'T — PW'4 I,)..( 4)N~I 14 1. SU N I D.1 I41 If Ii 1 C) 0 1 I m N I l1 \ I I A Te following tite is milssing pages or has Aon Exact duplicate coul not be found. Irregularites do exist. II 9. / V 14 S 0w D A BO L L' SCHOOLMASTER'S ASSISTANT, IMPROVED AND ENLARGED; BEIKG A PLAIN PRACTICAL SYSTEM OF ARIT H E T I C. ADAPTED TO THE UNITED STATES. BY NATHAN DABOLL. WITH THE ADDITION OF THE FARMERS' AND MECHANICS' BEST METHOD OF BOOK-KEEPING, DESIGNED AS A COMPANION TO DABOLL'S ARITHMETIC, BY SAMUEL GREEN. ITHACA, N, Y.: ANDRUS, GAUNTLETT, & CO0 1850. Entered, according to Act of Congress, in the year 1839, by MACK, ANDRUS, & WOODRUFF, in the office of the Clerk of th. District Court of the northern district of New York. RECOMMENDATIONS. Yale College, Nov. 27, 1799. I HAVE read DABOLL'S SCHOOLMASTER'S ASSISTANT..The arrangement of the different branches of Arithmetic is judicious and perspicuous. The author has well explained Decimal Arithmetic, and has applied it in a plain and elegant manner in the solution of various questions, and especially to those relative to the Federal Computation of money. I think it will be a very useful book to Schoolmasters and their pupils. JOSIAH MEIGS, Professor of Mathematics and Natural Philosophy. [Now Surveyor-General of the United States.] I HAVE given some attention to the work above mentioned, and concur with Mr. Professor Meigs in his opinion of its merit. NOAH WEBSTER. New-Haven Dec. 12, 1799. Rhode-Island College, Nov. 30,1799. 1 HAVE run through Mr. DABOLL'S SCHOOLMASTER'S A SSISTANT, and have formed of it a very favourable opinion According to its original design, I think it well " calculatec to furnish Schools in general with a methodical, easy, ant comprehensive System of Practical Arithmetic." I there fore hope it may find a generous patronage, and have ai extensive spread. ASA MESSER, Professor of the Learned Languages and teacher of Mathematics. [New President cf that Tnstihittn.l RECOMMEN1DAT IONS. PlainJield Academy, April 20, 1802 I MAKE use of DABOLL'S SCIOOLMASTER'S ASSISTANT in teaching common Arithmetic, and think it the best ca! culated for that purpose of any which has fallen within m y observation. JOHN ADAMS, Rector of Plainfield Academy. [Now Principal of Philips' Academy, Andover, Mass.] Billerica Academy, (JMass.) Dec. 10, 1807. HAVING examined Mr. DABOLL'S System of Arithmetic, I am pleased with the judgment displayed in his method, and the perspicuity of his explanations, and thinking it al easy and comprehensive a system as any with which I ar t acquainted, can cheerfully recommend it to the patronag of Instructers. SAMUEL WHITING, Teacher of Mathematics. from Mr. Kennedy, Teacher of Mathematics. I BECAME acquainted with DABOLL'S SCHOOLMASTER'{ ASSISTANT, in the year 1802, and on examining it atten tively, gave it my decisive preference to any other system extant, and immediately adopted it for the pupils under my charge; and since that time have used it exclusively in elementary tuition, to the great advantage and improvement of the student, as well as the ease and assistance cI the preceptor. I also deem it equally well calculated fc the benefit of individuals in private instruction; and think it my duty to give the labour and ingenuity of the author the tribute of my hearty approval and recommendation. ROGER KENNEDY New- York, March 20 1811 PREFACE. 'jaBE design of this work is to furnish the schools of tt btited States with a methodical and comprehensive systei of Practical Arithmetic, in which I have endeavourec through the whole, to have the rules as concise and fam liar as the nature of the subject will permit. During the long period which I have devoted to the ii struction of youth in Arithmetic, I have made use of variol j systems which have just claims to scientific merit; but ti,; authors appear to have been deficient in an importai point-the practical teacher's experience. They have bee ~ too sparing of examples, especially in the first rudiments I. in consequence of which, the young pupil is hurried throug to the ground rules too fast for his capacity. This objectic-. I have endeavoured to obviate in the following treatise. In teaching the first rules, I have found it best to el courage the attention of scholars by a variety of easy at 5 familiar questions, which might serve to strengthen the minds as their studies grew more arduous. The rules are arranged in such order as to introduce tl most simple and necessary parts, previous to those whi(< are more abstruse and difficult. To enter into a detail of the whole work would be t dious; I shall therefore notice only a few particulars, at refer the reader to the contents. Although the Federal Coin is purely decimal, it is as nearly allied to whole numbers, and so absolutely necessae. to be understood by every one, that I have introduced iL immediately after addition of whole numbers, and as. shown how to find the value of goods therein, imniediatci after simple multiplication; which maybe of giat advci'tage to many, who perhaps will not have,n opiorttii9 i learning fractions. In the arrangement of fractions, I have tatken Ln ~'' new method, the advartages and factlit -If whit"' - sufficiently npologize fr its not hilg w ^-h a i il I: a - IPREFACE systems. As decimal fractions may be learned much easier than vulgar, and are more simple, useful, and necessary, 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 learn decimals, which will be necessary in the reduction of currencies, computing interest, and many other branches. Besides, to obtain a thorough knowledge of Vulgar Fractions, is 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, ac cording to the method above hinted; and after going through the principal mercantile rules, have treated upon Vulgai 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 seve. ral new and concise rules; some of which are particularly designed for the use of the compting-house. The Appendiz contains a variety of rules for casting Interest, Rebate, &c. together with a number of the mosi easy and useful problems, for measuring superficies and solids, examples of forms commonly used in transacting business, useful tables, &c. which are designed as aids in the common business of life. Perfect accuracy, in a work of this nature, can hard!3 bi expected; errors of the press, or perhaps of the author may have escaped correction. If any such are pointed out, it will be considered as a mark of friendship and fa your, by our by The public's most humble and obedient Servant, r ATHAN DABOLL. TABLE OF CONTENTS. ADDITION, Simple, - of Federal Money, - - - Compound,. Alligation, Annuities or Pensions, at Compound Interest, Arithmetical Progression, - - Barter, - Brokerage, - - Characters, Explanation of, - Commission, Conjoined Proportion, Cooins of the United States, Weights of, Division of Whole Numbers, - Contractions in, - Compound, Discount, - - - Duodecimals, Ensurance, Equation of Payments, - - - Evolution or Extraction of Roots, - Exchange, Federal Money, - - - - - -- Subtraction of, - - Fellowship, - Compound, - -- Fractions, Vulgar and Decimal, Interest, Simple, - - -by Decimals, - - Compound, by Decimals, - Inverse Proportion, - - - Involution, Loss and Gain, Multiplication, Simple, Application and Use of, ---- - Supplement to, - - -.-..- Compound, Numeration, - Practice, Position, Permutatlon of Quantities - - - - - - 17 17t - 195 - - 182 - - 126 - - 113 14 112 137 - 229 36 53 123 216 - - 114 - 126 167 139 - 21 25 - 13 - 134 - 69,143 - 108 157 - 122 - 165 - - 1z5 166 128 - - 27 30 37 48 - - 15 -99 - - 188 - t I 1 I i VW 'IfLE nF CONTBNW Questions for Exercise,. - 191 Reduction, - 5 of Currencies, do. of Coin, - 82, 8t Rule of Three Direct, do. Inverse, - 90, 91 - Double, ---— 136 Rules for reducing the different currencies of the several United States, also Canada and Nova Scotia, each to the par of all others, - 8 -- Application of the precedinr, - - 89 --- Short Practical, for calculating Interest, - - 114 -- for casting Interest at 6 per cet, - - - - 203 ---- for finding the contents of Superfices and Solids, - 208 - to reduce tlhe currencies of the different States to Federal Monel, - - -20( Rebate, a short metihod of finding the, of any given sum, for months and days, - - -20' Subtraction, Simple, - - 23 ----- Compoundi, - - 43 Table, Nureration and Pence, - -9 -- Additi,,n, Subtraction, and Multiplication, - - 10 of Weight and Measure, - - 11 --- of Time and Motion, - -- showing the number of days from any day of one month to the same day in any other month, - - - - showing the amount of 11. or 1 dollar, at 5 and 6 per cent. Compound Interest, ftr 20 years, - - - ( -- showing the amount of 11. annuity, forborne for 31 years or under, at 5 and 6 per cent. Compound Interest, - 221 --- showing the presnt worth of 11. annuity, for 31 years, at 5 and 6 per cent. Compound Interest, - - 21 --- of Cents, answering to the currencies of the United States, with Sterling, &c. 224 -- showing the value of Federal Money in other currencies, 225 Tare and Tret, - - 103 Useful Forms in transacting business, - - 226 Weights of several pieces of English, Portuguese, and French Gold Cains, in dollars, cents, and mills, - - - 222 -— of English and Portuguese Gold, do. do. - 223 --- of French and Spanish Gold, do. do, - 223 DABOLL'S SCHOOLMASTER'S ASSISTANT. ARITHMETICAL TABLES. Numeration Table. Pence Table. fCe.4 0 0k o.Ce Ro.. rCe ~E-4 E-4 E- 4 9876543 987654 98765 9876 9 8 7 9 8 d. s. d. 20 is1 8 30 2 6 40 3 4 50 4 2 60 5 0 70 5 10 80 6 8 90 7 6 100 8 4 110 9 2 120 10 0 d. s. 12 is 1 24 2 36 3 48 4 60 5 72 6 84 7 96 8 108 9 120 10 132 1 1 Ce 2 3 4 5 6 7 8 9 4 5 6 7 8 9 make 4 farthings 1 penny, d. 12pence 'I shillin, s. 20 A~illings, I P'Dunt X E. 1 )II ')A R rIT MIOfTI('AI1. AHI Lf ADDITION AND I S UBTh"A CTiON TABLE I '21 31 4 1 5 1 61 7I j 9 110I11112 2 1 41J 5j 1 6 I1 8 9 110i11 [ 12 113 114 -3 5I 61 71 I 9 1 11I11112113114115 4 6 71 910 I I I 1I 131 14 1 5 1 16 KM 71 I 9110 11112 113114115116117 6 81 910111',),13 14 11511611711 18 7] 9110 11-12) 1814145116117118 1 19 8]) 10 1 1) 13 141151 16 117 18 19120 it 121 I.,1:3 1.4_115 116 17 IS 119 20 21 MULTJ1PLICAITI)N TABLE. 1121 1T 41 T 61 71 SI 9 101I111 f 'Z 2141 61 8110112114 161 181 2h.01 221 241 361 9112115 182 "I),1241 271 8 330 36 4 18112 1620 24 1281321 36 40j 441 4.8 15 10 15 120125 130 135 40 451 501 551 60 6 1 12 118 124 130 1 36 142 1481 54 1 60 1 66 1 72 7 1 14 1 21l28135 4'149 s6I 1 613 1 70 1 77 1 84 8 1 16 1 24 132 140 1 48 156 1 64 1 72 1 80 1 88 K 96 -91 18 1 27 1 36 1 45 154 1 63 1 72 I 811 I0 1 99 1 108 10] 20-130 10401 50 1 60 1 701801 901100 I 10 1 1201 11 1 22 I 33 I 44 I 55 I 66 j 77 1 88 1 99 1 110 1 121 I 132 12 1 24 I 36 1 48 1 60 1 72 I 84 I 96 1 108 120j 1 132 I 144 To learn this Table Find your multiplier in the left hand column, and the multiplicand a-top, and in the cornmon angle of meeting, or against your multiplier, along as the right hand, and under your multiplicand, you will fin" the product, or answer. ARITHMETICAL TABLES. 11 2. Troy Weight. 24 grains (gr.) make 1 penny-weight, marked pwt. 20 penny-weights, I ounce, oz. 12 ounces, I pound, lb 3. Avoirdupois Weight. 16 drams (dr.) make 1 ounce, oz. 16 ounces, 1 pound, lb. 28 pounds, I quarter of a hundred weight, qr. 4 quarters, 1 hundred weight, cwt. 20 hundred weight, I tun. T. By this weight are weighed all coarse and drossy goods, grocery wares, and all metals except gold and silver. 4. Apothecaries Weight. t0 grains (gr.) make 1 scruple, 9 3 scruples, I dram, 3 8 drams, 1 ounce, ~ 12 ounces, 1 pound, 1; Apothecaries use this weight in compounding their medicines. 5. Cloth Measure. 4 nails (na.) make I qualtex of a yard, qr. 4 quarters, 1 yard, yd. 3 quarters, I Ell Flemish, E. Fl. 5 quarters, I Ell English, E. E. 6 quarters, I Ell French, E. Fr 6. Dry Measure. 2 pints, (pt.) make 1 quart, qt. 8 quarts, 1 peck, pk. 4 pecks, 1 bushel, bu. This measure is applied to grain, beans, flax-seed, salt cats, oysters, coal, &c. I 12 ARITHIMETICA L TABLES, 7. WVine Measure. 4 gills (gi.) make 1 pint, pt. 2 pints, 1 quart, gt. 4 quarts, 1 gallon, gal 31 gallons, 1 barrel, bl. 42 gallons, I tierce, tier. 63 gallons, 1 hogshead, hhd 2 hogsheads, 1 pipe, p. 2 pipes, I tun, T. All brandies, spirits, mead, vinegar, oil, &c. are measul ed by wine measure. Note. 231 solid inches, make a gal Ion. 8. Long Measure. 3 barley corns (b. c.) make 1 inch, marked in. 12 inches, 1 foot, ft. 3 feet, 1 yard, yd. 5- yards, 1 rod, pole, or perch, rd. 40 rods, 1 furlong, fur; S furlongs, I mile, m. 3 miles, 1 league, lea. 69! statute miles 1 degree, on the earth. 360 degrees, the circumference of the earth. The use of long measure is to measure the distance of places, or any other thing, where length is considered, without regard to breadth. N. B. In measuring the height of horses, 4 inches make 1 hand. In measuring depths, 6 feet make 1 fathom or French toise. Distances are measured by a chain, four rods long, containing one hundred links. ARITHMETICAL TABLIS. IS 9. Land, or Square Measure. 144 square inches make 1 square foot. 9 square feet, I square yard, 30 square yards, or 1 square rod. t721 square feet, 40 square rods, 1 square rood. 4 square roods, 1 square acre, 640 square acres, I square mile. 10. Solid, or Cubic Measure. 728 solid inches make 1 solid foot. 40 feet of round timber, or 1 tun or load. 50 feet of hewn timber, 128 solid feet, or 8 feet long, ordof wood 4 wide, and 4 high, All solids, or things that have length, breadth, and depth, tle measured by this measure. N. B. The wine gallon o ntains 231 solid or cubic inches, and the beer gallon, 282.. bushel contains 2150,42 solid inches. 11. Time. 60 seconds (S.) make 1 minute, marked M. 60 minutes, 1 hour, h. 24 hours, 1 day, d. 7 days, I week, w. 4 weeks, 1 month, mo. 13 months, 1 day and 6 hours, 1 Julian year, yr. Thirty days hath September, April, June, and November, February twenty-eight alone, all the rest have thirty-one. N. B. In Bissextile, or leap year, February hath 29 days, 12. Circular Motion. 60 seconds (") make 1 minute, 60 minutes, I degree, ~ 30 dpees, I sign, S. 12 sins, or 360 degrees, the whole great circle of the?Zodiack B 14 e BARACTERS. Explanation of Characters used in this Book. = Equal to, as 12d. -1 s. signifies that 12 pence are equal to 1 shilling. + More, the sign of Addition; as, 5+7=12, signifies that 5 and 7 added together, are equal to 12. - linus, or less, the sign of Subtraction; as, 6-2=4, signifies that 2 subtracted from 6, leaves 4. x Alultiply, or with, the sign of Multiplication; as, 4 x 3=12, signifies that 4 multiplied by 3, is equal to 12. -- The sign of Division; as, 8-2=4, signifies that 8 di vided by 2, is equal to 4; or thus, ]-=4, each of whitc signify the same thing.:: Four points set in the middle of four numbers, denote them to be proportional to one another, by the rule of three; as 2: 4::: 16; that is, as 2to4, so is 8 to 16. V Prefixed to any number, stpposes that the square root of that number is required. ' Prefixed to any number, supposes the cube root of that number is required. V Denotes the biquadrate root, or fourth power, &c. ARITHMETIC. ARITHMETIC is the art of computing by numbers. and has five principal rules for its operation, viz. Numeration, Addition, Subtraction, Multiplication, and Division. NUMERATION. Numeration is the art of numbering. It teaches to express the value of any proposed number by the following characters, or figures: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0-or cipher. Besides the simp'e value of figures, each has a local ralue, which depend e upon the place it stands in, viz. any ligure in the place of units, represents only its simple value,,r so many ones; but in the second place, or place of tens, it becomes so many tens, or ten times its simple value; and in the third place, or place of hundreds, it becomes a hundred times its simple value, and so on, as in the following.'ote.-Although a cipher standing alone signifies nothing; yet when it Is placed on the right hard of figures it increases their value in a tenfold proportion, by throwing them into higher places. Thus, 2 with a cipher an. Vexed to it, becomes 20, twenty, and with two ciphers, thus, 200,two hundred. 2, When numbers consisting of many figures, are given to be read, it will be found convenient to divide them into as many periods, as we can, of six figures each, reckoning from the right hand towards the left, calling the first the period of units, the second that of millions, the third billions, the fourth trillions, &c. as in the following number: 8 0 7 3 6 2 5 4 6 2 7 8 9 0 1 2 5 0 6 7 9 2 4. Period of 3. Period of 2. Period of j1. Period of Trillions. Billions. Miltions. Units. 8073 626462 789012 506792 The foregoing number is' read thus-Eight thousand and seventy-three trillions; six hundred and twenty-five thousand, four hundred and sixtytwo billions; seven hundred and eighty-nine thousand and twelve millions; five.hundred and six thousand seven hundred and ninety-two. N.... Billions is substituted for millions of millions. Trilrons foe, millions of mtillions of millions. Quatrillions for millions of millions of millions of millions, 4&b. 16 N UMERATION. TABLE. o o o = o -?. —, S r l.. ~, I 1-One c, a, ' 2 1 -Twenty-one., I. X*, 3 2 1 -Three hundred twenty-one. ' ' 4 3 2 1 -Fourthousand 321. *..* 5 4 3 2 1 -Fifty-four thousand 321. *.. 6 5 4 3 2 1 -654 thousand 321. 7 6 5 4 3 2 1 -7 million 654 thousand 321. 8 7 6 5 4 3 2 1 -87 million 654 thousand 321. 9 8 7 6 5 4 3 2 1 -987 million 654 thousand 321 1 2 3 4 5 6 7 8 9 -123 million 456 thousand 789. 9 8 7 6 5 4 3 4 8 -987 million 654 thousand 348. To know the value of any number of figures: RULEI.-. Numerate from the right to the left hand, each figure I its proper place, by saying, units, tens, hundreds, &c. as in the Numr ration Table. 2. To the simple value of each figure, join the name of its plaso beginning at the left hand, and reading to the right. EXAMPLES. t Read the following numbers. 365, Three hundred and sixty-five. 5461, Five thousand four hundred and sixty-one. 1234, One thousand two hundred and thirty-four. 54026, Fifty-four thousand and twenty-six. 123461, One hundred and twenty-three thousand foul hundred and sixty-one. 4666240, Four millions, six hundred and sixty-six thou. sand two hundred and forty. NOTE. For convenience in leading large numbers, they may be divided into periods of three figures each, as follows < 987, Nine hundred and eighty-seven. 987 000, Nine hundred and eighty-seven thousand. 987 000 000, Nine hundred and eighty-seven million. 987 654 321, Nine hundred and eighty-seven million, si hundred and fifty-four thousand, three hun dred and twenty-one. IlttR ADDITfi To write numbers.,Re.,- 3Begin on the right hand, write units in the units place, tens hi ho tens place, hundreds in the hundreds place, and so on, towards the left hand, writing each figure according to its proper value in numeration; taking care to supply those places of the natural order with ciphers which are omitted in the question, EXAMPLES. Write down in proper figures the following numbers: Thirty-six. Two hundred and seventy-nine. Thirty-seven thousand, five hundred and fourteen. Nine millions, severity-two thousand and two hundred. Eight hundred millions, forty-four thousand and fifty-five. SIMPLE ADDITION. IS putting together several smaller numbers, of the same denomination, into one larger, equal to the whole or sum total; as 4 dollars and 6 dollars in one sum is 10 dollars. RULE.-Having placed units under units, tens under tens, &c. draw L line underneath, and begin with the units; after adding up every figure in that column, consider how many tens are contained in their um; set down the remainder under the units, and carry so many as Fou have tens, to the next column of tens; proceed in the same manter through every column or row, and set down the whole amount,f the last row. EXAMPLES. (1.) (2.) (3.) (4.) k.~3 J Cb oa 0 0 z U0 a t.. 59h ^ 2. E ~ D ES1E4 E E; 42 4 1 4 1756 552621 53 291 0432 346977 52 851 9478 413339 13 152 1666 321 0 12 89 698 7422 876543 Bo 18 SIMPLE ADDITION. (5.) 31485 67237 42719 97145 32851 14572 (s8.) 64273 17845 37256 25417 61723 38419 72843 -.. (6.) 64179 25712 84 1 94 32516 71432 32719 (9.) 84128 93714 37147 1 8321 71437 51726 72513,,. (7.) 37145 51714 6 0 8 4 5 60845 37857 61784 52 1 01 (10.) 52637 27 1 96 38419 5319 6 1 08 1 3 7 1 9, 2 9 1 4 (11.) (12.) 942317829 371845687 742106 108 51 1 704229 61 0042796 19466372 762314572 8340734 200041234 270155 704136053 36023 567809387 1950 (13.) (14.) 962430646 259004 46281451 34004 2160432 540443 It 87610425 3705532 ( 346214 4052 1 74 40309 406476269 9827 2068591 0L To prove Addition, begin at the top of the sum, and recko0 i e-figures downwards in the same manner as they were added ur SIMPLE ADDITION. 19 wards, and if it be right, this sum total will be equal to the first: Or out off the upper line of figures, and find the amount of the rest; then if the amount and upper line, when added, be equal to the total, the work is supposed to be right. 2. There is another method of proof, as follows:Reject or cast out the nines in each row EXAMPLE. or sum of figures, and set down the re- 3 7 8 2 | 2 m ainders, each directly even with the figures 5 7 6 6 6 in its row; find the sum of these remain- 8 7 5 5 ' 7 dors; then if the excess of nines in the - sum found as before, is equal to the excess 18 3 0 3 ~ 6 of nines in the sum total, the work is sup-. - posed to be right. 15. Add 8635, 2194, 7421, 5063, 2196, and 1245, together. Ans. 26754. 16. Find the sum of 3482, 783645, 318, 7530, and 678045. Ans. 10473020. 17. Find the sum total of 604, 4680, 98, 64, and 54. Ans. Fifty-five hundred. 18. What is the sum total of 24674, 16742, 34678, 10467,,rd 13439? Ans. One hundred thousand. 19. Add 1021, 3489, 28763, 289, and 6438, together. Ans. Forty thousand. 20. What is the sum total of the following numbers, viz. 2340, 1066, 3700, and 4005? Ans. 11111. 21. What is the sum total of the following numbers, viz. Nine hundred and forty-seven, Seven chousand six hundred and five, Forty-five thousand six hundred, Three hundred and eleven thousand, Nine millions, and twenty-five, Fifty-two millions, and nine thousand? Answ6r, 61374177 22. Required the sum of the following numbers, viz. Five lundred and sixty-eight, Eight thousand eight hundred and five. Seventy-nine thousand six hundred, PFDRRAL BONZEY. Nine hundred and eleven thousand, Nine millions and twenty-six. Answer, 9999999 QUESTIONS. 1. What number of dollars are in six bags, conltainilh each 37542 dollars I Ans. 22525~. 2. If one quarter of a ship's cargo be worth eleven thousand and ninety-nine dollars, how many dollars is the whole cargo worth I Ans. 443!9 dols. 3. Money was first made of gold and silver at Argos, eight hundred and ninety-four years before Christ; how long has money been in use at this date, 1814? Ans. 2708 years. 4. The distance from Portland in the Province of Main, to Boston, is 125 miles; from Boston to New-Haven, 161 miles; from thence to New-York, 88; from thence t. Philadelphia, 95; from thence to Baltimore, 102; fronm thence to Charleston, South Carolina, 716; and from thelnci: to Savannah, 119 miles-What is the whole distance from Portland to Savannah? Ans. 1407 miles. 5. John, Thomas, and Harry, after counting their prizi money, John lhd one thousand three hundred and seventy five dollars; Thomas had just three times as many as John; and Harry had just as many as John and Thomas both — Pray how many dollars had Harry 1 Ans. 5500 dollars. FEDERAL MONEY. NEXT in point of simplicity, and the nearest allied tc whole numbers, is the coin of the United States, or FEDERAL MONEY. This is the most simple and easy of all money-it in creases in a tenfold proportion, like whole numbers. 10 mills, (m.) make I cent, marked c. 10 crnts, 1 dime, d. 10 lio'es, 1 dollar, $. 0 j nirs, 1 eageo, E. ADDITION OF FEDIRAL MONEY, Dollar is the money unit; all other denominations being vrlued according to their place from the dollar's place.A point or comma, called a separatriz, may be placed after the dollars to separate them from the inferior denominations; then the first figure at the right of this separatrix is dimes, the second figure cents, and the third mills.* ADDITION OF FEDERAL MONEY. RULE.-1. Place the numbers according to their value; that is, dollars under dollars, dimes under dimes, cents under cents,.&c. and proceed exactly as in whole numbers; then place the separatrix in the sum total, directly under the separating points above. EXAMPLES. $. d.c.m. $. d. c.m. $. d.c.m. 365, 541 439, 3 0 4 136, 514 487, 060 416, 390 125, 090 94, 670 168, 934 200, 909 439, 089 239, 060 304, 006 742, 500 143, 0 0 5 111, 191 2128, 860 2. When accounts are kept in dollars and cents, and no other de-.ominatlons are mentioned, which is the usual mode in common reckuning, then the first two figures at the right of the separatrix or point, lay be called so many cents instead of dimes and cents; for the lace of dimes is only the ten's place in cents; because ten cents make i dime; for example, 48, 75, forty eight dollars, seven dimes, five cents, nay be read forty-eight dollars and seventy-five cents. If the cents are less than ten, place a cipher in the ten's place, or,ace of dimes.-Example. Write down four dollars and 7 cents. Thus, $4, 07 cts. * It may be observed, that all the figures at the left hand of the separatrix are dollars; or you may call the first figure dollars, and the other eagles, iSc. Thus any sum of this money may be read differently, either wholly in the lowest denomination, or partly in the higher, and partly in the lowest; for example, 37 54, may be either read 3754 cents, or 375 dimes and 4 cents,r S7 uollars 5 dimes and 4 cents, or 3 eagles 7 dollars 5 dimes and 4 centi. ADDITION OF FEE)gRAL MONEY. EXAMPLES. 1. Find the sum of 304 dollars, 39 cents; 291 dollars, 9 cents; 136 dollars, 99 cents; 12 dollars and 10 centd. 304, 39 Thus, 291, 09 136, 99 12, 10 Sum, 744, 57 Seven hundred forty-four do) lars and fifty-seven cents. (2.) (3.) (4.) $. cts. $. cts. $. cts. 0, 99 364, 00 3287, 80 0, 50 21, 50 1729, 19 0, 25 8, 09 4219, 99 0, 75 0, 99 140, 01 (5.) (6.) (7.) $. $. cts. $. cts. 2468 124, 50, 161 1900 9, 07, 99 246 0, 60, 86j 146 231, 01, 17 167 0, 75, 67j 46 24, 00, 72 19 9, 44, 99 8 0, 95,09 8. What is the sum total of 127 dols. 19 cents, 278 dols, 19 cents, 34 dols. 7 cents, 5 dols. 10 cents, and 1 dol. 99 cents? Ans. $446, 54 cts. 9. What is the sum of 378 dols. 1 ct., 136 dols. 91 cts., 344 dols. 8 cts., and 365 dots.? Ans. $1224. 10. What is the sum of 46 cents, 52 cents, 92 cents, anc 10 cents I Ans. $2. 11. What is the sum of 9 dimes, 8 dimes, and 80 cents S Ans. Pi. SIMrLE SUBTRACTION. 23 12. 1 received of A, B, and C, a sum of money; A paid nae 95 dols. 43 cts., B paid me just three times as much as A, and C paid me just as much as A and B both: can you tell me how much money C paid me? Ans. $381, 72 cts. 13. There is an excellent well built ship just returned fiom the Indies. The ship only is valued at 12145 dols. 86 cents; and one quarter of her cargo is worth 25411 dols. 65 cents. Pray what is the value of the whole ship and cargo? Ans. 113792, 46 cts. A TAILOR'S BILL. Hfr. James Paywell, To Timothy Taylor, Dr. 1814, $. cts. $. cts. \ pril 15. To 24 yds. of Cloth, at 6, 50 per yd. 16 25 To 4 yds. Shalloon, 75 3 00 To making your Coat, 2 50 To 1 silk Vest pattern, 4 10 To making your Vest, 1 50 To Silk, Buttons &c. for Vest, 0 45 Sum, $27 80 Er By an act of Congress, all the accounts of the United States, die salaries of all officers, the revenues, &c. are to be reckoned, in federal money; which mode of reckoning is so simple, easy, and con-,enient, that it will soon come into common practice throughout it the States. SIMPLE SUBTRACTION. -W Subtraction of whole Numbers, TEACHETH to take a less number from r er, of the same denomination, and thereby shows the difference, or remainder: as 4 dollars subtracted from 6 dollars, the reinainder is 2 dollars. RULE.-Place the least number under the greatest, so that units may stand under units, tens under tens, &c. and drvw ' urider thmnm. XT SIMPLE SUBTRACTION, 2. Begin at the right band, and take each figure in the lower line trom the figure above it, and set down the remainder. 3. If the lower figure is greater than that above it, add ten to the upper figure; from which number so increased, take the lower and Bet down the remainder, carrying one to the next lower number, with which proceed as before, and so on till the whole is finished. PROOF. Add the remainder to the least number, and if the sum be equal to the greatest, the work is right. EXAMPLES. (1.) Greatest number, 2 4 6 8 Least number, 1 3 4 6 r-~ l*... (2.) 62157 1 2 1 48 Vllnerence, Proof, (4.) From 4167883 Take 3154299 (3.) 8796475 1643489 (6.) 65432167890 12345697098 (5.) i9 918764520 )9 91243806 Rem. (7.) From 917144043605 Take 40600832164 Rem. (9.) (10.) From 100000 2521665 Take 65321 2000000 (8.) 3562176255002 1235271082165 (11.) (12.) 200000 10000 99999 1 Dif. - 13. From 3604L8, take 293752. Ans. 66666. 14. From 765410, take 34747. Ans. 730663. 15. From 341209, take 198765. Ans. 142444. 16. From 100046, take 10009. Ans. 90037. 17. From 2637804, take 2376982. Ans. 260822. 18. From ninety thousand, five hundred and forty-si take forty-two thousand, one hundred and nine. Ans. 48437. 19. From fifty-four thousand and twenty-six, take nine thousand two hundred and fifty-four. Ans. 4477%2 SUBrRACTION OF FEDERAL MONEY. 25 X, ~o.f 20. From one million, take nine hundred and ninetynine thousand. Ans. One thousand. 21. From nine hundred and eeighty-seven millions, take nine hundred and eighty-seven thousand. Ans. 986013000. 22. Subtract one from a million, and show the remainder. Ans. 999999. QUESTIONS, 1. How much is six hundred and sixty-seven greater nan three hundred and ninety-five 1 Ans. 272. 2. What is the difference between twice twenty-seven, and three times forty-five 3 Ans. 81. 3. How much is 1200 greater than 365 and 721 added,together? Ans. 114. 4. From New-London to Philadelphia is 240 miles. Now \' a man should travel five days from New-London towards I hiladelphia, at the rate of 39 miles each day, how far p Iuld he then be from Philadelphia. Ans. 45 miles. 5. What other number with these four, viz. 21, 32, 16, a id 12, will make 100? Ans. 19. 6. A wine merchant bought 721 pipes of wine for 90846 d >llars, and sold 543 pipes thereof for 89049 dollars; how many pipes has he remaining or unsold, and what do they stand him in 1 Ans. 178 pipes unsold, and they stand him in $1797. SUBTRACTION OF FEDERAL MONEY. Ru LE.-Place the numbers according to their value; that is, dollar rider dollars, dimes under dimes, cents under cents, &c. and subtract s in whole numbers. EXAMPLES. $. d.c.m. From 45, 4 7 5; ' Take 43, 4 8 5 Item. $,, 9 9 0 one dollar, nine dimes, and nine cents or one dollar and ninety-nine cents 1t SUBTRACTION OF FEDERAL MONEY. $. d. c. d. d. c. m. From 45, 7 4 46, 2 4 6 211, 1 1 0 Take 13, 8 9 36, 1 6 4 111, 1 1 4 iem. - $, $. cts. $. cs. From 4 2 8 4 411, 24 960, 00 Take 1 993 1, 09 136, 41 Item. $. cts. $. cts. $ cts. From 4106, 71 1901, 08 365, 09 Take 221, 69 864, 09 109, 01 Rem. _ 11. From 125 dollars, take 9 dollars 9 cents. Ans. 115 dells. 91 ctb 12. From 127 dollars 1 cent, take 41 dollars 10 cents. Ans. 85 dolls. 91 cts. 13. From 365 dollars 90 cents, take 168 dols. 99 cents Ans. $196, 91 cts. 14. From 249 dollars 45 cents, take 180 dollars. Ans. $69, 45 cts. 15. From 100 dollars, take 45 cts. Ans. $99, 55 cts. 16. From ninety dollars and ten cents, take forty dollana and nineteen cents. -Ans. $49, 91 cte 17. From forty-one dollars eight cents, take one dollai nine cents. Ans. $39, 99 cts. 18. From 3 dols. take 7 cts. Ans. $2, 93 cts. 19 From ninety-nine dollars, take ninety-nine cents. Ans. $98, 1 ct. 20. From twenty dols. take twenty cents and one mill. Ans. $19, 79 cts. 9 mills. 21. From three dollars, take one hundred and ninety-ninu cents. Ans. $1, 1 ct. 22. From 20 dols. take 1 dime. Ans. $19, 90 cts. 23. From i.k e dollars and ninety cents, take ninety-nine im~e r Ans. 0 remains. 21 Jai.. 'ze money was 219 dollars, and Thomnlr StMPLE MULTIPLICATION. t? received just twice as much, lacking 45 cents. How nuch money did Thomas receive? Ans. $437, 55 cts. 25. Joe Careless received prize money to the amount of 1000 dollars; after which he lays out 411 dolls. 41 cents for a span of fine horses; and 123 dollars 40 cents for a gold watch and a suit of new clothes; besides 359 dols. and 50 cents he lost in. gambling. How much will he have left after paying his landlord's bill, which amounts to 85 dols. and 11 cents? Ans. $20, 58 cts. SIMPLE MULTIPLICATION TEACHETH to increase or repeat the greater of two numbers given, as often as there are units in the less, or.ul!tiplying number; hence it performs the work of many dllitions in the most compendious manner. The number to be multiplied is called the multiplicand. The number you multiply by, is called the multiplier. The number found from the operation, is called the pro)uct. NOTE. Both multiplier and multiplicand are in general salled factors, or terms. CASE I. When the multiplier is not more than twelve. RULE.-Multiply each figure in the multiplicand by the multiplier;.arry 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. Ans. 1095 product. * Multiplication may also be proved by casting out the 9's in the two actors, and setting down the remainders; then multiplying the two remainders together; if the excess of 9's in their product is equal to the extess of 9's in the total product, the work is supposed to be right. Mult iplicaznd. Mlutit plier.. 2345 907J1 3 4 5 0 Product, 47094 71034 31261 8 9 4320 10 1432046 210240613 4684114 11 12 12 CASE If. When the -mAt-:Riplier cc-.-sists of several figures. RULE.-The mul'tiplier be~zig pX'ced under the multiplicand, unit a under units, tens un'!er tens, &c. multiply by each significant figure in the multiplier separatnly, placing the first figure in each product exactly under its mult~plicr; t~eu add the several products togethtif in the same order as they' sta q,!, and their sum will be the total product EXA-TPLES. What number is e-'.u- i ro 7tms35 Mutit plicand, 3 6 5 M~ultiplier, 4 7 2 5 5 5 1 46 0 Ans. 1 7 1 5 5 product Iifwliplicand, 37864 Multiplier, 209 34293 74 47042 91 Product, 825. 826 MGMO7 340776 75728 7913576 25203 402 5 101442075 2537682 4280822 2193 4072 8929896 9" 9876 9405 18837S6 iSIi'LE MULTIPLICATION. '19 2i 81 261986 40634 4'A9 7638 42068 L244038649 2001049068 1709391112 134092 918273645 87362 1003245 11714545304 921253442978025 14. Multiply 760483 by 9152. Ans. 6959940416. 15. What is the total product of 7608 times 365432. Ans. 2780206656. 16. What number is equal to 40003 times 4897685. Ans. 195922093055. CASE III. When there are ciphers on the rght hand of either or both of the factors, neglect those ciphers; then place the significant figures under one another, and multiply by them tnly, and to the right had of the product, place as many ciphers as were omitted in both the factors. EXAMPLES. 21200 31800 84600 70 36 34000 1484000 1144800 2876400000 35926000 82530 3040 98260000 10921S040000 8109397800000 7065000 x 8700=61465500000 749643000 x 695000=521001885000000 360000 x 1200000=432000000000 CASE IV. When the multiplier is a composite number, that A, when u, produced by multiplying any two number m mze ts ble together, multiply first by one of thcse figv "s ate c 2 1so SIMPLE MULTIPLICATION. product by the other, and the last product will be the totlJ required. EXAMPLES. Multiply 41364 by 35 7 x 5=35. 7 289548 Product of 7 -5 1447740 Product of 35 2. Multiply 764131 by 48. Ans. 36678228 8. Multiply 342516 by 56. Ans. 19180896. 4. Multiply 209402 by 72. Ans. 15076944. 5. Multiply 91738 by 81. Ans. 7430778. 6. Multiply 34462 by 108. Ans. 3721896. 7. Multiply 615243 by 144. Ans. 88594992 CASE V. To multiply by 10, 100, 1000, &c. annex to the muki plicand all the ciphers in the multiplier, and it will make the product required. EXAMPLES. 1. Multiply 365 by 10. Ans. 3650. 2. Multiply 4657 by 100. Ans. 465700. 3. Multiply 5224 by 1000. Ans. 5224000. 4. Multiply 26460 by 10000. Ans. 264600000. EXAMPLES FOR EXERCISE. 1.Multiply 1203450 by 9004. Ans. 10835863800. 2. Multiply 9087061 by 56708. Ans. 515309055188. 3. Multiply 8706544 by 67089. Ans. 584113330416. 4. Multiply 4321209 by 123409. Ans. 533276081481. 6. Multiply 3456789 by 567090. Ans. 1960310474010. 6. Multiply 8496427 by 874359. Ans. 7428927415293. 98763542 x 98763542=8754237228385764. Application and Use of Multiplication. In making out bills of parcels, and in finding the value of goods; when the price of one yard, pound, &c. is given (in Fideral Money) to find the value of the whole quaptity. SIMPLE MULTIPLICATION. 31 RULE.-Multiply the given price and quantity together, as in dhole numbers, and the separatrix will be as many figures from the ight hand in the product as in the given price. EXAMPLES. 1. What will 35 yards of broad- } $. d. c. m. cloth come to, at ) 3, 4 9 6 per yard? 3 5 17 4 8 0 104 8 8 Ans. $122, 3 6 0-122 dol[lars, 36 cents. 2. What cost 35 lb. cheese at 8 cents per lb.?,08 Ans. $2, 80=2 dollars 80 cents. 3. What is the value of 29 pairs of men's shoes, at 1 dol&x 51 cents per pair I Ans. $43, 79 cents. 4. What cost 131 yards of Irish linen, at 38 cents per Tard? Ans. $49, 78 cents. 5. What cost 140 reams of paper, at 2 dollars 35 cents per ream? Ans. $329. 6. What cost 144 lb. of hyson tea, at 3 dollars 51 cents per lb.? Ans. $505, 44 cents. 7. What cost 94 bushels of oats, at 33 cents per bushe1? Ans. $31, 2 cents. 8. What do 50 firkins of butter come to, at 7 dollars 14 ients per firkin? Ans. $357. 9. What cost 12 cwt. of Malaga raisins, at 7 dollars 31 ents per cwt.? Ans. $87, 72 cents. 10. Bought 37 horses for shipping, at 52 dollars per head: that do they come to? Ans. $1924. 11. What is the amount of 500 lbs. of hog's-lard, at 15.ents per lb.? Ans. $75, 12. What is the value of 75 yards of satin, at 3 dollars 5 cents per yard? Ans. $281,25. 13. What cost 367 Acres of land, at 14 dols. 67 cents er acre? Alns. $5383, 89 cents. 32 DIVISION OF WHOLE NUMBERS. 14. What does 857 bis. pork come to, at 18 dols. 89 cents per bl.? Ans. $16223, 1 cent. 15. What does 15 tuns of hay come to, at 20 dols. '7P cts. per tun? Ans. $311, 70 cents' 16. Find the amount of the following BILL OF PARCELS. New-London, March 9, 1814. Mr. James Paywell, Bought of William Merchant 28 lb. of Green Tea, 41 lb. of Coffee, 34 lb. of Loaf Sugar, 13 cwt. of Malaga Raisins 35 firkins of Butter, 27 pairs of worsted Hose, 94 bushels of Oats, 29 pairs of men's Shoes, Received payment in full, $. cts. at 2, 15per lb. at 0, 21 at 0, 19 at 7, 31 per cwt. at 7, 14perfir. at 1, 04 per pair. at 0, 33 per bush. at 1, 12 per pair. Amount, $511, 78. WILLIAM MERCHANT A SHORT RULE. NOTE. The value of 100lbs. of any article will be jus as many dollars as the article is cents a pound. For 100 lb. at 1 cent per lb.-100 cents=1 dollar. 100 lb. of beef at 4 cents a lb. comes to 400 cents=4 dollars, &c. DIVISION OF WHOLE NUMBERS. SIMPLE DIVISION teaches to find how many time one whole number is contained in another; and also wha remains; and is a concise way of performing several sub tractions. Four principal parts are to be noticed in Division: 1. The Dividend, or number given to be divided. 2. The Divisor, or number given to divide by. 3. The Quotient, or answer to the question, which show how many times the divisor is contained in the dividend. 4. The Remainder, which is always less than the diviso: and of the same name with the Dividend. DIVISION OF WIOLE NUMBEIS. 33 RULE.-First, seek how many times the divisor is contained in as wlany of the left hand figures of the dividend as are just necessary; (that is, find the greatest figure that the divisor can be multiplied by, to as to produce a product that shall not exceed the part of the dividend used;) when found, place the figure in the quotient; multiply the divisor by this quotient figure; place the product under that part of the dividend used; then subtract it therefrom, and bring down tho next figure of the dividend to the right hand of the remainder; after which, you must seek, multiply and subtract, till you have brought down every figure of the dividend. PROOF. Multiply the divisor and quotient together, and add the.'Lmainder, if there be any, to the product; if the work be right, the sum will be equal to the dividend.* EXAMPLES. 1. How many times is 4 2. Divide 3656 dollars rontaiped in 9391? equally among 8 men. Oinissi Div. Quotient. Divisor, Div. Quotient. 4)9391(2347 8)3656(457 8 4 32 13 9388 45 12 +3 Rem. 40 1) 9391 Proof. 56 16 56 31 3656 Proof by 28 addition. 3 Remainder. * Another method which some make use of to prove division is as follows: viz. Add the remainder and all the products of the several quotient figures multiplied by the divisor together, according to the order in which they stand in the work; and this sum, when the work is right, will be equal to the dividend. A third method of proof by excess of nines is as follows, viz. 1. Cast the nines out of the divisor, and place the excess on the left hand. 2. Do the same with the quotient, and place it on the right hand. 3. Multiply these two figures together, and add their product to the remainder, and reject the nines, and place the excess at top. 4. Cast the nines out of the dividend and place the excess atbottom.. ote. If the sum is right, the top and bottom figures rfJl-be alike 34 DIViSION OF WHOLE NUtMBERi:. Divisor. Div. Quotient. 29) 1o359(529 65)49640(1 36 145 365 Proof by ezcess of 9's. 85 1314 5 58 1095 2 X 7 279 2190 5 261 2190 Remains 18 0 Rem. Divisor. Div. Quotient. 95(85595(901 61)2S609(469 736)863256( 1172 472)251104(532 tlhere remains 664. 9. Divide 1893312 by 912. Ans. 207( 10. Divide 1893312 by 2076. Ans. 912 11. Divide 47254149 by 4674. Ans. 10110 -9.7 12. What is the quotient of 330098048 divided by 420'1 Ans. 78464. 13. What is the quotient of 761858465?ivided by 846 t1 Ans. 90001. 14. How often does 761858465 contain 90001? Ans. 8465. 15. How many times 38473 can you have in 119184691 Ans. 3097 '3348 1. 16. Divide 280208122081 by 912314. Quotient, 3071400 -WT-?T. MORE EXAMPLES FOR EXERCISE. Divisor. Dividend. Remainder. 234063)590624922( Quotient)83973 47614)327879186( ) 9182 987654(988641654( ) -- 0 CASE II. When there are ciphers at the right hand of the divisuz, cut off the ciphers in the divisor, and the same number of figures from the right hand of the dividend; then divide tht, remaining ones as usual, and to the remainder (if any) annex those figures cut off from the dividend, and you wi have the true remainder DIVI 1ON OF WHOLE NUMBERS. s$ EXAMP lES. 1. Divide 4673625 Iby 21400. a,4(00)46736)25(218 -?4-o true quotient by Restitution 428 -- 2. 3. 4. 5. 393 214 1796 1712 8425 true rem. Divide 379432675 by 6500. Ans. 583747- 1. Divide 421400000 by 49000. Ans. 8600. Divide 11659112 by 89000. Ans. 131 8-!0 Divide 9187642 by 9170000. Ans. 1 l MORE EXAMPLES. Divisor. Dividend. Remains. 125000)436250000( Quotient. ) 0 120000)149596478( ) 76478 901000)654347230( )221230 720000)987654000( )534000 CASE III. Short Division is when the Divisor does not exceed 12. RULE.-Consider how many times the divisor is contained in the irst figure or figures of the dividend, put the result under, and carry is many tens to the next figure as there are ones over. Divide every figure in the same manner till the whole is finished. EXAMPLES. Divisor. Dividend. 2)113415 3)85494 4)39407 5)94379 Quotient, 56707-1 6)120616 7)152715 8)96872 9)118724.. ^986197 12)14814096 12)570196382 36 CONTRAC(i. cr IN DIVISION, C'ontractions in Division. When the divisor is such a number, that any two figures in the Table, being multiplied together, will produce it, divide the given dividend by one of those figures; the qiotient thence arising by the other; and the last luotient will be the answer. NOTE. The total remainder is found by nultiplying the last remainder by the first divisor, and adding in the fir' remnainder. EXAMPL ES. Divide 162641 by 72 9)162641 or 8)18071-2 2258-7 8)162641 9)20330-1 2258-8 rue Quotient 2258.7 2. 3. 4. 5. 6. 7. 8. 9. 10. Divide 178464 by 16. Divide 467412 by 24. Divide 942341 by 35. Divide 79638 by 36. Divide 144872 by 48. Divide 937387 by 54. Divide 93975 by S4. Divide 145260 by 108. Divide 1575360 by 144. last rem. ' xi 6t first rem. +S True ren. C' Ans. 11154. Ans. 194751 -. Ans. 26924-1-, Ans. 2212-3-. Ans. 3018A. Ans. 17359A-. Ans. 1118]. Ans. 1345. Ans. 10940. 2. To divide by 10, 100, 1000, &c. RULE.-Cut off as many figures from the right hand of the dividen as there are ciphers in the divisor, and these figures so cut off are th remainder; and the other figures of the dividend are the quotient. 1. 2. 3. Divide 365 Divide 5762 Divide 763753 EXAMPLES. by 10. Ans. 36 and 5 remains by 100. Ans. 57 - 62 rem. by 1000. Ans. 763 - 753 rem. SUPPLEMENT To MULTIPLICATION. 37 SUPPLEMENT TO MULTIPLICATION. To multiply by a mixt number; that is, a whole number joined with a fraction, as 81, 54, 63, &c. AULE.-Multiply by the whole number, and take i, 1, 4, &c. of 3, i multiplicand, and add it to the product. EXAMPLES. Multiply 37 by 234. Multiply 48 by 23. 2)37 48 23- 21 18' 24=111 12=74 96 869- inswer. 132 Ans. 3 Multiply 211 by 50.. Ans. 10655X. 4. Multiply 2464 by 84. Ans. 205334. 5. Multiply 345 by 194. Ans. 6598-. 6. Multiply 6497 by 54. Ans. 334134. Questions to exercise Mlultiplication and Division. 1. What wil' 94 tuns of hay come to, at 14 dollars a i I? Ans. $136-. 2. If it take 320 rods to make a mile, and every rod c ntains 54 yards; how many yards are there in a mile? Ans. 1760. 3. Sold a ship for 11516 dollars, and I owned 4 of her; what was my part of the money? Ans. $8637. 4. In 276 barrels of raisins, each 3~ cwt. how many hundred weight? Ans. 966 cwt. 5. In 36 pieces of cloth, each piece containing 241 yards; how many yaids in the whole? Ans. 873 yds. 6. What is the product of 161 multiplied by itself? Ans. 25921. 7. If a man spend 492 dollars a year, what is that per.alendar month I Ans. $41. 8. A privateer of 65 men took a prize, which being equally divided among them, amounted to 1191. per man; what is the value of the prize? Ans. ~7 735. D A 38 COMPOUND ADDITION. 9. What number multiplied by 9, vill imake 225 1 Ans. 25. 10. The quotient of a certain number is 457, and the divisor 8; what is the dividend? Ans. 3656. 11. What cost 9 yards of cloth, at 3s. per yard? Anls. 27s. 12. What cost 45 oxen, at 81. per head.? Ans. ~360. 13. What cost 144 lb. of indigo, at 2 dols. 50 cts. o0 250 cents per lb. Ans. $360. 14. Write down four thousand six hundred and seventve_, multiply it by twelve, divide the product by nine, and add 365 to the quotient, then from that sum subtract five thousand five hundred and twenty-one, and the remainrde will be just 1000. Try it and see. COMPOUND ADDITION, IS the adding of several numbers together, having dif ferelt denominations, but of tile same generic kind, ti pounds, shillings and pence, &c. Tuns, hundreds, quar. ters, &c. RULE.-1. Place the numbers so that tihose of tlhe same denomnilr tion may stand directly under each other. 2. Add tile first column or denomination together, as in whole nui hers; then divide tile sum by as many of the same denomination T nake one of thle next greater; setting down the remainder under thl column added, and carry the quotient to the next superior denomina tion, continuing the same to the last, wlich add, as in simple addition.' 1. STERLING MONEY, Is tle money of account in Great-Britain, and is reckoned in Pounds, Shillings, Pence and Farthings. See tile Pence Tables. - ---- - -- * The reason of this rule is evident: For, addition of this money, as 1 in the pence is equal to 4 in the fart!ings; 1 in the shillings, to 12 in the pence; and 1 in the pounds, to 20 in the Fhillings; therefore carrying as di rected, is the arranging the money, arising from each column, properly in the scale of denominations: and this reasoning will hold good in the ad dition of conmbund numbers of any dletinin;atson whatever. COMPOUND ADDITION. 3) EXAMPLES. ~. $. d. What is the sum total of 471. 13s. 47 13 6 6d.-191. 2s. 9d. —14l. 10s. 11ld. Thus19 2 ind 121. 9s. 13d. 14 10 114 12 9 13 Answer, ~. 93 16 4j (2.) (4.) ~. s. d. ~. s. d.qr. ~. s. d. r. 17 13 11 84 17 5 3 30 11 4 13 10 2 75 13 4 3 15 10 9 1 10 17 3 50 17 8 2 1 0 1 1 8 7 20 10 10 1 3 9 8 3 3 3 4 16 50 4 6 3 1 s. d.. s. dr.. d. qr. 17 17 6 7 17 10 3 541 0 0 0 3 9 10 3 60 6 8 0 711 9 8 1 59 17 11 2 7 14 11 2 918 6 9 3 117 16 9 3 18 19 9 3 140 15 10 1?62 19 10 1 91 15 8 2 300 19 11 3 107 17 6 2 18 17 10 3 48 10 7 3 1 199 0 5 0 1 2 0 14 9 3 (8.) (9.) (10.) ~. s. d. ~. s. d. ~. s. d. 105 17 6 940 10 7 97 11 6~ 193 10 11 36 9 11 20 0 4 901 13 0 11 4 10 144 1 10 319 19 7 141 10 6 17 11 9 48 17 4 126 14 0 9 16 10^ 104 11 9 104 19 7 0 19 91 96 16 7 160 10 6 19 9 4 111 9 9 100 0 0 234 11 104 976 0 10 9 0 9 180 14 6 449 12 6 0 19 6 421 10 3i 29 10 4 120 0 8 341 10 4 11. Find the amount of the following ) ~. s. d. umsn, viz. 421. 13s. 5d. —11. lOs. —41. ITs. 8d.-131. Os. 7d.-19s. 4-d.-271. and 151. 6s.____ Ans.. 115 7 0~ 40 COMPOUND,I)DITION. 12. Add 3041. 5s. and Od. —341. 19s. 7d.-71. 18s. Sit -2471. Os. lld.-19s. 6d. lqr. and 451. together. Ans. ~. 640 3s. 51d. 13. Find the sum total of 141. 19s. 6d.-111. 4s. 9d.25. 10s. —41. Os. 6d.-31. 5s. 8d.-19s. 6d. and Os. 6d. Ans. ~. 60 Os. 5d. 14. Find the amount of the following sums, viz. Forty pounds, nine shillings, -—. s. d. Sixty-four pounds and nine pence, - - - Ninety-five pounds, nineteen shillings, - - Seventeen shillings and 4d. - - - - - Ans. ~. 201 6s. 1I a 15. How much is the sum of Thirty-seven shillings and sixpence, - Thirty-nine shillings and 4-d. - - - Forty-four shillings and nine pence, Twenty-nine shillings and three pence, Fifty shillings, ----------- Ans. ~. 10 Os. 10Od. 16. Bought a quantity of goods for 1251. 10s.; paid f( truckage, forty-five shillings, for freight, seventy-nine shil lings and sixpence, for duties, thirty-five shillings and tet pence, and my expenses were fifty-three shillings and nip pence; what did the goods stand me in? Ans. ~. 136 4s. ld. 17. Six men took a prize, and having divided it equally amongst them, each man shared two hundred and forty pounds, thirteen shillings and seven pence; how mucl money did the whole prize amount to I Ans. ~. 1444 is. 6d 2. TROY WEIGHT. Ib. oz. pwt. r. Ib. oz. pwt. gr. 16 11 19 l3 8 11 19 21 4 4 16 21 6 10 16 8 8 8 19 14 7 8 17 21 6 9 14 17 4 6 8 23 4 7 10 7 9 7 14 17 0 7 11 12 7 9 13 10 ----... —c COMPOUND ADDITION. 41 tt. r. lb. 2 27 1 1 17 4 2 26 6 1 13 3 3 15 6 2 16 3 9 gr. 9 1 17 3 2 9 6 1 17 4 0 16 5 2 12 6 1 10 rd. qr. na. 1 3 3 13 2 1 10 0 1 42 3 3 57 2 2 49 2 2 pk.qt. pt. 2 6 0 260 1 5 0 2 4 1 2 6 1 3 6 0 17 2 1 2 4 3 0 1 17 2 1 2 19 1 1 2 8 0 0 3 10 2 1 1 3. AVOIRDUPOIS lb. oz. dr. 24 13 14 17 12 11 26 12 15 16 8 7 24 10 12 11 12 12 A. APOTHECARIES S 3 9 gr. 10 7 2 19 6 3 0 12 76 1 7 9 5 2 12 6 1 0 16 9 3 2 19 5. CLOTH MEA E.E. qr. na. 44 3 2 49 4 3 06 2 3 84 4 1 07 0 0 61 2 1 6. DRY MEAS bu. pk. qt. 17 2 5 34 2 7 13 3 6 16 3 4 27 2 6 56 0 7?. WINE MEAS hhd. gal. qt. p 42 61 3 27 39 2 9 14 0 0 92 16 24 1 5 00 3 D 2 WEIGHT. T. cwt. qr. Ib. oz. dr 91 17 2 24 13 14 19 9 0 17 10 12 14 13 2 04 9 11 47 11 3 ]9 14 5 69 00 1 00 00 12 77 19 3 27 15 11 WEIGHT. t a 3 9 gr. 12 11 6 1 15 4 9 7 0 12 9 10 1 2 16 4 8 1 2 19 9 0 0 1 10 4 921 6 SURE. E. F. qr. na. 84 2 1 07 1 3 76 0 2 52 2 3 53 2 2 0o 2 3 URE. bu. pk. qt. pt. 25 3 7 1 64 2 6 1 43 0 4 0 52 3 5 1 94 2 3 0 54 3 7 0 SURE. t. tun.hhd. gal. qt 1 34 2 4 2 0 19 1 59 1 1 28 2 2 1 1 19 0 32 2 1 37 3 11 1 0 0 1 9 0 COMPOUND ADDITION. yds. ft. in. b.c 4 2 11 ' 3 1 8 ] 1 2 9 i 6 2 10 1 1 0 6 ] 3 1 7 ( 8. LONG MEASURE. A. m.fur. po. 1 46 4 16 I 58 5 23 S 9 6 34 l 17 4 18 l 7 3 15 ) 5 224 9. LAND OR SQUARE MEASUR acres. roods. rods. 478 3 31 816 2 17 49 1 27 63 3 34 9 3 37 T. JI. 41 43 12 43 49 6 4 27 acres. roods. rods. 856 2 18 19 3 00 9 1 39 1 3 00 0 i 27 10. SOLID MEASURE. cords. feet. 3 122 4 114 7 83 10 127 le. m. fur. pc 86 2 6 32 52 1 7 16 64 2 5 19 73 1 4 15 7 2 3 25 28 2 4 17 E. sq. ft. sq. tn 5 136 6 129 8 134 0 143 4 3a1 feet. inches. 13 1446 16 1726 3 8w 14 W2 Y. m. w. da. 57 11 3 6 3 9 2 3 29 8 2 5 46 10 2 4 10 7 1 2 12. CI S. 0~ f 3 29 17 14 1 6 10 17 4 18 17 11 6 14 18 10 11. TIME. Yr. da. h. m. sec 24 363 23 54 34 21 40 12 40 24 13 112 14 00 17 14 9 11 18 14 8 24 8 16 13 RCULAR MOTION. S. ~! / 11 29 59 50 0 00 40 10 9 4 10 49 4 11 6 10 COMPOUND SUBTRACTION. 43 COMPOUND SUBTRACTION, TEACHES to find the difference, inequality, or excess,.etween any two sums of diverse denominations. RULE.-Place those numbers under each other, which are of the imme denomination, ths less being below the greater; begin with the saust denomination, and if it exceed the figure over it, borrow as many units as make one of the next greater; subtract it therefrom; and to the difference add the upper figure, remembering always to add one to the next superior denomination for that which you borrowed. NOTE. The method of proof is the same as in simple subtraction, From Take Rem. Borrov Paid Remai unpa 1. (1.) ~. s. d. qr. 346 16 5 3 128 17 4 2 217 19 1 1 (4.) ~. s. d. red 44 10 2 36 11 8 no id EXAMPLES. Sterling Money. (2.) ~. s. d. r. 14 14 6 2 10 19 6 3 (3.) ~. s. d. 94 11 6 36 14 8 (5.) ~. s. d. gr. Lent 36 0 8 2 Received 18 10 7 3 Due to me From Ta e Rem. From Take Rem. (6.) ~. s. d. 5 0 0 4 19 11 (9). as. d. qr. 141 14 9 2 1f 13 10 2 (7.) ~. s. dqr. 7 11 1 2 4 17 3 1 (10.). s. d. 125 01 8 124 19 8 (8.) ~. s. d.qr. 476 10 9 1 277177 1 (11.) ~. s. d. qr. 10 13 7 1 0 963 44 COMPOUND SUBTRACTION. 12. Borrowed 271. 1 s. and paid 191. 17s. 6d. how nuc I remains due? Ans. ~7 13s. 6d. 13. How much does 3171. 6s. exceed 1781. 18s. 5 d.? Ans. ~138 7s. 6,d 14. From eleven pounds take eleven pence. Ans. ~10 19s. ld. 15. From seven thousand two hundred pounds, take 18i 17s. 6-d. Ans. ~7181 2s. 5-d. 16. How much does seven hundred and eight pound, exceed thirty-nine pounds, fifteen shillings and ten penc halfpenny? Ans. ~668 4s. 1 d. 17. From one hundred pounds, take four pence hall penny. Ans. ~99 19s. 7-d. 18. Received of four men the following sums of mone) viz. The first paid me 371. lls. 4d. the second 251. 16( 7d. the third 191. 14s. 6d. and the fourth as much as ti the other three, lacking 19s. 6d. I demand the whole sui received? Ans. ~165 5s. 4d. 2. TROY WEIGHT. lb. oz. piwt. From 6 11 14 Take 2 3 16 Rem. lb. oz. pwt. gr. 684 21 14 683 1 9 13 oz. pwt. r3. 4 19 21 2 14 23 lb. oz. pwt. g 44 9 6 ] 17 3 16 lb. oz. pwot.gr. 942 2 0 0 892 9 2 3 3. AVOIRDU lb. oz. dr. cwt. qr. lb, 7 9 12 7 3 1 3 12 9 5 1 15 T. cvet. qr. Ib. oz. dr. 810 11 0 20 10 11 193 17 1 20 12 14 POIS WEIGHT. T. cwt. qr. lb. oz 7 10 3 17 5 3 12 1 19 10 d? 1 T. cwt. qr. lb. or d 317 12 1 12 9 r 180 12 1 14 10 1 COMPOUND SUBTRACTION. 4. APOTHECARIES' WEIGHT. 45 fb 3 19 8 7 9 11 6 Vtd. qr. na. 3'5 1 2 19 1 3 Yd. qr.na. 813 3 1 174 1 0 Su. pk. qt. 65 1 7 14 3 4 ifaZ.t. pt. gi. 1 2 0 1 14 2 1 3 a B gr. 4 1 17 1 2 15 5. CLOTH MEASURE. E.E. na. 467 3 1 291 3 2 E.E. r. fna. 615 1) 1 316 226 2 2 6. DRY MEASURE. bu. pk. qt. 8 1 5 3 1 6 7. WINE MEASURE. hhd. al. g t. pt. 13 0 1 0 10 60 3 1 If s 3 gr. 35 7 3 1 14 17 10 6 1 18 E.Fl. qr. na 765 1 3 149 2 E.F. qr. na. 845 1 1 576 2 3 bu. pk. qt. pt 17 2 -3 0 6 2 6 1 1. hhd. gal. qt. pt 2 3 20 3 1 1 2 27 0 0 *Tgal. t. pt. 25 3 0 le. m. fur. po. 86 2 6 32 24 1 7 31 le. m.fur. po 9 2 0 1 11 8 hhd. Sal..pt. hh 612 3 0 52 75 37 1 1 25 8. LONG MEASURE. yd. ft. in. b.c. m. fur. po. 4 2 11 0 41 6 22 2 2 11 1 10 6 23 le. m. fur. po. 27 1 6 37 19 2 4 39 le. m.fur. po. 16 0 1 3 10 1 3 5 46 46 COMP~OUiND 61)'3IR1AC'TIUN. 9. A. roods. rods. 29 1 10 24 1 25 A. qr. rods. 540 0 25 119 1 27 LAND OR SQUARE MEASURE. A. r. po. 29 2 17 17 1 36 A. qr. rods. 130 1 10 49 1 1S 10. SOLID MEASURE. cords. ft. 72 114 41 120 I 1. TIME. q-. ft. sq. i 399 1: 143 125 tuns. ft. i 451 14 16 14 14 tuns.. 116 24 109 39 yrs. mo. wi. da. 54 11 3 1 43 11 3 5 2v. d. h. min. sec. 472 2 13 18 42 218 4 16 29 54 12. CIRCULAR S. S. ~ ' ft 9 23 45 54 3 7 40 56 yrs. days. h. min. sec. 24 352 20 41 20 14 356 20 49 19 w. d. h. min. sec. 781 1 8 23 21 197 3 12 42 53 MOTION. S. ~ ' f 9 29 34 54 7 29 40 36 QUESTIONS, Ahewing the use 'f Compound Addition and Subtraction NEW-YORK, MARCH 22, 1814. 1. Bought of George Grocer, 12 C. 2 qrs. of Sugar, at 52s. per cwt. ~32 10 t 28 lbs. of Rice, at 3d. per lb. 0 7 ( 3 loaves of Sugar, wt. 35 lb. at Is. Id. per lb. 1 17 11 3 C. 2 qrs. 14 lb. of Raisins, at 36s. per cwt. 6 10 ( Ans. 41 5 I QUESTIONS, &C. 47 2. U hat sum added to 171. 1 Is. 8-d. will make 1001. 1 Ans. 821. 8s. 3d. 3qr. 3. Borrowed 501. 10s. paid again at one time 171. 11s. and at another time, 9L 4s. Sd. at arJ:ther time 171. 9s., and at another time 19s. 61d. how much remains tinid? Ans. ~4 4s. 9'7d. 1. Borrowed 100/. and paid in part as follows, viz. at one ie 211. 11s. 6d. at another time 191. 17s. 4-d. at another me 10 dollars at 6s. each, and at another time two English ineas at 28s. each, and two pistareens, at 14',. each; vi much remains due, or unpaid? Ans. ~52 12. 8Sd. 5i. A, B, and C, drew their prize money as follows, viz. had 751. 15s. 4d. B hlad three times as much as A, king 15s. 6d. and C, had just as much as A and B both; ty how much had C I Ans.;302 5s. 10d. 3. I lent Peter Trusty 1000 dols. and a,',erwards lent n 26 dols. 45 cts. more. He has paid me at one time 1 dols. 40 cts. and at another time 416 dols. 09 cents,;ides a note which he gave me upon James Paywell, for 3 dols. 90 cts.; how stands the balance between us? Ans. The balance is $105 06 cts. due to me. /. Paid A B, in full for E F's bill on me, for 1051. 10s.. I gave him Richard Drawer's note for 151. 14s. 9d. ter Johnson's do. for 301. Os. 6d. an order on Robert.aler for 391. 11s. the rest I make up in cash. I want to ow what sum will make up the deficiency? Ans. ~20 3s. 9d. J. A merchant had six debtors, who together owed!ahim 171. 10s. 6d. A, B, C, D, and E, owed him 16751. 1. of it; what was F's debt? Ans. ~1241 16s. 9d. ). A merchant bought 17 C. 2 qrs. 14 lb. of sugar, of ich he sells 9 C. 3 qrs. 25 lb. how much of it remains und? Ans. 7C. 2 qrs. 17 lb. 10. Frm a fashionable piece of cloth which containes yds. ~ na. a tailor was ordered to take three suits, eace lds. 2 qrs. how much remains of the piece? Ans. 32 yds. 2 qrs. 2 na. \1. The war between England and America csmmenaed 48 COMi'OUND MULTIPLICATION. April 19, 175, an I a general peace took place January 20th, 1783; how Icng did the war continue? Ans. 7 yrs. 9 mo. 1 d. COMPOUND MULTIPLICATION. COMPOUND Multiplication is when the Multiplicail consists of several denominations, &c. 1. To Jultiply 'Federal Money. RuLE.-Multiply as in whole numbers, and place the separatrix 81 many figures from the right hand in the product, as it is in the rmui tiplicand, or given sum. EXAMPLES. $ cts. $ d. c. n. i. Multiply 35 09 by 25. 2. Multiply 49 0 0 5 by 9I' 25 97 17545 343035 7018 441045 Prod. $877, 25 $4753, 4 8 5 $. cts. 3. Multiply 1 dol. 4 cts. by 305 Ans. 317, 20 4. Multiply 41 cts. 5 mills by 150 Ans. 62, 25 5. Multiply 9 dollars by 50 Ans. 450, 00 6. Multiply 9 cents by 50 Ans. 4, 50 7. Multiply 9 mills by 50 Ans. 0, 45 8. There were forty-ole men concerned in the paymei of a sum of money, and each paid 3 dollars and 9 mills how much was paid in all? Ans. $123 36 cts. 9 mills. 9. The number of inhabitants in the United States five millions; now suppose each should pay the trifiin sum of 5 cents a year, for the term of 12 years, toware a continental tax; how many dollars would be raised thenr by? Ans. Three millions Dollars. 2. To IMultiply the denominations of Sterling Amone Weights, lMeasures, A ' ~ z~/"S —'' V LCI(AXAL F R A TI UN S. 81 piace of huudredths, by 5, it the shillings be odd; and the third plaes by 1 when the farthingrs exceed 12, and by 2 when they exceed 36. EXAM ILE S. I. Find the decimal of 7s. 93d. by inspection. 5 for tile odd slitIbII] rs. "9=the farthiugs in, 94d. 2 for the excess of 36. ~.,391=decimal required. 2. Find the decimal expression of [Cs. 411d. and 17s. S.'d. Ans. ~,819, anid L,SS5 3. Write down, ~47 13 101 in. a decimal expression. Ans. ~47,943 4. Reduce ~1I Ss. 2Ad. to ani equivalent decimial. Ans. LI,40 P r-OBLEM It. 41.r and easy nmctltod to f ind the value of any decimal of a pound by inspection. Ru.-Double the first figrure, or place of' tenths, for -shillings, antl (the svecnd figrure be 5, or more than.5, reckon another shilling; thenr, ifter this 5 is deduc:ted, call the figrures in~the second and third places o many farthings, abating I when they are above 12, and 2 when hGove 36, and the rcsult will be the answer. NOTE.-When the decimal has but 2 figures, if any thing emainis after the shillings arc takens out, a cipher must bo:-nnexed to the left hand, or supposed to be so. EXAMPLES. L. Find the value of LZ.,679 by inspection. 12Als=double of 6 1 for the a' in, the second place which is to be [deducted out of 7 dd 7 d 1-'d=,29 farthings remiain to be added. )e&ulw 4.fo the excess of 12.. 2. Otind the value of ~.,876 by inspection. Ans. 17s. 61d. 3. Findl thevalue of ~.,842 by inspection. Ans. 16s. 104. 4. Find the vslue of.C 097 by inspection. Ans.1is. I ld. 82 REiLtUC'1JON O.)F' ( L:(.iJ1iN(.;Eb. REDUCTION OF CURRENCIES. RULES for reducizng the Currellcies of the several United Statesc into Federal Money. CASE I. To reduce the currencies of the different states, where a dollar is an even number of shillings, to rederal Money. They are New-England, - rk, and Virginia, North Carolina. } ' Kentucky, and Tennessee. RULE.-1. When the sum consists of pounds only, annex a cipnez to the pounds, and divide by half the number of shillings in a dollar, the quotient will be dollars.t 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 de.. cimal of a shilling; annex said decimal to the shillings, with a decimal point between, then divide the vlwhole by the number of shillings coP. tained in a dollar, and the quotient will be dollars, cents, mills, &c. EXAMPLES. 1. Reduce 731. New-England and Virginia currency, c ederal money. 3)730 |- $ els. $243 —=243 33a '2. Reduce 451. 15s. 71d. New-England currency, to fed I, 20 [ral money. d. A dollar=6)915,625 12)7,500 $152,604+ Ans.,t'25 decirmas. - * Formerly the pound was of the same sterling value in all the colonies as in Great-Britain, and a Spanish Dollar worth 4s. 6d.-but the legisla tures of the different colonies emitted bills of credit, which afterwards deo preciated in their value, in some states more, in others less, &c. Thus a dollar is reckoned in.New-England, J1ew-Jersey ) SouthVirginia,. Pennsylvania, 7 6dCarolina, 4s 8 Kentucky, and ( Delaware, and * and Tennessee J aryland, Georgia, J NJew —YorIk, and } Xorth Carolina, 8sl t Adding a cipher to the pounds, multiplies the whole by 1!, riignl them into tenths of a pound then because a dollar is just thr 'e o pound N E. currency, dividing those tenths by $, brings them intodolla a &c. See Note, page 78. REDUCTION OF CURRENCIES 8 83 NOTE. I farthingis,25 w-hich annex to the pence, and 2 -, "5O divide by 12, you will have the 3 - 75) decimal required. 3. Reduce 3451. l0s. II1~d. New-Hampshire, &c. currencr, to Spanish milled dollars, or federal money. ~345 10 11-1 20 d. 6)6910,9375 12.)11,2500 nC3.y._. $1 151,8229~+ Ans. 4. Reduce lO5l. 14s. 31d. New-York and North-Caroli7.at currency, to federal money. ~105 14 31 d 210 12)3,7500 k dollar=8)21 14,3125,3125 decimal. $264,289 06 Atns. O r $ d cm. TqO4, 5. Reduce 4311. New-York currency to federal money. I'lis beilig poun1ds ony* 4)4310 - - $ Cts. Ans. $1077i 1077,50 6. Reduce 281. 1 Is. 6d. New-England and Virginia curency, to federal money. Ans. $93, 25 cts. 7. Change 4631. 10s. 8d. New-England, &c. currency federail nmoney..Ans. $1545, llcts. Inm.+ 8. Reduce 351. 19s. Virginia, &e. currency, to federal aon1ey. Alns. $119, 83 Lts. 3 mf. + 9. Reduce 2141. l0s. 71 d. New-York, &C. currency, to Federal money. Ans. $536, 32 cts. 8 m.+ 10. Reduce 3041. 1 Is. 5d. North-Carolina, &c. currency, federdl1 money. Ans. $761 42 cts. 7 m.+ 1i. Changre 2191. 11s. 73d. New-England and Virgiuia.irreapy, to federal mioney. Ans. $731 94 cts.+ A do~ar is 8s. II n tis currency-,4==4-1O of a pound; therefore, multi. y Ity 10,~ sad divide by 4. brings the pounds into dollars, &c. 6 I 81 REDL~LTION UF CL!IRENL"I&S. 1'2. Chiange, 2411. New-Engrltnd, &c. currency, into fe. deral money. Ants. $303, 33 cts. 13. Bring 201. 18s. 51d. New-E nglanrd currency, into dtollars. Ans. $69, 714 cts. 61 nm.-j-~ I.I. Pleduee 4681. New-York currency to federal money Ans. $11 70 Its. leducee 117s. 9.9d. lNew,-Yorlk, &c. currency, to dlollsc.i S. 2'2 22 t S. 6, 5) m. 16. Borrowed 10 English crowns, at Os. 8d. each, bo7 nvinv11 dollars, at 6s. eaeh, will pay the debt? iins. $11, 11 ets. I nt. NCTrE.-There are several shiort practical methods of re dnueinig New-England and New-Yorh currencies to F edera&, Money, for which see the Appendix. CASE II. To reduce the currency of New-Jersey, Pennsylvanita, Delaware, and Maryland, to Federal Money. RULtE.. —MUltipl1y the given sumn by 8, and divide the product by 2 and( the', quotient. will be dollars, &-e.* EXAMPLES. I.Reduce 2451. New-Jersey, &c. currency, to feder.i ~2,.1A5 x 3-iqoo and 19600~8'3$653j-=$653, 331cts. NoxiE. -When there are shillings, pence, &c. in the gi-T i snreduce themn to the decimal of a pound, then rmultil 1 ard dtivide as above, &C. R2 educe 361. 11s. 8(/. New-Jersey, &e. currency, ti redeial money. ~4(36,5S54 dccinzal vaiue. 8 3)292,6332(07,563106 Ans. ANSWERS. 3. ReduLce 240 0 0 to foderal money 640 00 4. Reduce 1251 3 0 334 40 5. 1W~duce 99 7 641 265 00 5+ 0, -duee 100Ot0 0 266 66 6+ 7. Re4duce 25 3 7 67 14 4 3. Reduce 0 17 9 2 36 6,6 *A dollar is 7s. 6d.==90d. in this currency=90.240-3-8 ofa pound; thee, froe multiplying by 5, nanddividing by 5, gives the dofllrs, cents, &c. RIEDUCTION ()F CURREN CIE$. 85 CASE III. To reduce tihe currency of South-Carolina and Georgia, to Federal Money. RULE.-IMultiply the given sum by 30, and dividoe the product by 7, the quotient will be the dollars, cents, &c.* EXAMPLES. 1. Reduce 1001. Soutlh-Carolina and Georgia currency, u federal money. 1001. x30-=, 000; 3000 -7=$428,5714 Ans. 2. Reduce 541. l6s -d. Georgia currency, to federal loney. 54,8406 decimal expression. 30 Ans. U. Reduce 14 1. Reduce 19 5. Reduce 417 6. Reduce 140 7. Reduce 160 8. Reduce 0 9. Redtuce 41 7)1645,2180 2.35,0311 S. d. 14 S to fed{ 17 61 14 6 10 0 0 0 11 6 17 9 I ANSWERS. $ cts. m eral money, 405 99 8+ 85 I8 7+ 1790 25.-.. 602 14 2+ 685 71 4 2 46 4+ 179 51 418I CASE IV. ro redtuce the currency of Canada and Nova-Scotia to Federal Money. RPuLE.-Multiply the given sum by 4, the product will be dollars. NoTE.-Five shiliings of this currency are equal to a d}!ar; consequently 4 dollars make one pound. EXAMPLES. 1. Reduce 1251. Canada and Nova-Scotia currency, to federal money. 125 4 Aim. $500 * 4s. 8d. or 56d. to the dollar=: -==~ of a pound; &erefore 30- 7. 86 86 ~REDULCTION OF COIN 2. Reduce 551. 10s. 6d. Nova-Scotia currency, to dollars, 55,525 decimal valute. 4 - c ts. Ans. $222, 1O00-222 10 ANSWERS,, 3. Reduce 2411. 18s. 9d. to federal money $967 75 4. ReduJce 58 13 61- 234 70 0. Reduce 5208 17 S 2115 53 6. Reduce 1 2 6 4 50 7. Reduce 224 19 (0 899 s0 8. Reduce 0 13 111 2 79 REDUCTION OF COIN. RULES for reducing the Federal Money to the currencies of the, scv -eral United States. To reduce Federal Money to the currency of New-England, RULE.-MlUltiply the given sum by,3 and ths! KeVigntuck, ad product will be pounds, and decimals of 9 ~1Ternnessee. p )ound. New-ork,~tnd BLLE.-Multiply the given sum by,4, and the 2. Newrihorklina. product wvill ba pounds, and decimals of (Aew.Jesei, ~RUL.-Multdiply the given sun, by,3, andd-. D clattreanda vide thle product by 8, and the qiotient w 11 Dclalawoand. be pounds, and decimai Ls of a pouni. South-Carolina, RuLE.-Multiply the given sum b;,7 ai'r S4nwe n ondannecmldo divide b~y 3, the quotient will be tJ~p ~ Geogia. pound. Examples in the foregoing Rules. I Reduice $1.520, 60 cts. to New-Enigland currency. ~45, 780 Ans.=Z45 15s5. 7 2d. 20 But the value of any decimal of -. a pound, mnay be found by inspec15, 600 tion. See Problem 11. page Si. 7, 200 REDUCTION OF COIN. 87 2. In $196, how many pounds, N. England currency,3 ~58,8 Ans. ~58 16 3. Reduce ~629 into New-York, &c. currency.,4 ~251,6 Ans.=-~251 12 I. Bring $110, 51cts. 1 m. into New-Jersey, &c. currency $110,511,3 Double 4 makes 8s. Then 39 farthings 8)331,53b are 9d. 3qrs. See Problem II. page 81. ~41,441 Ans.= —tl 8s. 93d. by Inspection. 5. Bring $65, 36 cts. into South-Carolina, &c. currency.,7 3),45,752 ~15,250=-~15 5s. Ans. ANSWERS. $ cts. ~ s. d. 6. Reduce 425,07 to N. E. &c. currency. 127 10 5 + 7. Reduce 36,11 to N. Y. &c. currency. 14 8 10-1+ 8. Reduce 315,44 to N. J. &c. currency. 118 5 9}-+ 9. Reduce 690,45 to S. C. &c. currency. 161 2 1,2 To reduce Federal Money to Canada and JNova-Scotia currency. RULE.-Divide the dollars, &c. by 4, the quotient will be pounds, %nd decimals of a pound. EXAMPLES. 1. Reduce $741 into Canada and Nova-Scotia currency. $ cts. 4)741,00 ~185,25=-~185 5s. 2 Bring $311, 75 cts. into Nova-Scotia currency. $ cts. 4)311,750 ~77,9375=-~77 18s. 9d. 3. Bring $2907, 56 cts. into Nova-Scotia currency. Ans. ~726 17s. 9_d. Reduce $2114, 50 cts. into Canada currency. Ans. ~528 12s. 6d. it RLES F~UR ItEDUCING CUfRRENC1ES. RULES for redtecing the Currencies of the several United States, ede Canada,.Nova Scotia, and Sterting, to the par of all the others jjj~ See. the given currency in the left tend coluinti, and then cast y' u eye to the rig-ht hand, till you cotni under the required currency and cvL -vill have the rue. icw- En- Wiew.Teey, lanId, Vir- Pennsylva- IWew-York. tSouth-Ca- Canada, giiKen nia, Dela jand J'rt~s rioina, an a IdSterling anda muare, and Carenoli. Geria.vna~cotta nninesste. Marvandt. S -g-Add one Add one Multiply the Muttiply the Deduct ont land, Vir-~ tburth to &thlhird to the fiven oem, given sum fourth front ginia, Kim-l given SUM, Igiven sum. by7, anid ti- I by 5, and di- the given Cocky, aindi vde the pro-I videdtepo un Tennessee. I duct by t. de yi NewTersew,i Deduct oral~ AI t~ tpyte Deduct oneitil vannjjy- ihfo' fifteenth to given sutm d rm~rc n ware, andisurn- 8ivenM di iet tile sum gVen e he Marvland. I Iio 45 otb.____Deduct onte Deduct one Multitply the Suluiply the Multiply 04 and Naorth- the New- from the N, Ey7 ht e d i- by 5. and di- liy 9, -and di Carolina. York, &e. York. Vode the pro- yule th epro- yule I hej tra dutby N'. duet by 8. ducl by. ~6. Multi-ply the Muttiply the' Multiply til Multiply t i- From lb Souotk-ca-Igiven u eismcv slim given 011,11 W ill SUTI(I'V)Sn roline, andl by 9, and di- by 45, antdtiy 12 used hy1,azid ii titu Georgta. vude the pro- idividte tteicvitide the 'uluvid'fe theniotyduct by7.- grad uei by prductbY.i, Product by eighth. Add one Add otto Multiply thae Deductone Deduct oa a Canada, fifth to the half to the geen sumt fifteenth ten th frc m and Canada,&c. Canada by 8, and d i- Ifrom the gi - the gfivabe,Vovus~cotia BUM. vuethe Vr- va uSUM. duct by a. To the En- Multiply thek~lultiply thelTo the Eng- Add one gush sum Eturglish mo-. English turn bos money ni orth to the Starling. jadd one ney by 5,and by1, anut adl oset gives sum. third. vie the twenty-seprdcIW rdutb9.vnh REDUCTION 0( COIN. 89 Application of the Rules contained in the foregoing Table. EXAMPLES. 1. Reduce 461. 10s. 6d. of the currency of New-Hampshire, into that of New-Jersey, Pennsylvania, &c. ~. s. d. See the rule 4)46 10 6 in the table. +11 12 7 -Ans. ~58 3 12 2. Reduce 251. 13s. 9d. Connecticut currency, to NewYork currency. ~. s. d. 3)25 13 9 By the table, +-, &c. +8 11 3 Ans. ~34 5 0 3. Reduce 1251. 10s. 4d. New-York, &c. currency, to South-Carolina currency. ~. s. d. Rule by the table, 125 10 4 x7, — by 12, &c. 7 12)878 12 4 Ans. ~73 4 4 -4. Reduce 461. 1Is. 8d. New-York and N. Carolina cur-,ency to sterling or English money. ~ s. d. 46 11 8 9 See the table. 16=4 X 4)419 5 0 X given sum by 4)104 16 3 9,-by 16, &c. Ans. ~26 4 0f To reduce any of the different currencies of the several Statesinto each other, at par; you may consult the preceding table, which will give you the rules. MORE EXAMPLES FOR EXERCISE. 5. Reduce 841. 10s. 8d. New-Hampshire, &c. currency, into New-Jersey currency. Ans. ~105 13s. 4d. 6. Reduce 1201. 8s. 3d. Connecticut currency, into NewYoik currency. Ans. ~160 ufs. Od. n2 90 RULK OF THIK RE&. LRKCT. 7. Reduce 1201. 10s. Massachusetts currency, into South Carolina and Georgia currency. Ans. ~93 14s. 51d. 8. Reduce 4101. 18s. lid. Rhode-Island currency, int4 Canada and Nova-Scotia currency. Ans. ~342 9s. Id. 9. Reduce 5241. 8s. 4d. Virginia, &c. currency, into ster ling money. Ans. ~393 6s. 3d. 10. Reduce 2141. 9s. 2d. New-Jersey, &c. currency, inte N. Hamp. Massachusetts, &c. currency. Ans. 1711. 1 is. 4d. 11. Reduce 1001. New-Jersey, &c. currency, into New York and North-Carolina currency. Ans. 1061. 13s. 4d. 12. Reduce 1001. Delaware and Maryland currency into sterling money. Ans. 601. 13. Reduce 1161. 10s. New-York currency, into Connec. ticut currency. Ans. 871. 7s. 6d. 14. Reduce 1121. 7s. 3d. S. Carolina and Georgia curren. cy, into Connecticut, &c. currency. Ans. 1441. 9s. 33d. 15. Reduce 1001. Canada and Nova-Scotia currency, into Connecticut currency. Ans. 1201. 16. Reduce 1161. 14s. 9d. sterling money, into Connecticut currency. Ans. 1551. 13s. 17. Reduce 1041. 10s. Canada and Nova-Scotia currency, into New-York currency. Ans. 1671. 4s. 18. Reduce 1001. Nova-Scotia currency, into New-Jer sey, &c. currency. Ans. 1501. RULE OF THREE DIRECT. THE Rule of Three Direct teaches, by lhaving three 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 same name or kind; one of which must be the first number in stating, and the other the third number; consequently the first and third numbers must al ways be of the same rame, or kind; and the other number, which is of the same kind with the answer, or thing sought, will always possess the second or middle place. 2. The third term is a demand; and may be k-ow by these or the like words before it, viz. What will! WIB lt 1 How many? How far? How long? or, How much?*. e. RULE OF THREE DIB 'CT. 91 RULE.-1. State the question; that is, pl'.ce the numbers so that the first and third terms may be of the sanm kind; and the second ermn of the same kind with the answer, or thing sought. 2. Bring thi first and third terms to the same denomination, and reduce the second term to the lowest name mentioned in it. 3. Multiply the second and third terms together, and divide their product by the first term; and tie quotient will be the answer to the question, in the same denomination you left the second term in, whichl may be brought into any other denomination required. The method of proof is by inverting the question. [NOT E.-The following methods of operation, when they can be used, perform the work in a much shorter manner than the general rule. I. Divide the second term by the first; multiply the quotient into the third, ind the product will be the answer. Or, 2. Divide the third term by the first; multiply the quotient into the second, mnd the product will be the answer. Or, 3- Divide the first term by the second, and the third by that quotient, and lhe last quotient will be the answer. Or, 4. Divide the first term by the third, and the second by that quotient, and lhe last quotient will be the answer.] EXAMPLES. 1. If 6 yards of cloth cost 9 dollars, what will 20 yards,ost at the sale s rate? Yds. $ Yds. Here 20) yards, which moves the 6: 9: 20 -nestion, is the third term; 6 yds. 9 he same kind, is the first, and 9 lollars the second. 6)180 2. If 20 yards cost 30 dols.:hlat cost 6 yards? Ydls. $ Yds. 20: 30:: 6 2,0)18,0 Ans. $30 3. If 9 dollars will buy 6 yards, how many yards will 30dols. buy? $ yds. $ 9: 6:: 30 6 9) 80 Ans. $9 Ans. 20y 4. If 3 cwt. of sugar cost 1l. 8s. what will 11 cwt. I q 4 lb. cost? 3 cwt. 8I. 8s. C. qr. lb. lb. s. 12 20 11 1 24 As 336: 168:: 12Q41b. --- 4 168 36 lb. 1 68s._ 45 [Carried up..)272 rd Ire 9t RULE OF THREE DIRECT. 45 10272 28 7704 -- 1284 364 (2, j 92 336)215712(64,2.-. 2016 1284 -- 321 2: 1411 Ans 1344 672 672 5. If one pair of stockings cost 4s. 6d. what will 19 do zen pair cost? Ans. ~51 6s. 6. If 19 dozen pair of shoes cost 511. b. what will cn, pair cost? Ans. 4s. 6d. 7. At 101d. per pound, what is the value of a firkin ( butter, weight 56 pounds? Ans. ~2 9s. 8. How much sugar can you buy for 231. 2s. at 9d. pec pound? Ans. 5 C. 2 qrs. 9. Bought 8 chests of sugar, each 9 cwt. 2 qrs. what d.. they come to at 21. 5s. per cwt.? Ans. ~171. 10. If a man's wages be 751. 10s. a year, what is that 'i calendar month Ans. ~6 5s. 10d. 11. If 44 tuns of hay will keep 3 cattle over the winter; how many tuns will it take to keep 25 cattle the same time( Ans. 374 tuns. 12. If a man's yearly icome be 2081. Is. what is that a day? Ans. 11 s. 4d. 3-jV qrs. 13. If a man spend 3s. 4d. per day, how rmuch is that a year? An*s., s. 8d. 14. Boarding at 12s. 6d. per week, lmw ' will 321. 10s. last me?.As. I year. 15. A owes B 34751. but B compoundswit him for 13s 4d. on the pound; pray what must he reeii fir his debt I A,.:~*' 13s. 4d. 16. Agoldrnith sold~a tankard for 128. at 5s. 4d. per oz.whatwas ne weight of the tankardl Ans. 2b. 8oz5pwt. 17. If 2 3wt. 3 qrs. 21 lb. of sugar cot 6lt l& 8d. whau cost 8S ewt Ass. ~73. aULt OF THREE DIRECT. 93 18. Bought 10 pieces of cloth, each piece containing 91 yards, at l s. 4-d. per yard; what did tlhe whole come tc I Ans. ~55 9s. 0dd, FEDERA.L MONEY. NOTE 1. You must state the question, as taught in the Rules foregoing, and after reJucing thle first and tlird terms to tle same name, &c. you raay multiply and divide according to the rules in decimals; or by the rules for multiplying and dividing Federal Money. EXAMPLES. 19. If 7 yds. of cloth cost 15 dollars 47 cents, what will 12yds cost Yds. $ cts. yds., 7: 15,47;: 12 12 7)185,64 Ans. 26,52 —=$6, 52 cts. But any sum in dollars and cents may be written down ts a whole number, and expressed in its lowest denominaion, as in tie following example: (See Reduction of Fedeal Money, page 62.) 120. What will I qr. 9 lb. sugar come to, at 6 dollars 45 i. percwt. qr. lb. Ib. cts. Ib. 1 9 As112 645::37 28 * 37 37 b. 4515 1935 - cts. 112)2:365(213 + Ans=:2,13. 224 146 112 345 336 9 '394 (34 aRULE UP TH1REE~ WriLECT. NOTE 2. When the first and third numbers are federa money, you mnay annex ciphers, (if necessary,) until yot make their decimal places or figures at the right hand o; the separatrix, equal: which will reduce them to a like denomination. Then you may multiply and divide, as in whol( numbers, and the quotient will express the answer in th( 1'ast denomination mentioned in the second, or middle term. EXAMPLES. 21. If 3 dols. will buy 7 yds. of cloth, how many yds. can I buy for 120 dols. 75 cts.? cts. yds. cts. As 300: 7:: 12075 7 yds. 300)84525(2S13 Ans. 22. If 12 lb. of tea cost 6 dols. 600 78 cts. and 9 mills, what will 5 lb. cost at the same rate? 2452 lb. mills. Ib. 2400 As 12: 6789: 5 5 525 - 300 12)33945 cts.m. 225 Ans. 2828+mills.-2,2,8. 4 900(3 ras. 900 $ cts. 23. If a man lay out 121, 23 in merchandise, and tnereby i)bs $39 51 cts. how much will he gain by laying out $12 -'it the same rate? Cents. Cents. Cents. As 12123: 3951:: 1200 1200 ts. $ eta. 12123)4741200(~91=3,91 Ans. 36369 110430 109107 Carried up, RULE, OF THREE DIRECT. 13430 12123 1107 24. If the wages of 15 weeXs come to $64 19 cts. what is year's wages at that rate? Ans. $222, 52 cts. 5m. 25. A man bought sheep at $1 11 cts. per head, to the nount of 51 (lols. 6 cts.; Ihow many sheep did he buy? Ans. 46. 26. Bought 4 pieces of cloth, each piece containing 31 irds, at 16s. 6d. per yard, (New-England currency;) what )es the whole amount to in federal money? Ans. $341. 27. When a tun of wine cost 140 dollars, what cost a lart? Ans. 13 cts. 8-,8- m. 28. A merchant agreed with his debtor, that if he would Ly him down 65 cts. on a dollar, he would give him up a te of hand of 249 dols. SS cts. I demand what the debtor ust pay for his note? Ans. $162 42 cts. 2nm. 29. If 12 horses eat up 30 bush. of oats in a week,how many ishels will serve -45 horses the same time? Ans. 112- bush. 30. Bought a piece of cloth for $4S 27 cts. at $1 19 cts. per.; how many yds. did it contain? Ans.40 yds. 2 qrs.-i3-. 31. Bought 3 hlids. of su,ar, each weighing 8 cwt. 1 qr. lb. at $7 2(6 cts. per cwt. what come they to? Ans. $182 1 cI. 8 m. 32. What is the price of 4 pieces of cloth, the first piece ntaininlg 21, the second 23, the third 24, and the fourth yards, at 1 dollar 43 cents per yard? Ans. $ 135 85 cts. 21 -+23 - 24 +- 27=95 yds. 33. Bought 3 lihds. of brandy, containing 61, 62, 62' lions, at 1 dollar 38 cts. per gallon, I demand how much -y amnount to? Ans. $255 99 cts. 34. Suppose a gentleman's income is $1836 a year, and spends $3 49 cts. a day, one day with another, Low much 1i he have saved at the year's end? Ans.$562, 15 cts. 35. If my horse stand me in 20 cts. per day keeping, at will be the charge of 11 horses for the year, at that et Ans. $803. RULTE (F TVtRlEE DIRECT. 36. A merchant bought 14 pipes of wine, and is allowet 6 months credit, but for r idy money gets it 8 cts. a gallon cheaper; how much did he save by paying ready motey i -in? s. $ 141, 12 cts. Exanzples prowmiscuousiy placed. 37. Sold a ship for 5371. anid I owned f of her; what was rmy part of tile money? A vs. ~201 7s. (id. 38. If -'? of a ship cost 781 dollars 25 cents, what is the whole worth? As 5: 781,25:: 16: $2500 Ans. 39. If I buy 54 yards of cloth for 311. 10s. what did il cost perEll English? Ans. 14s. 7d. 40. Bought of Mr. Grocer, 11 cwt. 3 qrs. of sugar, at 8 dollars 12 cents per cwt. and gave him James Paywell'a note for 191. 7s. (New-England currency) the rest I pay in cash; tell me how many dols. will make up the balance? Ans. $30, 91 cts. 41. If a staff 5 feet long cast a shade on level ground t feet, what is the height of that steeple whose shade at thit same time measures 181 feet? Ans. 1 13 ft. 42. If a gentleman have an income of 300 English gli. neas a year, how much may he spend, one day with ann ther, to lay up $500 at the year's end? Ans. $2, 46 ts. 5 m. 43. Bought 50 pieces of kerseys, each 34 Ells Flemish, a 8s. 4d. per Ell Elglish; what did the whole cost? Ans. ~425. 44. Bought 200 yards of cambrick for 901. but being da maged, I am willing to lose 71. 10s. by the sale of it; whn must I demand per Ell Eng!.lish? Ans. 10s. 33d. 45. How many pieces of Holland, each 20 Ells Flemisl, mayl hlave for 23l.8s. at 6s.6d. per Ell English? Ans. 6pcs. 46. A merchant bought a bale of cloth containing 240 yds at the rate of $7u- for 5 yds. and sold it again at the rate c $11- for 7 yards; did he gain or lose by the bargain, and hot much? Ans. He gained $25, 71 cts. 4 m. + 47. Bought ajpipe of wine for 84 dollars, and found it ho leaked out 12 gals.; I sold the remainder at 12c ets. a pin what did I gain or lose? Ans. I gained $30.:B. A gentleman bought 18 pipes of wine at 12s. 6( (New-Jersey currency) per gallon; how many dollars wi pay the purchase? Ans. $3780 It L OF. THREE INVERSE. 17 49. Boughlt a quantity of plate, weighing 15 lb. 11 oz. 18 wt. 17 gr. how many dols. will pay for it, at the rate of 1 s. 7d. New-York currency, per oz.? Ans. $301, 50, cts. 2k-m. 50. A factor bought a certain quantity of broadcloth and drug,,et, which together cost 811. the quantity of broadcloth was 50 yds., at 1Ss. per yd., and for every 5 yds. of broad cloth lie had 9 yards of drugget; I demand how many yds of drugget he lbad, and what it cost him per yard Ans. 90 yds. at 8s. per yd. 51. If I give 1 eagle, 2 dols. 8 dimes, 2 cts. and 5m. for675 t~ ps, how many tops will 19 mills buy? Ans. I top. 52 Whereas an eagle and a cent just threescore yards did buy, How many yards of that same cloth for 15 dimes had I? Ans. 8 yds. 3 qrs. 3 na.+ 53. If the legislature of a state grant a tax of 8 mills on the dollar, how much must that man pay who is 319 dols. '5 cents on the list? Ans. $2, 55 cts. 8 m. 54. If 100 dols. gain 6 dols. interest in a year, how much will 49 dols. gain in the same time? Ans. $2, 94 cts. 55. If 60 gallols of water, in one hour, fall into a cistern contaililng 300 gallons, and by a pipe in the cistern 35 gal-!,ls run out in arn lour; in what time will it be filled? Ans. in 12 hours. 56. A and B depart from the same place and travel the same road; but A goes 5 days before B, at the rate of 15 niles a day; B follows at the rate of 20 mile a day; what lisiance must he travel to overtake A? Ans. 300 miles. RULE OF THREE INVERSE. TI E Rule of Three Inverse, teaches by having three 'umbers given to find a fourth, which shall have the same Jroportion to the second, as the first has to the third. If mor requires more, or less requires less, the question ~elongs to the Rule of Three Direct. But if more requires less, or less requires more, the quesion belongs to the Rule of Three Inverse; which may al-;ays be known from the nature and tenor of the question. 'or exnampie: 98 RULE OF T7REE INVERSB If 2 men can mow a field in 4 days, how many dlys will it require 4 men to mow it? men days men 1. If 2 require 4 how much time will 4 reqaire Answer, 2 days. Here more requires less,.iiz. the more men the less time is required. men days men 2. If 4 require 2 how much time will 2 require) Answer, 4 days. Here less requires more, viz. the less the number of men are, the more days are required-therefore the question belongs to Inverse Proportion. RLE. —1. State and reduce the terms as in the Rule of Three Direct. 2. Multiply the first and second terms together, and divide the pro duct by the third; the quotient will be the answer in the same deno mination as the middle term was reduced into. EXAMPLES. 1. If 12 men can build a wall in 20 days, how many men can do the same in 8 days? Ans. 30 men 2. If a man perform a journey in 5 days, when the day is 12 hours long, in how many days will he perform it wnhen the day is but 10 hours long? Ans. 6 days. 3. What length of board 7-, inches wide, will make a square foot Ans. 19} inches. 4. If five dollars will pay for the carriage of 2 cwt. 150 miles, how far may 15 cwt. be carried for the same moneyl Ans. 20 miles. 5. If when wheat is 7s. 6d. the bushel, the penny loaf will weigh 9 oz. what ought it to weigh when wheat is 6s. per bushel? Ans. 11 oz. 5 pwt. 6. If 30 bushels of grain, at 50 cts. per bushel, will pay a debt, how many bushels at 75 cents per bushel, will pay the same? Ans. 20 bus7hls. 7. If 1001. in 12 months gain 61. interest, what principal will gain the same in 8 months? Ans. ~150. 8. If 11 men can build a house in 5 months, by working 12 hours per day-in what time will the same number of men do it, when they work only 8 hours per day? Ans. 71 monnths. Q W~ number of men must be employed to finish in 5 du wil 15 men would be o0 days about? Alns. i9 men. PRAt TIC 10. Suppose 650 men are in a garrison, and their proviL sions calculated to last but 2 months, how many men mugs leave the garrison that the same provisions may be sufficient for those who remain 5 months? Ans. 390 men. 11. A regiment of soldiers consisting of 850 men are to be clothed, each suit to contain 3- yards of cloth, which is [L yds. wide, and lined with shalloon 3 yd. wide; how many yards of shalloon will complete the lining? Ans. 6941 yds. 2 qrs. 21 na. PRACTICE. PRACTICE is a contraction of the Rule of Three Direct, when the first term happens to be a unit or one, and is a concise method of resolving most questions that occur in trade or business where money is reckoned in pounds, shilings and pence; but reckoning in federal money will render this rule almost useless: for which reason I shall no enlarge so much on the subject as many other writers have done. Tables of Aliquot, or Even Parts. I'arts of a shilling. Parts of a pound. J Parts of a cwt.,. a,. 7t -. d. S. 6 is - 4 I 3 3 I4 2 1 2 8 'arts of 2 shillings. Is. is - s. d. 10 0 6 8 5 0 4 0 3 4 2 6 1 8 is I == 1 4 8 T J. 10b. 56 28 16 14 7 is -t. I TI, 8d. = - The aliquot part of any number is 6d. I such a part of it, as being taken a cer4d. i tain number of times, exactly makes 3d. 3 that number. 1 T CASE I. When the price of one yard, pound, &c. is an even part f one shilling-Find the value of the given quantity at s. a yard, pound, &o. and divide it by that even part, and la Quotient will be the answer in shillings, &c. i00 F'RACTAClt:. Or find the value of the given quantity at 2s. per yd. &.eand divide said value by the even part which the given price is of 2s. and the quotient will be the answer in shillings, &c. which reduce to pounds. N. B. To find the value of any quantity at 2s. you need only double the unit figure for shillings; the other figures will be pounds. EXAMPLES. 1. What will 461} yds. of tape come to at lId. per vd. 1 s. d. i1d. I 1 461 6 value of 461} yds. at Is. per yd 5,7 81 ~2 17s. 8 d. value at I1Id. 2. What cost 256 lb. of cheese at 8d. per pound? 8d. I I ~25 12s. value of 256 lb. at 2s. per lb. ~8 10s. 3d. value at 8d. per pound. Yards. per yard. ~. s. d. 486~ at Id. Answers. 2 0 6~ 862 at 2d. 7 38 911 at bd. 11 7 9 749 at 4d. 12 9 8 113 at 6d. 2 16 6 899 at 8d. 29 19 4 CASE iI. When the price is an even part of a pound-Find tbh value of the given quantity at one pound per yard, &c. and divide it by that even part, and the quotient will be the aa swer in pounds. EXAMPLES. What will 1291 yards cost at 2s. 6d. per yard?. d. ~. s. ~. 2 6 I 129 10 value at 1 per yard. Ans. ~16 3s. 9d. value at 2s. 6d. per yard. Yd s. d. ~. s. d 123 at 10 0 per yard Answers. 61 10 0 7*7 at 0 -171 17 6 PRACTICE. 101 Yds. s. d. ~. s. d. 211 at 4 0 per yard. 42 5 0 543 at 6 8 181 0 0 127 at 3 4 - 21 3 4 461 at 1 8 38 84 NoTE.-When the price is pounds only, the given quantity multiplied thereby, will be the answer. EXAMPLE.-11 tuns of hay at 41. per tun. Thus, 11 4 Ans. ~44 CASE III. When the given price is any number of shillings under 20. 1. When the shillings are an even number, multip y quantity by half the number of shillings, and double the first figure of the product for shillings; and the rest of the product will be pounds. 2. If the shillings be odd, multiply the quantity by the whole number of shillings, and the product will be the an swer in shillings, which reduce to pounds. EXAMPLES. lst.-124 yds. at 8s. 2d.-132 yds. at 7s. per yd 4 7 ~49 12s. Ans. 2,0)92,4 ~46,4 Ans. Yds. ~. s. Yds. ~. s. 562 at 4s. Ans. 112 8 372 at 11s. Ans. 204 12 378 at 2s. 37 16 264 at 9s. 118 16 )13 at 14s. 639 2 250 at 16s. 200 00 CASE IV. When the given price is pence, or pence and farthings, md not an even part of a shilling-Find the value of the riven quantity at Is. per yd. &c. which divide by the greatist even part of a shilling contained in the given price, and ake parts of the quotient for the remainder; t 6e prcce, ind the sum of these several quotients will i t 'tr rver n shillings, &c. which reduce to pounds. i2 102 'RACTIC;lG. EXAMPLES. What will 245 lb. of raisins come to, at 9id. per lb. t s. d. 6d. a 245 0 value of 245 lb. at Is. per pound. 8d. ~ 122 6 value of do. at 6d. per lb. id. 61 3 value of do. at 3d. per lb. 15 33 value of do. at 2d. per lb. 2,0)19,9 01 Ans. ~9 19 01 value of the whole at 93d. per lb. lb. d. L. s. d. lb. d. ~. 3 a 372 at 13 Ans. 2 14 3 576 at 71 Ans. 18 0 6 325 at 21 3 0 111 541 at 94 20 17 0~ 827 at 4j 15 10 1' 672 at 11 32 18 0 CASE V. When the price is shillings, pence and farthings, and nco the aliquot part of a pound-Multiply the given quantity by the shillings, and take parts for the pence and farthings, as in the foregoing cases, and add tl'em together; the sunwill be the answer in shillings. EXAMPLES. 1. What will 246 yds. of velvet come to, at 7s. 3d. per yd.1.s d 8d.! ~1 246 0 value of 246 yards at Is. per yd. 7 1722 0 value of do. at 7s. per yard. 61 6 value of do. at 3d. per yard. 2,0)178, 3 6 Anm. ~89 3 6 value of do. at 7s. 3d. per yard. ANSWERS.. d. ~. s. d. t. What cost 139 yds. at 9 10 per yd.? 68 6 10 3. What cost 146 yds. at 14 9 per yd. 107 13 6 4. What eOst 120 cwt. at 11 3 per cwt.? 67 10 5. What cost 17 yds. at 9 8 peryd.? 61 12 11 i Whbat oost b,. at 3 ll per lb.? 9 15 ItI TARIE A D 'TRET. 103 CASE VI. Wlien the price and quantity given are of several denoinations-NMutltiply the price by the integers in the given lantity, and take parts for the rest from the price of an inger; which, added together, will be the answer. This is iplicable to federal money. EXAMPLES. 1. What cost 5 cwt. 3 qrs. L lb. of raisins. at 21. 11s. 1. per cwt. ~ 2 qrs. ~ ~. s. d. 2 11 8 5 12 18 4 1 5 10 12 11 6 5~ ~15 3 61 1 qr. 14 lb. 1 2 l 2 2. What cost 9 cwt. 1 qr. 8 lb. of sugar, at 8 dollars, 65 cts. per cwt. l $ cts. 1 qr. { 8,65 9 77,85 7 b. 1 2,1625 1 lb.,5406,772 Ans. $80,6303 ANSWERS. per cwt. $75, 61 cts. 3 m. rcwt. ~14 19s. 3d. per cwt. ~10 2s. 51d. per cwt. $76, 47 cts. 6 m. per cwt. $2,.55 cts. 2ro m. Ans. C. 7 5 14 12 0 qrs. 3 1 3 0 0 lb. 16 at $9, 58 cts. 0 at 21. 17s. per 7 at 01. 13s. 8d. 7 at $6, 34 cts. 24 at $11, 91 cts TARE AND TRET. TARE and Tret are practical rules for deducting cerin allowances which are made by merchants, in buying d selling goods, &c. by weight; in which are noticed the flowing particulars: 1. Gross Weight, which is the whole weight of any sort goods, together with the box, cask, or bag, &c. which ntains them. 2. Tare, which is an allowance made to the buyer, for e weight of the box, cask, or bag, &c. which contains the ods bought, and is either at so much per box, &c. or at much per cwt. or at so much in the whole gtots weight. 104 TARE AND TRET. 3. Tret, which is an allowance of 4 lb. on every 104 Ib. for waste, dust, &c. 4. Clof, which is an allowance made of 2 lb. upon every 3 cwt. 5. Suttle, is what remains after one or two allowances have been deducted. CASE I. When the question is an Invoice-Add the gross weights into one sum and the tares into another; then subtract tho total tare from the whole gross, and the remainder will be the neat weight. EXAMPLES. 1. What is the neat weight of 4 hogsheads of Tobaccc marked with the gross weight as follows: C. qr. lb. Ib. No. 1- 9 0 12 Tare 100 2- 8 3 4 - 95 3- 7 1 0 - 83 4 -6 3 25 - 81 Whole gross 32 0 13 359 total tare. Tare 359 lb.= 3 3 23 Ans. 28 3 18 neat. 2. What is the neat weight of 4 barrels of Indigo, No. and weight as follows: C. qr. Ib. lb. No. 1-4 1 10 Tare 36) 2- 3 3 02 29 3 -4 0 19 - 32 A wt.qr. lb. 4-4 0 0 - 35 Ans.15011 CASE Il. When the tare is at so much per box, cask, bag, &c.Multiply the tare of 1 by the number of bags, bales, &c, the product is the whole tare, which subtract from the gross, and the remainder will be the neat weight, EXAMPLES. 1. In 4 hhds. of sugar, each weighing 10 cwt. 1 qr. 151 gross; tare 75 lb. per hhd. how much neat? Cwt. qrs. lbs. 10 1 15 gross weight of one hhd. 4i [Carried up { TASX. AND '1TH ET. 105 41 2 4 gross weight of the whole. 75 X4=2= 2 20' whole tare. Ans. 38: 12 1eaf. 2. What is the nea;t weizLht of 7 tierces of rice, each Ie'ghing 4 cwt. 1 qr. 9 l). gross, tare per tierce 34 lb. 1 Ans. 28 C. 0 qr. 21 lb. 3. In 9 firkins of butter, each weighing 2 qrs. 12 lo. gross, tare 11 lb. per firkin, how much neat? Ans. 4 C. 2 qrs. 9 lb. 4. If 241 bls. of figs, each 3 qrs. 19 lb. gross, tare 10 lb. per barrel; how many pounds neat? Ans. A22413. 5. In 16 bags of pepper, each 85 lb. 4 oz. gross, tare per bag, 3 lb. 5 oz.; how many pounds neat? Ans. 1311. 6. In 75 barrels ef figs, each 2 qrs. 27 lb. gross, tare in the whlole 597 lb.; how much neat weight? Ans. 50 C. 1 qr. 7. rWhat is the neat weight of 15 lhhds. of Tobacco, each weighing 7 cwt. 1 qr. i3 lb. tare 100 Ib. per llhd.? Ans. 97 C. Oq-. 11 lb. CASE III. Whlen the tare is at so much per cvt. —Divide the gross weight by the aliquot part of a cwt. for the tare, which subract from the gross, and the remainder will be neat weight. EXAMPLES. 1. What is the neat weight of 44 cwt. 3 qrs. 16 lb. gross rare 14 lb. per cwt.? C. qrs. Ib. 14 Ib. 144 3 16 gross. 5 2 121 tare. Ans. 39 1 3- neat. 2. 'What is the neat weight of 9 hhds. of Tobacco, each.veighing gross 8 cwt. 3 qrs. 14 lb. tare 16 lb. per cwt.? Ans. 68 C. 1 qr. 24 lb, 3. Wihat is the neat weight of7 bls. of potash, each weighing '01 lb. gross, tare 10 lb. per cwt.? Ans. 1281 Ib. 6 oz. 4. In 25 bis. of figs, each 2 cwt. 1 qr. gross, tare per. cwt: 16 lb.; how much neat weight? Ans. 48 cwt. 24 lb. 5. In 83 cwt. 3 qrs. gross, tare 20 lb. per cwi t. what neat freight? Ans. 68 cwt. 3 qrs. 5 lb. 6. In 45 cwt. 3 qrs. 21 lb. gross, tare 8 lb. per cwt. how *tuch neat weight? Ans. 42 cwt. 2 qrs. 17j lb. 7. What is the value of the neat weight of S hhds. of su 1061 T' ARE AI 'NI T KP, T. gar, at $9, 54 cts. lcr cwt. caci wC;ghlillg 10 cwt. I qr. 14Ib gross, tare 14 11). per cwt.. Is. t69j2, 84 cts. 2 m. CASE IV. When Trct is alloxwed with the Tare. 1. Find the tare, which subtract from the gross, and call the remainder suttle. 2. Divide the suttle by 20, and the quotient will be tie tret, which subtract from the suttle, and the remainder will bz the neat weight. EXAMPLE S. 1. In a hogshead of sugar, weighing 10 cwt. 1 qr. 12 lb. gross, tare 14 lb. per cwt., tret 4 lb. per 104 lb.,* how much neat weight? Or thus, cwt. qr. lb. ctf. qr. lb. 10 1 12 14 ---=)10 1 12 gross. 4 1 1 5 tare. 41 26)9) 0 7 suttle 28 1 11 tret. 330 Ans. 8 2 24 ncat. 83 14=-li) 60 gross. 145 tare. 26)1015 suttle, 39 tret. Ans. 976 lb. neat. 2. In 9 cwt. 2 qrs. 17 lb. gross, tare 41 lb., tret 4 lb. pel 104 lb., how much neat 1 Ans. 8 cwt. 3 qrs. 20 lb. 3. In 15 chests.of sugar, weighing 117 cwt. 21 lb. gross, tare 173 lb., tret 4 lb. per 104, how many cwt. neat? Ans. 111 cwt. 22 lb. 4. What isfth' neat weight of 3 tierces of rice, each weighing 4 cwt. 3 qrs. 14 lb gross, tare 16 lb. per cart., and allowing tret as usual I Ans. 12 cwot. 0 p s. 6 Ib. 5. In 25 bls. of figs, each 84 lb. gross, tare 12 lb. per cwt., tret 4 lb. per 104 lb.; how many pounds neat? Ans. 1803 + * This is the tret allowed in London. The reason of divividing by 26 i because 4 lb. is 1-2t1 *f04 lb. but if the tret is at any other rate, other pat Must be taken. acoring to the rate proposed, &c. TA AND TRET. 107 6. What is the value of the neat weight of 4 barrels of Spanish tobacco; numbers, weights, and allowances as folows, at D9d. per pound? tq. qrs. lb. No. I (rs I 1 Tare 16 lb. per cwt. I 0 Tret 4 lb. per 104 lb. 4 0 3 21 Ans. ~17 16s. 3d. 4 o a 215 CASE V. When Tare, Tret, and Cloff, are allowed: Deduct the tare and tret as before, and divide the suttle )y 168 (because 2 lb. is the y- of 3 cwt.) the quotient will,e the cloff, which subtract from tlh suttle, and the remain-!er will be the nieat weiglt. EXAMPLES. 1. In 3 hogsheads of tobacco, each weighing 13 cwt. 3 qrs. 3 lb. gross, tare 107lb. per hhd., tret 4 lb. per 104 lb., and loff 2 lb. per 3 cwt., as usual; how much neat? cwt. qrs. lb. 13 3 23 4 55 28 443 112 1563 lb. gross. of 1 hhd. 3 4689 whole gross. 107 X 3=321 tare. 26)4368 suttle. 168 tret. 168)4200 suttle. 25 cloff. Ans. 4175 neat weight.!. What is the neat weight of 26 cwt. 3 qrs. 20 lb. gross, re 52 lb., the allowance of tret and cloff as usual? Ans. neat 25 cwt. 1 qr. 5 lb. 1 oz. nearl/; omitting further firact!:ns. 10P> I N'T Er. T INTEREST. INTEREST is of two kinds; Simple and Compound SIMPLE INTEREST. Simple Interest is the sum paid by the borrower to thm lender for the use of money lent; and is generally at I certain rate per cent. per annum, which in several of the UIni ted States is fixed by law at 6 per cent. per annum; that ip, 61. for the use of 1001. or 6 dollars for the use of 100 do) lars for one year, &c. Principal, is the sum lent. Rate, is the sum per cent. agreed on. Amrn-t is the nrincipal nndl interest added together. C: A S 1.I ro find the interest of any given sum for one year. RULE.-Multiply the principal by the rate per cent. and divide tht product by 100; the quotient will be the answer. EXAMPLES. 1. What is the interest of 391. 1Us. Sd. for one yeau, 61. per cent. per annuln? ~. s. d.?9 118; 6 2137 10 3 20 7150 12 6103 4 (t12 Ans. ~2 7s. 6d.4-,. 2. What is the interest of 2361. 10s. 4d. for a year, at 6 per cent I Ans. ~ 11 i6. 64. SIMPLE INTEREST. 109 3. What is the interest of 5711. 13s. 9d. for one year, at lIl. per cent.? Ans. ~34 6s. 0ld. 4. What is the interest of 21. 12s. 9-1d. for a year, at 61. leer cent.? Ans. ~0 3s. 2d. FEDERAL MONEY. 5. What is the interest of 468 dols. 45 cts. for one year, it 6 per cent.? $ cts. 468, 45 6 Ans. 28110, 70=$28, 10 cts. 7m. Here I cut off the two right hand integers, which divide by 100: but to divide federal money by 100, you need only rall the dollars so many cents, and the inferior denominalions decimals of a cent, and it is done. Therefore you may multiply the principal by the rate, ind place the separatrix in the product, as in multiplication n'ofederal money, and all the figures at the left of the separ itrix, will be the interest in cents, and the first figure on..ic right will be mills, and the others decimals of a mill, as.1 the following EXAMPLES. 6. Required thle interest of 135 dols. 25 cts. for a year at; per cent? $ cts. 135, 25 6 Ans. 811, 50==$8, 11 cts. 5m. 7. What is the interest of 19 dols. 51 cts. for one year, at 5 per cent.? $ cts. 19, 51 5 Ans. 97, 55=97 cts. 5-m. 8. What is the interest of 436 dols. for one year, at 6 per cert.? 6 Ans. 2616 cts.=$26, 16 cts. K 110 SIM PLE INTEREST. ANOTHER METHOD. Write down the given principal in cents, which multiply by 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. Wlat is the interest of $73, 65 cents for a year, at 6 per cent.? Principal 7365 cents. 6 Ans., 441,90=441i cts. or $4, 41 cts. 9 m. 10. Required the interest of $85, 45 cts. for a year, at " per cent.? Cents. Principal 8545 7 Ans. 598, 15 cents,-.=t$,98 cts. lam. 3.^, CASE II Td6find the simple interest of any sum of money, for ary number of years, and parts of a year. GENERAL RULE.-1st. Find the interest of the given sum for one year. 2d. Multiply the interest of one year by the given number of years, and the product will be the answer for that time. 3d. If there be parts of a year, as months and days, work for thp months by Hhe aliquot parts of a year, and for the days by the Rule df Three Direct, or by allowing 30 days to the month, and taking aliquot parts of the same.* * By allowing the month to be 30 days, and taking aliquot parts thereof, you will have the interest of a4Ordinary sum sufficiently exactlrcommo use: but If the sum be very lar, you may say, As 365 days: is to the intercst of one year: so is the given number o 4av s to the interest required % SIMPLE INTEREST. III EXAMPLES. 1. What is the interest of 751. 8s. 4d. for 5 years and 2. months, at 61. per cent. per annum 1 ~. s. d. 75 8 4 ~. s.d. 6 2 no.=-4)4 10 6 Interest for 1 year. 5 4152 10 0 20 22 12 6 do. 5 years. 0 15 1 do. for two months. 10150 12 ~23 7 7 Ans. 6100 2. What is the interest of 64 dollars 58 cents for 3 years,, months, and 10 days, at 5 per cent.? $ 64,58 5 322,90 nterest for 1 year in cents, per 3 [Case I. 968,70 do. for 3 years. 4 mo.. 107,63 do. for 4 months. I mo. 4 26,90 do. for 1 month. 10 days, - 8,96 do. for 10 days. Ans. 1112,19=1112cts. or $ 11, 12c. 1-.m. 3. What is the interest of 789 dollars for 2 years, at 6 per cent. 1 Ans. $94, 68 cts. 4. Of 37 dollars 50 cents for 4 years, at 6 per cent. pa? annum? Ans. 900 cts. or $9. 5. Of 325 dollars 41 cts. for 3 years and 4 months, at 5 per cent.? Ans. $54, 23 cts. 5 m. 6. O0j.l. 12s. 3d. for five years, at 6 per cent.? 7 Ans. ~97 13s. 8d. 7. Of 14/. 10s. 6d. for 3 and a l9lf years, at 6 per cent.? Ans. ~36 13s. 8. Of;501. 16s. 8d. for 4 years and 7 months, at 6 per eut. Ans. ~41 9s. 7d-.' t 112 COMMISSION. 9. Of I dollar for 12 years, at 5 per cent. Ans. 60 cts. 10. Of 215 dollars 34 cts. for 4 and a half years, at bJ and a half per cent. Ans. $33, 91 cts. 6m. 11. What is the amount of 324 dollars 61 cents for l years and 5 months, at 6 per cent.? Ans. $430, 10 cts. 8ms2-m. 12. What will 30001. amount to in 12 years and 10 months, at 6 per cent.? Ans. ~5310. 13. What is the interest of 2571. 5s. Id. for 1 year and 3 quarters, at 4 percent.? Ans. ~18 Os. ld. 3qrs. 14. What is th iterest of 279 dollars 87 cents for 2 years and a half, at 7 per cent. per annum? Ans. $48, 97cts. 7~m. 15. What will 2791. 13s. 8d. amount to in 3 years and a half, at 5- per cent. per annum? An. 33s. ~33s. 6d. 16. What is the amount of 341 dols. 60 cts. for 5 years and 3 quarters, at 7 and a half per cent. per annum? Ans. $488, 914 cts. 17. What will 730 dols. amount to at 6 per cent. in 5 years, 7 months, and 12 days, or -2 of a year? Ans. $975, 99 cts. 18. What is the interest of 18251. at 5 per cent. per an. num, from March 4th, 1796, to March 29th, 1799, (allow ing the year to contain 365 days?) Ans. ~280. NOTE.-The Rules for Simple Interest serve also to calculate Commission, Brokerage, Ensurance, or any thing else estimated at a rate per cent. COMMISSION, IS an allowance of so much per cent., to a for or correspondent abroad, for buying and selling goo tr his em-?loyer., EXAMPLES. 1. What will the commission of 84831. 1 s ome to at f -ir amt,. EROKEAGE. 113 ~. s. Or thus, 843 10 ~. s. 5 ~5 is -1)843 10 121 17 10 Ans. ~42 3 6 20.... 3150 12 6100 ~42 3s. 6d.:I Required the commission on 964 dols. 90 cts. at 2 per vent.? Ans. $21, 71 cts. 3. What may a factor demand on 12 per cent. commission ior laying out 3568 dollars? Ans. $62, 44 cts BROKERAGE, IS an allowance of so much per cent. to persons assisthig merchants, or factors, in purchasing or selling goods. EXAMPLES. 1. What is the brokerage of 7501. Ss. 4d. at 6s. Sd. per tent. I ~ s. d. 750 8 4 Here I first find the brokerage at I pound I per cent. and then for the given rate, -- which is ~ of a pound. 7,50 6 4 20 s. d. ~. s. d. qrs. -- 6 8s=)7 10 1 0 10,08 - 12 Ans. ~2 10 0 1, 1,00s 2. What is the brokerage upon 4125 dols. at - or 75 cents per cent.? Ans. $30, 93 cts. 71 m. 3. If a broker sell goods to the amount of 5000 dollars, what is his demand at 65 cts. per cent.? Ans. $32, 50 fs. 114 ENSURANCE. 4. What may a broker demand, when he sells goods to the value of 5081. 17s. 10d. and I allow him 1l per cent. 1 Ans.7 s. ~7 d. ENSURANCE, IS a premium at so much per cent. allowed to persons and offices, for making good the loss of ships, houses, mer. chandise, &c. which may happen from storms, fire, &c. EXAMPLES. 1. What is the ensurance of 7251. 8s. 1Od. at 12} per cent. 1 Ans. ~90 13s. 7~d. 2. What is the ensurance of an East-India ship and catr go, valued at 123425 dollars, at 15} per cent. 1 Ans. $19130, 87 cts. 5 m. 3. A man's house estimated at 3500 dols., was ensuril, against fire, for 1- per cent. a year: what ensurance did he annually pay? Ans. $61, 25 cts. Short Practical Rulesfor calculating Interest at 6 per cent. either for months, or months and days. 1. FOR STERLING MONEY. RULE.-1. If the principal consist of pounds only, cut off the unit figure, and as it then stands it will be the interest for one month, in shillings and decimal parts. 2. If the principal consist of pounds, shillings, &c. reduce it to its decimal value; then remove the decimal point one place, or figure, further towards the left hand, and as the decimal then stands, it will show the interest for one month in shillings and decimals of a shilling. EXAMPLES. 1. Required the interest of 541. for seven month# -deil days, at 6 per cent. &HEORT [RACTICAL RULE4. 116 S. 10 days==-)5,4 Interest for one month. 7 37,8 ditto for 7 months. 1,8 ditto for 10 days. Ans. 39,6 shillings-~1 19s. 7,2d. 12 7,2 2. What is the interest of 421. 10s. for 11 months, at 6 per cent. 1 ~. s. ~. 42 10 = 42,5 decimal value. Therefore 4,25 shillings interest for 1 month. 11 ~. s. d. Ans. 46,75 Interest for 11 mo. = 2 6 9 3. Required the interest of 941. 7s. 6d. for one year, five months and a half, at 6 per cent. per annum? Ans. ~S 5s. Id. 3,5qrs. 4. What is the interest of 121. 18s. for one third of a month, at 6 per cent.? Ans. 5,16d. II. FOR FEDERAL MONEY. RUL-.-1. Divide the principal by 2, placing the separatrix as usual, ad the quotient will be the interest for one month in cents, and decimals of a cent; that is, the figures at the left of the separatrix will be cents, and those on the right, decimals of a cent. 2 Multiply the interest of one month by the given number of months, or months and decimal parts thereof, or for the days take the eren parts of a month, &.R. lift SHORT PRACTIGL rULZS EXAMPLES. 1. What isthe interest of 341 dols. 52 cts. for 7I mouths 1 2)341,52 Or thus, 170,76 Int. for 1 month. 170,76 Int. for 1 month. X 7,5 months. 74 --- 85380 1195,32 do. for 7 mo. 119532 85,38 do. for 2 mo. --- ts. m. -- 1280,700cts. = 12,80 7 1280,70 Ans. 1280,7cts.=$12, 80cts. 7m. 2. Required the interest of 10 dols. 44 cts. for 3 ye6 months, and 10 days. 2)10,44 10 days=4l) 5,22 interest for 1 month. 41 months. 5,22 208,8 214,02 ditto for 41 months. 1,74 ditto for 10 days. 215,76 cts. Ans. — $2, 15 cts. 7 m. + 8. What is the interest of 342 dollars for 11 months 1 The - is 171 interest for one month. 11 Ans. 1881 cts.=$18, 81 ets. NOTE.-To find the interest of any sum for two months, at 6 per cent. you need only call the dollars so many cents, asd the inferior denominations decimals of a cent, and it is done: Thus, the interest of 100 dollars for two months, is 100 cents, or one dollar; and $25, 40 cts. is 25 cts. 4m. &c. which gives the following RULE II.-Multiply the principal by half the number of months and the product will show the interest of the given time, ia otoutad dceimals of cent, as above. FOR CAt CTLATIN-O INTEREST. 117 EXAMPLES. 1. Required the interest of 316 dollars for 1 year and 10 months. 11=- the number of mo. Ans. 3476 cts. ==$34, 76 cts. 2. What is the interest of 364 dols. 25 cts. for 4 months $ cts. 364, 25 2 half the months. 728, 50 cts. Ans. —$7, 28 cts. 5 m. III. When the principal is given in federal money, at 6,er cent. to find how much the monthly interest will be in New-England, &c. currency. RVLE.-Multiply the given principal by,03, and the product will be Ihe interest for one month, in shillings and decimal parts of a shilling. EXAMPLES. 1. What is the interest of 325 dols. for 11 months?,03 9,75 shil. int. for one month X 11 months. Ans 107,25 s.=~5 7s. 3d. 1. What is the interest in New-England currency of 31 dols. 68 cts. for 5 months I Principal 31,68 dols.,03,9504 Interest for one month. 5 Ans. 4,7520s.=-4s. 9d. 12 9,0240 118 SHORT PRACTICAL RULES IV. When the principal is given in pounds, shillings, &c New-England currency, at 6 per cent. to find how much th monthly interest will be in federal money. RULE.-Multiply the pounds, &c. by 5, and divide that product b 3. the quotient wi!l be the interest for one month, in cents, and deci males of a cent, &c EXAMPLES. 1. A note for ~411 New-England currency has been o0 interest one month; how much is the interest thereof in fe deral money ~. 411 5 3)2055 Ans. 685 cts.=$6, 85 cts. 2. Required the interest of 391. 18s. N. E. currency, fto 7 months? ~ 39,9 decimal value. 5 3)199,5 Interest for 1 mo. 66,5 cents. 7 Ditto for 7 mo. 465,5 cts.m$4, 65 cts. 5 m. Ans. V. When the principal is given in New-England and Vir ginia currency, at 6 per cent. to find the interest for a year in dollars, cents, and mills, by inspection. RULE.-Since the interest of a year will be just so many cents a the given principal contains shillings, therefore, write oown the shil lings and call them cents, and the pence in the principal made loss b; 1 if they exeaed 3, or by 2 when they exceed 9, will be the millse, nearly. JR.CALCULATINGI INTEREST. 119 EXAMPLES. 1. What is the interest of 21. 5s. for a year, at 6 per ct. I ~2 5s=45s. Interest 45 cts. the Answer. 2. Required the interest of 1001. for a year, at 6 per ct.? ~100=2000s. Interest 2000 cts.-$20 Ans. 3. Of 27s. Gd. for a yfar? Ans. 27s. is 27 cts. and 6d. is 5 m. 4. Required the interest of 51. 10s. lid. for a year? ~5 lOs.=l 10s. Interest 110 cts.=$1, 10 cts. 0 m. 1 pence.-2 per rinl leaves 9= 9 Ans.$1, 10 9 fl. To compute the interest on any note or obligation, en there are payments in part, or endorsements. luLU. —. Find the amount of the whole principal for the whole e.. Cast thF interest on the several payments, from the time they e paid,to the time of settlement, and find their amount; and lastly uct the aniA-dlt of the several payments from the amount of the Icipal. EXAFMPLES. suppose a bcnid or note dated April 17, 1793, was given 675 dollars, interest at 6 per cent. and there were paynts endorsed upon it as follows, viz. 'irst payment. 148 dollars, MIay 7, 1794. second payment, 341 dols. August 17, 1796. 'hird payment, 99 dols. Jan. 2, 1798. I demand how ch remains due on said note, the 17th June, 1798 $ cts. 148, 00 first payment, May 7, 1794. Yr. mo. 36, 50 interest up to-June 17, 1798.:4 14 184, 50 amount 341, 00 second payment, Aug. 17, 1796. Yr. mo. 37, 51 interest to —June 17, 1798. =1 10 378, 51 amount. [Carried over.] 120 sHORT I'RAC(TICAr. At,3LE cts. 99, 00 third payment, January 2, 1798. 2, 72 interest to-June 17, 1798.= 5 meo 101, 72 amount. 184, 50 } 378, 51 several amounts. 101, 72 664, 73 total amount of payments. 675, 00 note, dated April 17, 1793. Yr. me 209, 25 interest to-June 17, 1798. =5 2 884, 25 amount of the note. 664, 73 amount of payments. $219, 52 remains due on the note, June 17, 1798. 2. On the 16th January, 1795, I lent James Paywell 50M dollars, on interest at 6 per cent. which I received back it, the following partial payments, as under, viz. 1st of April, 1796 - - - - $ 50 16th of July, 1797 - - 400 1st of Sept. 1798 - - - - How stands the balance between us, on the 16th Noven ber, 1800 Ans. due to me, $63, 18 cts. 3. A PROMISSORY NOTE, viz. ~62 1Os. New-London, April 4, 1797. On demand, I promise to pay Timothy Careful, sixty-tw, pounds, ten shillings, and interest at 6 per cent. per annum till paid; value received. JOHN STANBY, PETER PAYWELL. RICHARD TESTIS. Endorsements. ~. s. 1st. Received in part of the above note, September 4, 1799, 50 0 And payment June 4, 1800, 12 10 How much remains due on said note, the 4th day of Do calnber, 1800. ~.. d. Ans. 9 12 6 FUO CAIL.:tLAT.IG INTER N(oTr.-Tl]- preceding Ruls, by jalstoml voptular, av,so;-ltch pri,- is;s and esteemed kccounit of ilF being sil ile aid concise, that I have:e-n,:" a place: it may answer for short periods of tihme,., in a long course of years, it will be found to be very erroneOILs. Although this method seems at first view to be '.poP -. gr'ound 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, completely expunge, or as it may be said, eat up the debt. For an explanation of this, take the following EXAMPLE. A lends B 100 dollars, at 6 per cent. interest, and takes is note of hand; B does no more than pay A at every c ear's end 6 dollars, (which is then justly due to B for the se of his money) and has it endorsed on his note. At the end of 10 years B takes up his note, and the sum he has to I ay is reckoned thus: The principal 100 dollars, on inte rest 10 years amounts to 160 dollars; there are nine eni'orsements of 6 dollars each, upon which the debtor claims interest; one for nine years, the second for 8 years, the 'hird for 7 years, and so down to the time of settlement; tile whole amount of the several endorsements and their incerest, (as any one can see by casting it) is $70, 20 cts. this subtracted from 160 dols. the amount of the debt, leaves in favour of the creditor, $89, 40 cts. or $10, 20 cts. less than the original principal, of which lhe has not received a cent, but only its annual interest. If the same note should lie 20 years in the same wmy, B would owe but 37 dols. 60 cts. without paying the least fraction of the 100 dollars borrowed. Extend it to 28 years, and A the creditor would fall in debt to B, without receiving a cent of the 100 dols. which qe lent him. See a better Rule in Simple Interest by decimals. page 175. I, IN2 O0M1'(UND INTEREST. COMPOUND INTEiiEST, IS when the interest is added to the principal, at tie end of the year, and on that amont the interest cast for another year, and added again, and so onl: this is called interest upon interest. RULE.-Find the interest for a year, and add it to the principal, which call the amount for the first year; find the interest of this amount, which add as before, for the amount of the second, and so on for any number of years required. Subtract the original principal from the last amount, and the remainder will be the Compound Interest for the whole time. EXAMPLES. 1. Required the amount of 100 dollars for 3 years at C per cent. per annum, compound interest? $ cts. $ cts. 1st Principal 100,00 Amount 106,00 for I year. 2d Principal 106,00 Amount 112,36 for 2 years. 3d Principal 112,36 Amount 119,1016 for 3 y;s. Ans. 2. What is the amount of 425 dollars, for 4 y ears, at X per cent. per annum, compound interest? Ans. 2516, 59 cts. 3. What will 4001. amount to, in four years, at 6 pei cent. per annum, compound interest ' Ans. ~504 19s. 93d. 4. What is the compound interest of 1501. 10s. for 3 years, at 6 per cent. per annium? Ans. ~28 14s. l1{d.4 -5. What is the compound interest of 500 dollars for 4 years, at 6 per cent, per aimulan Ans. $131,238+ 6. What will 1000 dollars ah4:.nt to in 4 years, at 7 per cent. per annum, compound interest? Ans. $1310, 79 cts. 6 m. + 7. What is the amount of 750 dollars for 4 years, at 6 per cent. per annum, compound interest? Ans. $946, 85 cts. 7,72 m. 8. What is the comporund interest of 876 dols. 90 cents far 3V years, at 6 per cent, per annum? Ans. $198, 83 cts.+ DIISCOUNT 113 DISCOUNT, IS an allowance made for the payment of any sum of money before it becomes due; o1 upon advancing ready money for notes, bills, &c. which are payable at a future day. What remains after the discount is deducted, is the present worth, or such a sum as, if put to interest, would at the given rate and time, amount to the given sum or debt. RULE.-As the amount of 100. or 100 dollars, at the given rate and time: is to the interest of 100, at the same rate and time:: so if ihe given sum: to the discount. Subtract the discount from the given sum, and the remainder is the present worth. Or-as the amount of 100: is to 100: so is the given sum or debt: to the present worth. PROOF.-Find the amount of the present worth, at the given rate and time, and if the work is right, that will be equal to the given sum. EXAMPLES. 1. What must be discounted for the ready payment of 100 dollars, due a year hence at 6 per cent. a year? $ $ $ $ cts. As 106: 6: 100: 5 66 the answer. 100,00 given sum. 5,66 discount. $94,34 the present worth. 2. What sum in ready money will discharge a debt of 9251. due 1 year and 8 months hence, at 6 per cent.? ~100 10 interest for 20 months. 110 Am't. ~. ~. ~. ~.. d. As 110: 100:: 925: 840 18 2+ Ant. 3. What is the present worth of 600 dollars, due 4 years hence, at 5 per cent.? Ans. $500. 4. What is the discount of 2751. 'Os. for 10 months, at 6 per cent. per annum? dns. ~13 2s. 4d. M4 ANNUITIES 5. Bought goods amounting to 615 dols. 75 c eel, at 7 months credit; how much ready money must I pay, discount at 4' per cent. per annum? Als. )600. 6. What sum of ready money must be received for a bill of 900 dollars, due 73 days hence, discount at 6 per cent. per annum? AIns. $SS9, 32 cts. S m. NoTE.-When sundry sums are to be paid at different times, find the Rebate or present worth of each particular payment separately, and when so found, add them into onl' sum. EXAMPLES. 7. What is the discount of 7561. the one half payable in six months, and the other half in six months after that, at 7 pccent.? Ans. ~37 1 Os. 2 d. 8 If a legacy is left me of 2000 dollars, of wvlich 500 dols. are payable in 6 months, 800 dols. payable in 1 year, and the rest at the end of 3 years; how much ready mnorey ought I to receive for said legacy, allowing 6 per cent. dis, count? Ans. $1833, 37 cts. 4 m. ANNUITIES. AN Annuity is a sum of money, payable every year, ci for a certain number of years, or for ever. When the debtor keeps the annuity in his own hands beyond the time of payment, it is said to be in arrears. The sum of all the annuities for the time they have been foreborne, together with the interest due on each, is called the amount. If an annuity is bought off, or paid all at once at the beginning of the first year, the price which is paid for it is called the present worth. To find the amount of an annuity at simple interest. RULE.-1. Find the interest of the given annuity for 1 year. 2 And then for 2, 3, &c. years, up to the given time, less 1. 3. Multiply the annuity by the number of years given, and ad4 the product to the whole interest, and the sum will be the amrnc cccght. ANNUITIES. i 25 EXAMPLES. I If an annuity of 701. be forborne 5 years, what will )e due fQr the principal and interest at the end of said erm, simple interest being computed at 5 per cent. per rnnun? Yr. ~. s. 1st. Interest of 701. at 5 per cent. for 1- 3 10 2 —7 0 3-10 10 4-14 0 2d. And 5 yrs. annuity, at 701. per yr. is 350 0 Ans. ~385 0 2. A house being let upon a lease of 7 years, at 400 lollars per annum, and the rent being in arrear for the.vhole term, I demand the sum due at the end of the term, simple interest being allowed at 61. per cent. per annum? Ans. ~3304. To find the present worth of an annuity at simple interest. RULE.-Find the present worth of each year by itself, discounting from the time it falls due, and the sum of all these present worthu will be the present worth required. EXAMPLES. 1. What is the present worth of 400 dols. per annum, to continue 4 years, at 6 per cent. per annum? 106 377,35849 = Pres. worth of 1st yr. 1 12 4 35714285 - --- 2d yr. 118 10 338,98305 3d yr. 124 322,58064 ---- 4th yr. Ans. $1396,06503 $1396, 6cts. 5Sm. 2. How much present money i, equivalent to an annuity of 100 dollars, to continue 3 years; rebate being made at 3 per cent.? Ans. $268, 37 cts. 3. What is 801. yearly rent, to continue 5 years, worht In ready m )nmey, at 61. per cent. Ans. ~340, Os 44., 1.4 1i EQUATION OF PA' MEAiSI',. EQUATION OF PAYMENTS, IS finding the equated time to pay at once, several del to die at different periods of time, so that no loss shall be sustained by either party. RULE.-Multiply each payment by its time, and divide the sum c1 the several products by the whole debt, and the quotient will be the equated time for the payment of the whole. EXAMPLES. 1. A owes B 380 dollars, to be paid as follows-viz. 100 dollars in 6 months, 120 dollars in 7 months, and 160 dollars in 10 months: What is the equated time for the payment of the whole debt? 100 x 6 - 600 120 x 7- 840 160 x 10 = 1600 380 )3040(8 months. Ans. 2. A merchant hath owing him 3001. to be paid as fol lows: 501. at 2 months, 1001. at 5 months, and the rest s.t K 8 months; and it is agreed to make one payment of the whole: I demand the equated time? Ans. 6 months. 3. F owes H 1000 dollars, whereof 200 dollars is to ba paid present, 400 dollars at 5 months, and the rest at 15 months, but they agree to make one payment of the whole; I demand when that time must be 1 Ans. 8 months. 4. A merchant has due to him a certain sum of money, to be paid one sixth at 2 months, one third at 3 months, and the rest at 6 months; what is the equated time for tho payment of the whole? Ans. 44 months. r ~ ~ --- —C -I i -s ~ BARTER, IS the exchanging of one commodity for another, and directs merchants and tracars how to make the exchange without loss to either party. RULE.-Find the value of the commodity whos, cuantity is given I ien find what quantity of the other at the proposed r!'e can be boight for the same money, and it gives the answer. BAR ER 27 EXAMPLES. 1. What quantity of flax at 9 cts. per lb. must be given in barter for 12 lb. of indigo, at 2 dols. 19 cents per lb.? 12 lb. of indigo at 2 dols. 19 cts. per lb. comes to 26 dols. 28 cts.-therefore, As 9 cts.: 1 lb.:: 2d,28 cts.: 292 the answer. 2. How much whetat at 1 dol. 25 cts. a bushel, must be given in barter for 50 bushels of rye, at 70 cts. a bushel I Ans. 28 bushels. 3. How much rice at 28s. per cwt. must be bartered for 31 cwt. of raisins, at 5d. per lb.? Ans. 5 cwt. 3 qrs. 9t 6b. 4. How much tea at 4s. 9d. per lb. must be given in barter for 78 gallons of brandy, at l,2s. 32d. per gallon? Ans. 201 lb. 1327oz. 5. A and B bartered: A a1d 8} cwt. of sugar at 12 cts. per lb. for whicl 3 gave him 18 cwt. of flour; what was the flour rated a, per lb. Ans. 5~ cts. 6. B delivered 3 hhds. of brandy, at 6s. Sd. per gallon, to C, for 126 yds. of cloth, what was the cloth per yard? Ans. 0s. 7. D gives E 250 yards of drugg,:t, at 30 cts. per yd. Ibr 319 lbs. of pepper; what does the pepper stand him in )er lb.? Ans. 23 cts. 5g-m. 8. A and B bartered: A had 41 cwt. of rice, at 21s. )er cwt. for which B gave him 201. in money, and the est in sugar at 8d. per lb.; I demand how much sugar B 'ave A besides the 201.? Ans. 6 cwt. 0O 191lb. 9. Two farmers bartered: A had 120 bushe rf wheat 1t 1- dols. per bushel, for which B gave himi 100 bushels if barley, worth 65 cts. per bushel, and the balance in oats t 40 cts. per bushel; what quantity of oats did A receive rom B? Ans. 2871 bushels. 10. A hath linen cloth worth 20d. an ell ready money; ut in barter he will have 2s. B hath broadcloth worth 14s. d. per yard ready money; at what price ought B to rate is broadcloth in barter, so a3 to be equivalent to A's bar_ring price? Ans. 17s. 4d. 32- qrs. 1MC t(lO SS AND) GAIN. 11. A and B barter: A hath 145 gallons of brandy al I dol. 20 cts. per gallon ready money, but in barter he will have 1 dol. 35 cts. per gallon: B has linen at 58 cts. per yard ready money; how must B sell his linen per yard in proportion to A's bartering price, and how many yards are equal to A's brandy? Ans. Barter price of B's linren is 65 cts. 2-m. and he must give A 300 yds. for his brandy. 12. A has 225 yds. of shalloon, at 2s. ready money pel yard, which he barters with B at 2s. 5d. per yard, taking indigo at 12s. 6d. per lb. which is worth but 10s. how much indigo will pay for the shalloon; and who gets the best bargain? Ans. 431lb. at barter price will pay for the shalloon, and B has the advantage in barter. Value of A's cloth, at cash price, is ~22 10 Value of 43l1b. of indigo, at 10s. per lb. 21 15 B gets the best bargain by ~0 1!i LOSS AND GAIN, IS a rule by which merchants and traders discover their profit or loss in buying and selling their goods: it also in structs them how to rise or fall in the price of their goodyf, so as to gain or lose so much per cent. or otherwise. Questions in this rule are answered by the Rule of Three. EXAMPL ES. 1. Bought a piece of cloth containing t5 yards, for 191 dols. 25 cts. and sold the same at 2 dols. 81 cts. per yard; what is the profit upon the whole piece? Ans. $47, 60 cts. 2. Bought 12- cwt. of rice, at 3 dols. 45 cts. a cwt. a'nm sold it again at 4 cts. a pound; what was the whole gain I Ans. $12, 87 cts. 5m.. 3. Bought 1I cwt. of sugar, at 6~d. per lb. b t could not sell it again for any more than 21. 16s. per cwt.; $did I gain or lose by my bargain? Ans. Lost, ~2 Uls, 4d. 4. Bought 44 lb. of tea for 61. 12s. and sold it again fw 81. O1s. 6d.; what was the profit on each nournd Ans. 10d L,05.i ANtD 4 t'N. 129.. Mpuglt a hihd. of molasses containing 119 gallons, at 52.cellts per gallon; paid for carting the same 1 dollar 25 c nts, and by accident 9 gallons leaked out; at what rate must I sell the remainder per gallon, to gain 13 dollars ill the whole? Ans. 69 cts. 2 m.+ II. To knovw what is gained or lost per cent. RtULE.-First see what the gain or loss is by subtraction; then, As the price it cost: is to the gain or loss:: so is 1001. or $100, to the gain or loss per cent. EXAMPLES. 1. If I buy Irish linen at 2s. per yard, and sell it again at 2s. 8d. per yard; what do I gain per cent. or in laying out 1001.: As: 2s. 8d.::: 100: ~33 6s. 8d. Ans. 2. If I buy broadcloth at 3 dols. 4- cts. per yard, and sell it again at 4 dols. 30 cts. per yard: what do I gain per ct. zI in laying out 100 dollars?. cts. ) Sold for 4, 30 | $ cts. cts. $ $ Cost 3, 44 A 3 4: 86:: 100: 25 I ls. 2 ) per cent. iained per yd. 86 3 3. If I buy a cwt. of cottol fir; 34 dols. 86 cts. and sell it igain at 41- cts. per lb. what do I gain or lose, and what )er cent.? $ cts. I cwt. at 41 cts. per lb. comes to 46,48 Prime cost 34,86 Gained in the gross, $11,61 As 34,86: 11,62: 100: 33- Ans. 33~ per cent. 4. Bought sugar at Sd. per lb. and sold it again at 41. 7s. per cwt. what did I gain per cent.? Ans. ~25 19s. 53d.. 5. If I buy 12 hhds. of wine for 2041. and sell the same gain at 141. 17s. 6d. per hhd. do I gain or lose, and what er cent.? Ans. I lose 122 per cent. 6/:At lid. profit in a shilling, how much per cent.? Ans. ~12 lOs. LOUS> AND)()i 7. At 25 cts,;. ar it In a 1oil 0,1lo lid( 41) per centA Ains. 25 per No~rE.-AVicii rgoodI are boiiaht or sold on credit, ou mI-ust calculate (Dy diicodulIt) tlie present wvorth of' their prcin order to had y~our trule ai or loss, &c. E MA1 I PL S. 1. B3oughit 164 yards of broadvlotb;vjt PIls. (3d. per yard rea(Iy mnoney, 'and sold the sani~am alit for 1.541. l0s. on 6 mnonths credit, ivxbat did I gain liy the -whole;allowing,, discount at 6 per cent. a year? ~ ~. ~ S. ~.. As 103: 100:,'5'4 I() I15P0 0,rveii wrth. 115 IS 'rhdco)St. Gainedl ~X3lt'44j vce i. 2. If I buy cloth at 4 dols. 16 cl-,. per yatd, onl eigh', months credit, and sell it again at -'o dols. 90. cts.% luer ydi re, noady rniiy, wvl ~t do I lose pcer ceit. allowing 6 per ceni discount on thje purchase pi-ice? A77s. 21 per' ccnt. 111. To know how a commiodity m1ust, be sold, to gall or lose so much per cent. R{ULE.-As 100:is to the purchase price so is 1001. or 10I dollars, with the profit added, or loss subtracted:to the sellinil price. EXAMPLES. 1. If I buy Irish linen at 2s. 3d. per yard; how must I sell it per yard to gain 25 per cent.? As 100i.: 2s. 3d.:1251. to 2s. 9d. 3 qrs. Ans. 2. If I buy rum at 1 dol. 5 cts. per gallon; how must I sell it per gallon to gain 30 pkr cent.? 3IfAs $100:$1705:$130: $1,361- cs. Ans. 3Iftea cost 54 cts. per lb.; how must it be sold per IL to lose 121- per cent.? As $100: 54 cts.::$87, 5-0 cts.:47 cts. 21 m. Ans. 4. Bought cloth at 17s. 6d. per yard, which not provinA so good as I expected, I am obliged to lose 15 per cent. ~, it; how must I sell it per yaAd? Ans. 14s. 1O0d LO)SS ANP GAIN. 131 5. If 11 cwt. I qr. 25 lb. of sugar cost 126 dols. 50 cts. how must it be sold per lb. to gain 30 per cent.? Ans. 12 cts. 8n. 6. Bought 90 gallons of wine at 1 dol. 20 cts. per gal.. 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 1.21 per cent.? Ans $1, 51 cts. S-mn, IV. When there is gained or lost per cent. to know what the commodity cost. RULE.-As 1001. or 10-ols. with the gain per cent. added, or loss ~er cent. subtracted, is. to the price, so is 100 to the prime cost. EXAMPLES. 1. If a yard of cloth be sold at 14s. 7d. and there is gain1,, 161. 13s. 4d. per cent.; what did Ohe yard cost? ~. s. d. s. d. ~. As 116 13 4: 14 7:: 100 to 12s. 6d. Ans. 2. By selling broadcloth at 3 dols. 25 cts. per yard, I:)se at the rate of 20 per cent.; what is the prime cost of aid cloth per yard? Ans. $4, 06 cts. 2-m. 3. If 40 lb. of chocolate be sold at 25 cts. per lb. aind I ain 9 per cent.; what did the whole cost me? Ans. $9, 17 cts. 4m. - 4. Bought 5 cwt. of sugar, and sold it again at 12 cents er lb. by which I gained at the rate of 25L- per cent.;,hat did the sugar cost me per cwt.? Ans. $10, 70 cts. 9m.+ V. If by wares sold at a given rate, there is so much lined or lost per cent. to know what would be gained or st per cent. if sold at another rate. R1ULE. —As the first price: is to 1001. or 100 dols. with the profit r cent. added, or loss per cent. subtracted:: so is the other price: to - gaia or loss per cent. at the other rate. N. 13. If your answer exceed 1001. or 100 dols. the cess is your gain per cent.; but if it be less than 100, it deficiency is the loss per cent. 132 FELl OW\5IIP. EXAM PLE S. 1. If I sell cloth at 5s. per yd. and thereby gain 15 pe1 cent. what shall I gain per cent. if I sell it at 6s. per yd.? s. ~ s. ~. As 5: 115:: 6: 138 iAns. gained 38per cent. 2. If I retail rum at 1 dollar 50 cents per gallon, and thereby gain 25 per cent. what shall I gain or lose per ceat if I sell it at I dol. 8 cts. per gallon 1 $ cts. $ $ cts. $ 1,50: 125:: 1,08: 90 Ans. I shall lose 10per cent. 3. If I sell a cwt. of sugar for 8 dollars, and thereby ose 12 per cent. what shall I gain or lose per cent. if I sell 4 cwt. of the same sugar for 36 dollars? Ans. I lose only 1 per cent. 4. I sold a watch for 171. Is. 5d. and by so doing lost 15 per cent. whereas I ought in trading to have clearer 20 per cent.; how mucl was it sold under its real value' ~. ~ s.d. ~. ~. ~. d. As 85: 17 1 5:: 100: 20 1 8 the prime cost. 100: 20 1 8:: 120: 24 2 0 the realvalue. Sold for 17 1 5 ~7 0 7 Answer. FELLOWSHIP, IS a rule by which the accounts of several merchant or other persons trading in partnership, are so adjustec that each may have his slare of the gain, or sustain hi share of the loss, in proportion to his share of the joit stock.-Also, by this Rule a bankllrupt's estate may be d vided among his creditors, &c. SINGLE FELLOWSHIP, Is when the several shares ef stock are continued i trade an equal term of time. RULE.-As the whole stock is to the whole gain or loss: so is ea, man's particular stock, to his particular share of the gain or loss. FEI.()WS H I 133 PRooF.-Add all tile particular shares of the gain or loss to-;ether, and if it be right, the sum will be equal to the whole gain or loss EXAMPLES. 1. Two partners, A and B, join their stock and buy a quantity of merchandise, to the amount of 820 dollars; inl the purchase of which A laid out 350 dollars, and B 47C dollars; the commodity being sold, they find their cleal gain amounts to 250 dolla:'. What is each person's share of the gailr? A put in 350 B - 470 -As f0 250 350: 106,7073n+A's share 470 143,2926+-B's share. Proof 249,9999+=$250 2. Three merchants make a joint stock of 12001. of tvhich A put in 2401. B 3601. and C. 600.; and by trading hey gain 3251. what is each one's part of the gain? Ans. A'spart ~65, B's ~97 10s. C's ~162 10s. 3. Three partners, A, B, and C, shipped 108 mules for Ahe West-Indies; of which A owned 48, B 36, and C 24; But in stress of weather, the -mariners were obliged to throw 45 of them overboard; 1 demiand how much of lhe loss each owner must susttin? Ans. A 20, B1 15, and C 10. 4. Four men traded witil 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 wiat each man gained by trading? Ans. D's stock weas $100, and A gained $53, 72 cts. 5 m. B $99, 77}L cts. C $115, 121 cts. and D $38, 37- cts. 5. A bankrupt is indebted to A 2111. to B 3001. and to C 3911. and his whole estate amounts only to 6751. 10s. which he gives up to those creditors; how much must each have in proportion to his debt? Ans. A must have ~158 Os. 33 d. B ~224 13s. 41d. and C J292 16s. 33tl. I31 tC(1M'OUUND FELI,()OWSHIP 6. A captain, mate, and 20 seamen, took a prize worti 3501 dols. of which the captain takes 11 slhares, and the mate 5 shares; the remainder of the prize is equally divided among the sailors; how much did each man receive $ cis. Ans. The captain received 1069, 75 The mate 486, 25 Each sailor 97, 25 7. Divide the number of 360 into 3 parts, which shall be to eacll other as 2, 3 and 4. Ans. 80, 120 and 160. 8. Two merchanlts have gained 4501. of which A is to have three tin-es as much as B; how much is each to have Ans. A ~337 1Os. and B ~112 10s.- 1+3=4: 450: 3: ~337 lOs. A's share. 9. Three persons are to share 6001. A is to have a certain sumn, B as much again as A, and C three times as much as B. I demand each man's part? Ans. A ~66-, B ~1339, and C ~400. 10. A and B traded together and gained 100 dols. A put in 640 dols. B put in so much that he must receive 60 dols. of the gain; I demand B's stock? Ans. $960. 11. A, B and C traded in company: A put in 140 dols. B o50 dols. and C put in 120 yds. of cloth, at cash price; they gained 230 dols. of which C took 100 dols. for his share of the gail: how did C value his cloth per yard in common stock, and what was A and B's part of the gain Ans. C' put in the cloth at 62o per yard. A gained $46 67 cts. 6 m. +and B $83, 33 cts. 3 m.+ COMPOUND FELLOWSHIP, OR Fellowship with time, is occasioned by several shares of partners being. continued in trade an unequal term of time. RuLrE.-Multiply each man's stock, or share, by the timb it was continued in trade: t1ttn, As the sum of the several products, Is to the whole gain or loss: So is each man's particliar product, To his particular share of the gain or loss. COMPOUND FELLOWSHIIP 135 EXAMIPLES. 1. A, B and C hold a pasture in common, for which they pay 191. per annum. A put in S oxen for 6 weeks; B 12 oxen for 8 weeks; and C 12 oxen for 12 weeks; what must each pay of the rent? ~.. d. 8x 6- 48 ( 48: 3 3 4 A's part. 12x 8 —96 96: 6 6 8 B's - 12X12=144 As288:191.::" 144: 9 10 0 C's - Sum 288 Proof 9 0 0 2. Two merchants traded in company; A put in 215 dols. for 6 months, and B 390 dols. for 9 months, but by misfor.ine they lose 200 dols.; how must they share the loss? Ans. A's loss $53, 75 cts. B's $146, 25 cts. 3. Three persons had received 665 dols. interest: A had put in 4000 dollars for 12 months, B 3000 dollars for 15 months, and C 5000 dollars for 8 months; how much is each man's part of the interest? Ans. A $240, B $225, and C $200. 4. Two partners gained by trading 1101. 12s.: A's stock was 1201. 10s. for 4 months, and B's 2001. for 6} months; what is each man's part of the gain? Ans. A's part ~29 18s. 37d. 3S 5 B's ~80 13s. 84 d. 472s 5. Two merchants enter into partnership for 18 months. A at first put into stock 500 dollars, and at the end of 8 months he put in 100 dollars more; B at first put in 800 dollars, and at 4 months' end took out 200 dols. At the expiration of the time they find they have gained 700 dollars; what is each man's share of the gain?. ( $324, 07 4+ A's shiare. Ans. $375,92 5+B's do. 6. A and B companied; A put in the first of January, 1000 dollars; but B could not put in any till the first of May; what did he then put in to have an equal share with A at the year's end? Mo. $ Mo. Asl2: 1000 8: 1000 X12=1500 Ans 8 (36 DOUBLE RUI.E OF THREE. DOUBLE RULE OF THREE. THE Double Rule of Three teaches to resolve at once such questions as require two or more statings in simple proportion, whether direct or inverse. In this rule there are always five terms given to find a sixth; the first three terms of which are a supposition, the last twc a demand. RULE.-In stating the question, place the terms of the supposi. tion so that the principal cause of loss, gain, or action, possess the first place; that which signifies time, distance of place, &c, in the second place; and the remaining term in the third p? ice Place the terms of demand, under those of the same kind in the supposition. If the blank place, or term sought, fall under the third term, the proportion is direct; then multiply the first and second terms together for a divisor, and the other threes for a dividend: but if the blank fall under the first or second term, the proportion is inverse; then multiply the third and fourth terms together for a divisor, and the other three for a di vidend, and the quotient will be the answer. EXAMPLES. 1. If 7 men can build 36 rods of wall in 3 days; hew many rods can 20 men build in 14 days ' 7: 3: 36 Terms of supposition. 20: 14 Terms of dsaand. 36 84 42 504 20 7 x 3=21)10080(480 rods. Aiw. 2. If 1001. principal will gain 61. intel 1st in 12 monthd what will 4001. gain in 7 months? Principal 1001. 12 mo.:. (At interest. 400 7 A^. '4/. CONJOINED PROPORTION. 137 3. If 1001. will gain 61. a year; in what time will 4001. gain 14l. ~. mo. ~ 100: 12: 6 400:: 14 Ans. 7 months. 4. If 4001. gain 141. in 7 months: what is the rate per cent. per annum 1:~. mo. Int. 400: 7: 14 100: 12 Ans. ~6. 3. What principal at 61. per cent. per annum, will give 141. in 7 months? ~. mo. Int. 100: 12: 6 7: 14 Ans. ~400. 6. An usurer put out 861. to receive interest for the same; and when it had continued 8 months, he received principal and interest, 881. 17s. 4d.; I demand at what rate per ct. per ann. he received interest? Ans. 5per cent. 7. If 20 bushels of wheat are sufficient for a family of B persons 5 months, how much will be sufficient for 4 pertons {21 months? Ans. 24 bushels. 8. If 80 men perform a piece of work in 20 days; how many men will accomplish another piece of work 4 times as large in a fifth part of the time? 80: 20:: 1 4:: 4 Ans. 600. 9. If the carriage of 5 cwt. 3 qrs. 150 miles, cost 24 dollars 58 cents; what must be paid for the carriage of 7 cwt. 2 qrs. 25 lb. 64 miles, at the same rate? Ans. $14, 08 cts. 6m. + 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 feet long, 8 feet high, and 6 feet thick? 8: 12:: 20x6x4 24: 200 x8 x6 80 days. Ans. I I II _ ~,.- Ir'l — -- _ I CONJOINED PROPORTION, IS when the coins, weights or measures of several counries are compared in the same question; or it is joining nany proportions together, and by the relation which M 2 138 CONJOINED PROPORTION. several antecedents have to their consequents, the propor, tion between the first antecedent and the last consequent is discovered, as well as the proportion between the others ir their several respects. NOTE.-This rule may generally be abridged by cancelling 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, are equal to a given quantity of the last. RULE.-Place the numbers altcrnately, beginning at the left hand, and let the last number stand on the left hand column; then multiply the left hand column continually for a dividend, and the right hand for a divisor, and the quotient will be the answer. EXAMPLES. 1. If 100 lb. English make 95 lb. Flemish, and 19 lb. Flemish 25 lb. at Bologna; how many pounds English are equal to 50 lb. at Bologna? Ib. Ib. 100 Eng.=95 Flemish. 19 Fle. =25 Bologna. 50 Bologna. Then 95 x 25=2375 the divisor. 95000 dividend, and 2375)95000(40 Ans. 2. If 40 lb. at New-York make 48 lb. at Antwerp, and 30 lb. at Antwerp make 36 lb. at Leghorn; how many lb. at New-York are equal to 144 lb. at Leghorn? Ans. 100lb 3. If 70 braces at Venice be equal to 75 braces at Leghorn, and 7 braces at Leghorn be equal to 4 Americajl yards; how many braces at Venice are equal to 64 American yards? Ans. 104-5. CASE IT. When it is required to find how many of the last sort of coin, weight or measure, mentioned iP te*, question, are equal to a giren quantity of the first. EXCHANOE. 139 RULE.-Place the numbers alternately, beginning at the left hand, and let the last number stand on the right hand; then multiply the fArit row for a divisor, and tiic second for a dividend EXAMPLES. 1. If 24 lb. at New-London make 20 Ib. at Amsterdam, and 50 lb. at. Amsterdam 60 lb. at Paris; how many at Paris are equal to 40 at New-London 1 Left. 'Rigrt. 24 =20 20 x 60 x 40 = 48000 50 60 40 Ans. 40 24 X 50 1200 2. If 55 ib. at New-York make 45 at Amsterdam, and 60 lb. at Amsterdam make 103 at Dantzic; how many lb. at Dantzic are equal to 240 at N. York? Ans. 278-,1 3. If 20 braces at Leghorn be equal to 11 vares at Lisbon, and 40 vares at Lisbon to 80 braces at Lucca; how many braces at Lucca are equal to 100 braces at Leghorn? Ans. 110. EXCHANGE. BY this rule merchants know what sum of money ought io be received in one country, for auy sum of different specie paid in another, according to the given course of exchange. To reduce the moneys of foreign nations to that of the nited States, you may consult the following TABLE: owing the value of the moneys of account, of foreign nations, estimated in Federal money.* $ cts. Pound Sterling of Great Britain, 4 44 Pound Sterling of Ireland, 4 10 Livre of France, 0 18' Guilder or Florin of the U. Netherlands, 0 39 Mark Banco of Hamburgh, 0 331 Rix Dollar of Denmark, 1 0 * Lawn V. S. A. t I( EXCHANGE. Rial Plate of Spain, 0 10 Milrea of Portugal, 1 24 Tale of China, 1 48 Pagoda of India, 1 94 Rupee of Bengal, 0 55i I.-OF GREAT BRITAIN. EXAMPLES. 1. In 451. 10s. sterling, how many dollars and cents? A pound sterling being=444 cents, Therefore-As 11.: 444cts.: 45,51.: 20202cts. Ans 2. In 500 dollars how many pounds sterling As 444 cts.: 11.:: 50000 cts.: 1121. 12s. 3d.+ Ans. II.-OF IRELAND. EXAMPLES. 1. In 901. 10s. 6d. Irish money, how many cents? 11. Irish=410 cts. ~. c cts. ~. cts. $ cts 'herefore-As 1:410: 90,525: 37115 —=371, 151 2. In 168 dols. 10 cts. how many pounds Irish? As 410 cts.: 11.:: 16810 cts.: ~41 Irish. Ans. III.-OF FRANCE. Accounts are kept in livres, sols and deniers. {12 deniers, or pence, make 1 sol, or shilling. 20 sols, or shillings, - 1 livre, or pound. EXAMPLES. 1. In 250 livres, 8 sols, how many dollars and cents. 1 livre of France =18-S cts. or 185 mills.. m. ~. m. $ cts. m. As 1: 185:: 250,4: 46324 -46 32 4 Ans. 2. Reduce 87 dols. 45 cts. 7 m. into livres of France. mills. liv. mills. liv. so. den. As 185: 1:: 87457: 472 14 9+ Ans. IV.-OF THE U. NETHERLANDS. Accounts are kept here in guilders, stivers, groats anj henmgs. t 8 phennings make 1 groat. f2 groats - 1 stiver. 20 stivers - 1 guilder or florin. A guilder is=39 cents, or 390 mills. EXCHANCGE 14t ' EXAMPLES. Reduce 124 guilders, 14 stivers, into federal money. Guil. cts. Guil. $ d. c. m. As 1: 39: 124,7: 48, 6 3 3 Ans. mills. G. mills. G. As 390: 1:: 48633: 124,7 Proof. V.-OF HAMBURGH, IN GERMANY. iccounts are kept in Hamburgh in marks, sous and de. L.- ros-lbs, and by some in rix dollars. I 12 deniers-lubs make 1 sous-lubs. 16 sous-lubs, - 1 mark-lubs. 3 mark-lubs, - 1 rix dollar. NiOTE. —A mark is = 331 cts. or just I of a dollar. AULE. —Divide the marks by 3, the quotient will be dollars. EXAMPLES. Reduce 641 marks, 8 sous, to federal money. 3)641,5 $213,833 Ans. But to reduce federal money into marks, multiply the -iven sum by 3, &c. EXAMPLES. Reduce 121 dollars, 90 cts. into marks banco. 121,90 3 365,70=365 marks, 11 sous, 2,4 den. Ans. VI.-OF SPAIN. Accounts are kept in Spain in piastres, rials, and mar. adies. 34 marvadies of plate make I rial of plate. 8 rials of plate - 1 piastre or piece of 8. To reduce rials of plate to federal money. Since a rial of plate is = 10 cents or 1 dime, you need _mnly call the rials so many dimes, and it is done. EXAMPLES. 485 rials=485 dimes=48 dols. 50 cts. &c. '142 EX 'IIANGlE, But to reduce cciets iito rials of 1la-te, divide by 10 Thus, 845 cents-: 1() —=,5=-S4 rials, 17 marvadies, &c. VII.-OF PORTUGAL. Accounts are kept throughout this kingdom in miF s, and reas, reckoning 1000 reas to a milrea. NOTE.-A milrea is - 124 cents; therefore to r iace milreas into federal money, multiply by 124, and th product will be cents, and decimals of a cent. EXAMPLES. 1. In 340 milreas how many cents? 340 x 124=42160 cents=$421, 60 ct Ins. 2. In 211 milreas, 48 reas, how many cents? NOTE.-When the reas are less than 100, place i cipher before them.-Thus, 211,048 x 124=26169,952 ctj. or 261 dols. 69 cts. 9 mills. + Ans. But to reduce cents into milreas, divide them by 124; and if decimals arise you must carry on the quotient as far as three decimal places; then the whole numbers thereof will be the milreas, and the decimals will be the reas. EXAMPLES. 1. In 4195 cents, how many milreas 4195 124=33,830+or 33 milreas, 830 reas. Ans. 2. In 24 dols. 92 cents, how many milreas of Portual 1 Ans. 20 milreas, 096 reas. VIII.-EAST-INDIA MONEY. To reduce India Money to Federal, viz. Tales of China, multiply with 148 Pagodas of India, 194 Rupee of Bengal, 55; EXAMPLES. 1. In 641 Tales of China, how many cents? Ans. 94868 2. In 50 Pagodas of India, how many cents? Ans. 9700 3. In 98 Rupees of Bengal, how many cents 1 Ans 5439 VULGAR P'RACTIONS 143 VULGAR FRACTIONS. I 1AVING briefly introduced Vulgar Fractions immeaiately after reduction of whole numbers, and given some Leneral definitions, and a few such problems therein as were necessary to prepare and lead the scholar immediateTy to decimals; the learner is therefore requested to read hose general definitions in page 69. Vulgar Fractions are either proper, improper, single, 1ompound, or mixed. 1. A single, simple, or proper fraction, is when the nu. nerator is less than the denominator, as I, 3, -, I5, &c. '2. An Improper Fraction, is when the numerator ex ~eeds the denominator, as -, 7,?1, &c. 3. A Conmpound Fraction, is the fraction of a fraction, 'oupled by the wold of, thus, 2- of, of 2 of 3, &c. 4. A Mixed Nunlber, is composed of a whole number and fraction, thus, 8, 14 —9, &c. 5. Any whole number may be expressed like a fraction sy drawing a line under it, and putting 1 for denominator, hus, 8 —, and 12 thus, 1r2, &c. 6. The common measure of two or more numbers, is hat number which will divide each of them without a renainder; thus, 3 is the common measure of 12, 24, and 30: -id the greatest number which will do this is called thl greatest common measure. 7. A number, which can be measured by two or inore umlbers, is called their common multiple: and if it be the east number that can be so measured, it is called the lea.% ommon multiple: thus 24 is the common multiple 2, 3 ana; but their least common multiple is 12. To find tlhe least common multiple of two or more num. ers. RULE.-1. Divide by any number that will divide two or more of ie given numbers without a remainder, and set the quotients, togejer with the undivided numbers, in a line beneath. 2. Divide the second lines as before, and so on till there are no two umbers that can be divided; then the continued product of the diiaore and qudtients, will givef t-he multiple required. 144 Ii EDUCTION OF VULGAR FRACTIONS. EXAMPIES. 1. What is the lca:t common multiple of 4, 5,6 and l10 Operation; 5)1 - 6 10; 6)4 1 G 2 X2 1x3 1 5 X2x2x3=60 Ans. 2. What is the common multiple of 6 and 8? Ans. 24. 3 What is the least number that 3, 5, 8 and 12 wil measure? Ans. 120. 4. What is the least number that can be divided by thr 9 digits separately, without a remainder? Ans. 2520. REDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form into another, in or der to prepare them for the operation of Addition, Sul) traction, &c. CASE I. To abbreviate cr 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 remainder, till nothing remains; the last divisor is the common measure.* 2. Divide both of the terms of the fraction by the common measure, and the quotients will make the fraction required. * To find the greatest common measure of more than two numbers, you must find the greatest common measure of two of them as per rule above; then, of that common measure and one of the other numbers, and so on through all the numbers to the last; then will the greatest common mea sure last found be the answer. RED)UCTJON OF VI;LAR FRACTIONS 145 Or, if u c you ch you lmay take that easy method in Problem I. p,.ge 09.) EXAMPLES. 1. Reduce - - to its lowest terms. i 8 )6( Operation. 4)u( R6 common measure, 8)41=-3 Ans. 2. Reduce,- to its lowest terms. Ans. {# 3. Reduce to2 to its lowest terms Ans. 13 4. Reduce 1"IR to its lowest terms. Ans. CASE II. To reduce a mixed number to its equivalent improper fraction. RULE.-Multiply tle whole number by'the denominator of the gi-, n fraction, and to the product add the numerator, this sum written 0-ove the denominator will form the fraction required EXAMPLES. 1. Reduce 45z to its equivalent improper fraction 45 x 8+7-37 Ans. 2. Reduce 19}2 to its equivalent improper fraction. Ans., ~4 3. Reduce 16ljOy to an improper fraction. Ans. leis 4 Reduce G61 5 to its equivalent improper fraction. Ans. 2 z.Q5 CASE III. To find the value of an improper fraction. R jL.I.-Divide the numerator by the denominator, and the quo ent will be the value sought. EXAMPLES ANSWERS. 1. Find the value of 48 5)48(9} 2. Find the value of 35_4 19 3. Find the value of 9-313 84'4. Find the value of2a 61 ' 5 Find the value of 7 I. 14$ 19/DLUCTION OF VULGAR FRACTIONS. CASE IV. To reduce a whole number to an equivalent fraction, haw 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 fraction required. EXAMPLES. 1. Reduce 7 to a fraction whose denominator will be 9. Thus, 7 X 9=63, and 3 the Ans. 2. Reduce 18 to a fraction whose denominator shall be 12. Ans. 2Ai 3. Reduce 100 to its equivalent fraction, having 90 for a denominator. Ans. 9 -o 0=9 0==1 CASE V. To reduce a compound fraction to a simple one of equal value. RULE.-1. Reduce all whole and mixed numbers to their eql iva lent fractions. 2. Multiply all the numerators together for a new numerator, ant all the denominators for a new denominator; and they will forwr thA fraction required. EXAMPLES. 1. Reduce } of 2 of 3 of -A to a simple fraction 1x2x3x4 =2-24-4 Ans. 2 4X0 10 4 x1 2x3x4x10 2. Reduce ] of A of 3 to a single fraction. Ans. Q,a Reduce j of -} - of -, to a single fraction. Ans., o 4. Reduce a of i of 8 to a simple fraction. Ans. W?=3 -5. Reduce { of '= of 42~ to a simple fraction. Ans. 12- 6 0 o-21 w t-.-If the denominator of any member of a com, ~A'i3,tction be equal to the pumeratnr of another mem REDUCTION OF VULGAR FRACTIONS. 141 ber thereof, they may both be expunged, and the other members continually multiplied (as by the rule) will produce the fraction required in lower terms. 6. Reduce 2 of 4 of 7 to a simple fraction. Thus 2 x 5 - fig_ Ans. 28 14 4x7 7. Reduce 3 of 1 of A of 'I to a simple fraction. Ans. Hi=2 CASE VI. Co reduce fractions of different denominations to equiva lent fractions having a common denominator. RULE I. 1. Reduce all fractions to simple terms. 2. Multiply each numerator into all the denominators except its own, for a new numerator; and all the denominators into each other continually for a common denominator; this written under the several new numerators will give the fractions required. EXAMPLES. 1. Reduce -, 2, 4, to equivalent fractions, having a comanon denominator. + 3- + - =24 common denominator. 1 2 3 x3 2 3 3 4 9 x4 4 2 12 16 18 new numerators. 24 24 24 denominators. 1. Reduce -, ~-, and 1, to a common denominator. Ans. 8 4, _ and Ae. 3. Reduce, i, j, and I, to a common denominator. Ans., a, 9,, and A4g I 8 REDUCTION OF VULGAR FRACTIONS. 4. Reduce 4, 4s, and —, to a common denominator 800 300 400 and =0 L ' and 44 1 — Ans. 1000 1000 1000 5. Reduce ~,., and 121-, to a comlmlon deniominiatoi. Gu * 7-? 27 " 7 2~ 6. Reduce 2,, and g of 1-, to a conmmon denominator A nIS. -37-~ 2 4a 9 3, 1 9 8 06. 34663, 345U 2, WI The foregoing is a general rule for reducing fractions tt a common denominator; but as it iyill save much labour to keep the fractions in the lowest terms possible, the follows ing Rule is much preferable. RULE II. For reducing fractions to the least common denominrator, (By Rule, page 143) find the least common multiple cf all the denominators of the given fractions, and it will hi the common denominator required, in which divide each particular denominator, and multiply the quotient lby itf own numerator, for a new numerator, and the new numi;m rators being placed over the common denominator, will ex press the fractions required in their lowest terms. EXAMPLES. 1. Reduce -, 3, and 5, to their least common deroiniiaka 4)2 4 8 2)2 1 2 1 1 1 4X2=8 the least coin. denominator. 8-2 x 1=4 the 1st numerator. 8-4 x 3=6 the 2d numerator. — 8 x 5=5 the 3d numerator. These numbers placed over the denominator, give the answer 4, -, equal in value, and in much lower terms Ihan the general Rule would produce 3q2, 8, 44. 2. Reduce 3, 4, and -7, to their least common denomina tor. Ans. i, 41.. REDUCTION OF VULGAR FRACTIONS. 149 3 Reduce - A i and 8 to their least 'common denominat(.. Ans. 29 A 16 I 4. Reduce 2 4 5- and - to their least common denominator. Ans. 18 1 f0 o 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 denomination you would reduce it to; lastly, reduce this coin iound fraction to a single one, by Case V. EXAMPLES. 1. Reduce 5 of a penny to the fraction of a pound. By comparing it, it becomes 5 of -rL of -0 of a pound. 5xl x 5 -- - = Ans. 6x 12x20 1440 2. Reduce T-54 of a pound to the fraction of a penny. Compared thus - o- of yo of 2d. Then 5 x 20 x 12 --, 1 2 0_ 0,. 1440 1 1 3. Reduce 2 of a farthing to the fiaction of a snilling. Ans. d 4. Reduce - of a shilling to the fraction of a pouna. Ans. h-='.o 5. Reduce { of a pwt. to the fraction of a pound troy. Ans. OTAr y=i6. Reduce v of a pound avoirdupois to the fraction of eft. Ans. ' cart. 7. What part of a pound avoirdupcis is T' of a ct. Compounded thus T ofI of aiI — 6= — Ars. 8. What part of an hour is Th of a week. N IBn REDUCTION OF VULGAR FRACTIONS. 9. Reduce 3 of a pint to the fraction of a hhd. Ans. r{l 10. Reduce 4 of a pound to the fraction of a guinea. Compounded thus, - of 2 of 8s.=4 Ans. 11. Express 5- furlongs in the fraction of a mile. Thus 54= 2' of — l Ans. 12. Reduce a of an English crown, at 6s. Sd. to the frao tion of a guinea at 28s. Ans. -9 of a guinea. CASE VIII. To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c. RULE. Multiply the numerator by the parts in the next inferiei denomination, and divide the product by the denominator and if any thing remains, multiply it by the next inferior de. nomination, and divide by the denominator as before, and sc on as far as necessary, and the quotient will be the answer NOTE.-This and the following Case are the same with Problems II. and III. pages 70 and 71; but for the scho lar's exercise, I shall give a few more examples in each. EXAMPLES. I What is the value of t_ _ of a pound? Ans &s. c d. 2. Find the value of I of a cwt. Ans. 3 qrs. 3.. 1 o2.2 8. Find the value of I of 3s.d. Ans. 3s. Od. 4. How much is -,g of a pound avoirdupois? Ans. 7 oz. 10 dr. 5. How much is 4 of a hhd. of wine? Ans. 45 gals 6. What is the value of 44 of a dollar? Ans. 5s. 7 Ld " What is the value of A of a guinea? Ans. 1Ss ADDITION OF VULGAR FRACTIONS. 151 8. Required the value of - 7 of a pound apothecaries. Ans. 2 oz. 3 grs. 9. How much is 7 of 51. 9s.? Ans. ~4 13s. 5+d. 10. How much is 2 of 2 of 3 of a hhd. of wine 1 Ans. 15gals. 3 qts CASE IX. To reduce any given quantity to tile fraction of any greater denomination of the same kind. [See the Rule in Problem III. page 71.] EXAMPLES FOR EXERCISE. 1. Reduce 12 lb. 3 oz. to the fraction of a cwt. Ans. i-s_, 2. Reduce 13 cwt. 3 qrs. 20 lb. to the fraction of a ton. Ans. 39 3. Reduce 16s. to the fraction of a guinea. Ans. 4 4. Reduce 1 hhd. 49 gals. of wine to the fraction of a n. Ans. 4 5. What part of 4 cwt. 1 qr. 24 lb. is 3 cwt. 3 qrs. 17 lb. oz.? Ans. 7 ADDITION OF VULGAR FRACTIONS. RULE. Reduce compound fractions to single ones; mixed num-;rs to improper fractions; and all of them to their least ammon denominator, (by Case VI. Rule II.) then the sum f the numerators written over the common denominator ill be the sum of the fractions required. EXAMPLES. 1. Add 51 4 and. of f together. 5-='- and 2 of — =4 a 8 24 hen Y, {, -1 reduced to their least common denominator by Case VI. Rule II. will become -2, 8, It Then 132+18+14:= d=6i or 61 Ans. 152 ADDITION OP VULGAR FRACTIONS. 2. Add I, 4, and 3 together. 3. Add, 4, and 4 together. 4. Add 124 32 and 4 4 together. 5. Add ~ of 95 and 8 of 141 together. ANSWERS. 1I 18 20H4 1 2 441 7 48 NOTE 1.-In adding mixed numbers that are not corn pounded with other fractions, you may first find the sum og the fractions, to which add the whole numbers of the given mixed numbers. 6. Find the sum of 53, 74 and 15. I find the sum of and 4 to be a l0-1 Then 1 — +5+7+15=281- Ans 7. Add 2 and 17- together. ANSWERS. 17,80 8. Add 25, 84 and - of f f of 33 -NOTE 2.-To add fractions of money, weight, &c. redu~s fractions of different integers to those of the same. Or, if you please, you may find the value of each fraction by Case VIII. in Reduction, and then add them in their proper terms. 9. Add 4 of a shilling to 3 of a pound. 1st method 2d method. of — = 4. 3~-=7s. 6d. Oqra. Then T14+-} 4 --- ~. 4s.-0 6 34 Whole value by Case VIII. is 8s. Od. 3- qrs. Ans. Ans. 8 0 33 By Case VIII. Reduction. 10. Add - lb. Troy, to 5 of a pwt. Ans. 7 oz. 4pwt. 134-grs. 11. Add 4 of a ton, to lo of a cwt. Ans. 12 cwt. 1 qr. 8 lb. 12A-&- oz. 12. Add 3- of a mile to — 7 of a furlong. Ans. 6fr. 2Spo. 13. Add 4- of a yard, - of a foot, and 4 of a mile together Ans. 1540 yds. 2ft. 9 in. 14. Add - of a week, l of a day, - of an hour, and I of a minute together. Ans 2 da. 2 ho. 30 mia. 45 c,. SUBTRACTION OF VIULGAR FRACTIONS. 153 SUBTRACTION OF VULGAR FRACTIONS. RULE.* Prepare the fraction as in Addition, and the difference tf the numerators written above the common denominator, will give the difference of the fraction required. EXAMPLES. 1. Fr6m i take 2 of - - of {-j -T-{7 Then 4 and a — = d7 Therefore 9-7= —2=- the Ans. 2. From. - take 4 Answers. 41 3. From 1-' take -7 -,4. From 14 take '-: 13-, 5. What is the difference of 94 and -7? I5 6. What differs; from '-? 7. From 144 take. of 19 1-2 8. From 3- take - 0 remains. 9. From -- of a pound, take 4 of a shilling. 3of -i~. Then from I1~. take - ~. Ans. 7~. NOTE.-In fractions of money, weight, &c. you may, if you please, find the value of the given fractions (by Caso VIII. in Reduction) and then subtract them in their pr per terms. 10. From i-7~. take 38 slhillings. Ans. 5s. 6d. 23 qrs. 11. From 3 of an oz. take 7 of a pwt. Ans. llpwt. 3gr. 12. From ~ of a cwt. take 1- of a lb. Ans. 1 qr. 27 lb. 6 oz. 10- dr. 13. From 32 weeks, take -' of a day, and I of 2 of 4:f:m hour. Ans. 3 w. 4 da. 12 ho. 19 mini. 174 sec. * In subtracting mixed numbers, when the lower fraction is greater than 'he upper one, you may, without reducing them to improper fractions, subract the numerator of the lower fraction from the common denominator,.nd to that difference add the upper numerator, carrying one to the unit's.lace of the lower whole number. Also, a fraction may be subtracted from a whole number by taking the numerator of the fraction from its denominator. and placing the remainder - rer the denominator, then taki'm one from the whole number 154 MULTIP'(IAT'IO)N, DIVISION, &tC. MULTIPLICATION OFl VULGAR FRACT'IONS RULE. Reduce whole anid m.ixed iinbics- to th e improper frac tions, mixed fra,; ions to simple oies, and those of different integers to the same; then multiply all the numerators to gether for a new numerator, and all the denominators to gether for a new denominator. X A11 'PL ES. 1. Multiply - by Answers. 12-= II 56 2. MIultiply 3 by: 6 3. Multiply 51 by - 4. Multiply 2 of 7 by A 3} 5. iMultiply 1- by i, 32 6. Multiply f of by of 5 131 7. Multiply 7- by 9'- 693 8. Multiply - of, by, of:7 '9. What is the continued proditct of oft ', 7, 54 and { of? Ans. 4- - DIVISION OF VULGAR FRACTIONS. RULE. Prepare the fractions as before; then, invert the divisot and proceed exactly as in Multiplication:-The product. will ba the quotient required. EXAMPLES. 1. Divide 4. by 3 4 x 5 Thus, — = Ans. 3 x 7 2. Divide 7 by - 3. Divide 4 of 4 by 4 4. What is the quotient of 17 by -? 5. Divide 5 by -7, 6. Divide - of- of 7 by I of 3 7. Divide 45 by 5 of 4 8. Divide 71 by 127 9. Divide 52054 by 4 of 91 Answers. 1 Aa d 74 31 2a 71} RULE OF THREE DIRECT, INVERSE, &cC. I 5 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 Three in whole numbers, and invert the first term in the proportion; then multiply all the three, terms continually together, and the product will be the answer, in the same name with the second or middle term. EXAMPLES. 1. If f of a yard cost 4- of a pound, what will -- of an Ell Einglish cost? yd.= of 4 of =2o or I Ell English. Ell. ~. Ell. s. d. qrs. As -: 3:: - l And - x -q X i=,5-?~.=10 3 1 Ans. 2. If 3 of a yard cost - of a pound, what will 403 yards m fie to? Ans. ~59 8s. 61 d. 3. If 50 bushels of wheat cost 17j1. what is it per bush el? Ans. 7s. Od. 1 3 qrs. 4. If a pistareen be worth 142 pence, what are 100 pista-:eens worth? Ans. ~6 5. A merchant sold 5- pieces of cloth, each containing 24- yards at 9s. - d. per yard; what did the whole amount to? Ans. ~60 10s. Od. 32 qrs. 6. A person having 3 of a vessel, sells - of his share for 3121.; what is the whole vessel worth? Ans. ~780 7. If v of a ship be worth 2 of her cargo, valued at 80001. what is the whole ship and cargo worth? Ans. ~10031 14s. 11 'd. — b INVERSE PROPORTION. RULE. Prepare the fractions and state the question as before. hen invert the third term, and multiply all the three terms together, the product will be the answer. I.66 RULE OF THREE DIRECT IN DECIMALS EXAMPLES. 1. How miuch sliallooii that is 3 yard wide, wvill line 5j yards of cloth which is 1L3 yard wvide? Yds. yds. yds. 4f ds. '2. If a man perform a journey in 3-1 days, when the day Ui.21- hoius long; in how mnany days will he do it whea -he day is but 9i hours? Ans. 4 —~ days. 3. If 13 men in 112- days, mnow 2111 acres, in how imany dhays wvill 8 men (10 the samc?1 An's. I8 S2days. 4. How much in lengfth that is 71 inches broad, wvill inake a square foot? Ans. 20 inches. 5. If 252s. will Pay for the carriage of a cwt. 145k miles; hiow far may 61 cwt. lbe carried for the same money?1 Ans. 22 —Qr mile. 6. How many yards of baize which is I1I yards wi jO,P "Il line ]ST yards of camnblet 3 yard wide? Ans. I11 ydIs. 1 qr. I1~ na.;'TJ[FE OF TiIRtE DIRECT IN DECIMALS. RULE. Reduce your fractions to decimals, and state your ques -fion as in whole numbers; multiply the second and third to.. gethler; divide by the first, and the quotient wvill be thie amr. swer, &C. EXAMPLES. I. If - of a yard cost -I7- of a pound; what will 15-1 yardsl,vomie to? 8-,875-f~=,583+ and4=7 Yds. ~. Y'ds. ~. ~. s. d. qrs. As,8705:,583: 15,75: 10,494=10 9 10 2,24 Ans 2. If 1 pint of wvine cost 1,2s. what cost 12,5 hhidsI Ans. ~378 -3. If 44' yards cost 3s. 4A d. what will 301 yards costt Ans. LI 4s. 3d. 3 qrs. + SIMPLE INTEREST BY DECIMALS. 157 4. If 1,4 cwt. of sugar cost 10 dols. 9 cts., what will 9 zwt. 3 qrs. cost at the same rate? cw't. $ cwt. $ As 1,4:: 10,09:: 9,75: 70,269=$70, 26 cts. 9 m. + 5. If 19 yards cost 25,75 dollars, what will 435- yards come to? Ans.. $590, 21 cts. 74-~ m. 6. If 345 yards of tape cost 5 dols. 17 cents, 5 m., what will one yard cost? Ans.,015=11 cts. 7. If a man lay out 121 dollars 23 cts. in merchandise, and thereby gains 39,51 dollars, how much will he gain in laying out 12 dollars at the same rate? Ans. $3,91=$3, 91 cts. 8. How many yards of riband cal I buy for 251 dols. if ~93 yards cost 4}- dollars? Ans. 178- yards. 9. If 178k yds. cost 251- dollars, what cost 293 yards? Ans. $4, 10. If 1,6 cwt. of sugar cost 12 dols. 12 cts., what cost 3,hids., each 11 cwt. 3 qrs. 10,12 1b.? Ans. 269,072 dols. —269, 7 cts. 2 m.+ SIMPLE INTEREST BY DECIMALS. A TABLE OF RATIOS. Jiate per cent.l Ratio. [Rate per cent.l Ratio. 3,03 i 5',055 2 -4,04 6,06 41,045 i,065 5,05 7 -,07 Ratio is the simple interest of 11. for one year; or in fe deral money, of $1 for one year, at the rate per cent. agreed on. RULE. Multiply the principal, ratio and time continually toge. ther, and the last product will be the'interest required. EXAMPLES. 1. Required the interest of 211 dols. 45 cts. for 5 years at 5 per cent. per annum? 158 SIMPLE INTEREST BY DECIMALS. $ cts. 211,45 principal.,05 ratio. 10,5725 interest for one year. 5 multiply by the time. 52,8625 Ans.==52, 86 cts. 2- m. 2. rWhat is the interest of 6451. 10s. for 3 years, at 5 pet cent. per annum? ~645,5 X 06 x3=116,190=-~116 3s. 9d, 2,4 qrs. Ans. 3. What is the interest of 1211. 8s. 6d. for 4~ years, at 6 per cent. per annum? Alns. ~32 15s. 8d. 1,36 qrs. 4. What is the amount of 536 dollars, 39 cents, for 1I years, at 6 per cent. per annum? Ans. $584,6651. 5. Required tlhe amount of 648 dollars 50 cents for 12| years, at 51- per cent. per annum-? AIns. $1103, 26 cts. CASE II. The amounnt, time and ratio given, to find the principal. RULE.-MIultiply the ratio by the time, add unity to the product for a divisor, by which sum divide the amount, and the quotient will be the principal. EXAMPLES. 1. What principal will amount to 1235,975 dollars, in 5 years, at 6 per cent. per annuml? $ $,;6 x 5+ 1=1,30, 1235,975(950,75 Ans. 2. What principal will amount to 8731. 19s. in 9 yeais, at 6 per cent. per annum? Ans. ~567 10s. 3. What pIrilcipal will amount to $626, 6 cents in I! years, at 7 per cent.? Ans. $340,45= 0340, 25 cts. 4. What principal will amount to 9561. 10s. 4,125d. In 83 years, at 5' per cent.? Ans. ~645 15s. CASE III. The amount, principal and time given, to find the ratio. RULE.-Subtract the principal from the amount, divide the re. mainder by the product of' the time and principal, and the quotient will be the ratio. EXAMPLES. 1. At what rate per cent. will 950,75 dollars amount to 1230,975 dollars il 5 years? SIMPLE INTEREST BY DECIMALS. From the amount = 1235,975 Take the principal = 950,75 950,75 x 5=-4753,75)2S5,2250(,06=6 per cent. 285,2250 Ans. 2. At whlat rate per cent. will 5671. 10s. amount to S731. 19s. in 9 years? Ans. 6 per cent. 3. At what rate per cent. will 340 dols. 25 cts amount a 626 dols. 6 cts. in 12 years? Ans. 7 per cent. 4. At what rate per cent. will 6451. 15s. amount to 9561. 10s. 4,125d. in 82 years? Ans. 5~ per cent. CASE IV. The amount, principa, and rate per cent. given, to find the time. RULE.-Subtract the principal from the amount; divide the remainder by the product of the ratio and principal; and the quotient vill be the time. EXAMPLES. 1. In what time will 950 dols. 75 cts. amount to 1235 iollars, 97,5 cents, at 6 per cent. per annum? From the amount $1235,975 Take the principal 950,75 950,75 X 06=57,0450)285,2250(5 years, Ans. 285,2250 2. In what time will 5671. 1Os. amount to 8731. 19s. at o per cent. per annum? Ans. 9 years. 3. In what time will 340 dols. 25 cts. amount to 626 Aols. 6 cts. at 7 per cent per annum? Ans. 12 years. 4. In what time will 6451. 15s. amount to 9561. 1Gs. l,125d. at 5~ per ct. per annum? Ans. 8,75=84 years, TO CALCULATE INTEREST FOR DAYS. RULE.-Multiply the principal by the given number of days, and that product by the ratio; divide the last product by 365 (the number of days in a year) and it will give the interest required. EXAMPLES 1. What is the interest of 3601. 10s. for 146 days, at 6 pr. ct.? A TABLE, shoowing rne number of Days from any aay of one month, to the same day of any other month. FROM ANY DAY OF i Je.I —ec.I -4 x a W Q cU7 Li: Z w 4 e C& -3:?4 W s4 ~~ t-;.; C,:) T3O i a: 11 t C r * C ":0 D2 Jan. JFeb. lMlar. [ Jan. 365 334 306 Feb. 311 365 337 Afar. 591 28' 365 Ap'l. 90 591 31 May 120 89 61 June 151 1201 92 Juiy 181 1501 122 Aug. 212 181 153 Sept. 243 212 184 Oct. 273 2421 214 |Nov. 304 2731 246 Dec.1 334 3031 275 4p'l. May June 275 245 214 306 276 245 334 3041 273 365 335 304 30 365 334' 61 31 365 91 61 30 122 921 61 153 123 91 183 153 1221 214 184 153 244 214 183 July IAug. Sept Oct. Nov.lDec.! 184 153 122 92 61 31 215 1841 153 123 92 62 243 212 181 151 120 90! 274 243 212 182 151' 1211 304 273 242 212 181 151j 335 304 273 243 2121 182j 365 334 3031 273 242 21 2 31 365 334 304 273 243 62 31 365 335 304 274 92 61 30 365 334 304j 123 92 61 31 365 3351 153 122 91i 61 30 365 C= C" I 1 I rC CV3 C U ---- --- - -- --- SIMPLE INTEREST BY DECIMALS. 161 When interest is to be calculated on cash accounts, &c. where partial payments are made; multiply the several balances into the days they are at interest, then muliply the sum of these products by the rate on the dollar, and divide the last product by 365, and you will have the whole interest due on the account, &c. EXAMPLES. Lent Peter Trusty, per bill on demand, dated 1st of June, 1800, 2000 dollars, of which I received back the 19th of August, 400 dollars; on the 15th of October, 600 dollars; on the 11th of December, 400 dollars; on the 17th of February, 1801, 200 dollars; and on the 1st of June 400 dollars; how much interest is due on the bill, rekoning at 6 per cent.? 1800. dols. days. products. June 1, Principal per bill, 2000 79 158000 August 19, Received in part, 400 Balance, 1600 57 91200 October 15, Received in part, 600 Balance, 1000 57 57000 December 11, Received in part, 400 1801. Balance, 600 68 40800 February 17, Received in part, 200 Balance, 400 104 41600 June 1, Rec'd in full of principal, 400 388600 Then 388600,06 Ratio. $ cts. m. 365)23316,00(63,879 Ans. = 63 87 9+ The following Rule for computing interest on any note, or obligation, when there are payments in part, or endorsements, was established by the Superior Court of the State of Connecticut, in 1784 U Z 162 QSIMPLE INTEREST BY DECIMALS. RULE. I' Compute the interest to the time of the first payment if tlht be one year or more from the time the interest commenced, add it to the principal, and deduct the payment from the sum total. If there be after payments made, compute the interest on the balance due to the next payment, and then deduct the payment as above, and in like manner from 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 interest hath accrued, then compute tho interest on the principal sum due on the obligation for omn year, add it to the principal, and compute the interest on. the sum paid, from the time it was paid, up to the end oi the year: add it to the sum paid, and deduct that sum from the principal and interest added as above.* " If any payments be made of a less sum than the interest arisen at the time of such payment, no interest is to be con, puted but only on the principal sum for any period." Kirby's Reports, page 49. EXAMPLES. A bond, or note, dated January 4th, 1797, was given foi 1000 dollars, interest at 6 per cent. and there were pay ments endorsed upon it as follows, viz. $ 1st payment February 19, 1798, 200 2d payment June 29, 1799, 500 3d payment November 14, 1799, 260 I demand how much remains due on said note the 24tl of December, 1800? 1000,00 dated January 4, 1797. 67,50 interest to February 19, 1798=13- months. 1067,50 amount. [Carried up.] * If a year does not extend beyond the time of final settlement but if it does, then find the amount of the principal sum due on the obligation, up to the tune of settlement, and likewise find the amount of the sum paid, from the time it was paid, up. to the time of the final settlement, and deduct this amount from the amount of the principal. But if there be several payments made within the said time, find the amount of the several payments, from the time they were paid, to the time of settlement, and deduct their amount from the amount of the nrincina. SIMPLE INTEREST BY DECIMALS. 163 1067,50 amount. [Brought up. 200,00 first payment deducted. 867,50 balance due, Feb. 19, 1798. 70,845 interest to June 29, 1799=1Q4 months. 938,345 amount. 500,000 second payment deducted. 438,345 balance due June 29, 1799. 26,30 interest for one year. 464,645 amount for one year. 269,750 amount of third payment for 7- months.* -2 194,895 balance due June 29, 1800. mo. da. 5,687 interest to December 24, 1800. 5 25 200,579 balance due on the Note, Dec. 24, 1800. RULE II. ~, tablished by the Courts of Law in Massachusetts for computing interest on notes, 4'c. on which partial payments have been endorsed. i Compute the interest on the principal sum, from me fin.e when the interest commenced to the first time when a payment was made, which exceeds either alone or in conjunction with the preceding payment (if any) the interest at that time due: add that interest to the principal, and from the sum subtract the payment made at that time, together with the preceding payments (if any) and the remainder dorms a new principal; on which compute ard subtract the payments as upon the first principal, and proceed in.his manner to the time of final settlement." $ cts. *260,00 third payment with its interest from the time it was paid, up to 9,15 the end of the year, or from Nov. 14, 1799, to June 89, w00f - which is 7 and 1-2 months. 269,75 amount. 164 SIMPLE INTEREST BY DECIMALS. Let the foregoing example be solved by this Rule. A note for 1000 dols. dated Jan. 4, 1797, at 6 per cent. 1st payment February 19, 1798, $206 2d payment June 29, 1799, 500 3d payment November 14, 1799, 260 How much remains due on said note the 24th of Decem ber, 1800? $ cts. Principal, January 4, 1797, 1000,00 Interest to February 19, 1798, (131 mo.) 67,50 Amount, 1067,50 Paid February 19, 1798, 200,00 Remainder for a new principal, 867,50 Interest to June 29, 1799, (161 no.) 70,84 Amount, 938,34 Paid June 29, 1799, 500,00 Remains for a new principal, 438,34 Interest to November 14, 1799, (41 mo.) 9,86 Amount, 448,29 November 14, 1799, paid 260,00 Remains for a new principal, 188,20 Interest to December 24, 1800, (131 mo.) 12,70 Balance due on said note, Dec. 24, 1800, 200,90 $ cts. The balance by Rule I. 200,579 Rule II. 200,990 Difference, 0,411 Another Example in Rule II. A bond or note, dated February 1, 1800, was given for 500 dollars, interest at 6 per cent. and there wvere payments endorsed upon it as follows, viz. $ cts. 1Ist payment May 1, 1800, 40,00 2d payment November 14, 1800 8,00 COMPOUND INTEREST BY DECIMALS. 155 3d payment April 1, 1801, 12,00 4th payment May 1, 1801, 30,00 How much remains due on said note the 16th of Sep %-nber, 1801 ' $ cts. Principal dated February 1, 1800, 500,00 Interest to May 1, 1800, (3 mo.) 7,50 Amount 507 50 Paid May 1, 1800, a sum exceeding the interest 40,00 New principal, May 1, 1800, 467,50 Interest to May 1, 1801, (1 year,) 28,05 Amount 495,55 Paid Nov. 4, 1800, a sum less than the interest then due, 8,00 Paid April 1, 1801, do. do. 12,00 Paid May 1, 1801, a sum greater, 30,00 50,00 New principal May 1, 1801, 445,55 Interest to Sept. 16, 1801, (4 mno.) 10,92 Balance due on the note, Sept. 16, 1801, $455,57 {I'The payments being applied according to this Rule, keep down the interest, and no part of the interest ever forms a part of the principal carrying interest. COMPOUND INTEREST BY DECIMALS. RUvLE.-Multiply the given principal continually by the Amount of one pound, or one dollar, for one year, at the ate per cent. given, until the number of multiplications are:qual to the given number of years, and the product will,e the amount required. Or, In Table I, Appendix, find the amount of one dollar,:r one pound, for the given number of years, which multiply I' the given principal, and it will give the amount as before. 16O INVOLUTION. EXAMPLES. 1. What will 4001. amount to in 4 years, at 6 per cent oer annum, compound interest? 400 X 1,06 x 1,06 x 1,06 x 1,06=~504,99+o [~504 19s. 9d. 2,75 qrs.+ Ans. The same by Table I. Tabular amount of ~1=1,26247 Multiply by the principal 400 Whole amount=~504,98800 2. Required the amount of 425 dols. 75 cts. for 3 years at 6 per cent. compound interest? Ans. $507,7~ cts. + 3. What is the compound interest of 555 dols. for 1 years at 5 per cent.? By Table I. Ans. 543,86 cts.+ 4. What will 50 dollars amount to in 20 years, at 6 pe cent. compound interest? Ans. $160, 35 cts. 6-m. INVOLUTION, IS the multiplying any number with itself, and that pro duct by the former multiplier; and so on; and the severa products which arise are called powers. The number denoting the height of the power, is call, the index or exponent of that power. EXAMPLES. What is the 5th power of 8 8 the root or 1st power. 8 64 =2d power, or square. 8 512 =3d power, or cube. 8 4096 =4th power, or biquadrate. 8 32768 =5th power, or sursolid. An. .-VOLCliCN OR EXTRACTIO(N OF ROOTS. 167 Ihtt is the square of i7,1? Ans. 22,41 What is the squlre of,085 Ans.,007225 What is the cubl of 25,4? Ains. 16387,064 What is the hiquadrate of 121 Ans. 20736 What is the square of 7} 1 Ans. 52-~ EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any 'power is required, the business f finding it is called the Extraction of the Root. The root is that number, which by a continued multipli ation into itself, produces the given power. Alth',ugh there is no nunmber but what will produce a erfect power by involution, yet there are many numbers of hich precise roots can nlever be determined. But, by the elp of decimals, we can approximate towards the root to iy assigned degree of exactness. The roots \which approximate are called surd roots, and ise which are perfectly accurate are called rational roots. A Table of the Squares and Cubes of the nine digits. oots~. l1 1 3 1 41 5 6 71 S 91 luan,',ls. 1 4 1 16 25 1 36I 49I 64 1 81 u6es. 1 8~27 64 125 ~ 216343 512 7291 EXTRACTION OF THE SQUARE ROOT. Any number multiplied into itself produces a square. To extract the square root, is only to filld a number, hjch being multiplied into itself shall produce the given Imber. RULE. —]. Distinguish tile given number into periods of 1o figures each, by putting a point over the place of units, lothcr over tile place of hundreds, -and so on; and if ere are decilmals, point tllem in the same manner, from tits towarlds tle right hand; wvlich )poiIts show the num*r of figures the root will consist of. 2. Find the greatest square number in tle first, or left Ind period, place the root of it at tlle right hand of tile 168 EVOIUTION, OR EXTRACTION OF ROOTS. given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number under the period, and subtract it therefrom, and to the re mainder bring down the next period, for a dividend. 3. Place the double of the root, already found, on the left hand of the dividend, for a divisor. 4. Place such a figure at the right hand of the divisor, and also the same figure in the root, as when rmuniplied into the whole (increased divisor) the product sidail 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 next period for a new dividend. 6. Double the figures already found m the root, for a new divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner till you have brought down all the periods. Or, to facilitate the foregoing Rule, when you have brought down a period, and formed a dividend in order to find a new figure in the root, you may divide said dividen) (omitting the right hand figure thereof) by double the root already found, and the quotient will commonly be thi figures sought, or being made less one or two, will generally give the next figure in the quotient. EXAMPLES. 1. Required the square root of 141225,64. 141225,64(375,8 the root exactly without a remainder; 9 but when the periods belongirg to any - given number are exhausted, and still 67)512 leave a remainder, the operation may 469 be continued at pleasure, by annexire periods of ciphers, &c. 745)4325 3725 7508)60064 60064 0 remains. EVOLUTION, OR EXTRACTION OF ROOTS. t16 2 3 4. 5. 6 7. 8. 9. 10. What is the square root of 1296 Of - 56644? Of - 6499025? Of - 36372961? Of - 184,2? Of 9712,693S09? Of - 0,45369? Of -,002916? Of - 45? AnSWERS. 36 23,8 2345 6031 13,57+ 98,553,6734,054 6,708+ TO EXTRACT THE SQUARE ROOT OF VULGAR FRACTIONS. RULE. Reduce the fraction to its lowest terms for this and all otfler roots; then 1. Extract the root of the numerator for a new numeratot', and the root of the denominator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root. I. 3. 4. 5. 6. 7. 8. EXAMPLES. What is the square root of -9-? What is the square root of AT2-4 What is the square root of 1{I? What is the square root of 20? What is the square root of 248? SURDS. What is the square root of 5? What is the square root of 42? Required the square root of 36? &NSWERSI. 1, 32 4 151 9128-r,7745+ 6,0207+ APPLICATION AND USE OF THE SQUARE ROOT. PROBLEM I.-A certain general has an army of 5184 men; how many must he place in rank and file, to form them into a square? I t10 EVOLUTION, OR EXTRACTION OF ROOTS. RULE. —Extract the square root of the given number. /5184=72 Ans. PROB. II. A certain square pavement contains 2073( quare stones, all of the same size; I demand how many are contained in one of its sides? V20736 =144 Ans. PROB. III. To find a mean proportional between tw, numbers. RULE. —Multiply the given numbers together and extract the square root of the product. EXAMPLES. What is the mean proportior.al between 1S and 72? 72 x 18=1296, and V/1296=36 Ans. PROB. IV. To form any body of soldiers so that they may be double, triple &c. as many in rank as in file. RUiE.-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 fil,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 ranit may be double the number in file. 13122 —2=6561, and x/6561=81 in file, and 81x2 -162 in rank. PROB. V. Admit 10 hhds. of water are discharged through a leaden pipe of 2- inches in diameter, in a certain time; I demand what the diameter of another pipe.nust be to discharge four times as much water in the same time. RULE.-Square the given diameter, and multiply said square by he given proportion, and the square root of the oroduct is &a answer. 2}? 2f i 2,5 X 2,5=6,25 square. 4 given proportion. /2f5.00-4 inch. di~a 1. Ans. EVOLUTION, 9)R1EXThAUTION Oh' RCOTS. t 171I PROB. VI. The sum of any two numbers, and their proeicts 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, gives 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. 43 X 43=1849 square of do. The product of do. 442 x 4=1768 4 times *he pro. Then to the I sum of 21,5 -numb. f and — 4,5 /81=9 diff. of the Greatest n- anber, 26,0 4, the i diff. Answers. east number, 17,0 EXTRACTION 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 figures each, by putting a point over the unit figure, and every 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 Mlace its root in the quotient. 3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, cang this the dividend. 4. Multiply the square of the quotient by 300, calling it the divisor. 172 EVOLUTION, (CR EXTRACTIONV (F HOR1'. 5. Seek how often the divisor may be had in the divl dend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product underthe dividend. 6. Multiply the former quotient figure, or figures, by the square of the last quotient figure, and that product by 30, and place the 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 for a new dividend; with which 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 co.requently cannot be subtracted therefrom, you must make the last quotient figure one less; with which find a new subtrahend, (by the rule foregoing,) and so on until you can subtra'Y the subtrahend from the dividend. IXAMPLES. 1, Required the cube root of 18399,744. 18399,744(26,4 Root Ans. 8 2 X2=4 x 300=1200)10399 first dividend. 7200 6 x 6-36 x 2=72 x 80=2160 6x6x6- 216 9576 1st subtrahend. 26 X 26=676 x 300=202800)823744 2d dividend. 811200 4 X 4=16 x 26=416 x 30= 12480 4X4x4= 64 823744 2d subtrahert EVOLUTION, OR EXTRACTION OF ROOTS. 173 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 ciphers, and continue the operation as far as you think it necessary. Answers 2. What is the cube root of 205379? 59 3. Of - 614125? 85 4. Of -41421736? 346 5. Of 146363,183? 52,7 6. Of -- 29,508381 3,09+ 7. Of -- 80,763? 4,32+ 8. Of -,162771336?,546 9. Of --,000684134?,088+ 10. Of - 122615327232? 4968 RULE.-1. Find by trial, a cube near to the given number, and call it the supposed cube. 2. Then, as twice the supposed cube, added to the given number, is to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to-it. 3. By taking the cube of the root thus found,for the supposed cube, and repeating the operation, the root will be had to a greater degree of exactness. EXAMPLES. 1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then-1I X 1,3 x 1,3=2,197=supposed cube. Then, 2,197 2,000 given number. 2 2 4,394 4,000 2,000 2,197 As 6,394:6,197:: 1,3: 1,i9root, which is true to the last place of decimas bumingt by repeating the operation be brought tc[.6n t ) xaftc e's. 2. What is the cube root of 684. '. ra A3 ' UVOLUTION, on EXTRACTION Or ROOTS. 3, Required the cube root of 729001101 Ans. 900,0004 QUESTIONS, Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solUd inches. I demand the side of a cubic box, which shall contain that quantity I / 21 50,425= 12,907 inch. Ans. NoTE.-The solid contents of similar figures are in pro portion to each other, as the cubes of their similar sides or diameters. 2. If a bullet 3 inches diameter weigh 4 lb. what will a bullet of the same metal weigh, whose diamnter is 6 in ches? 3x3x3=27 6X 6X6216. As 27: 4 *.: 216i 32 lb. Ans. 3. If a solid globe. of silver, of 3 inche diameter, tl worth 150 dollars; what is the value of another globe ol silver, whose diameter is six inches? 3X3x3=27 6x6x6=216, As 27: 10:: 216 $1200. Ans. The side of a cube being given, to find the side of thai cube which shall be double, triple, &c. in quantity to the given cube. RULE.-Cube your given side, and multiply by the given proper tion between the given and required cube, and the cube root ofthe product will be the side sought. EXAMPLES. 4. If a cube of silver, whose side is two inches, be wortl 20 dollars; I demand the side of a cube of like silver whose value shall be 8 times as much? 2 x 2 x 2=8, and 8 X 8=64 V64=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall con. tain 4 znes as much? 4x4 x 464, and 64 x 4=256 /256=6,349+ft. Ans. 6. A cooper having a cask 46 inches long, and 32 in &VOLUTION, OR EXTRACTION OF ROOTS. 17S chce at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; wha will be the bung diameter and length of the new cask? A0 X 40 x 40 X 2=128000 then V 128000=50,3 + length. 32 X 32 x 32 x 2=65536 and V65536=40,3+ bung diam. A General Rule for extracting the Roots of all Powers. RULE. 1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. 2. Find the first figure of the root by 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 wext 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 in the dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and subtract it (always) from as many periods of the given number 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 find a new divisor as before, and in like manner proceed till the whole be finished. NoTE.-When the number to be subtracted is greater thaL those periods from which it is to be taken, the last,uotient figure must be taken less, &c. EXAMPLES 1. Required the cube root of 135796,744 by the above general method. 176 EVOLUTION, OR EXTRACTION OF RtOC t 135796744(51,4 the roo. 125=1st subtrahend. 5)107 dividend. 132651=2d subtrahend. 7803) 31457=2d dividend. 135796744=3d subtrahend. 5 x 5 x 3-75 first divisor. 51 x 51 x 51=132651 second subtrahend. 51 x 51 X 3=7803 second divisor. 514 x 514 x 514=135796744 3d subtrahend 2. Required the sursolid or 5th root of 6436343. 6436343(23 root. 32 2 x 2 x 2 x 5=80)323 dividend. 23 x 23 X 23 x 23 x 23=6436343 subtrahend. NoTE.-The roots of most powers may be found by thc square and cube roots only; therefore, when any evec power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which re duces it to half the given power, then the square root ot that power reduces it to half the same power; and so on, till you come to a square or a c lbe. For example: suppose a 12t, power be given; the square root of that reduces it to a 6th power: and the square root Df * 6th power to a cube. EXAMPLES. 3. What is the biquadrate, or 4th root of 19987173376 Ans. 376. 4. Extract the square, cubed, or 6th root of 12230590 464. Ans. 48. 5. Extract the square Iquadrate, or 8th root of 72138 95789338336. Ans 96. ALLIGATION. 177 ALLIGATION, IS the method of mixing several simples of different quauties, so that the composition may be of a mean or middle quality: It consists of two kinds, viz. Alligation Medial, and Alligation Alternate. ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed or 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 bushe., l8 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a itushel of this mixture is worth? bu. cts. $cts. btw. $ cts. bu. 15 at 64=9,60 As 54: 25,38:: 1 18 55=9,90 1 21 28=5,88 -- cts. - 54)25,38(,47 Ans. 54 25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth? Ans. $1,20 cts. 3 A tobacconist mixed 36 lb. of tobacco, at Is. 6d. ter lb. 12 lb. at 2s. a pound, with 12 lb. at Is. lOd. per.b.; what is the price of a pound of this mixture? Ans. Is. 8d. ~ 4. A grocer mixed 2 C. of sugar at 56s. per C. and 1 J. at 43s. per C. and 2 C. at 50s. per C. together; I denand the price of 3 cwt. of this mixture? Ans. ~7 13s. 5. A wine merchant mixes 15 gallons of wine at 4s. -d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons t 6s. 3d; what is a gallon of this composition worth? Ans. 5s. 1 Od. 21i qrs. 178 ALLIGA'TION A ALTERNATE. 6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. aiiother sort at 9 dols. per cwt. a third soil at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. and he would mix an equal (qualtity of each together; I demand the price of 3l cwt. of this mixture? Ans. $34 12 cts. 5 m. 7. A goldsmith melted togethler 5 lb. of silver bullioin, of 8 oz. fine, 10 lb. of 7 oz. tine, and 15 lb. of 6 oz. fine; pray what is tile quality or fineness of this composition 1 Ans. 6 oz. 13pwt. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what it the quality or finneness of this mass? As. 19 carats fine. ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients wliose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Medial, and may be proved by it. CASE I. When the mean rate of the whole mixture, and the rates of all the ingredients are given, without any limited quaua tity. RULE. 1. Place the several rates, or prices of the simples, be. ing reduced to one denomination, in a column under each other, and the mean price in the like name, at the left hand 2, Connect, or link the price of each simple or ingredient, which is less than that of the mean rate, with one o0 any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num. her of the less. 3. Place the difference, between the mean price (or mit ture rate) and that of each of the simples, opposite to tlu rates with which they are connected. ALtIGATION ALTERNATE. 179 4. Then, if only one difference stands against any rate, it will be the quantity belonging to that rate, but if there be several, their sum will be the quantity. EXAMPLES. 1. A merchant has spices, some at 9d. per lb. some at Is. some at 2s. and some at 2s. 6d. per lb. how much of each srt must he mix, that he may sell the mixture at Is. Sd. per pound? d. lb. d. d. Ib. -9- 10 at 9 C 9 4) i. 12. 4 12 Gives the d. 12 10 1O 243 8 24 Answer; or 20 243 11 30- 11 303 30- 83 2. A grocer would mix the following qualities of sugar; iz. at 10 cents, 13 cents, and 16 cents per lb.; what quanity of each sort must be taken to make a mixture worth,2 cents per pound? tlns. 5 lb. at 10 cts. 2 lb. at 13 cts. and 2 lb. at 16 cts. per lb. 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. icr lb. how must he mix them so as to afford the composiisn for 12s. per lb.? Ans. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with -me of 19, 21, and 24 carats fine, so that the compound aay be 22 carats fine; what quantity of each must he take? Ans. 2 of each of the first three sorts, and 9 of the last. 5. It is required to mix several sorts of rum, viz. at 5s. s. snd 9s. per gallon, with water at 0 per gallon, togeler, so that the mixture may be worth 6s. per gallon; how iuch of each sort must the mixture consist of? Ans. 1 gal. of rum at 5s., 1 do. at 7s., 6 do. at 9s. and 3 gals. water. Or, 3 gals. rum at 5s., 6 do. at 7s., I do. at 9s. and I gal. water. 6. A grocer hatn several sorts of sugar, viz. one sort at 12 s. per lb. another at 11 cts. a third at 9 cts. and a fourth 8 cts. per lb.; I demand how much of each sort he must iit together, that the whole quantity may be afforded at 3 cents per poiind'? 180 ALTERNATION PARTIAL. lb. cts. lb. cts. lb. tt., 2 at 12 1 at 12 3at12 1 at 11 0 at 11 2 at 11 -t Ans. at 2d Alls. at 1 3d Ans. 2 at 9 l at 9 2 at 9 2at 9 2 at 8 1 at S 3 at 8 7th A7 n', 3 l6. (f tach/ sort.~ CASE II. ALTERNATION PARTIAL, Or, when one of the ingredients is limited to a certain quantity, thence to find the several quantities of the rest, in proportion to the quantity given. RULE. Take the differences between each price, and the meap rate, and place them alternately as in CASE I. Then, as thb lifference standing against that simple whose quantity ii given, is to that quantity: so is each of the other differ ences, severally, to the several quantities required. EXAMPLES. 1. A farmer would mix 10 bushels of wheat, at 70 centi per bushel, with rye at 48 cts. corn at 36 cts. and barley a' 30 cts. per bushel, so that a bushel of the composition nviy +e, sold for 38 ets.; what quantity of each must be taken? (70 — 8 stands against the given quan J48 2 [tity Mean rate, 88 36 1 Q t30- 32 2 2: 2 bushels of rye. As 8: 10: 10: 12- bushels of corn. 32: 40 bushels of barley. ~ These four answers arise from as many various ways of linking thi rates of the ingredients together. Questions in this rule admit of an infinite variety of answers: for after th< quantities are found from different methods of linking; any other numbers ix the same proportion between themselves, as the numbers which compose th~ aa"wer, will likewise satisfy the conditions of the question. ALTERNATION PARTIAL. 181 2. How much water must be mixed with 100 gallons of rum, worth 7s. 6d. per gallon, to reduce it to 6s. 3d. pet gallon? Ans. 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 cents per bushel; how much barley and oats must be mixed with the 20) bushels of rye, that the provender may be worth 41 cts. per bushel? Ans. 20 bushels of barley, and 619T bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other rlm at 6s. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon; I demand how much rum a id how much water I took? Ans. 95 gals. rum at 6s. Sd. and 30 gals. water. CASE III. ~ hen the whole composition is limited to a given quantity. RULE. Place the difference between the mean rate, and the sete 'al prices alternately, as in CASE I.; then, As the sum of thro quantities, or difference thus determined, is to the given qu xntity, or whole composition: so is the difference of each tate, to the required quantity of each rate. EXAMPLES. L A grocer had four sorts of tea, at Is. 3s. 6s. and 10s. ter lb. the worst would not sell, and the best were too dear, Se therefore mixed 120 lb. and so much of each sort, as to tell it at 4s. per lb.; how much of each sort did he take? 1 — 6 6: 60 at 1) 3 2 lb. Ib. 2:20 3 4 6J 1 As 12: 120: 1: 10- 6 Per lb. 10 — 3 3:30 -10 Sum, 12 120 182 ARITHIMETICAL PROGRESSION. 2. How much water at 0 per gallon, must be mixed witb wine at 90 cents per gallon, so as to fill a vessel of 100 galions, which may be afforded at 60 cents per gallon t Ans. 331 gals. water, and 662 gals. wzne. 3. A grocer having sugars at 8 cts. 16 cts. and 24 cts per pound, would make a composition of 240 lb. worth 20 cts. per lb. without gain or loss; what quantity of each must be taken? Ans. 40 lb. at 8 cts. 40 lb. at 16 cts. and 160 lb. at 24 cts. 4. A goldsmith had two sorts of silver bullion. one of 10 oz. and the other of 5 oz. fine, and has a mind to mix a pound of it so that it shall be 8 oz. fine; how much of each sort must he take? Ans. 41 of 5 oz. fine, and 7- of 10 oz. fine. 5. Brandy at 3s. 6d. and 5s. 9d. per gallon, is to be mixed, so that a hhd. of 63 gallons nmay be sold for 121. 12s.; how many gallons must be taken of each? Ans. 14 gals. at 5s. 9d. and 49 gals. at 3s. 6d. ARITHMETICAL PROGRESSION. ANY rank of numbers more than two, increasing by common excess, or decreasing by common difference,is said to be in Arithmetical Progression. So 2,4,6,S, &c. is an ascending arithmetical series. 8,6,4,2, &c. is a descending arithmetical series: The numbers which form the series, are called the terms of the progression; the first and last terms of which ari called the extremes.* PROBLEM 1. The first term, the last term, and the number of terms being given, to find the sum of all the terms. * A series in progression includes five parts, viz. the first term, last term. number of terms, common difference, and sum of the series. By having any three of these parts given, the other two may be found, which admits of a variety of Problems; but most of them are best understood by an algebraic process, and are here omit.td. ARITHMETICAL PROGRESSION LULE.-Multiply the sum of the extremes by the number tsms, and half the product will be the answer. EXAMPLES. 1. The first term of an arithmetical series is 3, the lasi term 23, and the number of terms 11; required the sum of the series. 23+3=26 sum of the extremes. Then 26 x 11 -2=143 the Answer. 2. How many strokes does the hammer of a clock strike in 12 hours. Ans. 78. 3. 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? Ans. $50-. 4. A man bought 19 yards of linen in arithmetical progression, for the first yard he gave Is. and for the last yd. 11. 17s. what did the whole come to? Ans. ~18 Is. 5. A draper sold 100 yards of broadcloth, at 5 cts. for the first yard, 10 cts. for the second, 15 for the third, &c. increasing 5 cents for every yard; what did the whole amount to, and what did it average per yard? Ans. Amount $252-, and the average price is $2, 52 cts. 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. 23 miles, 5furlongs, 180 yds. PROBLEM II. Tihe first term, the last term, and the number of terms given, to find the common difference. RULE.-Divide the difference of the extremes by the number ef terms less 1, and the quotient will be the common difference. 184 ARITHMETICAL PRUORESSION. EXAMPLES. 1. The extremes are 3 and 29, and the number of terim 14, what is the common difference? 29 ) ~3 } Extremes. -3 Number of terms less 1=13)26(2 Ans. 2. A man had 9 sons, whose several ages differed alike, the youngest was three years old, and. the oldest 35; what was the common difference of their ages 1 Ans. 4 years. 3. A man is to travel from New-London to a certain place in 9 days, and to go but 3 miles the first day, increasing every day by an equal excess, so that the last day's journey may be 43 miles: Required the daily increase, and the length of the whole journey Ans. The daily increase is 5, and the whole journey 207 miles. 4. A debt is to be discharged at 16 different paymenti (in arithmetical progression,) the first payment is to be 141 the last 1001.; What is the common difference, and th( sum of the whole debt Ans. 51. 14s. 8d. conmmon difference, and 9121. the whol debt. PROBLEM Ill. Given the first term, last term, and common difference, t( find the number of terms. RULE.-Divide the difference of the extremes by the commor difference, and the quotient increased by 1 is the number of terms EXAMPLES. 1. If the extremes be 3 and 45, and the common differ ence 2; what is the number of terms? Ans. 22. 2. A man going a journey, travelled the first day fivc miles, the last day 45 miles, and each day increased his journey by 4 miles; how many days did he travel, and how far I Ans. 11 days, and the whole distance travelled 275 niles GEOMIkTRICAL rROGRESSION. 18 GEOMETRICAL PROGRESSION, (S when any rank or series of numbers increase by one ~ommon multiplier, or decrease by one common divisor as, 1, 2, 4, 8, 16, &c. increase by the multiplier 2; and 27, 9, 3, 1, decrease by the divisor 3. PROBLEM I. The first term, the last term (or the extremes) and the ratio given, to find the sum of the series RULE. Multiply the last term by the ratio, and from the product subtract the first term; then divide the remainder by the ratio, less by 1, 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? 3x 1458-2 -- =2186 the Answer. 3 —1 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4; what is the sum of the series? Ans. 87381. PROBLEM II. Given the first term, and the ratio, to find any other term assigned.* CASE I. When the first term of the series and the ratio are equal.t * As the last term in a long series of numbers is very tedious to be found by continual multiplications,it will be necessary for the readier finding it:out, to have a series of numbers in arithmetical proportion, called indices, whose common difference is 1. f When the first term of the series and the ratio are equal, the indiea must begin with the unit and in this case, the product of any two terms is equal to that term, signified by the sum of their indices: Q2 186 GEOMETRICAL PROGRESSION. 1. Write down a few of the leading terms of the series, and place their indices over them, beginning the indicea with a unit or 1. 2. Add together such indices, whose sum shall make uj the entire index to the sum required. 3. Multiply the terms of the geometrical series belonging to those indices together, and the product will be the tern wight. EXAMPLES. 1. If the first be 2, and the ratio 2; what is the 13th term? 1, 2, 3, 4, 5, indices. Then 5+5+3=13. 2, 4, 8, 16, 32, leading terms. 32 x32 x8=8192 Ans. 2. A draper sold 20 yards of superfine cloth, the fivrst yard for 3d., the second for 9d., the third for 27d., &c. in triple proportion geometrical; what did the cloth come te at that rate? The 20th, or last term, is 3486784401d. Then 3+3486784401-3 — 5230176600d. the sum of all 3-1 the terms (by Prob. I.) equal to ~21792402, 10s. 3. A rich miser thought 20 guineas a price too much foi 12 fine horses, but agreed to give 4 cts. for the first, 16 cts. for the second, and 64 cents for the third horse, and so on in quadruple or fourfold proportion to the last: what did they come to at that rate, and how much did they cost Per head one with another? Anxs. The 12 horses came to $223696, 20 cts., and the average price was $18641, 35 cts. per head. Thu, 2 3 4 5, &c. indices or arithmetical series ThIus, } 2 4 8 16 32, &c. geometrical series. 3+2 = 5 = the index of the fifth term, and Now, 4x8 = 32 = the fifth term. GEOMETRICAL PROGRESSION. 187 CASE I1. When the first term of the series and the ratio are diffe. rent, that is, when the first term is either greater or less than the ratio.* 1. Write down a few of the leading terms of the series, and begin the indices with a cipher: Thus, 0, 1, 2,3, &c 2. Add together the most convenient indices to make an index less by 1 than the number expressing the place of the terms sought. 3. Multiply the terms of the geometrica. series together belonging to those indices, and make the product a dividend 4. Raise the first term to a power whose index is one less than the number of the terms multiplied, and make thle result a divisor. 5. Divide, and the quotient is the term sought. EXAMPLES. 4. If the first of a geometrical series be 4, and the ratio 3, what is the 7th term? 0, 1, 2, 3, Indices, 4, 12, 36, 108, leading terms. 3 + 2 + 1=6, the index of the 7th term. 108 x 36 x 12=46656 -- =2916 the 7th term required. 16 Here the number of terms multiplied are three; therefore the first term raised to a power less than three, is the 2d power or square of 4=16 the divisor. * When the first term of the series and the ratio are different, the ir-dices must begin with a cipher, and the sum of the indices made choice of must be one less than the number of terms given in the question: because I in the indices stands over the second term, and 2 in the indices over the third term, &c. and in this case, the product of any two terms, divided by the first is equal to that term beyond the first, signified by the sum of their indices. Thus, 0, 1, 2, 3, 4, &c. Indices. us 1, 3, 9, 27, 81, &c. Geometricalseries. Here 4+3=7 the index of the 8th term. 81 X27=2187 the 8th term, or the 7th beyond the 1st. 188 POSITION. 5. A Goldsmith sold 1 lb. of gold, at 2 cts. 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? Ans. $111848, 10 cts. 6. What debt can be discharged in a year, by paying I farthing the first month, 10 farthings, or (2-d) the second and so on, each month in a tenfold proportion 1 Ans. ~115740740 14s. 9d. 3 qrs. T. A thrasher worked 20 days for a farmer, and received for the first days work four barley-corns, for the second 12 barley corns, for the third 36 barley corns, and so on, in triple proportion geometrically. I demand what the 20 day's labour came to supposing a pint of barley to contain 7680 corns, and the whole quantity to be sold at 2s. 6d. per bushel? Ans. ~1773 7s. 6d. rejecting remainders 8. A man bought a horse, and by agreement, was to give a farthing for the first nail, two for the second, foul for the third, &c. There were four shoes, and eight nails in each shoe; what did the horse come to at that rate? Ans. ~4473924 5s. 33d 9. Suppose a certain body, put in motion, should move the length of 1 barley-corn the first second 'f time, (ne inch the second, and thiee inches the third second of time, and so continue to increase its motion in triple proportion geometrical; how many yards would the said body move iin the term of half a minute. Ans. 953199685623 yds. Ift. 1 in. lb. which is no less than five hundred andforty-one millions of miles. POSITION. POSITION is a rule which, by false or Wsupposed num bers, taken at pleasure, discovers the true ones required.It is divided into two parts, Single or Double. SINGLE POSITION IS when one number is required, the properties o!;xhicb are given in the question. s8IGLE 189 RUL..-1. Take any number and perforln t\.ie same operation with it, as is described to be performed in the question. 2. Then say; as the result of the operation: is to the 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 ques tion. 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 he? Suppose he had 12 As 37: 148: 12: 48 Ans. as many 12 48 -X as many = 6 24 3 as many = 4 16 - as many = 3 12 Result, 37 Proof, 149 2. What number is that which being increased by a, 3, and - of itself, the sum will be 125 Ans. 60. 3. Divide 93 dollars between A, B and C, so that B's share may be half as much as A's, and C's share three times as much as B's. Ans. A's share $31, B's $151, and C's $46-. 4. A, B and C, joined their stock and gained 360 dols. of which A took up a certain sum, B took 3- times as much as A, and C took up as much as A and B both; what share of the gain had each? Ans. A $40, B $140, and C $180. 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 7311. principal and interest together; what was the sum delivered to him at first? Ans. ~425. 6. A vessel has 3 cocks, A, B and C; A can fill it in 1 hour, B in 2 hours, and C in 4 hours; in what time will theyvall fill it together? Ans 34min. 171 sec. qheC1 VIOT, Ifec 90 DOUBLE POSITION. DOUBLE POSITION, TEACHES to resolve questions by making two suDp eitions of false numbers.* RULE. 1. Take any two convenient numbers, and proceed wi each according to the conditions of Lhe question. 2. Find how much the resuJts are different from ihe r, suits in the question. 3. Multiply the first position by the last error, and the la position by the first error. 4. If the errors are alike, divide the difference of the pr ducts by the difference of the errors, and the quotient wi be the answer. 5. If the errors are unlike, divide the sum of the pr( ducts by the sum of the errors, and the quotient will i: the answer. NoTE.-The errors are said to be alike when they ar both too great, or both too small; and unlike, when on 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 four dollars more thai A, and C 8 dollars more than B, and D twice as many a C; what is each one's share of the money? 1st. Suppose A 6 2d. Suppose A 8 B 10 B 12 C 18 C 20 D 36 D 40 70 80 100 100 1st error, 30 2d error, 20 * Those questions in which the results are not proportional to their poal tions, belong to this rule; such as those in which the number soeigh in, creased or diminished by some given number, which is no known prt of tI number reouired. DOUBLE POSITION. nI< The errors being alike, are both too small, therefore, Pos. Err 6 30 $ [A 12 JB 16 C 24' XD 48 8 20 Proof 100 240 120 120 10)120(12 A's part. 2. A, B, and C, built a house which cost 500 dollars, of hich A paid a certain sum; B paid 10 dollars more than, and C paid as much as A and B both; how much did ch man pay? Ans. J paid $120, B $130, and C $250. 3. A man bequeathed 1001. to three of his friends, after is manner; the first must have a certain portion, the se-!nd must have twice as much as the first, wanting 81. and e third must have three times as much as the first, wantg 15L; I demand how much each man must have? Ans. The first ~20 10s. second ~33, third, c46 10s. 4. A labourer was hired for 60 days upon this condition; at for every day he wrought he should receive 4s. and for,ery day he was idle should forfeit 2s.; at the expiration 'the time he received 71. 10s.; how many days did he ork, and how many was he idle? Jins. He wrought 45 days, and was idle 15 days. 5 What number is that which being increased by its ~, 4;, and 18 more, will be doubled? Jins. 72. 6. A man gave to his three sons all his estate in money z. to F half, wanting 501. to G one-third, and to H the st, which was 10/. less than the share of G; I demand e sum given, and each man's part? ts.. tie sum given was ~360, whereof F had ~ l30, G ~120 and H ~110. V'2 PPERMUTATION OF QUANTITIES. 7. Two men, A and B, lay out equal sums of money in trade; A gains 1261. and B loses 871. and A's money if now double to B's; what did each lay out? Ant. ~300. 8. A farmer having driven his cattle to market, received for them all 1301. being paid for every ox 71. for every cow 51. and for every calf 11. 10s. there were twice as many cows as oxen, and three times as many calves as cows; how many were there of each sort? Ans. 5 oxen, 10 cows, and 30 calves. 9. A, B, and C, playing at cards, staked 324 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 5th part of both their sums added together; how many did each get? Ans. A got 127k, 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 chitnges, 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 number' from one up to the given number, continually together, and the [tie product will be the answer required, EXAMPLES. 1. How many changes can be 1 a b c made of the first three letters of 2 a c b the alphabet? Proof, 3 ba c 4 b ca 5 cba 1X2X3-6 Ans.6 cab 2. How many changes may be rung on 9 bells? Ans. 36280. ANNUITIES OR PINSIONS. 19a 3 Seven gentlemen met at an inn, and wemre. well pleased with their host, and with each other, thai 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 fulfilled their agreement? Ans. 1104$j yedrs, ANNUITIES OR PENSIONS, COMPUTED AT COMPOUND INTEREST. CASE I. To find the amount of an Annuity, or Pension, in arrears, at Compound Interest. RULE. 1. Make I the first term of a geometrical pxograsaiorn, and the amount of $1 or ~1 for one year, at the givenriata: oer cent. the ratio. 2. Carry on the series up to as many terms as the given nuImber of years, and find its sum. 3. Multiply the sum thus found, by the given annuity, and the product will be the amount sought. EXAMPLES. 1. If 125 dols. yearly rent, or annuity, be forborne (or unpaid) 4 years; what will it amount to at 6 per cent. per annum, compound interest? 1+1,06+1,1236+1,191016=4,374616, sum of the series. ---Then, 4,374616x 125=$546,827, the amount sought. uOR BY TABLE II. Multiply the Tabular number under the rate, and opPoiite to the time, by the annuity, and the product wi be the amount sought. * The sum of the series thus found, is the amount of II. or 1 dollar an. nuity, for the given time, which may be found in Table II. ready calculabed. Hence, either the am:)unt or present worth of annuities may be readil found by tables for that surpose. I 1lo ANNUITIES OR PENSIONS. 2. If a salary of 60 dollars per annum to be paid yearly be forborne twenty years, at 6 per cent. compound interest what is the amount? Under 6 per cent. and opposite 20, in Table II., yot will find, Tabular number=-36,78559 60 Annuity. Ans. $2207,13540=$2207, 13 cts. 5 m. + 3. Suppose an annuity of 1001. be 12 years in arrears, it ii required to find what is now due, compound interest being allowed at 51. per cent. per annum? Ans. ~1591 14s. 3,024d. (by Table II.) 4. What will a pension of 1201. per annum, payable yearly, amount to in 3 years, at 51. per cent. compound interest? Ans. ~378 6s. II. To find the present worth of annuities at Compound In. terest. RULE. Divide the annuity, &c. by that power of the ratio sig. nified by the number of years, and subtract the quotient from the annuity: This remainder being divided by the ra tio less 1, the quotient will be the present value of the an nuity sought. EXAMPLES. 1. What ready money will purchase an annuity of 501 0 continue 4 years, at 51. per cent. compound interest? 4th power of } =1,215506)50,00000(41,13513+ From 50 Subtract 41,1313 D*. ' W -..: 86487 17,297 e177 5s. I I d. An". ANNULTIMS OR PENSIONS. in BY TABLE III. Under 5 per cent. and even with 4 years, We have 3,54595=present worth of 11. for 4 yeara. Multiply by 50=Annuity. Ans. ~177,29750=present worth of the annuity. 2. What is the present worth of an annuity of 60 dsls per annum, to continue 20 years, at 6 per cent. compound interest? Ans. $688, 191 cts.+ 3. What is 301. per annum, to continue 7 years, worth in ready money, at 6 per cent. compound interest? Ans. ~167 9s. 5d.+ III. To find the present worth of Annuities, Leases, &c. taken in REVERSION at Compound Interest. 1. Divide the annuity by that power of the ratio denoted by the time of its continuance. 2. Subtract the quotient from the annuity: Divide the remainder by the ratio less 1, and the quotient will be the present worth to commence immediately. 3. Divide this quotient by that power of the ratio denoted 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 2d power of 1,05.=1,1025)177,297(160,8136=~160 16s. 3d. 1 qr. present worth of the annuity in reversion. OR BY TABLE III. Find the present value of 11. at the given rate for the sum of the time of continuance, and time in reversion added together; from which value subtract the present worth of IU. for the time in reversion, and multiply the remainder by the annuity; the product will be the answer. AN04Uvflts OR PEICNSI0N6 Thw in Eamrple 1 Time of 4a81itiuance, 4 years. Yitto of reversion, 2 The sum, -6 years, gives 5,075692 Time in reversion, =2 years, 1,859410 Remainder, 3,216282 x 50 Ans. ~160,8141. 2. What is the present worth of 751. yearly rent, whicl is not to comnmeece until 10 years hence, and then to continue 7 years after that time at 6 per cent.? Ans. ~233 15s. 9d. 4. What is the present worth of the reversion of a lease of 60 dollars per annum, to continue 20 years, but not to commence till the end of 8 years, allowing 6 per cent. io the purchaser? Ans. $431, 78 cts. 2-2m. IV. To find the present worth of a Freehold Estate, or ay Annuity to continue forever, at Compound Intereet. RULE. As the-rate per cent. is to 100.: so is the yearly rent u the vifue tequired. EXAMPLES. I. What is the worth of a freehold estate of 401. per an. num, allowing 5 per cent. to the purchaser? As ~5: ~100 -: ~40: ~00 Ans. 2. An estate brings in yearly 1501. what would it sell for, allowing ie hJtriaer6 per cent. for his money? Ans. ~2500. V. To find the present worth of a Freehold Estate, in Re. versiwq, at Compound Interest. RItuL.-1. iiniht Pe pitdsint value of the estate (by the foregoing rule) -s thgh it wer to be entered on imrmediately, 'and divide the maid value by that power of the ratio denoted by the time of rever-?p. and the quotient will be the present worth of the estate in reEXAMPLES. ap ehold estate of 40L per annum to eom Menee years eIh,;bqput on sale; what is its value. allowing the purchaser 5 per cent.? AVEftSTVONS loft SIZXAiCISM f Mf As 6: 100:: 40 800=present worth if entered on Immediately. Then, 1,05=1,1025)800,00(725,62358=7251. 12s. 64d.=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 es tate in reversion. Thus in the foregoing example, l,859410=present worth of 1l for 2 years. 40=annuity or rent. 74,376400=present worth of the annuity or rent, fot [the time of reversion. From 800,0000=value of immediate possession. Take 74,3764-present worth of rent. ~725,6236~725 12s. 51d. Ans. 2. Suppose an estate of 90 dollars per annum, to commence 10 years hence, were to be sold, allowing the purchaser 6 per cent.; what is the worth? Ans. $837, 59 cts. 2 m. 3. Which is the most advantageous, a term of 15 years, in an estate of 1001. per annum; or the reversion of such an estate forever after the said 15 years, computing at the rate of 5 per cent. per annum, compound interest Ans. The first term of 15 years is better than the reversion forever afterwards, by ~75 18s. 7ad. A COLLECTION OF QUESTIONS TO EXERCISE THE FOREGOING RULES. I demand the sum of 1748S added to itself? Ans. 3497. 2. What is the difference between 41 eagles, and 4099 dimes? Ans. 10 cts. 3. What number is that which being multiplied byp.,l hde product will be 1365 t19 qUESTIA4S VAR ~iEP CS2. 4. What number is that which being divided by 19, the quotient will be 72 1 Ans. 1368. 5. What number is that which being multiplied by 15, the product will be 3 i Ans. - 6. There are 7 chests of drawers, in each of which there are 18 drawers, and in each of these there are six divisions, in each of which is 161. 6s. 8d.; how much money is there in the whole. Ans. ~12348. 7. Bought 36 pipes of wine for 4536 dollars; how must I sell it a pipe to save one for my own use, and sell the rest for what the whlle cost? Ans. $129, 60 cts 8. Just 16 yards of German serge, For 90 dimes had I; orow many yards of that same cloth Will 14 eagles buy? Ans. 248 yds. 3 qrs. 2 na. 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 1 Ans. 214. 10. A grocer bought an equal quantity of sugar, tea, and coffee, for 740 dollars; he gave 10 cents per lb. for the su gar, 60 cts. per lb. for the tea, and 20 cts. per lb. for (th coffee; required the quantity of each? Ans. 822 lb. 3 oz. 88 dr. 11. Botrght cloth at $14 a yard, and lost 25 per cent., how was it sold a yard? Ans. 933 cts. 12. The third part of an army was killed, the fourth par taken prisoners, a d 1000 fled; how many were in this army, how many killed, and how many captives? Ans. f400 tn the army, 800 killed, arid 600 taken prisoners. 13. Thonias sold 150 pine apples at 333 Aehts apiece, and received as much money as Harry received for a certain number of water-melons, which he sold at 25 cents apiece; how much money did each receive, and how many melons eltHkrry? Ans.Each rec'd$50, and Harry sold 20n melons. 14. Said Sohn to Dick, my purse and money are wofth 91 2s., but tiie money is twenty-five times 'as much as the purse; I demand how much money was in it I Ans. ~8 15J. qPEST1ONS FOR ~Xmlscis5a. Le9 15. A young man received 2101. which was I of his el. Cer brother's portion; now three times the elder brother's,ortion was half the father's estate; what was the value of,he estate? A ns. ~ 1890. 16. A hare starts 40 yards before a grey-hound, and is lot perceived by him till she has been up 40 seconds; she ticuds away at the rate of ten miles an hour, and the dog, o)n view, makes after her at the rate of 18 miles an hour: tlow long will the course hold and what space will he run over from the spot where the dog started? Ans. 60 2 sec. and 530 yds. space. 17. What number multiplied by 57 will produce just what 134 multiplied by 71 will do? Ans. 166gf. 18. There are two numbers whose product is 1610, the greater is given 46; I demand the sum of their squares, and the cube of their difference? Ans. the sum of their squares is 3341. The cube of their difference is 1331. 19. Suppose there is a mast erected, so that - of its 4ingth stands in the ground, 12 feet of it in the water, and j of its length in the air, or above water; I demand the whole length? Ans. 216feet. 20. What difference is there between the interest of 5001. at 5 per cent. for 12 years, and the discount of the same sum at the same rate, and for the same time? Ans. ~112 10s. 21. A stationer sold quills at lls. per thousand, by which he cleared - of the money, but growing scarce raised them to 13s. 6d. per thousand; what might he clear per cent. by the latter price? Ans. ~96 7s. 3-,ad. 22. Three persons purchase a West-India sloop, towards ws payment of which. A advanced -, B ", and C 1401. How much paid A and B, and what part of the vessel had C? Am. Apaid ~267-T, B ~305,T, and C's part of the Ve was it. 23. What is the purchase of 12001. bank stock, at 1034 Mt; cep. I Ans. ~1243 10s. 24 -Bought 27 pieces of Nankeens, each 1 yards, at 200 QUESTIONS FOR EXERCISE. 14s. 44d. a piece, which were sold at 18d. a yard; required the prime cost, what it sold for, and the gain. ~. s. d. { Prime cost, 19 8 $1 Ans. Sold for, 23 5 9 Gain, 3 17 7a 25. Three partners, A, B and C, join their stock, and buy goods to the amount of ~1025,5; of which A put in a certain sum; B put in...I know not how much, and C the rest; they gained at the rate of 241. per cent.: A's part of the gain is -, B's 1, and C's the rest. Required each man's particular stock. A's stock was 512,75 Ans. B's - 205,1 C's 307,65 26. What is that number which being divided by 2, the quotient will be 21? Ans. 15Z. 27. If to my age there added be, One-half, one-third, and three times three, Six score and ten the sum will be; What is my age, pray show it me? Ans. 66. 28. A gentleman divided his fortune among his three sons, giving A 91. as often as B 51. and to C but 31. as often as B 71. and yet C's dividend was 25841.; what did the whole estate amount to? Ans. ~19466 2s. Sd. 29. A gentleman left his son a fortune, I of which he spent in three months; 7 of the remainder lasted him 10 months longer, when he had only 2524 dollars left; pray what did his father bequeath him? Ans. $5889, 33cts. + 30. In an orchard of fruit trees, i of them bear apples, ~ pears, J plums, 40 of them peaches, and 10 cherries: how many trees does the orchard contain t Ans. 600. 31. There is a certain number which being divided 1y 7, the quotient resulting multiplied by 3, that product divided by 5, from the quotient 20 being subtracted, and 39 added to the remainder, the half sum shall make 65; can ygu tUll me the number? Am. 1400, QUETS''IONS FOR EXERCISE. 1UJ.' What part of 25 is - of a unit I Anis i. 33. If A can do a piece of work alone in 10 daysr, B in 2C days, C in 40 days, and D in 80 days; set all fout about it together, in what time will they finish it? Ans. 5~ (d tys. 34. A farmer being asked how many sheep he 1h l, answered, that he had them in five fields; in the first t o had [ of his flock, in the second a, in the third -, in the 'ourth 1J, and in the fifth 450; how many had he? Ans. 1!i)0. 35. A and B together can build a boat in 18 da3:, and with the assistance of C they can do it in 11 days; i what time w-ould C do it alone? Ans. 28 — d) ys. 36. There are three numbers, 23, 25, and 42; what 's the difference between the sum of the squares of the firsi and last, and the cube of the middlemost? Ans. 133 12. 37. Part 1200 acies of land among A, B, and C, sc that 6 may have 100 more Lhan A, and C 64 more than B Ans. A 312, B 412, C 47 i. 38. If 3 dozen pairs of gloves be equal in value to 2 pi ices of Holland, 3 pieces of Holland to 7 yards of satin, 6 y.trds of satin to 2 pieces of Flanders lace, and 3 pieces of Fl anders lace to 81 sfillings; how manv dozen pairs of gloves may be bought for 28s. 1 Ans. 2 dozenpairs 39. A lets B have a hogshead of sugar of 18 cwt., worth $ dollars, for 7 dollars the cwt. - of which he is to pay iiI eash. B hath paper worth 2 dollars per ream, which he gives A for the rest of his sugar, at 2- dollars per ream; which gained most by the bargain? Ans. A by $19 20 cts. 40. A father left his two sons (the one 11 and the otiher LC years old) 10,000 dollars, to be divided so that each share b ing put to interest at 5 per cent. might amount to equal tiums when they would be respectively 21 years of age. Required the shares? Ans. 54541i and 4545T- dollars. 41. Bought a certain quantity of broadcloth for 31. 2'2 20 QUE6TIOINS F'OR ESXERCISF.6 5s. and if the number of shillings which it cost per yard were added to the number of yards bought, the sum would be 386; I demand the number of yards bought, and at what price per yard? Ans. 365 yds. at 21s. per yard. Solved by PROBLEMI VI. page 171. 42. Two partners Peter and John, bought goods to the amount of 1000 dollars; in the purchase of which, Petej paid more than John, and John paid.....I know not how much: They then sold their goods for ready money, and thereby gained at the rate of 200 per cent. on the prime cost: they divided the gain between them in proportion to the purchase money that each paid in buying the goods; and Peter says to John, My part of the gain is really a handsome sum of money; I wish I liad as many such sums as your part contains dollars,I should then have $960,000. I demand each man's particular stock inl purchasing the goods. Ans. Peter paid $600 and Johinpaid $400. TIE FOLLOWING QUESTIONS ARI; PROPOSED TO SURVEYORS: 1. Required to lay out a lot of land in form of a lon: square, containing 3 acres, 2 roods and 29 rods, that shall take just 100 rods of wall to enclose, or fence it sound; pray how many rods in length, and how many wide, must said lot be? Ans. 31 rods in length, and 19 in breadth. Solved by PROBLEM VI. page 171. 2. A tract of land is to be laid out in form of an equal square, and to be enclosed with a post and rail fence, 5 rails high; so that each rod offence shall contain 10 rails. How large must this noble square be to contain just as many acres as there are rails in the fence that encloses it, so that every rail shall fence an acre? Ans. the tract of land is 20 miles square, and contains 256,000 acres. Thus, 1 mile=320 rods: then 320 x320-160=640 acres: and 320 x 4 x 10=12,800 rails. As 640: 12,800: I 12,800: 256,000, rails, which will enclose 256,000 acres20 miles square. AN APPENDIX, CONTAINING SHORT RULES, FOR CASTING INTERLEST AND REBATE: TOGETHER WITH SOMF USEFUL RULES, fii FINDING THE CONTENTS OF SUPERFICES, SOLIDS, &C. SHORT RULES, I OR CASTING INTEREST AT SIX PER CENT.. To find the interest of any sum of shillings for any number of days less than a month, at 6 per cent. RULE. i. Multiply the shillings of the principal by the number,f days, and that product by 2, and cut off three figures to he right hand, and all above three figures will be the interest n pence. 2. Multiply the figures cut off by 4, still striking off 'free figures to the right hand, and you will have the farhings, very nearly. EXAMPLIS. 1. Required the interest of 51. 8s. for 25 days. ~. s. 5,8=.108x 25 x 2 —5,400, and 400 x 4=1,600. Ans. 5d. 1,6qrs. 2. What is the interest of 21 1. Os. for 29 days? Als 1 s. Od. 2 qrs. 204S A P1'F.N DI X. bEDERAL MONEY. II. To fil ( the interest of any number of cents for ant ullll c;r of days less than a month, at 6 per cent. RULE. Multiplj the cents by the number of days, divide the pro duct by 6, and point off two figures to the right, and all the figures at she left hand of the dash, will be the ipterest in mills, neal y. EXAMILES. Req(niiice the interest of 85 dollars, for 20 days. $ cts. mills. 85=8500 X 20-6=283,33 Ans. 283 which is 28 cts. 3 mills. 2. lWhat is the interest of 73 dollars 41 cents, or 7341 cents, for 2 r dlays, at 6 per cent.? Ans. 330 mills, or 33 cts. III. When l ihe principal is given in pounds, shillings, &c New-England currency, to find the interest for any number of dai s, less than a month, in Federal Money. RULE. Multi| ly tl e shillings in the principal by the number ot lays, and dlit ide the product by 36, the quotient will be the interest ill ilills, for the given time, nearly, omitting fractions. EXAMPLE. Requi4ed tl e interest in Federal Money, of 271. 15s for 27 days,; t 6) er cent. s. s. Ans. 27 15=555 x 27-36=416 mills.=41 cts. 6m. 1V. Whea the principal is given in Federal Money, and you want the interest in shillings, pence, &c. New-England currency for any number of days less than a mont. APPENDIX. RULE. Multiply the principal, in cents, by the number pf days iad point off five figures to the right hand of the product which will give the interest for the given time, in shillings and decimals of a shilling, very nearly. EXAMPLES. A note for 65 dollars, 31 cents, has been on interest 25 days; how much is the interest thereof in New-England currency? $ cts. s. s. d. qrs. Ans. 65,31=6531 x 25=1,63275=1 7 2 REMARKs.-In the above, and likewise in the preceding practical Rules, (page 115) the interest is confined at 6 per rent. which admits of a variety of short methods of casting: and when the rate of interest is 7 per cent. as estak:lished in New-York, &c. you may first cast the interest at; per cent. and add thereto one sixth of itself, and the sum,rill be the interest at 7 per ct.,which perhaps, many times Vsill be found more convenient than the general rule of cast tg interest. EXAMPLE. Required the interest of 751. for 5 months, at 7 per cent 3. 7,5 for 1 month. 5 - ~. s.d. 37,5=1 17 6 for 5 months at 6 per cent. +6= 63 Ans. ~2 3 9 for ditto at 7 per cent. A SHORT METHOD FOR FINDING THE REBATE qF ANT GIVEN SUM, FOR MONTHS AND DAYS. RULE.-Diminish the interest of the given sum for the time by its own interest, and this gives the Rebate very nearly. EXAMPLES. 1. What is the rebate of 50 dollars, for 6 months, at 6 per cent.? s06 APPENDIL, cts The interest of 50 dollars for 6 months, is 1 50 And, the interest of 1 dol. 50 cts. for 6 months, is 4 Ans. Rebate, $1 46 2. What is the rebate of 156,. for 7 months, at 5 per cent.? ~. s. d. Interest of 1501. for 7 months, is 4 7 6 Interest of 41. 7s. 6d. for 7 months, is 2 6f Ans. ~4 4 11 nearly By the above Rule, those who use interest tables in their counting-houses, have only to deduct the interest of the in terest, and the remainder is the discount. A concise Rule to reduce the currencies of the diferent States, where a dollar is an even number of shillings, to Federal Money. RULE. I.-Bring the given sum into a decimal expression by in. spection, (as in Problem I. page 80) then divide the whole by j in New-England, and by,4 in New-York currency, and the quotient will be dollars, cents, &c. EXAMPLES. 1. Reduce 541. 8s. 31d. New-England currency, to fo leral money.,3)54,415 decimally expressed. Ans. $181,38 cts. 2. Reduce 7s. 11d. New-England currency, to federal aoney. 7s. 11d.=~0,399 then,,3),399 Ans $1,33 3. Reduce 5131. 16s. 10d. New-York, &c. currency, to federal money.,4)513,842 decimal. Ans. $1284,600 Al.PENDIX. S07 4. Reduce 19r. 53d. New-York, &c. currency, to Fedeeal Money.,4)0,974 decimal of 19s. 5"d. $2,431 Ans. 5. Reduce 6411. New-England currency, to Federal Money.,3)64000 decimal expression. $213,33- Ans. NOTE.-By the foregoing rule you may carry on the decimal to any degree of exactness; but in ordinary practice, the following Contraction may be useful. RULE II. To the shillings contained in the given sum, annex 8 times the given pence, increasing the product by 2; then divide the whole by the number of shillings contained in a dollar, and the quotient will be cents. EXAMPLES. 1. Reduce 45s. 6d. New-England currency, to Federal Monty. 6 x 8+2 = 50 to be annexed. 6)45,50 or 6)4550 -- -- S cts. $7,58} Ans. 758 cents.=7,58 2. Reduce 21. 10s. 9d. New-York, &c. currency, to 'ederal Money. 9 x 8+2=74 to be annexed. Then 8)5074 Or thus, 8)50,74 $ cts. - Ans. 634 cents.=6 34 $6,34 Ans. N. B. When tiere are no pence in the given sum, you must annex two ciphers to the shillings; then divide as be. fore, &c.:3. 1Rdu-, 31. 5s. New-England currency, to Federal Mo, ey l3 5s -6.s. Then 6)6500 Ans. 1083 cents. 208 APPENDIX. SOME USEFUL IRULES, FOR FINDING THE CONTENTS OF SUPERFICES AND SOLI&S. SECTION I.-OF SUPERFICES. The superfices or area of any plane surface, is compo sed or made up of squares, either greater or less, according to tie different measures by which the dimensions of tlh figure are taken or measured:-and because 12 inches ir length make 1 foot of long measure, therefore, 12X 12-144 tlhe 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 prcduct will be the area, or content. EXAML' I.ES. 1. HIow many square feet of boards are contained in thb floor of a room which is 20 feet square. 20 x 20=-100 feet, the Answer. 2. Suppose a square lot of land measures 2o rods oR each side, how many acres doth it contain? NOTE.-160 square rods make an acre. Therefore, 26x26=676 sq. rods, and 676- 160=4 a 36 r. tAe Answer. ART. 2. To measure a parallelogram, or long square. RULE. Multiply the length by the breadth, and the product vrill be the area, or superficial content. EXAMP LES. 1. A certain garden, in form of a long square, is 96 ftee long, and.54 wide; how many square feet of groulld are coltained in it? Ans. 96 x 54=-5164 square feet. 2. A lot of land, in form of a long slquare, is 120 rolls in tngth, and 60 rods wide; how maniv acies are in it? 120 x 60-=7200 sq. rods, tfen,io- _45 acres. Anrs; 3. If a board or plank he 21 feet long, and t1 incihe road; how many square feet are contained in it 18 inches=1,5feet, then, 21 1,5=31,5 Ans. APPENDIX. 209 Or, in measuring boards, you may multiply the length in 'aet by the breadth in inches, ind divide by 12, the quoier.t will give the answer in square feet, &c. Thus, in the foregoing example, 21 X 18-12=31,5 as nefore. 4. If a board be 8 inches wide, how much in length will make a square foot? RULE.-Divide 144 by the breadth, thus, 8)144 Ans. 18 in. 5. If a piece of land be 5 rods wide, how many rods in length will make an acre RULE.-Divide 160 by the breadth, and the quotient will be the length required, thus, 5)160 Ans. 32 rods in length. ART. 3.-To measure a triangle. Definition.-A triangle is any three cornered figure which is bounded by three right lines.* RULE. Multiply the base of the given triangle into half its perpendicular height, or half the base into the whole perpendicular, and the product will be the area. EXAMPLES. i. Required the area of a triangle whose base or longest side is 32 inches, and the perpendicular height 14 inches. 32 x 7-=224 square inches the Answer. 2. There is a trianudlar or three cornered lot of land whose base or longest side is 51 rods; the perpendicular from the corner opposite the base measures 44 rods; how many acres doth it contain? 51,5 x 22=1133 square rods,=7 acres, 13 rods. * A Triangle may be either right angled or oblique; in either case the teacher can easily give the scholar a right idea of the base and perpendicu km, by marking it down on the slate, paper, &c. s2 qih APPEND i. TO MEASURE A CIRCLE. ART. 4.-The diameter of a circle being given, to fims the circumference. RULE.-AS 7: is to 22::so is the given diameter: to the circum ference. Or, more exactly, as 113: is to 355:: &c. the diameter is found inversely. NOTE.I-The diameter is a right lin!e drawn across the circle through its centre. EIXAMI'LES. 1. What is the circumference of a wheel whose diameter is 4 feet — as 7: 22:: 4: 12,57 the circumference. 2. What is thle circuimferelce of a circle whose diamete'r is 35?-As 7: 22: 35: 110 Ans.-and inversely as 22 7: 110: 35, the diameter, &c. ART. 5.-To find the area of a Circle. RUiLE.-Multiply half the diameter by half the circumference, and the product is the area; or if the diameter is given wiithout the cir. cumference, multiply the square of the diameter by,7854, and tho product will be the area. EXAMPLES. 1. Required the area of a circle whose diameter is 12 inches, and circumference 37,7 inches. 1,851==lialf tihe circumference. 6=half the diamneter. 113,10 area in square inches. 2. Required the area of a circular garden wihose diame. ter is 11 rods?,7854 By the second method, 11X 11 = 121 Ans. 95,0334 rods SEiCTION c.-OF SOI,DS. Solids are estimated by the solid inch, solid foot, &c, 1728 of these inches, that is, 12 X 12 x 12 make 1 cublic 0 sGolid foot. APPENDIX. 211 ART. 6.-To measure a Cube. Definition. —A cube is a solid of six equal sides, each of which is an exact square. RULE.-Multiply the side by itself, and that product by the same ride, and this last product will be the solid content of the cube. EXAMPLES. 1. The side of a cubic block being 18 inches, or 1 foot and 6 inches, how many solid inches doth it contain? ft. in ft. 1 6=1,5 and 1,5 x 1,5 x 1,5=3~375 solidfeet. Ans. Or, 18 x 18 x 18-,=5832 solid inches, and - 8_-7.=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? 12 x 12 X 1-=1728 sold feet, the Ans. lRT. 7.-To find the content of any regular solid of threq dilnensions, length, breadthl aid thickness, as a piece of timber squared, whose length is more than the breadth and depth. RULE.-Multiply the breadth by the depth, or thickress, and that pioduct by the length, which gives the solid content. EXAMPLES. 1. A square piece of timber, being one foot 6 inches, od L18 inches broad, 9 inches thick, and 9 feet or 108 incheb 'long; how many solid feet doth it contain I 1 ft. 6 in.==1,5 foot 9 inches =,75 foot. Prod. 1,1 5 X 9= —I,125 solid feet, the Ans. in. in. in. solidi in. Or 18 X 9 x 108=17496 - 172S 0,1 25feet. But, in measuring timber, you may multiply the breadth }n inches, ale, the depth i.i-hcs, and tha' pr,,duct by 'he length in fee'. and divide ttie Iast product by 144, which 'will give the solid content in Cet, &c. 2J2 APPENDIX. 2. A piece of timber being 16 inches broad, 11 inches thick, and 20 feet long, to find the content? Breadth 16 inches. Depth 11 Prod. 176 x 20=-3520 then, 35~0~ 144=244 feet. Ans. 3. A piece of timber 15 inches broad, 8 inches thick, and 25 feet long; how many solid feet doth it contain I Ans. 20,8+feet. ART. 8.-When the breadth and thickness of a piece of timber are given in inches, to find how much in length will make a solid foot. RULE.-Divide 172t by the product of the breadth and depth, and,he quotient will be the length making a solid foot. EXAMP LES. 1. If a piece of timnber be 11 inches broad and 8 inches,,eep, how many inches in lenItlh will imake a solid foot? 11 x8S 8)172S(l9,6 inches. Ans. -2. If a piece of timber be 18 inches broad and 14 inchet deep, how many inches Mi legth wlill make a solid foot? 18 x 14=252 divisor, then, 252)17 6,1,8 inches. Ans ART. 9.-To lmea.sure a Cylinder. Definition.-A Cylinder is a round body lwhose bases art circles, like a round colutln or stick of timber, of equal bigness from end to end. RULE.-Multiply the square of tbe diameter of the end by,7854 which gives the area of the tb,;ae; then multiply the area of the base by the length, and the product will be the solid content, EXAM PLE. What is the solid content of a round stick of timber of equal bigness from end to end, whose diameter iY 18 inches, and length '20 feet? PPEND11. 213 18 i8c.l,5 ft. x 1,5 Square 2,25 x,7854=1,76715 area of the base; +20 length. Ans. 35,34300 solid content. Or, 18 inches. 18 inches. 354 X>,"54=254,4696 inches, area of the base. 20 length in feet. 144)5089,3920(35,343 solid feet. Ans. tRT. 10. To find how many solid feet a round stick of timber, equally thick from end to end, will contain when hewn square. RULE. Multiply twice the square of its semi-diameter in inches by the length in feet, then divide the product by 144, and the q'iotient will be the answer. EX A i MPL E. If the diameter of a round stick of timber be 22 inches ind its length 20 feet, how many solid feet will it contain,vhelii liew square? 11x 1I x2.20 144=33,G+ feet, the solidity when ietll s(quare. 'RT. II. To find how many feet of square edged boards of a given thickness, can be sawn from a log of a given diameter. RULE. Find the solid content of the log, when made square, by!e last article-Then say, As tle tlickness of tile board clludinlg the saw calf: is to the solid feet: so is 12 (in-:les) to the number of feet of boards. EXAMPLE. How many feet of square edged boards, 1:- inch thick, ieludiing the saw calf, can be sawn from a log 20 feet long tid 24 inches diameter? 12 x 12 x 2 x 20 1 44=40feet, solid content; As I: 40:: 12: 384 feet, theAns. 214 APPENDIX. ART. 12. The length, breadth and depth of any square bo being given, to find how many bushels it will contain. RULE. Multiply the length by the breadth, and that product b the depth, divide the last product by 2150,425 the soli inches in a statute bushel, and the quotient will be the ar cwoer. EXAMPLE. There is a square box, the length of its bottom is 5 inches, breadth of ditto 40 inches, and its depth is C inches; how many bushels of corn will it hold? 50 x 40 x 60 -2150,425=55,84 + or 55 bushels thr. pecks. Ans. ART. 13. The dimensions of the walls of a brick buildin being given, to find how many bricks are necessary build it. RULE. From the whole circumference of the wall measur, round on the outside, subtract four times its thickness, th( multiply the remainder by the height, and that product 1 the thickness of the wall, gives the solid content of t] whole wall; which multiplied by the number of briec contained in a solid foot gives the answer. EXAMPLE. How many bricks 8 inches long, 4 inches wide, and ' inches thick, will it take to build a house 44 feet loun, feet wide, and 20 feet high, and the walls to he 1 foot tlicl 8 x 4x2,5=80 solid inches in a brick, then 1728- Si 21,6 bricks in a solid foot. 44+40+44+40=168 feet, whole length of wall. -4 times the thickness. 164 remains. Multiply by 20 height. 3280 solid feet in the whole wall. Multiply by 21,6 bricks in a solid foot. Product, 70848 bricks. Ans. APPENDIX. 215 ART. 14. —To find the tonnage of a ship. RULE.-Multiply the length of the keel by the breadth of the beam, and that product by the depth of the hold, and divide the last product by 95, and the quotient is the tonnage. EXAMPLE. Suppose a ship 72 feet by the keel, and 24 feet by tbh beam and 12 feet deep; what is the tonnage? 72 x24 x 12 -95=218,2+tons. Ans. RULE II. Multiply the length of the keel by the breadth of the beam, and1 bat product by half the breadth of the beam, and divide by 95. EXAMPLE. A ship 84 feet by the keel, 28 feet by the beam; what is. It tonnage? 84 x 28 x 14 95=350,29 tons. Ans. U RT. 15.-From the proof of any cable, to find the strength, of another. RULE.-The strength of cables, and consequently the weights o, aeir anchors, are as the cube of their peripheries. 'herefore; As the cube of the periphery of any cablet Is to the weight of its anchor; So is the cube of the periphery of any other cable, To the weight of its anchor. EXAMPLES. 1. If a cable 6 inches about, require an anchor of 2t cwt, f what weight must an anchor be for a 12 inch cable? As6 6x6: 2 ~cwt.:: 12x12x12: 18cwt. Ans. 2. If a 12 inch cable require an anchor of 18 cwt. what. must the circumference of a cable be, for an anchor of 2~ rt.? cwt. cwt. 3 in. As 8: 12x12xx1:: 2,25: 216 V1-6 Ans. RT. 16.-Having the dimensions of two similar built shipa of a different capacity, with the burthen of one of them, to find the burthen of the other. 210 APPENDIX. RULE. The burthens of simnilar built ship~s are to each other, w.the cubes of their lihe (linteisiO is. If a ship of 300) tons ibirtheti be 75f feet long inj th~e ki.ee t de~mand the burthen of another ship, whose keel is 100 feetiou(?7. cwet. qrs. lb. As 75 X75)X75: 3iM) lO0X 0xlOxlOO711 -2o 0 244 DUODECIMIALS, OR CRLOSS MULTIPLICATION, IS a rule miade use of by workmien and artificers in easi tug1 Up thle coiitenets of tiCir wvork. 1. Under the rn~ui'phcaud wvrite the corresponding deno iniiations of thle multiplier. $2. ~lultiply each termn into thle 1multinlicand, beg~inning at te loestby the" hio glest (leloioluIatioul iii thle multipli]er and write the result of each under its respective terni; oh..serving to carry un unit for every 12, from each, lower do. nomination lo its iiext suicrlor. 3. In1 the Salle mnanner multiply all the multiplicand by the inches, or second~ demmotuination, In. the multiplier, amid set the result of echmi teruh one imlaee removed co the rmghin.huid of those in thle mlultil~l)icand. 4. Do the same with the seconds in the multipfier, %e-t* ting the result of each term two plalces to the right ban.) oi thlose ill the mullipli-cand, &c. EXAMPLES. P.1 P1. I. F.!1. I.1 Multiply 73 7 5 4 6 9 7 By 4 7 3 9 5 8 9 7 290".) 27 99 25 6 91 10!I Product, 33 2 9 APPENDIX. tM F.I. Multiply 4 7 By 5 10 Product, 26 8 10 F. '. Multiply 3 11 By 9 5 Product, 36 10 7 F. 1.L 3 8 7 6 27 6 F.. 6 5 7 6 48 16 F. L 9 7 3 6 32 6 6 F. I. 7 10 8 11 69 10 2 FEET, INCHES AND SECONDS. F.. L " Multiply 9 8 6 By 7 9 3 67 11 6."' =prod. by the i 7 3 4 6 " =ditto by the i 2 5 1 6 =ditto bythe s 75 5 3 7 6 Ans. F. ' F. I Multiply 7 1 9 5 6 By 7 8 9 8 9 Product, 55 2 9 3 9 48 11 [tiplier. Feet in the mulnches. econds. Iif 7 10 2 8 10 How many square feet in a board 16 feet 9 inches long, and 2 feet 3 inches wide? By Duodecimals. By Decimals. F. 1. F. I. 16 9 16 9=16,75 feet. 2 3 2 3=2,25 33 6 ' 8375 4 2 3 3350 3350 Ans. 37 8 3. F. 1. Ans 37,6&'587 8 3 2l1 APPENDIX. TO MEASURE LOADS OF WOOD. RULE.-Multiply the length by the breadth, and the product by t~h depth or height, which will give the content in solid feet; of wl ch 64 make half a cord, and 128 a cord. EXAMPLE. How many solid feet are contained in a load of wood. 7 feet 6 inches long, 4 feet 2 inches wide, and 2 feet '3 inches high t 7ft. 6 in.=7,5 and 4ft. 2 in. =4,167 and 2ft. 3 in,= 2.:25; then, 7,5 x 4,167=31,2525 x 2,25=70,318125 solid feet, Ans. But loads of wood are commonly estimated by the foot, allowing the load to be 8 feet long, 4 feet wide, and then 2 feet high will make half a cord, which is called 4 feet of wood, but if the breadth of the load be less than 4 feet, its height must be increased so as to make half a cord, wnichl is still called 4 feet of wood. By meastwing the breadth and height of the load, thi content may be found by the following RULE.-Multiply the breadth by the height, and half the produt will be the content in feet and inches. EXAMPLE. Required the. content of a load of wood which is 3 feelt f inches wide and 2 feet 6 inches high. By Duodecimals. By Decimals. F. in. F. 3 9 3,75 2 6 2,5 7 6 1875 1 10 6 750 9 4 6 9,375 - F. in. Ans 4 8 3 4,6875=48 or halfa cordand 81:> -inches ovir. The foregoing method is concise and easy to those who awe well acquainted weith Duodecunals, but the following table will give he sontent of any load of wood, by inspection only, suffieiently exact for O~mmon practice; whieh will be found ver nonvenient. A TABLE of Breadth, Height, and Contexrn, ' Breadth. Height wi feet. I nches. ft. in. 1213 j 2 3 4 51 61 71 8J 911011 2 6 1530 45601 1 2 41 5 6 71 9 10 1 12 1i 1 7 163147162 1 3 4 5 6 8 910 12 13 14 8 16 32 4864 1 3 4 5 7 8 9111 12 13115 9 17 33149166 1) 3 4 6 7 8 911 121415 10 1734'5168 2 3 4 6 7 91011 31416 11 18 35 2 5370 2 3 4 6 7 910 12 13115 16 3 T 1836 54 72 2 356 81 9 1I2417 1 193756174 2 3 5 6 8 1112 1416 17 2 19138 57 76 2 3 5 61 8 1011 13 14; 1617 3 19395978 3 51 7, 81011 13 15 1618 4 20406080 2 3 5 718101213151718 - 5 2116282 22 3 5 7 810112141617 19 6 2114263 84 2 4 5 7 911 1214 1618 19 7 221436486 2 4 5 7 9111131411611820 8 22144 6688 2 4 61 7 9 11 13i 1517 1820n 9 23 45 68190 2 4 6 7 911 1315171921 10 231466992 2 4 6 7 912 13)1517 1921 1 1 23477094 2 4 6 8 10,1214s16 18l2022 4 0 24148172 96 2 4 6 8 10o1221416 18 0O22 TO USE THE FOREGOING TABLE. First measure the breadth and hei ht of sour load to the nearest average Inch; then find the breadth in the left hand column of the table, then move to the right on the same line till you come under the height in feet, and you will have the content in inches, answering the feet, to whi&ch- dd the content of the inches on the right and divide the sum by 12, and you will have the true content of the load in feet and inches. N'ote.-The contents answering the inches being always smll, may oe added by inspection. EXAMPLES. 1. Admit a load of wood is 3 feet 4 inches wide, and 2 feet 10 inches nih, required the content.Thus, against 3 feet 4 inches, and under 2 feet, stands 40 inches; and un. der 10 inches at top, stands 17 inches: then 40+17=57, true content inr inches which divide by 12 gives 4 feet 9 inches, the answer. 2. The breadth being 3 feet. and height 2 feet 8 inches; required the content.Thus, with breadth 3 feet 0 inches, and under 2 feet atop, stnds 36 "0 APPENDIX. inches; atd under 8 inches, stands 12 inches: now 36 and 12 make 48, iAh answer in inches; and 48-12==4 feet, or just half a cord. S. Admit the breadth to be 3 feet 11 inches, and height 3 feet 9 inches required the content. Under 3 feet at top, stands 70; and under 9 inches, is 18: 70 and 18, make 688 12=7 feet 4 inches or 7 ft. 1 qr. 2 inches, the answer. TABAE I. Showing the amount of ~1, or $1, at 5 and 6 per cent. pe annum, Compound Inten.rst, for 20 years. Yrs. 5per cent. 6 per cent. Yrs. 5 per cent. 6 per cent 1 1,05000 1,06000 1 1 1,71034 1,89 529 2 1,10250 1,12360 12 1,79585 2,01219 3 1,15762 1,19101 13 1,88565 2,13292 4 1,21550 1,26247 14 1,97993 2,26090 5 1,27628 1,33822 1 5 2,07893 2,39655 6 1,34009 1,41851 16 2,18287 2,54727 7 1,40710 1,50363 17 2,29201 2,69277 8 1,47745 1,59384 18 2,40661 2,85433 9 1,55132 1,68947 19 2,52695 3,02559 10 1,62889 1,79084 2d 2,65329 3,20713 VII. The weights of the coins of the United States. pwt. g'rs. Eagles, 11 6 S d Half-Eagles, 5 15 SGtndad Quarter-Eagles, 2 19 -Dollars, 17 8 Half-Dollars, 8 16 Sta Quarter-Dollars, 4 8 Silver. Dimes, 1 17 f lver Half-Dimes, 20s! Cents, 8 16 Coper. Half-Cents, 4 8 The standard for gold coin is 11 parts pure gold, and one part alloy-the alloy to consist of silver and copper. The standard for silver coin is 1485 parts fine to 179 parts alloy-the alley to be wholly copper. FS9WDX. 2M ANNUITIES. TABLE IL I1 TABLE III. Showing the amount of ~ annui- Shwwing the present worth ty, forborne for 31 years or un- of ~1 annuity, to contider, at 5 and 6 per cent. cor- nuefor 31 years, at 5 and pound interest. 6 per cent. compound int. Yrs. 5 6 5.6 1 1,000000 1,000000 0952381 t943396 2 2,500000t 2,060000. 1594!0 133393 3 3,152500 3,183600 2,723248 2,673012 4 4,310125 4,374616 3,545950 3,46106 5 5,525631 5,637193 4,329477, 4,212364 6 6,801913 6,975319 5,075692 4,917324 7 8,142009 8,393838 5,786278 5,582381 8 9,5491091 9,897468 6,463213 6,209794 9 11,026564 11,491316 7,107822] 6,801692 10 12,577892 13,180770 7,72173557,360087 11 14,206787 14,971643 8,306414 7,886875 12 15,917126 16,869942 8,863252 8,38344 13 17,712982 18,88138 9,393573 '8,852683 14 19,59863221,015066 9,898641 9,294984 15 121,578564 23,275969 10,379658 9,712249 16 23,657492 25 27 10,837769 10,105895 17 25,840366 28,212380 11,274066 10,477260 18 28,132385t30,905653 1a 6587, 11,0 7603 19 30,5399004t33,75999 12,085321 11,158116 20 33,065954 36,78592 12,46221011,469921 -21 3,719252 39,9.727 12,821153 11,764077 22 38,505214143,3 91 13,163003 12,0141582 23 41,430475146,99828 13,488574 12,33380 24 - 44,501999150,8~5578 13,79864q{ 12,560357 25 47,727099 54,864512 14,03944312',73356 26' 51,113454 59,156382 14,375 13 31 27 54,669126 63,7057615 14,64304113',21i0534 28 58,402583 68,528112 14,898127113,406164 29 62,332712173,639 798 15,141073i,!590721 30 66,43884779,058186 15,372451 13 74 1 i70,760079,9,167 15,5928101 1,^Pe --------------------- ^ 4* APPENDIX. TABLES. THE 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 payment of all debts and demands, at the several and respective rates following, viz. The Gold Coins of Great Britain and Portugal, of their present standard, at the rate of 100 cents for every 27 grains of the actual weight thereof.-Those of France and Spain 272 grains of the actual weight thereof.-Spanish milled dollars weighing 17 pwt. 7 gr. equal to 100 cents, 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 several pieces of English, Portuguese and French Gold Coins. Pwat. Gr. Dols. Cts. lM. Johannes, - - - 018 1 0 Single ditto, - - 9 8 0 0 English Guinea,- - 5 6 4 662 Half ditto, -- - 2 15 2 33 French Guinea, - - - 5 6 4 59 8 Half ditto. --- 2 15 2 29 9 4 Pistoles, ----- -- 6 1 12 1 45 2 2Pistoles, - -. — 8 6 7 22 6 1 Pistole, - 4 3 3 61 3 aoidore, -. 6 22 6 14 8 !, TABLE V. [Weights of English and Portuguese Gold, in Dollars, Cents ana Mills. EGr. I Cts. Mills. Pwts.l Dols. Cts. Mills. 2 3 4 5 6 7 8 16 i 19 20 | 22 1 17 1 18 I I 19 I 21 123.... - - 3 7 11 14 18 22 25 29 33 37 40 44 48 51 55 59 63 66 70 74 77 81 85 7 4 1 8 5 2 9 6 3 7 4 1 8 5 2 6 4 1 2 5 2 2 3 13 4 1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 oz 1 2 3 0 1 2 3 4 5 6 7 8 8 9 10 11 12 13 14 15 16 16 17 35 53 89 77 66 55 44 33 22 11 89 77 66 55 44 33 22 11 89 77 55 33 7 6 5 4 3 2 1 7 6 5 4 3 2 1 5 5 IWeighi Grs. 2 3 4 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 1 22 9 23 TABLE VI. ts of French and Spanish Gold, in Dollars, Cents and Mills. Cts. Mills. \\Pwts.\ Dols. Cts. Mills. 7 11 14 18 21 25 29 32 36 40 43 47 51 54 58 62 65 69 73 76 80 83 3 6 2 9 5 2 8 5 1 8 4 1 6 4 0 7 3 6 3 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 oz 1 2 3 0 87 1 75 2 62 3 50 4 38 5 25 6 13 7 00 7 88 8 76 9 63 10 51 11 38 12 26 13 13 14 1 14 89 15 76 16 64 17 51 35 3 52 55 6 2 7 3 5 1 7 3 5 1 7 3 9 5 0 6 2 8 6 5 i>,,q Iw P. ta It I0 I5......w.... - 224 Al 'T'fND1 X. VII. TABLE of Cents, answering to the Currenctse of the United States, with Sterling, 4~c. NOTE.-The figures on the right hand of the space show the parts of a cent, or mills, &c. 6s. to 8s. toi7s.6d. 4s. d. 5s. to 4s.6d. 4s. 10d to the the to the to thel the to the the Doll. 1DoU. ll. Doll. Dol. oll oll. Dollar. eP. cn cennts cents. cents. cen ts. cents. I 1 13 1 0 1 1 17 16 18 1 7 \ 2 27 2 2 35 33 37 3 4 3 4 1 3 1 33 53 5 5 5 1 1 4 55 4 1 441 7 1 63 74 6 8 5 69 52 55 891 86 92 8 5 6 8 3 6 2 6 6 10 7 10 11 1 10 2 7 97 721 77 12 5 11 6 129 11 9 8 11 1 8 3 8 814 1413 8 13 6 9 12 5 9 3 10 16 15 16 6 15 3 10 13 8 10 4 11 17 8 16 6 18 5 17 11 15 2 11 4 12 2 19 6 18 3 20 3 18 S. 1 16 6 12 5 13 3 21 4 20 22 2 20 2 33 3 25 26 6 42 8 40 44 4 41 3 50 37 5 40 64 60 66 6 61 5 4 66 6 50 53 3 85 7 80 88 8 82 5 83 3 62 5 66 6107 1 100 111 1 102 5 6100 75 80 128 5120 133 3 123 7 11 6 87 5 93 3150 140 1155 143 5 8 133 3 100 106 6171 4 160 177 7 164 1 9 150 112 5120 192 8180 200 184 6 10 166 61125 133 3214 21200 222 2 205 1 11183 31137 51466235 7220 24 4 226 6 2 200 150 160 257 124: 266 246 1 131216 6162 5173 3!278 51 60 88 28 2:6 6 141233 31175 186 6300 280 311 1287 1 15250 187 5200 321 4300 333 3 307 6 161266 61200 213 3134 8320 355 5 328 2 17283 3212 6226 6364 2340 377 7 348 7 18300 225 I240 385 6j360 400 369 2 19316 6237 5`25 31407 1o380 1422 389 7 20333 250 266 6428 51400 444 410 2 - -_~, APPENDIX. TABLE IX. Shewing the value of Federal MIoney in other Currencies. t New Jersey, New Eng- New York Pennsylva- South-CarFederal land, Vir- and North nia, Dela- olina, and Money. ginia, and Carolina ware, and Georgia Kentuky currency. Maryland currency. _ currency.: _ currency. Cents. I s. d. s. d. s. d. s. d. 1 0 03 0 1 0 1 o0 0 2 n 0 1 0 2 0 14 0 1 3 0 21 0 3 o 0 23 0 4 0 3 0 3 0 3 0 21 5 0 32 0 43 0 4- 0 23 6 0 42 0 53 0 5- 0 34 7 0 5 0 63 0 61 0 4 8 0 53 0 71 0 71 0 41 9 0 6- 0 81 0 8 0 5 10 0 74 0 91 0 9 0 5~ 11 0 0 10 0 10 126 0102 0 0 6 13 0 91 1 0 0 9 1 7~ 14 0 10 1 4 1 70 15 0 103 1 21 1 1 0 82 16 0 14 1 31 1 21 0 9 17 1 1 4 3~ 0 92 18 1 1 1 51 1 4 0 10 19 1 1 1 6} 1 51 0 103 20 1 21 1 71 1 6 0 11I 30 1 91 2 43 2 3 1 43 40 2 41 3 21! 3 0 1 10 -50 3 0 4 0 3 9 2 4 60 3 74 4 9 4 6 2 91 70 4 21 5 7 5 3 3 31 80 4 94 6 4 6 0 3 83 90 5 43 7 2 6 9 4 21 100 6 0 8 0 7 6 4 8 262j.%\t I'-rf i t A. A FEW USEFUL, FORL,.S IN TRANSACTING BUSINESS. -g. AN OBLIGATORY BOND. KNOW all men by these presents, that I, C. D. ai in the county of am held and firmly bound tc H. W. of in the penal sum of to be paic H. W. his certain attorney, executors, and administrators to which payment, well and truly to be made and done I bind myself, my heirs, executors, and administrators firmly by these presents. Signed with my hand, an( sealed with my seal. Dated at this daj of A.D. The condition of this obligation is such, That if the above bounden C. D. &c. [Here insert the condition. then this obligation to be void and of none effect; other wise to remain in full force and virtue. Signed, sealed, and delivered, } in the presence of A BILL OF SALE. KNOW all men by these presents, that I, B. A. of for and in consideration of to me in hand paid bj D. C. of the receipt whereof I do hereby ac knowledge, have bargained, sold, and delivered, and, bi these presents, do bargain, sell and deliver unto the saik D. C. [Here specify the property sold.] To HAVE and t( HOLD the aforesaid bargained premises, unto the said ). C his executors, administrators, and assigns, forever. And ' the said B. A. for myself, my executors and administrators shall and will warrant and defend the same against all per. sons unto the said D. C. his executors, administrators, ant assigns, by these presents. In witness whereof, I havt hereunto set my hand and seal, this day of 1814. In presence of A SHORT WILL. I, B. A. of, &c. do make and ordain this my last will and testament, in manner and form following, viz. I giv APPENDIX. 227.nd bequeath to my dear brother, R. A. the sum of ten )ounds, to buy him mourning. I give and bequeath to ny son J. A. the sum of two hundred pounds. I give and,equeath to my daughter E. E. the sum of one hundred sounds; and to my daughter A. V. the like sum of one iundred pounds. All the rest and residue of my estate foods and chattels, I give and bequeath to my dear beovcd wife, E. R. whom I nominate, constitute and appoint ole executrix of this my last will and testament, hereby evoking all other and former wills by me at any time teretofore made. In witness whereof, I have hereunto et my hand and seal, the day ot in the year of our Lord Signed, sealed, published and declared by the said tesitor, B. A. as and for his last will and testament, in the Iresence of us who have subscribed our names as witnesses h treto, in the presence of the said testator. R. A. S. D. L. T. NOTE.-The testator, after taking off his seal, must, in i -sence of the witnesses, pronounce these words: "I t blish and declare this to be my last will and testament." Where real estate is devised, three witnesses are abolutely necessary, who must sign it in the presence of he testator. A LEASE OF A HOUSE. KNOW all men by these presents, that I, A. B. of I for and in consideration of the sum of reeived to my full satisfaction of P. V. of this ay of in the year of our Lord have demised ad to farm let, and do by these presents, demise and to farm let, nto this said P. V. his heirs, executors, administrators and asgns, one certain piece of land, lying and being situated in said bounded, &c. [Here describe the boundaries] with a welling house thereon standing, for the term of one year from lis date. To HAVE and to HOLD to him the said P. V. his heirs, Kecutors, administrators and assigns, for said term, for him the fid P. V. to use and occupy, as to him shall seem meet and roper. And the said A. B doth FvXnrtrER COVENANT with the 2A P It'. N'LD SX. said P. that he hath good right to let and demise the said letten and demised premises in manner aforesaid, and that ho the said A. during the said time will suffer the said P. quietly to HAVE and to HOLD, use, occupy and enjoy said demised premises, and that said P. shall have, hold, use, occupy, possess and enjoy the same, free and clear of all incumbrances, claims, rights and itles whatsoever. In witness whereof, I the said A.. have iereunto set my hand and seal, tihis (lay of Signed, sealed and delivered in presence of 5 A 13. A NOTE PAYABLE AT A BANK. $500, 60] HARTFORD, May 30, 1815. FOR value received, I promise to pay to John Merchant 9r order, Five Hundred Dollars and Sixty Cents, at Hartford Bank, in sixty days fiom the date. WILLIAM DISCOUNT. AN INLAND BILL OF EXCHANGE. [$83, 341 BOSTON, June 1, 1815. TWENTY days after date, please to pay to Thomas Goodwin or order, Eighty-Three Dollars and Thirty-Four Cents, and place it to my account, as per advice from your humble servant, Mr. T. W. IMerchant, S SIMON PURSE. New- York. 5 A COMMON NOTE OF HAND. [$1301 NEW-YORK, March 8, 1821. FOR value received, I promise to pay to John Murray, One Hundred and Thirty Dollars, in four months from this date, with interest until paid. JOHN LAWRENCE. A COMMON ORDER. NEW-YORK, June 10, 1822. Mr. Charles Careful, Please to deliver Mr. George Speedwell, the amount of iwenty-Five Dollars, in goods from your store; and charge the same to the account of Your Ob*t. Servant, E. WHITE. FINIS. THE PRACTICAL ACCOUNTANT, OR, FARMERS' AND MECHANICKS' BEST METHOD OF BOOK-KEEPING; FOR THE EASY INSTRUCTION OF YOUTH. DESIGNED AS A COMPANION TO DABOLL'S ARITHMETICK BY SAMUEL GREEN. INTRODUCTION. SCHOLARS, male and female, after they have acquired a sufficient knowledge of Arithmetic, especially in the fundamental rules of Addi. tion, Subtraction, Multiplication, and Division, should be instruct( d in the practice of Book Keeping. By this it is not meant to recoin - mend that the son or daughter of every farmer, mechanic, or she p keeper, should enter deeply into the science as practised by the me r chant engaged in extensive business, for such study would engross a great portion of time which might be more usefully employed in acquiring a proper knowledge of a trade, or other employment. Persons employed in the common business of life, who do not keel regular accounts, are subjected to many losses and inconveniences to avoid which, the following simple and correct plan is recommen( ed for their adoption. Let a small book be made, or a few sheets of paper sewed toge ther, and ruled after the examples given in this system. In the bool termed the Day Book, are duly to be entered, daily, all the transac tions of the master or mistress of the family, which require a charg to be made, or a credit to be given to any person. No article thu. subject to be entered, should on any consideration be deferred ti another day. Great attention should be given to write the transac tion in a plain hand; the entry should mention all the particulars nu cessary to make it fully understood, with the time when they too. place; and if an article be delivered, the name of the person to whoi, delivered is to be mentioned. No scratching out may be suffered; be cause it is sometimes done for dishonest purposes, and will weake l or destroy the authority of your accounts. But if, through mistake, any transaction should be wrongly entered, the error must be rectifiE I by a new entry; and the wrong one may be cancelled by writing ti D word Error in the margin. A book, thus fairly kept, will at all times show thc exact state of k persons affairs, and have great weight, should there at any time he a necessity of producing it in a court of justice. FORM OF A DAY BOOK. 3 *JEREMIAH GOODALE, Albany, January 1, 1822. ntered. Joseph Hastings, Cr. $ ct. 1 By 3 months' wages, at $6 a month, due this date, 18 00.... 5 Entered. Samuel Stacy, Dr. 1 To 2 weeks' wages of my daughter Ann, spinning yarn, at 75 cents a week, ending this day, 1 5G Entered Joseph Hastings, Dr. 1 To my order for goods out of the store of Anthony Billings,....... 11 50 tntered. Anthony Billings, Cr. 1 By my order in favour of Joseph Hastings,. 11 50.-.. 15 - fitered. Thomas Grosvenor, Dr. 1 To the frame of a house completed and raised this day on his Glover Farm, so called, 4000 feet at 2~ cents per foot,.... 100 00 -18 -. tntered. Edward Jones, Cr. 1 By his team at sundry times, carrying manure on my farm,......5 64 25 -- i ntered. Thomas Grosvenor, Dr. 1 To 48 window sashes delivered at his Glover Farm, so called, at $1,00.... 48,00 Setting 500 panes of glass by my son John, at 13 cents,..... 7,50 10 days' work of myself finishing front room, at $1,25 a day,.... 12,50 7~ do. of William, my hired man, laying ) the kitchen floor and hanging doors, at > 6,30 84 cents a day, ) - 4 30 -. 26 (ittered. Anthony Billings, Cr. 1 By 2 galls. molasses, at 36 cts. per gall. 0,72 4 yds. of India Cotton, at 183 cents, 0,74 2 flannel shirts to Joseph Hastings, 2,16 -- 3 62 Entered. Joseph Hastings, Dr. 1 To 2 shirts of A. Billings,.... 2116 * There put the name of the owner of the book, and first date. 4 FOhM-OF A I)AY HOCt. Albany, February 12, 1822. Entered.l Thomas Grosvenor, Cr. 57 1IBy my order in favour of Joseph Hastings, 3 5 Entered. Joseph Hastings, Dr. 1 l'o my order on T. Grosvenor, 3 54 - 16 Entered. Thomas Grosvenor, Dr. 1 To 3 days' work of myself on your fence at $1,25 per day,..... 3,75 3 days' do. my man Wm. on your stable and finishing off kitchen, at 84 cts... 2,52 2 pr. brown yarn stockings, at 42 cts. 0,84 7 11 18. Entered. Edward Jones, Cr. 1 By 4 months' hire of his son William at $10 a month, 4C 00 24 Entered. Edward Jones, Dr. 1 To my draft on Thomas Grosvenor,.. 3800 Entered. Thomas Grosvenor, Cr. 1 By my draft in favour of E. Jones,.. 38 0 ---— 28 1.. — Entered. Thomas Grosvenor, Dr. 1 To the frame of a barn,.... 75 0( Entered. Anthony Billings, Cr. 1 For the following articles, 14 lbs. muscovado sugar at $12 pr cwt. 1,50 1 large dish,..... 03 6 plates,...... 0,30 4 cups and saucers 0,20 1 pint French Brandy, 0,17 1 quart Cherry Bounce, 0,33 Thread and tape,.0,18 2 Thimbles,.. 0,04 1 pair Scissors,..... 0,17 1 quire paper,..... 0,25 Wafers, 4; ink, 6; 1 bottle, 8;.. 0,18 5& Entered. Peter Daboll, Dr. 1 To a cotton Coverlet delivered Sarah Bradford, by your written order, dated 14 Jan.. 5 li FORM OF A DAY BOOK. Albany, March 1, 1822, Cntered. Thomas Grosvenor, Cr. - ce 1 By cash paid me this date,....75 00 Entered. Anthony Billings, Dr. 1 To one barrel of Cider,.... $1 17 1 barrel containing the same, (from Tho mas Grosvenor,)... 0 58 - 1 76 7..... ----- 7 --- —---- Entered. Thomas Grosvenor, Cr. 1 By 1 barrel containing Cider sold and delivered to Anthony Billings,..... 0 58 - 10 -- Entered. Anthony Billings, Dr. 1 To cash per his order to George Gilbert,. 24 32 -15 Entered. Peter Daboll, Cr. 1 By amount of his Shoe account,.. $4 48 Yarn received from him for the balance of his account,..... 1 () 551 IEntered. Samuel Green, Cr. 2 By amount due for 12 months New-London Gazette,..... $2 00 4 Spelling Books, at 20 cents, for children, 0 80 1 Daboll's Arithmetic, for my son Samuel, 0 42 2 blank Writing Books, at 12~ cents,. 0 25 1 quire of Letter Paper,... 0 34 381.... -24 -Entered. Notes Payable, Dr. 2 By my note of this date, endorsed by Ephraim Dodge, at 6 months, for a yoke of Oxen bought of Daniel Mason, at Lebanon,... 48 00 - 28- --- Entered. Jonathan Curtis, Dr. 2 To an old bay Horse,....$23 00 A four wheeled Wagon, and half worn Harness,..... 42 00 -- 65100 CEtered. Samuel Green, Dr. t To cash in full,...... 381 U 2 FORM OF A DAY BOCB. Albany, April 6,1822.,ntered. Anthony Billings, Dr. * i 1 To 2 tons of Hay, at $11 25,.. $22 50 Amount of order dated March 26, 1822, in favour of Fanny White, paid in 1 0 54 pair yarn stockings,.. Hire of my wagon and horse to bring) sundry articles from Providence, 3d 3 00 of this month,.. 26 120..... --- —12 —....... ---- Entered. Thomas Grosvenor, Cr. 1 By his order on Theodore Barrell, New-London, for 68 dollars,...... 68 00 Entered. Anthony Billings, Dr. 1 To 1 hogshead Rum from Theodore Barrell, 100 gals. at 50 cents,... $50 00 Cash received from said Barrell for balance due on Thomas Grosvenor's order, 18 00 ' -- 68 06 -18 -Catered. Jonathan Curtis, Cr. 2 By a coat $14,75, pantaloons $5,00,.. 19 75 22 -'ntered. Thomas Grosvenor, Dr. 1 To mending your cart by my man William, $1 00 Paid Hunt for blacksmith's work on your cart,..... 0 58 Setting 6 panes of glass, and finding glass, 0 66 0 224, - - 2625 - ientered. John Rogers, Dr. 2 To a yoke of Oxen, at 60 days' credit,.. 60 00 -29 Entered. Anthony Billings, Cr. 1 By Garden Seeds of various kinds,.. $0 56 1 pair Boots, myself, $4,00, and 1 pair for John, $3,50,..... 7 50 1 pair of thick Shoes forJoseph Hastings, 1 25 Tea, Sugar, and Lamp Oil, per bill,. 0 68 - 9 99 Entered. Notes Payable, Cr. S By my note to Isaac Thompson, at 6 months, 9000 fORM OF A DAY BOOK. 7 Albany, May 3, 1822. entered. Theodore Barrell, New-London, Dr.. c. 2 To 16 cheese, 308 lbs. at 5 cents,.. $15 40 217 lbs. of butter, at 15 2-3 cts... 34 00 24 lbs. of honey, at 12~ cents,.. 3 00 8 -Entered. Joseph Hastings, Dr. 1 To 1 pair shoes, 29th April, from Anthony Billings, 1 25 12 tntered. Anthony Billings, Dr. 1 To 84 bushels of seed potatoes, at 33 1-3 cents,...... $28 00 8 pair mittens, at 20 cents,.. 1 60 Cash,...... 14 00,. 43 60.. 15 — Entered. Joseph Hastings, Cr. 1 By 4Q months wages, at 7 dollars, 31 5( — 20 Entered. Theodore Barrell, Cr. 2 By cash in full of all demands,... 52 4f 25 Entered. Thomas Grosvenor, Cr. 1 By his acceptance of my order in favour of Anthony Billings,...... 54 OC Ent3red. Anthony Billings, Dr. I To amount of my order on Thomas Grosvenor, 54 0Ot Sept. 24 &ntered. Notes Payable, Dr. 2 To cash paid for my note to D. Mason,. 48 00 The foregoing example of a Day Book, may suffice to give a good idea of the way tn which it is proper to make the original entries of all debt and credit articles. Ano-. 'her small book should next be prepared, according to the following form, termed the book of Accounts, or Leger. Into this book must be posted the whole contents of the Day Book; care being taken that every article be carried to its corresponding title he debt amounts to be entered in the left, and the credit in the right hand page. Thus, should it at any time be required to know the state of an account, it will only be neccs. sary to sum up the two columns, and to subtract the smaller amount from the greater the remainder will be the balance. When an article is posted from the Day Book into the Leger, it will be proper, op, psite the article, to note the same in the margin of the Day Book, by writing the word Entered, or making two parallel strokes with the pen; to which should be added the figure denoting the page in the Leger where the account is. On a blank page at the beginning or end of the Leger, an alphabetical index should be written, containing the names of every person with whom you have accounts in the Leger, with the number of the paoe where the rscwa ae. i FORM OF A LIOER. Joseph Hastings. Dr. 1822. i let. Jan'y 5 To my order on Anthony Billings for goods, 1150 26 2 shirts of Anthony Billings, - - - 216 Feb'y 12 My order on Thomas Grosvenor, - - 3150 May 8 1 pair shoes, 29th April, from A. Billings, - 1125 Dr. Samuel Stacy. 1822 # ct. Jan'y 5 To 2 weeks' wages of my daughter, at 75 cents a week, - - - - - - 1 50 Dr. Anthony Billings. 1822. $ (ct. March 4 To 1 barrel of cider, and barrel, - - - 75 10 Cash paid your order in favour of G. Gilbert, 24 3I! April 6 Sundries, - - 26 01 12 ditto, - - 68 00 May 12 ditto, - - -43160 25 My order on Thomas Grosvenor, - 54100 Dr. Thomas Grosvenor. 1822. $~ 7ct. Jan'y 15 To the frame of a house, - 100 00 25 Sundries, - - - - 74 30 Feb'y 16 Sundries, - - 7 11 28 The frame of a barn, - -75 00 April 22 Sundries, - - - - 224 Dr. Edward Jones. 1822. f I W ct. Feb'y 24 To my draft on Thomas Grosvenor, - 38 Dr. Peter Daboll. ~.by To ndies, Feb'y 128[To sundries, I 5t. -! 151 rORM OF A LEGER. I A hired lad, Cr. 1822. $jct. Yan'y I By 3 months' wages due this day, at $6, - - 18 00 May 15 4Q months'wages, at $7, - - - - 315 Farmer, Cr. I i_ _ Merchant, Cr. 1822. d "c. Jsn'y 5 By my order in favour of Joseph Hastings, 11 50 26 Sundries, - - - - 3 62 Feb'y 28 ditto, - - - - - 3155 April 29 ditto, - - - - 999 Judge of County Court, Cr. Feb'y 12 By my order in favour of Joseph Hastings, - $3 50 24 My draft in favour of Edward Jones, - - 38 00 March 1 Cash paid me this day, 75 00 1 empty cider barrel, - - - - 58 April 12 Amount of your order on Theodore Barrell, 68 00 May 2 My order in favour of Anthony Billings, - 5400 Labourer, Cr. 1822. J $ ct. Jan'y 18 By team hire at sundry times, - - - 5 64 Feb'y 18 4 months' hire of his son William, at $10, - 40 00 Farmer, Cr. March I15 By sundries in full, - I id.l - 1 5l51 2 rORM OF A LEGER Samuel Green. Dr. 1822. 1 1 I Ice March 123To cash in full of his account, - - 381 Dr. Notes Payable. 1822.1 1 cit Sept. |24lTo cash paid for my note to D. Mason, - - 48 06 Dr. Jonathan Curtis. 1822. * ci| March 28 To a bay horse, - 2300 A wagon and harness, - - - - 42100 Dr. John Rogers. 1822. $ ct. April 25 To 1 yoke of oxen at 60 days' credit, - - 60 00 Dr. Theodore Barrell. Mayl 3JTo 16 cheese, weight 308 Ibs. at 5 cents, - - 15 4 217 lbs. butter at 15 2-3 cents, - - - 3400 24 lbs. honey at 12 cents, 300 - |52 40 INDEX TO THE LEGER. B. PAGE H. PAGE Barrell, Theodore, - - 2 Hastings, Joseph, - - Billings, Anthony, - - 1. C. Jones, Edward, - - Curtis, Jonathan, - - 2 N. D. Notes Payable, - - 2 Daboll, Peter, - - - R......... Rogers, John,- - - 2 G. Grosvenor, Thomas, - I S. Gleen, Samuel, - - 2 Stacy, Samuel, - * I FORM OF A LEOaR. New-London. I Cr. 1822. ~ l ct. March 15 By sundries, - 81 _ *t I "" Cr. 1822. l * Ict. March l24IBy my note to Daniel Mason, at 6 months, endori sed by Ephraim Dodge, - - - 48 00 April 29| Do. Isaac Thompson, at 6 months, - - 90 OC Danbury. Cr. 1822. $ ct. April 18 By a coat, - 14 75 A pair of pantaloons, - - - 5 00 Hudson. Cr. -^ - -_-_ - t2. ll0 cf. New-London. Cr. 1822. $ ct. Mayl20 By cash in full, 52 40 52140 QUESTIONS TO EXERCISE THE STUDENT. What is the state of thefollowing Accounts? Joseph Hastings, Samuel Stacy, Anthony Billings, Thomas Grosvenor, Edward Jones, Notes Payable, Jonathan Curtis, John Rogers. J 'Due Mseph Hastings, Edward Jones, - Notes Payable, - - W Samuel Stacy owes, - - Anthony Billings owes, Thomas Grosvenor owes, Jonathan Curtis owes, - John Rogers owe, - - $31 09 7 64 90 00 1 50 189 05 19 57 45 25 60 00 1 S -F;FlUl, FORM.S.J Farmt': Ri/ll. or Accomuan ABUIiRNi, Oct. 21, 1822. Thomas Yates, Esq. To John Mornington, Dr. 1822. April 5. To 5 barrels Cider, at $2,00... #10,00 20 bushels Potatoes, at 0,25. 5,00 55 lbs. Butter, at 0,17... 9,35 June 6. 1 ton of Hay,.... 10,00 July 15. 0t lbs. Cheese, at 0,08.. 3.20 2 cords of Wood, at 4,00... 8,00 Received the amount. $37,55 JOHN MORNINGTON. N. B.-To prevent accidents, care should be taken not to receipt an account until it is paid. A negotiable NJote. New-Haven, March 21, 1822. Six months after date, I promise to pay to William Walter, or or der, (a'. ay house,) One Hundred Dollars, value received in two yokt of oxen. JAMES HILLHOUSE. ['It is best to mention where the note shall be paid, and for what it is given. Without the words, " or order," a note is not negotiable A Receipt in full. Received, Hartford, May 22, 1822, of Theodore Barrell, Esq. Fifty two Dollars, in full of all demands. GEO. GOODWIN. 03' If the payment be not in full, write " on account." N. B.-For other useful forms, see the Arithmetick. NO TE. The affectionate Instructor, who always feels a parental solicitude for the permanent welfare of his pupils, cannot in any way so much contribute to their success in life, with so little trouble, as to teach them to understand this abridged, complete and simple system of Book Keeping. It contains all the important principles of extended and expansive works on the science; all, in fact, that is necessary to be known by the Farmer, Mechanic, and Shopkeeper, relating to accounts; and yet with very little explanation and repeated copying and balancing the accounts, will be so fully understood and deeply impress sed on the memory of scholars of common mind, as never to be forgot ten; while their knowledge of common arithmetick and practical pn manship will thereby be greatly improved. FINIS. UNfVERSITY OF MICHIGAN 3i1 11111 I 2I I II 3 9015 04872 4291 ..1 - awr, —, ", - Y:7 1- AV - Ir 7 11 - I.74. -, - ''W? 1.0 iwsmw -Wnk.VAU