ? An.s. GJld. 4. What is the valuo of .0845 of s cwt. ? Ans. 2qis,, 20 lbs., 10 oz., 01^;;. 5. What is the valuo of .4 of 9a. -ikd 'i Wo liavc .4^'^ and 9s. 4id., niuUipliod by 4, and the product, divided by 9, gives 4s. 2d., the exact value. G. What is the value of .020 of 1° 15' ? Reducing .020 to a vulgar fraction, we get ^^*^^=^\, and multiplying 1° 15' by 2, and dividing by 75, wo And 2'. ;.. !)!■ RATIO ANX» PROPORTION. 17. — Ratio is the relation whicli one quantity bears to another of tlie same kind with respect to magnitude, or the number of times that the less is contained in the greater. Thus, the ratio 7 to 21 ia 3, because 7 is contained 3 times in 21, or 21 is 3 times 7. The same 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 ^ 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 be 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 liomogeneous, or of the same kind. ^ Proportion is the equality of ratios. w The ratio of 9 to 27 is 3, but we have seen that the ratio of 7 to 21 is also 3, therefore the ratios of 7 to 21 and of 9 to 27 are the ItATlO AN]) I'llOl'OUTION'. 93 same, or 7-:-21— 9-:-27, iuul tlioso ([Uiuititios :iro, thtirclbro, calltil proportionals. Tli<; si^in ( : : ) was foniicrly used fur o(|uality, and is htill ictaiiicd lor cMjiiulify ol' ratios, and llic sii^n ( ) is nscd lor tho actual otpiaiily of (juautltics, thou'.rli occasionally used lor e(juality of ratios. Ilcnc*, tlu; usual mode ui' ^vritiIl^• the cijuality of two ratios is 7 : I'l : : It : L'7. Such a statement is called a pro- portion, or an analo;iy, and is read — 7 is to 21 as !) to 27, ('. e., 27 exceeds S) as many times as 21 exceeds 7, and this is expressed hy .saying 27 is the same niu!tii)le ol" I) that 21 is of 7, or that 1) is tho same sub-multiple, measure, or aliijuot part of 27 that 7 is of 21. The lour (luantities are called tluj hrms of the proportion ; tlie first and last arc called the cxfrtmc.t, and second and third the means; also, the lirst and third an; called ItomohfjoKs, or of the same name, t. c, 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 lirst is to divide the second term by the lirst, 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 bo correct, the product of the extremes is equal to the product of the means. To prove this, let us resume the analogy, 7 : 21 : : 9 : 27. We have seen that 21-^7=^27->9, or 3m3. Now, if each be multiplied by 03, we 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, 90, and 132, written thus : 48 : 90 : : 132 : , the required quantity. Now. 132x90:i-12G72, the product of the means are therefore equal to the product of the extremes. ^Ve have, therefore, a product, 12072, and one of its factors, 48 , hence, dividing this product by the given factor, we find tlie other factor to be 264, which is therefore the fourth proportional, or fourtli term of tho proportion, and we (san now write the whole analogy, thus : — 4B : 96 : : 132 : 204. To prove the correctness of the operation, multiply 204 by 48, and 12072 is obtained, the same as before. Uence, 94 AUITfLMITIC. I T II K 11 i: I, E . t Divide the product of the second and thnd terms f}>/ thpjrmt, nnd the quotient will he the rerjuired fourth term. To show tilt; order in which (ho throo ^'ivcn quantities arc to be arranged, let it bo re(|uired to find how much 7oO yards of linen will cost at the rate of SUO for i)0 yards. It is plain that the answer, or fourth term, must be di^llars, for it is a price that is reijuircd, and in order that the third term may have a ratio to the fourth, the $uO must be the third term. Again, since TIJO yd.s. will cost more than 50 yds., the iburth term will be greater than the third, and therefore the second must be greater than the first, and therefore the statement is 50 : 730: : 3 : -ith proportioual, and by the rule -''-•■ L' ^^' •* - - Vir" =::438, the fourth term, and we can now write the whole analogy, 50 yds: 730 yds:: 830 : $438. This may be called the ascending scale, for th(! second is greater than the lirst, and the fourth greater than the third. If the ques- tion had been to find what 50 yards of linen will cost at the rate of $438 for 730 yards, we still find '" H the answer will be doUan-, aud that therefore, as before, dolla . ..t he in the third place, but we see that the ansv»'er will now bo o than 438, as 50 yards, of which the price is required, will cost much less than 730 yards, of which the price is given, and that therefore the second term must be less than the first. Hence the statement is 730 yds : 50 yds : ; $438 : h\ P., and by the rule -'J,^.,^,fi-Q=:30, the fourth proportional. We now have the full analogy 730 yds . 50 yds : : $438 : $30. As the second is less than the fii'st, and the fourth less than the third, this may be called the descending scale. If the first should turu 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 equality. RULE FOR THE ORDER OP THE TERMS. If the question implies that the consequent of the second ratio must he greater than the antecedent, mahc the greater term of the first ratio the consequent, and the less the antecedent, and vice versa. The questions hitherto considered belong to what is called Direct Proportion, to distinguish it from another kind called Inverse Pro- portion ; because, in the former, the greater the number given, the less will be the corresponding number required, and vice versa; IIATIO AND PROPORTION. 95 wliercas, in the latter, tho j^ruiitcr tlic number given, tlio less will he the nuntber required, and vice versa. To illustrate this, let it be required to find how long a staek of hay will feed 12 hor.si s, if it will feed I) horses for 20 weeks. Here the answer re(iuired iy time, and therefore 20 weeks will be the antecedent ol" the seeond 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 : !♦ ; and hence the name Inveiise, because the term 0, for which the time (20 weeks,) is given, and which therefore we should expect to be in tlu; first place, has to be put in the seeond ; and tl.e term 12, lor 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 ana- logy iri 12 : 9: : 20 : 15.* The principal changes that may be made in the order of the terms, will be more readily and clearly understood by the subjoined scheme, than by any explanation in words : Original Analogy : 8 : G : : 12 : 9 for 8x0—72^:^0X1:2. Alternately: 8 : 12: : G : 9 for 8x9^72^0x12. By Inversion : : 8 : : 9 : 12 for GX 11^^72^8X0. By Composition : 8-f G : G : : 12-^-9 : 9 or 14 : G : : 21 : 9 for 14X0=-12G:i--0x21. By Division : 8— G : G : : 12—9 : 9 or 2 : G : : 3 : 9 for 2X9-^ 18=0X3. By Conversion : 8 : 8— G : : 12 : 12—9 or 8 : 2 : : 12 :":i for 8X3^24=2X12. Simple transposition is often of the greatest use. Let us take an easy practical example. In calcula- ting what power will balance a given weight, when the arms of the lever arc known, let P be the power, W the weight, A the arm of power, and B the arm of weight. The rule is, that the power and weight are inversely as the arms. This solves all the four possible cases by transposition. * Inverse ratio is sonietiiin's spoken ol, but in reality there is no sue. thing. It is true that Inverse Proportion requires the terms of one of thi ratios to be inverted, but that is a mattei of analogy, iiot of ratio, for we hav< seen already that 7-J-2I expresses the vorv same relatiou as 2l-r-7.— (See in- 90 AI11T1I31ETIC. Ip A ; B : : W : I', j^ivcs tlio jxiwor when tlio otiirrs iin^ known, B : A : : I* : W ;;ive.s tlio w.'iijiit warn tliu kiIxm-s ihc kii iwn, W : 1': : A : If f^ivcallioariii ol'\vci;j,lil wlicn tin; (itlicrs.-iro kiwiwn, 1* : W : : 15 ; A<;ivcH t!ic iiriu«i|)K)Wi.'r wlion tho otliciSiin! known. The work iiijy often bu contr.ictc I li tlio lollowiiiLr iiiaiiiicr : — Resuming' our oxamplo 48 : S)i>: : I'M : lotiiih i)r()poitii)ii 1, we sec that Iir» is (louhlo of 48, : n.j lliorol'ore tlio ratio of 4.S to "j.i is the satnc as tli.il of any two imnibi'rs, the sccouij of wl,ic!i is doublo the first. Jiuil IS : I)'.) is tlio siiiic as I : Z, and wo rodiu".; llio any] ijry to the .siniploionn of I : *2 : : KI'J : 4 li |»i(>p., and wo liav • ' •• •■' ' - 2(14, tho toriii ro(juirod, as befurc. In tlio oxaniplo 5!) : 1'.)') : : .;(> : 4tlj term, we h.ivo ''••;;V3o=ll53.!>-^W;iA-. -^73x0^438. This is C(|uivalont to dividing the first and t^ocond by 10, and the first nnd third by .'). llonoawe may divide tho first ijiid secoiul, or first and thirl by any number that will nieasuro both. The siiiio ])riiioiple will also bo illustrated by tho considoratit^n that tho second and third arc multipliers, and the first a divii-or; and if wc first multiply, and then divide by the same (juantity, the one operation will manifestly neutralize the other. Thus : 48 : 9G : : 132 : F. P. may be written 1X48 : 2X48: : 132 : F. P. ; where it is plain that since by first multiplying 132 by 48, and then dividing by the same, tho ono 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 : 3(5, and 12 is called a mean proportional between 4 and 3G. To prove the correctness of this statement, we multiply 3G 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 8(iuare loot of 144, or that which produced it, or the root from which it grew ; hence, to find a mcun proportional between two given quaniities, we have tho following RULE M>d:iphj them together, and take the square root of the product. Thus, in the above examph, 4X3G-=144, the square rootof which is 12. Again, to find a mean proportional between 1) and 40, we mul- trodiic:ory iTiuarlis.) Tlio term lieciprocul limio'i^ lialilc ,o llic same ol)joe- tiou, for tlioiirrh 3 and i} are reciprocals, yet lliey express the j^anie i elation. When the expression Invcae I2atio is legitiniateiy used, it does not reler to a shujle ratio, hut means that : 21 : 40, or, writ- toti at full IcML'th, J) : 21 : :21 : 4l>. Proof: V.^ '[) Ml and 2rx2U-in. As Iho learner is not suppctscd, at this htai.'\ to know the niclhoil of lindiiii^ the roots of (|aantitics hcyontl the limits of the iimltiplieatioii table, we a])pouil a table ol'^icce of work in 5 days, hut after workinj^ 4 days t'-'y find it impossible to complete the job in less than ."> days mo.re, how many addiliomd men mu.st be cmj)loyed to i\o the work in the time agreed upon at iir«t 'r Ans. 10. 17. A watch is 10 minutes too last at 12 o'clock (noon) on 3Ion- day, and it i^ains \] minutes 10 seconds a day, what will be the time by the watch at a qnartor past 10 o'clock, A. M., on the following; Saturday ? Ans. 10 h. 10 m. 3^]^\ s. 18. A bankrupt owes $072, and his property, amountini^ to $G07.50. is distributed amoni^ his creditors ; what does one receive whose demand is $1 1 ..'j;}', ? An.s. $7.083H-. 10. What is the value of .15 of a hhd. of lime, at $2.1)0 per hhd. ? Ans. S.;]585. 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 reinibvccd by 400 men ? Ans. (iy montLs. 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 5i feet wide ? Ans. 150 rod.s. 22. A borrowed of JJ 3745, for 90 days, and afterwards would return the i'avor by lending B $1341 ; for Jiow long should he lend it? 23. If a man can walk 300 miles in successive day.s, how many miles has he to walk at the end of 5 days ? Ans. 50. ItATIO AND rnoPOliTlON. 00 24. li' 4'J.') gallons (if \\i\w cost i?o'.il ; how much will ^7- I'ay for ? Alls. !)0 '^'al. •J.'). If 1 1- head ofoatllo consume a rortain fjunntity (»f liay in '.» (hiys j how It'll',' will llio hamo (juaiitity last SI lu'ud? Aiis. lU ilay.s. 20. If 171 moil t;aii l)uil(l a hoi'so in IdSihiy.s; in what time will 1. A was sent with a warrant ; after he had ridden »!.') miles, II was sent after him to sti.j the exceution, and for every 10 miles that A rode, B rode 'Jl ; How far had each ridden when H overtook A ? Ans. 2711 miles. .'50. Find a fourth proportional to '.). lf> and HO. An-. 20f). .'Jl. A detective cha.sed a eulprit for L'OO miles, travelliiiu at the rate of K miles an hour, hut the eulprit had a start of 7.") miles ; at what rate ditl the latter travel ? Ans. 5 miles an hour. !52. Hew iiiueh rum may ho houtrht for 811i'.50, if 111 iruUons cost !{;8l).(J25 ? ■ Ans. 14S gallon? . 3:5. If 110 yards of cloth cost $18 ; what will 8t;3 pay for ? Ans. :>85 yards. u4. If a man walk from llochcster to Auburn, a distance ! fsay) 7!) miles in li7 hours, 54 minutes ; in what time will he ua.,v at tho same rate from Syracuse to Albany, suppoainj; the distance to be 152 miles ? 35. A butcher used a false wolirhi '. 1] oz., in.stcad of 10 oz. lor a pound, of how many lbs. did he defraud a customer who bou^^lit 112 just lbs. from him ? Ans. O'-i; Ib^. 30. If 123 yards of muslin co.st $205 ; how much will 51 yards aost? Ans. 885. 37. In a copy of Milton's Paradise Lost, containing 304 pr^r'^s, the combat of Michael and Satan commences at the 139tli page ; at what page may it be expected to cointucnce in a copy containing 328 pages ? Ans. The fourth proportional is 149jj|J ; and hence the passage will commence at the foot of page 150 38. Suppose a man, by travelling 10 hours a day, performs o 100 aritidietk;. Rn- h !l 'J !'■ journey in four weeks \vithnut dcsecratiug the Sabbath ; now many weeks would it take liim to perform the same journey, provided he travels only 8 liours per day, and pays no regard to the Sabbath ? Ans. -1- weeks, 2 days. 39. A cubic fool of pure fresh water weighs 1000 oz., avoirdu- '^pois; find the wciglit ot a rcsLcl of water containing 217^ cubic in. Ans. 7 lbs.", K-};;^':] oz. 40. Suppose a certain pa:; lure, in whicii are 20 cows, is sufficient to keep them (5 weeks ; how many muPt be turned out, that the same pasture may keep the rest G months ? Ans. 15. 41. A wedge of gold weighing 14 lbs., o 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, wh^n another came up and asked him how many feet ho had laid ; he replied, that the part he had finished bore the same proportion to one league which Y^^ does to 87 ; liow many feet had he laid ? 43. A farmer, by hia will, divides his farm, consisting of 97 •lores, 3 roods, 5 rods, between his two sons so that the share of the younger shal' be f the share of the elder; required the shares. Here the ratio of the shares is 4 : 3, and we have show n 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, wc must take the sum of 4 and 3 for first term, and either 4 or 3 for the second, and therefore 7 : 4 : : 97 acres, 3 roods, 5 rods : F.P., i. c, the sum of th.e numbers denoting the ratio of the shares is to one of them ai; the sum of tlie shares is to one of them. This gives for the elder brother's share, 55 acres, 3 rood'' 20 rods, and the younger's share is found either by repeating the operation, or by subtracting the share thus found from the whole, giving 41 acres, 3 roods, 25 rods. 44. A legacy of $398 is to be divided among thi'cc orphans, in parts which shall be as the numbers 5, 7, 11, the eldest receiving the largest share ; required the parts ? 23 : 5 : : 398 : 8G^§, the share of the youngest. 23 : 7 : : 398 : 121-v'^^, the share of the second. 23 : 11 : : 398 : 190-«,., the share of the eldest. 45. Three sureties on $5000 are to be given by A, B and C, so that B's share may be one-half greater than A's, and C's one-half greater than B's ; required the amount of the security of each? cosrrouND iTiOPonxioN. 101 Ans. A'.HlKa-c,S>1052.(J3,^5;IJ's,ei578.94;j;C'H,§23G8.42;^5. 40. Suppos^c that A starts from "Wasliinp^ton and vralks 4 miles an hour, and li at the same time starts from Boston, to meet liim, at the rate of 3 miles an liour, how far from Washington will they meet, the whole distance being 432 miles ? 47. A eortain number of dollars is to be divided between two persons, t'lie less share being f of the greater, and the difference of tln! shares $8(10 , what arc the shares, and what is the whole sum to be divided ? Ans. Less-share, i^KlOO ; greater, $2400 ; total, 64000. 4S. A certain number of acres of land are to be divided into two parts, such that the one shall be ^ of the other ; required the parts and the whole, the dia'erence of the part.'? being 71 G acres ? Ans. thr less part 537 aores; the greater, 1253 acres ; the whole, 1790. '!!>. A mixture is made of copper and tin, the tin being ^ of the copper, the difference of the parts being 75 ; required the parts and tiie whole mixture ? Ans. tin, 37^ ; copper, 112.1 ; the whole, 150. 50. Pure water consists of twogasses, oxygen and hydrogen ; tlie liydrogen is about ,-^ of the os)gcn; how many ounces of water will there bo when there avo 764J':oz. of oxygen more than of hydrf)gen ".'' Ans. 1000 cz. COMPOUND PROPORTION. Proportion is called simple when the (question involves only one condition, and compound 'vhcn the question involves more conditions than one. . : each condition implies a ratio, simple propv)rtion 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 uiven time, if OG men, working equally, can reap 40 acres in the same time? Here there is but one condition, viz , that 9o men can reap 40 acres in the given time, which implies but one ratio, and when the ;uestion has been stated 40 : G5 : : 9G : F.P., and the required term is found to be 15G, and the proportion 40 : 05 : : 90 ; 150, 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 w^dking at t!ic 102 ARITHMETIC. ^ .1 y'' m i*'' ii. ii;: : same rate, 10 hours each day, to travel 400 miles ? Here there arc two conditions, viz. : Jii'st, that, in the one case, he travels 12 hours a day, and in the other 10 hours ; and, secondly, that the distances are 250 and 400 dhIcs. The statement, as we shall presently show, would be 10:12 ] ..n.-i"? Here each condition im- 250 : 400 \ " ' '^^' plies one ratio, 10 : 12 and 250 : 400, and when the required term, which is ITJj, is found, there are four ratios, viz., the two already noted, and 9 : IT^''^, gives two more, one in relation to 10 : 12, and one in relation to 250 : 400. This will be evident, when we have shown the method of statement and operation. EXPLANATORY STATEMENT AND OPERATION. 11 : 33:: 12 : F.P. 1:3 : 12 : 36 18 1 5 :36 : 2 F.P. 10. PRACTICAL STATEMENT AND OPERATION. 1 1 : oo 1 , p 1^ p 18: 5| •l--^^- ■«"• " ■ ^ I • • 2 • F P 3: 5| •• -• ^•^• 1 : 5| •• " ■ ■^"• 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 ? \Vc 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 the question were — If 12 men can reap 11 acres in a given time, how many men will be re- quired to reap 33 acres in the same time. This, then, is a question in simple proportion, and by that rule we have the statement — 11 : 33 : : 12 : F. P., which, by contraction, becomes 1 : 3 : : 12 : F. P. ; and thus, we find F. P. to be 36, the number of men required, if the time were the same in both cases. The question is now resolved into this : How many men will be required to reap, in 18 days, the same quantity of crop that 36 men can reap in 5 days? This is obviously a case of inverse proportion, for the longer the time allowed the less will be the number of men required, and hence the siutemcnt, 18 : 5 : 36 : F. P., vhich, by contraction, becomes 1 : 5: : 2 : F. P., which gives 10 for the number of men. The Avork may be shortened by making the nvo statements at once, as on the right margin. We first notice that the last term is to represent a COMPOUND ritoroiiTioN. 103 certain nuiiiber of men, ;iik1, thcrctoro, we jJucc 12 in the third place; next, v^o scg thi\t, ofhcr (hiiufs hrimj days, and that therefore the fourth terni, 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 Ave get ., 1:3] : ') i : : 2 : V. V. Now, as 3 in the first term is to 11X18: 33x5 198 165:: 12: F. P. 105X12=10 198 be a multiplier, and 3 in the second a divisor, we may omit these also, and we obtain ^ ' ^ \ : : 2 : 10, the answer as before. The full uncontractcd oj^ration would be to multiply 18 by 11, which gives 198, then to multiply 33 by 5, which gives 1G5, then multiply 165. the product of the two second terms, by 12, and divide the result, 1980, by 198, the product of the two first f-erms, which gives 10 as before. Because in tlie atialogy 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 mtio. We 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, Avhole, fractional or mixed, say 5, |-, I, 19, 12, 1, 17, 11, I?, 25, then the ratio of the first to the last is said to be compounded of the ratio of the first to the second, the second to the third, the third to the fourth, &c., &c., &c., to the end. Now the ratio of 5 to 25 is --A---5, and the several ratios are in this .! order, X :■ in « \/-V;i V 1- V J •" A V A — " i- 1 ."> ■A..)AT-;X-i'Xl4X-i^Xj-|iwhieh leaving finally '-'A=5 as before. Tf we took them in reverse order, viz., ./',,= ', it is obvious that all therein could be cancelled, as each would in suc- cession be a multiplier and a divisor. Wo would also remark that compound proportion is nothing else than a number of questions in simple proportion solved by one opera- M 101 AniTIDIETIC. 11 li ■ ■ .--r '.'■■Ml ill!* iinl tion. This will be evident from our .second example by comparing the two opcrutioiis ou the opposite marqins. Af:;ain, we remarked that every condition implies a ratio, and that therelbrc the third and fourth terms of our lirst example really involve two ratios, one in relation to each of the prcccdinj^. 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 svritten once, the number of ratios appears to bo odd, but is in reality even. u i: 1. K : I'lacc, ((s ill ^iDipic jirojhji'tiun, in tJic third place the term tliat is the sime as the re'pdred term. Then consider each condition scjmralelj/ to sec ichich must he jdaced first, and uhich second, other things being equal. K X A yi V h K . 1. If i»;35.10 pay 27 men for 24 days; liow much will pay lb men 18 days ? Here we first o])serve that the an.swcr will be money, and therefore $35.10 must be in the third place. Again, it will take less money to pay IG men tl)an 27 men, and therefore, other tliinjzs bein;^- equal, the answer, as far as this is concerned, will be less than ^35.10, and therefore wv, put the less quantity, 10, in the second place. So also because it will take less to pay any given num- ber of men for 18 days tlian for 24 days, therefore we put the less quantity in tlie second place, which the statement shows in the margin. 27: IG:: $35.10 24 : 18 2 !) : 4 S35.10 4 9)140.40 Ans. $15.00 EXERCISES. 1. If 15 men, working 12 hours a day, can reap 60 acres in 10 days ; in what time would 20 boys, working 10 hours a day, reap 08 acres, if 7 men can do as much as 8 boys in the same time ? Ans. 203? days. 2. If 15 men, by working O5 hours a day, can dig a trencli 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 30 feet long, 12 feet broad, and 3 feet deep, in days ? Ana. 3?. noMPOUND rnoroRTiox. 105 -marked |iirtl and one ill [lubcr of |iubcr of is only reality ■111 tluxt ndition /, other pay 16 Iiat the 635.10 it will I men, :il, the bo less 10 less 'o also num- days, second aruin. in IG ip OS Jays. h4S ours :, 12 3^. ?>. If 48 men can build a wall SGi feet lonjr, feot lui^li, and \\ loot, wide, in vJG days; how many men will bo required to build a wall I}() feet lon^jT, Sleet hi.qh, and A foot wide, in 4 days? Ans. .'j2. •1. In what time would 2.'i men weed a (juantity of potato ground which 10 women would weed in days, if 7 men can do as much a> D women? Ans. SJ:^ days. 5. Suppose that 5i) men can di feet deep; how many days will 50 men require, working ."> hours, each day, to dig 2-1 cellars which arc each .JO feet long, 21 i'ect wide, and 20 foot deep ? Ans. 45 days. 0. If 15 bars of iron, each G ft. t) in. long, 4 in. broad, and 3 in thick weigh 20 cwt., 3 qrs., (28 lbs.) IG lbs. ; how much will G bar^s 1 ft. long, 3 in. broad, and 2 in. thick, weigh ? Ans. 2 cwt., 2 qrs., 8 lbs. 7. If 112 men can seed 4G0 acres, 3 roods, 8 '-o^s, in G days : how many men will be required to seed 72 acres in 5 days ? Ans. 21. 8. If the freight by railway of 3 cwt. for G5 miles be $11.25 : how far should 35/,- cwt. be carried for ^18. 75? 9. If a family of 9 persons can live comfortuuiy in Philadelphia for $2500 a year; what will it cost a family of 8 to live in Chicago, all in the same style, for seven months, prices supposed to be ^, of .vhat they would be in Philadelphia ? 10. If 126 lbs. of tea cost $173.25 ; what will G8 lbs. of a differ- 3nt quality cost, 9 lbs. of the former being equal in value to 10 lbs. of the latter ? 11. If 120 yards of carpeting, 5 quarters wide, cost^GO; what will be the price of 3G yards of the same quality, but 7 quarters wide ? Ans. $25.20. 12. If 48 men, in 5 days of 12^ hours each, can dig a canal 139J yards long, A^ yards wide, and 2^ yards deep ; how many hours, per day must 90 men work for 42 days to dig 491 /g yards long, 4 J yards wide, and 3! 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 ; mw, if sound travels 1142 feet per secoad, M m r • I! 11. ' 106 ARITHMETIC. .1 vk iind the pulse of a person in health beats 75 stroices in a minute, what is the breadth of the river ? Ans. 1 mile, 201 •! feet. 14. If 204 men. workin;.-; 12 hours a day, can make 240 yards of a canal, [> yards wide, and 12 yards deep, in 5 days ; 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 81G for 20 miles; how much will it be for 1250<^ lbs. for 100 miles? Ans. $92.^!:. IG. If §42 keep a I'amily of 8 persons for 10 days; how long, at that rate, will $100 keep a family of G persons ? Ans. 50^ !J days. 17. If a mixture of wine and v/ator, measuring 03 gallons, oon- sist of four parts wine, and one of water, and be wcrth §138.00 ; what would 85 gallons of the same wine in its purity be worth ? Ans. $233.75. 18. If I pay 10 men $02.40 for IS dayn work ; how much must I pay 27 men at the same rate ? Ans. $140.40. 19. If 00 men can build a wall 300 feet long, 8 feet hijrli, and feet thick, in 120 day.s, when the days arc 8 hours long ; in what time would 12 men build a wall 30 feet long, feet high, and 3 feet thick, when the da3-s arc 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^ feet long, 5? feet wide, and 3-^^ feet deep ; in how many days, of 11 hours each, will 490 men dig a trench of 7 degrees of hardness, 405 feet long, Sj 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 are each 30 feet long, 21 feet wide, and 20 feet deep ; how many men would be required to dig, in 27 days, working 5 liours each day, 18 cellars, which are each 48 feet long, 28 feet wide, and 15 feet deep ? Ans. 50. 22. If 15 men, 12 women, and 9 boys, can complete a certain piece of work in 50 days; what time would 9 men, 15 women, and 18 boys, require to do twice as much, the parts performed by each, in the same time, being as the numbers 3, 2 and 1 ? Ans. 104 days. 23. If 12 oxen and 35 sheep eat 12 tons, 12 cwt. of hay, in 8 days ; how much will it cost per month (of 28 days,) to feed 9 oxen and 12 sheep, the price of hay being $40 per ton, and 3 oxen being supposed to cat as much as 7 sheep '"' Ans. $924. MISCELLANEOUS EXERCLSES. 107 24. A vessel, whose speed was !.>]• miles per hour, lelt Belleville at 8 o'clock, a. ni., ftr Gaiianof|uc, a distance of 74 n.ilos. A second vessel, whose speed was to that of the first as S is to 5, start !!)}.•• from the same place, arrived 5 minutes before the first ; what timo did the second vessel leave Belleville? Ans. r>5 min. past 10 o'clock, a. m. 25. If 9 compositors, in 12 days, working 10 hours eac-h day. can eompore 3G sheets of 10 pages to a sheet, 50 lines to a page, and 45 letters in a line ; in how many days, each 11 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, 10 days. MISCELLANEOUS EXERCISES ON THE PIIECEDINO RULES. 1. What is the value of .7525 of a mile? Ans. G fur., rd, 4 yds, 1 ft., 21 in. 2. What is the value of .25 of a score ? Ans. 5. 3. Reduce 1 ft. in. to the decimal of a yard. Ans. .5. 4. What is the value of 11 yards of cloth, at $3,375 per yard ? Ans. §47.15. 5. What part of 2 weeks is j\ of a day ? Ans. -,-?;g. "j. What part of £1 is ISs. 4d ? Ans. ^. 7. Reduce ,". of a day to hours, minutes and seconds. Ans. 2 hours, 52 min., 48 sec. 8. Add I of a furloug to ^ of a mile. Ana. 7 fur., 31 rds, yd., 1 ft., 10 in. 9. What is the value of .S57i, of a bushel of rye ? Ans. 48 pounds. 10. Reduce 47 pounds of wheat to the decimal of a bushel. Ans. .7831. 11. Reduce 9 dozen to the decimal of a gross. Ans. .75. 12. Add ,", of? cwt. to ? of a quarter. Ans. 3 qrs., 10 lbs. 13. Subtract I jf a day from ^ of a week. Ans. 4 days, 3 lirs. 14. From [^ of 5 tons take '^ of 9 cwt. Ans. 2 tons, 17 cwt., 1 qr., •!:• lbs. 15. Bow many yards of cloth, at $3^ a yard, can be bought for $48| ? Ans. 13^^ yards. 16. A man bought | of a, yard of cloth for $2.80 ; what was the rate per yard? Ans. $3.20. 17. How many tons of hay, at SIO^ per ton, can be bought for $196^ ? Ans. 11 3| tons. 108 ARITHMETIC. i«'i K'lr : ■," 18. At $17^ por week, how many weeks can a family board for $7G5g ? Ans. 4;U weeks. 19. What number must be added to 20^, and the sum multipli- ed by 7f, that the product may be 49G ? Ans. 37 r. 20. A man owns J of an oil weH. lie sells J of hirf Kharc for $3500 ; (vhat part cf his sliare in the well has he still, and what is it worth at tho sumo rate ? 21. How lont; will llOi) hluls. of water la^t a company of 30 men., allowing ouch man V of a gallon a day ? Ans. 027 days. 22. licducc -^ of 2.1, ,% of r;;, and 3} of 2|, to equivalent IVac- tions having the least common denominator. Ans. 3 r> .1 .-1 3 (i Hi) 4 6' 4 iV • 23. From :^ of 21 of 4, take ,"^ ofGi of J. Ans. 21 24. What is the sum of ^, J, ^, >,, c. 7. d.and ^? I 1 I An.s. 1 •! R ".) '^ Z ..J I)' 25. Whatisthosumof J-r of3;i + Jl of85? Ans. 22a:({!^. 2G. How long will it take a person to travel 442 inilcs, if ho travels 3 J miles per hour, and 8^ hours a day? Ans. IG days. 27. Find the sum of 2| of j\, 3^ of I of ,'V of 4^^ and ^. Ana. 6.f^. 28. A has 2| times 8^ dollars, and B 6^ times 9^ dollars ; how much more has B than A? Ans. $44 g|. 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. 11' 7 horses cat 93 J bushels of oats in GO days ; how many bushels will one horse cat in 87 1 days ? Ans. 19^. 31. Bought 14/g yards of broadcloth for $102.90 ; what was tho value of 87^ yards of the same cloth ? Ans. $G12. 32. How many bushels of wheat, at $2f per bushel, will it re- quire to purchase lG8g^jj bushels of corn worth 75 cents per bushel? Ans. 47/',. 33. If in 82j^ feet there are 5 rods ; how many rods in one tuile ? Ans. 320. 34. Suppose T pay $55 for g of an acre of land ; what is that per acre ? Ans. $88. 35. If -^ of a pound of tea cost $1 .G6J- ; what will I of a pound cost? Ans. $1.55.1 -'. 36. Subtract tho sum of 2| and 1 /n, from the sum of I, 7^ and 3, and multiply the remainder by 3y\. Ans. 24J ^. 37. If I lb. cost 2'3/^ cents; what will 2\^ cost ? Ans. 7^rJ^^ cents. MISriXLANEOUS EXERCISES. luy 3S. What is tlio dlfforcnco brUvcon '2^\,> 3}, iwul i\',Xi>,',i V Ans ' (10 • Aus. !{y^ cents. :i9. If I lb. cost $^ ; what will | I lb. cost ? 40. What is the diflbrencc bitwocn ;^ ol" i+! + -JXi, and 41. If 4,",- yards cost $1..'.; , what will 2^ yards cost? Ans. 47 rj cents. 42. Bought 2 of 2000 yards of ribbon, and .«old rJ of it ; how much remains ? Anp. 285i yards. 43. Divide the sum of A-, ff, I, \l, i|.|,, §.^, jHa by the sum of ^, h s> t'(J> -rJ) (Vn I if.; ^"^ divide the-Ciuoticnt by t), 3^, and multiply the result by il of •;. Ans. ■^. 44. I bought 2 of a lot of wood land, consisting of 47 acres, 3 roods, 20 rods, and have cleared ^ of it ; how much remains to be cleared ? Ans. 20 acres, 3 roods, 31.^ rods. 45. What is the difference between 1 ,-^'^3 and l;:g ? Ans. i^}. 4G. If $l.j pay for a 1^ st. of flour; ibr how much will $^ pay? Ans. 1 f'^ St. 47. Mount Blanc, the highest mountain in Europe, is 15,872 feet above the level of the sea ; how far above the sea level is a clim- ber who is ,-'j of the whole height from the top, i. e., ~^\ of perpen- dicular bight ? Ans. 12896 feet. 48. What will 45.04375 tons cost if 12.796875 tons cost $54.04 ? Ans- $196.17. 49. If I gain S37.515625 by selling goods worth $324.53125; what shall I gain by selling a similar lot for 6520.663541 G. ? Ans. $60.1884. 50. If 52.815 cwt. cost $22,345 ; what will 192.664 cwt. cost at the same rate ? Ans. $81,512-1- 51. Required, the sum of the surfaces of 5 boxes, each of which is 5^- feet long, 2| feet higli, and 3^ foot wide, and also the number of cubic feet contained in each box. The box supposed to bo made from incli lumber ? 52. If I pay $ ,^„ for sawing into three pieces wood that is 4 ft. long; how muu3h more should I pay, per cord, for sawing into pieces of the same length, wood that is 8 feet long ? Ans. 22^- cents. 53. A sets out from Oswego, on a journcj, and travels at the rate of 20 miles a day ; 4 days after, B sets out from the same place, and travels the same road, at the rate of 25 miles per day ; how many days before B will overtake A ? Ans. 16. 4" 110 AiirriDiKTic. . i. ! . 1 i I :■ , 'ii I ;, ' . :i [ ' ' ' . . fcisi 54. A fanner luiviii^' ")(U tons of hay, sold [ of it nt 810^ per ton, and the rcuiaindor at ij,'.).'!^) jkt ton; liow luncli diil h<> receive for his hay? Ans. So80^;j. 55. If the sum of 87 } \ and 117!!^ is d-- " 'cd by their dillercnoe ; what will bo the (juoticnt ? Ans. C;il}-,'. 6G. If 8J 3-ards of silk make a drcb. and 1) dresses be made fronj a piece contain^nf* 80 yards; what will be the remnant left? Ans. 1 }- yards. 57. A merchant expended ^840 for dry {^oods, and then had re- maining; only ^Ij as much money as ho had at first ; how much money had he at first ? * Ans. $34:i(). 58. If a person travel a certain distance in 8 days and hours, by travelling 12 hours a day; how long will it take him to perform the same journey, by traveling 8,J hours a day ? Ans. 12 days. 51>. If 15 horses, in 4 days, consume 87 bushels, G qrts; of oats ; liow many horses will 010 bushels, 1 peck, 2 qrts, keep for the same time ? Ans. 105. GO. lleducc 1 pound troy, to the fraction of one pound avoirdu- pois. Ans. |:}^. Ans. J. Gl. lleduce ■^■ , > » ■ to a simple fraction. 4-Vot:5 G2. What will be the costof 8 cwt., 3 qrs., 12^ lbs. of beef, if 4 3wt. cost $o4 ? Ans. $75, ''j. G3. If 4 men, working 8 hours :i day, cau do a certain piece of work in 15 days; how long would it take one man, working 10 hours a day, to do the same piece of work ? Ans. 48 days. G4. Divide $1728 among 17 boys and 15 girls, and give each boy y'j- as much as a girl ; what sum will each receive ? Ans. Each girl, SGG^" ; each boy, $42^?. C". If A can cut 2 cords of wood in 12 J- hours, and B cau cut 3 cords in 17i't hours ; how many cords can they both cut in 24i- hours ? Ans. 8J^. GG. If it requires 30 yards of carpeting, which is ^ of a yard wide, to cover a floor ; how many yards, which is IJ yards vide, will be necessary to cover the same floor ? Ans. 18. G7. A person bought 1000 gallons of spirits for $1500 ; but 140 gallons leaked out ; at what rate per gallon must he sell the remain- der so as to make $200 by liis bargain ? 68. What must be the breadth of a piece of land whose length is 40^ yards, in order that it may be twice us great as another piece of ANALYSIS AND SYNTHESIS. Ill Inn J whose lenjjtli is 1-i'^ yards, and whoso breadth is 13,-, yards? Ans. 0^- yards. CO. If 7 men can reap a rectangular Gold whose length is 1,S0(I feet, and breadth 9G0 loot, in 9 days of 12 hours each ; liow loii;j will it take .') men, working 11 hours a day, to reap a tield whose length Is 800 feet, and breadth 700 feet ? Ans. 'A}j days. 70. 124 men dug a trench 110 yards long, .'I feet wide, and l feet deep, in f) days of 11 hours each; another trench was (h>g by one-half the number of men in 7 days of 9 hours eacli ; how many feet of water was it capable of holding ? Ans. 22(58 cubic i'eet. 71. If 100 men, by working G hours each day, can, in 27 days, dig 18 cellars, each 40 feet long, 36 feet wide, and ''2 feet deep ; how many cellars, that are each 24 feet long, 27 feet wide, and 18 feet deep, can 240 men dig in 81 days, by working 8 liours n day ? Ans. 2r)(».. 72. A gentleman left his son a fortune, !, of which he spent in 2 months, ]■ of the remainder lasted him .'} months longer, and j| fd what then remained lasted him 5 months longer, when he had only $895.50 left; how much did his father leave him ? Ans. $4477.50. 73. A farmer having sheep in two different fields, sold ^ of the number from each field, and had only 102 shec • remaining. Now 12 sheep jumped from the first field into tlio second ; then the num ber remaining in the first field, was to the number in the second field as 8 to 9 ; how many sheep were there in eaeli field at first ? Ans. 80 in first field ; 50 in second. 74. A and B paid $120 for 12 acres of pasture for 8 weeks, with an understanding that A should have the grass that was then on the field, and B what grew during the time they were grazing ; how many oxen, in eijuity, can each turn into the pasture, and how much should each pay, providing 4 acres of pasture, together with what g;rew during the time they were grazing, will keep 12 o.\en G weeks and in similar manner, 5 acres will keep 35 oxen 2 weeks ? Ans j A should turn into the tield 18 oxen, and pay $72,, " ( B should turn into the field 12 oxen, and pay $48. ANALYSIS AND SYNTHESIS. Analysis is the act of separating and comparing all the different parts of any compound, and showing their connection with each other, and thereby exhibiting all its elementary principles. ii 112 AUITHMKTin. \i % il I;! r '".'1 t'^'i is;i Tho convorst; of An;ily.si.s is SytitlicMis. Tlic moiuiiiit; and uso of thoHi! tiTins will probably bu most readily foiiiprchcmlod by lelcrcnce to (heir clurivation. They iiro both puro (Irofk word.s. AnalyHi« means hosiinj vp. Tho ^dneral rcauer would horo probably t'xpeet liHmnrj (loir)i, ns employed in most popular dotinitions- but wo may illustrate tijc Greek term, iinming up, l)y our own everyilay fihrnse, tin ring up, wliieh means irndiixj into x/trnh, the Knnli>li vj. eonveyinj^ tho Hume idea here as the Greek auii in analysis. The (jreek synthesis means literally plaeing to;^'ether ; that is, tljc con)ponent parts being known, the word synthesis indieatcs tho net oi'eondjininir them into one. We nn days, when B assisted liim to complete the job ; how long did it take them to finish the work ? s o I, u T I N . If A can do the work in 8 days, in one day he can do the -J- 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 ^-[""i^iTj of the work iu one day. But A works alone for 3 days, and in one day lie can do -J of the work, in I) days he would do IJ times -y— -^- of the work, and iis the whole work is equal to ^ of itself, there would be ^ — s-~ s of the work yet to be completed by A and B, who, according to the con- ditions of the question, labour together to finish the work. Now A and B working together for one day can do t>'^,- of the entire job, and it will take them as many days to do the balance ^ as t,^,- is contain- ed in §, which is equal #X-V-r='^l <^^y^- 22. A and B can build a boat iu 18 days, but if C a.ssists them, they can do it in 8 days ; how long would it take C to do it alone ? An«. 14| days. 23. A certain pole was 25i- feet high, and during a storm it was broken, when :]; of what was brokcu off, equalled 5 of what remained J how much was broken off, and how much remained ? Ans. 12 feet broken off, and 13?,- remained. 24. Tliero are ^5 pipes leading into a certain cistern ; the first will fill it in 15 minutes, the second iu 30 minutes, and the third in one liour ; in what time will they all fill it together ? x\.ns. 8 min., 34'| sec. ANALYSIS AND SYNTHESIS. 113 2'i. A. uiiil J». start. tn^otliiT by railw.iy train IVnui I'ull'cil) tc Krio ;i distaiici; nl' (s,iy ) IdO luiK's. A ljocs l)y fVc'i_u,i.t tniiii, at the rate of 111 miles \)v.v Iiotir, ami i> ])y mixed train, at. llu; into of l.~- miles jior hduv, (' loavcH Erio I'cr JJuIlalo at the same time )iy ox j)rcss train, wliifh runs at tin; rale dl' I'l miles jut lumr, liou' far, from IJulTalo will A and 1> each bo when C meets tlieni. 2(j. A cistern has two i)ipe.s, one will lill it hi IS minutes, und the other will empty it in 72 minutes ; what time will it require tc fill the cistern when both itre runnintjj ? Ans. 2 liours, 24 luin. 27. If a man spends ['.^ of his time in working, .V in fili-epinLT, ,'^ in eating, and 1^ hours each day in reading;; liow much lime will be left? Ans. 3 hours. 28. A v/all, which was to be built o2 feet high, was raised 8 feet by G men in 12 days; how many )nen must be employed to finish the wall in G days? Ans. oO men. 29. A and B can perform a piece of work in 5,'', dayn ; IJ and C in G§ days ; and A and C in G days ; in what time would each of them perform the work alone, and how long would it take them to do the work together ? Ans. A, 10 days ; B, 12 days; C, 15 days ; and A. 15, and C, together, in 4 days. 30. My tailor informs me that it will take 10]- square yards of cloth to make me a full suit of clotlics. The cloth I am al)out to purchase is 1 ^- yards wide, and on sponging it will shrink .,\^ in width and length ; liow many yards of this cloth must I purchase for my "new suit?" Ans. <),Ii3:! yards. 31. If A can do § of a certain piece of work in 4 hours, and B can do J of the remainder in 1 hour, and C can finish it in 20 min. ; in what time will they do it all working together ? Ans. 1 hour, 150 min. 32. A certain tailor in the City of Brooklyn bought 40 yards of broadcloth, 2.^ yds wide ; but on sponging, it shrunk in length upon every 2 yards, -/,t of a yard, and in width, l?,- sixteenths upon every 1^ yards. To line this cloth, he bought flannel 1 } yards wide which, when wet, shrunk ■?.- the width on every 10 yards in length, and in width it shrunk -?.- of a sixteenth of a yard ; liow many yards of flannel had the tailor to buy to line his broadcloth ? Ans. 71 ,'■., yards. 33. If G bushels of wheat are equal iu value to 9 bushels of j)i!r- ley, and 5 bushels of barley to 7 bushels of oats, and 12 bushels of U'.T Mil 116 ARITHMETIC. Ml oats to 10 bushels of ixsasc, and 13 bushels of pease to ^ ton of hay,, and 1 ton of hay to 2 tons of coal, how many tons of coal are equal in value to 80 bushsls of wheat ? SOLUTION. ,^ If G bushels of wheat arc equal in value to 9 bushels of l)arley, ■ or 9 bushels of barley to G bushels of wheat, one bushel of barley would be equal to ^ of G bushels of wheat, equal to 'j , or § of a bushel of wheat, and 5 bushels of barley would l)c equal to 5 times I of a bushel of wheat, equal to ^y^b=yf--=^'6^ bushels of wheat. But 5 bushels of barley are equal to seven bushels of oats ; hence, 7 bushels of oats arc equal to ?J^ bushels of wheat, and one bushel ol oats would be equal to 3^-^-7^=}^'^ bushels of wheat, and 12 bushels of oats would be equal to 12 times ^'^=z3r/f-=5-^ bushels of wheat.. But 12 bushels of oats are equal in value to 10 bushels of i^ease, hence, 10 bushels of pease are equal to 5| bushels of wheat, and one bushel of pease would equal b^~-10=1^ of a bushel of wheat, and 13 bushels of pease wou)d equal :*Xl3=^^--=:7| bushels of wheat. But 13 bushels of pease equal in value ^ ton of hay, hence, ^ ton of hay equals 7^ bushels of wheat, and one ton would equal 7|X2= 14^ bushels of wheat. But one ton of hay equals 2 tons of coal, hence, 2 tons of coal are equal in value to 14 1 bushels of wheat, and one ton would equal 14^-^-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 lOj^ tons of coal. NoTK. — This question belongs to that part of arithmetic usually called CoDJoined Proportion, or, by some, the " Chain Rule," which has each ante- cedent ot a compound ratio equal in value to its consequent. AVe have thought it best not to introduce such questions under a head by themselves, on account of their theory being more easily understood when exhibited by Analysis than by Proportion. Questions that do occur like this will most probably relate to Arbitration of E.xchange. Although they may all be worked by Compound Proportion as well as by Analysis, yet tlie most expe- ditious plan, and the one generaly adopted, is by the followiiij^ RULE. Place the antecedents in one column and the consequents m another, on the right, with the sign of equality hetween 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 the quotient will he the answer. ANALYSIS AND SYNTHESIS. 117 Let us now take our iast cxamnlo (No. I>^>), and solve it by tliis ♦^^le: G bushels of whcat=zl) bushels of barley. 5 bushels of barley^^T bushels of oats. 12 bushels of oats=10 bushels of pease. 13 bushels of pease=^ ton of hay. 1 ton of hayr::=2 tons of coal. — tons of coal=:80 bushels of wheat. 20 SI. 7. IQ. \-%, %. m ^, ^, '^, =3^4.o=:ioy. Ans. 34. If 12 bushels of wheat in Boston are equal in value to 12^ bushels in Albany, and 14 bushels in Albany arc worth 14^ bushels in Syracuse ; and 12 bushels in Syracuse are worth 12J 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 IG shilling's in New York, and 24 shillings in New York arc worth 22^ shillings in Pennsylvania, and 7^^ shillings in Pennsylvania are worth 5 shillings in Canada ; how many shillings in Canada are worth 50 shillings in Massachusetts? Ans. 41*. 36. 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 6 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 I" of 20 rods, and ^ of 20 rods is 5 rods. Now, if one man can build 5 rods in one day, 7 men would build 7 times 5 rods in one day, and 7 times 5 rods=35 rods. ]iastly, 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 G ; 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 3G days, of 10 hours each, build a wall 24 feet long, IG feet high, and 3 feet thick; in how many days, of 8 118 ARITHMETIC. \h4''''' «'■' I * hours each, would the same lot of men build a wall 20 feet long, 12 feet high, and 21- feet thick ? Ans. 23/,,. 38. If 5 men can i^erform 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 number of ' liours in a day, supposing that 2 of the second set can do as much work in an hour as 3 of the first set? Ans. GGJ men. NoTK. — Such questions ns Ibis, where the answer involves ii iraction, may frequently occur, and it may be asked how § of a man can do any work. The answer is simply this, that it requires G(5 men to do the work, and one man to continue on working § of a day more. 30. Suppose that a wolf was observed to devour a sheep in -^- of an hour, and a bear in £- of an hour ; how long would it take them together to eat what remained of a slieep after the wolf had been eating -^ an hour? Ans. 10/j min. 40. Find the fortunes of A, B, C, D, E, and F, by knowing that A is worth $20, which is \- as much as B and C are worth, and that C is worth ^ as much as A and B, and also that if 19 times the sum of A, B and C's fortune was divided in the proportion off, ^ and J, it would respectively give f of D's, ^ of E's, and ^^ of F's fortune. x\ns. 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 days turns and goes back as far aa ±» has travelled during those 9 days ; he then turns again, and pursuing his journey, .jvertakes B 22^^ days after the time they first set out. It is required to find the rate at which B uniformly travelled. Ans. 10 miles per day. 42. A hare starts 40 yards before a greyhound, and is not per- ceived by him until she has been running 40 seconds, she scuvS away at the rate of 10 miles an hour, and the dog pursues her at the rate of 18 miles an hour ; how long will the chase last, and wliat dis- tance will the hare have run ? Ans. GC^'^ sec. ; 490 yarc^. 43. A can do a certain piece of work in 9 days, and B can do the same in 12 days; they work together for 3 days, when A is taken sick and leaves, B continues on working alone, and after 2 days he is joined by C, and they finish it together in IJ days; liow long would C be doing it alone ? Ans. 12 days. 44. A, in u scuffle, seized on ■§• of a parcel of sugar plums ; B caught |- of it out of his hands, and C laid hold on -,-'y more ; D ran off with all A had left, except -j which E afterwards secured slyly for himself; then A and C jointly set upon B, who, in the conflict, lei" ITACTICI 110 fail h lio liad, wliicli Avoiu equally i)ickcd up by D aud E, v,\\o lay perdu. B then kicked down C's hat, and to \,'ork they all went iinew for what it contained ; of which A got :}, B it, D 7, and C and E C(|ual shares of what was left fif that stock. D then struck ;{■ of ■what A and B last acquired, out of tlieir hands ; they, with some difficult}', recovered -g^ of it in C(jual .shares aijain, but the other throe carried off -}■ a piece of the .same. Upon this, they called u truce, and agreed that the -?; of the whole left by A at first, sliould be equally divided among them ; liow many plums, after this distribu- tion, had each of the competitors ? Ans. A had 2863 ; B, G335 ; C, 2438 ; J), 10294 and E, 1950. PRACTICE The rule which is called Practice is nothing else man a particu- lar case of simple proportion, viz., whcu the first cerm 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. Here the first term being 1, the question becomes one of simple multiplication, but the answer, $19G, is. really the fourth term of an analogy. Again, to lind the price of 40 barrels of flour, at §7.62^ per barrel, we have only to multiply §7.G2A- by 40. In many cases, however, it is more conveni- ent to multiply the 46 by 7, wliich will give the price of 46 barrels at $7 each. Now, 50 cents being half a dollar, the price of 46, at 50 cents, will be $23, and 12 J cenis being ^ of 50 cents, the price at 12A- 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-}r. Hero the question stated at length would be, if 1 cwt. cost $4.87^-, what will 30 cwt., 2 qrs., 15 lbs. cost? The statement wouM be 1 : 36,, 2., 15: : $4.87^: F. r. This becomes a question of vuiki- $7.62* 46' 23 4572 3048 $350.75 30 121 i 46 7 000 23 5.75 8350.75 ^1 Mi ) );■]< ■<* : ,: ? i! t : r J{ U 1 ^::1, i rip 120 ARITHMETIC. plication because the tirst term is unity, and divided by 1 would not alter the product of the other two terms. Thus . 2 qrs, 10 lbs. 5 i- of 1 cwt, I of 2 qrs. h of 10 lbs. 36 18 2922 14G1 175.50 = price of 3 cwt., (2) ^4.87^ per cwt. 2.437= " 2 qrs. " " " " .487= " 10 lbs. " " '' 243;^ '' ^ " " '• '' 5 $178.GG7= " 3G cwt., 2 qrs., 15 lbs. We would call tlie 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 higber to find the price of the former, be sure you divide the line lohich is the 2>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=i- of 1 acre. 4.20 189 20 rods=J of 2 roods, 210 525 10 rods=J of 20 rods. 2G25 525 2 rod3=.i of 10 rods, 3780 33G0 420 Since the rent of 1 acre is $4.20, the half of it, $2.10, will be the rent of 2 roods, the rent of 20 rods will be . 525, the :|- of the rent of 2 roods, the half of that, . 2G25, will be the rent of 10 rods, and, lastly, .0525 will bo the rent of 2 rods, which is tbe ! of 10 rods. AVc tben multiply by 189, and set the figures of the product in the usual order, so that tbe 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. EXERCISES. 1. What is the price of 187 cwt. at $5.37^^ per cwt. ? Ans. $1005.12^. $696.74 •^/^^V TRACTICE. 121 2. "Wiiat is the value of 1857 lbs., at §3.87^ per lb.? Ans. $7195.87^ 3. What will 479G tons amount to at §14.50 per ton ? Ans. 621582 4. What is the price of 29 score of sheep, at $7.G2} eacli ? Ans. 84422.50 5. Sold to a cattle dealer 19G head of cattle at $18.75 each, find the amount. Ans. $3075 G. Sold to a dealer 97 head of cattle, at $10.12^ each, on the average; find the price of all. Ans. $1564.12^ 7. What is the price of IG tons, 17 cwt., 2 qrs. of coal, at $8.62^ per ton ? 8. What is the yearly rent of 97 acres, 3 rood.s, 20 rods, at $4.37^ per acre ? 9. If a man has $12.50 per week ; how much has he per year r Ans. 8050. 10. If a clerk has $2.12^ salary for every working day in the year ; what is his yearly income ? Ans. 6GG5.12^-. 11. If a tradesman earn $1.G4 per day ; how much does ho earn in the year, the Sabbaths not being reckoned ? . Ans. $513.32. 12. If an oflGicer's pay is a guinea and a half per day ; how much has he a year ? Ans. £574 17.s. Gd. 13. What is the price of 479 cwt. of sugar, at $17.90 per cwt. Ans. 88574.10. 14. Find the price of 879 articles, at 81-19 each. 15. Find the cost of 1793 tons of coal, at §7.87i per ton. 16. What is the value of 2781 tons of hay, at 88.G2^ per ton ? 17. What is the rent of 189 acres, 2 roods, 32 rods, at 84.20 per acre? Ans. 8795.74. 18. AVhat is the price of 879 hogs, at $4.25 eacli ? 19. Yv'hat will 3GG tons of coal come to at $S.12t per ton ? 20. AVhat is the price of 118 acres, 3 roods and 20 rods of cleared land, at $30.75 per acre ? Ans. 843G8.GG, 21. What is the price of 28G acres, 1 rood, 24 rods of uncleared land, at 87.25 per acre ? Ans 8207G.40. 22. A has 84 acres, 2 roods, 30 rods of cleared land, worth $24.00 an acre ; B has 208 acres, 3 roods, 24 rods of uncleared land, worth $4.40 anaci'c — they exchange, the difFerencc of value to be paid in cash ; which has to pay, and how much ? Ans. 13 $989.08. 122 AlUTITMETIC. ACCOUNTS AND INVOICES. AcciiiATH lire statciiu'iit.-t Iroiii tiicrcliiuils tn custoincrs that, have pur- chased good.H (ni credit, aihl are f^t'iierally mrule out )teiiodicalIy, imlew spechilly called lor. An invoice is Hiinpljr u Btiitemniit rendered })y tlio seller to the buyer, rt time of purchase, showing the articles bought, and the prices ofeuch. 1. Nkw Yokk, July l«t, ISCC. Mr. James Andkiisox, To FiiKNCit. WiiiTi: & Co,, Dr. 18G6. 2.85 I.3S 2.''** Jany. 4, To 2 lbs. tea, l.i"'- ; a lbs. coflee, JSc. ; 20 lbs. rice, lOc. . . '' 29. •' 2i yds. Amer. tweed, l.o" ; 1 vest \7S 2,0'' Feb. 10, " It lbs. Mus. sugar. r2,lc. ; 10 lbs. crus. white sugar, 20c. (iOc. ' 2Sc. 1."''' '• 22, " 1 lb. bk. i-oda, ; 1 lb. car. soda, ; t lbs. i-oflee, 45c.. IJ.oo 87, 1 e. Mar. 11, " 10 yds. print, 30c. : trimming, ic, per bill ' 1.8 s^c. 2.0 " 1!). '• 2 lbs. tobacco. nOc. ; 1 gal coal oil, : 2ga!s. symp, l.oo. Aprl 12. " 1, yd. blk. silk, ?>.-'0 ■ j yd. blk. velvet, C." -J^^ :5.3''5 40c. UOc. May (!, " 2 lbs. tea, 1.''"':^ ; 1 bottle pickles-, ; 1 lb. pepper, IJ.'ie. 1.00 i.,'.o " 20," 1 bag salt, ; 10 lbs. sugar, 10c. ; 1] lbs. vai.sin.s, 50c 75c. ' 2.=o " 31, " 3 lbs. currants, 25c. ; 10 lbs. Avhite sugar, 25c 1.^0 i2,ic. 2.00 June 10, ■' 2 lbs. tobacco, 75c. ; ^ Ib.B. soda, 25c."; 201bs. rice, 10c. . 40c. 10c. 30c. l.'^^ •• 17, " 1 Ib.cloves, : \ Ib.nutmegs, ; } cinnamon, ; 1 lb. tea, $17.Gl 2. Balti.mobe, Oct. 1st, ISfiG. .Mil. William Patterson-, To Moffat & Mukray, Dr. 18GG. July 3, To 14 yds. fancy print, 20c. ; 12 yds. coFd silk, 2^^ " 14, '• 2 ladies' felt hats, 2.oo; 2 prs. kid gloves, l.so " 22, " 4 prs. cotton hose, 40c. ; 3 yds. red flannel, 80c Aug. 19, " 2h yds. blk. casslmere, 2.»5 ; 2} yds. cotton, 20c " 27, " l|- yds. white flannel, 75o. ; buttons, 10c. ; twist, 15c Sept. 1, " 2 suits boys' clothes, 9.00; 2 felt hats, 1.8 '' 8," 2 prs gloves 80c. ; 2 neckties, 02^ c " 22, •* J doz. prs. cotton hose, 7.5 ; ^ doz. shirts, 2t).oo Contra, Cr. 20.00 ly.oo Aug.18, ByCash, ; 27. Cash, $36.00 " 25. " flrkin butter, 95 lbs., at 22c 20.90 55.90 Balance due $41.71J Received payment in full, MOFFAT & MURRAY. ACCOUNTS AND INVOICED. 123 RocuESTEii. Jail. 2n(l, ISCG. Mr. John' Deans. To Wood & Fuixjim!, Ifr. 18CC. July 4, To 12 lbs. sugar, 10c. ; 3 lbs. tea, l.a^ ; 2 lbs. tobacco, S'lc. " 11, " 1 bbl. salt, 295 . 2 lbs. indigo, 2')C. ; U lbs. pepper, :!0c. " 18, " 2 prs. socks, 'l.'jc. ; 1 neck-tio, ".Oc ; 2 scarfs. 2'k' " 25, " 10 lbs. sugar, lie; 201bs. dr'dnpples. lOc; 2 Ihs. colVoo 28c' " " " 18 lbs. dried peaches, 12Jc. ; 1 bush, onions, n^'i Aug. 4, " 12 lbs. rice, 7c. ; 2 gals, syrup, 75c. ; Itlbs. sugar, 12c.. . " '• •• 1.*} lbs. mackerel, 12c. ; 2 lbs. ginger, 20c. ; 2 lbs. tea, l.^-^ " 21. '• 2 prs. kid gloves, l"* ; 2 boxes collars, rJJc Sept. 12, " 10 1b.s.sugar, 15c. ; 2 lbs. coffoe, 35c. ; 1 lb. chocolate. lOe. Oct. 4," 2 folt hats, 125; shoo blacking, 25c " 21. " 2 lbs. pepper, 15c. ; soda, 40c. ; .sulpelre, 30c. ; salt, 75c. Contra. Or. 10.0 -,.0 Sept. 14. By Cash, ; Oct. -5, Casli Oct. 17. '• 2 bbls. winter apples, 2''''' $ 18.42 Boston-. Nov. Ist, '80(5. Mr. Wm. Reid. To CAMrBEix, LixN it Co., JDr. Aug. 4, To 2 prs. kip boots, 3-''' ; 2 prs. cobourgs, 2-^ " 17. " 7 yds. fancy tweed, 2=5 ; trimmings, po ; buttons 25c.. Sept. 4, '• 2 prs. gloves, 75c. ; 3 prs. socks, 35c. ; 2 straw hats, 40c. •' 2G. •' 10 yds. print, 35c. ; trimmings, l^^ ; ribbons, 75o Oct. 11, '' 3 neck-ties, G2^c.; 2 prs. boys' gaitors, 2'''5; shoe ties,12|Jc. •' 22, " 1 business coat, 14oo ; 2 felt hats, l'^^ . i umbrella, 2^0 "■ 27, '• 2 flannel shirts, 4=5 ; 1 pr. pants, fi^o . over-coat, 10°". " 30. • 2 laco scarfs, 2^5 ; 3 pj-y woollen mits 75c. ; pins, 25c. Contra. Cr. 1000 goo Sept. 12. By Cash ; Oct. 4, Cash, Oct. 24. " 300 lbs. cheese, 10c. ; 75 lbs. butter, 25c Balance duo $37.00 Received payment. ca:\ipbell, linn & co. i \i.4\'l :! ti I) .) . 124 AMTIIMETIC. AcurRN, Sept. let, 18GG. Aln. .9. SMirn 7<) WiusoN, Uay & Co.. JJ)\ 18GC. Jan. 15, To er annum, by the year. Thus, $G a year for every $100, is called six per cent, per IXTEllEST. 10,"! annum. The term is also extended to designate the return iiccruir.c from any investment, sucli as hliares in a joint stock company. To show the object and use of such tvansactions, wo may suppose a case or two. A person feels himself cramped or embarrassed in liis ciiciuii stances and operations, and he applies to some friendly party thai lends him ^[ilOO for a year, on the condition that at the expiration ol the year he i.s to receive $100, that is, the SlOO lent, and $»! irorc as a return for the use of the $100 ; or, if the borrower f:;ets $000. he pays at the and of the stipulated time not only the 8000, but alsc $30 (SO for each 8100) in return for the use of the 8000. By thij means the borrower gets clear of his difficulty, and maintains hi.- credit at a small sacrifice. Tlie sum on which interest is paid is called the i^rindpal. The sum paid for the use of money is called the intenst. The sum paid on each $100 is called the rate. The sum of the principal and interest is called the amount. When interest is charged on the principal only, it is called suni^c interest. When interest is charged on the amount, it i,s called compound interest. When a certain rate per cent, is established by law, it is called legal interest. When a higher rate per cent, is charged than is allowed by law. it is called imiry. The legal rate per cent, differs in different States and in different countries, so also does the mode of calculation differ. In some States it is considered le. !' 'l>^ dii 130 ARITHMETIC. Ill " i ti< V, I'! Ill ti|! I est, JuiJ tlie time, any tlircc of which boiug known, the Ibuithcan bo ibinul. The finding of the interest inchides by far the greatest number of c:ises. We shall first .show the genera! principle, and from it deduce an easy practical rule. Let it be required to find (he interest on $±G8 for one year, at G per cent. As 100 is taken as the basis principal in relation to which all calculations arc made, it is plain that 100 will have the same ratio to any given principal that the rate, which is the interest on 100, has to the interest on the given principal. Hence, in the question proposed, we have as $100 : $408: : $G : interest=$4G8XTS?5= $4G8X-0G=:$28.08. Now .OG is the rate ^>cr «?u7, and from thia we can deduce rules for all cases. CASE I . To find iLo interest of any sura of money for ono year, at any given rate per cent. II TT L E . Multiply the principal hy the rate per unit. EXERCISES. 1. What is the interest on $15, for 1 year, at 3 per cent. ? Ans. $0.45. 2. What is the interest on $35, for 1 year, at 5 per cent. ? Ans. $1,75. 3. What is the interest on $100, for 1 year, at 7 per cent. ? Ans. $7.00, 4. What is the interest on $2.25, fer 1 year, at 8 per cent. ? Ans. $0.18. 5. What is the interest on $G.40, for 1 year, at 8^ per cent. ? Ans. $0.54. G. What is the interest on $250, for 1 year, at OA- per cent. ? " Ans. $23.75. 7. What is the interest on $760.40, ior 1 yei.., at 7^- per cent. ? Ans. $57.03. 8. What is the interest on $9G4.50, for 1 year, at GJ per cent. ? Ans. $G2.69. 9. Wnat is the interest on $568.75, for 1 year,'at 7J per cent. ? Ans. $41.23. INTEREST. CASK I r . 137 To find the interest of any sum of money, for any number of years, a^ a given rate per cent. RULE. Find the interest for one year, and multiply by the number nj years. EXERCISES. 10. What is the interest of $-±.G0, for 3 years, at per cent. ? Ans. $0.83. 11. What is the interest of $570, for 5 years, at 7^ per cent. ? And. $213.75. 12. What is the interest of $460.50, for 3 years, at G.^ per cent. ? Ans. $80.34. 13. What is the interest of $17.40, for 3 years, at 8J per cent. ? Ans. $4.35. 14. What is the iucerest of $321.05, for 8 years, at 5J per cent. ? Ans. $147.G8. 15. What is the interest of $1650.45, for 2 years, at 9 per cent. ? Ans. $rD7.08. 16. What is the iutcjest of $964.75, for 4 years, at 10 pr r cent. ? Ans '3385.90. 17. What is the interest of $1674.50, for 3 years, tt 10^ per cent? Am. $527.47. 18. What is th. interest of $640.80, for 5 years, at 4| por cent. ? Ans. $152.19. 19. What is the interest of $9G5.50, for 7 years, at 5| per cent. ? Ans. $371.72. 20. What is the interest of $2460,20, for 4 years, at 7 per cent. ? Ans. $688.86. CASE III. To find the interest on any sum of money for any number of months, at a given rate per cent. RULE. Find the interest ft, one year, and take aliquot parts for the months ; or, Find the interest for one year, divide by 12, and multiply by the number of months. f f^^ '' i! ; 138 AHrniMETIC. K X K II C I S E H . 21. What is the interest on $G84.20, for 4 months, at (5 per cent.? Ans. 813.G8. 22. What is the interest on $7G0.50, for 5 months, at 7 per cent.? Ans. 822.18. 23. What is the interest on $899.99, for 2 months, at 8 per cent. V Ans. $12.00. 24. What is the interest on $964.50, for 4 months, at 9 per cent. ? Ans. 628.94. 25. What is the interest on 61500, for 7 months, at 10 per cent. ? Ans. §87.50. 2G. What is the interest on $1500, for 11 months, at 7i- per cent. ? Ans. 6107.25. 27. What is the interest on 61575.54, for 8 months, at G^ per cent. ? Ans. 6G5.G5. 28. What is the interest on 61728.28, for 9 months, at 8^ per cent. ? Ans. $110.18. 21'. What is the interest on 62G8.25, for 13 months, at 7 per cent. ? Ans. $20.34. 30. What is the interest on 615G9.45, for 1 year, 3 months, at 8 per cent. ? Ans. 615G.95. 31. What is the interest on 6642.99, for 1 year, 5 months, at 10 per cent. ? Ans. 691.09. 32. What is the interest on $500.45, for 1 yciir, G months, at dh per cent. ? Ans. 679.86. 33. What is the interest on 648.50, for 3 years, 9 months, atlO^* per cent. ? Ans. 619.10. 34. What is the interest on $560.80, for 2 years, 8 months, at llf per cent. ? Ans. $175.72. 35. What is the interest on $2360.40, for 19 months, at 12 per cent. ? ' Ans. $448.48. CASE IV , To find the interest on any sum of money, for any number o£ months and days, at a given rate per cent. RULE . Find the interest for the w nths, and take aliquot parts for the days, reckoning the month as consisting of 30 days. EXAMPLE. 36. What is the interest on $875.50, for 8 months, 18 days, at 11 per cent. ? SIMPLE INTEREST. 139 SOLUTION. Principal $875.50 Rate per unit .11 Interest foi 1 year DO.^OuO ' Interest for (> months ; or, \ of interest for 1 year 48.152.") Interest for 2 months ; or, J- of interest for months Kl.OoOS Interest for 15 (lays; or, ^ of interest for 2 months 1.0127 Interest for 3 days ; or, J, of interest for 15 days .8025 Interest for 8 months, 18 days 800.0185 We find the interest for 1 year to be §90.305, and as *> mouths are the ^ of 1 year, the interest for U months will bo the ^ oi the interest for 1 year ; likewise the interest Ibr 2 months will bo the ^ of the interest for months, and as 15 days are the \ of 2 months or GO days, the intercst^for 15 days will be the ^ of the in- terest lor 2 months, and likewise the interest for 3 days, will bo the I of the interest for 15 days. Adding the interest for the innnths and dai/s together, we obtain $09.02, the sum to be paid for the unc of $875.50, for 8 months, 18 days, at 11 per cent. EXERCISES. 37. AVhat is the interest on $468.75, for 4 months, 15 days, at 7 per ecnt. ? Ans. 812.30. 38. "What is the interest on $1654.40, for 3 months, 8 days, at 5 per cent. ? Ans. 822.52. 39. What is the interest on 8345.05, for 11 months, 25 days, at 6 per cent. ? Ans. 820.45. 40. What is the interest on |74.85, for 5 months, 22 days, at 9 per cent. ? Ans. 83.22. 41. What is the interest on 8073.75, for 8 months, 19- days, at 7J per cent. ? Ans. 836.35. 42. AVhai is the interest on 857.45, for 1 year, 2 months, 12 days, at 6 per cent. ? Ans. $4.14. 43. What is the interest on $2647, for 1 year, 5 months, 18 days, at G|- per cent. ? Ans. $242.64. 44. What is the interest on 8268.40, for 2 years, 1 month, 1 day, at 8 per cent. ? Ans. 844.79. 45. What is the interest on 82345.50, for 3 years, 7 mom is, 20 days, at 10 per cent. ? Ans. $853.50. 140 ARITHMETIC. » .-.f^ [I hv} 40. What is the intciest on f 11308.45, ibr 4 years, 11 months, 11 duyfl, at 11 j per cent. ? Ann. $2481.24. 47. What is the intere.st of $042.20, for 2 yearn, 7 months, 24 days, at 12 per cent. ? 48. What i.s the interest of $04.50, for 2 years, 11 months, 2 days, at 7 per cent. ? Ans. Si:}. ID. 41). What i-i the amount of 8740.25, for 1 year, 10 months, 12 days, at 5 per cent. ? 50. Wliat is the interest of §080, for 4 years, 1 month, 15 days, at per cent. ? Ans. $108.30. CASK V . To find the interest on any sum of money, for any number of days, at a given rate per cent.=" RULE. Find the interest for one year, and sai/, as one year (305 dnys,) is to the given number of d'lys, so is the interest for one year to thf, interest required ; or, Having found the interest for one year, multiply it by the given number of days, and divide by 305. EXERCISES. 51. What is the interest on $404, for 15 days, at per cent. ? Ans. $1.14. 52. What is the interest on $304, for 12 days, at 7 per cent. ? Ans. 84 cents. 53. What is the interest on $50.82, for 14 days, at 8 per cent. ? Ans. 17 cents. ,. t * To find how many years elapse between any two dates, we have only to subtratt the earlier from the later date. Thus, the number of years from 1814 to 18G5 is 51 years. To find mouths, we must reckon from the given date in the first named month, to the same date in each successive month. Thus, five months from the 10th of March brings us on to the 10th of August. To find days, we require to count how many days each month contains, for to consider every month as consisting of 30 days, in the calculation of inter- est, is not strictly correct, although for portions of a single month it causes no serious error. Thus, the correct time from March 2Dd to June 14th, would be 104 days, viz., 29 for March, 30 for April, 31 for May, and 14 for June. A very convenient plan for reckoning time between two given dates is to count the number of months and odd days that intervene. Thus, from June 14th to November 20th, would be 6 niontlis and 6 days. RIMPLK INTEREST. Ill 64. What is the interest on $75.50, for 18 cbys, at B\ per cent.? Ans. '.V2 ciMitH. 65. What is the interest on $125.25, for 20 day.<«, at 5 per tent. ? Ans. .'>4 cents. 6G. What is the interest on 8150.40, for 33 days, :it (5 per cent. ? Ans. 82 cents. 67. What is the interest on $50.48, for 45 days, at OA per cent. ? Ans. 45 ccMits. 58. What i.s the interest on ^75.75, for 05 days, at 7 per cent. ? Ans. 94 cents. 59. What is the interest on $268.40, for 70 days, at 7^ per cent. ? Ans. $3.80. 60. What is the interest on $464.45. for 80 days, at 8 per cent. ? Ans. $8.14. 61. What is the interest on $15.84, for 120 days, at 9 per cent. ? Ans. 47 cents. 62. What is the interest on $240, for 135 days, at 9A per cent. ? Ans. $8.43. 63. What is the Interest on $2400, for 145 days, at 10 per cent. ? Ans. $97.73. 64. What is the interest on $1568, for 170 days, at 11 per cent. ? Ans. $80.33. 65. What ia the interest on $2688, for 235 days, at llf per wnt. ? Ans. $203.35. 66. What is the amount of $364.80, for 320 days, at 11| per jont.? Ans. $401.58. C A 8 E V I . To find the interest on any sum of money, for any time, at 6 per cent. Since .06 would be the rate per unit, or the interest of $1 for 1 year, it follows that the interest for one month would bo the y'j of .06, or -fij of a cent, equal to ^ cent or .005, and for 2 montlis it would equal ^ cent, or .0G5x2=.01. Therefore, when interest is at the rate of 6 per cent., the interest of $1, for every 2 months, is one cent. Again, if the interest of $1, for one month, or 30 days, is ^ cent or .005, it follows that the interest for G days will be the J of .005 or .001. Tht;refore, when interest is at the rate of G per cent,, the interest of $1 for evciT G days is one mill. Hence the :;;t 't ■';^:-,: 142 ARITHMETIC. U U L E . Find the interest of 01 /or the (jivai time hi/ rcckoniixg G cents for every year, 1 cent for every 2 months, and 1 mill for every G days; then mrdtiply the given principal by the number denoting that in- terest, and the 2>roduct will be the interest required. NoTK — Tliis method can be adopteil for ivny rate per cent, by (irst finding the interest at (> per cent., then adding to, or subtracting from the interest so found, Bucli a part or parts of it, as the given rate e.Kceeds, or is less than G per cent. This method, although adopted by some, is not exactiy correct as the year is considered us consisting of SCO days, instead of 3G5 ; so that tho in- terest, obtaLied in this manner, is too large by ^-jy or ^j, which for every $73 interest, is $1 too much, and must therefore be subtracted if the exact amount be required. EXAMPLE. 07. What is the interest cf $24, for 4 mouths, 8 days, at 6 per cent. ? SOLUTION. The interest of $1, for 4 months, is 02 Thointercst of $1, for 8 dajs, is OOIJ- Ilcnce the interest of $1, for 4 months, 8 days, is 021^ Now, if the interest of $1, for the given time, is .021^, the inter- est of $24 will be 24 times .0211, which is $.512. EXERCISES. 68. What is the interest on $171, for 24 days, at 6 per cent. ? Ans. G8 cents. G9. What is the interest on $112, for 118 days, at G per cent. ? Ans. $2.20. 70. What is the interest on $11, for 112 days, at G per cent. ? Ans. 21 cents. 71. What is the interest on 50 cents, for 3G0 days, at G per cent. ? Ans. 3 cents. 72. What is the interest on $75.00, for 236 days, at G per cent. ? Ans. $2.95. 73. What is the interest on $111.50, for 54 days, at G per cent. ? Ans. $1.00. 74. What is the interest on $15.50, for 314 days, at 6 per cent. ? Ans. 81 cents. SIMPLE INTEREST. 143 75. What 7G. What i cent. 77. Wliat i 78. What i per cent. ? 79. What i 80. Whati at 7 per cent. ? 81. What i 82. What i 83. What i at 10 per cent. 84. What i per cent. ? 85. What i 7 per cent. ? 80. What i per cent. ? 87. What i 8 per cent. ? 88. What i per cent. ? 89. What i cent. ? 90. What i per cent. ? s 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 G per Ans. 8 cents. s tlio interest on $154, for 3 months, at per cent. ? Ans. $2.31. s the interest on $172, for 2 months, 15 days, at 6 Ans. $2.15. 3 the interest on $25, for 4 months, at G per cent. ? Ans. 50 cents. 3 the interest on $36, for 1 year, 3 months, 1 1 days, Ans. $3.23. s the interest on $500, for IGO days, at G per cent. ? Ans. $13.33. s the interest on $92.30, for 78 days, at 5 per cent. ? Ans. $1.00. a the interest on $125, for 3 years, 5 months, 15 days, Ans. $43.23. s the amount of $200, for 9 months, 27 days, at 6 Ans. $209.90. s the interest on $125.75, for 5 months, 17 days, at Ans. $4.08. 3 the interest on $84.50, for 1 month, 20 days, at 5 Ans. 59 cents, s the amount of $45, for 1 year, 1 month, 1 day, at Ans. $48.91. s the interest on $175, for 7 months, G days, at 5A- Ans. $5.78. s the interest on $225, for 3 months, 3 days, at 9 per Ans. -$5.23. s the interest on $212.60, for 9 months, 8 days, at 8^ Ans. $13.95. CASE VII. To find the interest on any sum of money, in pounds, shillings, and pence, for any time, at a given rate per cent. RULE. Multiply the principal hij the rate per cent., and divide by 100. 144 m 'I AEITHMETIC. EXAMPLE. 91. What is the interest of £47 ISs. 9d., for 1 year, 9 months, 15 days, at 6 per cent. ? SOLUTION. £ s. D. Interest for 1 year 2 17 4 Interest for '.'■> mos.^ or ^ of int. ibr 1 year, 18 8 Interest for IJ mos., or | of int. for G nios., iA 4 Interest lor 15 days, or*i of int. for 3 mos., 2 4h 2^80 14 G 20 Interest for 1 year, 9 months, 15 days.... £5 2 8^ 17;34 12 £ 8. D. 47 15 9 6 4;i4 92. What is the interest of £25, for 1 year, 9 months, at 5 per cent. ? Aus. £2 3s. 9d. 93. What is the interest of £75 12s. Gd., for 7 months, 12 days, at 8 per cent. ? Ans. £3 14s. 7^d. 94. What is the amount of £64 10s. 3d., for 3 months, 3 days, at 7 per cent. ? Ans. £G5 13s. 7d. 95. What is the interest of £35 4s. 8d., for 6 months, at 10 per cent. ? Ans. £1 15s. 2|d. 96. What is the amount of £18 12s., for 10 months and 3 days, .at G per cent. ? Ans. £19 10s. 9|d. CASE VIII. To find the PRINCIPAL, the interest, the 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 6 per cent., it will produce .075 interest. Now, if in this example, .075 be the interest on 81, the number of dollars required to produce $4.50, will be represented by the number of times that .075 is con- tained in $4.50, vhioh is 60 times. Therefore, $G0 will produce $4.50 interest in 1 year, 3 montiis, at 6 per cent. Hence the SIMPLE INTEREST. RULE. 145 Divide the given interest hy the interest of $1 for the given tinUf at the given rate per cent. EXERCISES. 98. What principal will produce 77 cents interest in 3 months, 9 days, at 7 per cent. ? , Ans. $40. 99. What principal will produce $10.71 interest in 8 months, 12 dayr. at 7^ per cent. ? Ans. $204. 100. What principal will produce $31.50 interest in 4 years, at 3^ per cent. ? Ans. $225. 101. What sum of money will produce $79.30 interest in 2 years, 6 months, 15 days, at 6 J per cent. ? Ans. $480. 102. What sum of money is suflScient to produce $290 interest in 2 years and 6 months, at 7J per cent. ? Ans. $1600. CASE IX . To find the rate per cent., the principal, the interest, and the time being given. EXAMPLE. 103: If $3 be the interest of $60 for 1 year, what is the rate per cent. ? SOLUTION. If the interest of $60 for 1 year, at 1 per cent, is .60, the re- quired rate per cent, will be represented by the number of times that .60 is contained in 3.00, which is 5 times. Therefore, if $3 is the interest of $60 for 1 year, the rate per cent, is 5. Hence the RULE. Divide the given interest hy the interest of the given principal at 1 per cent, for the given time. EXERCISES. 104. If the interest of $40, for 2 years, 9 months, 12 days, is $13.36 ; what is the rate per cent. ? Ans. 12. 105. If I borrow $75 for 2 months, and pay $1 interest ; what ia the rate per cent. ? Ans. 8. 146 ARITH3IETIC. I [*■:' iiii 100. If I give $2.25 for the use of $30 for 9 months ; what rate per cent, am I paying ? Ans. 10. 107. At M'hat rate per cent, will $150 amount to $1G5.75, in 1 year, 4 montlis, 24 days ? Ans. 1h. 108. At what rate per cent, must $1, or any sum of money, be on interest to double itself in 12 years ? Ans. Ans. 8A-. 109. At what rate per cent, must $425 be lent to gain $11.73 in 3 months, 18 days ? Ana. 9J . 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 produce $6, it is evident that the number of years required to produce $12 interest, will be lepresente.l 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 year, at the given rate 2>cr cent. EXERCISES. Hi!. In what time will $12 produce $2.88 interest, v.t 8 per cent ? Ans. 3 years. 114. In what time will $25 produce 50 cents interest, at 6 per cent. ? Ans. 4 months. 115. In what time will $40 produce 75 cents interest, at 6^ per cent. ? Ans. 3 months, 18 days. SIMPLE INTEREST. 147 116. In wliat time will any sum of money doublo itscU'. at G per cent. ? Ans. IG years, 8 months. 117. In what time will any sum of money quadruple itself, at 9 per cent. '? Ans. 33 yeans, 4 months. 118. la what time will S125 amount to $138.75, at 8 percent.? Ans. 1 year, 4 monthb, 15 days. 119. Borrowed, January 1, 18G5, $G0, at G per cent, to be paid as soon as the interest amounted to one-half the principal. AVlien is it due? • Ans. May 1, 1873. 120. A merchant borrowed a certain sum of money on January 2, 185G, at 9 per cent., agreeing to settle the account when the in- terest equalled the principal. When should lie pay the same ? Ans. Feb. 11, 1867. M E a C II A N T S ' TABLE For showing in what time any sum of money will double itself, at any mte per cent., from one to twenty, simple interest. Per cent. Years. Per cent. Years. Per cent. Years. Per cent. Years. 1 100 6 16ft 11 9tV 16 ^ 2 3 50 7 8 14| 121 12 13 17 18 5|f 5?, 4 25 9 in li 7| 19 5A 5 20 10 10 15 Gf 20 5 MIXED EXERCISES. 121. What is the interest on $64.25 for 3 years, at 7 per cent. ? Ans. $13.49. 122. What is the interest on $125.40 for G months, at G per oent. ?* Ans. 3.76. 123. What is the amount of $3G9.29 for 2 years, 3 months, 1 day, at 9 per cent. ? Ans. $444.16. 124. What must be paid for the use of 75 cents for G years, 9 months, 3 days, at 10 per cent. ? Ans. 51 cents. 125. What will $54 amount to in 254 days, at 10 per cent. ?=i^ Ans. $57.81. * This and the following exercises (marked with a *) are to be worked by Case VI. us ARITHMETIC. h- -:| 12G. What must be paid for the interctit of 845 for 72 clays, at 9 per cent. ?''• Ans. 81 cents. 127. ^y\v^t is the interest of ^240 from January 1, 18GG, to June 4, 18GG, at 7 per cent. ? Ans. $7.14. 128. What will $140.40 amount to from August 29, 1SG5, to November 29, 18GG, at Gh 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 840 amount to $44.40, at 8 per cent.? Ans. 1 yr., 4 mos., 15 days. 131. At what rate per cent, will $40 produce in 1 yr., 4 mos., 15 days, $4.40 inteicst ? Ans. 8. 132. What must be paid for the int-i-est of $145.50 for 240 days, at 9-^ per cent. ?* Ans. !;;'9.22.. 133. What will $1G0 amount to in 175 days, at G per cent. T''' Ans. $1G4.G7. 134. At what rate per cent, must any sum of money bo on interest to quadruple itself in 33 years and 4 months? Ans. 9. 135. In what time will any sum of money double it.self, at 10 per cent. ? Ans. 10 years. c A 8 lo XI, To find the interest on bonds, notes, or other documents draw- ing 7,^,, percent, interest. Since .07 ,^q or, .073 would be the rate per unit, or the interest of $1 for I year or oG5 days, it follows that the interest for 1 diy would be the ^^^ part of .073 which is .0002, equal to two tenths of a mill, hence the RULE. Multiplij the principal hi/ the number of days, and the product by two tenths of a mill the rcsidt will be the ansicer in mills. EXAMPLE. What must be paiu for the use of $75 for 3G days at 7-f^ per .sent. ? SOLUTION. The interest on $75 for 3G days would be the same as the inter- nist on $75X3G— $2700 for 1 day, and at f^ of a mill per day- would be $2700 X. 0002=54 cents. 2. What would be the interest on $118.30 for 42 days at 7fg per cent. Ans. 99cts. am COJIMERCLVL rAPEIt. 149 COMMERCIAL PAPER. Commercial paper id divided into two classes — negotiable and nox-ni;gotiaiih:. N K G () T I A IJ L E CO JI JI E II C I A L PAPER. Nvgot'aihUi commercial paper is that wliich may be freely trans- ferred from one owner to another, so as to pass the rij^dit of tiction to the holder, without beinjj; subject to any set-offs, or Ic^al or oquitiiblc djl'ences existing between the original parties, if transferred for a valuable consideration before maturity, and received without any defect therein. Negotiable paper is made payable to the payee therein named, or to his order, or to the payee or bearer, or to bearer; or some similar term is used ; showing that the maker intends to give the payee authoiity to transfer it to a third party, irce IVom all set-offs, or c(|uitablc or legal defences existing between hiniseH' and the payee. N O N - N E G O TI A B L E C O JI 51 E R C I A I. PA P i; R . Kon-iicgotiahle commercial paper is that which is made payable to the payee therein named, without authority to tiansi'er 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-off's, and legal or equitable defences existing between the original parties. II O W T II E T I T L E PASSES. The title to negotiau.'e paper })asses iVoui one owner to another by delivery, if made payable to payee or bearer, or to bearer. It passes by indorsement and delivery, if made payable to payee or order. The title to non-negotiable paper pa.sscs by a mere verbal assignment and delivery, or by indorsement and delivery. p ]l I :,[ A R V 1) E B T o il . In a promissory note there are two original parties — the maker and the payee. The obligation of the maker is absolute, and con- tinues until the note is presumed to have been paid under the Statute of Limitations. The maker is the primary debtor. In a bill of exchange tliere are three parties. When the drawer accepts the bill, he becomes the primary debtor upon the bdl of exchange. PROMISSORY NOTE NOT PAYABLE IN MONEY. When a promissory note is payable in anything but money, it does not come within the Statute. There is no presumption that it is founded upon a valuable consideration. A consideration must bo :*(( f 150 AIIITII5IETIC. m tiUcged in the complaint, .'md proved on the trial. The acknowledg- ment ol" :i coasideviition iu such promissory note, by inserting tho words " value received^" is sufficient to cast upon the defendant tho burden of proof that tlierc was no consideration. Tho acknowledg- ment of '• value received," raises tho presumption that the noto was given for value ; but this presumption may be rebutted by tho de- fendant. A negotiable instrument is a written promise or request for tho 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 lOt certain of fulfillment. The penson, to whose order a negotiable instrument is made payable, must bo ascertainable at the time the instrument is made. A negotiable instrument may give to the payee an option between the payment of the sum specified therein, and the performance of another act. A negotiable instrument may bo with or without date; with or without seal ; and with or without designation of the time or place of payment. A negotiable instrument may contain a pledgo of oollateral secu- rity, with authority to dispose Llicreof. A negotiable instrument must not contain any other contract than such as is specified. Two different contracts cannot be ad- mitted. Any date may be inserted by the maker ef a negotiable instru- ment, whether past, present, or future, and tho instr-'nieut is not invalidated by his death or incapacity at the time of the nominal date. There are several classes of negotiable instruments, namely : — 1. Bills of Exchange; 2. Promissory Kotcs ; li. Bank Notes; ' . Cheques on Banks and Bankers ; 5. Coupon Bonds ; 6. Certifi- cates of Deposit; 7. Letters of Credit. A negotiable instrument that doeo \iui specify the time of pay- ment, is payable innnediately. A negotiable instrument wiiich docs not specify a place of pay- ment, is payable wherever it is held at its maturity. An instrument, otherwise negoti.Lbi*, in form, payable to a person named, but adding the words, " or to liis order," or " to bearer," or equivalent thereto, is in the fomicr cu-c rtiyablc to the written order of such person, and in the latter case, payable to the bearer. A negotiable instrument, made payable to the order of the maker, or of a fictitious person, if issued by the maker for a valid considera- tion, without indorsement, has the same effect against him and all other persons having notice of the i'acts, as if payable to tho bearer. x\ negotiable instrument, made payable to tho order of a person obviously fictitious, is payable to the bearer. The signature of ever}' drawer, acceptor and indorser of a nego- COMMERCIAL rAl'Ei;. lol liable instrument, is presumed fo have been maac for a valuable considcraiion, before the maturity of the instrument, and in tho ordinary course of buniness, and the words ''value received," acknowlodcrc a consideration. One who writes his name upon a negotiable instrument, otherwi.so than as a maker or acceptor, and delivers it, with his name theieon, to another person, is called an indoiscr, and his act is called an indorsement. One who aj;;rees to indorse a negotiable instrument is bound to write his signature upon the back of the instrumchL, if there is sufficient space thereon for that purpose. When there is not room for a signature upon tlie back of a nego- tiable instrument, a signature equivalent to an 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 specifies the indorsee. A negotiable instrument bearing a general indorsement canuot be afterwards specially indorsed ; but any lawful holder may turn a general indorsement into a special one, by writing above it a direction for payment to a particular person. A special indorsement may, by express words ibr that purpose, but not otherwise, be so made as to render the instrument not negoti- able. Every indorser of a negotiable instrument warrants to every subse- quent holder thereof, who is not liable thereon to him : 1. That it is in all respects what it purports to be ; 2. That he has a good title to it ; 3. That the signatures of all prior parties are binding upon them ; 4. That if the instrument is dishonored, the indorser will, upon notice thereof duly given unto him, or without notice, 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 the same extent as in the ease of a transfer without indorsement. Except as otherwise prescribed by the last section, an indor.se- ment " without recourse" has the same effect as any other indorse- ment. An indorsee of a nnf!;otiablc instrument has the i^anie ridit against every prior party thereto, that he would have had if the contract had been made directly between them in the first instr.ncc. An indorser has all the rights of a guarantor, and is exonerated from liability in like manner. ill lltii'fl 152 rt AlUThMETIO. Pi m m > 1 *^] i . f , 'q 1-i 1 < 1 m m Olio wlio iiulorsos a ii0!j;oti;iblo instrument, at tlio request, and I'or till! *' acconiniodation" of auother party to the instrument, lias al! the 1 iiihts of a surety, and is exonerated in like manner, in respect to every one Ir.ivinu,' iioliee of tlio I'aets, except that lie is not entitled to contribution i'loni subse(|uent indorsers. Tlu! want of consideration for tlie undertaking of a maker, acceptor, or indorscr of a negotiable instruni'mt, docs not exonerate liiiii i'roni liability lliereon, to an indorsee in good I'aitli i'or a consid- eration. All indorsee in due course is one who in good faith, in the ordi- n;uy course of business, and for \alue, before its apparent maturity or presumptive dishonor, ac((uire.s a negotiable instruiiK'nt duly indorsed to him, or indor^^ed generally, or payable to the bearer. An indorser of ti negotiable instrument, in due course, acquires an absolute title thereto, so that it is valid in his hands, notwith- standing any provision of law making it generally void or voidable, and notwithstanding any defect in the title of the person i'rom whom ho acquired it. One who makes himself a party to an instrument intended to be negotiable, but which is left wholly or partly in blank, for the pur- pose of filling afterwards, is liable upon the instrument to an indorsee thi'reof in due course, in whatever manner, and at whatevor time it may be filled, so long as it remains negotiable in form. It is not necessary to make a demand of payment upon the principal debtor in a negotiable instrument in order to charge liim ; but if the instrument is by its terms payable at a specified place, and he is able and willing to pay it there at maturity, such ability and willingness are equivalent to an oiler of payment upon his part j'*resentnient of a negotiable instrument for payment, when necessary, mu^t be made as follows, as nearly as by reasonable dili- gence it is practicable : 1. The instrument must be presented by the holder, or his authorized agent. 2. The instrument must be presented to the principal debtor, if he can be found at the place where presentment should be made, and if not, then it must be presented to some other person of discretion, if one can be found there, and if not, then it must bo presented to some other person of discretion, if one can be found there, and if not, then it must be presented to a notary public within the State ; 3. An instrument which specifies a place for its payment, must be presented there, and if the place specified includes more than one house, then at the place of residence or business of the principal debtor, if it can be found therein ; 4. An instrument which docs not specify a jilace for its payment, must be presented at the place of residence or business of the prin- cipal debtor, or wherever he mav be found, at the option of the presenter ; and, cnt by be COMMEllAIAL VATEli. 153 a 5. Tho instrument must Lo presented upon the day of its nppiu-- cnt maturity, or, it' it is payable on demand, at any time l)(:lnro its apparent maturity, witliin reasonable liours, and, if it is payable at a banking house, within the Ubual banking liours of tho vicinity; but, by tho consent of the person to w!iom it should be prcsenled, it may be presented at any liour of the day. The apparent uiaturily of a negotiable instru.ncnt, payabK; at a particular time, i.s tho day on which by its terms it becomes due ; or, when that is a holiday, it should b^i paid the previous day. A bill of excliange, payable at a specified time after night, whicli is not accepted within ten days after its date, in addition to the time whicli would suffice, with ordinary diligence, to forward it for acceptance, is presumed to have been dishonored. The apparent maturity of a bill of exchange, payable at sight or crj demand, is : 1. If it bears interest, one year after its date ; or, 2. If it does not bear interest, ten days after its date, in addition to the time which would suffice, with ordinary diligeaco, to Ibrward it for acceptance. The apparent maturity of a promis.sory 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, six months after its date. When a promissory note is payable at a certain time after sight or demand, such time is to be added to the periods mentioned in tho last paragraph A party to a negotiable instrument may require, as a condition concurred to its payment by him : 1. That the instrument be surrendered to him, unless it is lost or destroyed, or tho holder has other claims upon it ; or, 2. If the holder has a right to retain the instrument, and does not retain it, then that a receipt for the amount paid, or an exonera- tion of the party paying, be written thereon ; or, 3. K 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 claim thereon ; or 4. If the instrument is destroyed, then that proof of its destruc- tion be given to him. A negotiable instrument is dishonored when it is cither not paid, or not accepted, according to its tenor, or presentment lor tiic purpose, or without presentment, where that is excused. Notice of the dishonor or protest of a negotiable instrument may be given : 1. By a holder thereof; or, 2. By a party to the instrument who might bo compelled to p;iy it to the holder, and who would, upon taking it up, liuvo a right to reimbursement from the party to whom the notice is given. 4 lo-Att Ai;nj!.ur.Ti(' li Mi: A nolico of dislioiKH' inay '>'! ^'ivni in any IVirtn which dcj-crihcs tho instniiiiotit vith rcasdiKililc ccrttiiiity, niul Mili:.t:iiiti:il!y inrornis llio j).»rty iccciviiiLC it that (ho iiistrmiiciit has Ikcm di>liui>niTd, A iHitico of dishonor may bo L'iven : 1. \iy delivfiiuix it to ihi; ])arty to 1k! chariird, ncreonnlly, ut any i)hicc; or, ii. l{y diilivoriiiLt it to some per.-ou of discretion at tho jiluco of rcsidcnci) or l)n>ino.ss of fsuch I'arty, apparently uctlti!^ for him ; or, ;*). IJy propcnly folilin;,' the notice, directing' it to tho party to bit char^'od, at hi:< place of residence, according to the best information that tho person ^ivinu; tho notico can obtain, depositing; it in tlio post-office most eonvenietitly accessibk! from the JjUicc where the presentment was made, and jjayinjj; the postage thereon. In case of the death off. party to whom notice of dislionor siiouhl otlierwise be uiven, tho notice must be j^iven to one of his personal representatives; or, if there are none, then to any member of hi.s family who resided with lam at liis death , or, if tlioro is none, then it must be mailed to hi.s Inst I'lace of residence, :is prescribed by ■subdivision .'J of the last paraLiraj)h. A notice of dishonor sent to a party ai'ter his dc-atli, but in i^^no- ranco thereof, and in j^ood i'aith, is valid. Notice of di,-li(iii(niir tlTiS !) Slfi'!in'j». Ah i'iiiijtt'.hi>i Dirvnihie jirovlntln. II inns phi'iin jmiirr pur iv. innmlut ii I'ordri' (lii HoustiiFtui's In smnmo. dc rent inminniiie linUhcres atcrlinijs 1) nhflliDi^H fakur en iKmstnCniin (t ipir ^iismrc^ Niiivanl iavis de, ■i Mes.sierirs " ii L> ndrtM, Xtirnheni, dun 28 ()rU,U'-r. 1H:;8. Vfo tloi) SlerUmj. Ziccl Dioiude nac.h d'do zahkn Ste t^'joi dlcsm Prima Wrrh.sd an die (hdrc. des llcrrn Kin llmulfvt \'finul ,' Werik rrhalten. >iV hrimjfa solche an/ Uechnnmj laut LerkUl von ilerrn ' " Londini. I ALUN, JAvovno, hi 2,') yi'llriiihrr. IMS. I'lr .ij.'.OO Slcrliue. A 3Vc mesi data ptojiUe per fpicsta primn de Oindilo ( mnj sd vnHa ) nlV ordlne . Id soninia dl J/ire c'dkiho rftilo shrlhie ndn'n frttii' hiata, e pontle in vindo M. >'. secnndn I'avviso Addi ■. Ji_. Limdra. Sl'AMSlI. Mukuja. d -H) de .V///' (k IMS. >'■;- t.'AW. A noventd dias/icltn sc sen-Iran T* mandur patjuv pur rsln ]iriinfr;i dc cmn- hio a la onleu de ios S''" Tros vuntas Uhrn.'i K.s(erl'nas en oro o phda valor recibldo dc dhon i'"' (ptr, anotaran valor en vaenta seijun aviso de , A Ios S"' Londrvs. PoUTCOfESK. £C00 Eslcrrt)ias. ^ Lisbon, aus K \ London. i'3l7 19.S:. 1a>/mcnts, and the interest on each, from the time they wcrejyaid to the time of settle ent, the remainder will he the amount due. 6. $500. Prescott, May 1st, 186-4. " One yeur after date, for value received, I itromise to pay Musgrove f& Wright, or order. Five Hundred Dollars, at their office^ in the city of Toronto, loith interest at G per cent., payable annually. Jajies Manning. There was paid on this note : May 4th, 1865 $150 Doc. 18th, '' 300 How much was due June 1st, 1866 ? SOLUTION. Face of note, or principal ^500.00 Interest on the same from May 1st 1864, to June 1st, 1866 '. 62.50 Amount of the principal at time of settlement 562.50 First year's interest on principal $30 Interest on the same from May 1st, 1865, to June 1st, 1866 $1.95 Second year's inte-est on principal $30 Interest on the same from May 1st, 1866, to June 1st, 1866 15 Amount of interest upon annual interest 2.10 Total amount of principal $564.60 First payment, May 4th, 1865 $150.00 Interest ou the same from May 4th, 1865, to June 1st, 1866 9.70 Second payment, December 18th, 1865 300.00 Interest on the same from December 18th, 1865, to June 1st, 1866 8.20 Payments and interest on the same 467.90 Amount due June 1st, 1866 $96.70 11 ' !!.,1i. ■\m ti! ■K I 'J "3 fill •I !S l! 158 ARITHMETIC. 7. §700. Cincinnati, January 2nd, 18G3. Eighteen months after (late, T promise to pmj to tht order of J. II. Wilson, Scoen Hundred Dollars, for vdlitc received, with interest at (y jx'r cent., payable annualli/. Tiios. A. Brvce, There was paid on thi.s note : January 15th, lSG-1 $350 July 2nd, 18G-t 300 What amount wa3 duo January 2n(l, 1SG5 ? Ans. S107.22. 8. 01 ', Indianapolis, Jan.Srd, 18G3. VV ■ ears afterdate, I promise to pay A. li. Tcnnison, or order, j\tf,L iMi'd 'd and Fifty Dollars, with interest at jicr cent., p)ciyahle annuauy, value received. Jajies S. Paumenter. The following payments were receipted on the back of this note : February 1st, 1SG4, received $500 3Iay 14th, " " 100 January 12, 18G5, <' 3C0 What was due May Gth, 18G5 ? Ans. $188.94. 9. $250. Mobile, January 2nd, 18G3. Three years from date, for value received, I jnvmisc to pay Michael Wright, or order. Two Hundred and Fifty Dollars.^ with interest, pay ahle annually, at Gj)er cent. Calvin W. Pearsons. At First National IJanJc here. What was the amount of this note at maturity ? Ans. $297.70, CONNECTICUT RULE. The Supreme Court of the State of Connecticut has adopted the following RULE. Compute the interest on the principal to the time of the first pay- ment ; if that he one year or more from the time the interest com- menced, add it to the principal, and deduct the payment from the sum total. If there he of tcr payments made, compute the interest on the halanve dnc (n the. next payment, and then deduct the payment as above, ami in iUci: iiuinucr from one p-jipnent t-^ "iH^thcr. t'P aV thf a noli then for sumf it ti intel 10. PARTIAL PAYMENTS. 159 prri/ments nrr. ahmrhed, provided the time hrtwecn one pai/vieut and aiiothc)' he one year or more. Jf 'ill!/ p!'i/mcitts ie made before one years interest has neenied, then eompii/e the. interest on the principal sum due on the oldigadon for one j/ear, add it to the principal, and compute the interest on the sum paid, from the time it was paid, up to the end of the year ; add it to the. sum petid, and deduct that sum from the jmncipal and interest, added as above. If any payments be made, of a less sum than the interest arising at the time of such jiaymcnt, no interest is to be comjmted, but only on the jirineipal sum for any period. NoTi:.- ll;i yi'iu- exiomls beyond tliu tiim-of i'e//;enieH/.Hticl tlin amount of thi! rcMniiiiiio},' priiicipiil to the time oi sdtkinent ; find also the amount of tho payment or jjiiyments, if any, from the time they were paid to the time of sottlemeut, and subtract their sum from the amount of t!> rincipal. I : A :« p L E s . iO- S^OO. KiNGST' J.nelst, 1802. On demand re promise to pay J. A Smith rf- Co., or order, nine hundred dollars, for value rccei 'l, with interest from date, at Qt per cent. JoNEs .^t WlllCIIT. On the back of this note wore receipted the following payments: June IGth, 18G3, received $200 August 1st, 18G4, " IGO Nov. IGth, 1864, " 75 Feby. 1st, 18GG, '• 220 AVhat amount was duo August 1st, 18G6 ? SOLITTION. Face of note or principal $900.00 Interest on the same from June 1st, 18G2, to June IGth, 1863 5G.25 Amount of principal and interest, June IGth, 1803 95G.25 First payment to be taken from this amount 200.00 Balance due 7 .') G . 2 .'> Interest on the same from June IGth, 1863, to August 1st, 1804 .jl.O-lG Amount due August 1st, 18Gi S07.29G M' * k H' I' J t' ,. i f 1 1 ( ■ ( 1 f »; 1 ! r 1 1 IGO AiurniiETic. Second payment to be taken from this amount IGO.OOO Balance due G47.29G Interest on ihe same for one year 38.837 Amount due August 1st, 18G5 G8G.J.33 Amount of 3rd payment from Nov. IGth, 18G4, to August 1st, 18G5 78.187 Balance due G07.94G Interest on the same from August 1st, 1865, to August 1st, 18GG 3G.476 Am.ount duo August 1st, 18G6 644.422 Amount of 4th payment from February 1st, 18G6, to August 1st, 1866 226.600 Balance due August 1st, 1866 $417,822 merchants' rule. It is customary among merchants and others, when partial pay- ments of notes or other debts are made, when the note or debt is settled within a year after becoming due, to adopt the following RULE . ■ Fi7ul the amount of the j^rificipal from the time it became due xintil the time of settlement. Then find the amount of each payment from the time it toas paid until settlement, and subtract their sum from the amount of the principal. E X A M t> L E . 11. $400. Maitland, January 1st, 1865. For value received, I promise to pay J. B. Smith & Co., or order, on demand, four hundred dollars, loith interest at 6 per cent. A. K. Cassels. The following payments were receipted on the back of this note : February 4th, 1865, received $100 MaylCth, " " 75 August 28th " " 100 November 25th, " " 80 What was due at time of settlement, which was December 28th, 18G5? I'AI.TIAL rAYMENTS. SOLUTION. ■>l i(;l Principal or face of note SiOO.OO Tiilercst on the same IVoni flan. 1st, 1805, to Doc. 2fitli, 1805 23.80 Amount of principal at scttlonicnt 123.80 First payment $100.00 Interest on the same Irom Feb. 4th, 1805, to Dec. 28th, 1805 5.40 Second payment 75.00 Interest on the same i'rom May 10th, 1805, to Dec. 28th, 1805 ii.77^ Third payment 100.00*^ Interest on the same from August 28th, 1805, to Dec. 25th, 1805 2.00 Fourth payment 80.00 Interest on tlic same from Nov. 25th, 1805, to Dec. 28th, 1805 44 Amount of payments to be taken from amount of principal ,%5.GU Balance due, December 28th, 1805 ^58.18* 12. $500. * Cleveland, January l.st, 18G5. Three months after date, 1 jmymisc to pnij James Man- ning, or order, five hundred dollars, for value received, at the First National Bank of Jhiffalo. Cyrus Kino. Mr. King paid on this note, July 1st 1805, $200. What was due April 1st, 1800, the rate of interest being 7 per cent? Ans. $:]24..50. 13. $240. PuiLADELPiiiA, May 4th, 1805. On demand, I j)roniise to j^^'!/ -^^- A". Frost <(• Co., or order, tico hundred ail forty dollars, fur value received, xcith in- terest at per cent. David Flook. The following payments were receipted on the back of this note ; September 10th, 1805, received $00 January 10th, 1800, '' DO What wn«J due fit, the time of settlement, which was ]M;iy -I tli, 1806? Ans. SiOO.44. \ IN 11 '!! 1G2 IJ. s:uo. ARITHMETIC. Lowell, Juiiii lOtli, IfiOi. 2'hrcc months vftrr (hih\ I prcmiKc to p'i;i Thnnxus Culrrrwil/, or onUr, three huttdrtd (Uinning receive as au equivalent, January 1st, 1862? Ans. 8653.40. 16. I buy a bill of goods amounting to $2500 on six months' credit, and can get 5 per cent, off by paying cash ; how much would 1 gain by paying the bill now, provided I have to borrow the money, and pay 6 per cent, a year for it ? Any. $53.75. BANKS AND BANKING-. General Principles of Banking. — Banks are commonly divided into the two great classes of banks of deposit and banks of issue. This, liowevcr, appears at first sight to be rather an imperfect classi- fication, inasmuch as almost all banks of deposit are at the same time banks of issue, and almost all banks of issue also banks of deposit. But there is in reality no ambiguity; for by banks of deposit are meant banks for the custody and employment of the money deposited with them or entrusted to their care by their customers, or by the public ; while by banks of issue are meant banks which, besides employing or issuing the money entrusted to them by others, issue money of Ihcir own, or notes payable on demand. The Bank of England is principally a bank of issue ; but it, as well as the other banks in the different parts of the empire that issue notes, is also a great bank of deposit. The private banking companies of London, and the various provincial banks, that do not issue notes of their own, are strictly banks of deposit. Banking business may bo con- ducted indifferently by individuals, by private companies, or by joint stock companies or associations. Utility and Functions of Batiks of Deposit. — Banks of this class execute all that is properly understood by banking business ; and their establishment has contributed in no ordinary degree to give security and facility to commercial transactions. They afford, when properly conducted, safe and convenient places of deposit for the money that wculd otherwise have to be kept, at a considerable risk, in piivato houses. They also prevent, in a great measure, the necessity of carrying money from place to place to make payments, and enable them to be made in the most convenient and least expen- sive manner. The objects of harikinff. — Correct sentiments beget correct con- duct. A banker ought, therefore, to apprehend correctly, the objects of banking. They consist in making pecuniary gains for the stockholders ty legal operations. The business is eminently :l 168 ARITIDrETIC. « bencfi'iial to society; but sonic bunkers bavc d emeu tJie gn^xA cf Fociety so much more worthy of regard tlian the pri.'aic t ;od of 'il.ookholders, that they liavo supposed all loans bliould be ui.speubed with direct reference to the beneficial cflfcet of the loariH on society, irrespective, in some dera'cc, of the pecuniary interests of the dis'pen- sing bank. Such a banker will lend to buUders, that houses or ships may bo multiplied; to. manufacturers, that useful iabrics may bo increased ; and to merchants, that goods may be seasonably replen- ished. He deems himself, cx-nfficio, the patron of all interest.'?) tliat concern his neighbourhood, and regulates his loans to these interests by the urgency of their necessities, ri'ther than by the pecuniary profits of the operations to the bank, or the ability of the bank to sustain such demands. The late Bank of the United States is a remarkable illustration of these errors. Its manager seemed to believe that his dutes comprehended the equalization of foreign and domestic exchanges, the regulation of the price of cotton, tho up- holding of State credit, and the control, in some pnrticulars, of Congress and the Preshlent — all vicious perversions of banking to an imagined paramount end. When we perform well the direct duties of our station, wo need not curiously trouble ourselves to cfFect, indirectly, snuie remote duty, llesults belong to i'rnvidence, and by the natural catenation of events (a system admirably adapted to our restric fed foresight), a man can usually in no way so efficiently promote the general wel- fare, as by vigilantly guarding tho peculiar Interests committed to his care. If, for instance, his bank is situated in a region dependent for its prosperity in tlic business of lumbering, the dealers in lumber will naturally constitute his most profitable oustomors ; hence, in promoting his own interest out of their wants, he will, legitimately, benefit them as well as himself, and benefit them more perinanently than by a vicious subordination <.[ his interests to theirs. Men wall not engage iKu'iiLiuf tly in any business that is not pecuniarily beneficial to tbe,:\ v: .onally ; iience, a banker becomes recreant to even the manufacturing and other interests that he would protect, if he so manage his bank as to make its stockholaers unwill- ing to continue the employment of their capital in banking. This principle, also, is illustrated by the late United States Bank, for the stupendous temporary injuries which its mismanagement inflicted on society, arc a smaller evil than the permanent barrier its mismanage- ment has probably produced against the creation of any similar institution. Banh of England' Notes Legal Tender. — According to the law as it stood previously to 183-1, all descriptions of notes were legally payable at the pleasure of the holder in coin of the standard weight and purity. But the policy of such a regulation was very question- able : and we regard the enactment of the Stats. 3 & 4, Will. 4, c. 99, which mak s Bank of England notes legal tender, everywhere B.iNKS AND BANKING. 169 J, IS a except at Uie Bark uud its brancliei'j, for all sums above great iniprovciiKnt. Savincrs Banks have been in use in f]urrpe over fifty year.-*, and in Canada and the United States, almost as long. Tbey aro established for the purpose cf receiving- from people in moderate circum^itances, small Kums of money on interest. In England the deposits arc held by the Government, and invested in the three per cent, funds. In !Ncw PjUgland, New York and other States, the deposits aro generally loaned on bond and mortgage at six or seven per cent, interest. Friendlij Societies. — Friendly Societies are associations, mostly in England, of persons chiefly in the humblest classes for the pur- pose of. making provision by mutual contribution against those con- tingencies in human life, the occurrence of which can be calculated by way of average. Tlie principal objects contemplated by such societies are the following: The insurance of a sum of money to be paid on the birth of a member's child, or on the death of a member 0' any of his family; the maintenance of members in old age and widowhood ; the administration of relief to membens incapacitated for labor by sickness or accident ; and the endowment of members or their nominees. Friendly Societies aro, therefore, associations for mutual assurance, but arc dlstingushed from assurance societies, properly so called, by the circumstance that the sums of money which they insure arc comparatively small. BANK DISCOUNT. The Bank Discount of a note is the simple interest on the sum for which it is given from the time it is discounted to tlio time it becomes due, including three days of grace. Suppose, for example, in getting a note of S200 discounted at a bank 1 am charged ijsl-i for discount, which being deducted, i receive but $188, so that I pay interest on $12 which I 'id not receive. From this it is clear that T am paying a higher .e of interest iii discounting a note at a bank, than I would pay wore I to borrow money at the same rate. As bank discount is tlie same as interest, we derive the following li 'II g|^ RULE. Find the inten-st on the sum specified in the note at the qic n rate, and for the given time, including three dai/s of grace, and thi' will he the bank diipCOUNT. Suhfract the discoimt from the face of the note, and the remain der will he the p"ocp:eds oh I'RESEnt worth. 170 AEITHMETIC. E XE R CISES $527. 1. What is the bank discount on a note, given for 60 days, for $350, at G per cent. ?* • Ans. $3,67. 2. What is the bank discount on a note of $495, for 2 months, at 5 per cent. ? Ans. 4.33. 3. What is the present value of a note of $7840 discounted at a bank for 4 months and 15 days, at G per cent. ? Ans. $7659 G8. 4. How much money should be received on a note for $125, payable at the end of 1 year, 3 months, and 15 days, if discounted at a bank at 8 per cent. ? Ans. S112. 5. A note, dated December 3rd, 1860, for $160.40, and having 6 months to run, was discounted at a bank, April 3rd, 18G1. at 6 per cent. ; how long had it to run, and v/hat were the proceeds ? Ans. 64 days ; proceeds $158.71, 6. On the first day of January, 186G, I received a note for $2405 at 60i3ays, and on the 12th of the same month had it discounted at a bank ut 7 per cent. ; how much did I realize upon it. Ans. $237.61. 7. A merchant sold 240 bales of cotton, each weighing 280 pounds, for 12|- cents per pound, which cost him, the same day, 10 cents per pound ; he received ''^ payment a good note, for 4 months' time, which he discounted immediately at a bank at 7 per cent, ; what will be hi^ profits? Ans. $1479 10, 8. I hold a note against Clemes, llice & Co., to the amount of $327.40 dated April 11th, 1866, having sis months to ran after date, and drawing interest at the rate of 6 ^er cent, per annum. What are the proceeds if discounted at the Girard liauk on the 10th of August, at 7 ,^y per cent. ? Ans. $332.99. NoTK. When a. note drawing interest, is discounted at a bank, the interest is calculated on the lace of the note liom its date to the time of maturity, auu added to the lace of the note, and this amount discounted lor the length of tiiii-i Iht; note has still to ruu. 9. Wliat will be the discount on the following note if discounted nt the Ciiv Bank on the 17th of Nouembcr, at 6 per cent. (360 days t.' a year), • ' Tht'ougLuut. oU the exercises, unless otherwise specified, the yi-iir is to be cousidered us consisting of 365 days. Since it^s customary in ))usine.=s when fraction of a cent occurs in and result to reject it. if less thuu luilf a cent, and il not lesp.to call it a cent, we have adopted this principal through- out tlie bool- IIAMK DISCOUNT. 171a $527.-,Vc. Oberltn, Oct. 4, .SCO. Ninctif oays a/ta' date fur value received, v:e prom'm to pay to the order of Smith, Warren it' Co., five, hiuuhrd twenty- seven and j"||'„ dollars at the City Bank, Ohcrlin, icilh interest m eight jjtr cait. Thompson & Burns. 10. What will be the discount at 7,''^ per cent, on u note for $227.41, drawing interest at 8 per cent., dated May 1st, 18G5, at 1 year after date, if discounted on ?Iarch 7th, 1866? 11. Wliat amount of money will I receive on the following note, if discounted at the First National Bank of Detroit on June 21st, at 9 per cent. ? $473.80. Detroit, May 17, 1866. Three months afte7-daie I promise to pay to the order oj J. li. Sing & Co., four hundred and seventy-three and -^^^^j Dollars, at the First National Banlc, Detroit, for value received with interest at7j\perce7it. RiciiAKD Dunn. 12. What must I pay for the following note on August 15th, 1866, so as to make at the rate of 30 per cent, interest per annum on the money I pay for it? Ans. §708.54. $746.75. Adrian, January 10, 1806. One year from date, for value received, u-c irromiae to •pay James Ames, or order, seven hundred andforfy-six -|•^j'^^ dollars, at the Commercial Banh, Adrian, icith interest at '^ f[, 2'<^>' cmf. per annum, AVilson k Cummino.'^. 13. A holds a note against B to the amount of $478.02, dated May 10th, 1865 at 1 year after date drawing 7,'',j- per cent, interest. I purchase this note from A. on August 18tL, paying for it .' nioiitlit, lit G per cent.? An,':;. $(5I8.e.S. G. A I'arnicr houuht :i larUi i*-r §5000 cuK-h. and l.tivin;^ only one- hali' of the sum on liuiid. ho wif-hcs to cbtain the baumco from the bank. For what sum niu.-t lie uive his note, to bo discounted for mouths, :it G per cont. ? Ans. 62G19.17. 7. If a mcrchiint vis-hosi lo obtain $r)r>0 of a bank, for what sum lauist lie give his note, payable in GO days, allowing it to be dis- counted at h per cent, per montli ? Ans. $535.75. 8. I sold A, Mills, merchandize valued at $;»1S.1G, for which he was to pxiy me cash, but bein;^ disappointed in reoeiving money ex- pected, he gave me his endorsed nt>te at 'Jo dnys, lor sueii an amount that when discounted at the bank at 7 pjr ociit. it would produce the p'ice of the merchandize. AVhat was tho face ol ilie note ? 9. I am owing i{. Harrington on accouui, now due, sj1GS.-15 ; liO also holds a note against nie for 8"J10. i.luo in M days, including days of grace ; he allows a disccunt of 8 per cent, on the note, and if I give him my note at GO day^^ ior an amount tliat wiil be suliicient if discounted at (5 per cent., to produce the amount cf account and note. What will be the face of new note ? 10. Samuel Johnson lias been owing me 8'J71.'18 for 84 days. I charge him interest at G per cent, per annum ibr this time, and he gives mc liis note at 90 days ibr such an amount that when dis- counted at the Girard Bank, at 8 per cent., the proceeds will equal the amount now due. AVhat is the face of the note ? 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 bnnk discount. i; X A .'\i I' h i: 1. What rate of interest is paid when a r'^te, payable in 3G2 days, is discounted at 10 per cent. ? 172 ARITHJIETIC. SOLUTION If WO discount 81 for the given tiiuc, and at the given rate, the proceeds will be .90, or 90 centx. Hence, the discount being 10 cents, we arc paying 10 cents for the use of 90 cents. Now, if wo pay 10 cents for the use of 90, for the use of 1 cent wo must pay y\j of 10 cents, or }j of a cent, and for $1, or 100 cents, wo must pay 100 times 1 of a cent, or Mj^^.Hi, and for $100, $11 J, or UJ 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., exjn'cssed decimally, or the rate per unit, hy the nvmher denoting the proceeds of $1 for the given time and rate. The quotient will he the rate of interest required. EXERCISES. 2. What rate of interest is paid when a note, payable in GO days,, is discounted at 7 per cent. ? Ans. Tvfj'iy. 3. What rate of interest is paid when a note, payable in 3 month.?, is discounted at G per cent. ? Ans. Gy'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^ 1 3*. 5. What rate of interest is paid, when a note of $200, payable in 70 days, is discounted at ;J per cent, a month ? Ans. 9 ,^,,'j . G. When a note of ^i5, payable in G5 days, is discounted at 7 per cent., to what, rat^- of interest does the bank discount correspond ? \ns f-^s:'- 7. A bank, by discounting a note at 6 per cent., receives for its money a discount equivalent to Gh per cent, interest ; how long must the note have been discounted before it was due ? Ans. 1 yr,, 3 mos., 12d^ COMMISSION. Commission 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 agint, to be sold on account and risk of another, are termed a consignment. COMMISSION. 173 The person to whom these goods are consigned is called the con- 8ip*'ev 07 correspondent. Thfc tcriii shipment is somctimcH used instead of consignment. E X A 31 IM. E . A «!^iiimission merchant soils for mo goods worth $1200, and cbartjcff t per cent. ; what have I to pay him ? SOLUTION. 4 per cent, of §1200 is equal to $1200 X .04—848. Ilonce 1 would have to pay 848, and from this wc deduce the following RULE . Find the i^crcentage on (he given sum at the given rate, which ivill he the commission. E X E U C I 8 E S . , 1. Consigned to A.K.Coomcr, Esq., Syracuse, by the Troy, N.I., foundry, agricultural implements which arc sold for >?1875.7r), what is the agent's commission at 2^ per cent. ? Ans. S4t).80. 2. Bought in Boston 12 chests of tea, containing {J4 lb.s. each, at SI. 12^ per lb., on a commission of If per cent. ; what was my com- mission ? An/-. S15.12. 3. My Toledo correspondent has bought for rnc 27G8 lbs. of bacon, at 12i- cts. a pound ; what is his commission at 3;}- per cent.? $11.25, 4. Bought a carriage and pair of horses, per the order oi' 8. Williams, Portland ; paid for the horses ^240, and charged 4^ per cent., and paid for the carriage SIGO, and charged 1^ per cent. ; how much did I earn ? Ans. $13.20. 5. A commission agent in a Southern State bought cotton worth $2284 for an English manufacturer, and charged 5^ per cent.; what is his commission ? Ans. $125.G2. 0. On another occasion the manufacturer gave 'the commission merchant $105.78, for purchasing for him cotton worth $3G84 ; what was the rate per cent ? Ans. 4^ 7. An English commission merchant buys for a Portland house, .£57G 10s. Od. worth of provisions, and charges 4^ per cent. ; what is his commission? Ans. £25 ISs. 10 hi. 8. A New York provision merchant instructs a Belfast (Ireland) commission merchant to purchase for liim £534 4s. Od. worth oi '*.« ■i:> m ha 111 I '1 i n IMAGE EVALUATION TEST TARGET {MT-3) h // y. f/i 1.0 I.I l^|28 |2.5 S ^ Ilia S lis ilM 6" 1.8 L25 IIIIIU IIIIII.6 Photographic Sciences Corporation ^V! i\ ^s^ ^ \\ 1> •«'. o\ 23 WEST MAIN STREET WEBSTER, NY. T^SSO (716) 872-4503 ^ 17-1 AlilTIDLETIC. I' i I |! i let ii'i i bacon and hums, iiivl oilers him 7:}- per cent, get? ; ^vhat does the agent Ans. X38 lis. 7d. i). A book agent in Cincinatti, sells §487.50 worth of books for Day k Co., of Montreal, and receives $72.05 for liis 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 S2G2.G0 ; 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 by the given rate per unit, and the quotient icill he the sum to he invested ; subtract this from the given amount, and the remainder will he the commission. EXAMPLE. If I send $1890 to a commission merchant, and instruct him to buy merchandise with what is left after his commission at 5 per cent. is deducted ; wliat will be the sum invested, .ind the agent's com- mission ? SOLUTION. It is plain that for every dollar of the proposed investment I must remit 105 cents, 100 towards the investment, and 5, towards the commission, and hence the number of dollars which can be in- vested from the sum remitted will be the same as the number of times that 1.05 is contained in 1890. Now, $1890-^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. Pauls, $988 to purchase flour for mo with the balance that remains after deducting his commission at 4 per cent. ; required the purcliaso nion"'y and percentage ? Ans. 8950 and $38. 2. Ileceived a commission to buy wheat with $779, less by my commission at 2^ per cent. ; required the price of the wheat and my commission. Ans. 87G0, and $19. DllOKERAGE. 1 «ewmarli('t, commissions W\ Orr, Portland, to lU'ocuro for liim a quantity of tine Hour, and remits 6!>17.01 ; how much fhnir can ho have, al'ter allowini^ 1;^ per cent., and wliat will the commission amount to ? Ans. 8H7(), and $11.01. 5. John ytalkcr, Jjondon, commi.ssions J. Fleming New York, to purchase for him as n^uch butter as lie can procure for the balance between $770.52, and his own commission at 1-^- per cent. ; how many pounds butter did he get at 25 cents per lb. ; what the whole price, and what was the commission? Ans. 3072 lbs., $708, and $11.52. 0, Dr. Gallipot is about to remove to England, and sends ^o a London cabinet maker $4005.45 towards getting his house furnished, ho is charged 3^ per cent, over and above the price of the furniture, for time and labour, what docs the furniture cost ? Aus. $3870. 7. Graham IJros., of Newbury, send to 11. White, Charleston, bacon and hams worth $1500, they charge 5^ per -cent, coramission, and the charge for lading is $75.15 ; how much does 11. White owe them ? Ans. $1720.95. 8. P. liobson, coramission 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 the price of the goods, and what was the amount of the commissimi ? Ans. $450.75, and $18.27. BROKERAGE. BuoKEaAGE is a per centage paid to an agent Tor ncgociating bills, collecting accounts, exchanging money, buying and selling shares and stocks, and all similar transactions. Such an aurcnt is called a Broker. A smaller percentage is usually allowed to a broker than to a commission merchant, because the work he has to do requires less time and labour. Like commission, brokerage is merely a particular case of percentage, and hence the 11 V ], E . To find the hrokeritr/e on ttuij sum, Jhul the pcrccntaijc on fue given sum at. the given rate, ichieh v.-iH hr the Itrol'erage i\ *i 176 ARITHMETIC. \i: 1. A broker in Buffalo has bouc^ht for me $1275 worth of Erie R, 11. stock ; what will bo the brokerage at 2^ per cent. All'* '^7 "^- 2. I pay a collector of accounts 2 per cent, for collcctini; $1 1H.50 ; how much docs it cost inc ? Ans. $J..'?7. '{. I pay a broker 1^ per cent, for selling $271().7r) government stock ; how much do T give him ? Ans. $50 04, nearly. 4. Advised 11. P., broker, to collect two bills amounting to $897, he has collected ^ of it, and I have given him 1^ per cent, on the amount collected ; how much have I paid him ? Ans. $8.97. 5. A. B. sent me $756 to purchase flour for him. 1 have charged 2.^ per cent, commission on the whole sum, and purchased flour with the remainder ; what is my commission, and how much do I vest in flour for A. B. ? Ans. 0738.90, and $17.01. 6. The school taxes on all the sections of a country amount to $1180, and collectors get 2y- per cent.; how much remains available for school purposes? Ans. $1140.03, 7. Instructed a broker in Syracuse to sell for me 200 shares of N. Y. C. 11. 11. stock, at 114^; what will bo my proceeds, broker's commission being ^ per cent. S. I am charged ^ per cent, by a broker in llakigh, for nego- ciating a draft for $750 ; what are the proceeds coming to me ? Ans. $748.12^. 9. Bought G. W. 11. shares to the amount of $578, and paid my broker 2\ per cent. ; how much did I give him ? Ans. $13.01. 10. Gave D. F. 81 per cent, for collecting accounts for me to the amount of $G39 ; how much did I give him ? Ans. $21.30. To find the sum that can be invested when the given amount includes both the brokerage and the investment. For example, if I wish a broker to invest for me $700, and his charge is 2 per cent., 1 must obviously remit to him $714, as $14 is 2 per [cent, on $700; conversely, if I send him $714, and instruct him to invest for me that sum, minus his own percentage, he will have to calculate how much he will have remaining to invest after deducting his own charge. Xow, since his percentage is $2 ou every $100, he should get from me $102 for every $100 he is to in- vest, and therefore the sum he can invest will be the 102nd part of what I remit, i. c, $714-^1.02:^$700. Hence the BROKERAGE. RULE. 177 Divide the given amount hy one, increased by the given rate per unit o/ brokerage, and the quotient will he the sum, to be invested; subtract this from the given amount, and the remainder will be the brokerage. EXERCISES . 1. A broker receives $574, with instructions to invest vrhat re- mains after deducting brokerage at 2^ per cent., in R. 11. shares ; how much has ho to invest ? Ans. $560. 2. The assessment on a certain district, together with the per- centage for collection at 2.\ per cent., is $1717.80 ; what is the amount of the assessment, and what the expense of collection ? Ans. eiG80, and $37.80. '6. A tax amounting to $3270.52, including collector's fees at 4 per cent., is levied on a certain town; what is the amount of the tax, and how much is the collector entitled to ? Ans. $3150.50, and $120.02. 4. A gentleman once invested in U. S. government bonds, a certain sum which, with the broker's fee at IJ per cent., amounted to $18,315 ; what was the amount of the investment ? Ans. $18,000. 5. A Portland broker negociatcs a draft for $1218 for a Hamil- ton merchant, at li^ per cent. ; what are the proceeds ? Ans. $1199.73. 6. A broker, after deducting his charge at 1:^ per cent., invests the balance of $2450.25 for his employer in bank stock ; how much docF he invest ? Ans. $2420. 7. My broker invests for me in oil well shares, at $83 each, what remains after deducting his fee at ^ per cent, from $8341.50 ; how much does he invest, and how many shares docs he purchase ? Ans. $8300, and 100 shares. 8. A broker's charge is $285, at \\ per cent., on a certain sum invested ; what is the sum ? — (Sec Percentage, Case II.) Ans. $1C000. 9. A broker sells slocks for me, and the sum which is realized, together with the brokerage at 4 per cent., amounts to $010 ; what is the sum procured, and what the brokerage ? Ans. $875 and $35. 178 /o VnTTITMF.'J'TC. :» !' $ !• 3I_y iiutiit i:i llichiuniid li;i> jiircliascd cotton IVir nu! to the aiiinuiit of SI "r'j.'O :uh1 f!i;ir^c.s iiic n couitni;^.sioii of ^ pci- cnt. ; how luufli liuvo 1 to rciiiit liiin to pay i'or the cotton and C(Miniii.s- Mon. Ans. i^l8l)1.4:iA. -. 1 h;iv(.! rcci'ived iVom a correspondent hi 'J'foy $ HSiJ.ll , whh instruct ions to inve^t the y,\\iu\ h' Five-twenties at 1(15.',, first deduc- ting my (;onnni.s,-ion of ['i per cent. AVhat is the coinniis.^ion, and what amount ol" Five-twenties can I }»urchase ? An.-:. Commis.'-ion ^S^.til ; invested in Five-twenties, f-'->;jOO. per ii. A collector receives $:J0 for eoUectinj^' S'JOU ; at what cent, is Ik- })aid ? An.-:. 2 4. A purchased jier the order of Andrew Camphell i': Co., Naslu'ille, Tenn., 1-1:^^72 lbs. ('. C bacon at i:J] cts. per 11),. clian- in,i;' a commis.^ion of \h ]wy cent. A wislies to draw on them lor re- imbursement ; wliat must be the ftcc of the draft if it co."t -.V jicr cent, to get it caslicd, and what is the c-ou;mission on i>urchaseV Ans. Face of dft. 8-<'l".l ■") : Coi;imi,ssion C2:).r<(h 5. A broker invests for me 81 Vf)!). and J p.iy him for !iis trouble ^43.75, at what rate jier cent, do 1 pay him ^ Ans. 2\. (■». An Au( tioueer valued the furniture of a decea.sod u'cntloman, and charging 4 per cent., he was paid ijo^J.Su ; what was the vaiue ol the furniture V Ans. SK'-til.T)!). 7. 1 sent to Taylor ^t Morrison, Cum. merchants, 'Nlvv Yerk, 2r>() firkiiis butter, containing on an average 51! lbs. each, at 15 cts. per lb. Tluy sold at ;in advance of 10 per cent. ; i'reight, Sic, de- ducted ;:;'I().!5, counnissiou 2A- per cent. They have remittcsd me a sight drai't ibr net proceeds, which they purchai>ed at ;; per cent, premium, charging [ i)er cent, commission on face of drai't. What amount of draft did [ receive, and what amount of commission charged V 8. A certain district pays §800 school taxes, the collector gets ^'38 for collecting; Avhat per eentagc docs ho get ? Ans. 4^. i). 1>. instructed a broker to sell for him lOG shares of the N. Y.C. 11. li. at 112^, how niucli would the broker's commis.sion be at :^ pcv cent. 10. An accountant , is entrusted to make schedules of the debts and assets of a bankrupt ; he charges only 2^ per cent, on the debts, on the principle that lie will have little trouble in getting the accounts due by the bankrupt sent in ; but as he knows very well that he will have trouble in getting correct statements sent in of accounts due to the bankrupt, he stipulates for 5-^- per cent, on these ; how much does ho get altogether, the debts being $2786, and the assets $018 ? Ans, $103.04. *'! T INSUR.VNCE. 179 J INSURANCE. Insurance is an engagement by which one party is bound, in consideration of receiving a certain sum, to indemnify another for something in case it should in any way be lost. The party under- taking the risk is seldom, if ever, an individual, but a joint stock company, represented by an agent or agents, and doing business under the title of an " Insurance Company, ^^ or " Assurance Com- pany,'' such as the "Globe Insurance Company," the "Mutual Insurance Company." Some companies are 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 callr'l the Policy of Insurance, or simply tho Policy. The party tliat undertakes to indemnify is called the Insurer, or underwriter after he has written his name at the foot of tlie 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 bo classified under three heads. Fire Insurance, including all cases on land where property is ex- posed to the lisk of being destroyed by tire, 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 be 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 lu- svrance will be applicable. This kind, as the name implies, insures against all accidents by sea. Life Insurance. — This is an agreement between two parties, that in case the one insured should die within a certain stated time, the other shall, in consideration of having received a stipulated sum annually, pay to the lawful heir of the deceased, or some one laen- ii' i IB: in m 180 ARITHMETIC. If I < w'i tioned in his \7ill, or some other party entitled thereto, the amount recorded in the policy. For instance, a man may, on the occasion of his marriage, insure his Hfe for a certain sum, so that should he die within a certain time, Lis widow or children shall bo paid that sum by the other party. Again, a father may insure the life of his child, so that in case of the child's death within a specified time, he shall 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 tisk that A may die before ho is able to pay B ; another party engages, for a certain yearly sum, to pay B in case A should fail to do so during his life time. In some instances, insurances arc effected to gain a support in case of sickness. Such a contract is called a Health Insurance. In- surances are now also effected for compensation in case of railway accidents. These we may call Jiailwai/ 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- eed upon are not regularly kept up, the policy lapses, that i im ,s null and void, so that the holder of it forfeits not only his v—^m 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. Tho date at which the system of insurance began cannot be clearly ascertained ; but, whatcv'>r its date, its origin seems to have been protection against the perils of the sea. We know that it was practised, in a certain way, by the ancient Greeks and Romans. If a Roman 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 the 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 ou board the vessel to look after the cargo, and receive the repayment on the safe delivery of the goods. This agent corros- pon'led pretty nearly to our more modern supercargo. As tho art of navigation advanced, and the securities afforded by law became INSURANCE. 181 more stringent, and also facilities of communication increased, this system 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 equitablencss of insurances, and their utility in promoting commercial exterprisc, we may remark that they make the interest of every merchant, the interest of every other. To show this, we may c )mpare an inmrancc office to a club. Suppose the 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 system causes that each epeculator, by insuring his own stake, contributes .>jo much to the funds of a company, which is bound to indemnify each loser. On the other hand, the insurer or insuring company, gains in this way, that the profits accruing from cases where no loss is sustained, far exceed the cases where loss is sustained, and the trifling expense of insuring is of no moment to the insi red, in comparison with the damage of a disastrous voyage, or consuming conflagration. By the insurance system, loss is virtually distributed over a large commu- nity, and therefore falls heavily on no individual, from which we draw our conclusion, that it is equivalent to a mutual mercantile indemnification club. Wc must now show the rules of the clnh, and principles on which its calculations are made. The principal thing to be taken into account, in all insurances, is the amount of risk. For example, a store, where nothing but iron is kept, would 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 are 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 insurance 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 i\ til 182 Ar.iTni^iETic. n" hi givcu rate, so that in cafe of loss, both the principal and premium may bo recovered. As the premium is reckoned as so much by the hundred, insur- ance is merely a particular case of percentage, llencc to find the premiuui of insurance on any }^ivcu amount, at a given rate per cent., ^c deduce the following u u L £ . MvXtiphj the ijivcii amount hij the rate per unit.* EXAMPLES. 1. To find the cost of insuring a block of buildings valued at $2G88, at C) per cent. ? Here we have .00 for the rate per unit, and $2G88X-0G=r$lG1.28, the answer. 2. What will bo the cost of insuring a cargo worth S3G79, at 3 per cent? The rate per unit is .03, and $3G79X-03r::.8110.37, the answer. 3. A gentleman employed a broker to insure his residence and outhouscH, valued at 827G0, the rate being 8 per cent., and the bro- ker's charge 1 \ per cent, ; how much had he to pay ? The cost of insurance is 827G0X.08==^$220.80, and the brokerage 6-41.40, which added to $220.80, will give $202.20, the answer. EXERCISES. "Wliat will be the premium of insurance on goods worth $1280, at 5^ per cent. ? Ans, $70.40. 2. A ship and cargo, valued at $85,000, is insured at 2\ per cent. ; what is the premium ? Ans. $1912.50. 3. A ship worth $35,000, is insured at 1^ per cent., and her cargo, worth $55,000, at 2^ per cent. ; what is the whole cost ? Ans. $1900.00. 4. What will be the cost cf insuring a building valued at $58,000, at 2^ per cent. ? Ans. $1450.00. * It is plain tliat tlie rate cau bo found, if the amount ami premium are given, and the amount can be found if the rate and premium are given. lu the case of insuring property, a professional surveyor is often employed to value it, and likewise in the case of life insurance, a medical certiflcatc is required, and in each case the fee must be paid by the person insured. As 100, the basis of percentage, is a consttrnt quantity, when any two of the other quantities are given, the third can be found. INSURANCE. 183 5. "What must I pay to itisuiv a house valued at SSOJ^.TiO, at '^ per cent. ? (I. A villapjo store was valued at $11S0 ; the proprietor insured it for six years ; the rato for the first year wj/S 3.f per eoiit., with a reduction of.]- each succeodin"^ year ; the stflck maintained an aver- age value of §1508, and was insured each of the six years, at 'J| per cent. ; how much did the [)ropriotor pay for insurance durint; tlio six years? Ans. $31)7.5:5. 7. A store and yard were valued at $1280, and insured at 1 J j>cr cent. ; the policy and surveyor's fee came to $2.25 ; what was tho whole cost (if insurin}^? Ans. 8H>.(J5. H. W. Smith, Port Huron, requests 11. Tomlinson, Toronto, to insure for him a buildinj; valued at $970 ; 11. Tomlinson efteets the insurance at 1^' per cent., and charjres •{ per cent commission ; how much hua W, Smith to remit to K. Tomlinson, the latter having paid *hc premium V Ans. $4<».3(). '.). The co.st of insuring a fictory, valued at 625,000, is $125; what is tho i to per cent. ? Ans. I.. 10. A 1| per cent, insuring my dwelling house copt me $50; what is tho value of tho liou.se ? Ans. $1000.00. To find how much must be insured for, so that in case of losa, both principal and premium may be recovered. II ore it is obvious that the sum insured for must exceed the value of the property in tho same ratio that 100 exceeds tho rate. E X A JI 1' L E . To find what sum must be insured for on property worth $000, at 4 per cent., to secure both property and premium, we have as $100_4:-$90 : $100:: $000 : F. P.^«-»-"X5U»-0:^-:$G25, the sum rcfiuircd. Taking the rate per unit we find -'/oo~^^7 00=^ • ^^• This gives tho RULE. Divide the value of (he pwperti/ hy 1, dinuaishcd bjj the rate jycr unit, and the quotient will he the sum required. E X A ;^i p L E s . 1. A foundry is valued at $87-1: for what sum at 8 per cent, must it be insured to secure both the value of tho property and the premium? One minus the rate or 1.00 — .08:^.92, and $871->-.92 ::=$950, the tmswer, \ ii.» 1«4 AiiniiMKiic;. The prciniacs of ii ;,'un8inith, who hcIIs {gunpowder, arc viiluod at $2G1H.H5: Ibr how much, at 15 per cent., must they be insured in order to recover tl»c value of the property and also the premium of insurance '( Subtract .15, the rate per unit, from I, and the remain- der is .85 and $:iG18.83-:-.85 give.i 83081, tlie Bum rc(juired. E X E u r I S E s 1. A chemist's laboratory and appurtenances are valueci ac 82G,25(), for what sum shoulj ho insure them at G.j- per cent., to secure both propt^rty and premium ? 828,000. 2. A New York merchant sent goods worth $1,180 by water conveyance to Chicag((; lie insured them from New York to IJufTalo at ]^ per cent., and from IJuffalo to Chicago at 2^ per cent., and in both ca.ses Ko as to secure the premium as well as the cargo; how much did tlie insur-xnce cost him ? Ans. $45.42. 'I. A person owned >•. flour mill, valued at 81840.05, which he insured at \\ per cent. He also owned a flax mill, valued at $840.30, which ho insured at 2i per cent., and in both cases at such a sum as to secure both proi^crty and premium. ^Vhich cost liim most, and how much more ? Ans. The flour mill cost him $1.07 more than the other. 4. Collins k Co., of Pliiladelphia, ordered a quantity of pork from G. S. Coates k Son, Cinciunati, which amounts to iij!2423.10. They insure it to Pittsburg at h per cent., and fniii Titt-sburg to Philadelphia at 3 per cent., and in all cases so as to secure both the price and premium. How much does the whole insurance come to ? Ans. 887.12. 5. In order to secure both the value of goods shipped and the premium, at ij per cent., an insurance is cftcctcd on 81520.72. What is the value of the goods ? Ans. $1500.00. 6. The Mechanics' Institute is valued at $18,000 ; it is insured at 1:J- per cent., so that in case of fire, the property and premium may both be recovered. For how mnch is it in.surcd ? Ans. 818,227.85. 7. How much must be insured ou a cargo worth 840,000, at j^ per cent.,' to secure both, the value of the cargo and the cost of insurance ? Ans. 840,20 1.00. LIFR IN'RUnANCr:. 185 8. Tho TXomn IIousi', Kinf:;-.strtct, Toronto, is valued at. nay, 8150,000, and \n insured at 1 J per cent, «o that in case of another conflat^ration, hoth the valuo of tin- jirojicrty and tlin pnniiiini < f insurance may lie recovered. I'or how much must it h>> insured ? Ans. .Sl'i^.tlTl.TO, nearly. 1>. A jail and court-house, adjoining; chr niical works, anrO' ducts by 30, or the number of days in any term agreed npon. The quotient will give the number of bushels, barrels, or other articles on lohich storage is to be charged for that term, 2. Wl: ', will bo the cost of storing salt at 3 cents a barrel per month, which was put in store and taken out as follows . — Put in, January 2, 186G, 450 barrels ; January o, 75 barrels ; January 18, 300 barrels ; January 27, 200 barrels ; Februj.ry 2, 75 barrels. Taken out, January 10, UO barrels; January 30, 150 barrels; February 10, 190 barrels; February 20, 300 barrels; JIarch 1, 250 barrels; and on March 12, the balance, 150 barrels? Ans. $39.44. 3. lleceived and delivered, on account of T. C. Musgrove, sundry bales of cotton, as Ibllows : — lleceived January 1, 18GG, 2310 bales ; January 16, 120 bales; February 1, 300 bales. Deli- vered February 12, 1000 bales ; March 1, GOO bales ; April 3, 400 bales ; April 1 0, 312 bales ; May 10, 200 bales. Required the num- ber of bales remaining in store on June 1, and the cost of storage up to that date, at the rate of 5 cents a bale per month. Ans. 218 bales in store ; 1321.18 cost of storage, 4. W. T. Leeming & Co., Commission Merchants, Albany, in account with A. B. Smith & Co., Oswego, for storage of salt and gunpowder, received and delivered as follows : Received, January 18, 18*^0, 400 kegs of gunpowder and 50 barrels of salt; January 25, 250 barrels of salt; February 4, 150 barrels of salt, and 50 kegs of gunpowder; February 15, 100 kegs of gunpowder ; March 5, 64 kegs of gunpowder ; April 15, 50 kegs of gunpowder, and 75 barrels of salt. Delivered, February 25, 15 kegs of gunpowder, and 40 barrels of salt ; March 10, 150 kegs of gunpowder, and 285 barrels of salt ; April 20, 200 kegs of gunpow- der; April 125, 50 barrels of salt, and 200 kegs of gunpowder. Required the number of barrels of salt and kegs of gunpowder in store May 1, and the bill of storage up to that date. The rate of 11 M I 'M IM I 190 ARITHMETIC. f* 8toraj»fi for salt being; II cents a burrcl per month, and for gunpowder 12 centa a keg per month. Ans. In store, 50 barrels of salt and 99 kegs of gunpowder ; bill of storage, $200.01. GENE L AVERAGE, Tins is the term used to denote tlie contribution of all persons interested in a ship, freight, or cargo, towards the loss or damage incurred by any particular part of a ship, or cargo, for the preserva- tion of the rest. This sacrilico of property is called yf.7/iso«, from the goods being cast into the se;i to save the vessel; although not only j)ropcrty destroyed in that way is the subject of general average, but also any damages or expenses voluntarily incurred for the good of all. For example, the expense of unloading the cargo that the ship may be repaired; masts or sails cut away and abandoned to save the ship. The only articles exempt from contribution are provisions, wear- ing apparel of passengers, and wages of the seamen. The owners contribute according to the clear value of the ship and freight at the end of the voyage, after deducting the wages of tlic crew and other expenses. In New York A, and in other States ^ of gross freight is some- times deducted for seamen's wages ; but as a general custom the exact amount is ascertained and deducted. Goods that have been subject to jettison, and a''c lost, are valued, when the average is calculated at the place of the ship's destination, at the price they could have sold for there ; but when the average is to be ascertained at the port of lading, the invoice price is the ^standard of value. In making an account of the articles which are to contribute, the property lost or sacrificed must be included, and its owners must suffer the same proportionate loss as the rest. The losses to the dif- ferent parties interested in the vessel, freight and cargo, arc paid by their insurers. When repairs have to be made to a ship — new sails, masts, or rigging, for example — one-third of the expeuse is deducted on account of vidlunUion, or the improved condition of the ship by these repairs. "When tljc .ship is new, and on her first voyage, the full amount of the expense of repairs is allowed in computation of the loss. GENEIJAL AVEUAiiE. 197 K X A M !• I, K . On tho 2Gth Juno, 18C5, tho Htcanier Cuba loft New York for Liverpool with a carji;o, as follows : Shipped by T. A. Collins, $7480; II. Evans & Co., ^.-jIJO-) ; II. (!. Wri-ht, 69218; W. Man- ning,' k Co., 811128 ; E. Carpenter, $7559. When olF Sandy Hook n heavy t^ule was experienced, durini; whioh cargo to the value ot $3498 was thrown overboard; of this $1123.40 belonged to 1{. Evans & Co., and tho balance to E. Carpenter. The necessary repairs of the steamer cost $87(5, and tho expenses in port, while getting repairo'l, were $253. The steamer was valued at 8100,000 , gro.ss freight, $4310. The seamen's wages were $800. What was the loss per cent., and what was the '"iS of each contributory in- terest ? SOLUTION. Loss for general hrnrfit. Contribntori/ interests. Cargo thrown overboard,$3498 Value of steamer $100,000 llcpairs to steamer less J 584 Invoice price of cargo.... 41,050 Expenses in port 253 Fr'ght, less seamen's wages 3,460 Total loss $4335 Total contrib. int.. ..$144,500 $4835-^144,500=^.03 1. ^s per unit, or 3 per cent. $100,000 X.03rr=$3000.00, steamer's share of less. 7,480X.03-- 224.40, T. A. Collins' share of loss. 5,3G5X .03=^ 100.95, 11. Evan & Co.'s share of loss. 9,218X.03=. 270.54, IJ. C. Wright's share of loss. 11, 428 X -03= 342.84, W. Manning & Co.'s share of loss. 7,559 X .03^ 220.77, E. Carpenter's share of loss. 3,450 X .03— 103.50, Freight's share of loss. $4335.00, Total loss. $3000.00— 837.00=.$21G3.00, balance payable by steamer. 1123.40—160.95— $962.45, balance receivable by U. Evans tt Co. 2374.60—226.77-^ 2147.83, balance receivable by E. Carpenter. Note.— It is evident that since the stoamor lost §!8u7 (S.")8t by repairs, and $253 by expenses), — that tho net amount reciiiired from tho steamer will bo $o009— 837=$21li3. 11. Evans & Co. luiviny lost by merchandize) bcinj;- thrown overboard $1123.40, a sum greater than their share of tho general loss, so that there must be due them $1123.40 -1C0.95=$9G2.45 ; go also tho .amount of E. Carpenter's share of the general loss must be deducted from his individual loss in order to Ond the balance duo him. 14 ■I I 108 ARlTirsrETIC. I I 'I I** :M' 11 i: I. E . Find thr ruff per unit of Ions, li/ which mvUipli/ the value oj :nih cnnlriljutorji intmst, and the product will he the share of lots to be sitataiitcd hi/ vich. K X E R C I 8 E 8 . 1. The flte.imsliip Ocean Qitrrn on her trip from rhilndelpliin to Liverpool, vraa crippled in a Btorni, in fonscciucncc of whicli the captain had to throw ovcrhoacd a portion of the cargo, atnouutinf^ in value to ^-1405.50, and the necessary repairs of the vessel cost'642i>. The contributory interests were as follows: —Vessel, $30, 000 ; gro^s frcii^ht, $0225 ; car^o shipped by .J. Jones &, Co., 83G50 ; by Henry Anderson, 0i;5OO; by (Jorge Millaii, 82000 ; by J. Foster & Hon, $550 ; by lirown Brothers, 1^5450 ; and by Wilnon & Carter, 68500. Of the cargo thrown overboard, thrre belonged to lltnry AnderHon the value of S'>000, and t(» IJrown IJrothers the remainder, S1405. 50. The cost (if detention in port in consecfuenco of repairs, was .Sllt5.50 j >eanian"s wages, $2075. Ilow ought the loss to be shared among the contributory interests? Ans. 8 per cent. 2. The strainer Persia left lioston for Halifax, June 30th, '.""aded with 7210 bushels of spring wheat, shipped by J. M. Mus- grovc, and invoiced at 95 cents per bushel; 4815 bushels of corn, shipped by Thomas A Brycc it Co., and invoiced at CO cents per bushel ; 2180 barrels of iiuur, shipped by A. I?. Smith il. Co., and invoiced at $5.50 per barrel. When near Halifax, the .steamer collided with the Bai/ State, and the captain Ibund it necessary to throw overboard 1000 bushels of wheat, 1280 bushels of corn, and 720 barrels of flour. On estimating the proportionat'^ loss, it was allowed that, the wheat would have sold in AMontrcal at an advance of 10 per cent., the corn at an advance of 15 per cent., and the flour for $5 per barrel. The contributory iutcrc.:ts were: — Steamer, 695,000; cargo, $ ; gro.^s freight, lir23G1.20. The cost of repairs to steamer was $2193.15; cost arising i'rom detention during repairs, S318; 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 7G00 bushels of wheat, valued at $1.25 per bushel, shipped by Dunn, Lloyd & Co., and insured in the Hartford Insurance Company at 1£- pcr cent., 9200 bushels of corn, valued at 75 cents per bushel, ![ TAXEH AND CUSTOM DUTIES. 199 ihippodlty J. W. Iloo.nnd in^urod in tlio ^Tiltn.i Tnsuranco Company it 1.] per (int.; 11,^00 Itiifilitl^ of oiif.^, vulucd at .'17 ,J ccntn per bushel, hliippiHi hy >Iorri;j. Wri^'htit Co., and insurodin Iho Mutual [nsurnncn Oonipany at \\ per cent. ; 1,H00 barrels of flour, valued it $5.25 per Inirrel, s^hippcd l»y Smith &, Worth, and in.sured in the Bcavir Insurance Company at \\ per cent. In conHcqaenco of a violent ;,m1o in the fiulfof l^Icxle(>, it wan found neces.-^ary to throw overboard th(! flour, 4,(500 bushels of oat.^, and 3,150 bushcl.s of wheat. Tlie propeller was valued at $45,000, and insured in the Beaver Insurance Company for $12,000, v.t 2 percent., and in the Western for 825,000, at 2j per cent. The gi'oss freight was $4950 ; seamen'H wages, 6:540, and nipuirs to tho boat, $3953.75 ; what was the loss sustained by each of the contributory iutercsts, the propeller l)eing on her first trip? TAXES AND CUSTOMS DUTIES. .1 t((.c is a money payment levied upon the subjects of a State or tho members ol' any community, for the support of the govern- ment. A tax is either levied upon the property or the per.«ons of indi- viduals. When levied upon the person, i* is called a poll tax. It may bo either dirrct or iivh'rrct. AV'^heu direct, it is levied from the individuals, or the properly in the )..inds of the ultimate owners. AVhcn indirect, it is in the nature ' i a ciis'oms' or excise dutji, which is levied upon imports, or nuinuiacturcs, before they reach the consumer, although in the end they arc paid by tho latter. Customs' duties are paid by the importer cf gouJi; at the port of entry, where a custom-house is stationed, with government employees called custo7n-housr. ajjicers, to collect these dues. Excise duties are those levied upon articles manufactured in the country. An invoice is a complete list of the particulars and prices of goods sent from one place to another. A Specific duty is a certain sum i)aid on a ton, hundred weight, yard, gallon, &c., without regard to the cost of the article. An ad valorem duty is a percentage levied on the actual cost, or fair market value of the coods in the eonntry from which they are imported. I 200 ARITHMETIC. £!^ I Gro6s weight is the weight of goods, upon which a specific duty is to be levied, before any allowances are deducted. Ket weight is the weight of the goods after all allowances are deducted. Among the allowances made are. the following : Breakage — an allowance on fluids contained in bottles or break' ablo 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, Qither to increase or decrease the rate per cent, of duty, as greenbacks fluctuated in value, compared with gold (the invoice price of tlie 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 value, the duties are made payable in the same currency. When goods are imported from any country which has a depre- ijiated 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 residing at the port/rowi which they are exported. EXAMPLES. To find the specific duty on any quantity of goods. Suppose an Albany Provision Merchant imports from Ireland 59 casks of butter, each weighing 68 lbs., and that 12 lbs. tare is allowed on each'cask, and 2 cents per lb. duty on the net weight. We find the gross is 59x68=^=4012 lbs. " tare is , 59X12== 708 lbs. Hence the net weight is 3304 lbs The duty is 2 cents per lb 2 The duty, therefore, is $66.08 TAXES AND CUSTOM DUTIES. 201 To find the act valorem duty on any quantity of goods. Suppose ;i Troy dry goods merchant to import from Montreal 43G yards of silk, atSl.75 per yard, and that 35 per cent, duty ia charged on them. Here wo find the whole price by the rule of Practice to be $763, then the rest of tlic operation is a direct case of percentage, aud therefore we multiply $703 by .."5, which gives $267.05, the amount of duty on the whole. Ilcncc we have the following RULE 1' O U SPECIFIC DUTY. ^uhtract the tare, or other allowance, and multipli/ tne remain' der hy the rate of duty per box, gallon, &c. M: RULE FOR AD VALOREM DUTY. Multiply the amount of the invoice hy the rate per unit. EXERC ISES. 1. Find the specific duty on 5120 lbs. of sugar, the tare being 14 per cent., and the duty 2'^- cents per lb. Ans. $121.09. 2. What is the ad valorem duty on a quantity of silks, the amount of the invoice being $05,800, and the duty 62-2- per cent? Ans. 859,875. 3. At 30 per cent., what is tlio ad valorem duty on an importa- tion of china worth $1260.? Aus. $378. 3. What is the specific dut}', at 10 cents per lb., on 45 chests of tea, each weighing 120 lbs., the tare being 10 per cent. ? Ans. .^■486. 5. What is the ad valorem duty on a shipment of fruit invoiced at $4500, the duty being 40 per cent.? Ans. $1824. G. What is the specific duty on 950 bags of coffee, each weighing 200 lbs., the duty being 2 cents per lb., and the tare 2 per cent? Ans. $3724. 7. What is the ad valorem duty on 20 casks of wine, each con- taining 75 gallons, at 18 cents a gallon ? Ans. $270. 8. A. ]i. shipped from Oswego 24 pipes of molasses, each con- taining 96 gallons; 2 per cent, was deducted for leakage, and 12 cents duty })cr gallon charged on the remainder ; how much was the duty? Ans. $270.05. 202 Ai;iTIIMETIC. h %: !). J\;(or tSinith 6c Co., Brooklin, import from Cadiz, 80 baskets of port wine, at 70 francs per basket ; 42 baskets of sherry wine, at 05 francs per basket ; 00 casks of cluinipa,qnc, containing 31 gallons each, at l i'rancs per gallon. The waste of the wine in the casks was reckoned fit 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. S77G.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. Od., duty 10 per cent. ; and shoe-lasting to the cost of £60, duty 4 per cent. Required the whole amount of duty, allowing the value of the pound sterling to be $4.84. Ans. $173.04. 11. John McMaster & Co., of CoUingwood, Canada West., bought of A. M. Smith, of Buffalo, N. Y., goods invoiced at 85440.50, which should have passed through the custom-house dur- ing the first week in May, when the discount on American invoices was 43^^ per cent., but they were not passed until the fourth week in May, when the discount was 36| per cent. The duty in both cases being 20 per cent. ; what was the loss sustained by McMaster 6 Co. on account of their goods being delayed ? Ans, $70.60. nil i 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, are usually £100, £50, or £10 each. In the United States they arc generally $100, 850, or 810 each. The management of incorporated companies is generally vested in officers and directors, as provided in the law or laws, who are elected by the stockholders or shareholders ; each stockholder, in most cases, being entitled to as many votes as the number of shares lie holds ; but sometinres the holder of a few shares votes in a larger proportion than the holder of many. The accumulating profits which are distributed among the stork- holders, once or twice a year, are called " dividends," and when '"declared," arc a certain percentage of the par value of the shares. In nuiu'iu.'. laii.l t^omo otl.cr companies, where the shares are onl}' a STOCKS AND ]50ND;-5. 2j3 fit IS few dollars each, the dividend is usually a fixca .sum '"por share." Certificates of stock arc issued by every company, .sipncd by the proper officers, indicating tlie number of sjiarci* cacli stocklioldcr is entitled to, and ;is an evidence of ownership ; these are transferable, and may be bought and sold like any other property. When tho market value ecjuals their nominal value they arc said to be " o« par." When they sell for more than their nominal value, or face, they arc said to be above par, or at a " ])remiura" ; -when for less, they are below par, or at a " discount." Quotations of the market value arc trencrally made by a percentage of their par value. Thus, a share which is $25 at par, and sells at $-2S, is (juoted at twelve per cent, premium, or 112 per cent. When states, cities, counties, railroad companies, and other corporations, borrow large amounts of money, for the prosecution of their objects, instead of giving common promissory notes, as with the mercantile community, they issue bonds, in denominations of convenient size, payable at a specified number of years, the interest usually payable semi-annually at some well known place. These arc usually payable to "bearer," and sometimes to the "order" of the owner or liolder. When issucu by Governments or States, these bonds are frequently called Government stocks or State stocks, under authority of law. To these bonds arc attached, what are called " coiqwns" or certificates of interest, each of which is a due bill for the annual or semi-annual interest on the bond to which it is attached, representing the amount of the periodical dividend or interest; which coupons were usually cut off, and presented for payment as they become due. These bonds and coupons are signed by the proper officers, and like certificates of capital stock, are nego- tiable by delivery. The loan is obtained by the sale of the bonds, with coupons attached, but Ihcy are .sometimes negotiated at par. Their market value depends upon the degree of confidence felt by capitalists of their being paid at maturity, and the rate of interest compared with the rate in tho market. Treasury notes are issued by the United States Government, for the purpose of effecting temporary loans, and for the payment of contracts and salaries, which resemble bank notes, and are made payable without interest generally, lleccntly such notes have been issued bearing one year or three years' interest. "Consols" is a term abbreviated from the expression "consoli- dated," the British Government having at various times borrowed money at different rates of interest and payable at different times, " consolidated" the debt or bonds thus issued, by issuing new stock, drawing interest at three percent, per annum, payable semi-annually, and redeemable only at the option of the Government, becoming practically perpetual annuities. V.'ith the proceeds of this, the old .stock v.'as redeemed. The quotations of these three per cent, per- petual annuities, or "consols," indicate ordinarily the ,':ain during one year $210; what is each man'.s bhare of the profit ? SOLUTION BY T U P O U T I N . A.'s stock, Z3Cyj B.'s " 400 Entire stock $700 : 300 : : $210 : $90 A.'s gain. " " 700 : 400:: 8210: 120 R's " S O L U T I O N HY P E 11 C K N T A (J E . Since the entire amount invested is $700, and the gain $210, the gain on every $1 of investment will be represented by the num- ber of times that 700 is contained in $210, which is .30 or 30 cents on the dollar. Now if eacli man's s'ock be muUiplied by .30 it will represent bis share of the gain thus : 8300 X -30-:-$ 90 A.'s sain. 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, we have the following RULE. As the whole stochis to each jyartncr' s stock, so is the whole gain or loss to each jmrtne/s gain or loss ; or, divide the whole gain or loss hy the number denoting the entire stock, and the quotient will he the gain or loss on each dollar of stock ; which midtipUed hy the number dciwting each j^o^'tna-'s sJtare of the mtire stock, will give Jus sliare of the entire gain or loss. EXERCISES. 1. Three persons. A., B., and C, enter into partnership. A. advances $500, B. $550, and C. $600 ; they gain by trade $412.50. What is each partner's share of the profit ? Ans. A.'s $125; B.'s $137.50; C.'s $150. rAIlTNEItSHIP. 211 2, A, U, (' and 1) pnrcliaso an oil well. A pays for (5 shares, 1] for 5, C for 7, and I) lor S. Their not profits at the md of three moQths have amounted to IjjTHOO ; what sum ouj^ht eaeli to reeeivo ? Ans. A, tUHOO; 1{, SIHOO; C\ I?2IU() ; D, 82KM). 3. A and IJ i)ureha,sed ii lot of land for $1500. A paid ^ of the price, and \i the remainder ; they <^ained by the sale of it 20 per cent. ; what was each man's share of the profit ? Ans. A, $;}()() ; IJ, 8G0(). '4. A captain, mate, and 12 sailors, won a prize of 82240, of which the captain took i4 shares, the mate <», and the remainder was equally divided amonj,' the sailors ; how much did each receive ? Ans. The captain, $980 ; the mate, $420 ; each sailor, 870. 5. A and 13 invest equal sums in trade, and clear 8220, of which A is to have 8 shares on account of transact in.u; tlie business, and li only .'{ sliares ; what is each man's gain, and what allowance is made A for his time ? Ans. J-lach mah's <^ain 8(iO ; A 8100 for his time. G. A, ]J, C and l> enter into })artnersliip with a joint capital of 84000, of which A furnishes 81000 ; JJ 8800 ; C 8i:!00, and D the balance ; at the end of nine months their net profits amount to $1700 ; what is each partner's share of the gain, supposing B to re- ceive 8100 for extra services ? Ans. A, 8400 ; JJ, 8320 ; (J, 8.')20 ; i>, 8:500. 7. Hi.K persons, A, 13, C, D, E and F, enter into partnership, and gain 87000, which is to be divided among them in the following manner: — A to have I, ; B, ^ ; C, j^ as much as A and 13, and the remainder to be divided between D, E and F, in the proportion of 2, 2A- and 3-t ; how much does each partner receive ? Ans. A,^81400 ; ii, 81000 ; C, 8800 ; D, 8950 ; E, 81187.50 ; F, 81002.50. 8. A, 13 and C enter into partnership with a joint stock of $30,000, of which A furnished an unknown sum; B furnished 11, and C 1|- times as much. At the end of six monllua their profits were 25 per cent, of the investment ; what was each man's share of the gain ? Ans. A's, 82000 ; 13's, 83000 ; and C's, $2500. 9. A, B, C an 1 I) trade in company with a joint capital of 83000 ; on dividing the profits, it is found that A's share is 8120 ; B's, 8255 ; C's, $225 ; and D's, $300 ; what was each partner's stock ? Ans. A's, 8400 ; B's, $850 ; C's, 8750 ; and D's, $1000. 10. Three labouring men, A, B and C, join together to reap a certain field of wheat, for which tliey agree to take the sum of J ■\ 31 ' 212 ARITHMETIC. $10.84 ; A and B calculate that they can do \ of tho work ; A and C f ; U and C j} of it ; how much should each receive according to thcBo estimates ? Ana. A, $8.32 ; B, 87.04 ; and C, S4.48. u tl, l:;.;.. To find each partner's share of tho gain or loss, when tho capital is invested for different periods. EXAMPLE Two merchants, A and B, enter into partnership. A invests $700 for 15 months, and B $800 for 12 months; they gain 8003 ; what is each man's fehare of tlio profits ? 80 LUTION $700xl5=Si0500 $800X12^. 9G00 20100 : 10500 : : $603 : ^315 A^s gain. 20100 : 9G00 : : $003 : $288 B's gain. The reason for multiplying each partner's stock by the time it was in trade, is evident from the consideration that $700 invested for 15 months would be equivalent to $700x15 equal to $10500 for one month, that is $1 0500 would yield, in one month, the same in- terest that $700 would in fifteen months. Likewise $800 invested for 12 months would bo the same as $9G00 for one month; hence the question becomes one of the previous case, that is, their invest- ments are the same as if (^!/^ncnt hy the time that must elapse before it becomes due, and divide the sum of these products by the sum of the 2)ayments. EXAMPLE. To find the equated time for the payment of three debts, the first for $45, due at the end of 6 months ; the second for $70, due at the end of 11 months, and the third for §75, due at the end of 13 inontlis. ^45 X G==$270 70X11= 770 75X13=^ 975 190 2015 and 2015-T-19=105^, so that the equated time will be 10 months and 18 days, the small remaining fraction being rejected. Let us suppose that nothing is paid until the end of the 13 jnonths, and all paid at once, then the amount to be paid will be, at G per cent.. For first debt overdue 7 months, $45-f-1.57^, interest for 7 months $46.57i For second debt overdue 2 months, §70-]-. 70, interest for 2 months 70.70 For third debt just due, $75, no interest 75.00 $192.27i EQUATION OF TAYMEXTS. 219 The work may often be somewhat shortened by counting the differences of time from the date at which the first payment becomes due, the mean time between the dates v.'hen the first and last become due being alone required. Ka person owes $1200 to bo paid in four instalments, $100 in 3 months; $200 in 10 months; $300 in 15 months, and SGOO in 18 months, then the excesses of time of the last three above the first arc 7, 12 and 15 months, and the work will stand as below. 3100 (no time.) 200X 7:^1400 300X12=:3G00 GOOX15==9000 1200) and ll5-XiJ^==14f months 14000(115 This gives the II U L E , Multiply each debt, except the one first due, by the difference be- tween its term and the term of the first ; divide the sum of the pro- ducts by the sum of the debts, the quotient with the term of the first added to it loill be the equated time. Another method, which is often convenient, may be illustrated by the example already given, as the two operations will give the same result. Interest on §300 for 4 months:3z:$ G.OO Interest on 500 for G " =:r 15.00 Interest on 400 for lOA- " -^ 21.00 Interest on 1200 for 1 month=6)42.00(7 months as boforo. K u L E . I^ind the interest on each instalment for the given time, and divide the sum of these by the interest of the whole debt for one month, and the quotient will be the equated time. As the sum of the instalments is equal to the debt, the result will be the same for any rate of interest. For the first instalment, 8300, overdue 3 months, A has to pay 84 50 For the second instalment, 8500, overdue 1 month, A has to pay 2 50 87 00 t 220 AEITHMETIC. HI >i !': II- \: For the third instainient, $400, not duo for Sh mouths, A has to get $7 GO ?o that the amounts of interest exactly balance, and the paying of the whole, at the end of 7 months, is precisely equivalent to the pay- ing of each instalment as it falls due. The only difference "that could arise is, that it might be inconvenion for the creditor to lie out of the first instalment for the three months. In all other respects tho settlement is strictly equitable, according to the xinderstdndbuj that exists amon^- business men. In the first place, the difference between this and what is called " tho accurate rule," is insignifi- cantly small ; and, in the second place, the '• mercantile rule" saves much time, and time is equivalent to so nmch capital in mercantile transactions. Independently, however, of any other consideration, we may remark that when the mode of reckoning is conventionalli/ understood, it becomes perfectly equitable, because every merchant knows the terms on which he can do business with any other, just £>3 bank discount becomes perfectly equitable, because every man, before going to a bank for the discounting of a note, knows perfectly well on what terms he can have it. Much warm discussion has been indulged in on this subject ; but, as we consider the discus?ion more subtle than profitable, we shall dis- miss the subject in a few >vords. We shall adopt the usual case, that A owes B §200, one-half to be paid at the presea'". time, and the remainder at the end of two years. It is perfectly obvious that, at the end of the first year, A should pay $100, that is. the principal, plus the interest agreed upon. Regarding the settlement of tho second instalment, if A proffers payment of the whole at once, he is clearly entitled to claim a reduction for the unexpired term. Now, the question is, what ought the reduction to be. By the mercantile rule he should pay $94, but the true present worth of $100, due at the end of the year, would be 94.335], so that he would have to pay $100 on the instalment over due, and $94.33|J- on the one not due, making $200.33|J, whereas the object is to find at what time inter- est should be chargeable to neither party. As a further illustratiou of the general rule, let us suppose that J. Smith owes II. Evans $1300, of which $700 are to be paid at the end of 3 months, $100 at the end of 4 months, and the balance at tjie end of 8 months, to find the equated time. We bhall suppose that J. Smith agrees to pay B. Evans the whole amount at the time the debt was contracted ; then J. Smith would EQUATION OF PAYMENTS. 221 owe 11. Evans $1300, minns the discount for the length of time the iiniount ^v'lls paid before it became due, viz., tliree months, equalling the discount on i;210 for 1 month ; $100, less the discount for 4 months, equalling the discount on $400 for 1 month ; $500, less the discount for 8 months, equalling the discount on $4000 for 1 month. This gives a total of $2100-l-$400+$4000r:=$G500, for 1 month. Now, it is evident that if J. Smith wished to pay the whole amount at sucli a time that there should be no loss to cither party, ho must retain this amount for such a length of time as it will take this amount to equal the discount on $G500 for 1 month, which will be , :j'„o of $6500, that is, for 5 months. To prove that 5 months must be the equated time, we have recourse to the principles laid down under the head of Interest. If a settlement is not made until the expiration of 5 months from the time the debt was contracted, then . Smith would owe II. Evans $700, phis the interest of that principal during the time it remained unpaid after becoming due, viz., two months, which would give an amount of $707. So also, $100, p/ihtccn dajs on the .second. But If payment be dehiyed till August 2, A would bo entitled to one month's interest on the first purchase, and B to the interest on the second for one montli and eighteen days, so that there would be in favour of B, on the whole, a balance of iutor- cst for cij:,hteen days. .\2;alii, supposing the settlement is not made till September 20, when all i.s due, no interest cum bo either charged or claimed on tlic second purchase, the term of credit having just then expired ; but as the first debt is two months and eighteen days overdue, A is entitled to interest on it for that period. If neither is paid till after September 20, A has a right to claim interest on each for the period it has been overdue. But this regulates only one side of the account. In order to settle the other, let us suppose that B lias, in the meantime, sold goods to A, it is obvious that B's claims on A must be settled on the very same principle, and that therefore the final result must bo simply the finding of the balance. It is more usual, however, in accounts current, to fix on a time such that the interest due by A shall exactly balance that due by B. To illustrate this, lot us suppose a case corresponding to a ledger account : R. EVANS. 1865. Dr. July 21, To Merchandise on 2 months' credit... $200 July 25, To Cash 150 Aug. 24, To Merchandise on 4 months' cred'.t... 100 Sept. 21, To Merchandise on 3 months' credit... 250 $700 1865. Cr. August 1, By Cash $100 August 20, By Merchandise at 22 days 110 Sept'r 30, ByCa.sh 180 Balance 310 $700 To find in this case at what time the account may be settled so that interest shall be chargeable to neither party. Equating the time, as in cqvtation of payments, wo have the following operation : AVEHAQINO ACCOUNTS. 225 Du. 1865 July 25 150X Sept. 21 200X 58. Dcor.24 250X152. Deer. 21 100x149. .IIGOO .38000 .14900 G4500 Cr. 18G5. August 1 lOOX Sept. 12 110X22= 2420 Sept. 30 180X71^12780 390 15r90 15200-i-390=..39 days. Duo 39 days from AuguBt 1, viz., on September 9. 700 G4500-j-700.r=92 days. Due 90 days from July 25, viz on October 25. Time from September 9, to October 25=:46 days. Excess of debit above credit 700—390=310. 390X4G=.17940, and 17940—310=58 days, nearly. Counting 58 days forward, from October 25, will bring \iz to December 22, the time required for a settlement, with interest chargeable to neither party. Here the time is counted forward from the average date of the larger side which becomes due last, but had it become due first, we should have counted backward. The first transaction on the debit side being two months' credit from July 21, is not to be taken into consideration till September 21. The second transaction, being a cash one, and therefore consid- ered as so much duo, will therefore mark the date from which all others shall be reckoned ; and, since there is no interval of time, wo write it without a multiplier. The next transaction has a term of credit extending to 152 days, and therefore we write 250x152= IIGOO. The term of the next extends from September 21 to December 21, a period of 149 days, and we write 100x149=14900. The sum of the debits is $700, and the sum of the results obtained by multiplying each item by the number of days it has to run from July 25 is $64500. Then 64500-f-700=92, the equated time in days for the debit side. Now, as already explained, the interest for $700 for 92 day^ will be the same as the interest of $64500 for 1 day. Hence, the debits are due 92 days from July 25, viz., on October 25. In like manner, on the credit side, the first transaction being a cash one, we start from its date, August 1, and, as there is no inter- val, we have no multiplier. The second being mi rchandiae, on 22 ,• k' K 220 ARITHMETIC. 1; m 11 'Pi^ (lays' crc'lit, wo uritc 11 0^/22 --2420. Tlio third is cash paid 71 days al'tLT August 1, and wo write iKOy 71:--127H(). Had tiiO account boon HCttle(l on Sopteudjor I), the debits would have bocii paid 4(» days befuro coniiii}:; duo, and tiic credit side would have j;aincd and the debit side lost the interest for that time. Aj^ain, wo must consider liow long it would take the balanro, 6310, to ])roduce tlio same interest that $3IM> would produce in 4G days. It is obvious that whatever interest 8.'>()() {i;ives in 4(5 days will require 4(J times $!J1J0 I'or 81 to produce the same interest, that is, 390X40:- 17040 days, and it will require 17940-: ;m)r=5S days, lor §310 to produce the ennic interest. If the settlement in made on October 25, the latest date, then the credit has been due 4(1 days, and therefore bearing interest; and in order that the debit side may bo increased by an equal amount, the time nmst le ex- tended beyond October 2."), that is, it must bo connicd fur ward. For tlic same reason, if the greater side had become duo first, then the balance must bo considered as duo at ix^n'cvious date, and therefore M'C must count hachward. An account may be averaged from any date, but cither the first or the last will bo found the most convenient. The first duo is generally used. On the principles- now explained may be founded the following 11 u L E . Find the equated time when each side becomes due. Multipltj the amount of the smaller side hij the number of days between the two average dates, and divide the product by the balance of the account. The quotient thus obtained will be the time that the balance becomes due, counted from the average date of the larger side, for- ward Khc7i the amount of that side becomes due LAST, but BACK- WARD when it becomes due FIRST. The cash value of a balance depends on the time of settlement. If the settlement be made before the balance is due, the interest for the unexpired time is to be deducted ; but if the settlement is not made till after the balance is due, interest is to be added for the time it is overdue. EXERCISES. In J. H. Marsdcn's Ledger, we find the following accounts, which, A\T.r.A(iIN( i AfCOUNTS. 227 on bcinj^ equiitcd, stanil as follows ; ut wliat timo should tho ro8j>cc- tivc balances commence to draw interest : 1. Br. J. S. TECivtrAM. Cr. May IGth, 18G3 S72t.4r). ] July i>9th, 18(m S4SG.80. Ans. December 15tli, 18G4. 2. Dr. Nelson Bostpoiid. Cr. November 19th, 1805 $G35. | December 12th, ISG.') 8950. Ans. January 27th, 186G. « 3. Dr. James Crow & Co. Cr. February 24th, 18GG... .$512.25. | June 10th, 18G5 $309,70. Ans. March 27tli, 18G7. 4. Dr. J. II. BuiiRiTT k Co. Or. March 17th, 18GG $145. | January 15th, 18GG .$095.00. Ans. December IJOth, 1805. 5. Dr. M. McDonald. Cr. August 27th, 1SC5 $341. | November 7(li, 1805 .^.'17. G. Dr. James I. Musgrove. Cr. July 20th, 18GG 8711. | April 14th, 180G !?120(). Ans. Dcecmbor 9th, 1805. 7. /;/-. Tiios. A. liuYCE k Co. Cr. June 24th, 1SG4 $1418. | September 7th, 1805 .S2:J40. 8. Dr. E. II. Cahpenteu. Cr. December 2nd, 1805.. .$1040.80. | Augu.st 13th, 18G5....,S1112.4U. 9. llequircd tho time when tho balance cf tho following aeeount becomes subject to interest, allowing the morehan'Jiso to have been on 8 mouths' credit ? Dr. A. 15. Smith & Co. Cr. 1804. May 1, July 7, Sep. 11, Nov. 25, Dec. 20, i |i 1805. To Mdse :$300.00iJan. 1 " '• 759.90:|Fcb. 18 " " 4l7.20'!Mar. 19 287.70 'April 1, 571.10 ;May 25, By Cash....i$500.00 '■■ Mdso. .| 481.75 " Cash.... I 750.25 " Draft... I 210.00 '^ Cash....; 100.00 Ans. Au";ust 5, 1805. i ym \i 4 "«.*. m i: ?. ff- ■^'• m W 228 ARITHMETIC. 10. When will the balance of the following account fall due, the merchandise items being on 6 months' credit ? Dr. J. K. White. Cr. 1865. May 1, May 23, June 12, July 29, Aug. 4, Sept. 18, To Mdse. Cash paid dft. , Mdse Cash 1865. $312.40 June 14, By Cash.... $200.00 85.70 July 30, " Mdse.... 185.90 105.00 Aug. 10, " Cash.... 100.00 243.80 Aug. 21, " Md.sc.... 58.00 92.10 Sept. 28, a a 45.10 50.00 Ans. January 12, 1866. 11. When docs the balance of the following account become subject to interest ? Dr. W. II. MUSGROVE. Cr. 1864. Aug. 10, Aug. 17, Sept. 21, Oct. 13, Nov. 25, Nov. 30, Dec. 18, 1865. Jan. 31, To Mdse 4 nios. " " 60 days " " 30 " Cash p'd dft. Mdse G mos. " 90 days " 2 mos. Cash $285.30 192.60 256.80 190.00 432.20; 215.25 08.90 1864. I Oct 13, By Oci. 26,1 " Dec. 15,! " Dec. 30, I 1865. Jan. 4, 'Jan. 21, J ash . Mdse 2 mos a 4 a 100.00 Cash. $400.00 150.00 345.80 230.40 340.30 180.00 12. In the following account, when did the balance l)ecomo due, the merchandise articles bein": on 6 months' credit ? Dr. R. J. Bryce in account with D. IIiCKS & Co. Cr. 1864. Jan. 4. Jan. 18', Feb. Feb. Feb. Mar Mar. 24. April 9, May 15, May 21, 4, 4 9 3 To a u u 11 t( a (( i( (( Mdse 1 $ 96.57 57.67 80.00 38.96 50.26 154.46 42.30 23.60 28.46 177.19 ! 1864. Jan. 30 By Cash... ■ • • $240.00 (( April 3, May 22, 48.88 Cash paid draft. Mdse 50.00 Cash paid draft. Mdse '< u (( (( Ane. December 22nd, 1864. AVERAGING ACCOUNTS. 220 13, When, in equity, should the balance of the following account be payable ? Dr. Jan. 3, Jan. 31, Feb. 8, Feb. 21, Mar. 10, Mar. ?A, Apr. 12, Juno 1, June 20, July 4, Sept. 27, Dec. 9, J. McDonald & Co. Cr. Cash.... $200 < 300 • • • • 75 100 350 25 40 80 125 268 250 100 1864. Sept. 20, By Mdse, 6 raos. . $583.17 Oct. 27, (( t( 4 " .. 321.00 Dec. 5, << (( 6 " .. 137.00 1865. Jan. 18, K U 60 days. 98.75 Feb. 26, (< (( 6 mos. . 53.98 Apr. 15, U t( 4 " .. 634.00 June 12, ii u 2 " .. 97.23 Sept. 21, (( li 6 " .. 84.00 Dec. 29, U (( 6 " .. 132.14 Ans. October 10, 1866. CASH BALANCE. To find the true cash balance of an account, wheu 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 & WllIGIIT. Cr. 1865. I Mar. 12, To Apr. 21, May 6, May 27, July 16, Sept. 10, Ont.. 19, 16 $340.00 150.00 165.00 215.00 100.00 310.00 " i 120.00 Merchandise.... Cash paid draft Mdsc Ca.sh Mdse 1865. Apr. 20 May 4 Juuc 15 Aug. 10 Sept. 23 Nov. 12 Dec. 15 JBy Mdsc... Cash.... Mdsc... Cash.... u $200.00 110.00 j 230.00 ! 180.00 ! 50.00 50.00 100.00 ■H|tf i 1 I! J ■i / ,4 it •'! 230 m^ t 1 i 11 111 m'' 1 1;! ARITHMETIC. SOLUTION Debits. Credits. Due. Due. June 12, $340X221= 75140 July 20, $200X183- 30000 July 21, 1.50X182— 27300 May 4, 110X260= 28600 May 6, 165X258=: 42570 June 15, 230X218— 50140 Aug. 27, 215X145— 31175 Nov. 10, 180 X 70= 12600 July 16, 100X187— 18700 Sept. 23, 50X118= 5900 Dec. 10, 310 X 40— 12400 Nov. 12, 50 X 68= 3400 Jan. 19, 120X 0= Doc. 15, lOOx 35— 3500 61400 6)207285 $920 0)140740 • $34,547 $23,456 The diflforent items on the debit and credit sides of the account being on interest from the date on which it becomes due until the time > f settlement, the total interest of all the debit items will be the same as the interest of $207285 for one day, or the interest of $1 for 207285 days, which is $34,547. So also, the total interest of all the credit items will be the same as the interest of 3140740 lor one day, or the interest of $1 for 140740 days, which is $23,456. Now, since each side of the account is to be increased by its interest, the cash balance will be represented by the number denoting the differ- ence between the two sides of the account, after the interest is added ; thus, $14004-$34.547=$1434.547, amount of debit side, and $920 -{-$23.450=$943.456, amount of credit side, then $1434.547— $943.456=§491.09, cash balance. SECOND METHOD. Debits. Credits. Days. Int. Days. Int. Int. on $340 for 221— $12,523 Int. on $200 for 183=: = $6,100 150 " 182— 4.550 (f 110 " 200r^ . 4.766 165 " 258= 7.095 K 230 " 218... . 8.353 215 " 145— 5.195 '( 180 " 70= = 2.100 100 " 187— 3.116 (.' 50 " 118:r. . .983 310 " 40— 2.066 (I 50 " 68=: . .566 120 " (( 100 " 35= $920 . .583 $1400 $34,545 $23,454 Now, $34,545 debit interest— $23,454 credit interest=$11.09. CASH BAL.VNCE. 231 the balance of interest, and $1400, amount of debit items-]- $11. 09 =$1411.09, and $1411.09— $920 amount of credit items:^$491.09 the cash balance, which is the same as obtained by the first solution. Hence from the foregoing we deduce the following RULE. Multiply each item of debit and credit by the number of days intervening between its becoming due and the time of settlement. Then consider the sums of the products of the debit and credit items as so many dollars, and find the interest on each for one day, which will he the interest, respectively, of the debit and credit items. Place the balance of 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 interests, on the debit and credit sides of the account will represent the balance of interest, lohich is placed on its own side of the account, and the difference then between the two sides will be the true balance. Note. — If any item abould nut come due until after the time of settlement, the side upon which it is, should he diminished, or the opposite side in- creased by the interest of such item from the time of settlement until due. EXERCISES. 1. What will be the cash balance of the following account if settled on January 1, 18G5, allowing interest at 8 per cent, on each item after it is due ? Dr. 11. Evans in account with John Jones. Cr. '•"Uii 1864. June 11, June 29, July 18, Aug. 25, Aug. 31, Sept. 3, Sept. 20, Oct. 14, Oct. 19. To (( (( (< (( a u a u i jl 1864. Mdse,4mos. $315.00,|Apr. 15, 6 Cashp'ddft. Cash Mdse,2 mos. Cash 180.00 200.00 75.00 80.00, " i 150.00; May 10 June 12, June 30, 50.0.0; July 15, 100.00, 'July 27, Aug. By Mdse, 3 mo&. Cash Mdse,as cash Cash Mdse, 3 mos. $350.00 120.00 240.00 100.00 90.00 80.00 lOO.Ort 175.00 75.00 2. A.B. as follows :- Aug. 20, Mdse,ascash 300.00 Aug. 30, Ans. $110.80. Smith is in account and interest with J. K. Amos & Co., -Debtor, January 1, 1865, to merchandise, on G months, >b 232 ARITHMETIC. m ■k«i Ij'i $156.10 ; February 3, to cash paid draft, $100 ; March 20, to mer- chandise, on 4 months, $316.90 ; March 30, to merchandise, on 4 months, $162; May 15, to cash paid draft, $100; August 20, to merchandise, on 6 months, $213. Creditor, February 1, by cash, $120; March 20, by merchandise, on 4 months, $420.16; May 1, by merchandise, on 6 months, $300 : July 1, by merchandise, on 4 months, $50; September 10, by merchandise, on 4 months. $99.84. Required, the true balance, if settled on December 1, 1865, interest being at 6 per cent. ? Ans. $01.36. 3. Required the true balance, March 25, 1865, on the following account, each item drawing 7 per cent, interest from its date. A. B. Lyman in account and interest with John Russell & Co. : — Debtor, July 4, 1864, to merchandise, $200 ; September 8, to mer- chandise, $300 ; September 25, to merchandise, $250 ; October 1, to merchandise, $600 ; November 20, to merchandise, $400 ; Decem- ber 12, to merchandise, $500; January 15, 1865, to merchandise, $100; March 11, to merchandise, $120. Creditor, July 20, 1864, by cash, $300; August 15, by cash, $350 ; September 1, by cash, $400; November 1, by cash, $320; December 6, by merchandise, $600; December 20, by cash, $100; February 1, 1865, by cash, $200; February 28, by merchandise, $150. Ans. $50.64. ALLIGATION. Alligation is the method of making calculations regarding the oompounding of articles of different kinds or different values. It is a Latin word, which means binding to, or binding together. It is usual to distinguish alligation as being of two kinds, medial and alternate. ^l ALLIGATION MEDIAL. m Alligation medial relates to the average value of articles com- pounded, when the actual quantities and rates are given. EXAMPLE. A miller mixes three kinds of grain : 10 bushels, at 40 cents a bushel ; 15 bushels, at 50 cents a bushel ; and 25 bushels, at 70 cents a buslicl ; it is required to find the value of the mixture. ALLIGATION. 233 10 bushels, at 40 cents a bushel, will be worth 400 cents., 15 bushels, at 50 cents a bushel, will be worth 750 cents., 25 bushels, at 70 cents a bushel, will be worth 1750 cents., giving a total of 50 bushels and 2900 cents, and hence the mixture is 2900—50=58 cents, the price of the mixture per busliel. Hence the RULE . Find 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 will he the price of the mixture. EXERCISES. 1. A farmer mixes 20 bushels of wheat, worth $2,00 per bushel, with 4^ 1 ushels of oats, worth 50 cents per bushel ; what is the price of one bu&»i(jl 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 CiO cents per bushel; and GO bushels of barley, worth 40 cents per bushel ; what was the value of 2^ bushels of the mixture ? Ans. $1.50. 5. A goldsmith mixes tog2ther 4 pounds of gold, of 18 carats fine ; 2 pounds, of 20 carats fine; 5 pounds, of 16 carats fine; and 3 pounds, of 22 carats fine ; how many carats fine is one pound of the mixture? Ans. 18 f. I (I ■■•mj m IMM ALLIG-ATION ALTERNATE. Alligation alternate is the method of finding how much of seve- ral ingredients, the quantity or value ff which is known, must be eombined to make a compound of a given value. CASE I . Given, the value of several ingredients, to make a compound of a given value. T» <«ii« ft 23-1 ARITHSIETIC. E X A 31 P L E How much sugar that is worth cents, 10 cents, arid 13 cents per pound, must be mixed together, so that the mixture may bo worth 12 cents per pound ? SOLUTION 12 cents. Gain. 1 lb., at G cents, is a gain of cents. ) Gaii 1 lb., at 10 cents, is a gain of 2 cents, j 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 Gain 8 Loss 8 It is evident, in forming a mixture of sugar worth 6, 10 and 13 cents per pound so as to be 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 ^rreaic?' value than Iho 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 pound, when in the mixture will bring 12, so that a gain of 2 cents is obtained on every pound of this sugar used in the compound. Again, sugar that is worth 13 cents per pound, on being put into the mixture will sell 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 wp find that in taking one pound of each of the diifercnt qualities of sugar there is a gain of 8 cents, and a loss of only 1 cent. Now, our losses must equal our gains, and therefore we have yet to lose 7 cents> and as there is only one quality of sugar in the mixture by which we can lose, it is plain that we must take as much more sugar at 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 be worth 12 cents per pound, we will require 1 pound at G cents, 1 pound at 10 cents, and 1 pound at the 13 cents4-7 pounds of the same, which must be taken to make the loss equal to the gain. By making a mixture of uny number of times these answers, it will bo observed, that the compound will be correctly formed. Hence wo can readily perceive that any number of ansAvcrs mav be obtamed to th( thi or ei pel pel 64 $1 pel 62 po bu bu wc eai fin ALLIGATION ALTERNATE. 235 to all exercises of this kind. From what has been said we deduce the fallowing RULE. Find how much is gained or lost hy taking one of each kind of the proposed ingredients. Then take one or more of the ingredients, or such parts of them as ivill make the gains and losses equal. EXERCISES. 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, GO cents, G7 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 ceats 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 corn, worth 75 cents per bushel ; oats, worth 40 cents per bushel ; rye, worth 65 cents per bushel, and wheat, worth $1 per bushel, so that the mixture was worth 80 cents per bushel; what quantity of each did he take? Give four answers, and prove the work to be correctly done in each case. C .A S E II, Whoi one or more of the ingredients are limited in quantity, to find the other ingredients. E X A JI p L E . How much l)arley, at 40 cents ; oats, at 30 cents, and corn, at 60 t'! i!'J •JV fi 236 AMTHMETIO. cents per bushel, must- be mixed with 20 bushels of rye. at 85 cents per bushel, so that the mixture may be worth GO 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 5.00 .50 5.00 9 at 40, gives .' 1.80 9 at 30, gives 2.70 $5.00 $5.00 By taking 1 bushel of barley, at 40 cents, 1 bushel of oats at 30 cents, and 1 bushel of corn at 60, in connection with 20 bushels of rye at 85 cents per bushel, we observe that our gains amount to 50- cents and our losses to $5.00. Now, to make the gains equal the losses, we have to take 9 bushels more at 40 cents, and 9 bushels more at 30 cents. This gives us for the answer 1 bushel -{-9^10 bushels of barley, 1 bushel-}-9=10 bushels of oats, and 1 bushel of corn. From this we deduce the KULE . Find how much is gained or lost, hy taking one of each of the l>roposcd ingredients, in connection with the ingredient which is limited, and if the gain and loss he not equal, take such of the j^ro- posed ingredients, or such parts of them, as will make the gain and loss equal. KXEROISES. 6. How much gold, of 16 and 18 carats fine, must be mixed with 90 ounces, of 22 carats fine, that the compound may be 20 carats fine ? Ans. 41 ounces of 16 carats fine, and 8 of 18 carats fine. 7. A grocer mixes teas worth $1.20, $1, and 60 cents per pound, 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. How much barley, at 50 cents per bushel, and at 60 cents per bushel, must be mixed with ten bushels of pease, worth 80 cents ALLIGATION ALTERNATE. 237 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 ceuts ; 2J bushels, at (JO cents. 9. How many pounds of sugar, at 8, 14, and 13 cents per pound, must bo mixed with 3 pounds, worth 9,| cents per pound ; 4 pounds, worth 10^ cents per pound ; and G pounds, worth 13^ cents per pound, so that the mixture may bo worth 12^ cents per pound ? Ans. 1 lb., at 8 cts. ; 9 lbs., at 14 cts. ; and 5^ lbs., at 13 cts CASE 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 p©K pound ; how much of each sort must he take ? SOLUTION. Gain. Loss. 1 lb., at 8 cents, gives 3 1 lb., at 10 cents, gives 1 1 lb., at 12 cents, gives 1 1 lb., at 14 cents, gives 3 4 lbs. 4 4 240 lbs.-^4i=rG0 lbs. of each sort. By taking 60 lbs. of each sort we have the required quantity, and it will be observed that the gains will exactly balance the losses, consequently the work is correct. Hence the RULE. Find the least quantity of each ingredient by CASH I., Then divide the given amount hy the sum of the ingredients already found, and multiply the quotient hy the quantities found for the propor- tional quantities. 10. What quantity of three different kinds of raisins, worth 15 cents, 18 cents, and 25 cents per pound, must be mixed together to £lk a box containing 680 lbs., and to be worth 20 cents per pound ? Ans. 200 lbs., at 15 cents ; 200 lbs., at 18 cents ; and 280 lbs., at 25 cents. 16 238 AJilTHMETlC. ij i\i r T' iiu m ~t 'I |:Ni 11. How miu'ii sugar, tit (> cents, 8 ccMts, 10 cents, and 12 cciitt per [)oiuk1, must be mixed togotlier, so as to form a compound of 200 pounds, worth 9 cents per pound ? Ans. 50 lbs. of each. 12. How much water must be mixed with wine, vorth 80 ccnt.s per gallon, so as to fill a vessel of DO gallons, which may be offered at 50 cents per gallon ? Ans. 50^ gals, wine, and 33^ gals, water. 13. A wine merchant has wines worth $1, $1.25, $1.50, Sl.T5,and $2. per gallon, and he wishes to form a compound to fill a 151 gallon cask that will sell at $1.40 per gallon ; how many gallons of each sort must ho 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 pauuds ; how much of each kind must he take, so that tlie mixture may be worth 15 cents per pound? Ans. 33.'. lbs. of 8, 10, and 12 cents, and 100 lbs. of 20 cents. 15. A grocer requires to mix 240 pounds of different kinds of raisins, worth 8 cents, 12 cents, 18 cents, and 24 cents per lb., so that the mixture shall be worth 10 cents per pound ; how umch must be taken of each kind ? Ans. 192 lbs. of 8 cents, and IG 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 the very rudest state of society, men have felt the necessity of liaving some article, of more or less intrinsic value, that can at any time bo exchanged for different commodities. No other substances were so suitable for this purpose as gold and silver. They are easily divisible, portable, and among the least imperishable of all substances. The work of dividing the precious metals, and marking or coining tlicni, is generally undertaken by the Government of the country. Money is a commodity, and its value is determined, like that of other commodities, by demand and supply, and co«t of production. When there is a large supply of money it becomes ciioaj> ; in other words, more of it is recpiired to purchase otiier articles. IT all the MONEY : ITS NATURE AND VAIX'E. 239 money in circulation were douljlcd, prlcos woultl bo doubled. The usefulness of money depends a {;reat deal upon tho rapidity of its circulation. A ten-dollar bill that ehnnj^cs hands ten times in a* monlli, purchases, during that time, ii hundred dollars' worth of t^oods. A small amount of money, kept in rapid circulation, does the same work as u far larger sum used more gradually. Therefore, whatever may bo 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, docs not act upon prices. An increase in the circulating medium, conformable in duration and extent to a temporary activity in busincs-s, does not raise prices, it merely prevents the liill that would otherwiso cn3UG from its temporary scarcity. PAPER CURRENCY Paper Currency may be of two kinds — convertible and incon- vertible. "When it is issued to represent gold, and can at any time be exchanged for gold, it is called convertible. When it is issued by the sovereign power in a State, and is made to pass for money, by merely calling it money, and from the fact that it is received in pay- ment of taxes, and made a legal tender, it is known as an inconver- tible currency. Nothing more is needful to make a person accept anything as money, than the persuasion that it will be taken from him on the same terms by others. That alone would ensure its currency, but wo'ild 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 par value. But as soon as gold and silver are driven out of circulation by the flood of inconvertible currency, prices begin to rise, and get higher with every additional issue. Among other commodities the price of gold and silver articles will rise, and the coinage will rise in value as mere bullion. The paper currency will then become proportion ably depreciated, as com- pared with the metallic currency of ether countries. It v>uuld be '^'fii;'^ 240 AlllTIIMETIC. quite impossible for these results to follow the issue of convertible paper fur which },'old could at any time bo obtained. ' All variations in the value of the circulatinji; niediuui are mis- chievous ; they disturb existinij; contracts and expectations, and the liability to such disturbin<^ influences renders every pecuniary engagement of long date entirely precariou.i. A convertible paper currency is, in many respects, beneficial. It is a more convenient medium of circulation. It is clearly a gaiu to the issuers, who, until the notes arc returned for payment, obtain the use of them as if they vcic a real capital, and that, without any loss to the counnunitr. THE CURRENCY OF CANADA. >*•* t wi 'I ¥ w* W-f Wif kT i » 11 1 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 head of Decimal Coinage. The e(|uivalent in gold of the pound currency is l()l.|}21 grains Troy weight of the standard of fineness prescribed by law for the gold coins of the united kingdom of (Ireat Britjiin 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 84.86ij. 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 legal tender is meant the proffer of payment of an account in the currency of any country as established by law. Copper is a legal tender in Canada to the amount of one shilling or twenty cents, and silver to the amount of ten dollars. The British sovereign of lawful weight passes current, and is a legal tender to any amount paid in that coin. There is a silver currency proper to Canada, though United States' coins are most in circulation. The gold eagle of the United States, coined before July 1, 18;>4, is a legal tender for $10.G6| of the coin current in this province. The same coin issued after that is a legal tender for 5510. % EXCUANQE. 241 EXCHANGE. It often becomes necessary to send money from ono town or country to another for various purposes, generally in pnyniont for goods. The usual modo of makinj^ and receiving payments between distant places is by bills of exchange. A merchant in Liverpool. , whom wc shall call A. B., has received a consignment of fiour from C. D., of Chicago; and another man, li. F., in Liverpool, lias shipped a quantity of cloth, ia value c(^ual to the flour, to G. IL in Chicago. There arises, in this transaction, an indebtedness to Chi- cago for the flour, as well as an indebtedness from Chicago for the cloth. It .is evidently unnecessary that A. B., in Liverpool, should send money to C. J), in Chicago, and that G. II., in Chicago, should send an equal sum to E. F. in Liverpool. Tlic ono debt may be applied in payment of the otnor, and by this plan the expense and risk attending the double transmission of the money may be saved. C. D. draws on A. B. ibr tl e amount whioh he owes to him; and G. 11. liaving an equal amount to pay in Liverpool, buys this bill from C. D., and sends it to E. F., who, at tho maturity of the bill, presents it to A. B. for payment. In this way the debt duo from Chicago to Liverpool, and the Jebt due from Liverpool to Chicago are both paid without any coin passing from ono place to tho other. An arrangement of this kind can always be made when the debts due between the difierent places arc equal in amount. But if there is a greater sum due from one place than from the other, the debts cannot be simply written off against one another. Indeed, when a person desires to make a remittance to a foreign country, he docs not make a personal search for some one who has money to receive from that country, and ask him for a bill of exchange. There arc ex- change brokers and bankers whose business this is. They buy bills from those who have money to receive, and sell bills to those who have money to pay. A person going to a broker to buy a bill may very likely receive one that has been bought the same day from a merchant. If the broker has not on hand any exchange that he has bought, he will often give a bill on his own foreign correspondent ; and to place his correspondent in funds to meet it, he will remit to him all the exchange which he has bought and not re-sold. ii; 't m H- m 242 ARITHMETIC. AVhen brokers find that tliey are asked for more bills than are offered to th'^m, they do not absolutely refuse to give them. To enable their correspondents to meet the bills at maturity, as they have no exchange to send, they have to remit funds in gold and silver. Tliero are the expenses of freight and insurance upon the specie, besides the occupation of a certain amount of capital involved in this ; and an increased price, or premium, is charged upon 'the exchange to cover all. The reverse of this happens when brokers find that more bills are offered to them than they can sell or find use for. Exchange on the foreign country then falls to a discount, and can be purchased at a lower rate by those who require to make payments. There are other influences that disturb the exchange between different countries. Expectations of receiving large payments from a foreign country will have one affect, and the fear of having to make lai'ger payments will have the opposite effect. AMERICAN EXCHANG-E 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 necessity for a conptaut cuiculation AMERICAN EXCHAXGi:. of the relative values of gold and greenbacks, and has generated an extensive business in that species of exchange. When the term '■ American currency" is used in the following exercises it is understood to be Greenbacks. CASE I. To find the value of $1, American currency, when gold is at a premium. EXAMPLE. When gold is quoted at l-iO, or 40 per cent, premium, what is the value of $1, American currency ? SOLUTION. Since gold is at a premium of 40 per cent., it requires 140 cents of American funds to equal in value $1, or 100 cents in gold. ITcnco the value of $1, American money, will bo represented by the number of times 140 is contained in 100, which is .71 ij or 71 ■] cents. Ilencc to find the value of .'pi ol' any depreciated currency rcckoncu in dollars and cents, wo deduce the following 11 u L 13 . Divide 100 cents hi/ l^f) plus the rate of inemiuni on gold, and the quotient loill he the value of SI. Subtract this from 81, and the remainder loili he the rate of discount on the given currency. CASE II. To find the value of any given sum of American currency when gold is at a premium. E X A :\i r L E s . AVhat is the value of 0280, American money, when gold is quoted at 140, or 40 per cent, premium ? s o i> u T I o N . We find by Case I. the value of $1 to be 71'] cents. Now, it is evident that if 71 i) cents be the value of $1, the value of $280 will be 280 times 71^ cents, which is $200, or $280-:- 1.40=28000-^- 140:=::;$200. Hcncc we have the following i t 2U ARITHMETIO. K r,l I'^J; '•■*! pi ! i| RULE. Multiply the value of $1 iy «Ae number denoting the given amount of American money ^ and the product will he the gold value; or, Divide the given sum of American money hy 100 (Jthe number of cents in $1,) plus the premium, and the quoti.nt will he the value in gold. ASE III. To find tho premium on gold when American money is quoted at a certain rate per cent, discount. EXAMPLE. When the discount on American money is 40 per cent,, what is the premium on gold ? SOLUTION. If American money is at a discount of 40 per cent., the discount on $1 would be 40 cents, and consequently the value of $1 would be equal to $1.00 — 40 cents, equal to 60 cents. Now, if 60 cents in gold be worth $1 in American currency, $1 or 100 cents in gold would be worth 100 times g'g of $1, which is $1.66§, from which if we subtract $1, the remainder will be the premium. Therefore, if American currency be at a discount of 40 per cent., the premium on gold would be 66f per cent. Ilenco we deduce the following erican ainder amount of RULE . Divide 100 cents hy the number denoting the go. American currency, and the quotient will be the val currency, o/$l in gold, from which subtract $1, an will he the premium. CASE I V . To find the value in American currency of an gold. EXAMPLE. Wl.at is the value of $200 of gold, in American currency, gold being quoted at 150 ? SOLUTION. When gold is quoted at 150, it requires 150 csnts, in American currency, to equal in value $1 in gold. Now, if $1 in gold bo worth 81.50 in American currency, $200 will be worth 200 times $1.50, which is $300. Hence the AMERICAN EXCHANGE 245 H U L E . Multiply the value of $1 hjj the number denoting the amount oj ■gold to he changed, and the jyroduct icill he the value in American currency ; or To tJie given sum add thejjremiwn on itself at the given rate, and the residt will he the value in American currency. EXERG ISES. 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 $750, American money, at a discount of 35 per cent, for gold ; how much did he receive ? Ans. $427.50. 4. Purchased a draft on Montreal, Canada East, for $1500 at a premium of G4| per cent. ; what did it cost me ? 5^ns. 5. If American currency is quoted at 33 J per cent, discount ; what is the premium on gold? Ans. 50 per cent. 6. Purchased a suit of clothes in Toronto, Canada West, for $35, but on paying for the same in American funds, the tailor charged me 32 per cent, discount ; how much had I to pay him ? Ans. $51.47. 7. What would be the difference between the quotations of gold, if greenbacks were selling at 40 and 60 per cent, discount ? Ans. 83^ per cent. 8. P. Y. Smith borrowed from C 11. King, $27 in gold, and wished to repay him in American currency, at a discount of 38 per cent. ; how much did it require ? Ans. $43.55. 9. J. E. Pekham bought of Sidney Leonard a horse and cutter for $315.50, American currency, but only having $200 of this sum, he paid the balance in gold, at a premium of G5 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. ; how much gold did it require to pay the balance ? Ans. $53.80. 11. W. H. Hounsfield &. Co., of Toronto, Canada West, purchased in New York City, merchandise amounting in value to $4798.40, on 3 months' credit, premium on gold being 79^ per cent. A.t the 1' 1 24:0 ahitilaietic. expiration of the three months they purchased a draft on Adams, Kimball and Moore, of New York, for the amount due, at a discount of 57| per cent, ; what was the gain by exchange? Ans. $G47.75. 12. A makes an exchange of a horse for a carriage with B ; tho horse being valued at $127.50. in gold, and the carriage at $210, American currency. Gold being at a, premium of 05 per cent. ; what was the difference, and by whom payable ? Ans. B pays A 23 cents in gold, or 37 cents in greenback.s. 13. A merchant takes §G3 in American silver lO a broker, and wishes to obtain for the same greenbacks which arc 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. §80.85. 14. I bought the following goods, as per invoice, from John 3IcDonald & Co,., of Montreal, Canada East, on a credit of 3 months : 1120^ yards Canadian Tweed at 05 cents per yard. 2100 " long-wool red flannel at GO '• " " 3-100 <' " white flannel at 55 " '' " Paid custom house duties, 30 per cent. ; also paid for freight, $37.40. Gold at time of purchase was at a premium of G3^ per cent. ; what shall I mark each piece at per yard to make a net gain of 20 per cent, on full cost ? Ans. C. tweed, $2.44; red flannel, $1.54 ; white flannel, $1.41. 15. A merchant left Toronto, Canada West, for New York City to purchase his stock of spring goods, taking with him to defray expenses $95 .in gold. After purchasing his ticket to the Suspension Bridge for $2.40, he expended tho balance in greenbacks, which were at a discount of 41-^ per cent. When in New York he drew from this amount $23.85 to "square" an old account then past duo. On arriving home he found that ho still had in greenbacks $10.40, whicii ho disposed of at a discount of 43f per cent., receiving in payment American silver at a discount of 3h per cent., which he passed off at 2h 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 ? Ans. Total expenses in gold, $71. 7G ; expenses in greenbacks, $118.04; silver received, $9.53. Wf EXCHANGE WITH GREAT J3EITAIN. 247 EXCHANOE WITH GREAT BRITAIN. In Britain money is reckoned by pounds, shillings and pence, and fractions of a penny, and is called Sterling money, the gold sovereign or the pound sterling, consisting of 22 parts gold and 2 alloy, being the standard, and the sliilling, one-twentieth part of the pound, a silver coin of 37 parts silver and 3 copper, and the penny, one-twelfth part of the shilling, a copper coin, the ingredients and size ("f which have frequently been altered. The comparative value of the gold sovereign in the United States previous to the voar 1834 was $-4.44|, but by Act of Congress passed in that year it was made a legal tender at the rate of 94-S^ cents per pennyweight, because the old standard was less than the intviLsic value and also because the commercial value, though fluc- tuating, was always considerably higher. Hence, the full weight of the sovereign being 5 dwts. 3.274 grs., it was made equivalent to 4 dollars and 8.Gf| cents. The increase in the standard value was,, therefore, equal to 9^- per cent, of its nominal value. The real par of exchange between two countries is thai by which an ounce of gold in one country can be replaced by aa idnce of gold of equal fineness in the other country. If the course of exchange at New York on London were 108J per cent. ; and the par of exchange between England and America 109J^ 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, the interest must be deducted from the above differ- ence. The general form for the quotation of exchange with England is : 108, 108^, 100, 100^ &c., which indicates that it is at 8, 8J, U, or Ot*- per cent, premium on its nominal value. EXAMPLE. What amount of decimal money will be required to purchase a draft on London for £048 17s. Gd. ? — exchange 108. The old par value or nominal value is 84.44^=''.j'::=i:^ of $40 248 UITIIMETIC. % I }i.r by reaucing to an improper fraction. Isow, tno quotation is 108, or 8 per cent, above the nomfnul value, wq find the premium on 840 at 8 per cent., which is $3.20, which added to $40 will give $43.20, and $43.20-7-9=$4.80 to be remitted for every pound sterling, and therefore i.648 17s. 6d. multiplied by 4.80 or 4.8 will be the value in our money. 17s. Gd.=.875 of a pound, and the operation is as follows • £648,875 4.8 %},t '■(■ 'S •*lli.lll ii 5191000 2595500 $3114.0000 R U L E . Vo $40 add the premium on itselj at the quoted rate, multiply the sum bi/ the number representing the amount of sterling money, and divide the result by 9, the quotient will be the equivalent of the sterling money in dollars and cents. Note.— If there be shillings, pence, &c , in the sterling money, they are 4;o 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 above its nominal value. Here we have exactly the •converse of the last problem, and therefore, liaving found the value of £1 sterling, we divide the given sum instead of multiplying ; thus the premium on $40, at 8 per cent., is $3.20, which added to §40 makes $43.20, and 43.20-f-9=4.80, and $4G5-v-4.80=£96.17.0. m m R ;if:c RULE Divide the given sum by the number denoting the value of one pound sterling at the given rate above j)cir, and if there be a decimal remaining reduce it to shillings andi^cnce. m ^i EX ERG ISE S. 1. "When sterling exchange is ouoted at 108, what is the valuo of £1 ? Ans. S4.80. EXCHANGE WITH GIIEAT lilllTAIX. 249 2. IfjGl sterling lio worth $-i.84:'^, what is tlic picniiuiu of ex- change between London and /_:iierica. Ans. 9 per cent. 3. At 10 per cent, above its nominal value, what is the worth of £50 sterling, in decimal currency ? Ans. $244.44. 4. When sterling exchange is quoted at 9|- per cent, premium, what is the value of $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. G. A merchant sold a bill of exchange on London for £7000, at an advance of 11 per cent ; what did he receive for it more than its real value? Ans. $46G.G6j. 7. Bought a bill on London for £126G 15s. at 9 J per cent, pre- mium ; what shall I have to pay for it ? Ans. 86164.85. 8. A merchant sells a bill on London for £4000, at 8 per cent, above its nominal value, instead of importing specie at an expense of 2 per cent. ; what does he save ? Ans. $122.G6|. 9. A merchant in Kingston paid $7300 for a draft of £1 500 on Liverpool ; at what per cent, of premium was it purchased ? Ans. dh. 10. Exchange ou London can be purchased in Detroit at 108J; in New York at \0^\, At which place would it bo 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 Dn New York are at a premium of g per cent. Ans., Detroit by $6.82. 11. A broker sold a bill of exchange for £2000, on commission, it 10 per cent, above its nominal value receiving a commission of j'(j per cent, on the real value, and 5 per cent, on what ho obtained for the bill above its real value ; what was his commission ? Ans. $11.95^. 12. I owe A. N. McDonald & Co., of Liverpool, $7218, net pro- eceds of sales of merchandise effected for them, which I am to remit Ihcm in a bill of exchange on London Ibr such amount as will close the transaction, less |: per cent, on the face of the bill for my com- mission for investing. Bills on London are at 8 per cent, premium. Required the amount of the bill, in sterling money, to be remitted. Ans. £1500. "■' 1l- I 250 ARITHMETIC. TABLE OF FOREIGN MONEYS, Cities and Cou.ntries. London, Liverpool, &C' Paris, Havre, kc Auisterdani, Hague, &c. Bremen Denominations of Monky. 20 shillings Hamburg, Lubec, &c... Berl' I, i^antzic ffCi; ium , St - 'ersbur"-. Copenhagen , Vienna, Trieste, &c.... Naples Venice, Milan, &c Florence, Leghorn, &c. Genoa, Turin, &c Sicily... Portuj;ii Spain Constantinople British India.. Canton Mexico Monte Video. Brazil. Cuba.. Turkey United States. New Brunswick. Nova Scotia Newfoundland..., 12 pence— j1 shilling; 1 pound :^ 1 00 centimcsr:^:! franc -- 100 cents-- 1 guilder or florin...:^ 5 swares--! groto ; 72. grotes=l rix dollar =: 12 pfennings;=l schilling; lGs.= 1 mark banco ::= 12 pfennings=:-l groschen ; 30 gro. =1 thaler = 100 centimes=l franc = 100 kopecks=::=l ruble r=: 12 rundstyck3=;16 skillings; 483. :^1 rix dollar specie = IG skillings=l mark ; G m.=l rix dollar = GO kreutzcrs=:::l florin = 10 grani=:l carlino ; 10 car.=l ducat = 100 ccntesimi=l lira =: 100 centesimi=;l lira = 100 centesimi=r=l lira = 20 grani=l tare ; 30 tari=:l oz.= 1000 reas=l millrea = J 34: maravedis==l real vellon= \ G8 maravedis=l real plate. . = 100 aspers=l jjiasier =: 12 pice=l anna; IG annas=:l rujyee = 100 candarincs=l mace ; 10 m.= 1 tael = 8 rials=l dollar r= 100 centesimas=l rial ; 8 rials=::=l dollar = 1 000 reas=l milrea = 8 reals plate or 20 reals vellon=;l dollar -— 100 asper3=:l piaster =; 10 mills=l cent ; 10 cents=l dime ; 10 dimes--=:l dollar. ...= 4 farthings==l penny; 12 pence r=l sliilling ; 20 shillings:=l pound.-'' = Value. $4.86f .18^ .40 .78f .35 .G9 .183 .75" LOG 1.05 .481 .80 .16 .16 .18f 2.40 1.12 .05 .10 .05 .44^ 1.48 1.00 .824 1.00 .05 variable. 4.00 * Tho Governmont of New Brunswick how issues i)Oi?tago stamps in the decimal currency, but so for as wo liavo been able to ascertaia, tlio currency of ATtBITEATION OF EXCILV^•GE. 251 ARBITRATION OF EXCHANGE Arbitration of Exchange is the method of findin;^ tlic rate J^ exchange between two countries through the intervention of ont. more other countries. The object of tliia i;. to ascertain what is I'l most advantageous channel through which to remit money to a foreljiu country. Three tilings have here to be considered. First, what is the most secure channel ; secondly, what is the least expensive, and thirdli/, the comparative value of the currencies of the different countries. Regarding the two first considerations no general rule can be given, as there must necessarily be a continual fluctuation arising from political and '>ther causes. We are therefore compelled to confine our calculation to the third, z., i comparative value of the coin current of different countries For this purpose we shall inves*' 'a-.v a riile, and append tables. Let us suppose an English mercha"^*' in London wishes to remit money to Paris, and finds that owing ij certain international rela- tions, he can best do it through I bi./g and Amsterdam, and that the exchange of London on Hamburg is 13^- marcs per pound ster- ling ; that of Hamburg on Amsterdam, 40 marcs for 3G^ florins, and that of Amsterdam on Paris, 56| florins for 120 francs, and thus the question is to find the rate of exchange between London and Paris. > Ml solution: Wc write down the ecjuivalents in ranks, the equivalent of the first term being placed to the right of it, and the other pairs below them in a similar order. Hence the first term of any pair will be of the same kind as the second term of the preceding pair. As the answer is to be the equivalent of the first term, the first term in the last rank corresponds to the third term of an analogy, and is there- fore a multiplier, it must be placed below the second rank. The these three Provinces is, as usual, in pounds, shillings and pence. It is to be hoped that Avhon the Confederation of tho British Provinces takes place, the decimal currency will bo speedily adopted in tho Lower Provinces, and that the efforts now being made in Britain to adapt the same currency will prove successful. I'i.l I W,l\ ■1, mil ••«*• 252 ARITHMETIC. terras being thus arranged, we 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 13i marcs. 40 marcs = 36:|^ florins. 5GJ florins =::120 francs £1 StK. As it IS most convenient to express the fractions decimally, wo have 1 a.fiXan.a/iXiaoxi :25.87 francs. » lX40Xo6.78 The foregoing explanations may be condensed into the form of a RULE. Write dotcn the first term, and its equivalent to the right of it, ana the other pairs in the same order, the odd term heing placed under the second rank, and then divide the product of the second rank by the product of the first, the quotient will he the required equivalent. Note. — The true principle onwliich tliis oporation is founded, is that eacli pair consists of the antecedent and consequent whidi are to each oilier in the ratio of equality in point op intrinsic value, though not in roirard to THE NUMBERS BY ^VHicu TUEY ARE EXPRESSED, and therefore the required term and its equivalent must liavo the same relation to each other, (liat is, they will bo an antecedent and a consequent in the ratio of equality ha regards their value, but not as regards the numhershy wliich thi-y are expreg.s(!d. EXERCISES. 1. If the exchange of London on Parif is 28 irancs per pound sterling, and that of America on Paris 18 cents per franc ; wliut is the rate of exchange of America on London, through Paris? Ans. $5.04 per £ sterling. 2. If exchange between New York and London is at 8 per cent, premium, and between London and Paris 25^ francs per pound sterling ; what sum in New York is equal to 7000 francs in Paris ? 3. When exchange between Portland and Hamburg is atSi cents per' mark banco, and between Hamburg and St. Petersburg is 2 marks, 8 schillings per ruble ; how much must be paid in St. Peters- burg for a draft on Portland for $050 ? Ans. 764 rubles, 70-j-!| kopecks. EXCHANGE. 253 Id 4. If a merchant buys a bill in London, drawn on I'nris, at the rate of 25.87 francs per pound stcrlinir, and if this bill be sold in Amsterdam at 120 francs for Sii^- florins, and the proceeds be in- vested in a bill on Ilamburj^, at the rate of 3G.^ florins for 40 marcs ; what is the rate of exahango between London and IIambur » . 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. .060893+. 7. What is the fifth power of 4 ? Ans. 1024. 8. Find the third power of .3 to three places ? Ans. .036963. Ans. 485809. Ans. 622835864. Ans. 19.070689. Ans. 31640625. Ans. 1.9738-f-. 343 1.800943. 9. What is the third power of 1 ? 10. What is the fifteenth power of 1.04 ?* 11. Raise 1.05 to the thirty-first power. 12. What is the eighth power of | ? 13. What is the second power of 4| ? 14. Expand the expression 6^. 15. What is the second power of 5J ? 16. What part of 83 is 26? 17. What is the difference between 5*^ and 4<' ? Ans. 11529. 18. Expand 35 X2'». Ans. 3888. 19. Express, with a single index, 473 X475 X47« ? Ans. 47' -i. 20. How many acres are in a square lot, each side of which is 135 rods ? Ans. 113 acres, 3 roods, 25 rods. Ans. Ans. Ans. 4.538039. Ana f>r)f>i Ans. 234|. Ans. .7776. Ans. -!-2irz=30J. Ans, |. Ans. .000001. Ans. .00000081. Ans. .1.2762815625. Ans. .000000001. Ans. .00001836. 21. What is the sixth power of .1 ? 22. What is the fourth power of .03 ? ' 23. What is the fifth power of 1.05 ? 24. What is the third power of .001 ? 25. What is the second power of .0014 ? The second power of any number ending with the digit 5 may be readily found by taking all the figures except the 5, and multi- <* This exercise will be most readily worlied by finding the sixteenth power, and dividing by 1.04. So in the next exercise, find the thirty-second power, and divide by 1.05. A still more easy mode of working such ques- tions will bo iound under the head of logaritlnus. IM Hii 't* m H 258 ARITHMETIC. lit;'" E '*m«» II plying th'.it 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 second power of 15. So also. '/ 625" 3,5 4 1225 05 4225 10,5 11 11025 215 22 57,5 58 46225 330G25 E X E U C I S E S ON THIS METHOD 2G. AVhat is the second power of 135 ? 27. AVhat i,-j the second power of 205 ? 28. What is the second power of 335 ? 29. What is the second power of 455 ? 30. What is the second power of 585 ? 31. What is the second power of 795 ? NoTio. — Thosquarorootof imy riuantity ending in 0, must end hi either o Ans. 18225. Ans. 42025. Ans. 112225. Ans. 207025. Ans. 342225. Ans. G32025. or (. No second power can end in S, 7, ['> or 2. The second root of any quantily ondinj^ in C>, must end in i or 0. The second root ol'any ([luintity tuidiug in 5, must end also in 5. The second root of any quantity ondinji; in -1, must end eitlicr in 8 or 2. The second root ot any quantity ending in 1, nuist end either in 1 or I'. The second root of any quantity eudiny in 0, must also end la 0. EVOLUTION. The root of any quantity is a number such that when repeated, as a factor, the specified number of times, will produce that quantity. Thus, 3 rep^M.ted twice as a factor gives 9, and therefore 3 is called the second root of 9, while 3 taken (lure times as a factor will give 27, and therefore 3 is called the third root of 27, and so also it is called i\io fourth root of 81. There are two ways of indicating this. First, by the mark -j/ Avliich is merely a modified form of the letter r, the initial letter of the English word root, and the Latin word radix (root). When no mark is attached, the simple quantity or first root is indicated. When the seco7id root is meant, the mark ]/ alone is placed before the quantity, bu' if the third, fourth,, &c., roots are to be indicated, 'K :U SECOND OH SQUARE ROOT. 259 tho figures 3, 4, &,g., are written ia the aiif^ukr space. Thus: 3=1/0=-^ 27=1/ 81=:..y 2-L:}, &c., &c. The other inetliud is to ■write the index as a, fraction. Thus, 9^ means the second root of the first power of 9, i a. 3. 8o also, 27- is the third root of tho 2 first power of 27. In the same manner G4^ means the third root of the second power of 01, or the second power of the third root of 04. Now the third root of 04 is 4, and the second power of 4 is 10, or the second power of 04 is 4090, and the third root of 4090 is 10, so that both views give the same result. Evolution is the process of finding any required root of a given quantity. n SECOND OR SQUARE ROOT. Extracting the square or second root of any number, is the find- ing of a number which, wlicn 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 less than 100, will in like manner be found to consist of tlirce or four digits ; and, universally, the second power of any number will consist of cither tv:lce the number of digits, or o^kj Less than ficicc the number of digits that the root itself consists of. Hence, if we begin at the units' figure, and mark off the given number in periods of two figures each, we :hall find that the number of digits contained in the root will be the same 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 be required to find the second root of 144. "Wo know by the rule of involution that 144 is the second power of 12. Now 12 may be resolved into one ten and two icnits, or 10-[-2, and 10-|-2 multiplied by itself, as in the margin, gives 100-|-40-(-4, and since 100 is the second power of 10, and 4 the second power of 2, and 40 is twice the product of 10 and 2, we conclude that the second 260 ABITHMr.TiO. : l.y § '•nkiMi 10+2 10+2 100+20 20+4 100+40+4 power of any number thus resolved is . 'jual to he suo of the s'^c!" o powers of the parts, plw:' twioe th. product of the parts. Hence tu und the ?^ccud i ouc of 144, let us resolve it into tao three pans IOC -|-40+4, and we find that the second root of the first part is 10, and since 40 is twice the product of thj parts, 40 divided by twice 10 or 20 will give the other part 2, and 10+2:zr:12, the second root of 144. We should find the same result by ^csolving 12 into 11+1, or 9+3, or 8+4, or 7-p5, or 0+6, but the n? st 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^:60, and 420H-60— 7, the second part of the root, and 30+7=:37, the whole root. Again, let it be required to find the second root of 15129. This may be resolved as below : 10000 is the second power of 100. 400 is the second power of 20. 9 is the second power of 3. 4000 is twice the product of 20 ai)d 100. 600 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 hundreds, 2 the root of the tens, and 3 the root of t)ie units. Generalizir? i;- '''".'^se investigations, we find that the second power of '". number c-'a.*^istl .g of units alone is tb.e product of that number by itself; that tiie second power of a number consisting o'i tens and units is the second power of the tens, jp/«s the second power of the units, ^)?ws twice the produce of 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, tlie tens, and the units, plus twice the product of each pair. Now since the complement of the full second power, to the sum of the second powers of the parts, is twice the product of the parts, it follows that, when the first figure of the root has been found, it must be doubled before used as a divi- sor to find the second term, and for the same reason each figure, when found, must be doubled to give correctly the next divisor. Hence the SECOND OR SQUAEE BOOT. 261 RULE. Beginning at the vnits' fiyure, mark off the. vhole line in p rlod'^' of ttoo figures each ; find the greatest power contained in the left hand period, and subtract it from that period ; to the remainder anney. fht: next period ; for a new dividend, place the figure thus ohtaine.r • • it quotient, and its double as a divisor, and find how often that qi iHf.'fij is contained in the second partial dividend, omitting the last fij arc . annex the figure thus found to both divisor and quotient, nmliu}'ii and subtract as in common division, and to the remainder annex the next period ; double the last obtained figure of the divisor, eind p>roceed as before till all the periods are exhausted, — // there be a remainder, annex to it two ciphers, and the figure thence obtained will be cr. decimal, as will eveivj figure thereafter obtained. EXAaiPLES. 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 tbe 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 64, in the quotient, and its double 16 as a divisor, ?nd tvv how often 16 is contained in 157, which wo find to be 9 times ; placing the 9 in both divisor and quotient 'C multip', and subtract as in common div/Ion, and find a remainder of 53, to wJ r'.h we amiex the last period 49, and procepding as before, we find 3, the last figure of the root, without remaisider, and now we have the complete root 893. 2. This operation may bo illusti jd as follows : S 1G9 1783 797449 64 1574 1521 5349 5349 To find the second root of 273529. 500 500x2^1000+20, or 1020 10004-2x20+3=1043 18 500+20+3=523 uTl fii . iMr '•'«i»l S^ 1^1 A*, fej' 2G2 AKITHMETIC. 3. To find the second root of 153G87. 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. C9 782 78402 784049 392.029-f- 230000 15G804 7319G00 705G441 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, and 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 392.029-4-. 2G3159 EXEKCISES. Ans. 529. Ans. 8G42. Ans. G78. Ans. 28.0r785-f. Ans. 41.569219-I-. Ans. 25.80G9-!-. 1. What is the second root of 279841 ? 2. What is the second root of 74G841G4? 3. What is the second root of 459G84? 4. What is the second root of 785 ? 5. What is the second root of L728 ? 6. What is the second root of 6GG ? 7. What is the second root of 12345G789 ? Ans. lllll.inOG+. 8. What is the second root of 5 to three places ? Ans. 2.23G. 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 M^sX^ and therefore ■i/^ower or rational quintltij, 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::r:8, ■^64=^4 and |/64:=:2, but it is irrational regarding any other root expressed by a whole number. But 64, with the fractional index ^, i. e., 6-i^, is rational, because it has an even root as already shown. Wc 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 10, and according to the second view, 64- is 4096, and the third root of 4096 is 16, the same as before. |/'81=:3 is rational, and |/81rz::9 is rational, but 81 is not rational regarding any other root ; while |/25 is rational only regarding the second root, and |,y8~=2 only regarding the tliird root. The second root of an even square may be readily found by re- solving tho number into its prime factors, and taking each of those THIRD ROOT OR CUBE ROOT. 265 factors once,— the product will bo the root. Thus, 411 lis TJ ;< o X 7 X 7 and each factor taken onco is 3x7=21, tiie socond root. Here let it be observed, that if wo used each factor twice wo should obtain the second power, but if we uso each factor half the number of times that it occurs, wc shall have the second root of that peer. 04 is 2X2X-X2X-X2=2«, i. c, 2 repeated six times as a factor gives the number 04, and therefore half the number of these factors will give the second root o' '34, or 2x2X2=8, and 2X2X2 multiplied by 2X2X2=8X8=04. As this cannot bo considered more than a trial method, though often expeditious, we would observe that the smallest possible divisors should be used in every case, and that if the number cannot 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 fractions. THIRD ROOT OR CUBE ROOT. As extracting the second root of any quantity is the finding of what two equal factors will produce that quantity, so extracting tlio 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 i^ the first power for each period in the third power. The left hand period may contain only one digit. From the mode of finding the third power from the first, wo can deduce, by the converse process, a rul'^ for finding the first power :!• ir u 2()G Ar.ITIIMKTlC. from the tliinl. W'c know by the rule ol" involution that the third power of 25 is 15G25. If wo rcsolvo 25 into 20-|-5, and perform tho uiuUiplicatiou iu that form, wc have 20 -|- 5 400-1-100 lOO-f-25 400+200-|-25==(20-f5)- 20+5 8000+4000-K)00 . 2000+1000+125 8000+G000+1500+125=(20+5)3r=:15C25 Now, 8000 is the third power of 20, and 125 is the third power of 5 ; also, GOOO is three times the product of 5, and the second power of 20, and 1500 is three times the product of 20, and the second power of 5. Let a represent 20 and b represent 5, then a 3:^20 3 ^. 8000 3a2 6 =3X203X5 -- 6000 3 a ^2^3^20X5- =^ 1500 2,3^53 ^ 125 15625 By using these symbols wo obtain the simplest possible method of extracting the third root of any quantity, as exhibited by the subjoined sclieme : Given quantity 15625 a3=203=20x20x20 = 8000 Eemainder 7625 3 a^ 6^.3X202X5 == 6000 Remainder 1625 3 « 62^3X20X52 ==. 1500 Remainder 125 63^53^5X5X5 = 125 From this and similar examples we see that a number denoted by more than one digit may be resolved into tens and units. Thus, 25 is 2 tens and 5 units, 123 is 12 tens and 3 units, and so of all numbers. TIIII;!-) KOOT on CUDK KOOT. 2G7 To find the tliinl root of 18G0SG7 : As this nmnber consist.^ of throe periods, tho root will consist of throe difiils. and tho first period IVoiu tho left will ^ivc hundreds, tlio second tens, and the third units, and so also in case of remainder, each period to the ri>rht will give one decimal place, the first being tenths, the (second hundredths, &c., &.c. We may denote the digits by o, b and c. «=100 i=3a^ 6-^3 a==-«/'AVb^=20+, and 30000X^0-^ 3ai2::^3XlOOX400= 18G08G7(100-|-20-}-3^123 1000000 ■ 8G08G7 remainder. GOOOOO 2G08G7 remainder. 120000 1^=^20^=^ 1408G7 remainder. 8000 9- 32G7 remainder. 3240 27 27 no remainder. K U LE. Now (a+/>):=120 . • . 3 (a-f i)2r-1328G7 remainder. 43200, which is contained 3 times-|- in 1328G7, . • . c^3, and 3 (a+6)-c- ^3X120«X3^ 129600 And 3 (a+i) c^ =3X120X9= And lastly, c3=3'= Mark off the given number in periods of three fgures each. Find the highest third j^ower contained in the left hand period, and subtract it from that period. Divide the remainder and next period bg three times the second power of the root thus found, and the quotient loill be the second term of the root. From the first remainder subtract three times the product of the second term, and the square of the first, PLUS three times the product of the first term, and the square of the second, PLUS the third jjower of the second. Divide the remainder bg three times the square of the sum of the first and second terms, and the Quotient will be the third term. M\ '^l I IMAGE EVALUATION TEST TARGET (MT-3) // h / i/.. ^ iP 1.0 I.I 1^128 |2.5 S Itf i|20 1-25 i 1.4 14 1.6 72 7 Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 \ "^ :1>^ :\ \ .A «* *> 1^ 6^ >^ '% 268 ARITHMETIC. ■ t*! i i1^ ^•i ',1 fill'' In From the lust remainder subtract three times the product of the term last found, and the square of the SUA! of the 2)reccdhig t(rms, PLUS the product of the square of the last found term hi/ the sum of the preceding ones, PLUS the third power of the last found term, and so on. KXERCISES. 1. 2. :j. 4. 5. G. 7. 8. Ans. 3G. Ans. G3. Ans. 120. Ans. 177. Ans. 21G. Ans. 359. Ans. 411. Ans. 501. Ans. 53G. Ans. G30. m What is the third root of 4GG5G ? What is the third root of 250047 ? What is the tliird root of 200057G ? What is the third root of 5'>45233 ? What is the third root of 10077G9G ? What is the third root of 4G2G8279 ? What is the third root of 857GG121 ? What is the third root of 125751501 ? 9. What is the tliird root of 153990G5G ? 10. What is the third root of 250047000 ? 11. What is each side of a square box, tLe solid content of which is 59319 ? Ans. 39 inches. 12. What is the third root of 92G859375 ? Ans. 975. 13. Find the third root of 44.G. Ans. 3.45G-|-. 14. What is the third rout of 9 ? Ans. 2.08008-^. 15. What is the length of each side of a cubic vessel whose solid content is 293G.4935G8 cubic feet? Ans. 1432 feet. IG. Find the third root of 5. Ans. 1.7099. 17. A store has its length, breadth and height all ecjutil ; 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 inchci-. 19. What is the third root of 1 ? Ans. 1. 20. What is the third root of 144 ? Ans. 5.241483. The third root of a fraction is found by 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. THIRD ROOT OR CORE ROOT. 209 P n Mar EXERCISES. I. Wluit is Ihc third root of ^] ? Otherwise : .421875. To find the third root of 27- 6i- this wc have 703rr. 3x70-:x5 =73500) 3X70 X5-= 5250 [ 53= 125 J: .-121875(.70+.05^.75 343000 78875 remainder. 78875 no remainder. The third root of a mixed quantity will be most readily found by reducing tlie fractional part to the decimal form, and applying the general rule. It has been already explained that the second root of an even power may be obtained by dividing the given number by the smallest possible divisors in succession, and taking half the number of those divisors as factors. The same [linfiple will apply to any root. If the given quantity is not an even power, it may yet be found approx- imately. If wc take the number 4GG50, wo notice that as the last figure is an even number, it is divisible by 2, and by pursuing the same principle of operation we find six twos as factors, and afterwards six threes ; and, as in the case of the second root, we take each factor half the number of times it occurs, so in the case of the third root, we take aich factor one-third the number of times it occurs. The same principle on which the extraction of the second and third depends may be applied to any root, the line of figures being divided into periods, consisting of as many figures as there arc units in the index ; for the fourth root, periods of four figures each ; for the fiftli, five, &c., &c. We may remark, however, that these modes are now superseded by th6 grand discovery of Logarithmic Computa- tion. 18 270 AIUTmiETIO. '. J PROGRESSION. l;^ A scries is a succession of quantities increasing or decreasing by a Common Difference, or a Common liatio. Progression hy a Comnwn Difference forms a series by the addi- tion or [subtraction of the same quantity. Thus 3, 7, 11, 15, 19, 23 forms a scries increasing by the constant quantity 4, and 28, 21, 14, 7, forms a series decreasing by the constant quantity 7. Such a progression is also called an equidifferent series.-'" Progression hg 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, IG, 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. PROGRESSION BY A COMMON DIFFERENCE. In a scries increasing or decreasing by a common dififerencc, the sum of the extremes is always equal to the sum of any two that are equally distant from them. Thus, in the first example 3+23:^^7-}- 19^11-1-15^26, and in the second 28-|-7=214-14=35. If the number of terms be odd, the sum of the extremes is equal to twice the middle term. Thus in the series 3, 7, 11, 15, 19, 3-)- 19=2 X 11^=^22, and hcnco the middle term is half the sum of the extremes. • The names ArUhmdieal Progression and Geometrical Progression aro often applied to quantities so related, but tlicse terms aro altogether inappro- priate, as they would indicate tliat the one kiiid belonged solely to arithmetic, and the other solely to {^^eometry, whereas, in reality, each belongs to both these branches of science. rROr.KESSlON V.{ \ comjion dttfi-jiexce. 271 In treating of jiro<:rc.' sions by difiorcnco or cquidiffercnt soiiofl, there arc five things to bo considered, viz., tlio llrst term, the last term, the connnon diflbrenco, the number of t.^rms, and the sum of the t,) I' ,» ;]. What i.s the oiic-thousandth term of tho scricd of tlic odd ii;,'uros? Ans. 1999. 4. ^\^Iat is tlic live-hundredth tonn of tlio scries of even dii^ita ? Ans. 1000. ;'). What is the sixteenth term of the dccroasinj^ series, 100, 9G, d'2, kc. ? Ans. 40. \m To find the sum of any equidifforent scries, when the number of terms, and cither the niiddk! term or tlic extremes, or two terms ('(|uidistant IVom them, arc ^ivcn. Wc have seen already that in any such scries the sum of the extremes is ccjual to the sum of any two terms that arc equidistant from them, and when the number of terms is odd, to twice tho mid- dle term. Ilcncc the middle tern), or half the sum of any two terms equi-distant from the extremes, will be equal to lialf the sum of tliosc extremes. Thus, in the series 2-|-7-i-12-|-17+22-|-27-|-:J2, we liave itJi:- ---liil^-17, the middle term. It is plain, therefore, that if we take the niiddlc 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 f^ivc 119, tho same as would be found by addini:; together the original quantities. The same result would bo arrived at when the number of terms is even, by takiiijz' half the sunt of the extremes, or of any two terms that arc equi-diritanl Iroiu them. From these explanations we deduce the HKhE (2.) Multiplij the mkliUc term, or half the sitm of the extremes, or of amj tiro terms that are equidistant from than, hi/ the numhcr oj terms. NoTK. — If tho sura o( the two terms bo an odd number, it is generally more convenient to multiply by the number of terms before dividing 1)y 2. EXAM 1* L E S . Given 23, the middle term of a scries of 11 numbers, to find tho sum. Here we have onlyto multiply 23 by 11, and wc find at once the sum of the scries to be 253. Given 7 and 73, the extremes of an increasing scries of 12 num- bers, to find the sum. The sum of the extremes is 80, the half of whieli is 40. and 40x12—480, the sum required. PROGRESSION BY A COMMON DIFFERENCE. 273 Two equidiatimt terms of a scries, IJ5 and 70, arc given. in a scries of 20 numbers, to find the sum of the scries. In this case, wo have 35+70=^.105, and 105x20=2100, and 2100-^2=^1050, tho sum required. K X E tt c I s E 8 . 1. Find tho 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 scries whose first term is 2, and twenty-first G2 ? Ans. G72. o. What is the sum of 14 terms of tho scries, tlie first term of which is }i and the lust 7 ? Ans. 52-J. 4. Find the sum to 10 terms of the decreasing scries, the first term of which is GO and the ninth 12. Ans. 360. 5. A canvasser was only able to earn SG 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 earn during tlie two year.s, supposing the increase to have been at a constant monthly rate ? Ans. $1248. (5. If a man 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. 1419G. 8. If 8:1 ^^ ^'i*^ fourth part of the middle term of a series of 99 number.-!, what is the sum ? Ans. 34G5. 1). In a series of 17 numbers, 53 and 33 are equidistant from the extremes ; what is the sum of the series? Ans. 731. 10. In a series of 13 numbers, 33 is the middle term ; what is tho sum ? Ans. 429. To find the number of terms when the extremes and common difToroncc are given : As in the rule (1), we found the difference of the extremes by multiplying by one less than the numbor of terms, and added the first term to the result, so now we reverse the operation and find the RULE (3.) Inv'ulc the difference of the extremes hy the common difference and :-dd 1 to the result. 271 I m > {' AniTIIMETir. E X A M P I- K . Given llic extremes 7 uml 100, and the common flificrcncc, 3, (o lind tlio niiinbor of terms. In this case wo have lO')— 7- 102, and H)2-:-U—'M, and [i4-|-l— 35, tho number of terms. 1 : X E 11 I s E H . 1. What is the number of terms when the cxtrcmca arc 35 and 333, and the common difference 2 ? An.s. 150. 2. Two equidistant terms are 31. and 329, and the common dif- ference 2 ; what is tho number of terms ? Aus. 150. 3. The first term of a .scries is 7, and tho last 142, and tlio com- mon difTorence }- ; what is the number of terms ? .\ns. 541. 4. The first and lust terms of a scries arc 2^-^ and 35^;, and tho common difTorcnce \ ; what is the number of terms? Ans. 10(1. 5. Tho first term of a .series is \ and hist 1'2\, and tho common differonco I ; what is tho number of terms ? Ans. 25. Givon one extronio, llio sum of tiio series and tlio number ol terms, to find tlid o*' " extreme. This case ma ilvd bv roversin-^; Itulo (2), for in it ihc data are the sam i'.'. 4. The sum of a series is r)70, tho number of terras 24. and tho greater extreme is 47 ; wliat is tho less extreme ? Ans. 1 . 5. Tho sujn of a scries is 30204.V, the greater extreme IJ12, and tho number of terms I'.Kl ; what is the loss extreme ? Ans. 1. Given the extremes and number of terms. t(» lind the common difference. As explained in tho introduction to llulo (I), the number of common differences must bo one less than the number of terms. It is obvious also, that tho sum of the.so differcncot- constitutes tho differ- ence between tho extremes, and that therefore tho sum of tho differ- ences is tho same as 1 less than the number of terms. Therefore the difference of tho extremes, divided by ///': sum of th' dijj'rrcnccn, will give one difference, i. r., the common difference. This gives us tho R IT L E (5. ) tSubtmct 1 from the numhcr of terms, and divide the dijfcraxce of the extremes hy the remainder. E X A -M P L E . If the extremes of an increasing scries bo 1 and 47, and the number of terms 24, wc can find the common difference thus : — 47 — l=^iG, and 4G-:-23=-2, the common difference. EXERCISES. 1. If the extremes are 2 and 30, 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 less extreme, and 1793 the greater, and 3C7 the number of terms, to find the common difference. Ans. 2^. 27G AniTUMETIC. 4. If and GO are tlio extremes in a Bcries of 10 numbers, what ia the common difference ? Ans. G. r». What h the common diffcrcnco in a dccrcasin;; flcricH of 4'2 terms, tlic extremes of which are 9 and 50 ? A.ns. 1. There are fifteen other cases, but they may all be deduced from the five here {jivon, Wo subjoin the Algebraic form aa it is more watisfactory and complete, and also more easy to persons acquainted with the symbols of that science. Lot a be the first term, d tho common difference, n tho number of terms, s tho sum of tho series ; tho scries will be represented by a^((i-}-il)-\-(a-\-2(l)-\-(a-\-M)-\-kc., to | a+(»i— 1)(?. | By iii- spectint^ this series it will hn seen tliut the co-efficient of tl is always 1 less than the number of terms, for in the second toriii 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 tho lust term is n — 1, and that term therefore is a~\-(n—l)d. Jf tho series were a decreasing one, that is, one formed by a succession of sub- tractions, the latjt term would bo a — (?t — l)d. To find tho sum of an equidifferent series. We have here «r=a+(a+(Z)+(a-[-2fZ) + (a4-3ci)+ &c -|- -j a-\-(n — l)d. (■ Since a-{-(H — l)d is tho last term, tho last but one will bo a-{-(n — 2)d, and the lust but two will be «-|-(n — 3)t/, &c., &e. But tho sura of any number of quantities is the same in "whatever order they may be written. Let us therefore write this series both as above, and also in reversed order : s==a-f-(a+rZj-)-(a+2i I + | 2a+ (n — l)d [■ + j 2a-\-(n — l)d [■ -}-&c., to n terms. rnooRESftioN uy a tommon DirrrnEXCE. 277 Id the last expression nil the tcnn.s arc the kuhio, but thcro aro n terms, and therefore the whole will bo 2«--n i 2i| (1.) • As wo have used no single symbol to represent the last term, w.» must now show how it may be obtained from the other data. Wo have seen that the last term is a4-(« — 1)'A which wo may donofti by /, which will give us the formula This formula, in the case of a decreasing scries, will bccomo l=a — (71 — 1 )(/, and generally l—a± (n—l)d. (2.) This formula is the same as llulo (1.) Wo may modily (1) by (2) by substituting I for a-^(n—l)d. Thus: 71 '=o(ia+l). (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, J, n, s, and, as already stated, any three of these being given, the other two can be found iVom (I) and (2.) To find d when a, I, n are given : By (2.) l.^-a'\~{n—\)d / in'* n—i, (4.) This finds the common difference, when the cxtremoa and num- ber of terms are given, and corresponds to Rule (5.) If a, n, s arc given, we have By (1.) 4l^« -,-{a-\)d\ } . • . 2s=2an-\-n (?i — l)d .' . dii (n — 1)= 2 (s — an) , 2(s— «») • . ii -j — ^ n (71 — I) 19 M '"i- % Mf I 278 AniTII.MKTIC. Il'u iti to be I'ound from n, , wo havo ^y(i.; *-|{2.,-i-(«-i)«/| . • . li.v 2'//t ..r//t- — dil And by solving this (jumlnitic oquution, we find 2d K X A .M 1" I. E 8 . Given o=^G, d—i, ;i -;20, to find «. First by (2) /—u\-(n—l)d : -G-|-(20— l)i —82 20 and hcnco by (.^) s--^-;^(G+82) •r.8*80. Uy (4) Givon a:i=3, / 300, n^^'3, to find d. dJ~" n—\ 297_ I) 5l: ;|J J\I I X K 1) K X K K C I S E S . 1. Given 70, the loss extreme, 10 the common difference, una 44 the number of terms, to find tlie sum. Ans. 12540. 2. What is the less extreme when the greater is 579, the common difference 5), and the sum of the series 18915 ? Ans. 3. 3. What is the series when s^ 143, d~2, n— 11 ? Ans. 3,5,7,9, 11, 13, 15, &c. 4. Given 4 and 49, the extremes, and fi the number of terms, to find the series. Ans. 4, 13, 22, 31, 40, &c. 5. If 120 stones arc laid m a straight line, on level ground, at a regular distance of a yard and a quarter, how fur must a person travel to pick them all up one by one and carry them singly and place them in a heap at the distance of yards from the first, and in the same line with the stones ? Ans. 10 m. 7 fur., 27 rds., \}f yds. G. Insert three means between the extremes 117 and 477. Ans. 207, 297, and 387. The other variations arc loft as e.xcrcises lor the student. rilOdUESSIONH IIY IIATIO. 271) 7. A courier agrcotl to rido 10(1 iiiili'.s on condition of bcin;^ priiil 1 cent Cor tlio lirst niiU', .'» lor tin; second, '.) lor tlio third, and so on; how much did he i:et per mile on an nvcru<^e, liow much lor the liut milo, and how much ulto;^ethcr? Ans. 61.!)!) i>or mile, 8:}.97 for the last, and $10!) lor all. 8. A man performed a journey iit 1 1 di. \)or day. 0. Vind the .series of which 72 is the sum, 17 the first term, and number of ter is (J. .\ns. 17, 15, 13, 11, D, 7. 10. The \'enetian clocks strike the hours for the whole day; how many stroked will one of these strike in :i year. Ans. lOi)5U0. 11. An MastiMii monareli bein^ threatened with invasion, oflered his conunander-iti-chiel' a reward equivalent to a mill for the first Boldier he would enlist within a month, two for the second, three for the tliird, and so on ; the ollicer enlisted !>!)!), OD!) men ; what was hi.s reward cijual to in our money. Ans. !f'l!)I),!)'J11.500. 12. One hundred sailors were drawn up in line at a distance from each other of 2 yards, inchulinj< the bri^adth of the body — the pay- master, seated a dititauee of two yards IVom the first, .sent a lieutenant to liand to the first a sum of prize money, then back again to the second, aud so on to each singly ; how i'ar had the lieutenant to walk ? Ans. 11 miles, 3 fur., 32 rods, 4 yds. •M k PROGRESSIONS BY RATIO. There arc in progression by ratio, as in progression by difference, the same five quantities to be considered, except that in place of a com- mon difference we have a common ratio ; that is, instead of increase or decrease by addition and subtraction, we have increase or decrease by multiplication or division. If any three of these are l.nown the other two can be ibund. We have noticed already that if any quantity, 2, be multiplied by itself, the product, 4, is called the square, or second power of that 'm 280 AniTIIMETIC. ■S: quantity ; if this be again multiplied by 2, the product, S, is called the cube, or third power of that quantity ; if this again be multi- plied by 2, the product is called the fourth power of that quan- tity, and so on to the fifth, sixth, &c., powers. To show the short mode of indicating this, let us take 3X'^X3x3x3--r=243. For brevity this is written 3'', which means that there arc 5 factors, all 3, to be couiuually multiplied together, and 5 is called the index, because it indlcntcs the number of erjual factors. Given the first term and the coiumou ratio to find the last pro- posed term. Let it be required to find the sixth term of the increasing series, of which the first term is 3 and the ratio 4. This may obviously bo found by successive multiplications of the first term, 3, by the ratio, 4, — thus : — 3^1 St term. 3X4= 12=r2nd term. 12X4=: 48^3rd term. 48X4== 192z=4th term. 192x4== 7()8=-.5th term. 768x4=3072=6th term. The series, therefore, is 3, 12, 48, 192, 7G8, 3072. From this, it is plain, that as to find the last of G terms, only 5 multiplications of the first arc required, in all cases the number of multiplications will be one less than the number of terms. But to multiply five times by 4 is the same as to multiply by 1024, the fifth power of 4, for 4X4X4X4X4=1024, and 1024X3=3072.-!^ This gives us the general RULE (1.) Mult'qAy the first term hy that jjoicer of the given ratio lohich is a unit less than the numhcr of terms. If the series be a decreasing one, divide instead of viudtijilying. EXAMPLES. Given iu a series of 12 numbers, the first term 4 and the ratio 2, to find the last term. Since 11 is one less than the number of terms, we find the 11th power of 4, which is 2048, and this, multiplied by the first term, 4, gives 8192 for the twelfth term. * For the abbreviated mode see Invohilion. rRoor.ESsioNS by ratio. 281 Given the ninth term of a decreasing series, 393GC, and the ratio 3, to find tlie first term. As there are 9 terms, we take the 8th power of the ratio, 3, which we find to be G5G1, and the first term 393GG-^-05Gl;^l3, the first term. EXERCISES. 1. What is the ninth term of the increasing scries of which 5 is the first term and 4 the ratio ? Ans. 327680. 2. What is the twelfth term of the increasincr series, the first term of which i.s 1 and the ratio 3 ? Ans. 177147. 3. In a decreasing series the first term is 78732, the ratio 3, and the number of terms, 10 ; what is tlie hist term ? Ans. 4. 4. What is tho 2()th term of an increasing scries, tlie first of Vhich is LOG, and also the ratio LOG ? Ans. 3.207135. 5. In a decreosi.Tr series the first term is 12G.2477, the ratio LOG; what is the last of 5 terms? Ans. 100. Given the extremes and ratio, to find the sum of the scries. It is not easy to give a direct inoof of this rule without the aid of Algebra, but the following illustration may be found satisfactory, and, in some sort, be accounted a proof. Let it be required to find the sum of a scries of continual pro- portions, of which the first term is 5, the ratio 3, and the number of terms 4. Since 3 is the common ratio, wo can easily find the terms of the series by a succession of multiplications. These are — 5- 15+454-13.), and the sum is 200 15-[-4 5-i-135 -;-405 " 400 Let us now multiply each term by the ratio, 3, and, for conve- nience and clearness, place each term of the second line below that one of the first to which it is equal. Let us now subtract the upper from the lower line, and we find that there is no remaii.Jer, except the difference of the two extreme quantities, viz., 400. Now, it will be seen that this remainder is exactly double of the sum of the series, 200, and consequently 400 divided by 2, will give the sum 200. Also, 405 is the product of the last term by the ratio, and 400 is the difference between that product and the first term, and the divisor, 2, is a unit loss than the ratio, 3. Hence the 282 AniTIIMETIC. ' in, 1 "ji HULK (2.) MuUiphj (ha hist term hij the ratio, from this product suht'i'act the first irnn, '"~"^, calling this /, we have from (1.) s^^ (2.) If r is a fraction, r" and ar" decrease as n increases, as already shown under the head of fractions, so that if n become indefinitely great, «>•" will become unassignably small, compared with any finite quantity, and may bo reckoned as nothing. In this case 1 j will become s=~--::zr.~~ (3. ^i Ey this formula wc can find the sum of any inlinitc series so closely as to differ from the actual sum by an amount less than any assignable quaii ity. This is called the la7iit, an expression more strictly correct than the suin. From the Ibrmnla s=^-:^'^-, any three of the quantities a, r, 7, s being given, the fourth can be found. Let it be required to find the sum of tho scries l-rJ-]-i+^-r &c., to infinity. Here a=l and i—h . • . .s ^ =:1 — ^=-3; ==-1 X 2:^2. Thereferc, 2 is the number to which the sum of the series continually approaches, by the increase of the number of its terms, and is the limit from which it may be made to differ by a quantity less than any assignable quantity, and is also the limit beyond which it can never pass. rnOGRESSIONS BY RATIO. 285 By adding the first two terms, wo find l-)-^:=:^=2 — ^=^h. By adding the first three terms, we find 34-|-=4=:2 — :^=1|. By adding the first lour terms, we find 1-\-^:=J^^'-=2 — ^=1^- By adding the first five terms, we find 'A_[- Jgrr:Ji=r2 — yc=^ By adding the first six terms, we find ^J J+g'2=^5=2 — ^^j^ 13 1 It will be observed here that the difference from 2 is continually decreasing. The next term would differ from 2 by ^^^, and the next by -^^g, &e., &c. Thus, when the series is carried to infinity, 2 may be taken as the sum, because it differs from the actual sum by a quantity less than any assignable quantity. EXAMPLES. To find the sum of the first twelve terms of the series l4-3-f9-|- 27-1-&C. : Here a^=l, r=3, And S^^""— a-'i -» 3X177 1 47 -1 r—i — 3—1 a -265720. To find the sum of the series 1, — 3, 9, — 27, &c., to twelve terms, 11 _-8X-3 -1 ■'.X— 177117—1 - J r::— 1328G0. In the case of infinite series, if a is sought, s and r being given, we have from (3) a-^s (I — r), and iT , is sought, a and s being civen, we have ?•=—"' or 1 . EXERCISES. 1. Find the sum of the series 2, G, 18, 54, &c., to 8 terms. Ans. G560. 2. Find the sum of the infinite series 1 — J+-i'3 — ^V Observe here r= — h. Ans. ^. 3. What is the sum of the scries 1, ^, I, &c., to infinity ? Ans. {;. i. Find the sum of the infinite scries 1 — ^j-\-'}j — :J\f-\-Scc. Ans. 3. 5. What is the sum of nine terms of the series 5, 20, 80, &c. ? Ans. 43G905. 6. Find the sum of i/-|-j-J-{-|/^+&c., to infinity. Ans. v'^— 1 . 7. What is the limit to which the sum of the infinite series f, ^, J, I, &c., continually approaches ? Ans. -|. m ; in % r i ' ;.4, f i- ■■ , n h «i i w 280 ARITHMETIC. 8. What is the sum often terms of the series 4, 12, 3G, &c. ? Ans. 11809G. 9. Insert three terms between 39 antl 3159, so that the whole shall be a series of continual proportionals. Ans. 117, ..351 and 1053. 10. Insert four terms between ^, and 27, so that the whole shall (brm a series of continual proportionals. Ans. ^, 1, 3, 9. 11. The sum of a series of continual proportionals is 10^, the first term 3^ ; what is the ratio ? Ans. §. 12. The limit of an infinite series is 70, the ratio ^ ; what is the first term ? Ans. 40. ANNUITIES In Tho word Annuity originally denoted a sum paid mimcalli/, and though such payments arc often made half-yearly, quarterly, &e., still the term is applied, and quite properly, because the calculations are made for the year, at what time soever the disbursements may be made. By the term annuities certain is indicated such as have a fixed time for their commencement and termination. By the term annuities contingent is meant annuities, tho com- mencement or termination of which depends on some contingent event, most commonly the death of some individual or individuals. By the term annuity in reversion or deferred, is meant that the person entitled to it cannot enter on the enjoyment of it till after the lapse of some specified time, or the occurrence of some event, gener- ally the death of some person or persons. An annuity in perpetuity is one that " lasts lor ever," and there- fore is a species of hereditary property. An annuity forborne is one the payments of which have not been made when due, but have been allowed to accumulate. By the amount of an annuity is meant the sum that the principal and compound interest will amount to in a given time. The present worth of an annuity is the sum to which it would amount, at compound interest, at the end of a given time, if forborne for that time. Tables have been constructed showing the present and final values per unit for dlfFcrcnt periods, by which the value of any annuity may be found according to the following n ANNUITIES. 2S7 RULES. To .Ind either the amount or the present value of an annuity, — MuUiplij the value, of the vnif, (is found in the tables, by the iinmbcr denoting (he. (innnifi/. If the annuity Ijo in perpetuity, — Divide the (innnltt/ hi/ the numbtr ihuotlng tin: intercut of the unit for one year. If the annuity be in reversion, — Find the value of the xinit }ip to the date of coiwienccmait, and also to the date of termination, and multiply their difference by the number denoting the annuity. To find the annuity, tlie time, rate and i)rcscnt wortli being given. Divide the 2^ resent worth by the worth of the unit. Tables arc appended varyinc; from 20 to 50 years. EXAMPLES. To find what an annuity of $400 will amount to in 30 years, at 3^ per cent. Wc find by tlic tables Jio amount of $1, for 30 years, to be $51.622677, whicli multiplied by 400 gives $20649.07 uearly. 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, yielding $600, at 3^ per cent. The rate per unit for one year is .035, and 600 divided by this gives $17142.80. To fiid the present worth of an annuity on a lease in reversion, to commen>;c at the end of three years and to last for 5, at 3A- per cent. By the table we find the rate per unit for 3 years to be $2.801637, and for 8 years, the time the lease expires, $6.873956 ; the differ- ence is $4.072319, which, multiplied by 300, gives $1221.70. Given $207.90, the present worth of an annuity continued for 4 years, at 3 per cent., to find the annuity. By the tables, the value for $1 is $3.717098. and $207.90, divided by this, gives $55.93. .1! il 1 1 i}t.8 288 ARITIIJIETIC. T A lihE, r t^ ;;? SnOWIXO TIIK AMOt *:T op an ANMITV or ONI-; DOMJIU rtll ANNUM, IMl'KOVEC AT COMPorNM INTKIIKST FOIl A'.V NCMHKK OF YKAIW NOT KXCKKDINO KIl-TT. i 3 per cen!. .'U per cent. 4 per cent. 5 per cent. 1 6 per cent. / per cent. 1 1.01)0 000 1.000 OOli i.iioo 0011 1.000 00" 1.000 000 1.00 t 001 2 2.03 ) (JOO 2.035 00(1 2.010 00.)' 2.050 00(1 2.<'(W) 00;; 2.070 00(1 3 3.090 {)0;I1 3.106 -.l-. 3.121 6111. 3.152 50(1 :;.'83 600 3.-14 90(. 4 4.U-3 6271 4.211 9.;; 4 246 461, 4.310 12.". 4.: 7 1 61(. 4.4:19 94:, S ■).3()9 13(i a.:'.(i2 4i.(, 5.416 3231 5.5J5 (i3l 5.637 093 5.750 73il (1 6.i(i.s4io (;..-.,-)0 1.".: (;.('.;;2 975 6.8.11 913 6.975 319 7.153 291 7 7.662 462 7.779 !;)^ 7.>^;;8 294 8.142 OU;- .-> :<.K', K\f- 8.65 1 021 6 RS92 33..; 9.051 687 9.214 226 l.'.549 I'iy !).8i)7 468 10.2.59 8O;; 9 10.1.VJ 106 1(>.:',(;8 4!)(; 1('."S2 795 1 1.0.6 564 11.491 316 11.977 9S!.' 10 11. 4(;:' '79 11.731 393 12.00() 107 12.577 8!i;' i;i.i80 795 1:1.8 11; .11)- 11 12.807 -96 13.141 992 13.486 351 14.206 7.^i 14.971 64; 15.78:1 59! 12 14.192 03' U.601 96: 15.025 80,"> 15.917 127 16.869 911 17.888 45 i 13 ir).617 791) 16.113 0:1! 16.626 838 17.712 983 18.882 138 20.140 64;' 11 17.086 324 17.076 98> 18.291 911 19.598 6.32 21.015 066 22.550 488 Ij lS.r.98 914 19.295 68] 1^0.023 588 21.578 564 2:i.275 970 2.5.129 022 16 3 l.'-)6 8S1 20.971 031/ 21.S24 53! 23.657 492 25.670 528 27.888 054 17 21.761588 22.705 016 23.697 51: 25.840 366 28.212 880 :i0.840 217 18 23.414 43'. 24.499 691 25.615 413 28.132 385 30.905 653 ;!:i.999 033 19 2 -..11 6 8nt> 26.357 18;) 27.671 22ti 30.539 004 3:1.759 992 37.:i78 965 20 26.870 374 28.279 682 29.778 07! 33.01)5 954 36.785 5!)1 40.995 492 21 28.676 486 39.269 471 31.969 20: 35.719 252 39.992 727 44.8(if> 177 22 30..'-,36 780 32.328 902 34.247 97( 38.505 211 4;;.392 290 49.005 739 23 32.4.")2 884 31.460 414 36.617 88!) 41.130475 46.995 828 53.4:16 141! 21 31.426 470 36,666 528 39,082 6;)'l 44.501 99!) 50.815 577 58.176 671 2;-) 36.459 264 38.949 857 41.645 90c- 47.727 099 54.^J6151: 6:5.249 030 26 3S.-i:)3 042 41.313 102 44.311 74,- 51.113 454 59.156 383 68.676 470 27 40.70!) 634 4::.759 061' 47.081 211 51.669 126 63.705 766 74.483 823 28 42.931) 92:i 46.29;) 627 49.967 5,'sl 58.402 583 68.528 \l: 8.).697 691 29 4r).218 SJO 48.910 799 52.966 28(i 62.322 712 7;i.6;;!) 798 87.346 529 30 47.r>75 416 51.622 677 56.084 93S 66.4,38 ^\i< 7i).058 18(: 94.460 7S6 31 r);).U02 678 54.429 471 59.328 33.". 70.760 790 81.801 677 102.073 041 32 rj2.r.02 7,J9 57.334 502 62.701 46;i 75.298 82!» 90.8:9 778 110.218 154| 33 55.077 841 60.341 210 66.209 527 80.063 771 97.313 165 118.933 425; 31 57.73:) 177 63.153 l.V.' 69.857 9Uli 85.066 1)5!'. 104.183 755 128.1^58 765| 3j 60.4 :i2 USl 66.671 013 73.652 225 !)().320 307 111.43478' 13S.2;16 87S; 36 63.271 944 70.007 603 77.598 314 95.836 32: 119.120 HCi 148.913 460j 37 66.174 223 73.457 869 81.702 24 (i 101.628 13!i 127.268 119 160.3:37 400 172.561 0201 38 69.1,)9 449 77.028 895 83.970 336 107.709 54 ( 1:15.90 1 206 39 72.234 233 80.724 906 90.409 15( 114.095 02:; 115.058 458 185.6 10 292i •10 75.401 260 84.550 278 95.025 SIC 120.799 77 1 154.761 966 199.635 112j 41 78.663 298 88.509 537 99.826 53(, 127.839 76;, l(i5.047 684 214.609 570! 42 82.023 196 92.607 371 104.819 598 135.231 751 t75.950 645 230.6:52 240i 43 85.4 S3 892 96.848 629 110.012 381 142.99;j 33!) 187.507 577 247.776 496 41 89.018 409 101.238 331 115.412 877 151.143 00( 199.75S 032 266.120 851 45 92.719 861 105.781 673 121.029 ;'.92 15!).700 156 212.743 514 285.749 311 46 96.501 457 110.484 031 126.870 568 ICiXA'tS:) 16-1 226.508 125 :506.751 763 47 100.396 501 115.350 973 132.9 15 39(J 178.119 42: 241.0!)8 612 329.224 386 48 104.408 396 120.388 297 1.39.263 20(; 188.025 39; .'56.56 1 529 :15:5.270 093 49 108.540 048 125.601 846 145.833 73; 198.426 66;j 272.!)58 401 378.999 000 oO 112.796 867 130.999 910 152.6(;7 084 209.317 97( 290.3:15 905 106.f,'j8 929 m ANNUITIES. T A B L K 289 8U0W1SO TUK PllKHENT WOIITII OF AN AXXriTT OK OXR POI.LAIl TKU ANNTM, TO CONTINUE FOn ANY NUMBKIl OF VEAUS NOT KXCKKUINO FIJTY. 1 3 per cent. 3J per cent 1 per cent "' 'icrcc'iit. por cent. 7 pfT cent 1 0.970 874 0.9(;0 J8! 0.901 5::8 0,952 ;isi 0,!)43 390 0934 57!. 2 1.913 47(1 1899 094 1.8.S0 ()9.- 1.P59 41(1 1.833 3!i;; l,^()8 017 2 3 2.8:8 Gil 2.801 037 2.775 091 2.723 218 2.(;7;{ (111 2.01: 1 ;ii! 3 4 3717 098 3.07;i 079 3.0 J 9 89." ;j.545 951 :i.4(i5 l(l( it.Oi"* i 1:01 4 r> 4.r.79 707 4.515 05v 4.4.M v; >• 4.329 477 4.212 :i04 4,100 lor 5 (\ 5.417 191 5.32.> 55.; 5,242 i;;; 5.075 (-91 4.:ii7 32' 4.7(;0 5;»7 6 7 ().2;;{) 283 0.114 544 0,002 05; 5.780 37;. 5,582 3M 5.;;.s9 28( 7 8 7.019 092 0.873 950 0,732 74,- 0.4ti3 2i:; 0,20!) 741 5.97 I 2!),' 8 9 7.78(J lO'i 7.007 087 7,4 ;i5 3.i'. 7.107 82': 0,8 J 1 (,91' 0.515 2-8 9 10 fi.530 2... 8.31 li 005 8.110 ^9( 7.721 7;}5 7.300 0,S7 7.023 577 10 11 9.252 C24 9.001 551 8.700 477 8.30(i 414 7,880 87." 7.498 009 11 12 9 954 004 9.003 331 9.385 07 ■■ 8.^03 251: 8,3S3 84 1 7,942 (.71 12 13 10.(J34 955 10.302 738 9.985 04f 9.393 57;! 8.852 08; 8,;)57 (;:i,") 13 14 11. '.'!)(; 073 10.920 520 10.503 12; 9,898 041 9.294 98 1 8,745 45-,: 14 15 11.!/. ,7 935 11.517 411 11.118 387 10, .37 9 05K 9.712 24!- 9.107 898 15 1(1 12.501 102 12.094 117 11.052 29( 10.8;)7 770 10.105 89,- 9.410 032 10 17 13.1(iG 118 12.051 321 12.105 001 1 1.274 000 10.477 2Gv 9.7(;3 20( 17 18 13.753 513 1.3.189 C8v 12.059 297 1 1 .089 5^7 10.827 (;o:i 10.059 07(1 18 ID 14.;'.23 799 13,709 837 13.133 939 12.085 321 11,158 IK 10.335 578 19 20 14H77 475 14.212 403 13,590 320 12.402 210 11,409 421 10.593 997 iO 21 15.415 024 14.097 974 14,029 100 12.821 15;; 11,704 077 10.835 527 21 22 •1.5.93(5 917 15.107 125 14.451 115 13.103 003 12.041 581 11.001 241 22 23 10.413 008 15.020 410 14 850 842 13.488 574 12.303 ;579 11.272 187 23 24 1().9:55 542 1().058 308 15.240 903 13 798 042 12.550 1^,8 11,409 3;i4 1:4 25 17.413 148 10.481 515 15.022 08( 14.093 945 12.783 35(; 11.053 58C 25 2(; 17.87(! 842 10.890 35l' 15.982 70! 14.275 185 13 003 10( 11,825 779 20 27 18.327 031 17.285 3()5 10.329 58( 14.043 034 13.210 5;j4 11.980 709 27 28 18.7(;4 108 17.007 019 1(>.003 oo: 14.898 127 13.400 104 12.137 111 28 29 19.1^8 455 18.035 707 10.983 7ir 15.141 074 13.590 721 12.277 074 29 30 19.000 411 18.392 015 17.292 03; 15,372 451 1.3.704 831 12.409 041 30 31 20.000 4.'S 18.730 270 17.588 49-; 15,592 811 13.929 08(' 12.531 814 31 32 20.338 700 19.008 805 17.873 55: 15.802 077 14.084 04:}| 12.040 555 32 33 20.705 792 19,390 208 18.117 G4(. 10.002 54!; 14.230 23( 12.753 790 33 34 21.131 837 19.700 084 18.411 198 10.192 204 14.308 141 12.854 009 34 3) 21.487 220 20.0 JO 001 18.004 Cl.l 10.374 194 14.498 240 12.947 072 35 3G 21.832 25 i 10.290 494 18.908 281 16.540 851 '4.020 987 13.0;i5 208 3(1 37 22.107 235 20.570 525 '9.142 57l; 10,711 287 14.730 780 l;j.ll7 017 37 38 22.492 4i;2 20.811 087 19.307 80! 10.807 89:; 14.846 019 1:5.193 473 38 39 22.8;)8 215 21.102 500 19.581 485 17.017 Oil 14.919 075 13.204 928 39 40 23 114 772 21.355 07: 19.792 774 17.159 08( l;i.046 297 1:5.331 709 40 41 23.412 400 21.599 1 4 19.993 052 17.294 3(i8 15.138 010 I3.:J94 120 41 42 23.701 359 21.834 883 20.185 027 17.423 208 1,5.224 543 i;5,452 449 4-i 43 23.981 902 22.002 089 20.370 795 17.545 912 15.300 173 1:5,500 902 43 44 24.254 274 22,282 791 20.518 811 17.002 773 15.383 18- 13,557 908 41 4-) 21.5)8 713 22,4 !.5 450 J0.720 040 17.774 07 1,5.455 83. 13.005 5J2 45 4(1 24.775 449 22.700 918 20.884 054 17.880 007 15 524 370 13.050 020 40 47 25.0. >4 708 22.899 438 21.042 930 17,981 010 15,589 028 13.091 008 47 48 25.200 707 23.091 244 21.195 131 18077 158 15.050 027 l;i.730 474 48 49 21.5;!l 0-)7 2,3.270 504 21.341 472 18,108 722 15.707 572 13,700 799 49 r)() 25.729 704 23.455 018 21.482 185 18.255 925 15.7(;l 801 13.8 740 50 '1 ■ i ,11 290 ARITILMETIC. PARTNERSHIP SETTLEMENTS. 1% h TIjo (^ircuinstaiicos imdor which partnoi'Hliips nro Tormcd, the conditions on wliicli they :iro in' ' and the causi's that load to their dissolution, aro siv varied tli' s impossible to do nioro than <^ive jrencral directions dodixccd i om the cases of most connnon occnr- rcncf. Fn forniinj; a partnership, the prcat requisite is to luive the terms of agreement cxpresse 1 in the most clear and yet concise lan- gua^e possible, setting forth the sum invested by each, the duration of partnership, the bhare of gains or losses that fall to each, the suni that each may draw from time to time for private purposes, and any other circumstances arising out of the peculiarities of each case. The ease and satisfaction of making an C(iuitablo settlement, in case of dissolution, depends mainly on the clearness of the original agree- ment, and hence the necessity for its being distinct and explicit. Even when no dissolution is contemplated, settlements should bo frecjuently made, in order that the parties may know how they stand to each other, and how the business is succeeding. This is of great importance in preserving unanimity and securing vigour and regu- larity in all the transactions of a mercantile houi^e. A dissolution may take place 'from various causes. If the part- nership is formed for a term of years, the expiration of those years necessarily involves either a dissolution or a new agreement. The death of one of the partners may or may not cause dissolution, for the deceased partner may liave, by his will, left his share in the business to his son, or some other relative or friend. In no case, liowever, can an equitable settlement be made, except by the mutual consent of the parties, or else in exact accordance with the terms of agreement. It is also necessary that when a dissolution takes place public notice should be given thereof, in order that all jiartics liaving dealings with the firm may bo apprized of the change, and have their accounts arranged. For the same reason, it is necessary that some one of the partners, or some trustworthy accountant appointed by them, should be authorized to collect all debts due to the firm, and pay all accounts owing by it. Partnerships arc sometimes formed for a specific speculation, and therefore, of course, cease with the completion of the transactiof, and a settlement must necessarily be then made. No matter for what r.vnTNr.iisH u' hi:ttli:menth. 'jyj time the partncrsliip liu.s hcoii made, any partner is at liberty, at any time, to withdraw, on showing Huflicicnt cause and f^ivin;^ proprr notice. This is u jusi provision, I'or tiio circuinHlanccs ol" any part- ner may >so chanj^c, I'roni various causes, as to maiie it undosirable for him to remain in the business. If one partner is doputcd to settle the accounts of the hou.se, it would bo reckoned fraudulent Ibr any other partner to collect any moneys due, except that on receipt of them he hands tliem directly over to tiie person so deputed. The resources and liabilities, with the net investment on com- mencing busiuesH, being given, to find the net gain or los.s. 1. W. Smith and It. Evans are partners in business, and invested when commencing $1000 each. On dis-'^Uing the partnership, the assets and liabilities arc as follows: — Merchandi.so valued at $1-95 ; cash, So-44 ; notes against sundry individuals, S790 ; W. II. Monroe owes on account $SG.-10 ; l). R,. Carpenter owc^ $132.85, and C. V. Musgrove owes $G7.50. They owe on sundry notes, as per bill book, $212.40 ; E. G. Conklin, on account, $20.45, and II. C. Wright, on account, $41.30. What has been the net gain ? SOLUTION . Assets. Merchandise on hand... $1295.00 Cash on hand o44.00 Bills Receivable 700.00 Amt. due from W. H. Monroe 8'J.40 Amt. due from E. K. Carpenter 132.85 Amt. due from C. F. Mu.s";rove 07.50 Total amount Assets $2715.75 " " Liabilities, 2283.15 Net gain $432.G0 TAahUUics. Bills Payable $212.40 Amt. due E. G. Conkhn. 29.45 Amt. due II. C Wright. 41.30 W. Smith's invcstmcut... 1000.00 11. Evan's investment.... 1000.00 82283.15 ■! 1! R U L E . Find the sum of the assets and UahiUtks ; from thu assets subtract the liabilities, (including the net amount invested) and the differ- ence will be the net gain ; or, if the liabilities be the larger, subtract the assets from the liabilities, and the difference loill be the net loss. 'M 292 AurriiMKTic. M 'J. Ilarvcy Miller uiiil .Jamos Caroy nro partners in a dry poods htniii'si; llurvoy Miller iiivi'sliti;^ $1 KM), an I .liuuos I'aroy 8lli.')0. WluMi i'Idsih.; tlio books, (hey have! on hand — cash, 8ir-''>.-{0; uwr- chandiso as por inventory boitk, 81853.75; amount dopositod in I"'ir«t Nutiuiial U.ink, $1l!I)(); lunoiuit invested in oil lands, 8'.)i).'l; a nitc of la;id lor building purposes, valued at SlUDO ; Adam Dudgeon owes thorn, on acjjiint, $ll)t.I>J ; NViUiam Floiuin.; oWim 8J K).S;) , a not! a'^iinst AUVod Mills Tor $ii'.) 1 J, and a duo bill lor $'{ ), dr.uvii by JamiM iiiiii'.,'. Thi-y owe \\^ S. IIopj iV C)., on account, $Sl!).21 ; 11. J. Kin^'iSi Co., S(JI):-3.12, and on notes, $132lj.ld. What has been the net^ain or lo.ss ? Ans. 017G1.7;l f,'ain. l\. Jamoa Ilonninv; and Adam Mannin;..; liavo formed a co-part- ncr-ship for the purpjs3' of conducting a tjjncr.il dry good.s and {grocery busincs.s, each to Hhare gain.s or losses equally. At the end of one year they closo thobook.s, having S12S0 worth of merchandise on hand ; ca.sli, S71d.27 ; (Jirard Bank stock, 8500 ; deposited in Merchants' JJank, S.'J20.G() ; store and li.vlures valued at $.'>10(); am')unt due on iiotivs and b ) )k accounts, §.'>i7 1. d!). The firm owes on notes 83400, and on open accounts $717.10. eJamcs Ilonninj^ invested $1200, and Adam Manning, $1000 ; what is each partner\s interest in the business at closing ? Ans. James llcnning's interest, i";27 19.(53. Adam Manning's interest, 82519.G3. NoTK. — Where tho iiitcrost ol I'.ieh p.xrtiuT a; closuig isroquired the pjaui or lo:, mvI Ins drawn out $51)0 ; .1. M. Mmi^rovo invested $'MM, nnd has dr.iwn out $750. Wliit h:is hi;en the net gain or loss, and what is each pirtner's interest in the business ? Ans. NetloHs. 8550. '-N ; V. A. CMarkn's interest, $I5'J5,:J(J ; W. II, Mir.-idon's interest, 8180(J.18; .[, M. IMus^Tove's inter- est, 8:i II 0.1 8. NoTK. — III tills ami siicceoillnR ox iini>l»M, no IntcnMt is to In* allowi* 1 >':i lavi'sttn'Mit, or c'l irgijil oa aiujuaU witli'liMwa, uuIlm.'* su Hpccilltsl. 5. A, and C urj pirlnors. Thu .i^iini a;»d 1 )ssos are to bi; aharod as I'ollow-i: A, ,\ ; B, ,■', ; and 0, ,'.. A invested SiJOOO, and has withdrawn SJjOO, with the consent of 13. and C, up>u wliich no interest is t;) bj churi^od ; U invested 8J70I), and has withdrawn $1150; C invested $2500, and has witlidrawn ^dliO. After doiiiuj business H months, retires. Their assets consist ot bills receivable, S2:):J7.20 , morchuadise, $11)70 ; cash, $1210.80; 50 shares of tlrj Chic ii^o Permanent Buildin,:^ and Saviii,:^s' Society iStocI:, the par value of which is 850 pcr-hare; casli deposited in the Third National Bank, $1850; store and furniture, $3130 ; amount due from W. Smith, 83G0.80 ; Q. S. Brown, $2-10.10; and E. 11. Cirpcnter, $07.12. Their liabilities are as i'oUowa : Aiuount due Samuel Harris, $1G75 ; unpaid on .store and furniture, $933 ; and notes unredeemed, $3388.70. The Savings' Society stock is valued at 10 per cent, premium, and C in retiring takes it as part payment* What is the amount due C, and what is A's, and what is B's interest in the business ? Ans. Due C, $815.52; A's interest, $2350.90; B's interest, $2004.14. 0. E, F, Or and II are partners in business, each to share ^ of profit and losses. Th'^ business is carried on for one year, when E and F purchase from Or and II their interest in the business, allow- ing each $100 for his good will. Upon examination, their resources arc found to be as follows : Cash deposited in Glrard Bank, $3045 ; cash on hand, $1422 ; bills receivable, $1035 ; bonds and mortgages, $2740, upon which there is interest duo $100 , Metropolitan Bank stock, $1000; Girard Bank stock, $500; store and fixtures, $3500; house and lot, $1800; span of horses, carriages, harness, &o., $495; outstanding book debts duo the firm. $4780. Their liabilities arc : Notes payable, 82315 ; upon which there is interest due. >t;57 ; duo on book debts. $1500. E invested $5000 ; F $4500 ; n 29-1 AnrriDiETic. K'i • ^•' G, $4000 ; and II, $3000. E lia,s di-awn from the business S?1200, iipon which he owes interest $32 ; F has drawn 61000 — owes interest $24.50 ; G has drawn §950 — owes interest $12 ; and If has drawn nothing. In the scttlemcnL a discount of 10 per cent., for bud debts, is allowed, on the book debts due the firm and on the bills receivable. G takes the Metropolitan Bank stock, allowing on the same a pre- mium of 5 per cent. ; and II takes the Girard Bank stock, at a premium of 8 per cent. ; E and F take the assets and assume the liabilities, as above sLatcd. What has been the net gain or loss, the balances due G and H, and what are E and F each worth after the settlement ? Ans. Dixe G, $3057.75 ; due II, .i^3529.75 ; E's net capital, 64637.75 ; F's net capital, $4345.25. 7. II. C. Wright, W. S. Samuels, and E. P. Hall, arc doing business to;;et]ier — II. C. W. to have -?.- gain or loss; W. 8. 8. and E. P. H; each :}-. After doing business one year, W. S. S. and E. P. II. retire from the firm. On closing:' the books and takin" stock, the following is found to be tlic result : merchandise on hand, §3210.50; cash deposited in Sixth National Bank, $1627.35 ; cash in till $134.16; bills receivable, 8940.60; G. Brown owes, oa ac- count, 8112.40; Thos. A. Brycc owes 894.12; W. McKcc owes ^143.95; J. Anderson owes 854,20 ; 11. II. Hill owes $43.60 ; and S. Graham owCs 8260.13. TUcy owe on notes not redeemed 81804 ; II. T. Collins, on account, ;i.U24.45; and W. F. Curtis, $79.40. II. C. Wright invested $3200, and has drawn from the business $350. \V. S. Samuels invested $2455, and has drawn $140; E. P. Hall invested $2100, and has drawn $2000. A discount of 10 per cent. is to be allowed on the bills receivable and book accounts due the firm for bad debts. 11. C. Wright takes the assets and assumes the liabilities as above stated. What has been the net gain or loss, and what does II. C. AVright pay W. S. Samuels and E. P. Hall on retiring ? 8. T. P. Wolfe, J. P. Towlcr and E. B. 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 efibcts without regard to the proper pro- portion each should take. The following is tho result according to their ledger :— T. P. Wolfe invested $3495, and has drawn $2941 ; J, P. Towlcr invested 82900, and has drawn $2200 ; E. U. Carpenter rARTK.j?.RmP SETTLEMENTS. 295 invested $31. "30, and lius drawn §3000. How will tno partners settle with each otlier ? Ans. E. R. Carpenter pays T. P. Wolfe 88G, and J.P. Towler,«;2.'J3. 9. I, J, K, L and lu hav3 entered into co-partnership, a?;reclnt; to share tlie gains and lo.^ses in the followin:^ proportion : — T, ,^ ; J, -^^'g ; K, ,'- ; L, jV ; and 31. ,V . When dissolvini:^ the partnership the resources consisted of casli $4700 ; merchandise, $9355 ; notes on hand $70SD ; debentures of the city of Albany valued at $0780, on whioh thore is interest due, $123 ; horses, waggons, &c., $1280 ; Merchant's bink stock, $5000; First National bank stock, $5000; mortgages and bonds, §3G00 ; interest due on mortgages, $345.80 ; store and fixtures, $3000 ; anaount due from W. P. Campbell & Co., $2418; due from II. B.Smith, $712.00; due- from J. W.Jones, $1000. The liabilities are : — Mortgage on store and fixtures, $5000 ; interest due on tlie same, $212.25 ; due the estate of R. M. Evans,. $14075; noto.i and acceptances, $11940, on wliich interest is due, $85 ; sundry other boolc debts. $7500 ; I invested $7800, interest on liis investment to d.itc of dissolution. $702; J invested $6400, interest oia invostmcui, $5715; IC invested $G100, interest on invest- ment, $549 ; L invested $5300, interest on investment, $522 ; IM invested $5000, interest on investment, $450. I has withdrawn from the firm at different timos, $2425, upon wliich tho interest calcu- lated to time of dissolution is $183.40 ; J has drawn $29G0, interest, §267.85; K has drawn $1850, interest $37.30; L has drawn $3000, interest, $460 ; 31 has drawn $895, interest, $03.45. AVhat is the net gain or loss of each partnei", and what is the net capital of each partner ? Ans. I'e nee loss, $1233.29 ; I's net capital, $4660.31. J's net loss, $924.97; J's net capital, $2823.18. K's net loss, $016.05 ; K's net capital, $4095.05. L's net lo.s3, $1541.62 ; L's net capital, $1320.38. Ws net loss, $308.32 ; ^I'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 efiects, and upon examination of their ledger it shows the followiu'i- result: — A Ins drawn from the business $3465, and invested on commcncenient 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 ^ ^^' I 296 AEITHMETIO. proportion to the original investment. "Wliat has been each partner's gain or loss, and how do the partners settle with each other ? Ans. A's net gain, $368.43 ; B's net gain, 6330.20 ; C's net gain. 6278.00; D's net gain, $243.31. B has to pay in $4(J4.80 ; C has to pay in 61521.94. A receives $1143.43 ; D receives 6843.31. 11. Three mechanics, A. W. Smith, James Walker and P. Kanton, are equal partners in their business, with the understanding that each is to be charged $1.25 per day for lost time. At the close of their business, in the settlement it was found that A. W. Smith had lost 14 days, James Walker 21 days, and P. llanton 30 days. How shall the partners properly adjust the matter between them ? Ans. P. llanton pays A. V/, Smith, $9.58?;, and James Walker, 83J cents. 12. There are 5 mechanics on a certain piece of work in the following proportions : — A is -^^ ; B, rf(j ; C, ^f^ ; D, 375, and E, rfg. A is to pay $1.25 per day for all lost time; B, $1 ; C, $1.50; D, 61.75, and E, $1.62^. At settlement it is found that A has lost 24 ; B, 19 ; C, 34 ; D, 12 ; and E, 45 days. They receive in pay- ment for their joint work, $2500. What is each partner's share of this amount according to the above regulations ? Ans. A's share, $374.12 ; B's, $250.41 ; C's, $487.83 ; D's, $787.24; E's, $600.40. 13. A. B. Smith and T. C. Musgrove commenced business in partnership January 1st, 1804. A. B. Smith invested, on com- mencement, $9000; May 1st, $2400; June 1st, he drew out $1800; September 1st, $2000, and October 1st, he invested $800 more. T. C. Musgrove invested on commencing, $3000 ; March 1st, ho drew out $1600; May 1st, $1200; June 1st, he invested $1500 more, and October 1st, $8000 more. At the time of settlement, on the 31st December, 1804, their merchandise account was — Dr. $32000 ; Or. $27000 ; balance of merchandise on hand, as per inventory, $10500 ; cash on hand, $4900 ; bills receivable, $12400 ; R. Draper owes on account, $2450. They owe on their notes, $1890, and G. Hoe on account, $840. Their profit and loss account is, Dr. $860 ; Cr. $1520. Expense account is, Dr. $2420. Com- mission account is, Cr. $2700. Interest account is Dr. $480 ; Cr. $950. The gain or loss is to bo divided in proportion to each partner's capital, and in proportion to the time it was invested. Beauired each partner's share of the gain or loss, the net. balance PROrERTIES OF NUMBERS. 297 due each, and a ledger specification exhibiting the closing of all the accounts, and the balance sheet. Ans. A. B. S.'s net gain, SCG71.73; his net balance, $15071.73. T. C. M.'s net gain, $2748.27; his net balance, §124-18.27. PROPERTIES OF NUMBERS. The term Integer, or Whole Namhcr, is used in contradistinctiou to the term Fraction. All numbers expressed by the natural scriea 1, 2, 3... 10... 20... 100, kc, arc called integers, bo that .3 and 4 are integers, but J is a fraction. All numbers in the natural scries 1, 2 3, &c., that can be resolved into factors, are called Comiiosite, while those that cannot be so resolved are called Prime. Since 4=2X2, it is called composite, and so 0, 8, 9, 10, &c., but 1, 2, 3, 5, 7, 11, &c., are called prime because they cannot be resolved into factors. Thus, 11 can only be resolved into 11X1, or IXH, and these are not factors in the strict meaning of the word. A Prime Factor is a prime number, which is a factor of a com- posite number. The factors of 10 are 2 and 5, both prime numbers. A composite number may have composite factors, as rUi. which has 4 and 9 as factors, and both of these are composite. When any number will divide two or more others, it is called a Common Factor. Thus, 3 is called a common factor of 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 a composite number into its prime factors, divide it by the least possible factor that it contains, and repeat the process till a prime number is obtained. E X A >i i' L E s . 2)9G 2)48 2)24 2)12 2) G 3 BO that the prime factors of 96 arc 2X2X2x;2X2X3. ii #41 : I i' . "■ '■'■ % I' -. ■( 298 Ai:rniJiETic. fei S Also, because uX^XH^^^-jSr)," wo .sec that 5, 7 and 11 are the prime iactors of 385. E X E U I S E S 1. What arc tlie prime factors of 2310? Ans. 2, 3, 5, 7, 11. 2. What are the prime factors of 17G4 V Ans. 2, 2, 3, 3, 7, 7. 3. What arc 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 f\ictors of 51 ? G. AVhat arc the prime factors of 99 ? 7. What are the prime factors of 651 ? 8. AVhat arc the prime factors of 362880 ? Ans. 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 5, 7. 9. What factors arc 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 ending in the digit 5 is 5 units, and all other numbers ending in either 5 or are multiples of 5. .ADDITIONAL EXERCISES. Ans. Prime, It. Is 101 prime or composite? 12. Is 198 prime or composite ? Ans. It has the factors 2, 3, 3, 11. 13. Is 171 prime or composite ? 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. la 674041 prime or composite ? 18. Is 190 prime or composite ? 19. What are the prime factors of 210 ? 20. What arc the prime flictors of 51051 ? Ans. Prime. Ans. Composite. Ans. Composite. Ans. Composite. Ans. Prime. Ans. 5, 6, 7. Ans. 3, 7, 11, 13, 17. NoTK. — We have thought it Bufficiont under this liead to give onlj the leading and most useful principles. QUESTIONS roil COMMEKCUL STUDENTS. 299 QUESTIONS FOR COMMERCIAL STUDENTS. 1 . The following questions may be found interesting and instructive to young men preparing for the practical duties of accountants. On the 1st of May I purchased for cash, on a commission of 2h per cent., and consigned to lloss, Winans & Co., commission mer- chants, Baltimore, Md.; 380 bbls. of mess pork, at $27.50 per bbl., to be sold on joint account of himself and myself, cacli one half. Paid shipping expenses, $7.40. July 7th, I received from lloss, Winans & Co. an account sales showing my net proceeds to bo ^i5ol9,79, due as per average, August 12th. At'gust 8th, 1 draw on them at sight for the full amount of their account, which I sell at ^ per cent, discount for cash ; interest 7 per cent. What amoun.. of money do I receive and what are the journal entries ? 2. B. Empey, a merchant doing business in Montreal, Canada East, purchased from A. T. Stewart, of New York city, on a credit of three months, the following invoice of goods : 845 yds. Fancy Tweed, @ $1.90 per yd. 1712 " Amer. black broadcloth, © .3.85 " " 423 " Blue pilot, @ 2.75 " '' 700 '' Black Cassimere. @ 2.10 " " When the above goods were passed through the custom-house,' a discount of 27h per cent, was allowed on American invoices ; duty 25 per cent., freight charges paid in gold, §29.35. What must each piece be marked at, per yd., to .sell at a net profit of 15 per cent, on full cost ? What would be the gain or loss by exchange, if at the expiration of the three montlis B. Empey remitted A. T. Stewart, to balance account, a draft on Adams, Kimball & Moore, bankers. New York city, purchased at '62\ per cent, discount, and what are the journal entries ? 3. I purchased for cash, per the order of J. P. Fowler, 70 boxes C. C. bacon, containing on an average' 400 lbs. each, at 13f cents it ! '•!♦ 300 ARITHMETIC. per lb., and 140 firkius butter, 831-2 lbs., at l7-\- cents per id., on a commission of 2 J per cent; paid shipping and sundry expenses in cash $13.40. For reimbursement I draw on J. P. Fowler at sight, which I sell to the bank at -J- per cent, discount ; what is the face ol' drul't, and what are the journal entries ? ^ • Ans. Face of draft $5479.05. 4. Sept. 27th, I received from James Watson, Leeds, England' a consignment of 1243 yards black broadclotl:, invoiced at Icls. Gd. 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 Iv. Duncan, for cash, 207 yards, at $0.10 per yard ; Oct 24th, sold 317 yards to James Grant, at 6G.25 per yard, on a credit of 90 days ; Nov. 13th, sold E. G. Couklin, for his note at 4 months, 400 yards, at $G.30 per yard; Dec. 12th, sold the remainder to J. A. Mu.^grovc at $0.00 per yard, half cash and a credit of 30 days for balance; charges ibr storage, advertising, &c., $13.40; my com- mission, with guarantee ol' sales, 5 per cent. What would be the average time of sales ; the average time of James AVatson's account ; and what would be the face of a sterling bill, dated Dec. 15th, at CO days after date, remitted James AV'^atson to balance account purchased at $108ij[, money being worth 7 percent, and gold being 70 per cent, premium ? 5. Buchanan &, Harris of Milwaukee, Wis., are owing Vi. A. Murray tt Co. of Washington, $1742.75, being proceeds of consign- ment of tobacco sold for them, and Simpson & Co. of Washington, arc owing Uuchanan I'c Harris $2000 payable in Washington. Buchanan k, Harris wish to remit W. A. Murray & Co. the proceeds of their consignment and they do so by draft on Simpson & Co., but \Vashington funds are 2 per cent, premium over those of Chicago, llcquired the face of the draft and the journal entries. G. A. Cummings, of L'^ndon, England, is owing me a certain sum, payable there, and I am owing Charle.-, Massey, of the same place, $1985.42, being proceeds of consignment of broadcloth sold for him. I remit C. Massey in full of account, after allowing him $21,12 for inserest, my bill of exchange on A. Cummings at GO days' sight ; exchange 109|, gold 42 per cent, premium. What is the fiice of the drai't, and what arc the journal entries ? QUESTIONS FOR COMMERCLVL STUDEI^'TS. 301 7 . 3Iarch 10, I shippoil per steamer VdndrrbiU ana confumicd to Saixiuc! Vestry, Liverpool, Eng'.ani), to be sold on joint account of cor aLTnco and consif;nor, each one luill', (consignee'H half as cash), 27894 lbs. prime American Cheese at 15.^ cents per pound. Paid shippiu!:; expenses $18.30. InsuraUoo 1;^ per cent, and insured for suoh an amount that, in case the cheese was lost, the total cost would bo recoverable. May 19, I received from Samuel Vestry an account sales, showing my net proceeds to be ^^298 14 9.^, duo as per average August 21. June 1, I drew on Samuel Vestry at the num- ber of days after date that it 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,^. 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. I am doing a commission business in New York, and on Sept. 14, I received from A. J. Rice, of Hudson, to be sold on joint ac- count of liimself §, A. II. Peatman, of Newburg, -J, and myself^; merchandise invoiced at $1202.40, paid freight $14.20. The same day, I received from A. II. Peatman to be sold on joint account of liimself z, A. J. Rice -J-, and myself -^j merchandise invoiced at $1102.12; paid freight §10.00. I also invest to be sold on joint account of A. J. Rico J, A. II. Peatman J, and myself §, merchan- dise valued at 8745.35. The shares of each are subject to average sales. October 29th, I sold ^- of the mcrchandiKo received fronx A. J. Rice to 8. King at an advance of 20 per cent., on a credit of 90 days. November 9th, sold for cash one half of the remainder at 15 per cent, advance, closed the company, and rcLdered account sales; storage $3.50, commission 2^ per cent. November 12, sold to A. M. Spafford, on a credit of 30 days one half of goods received from A. H. Peatman, at an advance of 25 per cent. November 23, sold for cash the merchandise that I invested at an advance of 15 per cent. ; closed the company and rendered account sales ; storage $2.75, commission 2J per cent. December 4th, sold J of the remainder of merchandize received from A. II. Peatman to G. W. Wright, on a credit of GO days, at 33^ 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 2J per cent, December 23rd, I wish to settle with A. J. Rice, and A. H. Peatman, iu full ; I take to my own account, 002 Ai!ITlIM!:iii.' Ill il as casli, tlie balance of iiuTclKiiKli.sciinsuM ; t, ;iii ailvaiicc of 8 per cent.. What is the uvcrago lime of sales of vmU !M(1so. Co., the average tunc of A. J. 11. unil A. II. P's. accounts, the amount of money I shall have to pay them on December 2'), how do A. J. 11. and A. II. 1*. stand will; each other, and wliataro the journal entries? 9. E. II. Carpenter, 8. Northrup and Levi AVilliainf4, commenced business together as partners under the name and style of ]']. 11. Carpenter & Co., on January 1st, 1SG5, with the following effects: merchandise, $7841 ; cash, 85000 ; ston^ and fi.Kturos, $3084 ; bills receivable, 81732.50; of this amount there belonged to ]']. 11. Car- penter, as capital, 88000; S. Northrup, 80000; Levi AViiliums, 845G1.50. The lirm assumed the liability of Levi AVilliams, which was a note to the amount of $425.80 ; this note was paid on March 10th. The loss or gain is to be shared equally by the partners, but interest at the rate of 7 per cent, per annum is to be allowed on in- vestments, and charged on amounts withdrawn. E. 11. Carpenter is to manage the business on a salary of 81000 a year, payable half yearly (the time of the other partners not being required in the business). March 14th, S. Northrup draws cash, $300 ; Levi Williams, 8200; April 19th, E. 11. Carpenter draws cash, 8500; S. Northrup, $100. On the 1st of May, they admit Geo. ymith as a partner, under the original agreement, with a cash capital of $4000. Tiie books not being clcsed, he pays each partner for a participation in the profits to this time 8450, which they invest in the business. 3Iay 14th, E. 11. Carpenter draws cash, $100; 3Iay 24th, Levi Williams draws cash, 8100 ; June 12th, S. Northrup draws cash, 8250, and E. .11. Carpenter, 8200 ; July 1st, Levi Williams draws cash, $300, and S. Northrup, 8450; July 21st, Levi Williams draws cash, 8180 ; August 1st, Levi Williams retires from the partnership, the firm allowing him 8500 for his profits and good-will in the busi- ness, this amount, together with his capital, has been paid in cash. Oct. 14th, Geo. Smith draws cash, 8340; E. R. Carpenter, $725. November 1st, with the consent of the firm, S. Northrup disposes of his right, title and interest in the business to J. K. White, who is admitted as a partner under the original agreement. J. K. White is to allow S. Northrup 8000 for his share of the profits to date, and his good-will in the business. J. K. White not receiving funds' an- ticipated, is unable to pay S. Northrup but 81500, the firm therefore assumes the balance as a liability. December 10th, received from QUKSTIONH FOR COJIMERCLU. STUDEN'J'.S. co;j J. K. White, ami paid over to S. Northrup, the lull amount due him (S. N.) to date. December lilst, the books are closed, and the fol- lowin-,' clTccts :ire on hand:— Mdso, 81 10 IM. 75; cash, $2110.1J; bills receivable, SO-KH) ; store nud fixtures, 8-^850 ; personal accounts Dr. 814987.50; personal accounts Cr. 810711 ; BilLs I'ayable unre- deemed, 8 1000. ^Vhat has been the net gain or loss, the net capital of each partner at the end of the year, and what arc the double entry journal entries on commencing business, when Levi Williams retires, when Geo. Smith is admitted, when S. Northrup .seHs his interest and right to J. K. AVhite, for E. 11. Carpenter's salary, and tlie interest due from and to each partner ? The student may also, ia the above example, after finding the interest on the partners investments, and on the amounts withdrawn, give a journal entry that will adjust the matter of interest between the partners without opening any profit and loss account. « U I yoi iUUTUMETIC. MENSURATION.'* m ii ^Vc have already observed tliat no solid body can iiavu move than three dimensions, viz. : length, breadth, and thiekncss, or dcptli, and that a lino is length, or breadth, or depth, or it is iv lino or unit repeated a certain number of times. A loot iri Icngtli is a lino mea- sured by repeating the linear unit called an inch 12 times, and a yard is the linear unit called a ibot, repeated li times, and so on. Thus, 1 ft. 1 ft. 1 ft. T -3 feet. But there may be two such lines and c are 3 square feet, but the whole figure is 'iifcet square, and therefore three I'eet square must be equal to i) square feet. Three feet square, then, is a square, each of v.'hoso sides measures 3 linear feet ; but 3 square feet would denote 3 squares, each side of each measuring one Unear foot. The space tlius inclosed is called the a)'ea. This is the principle on which surflxccs are measured. PROBLEM I. To find the area of a paralellogram : RULE. IfuUipl^ the length Inj the jjefpendtcular breadth. Tf the figure he rectangular, one of the sides will he the perpendicular hreadth. * We have taken for granted that those studying mensuration have learned, at least, the elementary principles of geometry. We have, there- fore, only given tho rules, as oiu- space Avould not admit of o>ir giving demonstrations as this would reipiiro a separate treatise MENSURATION. 305 rt the fijurc be not rectanjultr, v.ithcr the pcrpcndiculxr breadth must be given or data from which to find it. EXERCISES. 1. How many uoroi uro tliero in a sijuarc, each sido of which la 24 rods? Ans. 3 acres, 2 roods, IG rods. 2. What is the area of a .sr^uaro picture frame, each side of which is 5 ft. 9 in. ? Ans. 'Si ft. 9 in. iJ. How many acres are there in u rectangular field, the length of which ia 13^ chains, and the breadth 9\ ? Ans. 130. G25 square chains, or 13 acres, roods, 10 rods, 4. What is the area of a rectangle, whose sides are 14 ft. G in. and 4 ft. 9 in. ? Ans. G8 ft., 12G 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. G. What is tho area of a rhomboidal field, tho length of which is 10.52 chains and the perpendicular breath 7.G3 chains ? Ans. 8 acres, roods, 4.281G rods. 7. What is tho area of a rhomboidal field, the length of which ia 24 rods and the perpendicular breadth 24 rods ? Ans. 3 acres, 2 roods, IG rods. 8. Wiiat is tho 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 5G yards; what is tho other ? Ans. 33 yards. 10. The area of a rhomboidal pavement is 205, and the length is 20 feet ; what is the perpendicular breadth ? Ans. 10^ feet. PROBLEM II. To find the area of a triangle. 1. If the base and perpendicular, or data to find them, be given, wc have the RULE. Multiply the base by the perpcndicula:', and take half t/ie pro^' duct ; or, multiply half the one by the other. 2. If the three sides are given RULE. From half the sum of the sides subtract each side successively, and the square root of the continual product of the half sum, and these three remainders loill be the area. 300 AltlTITMKTIC. Kxprcssod iilj^cbraically tliiH :»rc!i=:j/«(j» — fi)(« — &)(*— c). K X !■; i: (• I s i; s . 11. Wluit is tlio iiruii of a tiiaii;:;k', the base of wliieli u 17 inches, ftml the ultituili! 12 iiiehes? Ana. 10- H(juare inches. 12. What is the urea of a trlani^iihir ".'urden, the length of wliich is -lli rods, und tlie breadth ID rods? Ans, l.'iT .scjuare rods. l;{. Find how many aeres, &o., are in a trianguhir held, tjjc length of which is 4'J.75 rod.s, and the breadth '.il^ rods. Aus. 5 acres, 1 rood, 18,''j rods. It. The area of a trianguhir inclosurc is 150 square rods, and the base is 30 linear rods; what is the altitude ? Ans. 10 rods, 15. The urea of a triangle is 400 rods, and the altitude 40 rods, what is the base ? Ans. 20 rods. 10, Three trees are so planted that the linos joining them form a right angled triangle ; tlie two sides containing tiie right angle are 33 and 50 yards; what is the area in square yards ? Ans. 1)24, 17, Lot the position of the trees, as in the last example, be represented by the tri- angle A B C. and let the distance from A to IJ be 50 rods, and from 1) 'a> C 30 rods. lloquircd the area, — (See l']uelid I. 47.) Aus. 000 square rods. 18, In the figure annexed to 17, suppose A I> to represent the pitch of a gallery in a church, inclined to the ground at an anglo o[ 45" ; how many more persons will the gallery contain than if the seats were made on the flat B C, suppos^ing B C to be 20 feet and the froutagc GO feet in length ? Ans. None. ' luestion and the next to correct a common misapprehension on this point. Because the distance IVft'ni 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 pi'cciscly the same number of seats can bo made, and the same number of per- sons accommodated on B C t>s on A B. We have introduced th' >[KNSLll.VilUN. 'M'i 19. If B C bo halt' llio bxso ot ti hill, uiul A IJ one of its slupin^ Bidcfl, and H (y • -i<> yanl^i. ami A IL -.')() yards ; how many luoro rcnvH of trees cun bo planlcd on A It, than (', :it 1 yard aitart? Ana. None, because the trees bohii^ id! iK'rpendiciilar to tin; horizon, are puralli'l to eacdi otiii'i as rcp'.i'sonted by tlio vertical lines in tlu; last figure. 20. How many acres, «S:i'., are tli(M'e in a tri mj^ular lii'ld ol' which tho perpendicular len;^th and breadth are lli chains, 7i» links and !• chains, I'.l links? Atis. G acres, roods, -h rods. 21. A ship was stranded at a distance of 10 yards from tlus base of a cliff l{<) yards hij^h ; what was the length of a cable which reached from the top of the cliff to the ship? Ans ;")<• yds. -2. A cable 1*)0 yards \(Jn<^ was passed from tlie how to the stern of a ship tiirou^h the cradle of a mast placed in midships at the height of .'>() yards ; what was tho length of the ship ? Ans. SO yards, 23. A man attempts to row a boat directly across a river 20') yards broad, but is carried SO yards down the stream by tho current ; through how m:my yards was ho carried ? Ans. 215.4-1 yards. 2i. Let the three hides of a triangle bo 30, 10, 20; to tind tho area in square feet. Ans. 290.17.'»7 S(juarc feet. 25. What is the area of an isosceles triangle, each of the crjual s.idcs being 15 feet, and tho base 20 feet ?'^- Ans. llLSOii hij Itx*. 20. What is the area of a triangular space, of which tho ba.se is 50, and the hypotcrmsc 05 yards ? Ans. 924 square yards. 27. What is tho area of a triangular clearing, each side of v.liieh is 25 chains ? Ans. 27.0032 acre-'. 28. What is the area of a triangular clearing, of which the three sides arc oSO, 420 and 705 ? Ans. 9 acres, 3'7-?r perches. 29. A lot of ground is represented by the three sides of a right nngled triangle, of which tho hypotenuse is 100 rods, and the base CO rod."! ; what is the area? Ans. 15 acres. 30. What is tho'area of a triangular field, of which the sides are 49, 34 and 27 rods respectively ? Ans. 2 acres, 3 roods-j . 31. AVhat is the area of a triangular orchard, the sides of which arc 13, 14 and 15 yards ? Ans. 84 square yards. 32. Three divisions of an army arc placed eo as to be represented * This question, and some others may be solved by either rule, aad it Wi!l be foimd a good exercise to solve by botli, I ' H: l^'t 308 ABITHMETIC. m lib' by threo sides of a triangle, 12, 18 and 24 ; how many square miles do they guar*', 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 side 30 feet high ; what was the breadtli of the street ? Ans. 76. 1 23 -f feet. rnoBLEM III. To find the area of a regular Polygon. 1. When one ot 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 by half the other. 2. When a side only is given. Multiply the square of the side by the number found opposite the number of sides in the subjoined table. Note. — This table shows the area when the side is uniy ; or. Avhich is the dame thing, the square is the imit. 3 4 5 6 7 8 9 10 11 12 Triangle Square Pentagon Pexagon Heptagon Octagon Nonagon Decagon Ileredecagon. Dodecagon... 0.4.330127. 1.0000000. 1.72:4774. 2.59S07G2. 3.0339125. 4.8284272. G.1818241. 7.0942088. 9.3G5G395. 11.1901524. 34. If the side of a pentagon is 6 feet and the porpendicular 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 line dr-awn from the centre to any angle be 5 feet, what is the area ? Ans. 72 sq. feet. MENSURATION. 309 37. The side of a decagon is 20.5 rods ; what is the area ? Ans. 20 acres, roods, 33.5 rods, nowly. 38. A hexagonal table has each side GO inches, and a line from Ihe centre to any corner is 50 inches ; how many square feet in the surface of the tabic ? Ans. 38 feet, 128 inches. 39. What is the area of a regular heptagon, the side being 191 g and the perpendicular 10 ? Ans. G78.3. 40. An octagonal enclosure has each side G yards, what is its area ? Ans. 3 acres, 2 roods, 14 rods. 19 yards. 41. Five divisions of an army guard a certain tract of country — each line in 20 miles ; how many square miles are guarded ? Ans. G88.2, nearly. 42. Find the same if there are 6 divisions, and each line extends 5 miles ? Ans. 64.95+ miles. 43. The area of a hexagonal table is 73f feet; what is each side ? Ans. 6J feet. PKOBLEMIV. To find the area of an irregular polygon. Divide it into triangles hij a perpendicular on each diagonal from the opposite angle. Find the area of each triangle separately , and the sum of these areas toill be the area of the trr.jjrzium. Note. — Either the diagonala and perpendiculars must be given, or data from which to And Ihein. 44. The diagonal extent of a four-sided field is G5 rods, and the perpendiculars on it from the 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 is the area ? Ans. 17G.03125 sq. feet. 46. Required the area of a six-sided figure, the diagonals of which are as follows : the two extreme ones, 20.75 yards and 18.5, and the intermediate 27.48 ; the perpendicular on the first is 8.G, on the second 12.8, and those on the intermediate one 14.25 and 9.35? Ans. 531.889 yards. 47. If the two sides of a hexagon be parallel, and the diagonal parallel to them be 30.15 feet, and the perpendiculars on it from 21 ( ■■ i 310 iUIITHJIETIC. the opposite anglos .-^re. on the left, 10. 5G, and on the riglit 12.24 and the part of the diagonal cut off to the left by the first pcrpendi- cultf?, 8.2G, and to the right by the second, 10.1-1; on the other side, the perpendicular and segment of the diagonal to the left are 8.5G and 4.54, and on the right 9.26 and 3.93 ; what is the area? Ans. 470.4155 sq. feet. TROBLEJI V. To find the area of a figure, the boundaries of which arc partly right lines and partly curves or salients. Find the average breadth hi/ taking several 2'^crpendlculars from the nearest and most remote 2^ bits, from a fixed hase, axd dividing the sum of these hi/ their numhcr, the quotient, multiplied hi/ the length, will he a close aj^proximation to the area. Let the perpendiculars 9.2, 10.5, 8.3, 9.4, 10.7, their sum is 48.1, then 48.1-^-5=9.02, and if the base is 20, we have 9.02X20=. 192.4, the area. When practicable, as large a portion of the space as possible should be laid oif, so as to form a regular figure, and the rest found as above. A field is to be measured, and the greater part of it can be laid off in the form of a rectangle, the sides of which are 20.5 and 10.5, and therefore its area is 215.25, and the offsets of the irregular part ;;re 10.2, 8.7, 10.9, and 8.5, the sum of which, divided by their number, is 7.G6, and 7.GGX-0-5=157.03, the area of the irregular 2)art, and this, added so the area of the rectangles, gives 372.28, the whole area. 48. The length of an irregular clearing is 47 rods, and the breadths at G equal distances are 5.7, 4.8, 7.5, 5.1, 8.4 and 6.5; what is the area ? Ans. 1 acre, 1 rood, 29.86 rods. p R, B L E JI V I . To find the circumference of a circle when the diameter is known, or the diameter when the circumference is known. -•'- The most accurate rule is the well-known theorem that the diameter is to the circumference in the ratio of 113 to 355, and * In strictaoss the circnmfcrcnco and diameter are not like quantities, but we may suppose that a cord is stretched round the circnml'orenco, and then drawn out into a straight line, and its linear units compared witli those of the diameter. : -- 't ; ■- .- ^ ^ j^' '■' . ,;-fll»NlJ.«J: PILINa OF BALLS AND SHELLS Balls are usually piled on a base which is either a triangle, or square, or rectangle, each side of each course containing one ball less than the one below it. If the base is an equilateral figure, the vertex of a complete pile will be a single ball ; but if one side of the base be greater than the contiguous one, the vertex will be a row of balls. Hence, if the base be an equilateral figure, the pile will be a pyramid, and as the side of each layer contains one layer less than the one below it, the number of balls in height will be the same as the number of balls in one side of the lowest layer. It the pile be rectangular, each layer must also be rectangular, and the number of balls in height will be the same as the number in the less side of the base. If the base be triangular, we have the \i 31i ARITHMETIC. RULE. MuUij)hj the number on one side of the bottom row by itself VLXJS one, and the product by the same base row PLUS tioo^ and divide the result by six. For iv coinploto square pile we have the RULE . Multiply the number ../balls in one side of the lowest course by itself 2Li:s one, and this jyroduct by double the first mxdtiplicr plus one, and take one-sixth of the rcsxdt. If the pile be rectangular, we have the R u l e . 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. If the pile be incomplete, find what it woidd be if complete ; find also what the incomplete one xcoxdd be as a separate pile, and sub- tract the latter from the former. ex ercises. 64. In a complete triangular pile each side of the base is 40 ; how many balls are there ? Ans. 11480. 65. In each side of the base of a square pile there are 20 shells ; how many in the whole pile ? Ans. 2870. 66. In a rectangular pile there are 59 balls in the length, and 20 in the breadth of the base ; how many are in all ? .ins. 11060. 67. In an incomplete triangular pile, each side of the lowest layer consists of 40 balls, and the side of the upper course of 20 ; what is +ho number of balls ? • Ans. 10150. Note.— Since the upper course is 20, the first row in the wanting part woulfl be 19. JIEASLTiEMENT OV TDLDEll. 315 MEASUREMENT OF TIMBER. Timber is measured sometimes by the square foot, and sometimes by the cubic foot. Cleared timber, such as planks, beams, &c., are usually measured by the square foot. What is called board measure is a certain length and breadth, and a uniform thickness of ohc inch. Large quantities of round timber arc often estimated by the ton. To find either the superficial extent or board measure of a plank, &c. RULE. Multiply/ the length in, feet hy the breadth in inches, and divide by 12. Note. — The thickness being taken uniformly as one inch, tlie rule for find- ing the contents in square feet becomes the same as that for finding surface. If the thickness bo not an inch, — Multiply the board measure by the thickness. If the board be a tapering cue, take half the sum of the two extreme widths for the average width. If a one-inch plank be 24 feet long, and 8 inches thick, then we have 8 inches equal § of a foot, and § of 24 feet=:lG feet. A board 30 feet long is 26 inches wide at the one end, '^ud 14 inches at the other, hence 20 is the mean width, i. e., 1§ feet, and 30X1^=^0 ; or, 30x20=000, and 000^12=50. To find the solid contents of a round log when the girt is known. RULE. Mxdtiply the square of the quarter girt in inches by the length in feet, and divide the jyrodtcct by 144. If a log is 40 inches in girt, and 30 feet long, the solid contents will be found by taking the square of 10, the quarter girt in inches, which is 100, and 100x30=3000, and 3000-f-144=20g. To find the number of square feet in round timber, when the mean diameter is given. It: i *.i 816 ARITHMETIC. RULE. Multiply the diameter in inches h\j half the diameter in inches^ and the nroduct by the length infect, and divide the result by 12. If a log is 30 feet long, and 5G inches mean diameter, the number .-of square feet is 50x28x30-^-12:^3920 feet. To find the solid contents of a log when the length and mean diameter arc given. RULE . Multiply the sqiiare of half the diameter in incites by 3.141 G, and this product by the length in feet, and divide by 114. 68. How many cubic feet arc there in a piece of timber 14x18, and 28 feet long ? Ans. 49-|-cubic feet. 69. How many cubic feet are there in a round 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 4-cubic feet. 71. How many feet, board measure, are there in a log 23 inches in diameter, and 12 feet long ? Ans. 264^. 72. How many feet, board measure, arc there in a log, the diameter of which is 27 inches, and the length 16 feet. Ans. 480. 73. What are the solid contents of a round log 36 feet long, 18 • inches diameter at one end, and 9 at the other ? i bo api so ^!/ 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 ? An.=5. l^^g cubic feet. 70. What arc the solid contents of a board 20 feet long, 20 inches broad, and 10 inches thick? Ans. 27| feet- 77. What is the solid content of a piece of timber 12 feet long, 16 inches broad, and 12 inches thick ? Ans. 16 feet» 78. How many cubic feet are there in a log that is 25 inches in diameter, and 32 feet long ? 79. How many feet, board measure, does a log 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 thai, is 12 by 16 inches, and 47 feet in length ? Ans. 62|.. MEASUREMENT OF TIMBER. 317 81. IIow many feet, board measure, arc there ir. 22 one-inch boards, each being 13 inches in width, and IG feet in length ? Ans. 3811. BALES, BINS, iO. As bales are usually of the same form as boxes, the same rule applies. 82. Ilencc, a bale measuring 4J inches in length, 33 in width, and 3^ in depth, is, in solid content, 37^ feet. 83. A crate is 5 feet long, 4.J broad, and 3^.^ deep, what is the solid content ? To find how many bushels are in a bin of grain : Ans. 85/g. RULE. Find the 2>^'oduct of the length, breadth and depth, a\d divide by 5150.4. 84. A bin consists of 12 compartments ; each measures C feet 3 iijches in length, 4 feet 8 inches in width, and 3 feet 9 inches in depth ; how many bushels of grain will it hold ? Ans. 1055, nearly. To find how many bushels of grain are in a conical heap in the middle of a floor : •I RULE. Multipli/ the area of the base by one-third the height. The base of such a pile is 8 feet diameter and 4 foot 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 agaiuKt a wall take half the above result. If it be heaped in a cornei-, take one-fourth the above result. if 21 i 318 AKITILMETIC. MISCELLANEOUS EXERCISES. 1. What number is that ^ and I of which make 255 ? Ann. 201-i'^,j. 2. What must be added to 217^^, that the sum may be 17.^ time.s 10.V? Ans. IISJ. i]. What sum of money must be lent, at 7 per cent., to accumu- late to S-155 interest in :J months? Ans. $26000; 4. Divide SIOOO among A, B and C, so that A may have S15G more tlian B, and B $02 less than C Ans. A. 641G5 ; B, 8200^; C, $322§. 5. Where shall a pole 00 feet liigh be broken, that the top may vest on the ground 20 feet from the stump ? Ans. 26§ feet. 0. A maiv bought a liorse for $08, which was •) as much again as he sold it for, lacking 81 ; how much did he gain by the bargain? Ans. 012.50. 7. A fox is 120 leaps before a hound, and takes 5 leaps to the hound's 2; but 4 of the hound's leaps equal 12 of the fox's ; how many leaps must the hound take to catch the fox ? Ans. 240. 8. A, B and C can do a certain piece of work in 10 days ; how long will it take each to do it separately, if A does 1-V times as much as B, and B docs ^ as much as C ? Ans. A, 30 days ; B, 45 ; C, 22^. 9. At what time between five and six o'clock, are the hour and minute hands of a clock exactly together ? Ans. 27 min., IGy*!- sec. past 5. 10. A courier has advanced 35 miles with despatches, when a second starts with additional instructions, and hurries to overtake the first, travelling 25 milos 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 ?, — ,'s+r,^?— t/s+^Vs— &«. ? Ans. ^fg. 12. If a man earn $2 more each month than he did the month before, and finds at the end of 1 8 months that the rate of increase will enable him to car: the same sum in 14 months ; how much did he cam iu the whole time ? Ans. $4032. 13. How long would it take a body, moving at the rate of 50 MISCELLANEOUS EXEUCISES. nis milga an hour, to pass over u wpaco equal to tho distaneo of tho earth Jroui tho sun, /. c, Do millions of miles, a year beinjj; oG.') days ? Ans. 21(5 years, .'i2() days, IG liours. 14. Two soldiers start to^ctlicr for a eertain fort, and one travels 18 miles a day, and after travellini^ days, turns back as fi;r as tho second had travtHcd duriii;^ those I) days, ho then turns, avd in 21^ days from tho tini(> they started, arrives at the ibrt at tho saiii(> tine iis his comrade ; at what rate did the second travel ? Ans. 18 miles a day. 15. What quantity must "oe subtracted from tho square of 48, .so that tho remainder may be tho product of 54 by 10? Ans. 1440. IG. A father gave j] of his farm to his son, tho son sold § of his share for $12G0 ; what was tho value of tho whole farm ? An.s. §5040. 17. There were ^ of a flock of sheep stolen, and G72 Avcrc left : how many were there in all ? Ans. 1792. 18. A boy gave 2 cents each for a number of pears, and had 42 ceuts left, but if ho had given 5 cents for each, ho would liavo had nothing left, llequired the number of pears. Ans. 14. 1 19. Simplify 1+.- 2H-i 2' Ans. 20. A man contracted to perform a piece of work in GO days, he employed 30 men, and at tho 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 tho 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 5G43, 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 G inches in circui!iference, and the fore wheels 5 feet 7^- inches ? Ans. 2371 G. 24. What is the area of a right angled triangular field, of which the hypotenuse is 100 rods and tho ba.sc GO? Ans. 2400 sq. rds. r,) oi Al \ r,iy ; what are the numbers? Ans. I]G(i4^ and 1GG5L .10. A person bcinj; asked tiie hour of tlio day, replied that the time past noon was equal to one-tifth of tiie time p.ist midnight; what was the time? Ans. .'{ P.M. 31. A snail, in getting up a polo 20 feet high, climbed up S feet every day, but slipped back *1 feet every night ; in what time did he rcacii the top ? Ans 4 days. 32. What number is that who.so ^, -J, and \ parts make 48? Ans. 44,\j. 313. A merchant sold goods to a certain amount, on a commission of 4 per cent., and, liaving remitted the net proceeds to the owner, received ^ per cent, for immediate payment, which amounted to ^15.G0; what was the amount of his commission ? Ans. $260. 34. A criminal has 40 miles the start of the detective, but the detective makes 7 miles for 5 that the fugitive makes ; how far will the detective have travelled before ho overtakes the criminal ? Ans. 140 miles. 35. A man sold 17 stoves for $153; Ibr the largest size he received $19, for the middle size $7, and for the small size 6G ; how many did he sell of each size ? Ans. 3 of the large size, 12 of the middle, 2 of the small. 30. A merchant bought goods to the amount of $12400 ; S40G0 of which was on a credit of 3 months, $41 GO on a credit of 8 months and the remainder on a credit of 9 months ; how much ready money would discharge the debt,, money being worth G per cent. ? An.s. $12000. 37. If a regiment of soldicr.s, consisting of 1000 men, are to be clothed, each suit to contain 3f yards of cloth that is If yards wide, and to be lined with flannel 1|- yards wide ; how many yards will it take to line the whole ? Ans. 5G25. 38. Taking the moon's diameter at 2180 miles, what are the solid contents? Ans. 5424G17475-J- sq. miles. MISCELLANEOUS EXERCISES. :j21 39. A certain island h 7'1 miles in circumference, and if two men start out from the Humc point, in the Humu direction, the one walkinj^ ut the rate of 5 and the other ut the rate of li miles an hour; in what time will they come toj^ether ? Ana. 30 hour.s, uO minutes. 40. A circular pond measurcH half an acre ; what lenj,'th of cord will bo required to reach from tlie edge of the pond to the centre i* Ans. 832(J:}-|- feet. 41. A gentleman has deposited $150 for the benefit of hi>i son, in a Savings' Bank, at compound interest at a half-yearly rate of IJJ^ per cent. Ho is to receive the amount as soon as it becomes $1781.06^. Allowing that the deposit was made when the aon was 1 year old, what will be his ago when he can come in possession of the money ? Ana. 21 years. 42. The select men of a certain town appointed a li(juor agent, and furnished him with liquor to the amount of $825.00, and cash, $215. The agent received cash for liquor sold, ^1323.40. He paid for liquor bought, $937 ; to the town treasurer, $300 ; sundry ex- penses, $29 ; his own salary, $265 ; he delivered to indigent persona, by order of the town, liquor to the amount of $13.50. Upon taking stock at t'le end of the year, the liquor on hand amounted to $G1G.50. Did the town gain or lose by the agency, and how much ; has the agent any money in his hands belonging to the town ; or does the town owe the agent, and how much in either case ? Ans. The town lost $103.20 ; the agent owes the town $7.40. 43. A holds a note for $575 against B, dated July 13th, paya- ble in 4 months from date. On the 9th August, A received in advance $G2; and on the 5th September, $45 more. According to the terms of agreement it will be 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; ho\v much must B pay on the 3rd of October ? Ans. $111.43. 44. A accepted an agency from B to buy and sell grain for him. A received from B grain in store, valued at $135.60, and cash, $222.10 ; he bought grain to the value of $1346.40, and sold grain to the amount of $1171.97. 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. I 322 ARITHMETIC. fl 45. A gcncrf^l ranging liis men in the form of a square, had 59 men over, but luiving increased the side of the s([uarc by one man, he hicked 84 of completing the square ; liow many men had ho ? Ans. 5100. 46. What portion, expressed as a common fraction, is a pound and a half troy weight of three pounds avoirdupois ? Ans. -pTfj. 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 Gj acres of wheat in 2^ days, by working 8^ hours per day, how many acres will 15 men, working equally, reap in 3| days, working 9 hours per day ? Ans. 40 j ^ days. 49. Out of a certain quantity of wheat, ^ was sold at a certain gai. per cent., ^- at twice that gain, and the remainder at three times tlie gain on the first lot ; what was the gain on each, the gain on the whole being 20 per cent.? Ans. O?, 19 ^ and 281 per cent, 50. If a man by travelling G hours a day, and at the rate of 4^ miles an hour, can accomplish a journey of 540 r^'les in 20 days ; liow many days, at the rate of 4 j miles an hour, will he require to accomplish a journey of GOO miles ? Ans. 21^. 51. Smith in Montreal, and Jones in Toronto, a2;rec to exchange operations, Jones chiefly making the purchases, and Smith the sales, the profits to be equally divided ; Smith remitted to Jones a draft for $8000 after Jones had made purchases to the amount of $13682.24; — Jones had sent merchandise to Smith, of which the latter had made sales to the value of $9241.18 ; Jones had also made sales to the worth of $2836.24 ; Smith has paid $3C4.16 and Jones $239.14 for expenses. At the end of the year Jones has on hands goods worth $2327.34 and Smith goods worth $3123.42. The term of the agreement having now expired, a settlement is made, what has been the gain or loss ? AVhat is each partner's share of gain or loss ? What is the cash balance, and in favor of which partner ? 52. In a certain factory a number of men, boys and girls arc 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 thiin each boy, and each boy a third more than each girl ; the sum paid each day to all the boys is double the sum paid to all the girls, and f<»r every five shillings earned by all the boys each day, twelve shillings are earned by all the men ; it a^/^' fe MISCELLi'i^'EOUS EXERCISES. 323 is required to find the number of menj the number of boys and the number of girls, the whole number being 59. Ans. 24 men, 20 boys and 15 girls. 53. A holds B's note for §575, payable at the end of i months from the 13th July ; on the 9th August, A received $G2 in advance, as part payment, and on the 5th September 64:5 more ; according to agreement the note will not be due till IGth November, three days of grace being added to the term ; but on the iJrd October B tenders such a sum as will, together with the payments already made, ex- tend time of payment forty days forward ; how much must B pay on the 3rd of October ? Ans. $111.43. 54. If a man comnjcnce business with a capital of $5001) 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 yeiirs ? Ans. $27910. 55. A note for $100 was to come due on the 1st October, but on the 11th of August, the acceptor proposes to pay as much in ad- vance as will allow him GO days after the 1st of October to jniy tlio 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 one he gave one half of the whole and half a dollar ; to a second half the remainder and half a dollar ; to a third half the re- mainder and half a dollar; and also to the fourth half the reiauinder and half a dollar, together with one dollar that was left ; how much did he give to each ? ■ Ans. To the first, 81G ; to the second, $8 ; to the third, §4 ; to the fourth, $3. 57. A farmer being asked how many sheep he had, replied that he had them in four different fields, and that two-thirds of the num- ber in the first field was equal to three-fourths of the number in the second field ; and that two-thirds of the number in the second field was equal to three-fourths of the number in the third field ; and that two-thirds of the number in the third field was equal to four- fifths of the number in the fourth field; also that there were thirtv- two sheep more in the third field than in the fourth ; how many sheep were in each field and how many altogether ? Aus. First field, 243; second field, 21G; third field, 192) fourth field, IGO. • Total. 811. 'i (i 324 ARITHMETIC. 58. How many hours per day must 217 mea work lor 6^ aays to dig a trench 23J^ yards long, 3^ yards wide, and 2J deep, if 24 men working equally can dig one 33| yards long, 5| wide, 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 remainder, together with $975 ; to the youngest one-fourth of the remainder and 01400 ; he gives his wife a life interest in the remainder, and her share is found to be one-fifth of the whole ; what was the amount of the property ? Ans. $20,000. GO. Five men formed a partnership which was dissolved after four years' continuance ; the first contributed $60 at first and $800 more at the end of five months, and again $1500 at the end of a year and eight months; the second contributed $600 and $1800 more at the end of six months ; the third gave at first $400 and $500 every six months ; the fourth did not contribute till the end of eight months ; he then gave $900, and the same sum every six months ; the fifth, having no capital, contributed by his labor in keep- ing the books at a salary of $1.25 per day ; at the expiration of the partnership what was the share of each, the whole gain having been $20000 ? Ans. 1st, $2019.65 nearly; 2nd, $4871.81 nearly; 3rd, $4815.81 nearly ; 4th, $646.74 nearly; 5th, $1825.00. 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 per cent, more than C, and C is to have 8 par cent. more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 feet high, and 12 feet square at its base, it being an exact pyramid ; and it is agreed that A shall take his share first from the top of the stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sum shall each pay ? Ans. A. takes 13.22-|-ft., and pays $28.93^; B takes 3.14+rt., and pays $25.83^; C takes 2.06+ft., and pays $23.18^ ; D takes 1.58+ft., and pays $21.74J. 62. A merchant boudit 500 bushels of wheat and sold one half of it at 80 cents per bushel which was 10 per cent more than it )f MISCELLANEOUS EXERCISES. 325 <50st ; m nil 5 par ant. Ic-s^ than ho asked for it. ITo sold tlic rema idei at 12i- par coat, moro tlian it cost him. What was his askiiipc pnc3 for both lots? What did ho receive for the last lot, ati' ' much did he gain on the wliole ? • 63. May 1st, L882, I got my note for §2000 payable in 4 months discounted at a binlc, and imraadiately invested the money received in woodland. November 9th, I sold the land at an advance of 15 per cent., receiving ': of the price in cash, and a note for the remainder, payable August 10, 18G4, without grace, and to be on interest after January 1, 1834, at 7 per cent. I lent the cash re- ceived at G par cent. When my note at the bank became due I renewed it for the same time as before, and at the proper time renewed it again, and finally renewed it for such a time that the note would become duo August 10, 1804. Now, if I paid G per cent, on the money borrowed at the bank, and made a complete settlement 'Vugust 10, 1864, what was the amount of my gains ? G4. My agen.. at Mobile buys for me 500 bales of cotton, avci aging 500 lbs. \jcr bale, at 1.0 cents per pound. I pay him 1-^ per cent, on the amjunt paid for the cotton, and shipping charges at 60 cents from January 1 for an amount sufficient to pay for the cotton, charges and c mmlssion including also 2 per cent, discount on the draft. On the receipt of the invoice, I insure for the amount of the draft plus 10 per cent. ; I pay 1^ per cent, premium on the amount insured, an from the amount of the premium is discounted 1\ per cent, for cash. On the arrival of the cotton I pay ^ of a cent per pound for freight, and 5 per cent, primage to the captain on the freight money, and also 4 cents per bale for wharfage. I sell it on the wharf, January 20, at $1 per bale profit, and agreed to take in payment the note of the purchaser for G months from January 20. What amount would bo received on the note when discounted at a hfrk at 7 per cent. ? 22 320 AlUTHMETIC. FOREIGN GOLD COINS, MINT VALUE. COCfNTRY. DE.VOMINATIOXS. Anstrulla . . . .( Austria lu'l^^ium . . . IJolivia ]]rnzil Ceuirl America Cliili DiMimark. , . Kqiuulor.. . . England . . . France Germany, North Germany, Sontli Greece Ilindostan Italy Japan Mexico., Naples Netherlands. . . New Granada. Peru Portugal Prussia .. Rome ... !'.lussia . . Spain . . . , Sweden . Tunis . . . Turkey . Tuscany jPoundof 1852 [Sovereign 1855-GO iDucat iSouverain iNewUnionCrown (ossumcd) iTwenty-flve francs {Doubloon 20 Milreis Two cscudos Old doubloon Ten Pesos Ten thaler Four eseudos Pound or Sovereign, now. . Pound or Sovoriou. average Twenty Irancs, new Twenty Irancs, average. . . Ten thaler Ten thaler, Prussian Krone [crown] Ducat Twenty drachms Mohur 20 lire Old Cobang New Cobang Doibloon, average '• new Six ducati, new. Ten fjuilders Old Doubloon, Bogota Old Doubloon, Popaj'an. . Ten ])esos, new Old doubloon Gold crown NewUnion Crown [araumcd] 2^ .scudi.new Five roubles 100 reals SO reals Ducat 25 piastres 100 piastres Sequin TVEionr. FI.NE- NK8& Oz. Deo. Tnors. 0.281 916.5 0.256.5 916 0.112 986 0.363 900 0.357 900 0.254 899 0.867 870 0.575 917.5 0.209 853.5 0.867 870 0.492 900 0.127 sf)r> 0.433 841 0.25G.7 916.5 0.256.2 916 0.207.5 899.5 0.207 899 0.427 895 0.427 903 o.:;57 900 0.112 986 0.185 900 0.374 916 0.207 898 0.362 568 0.2S9 672 0.867.5 866 0.867.5 870.5 0.245 996 0.215 899 0.868 870 0.867 S58 0.525 891.5 0.867 868 0.308 912 0.357 900 0.140 900 0.210 916 0.2G8 896 0.215 869.5 0.111 975 0.161 900 0.231 915 0.112 999 VALUE. $5.32.37 4.85.58 2.28.28 6.75.35 6.64.19 4.72.03 15.59.25 10.90.57 3.68.75 15.59.26 9.15.35 7.90.01 7.55.46 4.86.34 4.84.92 3.85.83 3.84.69 7.90.01 7.97.07 6.C4.20 2.28.28 3.44.19 7.08.18 3.84.26 4.44.0 3.57.6 15.52.98 15.61.05 5.04.43 3.99.56 15.61.06 15.37.75 9.67.51 15.55.67 5.80.66 6.64.19 2.60.47 3.97.64 4.96.39 3.86.44 2.23.72 2.99.54 4.36.93 2.31.29 Valno after Ocductiou. §5.29.71 4.83.16 2.27.04 6.71.98 6.60.87 4.69.67 15.51.40 10.85.12 3.66.91 15.51.47 9.10.78 7.86.06 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 15.45.22 15.53.25 5.01.91 3.97.57 15.53.26 15,30.07 9.62.68 15.47.90 5.77.70 6.60.87 2.59.17 3.95.60 4.93.91 3.84.51 2.22.CI 2.98.05 4.34.75 2.30.14 FOIJEION SILVER COiXS. 327 FOREIGN SILVER COINS MINT VALUE. COUNTRY. DENOMISATIOSS. WKIGHT. i FI.NEJfESS. VALL'E. 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 0.927 0.182.5 0.178 0.800 0.712 0.595 0.340 0.3f0 0.719 0.374 0.279 0.279 0.867.5 0.866 0.844 0.804 0.927 0.803 0.866 0.766 0.433 0.712 0.595 0.864 0.667 0.800 0.166 1.092 0.323 0.511 0.770 0.220 TlIOUL. 833 902 833 900 900 838 897 903.5 GG7 918 5 925 850 90S 900.5 877 924.5 925 900 750 900 900 90,) 900 916 991 890 903 901 830 944 877 896 901 909 650 750 900 90O 875 900 89!) 750 809 8!)S.5 830 925 $1.02.27 II Old scudo 1. 02.64 II Florin before 1858 New florin 51.14 II 48.G3 II New Union dollar Maria Theresa dol'r,1780 73.01 11 1.02.12 Bel"i Til. . Five francs 98.04 Bolivia , . Xew dollar 79.07 I. Half dollar 39.22 Brazil Double Alilrii.'' 1.02.5;? Canada, 20 ceatii 18.87 Ceniral America. . . Dollar 1.00.1!) Chili . . . : OldDtollar 1.06.79 II New Dollar 9,S.17 Denmark Two rigsdaler 1.10.65 England Shilling, new 22.96 II Shilling, average Five franc, average Thaler. Ijolbro 1857 New thaler 22.41 France Germany, Norlli. .. II 98.00 72.67 72.89 Germany, South. . . Florin, before 1857 Xew florin [assumed] . . . Five drachms Rupee 41.65 41.65 Greece Hindostan. . . . 88.08 4'!.62 Japan Itzebii :!7.63 ti New Itzebu 33.80 Mexico , Dollar, new Dollar, average Scudo 2 J guild 1.06.G2 II 1. 00.20 Naples Netherlands 95.34 1.03.31 Norway Specie daler Dollar of 1857 1.10.65 New Granada 97.92 Peru Old dollar 1.06.20 11 Dollar of 1858 94.77 .; Half dollar, 1835-38.... Thaler before 1857 New thaler 38.31 Prussia 72.08 « 72.,«9' Rome Scudo 1.05.84 Russia Rouble. .• Five lire 79.44 Sardinia !)S.0O Spain New ])istareeu 20.31 Sweden Rix dollar 1.11.48 Switzerland Two francs 39.52 Tunis Five piastre Twenty piastres 62.4 !• Turkey 86.98 Tuscany Florin 27, GO LAWS OF THE UNITED STATES IlELA.rrNO TO INTEREST, DAMAGES ON BILLS, A.ND TIIK COLLl'XTION OF DEBTS Tlio following brief sketches uftlio laws of the riifloirnt .States of ;lio t'liinii, will he louuil useful, iioi, only to biisiucss inei but also to piivato Individuals. Tlio inl'oriiuition on which they iiris Ibiiuucil, has been derived from authcDlic sourc s and condensed into :i convenieut epitome, which may bo relied upon as correct r . regards tl;o present state < t' tho law, which is all that any ono can bo nnswerable for, its nltorations may hereafter bo niado ou sorao points. ALABAMA. Inlerci!. — Tho rate of interest in A\ bania i ! ei:iht per cent, per annum. Vcnailij fur Viutdtinn (if the I'f.iirij I.au-.i — .Ml <'nnt|ar.ls niado at a higher rat'< of intere :t tiian i'i;4ht jier <:eut. aro u.=:uriuus, and canu'it br- enlnrci'd exc I'pl. a;; to tlio priiinpal. Damages cii lUlls. — Damages on iulan 1 bills of ex ,,anL;c5 protesicd tor iiou-piiyniont, uro 5 per Vent. ; on foreign biUs of e.\Tluin«e 10 percent, on tho sinn drawn lor. All bills drawn and p.iyablo within tlii.s Maio aro termed inland bilLs ; thnso drawn in this Stato and pa\abio elsewhere, are con; idcrcd Corel jn bills. Sig!it Biiln. — Grace i.s allowed on bills, drafts, etc., payable at .'-ight. Cuiltctioii of Debli. — Original uttaclnr.ont.s, foreign and donie.^ ic, aro issued by judges of th'j circuit or county courts, or .justices of tho peace. An atlaclinient may issue, allhoujjh tlio tU^bt or demand of iho pluintilf be not dun; and .'•hall bn a lien on the iiropiriy all. 'died until tho debt ordeinaud becomes due, when .jud 'mont .sh dl bo rendered and e.\e iilion issued. A nourc.'-idi'nt plain. ilfmay have an ailacliinent a.Liiin-t the inoperty of a nnn-iTsident defend nnt, provi'ied he jjve..- L;ood a:ul .^utlicieut resident security in tbo reriuircL bond. n':alcinf.; oaih tliat the (lelou'lant has i;ot sulllcient property within tlie Slate of delendant'.-i icsideuco to gatisfy tlio debt or demand. ARKANSAS. Ininrst. — The legal rate of interest in Arkan.sas is six per cent. Pjiecial contracts in writing ■will admit an iiitiest not to exceed ten perieir. All Jud.gmenls i/r decrees upou contracts bearing more than six per cent shall bear the same rate of interest originally agreed upon. — (Gould'.s Digest, chap. 1)2, sec. 1, 2, kc, ISM.) rcnaltjijor Viotation of the Usury Laivs. — All contracts for reservation of a „reaier ra'.o of interest than ten per cent aro void. Tho excess taken or charged beyond ten per cent, way bo recovered back, jirovided tho action for recovery shall be b; ought witlnn oiio year afterp;iy- niout. (lb. sees. C & 7 ) Vanwpi^s on IjUIs — Tho damages on Bill.s of Exchange drawn or n:'got;aied in Arkan.sas, expresses to bo for value recei\ od, and protested for uou-acccptanco, or fur non-payment af^.er uou-acceptance, aro as follows. — (lb. chap. 25.) 1. If payable within tho State, 2 per cent. 2. If iwyablo in Alabama, Louisiana, MLssi-ssippI, Tennesson, Kentucky, Ohio, Indiana, Illinois or i.issouri, oi .U any point on tlie Ohio Uiver, 4 per cent. 3. If payable in any other . ta o or territory, 5 per cent. 4. If payable wiihi i either of tho United Stales, and protesicd for non-payment, after acceptance, percent. Foreign Jh Is. — The d linages on bills of exchange, expressed for value received, and pay.iblo beyond tho irnits of ibo Uniteil Slates (!b. chap. 2J), aro 10 p.T cent. Sight Bills — .hero is no slauito in f i\:c in Aruansas in ref'ieiicc !o grace or .=ight bills. Section 15, Gc nld's Mgest, says '■ Ftjreign and inland bills shall be governe I by the law- morchant as to days oi' grace, protest and notices." Colkct.iim. if Debts. — An at acliment may bo issued again.'t the protierty of a non-resiilcnt, and al.so against a resident of ilio tate wiien tJK- latter is about to lomove ul of ilie Slate; or is about to remove his goods or etl'ect.s, or about to sccreto himself, f-o tliat the ordiinry pr>)- cess of law cannot be sewed on hini. 329 CALIFORNIA. Inti-rcM. — Tim lo;:aI ralo of intcrost in Califomiii i^, ny Ptiitiito, fixed at tk» per cent. On special cniuracts any ralo of iaicru.sl may liu a^reuU upou or paid. I't'nntl'j fnr Vimatinn of t'fi Tiileresl f.nto. — Tlioro is no law in Calirornia fixing any pnnaliy liir c.liarsin;! r.ny ralo (jf interest aljovi) ten [lor cent. Tlio waiter is tliu.s lull outircly Ireo betwei ■ tiio cuntractiny partie.^. Dainaijci on liills. — i'lio Uamagcs on liillK of cx('liani:;o drawn or nci^otiated iu California payable in any .^laiM cast nt' tlio UolUv >1 uiilaln.'^, and ruiurncd under protoit for non-accept- ance or tion-pd fluent, aro unilunnly, Ij per cent. Foreign l)ill.<. — Tlio dama;,'cs on loreign of cxeliungo returned under jirotetit, av~ '-0 per cent. Sir/ltt Hills. — CiracQ is not allowed by iho bankm on bills, c' I'ks, drafts, etc., payable ul si'jht. 'fbo notarial fees for protesting a bill of excbango or pronii-.-iory note aro $5 or luoro, uceording to tlio number of noUcun sent. Act March \'i, 1850. CuUcc'wn of Debts. — 1. Creditors may proceed by attachment when tho defendant has absconded, or iH about to abscond from tlio State, or is concealed therein to tlio injury of his creditors. 2. When tlio defendant has removed or is about to remove any of his p:opcrty out of tlio [State, Willi intent to defraud his cicdito'\s. 'i. When iho defendant fran(iulenlly con- tracted tho debt or incurrod tho obligation, retipecting to which tho Huiti^ilirought. 4. When tho defendant is a non-resident. 5, Wlien ho has fraudulently conveyed, disposed of or con- cealed his property, or a part of it, or intc;ids to convey tho Bamo to defraud his creditors. In California tho real estate shall bo bound, and tho attachment shall bo a lien thereon, although tho debt or demand duo tho plaintilf bo not duo — in caso tho defendant is about to rcmovo liimself or his property from the State. Tho law of attachment applies in California when the contract has Lccn made in that Stale, or when viaUe payable in that Slate. I Y CONNECTICUT. Interest. — Tho le,';!al rate of intcrost in Connecticut is six per cent., and no Iilgher rate is allowed on special co.itracl.s. Banks aro forbidden, midcr penalty of $500, from taking directly or indirectly over per cent. Law passed ilay, 1854. Penalhj for Violation of the. Usury Laws. — Forfeiture of all tho interest received. In suits on usurious contracts, judgment is to bo rondend for the amount lent, without interest. Daniafies on liills. — The daiTiagos on bills of exchange n(?gotiated in Cmiiiecticut, pay- able in other Males', and returned under protest, are as follows: 1. llaino, Xew Ilamiisbirc, Vermont, Massarluisotls, Kliodo Island, Now York (interior^ New Jersey, renni(|lvaiiia, Delaware, Maryland, Virginia, District of C.iluinbia, o per cent. 2. New York Cily, 2 jii'r cent. 3. North Carolina, South Carolina, Ceorgiataii Oliin, 5 ]jer cvut 4. All tho other i: tales and Territories, 8 per cent. Foreion Jlills. — There is no statute in force in Co.iiioiiicut in rcferenc o to damages on foreign bills of e\c'iange. Sijlit liills. — Ci'ai.j id iiol allowed by statute or u.-^ago on checks, bills, ec'., payable aZ sight. Collection af Delils. — Attachment may bo granted against tho goods and chattels ;uid land of the defendant ; and likev,-iso against his" person when not oxomptod from imprisonment on tlio execution in the suit. The plaintilf to give bonds to prosecute his claim tu cUccl. DELAWARE. Inlcred.—Tho, legal rate of interest is six I'or cent , and no more is allowed on direct or indirect contract.". Penalty for Violation of the Uiisury /.lOd.t.— Forfeiture of tho money !.■:,'. other things lent, one liiilf to the Covernor for the support of government, tho other half payable to the person sueing for the same. Damages on liills. — There is no statute in force in Delaware is reference to damages on domestic or inland bills of exchange. Foreign Hills.— The damages upon bills of cxcnange drawn upon any person in Eng- land, or other parts of Europe, or beyond the seas, and returned under protest, are 20 per cent. Sight flito.— There is no statute with reference to bills, drafts, etc., at sight. They are not, by usage, entitled to grace. Collection of Debts. — A writ of domestic attachment issues against an inhabitaiit of Dela- ware when the defendant cannot bo found, or has absconded with intent to defraud his creui- toi-s; and n writ of foreign attachment when tho defendant is not an inhabitant of this State. This attachment is dissolved hy the dcfeudaiil's appearing and putting in special bail at aiiv time heforo judguicnt. 330 FLORIDA. Intfri'st. — Tlio l(';;il niio nl' intnrcst Is six I'or cent. On fipocial CODtractH eigiit per cent, may lio cliar.'cil. Pfnnlhj for VioMinn nf llic. ('.im-y /.iiit's— ForCoitr.r" nf tlio wholn iutorost puid. JJaiiKij/i.i oil liitis. — 'liio (liiiiiiiKt'^ oil lulls (if oxclmiiKc, iiL'^^oihiicd In KiDiidu, payable ill iillicri-wtrs 1111(1 Mniiiiii'il iiiuUrimiti'si I'it iioii-piiyuioiii, ,'iro iiulli)rnily u per cent. /•'())•«•((//( liitls. — Ditiuiig s on Idit'lKn bills ol' oxcliuun'' 5 P''r <'<'iit. iSiQid liilts. — (Jraco is not iillowoil on bilJH, driil'it^, elf., i)avaiili! at siglit. There ia no Ktatiili" ill l''liii;ila (ipnii tills Milpji-ct. ('■jl.'ii.liim ii/ Jti'liis. — All iiliai-linii'iil i«sllc;^ wlii'ii tin' amoiinl Is acliinlly diip, nnd tlio ilcl'cndani Is actually iimviug o it of llio SLilc, or abjcnuds or couccald liliiiscl!'. GEORGIA. Inlfrcft. —Tho legal raio of inli rost in Georgia i.s Foven i)er tont., nnd no htglier rato ia allowed on s|)oclid cimtracls. Open iiccoiints, unlliinldated, do not bear interest. J'luallii j(,f Vinldtion of Hii'. rituri/ I.tifs. — l-'oilelliiro ( 1" only the excels td' Inti'rest over .sjveii per cent. I'linclpiil a'' I lei^al interest are recoverable. (.Vets ol' 1805-0, jiage 2.09. ) JJamaiifs on liill.i. — Tlio ...images on lills of cxehango, negotiated in Georgia, payable in other iStaies, and r«tuiucd under protest, are uiill'onnly i> per eciit. Furiiijii Jiills. — Tho dumage.s on foreign bills ol' exchange, returned under protest, are 10 pel- cent. Siyht Ijill.i. — ••'flireo day.i, I'oinmonly ealleil the three days' of grace, shall not bo allowed upcii any sight dr.iMs or bills orexcliaiigi! diawn payable atsiglit, alter thJ pa.ssagoo( this Act; bii. the tanio shall bo payable on pre.senliitiiai thereof, .subject to tho provl.sions (d' tho first .•■cctioii of this Act. The llrs-t s-ectioiulosignatcstlio holidays." Act pashcd i<>b. 8, 1850. (.-co (..'obb-s New Digest of tho 1 iwh of Georgia, pp. 519, 522.) Endnrsi'i-s. — Endorsers are not entitled to notice of dishonour, except iipi n notes and bill.s payalilo at bank, or negotiated in bank, or placed in bank in eollcctioii. ' C'jlUvHuii of JJi'bts. — .V judge of tho riiperior Court, orajiistico of tho inferior cotirt, or a justice (if the iieace, may grant an attachment against a debtor whether tho debt bo matured or nor, when the latter is removing without tho limits of the State, or any county, or conceals himsell'. Tho remedy by attachment may bo resorted to by iion-resldentas well as by resident creditors. Tho nei'cssary aUldavit May bo made before any commissioner appointed by tho •State to take utndavit.s. Indorsers 6;,'!.— Attaclinieiits are LssueU by the clerks of the Circuit Court, when affidavit is (lied that the deleiidaiit lia.s departed, or is about to depart, out of the State, or conceals Iiiinself ko that tlic process caiinoi be sorv -d iip'in h"" ooL r; DIANA. Inlirfi't.—TUi^ l,.;.iil int'Tcst in Iiuliaivi is .-ix pc't rent., wlii'h nviy lio tnUpii In mlvjiu'O, ifHd cxprcs.ly ii;4i'(Mi. I'duillij I'iif Vivti.liiin cf l!i<: I'lru)!/ I.iuff.—ll' it Rp'iilrr nili) nr iiilcrc.-l thiin nH nliovo shall Ik; conlriirU'il lor. rfrcivcd or rc-eivoil, llui co.ilriuaijliull iioltlu'rcriin', Im vnid ; hiil IT 11 Is jmiv. il ill any iitliciu lliat a nr< iilor rutn tliaii nix jior cc.it. per niiimi;i hiiK bi'rii cdiitractoU lor, I 111! pl:iiiitill '^lia liiiily rcroviT his priiii'ipal Willi six por criit. interest ami ousts; ami iniio ilrlcnilant, Ikm paiil iIkilmih uvor bix iior >t'nt. inti^rest, Mieh cxcesa ol'InlorcslKhull bu UoUuctuU Innii iho 1 laintiirs ri!> i(vt'r\ , If any act i( in for a rinvivcry of n dcht, it is i.rovoil thit provions to tho coiumoncomont oflho .suit the (Icl'i'iiilanl li.is ti'mlcroil tlio atnount lUiP, with loyal iutorost, the ilffoiidunt shuU rocovcr losis, ami tho plaintilf shall only reoovor ihoainuunt tomloroU. Pamarifs on Jlills. — Damages, i)ayablo oi» protest for iioii-paymoiit or iion-acopUinco of iv bill of cxi-hango, drawn or liogotiatod within tbo State of Indiana, if drawn upon any person at any plaoo out of ildH Statu, aro at 5 per cent. Ucyond .siioh damages no Interest or charges aooruing prior tu iirotoat Khali bo uliowod, and tho rate of exchungo Khali not bo taken into account. Foreign liilh. — '1 ho damages payable on protest for non-payment or non-acceptance of a bill of oxchange, drawn on any iilace not in tho United Slates, arc, on tho principal of such bill, 10 por cent. No damages beyond the cost of protest are chargeable against the driwer or tho cndoi-fierofoithor species of bill, if upon notice of protest and demand of the principal sum, tho same is paid. Sight liiils.—drace is allowed on all bills of exchange payable in Indiana, whether sight or time bill.3. CoHec/iOrt o/" /)e/;t».— Tho projicrty of an Inhabitant of the State may bo attached, when- ever ho is secretly leaving tho State, or shall have left tho State with iiiteut to do fraud Lis crcditOLs. Tho property of a non-resident is liable to attachment as in other states. lOV/A. Interest— ThQ legal rate cf interest in Iowa is six per cent. Ten per cent, may bo charged on special contracts. On judgments, interest is chargeable tuon tho contract. I'enalty fur Violation of the Vsniij Laws. — Forfeiluro of tho (excess of interest paid for the bene.t of the School Fund. The borrower is by law a competent witness to prove usury. Damages on Bills. — Tho rates of damages allowed on non-acceptance or non-payment of bills drawn or indorsed in this State, aro as follows : II' drawn upon a person at a place out of tho United States, or in California, or in the Territories of Oregon, UUih, or Now Mexico, ten per cent, \ipon principal, expressed in the bill, with interest from time of i)rote.st. If drawn upon a p;rson at a jdace in Iowa, Jlis.iouri, Illinoi.s, Wisconsin, or in Minnesota, three por cent., with interest. If upon a person at a place in Arkansas, Louisiana, Mississippi, Tennes- see, Kentucky, Indiana, Ohio, Virginia, District of C'uhiiubia, Peinisylvania, Maryland, New Jersey, New York, Massachusetts, Ithode I.slaiid, or Connecticut, llvo per cent., with interest. If drawn ujiou a person at a i)lace in any other State, 8 per cent., with interest. (Code, §965.) Sight Bills. —Grace is allowed on bills and notes, locordlng to principles of tho law mer- chant, and notice to indorsers, etc., according to the rule.^ of the oommercial law. (Liiws, 1852-3.) Collection of Debts. — The idiintiir may cause .iny property of the defendant, which is not subject to execution, to bo attached at tho commencement, or during tiio progress of tho pro- ceedings, wholhar tho claim bo matured or not ; j)rovid d that an allldavit is filed to tho eU'.'Ct that tliQ defsndant is a foreign corporation, or acting as such, or that ho is a non-resi- dent of tho State, or (if a resident) that lie is in some manner about to di.spose of or removo his property out of tho State. KENTUCKY. Interest.— Tho le .al late of interest in Kentucky is six per cent. No higher rato of intero.st is allowed even on special contracts. All contracts made, directly or i;:directly, for the loan, or lorbearanoe of money, or other thing, at a greater rato than legal interest (6 per cent, per annum), shall be void lor tho excess of legal interest. lenallyfor Violation of the t'sury Laws. — If any discount or interest greater than the legal intorest or discount is taken by any ban'.<, or otl»' corporation, auUiorized to loan money, tho whole contract lor interest shall bo void, and any thing iiaid thereon for interest may be recovered back by the person paying the aiine ; or any creditor of his may recover thj same by bill in eipiity. Bank.s, or other nionied corporations, or individuals, arc not prevented, in dLscounting bills of exchange, from taki!ig a fair rato of exchange bctwee:i tho place where it is bought, and the plac • wiiere it ir. ,.i.yable, in addition to the discount lor intorest. But such privilege of buying bills of exchange at less than par value, shall not bo used to disguise a loan of money iit a greater rale of discount than the legal intorest or discount. Vamcig. s on liills.—Soi-Vdluie U in force in Iv utuckv ui.ou the subject of damages ou inland lii!!.: dle.'irhangc. 332 Foreign HHta.—W'hcxc. any l.i. of oxcimnsc, drawn on iiiiy pcwou out of tlio Uuitcl FIntcs, Klmll U: iirt)tcstpil I'cr tidii jnyini'iit nr iioii-icin-ptmn'o, It hIiiiII bo.ir ton iior cent, per yonr In- Icrost Iroiu the (l:iy ol' prolfst, (or not I inter lliau ciKlitiTii iiumtlis, iinlofs payiiioiit l>o sooiuT (Icriianclfil li-oin llio parly tu ho (;harj;i'(l. Suili iiUcri'st i^liall Ijo rocovcroil up to llio liiuo ot thoJii(l«iiR'iit, anil tlio jiiil^'niont Khali hear losal Inlerpsl tlirroullor. nmnugcH on all other bill! ar ■ (lisallov.ctl. [IJuvl.sud Matutcs. payea 19;J anJ 194] SiijH liiH.t.—CrMi- is allowed, hy fomio l)anUs on bills", tlrnfls, etc. . poyalile cif. siylit, but llio point \!i not yt liilly sutiiuil in tlilH >tate. Coltectioii nf jM)l.i.—l. The p'uintilV may liavo an attachment ogalnst Hi i)roperly of tlio (lefendalit when the lattur is a foreign corporation, or a non-resident of this State ; or, 2, who ha.s been ab.sont thercfrora four inonthH ; or, it, lia.s departed from tho .' tato with Intent to du- IrauU his creditors ; or, 4 has lolt tlio county of hi.s residenco to avoid (ho Bcrvico < f a sum- niona, or conceal.s him.>;elf that a HuinmouH cannot reach him ; or, 6, la about to r movo his property, or u inateri.il part thereof, out of tho Mate ; or, hag sold or conveyed lil.s i ropcrty with tho intent to defraud his creditors, or Is about so to sell or convey. 8uch atti'y;hmunt 1« bindiiiKUDon tho defcndaLt s property in tho county from tho titno of tho dcllvci/ of the order to the Sheritl'. LOUISIANA. In(;rest.—l. All debts shall bear interest at tho rate of fivk per cent, from tho time they bocomodue, unless otherwise .stipulated. (Act March 15, 1856.) 2. Conventional interest not exceeding eight per cent, per annum may bo contracted foi. — Ibid. 3. Tho owner of nny ))romi.=sory note, bond, or written obligation, for tho p.iyment ol money to order or to i)earcr, or tn'nsferablo by assignment, shall havo the rljjht to collect tho whole amount of such promis.sory note, bond, or written obligation, notwithstnnding such ])roral.ssory note, bond, or written obligation may include a greater rato of intore.^t or discoinit llian eight per cent iutorcst per annum. Provided that such obligations shall not bear raoro than eight per cent, interest per annum after their maturity until paid. (Act of March 2d, 1800.) Damages on JiiU.i. — Tho damages on bills of exchange, negotiated in I,oulsian,'\, p.iya' Ic in other State,-;, are uniformly 5 per cent. Foreign I>iU\ — Tho damages on foreign bills of exchange, returned under i rot est, arc uni formly (btatuto of ISoSI . . . ... • • lOp.erccnt. Sight nuix. — There is no statute uiion Ihi.s subject in Louisiana. A (loci.sion has been mado in one of the luferitir cuuria allowing threu days' gracj on sight bill.-', but llio u.sago i.s to pay on presontalion Colleclio.i of Dehln—X creditor may obtain an attachment against tho iiroperty of his debtor upon adUIavit: 1, when tho latter is about leavit;g iicrmaiiently tlio State before obtaining or executing .iiulgnient again.-t him; 2. when tiio debtor resides out of tho State; :j, when ho coiiccals himself Id avoid beiug cited to answer to a suit, and provided tho term of payment lias niTivcil. In the ab.'-.eiicfi of tlie creditor, tho oath may bo mudo by his agent or attorney, tu tho be.-?t (if his knowledge and belief. I.TAINE. Intern '..~'\'\\o legal rato of interest in Jliiiiio is ."-'ix per cent., anil no higher nde i= allowed ou (-pecial tonlracls. [U. S. 322. Cup. 45, .sec. 2.] renall'j fi Unit dm r.uo (if lntl o > kihOi hiiui or Value, wlilcli loiU'ltun! Kliall ciinro to llin biiH'lit oi any ilrlVMidunl who Hhall |ih. Tin! pli'a imi-it, liowov r, Hate I ho sum orainounl /(C7 /,<(i",<. — I'liriH H miK ,< ii|i(>ii niiitrnclH riv-crvlii;; ovi-r li'ii ('<'i' i(-iil. tuicr.-'i. liiny ii'idm r.|iiil«iiii nl lur llii> |iriiii i|.;il iiiiil lr;;nl niliinl liircri'.,il illliTt^a palil, mil im |ic'ii;i||y H,|' IVdcisJMK It. JloiM /!(/.■ liolciciH (■I'li-iii I'liiH n('«i)tiiiMi'|i.,|HM' laki'ii bi'liii, Illinois, liuliuim, Ohio, ri7/.i.— (inico i.M iillowcil iin all jiapor nut payaliUi on ilemaml. CoHirliiiHKj' ]>i'lili.—'X\w Ki'ouiiil.H of iitlaihnu'Ut in lliis .'•tato uru : 1, that tlio dpfondant Ii;h iili-iNiDiU'il, or \a iilxml to aliscninl, or lias concoalcil lilni'^plf ; 2, lli;iUio has iisslunoil or conci^aU'd, or is alioiit to ri'tnovo Ins property with a view to ilcfrau:! : ;), thai lio fraudulently ('oiitractcil tliodi'lit, or ini'iirri'd tho obligation about which llio Mull Is lirnii;.'hl ; 4, that lio lii not !i ri'Mdi'iit <'f tho stiiic, or has not nvidcd tlioro thrco mouihs inuuedlatcly procuUlnis tLip Miilt ; b. tluLi iho defendant is a foreign curporutioii. MINNESOTA. /(i/t'j'.'.v.— Intrroi^t for any lc;.'al indebtedness slmll bo at Iho rato off' for |100 for a your iiiile.^s 11 dillereiil late be contracted for in writing, but no agreement or rontni.'t f t a greato rato of Interest than fl'J for every $100 for a year Khali be valid for tho pxcosh of lotens' over twelve per cent. ; and all agreements and coutriicts sliall bear tho samo rato of interes' after they become due as before, if the rato bo clearly expressed therein . I'roriiled, tho samo thall not r.xcoed twelve per cent, per annum. Alljiidgments or decree,-!, in.ido by any court In tin.'? State, «hall draw Intorcst at tho rato of six (il) percent, per nnuni. (Laws oflStlO, j). 220.) J'i:nallii/iir Violatinn of inleirsl Lair. — Excess ot interest over 12 per cent, forfellnd. Oay^ li/Oniri:. — On all bills of exchange payable at sight, or at a future day certain within this Mate, and on all negotiable promi^■.sory notes, orders and draft?, payable at a future day ci rtain within this State, in which there is not an expres:i stipulation to tho contrary. U'/ic/i Grace not alluwccl—liw bills of exchange, note or draft, jiayablo on demand. When prfsntli'.d for i'aymaity dc. — bills of exchange, b;ink checks and promis.sory notes falling due, or the prescninient for acceptani'e or iiayiiient whoroof should be made on tho l.st day of Jiinuary. the 4th day of July, the 25th day of December, tho 22d day of rebrimry, ami every day appointed by the I'rcfident of the United States or tho (iovernor of tho State aa a day of fasting rir tlianksgiving. shall bo iiresentcd foracce|itai:co or payment on the i\ay prfcnl- ing. Such days [above eiui'.iiiatedj shall be treated and considered as tho llrst day of tho week, commonly called Sunda;. . [I'ol. Law.s, H'Ci.] Amplaiuc »/ Hill.1 of Excltinge — No person within thi.s State shall bo charged aa an ac- ceptor on a bdl'of exchange, unless his acceptance shall bo in writing, signed by himseiror hlii lawful agent. Damages on liilU of Kxihangi'. — On any bill of exchingo drawn or endorsed within thia State, and payable wlihlint the limits of tho I'nited States, which shall bo duly protested for non-acceptance or nonpayment, the party liable I'or tho contents of such bill shall, on duo tiotico and demand thereof, pay the .sameat tb ' current rato of exchange, at tho time of tho demand, and ilamagea at tho rato of ten per cent, upon tho contents thereof, together ./ith lntere,?t on said contents to lie comiiutcd from the date of tho protest; and suid amount ot contents, damages and interest shall be in full of all damages, charges and expenses. On all bill.-! drawn on any person, body politic or corporation out of thi.s State, but within ,=oino State or Territory of the United ,-tatcs, and iirotestod for non-acceptance or uou-paynacut live iier cent, diinages and interest, and costand charges of protest. Colli itinii of Vrhls.—A warrant of attachment may be issued against the property of a defendani when a foreign corporation; or, when not a resident of this Territory; or, has left tho Ti'iTiiOry with iiteiit to defraud his creilitors. Tims it will be i-cen that in all tho States the property of non-residents and foreign corpora- tions is liable to attachments at the suit of creditors, before judgment is rendered; likewi.so against domestic debtors when they have nb.sconded from the State, or havo fraudulently con- veyed, or are about to convey, sell, a.ssign or secrete tlieir cfTecta In sorao few States, how- ever, even this condition is not es.uentiai before a writ of attachment will issue. In IhoKtiitcs of Alabama, .MiKsachuselts, Connecticut, Maine, Now Hampshire, Vermont and lUiodo Island, the creditor may have a writ of attachment against the property of tho debtor at tho lir.sl institution of :i suit — and witho ;t any ground of fraud or frauduloiit intent — such properly being held by the attachniout until the termination of tho suit, or until judg- rneiit; the plaintilf in such cases giving bond or security to indemnity tho defendant for any loss or damage sustained, slKnild the case bi^ decided in favor of the latter. Cioncrally, the iii'openy is liable only when actually levied tijion; but in the State of Kentucky the pro- perty is liable from the time of dolivery of tin; order to the sherilf. I OO" MISSISSIPPI. Iiittrut.—'\i\c Ipgul riloof liitorcHlIn Mis.-nf^lpiii iHhlxiirrrcnt. Iiy llionrt pns.wd in Murili, 186a. Ike^ni)'-!! ,in liilln.—So ilamn^'p' tin' iillnwfil \«r ilcflmlt In llif iniynu'tit of any Mil nf o\ rlmnjio ilrnvn hy iiriy iicr-iiui or iicr-oiiH wiihiii ilio Maig on iiuy i vrxm it ixm-imim In iii>' c r Ihlanil liill.-i [ilniun i>n iicivims wulmi tint t^uui'], ami pruif.-iii'il I'ariMiii imynirnr, iIvi'imt cut. (Sci- iici nr Mny II, l>;'.7. | '■'iiniiiii /^tut f, icuiincil iMKl'i' |ii<>ti>i, lui! 10 |ic'i'C(iil., \N nil nil Inrlilontiil cliin-KrM unit l.iwhil jntorur'i. iSig/il lliil.i. — (Jnicc n niil. ullnwi'il mi IiII.j or i\ilmn).!i', (Iriil.H, cii; , |iiiyiil)lit tU .lii/lil. Ciillirt/iin of /Jthln, — An iilliulina'iit ii;iali;.-i lli<' csiaui. inclnillii;; iciil ("stiitc, t'lmci^, clmtlols A:c., da ilrliti r, wlii'ii liH .-Imwii iliiii Iri lias ivnin. mI, or l.s nliniii ri'inDviii;; ur ai>-riiniiiii;j JVoiii lli<( i^iali', iir piivalily fdnrcal-i lilin-cll'. Allarlinicnt iil.-l(lriil ili'.-'iciU'nt.-'. it may Im ililiini'il Ik loio i|ii> di'la Ih ill • icii' wlitcli it l.v-in-i, wIhmi iK'Ci'iHlilDf lia.'i uruniiil m licliuvc lliul llic rlubUir will irlimvu Willi liifj ilIocU out ul llic SiuH', or hut) icuiuvfd. MISSOUli:. /»ir intori'st In Minfouri In .-^Ix jior fciit. wlmn im other rato U ii!;r("d upon. I'artio.-f may unri'u in writing lor any lar^'cc r.ito, not fxcouUiiic tun li^'i' ii'i't. rartici lilay Hiirontrart as to i iiiiipiiMnd tlio IntDrcst annnally. I'enalli/J'df Yinhttinn <;/ titi- r./ Debt.i. — An attachmont mny bo l.ssin'd horn wlion the de'itor is not a resident of tlio t^tato; or if are.sidi'iil, when lio abscond:', iiljseiUs or conceals hiiiiseU", or is about to remove his properly or fraudulently convoy it, with a vi 'W lo hinder or delay his creditors, or when the debt was contracted out of the Stale, and 'lo debtor has secretly roinovod liiii cU'ccis into this Slate with intent to dofrand. NEW HAMPSHIHE. Interest. — Tho legal rato of Interest in Now Hampshire is six p'r cnat. ,and no more i.s allowed on contracts, direct or indirect. Penally ^or Vinlatinnofthe Usury Lawn. — Tho person receiving interest nt a liighr than tho legal rato, shall forfeit lor every such oll'enco three times tho sum so received. Damages on liilh. — No statute in force in Now llanipsldro. foreign liilts.—So statute in force in Now Uaini>siiire allowing damages on foreign bills returned under protest. Sight Hills. — No bill of exchange, negotiable promi.'^sory note, order or dr.ift, except such as are p.tyablo on c/emanci, shall bo payable until days of grace have been allowed thereon, unle.^s it appear in the instrument that it was the iutentioii of tho jiarliej tliat days of graco should not be allowed. [Keviscd H. 389, § 10.] Collection of Debts. — in this State a writ of attachment may bo issued upon tho in.stitutioia of any personal action ; anl lielwfcii ijoi-bons uctiiully localuU in cillun' Kiiul city or loWMsli I), or iiy iierson.s iiol ro., Ta.ssali; t'oaniy. Act, Kcbrnary 6 1S,')S, lipr^cn ("iHUity. A'.t, Ffljriia y 18, lii,)8, Uni nCoinily. Act, Maicii IS, 18o8, I ity ijl Tuliwav. Act -Marcli 120, 18.')', to all Havings Institutions in tin; "talo. " ' l)y Act ol'. March 28. loO'J, tho logislatnro aiitliori/.c I coniracts at s.'Vcii pir ccn;, i :lerc t by p.iriios residing in lliddlcse.K County. Penalty for Violation, of the Uiury Liiws. — Tho co:tr.iit i.s void, and tlic wliolc t^iuii i.>l',i!. f'oited. Vamagns on BilU of Exchange.— tlicro is no statute in Ibrcc in nd'creuce to d^onagoson bills of oxchango. Foreign ISills. — Tliero I.s likewise no statute in loice in rclercnco to aaina.L'i-.^ on iiroustod foreign billji of c.\chaiigo. i^ighl liills. — That all bills of cxcliango or draft.s drawn payablo at Pight, at any idrico wlihin this State, other than those upon banks or banking a.s.soi-iations, .shall lie deonied dao anil jiayablo at tho cxi)iration of tlircu days' grace alter llio sumo aliall Lu pre;i;ulcd lor acceptanco. Collection of Dcltls. — .Vn attachment may issue at the instance of a crt'dilor (or, in his absence, of bis agent or attorney), against tho property of a debtor when the latter is about t<* abscond from tho ttatc, or is not a losidont of the Stat ■, or i.s a foreign co. porutiou. able I^EW Yor-iic. //ifcrcsi.— Tho legal rate of interest in New York is .st'ven per co.it., and no higher rate is allowed on special contracts. I'enaUij for Violation of the. Vsunj Lan.s. — Forfeiture of the contract in civil r.clion.s. In criminal actions, ii lino not exjeoding one thousand dollars; or iniprisonnient noicx.ecdii]^ SIX months; or both. All bonds, bills, notes, assurances, couveyai\ce.s, all other contracts or securities whatsoever (except bottomry and respomlenlia bonds ami contracts), and all deposits of good.s, or other things whatsoever, whereupon or whereby there shall bo reservet! or taken, or ."ocured, or agreed to bo reserved or taken, any greater sum, o- greater value liir tlieloan or forbearance of any money, goods or other lliings in action than seven per cent, shall 1)0 void. (Hi'V. Htat. Vol. II., p. 1S2). For the purpo.so of calculating interest, a mnntli •shall be considered tho Iwell'tli i)art of a year, and as consisting cf thirty days; and intere.--l f )r any number of diys les-i than a month shall bo estimated by tho proportion whieti such inimbi'r oldays sliall bo;ir to thirty. Damagison Jiills. — The diimages en bills of exchange, ncgoti itod in Xew York and payable in other State sand return neoundi'r protest for non-ac<;i",)tancoor non-i)aymenl, areas follows: 1. JIaine. New ll.amp.-hire, N'ermont, Ma-sac!iusett.s, Kiiode Island, (.■oaueciicui, Neiv .ie.sey, I'eun-^ylvania. Delaw.irc, .Maryland, \'ir;,'i:iia, Hisui^a ofOolnmbia, andOhio, U p.er cei.t ■J. Xortli llarolina, ^o:,th Carolin i, (lcor;;i;i, JCenlucky, and Tennes ee, o per com. ;;. If drawn upon piu'ties in .any oUii>r Mate, 10 p 'v ceni. Tho following days, namely, tlu! first day of .lanuary, commonly calli'd Ni'W Year's day; the fourth day of ,Iuiy; the twenty-liftli day of Doce i bcr, cotnmouly called ( lirislnias diy ; and any day appointed or rocominended by tho (lovernor of the State, or tlie I'.'csideiit of lin- rnited .-late.=, as a day of fast or thanks giving, .sha I, for all purpo.ses wliaisoever, as re;;ards llio presenting for payment or accoptjnce, and of tho protesting and giving notice of the dis. honourof bills of exchange, bank chocks and promissory notes, made after the ji.i.ssage ofiliis act, 1)0 treated and considered as is the llrst day of tho week, commonly called Sunday, f i>i4;>, ch. 'JOl.) Foreign lUUs. — The damages on foreign bills of exchange, returned under protest, av 10 per cent. Sight nHh, — Grace isnot allowed by tho banks of the city of New York and of 'he intevii;-, upon bills, drafts, checks, &c. , payablo at sight. Co'dection of Ddils. — Any creditor to the amount of $25 may compel Iho assignment of their estates by debtors imprisoned on execution in civil causes for more than Oi) years, u tho debtor refuses to be examined, and to discloso his alfair.s, ho is liable to be conimitlcMl lu close conllnemcnt. If ho refu.sos to render an account iinentory, and make an as-ignment. ho will not bo entitled lo his discharge ; though tho ofllcer having jurisdiction in I ho case is authorized to make the assigument lor him. The proceedings and tho ell'ect of the discharge. when duly obtained, and the duties of the debtor, and tho riglitsof tho creditors, aie('.>-senlially tho sanio as in tho case of i)roccodiu.'s with tho as-ent of two-thirds of tho creditors. Dvoiv insolvent debtor may ttl.so | etition tho proper olUcers lor leave voluntarily to asSight Jiills. — By virtue of an net of tho Legislature, passed In January, 1849, grace i.i allowed on liiilsni sight, unless there is a stipulation to thecontrar.v. Prior to that date tho usage was, not to allow grace on such bills. Coll' clwn of Debts. — An attachment may i.-sue on tho complaint of a creditor, his 'igent, attorney or factor, against the property of a debtor when he has removeil or is about to remove, privately from tho State, so that tho ordinary process of lawwill not rcacli him. i-alil(> ,ows: (hiv; d.iv; ■llu- : (lis. ■ihis . S-l;i, !;■■• 1') OHIO. Jnterest.—lhc law allows interest at six per cent. i)er annum u\\ ail money duo, and no more. (Tho law allowiuj? 10 per cent, on special coutiacis was roin'aled April l.^l, ISo'J. but the repeal does not ulfect contracts entered into prior to lliis date.) Itaihoad (.'onipauies aro authorized to borrow at tho rate of 7 per cent. Penalties. — Thero aro no ])onalties onlinarily for Usury. Contracts fur itronter rales aro void as to the excess only ; and if interest beyond six per cent, has been i)aiil, tiio debtor has a right to have such excess applied as paymeut on the principal. An exce.-vi of interest taken iiv Banks invalidates tho debt. Hills of Pxcliange. — ''Damages on protested bills of exchange, drawn by a person or cor- poiaiiou in Ohio, are not recoveralilo on any contract entered intoafter the pa.^s.igeofthisact." (Pas=cd and took ellect April 4th 1809.) A check is not entitled to grace; but a check " payable on a future specilled day is a bill of exchange," and entitled to grace. (5 Oliio .Slate Bep. lu.) " Tho usage of banks in any jiarticular place, to regardjlrafts upon them, payable at a day certain after date, as check.s, and not cnlitleil to days of g'ace, is inadniicsibf; to c.uirol tho rules of law in relation to such i)aper." — (lb. ) Sight Bills. — By an act of tho legislature, approved Februaiy 22nd, 1801, it is enacted that "no note, check, dralt, bill of exchange, order or other negotiable <;r commercial instrument, payable at sight or on deiiiaiul, or on presentation, shall be entitled to days (d' grace but shall bo absolutely i)ayablo on presentment. ^All other notes, drafts or bills of cxcliauge .shall be eulitled to Uio usual days of grace. This act is in fore; from its passage. No grac is allowed on bank checks payable at sight. A statute is in force providing that "all bonds, notes or bill.s, negotiable by this act, .shall be entitled to three days' grace m tho time of payment.'' Collection of Debts — A creditor may procure, before or after the maturity of the claim, an attachment against the i)ioperty of lla! debtor, wheio the latter is a foreign coriioration or a non- resident ; or, if a resident, when ho has aUsconiled, or lelt the county of his residence, or ciiuceals himself, or is about to remove or convert his property, with a view to defraud his creditors. 2. When the debtor fraudulently contracted tlio debt or incurred the obligation. lit ol if Ml tu ueni. >e i.-: lariic. lially ')voiy HI ate pre- liit> eicil, iiiucv 1. Ill ' e as- PENNSYLVANIA. /ntercgt —Tho legal rate of interest in Pennsylvania is six per cent., except as provided in the fol. owing acts : Sue. 1 lie it oinctcil, dc., Thai commission merchants and cgents of parties not residing in this commonwealih be, and liiey are hereby authorized to enter into an agreement to retain the balances of money in their hands, au\s: 1. On bills on any part of North America other than tho United States and on the West In dies, i'2}2 per cent. 2. On bills drawn on any other part of the world, lo per cent. Sight Bills. — Tho statute of 18-18 enacts that "bills of exchange', foreign or domestic, pay able at sight, shall bo entitled to tho same days of graco as now allowed by law on bills ol exchange payable on lime." By a statiito pas.sod in 1831, it is enacted that if money or other commodity bo lent or advanced upon unlawful interest, the idaintilf shall be allowed to recover the amount or value actually lent, l...t without interest or cost. By an act passed in 1839, it is enacted that a debtor by bond, note, or otherwise, about to le.ivo the Slate, the debt not being yet due, may bo sued and held to bail. Tho plaintilf must swear to the oobl, and that ho d d not know tho del)tor meant lo remove at the time tho con- tract was made. But tho writ must be made returnable to tho term next succeeding tho maturity of tho note, etc. Collection ofVeOts. — A writ of attachment will issue atthoinstanco of the creditor wherever residing, against a debtor when ho is a non-resident— or against a citizen who has been absent more than a year and aday; or when ho absconds or is removing out of llie county; or cou- \ ccal,'? him.'self so that the ordinary [U'ocess of law cannot reach him. \ 839 TiCNNEaSEE. Interesl.~Thc losal rate jrinlorPRt in Tonnesseo is six jicr eoiit., and no liigluT rate can be recovered at law. Contiaota at ii groator rate of Interest are vo il a.s to tlie excess, and llio lender is liable to a One of $10 to f 1,000. I'enaUyfor Violation of the. I'sury Laws.--\,\ah\(^ to an indictment for uii>'demoanor. 11 convicted, to bo fined u Kura not less than tlio whole usurious interest talcea and rei'i'ivcd, ami no lino to be loss tlian ten dollar.". The borrower and his judsment cr diiors may also, at any time within .six years alter usury jiaid, recover it ))acl\ Iroui Ihn lender. Damages on lulls. — The damages en bills orexchango negotiated in Tennessee, iiiiyable in olhc'r fctates, aiul protcsicd for non-payment, are o iier cent. Foreign BiUs.—'XhQ damages allowed on foreign bills of exchange, returned und'r protest, arc as follows : 1. If upon any person out of the United State?, and in North America, bordering upon tho Gulf of Mexico, or in any part of the West India Islands, 15 per cent. 2. If payable in any other part of the world, 20 per cent. Sight Bills .—Tho legislature has passed an act providing that bills at sight shall not bo entitled to days of grace. By law, all negotiablo jiaper duo July 4, nooeiuber 'J.'i, .lanuary 1, or on any day appointed by tho Governor as a day of Thanksgiving, or as a public holiday, shall bo payable tho day preceding either of those days. Colleclion of Debts. — When a debtor has removed, or is aliout to removft out of the county privately, or absconds or conceals liiin.self. an attachment may be obtainedagaiust his pnn'Oity at the suit of a creditor, or his agent, attorney or factor. In iho caso of non-resident debtors, having any real or persuiial property in tho State, it is requir d. in ortler to obtain an attach- ment, to lilc a bill in cliaucery. TEXAS. Interests. — Onall written contracts ascertaining tho sums ilue, when no rat • of interest i.-; expressed, intoicst may be reccivered at the rate of eight \k\' cen;. per annum. The parties to any "viiitton contract may stipulate fjr any rate of interest, not (;xceediiig twelve per cent, per annum. Judgments bear ei;,dit jut cent, interest, except where lliiy are recovered on a 'oiitract in writing which sllpulateil lor more, nut exceeding twelve, in" which case they bear llie ralo contracted for Ko interest on account.-;, unless there be an express contract ; but only eight per cent, can be recovered on a verbal contract. Contracts to pay interest erson cr persons living beyond tho limits of this h'tat,', shall, after having lixed the llal)ility of tho drawer or endorser of any sucli draft or bill (if exchange, as ])rovide(l for in tlic act > 1, J:- . 1 .■..:■• . ,..,.■- f ' UPPEP- AND LOV/ER CANADA. ^ ' ■ • ■• . Interest. — fix per cent, is tho legal rate of into -est, but any r ito agreed upon can bo recov- ered. .Tudgments bear six per centum per ininum interest from the dato of entry. lianUsaro not allowe I a higher rate than seven per cent. Corporations and associations authorized by law to borrow and lend money, uuloss specially allowed by somo Act of Parliament, arc prohi- bitod from taking a liigher rat:; of interest than six per cent. Insurance) Compnnies, however, aro authorized to iako eight per cent. Bills nf Exchange and Promissorg JVotrs. — Three days of grace are allowed on all bills and notes payable within Upper or Lower Canada, except when drawn on demand. When the last day of grace I'ails on Sunday, or a legal holid.iy, it is payable the following day. Acceptances must bo in writing. No iior.son or corporation in Upper Canada can iss.o notes lor less than one dolhr. I'rotcsl may bo made, and tho ;iartiea to the bill ( r note not liod on the .same day tho bill or noto is dishonoured ; but, in c iso of non-jMiyiiior.t in Upper Canad'i, not before threo o'clock, p.m., and in Lower Canada any time alter the lorcnoon of tho la-tc ay of grace. Dishonoured inland bills or notes in I'jipcr Canada, when prolef^ied, and in Lower Canada without protest, bear interest at the rale nt si\- per cent, from dato of protest, or in Lower Canada f om matmily to time of payment ; but if interest is expros.sed to l.o payable Irom a part, cu'ar period, then irom the tim ; of such ncviod to tho time ol' )iaymc;5t. Tho damages allowed upon protested foioign bills drawn, soM or negotiated within L'pper or Lower Canada, 341 If clrnwmipon nny person in Europe. West Indica or in any part of America not within the tho i'ro\Miioo or ai.y oihor Biitisli Is'ortli American • ti|ps ofAmenm, Ten percent, upon tho pr nopal sum ppefilli i| in the lull. It'drawn in Lower Canad i ou pcreon.s in Upper Caniula, or ifdrawn In c!;Ik;' r(>per or l.mverCunadii on any person m any other of ilio Urii;sh North Amorican col n. .■) or I'nitoil ^inres of Amcrita four portent on tho principal sum f^peciiieU in the bill. Thoabovofor ign hill' aro also ku )■ jcct to six i)er c ntiim per annum of interest on tho amount for which tho bi'l was drawn, to borockoncd from tho dato of pruiest to day of repayment, tofreiher with ilio current rato of exchange of tho day when repayment is demanded, and tho e.^iienscs of nolng and jirotpsiins tho bill. Promisso'-y note.") niado in Upper Canada, payable in tho I nited States of America < r Briiisi Norlli Ainerican Colonies, not being Canada, and not othenuise or fhewherr, and pro- tested, ill addition to tho principal sum, aro liablo to dainayes at thi. rato of (our per cent, on such priuc pal Bum, and interest at tho rato ofsix per centum per annum, to be reckoned IVom the day of protest to tho day of repayment, tdgetlier with tho current rato cf c.xchanpo of tho day wiicn rcpiyment is demanded and tho expcnsos of protostmi; tho nolo. Tho (Statute of Limitations but 8 tho rig^ht of action on bill.sof ixcLungu and preinissory notes, in Upper Canada in six years, and in Lower Canada in live year.?. Colkctinn nf Debts. — Debts may bo recovered in Lower Canada by act ions at law, and in Upper Canada by aclii ins at law ir suits in equity. Debtors may bo arrested and held to bail in Upper Canada upon a i nindavit of the creditor, or ofsomo o'.her indiuidual, .shewing that ho lias a cause cf actiiin to tho amount of $100. or upwards, and has fiutlcred damages to that amount, and shows facts a'ld circumstances to satisfy tho .judge that th' ro is good and probablo causo lor believing that, such person unlcsho i.^ Ibrilnvitii apprehended, i.s about to quitCanada with intent to dc- frau 1 his creditors pencrally or tho deponent i;i particu'ar. In Lower Canada debtors may bo ar- restedand held to bail upon aHldavit of tho I'lnintiir, his book koe:c;', clerk, or le^al attorney, that tho Defcndai.t is; persontlly indebted to Iho riaintid'l'or a sum ainnuating to or exceeding S40, and that depon' nt bclicv , upon gro-nds .sot forth in aflldavit, that Pcicdant is imm-diately aboit to leave Iho Province with intent to defraud his creditors generally, or tho riaintill in particular, a:,d that such departure would deprive tho Flaintin of his remedy agai st tho Pefendarit, or that tho Dcfondiinl has sscreted or is about to sccroto his progeny with such Intent. A resident of Upper Canada, cannot in Lower Canada, bo arresicd nt Iho suit t f any person residing in Upper Canada, unless, in addition to tho ab vv, thoPlaintiffor somo ot'.:er person, makes oath bcforo a Judgo or s:mc other authorized officer that iho ncfcndant is imrac- dately about to resort to somo country or plac ! without the limits of the Province, and hath not within Upper Canada any lands or other real cstat.- out of which tho Plaintifl'can reason- ably expect to bo paid Iho amount of his debt. In Lower Canada any debtor Imprisir.od or held to bail, in a causo wherein judgment for a sum of $80 or upwards is rendered, is obliged to raako a statement under oath, and a declaration of abaudonmeL of all his property, lor thobeneflt of his creditors, according to tho rules, and subject to tho penalty of im'iisonment in certain cases. When such statement and declaration aro mado without traud. Iho ('.ebtor is exempt from arrest and imprisonment by reason of a.;y causo of action cxistii;-^ bei'ro tho making of such statement and declaration. In Upper Canada tho property i -edits and edecta of an absconding debtor, that Is to say, — any person resident in Up' cr Cii;b '• Indebted to any other person departing from Upp.'r Canada, with into t to defraud hiscr. dm/rs, and at tho time of his so departing is possessed to his own use and benelit of any real or personal eflfects therein — n;ay bo seized by a writ of attachment, provid d the debt exceeds $100. Judg- ment debtors may bo examined as to what debts aro duo to them, and such debts may bo attached upon affidavit, showinT that a judgment was reonvered and is still unsatLsflcd. In Lower Canada a writ of attachment may issue bel'oro .judgment upon proof on oath that Defendant is indebted to tho PlaintiiT in a sum exceeding S40 and is about to secrete tho same or doth abscond or doth suddoi^ly intend to depart fi-om Lower Canada with i tent to defraud bis creditois, !:nd that tho deponent believes without tho benefit of such attachment tho PlaintiiT would lose his debt or sustain damage. A trader's goods may bo at ached in Lower Canada, (a::d if the suit bo brought in tho Superior Court ho may bo arrest d. ) if, in addiUon to tho allegation, tho Defendant is indebted to tboPIaiutilTin tho sum rcq -.ired, it is r.ll god, that he Is a trader, that ho is notoriously insolvent, and has refused to compromi.'o orarrango with his creditors but still continues his tiado. Tho estate of insolvent debtor.^ may bo olso ottacbod by credtors lor sums