. < - * .^?vi ? - ! . - ; -' V r SCHOOLMASTER'S ASSISTANT: IMPROVED AND ENLARGED, BEING A*PLAI PRACTICAL SYSTEM ** OF ARITHMETIC :* . % ADAPTED TO THEJNITED RTATEsT EDITION. BY NATHAN DABOLL. RH1TED AND PrBLISHEB BY SAMUEL GREEN, FROFftlBTOfc OF THE COPT RIGHT. C Siarr, Stereotype Founder* A*. T, DISTRICT OF CONNECTICUT, SS.j t g BE IT REMEMBERED, That on the twentr- first day of October, in the thirty -sixth year of the Independence of the United States of America, SAMUEL GREKN, of said District, hath deposited in tWs office the title of a Book, the right whereof he claims al proprie- tor, in the words folUjiving, to wit : " Daboll's School- master's Assistant T improved and enlarged. Being a plain practical system of Arithmetic : adapted to the United Sfates. Stereotype Edition. By NATHAN DA- BOI.L." ." In conformity to the Act of the Congress of the United States, entitle9, " An Act for tMe encouragement of learn- ing, by securing the copi$y of Maps, Charts and Books, to the Authors and proprietors of them during the time* therein mentioned." HENRY W. EDWARDS, Clerk of the District of Connecticut. A true copy of Record : Examined and sealed by me, H. W. EDWARDS, Clerk ofthe'JHst. of Conn. RECOMMENDATIONS. YALE-COLLEGE, OV. 27, 1799. 1 HAVTS read DABOLL'S SCHOOLMASTER'S ASSISTANT. The arrangement of the different branches of Arithmetic is judicious^md perspicuous. The author has Well ex- plained Decimal Arithmetic, and has applied it in a plain and elega'nt manner in the solution of various questions, und especially to those relative to the Federal Conjputa- tion of money. I think it will be a very useful book to Schoolmasters and their pupils. JOS1AH MEIG$, Professor of Mathematics and Natural Philosophy* [Now Surveyor General of the United States.] 1 HAVE given some attention to the work above men- tioned, and concur with Mr. Professor Meigs in his opin- ion of its merit. NOAH WEBSTER. New-Haven, December 1, 1799. ^ RHODE^|pND COLLEGE, NOV. SO, 1799. I HAVE run through Mr. DABOLL'S SCHOOLMASTER'S ASSISTANT, and have formed of it a very favourable opin- ion. According to its original design, I think it well " calculated to furnish Schools in general with a method- ical, easy and comprehensive System of Practical Arith- metic." I therefore hope it may find a generous patron- age, and have an extensive spread. ASA MESSER, Professor of the teamed Languages, and Teacher of Mathematics \ Now President of that Institution."] 2030032 , RECOMMENDATIONS. FLAISJFIELD ACADEMY, APRIL 20, 1802. I MAKE use of DABOLL'S SCHOOLMASTER'S ASSISTANT, in teaching common Arithmetic, and think it the best calculated for that purpose of any which has fallen within my observation, JOHN ADAMS, Rector of Plttinfield Academy { [Now Principal of Phillips Academy, Andover, Mass. J BlLLERICA ACADEMY, (MASS.) DEO. 10, 1-807. HAVING examined Mr. DABOLL'S System of Arith- metic, I am pleased with the judgment displayed in his method, and the perspicuity of his explanations, and thinking it as easy and comprehensive a system as any with which I am acquainted, can cheerfully recommend it to the patronage of Instructors. SAMUEL WHITING, Teacher of Mathematics. TROM MR. KENNEDY, TEACHER OF MATHEMATICS. * I BECAME acquainted with DABOLL'S SCHOOLMAS- TWt's ASSISTANT, in the year 1802. and on examining it attentively, gave it my decided preference to any other ystem extant, and immediately adopted it for the pupils under my charge ; and since that time have used it exclu- sively in eJementary tuition, to the great advantage and improvement of the student, as well as the ease and as- sistance of the Preceptor. I also deem it equally well calculated for the benelit of individuals in private in- struction ; and think it my duty to give the labour and ingenuity of the author the tribute of my hearty approval and recommendation. ROGFK KENNEDY, New-York, March 0, 1811, PREFACE. 1 HE design of this work is to furnish the schools of the United States with a methodical and comprehei)sive system of Practical .arithmetic, in hich I have endea- voured, through the whole, to have the rules as concise and familiar, as the nature of the subject will permit. During the Jong period which I have devoted to the instruction of youth in Arithmetic, I have made use ot various systems which have just claims to scientific mer- it ; but the authors appear to have been deficient in an important point the practical teacher's experience. They have been too sparing of examples, especially in the first rudiments ; in consequence of which, the young pupil is hurried through the ground rules too fast for his capacity. This objection I have endeavoured te obviatt in the following treatise. In teaching the first rules, I have found it best to en- courage the attention of scholars by a variety of easy ami familiar questions, which might serve to strengthen thei; mim:s as their studies grow more arduous. The rules are arranged in such- order as to introduce the most simple and necessary parts, previous to those which are more abstruse and difficult. To enter into a detail of the whole work would be <- dious ; I shall therefore notice onlj a few particulars, anti refer the reader to the contents. Although the Federal Coin is purely decimal, it is f* nearly allied to whole numbers, and so absolutely neces- sary to be understood by every one, that I have intro- duced it immediately after addition of whole number-,, and also shown how to find t!.c value of gotx! immediately after simple multiplication; which may b of great advantage to many, who perhaps will not hav an opportunity of learning fractions. In the arrangement of fractions, I have taken an entire new method, the advantages and facility of which wili sufficiently apologize for its not beii^ rarirtfcatg to othar VI PREFACE. s y stems. As decimal fractions may be learned much easier than vulgar, and are more simple, useful, and neces- sary, and soonest wanted in more useful branches of Arithmetic, they ought to be learned first, and Vulgar Fractions omitted, until further progress in the science shall make them necessary. It may be well to obtain a general idea of them, and to attend to two or three easy problems therein : after which, the scholar may \earn decimals, which will be necessary in the reduction of cur- rencies, computing interest and many other branches. Besides, to obtain a thorough knowledge of Vulgar Fractions, it generally a task too hard for young scholars who have made no further progress in Arithmetic than Reduction, and often discourages them. I have therefore placed a few problems in Fractions, according to the method above hinted ; and after going through the principal mercantile rules, have treated upou Vulgar Fractions at large, the scholar being now capable of going through them with advantage and ease. In Simple Interest, in Federal Money, I have given several new and concise rules ; some of which are par- ticularly designed for the use of the compting-house. The Appendix contains a variety of rules for casting Interest, Rebate, &c. together with a number of the most easy and useful problems, for measuring superficies and solids, examples of forms commonly used in transacting business, useful tables, &c. which are desigaed as aids in the common business of life. Perfect accuracy, in a work of this nature, can hardly be expected ; errors of the press, or perhaps of the au- thor, may have escaped correction. If any such are point- ed out, it will be considered as a mark of friendship and favor, by The public's most humble and obedient Servant, NATHAN DABOLfc. TABLE 0V CONTENTS. Pagt. ADDITION, simple . . 17 of Federal Money ... 22 Compound .... 40 Alligation . 189 Annuities or Pensions, at Compound Interest . 2.P.5 Arithmetical progression .... 194 Barter 138 Brokerage 125 Characters, Explanation of . . . 14 Commission 124 Conjoined Proportion ..... 149 Coins of the United States, Weights of . . 32 Division of "NY hole Numbers S-* Contractions in .... 57 Compound 57 Discount 135 Duodecimals 228 Equation of Payments . . . . 138 Evolution, or Extraction of Roots . . 179 Exchange , . . 151 Federal Money . ! Subtraction ot . 27 Fellowship . . i44 Compound ... . 14f> Fractions, Yulgar and Decimal . . 74, Insurance Interest, Simple 120 by Decimals . 169 Compound . 134. by Decimals . . . MT Inverse Proportion Involution . 17 8 ;uid Gain 140 Multiplication-, Simple 3 Application and vse of Supplimeut to . Compound .... 51 Numeration ....... 15 Practice . . * . . . .109 Position ....... 200 Permutation of Quantifra* 20.4 \'y"l TABLE OF CONTENTS/ Page. Questf y is lor exercise 209 Reduction 63 of Currencies, do. of Coin . 89, 93 Rule of Tliree Direct, do. Inverse . 100, 108 Double . 148 Rules, for reducing the different currencies of the several United States, also Canada and No- va-Scotia, each to the par of all others 96, 97 Application ef the preceding ... 98 Short Practical, for calculating Interest 126 for casting Interest at 6 per cent. . . 215 for finding the contents of Superfices & Solids 220 to reduce the currencies of the different States, to Federal Money . . . 218 Rebate, A short method of nnding the, of any giv- en sum for months and days . . 217 Subtraction, Simple- 5 Compound .... 45 Table, Numeration and Pence .... 9 Addition, Subtraction, and Multiplication 10 . of Weight and Measure . . . 11 of Time and Motion .... 13 showing the number of days from any day ftf one mouth, to the same day in any other month 172 " showing the amount of ll. or 1 dollar, at 5 & 6 per cent. Compound Interest, for 20 years 253 showing the amount of ll. annuity, forborne for 31 years or under, at 5 and 6 per cent. Compound Interest . . . . 335 showing the present worth of It. annuity, far 31 yrs. at 5 & 6 per c. Compound Interest ib. of cents, answering to the currencies of the United States, with Sterling, &c. . . 36 showing the value of Federal Money in other currencies ... . 037 Tare and Trett ^14 Useful Forms in transacting business . " . 233 Weights of several pieces oi English, Portuguese, & French, gold coins, in dollars, cts. & mills 234 ot English & Portuguese gold, do. do. 285 of French and Spanish gold, do. do. DABOU/S SCHOOLMASTER'S ASSISTANT ARITHMETICAL TABLES. Numeration TgbU P*-< us 1 8 O 60 5 o m ts i en C w e P M T3 c 3 70 5 10 80 6 8 a A 3 S w H S a> c IT* GB C H i 1 V fc. -a 3 tr 1 . 'S 90 7 6 100 8 4 110 9 2 ,9 8 7 "5 5 4 3 2 1 120 10 9 8 7 6 5 4 3 2 9 8 7 6 5 4 3 9 8 7 6 5 4 ^^ 9 8 7 6 5 9 8 7 6 mac 9 8 7 4 farthings 1 9 8 12 pence, 1 s 9 20 shillings, 1 . d. i 12 Is I 24 S6 43 60 72 84 96 108 S 4 5 6 7 8 9 120 10 132 11 10 ARITHMETICAL TABLKS. ADDITION AND SUBTRACTION TABLE. 1 2 s 4 5 6 7 8 9 10 11 12 j 2 4 5 6 7 8 9 10 11 12 13 14 3 5 6 I 8 9 10 11 12 13 14 15 4 6 7 8 9 10 11 12 13 14 15 16 5 7 8 9 JO 11 12 IS 14 15 16 17 ^ 8 9 10 11 12 13 14 15 16 17 18; 7 9 10 11 12 13 14 15 16 17 18 19 8 10 H 12 \j 14 15 16 17 18 19 ill) 9 11 12 13 14 la 16 17 18 19 20 21 10 12 13 14 15 16 17 18 19 20 21 22 MULTIPLICATION TABLE. 1 1 2 3 4 5 6 7 : 8 9 10) 111 12 2 hr 4 6 Q 10 12 1-1 16 18 20) 22| 24 6 9 12 15 18 21 24 27 S0| 3S| 36j B 8 12 16 20 2.4 23 32 36 40| 44) 48' 10 15 20 25 30 35 40 4j 50] 65 1 60 6 7 12 Lb 24 30 42 48 0() CO I Goj 72 ii 2l 28 35 70j 771 84 24 !S2 40. o4 72 80| 88j 96 IfO 9~ 36 45 8! OOJ 991108 30 40 60 TO 80 90 IOOI110J120 22 ,> "S l>,'> 44 55 66 -7 a8 99 110|12t|13C [IST] 24 | 3C 48 60 r& 84 96 10S 120J132|1 -'4 To learn this Table fcand column, and the common angle of nice a.lon^ at the right han< you will find the prodi Find your multiplier in the left multiplicand a-top, and in the ting, or against your multiplier, 1, and under your multiplicand, ict, or answer." ARITHMETICAL TABLES. 1* 2. Troy Weight. 24 grains (.gr.) make 1 penny-weight, marked pwt, penny -weights, 1 ounce, oz. 12 ounces, 1 pound, Ib 5. Avoirdupois Weight. 16 drains (dr.] make 1 ounce, oz. 16 ounces, 1 pound, Ib. 28 pounds, 1 quarter of a hundred weight, qr. 4 quarters, 1 hundred weight, cwt. 20 hundred weight, 1 ton, T. By this weight are weighed all coarse and drossv grocery wares, and all metals except gold and silver. 4. Apothecaries Weight. 20 grains (gr.) make 1 scruple, 9 3 scruples, 1 dram, 5^ 8 drams, 1 ounce, 3 12 ounces, 1 pound, ffc, Apothecaries use tms weight in compounding their medicines. 5. doth Measure. 4 nails (na.) make 1 quarter of a yard, qr. 4 quarters, 1 yard, yd. 3 quarters, I fell Flemish, >, 5 quarters, 1 Ell English, E. E. 6 quarters, 1 Ell French, E.Fr. 6. Dry Msasure. Q pints (pt.} make 1 quart, qt, 8 quarts, 1 p. j,k. 4 pecks, 1 bushel, bu. This measure is applied to grain, beans, flax -seed, saU. oats, oysters, coali &c .? ARITHMETICAL TABLES. 7. Wine Measure. '<* (gi.) make 1 pint, ft. C j. 1 quart, qt. 4 qi . 1 pjallon, gal. 51} 1 barrel, ll. 1 tierce, tier. 1 hogshead) hhd. 2 hogsheads, 1 pipe, p> 1 tun, T. All brandies, spirit^, nif?:i'l. vinegar, oil, &c. are mea- sured by wine measure. +\'ote. '231 solid inches, make a gallon. 8. Long Measure. leycorni (ft. c.) make 1 inch, marked in. 1 font, ft. 1 yard, yd. 1 rod, pole, or perch, rd. I furlong, fur. 1 mile, m. 1 lf,i lea. \ deujMN-, on tine earth. * circumferem < irth. the distance of .^th is considered, -t of hordes, 4 inches make In mc^ -ke 1 lathoin, I bj a chain, .-. conuiuiug one hundred links. ARITHMETICAL TABLES. 1$ 9. Land, or Square Measure. 144 square inches maka 1 square foot. 9 square feet, 1 square yard. SOI r square yards, or > d 2/2$ square ieet, 5 40 square rods, 1 square rood* 4 square roods, 1 square acre. 640 square acres, 1 square mile. 10. Solid or Cubic Measure. 1728 solid inches make 1 solid foot 40 feet of round timber, or? . . ~ n f 4 c i ,- , > 1 ton or load., oO feet ot hewn timber, ^ 128 solid feet or 8 feet long, > 1 ^ of w(j()i]> 4 wide, and 4 high, j All solids, or things that have length, breadth and de]>-', are measured by this measure. N. B. The wine gaiiou contains 231 solid or cubic inches, and the beer g; 282. A bushel contains 2150,42 solid inches, 1 1. Time. 60 seconds (S.) make 1 minute, marked ft. _Y 60 minutes, 1 hour, 24 hours, 1 dav. 7 days, 1 wi-rk, 4 weeks, 1 month, 13 months, 1 day and 6 hours, 1 Julian year, yr. Thirty days hath September, April, June, and Noveii February twetity~eight alone, all the rest have ihirtv- N. i?. lu bissextile, or leap year, Ftbnnt; T2. circular Motion. uls ('') make 1 minute, us, I decree, 1 sin, 1) degrees, the whole great CHARACTERS. Explanation of Characters tised in thin Hook. * Equal to. a> !?;.e another, by the rule as 2 : 4 : : ii : i G j that is, as 2 to" 4, so is 8 to x / Pif'ired to anv number, supposes that the square root number,*. i:.e cube root of that ed. the biquadrute ronf, at ionrtli po\vc> 15 ARITHMETIC. ARITHMETIC is the art of commuting by numbers, and lias five principal rules for its operation, viz. Nume- ration, Addition, Subtraction, Multiplication, and Divi- sion. NUMERATION. Numeration is the art of numbering. / It teaches to express the value of any proposed number by the follow- ing characters, or figures : 1 , 2, 5, 4, 5, 6, 7, 8, 9, or cypher. '* the simple value of figures, each has a local value, which depends upon the place it stands in, vi/.. any figure in the place of units ; represents only its sim- ple value, or so many ones, but in the second place, or NOTE. Although a cypher standing alone signifies noth- ing ; yet when it is placed on the rij;ht hand of figures, it in- ereases their value in a tenfold p . l>y throwing them into higher places. Thus 2 with a cypher annexed to it, becomes 20, twenty, and with two cyphers, thus, 00, two hundred 2. "When numbers consisting of m.iny figures, arc given to he read, it will be found convenient to divide them into as many periods 'as we can, of f>ix figures each, reckonirg from tho ri^lit hand towards the left, coi'm- the first the period of units, the. second that of millions, the. third billions, tile fourth tiTtiions, &tr.. as in the following number : 7 S 6 5 4 6 2 7 JJ 9 I 2 5 6 7 9 2 4. Pciind of 1 S. rfr'od of Trillions. I Billions. 8073 625462 2. Pf, ions. 1. Period (f Ur The foregoing number is read thus Ei; seventy-three trillions; six hundred and twenty-five thou- sand, four hundred and sixty-hvo hillions ; s^v- eighty-nine thousand and twelve i;i idred and six thousand, seven hundred and ninety- f- N. B. Billions is substitute \ for millions of millions. Trillions for millions of mi-lions of millions. Qaatrillions for millions of millions of mil'iion of r be. 16 NUMERATION. place o! tens, it becomes so many tens, or ten times its simple value, and in the third place, or place of hundreds, *ht becomes an hundred times its simple value, and so on, as in the following TABLE: 98765 - One. - Twenty-one. - Three fiu rid red twenty-one. - Four thousand 521. - Fifty-four thousand 32*. - 654 th54 thousand 321. - UK7 million 654 thousand 321. - 123 million 456 thousand 789 - 987 million 654 thousand 348. To kno.v the value of anj number of figures. BULK. .merale from the rijjht to the left hand, each fig. n its proper place, by saying, units, tens, hundreds, 'MI mcration Table. U- value of each figure, join the name of ihc left hand, and reading to the right. EXAMPLES. .' the folluicing numbers. Hundred and sixtv-on*. -four. ;y-fifuv thousand and twentjr-ix\ MMl'LE ADDITION. 17 4G1, One hundred and twenty -three thousand four hundred and sixty -one. ;MO. Four millions, six hundred and sixty-six thou- sand two hundred and forty. :.- For convenience in reading large numbers, M - as follows : 987, Nine hundred and eighty-seven. . Nine hundred and eighty-seven thousand. %7 000 ooo, Nine hundred and eighty-seven million. 987 634 3-M. Nine hundred and eihty-sr-ven million, Fsix hn tulied and fifty-four thousand, three hundred and twenty -one. To u-rite numbers. RfJtE. n on the ri^ht hand, write units in t!i? uri's place, tens i.i the tens place, hundreds in the hundreds place, ., towards the left hand, writing each figure ac- conlin^to its proper value in numeration; tal: to sui-piy those places of the natural order with cyphers v.hich are omitted in the question. F.S. Write down in proper figures the fallowing numbers Thirty-six. Two hundred and seventy -nine. Thirty -seven thousand, five hundred and fourteen. Nine millions, seventy-two thousn: Eight hundred millions, forty- i^iud and iii'tv- !i\v. SIMPLE ADDIT! J.S putting together several smaller .of the aame denomination, into one la:^, 1 --. equal to the whole or sum total ; as 4 dollars and six dollars ii. is 10 collars. ]3 SIMPLE RULE. I la vino; placed umts under units, tens undtr-tens, See, a line underneath, and be.uiu with the units; after Adding MI j ire in tliat column, consider how ma- :.t:iinedin their sura: set down the remain - . the units. a!i. x tarry so many as you have tens, lunin of tons; proceed in the same manner t olrniin, or row, and set down the \vhol ! PI.KS. (2.) (3.) "= ^ ^ ~ I - .- r- r^ J.^' = r- 1 \ -J 1756 < 1 0452 7 8 2 G i 698 7 4 (11 64179 5 ; 7 1 . 1 71432 45? 3 2 : 5 5 2 G 2 1 546977 4 1 3 r, 521012 876545 (7.) S 7 1 4 5 51714 60845 57857 61784 52501 ADDITION. 19 ( 2 8 'i (9.) (too 6 4 5 S 4 1 2 S 5 2 6 3 f 1 7 8 4 5 9 3714 7 1 9 6 5 7 2 5 6 3 7147 3 8 4 I 9 2 5 4 1 7 i 8321 5 3 1 9 2 5 1 .3 8 7 2 4 1 S 9 7 5 1457 1726 6 3 1 7 1 8 i 9 o 7 2 8 4 3 7 2513 2 9 1 4 7 ,(1 ) \ (12.) 9 4 2 S 1 7 8 2 9 571845 6 8 7 7 4 2 1 6 1 8 511704 2 2 9 ti 1 4 o 7 9 6 19460 3 7 2 7 6 o S 1 4 5 7 2 8340 7 3 4 6 4 1 2 3 4 270 1 5 5 7 4 1 5 6 5 3 S 6 2 3 5 6 j* 8 9 3 8 T 1 9 5 5 G 2*4 (1 3 3.) 6 4 6 (14.) 5 9 4 6 2 8 1 4 5 1 S 4 4 5 2 1 G 4 3 o 5 4 4 4 3 3 8 7 6 1 4 5 370 5 5 3 2 6 3 4 6 o I 4 4 5 2 1 7 4 4 3 9 4 6 4 7 6 2 6 9 9 8 2 J^ 2 6 8 5 9 J |i t?*To prove Addition, i/egin at the top of the sum, and reckon the figures downwards in the same manner as they were added upwards, and if it be right, this .sum total will be equal to the first: Or cut off the upper lino of figures, and find the amount of the rest j then if the/u mount unper'Iine, when added, be equal to the total, the is fctipposcd to be right. SlMl'I.K ADDITION. i here is another method of proof, as follows : Roject or cast out the ninef; in each EXAMPLE. nm- or sum of figure*, and set down the 3782 . 2 u-H i directly even with the 5766 * 6 its row: find the sum (if these 8755 : 'icn if the ! nines * in the $'im found as before, is IMJIKI! to the 18503 i.iiu-s in t'u- u m total, the work .! to be ri^ht. . 7421, 5063, 2196, and 1245 ior Jlns. 6754. 16. Find the sum of 3482, 783G45, 318, 7530, and 45. Jus. 10473020. 17. Find the sum total of 604, 4680,98, 64, and 54. #NS. Fit? y-tivc hundred. 'A'hat is the sum total of 24674, 16742, 34G78, .'//:<. One hundred thousand. Vdd 1021, 3489, 28765, 289, and 6438 together. J/is. Forty thousand. is the sum total of the following numbers, viz* JO, and 4005 ? Jns. 11 111. >utn total ofthefollowmgnumbei (ircd and forty-seven, j sand six hundred and five, thousand six humlm!. liunilred and c.levcn t!ioi)s,and, ami twenty-;' . and nine thousan Jlnswtr, Gl 374 177 the sum of the following numbers, viz. (1 and i- 'ininlird and five, -ix hundred, mdrrd and eleven tluiusand, .iiid twenty -six 9999999 FEDEUAL MO.VKV. 21 QUKSTIONS. 1. "What member of dollars are in six bags, containing each 375-42 dollars ? An*. 22; 2. If one quarter of a shin's cargo be north eleven thousand and ninety -nine dollars, how many dollars is the whole cargo worth ? Jins. 44396 dols. 3. Moacy was first mcde of gold and silver at Argos, eight hundred arid ninety -f$<'- ;--tme Christ; ho\v long has money been in use at this date, 1314? . 2708 year?. 4. The distance from Portland in the Province of Maine, to Boston, is 125 miles; from Boston to !' Haven, 162 miles; from thence to Neu -York, 88 ; from thence to Philadelphia, 95; from thence to Baltimi 102: from thence to Charleston, South-Carolina, 716; and from thence to Savannah, 119 miles What is the whole distance from Portland to Savannah ? Jns. 1407 miles. 5. John, Thomas, and Harry, after counting their prize money, John had one thousand three hundred and seventy-five dollars : Thomas had just three times as ma- ny as John : and Harry had just as many as John aud Thomas both Pi-ay how many dollars had Harry ? Ans. 5500 dollars. FEDERAL MONEY. JN EXT in point of simplicity, and the nearest allied to whole numbers, is the coin of the I'nited States, or FEDERAL MONEY. This is the most simple and easy of all monev it in- creases in a tenfold proportion, like whole numbers. 10 mills, (m.) make 1 cent, marked c. 10 cents, 1 dime, d. 10 aratri\ or pt>inr, may be called so Many rents instead ot dimes and cents; for the purr uf "idv the. ten's place in cents ; because ten rents make a dime; for example, 48, 75* forty-eight dollars, :i climes live cents, may be read forty-eight dollars :its. dti^rrvod tint all tho figures at the left hand ix nn- dollars ; or you may c;ill the first figure "r t\-iu;!f*. kc. Tims nny sum of this differeothr. either wholly in tl ly in the higher, and partly in thr low- S7 Jit, may !) *-ithor rrnd 5>7. r >4 ecu' ' lollars 5 dimes and 4 cents, or ;!c. 7 (lollari r j dimes and 1 co.nta. A.DDIT16N Of FEDERAL If the eeuts are leas than ten, place a cypher in (he ten's place, or place of dimes. Example. Writa down four dollars and 7 cents. Thus, g4, 07 ots. EXAMPLES. 1. Find the sum of 304 dollars, 39 cents; 291 dollars, 9 cents; 136 dollars, 99 cents; 12 dollars and 10 cents, 39 T304, I 291, I 136, ,. j n, 09 Thu3 '} 13( 99 L 12, 10 Sum, 744, 57 Seven hundred forty-four doi lars and fifty-seven cants. g. cts, g. ct S64, 00 3287, 80 21, 50 1729, 19 8, 09 4249, 99 0, 99 140, 01 (6.) g. cts. 124, 50 9, 07 0, 60 231. 01 0, 75 24, 00 9, 44 0, 95 S Vv'hat is tneaum total of 127 dols. 19 cents, 78 Sj cwnts ? j?ns. S446, 54 ct-\ S4 ADDITION OF FEDERAL MONET. 9. What is the sum df 378 dols. 1 ct. 156 dels. 91 cts. 844 dols. 8 cts. and 365 dols. ? Jus. 1224. 10. What is the sum of 46 cents, 52 cents, 92 cents and 10 <*! .*N. g2. 11. What is the sum of 9 dimes, 8 dimes, and 80 cents? JHS. 82 A. 12. I received of A B and C a sum of money ; A paid me 9 > dols. 43 cts. B paid me justf three tint- much as A, and C paid me just as much us A and li both j can you tell me how much money C paid me r Jinn. $381, 72 cents. IS. There is an excellent well built ship just returned from the Indies. The ship only is valued at 12145 dols. 86 cents; and one quarter of her carjjo is worth 25411 dols. 65 cents. Pray what is the value of the whole ship and cargo? J/ Timothy Taylor, Dr. 1814. S. April 15. To - yds. ofClnth, lit ti, 30 p, oat, To 1 To in, I',;, MI n t ol Conges*, all the acr<-' SIMPLE SUBTRACTION. 2S SIMPLE SUBTRACTION.] Subtraction of whole Numbers, 1 EACHETH to take a less nu^er from a greater, f the same denomination, and thereby shows the difference, or remainder : as 4 dollars subtracted from 6 dollars, the remainder is two dollars. RULE. Place the least number under the greatest, so that units may stand under units, tens under tens, &c. and draw a line under them. 2. Begin at the right hand, and take each figure in the lower lire from the figure above it, and set down the re- mainder. 3. If the lower figure is greater than that above it, add ten to the upper figure ; from which number so in- creased, take the lower and set down the remainder, car- rying one to the next lower number, with which proceed a* before, and so on till the whole is finished. 1'iiooF. Add the remainder to the least number, and if the sum be equal to the greatest, the work is right EXAMPLES. (I.) (2.) (3.) Greatest nwnter,2 468 62157 879647:* L-ast number, 1346 12148 1643489 Difference, (4.) (5.) (6.) 41G78S39 918764520 5452167890 Take S154C999 91213806 12345697098 SIMPLE SUBTRACTION. (7.) f8.) From 917144043G05 3562176255002 Take 40600832164 1235271082165 Rem. (9.) (10.^ (11.) From 100000 2521 OT5 200000 Take 65321 2000000 99999 13. From 560418, take 29r>752. 3ns. 66666. 14. From 765410, take 347-47. dns. 730663. 15. From 341209, take 198765. 3ns. 142444. 16. From 100046, take 10009. 3ns. 90037. 1 7. From 2637804, take 2576982. 3ns. 260822. 18. From ninety thousand, five hundred and forty -six, take forty-two thousand, one hundred and nine. 3ns. 48437. 19. From fifty -four thousand and twenty-six, take nine thousand two hundred and fifty -four. .'Lis. 44772. CO. From one million, take nine hundred and ninety- Trine thousand. 3n$. One thousand. 51. From nine hundred and eighty-seven millions, *ake nine hundred ami eighty-seven thousand. 3ns. 986013000. .12. Subtract one from a million, and shew the remain- ,'JHs. 9P9999. QUESTIONS. 1. How much is six hundred and sixty-seven, reati:r hum] red and ninety-five ? \ What is the difference between twice tu tiiree, times fortv -five ? ./MS. HI. 5, llov. much r than 565*iil 72[ iuljeit 1 14. 01:1 New-T.ondon to PKIladelpKia is C:o miles. -London ward? 1 ia, at the rate of .'>!) miK-s vach day, u- would ' frm Philadelphia. 45 miles. SUBTRACTION Ol? FEDKUAL MONEY. 27 5. What other number with these four, viz. 21, 52, 16, and 12, will make 100 ? wins. 19. 6. A wine merchant bought 721 pipes of \vine for 90846 dollars, and sold 543 pipes thereof for 89049 dol- lars ; how many pipes has he remaining or unsold, and what do they stand him in ? . 178 pipes unsold, and they stand him in g!7~. SUBTRACTION OF FEDERAL MONEY. RULE. Place the numbers according to their value ; that is. dollars under dollars, dimes under dimes, cents under cents, &c. and subtract as in whole numbers. EXAMPLES g. d. c. TII. From 45, 475 Take 43, 485 Rem. gl, 990 one dollar, nine dimes, and nine cent*. or one dollar and ninety-nine cents. g. d. c. g. d. c. TII. g. of. c. m. From 45, 74 46, 2 4 6 211, 1 1 Take 13, 89 S6, 1 6 4 111, 114 g. g. cts. g. eta. From 4284 411, 24 960, 00? Take 1993 16, 09 136, 41 Rem. ' ~_ " g. cts. g. cts. g. cts. From 4106,71 1901,08 365,00 Take 221, 69 864, 09 109, 01 4l 11. From 125 dollars, take 9 dollars 9 cents. Ans. SH5, 9 lets. .*. From 127 dollars 1 cent, take 41 dollars 10 cents. Jlns. g85, 91 cts. IM.K Ml.' I. 1 1PL1CATJON. dollars 90 cents, take 10: Jns. i i is. 1 1. From 49 dollars 43 cents, take 18!) dollar-. 15. From TOO dollars, take 45 cts. I-'rom ninety dollars and ten i eat*. 17. From forty-one dollars eight i nine c J/ - rroin 3 dols. t:ik.- 7 rts. 19. From ninetv-nine dollars, take ninetv-nine cents. Jthts. S98. l ct. 30. From twenty do!?, take twenty 1 1 i:N and one mill J/s. 8*10, 7<)rts. mil Is. 21. From three dollars, take one hundred and ninety- nine ( J/2S. gl, 1 ct. From 20 dols. take 1 dime. Jis. 10, 90 ct?. iiine dollars and ninety cen*s, take niuet.v- nincd. Js. remains. '- pri /.e money was -219 dollars, and Tiioma* mucli, lacking -45 cents. How ! Thomas n-ceiyi- : Jjis. Sy4;3T. 55 cts. ' I pri/.c money to the amount of lays out -111 dols. 41 i '. dollars ; i a suit of new clothes: besides 350 dol~. s gambling. How much will ]\v. :'.0-r pa>iiii !>i.-. landlord's bill, which .inm to 85 dols. and H . JJns. g20, 58 M 1' 1 . i: M U L T I P L 1 C A T I O N, E TH to increase, or repeat the greater of iuo ten as there are units in the I : hence it performs the work oma- ' compendious manner. ! inulti])! i .is called the mulii number fouiu ; -:i, is call-- SIMPLE MULTIPLICATION. NOTE. Both multiplier and multiplicand are in gene- ral called factors, or terms. CASE I. When the multiplier is not more than twelve. RULE. Multiply each figure in the multiplicand by the multi- plier; carry one for every ten, (as in addition of whole numbers) and you will have the product or answer. PROOF.* Multiply the multiplier by the multiplicand. EXAMPLES. "What number is equal to 3 times 365 ? Thus, 365 multiplicand. 3 multiplier. Multiplicand 74635 Multiplier 5 s. 1095 product. 5432 2345 9075 456 Product 71034 1432046 11 240613 12 4684114 12 CASE II. When the multiplier consist* of several figures. RULE. The multiplier being placed under the multiplicand units under units, tens under tens, &c. multiply by i significant figure in the multiplier separately, placing first figure in each product exactly under its multiplier; * Multiplication may also he proved by casting out the 9's in th.> two factors, and setting down the remainders ; then multiplying the two remainders together ; if the excess of 8V in then product is equal to the ^excess oi i, the work is supposed to be right. 3* 30 f SIMPLE MULTIPLICATION. then add tin; several products togethernn *Hc same ordin as they stand, and their sum will be the total product. EXAMPLES. \Vhat number is equal to 47 times 365 ? Multiplicand 365 Multiplier 4 7 2555 1460 Ans. 17155 product. Multiplicand, 37864 34293 47042 Multiplier, 209 74 91 340776 75728 Product, 7913576 2537682 4280822 8253 25203 2193 9876 826 4025 4072 9405 6816978 101442075 8929896 92883780 269181 261986 40634 4629 7638 42068 1246038840 2001049068 1709391 IS-? 134092 918^ 87362 100 1171 . .iat is 1he loUl product of 7 i. at number is ctjual to 4(," .in*. SIMPLE MULTIPLICATION. SI CASE III. 'When thei'2 are cyphers on the right hand of cither or both of the factors, neglect those cyphers; then place the significant figures under one another, and multiply by them only, and to the right hand of the product, place as many cyphers as were omitted in both the factors. EXAMl'tBS. 21200 51800 4600 70 36 54000 1484000 1144800 2876400000 55926000 82530 . 5040 9860000 I092150400CO 8109397800000 7065000x8700=61465500000 7496431300x695000=521001885000000 360000X 1200000=452000000000 CASE IV. When the multiplier is a composite number, lhat ', when it is produced by multiplying any two numocrs m the table together ; multiply first by one of those figures and that product by the other; and the last product WJH be the total required. EXAMPLES. Multiply 41564 by 55. S5. 7 289548 Product of 7 5 1447740 Product of 35 2. Multiply 764151 by 48. Jlns. 56678288. 3. Multiply 542516 by 56. .ins. 1018089G. 4. Multiply 209402 by 72. .4ws. lo07GJ>44. . Multiply 91738 by 81. *lns. 7430778. V). Multiply 34462 by 108. .Ins. 5721896. 7. Multiply 615243 by 144. Anf. 885S4992. $5 SIMPLE MULTZFLICATIOW. CASE V. To multiply by 10, 100, 1000, &c. annox to the mul- tiplicand all the cyphers in the multiplier, and it will make the product required. EXAMPLES. 1. Multiply 365 by 10. Ans. 3650 2. Multiply 4657 by 100. jJns. 465700 3. Multiply 5224 by 1000. ' ^flns. 1 5224000 4 Multiply 26460 by 10000. ^K.. 264600000 EXAMPLES FOR EXERCISK. 1. Multiply 1203450 by 9004. ^JiS. 10835863800 2. Multiply 9087061 by 56708. Jins. 515309055188 5. Multiply 8706544 by 67089. Jliis. 584113530416 1. Multiply 4321209 by 123409. Ms. -533276081481 j. Multiply 3456789 by 5C7090. Jus. 1960310474010 6. Multiply 8496427 by 874359. Jins. 74289274] 5293 98763542x98763542=9754257223385764 Application and Use of Mult iplicailon. In making out bills of parcels, and in finding the value of goods ; when the price of one yard, pound, Sec. is giv- en (in Federal Money) to find the value of the whole quantity. RULE. Multiply the given price and quantity together, as iu whole numbers, and the separatrix will be as many figures the right hand in the product, as in the given price. EXAMPLES. 1. What will 35 yards of broad- > g. d. c. m. cloth come to, at S 3, 4 9 6 per yard ? 5 5 Ans. SI 22, .3 6 08=122 doi- [lars, 36 cents. What cost 55 Ib. cheese at 8 cents per Ib. ? ,08 . 22, 80=2 dollars, 80 SIMFtE MLI.TIPLICAT10N. 58" hat is the value of 29 pairs of men's shoos, at 1 51 cents perpa.il 1 ? Ms. &43, 79 cents. 4. What cost 151 yards of Irish linen, at 38 cents per yisi-J ? Ms. 49, 78 cents. 5. What cost 140 reams of paper, at 2 dollars 35 cents per ream ? Ms. g329. 6. What cost 144 Ib. of hyson tea, af 3 dollars 51 cents per Ib. ? Ms. 8505, 44 cents. 7. What cost 94 bushels of oats, at 33 cents per bush,; el ? Ms. 31, 3 cents. 8. What do 50 firkins of butter come to, at 7 dollars 14 cents per firkin t Ms. 357. 9. What cost 12 c\vt. of Malaga raisins, at 7 dollars 31 cents per c\vt. ? Ms. SS7, 73 cents. 10. Bought 57 horses for '.hipping, at 52 dollars pot head ; what do they conn Ms. S1924. 11. What is the amount of 500 !bs. of hog's-lard, at 15 cents per Ib. r Ms. g75. 12. What is the value of 75 yards of satin, at 3 dollars 75 cents per yard? Mt>. 8281, 25 cents. 13. What cost 367 acres of land, at 14 dols. 67 cents per acre ? Ms. S5385, 89 cents. 14. What does 857 bis. pork come to, at 18 dols. 9S cents per bl. ? Jns. SI 0223, 1 cent 15. What does 15 tons of Hay come to, at 20 dols. 78 cts. per ton ? * Ms. S3 11, 70 cents. 1G. Find the amount of the following BILL OF PARCELS New-London, Marcn 9, 1814. Mr. James Paywell, Bought of William Merckaxt, S. cts. 28 Ib. of Green Toa, at 2, 15 per Ib. 41 Ib. of Coffee, at 0, 21 54 !b. of Loaf Sugar, at 0, 19 13 cwt. of Malaga Raisins, at 7, 31 per cwt. '35 firkins of Butter, at 7, 14 per fir. .airs of worsted Hose, at 1, 04 per pair. !U bushels of Oats, at 0, 33 per bush.' "33 pairs of men's Shoes, at 1 , 1 2j?pr pair. Wm rcclved payment in full, WILMAM or WHOLE NUMBBII&. A SHORT RULE. NOTE. The value of lOOlbs. of any article will be just as many dollars as the article is cents a pound. Fir 'lOO Ib. at 1 cent per Ib. = 100 cents = 1 dollar. 100 Ib. of beef at 4 cents a Ib. comes to 400 cents=4 dollars, Sec. DIVISION OF WHOLE NUMBERS. SIMPLE DIVISION teaches to find how many ttmes one whol* number is contained in another ; and also what remains; and is a concise \vay of performing seve- ral subtractions. Four principal parts are to be noticed in Division : 1. The Dividend, or number i> |i:o. ....-; place the product under that part of the dividend used ; then subtract it there- from, aiulbrin^ down the next figure of the dividend to Kind of i .der ; after \vliich, you must :.iiply and subtract, till you ha\e brought down ihe dividend. Multiply the divisor and quotient together nnd add the remainder if thn-e be any to t!ie pryduo 4 - ; if mi will be equal to the divide;, id.* * Anot; '1 \\hich some r.o prove divi li.in is ;i.-> -'i?,. Add '! Vhat is the quotient of 330098048 divided bj 4-207 ? .'!??.. 78464. 13. What ts the quotient of 701Sj8463 ilivided bv 8465 ? 14. How often does 7618584G5 contain 90001 ? J/ 15. IIo\v many times 58473 can you have in 1 19l84v' Ans. SO 1 .- 16. Divide 80208122081 by 912:314. quotient 307140. MORE EXAMPLES FOR KXKUriSK. Divisor. Dividend. >( 4761 4)327879 1. S6( ) 9! 987654)98864 1G54( ) ...0 CASE II. When there nre cyphers at tin* right liaiiil r- v pliers in the divisor, and ; ii-nm tiie. ri^ht hand o!' remaining ones ;is usual, a..d to y"; annex those li^ui'es tut otf and yu will liave the true i - EN 1. Divide 467:,6-.M b . truequoti \7f\ COXTHACTIOXS IV DIVISION. 3* 2. Divide 57P43C6~fl by 6500 5. Divide 4<2 i4i!0 00 hv 49000 ^fl*. 8600 4. Divide 11659112 by 890000. Jin*. 1SUAVW 5. Divide 9187642 by 9170000. Ans. MOtE EXAMPLES. Divisnr. Dividend. Remains, 125000)436250000fQofiet.) 1 20000) 149596478( )7'*47S 90 1 000) 0543472 , r ;l ( V2 1 530 720000)987654000( )534000 CASK III. Short Division is when the divisor docs not exceed 1 RULE. Consider how many times the divisor is contained ill the 1ir*t figure or iiiriires of the dividend, put the result under, and carry as many tens to the next figure as ill ere are ones over. Divide every figure in the same manner till the whole i% finished. EXAMPLES. Divisor. Dh'idend. 2)113415 3)85494 4)39407 5)94379 Quotient 567071 S)120G16 7)152715 8)96872 9)118724 11)6986197 12)14814096 12)570196582 t Contractions in Division. "\Vhen the divisor is such a number, that any tires in the Table, beinj; multiplied together will produce ft. divide the given dividend by one of those figures ; the fj!iiti'nt thence arising by the other ; and ifiie last quo- tient will be the answer. NOTE. The total remainder is fotiml !r rsurtir) ving ilie last remainder bjthc first dtvigor, and j^ciog a the SITPLEMENT TO MULTIPLICAT1OK. EXAMI'LKS. by 7-2. or 8)16:2641 last rein. T i .''>.?() I X 7 C-23 8 8 GS ' . fr*i ran. -f2 Tru* Quotient xM- 2. Dh-itlp irS4i"54 by Hi. 3. n . 6. Pivi-le I l-l^ro by 48. 7. J/i.s. 8. Divi.; i /, 9. I- lv 103. 10. i)ivi'. i44. J 2. To divide by 10, 100, 1COO, &c. MILE. Cat of!' as ma;r ' .if tluxtivt- t 'S ill lli(' (I1-, l-ui". ;ii der ; and the other figure* oi' tliC div: i;t. \ Ml'I.ES. 1. Divide by ' 36 and 5 rr 2. ' 3. .^KNT TO MULTU'LH'AT! To mn' f -,!>.- I; jn. . ati;l tukc i, i j &. SUPPLEMENT TO MULTIPLICATION*. EXAMPLES. Multiply 57 by23i. Multiply 48 by 2$. 2)37 48 23$ . 22 13 5 24 = 111 J'=i 74 9t> j Jinvtcer. .?*. ^ S. Multiply 11 by 50Jv 4. Multij.i'v '-2404 by 8. 5. Multiply 34J by wSws. ( 6. Multiply 6497 by wJs. SS41 $ Questions tr> E.vcrt-isf J/ tlti;ni -ation ami J)ifi*ion. 1. \Vhat will 1)4' tous of hay ct*Kie to, at 14 aircis uf raisins, each 3 i cwt !>;.-. iiiaoy hundred \\oiv.htr Ait*. o. In 3I) j.ieces of cloth, each piece con; yards: h-\v inary yaids in the \.hole? dn>: 6. What Is the product "f 161 iiHiitipiietl bv i r 7. K* a man spends 482 the val'ie of V\>> !>ri/ 9. What, uuii'ber multiplied Uy 9, \\ill mu J/s. 25. 10. '!' e ! >iicnt of a certain number > -4 ; r. a i- ii!"ul : .1 11 \Vua- , af 3.s. pri v : . IS. Vi in* cost 45 oxen, at3A per hem 40 'COMPOUND ADDITION. 13. What cost 144 Ib. of Indigo, at 2 dols. 50 cts. or 250 cents per Ib. .0 w. gSGO. 14. Writedown four thousand six hundred and seven* teen, multiply it by twelve, divide the product by nine, and add 36j to the quotient, then from that sum subtract five thousand five hundred and twenty-one, aJid tiie re- mainder will be just 1000. Try it and see. COMPOUND ADDITION* IS the adding of several numbers together, having dif- ferent denominations, but of the same generic kind, as pounds, shillings and pence, &c. Tons, nuudreds, quar- ters, &.C. RULE. 1. Place the numbers so that those of the same denom- ination may stand directly under each other. 2. Add the first column or denomination together, as in vvh.i'e m: nber.-i : tiien divide the sum by as many of the same denomination as make one of the next Crater; sottiij^ iK-.\ n tin- remainder under the column added. v the quotient to the next MI;M- ior denominp nuing the same to the last, which add, as in simple addition. I. STERLING MONEY, Ji? the money of acrout't in Great-Britain, and is reck- oned in t'.i'.inds, fcluiliu^, Pence and R. ^ee the Police Tables. * T 'Mltion of tliis itir J/IM';' i -I in thr failliin^s ; 1 in \li>- - l'm iht- .'I I in tlu- pounds, to 20 in the l.hii.;^s ; tin ! ar- i ^ P5 (6) ) 4 s. d. - - rids, 111 1 : , - . , . . -.-.---- ,1 6 !.>. How much is the sum of ;i!id six pence, * .,;..... e j>cncc, - - . . .:, - . Jim. . If/. Bo^'Jit aquarv' '.'. lO.s. paid for mty-aine ' u-u -iuHiij^s aul i:at iliii l and jr.e in ? . 1;>(i 4s. Id. n took a pri/.e, and i\n\ it ecjnally ami. ; ?\vu liundied and lorrv : :-,'\*-n jH-iice ; how rnucli . M-14 IS. $$. 00VPOUKB ADDIYJAii, 43 2. TUOV WEIGHT. tf. 16 4 8 6 ' 4 OK. pict. r. t'i. os. p'iti. 9 17 10 12 6 12 15 14 13 2 04 9 11 16 8 7 47 11 3 19 14 5 24 10 12 69 00 1 00 00 12 11 12 12 77 19 3 27 15 11 5 3 *rr. 9 1 17 5 2 6 1 17 4 16 5 2 12 6 1 10 4. Al'OTHRCAHIES WEIGHT. 5 5 ; r - fc 5 3 gr- id 7 2 19 12 11 6 1 15 6 3 12 4970l;i 7617 9 10 1 2 16 9 5 2 12 4-812 iy 6 .1 16 9 001 9 S 2 19 49216 yd. qr. nm. 71 3 3 IS 2 1 10 I 42 3 ft 57 2 2 49 2 3 C. CLOTU MF.ASUttP^. * JK. E. (jr. na. IS. F. qr. rut. 44 3 2 '21 49 4 3 07 1 6 06 2 3 76 G 84 4 i ,v: 2 r, 07 53 2 e 61 2 1 09 2 ii 44 f. pi 1 7 1 2 i 1 5 2 261 300 COMPOUND ADDITION, 6. OKY MEASURE. bu. pk. t]t. 17 2 5 34 2 7 33 S G 16 S 4 27 2 6 50 7 71 261 64 4.1 4 52 3 5 1 94 2 3 34 3 7 gat. ijt. pt. ?i. 39 5 I 17 2 1 2 H 4(1 2 I * I 2 , . V TXi: MEASURE. hint. gal. tjt. pt. 42 (SI 3 1 27 59 2 9 14 1 921- 1G G4 1 1 4 00 3 tun.klul.gnl.ijt. 2 3-', 2 1 59 I 2 52 2 17 3 11 1 i) 1 9 54 19 23 19 /''. in. I.e. -i 2 II 2 3 1 8 I 1 2 9 S 6 2 10 1 1 6 8 3 1 7 8. LOXli MF.AfWRE. 4ti % 58 16 9 ( 17 4 18 7 5 i.; fr. m.fur. po. 86 2 6 x 3G 52 1 7 i 19 1 4 15 2 3 . o 7 ' :r 9. I.AXD Oil SqVA: .rii 19 9 1 5 6 8 4 143 34 10. SOLID MEASUKX. &'. ft. vit!i t!i* j le-i.->t denomination, and ;!" it eitce- fiyinv O V IT it, b"i-;-u\v as mauv units a# m~'. t'.e );t'^t .;' cater : siibtraet it therefrn v:ij*'er*u a(3w nextcuprior rienouiinatiou fertiat wliisbyon &r:'fl 4u COMPOUND NOTE. Tiie iiietuuJ of proof is the same as in simple subtraction. EXAMPLES 1. lfrling Mtniey. 0) ' (2) f.. .s. d. qr. . s. ' f/. . rf. grr ' 10 2 , J.ent 082 11 8 Iteurivcd IS 10 7 3 ' .ins -^ id Due to me (C) (*) (B) . s. rf. . 8 . J. qr. ,. d. qr. i 5 71112 476 10 9 1 Take 4 10 11 4 17 S 1 277 17 7 1 (D) (10) (11) ( / r ' y - " -* " 9 '18 ', 7 1 1 1 ; 1 9 8 0963 7l. 11s. and paid l^/. 17;:. fx,'. - 13. i 14. v:usand t;vo liundi . 7131 COMPOUND SLBTRACTIOV. 47 16. How much d-.'es seven hundied ami tlyht pounds, exci'tM tisirfv-nirie pound?;, til'ttreh shil!' -ti pence JK.S. 17. From one hundred pounds, take lour miiic-* per . 18" Received of four inen,t urns of money, vj/. Tin* first paid me 37/. IK - 7 third ID/. ! s mwch .is ;.H the of'icr t! !--'0, lacking 19. UtY. I d ; io'.e sum received'? wj?2i\ 'ltij 5s. 4rf. 53. THOY WRIGHT. Ib. oz, pet. oz. pu-t. sr. /'- o-r. From (i 11 14 4 19 'sJl 44 9 ].- Take 2 5 1G 2 14 5.3 17 3 16 IS Rcm. bap. pict. s;r. it), os. pivt. %r 2 10 14 942 2 "0 G83 1 9 13 892 9 2 3 3. AVOIHOUPOISE WF.IOHT. V). oz. dr. C. . oz. dr. 7 9 12 7 3 13 7 10 3 IT 5 12 3 12 9 5 1 15 5 13 1 19 10 9 T. cirt. qr. Ik. oz. dr. T. curt. qr. Ib. o.z. dr. 610 11 20 II) 11 S17 12 1 12 9 12 193 17 1 20 12 14 180 12 1 14 10 14 4. AVOTHECAIUES' WEIG.H'I. ih $ 5 39 gr- tb 5 5 D gr. 1'9 87 4 I 17 55 7 3 1 14 9116 1 15 17 10 6 1 18 5. TT.OTH 43 I"/. (?' I?./'. ' ... gt '. ,-jf. flf. T. !.' I O 2 > \ .4 * 1 .*> ]{) fO 3 I f ( 2 27 '. (ft. pf. /?'''. (;/. ',. /''. or: C5 i n 75 :>7 i i a.-; s o 8. LO\G 6.r. m. fur. pn. fr. ir.fir.fn. 11 41 6 f(> 2 2 11 I ID G 4 1 7 SI /<. m.fur ]*>. If. n., fur. p*. If. m. f I 6 7 l(i )' I 3 9 2 " 7 1Q 2 4 9 Id ! 3 5 1118 GO.MPUL'XD SUBTRACTION. 41 9- LAND OR SqUARE MEASURE. J). roods, rods. Ji. r. po, 5.7.^. s^.m. 3 I 10 29 2 17 S99 131 4 1 25 17 I 36 19 15* .#. ??*. /?. ??. llfi '24 72 114 45 *18 140 109 59 61 i:o 16 14 14* 11. TIME.. r/rs. wo. . r/(7. yrs. rfay.c. ft. win. s?c. :?-* II S 1 '24 S52 20 41 2fl 43 11 3 5 14 a-5f) 20 4i> 19 ?r. rf. A. min. SP liO 34 54 S 7 40 56 7 29 40 S6 59 QUESTION'S, &G.. QUESTIONS, Showing the use of Compuuntl Addition and Subtraction. M:\V-YOKK, MARCH L 2, 1814. 1. Bought of George Grocer, 19 C. 2 firs, of Si!!rar. at 52s. per cvvt. 52 10 23 lh.. of -i- 1!). 070 S lonvos (if v iiLar. \vt. /Uib. at Is. !nl. IDs. pakl again at tme time 17'. 11.9. 9ii. iio\v r.ins unpaid? !/?7i.s. 15 -U. 9jrf. 4. Hornnved !()('/. ni;d paid in p: 1 oiM- ri me H. lls. 6/7. ;U anothel* titne 19/. 17 1 -'. ^!^. at time :>i| fliil'MS at ... i.'Cils^l't . I'uc'i ; iii)vv much remains doe, or '//. 5. A, B, and C, drew their pri'/,c tni -:, \\T~. A had 7.)/. l.)s. 4.i'. 1 - A. tacking 15s. 6cF. and (% hat) ji -thj | I !< it I'pfcr 'i Mils lent bini e. lit' lias paid me :u or.r tiir.e f/ i d. i' I. 40 i . be- s'ulf- . ; '. tor ]}.) dols. 'JO ct ' 7. r. . ios. '.W. i order < !): to know what stinv^i uiaxc ut. Lui "JST:J MULTIPLIffiATTOX. 51 S. A merchant had . owed lira. S917/. ICs. &'. V. 9rf. of it; v, r . \ \(\s. 9tl. 9. A merchant b<;:< .'' sa^ar, of "which lie sells 9C. 3qrs. 2:3lb. how much of it remains unsold ? s-. i :-lb. 10. From a fashionable piece of cloth which < -'i'aimul 52yds. 2na. ataylor was o.-dcred (:<> take three suits, each 6yda. qrs. how uiucii reiaaius .{' tin- piece ? 'J -. '.yds. %r. 2a.. 11. The \var between K< i .Vmerica coalmen-, eed April 19, 17T5, aur J jjcacc took place Jan- uary 20tii, 1785 ; hw\y lon^ flitLt u - ci/iitiriup : I '/. COMPOUND MULTIPLiCATl N. COMPOUND Multiplicutlon is when the Multiplicand consists of several denominations, &i.c. 1. To Multiply Federal Muney. RULE. Multiply as in whole numbers, and place the separa- trix a* many figures from the li^ht hand in the product, as it is in the multiplicand, or given sum. EXAMPLES. S cts. g d.c.m. 1. Multiply 55 09j>y25. Z Multiply 49 5 by 9f. 25 * 97 17545 3-T>n35 7ulB 4-410-45 Proa'. g877, 2? S 5 S /*. 3. Multiply 1 cloi. 4 ct5. by f^.l 7, .:<5 4. Multiply 41 cts. 5 mills by fi. Multiply 9 dollars by tiply 9 Cfifs by 7. Multiply 9 mills by 50 Jins. 0, 45' 'T& COMPOUND MULTIPLICATION. 8. There were forty-one men concerned in the pay ment of a sum of money, and each paid 3 dollars and 9 wills; how much was paid in ail? 3ns. 2123 S6cfs. Smith. 9. The number of inhabitants in the United States is five million*; now suppose each should pay the trifling sum of 5 cent* a year, tor the term of 12 years, towards a to tinental tax j how many dollars would be raised fLereby ? three millions Dollars. 2. To Multiply the Denominations of Sterling Monty, Weights, Measures, <'c. RULE.* "Write down the Mu!ti])licand, and place the quantity 4indern much p-r ^ ttraJ il being to multiply the ^iren price iiid ppnc<>. inftnod of .11^}, Ib. tr. t)ii by die quantity COMPOUND MULTIPLICATION. f>& 31 16 8 12 17 10 14 10 7$ 8 9 10 S2 12 10 6. 19 1 68 4.} 11 12 12 Prr ','*! inns. "What cost nine yard., ofcKitii at 5s. 6<1. per yard ? 5 6 price of one yard. Multiply by 9 yards. ..0ns. 2 9 6 price of nine yards. QUESTIONS. ANSWF.ttS. . s. rf. . K. d. 4 gallons of wine, ai 8 7 per gallon. 1 14 4 5 u. Malaga Raisins, at 1 2 cj per cwt. 5 11 3 7 reams of paper, at 17 9 J- per ream. 6 4 6$ 8 yds. of broadcloth, at 1 7 9ipe--yard.il 24 9 Jb. of cinnamon, at Oil 4^ per 1ft. 5 2 2J 11 tons of hay, at 2 1 10 per ton. 23 2 12 bushels of apples, at 1 9 per bush. 1 10 12 bushels of wheat, at 910 per bush. 5 180 2. When the multiplier, thnt is, the <|ua,itity, is a com- posite number, ami greater than 12, take any two such numbers as when multiplier r, will exactly pro- Huce the given quantity, and multiply first by one of those figures, and that product by the other; and the last pro- duct will be the answer. EXAMPLES. What cost 28 yards of cloth, at 6s. lOd. per yard ? " . s. d. 6 10 price of one yard. Multiply by 7 Produces 2 7 10 price of 7 yards. Muli;;>ly by 4 Answer, 9 11 4 price of 28 vjyd?. S 51 60MPOUV& iii'LI'IH.ICATiOX. 4UE*TiOM AXSUTRH 10 prrcwt. 137 9 j vi!*. of cUitli, at 5 !* per vd. 14 ID d 19 Ib*, of indigo, at H 6 per'lb. M is ', L 13 7 per yd. 19 ir> 1 : lit vJs- broadcloth, at 1 2 6 per yd. 124 I ?i Eteavcr luta, at 1 4 apiece. 157 IT COMPOUND MUJLTlrLIOATlON. 55 4. To find the value of a hundred weight, by having the price of one pound. i f the price be far! .icgs, multiply 2s. 4d. by the hijrs iu the price of '.me lb. Or, if tin; price be pe 7JiuIti;iI\ 9s. -id. byf e j-ence In the price of one lb. and fcXAMPLES. What will 1 cvvt. of rice come to, at 2 id. per lb. ? s. J. 2 farthings=2 4 price 1 cut. at id. per Ii>. 9 fart!.i/^3 in the price of I lb. #/<*-. 1 1 price of I cwt. at 9 -\ per lb. What will I cwt. oi lead come to at ;\>. j.er lb. ? s. d. 9 4 J3ns. 5 5 4 Questions. Jnsiyers. 1 cwt. at 2i" per lb. i 3 4 1 din*, at 2;-d. = 158 1 ditto, at 3d = 180 1 ditto, ut 'M = 18 8 1 ditto, atakl sa i IxJ 3 E.^amplfs of Wrights. Measures, . r. /? r. Multiply 41 2 11 by 18 748 38 j/r. 7/J. z. ;/r. 7. . 8. Multiply CO 5 3 6 by 14 ^ o ; - ' o . . Multiply 1 15 48 24 by 5 7 10 2 cds. ft. ft. IP. Multiply 3 b7 by 8 29 56 'Practiced Questions in WEIGHTS & MEASIT. 1. What istlic Moi^it of 7hlids. of sugar, on \Vhat is ' of 6 chests of tea, How much bra..uy in P ing 41 -. 1 pt. ? 4. 5. In ' IK!, and . ' u\v man\ f wood, i. and 96 ! : . ? 7. . 21 o^. 15 pv. *. 1 v.liole? 4-2s. .'6. lOo^. 2pu.'t. * COMPOUND DIVISION. 5f COMPOUND DIVISION, 1EACHES to find how often one nvunber is contained ift another of different denominations. DIVISION OF FEDERAL MONET. $'3 s Any sum iu Federal Money may be divided as a whole number; for, if dollars and cents be written down as a simple number, the whole will be cents : and if the urn consists of dollars only, annex two cyphers to the dollars, and tfee whole will be cents ; hence the follow- ing . ' GENERAL RULE. * 'Writedown the given sum in cents, and divide as im whole numbers ; the quotient will be the answer in cents. NOTE. If the cents in the given sum are less than 10, you must always place a cypher on their left, or in the ten's place of the ceuts, before you write them down. EXAMPLES. 1. Divide 35 dollars 68 cents, by 41. 41)3568(87 the quotient in cents ; and when there 328 u any considerable remainder, you - - may annex a cypher to it, if vo ; cii-ase, 2oS and divide it again, and you will have 287 the milla, &c. Rem. 1 2. Divide 21 dollars, 5 rents, by 14. 14)2105(150 cents=i dol. Jo ct. bat to bring centi 14 int.- fig 1 '; '.>, and TO the rest will be doilais, &.C, rc 5 5. Divide 4 dols. 9 cts. or 409 cts. by 6. tins. 68 c^s.-}- 4. Divide 9 dola. 24 eta. by 12. *fl;w. 77 cii. 59 COMPOUND DIVISION. ' f. Divide 97 dols. 43 cts. by 85. Ms. gl l-lctt. Cm. 6. Divide 24b dols. 54 tts. b"\ JHS. lOHrfs. 8m.=Sl OMs. 8m. ' T. Divide 24 dols. 65 cts. by '248. Jns. 9<-fs. 9m. 8. Divide 10 dols. or 1000 cts. by 25. . 9. Divide 125 dols. by 500. JjD. Divide 1 dollar into 33 equal parts. J/is. SoJs.-f- PRACTICAL QUESTIONS. 1. Bought 25lb. of coffee for 5 dollars; \\hatisthat a pound r "2. If 131 yards of Irish linen cost 49 dols. 78 cts. \\hat i* that per yard r JJii-. 5. If an c\vt. of sugar cost 8 dols. 96 cts. what is that per pmiml r s. 8c/s. 4. If 140 reams of paper cost 529 dols. \vhat is ihat per ream ? 5. If a reckoning of 25 dols. 41 cts. be paid equally among 14 persons, \\hat do tiiey pay a !. gl 81V/S. 6. If a man's wajjes are 235 dols. 8' .hat is that a calendar month ? 7. T!.o s.iui-y of the President, of the 1'niU'd :.-r;'.ft's is |venty-five tfaovsand dollars a year ; v. hat is thai a day ? s. 68 2. To divide the (Ipnnminations n/ Sterling JIuney, ' RULE. vit.h ^ ., cU>nomination as ii Fthe reinainde >-:iiation: llu-n ihv iidcr, if any, as, before; and bu on, till PROOF The same as in Simple Division. (JOMFOUND DIVISION'. 53 EXAMPLES. . s. d. qr. . s. d. . te 97 3 12 2 by 5. 8)27 13 6 Quo't. 19 892 3 9 9f . s. cL . .v. t quotient will be the answer. EXAMPLES. . s. (I- . f.tl. 1. Divide 29 15 by 21 Jlns. I 8 4 2. Divide 27 16 by 32 17 44 3. 'Divide 67 9 4 by 44 1 10 8 4. Divide 24 16 6 bv 36 13 9$ 5. Divide 128 9 by 42 312 (5. Divide 2G9 12 4 by 56 4 16 Si 7 Divide 248 10 8 by 64 3173 8. Divide 65 14 by 72 018 3 *>. Divide 5 10 3 by 8i 1 41 COMPOUND DIVISION. f.sd -C. s tf. 10. Divide 115 10 by 90. 1 5 S ]!. 16 !>v KM. 154 ; G IV Kl. 1 13 6 13. . 4 4 by 144. 4 S r . When tli<* dr. i?,;>r is larj:?, and not a composite nu!i::;ei, you IH.IV divide by the whole divisor at once* alter moaner of long division, as follows, vi2. EXAMTLFS, Divide V28/. 13.. 3d by 47. f.. a. d. * s. (1. 47)1*8 13 5(2 14 9 quotient 94 54 pounds rewninmrr. Multiply by 20 and add in the : produces 693 shillings, which divided V 47 [Ks. in the quotient 223 188 Multiply by 12 and add in the produces 4C.1 pence, which divided as above, girts [9d. in the quotient. * rf - ' 2. Divide iffl 15 4 by 51. A:>. 3. Divide 85 6 5 bv r.^. 1 4. D ; 5 10 ;. 5. Divide IW 8 liV C. Divi:! 740 T 7. Divide 88S 13 10 by 9 COMPOUND D1VISIOX. EXAMPLES OF WEIGHTS, ME AS I- ^^^ T. Divide 14 cut. 1 qr. 8 Ib. f si^^^mjr aroorrg 6 men. C. 0r. Vo. ox. 8)14 1 8 I S 4 8 Quotient. 8 14 1 8 Proof. . Divide 6 T. 11 c\vt. S qrs. 191b. by -1 Jns. IT. 12r;rf. 3r/r 3. Divide 14 cwt. 1 qr. 1C IK by 5 Jlns. Zcvtt.Sfrs. I3lb. Dox. 9a 7 r.~f- 4. Divide 16 Ib. 13 ox. 10 dr. by 6 Jlns. -7(7;. r:Vr. I5 t /r. 5. Divide 56 I b. Go/,. lT|nvt. of silver into 9 equal parts. Ans. 6/0. Soz. Kpict. l^grs.-\- 6. Divide 26 Ib. t 07. 5 p\vt. by 2 I Js. ilb. lor. !/:.'. 1, 7. Divide 9 hlids. 28 jr a l s . 2 qts. by 12 Js. bhhds. 49gols. 2q!s. 1 8. Divide I63bu. 1 pk. 6 qts. by S3 Jlnt. 4bu. Spkf. 2qts* 9. Divide IT lea. 1 m. 4 fur. 21 po. by 21 .4ns. 2 HI. 4fur. \po. 10. Divide 43. yds. 1 qr. 1 na. by 11 2.. 5yds. Sqrs. Sna. 11. Divide 9rE.E. 4 qrs. 1 na. by 5 Jliis. 19yds. Qqrs. Sna.+ 1C. Divide 4J gallons of brandy equally am on"; 144 solilic.rs. Jlns. ]j-ll a-pii-sp. \\ Bought a dozen of silver spoons, wlrteh tocetljer weired Sib. 2 07.. 15 pwt. 12 grs. how much sihor did ac!\ spoon contain ? Atis. Soz. 4pwt. ll^i*. 14. iJn'^ht 17 c'.vt. 5 qrs. 19 Ib. of sugar, and sold eut eneUiirdui itj how much remains unsold ? . llcwt. 3m, S COMPOUXD DIVISION. 15. From ajArce of cl^th r v.; \Vs 2 na. a r; jfllfc 8 ^ * :| '' tool. A Bhe *vhoL ^i\v many C;Ull Cfll . %!*S. i PR 110NS. 1. If 9 vrrds of ck>th cost 4Z. 5s. 7i^. 'at per f.. s. rf. <7r. 9)4 3 7 2 2. If 11 tons of 'nay cost3/. ' ton ? 3. If I 1 2 gallons of brandy cost 4/. 15s. (jr. -Jiat 7s. 1 1 .7. 4. If 84 Ibs. ofchcese cost ll. 16s. ( per ', ? 5. Jioiv s of stockings for 11^. a pair dn I '1 me in ? 6. I! ; 51. 8s. lOi-J. b? rai v a-' t )iere r . contuiai: ; yard ? .;.l of vine ci gallon ? 0. If 1 c\\ t. of sugar cost S/. IDs. u hat " U ^ *' 10. If a man sjionds 7ll. 14s. CJ. a year, ptrcaliMit!;: 11. 't Wales' s ifi Liuf a (1 ' ii the ow am 1 the rmiaindcr is equally oi\ : ar liJ ia number j lio\v m\t REDUCTION. G3 ^.j. 13. Three merchants. A, .15, .and t^ve a ship in tompany. A hath -*, B , arid C . aruM*' for freiht 2281. ILS. 8d. It is reu: ide it amon the owners according to their respecthe s dns. J's share 143 Os. 5d. 11'* share 57 4s. &f. C's s/Mzre 28 12s. id. }'. \ [Jrivateer ha\ing taken a prixc wnrtli GS50, it is divideil into one hundred smarts: of which the cap- tain is to have 11; 2 lieutenants, each 5; l l i tnidship- men, each 2 ^ and the remainder isttibe divided equally amon^ the' .sail ; lijj in J.'2i. Capt-iin's share 753 5!)cf>'. ttezrf's, ^3-4 i! 50cf*. a midshipman.' a 5S137, aji.'t a sa/'f. REDUCTION, TEACHES to bring or chains HUT..! rs frori one name . to another, without altei ; Reduction i, either LK- Descend; small, as pounds into ' ,,,, . . . . - , > -uion. Inis is done by .Vliiltn . , Ascendin>- performed by Uiv' ' RE UCTKiX DESCENDING. RULE. Mu!t : nlv i' 6 '"^ iest ^""'ojnaticn c-vpn, bv so of 1 M.J !i;ve Ui-i.:.;^ht it do.- ' - erclcroftheo 1 KXY.MVLKS. ^ r,, , - 15 - s - 2 "- 2qrs. how many f rf. qrs. Proof. 4)24758 515 shillings. 12)6189 12 ^_ {0)5 115 9d. 61 S9 penca. .4 25 15 9| -4758 farthings. NOTE. In multiplying by 20, 1 added in the 15s.-4> f the Od.-uud by 4 the Sqrs. which must always be done in like cases. 'Ms. lOd. 1 jr. how many farthings ? Tn 4' T* \ 50329 t: -. J -i pence ? 6. In lSs.9, hoy/ mauj French crowns, at 6s. d. ? .ftu. 181. Reduction Ascending and Descending. 1. MONEY. I. In 1C I/. Os. 9d. how many half-rvrro P B09f . In 58090 half-pence, how many pouri V/;.s-. l-l/. Oi S. Bring 23760 half-pence into pounds. viVs. -J9 10s. 4. In 21-I/. Is. S(!. how many sliillin.. '.ccs, sia-- ;;c'?s, ;ui 2ft550Q farthings, 5. In t ; e, and Kn1is!i or French h? . . -MS. 6. : . ho\v many pence anif tig? .)/. ,:j? -_ 7. -v; laiinv gnw b. [n 48 guineas, < -4 : d. 9. In 81 gv.iucaSj at xirs. 4d. each how ir. .Is ? ii) 145. 10. In C450G ponce, }u\v . -;nd . .:S ? '. II. T.i ^i-2 nioitloriv*, at ;">t*. Ciicli. i at 2! 1 \ Li !./) Dutch guilders, at -;.nj 1;'j. i' how many ri- 14. In 50J. how many \iul!in;.>. nin*' -: CS, tbar-penct's. ;MU< ;; ier ? -i./. ^ v. -.'.' ,<,,,/ y;.-;o =, EXAMPLES IN REDUCTION OF FEDERAL MONEY. T. Reduce 745 dollars into cents. 745 dollars 1 Here 1 multiply by 100, the 100 1 cent* in a dollar: but dollars are >readi!y brought into ceutn JUns. 274500 I noting twj ciphers, and" into - J mills uy annexing three cyphers. Also, any sum in Federal money may be written dov. :-. a- a whole [lumber and expressed ill it< !;>>. tion : f>r, wlieti dollars and cents are joined t> a whole number, without a sepnratrix, they viil ? how many cents the given sum contalna: and vlien dol- lars, cents, and mills are so joined ogetoer, tiiey \vi'l shew the whole number of mills iii ii.e given sum Hence, properly Bpeakin^ there is no reduction ^ money : tor cents are readily turned into dollars by cut- ting oii' the two ri^lit hand liinirt.-, and nulls by poiii'ir..- off three figures with a dot : leit'nand of the dot, are dollars j uud the figures cut oft' are CCJHS, vr cents and mil Is. ~. In 34J dollars, how man. ' ;1 mills? :tft. 3. Reduce 48 do 1 .-;. 78 cts. i:::,, -ent.s. *i.s. 4. Reduce 5 dols. 8 cts. ii;to c .>. xlo(>S 5. Reduce 54 dols. S6ct^. 5m. iu to mills. J/-s.j 6. Reduce 9 dols.'9 cts.'9m. into mills. J/is. '. g cts. 7. Reduce 41925 cents into do!!a.~. A . 8. Change 4896 cents into doll.irs. 9. Change 45009 cents into doiiavs. 10. Bring 4625 mills into dollars. ; 02 5 2. THOY AVKIOKT. 1. How many grains in a silver tankard, lib. 11 Oi. 15 pwt.? 63 REDUCTION. if), or. ; I It 15 12 ounces in a pound. ounces. ennwelV-is in one ounce. 4r.) pern: ins iu one penny v/c" 950 J. 8 rains. *3ns. 15 pu-t. : ' .-. 11 oz. 15 p\\t. 2. In l --!o oz. how many p- i. U808P. 5. V -.8 4 i;. :uany pounds? . .;?'.?. .-.-f gold, i -;v. t. 1 |)wt. of silver, li'jw inair^ table pvv't. .al luu-bei of each . . . ju/. Tlie;efi>re 23544 3. AYOIIIDUI'OIS AVKIGHT. !b. 12 oz. liow many ounces ? [Canie REDUCTION. 569 quarters. Proof. fi8 16)101068 2bTG 28)1006C 719 _ , 4)359 14ft). 10066 pounds. 16 89crt. S^rs. Ulb. 60598 10067 1G1068 ounces. Answer. SI In 19 Ib. 14 oz. 11 dr. how many drams ? s. 5099. S. In 1 ton how many drams ? *5ns. 57S449. 4. In 24 tons, 17cwt. Sqrs. 17lb. 5 oz. how mir.j ounces? Jins. 892245. 5. Bring 5099 drams into pounds. . 19M. 14or. 11 dr. 6. Bring 573440 drams into tons. tins. 1. 4 7. Bring 892245 ounces into tons. S.'is. 24 /on*, 17e?rf. Syrs. 17Zft. . 8. In 12 hhds. of sugar, each 11 cwt. 2.lb. how many pounds r 9. In 42 pigs of lead, each weighing 4cv. t. r many father, at 19c\vt. 2nrs, ? .^js. IQfotki 10. A gentfeinan has 20 hhds. of tobacco, e Sqrs. 14lb. and wishes* tw put it info boxes c- . ach, I demand the number of boxes he must A*s. 4. APOTHECARIES' WKTGIIT. 1. In 9!f^ 8^ 13 2^ I9grs. how many grains. .071*. 55799. & 'I* 55T99 grains, how many pounds f ,n. 9fe 83 13 29 I9r. i TO REDUGTIOJT. 5. CLOTH* MKASUKE. 1. In 95 yards, h<>\\ many quai tors and nails ? fir. Ml. 2. In 541 vardi, oqrs. Imi. how many n; ; <. 5469. . 3. In 5783 nails, how many vakils ? Is. Iqr. Sna. 4. In 61 Ells English, how many quarters and nails ? >na. 5. In 56 Ells Flemish, how many (|i:artci> and jiuils? LML 6. In 148 Ells English, how ma;iv Ells rlemi rs. 7. In 1920 nails, how many yards, Ells Flemish, and Ells English ? . IfipJB. F. find96.E. 8. Flow many coats can Li -. of broadcloth, aliowiti- l^ yards to a cuut ? . xil. C. DRY MF.ASVHK. 1. In 136bushel." n'i arts and pints ? 2. In49 bush. SpkSr 5qts. hu\v ma:ivt;ii:. 3.. Iii -; : .r04 }.inK how many bu>V.d-, ? 9 5. ' i % !r, ". 1. . gallon! i/iilt OJ' > KEDt/CTIOS. 71 5. In 1T89 quarts of cider, how many barrels ? . 146/s. 25qts. 6. What number of bottles, containing a pint half each, can be filled with a barrel of cider? !/ns. 168. 7. How many pints, quarts, and two quarts, each an equal number, may be filled from a pipe of wine ? Jins. 144. 8. LONG MKASUHK. 1. In 51 miles, how many furlongs and po' *.(> poles. 2. In 49 yards, how many feet, incites, and barley- corns r . 14rjV. . . rr292*.c. 3. How many inches from boston to New-York, it being 248 miles ? Jn.-;. U7l3280iic/t. 4. In 43 5-2 inches, how many yards ? . 120?/rfs. 2/6. Sin. 5. In 68.2 yards, how many r x2-r-ll = 124?-oft's. 6. In 15840 yards, how many miles and leagues? 3lea. 7. How many times will a oarrif^ "'hoci, 16 feet and 9 inches in oircu i-.l in ^oi.'itr fi-jin New-York to Phila .^;. ; 8. ITov/ many bar'. it being SCO degrees ? Jins. 47o 9. LAND 1. In -U a rods or perches r .2. In C 2JG9~ scjiiare poles, how iria;v. j S. li apiece oT /.-M! ;i n -in of 17 ;ic:-cs, 3 roods, an. :;.utoi' it, many perciie^are therein the re . 4. T ; contain, the '' aero acr.-.s, I rood : in>. - c.aa Iliey bo duided irito, each s': >> r.-ds ? s. 61 shares, and 44 rods ftEDUCTIOW, 10. SOLID MKASl'RE. 1. In 14 tons of hewn timber, how many solid inches* f . l4xo(>x!r'>8=il'20%(W>. 2. lu 19 tons of round timber, how nianv inc' 13I55SO. 5. In 21 cords of wood, ho\v many solid feet ? 4. In 12 cords of wood, how man v .-olid terta;id inc' JH*. 163- '.;>. and 5. In 4603 solid feet oi' wood, Iv.nv nuuivcoiV * I 1. ! 1. In '! . hours, minutes, and seco; ; B888A. 41 f 2. .' 4. I. , how ma: . nfi.anvda- . . -. 6 ho'U'H. JNS. bow ^ur* 7. . to Nover . ia- -, huw in Jj. ' I. ' . 2. Brin:, . 13 25 r qr = 1. TP t.-.liof I'.; .-..MCJls Ut REDUCTION; 73 2. Borrowed 10 English guineas at 28s. each, and 24 English crowns at Gs. and 8d. each; how many pistoles each will pay the debt ? wins. 0. 3. Four men brought each 17/. 10s. sterling value in gold into the mint, how many guineas at 21s. each must they receive in return ? ctfns. 66 %nin. 14s. 4. A. silversmith received three ingots of silver, each weighing 23" ounces, with directions to make them into spoons of 2 oz. cups of 5 <>/,. salts t/f 1 07,. and snuffboxes of 2 <;z. and deliver an equal number of each ; what the number,? Jus. & of cacti, and I oz. > . 5. Admit a ship's cargo front Bordeaux to i pipes, ISO hhds. and 150 quarter casks [i hhds.~| many gallons in all ; allowing every jnnt to be a p;> what burden was the ship of.'' flns. 444 i the. ship's bunlen was 153 tons. 12e. ; 6. In 15 pieces of clot!!, each piece 20yds. how French Ells ? 7. In 1C bales of cloth, each bale 12 pieces, and eactt piece 25 Flemish Ells, how many yards ? .Int. 2250. 8. The forward wheels of a waggon are 14^ feet in circumference, and the hind many more times will the forward wheels turn r the hind wheels, in running from. Boston to lNe\v-V it being 248 miles ? shi?,. 7167. 9. How many tiir.o^ will a .ship 97 feet 6 inches long, sail her length m the distance ot 12800 leagues yan' Ans. . 10. The sun is 95,000,000 and a cannon ball at ib ::ou hall be, -"t tiiAt rate in, flying ft tin r Jns. 11. The Sun travels tin us of the half a year;! luteg and secondsl .35."l863^ !08.. kes does a J*w 13. How lon^ will it take to couir am: ; .V?:s.S03A. 20rr" or larf "Z\i:. -') m. FRACTIONS 14. The national debt of England amounts to about "9 millions of pounds sterling ; how long would it take to count this debt in doHars (4d. 6d. sterling) reckoning \vithout intermission twelve hours a day at the rate of 50 dollars a minute, and 365 days to the year r dns. 94 years, 1*34 days, 5 hours, min. FRACTIONS. JT RACTIONS, or broken numbers, are expressions for ajiv assignable part or an unit or whole number, and (in general) are ol two kinds, viz. VULGAR AND DECIMAL. A Vulgar Fraction, is represented by two numbers pla- ced one above another, with a line drawn between them, thus, 3, f,&c. signifies three-fourths, live-eights, &c. The figure above the line, is called the numerator, and that below it, the denominator, Tl 5 ^ Numerator. 1 IUS ' J {^Denominator. The denominator (which is the divisor in division) hows iiow many parts the integer is divided into : and the numerator (which is the remainder after divi*iou) t' how many of those parts are meant by theiractioii. A fraction is said to be in its k-^st or lowest terms, when it is expressed by the lea>t numbers po.oibic. when reduced to its lowest terms will be i, and equal to J, &c. PROBLEM 1. To abbreviate or reduce fractions to their lowest k LE. Divide the terms of the given fraction by any number r hich will divide them without a romaind icnts again in the same . that there is no number greater than i . them, and the fraction will be in if FRACTION'S. 75 EXAMPLES. 1. Reduce $ to its lowest terfns. 8 144 J9J?- tl >e Answer 2. Reduce ^ff to its lowest terms, rfnsuvrs $ S. Reduce f^f to its lowest terms. $ 4. Reduce 7 \ S T to its lowest terms. 5. Abbreviate $% as much as possible. 4i (\ Reduce ||-| to its lowest terms. 14 7. Reduce -i-f*- to its lowest terms. 8. Reduce -# to its lowest terms. 9. Reduce -]-|J- to its lowest terms. ^' 10. Reduce |4-r| to ^ s lowest terms. PROBLEM II. To find tlie value of a fraction in the known parts of tlie integer, as to coin, weight, measure, &c. RULE. Multiply the numerator by the common parts of the integer, and divide by the denominator, &c. EXAMPLES. What is the value of f of a pound sterling ? Numer. 2 20 shillings in a pound. JQenom. S)40(13s. 4rf. Ans. 3 10 9 3)12(4 12 t. What is the value of of a pound sterling ? Jns. 18s. 5d. Zfy 8. Reduce $ of a shilling to its proper quantity. Ans.' 4. What is the value of | of a shilling ? Jlr 5. What is the value <;f if of a pound troy ? 76 TKACTIOXS. 6 Ho\v rri'ict! is 7 r of an hundred \veight ? Sqrs.Tlb. W-^oz. 7. What is the Value of ; of a mile r . (fur. 6^0. lift. 8. How much . 3yv. S/&. lor. I l 2*dr. 9. Red'- . fcs proper quantity. iln*- ,na. of ahlul. of v ^nJ. 5 5 A- sec. PROBL To red e fraction of any >f the same kind. '.E. Tied'. -t term mention* el i- > the r or: ^.vhich uill be the frac- tion required. ,Bg, 1. Reduce 15 und. Iiiiogral {/art ven sum. |0 4 4 ' N'una. .7, Si. v > !!>. ? r cm| . iia. ? .4j'- 4. ^ 5. ^ s --& ft. ? 7. Ri'ilu, 8. What jwt of au acre u 2 roods, 20 pu! 9. Reduce 54 gallons to the fraction of a hogshead of wine. Jlns. f. JO. What part ef a hogshead is 9 gallons ? Jtos. | 1 1. What part of a pound troy is lOoz. lOpwt. IQgrs. ? Jiu. ft* BECIMAL FRACTIONS. A Decimal Fraction is that whose denominator is an unit, with a cypher, or cyphers annexed to it, Thus, -/y, The integer is always divided either into 10, 100, 1000, &c. equal parts ; consequently the denominator of the fraction will always, be either 10, 100, 1000, or 10009, &c. which being understood, need not be expressed ; for the true value of the fraction may be expressed by writing the numerator only with a point before it on the left hand thus, T j, is written ,5 ; T y T ,45 ; j-ffo ,725, &c. But if the numerator has not so many places as the denominator has cyphers, put so many cyphers before it, viz. at the left hand, as will make up the defect ; so write yfo thus, ,05 ; and rf^ thus, ,006, &c. NOTE. The point prefixed is called the separatrix. Decimals are counted from the left towards the right hand, and each figure takes its value by its distance from the unit's place : if it be in the first place after units,, (or separating point) it signifies tenths ; if in the seeond, hundredths, &c. decreasing in each place in a tenfold pro- portion, as in the following NUMERATION TABLE. C as rj f a: ai -n i* 7654321 254567 Numbers. 78 DliCHuAL TRACTIONS. Cyphers placed at the right hand of a decimal fraction do :f.)t alter its \o.lue, sitoce every significant figure con- tinui 'is the sa:np place: so ,5 ,50 and ,500 are all I' fa or i- But cyphers placed at the U'.l't hand of decimals, de : cvca :!ue in a tenfold pi.^ortioii, by minting them furti'.; 1 !- i'ro.n the decimal pi-Hit. 'I'hus, ,5 ,05 ,GU5, &c. .: hundredth p;i. thou- saiM i'espoctively. It is therefore evident that .Hide of a decimal fraction, compared with id upon the number of its figures, alue of its first left hand figure: tor in - atar- . any figure less than .9 sue! i c. if extended to an infinite number of figures, will not equal ,9. ADDITION OF DECIMALS. RULE. 1 . Plnrc the nn-i'bcr^, \\hether mixed or pure decimals, under each other, recording to the value of their places. 2. Fi;ifl tlieir r.ii-.i ns ia v.hole numbers, and poii. BO many place- 'ccimals. as are equal to the great- est number of decimal parts in any of the given nuiv.'- EXAMPLES. 1. Find the sum of 41,G53-f-S6,05-f 24,009-f 1,6. . '53 &'>' \ parts of .un- nr 1. RAL ;ie law of notation, DECIMAL TRACTIONS. . J For since dollar is the money unit; and a dime being the tenth, a cent the hundredth, and a mill the thousandth part of a dollar, or u:uf., it is evident that any number of dollars, dimes. d mills, is simply the expression of dollars, and ; \irts of a dollar: Thus, 11 dollar*, dimes, 5 cents, =11, 65 or H T 6 /o dol. &c. 2.' Add the following mixed numbers together. (2) (3) (4) lards. Out. Dollars. 46,23456 4o6 48,9108 24,90-400 7,891 1,8191 17,004 11 2,54 3,1030 3,01111 5,6 ,7012 5. Add the following; sums of Dollars together, viz. SI 2,345 65 + 7,891-1-2,34 +14,-f ,0011 - An*. S36.57775, or gS6, 5dL Tcts. T^mills, 6. Add the following parts of an acre together, viz. ,7569 +,23 +,654 +,199 Jlng. 1,8599 acres. 7. Add 72,5-f 32,071 4-2,1574+371,4+2,75 Ans. 480,8784 8. Add 30,07+200,71+59,4+3207,1 Ans. 3497.28 9. Add 71,467+27,94 + 16,084+98,009+86,5 Ans. 300 10. Add .75C9+,C074+,C9+,8408+,6109 Ans. 2,9 11. Add ,6+,099+,37+,905+. Ans. 2 12. To 9,999990 add one millionth part of an unit, and the sum will be 10. 1.5. Find the sum of Two::' ._.... Three hundred and Mxty-five thoysandths, Six teiitlis, and nine Hitllionths, - - - - Answer, 1,215009 CO DECIMAL TRACTIONS. SUBTRACTION OF DECIMALS, RULE. "Place the numbers according to their value ; then sub- tract .is in whole numbers, and point off the decimals as ix Addition. EXAMPLES. Dollars. Inches. I. From 125,64 2. From 14. Take 95,58750 TaJce 5,91 S. From 761,8109 719,10009 27,15 Takte 18.9113 7,121 1,51679 6. From 430 take 245,0075 ,?H.V. 54,9925 7. From 23G dols. take ,549 dols. Jns. &235,451 8. From ,145 take ,09634 Jrw. ,04816 9. From ,2754 take ,2371 .7ns. ,0383 10. From 271 take 215,7 Ms. 55,3 11. From 270,2 take 75,4075 J*s. 194.7925 12. From 107 take ,0007 -. 106,9993 15. From an unit, or 1, subtract the millionth part of itself. . ,9995)99 MULTIPLICATION OF DECIMALS. RULE. 1. Whether they be mixed numbers, or purr from the pn>< ; are decimal places in botii the I actors : and it" * not so many places in tlie p'-odurt t supply the defect K Vefixidg cjpucrs to tJe left hand. DFCIMAL FRACTIONS, 81 EXAMPLES. I. Multiplv 5. 2. Multiply S,0fi4 by ,008 by Product ,041888 6,74552 3. Multiply 25,233 by 12,17 Answers. 307,14646 4. Multiply .2461 bv ,:529 130,1869 5. Multiply 785S by 3,5 27485,5 B. Multiply ,007853 by ,035 ,000274855 ,004 by .004 ,000016 8. What cost 6,21 yards of cloth, at 2 dols. 32 c- mills, per yard? ,1ns. $14, 4d. 5c. B-^.m. . 9. Multiply 7,02 dollars, by 5,27 dollars. .-.. 36,9954rfote. or g36 99cfs. 5-fom. 10. Multiply 41 dols. 25 cts. by 120 dollars. Ans. g4950 II. Multiply 3 dols. 45 cts. by 16 cts. Ans. S0.5520==55cfs. Zmills. 12. Multiply 65 cents, by ,09 or 9 cents. x. gO,0585=5cte. Squills. 13. Multiply 10 dols. by 10 cts. Jins. gl 11. Multiply 341,45 dols. by ,007 or 7 mills. Ans. g2,39-f To multiply by 10, 100, 1000, &c. remove the separa- ting point so many places to the right hand, as the mul- tiplier has cyphers. f Multiplied by 10, makes 4,25 vSo ,425 -j by 100, makes 42,5 (^ by 1000, is 425, For ,425x10 is 4,250," &c. DIVISION OF DECIMALS. RULE. . 1. The places of the decimal parts of the divisor and Quotient counted together, must always be equal to those 82 DECIMAL FRACTIONS. in the dividend, therefore divide as ift whole numbers, and from the right hand of the quotient, point off so ma- ny places for decimals, as the decimal places in the divi- dend exceed those in the divisor. 2. If the places in the quotient are not so many as the rule requires, supply the defect by prefixing cyphers to the left hand of said quotient. NOTE. If the decimal places in the divisor be mere than those in the dividend, annex as manv cyphers to the dividend as you please, so as to make it equal, (at least) to tho divisor. Or, if there be a remainder, you may annex cyphers to it, and carry on the quotient to any de- gree of exactness. EXAMPLES. 9,51)77,4114(8,14 S,8),21318(,0561 76,08 190 231 228 3804 38 3804 38 00 00 3. Divide 780,517 by 24,3 Answers. 32,12 4. Divide 4,18 by ,1812 ,23068 -f 5. Divide 7,25406 by 957 .00758 6. Divide ,00078759 by ,525 ,00150-f 7. Divide 14 by 365 ' ,058356+ 8. Divide 8246,1476 by g604,25 ,40736-f 9. Divide g!8651S,239 by gS04,81 6ll,9-f 10. Divide gl,28 by g8.3l ,154-f- 11. Divide 56cts. by 1 dol. 12cts. ,5 1. Divide 1 dollar by 12 cents. 8,335 -f 13. If 21^ or 21,75 yards of cloth cost 54,317 dollars, what will one yard cost ? SI, 577 NOTE. When decimate, or whole numbers, are to be divided by 10, 100, 1000, &c. (viz,, unity with cyphers) f DECIMAL FRACTIONS. 83 , it is performed by removing the separatrix in the divi- dend, so many places towards the left hand as there ave cyphers in the divisor. EXAMPLES. {10, the quotient, is 57,2 100, - - 5,72 1000, - - ,572 REDUCTION OF DECIMALS. CASE I. To reduce a Vulgar Fraction to its equivalent Decimal. RULE. Annex cyphers to the numerator, and divide by the denominator ; and the quotient will be the decimal re- quired. ' 4 NOTE. So many cyphers as you annex to the given numerator, so many places must be pointed in the quo- tient; aiui it there be not so many places of figures in the quotient, make up the deficiency by placing cyphers to the left hand of the said quotient. EXAMPLES. 1. Reduce -J to a decimal. 8)1,000 Jns. ,125 2. What decimal is equal to {. ? Answers. ,5 5. What decimal is equal to ? - - - - ,75 4. Reduce ^ to a decimal. ------ , 5. Reduce -] to a decimal. ----- ,6875 f>. Reduce I ; J to a decimal. ,85 7 Jl'.-ijig^ toa decimal. ,09375 8. What decimal is equal to ^I - - ,037057+ 9. Reduce y to a decimal. - - - - ...33333 + 10. Reduce 17 \ T to its equivalent decimal. - ,008 11. Reduce & to a decimal. - - - ,1923076+ 84 DECIMAL TRACTION'. SE II. To reduce quantities of several denominations to a Decimal. RULE- Bring the given denominations first to a vulgar fraction by Problem TIL page 76 ; and reduce said vulgar frac- tion to its equivalent decimal ; or RULE 2. Place the several denominations above each other, letting the highest denomination stand at the bot- tom ; then divide each denomination (beginning at the top) by its value in the next denomination, the last quo- tient will give the decimal required. EXAMPLES. 1. Reduce 12s. 6d. Sqrs. to the decimal of a pound. 12 150 4 960)603,000000(,628125 Answer. 5760 2700 1920 7800 7680 100 960 2400 1920 By Rule 5. 6,75 12,5625 ,628125 6 4800 4800 DECIMAL fRAcrioNS. 'J2. Reduce 15s. 9d. 3qrs. to the decimal of a pound. .Ins. ,790625 S. Reduce 9d. Sqrs. to the decimal of a shilling. Ms. ,8125 4. Reduce 3 farthings to the'decimal of a shilling. Ms. .0625 5. Reduce. 3s. 4d. New-England Currency, to the dc eimalof a dollar. Ms. ,555555 -f- 6. Reduce 12s. to the decimal of a pound, .ins. .0 NOTE. When the shillings are even, half the numbci with a >oim prefixed, is their decimal expression ; but if tlie number be odd, annex a cypher to the shillings, and then by halving them, you will have their decimal ex- pression. 7. Reduce 1, 2, 4, 9, 16 and 19 shillings to decimals Shillings 1 2 4 9 16 19 Msu'ers. .05 ,1 .2 ,45 ,8 ,95 8. What is the decimal expression of 4/. 19s. G.Jd. ? Ms. 4,97708-1 9. Bring S4. 16s. 7$d. into a decimal expression. Ms. 34,83229 1G-J- 10. Reduce 2,51. 19s. 5Jd. to a decimal. Ms. 25,9729 JG-f 11. Reduce Sqrs. 2na. to tlie decimal of a yard. Ms. ,875 12. Reduce 1 gallon to the decimal of a hogshead. Mi. .015873^;. 13. Reduce 7oz. 19pwt. to the decimal of alb. tn>y. Jin s. .(562.5 14. Reduce Sqrs. Sllb. Avoirdupois, to the doc'nvnl c.f an owt. .-ins. ,937.5 15. Reduce 2 roods, 1G perches to tlie decimal of an acre. Jin' 16. I'cJuce 2 feet G inches to the decimal of a ; irfvio juris, ,>>o.-o.>o- j- 17. Reduce ofur. iGpo. totlic dcciiKal o! a IS. Reduce 4^ calendar months to the d 8. 86 DECIMAL FRACTIONS. CASE III. To find the value of a decimal in the known parts of th integer. RULE. 1. Multiply the decimal by the number of parts in the next less denomination, and cut oft' so many places for a remainder, to the right hand, as there are places in the given decimal. 2. Multiply the remainder by the next inferior denom- ination, and cut off' a remainder as before ; and so on through all the parts of the integer, and the several de- nominations standing on the left hand, make the answer EXAMPLES. 1. What is the value of ,5724 of a pound sterling? ,5724 20 11,4480 12 1,5040 Jlns. 11s. 5d. l,5qr. 2. What is the value of ,75 of a pound ? J)ns. 15*. 3. What is the value of ,85251 of a pound ? Ans. 17s. Od. 2,4gra. 4. What is the value of ,040625 of a pound ? Jlns. 5. Find the value of ,8125 of a shilling. Jlns. 6. What is the value of ,617 of an cwt. J/i.s. 2f a mile. .flns. 28 !//.. 11, Win. 11. . '..T quantity of ,90 ."5 of an acre. J//s. Sr. 25, l 2j,'/>. 12. What is the value of ,569 of aye;ir of 565 days- ? 13. What is the proper quantity of ,002084 of a pound troy : Ms. 12,00384 -r. 14. What is the value of ,046875 of a pound avoirdu- pois P Jlns. I2,dr. 15. A V hat is the value of ,712 of a furlong? 16. Wnatis the proper quantity of .142465 of a year r CONTRACTIONS IN DECIMALS. PROBLEM I. A CONCISE and easy method to find the decimal of any number of shillings, pence and fax-things, (to three places) by INSPKCTIOX. RULE 1. Write half the greatest even number of shillings for the first decimal figure. 2. Let the farthings in the given pence and farthings possess the second and third places ; observing to increase the second place or place of nundredths, by 5 i!' the shil- lings be odd; and the third place by 1 "when the far- tilings exceed 12, and by 2 when they" exceed 36. EXAMPLES. 1. Find the decimal of 7s. 9$d. by inspection. ,3 =* 6s. 5 for the odd shillings. 39=the farthings iii 9jd. 2 for the excess of 36. . ,391 =dccimal required. DECIMAL FRACTIONS. 2. Find the decimal expression of 16s. 4$d. and 17 8id. Jus. . .819, onrf . ,885 3. Write down 47 18 10 J in a decimal expression. ! dns. 47,945 4. Reduce 1 Ss. 2d. to an equivalent decimal. Ms. 1,408 PROBLEM II. A short and easy method to find th^ value of any deci- mal of a pound by inspection. RULE. Double the first figure, or place of tenths, for shillings, and if the second figure be 5, or more than 5, reckon another shilling; then, after this 5 is deducted, call the figures in the second and third places so many farthings, abating 1 when they are above 1 2, and 2 when above 36, and the result will be the answer. NOTE. When the decimal has but 2 figures, if any tiling remains after the shillings are taken out, a cypher roust be annexed to the left hand, or supposed to be so. EXAMPLES. 1. Find the value of . ,i79, by inspection. 12s.t=doublcot 6' 1 for the 5 in the second place which is to [be deducted out of 7. Add 7$d.=29 farthings remain to be added. Deduct id. for the excess of 12. Ms. 13s. 7d. 2. Find the value of . ,876 by inspection. Jw.. 17s. 6irf. 5. Find the value of . ,842 by inspection. Jus. 16s. lOrf. 4. Find Hie value of t ,097 by inspection. OF CURRENCIES 89 :iEI)UCTK)N OF CURRENCIES. RULES, I 1 OR reducing the Currencies of the several United States* into Federal Money. CASE L To reduce the currencies of the different states, where a dollar is an en: n number of shillings, to Federal Money. They are f~ New -England^ JVfcw- Fort, and ) j Virginia', North-Carolina, $ ^j Jti'iitucky, and \JKnnessee. RULE. 1. \Vhen the sum consists of pounds only, annex a cy- pher to the pounds, and divide by half the number of shillings in a dollar ; the quotient will be dollars. f 2. But if the sum consists of pounds, shillings, pence, &c. bring the given sum into shillings, and reduce the pence and farthings to a decimal of a shilling ; annex said decimal to the shillings, with a decimal point between, then divide the whole by the number of shillings contained in a dollar, and the quotient will be dollars, cents, mills,:&c. * Formerly the pound w;is of the same sterling value in all tlie colonies as in Great-Britain, arid a Spanish Dollar worth 4sG but the legislatures of the different colonies emitted bills of credit, which afterwards depreciated in their value, in some states more, in others less, &c. Thus a dollar is reckoned in JWu- En gland, ,""] Virginia, ' J&ntucky, and J ,A'. Carolina." > Jfete-Jersey, Pennsylvania, 1 ^ <, Delaware; and Maryland. South* Carolina -I S ia. j 4s8 a cypher to the pounds, multiplies the whole by 10, bringing them into tenths of a pound ; then because a dollar is just three-tenths of a pountl N. E. currency, divi- fVin those tenths by 3, brings them into dollars, kr. See Note, pa^e 85. * 8 REDUCTION OF CURRENCIES. EXAMPLES. ). Reduce 737. New-England and Virginia t'flrrencj, :. Federal Monev. 5)730 g cts. g24S->=243 33^ 2. Reduce 45l. 15s. 7 Id. New-England currency, to 20 ^federal money. AdoUar=G)915,623 12)7,500 152,604+ ^ns. 5 625 decimal. NOTK. 1 farthing is ,251 which annex to the pence, 2 = ,50 i. and divide by 12, you wfll 3 = ,75 J liave the decimal required. 3. Reduce 345/. 10s. lljrf. New-Hampshire, &c. cur rencv, to Spanish milled dollars, or federal money. 345 10 11* 20 d . ' 12)11,2500 6)6910,9375 J927 1 * decimai. SI 151, 8229+ .Ins. ' 4. Reduce 105/. 14s. S|rf. New-York and North-Caro- lina currency, to federal money. 105 14 3$ d. 20 12)3,7500 A dol!ar=8)21 14,3125 ,3125 decimal. $264,289 06 Jns. , Or g rfc7n. o V 5. R. in this currency -,4= -* s of a pound ; Iherrfurt* multiply by 10, and divide by 4, brings ike puunds intu dollars, REDUCTION OP OURHEMOIES. 91. C. Reduce 28Z. 11s. 6d. New-England and Virginia currency, to federal money. Jins. $95, 25cs. 7. Change 46S/. 10s. 8d. New-England, Sec. currency, -to federal money. . Jins. 81545, llcfc. Iw.-f- 8. Reduce 351. 19s. Virginia, &c. cun-eiiry, lo ilideral money. Jins. 119, BScfs. 3?n.-|- 9. Reduce 21 4i. 10s. 7d. New-York, &c. currency, to federal money. Jins. S536, 32rte. 8m. + 10. Reduce 304/. 11s. 5d. North -Carolina, &c. cur- rency, to federal money. Jins. g76l, 42cs. 7w.+ 11. Change 219i. Us. 7|d. New-England and Vir- ginia currency, to federal money. Jins. g7Sl, 9-.r^.+ 12. Change 241/. New-England, &.c. currency, into federal money. Jins. g803, 33cfs.+ 15. Bring 20/. 18s. 5d. New-England currency, into dollars. , Jins. S69, 74cts. GA?.+ 14. Reduce 468 Z. New-York currency, to federal mo- ney. Jins. 81170 15. Rfyiuce 17s. 9^d. New-York, &c. currency, to dollars, &c. J?is. 2, 22cfs. 6,5>;i.-f 16. Borrowed 10 English crowns, at 6s. 3d. each, how many dollars at 6s. ean, will pay the debt ? Jins. gll, llcfs. 1m. wt NOTE. There are several short practical methods ^ reducing New-England and New-York currencies Federal Money, for which see the Appendix. CASE II. To reduce the currency of New-Jersey, PennsyJ Delaware and Maryland, to Federal Mbnc ' RULE. Multiply the given sum^by 8, and divide j^anu the quotient, wi^l be dollars, ^cc. ie ,- ? ^ federal 1. Reduce 245J. New-Jersey, &c. curren"" 7nonc J- ; .'S3ic/s. 245x8=1960, and 1960-7-3 =gG5:U=g^ c> in the NOTE. When there are shillings, pence, _: - -- , - . *J1 dollar is 7s. 6 <* * 8 cts. in. 4 Reduce ^ 4 14 8 to federal money, 405 99 84- 5! Reduc^ 19 17> 6 ^ - 85 18 7~ C. Rediv* 417 14 6 - 179 ^ e 14 10 - 602 14 2 ,-Ar.B 160 00 685 71 4 - or 56rf. 2 \ ^d the product will bc F ' decimals of a 94 REDUCTION OF COIN. LV. .vl * Ii lu , h i ply lh , e ? ve .v, r um by / -' 1 A*. Carolina: ( 3 ] ^ the product will be pounds, J p; (^and decimals ot a pound. .Wir-Jpr-vv, "^1 . f Multiply the given sum by 3 \ Pennsylvania, ! ^ J and divide the product by 8, & '\ Delaware, 8{ j ^ j the (juotient will be pounds, \jMarijland. " [^atnl decimals of a pound. 3 J ft, ., /n, ,. "1 fMultiplv thee the ansver in pounds. Georgb. J P: Laud decimals of a po^d. EXAMPLES, In the foregoing Rules. 1. lieduce gl52, GO cts. to New-England currency. 3 45, 780 Jlns.=45 15s. 7,2rf. ^0 But tlie value of auy decimal of a pound, may be found by inspeo 15, 600 tion. 8ec problem II. page 88. 7, 200 2. In $196, how many pounds, N. England currency. S 58,8 flrts.=58 16 r>. Reduce 629 into No \v-York, &C. currency. ,4 !.t> .7">-.=/;-i'i 12 4. Bring S110, 5i cts.^i in. into New-Jersey, &c. t UITU11CV. REDUCTION OF COIN. 95 8110,511 S jJouble 4 makes 8s. Then 59 far- 8)331,533 tilings is 9d. Sqra. See Problem II. page 88. 41,441 fl?js.=41 8s. 9|fZ. bu Inspection. 5. Bring 65, 36 cts. into South-Carolina, &c. cuu- rency. ,7 3),45, 752 5s. 3s^ ANSWBRS. S cfs. . S. d. 6. Reduce 425,07 to N. E. &c. currency. 127 10 5 + 7. Reduce 36,1 1 to N. Y. &c. currency. 14 810*4- 8. Reduce 315,44 to N. J. &c. currency. 118 5 9i+ 9. Reduce 690,45 to S. C &c. currency t 161 2 1,2 To reduce Federal Money to Canuaa and Nova-Scotia Currency. RULE'. Divide the Dollars, &c. by 4, the quotient wm ue. pounds, and decimals ot a pound. EXAMPLES. 1. Reduce g741 into Canada ami Nova-Scotia cur- rency, g cts. 4)741,00 185,25 =185 5s. 2. Bring $311, 75 cts. into Nova-Scotia currency. g cts. 4)311,750 ' 77,9375=77 18s. 9d. 3. Bring g290~, 56 cts. into Nova-Scotia currency. Jhif. 726 17s. 9id. 4. Reduce S2114, 50 cts. into Canada currency. Ans. 528 ls. fi-L > FOR REDUOIXO, OU HU' ing the currem ial I'm- - others. currency od currency, . land, inia, ttcfcy, and Trn-tesffe. Dtlawure, and Maryland. J>V;r-I"orfc, 7?J cA*. Carolina. 'rlaitd, Kentucky, Ten), Add one 4tli given sura. Add one 3d to the given sum. 'AV/r-. Penn?y- < Delaware. ail . q,'.ven ? Add one 5f- tn-ntls to the -urn. ' - Ac Deduct one ' r '.i ft '< rk. n.-duct one itn tlio New-York. | ."id. -|yt!:o ; rodnct in- 7. Multij.ly the ^ivi-n .Miin by divide tiio product by 2S. Mulfij.ly the ur-i'ii sum hv i.i di- c pro- duct bv 7. 1 Ai'-l onc5t!i c. ___ Add nnchuil' ; ly the divide the ji :d di- vidv the pru- ' duct !> FOR KKDUCIN'O, &C. 97 ted States, also Canada, Nova-Scotia, and Sterling, eacji in the left hand column, and then cast your eye to thfc and you will have the rule. South-Caroliiui, and Georgia, Canadj, and Nova-Scotia. Sterling. Multiply the giv- en sum by 7, and divide the product by 9. Multiply the giv- en sum by 5, ami divide the product by 6. Deduct one fourth from the given sum. Multiply the giv- en sum by 28, and divide the product by 45. Deduct one third from the given sum. Multiply the giv- en sum by 5, and divide the product by 5. Multiply the giv- en sum by 7, and divide the product by 12. Multiply the giv- en sum by 5, and divide the product by 3. Multiply the giv- en sum by 0, and divide the product by 16. Multiply the giv- en stun by 15, and divide the product by 14. From the given sum, dedu. twenty-eighth. Deduct one fif- teenth from the given sum. Deduct one ienth tVom the given sum. Add one ninth to the given sum. X> the English mutiny add one tvvccty -seventh. * REDUCTION OF COIN. APPLICATION Of the Rules contained in the foregoing Table. EXAMPLES. 1. Reduce 461. 10$. &/. of the currency of New-Hamjj; shire, into that of New-Jersey, Pennsylvania, &c. . s. d. See the Rule 4)46 10 6 in the Table. -f 11 12 7* Jns. 58 3 H 2. Reduce 25Z. 135. 9\ lid. "Rhode-Island currency, in* to Canada and > ova -Scotia currency. >,zs. 342 9s. Id. 9. Reduce 5241. 8s. 4d. Virginia, &c. currency, into Sterling money. Jlns. 393 6s. 3d. 10. Reduce 214Z. 9s. 2d. New-Jersey, &c. currency, into New -Hampshire, Massachusetts, &c. currency. .flns. 171 lls. 4d. 11. Reduce 100Z. New-Jei-sey, &c. currency, into N. York and North-Carolina currency. .JJns. 106 13s. 4d. 12. Reduce WOl. Delaware and Maryland currency, into Sterling money. Jlns. 60. 13. Reduce 1162. 10s. New-York currency, into Con- necticut currency. w2ns. 87. 7s. 6d. 14. Reduce llQl. 7s. Sd. S. Carolina and Georgia currency, into Connecticut, &c. currency. Jns. 144 9s. 3 Id. 15. Reduce 100Z. Canada and Nova-Scotia currency, into Connecticut currency. Atis. 120. 16. Reduce 116J. 145. 9d. Sterling money, into Con- necticut currency. Jlns. 155 13s. 17. Reduce 104J. 10s. Canada and Nova-Scotia cur- rency, into New-York currency. Jlns. 167 4s. 18. Reduce WQL Nova-Scotia currency, into New- Jersey, &c. currency. Jlns. 150 RULE OF THREE DIRECT. RULE OF THREE DIRECT. r l* i HE Rule of Three Direct Teaches, by having tliree numbers given to find a fourth, which shall have the same proportion to the third, as the second has to the first. 1. Observe that two of the given numbers in your question are always of the ?; le, or kind; one of which must be the first number in stating, and the other the third number : consequently, the first and third num- bers must alwa>> ')e of the same name, or kind : and the other number, which is of the same kind with the answer, or thing sought, will always ; ic second or middle place. l -. The third term is a demand^and may be known by or the like words before -t. > i/.. What will ; "What cost? Kow mair ? JIuw for? How long? or, How much ? &c. 1. State the question: ; lice the numbers s that the first and i ie kind; and the second term of the. same kind with the answer, or thing sought. 2. Bring the ir-t and third terms to the same denom- ination, and reduce T;rm; the quotient will be the an- .:me denomination ,e second ten .;iy be brought into . T! iiestion. icn they 'rform tl; .: a much shorter mrmnar '.era I rule. ';ip!y the. quo- Or Or '1 oy Or first term ;j the last' oi the answer,. RULE OF THREE DIRECT. 101 EXAMPLES. 1 . If 6 yards of cloth cost 9 dollars, what will 20 yards cost at the same rate ? Yds. g Yds. " Here 20 yards, which moves 6 : 9 : : 20 the question, is the third term : 9 6yds. the same kind, is the first, and 9 dollars the second. 6)180 Jlns. gSO 2. If 20 yards cost SO dols. S. If 9 dollars will buy 6 what cost 6 yards ? yards, how many yards will Yds. 8 Yds 30 dollars buy ? , 20 : 30 : : 6 . g yds. 6 9 : 6 : : 30 g 2,0)18,0 Ans. &9 9)180 JJns. ZQyds. 4. If 3 cwt. of sugar cost 81. 8s. what will 1 1 cwt. 1 qr. 24 Ib. cost? 3 cwt. 81. 8s. C. qr. Ib. Ib. s. 112 20 11 1 24 As 53G : 168 : : 1284M. 4 r 168 336 Ik. 168s. 45 28 564 92 556)215712(64,2 2016 1284/6 S2J.2s. 1411 Jlns. 1344 672 672 *ULK OF THREE DIRECT- 'ir of stockings cost -4s. 6d. whnt will 19 -t ? iir of shoos rest 51/. 6s. what \vili one [fair t . 4s. G: 'I. a pound ? 5;/s. 5' Bmight Schcsls c.f su^ar, each 9 c\vi. > hat mo to at 2l. 5s. per cwt. ? 10. If a man's \vagcs are 7C>[. 10s. a yt i : atcahettdar 11. Jf 4i tons df hay will kocp S how many tons will it take to keep 5 cattle tin* same If a man's yearly income 1. that V ? . If a man spends 5s. 4d. per da-- . hat 14. , at iCs. Gd. per wt , 10s. last me ? 15. A owes B 3475^. out B cnmpn; >. pound j pra his (1. IT. nith sold a tankard for \vliat was the weight (-i . EXAMPLES. 19. If 7 yds. of cloth cost 15 dollars 47 cents, ^hst will 12 yds. c< Yds. $cts. yds. < 7 : 15,47 : :'. 12 7)185,64 JNS. 26,52 ==S2G, 52ete. Cut any sum in dollars and cents may beWritter as a whole number, and expressed in its lowest den-, nation, as in the following example: (See Reduction Federal Money, page 67.) 20. What will 1 qr. 9 Ib. sugar come to, at 6 dollars 45 cts. percwt. ? qr. Ib. Ib. cts. Ib. 1 9 As 112 : 645 : : 37 28 57 37 Ib. 4515 1935 . cts 112)23865(213+ .3ns. =g2, 1. 224 146 112 345 9 NOTE 2. When the first and third numbers are fede- r;;l money, you m-.iy annex cyphers, (if necessary) unlil you . .-.Mires at the right ! of the sep:ir;;trix, . e them t*> M li - a. Theii yon may multiply and di- i-i.-l tii^ (inotient will expres-! the least denomination mentioned iu ths 104 RULE OF THREE DIRECT/ EXAMPLES^ 21. If 3 dolurs \vi!I buy 7 yards of cloth, how many yards can I buy for 12C dollars, 75 cents? cts. yds. cts. As 300 ; 7 : : 12075 7 --- yds. 500) 84525(28 l}Mt. If 12 Ib. of Tea cost 6 dols. 600 78 cts. and r ' mills, what will 5 Ib. - cost at the same rate ? 2452 Ib. milis. Ib. 2400 As 12 : 6789 : : 5 - 5 525 - SOO 12)53945 - gcfs.?JT. 225 Ms 282a-f.jni//s,=r2,82,S. 4 900(3yr5. 900 g cts. - 23. If .1 man lays out 121, 23 in merchandize, and :')> gains r>0 dollars, 51 cts. how much will he gain by laying out 12 dollars at the same rate ? (H>i. 25. A man bought sheep at 1 dol. 11 cts. per head, to the amount of 51 dols. 6 cts. j ho\v many sheep did he buy ? Jlns. 46. 26. Bought 4 pieces of clot!), each piece containing 31 yards, at 16s. 6d. per yard, (New-England currency) what does the whole amount to in federal money ? Jlns 27. "When a tun of wine cost 140 dollars, v quart ? .,'//-. 13r. , 28. A merchant agreed with his debtor, that if he would pay him down 65 cents on a dollar, he v.oulii him up a note of hand of 249 dollars, 88 cts. I demand what the debtor must pay for his note ? Jlns. g!62. 4&-/S. 2?tt.' 29. If 12 horses eat up 30 bushels of oats in a week, how many bushels Avill serve 45 horses the same lit) Jin*. 30. Bought a pieca of cloth for g48 7' 19 cents per yard j how many yards did it contain ? " Jin*. 40yds. 2 . 31. Bought 3 hhds.of sugar, each weighintr 8 c\\; 12 Ib. at 7 dollars, 26 cents percwt. what come the, Jlns. . 32. What is the price of 4 pieces of cloth, tlie first piece containing 21, the second 23, the third 24, unu fourth 27 yards at 1 dollar 43 cents a yard ? Jlns. 135 %5cis. 2l-j-23-f 24-J-27=95y(/.s. 33. Bought 3 hlids. of brandy, containing (" gallons, at 1 dollar, 38 cents per gallon, 1 denuuui much they amount to ? 34. Suppose a gentleman's income r,, 1830 dollars a year, and he spends 3 dollars 49 cents a day, one with another, now much will he have sa nd ? ..Jus. j?.k 35. If iny horse stands me in 20 cents ptr ing, what will be the charge of 1 1 horses for th.- that rate ? Jlns. : 106 RULE OF THREE DIRECT. Sf\ A merchant bought 14 pines of wine, and is allow- ed 6 months credit, but for ready money gets it 8 cents a gallon cheaper; how much did he save by paying ready money? Ana. gl41,"l2cenis. EXJIJIPLES Promiscuously placed. 57. Sold a ship for 5371. and I owned | of her; what was my part of the money ? JHS. 201 7s. 6d. 38. If -, 5 6 of a ship cost 781 dollars 25 cents, what is the whole worth r S As 5 : 781,25 : : 16 : 2500 Ans. 39. If I buv 54 yards of cloth for 31f. 10s. what did it cost per Ell English ? Jin*. 14s. 7d. 40. Bought of Mr. Grocer. 1 1 cwt, 3 qrs. of sugar, at 8 dollars 12 cents per cwt. and gave him James Pavwell's note for 19.'. 7s. (New > urrci.cy) the rest i pay in cash : tell me how many dollars will make up the balance ? Jins. gSO, 91 ct*. 41. ll as. 4d. per Ell-English; what did the whole 44. Bo'-iaht 0i- v.inK of cambiirk for '."-0^. hu( beius; :^ed, lam willing to iojp 71. 1()s. by f it; .-t I demand per Ell-En 45. How many pieces ot Holla] i-Kleiu- ish, may 1 have for 23/. 8s. at 6s. 6d. pi:r E:!-l \ merchant bought a l>a! ;d it it 1 1} dollars i--< by the bargain, and how St,is. He gained fc RULE OF THREE DIRECT. 107 47. Bought a pipe of wine for 84 dollars, and found it had leaked out 1& gallons; Isold the remainder at 12J cents a pint; what did I gain or lose ? 2ns. I gained SSO. 48. A gentleman bought 18 pipes of wine at 12s. 6d. (New-Jersey currency) per gallon ; how many dollars will pay the purchase r Jins. gS780. 49. Bought a quantity of plate, weighing 15 Ib. 11 oz. 13p\vt. 17 gr. how many dollars will pay for it, at the rate of 12s. 7d. New -York currency, per ounce ? .Ins. gSOl, 50cfs. 2jV"- 50. A factor bought v a certain quantity of broadcloth and drugget, which together cost 81/. the quantity ot broadcloth was 50 yards, at 1 8s. per yard, and for every 5 yards of broadcloth he had 9 yards of drugget; I demand now many yards of drugget he had, and what it cost him per yard ? .Ins. 90 yards at 8s. per yard. 51. If I give 1 eagle, 2 dollars 8 dimes, 2 cents and 5 mills, for 675 tops, how many tops will 19 mills 1 . 52* Whereas an eagle and a cent just three scy: e yards did buy, How many yards of that same cloth for 15 dimes ha:l I ': 53. If the Legislature of a SMe grant a tax m' an the dollar, how much must that man pay . how much will 49 dollars gain in t 55. Jf GO gallons of water, in one ho-,;. tern con 35 gallons '-uu out filled ? iib. A ;>.r d R Hryiart fr- the same roail of 1 7 6J. r i x El . . . or le.ss >v;".'iT3 ! ;es- Kule of Throe Direct : requires less or / reej A hkii >\vii from the nature and tenor o! a'Mj'lt 1 : 5, limv many day* v ill it require 4 men to mo>\ 7/u-ji require 4 how rniuh time will 4 re- .'.sv/er, 2 tlays. Here more ir^ii! \i/.. days re- - I'UHie, tl'.c more (L . rse Pioportion. RULE. 1. State and rediu c t';e term? as in the Rule < Dire nd divi, ... diMioirii-i.ititfu as tne middle teiiu 1. 1 . . N X 'M* U-::{ r l!l Oi boa!'. r. 't ^ 10. 4. If five dollars will pay for the carriage of 2 cwt. 150 miles, how far may 15 cwt. be carried for the same mo- ney ? Jinn. 20 miles. 5. If when wheat is 7s. 6d. the bushel, the penny loat will weigh 9oz. what ought it to weigh when wheat is 6s,. per bushel ? Jlns. 1 1 ox. Cpwt. 6. If 30 J'ushels of grain, at 50 eta. per bushel, will pay a debt, how many bushels at 75 cents per bushel, will pay the same ? Jlns. 20 bushels. 7. If 100J. in 12 months gain 61. interest, what princi- pal will gam the same in 8 months ? Jlns. 150. 8. If 1 1 men can build a house in 5 months, by work- ing 12 hours per day in what time will the same num- ber of men do it, when they work only 8 hours per day ? Jlns. 7\ montfis. 9. What number of men must be employed to finish in 5 days, what 15 men would be 20 days about ? Jlns, 60 men. 10. Suppose 650 men are in a garrison, and their pro- visions calculated to last but two months ; how many men must leave the garrison that the same provisions may be sufficient for those who remain five months ? Jlns. 390 men. 11. A regiment of soldiers consisting of 850 men are to be clothed, each suit to contain S$ yds. of cloth, which is 1 2 yards wide, and lined with shalloon | yard wide; how many yards of shalloon will complete the lining ? Jlns. 6941yds. Zqrs. Sfwa. PRACTICE. PRACTICE is a contraction of the Rule of 'Hue? Direct, when the first term happens to be aa unit or (.MI , and is a concise method of resolving n.opt qacs occur in trade or business when* money i.-= reckoned in Bounds, shillings and pence: but reckoning ir- Money will render ij)i;. rule al-nost uncles* : t reason ! shall not enlarge so much on the subj^v. ry ether writers have dihie. 110 1WACTK11. Tables of Jlltquot, or Even Tarts. Paris of a Shilling. i 6 8 r. j 50 i 40 34 j.- 26 1 8 T V - uf a cwt. Ib. cict. 56 is i 28 = i 16 * The aliquot part of any number, is such a part of it. as being taken a certain number of ^times, exactly makes that number. CASE I. the price of one yard, pound. &c. is an even part of one shilling. Find the value of the given quantity at Js. a yard, pound, &c. and divide it bv that oven part ;.!jrl the Quotient will be the answer in shillings, &c. Or find the value of the given quantity at 2s. per yard, &o. and divide said value by the even part which the pvtn price is of s. and the quotient will be the answer in shi 1 lings, c. \\liichreducetopounds. N. B. To find the value of any quantity at 2s. you need only double the unit figure for shillings; the other fig- ures will be pounds. EXAMPI.KS. J. What will 461$ yards of tape come to. at l}d ] 01 J$d. 1 1 | 461 b value of 4d 1 ; vd. 5,7 8* 2 l?s. 8|J St. WHiat cost 256lh. i.i I" ( uul ." 8d. | || 25 12s. value ol 8 10s. feJ. jnd PRACU.Itf. lit Tarn's, per y-' s - & 4otU at lil. .Insurers, 206$ : tit 2d. 738 911 at 3d. 11 7 9 749 at 4d 12 9 8 113 at 6d 2 16 6 891> at 8d. 29 19 4 CASE II. When the price is an even part of a pound Find the value of the given quantity at one pound per yard, c. and divide it by that even part, and the quotient will be the answer in pounds. EXAMPLES. Wfcat will 129$ yards cost at 2s. 6d. per yard ? s.d. . s. . 2 G | j | 129 10 value at 1 per yard. s. 16 3s. 9ef. value at 2s. 6d. per yard. Yds. s. d. . s. rf. 123 at 10 per yard. Answers. 61 10 C87i at 5 " 171 17 6 21U at 4 42 5 543 at 6 8' 181 127 at 3 4 21 3 4 461 at 1 3 i. 38 8 4 NOTE. When the price is pounds only,4he given quan- tity multiplied thereby, will be the answer. EXAMPLE. 11 tons of hay at 4l. per ton. Thus 11 '4 Ans. 44 CASE III. When th given price is any number of shillings u ;u Ls of tiie quotif.iT i\\r the remainder of and the&Min i' ti. cjt.'ifiout.s will be iswer in shillings, &c. wl ce to po: KXAMF; Wl-.at will 245 Ib. of raisins come to, nt 9?d. pp.r Ib. F 5. rf. 4 245 value of 245 Ib. at 15. per pou' Jaeof do. at 6d. pci Ib. 4 61 S value of do. at 5d. JUT Ib. 15 3| value of do ,0)19,9 OJ ^n. 9 19 Oj value of the whole at S. PRACTICE. US 372 at 1J Ms. 2 14 3 35 at 2i .30 Hi 827 at 4i 15 10 1} 576 at 7* Ms. 18 541 at 9k 20 17 0* 672 at 112 32 18 CASE V. When the price is shillings, pence and farthings, and not the aliquot part of a pound Multiply the given quan- tity by the shillings, anu take parts for the pence and far- things, as in the foregoing cases, and add them together; the sum will be the answer ia shillings. EXAMPLES. 1. What will 246yds. of velvet come to, at 7s. 3d. per yard ? s. d. 3d. | i ( 246 value of 246 yards at Is. per yd. v 1722 value of do. at 7s. per yard. 61 6 value of do. at 3d. per yard. 2,0)178, 3 6 S/;s. 89 3 6 value of do. at 7s. per yard. ANSWERS. s. rf. . s. rf. ; Z What cost 159 yds. at 9 10 per yd. ? 68 6 10 3. What cfst 146 yds. at 14 9 per yd. ? 107 13 6 ' 4. What cost 120 cwt. at 11 - 3 per cwt. ? 67 10 5. What cost 127 yds. at 9 8$ per yd. ? 61 12 11$ 6. What cost 49| lb at S 11J per Ib. ? 9 15 Hi CASE VI. When th" price and quantity given are of several de- n-ominations-^Multiply the price by the integers in the i^ ven quantity, and take parts for the rest from the price of an integer ; which added together will be the This is applicable to Federal Money, 10* 114 TARE AND TRETT. EXAMPLES. 1. "W L41b. of Id. per c 2qrs iqr. 14 Ib. Ai IC.gr 17 S 5 1 14 S 12 tatc rais wt. i 4 4 * : S. 4 1 2 ost 5cwt. 3qrs. ins, at 2. 11s. ? . s. A 2 11 8 5 2. W! 8lb. of s 65 cts. p Iqr. 7 Ib. 1 Ib.t lat uga ere i i f cost 9c\vt. Iqr. , at 8 dollars, wt.? gets. 8,65 9 77,85 2,1625 ,5406 ,772 12 18 4 1 5 10 12 11 6 54 15 S 6* , Jtns. 280,6303 'O. ANSWERS. 6 at g9, 58cts. per cwt. g75, Glcfs. Swr. at 2/. 17s. per cwt. 14 19s. Sd. 7 at 01. 13s. 8d. per cwt 10 Qs - 5 &- 7 at g6, 34cts. per cwt. g76, 47cts. 6m, 4 at gll, 91ct8. per c\vt. g2, 55cte. a^iu. TARE AND TRETT. .1 ARE and Trett arc practical Rules for deducting certain allowances which are made by merchant?, in buying and selling goods, &c. by weight ; in which are noticed the following particulars : 1. Gross Weight, which is the whole weight of any sort of goods, together with the box, c-^k, or bag, &. which contains them. Tare, which is an allowance made to the buyer tor the weight of the box, cask, or bag, &c. which con- tains the goods bought, and is either afcr'somuch per box &C. or at so much per cwt. or at so much in the whole ross weight. 3. Trett, which is an allowance of 4 Ib. on every 1 tl!> for waste, dust. &c. TARE AND 1HETT. H5" 1. C'lfff, which is an allowance made of 2 Ib. upon every 3 c\vt. 5. Sutt'tti is what remains after one or two allowance? have been deducted. . CASE !. When the question is an Invoice. Add the gross weights into one sum and the tares into another ; then subtract the total tare from the whole gross, and the re- mainder will be the neat weight. EXAMPLES. 1. What is the neat weight of 4 hogsheads of Tobacco marked with the gross weight as follows : C. tjr. Ib. Ib. Tare 100 95 83 Si 359 total tare. J$o. 1 9 12 2 3 3 4 3 7 1 4 6 3 25 Whole gross 52 13 Tare 359 lb.=3 23 s. 28 3 IS seat. 2. "What is the neat weight of 4 barrels of Indigo, No Mvl weight as follows: C. (jr. Ib. Ib. No. t 4 1 10 Tare S6"| 2 3 3 02 29 [ 3 40 10 cwt. qv. Ib. 4 40 35 j Jlns. 15011 CASE II. When the tare is at so much per box, cask, bag, &c. Multiply the tare of 1 by the number of bags, bales, &c. Hie product is the whole tare, v.hich subtract from fhc and the remainder will be tke neat weight. EXAMPLES. i 1. In 4 hhds. of sugar, eacli weighing lOcwt. Iqr. loU). gvuss ; taie 75lb. v-cr njid. how much neat ? 116 , YARK AKB TASTT. tii't. grs. Ib. 10 1 15 gross iveig!;t of one Khd. 4 41 2 4 gross weight of the whole. 75x4=2 2 20 whole tare. Jns. 38 3 12 neat. 2. \Vhat is the neat weight of 7 tierces of rice, t weighing 4cwt. Iqr. 9lb. gross, tare per tterce S41' Ans. 28 C. Oqr. Zllb. 5. In 9 firkins of butter, each weighing 2qrs. J gross, tare 11 Ib. per firkin ; how much neat ? Jjn. -1C'. %r>. 4. In 9A\ bis. of figs, each Sqrs. 19lb. gri . '!b. per barrel; how many pounds neat? Jns. 2241.'. 5. In 1 6 bags of pepper, each 85lb. 4oz. gross, tare per '.i>. 5oz. ; how many pounds neat ? /<>. 1311. 6. In 75 barrels of figs, each Crjis. i!lb. gross tare in the whole, 597lb. ; how much neat weight : sins. 506'. Iqr. 7. "What is the neat weight of 15 hluls. of Tol)acro.. each weighing 7cwt. Iqr. 13lb. tare lOOlb. ppr lihd. ? jffns. 97 C. Oqr. 1Mb. CASE III. "SVhen the tare is at so much per cui. Divide the gross weight l>v the aliquot part of a rv T. I'm which subtract from the gross and tin- '-r will be neat weight. EXAMPLES. 1. What is the neat weight of 44cwt. Tqr?. grow, tare 14lb. per cwt. ? . C. qrs. Ib. I 14lb. | J | 44 3 1G greso. 5 2 J neaf. TAUE AXD TIIRTT. 117 2. What is the neat weight of 9 hhtk. of tobacco, each weighing gross Scwt. Sqrs. 14lb. tire I61b. per cwt. ? .flws. 68czt-. l//r. 24i'&. 3. What is the neat weight of 7 bbls. of potash, each \vcighing 29llb. gross, tare 10'ib. per wt. ? jJns. 1281ft. 6oz. 4. In 25 barrels of figs, each Scwt. Iqr. gross, tare per cwt. I61b. j how much neat weight? Jkns. 48cwt. 24ft. 5. In 83cwt. 3qrs. gross, tare 20lb. per cwt. what neat weight ? Jlns. GScwt. Sqrs. 5lb. 6. In 45cwt. Sqrs. 2llb> gross, tare 8lb. per c\vt. how much neat weight r Ans. 42cw>. 2qrs. i7ift. 7. What is the value of the neat weight of 8 hhds. of sugar, at g9, 54cts. per cwt. each weighing lOcwt. Iqr. 141b. gross, tare 14lb. per cwt. ? Jlns. R092, S4c#s. 2Jtn. CASE IV. When Trett is allowed with the Tare. 1. Tint] the tare, which subtract from the grass, anil call the remainder suttle. 2. Divide the suttlo by 26. and the quotient will be the trett, which subtract from the suttle, and the remainder wiH be the neat weight. EXAMPLES. 1. In a hogshead of sugar, weighing lOcwt. Iqr. 12ft). gross, tare 14lb. per cv. t. tiett 4lb. per 104lb.* how much neat weight ? * This is the Irett allowed in London. Tketreason of dividing by 26 is because 4ft. is ^ of 104Z&. but if the trett <'.- at any other rate, other parts must be taken, ac- cording to tf.e rate proposed, Sfc. J1& TARF. AND Or thus cwt. qr. Ib. cict. gr. fti. 10 1 12 14ll>=})10 1 1C c; 4 115 tare. 26)9 7 sultle. 111 trett. 3ns. 8 2 24 neat. M=J.)1160 gross, 145 tare. 26)1015 suttle. 39 trett. Stts. 97&lb. neat. 2. I?i 9 c\vt. 2 qrs. IT Ib. gross, tare 41 Ib. trett 4 Ib. per 104 Ib. how much neat? Jlns. 8c?rf. S^r.s. 20/6. 3. In 1J chests of sugar, weighing 117c\vt. 21 Ib. gross, tare 1T3 Ib. trett 4 Ib. per 104, how many cwt. neat ? Jlns. lllctct 22/6. 4. "What is the neat weight of 3 tierces of rice, each weighing 4 cwt. S qrs. 1-4 Ib. gross, tare 16 Ib, per cwt ing trett as usual ? Jlns. IQcict. Qqrs. 6lb. 5. Tri 25 iiarri'ls <>t figs, each 84 Ib. gross, tare 12 Ib. per < .. t. trett 4 Ib. jcr 104 Ib. ; how many pounds neat r '3 + value of the neat weight of 4 b.incU >'; uumbers, weights; anil allovanrv* as ii per pouii'i f 1 cict. /JTS, /A. No 1 (i.n^ I 2 l.n 2 1 Tare. 10 Hi. ] S 1 09 f Trett 4 !b. ,MT 10- 4 tt r? 21 TARE 4ND TRETT. 119 CASE V. When Tare, Trett, and Cloff' are allowed : Deduct the tare and trett as before, and divide tli-e sut- tle by 168 (because 2 Ib. is the T ]-g of 3 cut.) the quo- tient will be the cloflf, which subtract from the suttle, and the remainder will be the neat weight. EXAMPLES. 1. In 3 hogsheads of Tobacco, each weighing 13 cwt. 3 at is the interest f 571?. 1 3s. 9d. for one year, at 62. per cent. ? .fln?. 34 6s. Oirf. 4. What is the interest of 2/. 12s. 9$d. for a year, at 6^. per cent. ? tins. 3s. 2rf. FEDERAL MONEY. 5. \Vhat is the interest of 468 dols. 45 cts. for one ye&r at 6 per ce^t. ? $ cts. 468, 45 6 281,10, 70=g28, lOcfs. 7m. Ansf Here! cut off the two right hand integers, which di- vide by 100 : but to divide federal money by 100, you need only call the dollars so many cents, and the inferior denominations decimals of a cnt, and it is do:ie. Therefore you may multiply the principal by the rate, and place the separatrix in the prociuct, as in multiplica^ tion of federal money, and all the figures at the left of the separatrix, will be the interest in cents, and the first figure on the right will be miffs, and the others decimab of a mill, as in tbe following EXAMPLES. 6. Required the interest of 155 dels. 25 cts. forayeai at 6 per cent. S cts. 155, 25 6 811, 50=8, llcfr. 5."i. Jlns. 7. "\Vhatis tin- interest of 19 dulars 51 cents t year at 5 per cent. ? $ cts. 19, 51 5 97, 55=f>7rf3. 5$ir 8. What is the interest of 436 doi.at? for or.o v. 6pei*rcnt. f 6 Jlns. 1-22 SIMPLE INTEREST ANOTHER METHOD. Write down the nvep principal in cents, which multi- ply b) the rate, and divide by 100 as before, and you will have the interest for a year, in cents, and decimals of a cent, as follows : 9. What is the interest of ,g73, 65 cents for a year, at 6 per cent. ? Principal 7365 cents. 6 * * * .5ns. 441,90cte.:=44l T yte. or.g4,41cfs. 9m. 10. Required the interest of 80, 45cts. for a year, at 7 per cent. ? Cents. Principal 8545 7 Arts. 598, 15 crnfs,=S5,98cfs. Ijm. CASE II. To find tlie simple interest of any sum of money, for any number of years, and parts of a year. GENERAL -RULE. 1st. Find the interest of the given sum for one year, iid. Multiply the interest of one year by the* given number of years, and tiie product will be the answer for that time. Gd. If there be parts of a year, as months and days, 'lie moiii' (tnd of Three iiiroct. ing SO days to the mouth, ; ./.juut part.: uf the -same.* * By a!!mvii .i*juot , r.rJ.iua.y sum :u be very .f tluvi : ' t\ . SIMPLE INTEREST. 123 EXAMPLES. 1. What is the interest of 751. 8s. 4d. for 5 years and 2 months, at 61. per cent, per annum ? . s. d. 75 8 4 4)5.2 10 20 . s. d. 2mo.=|)4 10 G Interest for 1 year. 22 12 6 do. for 5 years. 15 1 do. for 2 months. 23 77 Ms. 2. What is the interest of 64 dollars, 58 cents, for 3 years, 5 months, and 10 days, at 5 per cent. ? 864,58 5 322,90 Interest for 1 year in cents, pw 3 [Case I. 4 mo. ^ 1 mo. J lOdays,^ 968,70 do. for 3 years. 107,63 do. for 4 months. 26,90 do. for 1 month. 8,96 do. for 10 days. Ms. 1112,19=1112cs. or gll, 12c. 1 T V. S. What is the interest of 789 dollars for 2 years, at 6 per cent. ? Ms. 94, tiScfs. 4. Of 37 dollars 50 cents for 4 years at 6 per cent, per annum ? Ms. flGUcfcj. or g9 5. Of 325 dollars 41 cents, for 3 years and 4 months, at 5 per cent. ? Ms. 54, 23cfe. 5;n. 6. Of 325Z. 12s. 3d. for 5 years, at 6 per cent, f Jus. 97 13s. Sd. 7. 'Of 174?. 10s. 6d. for 3 and a half years a; cent. ? Jliis. 8. Of 150?. 16s. 8d. far 4 years and 7 months, ai cent? Ms. 41 9s. ~d. SlMl'LE INTEREST. !\ Of 1 dollar ft'i- 1 years at 5 per eent. ? ..3ns. 6Qcts. 10. Of -2] 5 dollars 34 cts. for 4 and a half years, at 3 ami a half per cent. ? .ins. g33, 9 Ids. 6m. 11. What is the amount of 324 dollars, 61 cents, far 5 % ears and 5 months, at 6 per cent. ? 0ns. 8430, Wets, bf-fom. 12. "What will SOOOJ. amount to in 12 years and 10 months, at 6 percent.? .i/zs. 5310. 13. What is the interest of 25 7/. 5s. Id. for 1 year and :> quarters, at 4 per cent. ? jlns. 18 05. id. Sqrs. 14. W hat is Hie interest of 27'.) dollars, 87 oents for 2 years and a half, at 7 per cent, per annum ? fns. 48, 9rr/a. 7im 15. What will 279Z. 13s. 8d. amount to in 3 years and .-. half at 5} per cent, per annum ? Ans. 331 15. 6rf. If-. What Is the amount of 341 dols. 60 cts. for 5 yeas* a:ul 3 quarters, at 7 and a half per cent, per tinnum ? .flKs. S488, 9lic*s. 17. What will 730 dols. amount to at 6 per cent, iu .5 years, 7 months and 1C Jays, or ^ 2 T of a year ? .dtos. 8975, 99cte. 18. What is the interest of 1825/. at 5 per cent, per annum, from March -Uh, 179(5, to March 29th, 1799, (ol r lowing thc^-ear to contain 363 days ?) .'] zs. 280. No --E. The Rules for Simple Interest serve also to calculate Commission, Brokerage, Insurance, or any thing else estimated at a rate per cent. COMMISSION, IS an allowance of so much per cent. t a factor or cor- respoadoni abroad, for buying and selRng goods for hi ycr. KXAMPLES. 1. What will the commission of 843/. 10s. come to at 5 per cent. ? $iMPL.E INTEREST. 125 Or thas, . s. 5 is ^43 10 421 17 10 .fas. 42 3 6 20 5|50 12 6|00 42 3s. 6d. 2. Required the commission on 964 dote. 90 cte. at 2$ percent? .8ns. 821, 71cte. 5. What may a factor demand on 1 1 per cent, commis- sion, for laying" out 3568 dollars ? Jlns. g62, 44cfs. BROKERAGE, - IS an allowance of so much per cent, to persons assist- ing merchants, or factors, in purchasing or sell ing goods. EXAMPLE'S. 1. What is the brokerage of 750?. 8s. 4d. at 6s. 8d. per cent ? . s. d. 750 8 4 Here I first find the brokerage at 1 1 pourvi per cent, and then For the given rate, which is - t>f r pound. 7,50 8 4 20 S. d. . s. d. qrs. 6 8=)7 10 1 10,08 12 Ans. 2 10 Ij 1,00 2. \Viiatistrtebrokerageupon 4125 dols. at or 75 eents per cent. ? Jlns. S530, 93ct$. 7im. S. it" a broker sells goods to the amount of 5COO dels, what is his demand at 65 cts. per cent, r Ans. g3i2, 50cf 1) * 'i2G SIMPLE INTEREST^ 4. What may a broker demand, when he sells goods to the value of 508f. ITa. lOd. and I allow him 1$ per cent ? 7 13* 8& IS a premium at so much per cent, allowed to persons and offices, for making good the loss of ships, houses, mer- chandize, &c. which may happen from storms, fire, &c. EXAMPLES. 1. What is the insurance of 7251. 8s. lOd. at 12$ per cent, r Ans. 90 13s. 7d. 2. \Vhai is the insurance of an East-India ship and cargo, valued at 125425 dollars, at 15$ per cent. ? Ans. g!9130, ZTcts. 5m. 3. A man's house estimated at 5500 dollars, was insu- red against fire, for 1$ per cent, a year : what insurance djil lie annually pay? Jbis. &61, 25cfs. SHORT PRACTICAL RULES, Anting Interest at G per cent, either f or months and days. I. FOR STERLING MONEY. RULE. 1. If the principal consists of pounds only, cut off tk unit figure, and as it then stands it will be the interest for , &c. re- decimal point one place, or figure, further towards the left hand, and as the decimal then stands, it will shew the interest e month, in .-hillings, and decftnals of a shilling. EXAMPLES. i. Required the interest of 541, for seven months ana '.y, ;ir 6 per cent -SIMPLE, INTEREST, 127 S. 10 dajs=fj5,4 Interest for one month. 7 57,8 ditto for 7 months, 1,8 ditto for 10 days. ' [Ans. 39,6 shillings =1 19s. 7,2rf. 12 7,2 fi. What is the interest of 42/. 10s. for 11 months, at 6 per cent. ? s- . 42 10 = 42,5 decimal value. m Therefore 4,25 shillings interest for 1 mouth. 11 . s. J. Ms. 46,75 Intercstforllmo. = 269 3. Required the interest of 942. 7s. 6d. TITOL five months and a half, at 6 pe :am. 4. What is the interest of 19.1. lt>s. for one third of a month, at 6 per cent. ? -. 5,}6d. II. FOR FEDERAL MONEY. RULE. 1. Divide the principal by 2, placing the sepaiatrr<; a* usual, and the quotient will be flic interest for oneirr in cents, anu decimals of a cent; that is, die I'-. the left of the separatrix will be cents, and those u:. right, decimals of a cent. . Multiply the interest of one month by thegiven r\\>r.\. ber of months, or months, and decimal pits thereof, or fur tl^ tirjys tike the eveu parts of a month, &c. 1-28 SIMPLE INTEREST: EXAMPLES. 1. What is the interest of 341 dols. 52 cts. for 74 months ? 2)341,52 Or thus, 170,76 Int for 1 month. 170,76 Int. for 1 mouth. x7,5 months. 853SO 1 195.32 do. for 7 mo. 1 19552 85,58 ilo. for * mo. & cts.m. 1280,700ds. =12,80 7 1*280,70 ,9Ms. 1280,7cL>;.=S12, SOcfs. 7m. 2. Required tlve interest of 10 dols. 44 cts. for .3 years, 5 months anil 10 days. 2)10.44 10 days :) 5,22 Interest for 1 month. 41 months. 5,22 208,8 214,02 ditto for 41 months. 1,74 ditto for 10 days. 2f5,76cf.. .<;. =g2, 15cfs. 7m. -f < i.ai. is the interest of 342 dollars for 11 months ? The J is 171 Interest for one month. 11 JRS. 1881cfs.=gl8, 81c<*. No IK. To find the interest of any sum for 2 months, ic.r cent, you need only call the dollars so many cents, :thd Lhe inferior denominationsi decimals of a cent, and it i douo : Thus, the interest of 100 dollars for two months, is 100 ronts, or 1 dollar: anrl S^J, 40 cts. j* 25 cts. 4 in. &c. v. Ill- li tnvL's the following RULE II. Miiltijily the principal by half the number pf months, i in- product will shew the interest for the given time. ufs and decimals of a cent, as above. SIMPLE INTEREST. 129 EXAMPLES. 1 Required the interest of 3 16 dollars fer Ij'ear and 10 months. 1 1 =half the number of mo. Ans. 3476cs.=gS4, 76c*s. 2. What is the interest of 364 dols. 25cts. for 4 months ? 8 cts. . S64, 25 2 half the months. 728, 50c*s. ^ns.=gf, 28cs. 5m. III. When the principal is given in federal money, at 6;per cent, to find how much the monthly interest v, ill be in New-England, &c. currency. RULE. Multiply the given principal by ,03 and the product will be the interest tor 'one month, in shillings and deci- mal parts of a shilling. EXAMPLES. t. What is the iaterest of 325 dols. for 11 months ? ,03 9,75 shil. int. for 1 ITK Xll months. . 107,25s. =5 7s. Srf. & What is the interest in New-England currency, of 31 dols. 68 cts. for 5 months ? Principal 31,68 dols. ,03 ,050-i Interest for one month. 5 Ms. 4,7520s. =4s. 9d. 12 90840 SIMl'LE i:,'TE,RKST. IV. \ principal is given in pounds, shillings, tr-Knglanu < im-ency, ;il G per cent, to find how .aoiithiy interest will be hi federal money. RULE. lv (he pounds. &r. by 5, and divide that pro- due 1 <[unfifMt will be the interest for one month, It'ciiuals of a cent, &c. EXAMPLES. i. A note Ink- 411 New-England currency has beea. on interest one month ; how much is the interest thereof in federal nionr-. r ,'\ 411 )2055 .2. Required the interest of 39/. 18s. N. E. curretfcj, for 7 montln ? . 59,9 decimal value. 5 Interest for 1 mo. 60,5 cents. 7 Ditto for 7 months, 465,5cfs.=g4, 65c/s. 5m. 3ns.. V. When tlw principal i- > Nr.w-En^hind and Virginia cmrency, at 6 per cent, to iind Ihe rVjterest for , in dyllur.s, cents and mills, by inspection. RULE. Since Ike interest of a year will be ju^t so many cents ^ivcTi prinripal contains shillings, therefore, \vrite . Third payment, 99 dols. Jan. 2, 1793. I domaini lio\v much remains due un said note, the 17th of June, 1798 : 8 cts. 14S, 00 fi!-;t payment. May 7, 1. JV. 36, 50 interest up to June 17, ITt'S. -i 14 184, 50 amount 341, 00 second pay me , AM-.:. 17. 57, 51 Interest to June If, S*8, 51 amount. SIMPLE INTEREST. s. 99, 00 third payment, January 2, 1798. 2, 72 Interest to June 17, 1798.=5imo. 101, 72 amount. 184, 50"| 378, 5 1 v several amounts. 101, 72 J 664, 73 total amount of payments. 075, 00 note, dated April 17, 1795. Fr. ?n. 209, 25 Interest to June 17, 1798. =5 2 884, 25 amount of the note. 664. 73 amount of payments. J&219, 52 remains due on the note, June 17, 1798. 2. On the iCth of January, 1795, 1 lent James Paywell 500 dollars, on interest at 6 per cent, which I received back in the fol lowing partial payments, as under, viz. 1st of An; -U, l/9'.i - g 50 16th of July, 1797 - - 400 1st of Sept. 1798 - - 60 How stands the balance between us, on the 16th No- VCiriber, 1800? . due to me g63, 18cfs. 3. A PKOMISSORY NOTE, VIZ. 62 10.;. .Yrir- London, April 4, 1797. On demand I promise to pay Timothy Careful, sixty- two pounds, ten shillings, and interest at 6 per cent, per annum, till paid; value received. JOHN STANUY, PKTKtt PAYWELL. RICHARD TESTIS. Indorsement*. . s. 1st. Received in part of the above note, Sep- tember 4, 1799. 50 And payment .^ une -1. 1800, 12 10 How much remains due or, said note, the fourth day of December, J800 ? . s." ; . ^n, 9 19. <> SIMPLE INTEREST. 133 NOTE. Tha preceding Hula, by custom is rendered so popular, and so much practised and esteemed by many on account of its being simple and concise, that I have given it a place : it maij answer for short periods of time, but in a long course of years it will be found to be very erro- neous. Although this method seems at first view to be upon the ground of simple interest, yet upon a. little attention the following objection will be found most clearly to lie against it, viz. that the interest will, in a course of years, com- pletely expunge, or as it may be said^at up the debt. For an explanation of this, take the following EXAMPLE. A lends B 100 dollars, at 6 per cent, interest, ami takes his note of hand ; B does no more than pay A at every year's end 6 dollars, (which is then justly due to B i'or the use of his money) and has it endorsed on his note. At the end of 10 years B takes up his note, am? the sum he lias to pay is reckoned thus : Th<4 principal 100 dollars, on interest 10 years amounts to 160 dollars; there ate nine endorsements of 6 dollars each, upon which the debtor claims interest ; one for 9 years, the sec uid for 8 years, the third for 7 years, and so down to the time of settlement; the whole amount of the several endorsements and their interest, (as any one can-see by casting it) is S^O, 20cts. this subtracted from 160 ' the amount of the debt leaves in favour of the creditor, 889, 40cts. or glO, 20 cts. less than the original princi- pal, of which he has net received a cent, but only its an- nual interest. If the same note should lie years in the same way, B v/ould owe but 57 dols. 60 cts. without paying tlie least fraction of the 100 dollars borrowed. Extend it to 8 years, and A the creditor would full in debt to B without receiving a cent of the 100 do! which lie lent him. Seo a better Ilule in Simple Inter - wt by Decimals, page 175. C .NJ I 1 L N I XT H E S T. COMPOUND INTEREST, Ih \vlicu the interest :s added to the principal, at the end of the year, and on that amount the interest cast for anoth- er year, and added again, and soon : this is called Inter- rst upon Interest. % RULE. ind ir.c interest for a year, and add it tor the principal, . hich.ca!l the amount Tor the first year ; find the interest ,is amount, which add as before, for the amount of'the :ul, and so on for auv numb?.r of years required. -Subtract the original principal from the lust amount. and the remainder will be the Compound Interest fu. \vliole time. EXAMI'LES. 1. Required the amount of 100 dollars for 3 years at 6 per cent, per annum, compound interest ? 8 c/s. g cts. 1st Principal 100,00 Amount 1(() 5 ')0 for 1 year. c .d Principal 106,00 Amount i 1:2,36 for 2 years. 3d "Principal 112,36 Amount 119,ioi6ior 3yns. Jlns. 2. What is the amount of 425 dollars, for 4 years, at 5 per cent, per annum, compound interest ? 5. What will 400^. -, in 4 yeais, at 6 per cent. liar annum, compound iin.e*i ;'.< r .--Ins. /,'.>H4 \i)s. 9$d. 4. What is the coir- pound interest of liiO/. 10s. for S yen, ct. per lusnum ? i/^. 28 14s. 11]^.+ 5. \Viiat is the cnmpouftd interest ot 500 dollars for 4 years, at < per cent, per anr.um r Jlns. gli>l,258-f- h; t t will '1000 dollars amount to in 4 years, at 7 per cent pi-r annum, c-> ;310, T9ds. fiia.-f 7. What is the amount of 7.: ) ,- 4 j> eats, at 6 ,j r annum, compound inter. Jin: 8. Wlint is the compound isii.fi est of 870 , for Si years, at G per vei.t. nei annum r 01! DISCOUNT, % l^> an allowance Tingle for the payment of any sum of money before it becomes due; or upon advancing ready Money for notes, bills, ,t : to the present wort!). PROOF. Find the amount of -i>t worth, at the given rate and tune, and if the work is right, that will be equal to the given sum. 1 . What must be discounted for tlie-ready payment )() dollars, due a year hence at (J per cent, a year ? S S S S e.ts. As 106 : (i : : 100 : 5 66 the 100,00 given sum. 5,06 discount. v-t the present worth. 2. "What sum in ready money will discharge a debt <>: 925?. due 1 vearand b montlis hct'ie. at G [-er c 100 10 Interest for mo:: 110 Ain't. /;. . . . s. d. As 110 : 100 : : \l-25 : 840 18 2+Jns. f>. AVIiat is tlie present worth of 600 dollars, due 4 years hence, at 5 per ceat. ? 2ns. g500 4. What is the discount of 275/. 10s. for 10 m< at ^5 m-r tent, per annum r I 3 2s ANSUIilES. 5. BougLi goods amounting to G15 dols. 75 cents, at 7 months creuit; how much ready money must I pay, per cent, pfer annum ? " 3/i5. &S89, 32cte. 8'n. NOTE. When sundry sums are to be paid at different times, find {he Rebate or present worth ot each particular payment separately, and when so found, add them into one sum. EXAMPLES 7. What is the discount of 7.'>0/. the one half payable in six months, and the other half in six months after that, at 7 per cent. ? .flits. 37 IQs. Z$d. 8. If a legacy is left me of 2000 'dollars, of which 500 dols. are payable in 6 months. 800 dols. payable in 1 year, ami the rest at the end of 5 years ; how much ready money ought \ to receive for said legacy, allowing 6 per cent, discount? Ans. 81833, 37c?s. 4m. ANNUITIES. AN Annuity is a sum of money, payable every year, or for a certain number of years, or forever. \VI:cn the debtor li the annuity in his own hands, beyond t ! to be in arrears. Th" 'rliic li;ue thi-y h;n .teix>st due on each, is called th:' ; (fan o!l', or pai;l all at o . of tin- !; - :he puce v.hich is ^lid i ii c,: led liic prtv.ent \vorth. To find t'ue amount of an annuity at - RU I. Fin-.l ilie' interest oi' the givrn an'Hiity 1 n .. \nd tlien for 2, 3, &.c. yi'.: , time, '".'tip! y the annuity by the n'n.ibi i l;l OIL- product t> fontinue -I- years, at t ; per rent, per annum ? 5S49 = Pres. worth of 1st v,-. 1 1 2 100 ' - 400 : S57,l4285 = -- 2d yr. L18f; '538,98305= - 3d yr. 1 24 j 322,58064 = - 4th yi . Jns. S1396,OG503=S1396, Gets. 5m, 2. lieu- much present money is equivalent to an an- nuity of 100 dollars, to continue 3 years j rebaiti being made at 6 percent. ? .flus. g268, 37cs. li. R. N'.'hat is BO', yearly rent, to continue 5 year*, worth in ready money, at (V. per cent. ? Ans. 340 15s. -f- 12* 1J8 EQUATION' Or 1'AYMENTX EQUATION OF PAYMENTS, IS finding the equated time to pay at once, se^ debt* due at different periods of tinu... so that uo lo.ss shall be sustained by either party. RULE. Multiply each payment by it* time, and divide t! gf the several products by the \vholc debt, and the : will be the equated time for the payment of the whole. , XXAMl'LICS. 1. A owes B 3PO dollars, to be paid as follows vi/.. 100 dollars in 6 months, 1*0 dollars in 7 months, and 160 dollars in 10 months : What is the equated time f.>. pxyinunt of the whole debt ? 100 x G =. GOO 120 x 7 = 840 160 x 10 = 1600 580 )5040(8 months. Jus. :l. A. merchant hath owing him 3001. to be paid as fol- : .'>'<')/. at. ^ :.<,!)Mths, lOOt. at 5 months, and the rest at 8 months ; and it i.-> agreed to make one payment of the \vhole ; I demand the equated time ? Jlns. 6 months. 3. F owes II 1000 dollars, whereof 200 dollars is to be puhi 400 dollars at 5 months, and the rest at 15 months, but they v\ oc to make one payment of the whole : J demand \ finie must be ? Jius. 8 month*. chant lias due to him a certain sum of money, to be | .i : i on ; months. .ini>. thir(fat3 months, .mi! ; is j A\lut is tiif time for ;he j . .ie ? jli'.s. 4\- months. . BARTER, J S ti of one; commodity for . 12. A has 225 yds. of shalloon, at 2s. ready money, per yard, which he barters Avith B at 2s. 5d. per yard, takn indijro at 12s. fid. per Ib. which is worth but 10s. how Tiiucii indigo will pay for the shalloon; and who gets the best bargain ? .0//S. 43/6. at barter price will pay for the shalloon, and B has the advantage in baiter. Value of A's cloth at cash price, in : 10 Value of45^&. of indigo, at 10s. per Ib. B get* th e ! >e > t ba rga M i by 15 LOSS AND GAIN, AS a rule by which merchants and traders discover tl profit or loss in buying and gelling their goods : it also in- >f.rur*s them how to rise or full in the. price of tlwir good-, so as to gain or lose so much per cent, or otherv, , Questions in this rule are answered by the Rule of Three. 1. Bought a piece of cloth containing SS yard-. '<>r 191 dols. 25 cts. and sold the same at "2 d,> yard; what is the profit upon the wholf pi- . 2. Bought 12* cwt. (>f rice, at T. dols. 45 c.<->. and sold it again at 4 cts. a pound ; what was the whole gain? Jlns. g!2, XTcl*. 5>n S. Bought 11 cwt. of sugar, at f>id. per Hi. but <-.nld r.'it sell it nrnn for ir\y ihore than 2/. iCs. per cwt. : did v my bargain ? .'/... L'Kt.f^Z 11s. 4d. 4. Bought 44 Ib* of tea fo . rinlbr ii/. lf>s. M. ; what was the profit, on tu-Ji pound ? TOSS ANT> GAIN 141 5. Bought a hhd. of molasses containing 119 gallons, f,i 52 cte. per gallon ; paid for carting the same 1 dollar 25 cents, and By accident 9 gallons leaked out ; at what rate must I sell the remainder per gallon, to gain 13 dol l^i's in the whole ? Arts. 69cte. 2/n.-f II. To know what is gained or lost percent. RULE. Fir.*t see what the gain or loss is by subtraction; then As the price it cost : is to the gain or loss : : so Is 100Z. 01- S 1( '0> to the gain or loss per cent. KXAMPLES. 1. If I buy Irish linen at 2s. per yard, and sell it again It 2<*. fid. per vard ; what do I gain per cent, or in laying out lOtif.? As : 2s. 8rf. : : 100J. : 33 6*. 8d. Ans. 2. If I buy broadcloth at 3 dols. 44 cts. per yard, and sell it again at 4 dols. 30 cts. per yard ; what do I gain per cent, or in laying out 100 dollars ? S 'cts.-} Sold for 4, 301 g cts. cts. $ $ Cost 3, 44 J> As 3, 44 : 86 : : 100 : *? I Jins. 25 per cent, Gained per yd. 86J 3. If I buy a cwt. of cotton for 34-doIs. 86 cts. and sell it again at 41 cts. per Ib. what do I gain or lose, and what per cent. ? . g ots. I cwt. at 41$cts. per Ib. comes to 46,48 Prime cost 34,86 Gained in the gross, $11,62 As 34,36 : 11,62 :: 100 : 33| Jlns. Sty per cent. 4. Bought sugar at 8$d. per Ib. and sold it again at 4l. iTs. per owt. what did 1 gam per cent. ? Ms. 25 19s. 5$d. 5. If I buy 12 hhds. of wine for 204J. and sell the same again at 14. 17s. 6d. perhhd. do I gain or lose, and what per cent. ?. Jlns. I lose 12 per cent. 6 At 1 Jd. profit in a shilling, how much per cent. ? Ans. 12 1*. >. ND GA1.V. 7. At 5 cts. profit in a dollar, how much per cent. ? .ins. 25 per cent. NOTE. When good- are bought or sold on credit, you must calctilui' -count) tlie present worth of their 111 order to rind your tiue gain or los?, &c. KKAMPI.F.S. 1 BousjM 164 yards of broadcloth, at 14s. 6d. per yd. ready money, anil ^r>!.i i 1J4/. 10*. on 6 months c'fdit: what did I :iin by the whole; allow- ing discount ar ; * As li'3 : ICO : : i^-t 10 : loO u , orth. Gi ' 'T. 2. If I buy rl.Mh at - eight ms creii/t. . . \hat do I ' ii'i- cent. . cut. 'Id, to gaiu RULK. :o : : so is 100/. Or ti:e profit .'idcicii. or 1 -.cted : to the - ,c. . . 1. If I buy Irisli Hi. ird ; now must I afll . I buy Rum at 1 dn 30 per t 3. 1 1 - id [tr As Rl.:0 : 54cfs. : : g '.ot pvdving . - I expected, 1 a;- 15 |j'r " J;s. 14<. K>i.-/. LOSS AND GAIN. 145 5. If 11 cwt. 1 qr. 25 Ib. of sugar cost 126 doJs. 50 cts. how must it be sold per Ib. to gain SO per cent. ? Jus. IZcis. 8m. 6. Bought 90 gallons of wine at 1 dol. 20 cts. per gall, but by accident 10 gallons leaked out, at what rate must I sell the remainder per gallon to gain upon the whole prime cost, at the rate of 12A per cent. ? dns. Sl> olcfs. 8 r r ff H. IV. \Vhen there is gained or lost per cent, to know what the commodity cost. RULE. As 100/. or lOOdols. with the gain per cent, added, or loss per cent, subtracted, is to the price ; so is 100 to the prime cost. EXAMPLES. 1. If a yard of cloHi be sold at 14s. 7d. and there is gained 161. I3s. 4d. per cent. 5 what did the yard cost ? . *. d. s. rf. . As 116 13 4 : 14 7 : : 100 to 12s. 6 per ceru. added, or other p:ice : t - or loss per cei N. K. If your anbsver e.Kceed 1001. o, i;K) do! xce.ss is your gain per cent, j but if it be L U'ss per cent. 144 FELLOWSHIP. EXAMPLES. 1. If I sell cloth at 5s. per yd. and thereby gain 15 per cent, what shall I gain per cent, if I sell it at 6s. per yard ? s. . 8 : . As 5 : 115 : : 6 : 138 Jlns. gained 38 percent. 2. If I retail rum at 1 dollar .'>0 cents per gallon and thereby gain 25 per cent, what shall I gain or lose per cent, if I sell it at 1 dol. 8cts. per gallon ? 8 cfs. 8 g cts. 8 1,5(1 : 125 : : 1,08 : 90 Ans. I shall lose 10 per cent. 5. If I sell a cwt. of sugar for 8 dollars, and thereby lose 12 per cent, what shall I gain or lose per cent, if I sell 4 cwt. of the same sugar for 36 dollars . Jlns. Hose only 1 per cent. 4. I sold a watch for IT/. Is. 5d. and by so doing lost 15 per cent, whereas I ought in trailing to have cleared 20 per cent. ; how much was it sold under its real value"? ** fr** As 85 : 17 1 5 : : 100 : 20 1 8 the prime cost 100 : 20 1 8 : : 120 : 24 i the real value. Sold for 17 1 5 707 Answer. FELLOWSHIP, IS a rule by which the accompts of several mcrt.har other persona, trading in partnership, are so adj.; that cadi may ha'. ihe gain, or sustain hit share of the los. in proportion to his share of the stock. Also by this Rule a Ixmkriipt's estate may !.- vided among hi^ creditors. SIMJLK Is when the s* -ti H trade an equal term BUJ As the whole stocl< vhoie gain or lost: o i tach man's particular ktock, to liit> pu ticui*r shtrewf tH or low. JILtOWSHlP. PROOF. Add all the particular shares of the gain or loss together, and if it be right, the -sum will bo equal to the whole gain or loss. EXAMPLES. 1. Two partucrs, A and B, join their stock and buy a quantity of merchandize, to the amount of 820 dollars ; in the purchase of which A laid out S50 dollars, and B 470 dollars ; the commodity bein* sold, they find their clear gain amounts to 250 dols. What is each person's share of the gain ? A put in 350 B 470 * A* ao -2-- 5 550 : 106,7073 + A's share. 470: 143,3920+ ITs share. Proof 249,9999 -{- =g50 2. Three merchants make a joint stock of 1 200/. of which A put in 240 J. B 360/. and"C 60G/. and by trading- they gain 325J. what is each one's part of the gain ? Ans.A>spart65. /Ts 97 105. C'162 10s. 3. Three partners, A, B, and C, shipped 108 mules for the West-Indies; of which A owned 48, BS6,andC 24. But in stress of weather the mariners were obliged to throw 45 of them overboard ; I demand how much of tho loss each owner must sustain ? Jus. A 20, B 15, and C 10. 4. Four men traded with a stock of 800 dollars, by which they gained 307 dols. -A 's stock was 140 dols. B's 260 dols. C*s 300 dols. I demand D's stock and what each man gained by trading ? Jlns. D's stock was$lQV,and Ji gained g53, 72cs. 5m. B 899, 77ics. 6 y jgll5, lZ}cts. and D S38, STicts. 5. A bankrupt is indebted to A 21 ll. to B 3(Ktf. and to C 391/. and his whole estate amounts only to 6751. 10s. which he gives up to these creditors; how MUCH must each have in proportion to his debt ? Jlns. Ji must have 158 0*. 3|rf. B 224 13s. *j(* ana C 392 16s 3|rf 13 148 COMPOUND FELLO.WSIUJ?. 6. A captain, mate and seamen, took a prize wdrth 5501 dols. of which the captain takes 11 shares, and the mate 5 shares ; the remainder of the prize is equally di- vided among the sailors ; how much did each man re- ceive ? $ cts. ins. The captain received, 1069, 75 The mate 486, 25 Each sailor 97, 25 7. Divide the number of 350 into 5 parts, which shall be to each other as 2. 3, and 4. Jlns. 80, 120, and 160. 8. Two met chants have gained 450^. of which A is t* have 3 times as much as B ; how much is each to have ? Jlns. .1 337 10s. and B /: 112 10s. l-fS=l : 450 : : 3 : 337 10.>. J's . 9. Three persons ate to share GiXtf. A is to hav;i - tain sum, lias much again :.- much as B. 1 demand cacli man's part ? .tois. J <;>.<;., V l.:3-. unl ' 10. A nnd B tiado'.! t>^cther arid ."aiiunl 100 d( :; t ; -!0 dols. B |'iit in ...nd B's stock? 11. A, !?, and C,traled \ >>do!-;. ;tiid (' put i>> 1:20 ydi ; . of < il< I -. uf which (..' ti.ik Kid dn ; in : how did C value his cloili per v-a: jrk. ;u:'l \v! i at was A and B's part of i . C }>ut hi Hie clutk at . R46, G7As 288 : 19/. : :--. 144 : 9 10 C's gj -oof 19 : '.' A put in 215 ih>!s. f>r ': .ninulis, but by 'oitune t'u-v lose xl 1 d dols. : 1-, MV nvist they nhavotiiC . 'a SI 46, 2 3. Three persons liral received CQ5 dr.ls. interest: A iiacl put in.-'(;f*0 do's, for 12 mouths, B 3000 dels. f>,r 15 months, and C : ho\v much is each man's part of . t ? i, 5 &25 and C S200 ^ - 1 ; bv trading; 110/. 12s.: A r * ock w.?- ths, and F/s 200Z. for 6i months; \v',i;r MI'S p.irt of the p;ain ? *Jw. *i J 5j yj x s /;60 l;ls. 8 ] ^..^V J. TV. o men Jiants CI.UT into pnrtnersliip for 18 months. A at first put iut'> stock 500 dollars, and at the end of S months he put i;i 100 dollars more ; li at first put in 800 dollars, and at 4 month's end took out 200 dols. At the expiration of the time they find they have gained 700 dol- lars ; what is each man's share of the gain ? $8324. 07 4-M's share. 92 5+JSPs. do 6. A and 15 companied ; A put in the first of January, 1000 dols. ; but B could not put in any till th first of May: what did he then put in to have an equal share . A at the year's end ? J/o. g Jjb. S !' demand, under ; of the same kind in the supposition. If the blank place or i ,ut, fail under the third term, the pro- portion is direct; then multiply the first and second .s together for a divisor, and the other three for a lend : but if the blank fall under the first or second proportion is inverse ; then multiply the third i terms together for a divisor, and th.e other three for a dividend, and the quotient will be the answer. EXAMPLES. 1. If 7 men can Iniilit S6 roUs of wall in 5 tkysj liow rn.-ny rods can 2u aien build in 14 days? : 36 Terms of supposition. Terms wf demand. 7x^=^)10080(480 wda Jns,. 2. If 101 ,ain 6/. interest in 12 months, in 7 mo: : U'7/ 4 o. : : 6/. Int. 400 : 7 Ana. 14L GOXJOINKD PROrORTIOV. 149 S. If 100/. will gain 61. a year ; in what time will 400/. gain 14/. . mo. . 100 : 12 : : 6 400 : : : 14 Jlns. 7 iiWHf/&. 4. If 400/. gain 14L in 7 months ; what is the rate per otat. per annum P . wo. Int. 400 : 7 : : 14 100 : 12 Jlns. .6. 5. "What Principal at GJ. per cent, per annum, will gain 141. in 7 months,? mo. Int. 100 : 12 : : 6 7 : : 14 Jn*. 400. 6. An usurer put out 861. to receive interest for the game : and u^on it had continued 8 months, he received principal and interest, 881. 17s. 4d.; 1 demand at what rate per ce?if. pei-aiui. he received interest ? Jlns. 5 per ct. 7. If 20 bushels of wheat are sufficient for a fUisily of 8 persons 5 months, how much will be sufficient for 4 per- sons 12 months ? dns. 24 bushels. 8. If 50 men perform a piece of work in 20 days ; how many men will accomplish another piece of work 4 times 83 large in a fifth part of the time ? 30 : 20 : : 1 4 : : 4 Jns. 600. 9. If the earriage of 5 cwt. 5 qrs. 150 miles, cost 24 dollars 58 cents ; what must be paid for the carriage of 7 cwt. 2 qrs. 25 Ib. 64 miles at the same rate ? Jlns. g14, 08cs. G??i.-f 10. If 8 men can build a wall 20 feet long, 6 feet high and 4 feet thick, in 12 days ; in what, time will 24 men build one 200 fet long, 8 feet high, and 6 f*et thick ? 8 : 12 : : 20x6x4 S4 : 200x8x6 80 days, CONJOINED PROPORTION. IS wliei' the coins, weights or measures of several coun- tries are compared in the same question; or it is joi-iing proportions together, and by t)> relation which 13* i50 CONJOINED PROPORTION. several* antecedents have to their consequents, the pro- portion between the first antecedent ano the last conse- ouent is discovered, us well as the proportion between Uie others in their several respects. NOTE. This rule may generally be abridged by can- celling equal quantities, or terms that happen to be the same in both columns : and it may be proved by as many statings in the Single Rule of Three, as the nature of the question may require. CASE I. When it is required to find how many of the first sort of coin, weight or measure, mentioned in the question, arc equal to a given quantity of the last. RULE. Place the numbers alternately, beginning at the lei;. hand, and let the last nnmlvr stand on the left hand cot umn ; then multiply the left hand column continually for a dividend, and the right hand for a divisor, and the quo- tient will be the ui;:-;wt>r. EXAMPLES. 1. If lOOlb. English make 5lb. Flemish, ami 19lb- Flemish how m;ir:y pounds English are equal to 50lb. at Bologna ? Ib. Ib. 100 Eng.=95 Flemish. ID Fie. ^=5 Bologna. 50 Bologna. Then 95x2.5 =23 T5 the dh 95000 dividend, and 2375)95000(40 .. 2. If 4Glb. at New- York, make 48lb. at Antwerp, and SOlb. at Antwerp, make SGlb. at Leghorn ; how many ih. at New-York are eqnal to 14411). . :i ? V/HS. lOOlb. 3. If 70 braces at Venice be equal to 75 braces at Leg- horn, and " braces at Leghorn be equal io 4 Amor- yards; ho.v many braces at N'enicc arc eciu;il io (J4 Ame- lt)4 A CASE II. When it is required to find how many of the last sort of coin, weight or measure, mentioned in the question are equal to a given quantify of the first. EXCHANGE. 15 RULE; Place the numbers alternately? beginning at the left hand, and let the last number stand on the right hand; then multiply the fii'st row for a divisor, and the second for a dividend. EXAMPLES. 1. If 24lb. $ New -London inake 20!b. at Amsterdam, and r>'>:\j. at Amsterdam ';')!!>. at Pni'is ; how manylat Paris are equal to 40 at New-London ? L',"t. Right. 24 == 2ft 20 x 60 X 40 = 48000 50 = GO = 40 Ans. 40 24 x 50 = 1200 '2. If 501 b. at New-York make 45 at Amsterdam, ami 80!b. at Amsterdam snake 103 at Dantzic ; how many ft. at Dant/.ic arc equal to 240 'it N. York ? .2ns. 278-^ 3. Jf W bi-aces at Lowborn be equal to 11 vares at Lisbon, and 40 varcs at Lisbon to 80. braces at Lucca; how many braces at Lucca are ^qiial to 100 'braces at Leghorn r Ans. 110 EXCHANGE. _L Y tin's rule merchant 'iat sum of money ou^ht to l>e received in one country, for any sum of different specie paid in/fcaother, according to tlie given course of exchange. To reduce the monies of foreign nations to tRat of tht United States, you may fons;;.lt the follov;ing TAli Showing the value o r the monies of account, of foreign nations, estimated in Federal Money.* g cts. Pound Sterling ot Great-Britain, 4 44 Pound Sterling of Ireland, 4 10 Lpre of France, " 18i Guilder or Florin of tie U. Netherlands, u .39 Mark Banco of Kmnburglv, <> 351 Rix Dollai- of Denmark, 1 *Laws U. 3. Jt.j 152 EKCH.AX0K. Rial Plate of Spain, 1 Milrea of Portugal, t 24 Tale of China. 1 48 Pagoda of India, 1 94 Rupee of Bengal, 55 J I. OF GREAT BRITAIN. EXAMl'I.KS. 1. In 45/. 10s. sterling, how many dollars and . A pound sterling being^444 cents Therefore- As I/. : 444cfs. : : 45,5/. : 20202<*fs. Jlns. 2. In 500 dollars how many pounds sterling? As 444cte. :!/.:: 50000et*. : 11&. 12s. 5 general definition^, and a few such problems therein as were necessary to prepare and lead the scholar r.umedi- :-.i.elv to decimals: the 4earner is therefore requested to read those general definitions in page 74. Vulgar Fractions are eitiver proper, improper, single-. compound, or mixed. 1. A single, simple, or proper fraction, is when the nu- merator is less than the denominator, as 4 $ f 4-|? &c. 2. An Iwpvoipstv Fraction, is when the numerator cx- . .Icnoimnator, as -| % L 2 , occ. .'j. A Cumpou'itd Fraction, is the fraction of a fraction, (;ii:ned bv the \vord of, thus. f of ~j l 2 , A of of 3, &c. 4. A JJi.vcd Nuinber. is composed of a whole r. umber and a fracti.iii, lluis, 'Si, 14/ 3 -, i^c, 5. Any whole number may be expressed like a fraction . line under it. and puKing 1 for cienoinina- . thus, 8=%, and 1;" &c. he coiWncu measure of t;vo or mare numbers, is that number \\hich will divide each ol thi-.m witl^mt .1 ."imler; thus, 6 is thecoimun measure of H, ^4-jn(l .- which will do this, is called Lie greatest common measure. r. A number, which can be measured bv two or more numbers, is called their com-iiwH inulfipie : and if it l< least Uttmber that can be so measured, it is called the I- 't ''"uunun multiple: tlius, 24 is the common multiple of ^ 5 and 4 _; but their least common multiple i-, To (ind tl)e least common multiple of two or more numbers. RULE. 1. IVivi'lc bv any number that will divkl* hvo or of the a,ivci nuaiijers w;tiifiut a rtnutim'uT, r.;' (jiioticnts, tugetlier with the undivided numbers. beneath. 2. Divide the second lines ag before, an: tb'.c aie no Ivy mmmers 1ht caa be divided ; 156 REDUCTION OK VULGAR FRACTIONS. continued product of the divisors and quotients, will give the multiple required. EXAMPLES. 1. What is the least common multiple of 4, 5, 6 and 10 ? Operation, x5)4 5 6 10 X2)4 162 X2 1x3 I 5 X2x2x3=6p .tow. '. What is the least common multiple of 6 and 8 ? ..ins. 24 3. What is the least number that 3, 5, 8 and 12 will measure? Jins. 120 4. What is the least number that can be divided by the t) digits separately, without a remainder ? Jkis. 2520 REDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form uito another, in order to prepare them for the operation of Addition, Sub- traction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE. 1. Find a common measure, by dividing the greater term by the less, and this divisor. by the remainder, and so on, always dividing the last divisor by the last remain- der, till nothing remains, the last divisor is the common measure.* 2. Divide both of the term 1 * of the fraction by the com- mon measure, and the quotients will make the fraction required. * To find the greatest, cmnnwn measure uf jnorf than t !'') numbers, you must find the greatest common measure of two oftlu-masper rule abnvr ; then, nf that commqn measure and out of the other numbers, and so on through all tiw numbers to the ltrt $ then will tke greatest common measure last found be tfie answer. OF VULGAR FRACTIOUS. 157 OR, If you choose, you may take that easy method in Problem I. (page 74.} EXAMPLES. Jr. Reduce $$ to its lowest terms. 48)||(i Operation. fv*i(6 common mea. S)?=f Ans. J^ Bern. 2. Reduce f to its lowest terms. 3ns. 5 3. Reduce |ff- to its lowest terms. fins. | 4. Reduce f|| to its lowest terms. 3ns. $ CASE II. T reduce a mixed number to its equivalent improper fraction. RULE. Multiply the whole number by the denominator of the tiven fraction, and to the product add the numerator. this sum written above the denominator will form fraction required. EXAMPLES. 1. Reduce 45| to its equivalent improper fraction. 45x8+7 = -' 2. Reduce 19} J to its equivalent improper frsction. 3ns, V/ 5. Reduce iCvj^ to an improper fraction. 3ns. VA* 4. Reduce Cmj to its equivalent improper fraction J.,0 2 tO Si "*' Ti CASK III. To find the value of an improper fraction. RULE. Divide the numerator by the denominator, . quotient will be the value sought. EXAMPLES. 1. Find the value of V . 5)4,- 2. Find the value uf 3. Find the v;il'M 4. Find the value of ^V s . fii^jrj *. Find the va4ue of V 158 REDUCTION OF VULGAR FRACTIOUS. CASE IV. T reduce a whole number to an equivalent fraction, hav- ing a given denominator. RULE. Multiply the whole number by the given denominator ; place the product over the said denominator, and it will form the traction required. EXAMPLES. 1. Reduce 7 to a fraction whose denominator will be 9. Thus, 7x9=63, and 6 ,/ ike answer. 2. Reduce 18 to a fraction whose denominator shall be 12. Jns. 2 T V 6 3. Reduce 100 to its equivalent fraction, having 90 for a denominator. .0ns. CASE V. To reduce a compound fraction to a simple one of equal value. RULE. 1. Reduce all whole and mixed numbers to their equi valeut fractions. 2. Multiply all the numerators together for a new nit merator, and all the denominators ior a new denomint tor; and they will form the fraction required* EXAMPLES. 1. Reduce of f of of -^ to a simple question. 1X2X3X4 =^= rV * 2X5X4X10 2. Reduce 4 of of jto a single fraction. Jlns. / f 3. Reduce of | of ^ to a single fraction. 4. Reduce $ of | of 8 to a simple fraction. Jlns. VV 4- Reduce } f ^| of 42} to a simple fraction. NOTE. If ine denominator of any member of a com- p4uml fraction b qi*J to ti< cuiuerator of another 159 member thereof, they may both b* expunged, and the other members continually multiplied (as by the rule) will produce the fraction required in lower terms. 6. Reduce of | of j. to a simple fraction. Thus, 2x5 1 ft . . 5 Q+mf i"i T3T ** ^^* 4x7 T. Reduce |*of f of f of f to a simple fraction. CASE VI. To reduce fractions of different denominations to equiva- lent fractions having a common denominator. RULE 1. 1. Reduce all fractions to simple terms. 2. Multiply each numerator into all the denominators xceptitsown, fora new numerator; and all the denom- inators into each other continually for a common denom- inator; this written under the several new numerators, vnll give the fractions required. EXAMPLES. I. Reduce | | to equivalent fra&tions, having a common denominator. i 4- + 1=24 common denominator. 1 2 3 X3 2 3 3 4 9 X4 4 2 12 16 18 new numerators. 24 24 2. Reduce 1 24 A ai denominators. ,' id 44 to a commc . 5. Reduce f * and to a common denominator. S tt and 100 REDUCTION OF VULGAR FRACTIONS. 4 Reduce ^ and -^ to a common denominator. 800 300 400 - - and - = T V ,V and ^ = 1-ft Ana. 1000 1000 1000 5. Reduce > and 124 to a common denominator. /J W c 54 883 ^ S ' Ti 7"? TI 6. Reduce % % and of -fi to a common denominator. ) /? MS 758 2592 1989 tlln!> ' 1STXS 7t? 7TJS The foregoing is a.general Rule far reducing fractions to a common denominator; but as it will save much la* iioar to keep the fractions in the lowest terms possible, the fallowing Rule is much preferable. RULE II. For reducing function! to the least common denominator? (By Rule, page 155) find the least common multiple of all the denominators of the jiiven fractions, and it will be the common denominator required, in which divide each }>a; iciriar deuoininatov, ui:d multiply the quotient by its o\vn numerator for a new numerator, and the new nume- rators being placed over the common denominator, will" the fraction? required in their lowest terms. EXAMPLES. 1. Reduce 1 1 and f to their least- common denemina* tor. 4)2 4 8 2)2 1 2 1 1 1 4x2=8 the least cm. denominator 82x1=4 the 1st. numerator. 8 4xS=l the 3d. numerator. 88x5=5 t'ie 3d. numerator. These numbers placed over the denominator, give the answer ff equal in value, and in much lower terma than the general Rule, which would produce f| ! TT 2. Rorluce ^^ and T 7 j to their least common uenomi$ ii a tor. O7 VULGAK. FRACTIONS. l6l 5. Reduce ^ f f and T y to their least common de- nominator. ' Am. || T \ | || 4. Reduce | f and T 'y to thtir least common deiftm- || if T V ir-ator. Ans. CASE VII. To reduce the fraction of one denomination to the fraction of another, retaining the same value. RULE. Reduce the given fraction to such a compound one, as will express the value of the given fraction, by comparing it with all the denominations between it and that denomi- nation you would reduce it to ; lastly, reduce this com- pound fraction to a single one, by Case V. EXAMPLES. 1. Reduce _ of a penny to the fraction of a pound. By comparing it, it becomes f of $ of ^V of a pound. 5X1X1 5 = Ans. 6 x 12 x 20 1440 2. Reduce -^5- of a pound to the fraction of a penny. Compared thus, -j^Vtr f 2 i ^ V*^- Then 5 x 20 x 12 ._ _ noo _.. j JJfts 440 1 1 3. Reduce $ of a farthing to the fraction of a shilling. Ans. jyS. 4. Reduce f of a shilling to the fraction of a pound. 5. Reduce f of a pwt;. to the fraction of a pound tru^. 6. Reduce | of a pound avoirdupois to the fraction of it cwt. Ana. T i T c'-i. 7. What part of a pound avoirdupois is T -| ? of a cwt. ? Compounded thus, ^ e of 4 of S T !] f Ans. 8. What part of an hour is j| T of a week r 14* 162 REDUCTION O? VULSAR FRACTIONS. 9. Reduce J of a pint to the fraction of a hhd. W. Reduce -f of a pound to the fraction of a guinea. Compounded thus, -$ of 3 T of ^.=4 Ans. 11. Express 5$ furlongs ;n the fraction of a mile. Thus, 5}=V of !=! Ms. 12, Reduce |- of an English crown, at 6s. 8d. to thfc fraction of a guinea at 28s. flns. 2 \ of a guinea. CASE VHI. i To iind tlie value of a fraction in tlie known parts of the integer, as of coin, weight, measure, occ. RULE. Multiply the numerator by the parts in the next inferi- u denomination, and divide the product by the denomina- Kr ; aad if any thing remains, multiply it by the next in- tv rior denomination, and divide by the denominator as before, and So on as far as necessary, and the quotient will bo. the .1:1- Norii. This and the following Case are the same \ritli Problems 11. and III. pages 75 and 76 ; but for tlie scholar's exerci.-o, I shall give a few more examples in out 1 :. KXAMTLES. ,'. AVlvat is the value of -|'- of a pound ? 3ns. 8s. 2* Find the valu*e of | of a cvvt. ^ns. Sqrs. 3lb. loz. 3. Find tiie value of | of 5s. 6d- .Ins. 3s. Q$d. \. Ilmv mucli is T V^- of a j)ound avoirdupois ? Jlns. 7 ox. Wdr. nw much is 4 of a hhd. of wine ? .flns. 45^c/s. >\ \V !iat is the value of \\ of a dollar ? AKS. 9$. 7d. 7. What is tlie value of f \. of a guinea? 0K* ADDITION OF VULGAR FRACTIONS. 163 9. Required the value of ff of a pound apothecaries. jins, 2oz. Sgrs. 9. How ranch is f. of 5l. 9s. ? Jins. 4 to,-. 5 |rf. 10. How much is of f of of a hogohead tins. ISgalm. 3qts. CASE IX. To reduce any given quantity to the fraction of any great- er denomination of the same kind. [See the Rule in Problem III. page 75.] EXAMPLES FOR EXERCISE. 1. Reduce 12lb. Soz. to the fraction of a cwt. u lie- nominator, will be the s,am of the fractioi** required. EXAMPLES. i. Add 5i 5 and f of { together. 5i^=V a Tlieu y ^ l reduced to their least LOU by Gas* VI. Rule II. will becmne '- i 132-fl8+14=\fj i =6^ or 164 ADDITION OF VULGAR TRACTIOKS. 2. Add 4 |- and together. Ms. 1 S, Add 1 1 and together. .ins. !{ 4. Add 12$. 34 and 4| together. .5ns. 20}$ 5. Add } of 95 and I of 14$ together. .fl/ts. 44^| NOTE 1. In adding mixed numbers that are not com- pounded vith other fractions, you may first find the sum of the fractions, to which add the whole numbers of the girea mixed numbers. f. Find the sum of JA T* and 15. I find the sum of A and -*- to be }= Then 1 + o-f r-HJ=-28.l Ans. r. .\. "of .; of t V Jns. 3S T J T NOTE 2. To add fr.iciii'ns of money. \\i':g];t, &c. re lions of iliilerciit integers to those of the same. Or. if you please you may And the value of each frac- VIII. in reduction, and then add them in pn'jier terms. 9. Acid of a shilling to |- of a pound. Method. *=r1< value by Case V III. , Ud. 2d Method. |.=73. 6d. Oqrs. ^s. =0 (5 3| By Case VIII. Reduction. 10. Aia I Ib. Troy, to f of a mvt. tins. 7o: 1 . Add ^ of a ton, to T ' ff . of a cw t. Ms. IZcwt. Iqr. Sib. IQ^oz. } :1. Add | of a mile to ^ of a furlong. ir. 8o. IS. Add ^ of a v;i:-;l. ] of a foot, and J of a mire to- .ins. 1540^c?s. //. 9in. .!! '. cif a work, ^ of aday, $ of an hour, and $ of a miiictc together. .jns. 2aa. 2/to. SOmtn. SUBTRACTION OF VCLGA.R FRACTION'S. SUBTRACTION OF VULGAR FRACTION?. RU1 PREPARE the frz in Addition, and the dif- ftjrenceof the numerators written above the coir.wr. de- nominator, mil give the difference of the tra. . ed. EXAMi'LES. 1. From $ take 4 of I of 1=^-^1 Tfc" -A - r r Therefore 9 7= T 2 j = J Me jJns. 2. From ^ take * J4 3. From |j. take ft 4. From 14 take -}} 13^ 7 5. What is the difference < ." ? 7 * ? 6. What differs -^ from ^ 7. From 14$ take f of 19 l T 7 j 8. From take remains. 9. From-j-i- of apcund, take -J of a shilling. fofiV^ifc^'^enfrom^.tal No'iE. In tractions oi mo:iey, weight, &.c. you inav, i^ xtou pleasejiind thevalu .veu iractidiis (by Gis^ Vlll. in Reduction) and then subtract them iu their pro- per terms. 10. From T 7 T . take 3| shilling. Jns. 5s. M. ^}qrs. 11. From | ot* an oz. take | of a ; . llpit't. ?^? % . 1. Frora A of a cwt. take -^ of ;i II). s. for. QTlb. Coz. Wfclr. 13. From 3| \veeks, take I ot a day, and i of f of ^ of an hour. .is. Gtc. -Ida. 12/io. 19.'/it/j. ir.}sec. *In 3ubtracting mixed numbers, when the lower fraction ia greater than !l:e upper one, you may, without reducing them to improper fractions, subtract the .numerator of the lower fraction frojn the common denominator, and to that <]i[iereuce add the upper numerator, carrying one to the suit's place of the lower whole number. Also, a fraction may be subtracted from a whole number by taking the numerator of the fraction from ina- tor, and placing the remainder over the denominator, then taking one from the whole number. 166 MULTIPLICATION, DIVISION, StC. MULTIPLICATION OF VULGAR FRACTIONS. RULE. REDUCE whole and mixed numbers to the improper fractions, mixed fractions to simple, ones, and those ot lifferent integers to the same ; then multiply all the nu- merators i for a new numerator, and all the de- nominators together for anew denominator. EXAMPLES. 1. Multiply $ by -*- Answers. }|=4 2. Multiply $ by 2- } . Multiply 5i by. $ 4. Multiply 5. of"7 by * 3}$ :. Multiply j^.?- by -> 5 G. Muli' : S by"] of 5 is| 7. Multiply 7j by 9* 69| 8. Multiply I of % by g. of S? |^ 9. What is the continued product of of ^, 7, 5 4 and /> DIVISION OF VULGAR FRACTIONS. RULE. PREPARE the fractions as before; then, invert the ufvisor and proceed exactly a* in multiplication: TTfe products will be the quotient required. EXAMPLES. 4X5 1. Divide \ by j Thus, =|? An$. :> x r 2. Divide ', 7 r In- ? Jlnsicers. l- 2 ^ 3. Divide -jj of> I | v 1. ^"l>at is the quotient of 17 by I ? 59| 7| ivide * of of i by 1 of } 7. Divide 4.?,- by' f of 4 8. Divide 71 by 127 T 7 f V 9. Divide 5*o4 b ^ I of 51 BTJI.E O7 THUEB DIRKCT, INVKRSfc. &C. RULE OF THREE DIRECT IN VULGAR FRACTIONS. RULE. PREPARE the fractions as before, then state your question agreeable: to the Rules already laid down in the Rule of 'i'h roe in \vlxm . , and invert the fi'-st term- in t'.ie proportion ; tlien multiply all the three terms con- tinually together, and the product will be the answer, iu the sau.e name with the second or middle term. 1. If | of a yard cost 2- of a pound, what will ? s of an Ell English cost .- |vd.=Jof f er bas!- 4. If a pistareea be wortii 1-1^- ^.ence, whut are 1 tarcens worth ? tf;j .-. 5. A merchant sold JA j-iecc-sof cloth. .minj iM^. vd.->. at ( Js. Id. [>er yard : v/hat did the whole ;, to? Jus. /JGO 10>-. 08 C. A person having ; o'f n \es-ei.'relU | of i . : what is iho \\ ' i worth ? ..*/.?. / 7. II' 7 (tf.as!'.;p bo worilj f t of her cargo, valued a* \k!uit is Uie whole ship and car^o worth? . /MOOSl 14b'. i! SE PROPORTION. IliM.K. PREl'ARE the fractions a:i'l ^.< IK-- fore, then invert the third term, ;I;H! terms together, tl.e product will b ., t:-. J68 RULE OF THn"EE DIRECT Itf DECIMALS EXAMPLES. 1. l!o\v much si i a) loon that is |yard wide, will line 5 ol cloth which is 1 } yard wide r Yds. 7/i's. yds. Yds. As 1| : ;xVxf = V4 5 = 1G :jV^" s - 2. If n ; i^n perform n jc-nm-y in S| days, when the . ; carriage of an c\vt. 145$ ovt. be carried for the same mo- ney r Jins. 22^ miles. 6. How pi.-xny yaids of baize which is U yards vide, will line 18 yanls of camblet g-yd. wide? is. 1 1^/i/s. Igr. 1} no. RULE OF THREE DIRECT IN DECIMALS HULL;. IsEDl.TE your ' .decimals, and state your ijuo'io:. as in uin;>ly the second and third together ; divide by the iirst, and the quotient will be the answer, c. vMl'LES. 1. If I of a yd. cost /? of n pound ; what will 15$ yds. come to? | =,875 y r =,53-fand 3=,75 J'' s - L- s - d- H rs - As ,K75 : ,583 : : 13,75 : 10,494 = 10 9 10 2,24 Ans 2. If I pint of wine cost 1,2s. what cost 12,5 hhd. ? Ans. .378 5. If 4*ydi, cost 3. 4>d. >vhat will 30J yds. cost ? SIMPLE INTEREST BY DECIMALS. 169 4. If 1.4cwt. of sugar cost lOdols. 9 cts. what will 9 cut. 3 qrs. rost at the same rate ? eu-t. S <*K-f. 8 As 1,4 : t 10,09 : : 9,75 : rO,S69=g:0, 26cs. 9m. -j-; 5. It' 19 yards cost 25,75 dols. what will 435$ yards come to r .3ns. 8590, Slcte. 7, a 5 m. G. If 545 yards of tape cost 5 dols. 17 cents. 5m. what will 1 yard "cost? .fins. ,015=lJcJs. 7. if a man lay s out 121 dols. 5 cts. in merchandize, and thereby gains 39,51 dols. how much will he gam in laving out 1 dollars ui the >ame rate ? Jns. ?,<>i dk.*=g3, 91 c. Howiuany yards of ribbon can 1 buy for 25; if x9^ yds. cost 4} dolla Jn-f. 178J?/ajv/s. 9. It 178i yd:-.c<^ 5] dollars, what cost 9$yaids? Jus. g4i 10. If 1,6 cwt. of sugar cost 12 dols. 12 cts. what cost S Uhds. each 11 cwt. 3 qrs. 10,1 '2 Ib. ? J.-iS. 269,07 M M P L E 1 N T E R E S T Ti Y 1) E C I M A L ,S. A TABLE OF RATIOS. Hate per cent. f io. j jKa^e per cent. \ Re 5 4 4J 5 ,"03 ,04 ,045 ,05 6 $1 . ,06 ' ,065 ,07 Ratio is the simple interest of 1^. for ono year : or in federal money, of gl ; OUR year, at the rate per cent, agreed on. RULE. Multiply the Principal. Ratio and time con gether, aud the last product will be the interest rc(;u.. EXAMPLES. 1. liccjuired the interest of 1 1 doh. 4i cts. far 5 ; at 5 per cent, per annum ? 'SIMPLE INTEREST BY DECIMALS. 3 cL;. 211,45 Principal. ,05 Ratio. 10,5725 Interest for one year, o Multiply by the time. 52,86'2.i Jns.=g52, 86c?s. 2 2. \Vhat is the interest of 6451. 10s. for 3 years, at 6 per cent, per annum ? xOxf>=l IO,190=11G Ss. 9<7. 2,4grs. .Jns. What is the interest of \-2lL 8s. 6d. for 4i years, at 6 per cent, per annum? l.GGnrs. 4. \Vh;it is tliii amount of 506 dollars 39 rents, for 1$ years, at G per cent.. per annum ? -. 8584.6651 5. Required the amount of (>43 tloU. 50 cts. for 12jyrs. at 5 A nor c^ut. per annum ? Jns. gl!03, 26c/s.-f i; ii. Tlie amount, time and ratio given, to find the principal. RULE. Multiply the. ratio by the time, add unity to the product fur a divisor, by which sum divide the amount, and th quotient \\ill be the principal. F.KAMPI.F.S. 1. What principal will amount to 1235,975 dollars,in 5 years, at 6 per cent, per annum ? g 55 ' ,06x5 + 1 = 1 ,30) 1235,975(950,75 Jtwt. 2. \Vhatprincipal will amount to 87o/. 19s. in 9 years, at fi per cont. per annum r */ns. 567 10s. ... \\'li:le thevema: pi-odiict ; of the tiine and pal, and the quotient will be the ratio. KXAMPLK3. 1. At what rate per c;rnt. will 950>75 doll, amount 1235,975 dols in 5 yours? SIMPLE INTEREST BY DEC1MAM- i7\ From the amount = 1235,97.; Take the principal = 950,75 950,75 X5=4753,75)85,225()(,06=tj per cent 285,2250 J/iS. 2. At what rate per cent, will j'GTY. 10s. amount to 87S/. 19s. iu 9 years ? Jtns. 6 ptr ant. 3. At what rate per cent, will 340 dols. 25 cts. amount to 626 dols. 6 cts. in r" voars? vins. 7 per cent. 4. At what rate per cent, will G45/. 15s. amount to 10 ~ 00 0:* K> CO o t'j co o to Oi 10 oo 00 en 1 CO J to 10 Oi C7! ir> c> K> ho o o to CO co Oi 00 si -J 4- OQ '_. i > Ki 'O O CO C 8* c-- Ol J-. 09 C st to g - <> K) -- '-1 1 co o .'i c-. oo ui to O CO . 13 s iil" o s? c.o g CO 00 1 2 Oi -? f'T > 2 Ui OJ i. O' _ C.-J p " 00 g Oi C7> OJ J- 0> - 1 ,- to l-l 00 C.-J si '_! 00 o J- OJ *. > 0; iO 1 1 iO o ^ O U' 4k C s (V - CO (^ -; to ^ oc' -4 Ju. t^ -P ^-" 1 I co en 10 ^=- o .1 SIMPLE INTEREST BY DECIMALS. When interest is to be calculated on cash accounts, &c. where partial payments are made 5 multiply the several balances into the days they are at interest, then multiply the sum of these products bv the rate on the dollar, ami divide the last product by 365, and you will have the whole interest d-.e on the account, otc. . \ MPLES. Lent Peter Trusty, per bill on demand, dated 1st of June, 1800, 2uOO dollars, of which I received back the 19th of August. 400 dollars ; on the 15th of October, CiK- dollars: on the llth of ijecember, 400 dollars : on the 17th of February, 1801, 200 dollars; and on the 1st of June, 400 do .'-lUch interest is due on the bill, reckoning at G ..-cr ci i.i. ? 1 800, dulls, days, products, June 1, Principal per bill, August 19, Received in' part, Balance. October 15, Received in purt, Balance, December 11, Rcceued iii part, 1801, l:.tlanoe, February 17, Received in part. Balance, Jiir>e 1, Ree'd in full of prim : Then 333600 ,U6 Ratio. 2000 400 79 57 57 104 i 15800U 91200 57000 40^00 41 GOO 1C- iOOO 400 600 200 400 400 | S88600 3U.5V2331 6,00(63,879 .IMS. = 63 87 9 -f- iH'iirh/r llrh f;i tiu*re are payment* ui purl. i>r >-iul'tp*se- mei; '/ Lite tiiijwiur Co; ''itai-e, 'of Cunneciict;!, in 1 78 \. RULE. " Compute ti.ji i.iierest to the time of the iir-t ::ftv- ' 174 'SIMPLE IXTEREST BY DEOIMA. mcnt ; if that he one. vear or more from the time tiie in- terest commenced', add it to the principal, an.' deduct the payment from the sum total. If there be afterpayments made, compute the interest on the balance due t next payment:, and then deduct the payment as above; and in like manner fnnn one payment to another, till all the payments are absorbed; provided the time between one payment and another be one year or more. But if any payment be made before one year's in; i ac crucd, then compute the interest on the ; due on the obligation for one year, add it to ihe principal, and compute t!iO interest on the sum paid, from the time. it was paid, up 10 the end of the year; add it to'the sum pau!, and deduct that sum from the principal and interest added as above.* If any payments he made of a lass sum than the in- terest arisen at tiie time *.(' such. payment, no interest U to be computed but only <,u the principal sum fur a^r period." K'irht/'s li p:>rt*. /v/^r AMi-i.r.s. * A bond, or : d January -4t!i, 1707, was j^ivcn for- 1600 dollars, Mite rest at i per cent, and there were payments emlorsed upon it a.-> follows, > 8 1st payment }V:,mai v 1J, i ^UO 2d pa/meht June ^',' ir'Ji>. 500 3d payment November 14, 1799. 260 I demand how 111110:1 remains due on said iutG the 24tl) of December, J 1000,00 dated January -1, 1797. 67, 19, 1798ir=lSi months^ 1067,50 amount. [Carried tt| *lf;i y c.-'r (lues IMP! cxtei ii l-i-yond l^c t : ;. < 'tle- mri.l : ,. ;i,iiuunl i,. U. due on ;!if ol)li^, .me of s.-Mlcm ; I'lMtJ the ainiHt iftliere be Hevi-ra! payments made \\ithiutlie . find the amount of the several \ iwi.l , tVom the time t!i-y \\crt- paid, to tiie tiine ol'selUeiiifiit, and deduct their amount .mount of ll.e , SIMPLE INTEREST BY DECIMALS. 175 I06f ,50 amount. [Brought Up. 200,00 first payment deducted. 3jti7,50 balance due, February 19, 1798. 70,845 Interest to June 29, 1799 s 16$ month*. 958,345 amount. 00,000 second payment deducted. 458,545 balance due, June 29, 6,30 Interest for one year. 464,645 amount for one year. 69,750 amount of third payment for 7\ months.* 194,895 balance due June 9, 1800. mo. Aa. 5,687 Interest to December 4, 1800, 5 25 200,579 balance due on the Note, Dec. 24, 1800. RULE II. Established by the Courts of Law in Massachusetts for computing interest on notes, <"c. on which partialpay- mtnts have been endorsed. " Compute the interest on the principal sum, from the time when the interest commenced to the first time when a payment was made, which exceeds either alone or ia conjunction with the preceding payment (if any) the in- terest at that time due : add that interest to the princi-* pal, and from the sum subtract the payment made at that time, together with the preceding payment (if any) aad the remainder forms a new principal ; on which compute and subtract the payments as upon the first principal, and pmceed in this manner to the time of final settle- ment." S cts. *260,00 third payment with its interest from the time it 9,75 was paid, up to the end of the year, or front ./You. 14, 1799, to June 29, ?800, which is 7 176 SIMPLE INTEREST BY DECIMALS. Let the foregoing example be solved by this Rule. A note for 1000 dols. dateu Jan. 4, 179", at 6 per ceut 1st payment February 19, 1793. g200 2U payment June 29, 1799. 500 3d payment November 14, 1799. 260 How much remains due oil said note the 24th of Do- :cmber, 1800? 8 cl*. Principal, January 4, 1797, 1000,00 Interest to Feb. 19, 1798, (IS* wo.) 07,50 Paid February 19, 1798, Amount, l f ifS7,.v> 200,00 Remainder for a new principal, 867,50 Interest toJune 9, 1799, (lfi me.) 70^84 Paid June 29, 1799, Amount, 938,54 500,00 Remains for a nt Interest to November J4, 1799, (4} mo.) 438,34 9,86 November 14, If96, paid Remains a new principal Amount, 448,20 260,00 ' 188,20 Interest to December 24, 1800, (IS* mo ) 12,70 Balance due on said note, Dec. 24, 1800, 00,%< g rN. The balance by Rul I. 200,579 Byllule IJ. 200,990 Difference, 0,41 1 Anotler Example in Rule H. A l>*>nd or note, dated February 1, ItiOO, \va^i>*a h" J(> dollars, interest at G per cent, and ihc, upon it as follows, vi/,. fc Cl l>; j.aviueut May 1, 1800, 2d ( " urcmber 14, 1 COMPOUND INTEREST 15Y DF.C'IMALS. 177 3d payment April 1, 1801. 12,00 4th payment May 1, 1801. 50,00 How much remains due on said note the 16th of Sep. tomber, 1801? S ets. Principal dated February 1, 1800, 500,00 interest to May 1, 1800, (5 mo.} 7,50 Amount, 507,50 Paid May 1, 1800, a sum exceeding the intu -~i,t. 40,00 New principal, May 1, 1800, 467,50 futerest to May 1, 1801, (1 yenr.) 28,05 Amount, 495,55 Paid N.v. 4, 1800, a sum less than the interest then due, 8,00 Paid April 1, 1301, do. do. 12,00 Paid May 1, 1801, a sum greater, 30,00 50,00 New principal May 1, 1801, 445,55 Interest to Sep. 16, 1801, (4} mo.) 10,02 Balance due on the note, Sept. 16, 1801, $455,57 |Cp The payments being applied ace or ding to this Rult } keep down the interest, and no part of the interest ever forms a part of the principal carrying interest. COMPOUND INTEREST BY BECIMALS. RULE. MULTIPLY the given principal continually by the amount of one pound, or one dollar, for one year, at th rate per cent, given, until the number of multiplication* are equal to the given number of years, and the product will be the amount required. OK, In Table I. Appendix, find the amount of one dol- lar, or otie pound, for the giren number of years, which multiply by the given principal, and it will give the amount as before. 1*8 EXAMPLES. 1 . What will 400Z. amount to in 4 years, at 6 pet cent. per annujn, compound interest ? 400xl,06xl,06xl,06xi,06=:504,99-f or [504 19s. 9d. 2,75jrs.+ Ms. Tlie same by Table I. Tabular amount of 1 = 1,26247 Mullipiy by the principal 400 Whole amount 5t)4,98800 2. Required the amount of 4i.5 dol. 75 cts. for 3 yearf, at 6 per cent, compound interest. 3?w. $507,7ic/s.-f 3. Vv hat is the cump -iT.nl interest of 555 dols. for 14 years, at 5 percent., r By Table I. dns. $$548,86cs.-}- 4. What will 50 dollars amount to in 20 years, at 6 per cent, compound interest .' dns. fc>!60 Sects. G^m. INVOLUTION. IS the ini'ltiplyin^; any number with itself, and that pro- duct ty tK*; farmer multiplier ; and so ou j and the several pro'luct* which arise are called powers. The number denoting the height of the power, is called the index, or exponent of that power. EXAMPLES What is the 5th power of 8 ? 8 the root or 1st power. 64 =at d power, or square 8 512 =a 3d power, qr cube. 8 4096 = 4th power, orbiquadrate. 8 CrCS s 5th power, or siu-salid. A**. 179 What is the square of 17,1 ? Ms. 292,41 What is the square of ,085 ? Ans. ,007225 What is the cube of 25,4 ? Ans. 16387,064 What is the biquadrate of 12 ? Ana. 20736 What is the square ot'Ti ? .5ns. 52^ EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the busi- ness of finding it is called the Extraction of the Root. The root, is that number, 'which by a continual multipli- cation into itself, produces trie given p'>wer. Although there is no number but what will produce a perfect power by involution, yet there are many numbers of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root to any assigned degre^ of exactness. The roots which approximate, are called surd root?, and those which are perfectly accui ate are called rational roots. A Table of the (Squares mid Cubes of the nine digits. Roots. \ 1 1 2 3 u 5 61 7 8 9 Squares. M 1 4 9 16 J 36 | 49 64 81 Cubes. M l 27 64 | 125 216 | 343 512 729 EXTRACTION OF THE SQUAPE ROOT. Any number multiplied into itself produces a square. To extract the square root, is only to nnd a number, which being multiplied into itself, shall produce tne given number. RULE. 1. Distinguish the given number into periods of two figures each, by putting a point over the plan of units, another over ti 1 * r/.aoe >ds, an-1 - ui if there are dyirnals, point th^rv in the same manner, from units towards the M hand; which points bhow tha number of figures the root will consist of. 2, Find the greatest *MU are number in the fiist, orient 180 liand period, place the root ot it at the right hand of the given number, (after the manner of a quotient in division) tor the first figure of the root, and the square number under the period^ and subtract it therefrom, and to the remainder bring down the next period for a dividend. 3. Place the double of the root, already_found, on the left hand of the dividend fora divisor. 4. Place sch a figure at the right hand of the divisor, and also the same figure in the root, as when multiplied into the whole (increased divisor) the product shall be equal to, or the next less than the dividend, and it will be the second figure in the root. 5. Subtract the product from the dividend, and to the remainder join the nexJ, period for a new dividend. G. Double the figures already found in the root, for a ilexv divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner, til! you h I own nil the periods. Oi,to facilitate the foregoing Rule, wla-M you have brought d i. wa a period, and forme I a dividend, in order to find a new figure in the root, you may divide said divi- dend, (omitting the right hand iigure thereof,) by double the root already founn, and the quotient will commo !y be the figures sought, o,- heing ::ntU: les o:ie. or two, will generally give the next figure in the quotient. EXAMPLES. 1. Required the square root of t -11 225,64. 141225,64(375,8 the root exactly without a remainder ; 9 but when the periods belonging to anv given number are exhausted, and still 67,512 leave a remainder, the operation may 469 be continued at pleasure, bj periods oi cyphers, c. 745)4325 5725 7508)60064 60064 remain* EVOLUTION, Oft EXTRACTION OF ROOTS. 181 Answers. 2. What is the square root of 1296 ? 36 3. Of 56644 ? 3*8 4. Of 5499025 ? 345 5. Of 36372961 ? 6031 6. Of 184,2? 13,57-f 7. Of 9712,695809 ? 98,553 8. Of 0,45369? j673-f 9. Of ,002916? ,054 10. Of 45 ?" 6,708-ji TO EXTRACT THE SQUARE ROOT OF VULGAR FRACTIONS. RULE. Reduce the fraction to its lowest terms for this and all other roots ; then 1. Extract the root of the numerator for a new nume- rator, and the root of the denominator, fora new denomi- nator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root. EXAMPLES. 1. What is the square root of -^ ? Jlns. 2. What is the square root of /^ ? Jlns. %% 3. What is the square root of -*4| ? . *flns. 4. What is the square root of 20| ? Jltis. 4| 5. What is the square root of 248 T V ? Jns. ISf SURDS. 6. What is the square root of ? Jlns. 9128-f- 7. What is the square root of ^| ? Jlns. ,7745 4- 8. Required the square root of 361 ? Jhs. 6,0207+ APPLICATTON AND USE OF THE SQUARE ROOT. PnoitLF'-f T. A certain General nets nnai-myot 5184 men; how many must he place in rank ant $e, to form '""Ti -nto a square? 1 P 183 EVOLUTION, OH EXTRACTION OF ROOTS. RULE. Extract the square root of the given number. v/5184="2 Jim. PHOB. II. A certain square pavement contains 20736 square stones, all of the same size ; I demand how many are contained in one of its sides ? ^20756=144 Ans. PROB. III. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product. KXAMPLF.?. What is the mean proportional between 18 and 72 ? 72x18=1290, and v/ 1296 =36 Ms. PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as manv in rank as in file. HULK. ' Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men in file, which double, triple, &c. and the product will be the number in rank. EXAMPLES. Let 13122 men be so formed, as that the number in rank may be double the number in file. 13122-5-2=6561, antl v/GoGl^Sl in file, and 81x2 = 162 in rank. PROB. V. Admit 10 hlids. of water arc discharged through a lead en pipe of 2.} inches in diameter, in a cer- tain tnr.e; I demand what the diameter of another pipe must be, to discharge four times as much water iu the same time. HULK. Square the given diameter, and multiply said square by the given proportion, and the square root of the pro- duct is the answer. 2J2,5, and 2,1x2,5=6,25 square. 4 given proportion* v/25,00=5 inch, cliam. .'7 KVOLWTIOK, OR EXTRACTION OF ROOTS. 18S PROB. VI. The sum of any two numbers, and their products being given, to find each number. RULE. From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers ; then half the said difference added to half the sum, ;ives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number. EXAMPLES. The sum of two numbers is 43, and their product is 442 ; what are those two numbers ? The sum of the numb. 43x43=1849 square of do. The product of do. 442x 4=1768 4 times the pro. Then to the 4 sum of 21,5 [numb. +and 4,5 /81=9 diff. of the Greatest numb. 26,0 ") 4* the diff. E S- Answers. Least numb. 17,0 J EXTACTION OF THE CUBE ROOT. A Cube is any number multiplied by its square. To extract the cube root, is to find a number, which, being multiplied into its square, shall produce the given number. RULE. 1. Separate the given number into periods of three fig- res each, by putting a point over the unit figure, and very third figure from the place of units to the left, and if there be decimals, to the right. 2. Find the greatest cube in the left hand period, and place its root in the quotient. 3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, calling this the dividend. 4. Multiply the suqare of the quotient by SOO, calling it the divisor. 184 EVOLUTION, OR EXTRACTION OF ROOTS. 5. Seek how often the divisor may be had in the divi- dend, and place the result in the quotient ; then multiply 4 the divisor by this last quotient figure, placing the pro- duct under the dividend. (.. Mu]t|ply the former quotient figure, or figures by the square of the lust quotient figure, and that product by SO, and place thf! product under the last ; then under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend. 7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period fora new divi- dend ; with uhich proceed in the same manner, till the whole be finished. NOTE. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequent- ly cannot be subtracted therefrom, you must make the last quotient figure one less; with which find a ue'v sub- trahend, (by ttie rule foregoing) and so on until you can subtract the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18599,744(26,4 Root. Ant. 8 2x2=4x300 = 1200)10599 first dividend. 7200 6x6=S6x2=r2xSO=2lGO 6x6x6= 216 9576 1st subtrahend. 26x26=676x300=202800)823744 2d dividend. 811200 16x26=416x50= 12480 G4 823744 2d subtrahend. BVOLUTIOtf, OK EXTRACTION OF ROOTS. 185 NOTE. The foregoing example gives a perfect root ; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of cyphers, and sontinue the operation as far as you think it necessary. Answer.*. 2. What is th'e cube root of 205579 ? 59 S . Of 614125 ? 85 4. Of 41421736? 346 5. Of 146363,183? 52,7 6. Ot 9,503629 ? 5,09-f 7. Of 80,763 ? 4,52+ 8. Of ,162771356? ,546 9. Of ,000684134? ,0884- 10 Of 122615327232? . 4968 RULE II. 1. Find by trial, a cube near to the given number, and. tall it the supposed cube. 2. Then as twice the supposed cube, added to the giv- ?n number, is to twice the given number added to the supposed cube, so is th root of the supposed cube, to Lhe true root, or an approximation to it. 3. By taking the cube of the root thus found, forth* supposed cube, and repeating the operation, the root will >e had to a greater degree of exactness. , EXAMPLES. Let it be required to extract the cube rooWbf 2. Assume 1,3 as the root of the nearest cube; then !,3xl,3xl,3=2,l97=supposed cube. Then, 2,197 2,000 given number. 2 4,594 4,000 2.000 2,197 As 6,594 : 6,197 : : 1,5 : 1,2599 root, ivhich is true to the last place of decimals ; but might by epcating the operation, be brought to greater exactness. -2. What 5s the cube root of 584,arrW* t c , to- ISb EVOLUTION, OR EXTRACTION O? ROOTS. 5. Required the cube root of 729001101 ? Jns. 900,0004 QUESTIONS, Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity ? .3/2150,425=12,907 inch. Ans. Note. The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diamei^rs. 2. If a bullet 5 inches diameter, weigh 4lb. what will a bullet of the same metal weigh, whose diameter is 6 inches ? 3xSx3=27 6x6x6=2i6 As 27 : 4lb. : : 216 : 3211).' Jlns. 3. If a solid globe of silver, of 3 inches diameter, be worth 150 dollars; what is the value of another globe of silver, whose diameter is six inches ? g 5x5x3=27 6x6x6=216 As 27 : 150 : : 216 : SI 200. Jlns. The side of a cube being given, to find the side of that cube wich shall be double, triple, &c. in quantity to the given cube. RULE. Cube your */ven side, and multiply by the given pro- portion between the given and required cube, and the ube root of the product will be the side sought. 4. If a cube of silver, whose side is two inches, be worth 20 dollars ; I demand the side of a cube of like silver. 'vhtse value shall be 8 times as much ? 2x2x2=8 and 8x8=64^/64=4 incfes. Jlns. 5. There is a cubical vessel, whose side is 4 feet ; I demand the side of another cubical vessel, which shall contain 4 times as much ? 4x4x4=64 and 64x4 =256 ^2566,349-f-/t. Ans. 6. / cooper having a cask 40 inches long, and 3:-' in- EVOLUTION OR EXTRACTION OF ROOTS. 1&7 chca at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much j what will be the bung diameter and Length of the new cask ? 40x40x40x2=128000 then & 128000= 50,3 -f length. 32x32x32x2=65530 and ^/G5536=40,3-f bung diam. f General-ltulefor Extracting the Roots of ail Poirers, RULE. 1. Prepare the given number for extraction, by point ing oft' from the unit's place, as the required root directs 2. Find the first figure of the root bv trial, and subtract its power from the left hand period of the given number. 3. To the remainder bring down the first figure in the next period, and call it the dividend. 4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor. 5. Find how many times the divisor may be had iii the dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and sub- tract it (always) from as many periods of the given num her as you have found figures in the root. 7. Bring down the first figure of the next period to the remainder for a new dividend, to which fuid a new divi- sor, as before, and in like manner proceed till the whole be finished. NOTE. When the number to be subtracted is greater than those periods from which it is to be taken, tne lat quotient figure must be taken less, &c. EXAMPLES. 1. llequired the cube root of 135796,744 by the ai>ov jent'ial method. 188 EVOLUTION, OH EXTRACTION OF ROOTS. 135796744(5 1,4 the root. 125=lst subtiaheml 75)107 dividend. 152651 =2d subtrahend. 7803) 31457=2d dividend. 1 35796744 =3d subtrahend. 5x5x3=75 first divisor. 51x51x51 = 132651 second subtranend. 51x51x3=7803 second divisor. 514x514x514=.135796744 third subtrahend, 0. Required the sursolid, or fifth root of 6436343, 6436343)23 root 32 2x2x2x2x5=80)323 dividend. 23 X23 x23 x23 X23 =6436343 subtrahend^, NOTE. The roots of most powers may be found by the square and cube roots only ; therefore, when any even power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which reduces it to half the given power, then the squire root of that power reduces it to half the same power ; and so on, till you come to a square or a cube. tor example: suppose a 12th power be given: the square root of that reduces it to a sixth power : aud the square root of a sixth power to a cube. KXAMM.V.S. 3. What is the biqiiadrate, or 4th root of 19987173376 .- Jns. 376. 4. Extract the square, cubed, or 6Ui root of 12230590 464. Ans. 48. 5. Extract the square, biquadrate. or 8th root of 721 SS 95789388336. Jlns. 96 ALLIGATION 189 ALLIGATION, IS the method of mixing several simples of different qual- ities, so that the composition may be of a mean or middle quality : It consists of two kinds, vi/,. Alligation Medial,, and Alligation Alternate. ALLIGATION MEDIAL, Is when the quantities and prices of Several tilings are given, to find the mean price of the mixture composed of those materials. RULE. As the whole composition : is to the whole value : : so is any part of the composition : to its mean price. EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushel, 1 8 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a bushel of this mixture is worth ? bu. eta. facts. bii. g cfs. bu. 15 at 64=9,60 As 54 : 25,38 : : 1 18 5.7=9,90 1 21 28=5.88 cts. 54)25,38(,47 'Answer. 54 25,38 2. If 20 bushels of wheat at 1 dol. 55 cts. per bushel, be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth ? Jns. gl, 20cte. 3. A Tobacconist mixed 36 Ib. of Tobacco, at 13. 6d. Ker Ib. 12 Ib. at 2s. a pound, with 12 Ib. at Is. lOd. par ). ; what is the price of a pound of this mixture ? Jns. Is. 8rf. 4. A Grocer mixed 2 0. of sugar, at 56s. per C. and 1 C. at 43s. per C. and 2 C,'. at. 50s. per C. together; I de- mand the j)rice of 3 cwt. of this mixture? Jlns. 7 13s. 5. A "Wine merchant mixes 15 gallons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons, at 6s. 3d. ; what is a gallon of this composition worth ? Ms. 5s. Wd. I'.'fl ALLIGATION ALTERNATE. t>. A grocer lialh several sorts of suar, viz. one son at 8 tlol.s. percwt. another .sort at 9 dots, percwt. a third sen*, at 10 do's, ^or cv.f. and a fourth sort at 12 dols. per cwt. and ho uould mix an equal quantity of each togeth- er ; I demand the price of Si cwt. of this mixtii' Jus. 34 IZcts. .Int. 7. -\ ') molted together 5 Ib. of silver bullion, of 8 07.. line, 10 !b. of 7oz. fine, and 15 Ib. of 6 07.. line ; pray \vlwit is the quality, or fineness of this composition r Jlns. Goz. ISpwt. Sgi-.Jinc. Si Suppose 5 Ib. of old of 2 carats fine, 2, Ib. of 21 carats fine, and I Ib. of alloy be melted together ; what la the quality, or fineness of this mass r 19 carats fine. ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients, whose rates are given, will compose a mix- ture of a given rate ; so that it is the reverse of alligation ntedial, and may be proved by it. CASE. I. "When the mean rate ot the whole mixture, and the rate* of all the ingredients are given without any limited quantity. RUI1J. 1. Place the several rates, or prices of tho simples, be- in: 1 ; reduced to one denoiad nation, in a column under eath 1 the mean price in the like name, at the left kind. -2. Connect, or link, the price of each simple or ingre- dient, which is loss than that of the mean rate, with one or any number of those, which are jp-eator than the mean rate, and each greater rate, or price with one, or any number of the less. 3. Place the difference, between the mean price (or mixture rate) and that of each of the simples, opposite to ine rotes with which they are connected. ALLIGATION' ALTERNATE. 191 4. Then, if only one difference stands against any rate, it will be the quantity belonging to that rate, but it there be several., their sum will be the quantity. EXAMPLES. 1. A merchant has spices, some at 9d. per Ib. some at Is. some at 2s. and some at 2s. Gd. per Ib. how much of each sort must he mix, that he may sell the mixture at Is. oil. per pound ? (L (I Ib. d. Ib. 9 ,10 at 9"| f 1 q ^ 4 12 \Gives the d.J 12-;- 10 i 8 24 f Answer, or 2(H 24 J 11 f 2 11 30 j 1.30 ' 8J "S 2. A grocer would inix the following quantities of su- jVti" ; vi/.. at 10 cents, 13 cents, and 1(1 cts. per Ib. ; what 'juantity of each sort must be taken to make a mixture v.-orth fc cents per pound ? .ins. 5lb.at Wets. Qlb. at iScts. and 2lb. at 1C els. jier Ib. 5. A grocer has two sorts of tea, viz. at 9s. and at 15s. per Ib. how must he mix them so us to afford the compo- sition for 12s. per Ib.? Ans. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with ftb<> 'ii'p^i^r. PARTIAL. LU3 2. How much waier must be mixed with 100 gallons of rum, worth 7s. 6d. per gallon, to reduce it to 6s. 3d. per gallon ? ./fos. 20 gallons. 3. A fanner would mix 20 bushels of rye, at 65 cents per bushel, with barley at 51 cts. and oats at 30 cts. per bushel; how much barley and oats must be mixed with the 20 bushels of rye, that the provender may be worth 41 cents per bushel ? Jlns. 20 bushels of barley, and 61 T 9 T bushels of oats. 4. With 95 gallons of rum at f>s. per gallon, I mixed ether rum at 6s. 8d. per gallon, and some water; then 1 found it stood me in 6s. 4d. per gallon ; I demand how much rum and how much water I took ? flnj?. 95 gals, rum at 6s. Sd. and 50 gals, renter. CASE III. When the whole composition is liiftited to a given quantify . RULE. Place the difference between the mean rate, and the- several prices alternately, as in CASK 1. ; then, As the sum of the quantities, or difference, thus determined, is to the given quantity, or whole composition : so is the dirte- rcucc of each rate, to the required quantity of each rat.i.-. EXAMPLES. 1. A grocer had four sorts of tea, at Is. ;1s. fi*. and 1G. per Ib. the worst would not, sell, and the best we're too dear; he therefore mixed 120 h>. and so much tf e.icli sort, as to sell it at 4s. p-:r U>. ; how much of each sort di. that it shall be 8 oz tine; how much of each sort must he take ? JMS. 4* of 5 oz.f.n?, and 7\ of 10 oz.fine. 5. Hramly at 3s. (Jil. and js. Od. per gallon, is to be mixed, so that a hhd. o! Cms mm- !)e sold for 12/. 12j. ; how many gallons must I.e taken of each ? *i.'s. 14 /??. af '5s. ( Jrf. anc/ 49 -a/s. rti 5s. 6J' Problems : but ' ma ' r ARITHMETICAL PROGRESSION. 195 . RULE. Multiply the sum of the extremes by the number ot terms, and half tho product will be the answer. EXAMPLES. 1. The first twin of an arithmetical series is 3, the last term 23, and the number of terms 11 ; required the sum of the series. 23-j-3=25 sum of the extremes. Then 26x11-4-2=143 the Answer. 2. How many strokes docs the hammer of a clock strike, in twelve hours ? .flns. 78 S. A merchant sold 100 yards of cloth, viz. the first yard for 1 ct. the second for 2 cts. the third for 3 cts. &c. I demand what the cloth came to at that rate ? Ms. 8504. 4. A man bought 19 yards of linen in arithmetical pro- gression, for the first yard he gave Is. and for the last yd. I/. 17s. what did the whole come to? Jlns. 18 Is. 5. A draper sold 100 yds. of broadcloth, at 5 cts. for the first yard, 10 cts. for the sec.md, 15 for the third, &c. increasing 5 cents for every yard : What did the whole amount to ; and what did it average per yard ? Jlns. Amount, g252 }, and the average price is g2, 52cfs. 5 mills per yard. 6. Suppose 144 oranges were laid 2 yards distant from each other, in a right line, and a basket placed two yards from the first orange, what length of ground will that boy travel over, who gathers them up singly, returning with them one by one to the basket ? Ans.ZS miles, 5 furlongs, 180 yds. PROBLEM II. The first term, the last term, and the number of terete given, to find the common difference. RULE. Divide the difference of the extremes by the number of terms less 1, and the quotient will be the common dif- ference. 195 ARITHMETICAL PROGRESSION. I EXAMPLES. 1 . The extremes are 3 and 29, and the number of terms 14, what is the common difference ? Extremes. Number of terms less 1=13)26(2 .0ns. 2. A man had 9 sons, whose several ages differed alike, the youngest was 3 years old, and the oldest 35; what was the common difference of Iheir ages ? Ans. 4 years. 5. A man is to travel from New-London to a certain place in 9 days, and to go but 3 miles the first day, in- creasing every day by an equal excess, so that the last day's journey may be 43 miles : Required the daily in- crease, and the length of the whole journey ? Jlns. 'Die daily increase is 5, and the "whole journey 07 miles. 4. A debt is to be discharged at 16 differt^i. nayments (in arithmetical progress! on.) the first payment is to be 141. the last lOOJ. : What is the common difference, and the sum of the whole debt ? Jins. 5l. 14. 8rf. common difference, and 9121. the whole debt. PROBLEM III. Given the first term, last term, and common difference, to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and the quotient increased by 1 is the number of terms. EXAMPLES. 1. If the extremes be 3 and 45, and the common dif- ference 2 ; what is the number of terms ? Jlns. 22. 2. A man going a journey, travelled the first day five miles, the last day * miles, and each day increased lii* journey by 4 miles} how many days did he travel, and how fur ? Jus. 1 1 days, and tlie whole distance travelled 275 mrks GEOMETRICAL PROGRESSION. 197 GEOMETRICAL PROGRESSION, IS when any rank or series of numbers increased by one common multiplier, or decreased by one common divisor , as 1, 2, 4, 8, 16, Sec. increase by the multiplier 2; and 7, 9, 3, 1, decrease by the divisor 3. PROBLEM I. The first term, the last term (or the extremes) and tht ratio given, to find the sum of the series. RULE. Multiply the last term by the ratio, and from the pro- duct subtract the first term ; then divide the remainder by the ratio, less by l,and the quotient will be the sum oF all the terms. EXAMPLES. 1. If the series be 2, 6, 18, 54, 162, 486, 1458, and the ratio 3, what is its sum total ? 3x14582 : =2186 the Answer. 31 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4; what is the sum of the series? Atis. 87381. PROBLEM II. Given the first term, and the ratio, to find any other t?radin; terms. S + 2+l*=6, the index of the 7th term. 108x36x12=40656 -- =2916 the 7th term required. 16 Here the number of terms multiplied are three; there- fore the first term raised to a power U:>s than three, is the 2d power or square of -4 = 16 the divisor. * When the first term of the scries and the ratio are dif- ferent ..the indices must begin with a cypher, and the sum of the indices made choice of i.iust be one leas than the. num- ber of terms given in the question : because I in the indices stands over the second term, and in the indices over the. third term, <'c. a?;;l in this caw, the product of amj ftco terms, divided % the first, is equal to that term beyond the first, signified by the sum of ikeir indices. ' T/i 5' ! 2 ' S ' 4 ' ' I 1, 3, 9, 27, 81, cyr. Geomet ricnl srrie.s. Here 4 + 3=7 the inde.v of the 8th term. 81 x 27=21 37 the 8f/i term, or the 7lh beyond the 00 vosmojr. 5. A Goldsmith sold 1 Ib. of gold, at 2 cents for the first ounce, 8 cents for the second, 32 -cents for the third, c. in a quadruple proportion geometrically; what did the whole come to ? rfns. gl 11848, Wets. 3. "What debt can be discharged in a year, by paying 1 farthing the first month, 10 farthing?, (or 2^x1.) the se- cond, and so on, each mryitlrin * ^nsfold proportion ? 11 5740740 14s. 9d. Sqrs. 7. A thresher worked 20 days for a farmer, and receiv- ed fov the first day's work four barley-corns, for the second 1 2 barley -corns, for the third ?(5 barley-corns, and so on in triple proportion geometrical. 1 demand what the 20 lays' labor. came to, supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel ? flns. 1773 7s. 6d. rejecting remainders. 8. A man bought a horse, and by agreement was to jjive a farthing for the first nail, two for the second, four lor the third, obc. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate ? Jns. 4473924 5s. Sjrf. 9. Suppose a certain body, put in motion, should move ilu- length of one barley-corn the first second of time, one inch the second, ;uul three inches the third second of time, and so continue to increase its motion in triple pro- portion geometrical : how many yards would the said body move in the term of half a minute ? .flns. 953199G85G23 yds. }ft. lin. Ib.c. irhich is no less than jive hundrzd and forty-one millions of miles. POSITION. JT OS IT ION is a rule which, by false or supposed num- bers, taken at pleasure, discovers the true ones required. K Is divided into two parts, Single or Double. SINGLE POSITION, Is when one number is required, the properties of which are given in the question. SINGLE rOSITION. ('!] RULE. 1. Take any number and perform the same operation with it, as is described to be performed in the question. 2. Tken say; as the result of the operation : is to th given sum in the question : : so is the supposed number : to the true one required. -The method of proof is by substituting the answer in the question. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, If I had as many more as I now have, half as many, one-third and one-fourth as many, I should then have 148: How many scholars had lie ? Suppose he had 12 As 37 : 148 : : 12 : 48 Ans as many = 12 48 $ as many =6 24 as many =4 16 i as many = 3 12 Result, 37 Proof, 148 2. What number is that which being incrtasedby A, ^, aid i of itself, the sum will be 125 ?' 3. Divide 93 dollars between A, B and C. so that B's share may be half as much as A's, andC's ^::ire three times as much as B's. 4ns. A's share 51, B's 15$, and C's 46 J dolls.. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 5 A times as much as A, and C took up as much as A and B both ; vrhat share of the gain had each ? Ans. A 840, B g!40, and C g!80. 5. Delivered to a banker a certain sum of money, to receive interest for the same at 61. per cent, per annum, simple interest, and at the end of twelve years received 7SH. principal and interest together : What was the sum I delivered to him at first ? Ans. 425. 6. A vessel has 3 cocks, A, B and C ; A can lill U in 1 hour. B in 2 hours, and C in 4 hours ; in what time wilt they all fill it together ? Ans. 34?nin. I7sc. 20hi DOUBLE POSITION DOUBLE POSITION, AEACHES to resolve questions by making two suppft- 5 itions of false numbers.* RULE. f I. Take any two convenient numbers, and proceed v.itli each according to the conditions of the question. Find how much the results are different from the Its in the question. Multiply the first position by the last error, and the Jast position by the first error. 4. If the errors are alike, divide the difference of the products by the difference of the errors, and the quotient will be the answer. j. If the errors are unlike, divide the sum of the pro- ducts by the sum of the errors, and the quotient will be tlie answer. NOTE. The errors are said to be alike when they are both too great, or both too small : and unlike, when one is too great, and the other too small. EXAMPLES. 1. A purse of 100 dollars is to be divided among 4 men, A,B, C and D, so that B may have 4 dollars more than A, and C 8 dollars mo, c nan B, and D *wi many as C : what is each one's share of the money ? 1st. Suppose A G 2(1. Suppose A S B 10 B 12 C 18 C 20 D 5G D 40 70 80 100 100 1st. error 2d. error 20 * Those question*, in ichich the results tire not propor- tional to their position*, belong to this rule ; such as those, in u'hich the number sought is increased or diminished by some given number, which is no known part of the number required DOUBLE POSITION. 203 The errors being alike, are both too small, therefore, Pos. Err. 6 SO X 8 20 240 120 120 10)120(12 A'spart. 2. A , Band C, built a house which cost 500 dollars, ot which A paid a certain sum ; B paid 10 dollars more than A, and C paid as much as A and B both j how much urn each man pay ? Jins. A paid 120, B 130, and C 250 dots. 5. A man bequeathed 100/. to three of his friends, afte, this manner: the first must have a certain portion : the second must have twice as much as the first, wantin'o- SI. and the third must have three times as much as the lust, wanting 151. : I demand how much each man must have ? Jins. TheJlr*t2Q lO.s. second 33, third 46 10s. 4. A laborer was hired for CO days upon this condition : that for every day he wrought he should receive 4s. and tor every to he was idle, should forfeit 2s.: at tho ex- piration ot the time he received 7 1. 10s.; hov/manvdav* aid he work, and how many was he idle ? Jlns. He wrought 45 'days, and was idle 15 r/ays. o. What number is that which being increased by its $, its i, and 18 more, will be doubled ? Jlns. 72. 6. A man gave to his three sons all his estate in monov, '17.. to F half, wanting 501. to G olie-thinl, and to II tfie est, which was Wl. less than the share of G ; [demand fie sum ivon,and each man's part? ins. Thesnm given ivas 3GO, whereof Fhad /ISO. 204 PKaMUTATlON OF QUANTITIES. 7. Two men, A and B, lay out equal sums of money in trade ; A gains 126. and B looses 8?/. and A's money is now double to IVs : \\hat did each lay out ? Jilts. 500. 8. A farmer having driven his cattle to market, reciv- ed for them all ISO/, being paid for every ox 71. for every cow 5l. and for every calf I/. 10s. there were twice as many cows as oxen, and throe times as many calves as eows ; how many were there of each sort ? Jlns. 5 oxen, 10 cows, and 30 calves. 9. A, B and C, playing at cards, staked 524 crowns ; but disputing about tricks, each man took as many as he could : A got a certain number ; B as many as A and 15 more ; C got a fifth part of both their sums added togeth- er : how many did each get ? Jlns. A 127J, B 142}, C 54. PERMUTATION OF QUANTITIES, IS the showing how many different ways any given number of things may be changed. To find the number of Permutations or changes, that can be made of any given number of things, all different from each other. RULE. Multiply all the terms of the natural series of numbers, from one up to the given number, continually together and the, last product will be the answer required. EXAMPLES. 1. How many changes can be made of the three first letters u the alphabet? Proof; Ans. a b c a c b b a c b c a c b a cab 3. IIww maTiv changes may be rung en 9 bells? 362S80. ANNUITIES OR PENSIONS. 05 3. Seven gentlemen met at an inn, ami \vcrc so well pleased with their host, and with each other, that they agreed to tarry so long as they, together with their host, could sit every day in a different position at dinner ; how long must they have staid at said inn to have i'u Hilled their agreement : dits. llO^j year*. AXNUITIKS OR PKNSIONS, COMPCTKD AT COMl } U.VI) LVTEREST. ("ASK I. To find the amount of an annuity, or Pension, in arrears, at Compound Interest. RULE. 1. Make 1 the first term of a geometrical progression, and the amount of gl or 1 tor one year, at the given rate pr cent, the ratio. 2. Carry on the series up to as many terms as the given number of years*, and find its sum. 3. Multiply the sum thus found, by the given annuity, and the product will be the amount sought. KX \MIT.KS. 1. If 125 duls. vearly rent, or annuity, be forborne, (or unpaid) 4 years; what will it amount to, at G per cent, per annum, compound interest? 1 + 1,06+1, 1236+1, l910lG=4.Sr46lG sum of the scries** Then, 4,374Cloxl2o=S54G,827 the amount sought. OR BY TABLK I Multiply tMe Tabular number under the rate and op- posite to the time, by the annuity, and the product will be the amount sought. *Tlte sum of the series thus found, is the ancnntt of \l. or 1 dollar annuit;,'. far the given time, iruich mat/ be fuuntlin Table. JL ready calculated. Hence, either the a mount or present worth of nnnuitiea m.7y be readi!:/ found by Tables for that 206 ANN CITIES OR PENSIONS. 2. If a salary of GO dollars per annum to be paid year- ly, be forborne '20 years, at per cent, compound in- terest j what is the amount ? Under 6 per cent, and opposite 20, in Table II, you will find, Tabular number =36,78559 60 Annuity. -9ns. S2207,13540=g:220r, IScts. 5?n.+ . 5. Suppose an Annuity of 100J. be 12 years in arrears, it is required to find what is now due, compound interest being allowed at 5/. per cent, per annum ? .iiis. 1591 i4s. 3,024rf. (by Table II.) 4. What will a pension of 120Z. per annum, payable yearly, amount to in 3 years, at 5/. per cent, compound interest? .ins. 578 6s. II. To find the present worth of Annuities at Compound Interest. RULE. Divide tl>e annuity, &c. by that power of the ratio sig- nified by the number of years, ana subtract the quotient from the annuity: This remainder being divided by the ratio less 1, the quotient will be Iho present value of the Annuity sought. EXAMPLES.' 1. Whatready money will purchase an Annuity of 50J. to continue 4 years, at 5l. per cent, compound interest? =-1,215506)50,00000(41,13513 + From 50 Subtract 41,13513 ii.' 1 ,051 05) 8.86487 ANNUITIES OR TENSIONS. 07 BY TABLE II Under 5 per cent, and even with 4 years, We have 3,54595 =present worth of ll. for 4 years. Multiply by 50=Annuity. *flns. 177,29750=present worth of the annuity. 2. What is the present worth of an annuity of 60 dols. per annum, to continue 20 years, at 6 per cent, compound interest? Jitis. g688 J9i cts.+ 3. What is SOZ. per annum, to continue 7 years, worth in ready money, at 6 per cent, compound interest ? Jlns. 167 9s. 5d. + - III. To find the present worth of Annuities, Leases, &c- taken in REVERSION, at Compound Interest? 1. Divide the Annuity hv that power of the ratio deno- ted by the time of its continuance. 2. Subtract the quotient from the Annuity : Divide the remainder by the ratio less I, and the quotient will be the present worth to commence "immediately. 3. Divide this quotient by that power of the ratio deno- ted by the time of Reversion, (or the time to come before the Annuity commences) and the quotient will be the present worth of the Annuity in Reversion. EXAMPLES. 1. What ready money will purchase an Annuity of 501. payable yearly, for 4 years : but not to commence till two years, at 5 per cent. ? 4th power of 1,05=1,215506)50,00000(41,13513 Subtract the ,quotient=41, 13513 Divide by 1,05 1 =,05)8,86487 d. power 01 1,05=1, IQ25}177,297(160.8136=1GQ 16s. Sd. 1 jr. present worth of the Annuity in Reversion. OR BY TABLE III. Find the present value of \L at tfie given rate for the sum of the time of continuance, and time in reversion added together ; from which value subtract the present worth of I/, for the time in reversion, and multiply the re- mainder bv the Annuity ; the product will be the answer. 808 ANNUITIES OR FKXS1ONS. Tims in Example 1. Thv.c of continuance, 4 years. Ditto of reversion, 2 The sum, =6 years, gives 5.075695 Time in reversion, =2 years, 1,859410 Remainder, 3,216282x50 .flns. 160.8141 2. \Vliat is the present worth of 75l. yearly rent, which is not to commence until 10 years hence, and then to con- tinue 7 years after that time at 6 per cent. ? .'Ins. 235 15s. 9rf. 5. What is the present worth of the reversion of a lease of GO dollars per annum, to continue years, but not to commence till the end of 8 years, allowing G per cent, to the purchaser? Jns. S4S1 78efs. ~ f V"' IV. To find the present worth of a Freehold Estate, or an Annuity to continue forever, at Compound Interest. HULK. As the rate per cent, is to 100/. : so is the yearly rent to the value required. EXAMPLES. 1 . What is the worth of a Freehold Estate of 40/. per annum, allowing 5 per cent, to the purchaser ? As 5 : 100 : : 40 : 800 Ans. 2. An estate brings in yearly 150/. what would it sell for, allowing the purchaser G per cent, for his money? Jns. 2500 V. To find the present worth of a Freehold Estate, HI Reversion, at Compound Interest. RULE. 1. Find the present value of the estate (by the fore^n- iii2; riih } as though it were to be entered on immediately, and di ii.e the said value by that power of the ratio de- noted by the time of reversion, iirul the quotient will be the present worth of the estate in Kevei KXAMl'I 1. Suppose a freehold estate of 4l)f. per annum to com- mence two \ears hence, be put on sale ; what is its value, allowing the purchaser 5l. j;er ceut. r qUESTIOKS FOR EXERCISE, 09 As 5 : 100 : : 40 : 800 =prcsent worth if entered on immediately. Then, 1,05 = 1.1025)800,00(725.62S58=725/. 12s. 5irf.=present worth of 800 in two years reversion. Ans. OR BY TABLE III. Find the present worth of the annuity, or rent, for the time of reversion, which subtract from the value of the immediate possession, and you will have the value of the estate in reversion. Thus in the foregoing example, l,859410=present worth of ]/. for 2 years. 40=annuity or rent. 74,376400 =present worth of the annuity or rent, for [the time of reversion. From 800,0000 =value of immediate possession. Take 74,3764 =present worth of rent. 725,6236=/;725 12.. 5Arf. 2. Suppose an estate of 90 dollars per annum, to com mence 10 years hence, were to be sold, allowing the pur- chaser 6 per cent. ; wiuit is it worth ? .. $837, 59c/.s. 2m. 3. "Which is the most advanlr. -cms, a term of 15 years, in an estate of IQQl. per annun ; or the reversion of susb an estate forever after the ;iiu i 5 years, computing at the rate of 5 per cent. JVM- annum, cr-n^ound interest? .5ns. The first term of lo years is better than the re- vgrsiou forever afterwards, by /*75 18s. 7 id, ' Jv /w A COLLECTION OF QUESTIONS TO EXHRCISB THE FOREGOING RULES. 1. I demand the sum of 1748 added to itsolf ? Jlns. 3497. 2. What is the difference between 41 eagles, and 4099 dimes ? .fltzs. lOcfs. 3. What number is that which being multiplied by 21^ the product will be 1365 ? Aus. 65. 19* $10 QUESTIONS FOR KXURCISE. A. What number is that which being divided by 1,9, the quotient \vill bo 7-2 : Jlns." 1363. 5. What number is that which being multiplied by !, the product \vil! be r Jns. ^. (>. There are 7 chests of drawers, in each of which there arc IK drawers, and in each of these there are six divisions, in each of which is 161. 6s. 8d.; how much money is there in fhe whole ? Jns. 12348. 7. Nought SG pipes of wine for 453G dollars : hw must I sell it a pipe to sue one for my own use, and sell the. rest for what the whole cost? Ans. S129, GOc/s. 8. Just 16 yards of German serge, For 90 dimes had 1 ; How many vards of that same cloth Will 14 eagles buy ? Jus. 248yJs. Sqrs. 2fna. 9. A certain quantity of pasture will last 963 sheep 7 weeks, how many must be turned out that it will last the remainder 9 weeks? llns. 214. 10. A grocer bought an equal quantity of sugar, tea, and coffee, for 740 dollars; he gave 10 cents per Ib. for the sugar, 60 cts. per Ib. for the tea. and cts. per Ib. for the coneej required the quantity of each? .4ns. 83d*. f>or. 8{jrfr. 11. Bought cloth at glj a yard, and lost 2.5 per cent. how was it sold a yard ? .#?/>. '.'.i ; put in. ...I know iu.i lio\v much, and C the rest : Vhey gained .it the rate of 2-/. per cent. : A's part of th. gain is {, B's A, and C's the re^t. Required each man*! particular stock. . {Ws stack was 5 12 JS /r.s- __ 205.1 6"5 S07,C5 26. AVhat is 1'iat number M hich being di^ ided by ^, the quotient will be 21 ? *dns. 15$. 27 If to my a;j;e there added be, One-half. ~- '|i!oticnt rc^u'iin^ multiplied by 3, that product iiivided by 5, I'rom tlic quotient 20 being subtracted, and SO added'fo the remainder, the half sum shall make 65 j -II me the number ? /?$. 14Q>. QUESTIONS FOB. EXERCISE. SIS 32. What part of 25 is | of an unit ? Jlns. ^V 33. If A can do a piece of work alone in 10 days, B in 20 days, C in 40 days, and D in 80 days ; set all four about it together, in what time will they finish it ? Jlns. 5-J- days. 34. A farmer being asked how many sheep he had, an- swered, that he had them in five fields, in the first he had i of his flock, in the second -J-, in the third |, in the fourth \, and in the fifth 450; how many had he P Jlns. 1200. 35. A and B together can build a boat in 18 days, and with the assistance of C they can do it in 1 1 days ; in what time would C do it alone ? Jins. 28f- days. 36. There are three numbers, 23, 25-, and 42 ; what is the difference between the sum of the squares of the first and last, and the cube of the middlemost ? Jlns. 13332. 57. Part 1200 acres of land among A, B, and C,* so that B may have 100 more than A, and C 64 more than B. Jlns. Jl S12, B 412, C 476. 38. If 3 dozen pairs of gloves be equal in value to 2 pie ces of Holland, 3 pieces of holland to 7 yards of satin, 6 yards of satin to 2 pieces of Flanders lace, and 3 pieces of Flanders lace to 81 shillings ; how many dozen pairs of gloves may be bought for 28s. ? Jlns. 2 dozen pairs. S9. A lets B have a hogshead of sugar of 18 cwt. worth 5 dollars, for 7 dollars the cwt. $ of which he is to pay in cash. B hath paper worth 2 dollars per ream, which he gives A for the rest of his sugar, at 2J dollars per ream. Which gained most by the bargain ? Jins. Ji by g!9, 90cte. 40. A father left his two sons (the one 1 1 and tht other 16 years old) 10000 dollars, to be divided so that each share, being put to interest at 5 per cent, might amount to equal sums when they would be respectively 21 yean of age. Required the shares ? Jlns. 5454^- and 4545 T T dollars. 41. Bought a certain quantity of broadcloth for 383? 14 QUESTIONS ro:i .EXERCISE. 5=. and if the num -lings which it cost per yard were added to the , yards bought, the sum would be 586 ; i oer of yards bought, and at what price r*r ;, :uu r .foj. SG5j/rfs. at 21.-;. per yard. Solved by PROBLEM VI. page 183. 42. TV/-.) partner;";. Peter and .luhn, bought goods to the amount of iOOO dollars; in the purchase of whih, Peter paid more than John, and John paid. ...I know not how much : They i their goods lor ready money, and thereby gained at f 200 per cent, on the prime cost: t!'.-y (ii- 1-k-u ihc ^n\n between them in proportion to the purcha.ii- i.vjney ' . aid in buying the goods ; and Pet<-i John, My part of the gain is really a handsome sum of money ; i wish I hod a.s many such sums as your part contains dollars, I should then have 4)60000. 1 demand each man's particular stock in purchasing tho goods. dns. Peter paid GOO dollars, and Joint paid 400. TKTC FOLLOWING QUESTIONS ARE PROPOSED TO SURVEYORS. 1. Heqiv.rcd to lay ort a lot of land in form of a long square, containing 3 a-.re.s, 2 rood.", and 29 rods, that shall take just 100 rods of wall i > enclose, or fence it round j prav how mar.y rod a in length, and how many wide, must saiu lot be ? Jin.;, r-l rr, '.-; in /.';.;;''''. (')'d 19 in breadth. :\eu !>y PUOHI.KM V.I. pag^ ' tract of !:.nd is (: be iait! o:if in foi .n of an equaJ squr: . ("lice. 5 i' fence slijili contain 10 : IIov/ lar^e n:u>t t . square \K- many acres us tliercarc rails in tin- fi-: icloses it. so tliat e\ . nco a:i a .. Me fra; ';rad co. 1 ;/ Thus, J mile=G20 rods: then 3S20Xo:K-:-l60r=.640 acres : and 320x4x10 = 12800 rails. AsG40 : 12SO 12800 : 2JGOOO rails, which will enclose JGOOO acreg=^ 20 miles AN APPENDIX, CONTAINING SHORT RULES, FOR CASTING INTEREST AND REBATE? TOGETHER WITH SOME USEFUL RULES, 7OR FINDING THE CONTENTS OF SUPERFICIES, SOLIDSj &C. SHORT RULES, FOR CASTING INTEREST AT SIX PER CENT. I. To find the interest of any sum of shillings for any number of days less than a month, at 6 per cent. HULK!' 1. Multiply the shillings of the principal by the num- ber of days, and that product by 2, and cut off three figures to the right hand, and all above three figures will be the interest in pence. 2. Multiply the figures cut off by 4, still striking off three figures to the right hand, and you will have the farthings, very nearly. EXAMPLES. 1. Required the interest of 51. 8s. for 25 days. . .. 5,8=108x25x2=5,400, and 400x4=1,600 Jlns. 5d. 2. \Vhatistheinterestof 1J. Ss. for 29 days? Ants, 2s. Od. APPENDIX- FEDERAL MONEY. U. To find the interest of any number nf cents for any number of days less than a month, at per cent. RULE. Multiply the cents by the number of days, divide the product by 6, and point ott' two figure s to the rie;ht. and ail the figures at the left hand of the dash, will be the mterest in mills, nearly. EXAMPLES. Required the interest of 85 dollars, for GO days. g cts. mills. 85=8500x20-i-Ga=283,33 Jns. 283 which is 28cis. 3 HU//S. 2. What is the interest of 75 dollars 41 cents, or 73-11 cents, for 27 days, at 6 per cent. ? 330 TJU//S, or 33c/s. III. When the principal isgiven in pounds, shillings, &c. New-England currency, to find the interest for any number of days, less than a month, in Federal Money. RULE, Multiply the shillings "m the principal by the number of days, and divide the product by 36, the quotient win be the interest in mills, for the given time, nearly; omit- ting fractions. EXAMPLE. Required the interest, in Federal Money, of 27/. 15?. for 27 days, at (i per cent. . s. s. 27 15=5y5x27^-3Gr=4lCmi^s.=r41cKGjH. IV. When the principal is jiven in Federal Money, and jou want the interest in shillings, pence, &c. New-Eng- land currency, for any number oj" days less fhan a " sir RULE. Multiply the principal, in cents, by the number of day^ and point oft' five figures to the right hand of the product, which will give the interest for the given time, in shil- lings and decimals of a shilling, very nearly. EXAMPLES. A note for 65 dollars, 31 cents, has been OH interest 25 days ; how much is the interest thereof, in New-England currency ? $5 cts. s. 5. d.qrs* Ans. 65,3 1=653 1x25=1,63975=1 '7 2 " REMARKS. In thfc above, and likewise in the preced- ing practical Rules, (page 127) the interest is confined at six per cent, which admits of a variety of short methods of casting : and when the rate of interest i 7 per cent, as established in New-York, &c. you may first cast the in- terest at d per cent, and add thereto one sixtn of itself) and the sum will 1 the interest at 7 per cent, which per- haps, many times, will be found more convenient than the general rule of casting interest. EXAMPLE. Required the interest of 75t. for 5 months at 7 per cent. s. 7,5 for 1 month. 5 37,5=1 17 6 for 5 months at 6 per cent. + 1** 63 Jlns. 2 39 for ditto at 7 per ce^. A SHORT METHOD FOR FINDING THE REBATE OF Arfft GlVF.N SUM, FOR MONTHS ANQ DATS. RULE. Diminish the interest of the given sum for the time by its own interest, and this gives the Rebate very nearly. EXAMPLES. 1. Wbat is the rebate of 50 dollars for sii month*, at 3 per eenk ? 1 9 118 ' S cffi The interest of 50 dollars for 6 moaths, it 1 50 And, the interest of 1 dol. 50 cts. for 6 months, is 4 Int. Itetate, 81 46 . "What is the rebate of 150J. for 7 months, at 5 per cent. ? Interest of 150J. for 7 months, is 476 Interest of 41. 7s. Gd. for 7 mouths, is 2 6* Jlns. 4 4 Hi nearly. By the above Rule, those who use interest tables IB their counting-houses, have onlv to deduct the interest ot* the interest, and tlie remainder is the discount. Jl concise Rule to reduce the cur r end -s of the different State*, tckere a dollar is an even number of sfiitlir^gSf to Federal Money. RULE I. Bring the given jum into a decimal expression by in jyection, (as m Pi-ob!rac- be, the following Contraction may L>e useful. RULE II. To t'le shillings contained in the given sum, annex * times the given pence, increasing the product 'r; 'Z } t:ira- divide the whole by the number of shillings contained in a dollar, and the quotient will be cents. F.XAMPLK5. 1. Reduce 45s. Gd. >V,\ -England currency, to Fede- ral Money. 6x8-f2 = 50 to be annexed. ' 6)45,50 or 6)1550 7,58! .9ns. T53 f^Jif*. =' 2. Reduce C/. 10s. Ud. New-Yoj;k, &.c. currency, to federal Money. 9x8+2=74 to be annexed. Then 8)5074 Or thus, 8)^,74 ,Jw. 634 ce??fs.=6 34 26,34 N. B. When there are no pence in the given sum, you must annex two cyphers to the shillings ; theu divide as before, c. 3. Reduce 31. 5s. New-England currency, to Federal Money. Si 5s.=>65s. Theu 6)6500 1U86 ce*U.' S20 APPENDIX. SOMtf USEFUL RULES, TOE T1NDING THE CONTENTS OJ SUPERFICIES AND SOLIDS. SECTION I. OF SUPERFICIES. The superficies or area of any plane surface, is com- posed or made up of squares, either greater or legs, ac- cording; to the different measures by which the dimen- sions or the figure are taken or measured : and because 12 inches in length make 1 foot of long measure, there- fore, 12x12^=144, the square inches in a superficial foot, &c. ART. I. To find the area of a square having equal sides. RULE. Multiply the side of the square into itself, and the pro- duct will be the area, or content. EXAMPLES. 1. How many square feet of boards are contained m the floor of a room which is 20 feet square ? 20x20=400 feet, the Answer. 2. Suppose a square lot of land measures 26 rods on each side, how many acres doth it contain ? NOTE. 160 square rods make an were. Therefore, 26x9-f=-676 sq. rods, and 676-r-l60=4a. S6r. the Answer. ART. 2. To measure a Parallelogram, or long square. RULE. Multiply the length bv the breadth, and the product be tne area or superficial content. EXAMPLES. 1. A certain garden, in form of along square, is 96 ft. long, and 54 wide ; how many square feet of ground are contained in it? Jns. 96x54=5184 square feet. 2. A lot of land, in form of a long square, is 120 rod? in length, and 60 rods wide ; how many acres are in it ? 120x50=J200 sq. rods, tiien, 7 T gV=^ 5 acres, Ans. 3. If a board or plank be 21 feet long, and 18 inches eroad ; how many square feet are contained in it ? 18 inches^ 1,5 feet, HJs are estimalfuj by the solid inch, solid ibyt, &c. .8 of Uiese incKesTthat 12x12x12 make I oihic ' olid foot. APPENDIX. 22S ART. 6. To measure a Cube. Definition. A cube is a solid of six equal sides, each of \vhich is an exact square. RULE. Multiply the side by itself, and that product by the same side, and this last product will be the solid content >f the cube. EXAMPLES. 1. The side of a cubic block being 18 inches, or 1 foct and G inches, how many solid inches doth it contain ? ft. in. ft. 1 6=1,5 and 1,5x1,5x1,5=3,575 solid feet , 3ns. Or, 18x18x18=5832 solid inches, and {fff =3,375. -2. Suppose a cellar to be dug that shall contain 12 feet every way, in length, breadth and depth ; how many solid feet of earth must be taken out to complete the same ? 12x12x12=1728 solid feet, tte Answer. ART. 7. To find the content of any regular solid of three dimensions, length, breadth and thickness, as a piece ot timber squared, whose length is more than the breadth and depth. RULE. Multiply the breadth by the depth or thickness, and that product by the length, which gives the solid content. EXAMPLES. 1. A square piece of timber, b<>i;iu; I !;><>! *3 inches, or 18 inches broad, 9 inches thick, and \> 1'ivt ur 108 inches leng: how many solid feet doth it contain : 1 ft. Gin. = 1,5 foot. 9 inches = .75 foot. Trod. 1,125x9=10,185 solid fed, the AK . in. in. in. solid in. Or, 18x9xl08=17496-M728=10,125 feet. But, in measuring timber, you may multiply the bread tk in inches, and the depth in inches, and that product by the length in feet, and divide the last product by 144, which will srive the salidfcontcnt in feet, *". Sfi4 A A piece of timber being 16 inches broad, li inches thick, and 20 feet long, to fuid the center* ? Breadth 16 indie. Depth 11 Prod. 176x20=5520 then, S52C+ 144 =<24,4 /t!i <.r a:iy square box being given, to find how nu.ij uu^iieis a \vtii cuutatu. ROLE. Multiply the length by the breadth, and thru by the depth,, divide the last jvcc'i-ici !>y olid inches in a statute bushel, and tiie <;'iotiejat will be ' the answer. KXAMIM K. There is a square box, the I ugth of .n is 5 inches, breadth of ditto -10 inci.e*. a.ui if,s di-pth is 60 inches ; how many bushejs of cMru will it ).,;>',f the wall measured round on the outside, subtract f'rj r times its thickness, then multiply the remainder by t b c height, and that pro- duct bv t\ e thickness of the wa'J 4 . jjives the solid content of t.he.*v,hole wall; which mo^Jphed by the number uf bncka contained in a solid foo/., give* the ans cr. How man/ bricks 8 inc'n es lon, 4 inches wide, ami 2$ inches thick, will it tak.c to build a house 44 feet long, 40' fee 1 wide, and 20 fest. high, and tke walls to be one foot t' 8x4x2,5=80 solid inches in a brick, then 175 c=21.6 bricks in .1 sn'm'. foot. 44_i_40-J-44-f40=i7.bS fret, \\hulo leiij ' * of wall, 4 four times the thickn*. H-4 remains. Multiply;' by 20 height. 3280 solid feet in the whove wa& Multiply by 21,6 bricks in a solid foot. Product, 70848 bricks. .1ns. APPENDIX. 37 AXT. 14. To find the tonnage of a ship. RULE. Multiply the length of the keel by the breadth of toe beam, and that product by the depth of the hold, and di- vide the last product by 95, and the quotient ii the tn- Mge. EXAMPLE. Suppose a ship 72 feet by the keel, and 24 feet by the beam, and 12 feet deep ; wkat i.t the tonnage ? 72x24xl3-f-P5==2l8,2-ftos. Jns. KULK II. Multiply the length o& the keel by the breadth of the beam, and that product by hulf the breadth f the bean, and divide by 9j. EX. \MPI.E. A ship 84 feet by the keel, 28 feet by the beam ; what is the tonnage ? 84x28xl4-r-95=350.29 tons. Jn.. ART. 15. From the proof of any cabl*, to fuid the strength of another. RULE. The strenii;tl\ of cables, aiul consequently the weights ef their anchors, are as the cube of their peripheries. Therefore; As the cube of the periphery ol any cable, Is to the weight of ita anchor ; So is iho cuijcof tli e periphery of any other cabl, To tlie weight of its anchor. EXAMVI.ES. 1. If a cable 6 inches about, require an anchor of 2$ wt. of what vrii^ht niust an anchor be lor a 1 2 inch cable ? As 6xfixfi : -2}cirt. : : ls2xl2xH : . ins. 2. If a 12 iiich cable ref|uir-e an anchor of IS c\vt. what must the circumference oi' a cable be, for an anchor of Si cwt. ? ctt-t. cwf. in. As 18 : 12x12x12 : : 2,2J : 2If. v ^!f- ART. Ifi. Having the dimensions of t,vo similar built ship* of a different tipacitv, with the burthen ' fcf Uietw, to find tl'.t; burthen of the ot!:cr 23 APFEXDIX. RULE. The burthens of similar built ships are to each other, as the cubes of their like dimensions. EXAMPLE. If a ship of 500 tons burthen be 75 feet long in the keel, I demand the burthen of another ship, whose keel is 100 feet long ? T.cwt.grs.lb. A 75x75x75 : 500 : : 100x100x100 : 711 2 24-f DUODECIMALS, OR CROSS MULTIPLICATION, IS a rule made use of l>y workmen and artificers in eat* ing up the contents of th^ir work. RULE. 1. Under the nvilti;)lr.;nd .vrite the correspondingde' Bonunations of the multip'Vr. 2. Multiply each len.: into -iiintion in the" multi- plier, and write the result of each undtr its respective term ; observing to carry an uuk ibr eveiy 12, from each lower denomination to ; -nerior. 3. In the same mannf : -It, 'y ail the multiplicand fay the inches, or second denomination, in tlic multiplier, and set th n - v!i term on-.- nlace removed to the right band of those in the ini.ltipliriitn^. 4. Do the same with the seconds in the multiplier, let- ting the result of each term two places to the right hand f those in the multiplicand. &c. EXAMPLES. F. /. F. I. F. I. Multiply 73 75 4 f 15y 47 39 58 29 " 27 9 9 25 6 91 10 1 429 Product, .13 2 'J AWKNDIX. 9 t F. I ' Multiply 4 7 By ' r 5 10 ______ Product, 26 8 10 F. /. 3 8 7 6 27 6 [F. / 9 7 S 6 r _ 32 6 -------._-- F. /. 7 10 8 n Multiply By F. /. 3 11 9 5 "fTI ~f 6 5 7 6 Product, 36 10 7 48 1 6 69 10 2 FEET, INCHES AND SECONDS. F. /. " Multiply 986 By 793 [tiplier. 67 1 1 6 '" =prod. by the feet in the mut- 734 6 N ""=dittoby the inches. 251 6=ditto by the seconds. 75 5 3 7 6 j? F. /. " 567 8 9 10 F. 1. " 7 1 9 Z 8 9 w "// Multiply By Product, 55 2939 481128 10 How -many square feet in a board 16 feet 9 iuchct long, and 2 feet 3 inches wide ? By Duodecimal*. By Decimals. F. I. F. I. 16 9 16 9=16 5 75feet. 23 23= 2/25 S3 6 . 8375 423 3350 S350 37 8 S F. /. * Oft *fns. 37,68751-37 8 9 230 APPEXDIX. TO MEASURE LOADS OF WOOD. Kr MuNipljrthe length by the breadth, and the product by the depth r height. ,vhtc!i v/ill j^'m: the. content in solid i'eet; 01 wiiicu '.,ilt a ccrtl, and 128 a cord. VMPI.I:. Ilinv many solid feet are contained in ;i load of wood, 7 fee; ( :g, 4 feet 2 inches wide, and 2 feet 3 7ft. r nnil 2ft. f> tH=a C,5 : (hen, r,5x4,l67=51.io2.>x 25==rO,318l25 so/ui IJnt '-tads <;f wood are commonly estimated by the footj allowing; tlie load fo be 8 feet long, 4 fe^t wide, and (hea feet high wiil make half a cord, which is called 4 feet of wood ; !);i(- if the breadth of the load be loss titan 4 fect| its hi -t l>e increased SD as to make l>alf a cordj \vl;ic;i i. till called 4 feet of wood. By r ihc bre-ultli and hei^hthof the load, tin content may be found by the following "RULE. Multiply the broa:l!!i by thii hei^l'it. and half the pro- duct will be t'te content in feet and inches. EX A 51 PL. B. Required the content i-i a load of wo'id u-hicli is .3 feet 9 inches ^\ide and ^ ioet () inciios high. By Duodecimals, Ly F. in. F. F. In. =4 8, or liti[f a cyr..' 8 \ iuchr* vfer. tirf - i-c! ' . h't the follpwlng Tnlile M of nny 1AA.) uf w ">l. by ii^|odi<'i' -,taU . iirnctict! ; ^'.'cti wal! lie < /flC/JM. 'ft. in. 1 o S 4 3 4 5 6 |7|H 9 flOfll 2 6 Ju 3U 4o 1 4 5 6 7 9 10 11 12 14 7 3 1 47 1 3 4 5 6 8 9 10 12 13 14 8 u. 48 64 1 3 4 j 7 8 9 11 12 13 15 9 i; 33 49 1 314 6 7 8 9 li 12 14 10 17 54 51 2 3 4 6 7 9 10 LI 45 14 16 11 IS 3Ji ru 2 3 4 6 7 9 10 12 13 15 16 5 18 72 2 5 6 8 9 11 12 17. 1 19 37 .76 74 2 3 5 6 8 9 il 141'.: 17 2 19 70 S 5 6 8 10 1 ! 13 17 3 19 78 o 3 5 7 8 V 11 13 15 18 4 20 40 80 a 3 g y 8 10 12|Io 15 17(18 5 1 41 62 8:! 3 5 i 10 17|19 (5 63 84 4- 7 9 ir2 7 43 64 o 4 5 7 9 11 16 20 L 8 22 66 7 9 11 15 17 18 eo ' 9 23 68 4 7 9 15 15 17 19 21 10 23 46 69 ^ 4 6 7 9 1!) 21 11 23 70 (> 3 10 16 (b 20 22 4 24 4S 72 96 6 8 10 12 29 TO ' GOING TA< First measure thr, u;e,ul..:i and he^a. of yuur load to the nearest average inci. -.1 the bre:- hand column of the table ; t'.ion movett tli'i right on the s^mj line till you come under the height in fret, a. id yflu ui:i '.nvo the content in inche j, ru. , IM i,I.'. con- tent of the inriies on tli ide, and 2 feet 10 inches hi^h ; rf-r;uired t' ? r'.ntent Thus, against^ fl. 4 inche?. aH is'.i.li- ) inch- es ; and under 10 inches .- ;04~ 17=57 true content in inches, which divide by li givvs 4 fert 9 inches, the answer. . The" breadth being 3 feet, and height 2 feet S inches; Kequired the content. Thus, with brsaJth 3 feet inches, and under 2 feet AF9EKDIX. Atop, stands 36 inches ; and under 8 inches, staidif It inches : now 56 and 12, make 48, the answer in inches $ and 48-j-12=4 feet, or jwst half a cord. 3. Admit the breadth to be 3 feet 1 1 inches, and heigbt 3 t'-et 9 inches; required the content. I'udei- 3 feet at top, stands 70 ; andundtr 9 inches, is .'0 and 18, make 88-i-12=7 feet 4 inches, or 7 fit. 1 -jr. ;? inches, the answer. TABLE I. the amount of 1 . or gl, at 5 and 6 per cent, per annum, Compound Interest, for 20 years. FT* 5 per cent.\6per cent.\I'rs. 15 per cttnt.\6 per cent. 1 1,05000 1,06000 11 1,71034 1,89829 2 1,10550 1,12360 12 1,79585 2,01219 3 1,15762 1,19101 13 1,88565 2,13292 4 1,31550 1,26247 14 1,97903 2,26090 5 1 ^7G'28 1,33822 15 2,07893 2,39655 6 1 ,34009 1,41851 If) 2, is ;727 i 7 1,40710 1,50363 17. 2,29 1277 8 1,47745 1,59384 18 661 9 1,55132 1,68947 19 >95 , 10 1,62889 1,70084 20 2,G. 11 5 2 17 8 4 1 8 4 G l 15 L 10JJ 16 1 8 r I7f Sta;v (,: ..fard Silver. VII. The weights qf the coins of the Unite J Baffles, Half-Eagles, Quarter-Eagles, Dollars, Ha!M)Ilais, Quarter-Dollars, Dimes, Half-Din Cents Half-Cents, Tie standard fur gold cc n is II parts pure gold, ajid one part al- ley. the (Hoy to x onsivt ol' iriver and copper. The standard for *iiuT coin ii I486 parU fine to 179 parts alloy the alloy to be whol- ly ooppr. V.D1X. ANNUITIES. TABLE II. TABLE 111. ; jjg th?. ammwt of 'he present jfl annuity, fur' Vjorth of 1 annuity, ". I pars or i to con,in;ts fin- 5ll at 5 and 6 per yci?*, cd 5 and 6 per compound in??mst. I'rs. 5 5 1 1,OG< 1,OC J581 2 1.85 3,15 oV 2; r sois 4 L0125 4616 3,545950 .HOG 5 4,2i23G4 6 4,917524 f 8,1- . :!3838 6278 5,5' s . .'Jill!) 6,463213 6,20 9 11,026564 11,491516 7,10 6,80 1 692 12,577892 15,180770 7,721735 7,360037 11 14,206787 14,9716-13 8.506414 T Q < - 4 **Q\-'D i / 12 15,917126 16.869942 . -J3252 B,S83844 IS 17,712982 18,8821 .IS 9,593575 8,852683 14 19,598632 21,0150JC 9,898641 9,2: 15 21,578564 25,27 10,379658 9.712249 16 23,657492 25,672526 10,837769 10,105895,! 17 25,840366 28.212580 ll,27406u 10,47? 18 28,152585 30,905653 11,689587 1 0,827603 1 19 30,539004 33,75990-2 12,085321 11,158116 20 35.065954 36,785592 12,462210 11,46! 21 35,719252 59,992727 ! 12,821 155 i 1,764077 22 38,505214 45,592291 15,165005 12,041582 23 41,430475 46,995823 ! 15,488574 12,30 l -4 44,501999 50.8I557.S il3.79S642 12.550557 25 47,727099 54,8645 12J 14,09394-iji2,7So35ti 26 51,115454 59,1563821 14,375185 13,003166 27 54,6691 6 63,705765 14,645054 13,210534 28 58,402583 68,528112 M.M- i .: 1.3,406164 29 62,322712 73,659798 15,K j 5.590721 SO 66,438847 79,058186 15,57?451 !.i.r64851 *31 70,76079084,8016771 15,592810 13.929026 20' 254 APV1VBIX. TABLES/ 1 HE three following Tables are calculated agreeable to an Act of Congress passed in November, 1792, making foreign Gold and Silver Coins a legal tender for the pay- ment of all debts and demands, at the several and respec- tive rates following, viz. The Gold Coins of Great-Bri- tain and Portugal, of their present standard, at the rate of 100 cents for every 2^ grains of the actual weight there- of. Those of France and Spain 27| grains of the actual weight thereof. Spanish milled Dollars weighing 17 pwt. 7 gr. equal to 100 cen-ts, and in proportion for the parts of a dollar. Crowns of France, weighing 18 pwt. 17 gr. equal to 110 cents, and in proportion for the parts of a Crown. They have enacted, that every cent shall contain 208 grains of copper, and every half-cent 104 grains. TABLE IV. Weights of srreral pieces of English, Portuguese, and French Gold Coins. Pwt. Gr. Dols. CtS. Jlf; 18 16 9 8 tintlish Guinea, .... Half, ditto, 5 a 6 15 4 66f 2 33$ French Guinea, .... H.\lf, ditto, ..... 5 6 15 4 59 R 2 29 9 4 Pistoles, 16 12 14 45 2 2 Pistole*, 8 6 7 22 6 4 Pistole, w . 4 8 S 61 S Moitlore, . 6 22 6 14 8 APPENDIX. > ;js 55 3g 2 2: coo o .. O O -- ; j - to * c r~ (": * oo c o ?> tr> -. o *o *o ?c o o c^ <^ etw^oscr-esc; o or:-*O'jr-o(? . s- _ _ ^, c , .^ ^_ ( O -* G f5 * o cs c~ C3 ci O o (?) 1-3 -? o c: :o t~ o -i i C7 O ' iS r i- ct o T ct sr J"- O ~* ^ I ^ t) ,. 2:-. 6 Mil. T3BLE of Cants, ansa-cnnc; to tlie Currencies of tkt> United ftt,''!rs. K-ith Sterling, <$'c. No rv.. The fi<;;:;c>3 o-i 1:>e right hand of the space, show the parts of :i cent, or mills, &c. 6a. r&lKs. f(.> is. / 4s.Gd. 4s. 10}//. the !<> fin- the !o ilt- to the. Doll. Dull. \ JJolL Dollar. r. rente. (. cents. cent*. < \ 1 5 1 I I J 7 1 C 1 S 1 7 & C 7 2 2 ii 3 5 5 7 3 4 4 1 5 1 3 5 5 S 5 6 5 5 1 4 5 -5 4 1 4 -1 7 1 7 4 6 8 5 6 9 5 2 5 5 8 9 9 2 8 5 C 8 3 (i ii 6 G 10 7 10 11 1 10 2 7 9 7 7 2 7 T 12 5 11 C 13 9 11 9 8 11 1 8 3 8 8 1-1 2 13 S 14 8 15 6 9 12 5 10 16 15 16 r 15 3 10 13 8 10 4 1! ! 17 16 C 18 5 17 11 15 2 11 4 12 2 19 6 18 S 20 5 18 S. 1 16 G 1. 5 13 5 21 4 20 22 2 20 2 33 5 25 i,6 f 42 8 40 44 4 41 - 37 5 40 64 2 60 66 6 61 5 4 66 6 50 53 3 85 7 80 88 8 82 5 83 2 li'2 5 66 6 107 1 100 111 1 102 5 ti 100 75 128 5 120 133 5 125 . \\i- ; l ai.il : .. : ; to l.t- r ..rie ;:nd -.lone, 1 br T- MU! admtnistratMS, d v.-.tli my baud, aud i. 1-utod at tliis day f T//e conchii i "f t'l't obtigato . h. That if the above bound* a (.'.'I). &c. i tht c->}idiiio)i.] Then thi- in t be v.,u! ami of T'.or.e effect; otlier- Mi-,0 : .d \ i.fuc. lied and delivered 7 in the pretence of 5 A BILL OF SALE. KNOW all men by \ . iliat I,B. A. of iid in ci<:. hand ]>aid bj 1). C. tit' tlic ro; :ipt \vlifc;vof I an iicreby ac- knr. . . eld a.-d d. iivered, aud, by ^nts. ii<. bargain, :- i'.i er unto the said O v ; >:Wrf.] To HA\E and to HOLD the aforetaidbarraiaed prei .. -. ur ,tl I). 1\ bis execut-x--. - -c ' Ufl .'i , forever. And I, the ;-aid 1$. A. f.r inytell. my exert vrs and ad- iliii:'r -'iali :i!iexec ''us, ad- 'i 4 : *. In -uitnest >vfct- \c her^"nto set :..) hand aud s<^al, this ; d. 1814. In jirestncc t>f A BHOKT WIZX. I, R. A. of, &c. dn)..h.e rncL i.t^ismv last will ard testamcat, in manner and form follcnviiig, viz. I giv APPENDIX. G9 and bequeath to my dear '>:<. thcr, R. A. the, sum of ten pounds, to buy him mourn' ng. 1 give and bequeath to ray son, J. A. the sum of two hundred pounds. I jrive and bequeath to my daughter. K. II. the sum of otic hun- dred pounds; and t. in r A. V. the iik<> s;.;n of one hundred ;ioun<; ,. AM the rest and residue of inv estate, goods and chattels. I givo; and bequeath to u;y dear beloved wife, E. R. whom I nominate, cutibtitute and appoint sole executrix of this mv last v ill and tes- tament, hereby revoking all other and* former wills by me at any time heretoiu. e made. In -witness whereof, 1 have hereunto iet my hand and seal, the day of in flie year of our Lord Signed, sealed, published and declared by the said testator, B. A. as and lor uis last will and testament, in the presence oi us who have subscribed our names as wit- mcbsesi thereto, in the presence of the said testator. R. A, S. D. L. T. NOTE. Th testator after taking off his seal, must In presence of the witnesses pronounce these words. " I pub- lish and declare this to be my last will and testament." Where real estate is devised, three v i ,!ie---"s are abso- Intel v necessary, who must sign it in the presence of the testator. % A LEASE OF A HOUSE. KNOW all men' by these presents, ujat J, A. B. ot in for ud in consideration oi' the sum of received to my full satisfaction of P. V. of this day of in the j ear of our Lord, have demised and to farm let, am! "do by these pie. ert.= , de- mise and to farm let. unto the saul P. V. his heirs, esecu- tors, administrators and asstigns, one certain pi'-ceof 'and, lying and be'nuj; -i;nn:-d in -aid [Here describe the boundaries] with a duelling-huii .Q *horeon standing, for the term of one yea; frou t! ,3 (lai.e. To UAVB ;ind to H.n.n r him t: c said P. V. his hens?, ivecutors, linl-u? - !.;. , ,s lr ?]! r^vm, tor 240 lim the said P. V. to use and occup 1 - at to i.iin shall seem meet u:i \ proper. And the said A. B. doth FURTHER COVENANT with \i P. that he hath good ri^ht to let and demise, the said l"ttcn and demised prjmise.s ift manner aforesaid, aud that JMJ 1 A. during: the said time will suffer the s^iid P. quietly f.i H.\vr -md to HOLD, use, occupy and enjoy said demised pre- ..it ^ui 1 P. shall haw, hold, use, occupy, possess and < nioy the same, free ami clear of nil incumnmnces. claims, In witness whereof, I the said .\. Ji. i.uve hereunto set my hand ami seal this SigntJ, stnlfil and delivered ) 4 R $ In presence of A NOTE PAYABLE AT A BANK. [$500, 60] HARTFORD, May 30, 1815. FOR value rrcc-irod, I promise to pay to John Merchant^ r order, Fivo Hundred Dollars and Sixty Outs at HaxtfoTd Dank, iu sixty davs from the ua'.f. WILLPAJI DISCOUNT. AN INLAND BILL OF EXCHANGE. [483, 34] BOSTON, June 1, iai.'. i;NTY days 7tftcr date, ploase to pay to Thuma* iii or order, r.ijlity-Tliree Dollars and Thirty- Fonr : ud place it to my account, as per advice from your . > sen-ant, SIMON PI f k Mr. T. W Merchant, ) Wat-York. A COMMON NOTE OF HANI). , ;i rrh t;, i;;21. to pay to .lulu; Murraj, in four cionths from th JO; A ( T,R. ! 'l.nrle* <"; TV. rut . ' THE PRACTICAL ACCOUNTANT .KKST METHOD 01" INSTRUCTION OF YOUTH. AS A COMPANION TO DABOLL'8 ARITHMETIC. IJY S \MUEL GREEX. % * rUBUSIIED BY SAMUEL GREEN, XEW-LOffDON. INTRODUCTION. . , :ul fomalo, - T'lim-fic, ospcrinily in the r<( .iliil rn!.-s .->f AMI'*'-- >!i. MI!, tract ion, Multiplication, n:- 1 Di- -!if)nll be instni noeof BooA- A ' fiuj fitter of . ir-nly i- - i ' by t}? men-' i[>!o ed in :. -quiring a rat. 't' ill"' 1 , \vbo ' ;UlJ COTTC'Of Le ;-:' ]>:i]-pr ml'-d aiV , -y-tom. Ii> ' .)!., :n-i: duly to * > . .1 . all tin-" " which - on ;iny ron- i shoultl i .. , liia'i ; thfi rn<-> ' . n to whom rsJ : he" . 1 . i Entered. 1 Fnleral. 1 Entered. 1 Entuml. i Entered. 1 Entered. 1 Entered. 1 Entered. Cr. date, ..:... r 5 18 1 11 U 100 '> :i 3 2 ,-iO .30 -.!! G4 :10 Sarmicl^:. . . .Dr. :i',v\rk this ,h'.v, ..... Ur. To n is out of the stuiij ot'j All! An;'. of) v Uillir . . Cr. H\ iiiv urv!(v in !... .,-r ol J< i^s, i ~ _ 1).-. completed and i tlii= lay <"i l;i;> (- i ; sj called, 4000 foet at 2J cents per foot, . -i }.> Et>. . . . Cr. liy bis team ut sund'-y tiaiv nure on my lan.i. .... or Thornas Grosvcnor, . . Dr. To 48 \vindo-.v sishcs delivered Pt hi 1 * C.lovrr Farm, ?o called, at 4*1 00. . . ^ 1 | ...eaoi'ijlass by in. Job'i, :-.t li cents, . . 7,50 10 daj s' *vork ui'imself finishing front room, at jgl ,~j a day, . . 1J..OO; 7i Io. of WiMiam, my hired man. i laying' the kitchen floor i-iul liauvj- > 6,30 iug- doors, at C4 cci't^ a d-i v, ) An* . . . Cr. By 2 's. pfM-pv.ll. 0,7 j ds. of lii.li:i Cotton, a! I.';/, cents^ 0,74 2 flannel cph li^tnip', ^,li^ Dr. Toii Thtre put tin, name if iht wrier ij~ the /,:'_ FORM OF A DAY BOOK. Albany, February 12, Entered. 1 Entered. 1 Entered. 1 Eutered. 1 Entered. 1 EnJowd. 1 Edward Jones, . . . Cr. By 4 months' hire of his son William at $10 a month, -24 Edward Jones, To my draft on Thomas Grosvenor, Thomas Grosrenor, . . c'r. By my order in favor of Joseph Hastings, Joseph Hastings, To my order, on T. Grobvenor, Dr. Thomas Grosvenor, . . Dr. To 3 days' tvork of myself on your fence at gl',25 per day, . . . 3,75 Jo. my man Wm. on your stible and finishing off kitchen, at 84 cts. 2.52 2 pr. brown yarn stocking?*, at 42 cts. 0,34 13 ~ Dr. Thomas GrosvcDor, By my draft in favor of E. Jones, 23 Cr. Thomas Grosvenor, . . Dr. To part of u dav's work of my son John .-nan William, on bis barn, Anthony T .iUinpj, . . . Cr. For :.je following urti^les, 1 4 Ihs. n rado sugar at $12 pr cwt 1 ,50 dish, .... 0,23 C pi;ue,, 0,3 ucers, . . . 0,20 1 pint French Brandy, . . .0,17 i 'juart Cherry Bounce, . . 0,33 31>e, .... 0,18 bles, 0,04 1 piir Scissors, . . . .0,17 1 qnire paoer, . . . . 0, \Vafw, 4 \ ink, 6 ; 1 botUe, 8 ; . 0. 1 8 Jtatored. I'et ,-r Daboll, .... Dr. 1 JTu a cotton Covtrit>t deLvcrcd Sarah Brad- 1 ford, by your written order, dated 14. Jan. FORM OP A DAY COOK. L, Entered, i Tliunius Crtisvcuor, Cr. i Entered. 1 Entered. 1 Entered. 1 Entered. i IkulereJ. . Dr. ii v BilHi'L---, ' . pw 1 of Ciller, . . $1 ; 17 1 barrel containing the r.'iiie (from TiKHir.ii Grosvenor.) . . . 0,f>f; Tiiomas G: . Cr, Uv i biin-el _ iider aold aa0 1 Daboll's Aritliioctic, for my son Samuel, , 0, 1'-' Blai.h Writiog books at 12J ccnt, 0,'25 ) quire of Letter Paper, . . O..M .N. :..\;it)ie, . . . Cr. e of viii iatc endorsed by Ephraim Dodg'o, ..t C niontl ^, lor a yoke of Oxen bought of Lujaiol Mason, at Lebanon, Joiiuthan Curtis, . . . Dr. To an old bay horse, . . $93,OOJ a four wheeled w;!gon, and half worn harness, . . . 42,00' Entered. Samuel G reru, 2 To cash in f 3*81 FORM OF A DAY BOOK. ""Albany, AprU 6, 1822. Anthony billing, . . . Dr. : $11,25, . . 22,50 Amount of order dated March 2Cth, i iniuvow of I aiuiv White. >0,o4 "uir yarn .stockings, j Hire v.f i!y waggon and horse to i brii;. :Vo:jn Provi- >3,00 ut Uiis jnont]>, . N Thomas Grcivenor. . . . C'r. 1 |Ry iiis order oa Tln'odorc 15arrcJ3, ? a \>r Go d< ItntcrcJ. 26*04 'jj 08 . Dr. To 1 ills. alSOceuts. . . $60,OOJ| ..mi s;uj Barrel 1 fur balance due 0:1 Thomas (Irosvc- nor's order, . . . . 18,0fl 18- 2 By coat lutercil. Entfrcd. 1 Jou;i(li:a ('urtis, . . . Cr. 7i, p:inta)oon^ g">,00, Thomas ( rosvcrtor, Dr. To mending 1 vour cart by my man Wil- liam, '. . . . Paid Hunt, for blacksmith's work on your cart. .... St-itin;r '.; pfllc* i>f cla-.s, and finding glass, O.GG Oj H Dr. . Antliony Bv pa- . 1 . 7,a^ ilas- Oil, per t/. " >' ... i . or A BAY BOOK. Albany, Muv3, 182-.?. Kntercd. ._> Entered. ] Kntorcth 1 Etitoral. 1 Entered. 2 Entered. 1 Kntercd. 1 Knterod. 'I Theodora Jiarrcil, New-.Lon Jou, Dr. To 1.) die? , '>'! Ibs. at 5 cent.., L'17 Ibs. of butter, at 13 2-3 ccuts, u 34,GO 24 Ibs. of h<)ii'\v, at 12J cents, i 52 1 * 43 SI 52 54 48 c. -10 25 60 50 40 00 no oe Joseph Hustings, , . . Dr. :\' 1 i\iir t-liocs, :J9th Apnl, frofft Anthony i - - 12 Anthony B&iags, . . . Dr. 'l.'o !M bushels of seed potatoes, at 33 l-'3 <\-uts, ... . f, pair mittens at 20 cents, . .1,60 (. i'-h, 14,00 Joseph 1 . ... Cr. iJy 4i months wages at 7 dollars, "0 Theod >ri> Karrell, , , Cr. By cash in full uf all demands, . Thonv.'s Grosvenor, . . Cr. by his acceptance oi' my order in favor of Anthony JJillincrs, .... Anthony Billiu.'^, . . . Dr. To amount of my order on Thomas Orosvc- nor, ...... Si nt " 1 NoN-s pjn-abk-, . . . Dr. To cash paid for my note, to D. Mason, The foregoing exntnple of a Day Book, may suffice "to give a good ideaoi' *}]f w iv iii which it is proper to make iho original entries of all dobt and cTcdit articles. Another small book should next be prepared, according t the followin};- lonn, termed the book of Accounts, or Legor. Into this book must bt: posted the whole contents of tin D;iy Book; care being taken that vi TV article be carrii^d to its com-,ponlinc; title ; the debt anioiini* 'o b entered in the left, aru the credit in the right hand page 1 . Thus, should it st nny time be required to Un*w tlie staff c>f an nrivint. i* "'ill only l>e n- o -*-iiy olamus, and ay Book, "V ^vritiIl^ the word F-nlernl, or making two parallel stiok'.-s v.itl. the pen ; to \vhieh should be ad'ii d the figure denoting the page iu t'ac Le,er, where nnt in. On a blank p;ie at the Winning, or opd of the Legcr, an alphabetic*'. >IK!CX >houliJ br \\ritti-n. , .witaming the names >->f every rx-r^i-.i v :h whona 'i l-.iv.' :., !-'.;:nt, in .b I. per. ivhh the number oi the page v,*:sre 'he M OF A LE< . Dr. | C_ <>ds, ll ... . 50 8 B i 1 Dr. Sam;; JauY. .'> To ? -Vcr at ccuin a wi-?k. Dr. Anthony BiiS . $ c. Barrel of ( .rrel, 1 10 paid } our onlor in Savor of G. Gilbert April 6 U ditto -43 MY oirlrr on Thorn 11 Gro^vcuor, 54 JO Dr. Thomas Gro^vciior, Jan'y. "ame of a IIOUKP, irii'S. ...... 1 FebV 10 . iVamc of a iaru, .... April Sun; ..... Dr. Edward Jones, 24 To ray draft on Thomas Gros'venor, Dr. i- 185?. I Feb'y. I -,'0 1 To sundries, FORM OF A LEGER. A hired lad, Cr. 188*. i C. Jan'y. May 1 15 By 3 months' 1* months' wages due this day at $G, . wages at $7, .... 18 31 00 so Farmer. Cr. Merchant, Cr. 1822. $ C. Jan'y. Jfl By ray order in favor of Joseph Hastings, 11 50 Sundries, SH 62 Feb'v. ?!! ditto 3 S r i 4.pril o>0 ditto. c, O f Judge of County Court, Cr. Fy. V 22 12 By my order iu favor of Joseph 11-as.tiajjs, | 24 My draft in favor of Ed^rard Jontfi, 00 March 4 Cash paid me this day, . . 75 00 7 1 empty cider barrel, .... 58 April I2| Amount of your order on Theodore Barrel!, 68 00 May 25| My order in favor of Anthony Billings, 54 00 Labourer, Cr. 1822. | I JanV. j28jBy team hire at sumlry times, . . . "5J64 Feb'y. loj 4 ri>'.'!.1hH' hire of his son William at $10, Far; 1822. March 1 ;.|By sundries in full. Cr. TIFc. 551 T.r.ER. Dr. Samuel Green, May 128 Tocas>'= count. Dr. Sept. To cash paid for my cole to D. Mason, Dr. Jonathan Cu 1822. I Marchpl To a bay horse, I i Dr. April ttSiTo 1 ' . > Dr. The od re B.irrell, May 3ITol6ri. . ibs. at 5 < , . 24 Ibs. h< ,..OE\ TO THE I.EC::K. B. H, FADE FA6K. Br.rrHl Theodore, IlriMirv I Billn . I ! 1 Curtis Jonathan, . Notes Payable, . . a Daboli Peter, . 1 R. . nor Thorn . 1 S. . . a Stacy Samuel, . . i FORM v^T A J New Londo Cr. i ulries, April [28 my note to T) .:. a eti'i'. , .- . -.tl-v Dan 1 tic. . 90|00 ". ^ Cr. April 18 By a coat, I ! A pr\ir 01 Sic. Cr. New Lor 18??. May Cr. 20 By caib in full, \^.'. 10 liUESTIO.VS TO EXERCISE THE STUDENT. Jonathan ' Joho Ko-i- >t f/ic state of the following j !>ne Ju-fph l-1-..stiiv;-!, . . g . ,. , ... r ' 1 1 Joho ; j. ' tiO.OO Joseph f l'i~* i - 'i'V, Anthon Thoma Edn ? A .FanwrV BM. or Art<>ant. All': Thomas Yates, Esq. To John Moniington, Dr. last, Aprils. To :> barrels Cider r.t. M . . 10,00 2() bushcb Potatwec, at 0,J;> ... 5.> Ibs. Butter, at 0,17 JUDJ G. itoaofHay, Julv li. rda ol' AVi.ii.!, ;.: 4,00 . Received th amount. JOHN ArOFxMNtTON. IV. B. To prcvont accidents, cai-e shuukl he tukcu not to re- eipt an accouat uulil it is pni.l. tlal ' Six months after tlato, I promi^ to y. ' order, (at my house,) One llumirc'i Dollars. - ,vrl iu two yoke of "oxen. J A ML S IULI jj^7-It is bt -t to mention where : wlral it is ;ivun. U'ithjat. ihc l Bot negotiable. .? / '~JL Fifty tw., Dollars in full of rD demands.