HER BOOK i * DABQLL'8 SCHOOLMASTER'S ASSISTANT. IMPROVED AND. ENLAK9*D. ' ^V' BEIWG A *>J^ TACTICAL SYSTEM OF ARITHMETICS. ADAPTED TO THE UNITED STATES. BY NATHAN DABOLL. WITH THE ADDITION OF THE FARMERS AND ME^HANICKS' BEST IETHOD OF B00M=KEBP DESIGNED AS A COMPANION TO DABOLL'S ARITHMETICK. BY SAMUEL GREEN. MIBDLETOWN, (Con.) PUBLISHED BY WILLIAM H. NILE.S. Stereotyped by A. Chandler, New- York. District of Cennecticutj ss. BE IT R&MEMBKRED, That ton Hie tw*my-eij;Uih day of September, in the lor ty- ' ., ~ fifth year of the IpdtpeuikuM ol' ih* -Uii4*tf SktU* of AntMriea, Samuel Grli, '' of aitl District, hath deposited ia ^iia offie* the utl of a Book, th right \vneieofhe ctilms a yropritor } in tfie wocds fiDUowiiig, to wU : (( Daftoll's Sclipoltnastec's Assistant, improved and cularged. Being a plain prac- tical system of Arithmetick. Adapted to the United States. By Nathan Daboll." In conformity to the Act of the Congress of the United States, entitled, " An Act for the encouragement of learning, by securing the Copies of Maps, Charts, and Books, to the authors and proprietors of then), during the times therein mentioned." HENRY W. EDWARDS, Clerk of the District of Connecticut. A true copy ef Record : Examined and sealed by me, H. W. EDWARDS, Cftrk of the District of Connecticut. PSVCH. I.JRRARV GIF? RECOMMENDATIONS, EDUC.- Yak GoUege, &. 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 ex- plained 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 School- masters 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 men- tioned, and concur with Mr. Professor Meigs in his opinion of its merit. NOAH WEBSTER. New-Haven Dec. 12, 1799. Rhode-Island College, Nov. 39, J799. 1 HAVE run through Mr. DABOLL'S SCHOOLMASTER'S ASSISTANT, and have formed of it a very favourable opinion. According to its original design, I think it well " calculated to furnish Schools in general with a methodical, easy, and comprehensive System of Practical Arithmetic." I there- fore hope it may find a generous patronage, and have an r xtensive spread. ASA MESSER, Professor of the Learned Languages, and teztchcr of Dfathc/natirs. r \ow PreSirt*rft of that Tn^itutian.! 053 ft r-: c o - j M K x D A T r o x s . 40 Piainfield Academy, April 20, 1802. l< MARK use of DABOLT/S SCHOOLMASTER'S ASSISTANT, ia teaching common Arithmetic, and think it the best cal- culated for that purpose of any which has fallen within mv observation. JOHN ADAMS, Rector of Plain field Academy. [New Principal of Philips' Academy, Andover, Mass.] ELilkrlca Academy, (Mass.) Dec. 10, 1807. HAVING examined Mr. DABOLL'S System of Arithmetic, ]' am pleased with the judgment displayed in his method, and the perspicuity of his explanations, and thinking it as easy and comprehensive a system as any with which I am acquainted, can cheerfully recommend it to the patronage of rnstnirtw. SAMUEL WHITING, Teacher of Mathematics. Prom Mr. Kennedy, Teacher of Mathematics. I BECAME acquainted with DABOLL'S SCHOOLMASTER'S 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 improve- ment of the student, as well as the ease and assistance of the preceptor. I also deem it equally well calculated for 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 90, PHM. THE design of this work is to furnish rh;.' ->f UH; United States with a methodical and comprehensive .v, of Practical Arithmetic, in which 1 have endeavoured, through the whole, to have the rales as concise and fami- liar as the nature of the .subject will permit. During the long period which I have devoted to tho in- struction of youth in Arithmetic, I have made use of vanou ; systems which have just claims to scientific merit; but \\i\>. authors appear co have been .deficient in an important point the practical teacher's experience. They have b too sparing of examples, especially in the first rudiments; in consequence of which, the young pupil is hurried through the ground rules too fast for his capacity. This objection I have endeavoured to obviate in the fallowing tronti.se. In teaching the first rules, 1 have found it best to en- courage the attention of scholars by a variety of easy and familiar questions, which might serve to strengthen their minds as their studies grew more arduous. The rules arc arranged ia such order as to in!. most simple and necessary parts, previous to those which are more abstruse and difficult. To enter into a detail of the whole work would tlious ; I shall therefore notice only a fo\v particulars, refer the reader to the contents. Although the Federal Coin is purely decimal, it nearly allied to whole numbers, and so absolutely u to be understood by every one, that I have intro immediately after addition of whole nurnboivi, tiud shown how to find the value of goods therein, im?;; after simple multiplication ; which niny b tage to many, who perhaps will not 1\ learning fractions* In the arrangement of fraction new mwhod, the advantages and facility of which sufficiently npotoo-i/o for its not her: PREFACE. systems. As decimal fractions may be learned much easier than vulgar, and are more simple, useful, and necessary, rind 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 ne- cessary. 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 ne- ressary in the reduction of currencies, computing interest, and many other branches. Besides, to obtain a thorough knowledge of Vulgar Frac- tions, is generally a task too hard for young scholars who have made no further progress in Arithmetic than Reduc- lion, 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 Vulgar Fractions at large, the scholar being now capable of going through them with advantage and ease. In Simple Interest, in Federal Money, I have given seve- ral new and concise rules ; some of which are particularly designed for the use of the compting-house. The Appendix contains a variety of rules for casting Interest, Rebate, &-c. together with a number of the most easy and useful problems, for measuring superficies and solids, examples of forms commonly used in transacting business, useful tables, &c. which are designed as aids in the common business of life. Perfect accuracy, in a work of this nature, can hardly be expected ; errors of the press, or perhaps of the author, may have escaped correction. If any such are pointed out, it will be considered as a mark of friendship and fa- veur, by The public's most humble and obedient Servant, NATHAN DABOLL. TABLE OF CONTENTS. ADDITION, Simple, - - - - - - -- - of Federal Moray, ..... - Compound, ------- Alligation, Annuities or Pensions, at Compound Interest, - Arithmetical Progression, ------ Barter, - - - - - Brokerage, - ... - 113 Characters, Explanation of, ------ 14 Commission, - ------ --1 12 Conjoined Proportion, ------- Coins of the United States, Weights of, - - - '- 220 Division of Whole Numbers, 32 - -- Contractions in, 36 - Compound, S3 Discount, - ........ 123 Duodecimals, -------- 216 Ensurance, ---------H4 Equation of Payments, ------- 126 Evolution or Extraction of Roots, - - - - - 167 Exchange, _--_--_-. 139 Federal Money, 21 -- Subtraction of, - - - - . 55 Fellowship, ------.-_ \<%fc - Compound, ------- 134 Fractions, Vulgar and Decimal, - 69, 143 Interest, Simple, ---.... 593 by Decimals, ------ 157 Compound, by Decimals, * Inverse Proportion*, Involution, --------- jgg Lose and Gain, -----_.. Multiplication, Simple, ------- yj - - Application and Use of, - ... 30 - Supplement to, - . _ 37 - Compound, - - - - . 43 Numeration, ------.. I?* 06 * ......... 99 Position, --------- rgg Permutation of Quantities, - - - - - .. 195 VlU i \iJLK OF i 0\ TK.N i'.-; . Questions for Exorcise, ---.__ Reduction, -_._....._. 59 of Currencies, do. of Coin, - - - -82,86 Rule of Three Direct, do. Inverse, - - 90,97 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 preceding, ----- 89 :- Short Practical, for calculating Interest, - 114 for casting Interest at 6 per cent, - 203 for finding the contents of Superfices and Solids, - 208 to reduce the currencies of the different States to Fede- ral Money, -------- 200 Rebate, a short method of finding the, of any given .sum, for months and days, ----__* 205 Subtraction, Simple, 23 Compound, - 43 Table, Numeration and Pence, - 9 Addition, Subtraction, and Multiplication, - 10 of Weight and Measure, - if of Time and Motion, - - - - " 13 showing th-o number of days from any day of one month to the same doy in any other month, - 160 r- showing the amount of I/, or 1 dollar, at 5 and 6 per cent. Compound Interest, for 20 years, - 220 showing the amount of I/, annuity, forborne for 31 years or under, at 5 and 6 per cent. Compound Interest, - 221 showing the present worth of II. annuity, for 31 years, at 5 and 6 per cent. Compound Interest, - 221 i of Cents, answering; to the currencies of the United States, with Sterling, fcc. ----- 224 . showing the value of Federal Money in other currencies, 225 Tare and Tret, - - 10S Useful Forme in transacting business, - 226 Weights of several pieces of English, Portuguese, and French Gold Coins, in dollars, cents, and mills, - of English and PortuffUftse Gold, do. de. - fS ^ of French and Spanish Gold, do. do. DABOLL'S SCHOOLMASTER'S ASSISTANT. ARITHMETICAL TABLES. Numeration Table. Pence Table. en d. s. d. d. 5. j3 20isl 8 12 is 1 C/3 03 30 2 6 24 .2 . . O 1 40 3 4 36 3 i c o EH I 50 4 2 48 4 . >H 60 5 60 5 QB o EH en CO 70 5 10 72 6 *-! o QQ c J o & ctf T3 2 80 6 8 84 7 "O 01 o co 3 QQ 03 90 7 6 96 8 a a 9 1 8 9 3 g 7 8 c a 6 7 a 5 6 O EH 4 5 3 a 3 4 fl H 3 1 2 100 8 4 110 9 2 120 10 108 120 132 9 10 U 9 8 7 8- 6 5 4 3 9 7 8 7 5 6 4 5 9 8 7 6 make. 9 Q 7 9 8 9 4 farthings 1 penny 12 pence 1 shilling, 20 shilling. 1 pound. .f! 10 ARITHMETICAL TABLES. ADDITION AND SUBTRACTION TABLE. 1 * 3 4 5 | 6 | 7| 8 | 9 | 10 11 12 2 4 5 6 7 | 8 | 9 | 10 | 11 ! 12 13 14 3 5 6 7 8 | 9 | 10 | 11 | 12 | 13 14 15 4 6 7 8 9 | 10 | 11 | 12 | 13 | 14 15 16 5 7 8 9 10 | 11 | 12 | 13 | 14 | 15 16 17 6 8 9 10 11 | 12 | 13 | 14 |,15 | 16 17 18 7 9 10 11 12 | 13 | 14 | 15 | 16 | 17 18 19 8 10 11 12 13 | 14 | 15 | 16 | 17 | 18 19 20 9 11 12 13 14 | 15 | 16 | 17 | 18 | 19 20 21 10 12 13 14 15 | 16 | 17 | 18 | 19 | 20 |21 |22 MULTIPLICATION TABLE. I "2~ 2 4 3| 4 5| 6 7 8 9 10 11 12 6| 8 10 | 12 14 16 18 20 22 24 3 6 9 | 12 15 | 18 21 24 27 30 33 36 4 IT 8 12 | 16 20 | 24 28 32 36 40 44 48 '60 10 15 | 20 25 | 30 35 40 45 50 | 55 6 12 18| 24 30 |36 42 48 54 60 <*) 7* 7 14 21 | 28 35 I 42 49 56 .63 -75- 70 77 84 8 I" 5 " 16 T8- 24 I 32 40 | 48 5(5 64 80 88 96 27|36 45 | 54 63 72 81 90 99 108 10 20 30 | 40 50 (60 70 bO 90 100 111) 1201 11 22 33|44 55 | 66 yy 88 ^ 110 | 121 132 112 24 36|48 60 | 72 84 96 108 120 | 132 144 To learn this Table : Find your multiplier in the left hand column, and the multiplicand a-top, and in the com- mon angle of meeting, or against your multiplier, along at the right hand, and under your multiplicand, you will find the product, or answer. ARITHMETICAL TABLES. 2. Troy Weight. 24 grains (gr.) make 1 penny-weight, marked pwt+ 20 penny- weights, 1 ounce, oz. 12 ounces, 1 pound, lb 3. Avoirdupois Weight. 16 drams (dr.) make 1 ounce, oz. 16 ounces, 1 pound, //;. 28 pounds, 1 quarter of a hundred weight, qr. 4 quarters, 1 hundred weight, cwt. 20 hundred weight, 1 tun. T. By this weight are weighed all coarse and drossy goods, grocery wares, and all metals except gold and silver. 4. Apothecaries Weight. 20 grains (gr.) make 1 scruple, B 3 scruples, 1 dram, 8 drams, 1 ounce, 12 ounces, 1 pound, Apothecaries use this weight in compounding their me- dicines. 5. Cloth Measure. 4 nails (na.) make 1 quaiter of a yard, qr. 4 quarters, 1 yard, yd. 3 quarters, 1 Eli Flemish, E. FL 5 quarters , 1 Ell English, E. E. 6 quarter, 1 Ell French, E. Fr, 6. Dry Measure. 2 pints, (pt.) make 1 quart, qt. 8 quarts, 1 p ,ck, pk. 4 pecks, 1 hushel, bu. This measure is applied to grain, beans, flax-seed, salt, oats, oysters, coal, fyc. ARITHMETICAL TABLES. Wine Measure. 4 gills (gi.) make 2 pints, 4 quarts, 3l gallons, 42 gallons, 63 gallons, 2 hogsheads, 2 pipes, pint, quart, gallon, barrel, tierce, hogshead, pipe, tun, pt. q t gal. bl tier. Jihd f. All brandies, spirits, mead, vinegar, oil, &c. are measur- ed by wine measure. Note. 231 solid inches, make a gal- lon. 8. Long Measure. 3 barley corns (b. c.) make 1 12 incites, 1 3 feet, 1 5J yards, 40 rods, 8 furlongs, 1 3 miles, 1 69 J statute miles, 1 inch, marked foot, yard, 1 rod, pole, or perch, furlong, mile, league, degree, on the earth. ifi. fl. yd. rd. fur. in. lea. 360 degree?, 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, with- out 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 TABLES. 13 9. Land, or Square Measure. 144 square inches make 1 square foot. 9 square feet, 1 square yard. 30 square yards, or ) ^ o^ot i < * square rod. 272 square feet, f 40 square rods, 1 square rood. 4 square roods, 1 square acre, 640 square acres, 1 square mile. 10. Solid, or Cubic Measure. 1728 solid inches make 1 solid foot. 40 feet of round timber, or > Kt\ C 4 f U ' U 1 tU11 O1 * lOad. 50 feet of hewn timber, ) 128 solid feet or 8 feet long, . l rf f d 4 wide, and 4 high, J All solids, or things that have length, breadth, and depth, are measured by this measure. N. B. The wine gallon contains 231 solid or cubic inches, and the beer gallon, 282. A bushel contains 2150,42 solid inches. 11. Time. 60 seconds (S.) make 1 minute, marked M. 60 minutes, 1 hour, k. 24 hours, 1 day, d. 7 days, I week, ?/ 4 weeks, I 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 degrees, . I sign, #. 12 signs, or 360 degrees, the whole great circle of the 7odiack. 14 CHARACTERS. Explanation of Characters used in this Book. = Equal to, as 12df. = Is. 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. Minus, or less, the sign of Subtraction ; as, 6 2=4, sig- nifies that 2 subtracted from 6, leaves 4. X Multiply, 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, 8H-2=4, signifies that 8 di- vided by 2, is equal to 4; or thus, f =4, each of which 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 : : 8 : 16 ; that is, as 2 to 4, so is 8 to 16. ^ Prefixed to any number, supposes that the square root of that number is required. V Prefixed to any number, supposes the cube root of that number is required. V Denotes the biquadrate root, or fourth power, 14. From 249 dollars 45 cents, take 180 dollars.* Ans. $69, 45 cts. 15. From 100 dollars, take 45 cts. Ans. $99, 55 cts. 1 t>. From ninety dollars and ten cents, take forty dollars and nineteen cents, Ans. $49, 91 cts. 17. From forty-one dollars eight cents, take one dollar 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-nine 'ents. Ans. $1, 1 ct. 22. From 20 dols. take 1 dime. Ans. $19, 90 cts. 23. From nhie dollars and ninety cents, take ninety-nine dimes. Ans. remains. 2*4'. Jack's prize money was 219 dollars, and Thomas SIMPLE MULTIPLICATION, %. received just twice as much, lacking 45 cents. How much money did Thomas receive 1 Ans. $437, 55 cts. 25. Joe Careless received prize money to the amount of 1000 dollars; after which he Jays out 411 dolls. 41 cents for a span of fine horses ; and 123 dollars 40 cents tor a gold watch and a suit of new clothes ; besides 359 < and 50 cents he lost in gambling. How much will he imv . left after paying his landlord's bill, which amounts to " dojs. and 11 cents? Ans. $20, 58 ct*. SIMPLE MULTIPLICATION TEACHETH to increase or repeat the greater of tw< numbers given, as often as there are units in the less, or multiplying number ; hence it performs the work of man} additions 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- duct. NOTE. Both multiplier and multiplicand are in srenerni called factors, or terms. CASE I. When the multiplier is uotiiifore than twelve. RULE. Multiply each figure in the multiplicand by themul; carry one for every ten, (as in addition of whols numbers,) and will have the product or answer. PROOF Multiply the multiplier by the imiltiplica;- EXAMPLES. What number is equal to 3 times 365 ? Tbus, 365 multiplier' 3 multiplier. Ans. IQ95 product. * Multiplication may also be proved by casting out the 9's in the two factors, and setting down the remainders ; then multiplying the two re- mainders together ; if the excess of 9's in their product is equal 1o f 9's in the total prod.nct, the voH,- H SIM I'LE M L- LTIPLIC ATJON. Multiplicand^ Multiplier, Product, 47094 7 71085 5432 4 2345 9075 5 6 71034 8 31261 9 4320 10 2240613 4684114 12 1432046 11 CASE II. When the multiplier consists of several figures. RUJ.E. The multiplier being placed under the- multiplicand, units under units, tens under tens, &c. multiply by each significant figure in the multiplier separately, placing the first figure in each product exactly under its multiplier ; then add the several products together in the same order as they stand, and their sum will be the total product, EXAMPLES. What number is equal to 47 times 365 ? Multiplicand, 365 Multiplier, 4 7 2555 460 1 Ans. 17155 product. Multiplicand, 37864 Multiplier, 209 340776 75726 Prodi 7913576 34293 74 2537682 47042 91 820 6816978 25203 4025 4280822 2193 4072 9876 9405 101442075 8929896 92883780 269181 4629 SlMl'LK Ml' ;>61986 42068 1246038849 134092 87362 2001049068 1709391112 918273645 1003245 11714545304 921253442978025 14. Multipl^r604S3 by 915;2 Ans. 6959940416. 15 Wh^ ; 4l tal product 01 7608 times 54d Ans. 278020665(5. 16. What number is equal to 40003 times CASE III. When there are ciphers on the right hand of either or both of the factors, neglect those ciphers ; then place significant figures under one another, and multiply by tJ only, and to the right hand of the product, place as mai ciphers as were omitted in both the factors. 21200 70 1484000 EXAMPLES, 31800 36 1 144800 84600 34000 2876400000 3040 109215040000 98260000 8109397800000 7065000 X 8700=61465500000 749643000 x 695000^521001885000000 360000 x 1200000^432000000000 CASE IV. When the multiplier is a composite number, that is, when it is produced by multiplying any two numbers in the ta together; multiply first by one of those fiffin 30 [MPIrE MULTIPLICATION. product by the other, and the last product will he the total required. EXAMPLI Multiply 41364 by 35. 7x5=35. 7 289548 Product of ,__ 1447740 Product of 35. 2. Multiply 764131 by 48. Ans. 36678288. 3. Multiply 3425)6 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 multi- 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 2646Q 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. 5. Multiply 3456789 by 567090. Ans. 1960310474010. 6. Multiply 8496427 by 874359. Ans. 7428927415293. 98763542 x 987635428754237228383764. Application and Use ^f Multiplication. In making out bills qf parcels, and in finding the value of goods ; when the price of one yard, pound, &c. is given (in Federal Money) to find the value -of the whole quantity. SIMPLE MULTIPLK ATIO.V. 31 RULE. Multiply the given price and quantity together, as in whole numbers, and the separatrix will be as many figures from the right hand in the product as in the given price. EXAMPLES. 1. What will 35 yards of broad- > $. d. c. m. cloth come to, at J 3, 4 9 6 per yard ? 3 5 17 4 8 104 8 8 Ans. $122, 3 6 0=122 dol- [lars, 36 cents. 2. What cost 35 Ih. cheese at 8 cents per Ib. 1 ,08 Ans. $2, 802 dollars 80 cents. 3. What is the value of 29 pairs of men's shoes, at 1 dol- lar 51 cents per pair? Ans. $43, 79 cents. 4. What cost 131 yards of Irish linen, at 38 cents per yard ? Ans. $49, 78. cents. 5. What cost 140 reams of paper, at 2 dollars 35 cent- per ream ? Ans. $329. 6. What cost 144 Ib. of hyson tea, at 3 dollars 51 cents perlb. 1 Ans. $505, 44 cents. 7. What cost 94 bushels of oats, at 33 cents per bushel ? Ans. $31, 2 cents. 8. What do 50 firkins of butter come to, at 7 dollars 14 cents per firkin ? Ans. $357. 9. What cost 12 cwt. of Malaga raisins, nt 7 dollars 31 cents per cwt. ? Ans, $87, 72 cents. 10. Bought 37 horses for shipping, at 52 dollars per head : what do they come to ? Ans. $1924. 11. What is the emoiiT>t of 500 Ibs. of IKo.o-V-larcl, at 15 cents per Ib. ? , Ans. $75. 12. What is the value of 75 yards of ^tin, at 3 dollars 75 cents per yard ? Ans. $281 , 25. 13. What cost 307 acre* of land, at 14 dols. 67 cents per acre ? .4???. 45383, 89 cei 32 DIVISION OF WHOLE NUMBERS. 14. What does 857 bis. pork come to, at 18 dols. 93 cents per bl. ? Am. $16223, 1 cent. 15. What does 15 tuns of hay come to, at 20 dols. 78 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. $. cts. 28 Ib. of Green Tea, at 2, ISperlb. 41 Ib. of Coffee, at 0, 21 34 Ib. of Loaf Sugar, at 0, 19 13 cwt. of Malaga Raisins, at 7, 31 per cwt. 35 firkins o (1 V-utter, at 7, 14 per fir. 27 pairs oi worsted Hose, at 1, 04 per pair. 94 bushels of Oats, at 0, 33 per bush. 29 pairs of men's Shoes, at 1, 12 per pair. Amount, $511, 78. Received payment in full, WILLIAM MERCHANT. A SHORT RULE. NOTE. The value of lOOlbs. of any article will be just as many dollars as the article is cents a pound. For 100 Ib. at 1 cent per lb.=100 cents 1 dollar. 100 Ib. of beef at 4 cents a Ib. comes to 400 cents=4 dollars, &c. DIVISION OF WHOLE NUMBERS. SIMPLE DIVISION teaches to find how many times on* whole number is contained in another ; and also what 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 shows how many times the divisor is contained in the dividend. 4. The Remainder, which is always less than the divisor, and of the same name with the Dividend. DIVISION OF WHOLE LUMBERS. LE. First, seek how many times the divisor is contained in as many 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, so as to produce a product that shall not exceed the part of the divi- dend 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 the 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 remainder, 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 contained in 9391 1 equally among 8 men. Divisor, Div. Quotient. Divisor, Div. Quotient, 4)9391(2347 8)3656(457 8 4 32 13 9388 45 12 +3 Rem. 40 19 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 fol- lows : 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 re- mainder, and reject the nines, and place the excess at top. 4. Oast the nines out of the dividend, and place the excess at bottom. Vo/f. If the sum is right, (he top and bottom figures will be aiiko. \ 04 DIVISION OF WHOLE LUMBERS. Divisor. Div. Quotient. 29)15359(529 365)49640(136 145 365 Proof by excess of 9's. 85 1314 5 58 1095 279 2190 261 2190 Remains 18 Rem. v Divisor. Div. Quotient. 95(85595(901 61)28609(469 736)863256(1172 472)251 104(532 there remains 664. 9. Divide 1893312 by 912. Ans. 2076. 10. Divide 1893312 by 2076. Ans. 912. 11. Divide 47254149 by 4674. Ans. 10110 T / T7 . 12. What is the quotient of 330098048 divided by 4207 ? Ans. 78464. 13. What is the quotient of 761858465 divided by 8465 ? . Ans. 90001. 14. How often does 761858465 contain 90001 ? Ans. 8465. 15. How many times 38473 can you have in 119184693 ? Ans. 3097f|fif. 16. Divide 280208122081 by 912314. Quotient, 307140y T VV r T- MORE EXAMPLES FOR EXERCISE. Divisor. Dividend. Remainder. 234063)590624922( Quotient)S3973 47614)327879186( ) 9182 987654(988641654( ) - - - CASE II. When there are ciphers at the right hand of the divisor, cut off the ciphers in the divisor, and the same number of figures from the right hand of the dividend ; then divide the remaining ones as usual, and to the remainder (if any) an- nex those figures cut off from the dividend, and you will have the true remainder. DIVISION OF WHOLE .NUMBERS. 35 EXAMPLES. 1. Divide 4673625 by 21400. w true quotient by Restitution. 428-- ~393 214 1796 1712 8425 true rein. 2. Divide 379432675 by 6500. Ans. 58374f |f . 3. Divide 421400000 by 49000. Ans. 8600. 4. Divide 11659112 by 89000. Ans. 131 ^| 7 . 5. Divide 9187642 by 9170000. Ans. M^ffo. MORE EXAMPLES. Divisor. Dividend. Remains. 125000)436250000( Quotient. ) 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 first figure or figures of the dividend, put the result under, and carry as 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, 567071 6)120616 7)152715 8)96872 9)118724 11)6986197 12)14814096 12)570196382 CONTRACTIONS IN DIVISION. Contractions in Division. Whea the divisor is such a number, that any two figures in the Table, being multiplied together, will produce it, di- vide the given dividend by one of those figures ; the quo- tient thence arising by the other ; and the last quotient will be the answer. NOTE. The total remainder is found by multiplying the last remainder by the first divisor, and adding in the first remainder. EXAMPLEvS. Divide 162641 by 72 9)162641 or 8)162641 last rem.. 7 X9 8)180712 9)203301 22587 2588 63 first rcm. +2 True rem. 65 Ans. 11154. Ans. 19475if . Ans. 26924^- Ans. 2212^- Ans. 3018 T V Ans. 17359^- Ans. 1118ff. Ans. 1345. Ans. 10940. True Quotient 2258ff. 2. Divide 178464 by 16. 3. Divide 467412 by 24. 4. Divide 942341 by 35. 5. Divide 79638 by 36. 6. Divide 144872 by 48. 7. Divide 937387 by 54. 8. Divide 93975 by 84. 9. Divide 145260 bv 108. 10. Divide 1575360 by 144. 2. To divide by 10, 100, 1000, &c. RULE. Cutoff as many figures from the right hand of the dividend as there are ciphers in the divisor, and these figures so cut off arc the remainder ; and the other figures of the dividend are the quotient. EXAMPLES. 1. Divide 365 by 10. 2. Divide 5762 by 100. 3. Divide 763753 by 1000. Ans. 36 and 5 remains. Ans. 57 62 rcm. Ans. 763 753 rem. SUPPLEMENT MY; MULTIPLICATION, 3T SUPPLEMENT TO MULTIPLICATION. To multiply by a rnixt number ; that is, a whole number joined with a fraction, as 8|, 5i, 6J, &c. RULE. Multiply by the whole number, and take $, i, f, &c. of the multiplicand, and add it to the product. EXAMPLES. Multiply 37 by 23i. Multiply 48 by 2J. 2)37 48 18i 74 8691 3. Multiply 4. Multiply 5. Multiply 6. Multiply Answer. 211 by 2464 by 345 by 6497 by 50i. 96 132 Ans. Ans. 106551. Ans. 20533 J. Ans. 6598-1. Ans. 334131. Questions to exercise Multiplication and Division. 1. What will 9| tuns of hay come to, at 14 dollars a tun ? '_4n*.-$136. 2. If it take 320 rods to make a mile, and every rod contains 51 yards ; how many yards are there in a mile ? Ans. 1760. 3. Sold a ship for 11516 dollars, and I owned f of her ; what was my part of the money 1 Ans. $8637. 4. In 276 barrels of raisins, each 3i cwt. how many hundred weight ? Ans. 966 cwt. 5. In 36 pieces of cloth, each piece contain! n- yards ; how many yards 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 calendar month ? Ans. $41. 8. A privateer of 65 men took a piize, which being equally divided among them, amounted to H9/. per man : what is the value of the prize ? COMPOUND ADD1TIOA. 9. What number multiplied by 9, will make 225 1 Am. 25. 10. The quotient of a certain number is 457, and the divisor 8 ; whr?t is the dividend 1 Ans. 3656. 11. -What cost 9 yards of cloth, at 3s. per yard 1 Ans. 27s. 12. What cost 45 oxen, at 8J. per head 1 Ans. 360. 13. What cost 144 Ib. of indigo, at 2 dols. 50 cts. or 2-SO cents per Ib. Ans. $360. 14. Write down four thousand six hundred and seven- teen, multiply it by twelve, divide the product by nine, and add 365 to the quotient, then from that sum subtract five thousand five hundred and twenty-one, and the remainder will be just 1000. Try it and see. COMPOUND ADDITION, IS the adding of several numbers together, having dif- ferent denominations, but of the same generic kind, as pounds, shillings and pence, &c. Tuns, hundreds, quar- ters, &c. RULE. 1. Place the numbers so that those of the same denomina- tion may stand directly under each other. 2. Add-the first column or denomination together, as in whole num- bers ; then divide the sum by as many of the san-o denomination as make one of the next gn.-ater ; setting- down the remainder under the column odded, and carry the quotient to *he next superior denomina- tion, continuing the same to the last, which add, as in simple addition.* 1. STERLING MONEY, Is the money of account in Great-Britain, and is reckon- ed in Pounds, Shillings, Pence and Farthings. See the Pence Tables. * The reason of this rule is evident : For, addition of this money, as 1 in the- pence is equal to 4 in the farthings j 1 in the shillings, to 12 in the pence ; and 1 in the pounds, to 20 in the shillings ; therefore currying as di- rected, is the arranging the monov, arising from each column properly in the scale of denominations : and this reasoning will hold good mthe ad- .ditionof compound numbers of any denomination whatever. COMPOUND ADDITION. EXAMPLES. S. d. What isfthe sum total of 47/. 13s. C 47 13 6 6d.~19/. 2s. 9id.- 14?. 10s. ll\d. r j 19 2 9i and 12/. 9s. l$d. IS 1 14 10 ll| 1 12 9 If Answer, .93 16 4J (2.) (3.) . s. d. . s. d. qr. 17 13 11 84 17 53 13 10 2 75 13 4 3 10 17 3 50 17 8 2 87 20 10 10 1 334 16 5 ,0 (4.) . s. d. qr. 30 11 4 2 15 10 9 1 1011 3983 4631 (5.) . 5. d. qr. 47 17 6k; 3 9 10 3 59 17 11 2 317 16 9 3 762 19 10 1 407 17 6 2 1 19 9 (6.) . s. d. qr. 7 17 10 3 60 6 8 7 14 11 2 18 19 9 3 91 15 82 18 17 10 3 5012 (?) . s. d. qr. 541 000 711 9 8 1 918 6 9 3 140 15 10 1 300 19 11 3 48 10 73 14 9 3 (8.) . s. d. 105 17 6 193 10 11 901 13 319 19 7 48 17 4 104 11 9 96 16 7 111 9 9 976 10 449 12 6 29 10 4 (9.) . s. d. 940 10 7 36 9 11 11 4 10 141 10 6 126 14 104 19 7 !(>{) 10 6 100 909 19 6 120 8 (10.) . s. d. 97 11 6A 20 4 144 1 10 17 11 9 16 1(H 19 9-1 19 9 4 234 11 10* 180 14 6 421 10 3i 341 10 4 sum 17s. 8rf. 137. Os. 7d. 19s. and 15/, 6s. 11. Find the amount of the following^ . ms, vb. 4ftl. 13s. 5,/. 1U 105. 41. \ .27L ( 3 Ans. . 115 7 0. T 40 COMPOUND ADDITION. Add 304Z. 5s. and Os. lid. 19s. 6d. Iqr. and 45/. together. Ans. . 640 3s. 5J-rf. 13. Find the sum total of 14/. 19s. 6d.~llL 4s. 9<7. 25/. 105. 4/. Os. 6d.3l. 5s. 8d. 19*. 6< 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 Ans. . 201 6s. Ud. 15. How much is the sum of Thirty -seven shillings and sixpence, - Thirty-nine shillings and 4jr/. - - - - ' Forty-four shillings and nine pence, - Twenty-nine shillings and three pence, Fifty shillings, Ans. . 10 Os. 16. Bought a quantity of goods for 125/. 10s. ; paid for truckage, forty-five shillings, for freight, seventy-nine shil- Mngs and sixpence, for duties, thirty-five shillings and ten ence, andTny expenses were fifty-three shillings and nine >ence ; what did the goods stand me in 1 Ans. . 136 4s, Id. 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 much money did the whole prize amount to ? Ans. . 1444 Is. 6d. 2. TROY WEIGHT. //;. oz. pwt. gr. Ib. oz. pwt. gr. J(i 11 19 23 8 11 19 21 4 4 16 21 10 16 8 8 8 19 14 7 8 17 21 9 14 17 468 23 4 7 10 7 9 7 14 17 7 11 12 7 9 13 10 POUND ADDITION. 41 3. AVOIRDUPOIS WEIGHT. cwt. qr. lb. lb. oz. dr. T. cwt. qr. lb. oz. dr. 2 3 27 24 13 14 91 17 2 24 13 14 1 1 17 4 2 26 6 1 13 3 3 15 6 2 16 3 9 gr. 9 1 17 329 6 1 17 4 16 5 2 12 6 1 10 17 12 11 19 9 17 10 12 26 12 15 14 13 2 04 9 11 16 8 7 47 11 3 ]9 J4 5 24 10 12 69 00 1 00 00 12 11 12 12 77 19 3 27 15 1J ^RIES 4. APOTHEGM WEIGHT. g 3 9 gr. fe 3 3 9 gr. 10 7 2 19 12 11 6 1 15 6 3 12 4 9 7 12 7 6 1 7 9 10 I 2 16 9 5 ^j 12 j 8 1 2 19 6 1 16 9 1 10 9 ;> t> 19 4 9 2 1 6 5. CLOTH MEASURE. yd. qr. na. E. E. qr. na. E. F. qr. na. 71 3 3 44 3 2 84 2 1 13 2 1 49 4 3 07 1 3 10 1 06 2 3 76 2 42 3 3 84 4 1 52 2 3 57 2 2 07 53 2 2 49 2 2 61 2 1 09 2 3 G. PhY MEASURE. ?k.qt.pt. lu. pk. at. lu. vie. qt. pt. 7 1 17 '2 V 2i>'3 7 1 260 34 27 64 261 150 13 3 6 43 4 241 16 3 4 52 3 5 1 261 27 26 94 230 360 56 07 54a70 7. WINE MEASURE. gal. at. pt. gi. hhd. gal qt. pt. tun.hhd. gal. qt. 39 3 1 ^ 42 61 31 34 2 %4 2 17 2 1 2 27 39 2 19 1 59 J 24 3 I 9 14 1 28 2 2 1 19 1 1 2 0921 19 32 2 8003 16 24 1 1 37 3 11 1 40 2 I 1 5 00 3 0190 42 ' K)U N D ADD! T i > \ . 8. LONG MEASURE. yds. ft. in. b.c. '4 '2 11 2 in. fur. 46 4 po. 16 le. m. 86 2 fur. pu. 6 32 3 1 8 1 58 5 23 52 1 7 16 1 292 9 6 34 64 2 5 19 6 2 10 1 17 4 18 73 1 4 15 1 061 7 3 15 7 2 3 25 3 170 5 2 24 28 2 4 17 9. LAND OR SQUARE MEASURE. (icres. roods, rods. acres. , roods . rods sg. ft- sg. in. 478 3 31 856 2 18 5 136 816 2 17 19 3 00 6 129 49 1 27 9 1 39 8 134 63 3 34 1 3 00 143 9 3 37 2 27 4 34 SOLID 10. MEASURE. T. ft- cords. /fee*. feet. inches. 41 43 3 122 13 1446 12 43 4 114 16 1726 49 6 7 8'} 3 866 -4 27 10 127 14 284 11. TIME. Y. m. w. da. Yr. da. &. w. sec. 57 11 3 6 24 363 23 54 34 3 9 2 3 21 40 12 40 24 29 8 o rr 13 112 14 00 17 46 10 2 4 14 9 11 18 14 10 7 1 2 8 24 8 16 13 12. CIRCULAR MOTION. o in o o / // 3* 29 17 14 11 29 59 50 1 6 10 17 00 40 10 4 18 17 11 94 10 49 6 14 18 10 4 11 6 10 COMPOUND SUBTRACTION. COMPOUND SUBTRACTION, TEACHES to find the difference, inequality, or excess, between air. two sums of diverse denominations. RUL-:. Place those numbersxunder each other, which are of the same denomination, the less being below the greater ; begin with the least denomination, and if it exceed the figure over it, borrow as many units us make one of the next greater ; subtract it therefrom ; and to the difference add the 'ipper figure, remembering always to add one to the next superior d- , omination for that which you borrowed. NOTE. The method of proof is the same as in simple subtraction, EXAMPLES. Sterling Money. '(2.) (3.) . s. d. qr. . s. d. 14 14 6 2 94 11 6 10 19 6 3 36 14 8 1. . s! d.qr. From 346 16 5 3 Take 128 17 4 2 Rem. 217 19 1 1 (4.) . s. d. Borrowed 44 10 2 Paid 93 11 8 Remains unpaid Lent Received Due to me (5.) . s. d, qr. 36 082 18 10 7 3 (6.) . s. d. From 500 Take 4 19 11 Rem. From Take Rem. (9.) . s. sum received ? Ans. 165 5s. irf. 2. TROY WEIGHT. Ib. oz. pwt. oz. pwt. fir. Ib. oz. piut. p*r. From 6 11 14 4 19 21 44 9 6 12 Take 2 3 16 2 14 23 17 3 16 18 Rem. Ib. oz. pwt. gr. Ib. oz. pwt.gr. 684 2 10 14 942 200 683 1 9 13 892 9 2 3 . 3. AVOIRDUPOIS WEIGHT. Ib. oz. dr. cwt. or. Ib. T. cwt. or. Ib. oz. dr. 7 9 12 7 3 13 7 10 3 17 5 12 3 12 9 5 1 15 3 12 1 19 10 9 T. cwt. qr. Ib. oz. Ir. T. cwt. or. Ib. oz. dr. 810 11 20 10 11 317 12 I 12 9 12 193 17 1 20 12 14 180 12 1 14 10 14 COMPOUND SUBTRACTION. 45 4. APOTHECARIES' WEIGHT. fc 3 3 19 8 7 9 11 6 3 B gr. 4 1 17 1 2 15 ft g 3 9 gr. 35 7 3 1 14 17 1U 6 1 18 Yd. qr. na. 35 1 2 19 1 3 5. CLOTH MEASURE. E.E. qr. na. 467 3 1 291 3 2 E.Fl. qr. na. 765 1 3 149 2 1 .E.l^. qr. na. 845 1 1 576 2 3 Yd. qr. na. 813 3 1 174 1 E.E. qr. na. 615 1 226 2 2 bu. pk. qt. 65 1 7 14 3 4 6. DRY MEASURE. bu. pk. qt. 8 1 5 316 17 23 6261 ff*l- 14 2 1 3 7. WINE MEASURE. hhd. gal. qt. pt. 13 1 10 60 3 1 T. hhd. zal. qt. pt. 2 3 20 3 1 1 2 27 hhd. gal. 612 23 75 '37 qt. pt. hhd. 1 521 1 1 256 gal. qt. pt. 14 \ I 25 3 yd. ft. in. b.c. 4 2 11 2 2 11 1 8. LONG MEASURE. m. fur. po. 41 6 22 10 6 23 le. m. fur. po. 86 2 6 32 24 1 7 31 le. in. fur,po. 27 1 6 37 19 2 4 39 le. m. fur. po. 16 U^ i 3 10 1 3 5 fo. m. fur. po. 9 2^0 7 1118 46 COMPOUND SUBTRACTION. 9. LAND OR SQUARE MEASURE. A. roods, rods. A. r. po. 29 1 10 29 2 17 24 1 25 17 1 36 or. r 540 25 119 1 27 tuns. ft. 116 24 109 39 A. or. rods. 130 1 10 49 1 11 10. SOLID MEASURE. cords, ft. 72 114 41 120 19 143 131 132 so. in. 125 tuns. ft. in. 45 18 140 16 14* 145 yr 54 rs. mo. i. da. 11 3 1 43 11 3 5 11. TIME. yrs. days. li. min. 24 352 20 41 20 14 356 20 49 19 w. d. h. min. set. 472 2 13 18 42 218 4 16 29 54 w. d. h. min. sec. 781 1 8 23 21 197 3 12 42 53 12. CIRCULAR MOTION. o o / // S ' " 9 23 45 54 9 29 3-!-- 3 7 40 56 7 29 40 36 QUESTIONS, Shewing the use of Compound Addition and Subtraction. NEW- YORK, MARCH 22, 1814. 1. . Bought of George Grocer, 12 C. 2 qrs. of Sugar, at 525. per cwt. : 10 28 Ibs. of Rice, at 3d. per Ib. 3 loaves of Sugar, wt. 35 Ib. at Is. \d. per Ib. 1 3 C. 2 qrs. 14 Jb. of Raisins, at 36s. per cwt. 6 10 Ans. 41 0-. 47 2. What sum added to 17/. lls. 8d. will make 100?. ? Ans. 827. 85. 3d. 3qr. 3. Borrowed 50/. 10s. paid again at one time 17/. 11 s. \d. and at another time, 91. 4s. 'Sd. at another time 171. 9s. W and at another time 19s. tid. how much remains un- id ? ^w*. 4 4s. 9\d. 4. Borrowed 100/. and paid in part as follows, viz. atone hne 211. Us. Qd. at another time 19/. 17s. 4jd. at another ime 10 dollars at 6s. each, and at another time two English guineas at 28s. each, and two pistareens, at 14^6?. each; low much remains due, or unpaid ? Ans. 0% 12s. S^d. 5. A, B, and C, drew their prize money as follows, viz. had 75/. 15s. 4(/. B had three times as much as A, acking 15s. 6d. and C, had just as much as A and B both ; iray how much had C ? Ans. 302 5s. Wd. G. I lent Peter Trusty 1000 dols. and afterwards lent lim 26 dols. 45 cis. more. He has paid me at one time 361 dols. 40 cts. and at another time 416 dols. 09 cents, resides a note which he gave me upon James Paywell, for lols. 90 cts. ; how stands the balance between us 1 Ai&. The balance is $105 06 cts. due to me. 7. Paid A B," in full for E F's bill on me, for 105Z. 10s/ iz. I g.ive him Richard Drawer's note for 15J. 14s. 9d. ?eter ''Johnson's do. for 30/. Os. 6d. an order on Robert Dealer for 39/. Us. the rest I make up in cash. I want to iow what sum will make up the deficiency I Ans. 20 3s. 9<2. 8. A merchant had six debtors, who together owed him 2917/. 10s. 6d. A, B, C, D, and E, owed him 1675 J. 13s. 9d. of ii ; what was F's debt ? Ans. 1241 16s. 9d. 9. A merchant bought 17 C. 2 qrs. 14 Ib. of sugar, of which he sells 9 C. 3 qrs. 25 Ib. how much of it remains un- Ans.7C.%qrs. 17 Ib. 10. From a fashionable piece of cloth which contained 2 na. a tailor was ordered to take three suits, each . 2 qrs. how much remains of the piece ? Ans. 32 yds. 2 qrs. 2 na. 11. The war between England and America commenced 48 COMPOUND I.iULTIPLii:ATIOX. April 19,1775, arid a general peace took place January 20th, 1783 ; how long did the wa^ continue ? Ans. 7 yrs. 9 mo. I cL COMPOUND MULTIPLICATION. COMPOUND Multiplication is when the Multiplicand f consists of several denominations, &c. 1 . To Multiply Federal Money. RULE. Multiply as in whole numbers, and place the separatrix as many figures from the right hand in the product, as it is in the mul- tiplicand, or given sum. EXAMPLES. $ cts. $ d. c. m. 1. Multiply 35 09 by 25. 2. Multiply 49 5 by 97. 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 nulls by 50 Ans. 0, 45 8* There were forty-one men concerned in the payment of a sum of money, and each paid 3 dollars and 9 mills ; how much was paid in all 1 Ans. $123 36 cts. .9 mills. 9. The number of inhabitant* in the United >'.ates is five millions; now suppose each should pay \ he trifling sum of 5 cents a year, for the term of 12 years, towards a continental tax ; how many dollars would b-,> raised there- by 1 Ans. Three millions Dollars. 2. To Multiply the denominations of Sterling Money , Weights, Measures, fyc. RULE. Write down the Multiplicand, and place the quantity un- derneath the least denominatipn, for the Multiplier, and in multiply- COMPOUND :Il 'vnPLlCATiOX. 4^ ing by it, observe the same rules for carrying from one denomination to another, as in compound Addition.* INTRODUCTORY EXAMPLES. /. s. d. q. .s. d. Multiply 1 11 6 2 by 5. How much is 3 times II S> Prod. 7 17 8 2 1 15 3 5. d. . s> d. . . \\ LK > , . s. d. . s. d. 4 gallons of wine, at 8 7 f>er gallon. 1 14 4 5 C. Malaga Raisins, at 1 2 3 percwt. 5 .11 3 7 reams of Paper, nt 17 9^ per ream. 6 4 6} * When accounts are kept in pounds, shillings, and pence, this kind of mul- tiplication is a concise and elegant method oi finding thft value of goods, at so much per yard, Ib. fcc. the general rule being to multiply the siren price quantity. E 00 CO&flQUXD MULTIPLICATION, f 8yds. of broadcloth, at 1 7 9' per yard. 11 24 9 Ib. ofj cinnamon, at 11 4j per Ib. 5 2 2} 11 tuns jbf hay, at 2 1 10 per tun. 23 2 12 bush/als of apples, at I 9 per bush. 110 12 bushels of wheat, at 9 10 per bush. 5 18 2. When the multiplier, that is, the quantity, is a com- posite number, und greater than 12, take any two such numbers as v/hen multiplied together, will exactly produce the given quantity, and multiply first by one of those figures, and that product by the other ; and the last product, will be the answer. EXAMPLES. What cost 28 yards of cloth, at 6s. IQd. per yard ? . s. d. 6 10 price of one yard. Multiply by 7 Produces 2 7 10 price of 7 yards. Multiply by 4 Answer, $ 114 price of 28 yards QUESTIONS. ANSWERa. 5. d. qrs. . s. d. 24 yards fit 7 4 3 per yard, = S 17 6 27 at 9 10 = 13 5 6 44 at 12 4 2 ~_ 27 4 6 55 at 8 3 1 = 22 14 10J- 72 at 19 11 , = 71 14 20 4- at 3 6 2 = 3 10 10 84" at 18 4 .2 = 77 3 6 96 at 11 9 = 56 8 63; atl 17 6 = 118 o 6 *'44 at 1 4 2 = 174 3. When no two numbers multiplied together will exactly make the multiplier, you must multiply by any two whose product will come the nearest ; then multiply the upper lie by what remained ; which, added to the last product, .giyes the answer vOMl'tHJND MULTU'I.K \Tli>\ .') ! EXAMPLES. What will 47 yds. of cloth come to at JLf s. 9df. per yd. f i . s. d. 17 9 price of 1 yatd. Multiply by 5 Produces 4 89 price of 5 yards, Multiply by 9 Produces 39 18 9 price of 45 yards. 1 15 6 price of 2 yards. Amvscr, 41 14 3 price of 47 yards. QUESTIONS. ANSWERS J s. d. . s. d. 23 ells of linen., at 3 6* per ell. 4 1 5i- 17 ells of dowlas, at 1 6| per ell. 1 6 oT 39 cwt. of sugar, at 3 19 6 per cwt. 137 9 (T 52 yds. of cloth, at 5 9 per yd. 14 19 19 Ibs. of indigo, at 11 6 per Ib. 10 18 <} 29 yds. of cambric, at 13 7 per yd. 19 13 11 ill yds. broadcloth, at 1 2 per yd. 124 17 <) 1*4 beaver hats, at 1 9 4 u piece. 137 17 4 4. To find the value of a hundred weight, by haying the price of one pound. If the price be farthings, multiply 2s. 4d. by the farthings in the price of one Ib. Or, if the price be pence, multiply 9s. 4d. by the pence in the price of one Ib. and in (Mtlu^r casrj the product will be the answer. EXAMPLES. 1. What will 1 cwt. of rice come to, at %\d. per Ib. ? s. d. 112 farthings=2 4 price of 1 cwt. at ] d. per Ib. 9 farthings in the price of 1 Ib. Ans. 1 1 price of 1 cwt. at 2-Jrf. per !b. '2 < ' MULTIPLICATION. 2. What will 1 owl. of lead come to at 7d. per Ib. ? * .";. d. 9 4 3 5 4 Questions. Answers. I cwt. at 2J-d. per Ib. = 1 3 4 I ditto, at 2f d. 1 5 8 1 ditto, at 3d. = 1 b 1 ditto, at 2d. 18 8 1 ditto, at 3 id. = 1 12 8 Iltumples of Weights, Measures, fyc. \. flovv- much is 5 times 7 cv/t. 3 qrs. 15 Ib. ? f/#/. ^-5. 76. 7 3 15 ]^v. Cwt. 39 1 19 /6. oz. pwt. gr. cwt. qr. Ib. oz. :>. Multijily <2Q 2 7 13 by 4. (3) 27 1 13 12 4 6 Product Ib. 80 9 10 4 Ib. 164 26 8 ANSWEf yds. (jr. na. yds. qr. no. 4. Multiply 14 3 2 by 11 'l63 2 2 lilid. g. qt.pt. lihd. g. qt.pt. :>. 3Iultiply 21 15 2 1 by 12 254 61 2 /'. tn.fur. po. le. m. fur. po. <>. Mulriply 81 2 6 21 by 8 655 1 4 8 a. i\ p. a. r. p. 7. Multiply 41 2 11 by 18 748 38 ?//*. m. 2". d. yr. m. w. d. 8. Multiply 20 5 3 6 by 14 286 5 2 . ' >S'. ' '' 7 19 2 COMPOUND DIVISION 63 cds. ft. cds. ft. 10. Multiply 3 87 by 8 29 56 Practical Questions in WEIGHTS AND MEASURES. 1. What is the weight of 7 lihds. of sugar, each weigh- ing 9 cwt. 3 qrs. 12 Ib. 1 Ans. 69 cwt. 2. What is the weight of 6 chests of tea, each weighing 3 cwt. 2 qrs. 9 Ib. ? Ans. 21 cwt. 1 qr. 26 Ib. 3. How much brandy in 9 casks, each containing 41 gals. 3 qts. 1 pt. ? Ans. 376 gals. 3 qts. Ipt. 4. In 35 pieces of cloth, each measuring 27J yards, how many yards 1 Ans. 971 yds. 1 qr. 5. In 9 fields, each containing 14 acres, 1 rood, and 25 poles, how many acres ? Ans. 129 a. 2 qrs. 25 rods. 6. In 6 parcels of wood, each containing 5 cords and 96 feet, how many cords ? Ans. 34 * cords. 7. A gentleman is possessed of 1 J dozen of silver spoons, each weighing 2 oz. 15pwt. 11 grs. 2 dozen of tea-spoons, each weighing 10 pwt. 14 grs. and 2 silver tankards, each, 21 oz. 15 pwt. Pray what is the weight of the whole ? Ans. S Ib. 10 oz. %pivt. 6 grs. COMPOUND DIVISION, TEACHES to find how often one number is contained in another of different denominations. DIVISION OF FEDERAL MONEV. fty Any sum in Federal Money may be divided whole number ; for, if dollars and cents be written down GS Q simple number, the whole will be cents ; and if the sum consists of dollars only, annex two ciphers to the dollars, and the whole will be cents ; hence the following GENERAL RULE. Writedown the given sum in cents, and divide as in whole numbers ; the quotient will be the answer in cents. NOTE. If the cents in the given sum are less than 10, you raiuri always place a cipher on their left, or in the ten's place of the cen*i before you write them down. K 5 \l) DIVISION. EXAMPLES. 1. Divide 35 dollars G8 cents, by 41. 41)3568(87 the quotient in cents ; and when there 328 is any considerable remainder, you may annex a cipher to it, if you please, and 288 divide it again, and you will have the 287 mills, &c. Hem. I 2. Divide 21 dollars, 5 cents, by 14. 14)2105(150 cents 1 dol. 50 cts. but to bring cents 14 into dollars, you need only point off two figures to the right hand for cents, and 70 the rest will be dollars, &c. 70 3. Divide 4 dols. 9 cts. or 409 cts. by 6. Ans. 68 cts.+ 4. Divide 9 dols. 24 cts. by 12. Ans. 77 cts. 5. Divide 97 dols. 43 cts. by 85. Ans. $1 14 cts. 6m. Divide 248 dols. 54 cts. by 125. Ans. 198 cts. 8m.=$l 9S cts. 8m. 7. Divide 24 dols. 65 cts. by 248; Ans. 9 cts. 9m. 8. Divide 10 dols. or 1000 cts. by 25. Ans. 40 cts. 9. Divide 125 dols. by 50C. Ans. 25 cts. 10. Divide 1 dollar into 33 equal parts. Ans. 3 ctfs.+ PRACTICAL QUESTIONS. 1. Bought 25 Ib. of coffee for 5 dollars ; what is that a pound ? Ans. 20 cts. 2. If 131 yards of Irish linen cost 49 dols, 78 cts. what is that per yard ? Ans. 38 cts. 3. If a cwt. of sugar cost 8 dols. 96 cts. what is that per pound 1 Ans. 8 cfs. 4. If HO reams of paper cost 329 dols. what is that per ream ? Ans. $2 35 cts. 5. If a reckoning of 25 dols. 41 cts. be paid equally among 14 persons, what do they pay apiece? Ans. $1 81^ cts. 6. If a man's wages are 235 dols. 80 cts. a year, what is that a calendar month? Ans. $19 65 eta. 7. The salary of the President of the United States, i. twenty-five thousand dollars a year ; what is that a day ? Ans. $68 49 cts. To divide tlie denominations of Sterling Money, Weights, Measures, fyc. RULE. Begin with the highest denomination as in simple division ; ind if any thing remains, find how many of the next lower denomi- nation this remainder is equal to ; which add to the next denomina- tion : then divide again, carrying the remainder, if any, as before ; and so on till the whole is finished. PROOF. The same as in simple Division. EXAMPLES. s. d. qr. - s. d. Divide 97 3 11 2 by 5 8)27 18 6 Quo't. 19 892 39 9J s. d. s. d. 3. Divide 31 11 6 by 2 Ans. 15 15 9 4. Divide 22 3 9 by 3 7 7 11 r>. Divide 70 10 4 by 4 17 12 7 6. Divide 56 11 5.V by 5 11 6 3} 7. Divide 61 14 8 - by 6 10 5 9[ 8. Divide 24 15 6 by 7 3 10 9J- 9. Divide 185 17 6" by 8 23 4 8} 10. Divide 182 16 8 by 9 20 6 3i 11. Divide 16 1 11 by 10 1 12 2J 12. Divide 1 19 8 by 11 3 7 13. Divide 6 6 6 by 12 10 61 14. Divide 126 by 9 026 : 15. Divide 948 11 6 by 12 79 11^ 2. When the divisor exceeds 12, and is the product of two or more numbers in the table multiplied together. RULE. Divide by one of those numbers first, and the quotient by the other, and the last quotient will bo the answer. EXAMPLES. s. d. s. d. 1. Divide 29 15 by 21 Ans. 1 8 4 2. Divide 27 16 by 82 17 4 3. Divide 67 9 4 by 44 1108 s. d. s. a'. 4. Divide 24 16 6 by 36 13 91. 5. Divide 128 9 by 42 31 2* 6. Divide 269 12 4 by 56 4 16 3< 7. Divide 248 10 8 by 64 3 17 8 8. Divide 65 14 by 72 18 3 9. Divide 5 10 3 by 81 O 1 4j 10. Divide 115 10 by 90 158 11. Divide 136 16 6 by 108 154 12. Divide 202 13 6 by 121 1 13 6 13. Divide 34 4 by 144 049 3. When the divisor is large, and not a composite num- ber, you may divide by the whole divisor at once, after man- ner of long division, as follows, viz. EXAMPLES. 1. Divide 128/. 13s. 3d. by 47. s. d. > s. d. 47)128 13 3(2 14 9 quotient 94 34 pounds remaining. Multiply by 20 and add in the 13s. produces (>93 shillings, which divided by 47, gives- 47 [14s. in the quotient. 223" 188 35 shillings remaining. Multiply by 12 and add in the 3d. produces 423 pence, which, divided as abov-e, 423 [gives 9d. in the quotient. s. d. s. d. 2. Divide 113 13 4 by 31 Ans. 3 13 4 3. Divide 85 6 3 by 75 129 4. Divide 315 3 10J by 365 17 3{ 5. Divide 132 8 by 68 1 18 10 6. Divide 740 16 8 by 100 7. Divide 888 18 10 by 95 9 7 ji v.O-YlPOUA'D DIVISION. 57 Examples of Weights, Measures^ fyc. 1 . Divide 1 4 cwt. 1 qr. 8 Ib. of sugar equally among 8 men C. qr. Ib. oz. 8)14 1 8 1348 Quotient. 8 14 1 8 Proof. 2. Divide 6 T. 11 cwt. 3 qrs. 19 Ib. by 4. Ans. 1 T. 12 cwt. 3 qrs. 25 Ib. 12 oz. 3. Divide 14 cwt. 1 qr. 12 Ib. by 5. Ans. 'i cwt. 3 qrs. 13/6. 9 oz. 9 dr.+ 4. Divide 16 Ib. 13 oz. 10 dr. by6.Ans.2lb. 12 oz. 15 dr. 6. Divide 56 Ib. 6 oz. 17 pwt. of silver into 9 equal parts. Ans. } Ib. 3 oz. 8 pwt. 13 grs.-f- 6. Divide 26 Ib. 1 oz. 5 ,>t. by 24. Ans. 1 Ib. 1 oz. 1 pwt. 1 gr. 7. Divide 9 hhds. 28 gals. 2 qts. by 12. Ans. hhd. 49 gals. 2 qts. 1 p/. 8. Divide 168 bu. 1 pk. 6 qts. by 35. Ans. 4 bu. 3 pks. 2 <^,?. 9. Divide 17 lea. 1 m. 4 fur. 21 po. by 21. Ans. 2 m. 4 fur. 1 jw. 10. Divide 43 yds. 1 qr. 1 na. by 11. y^s. 3 yds. 3 tfrs. 3 na. 11. Divide 97 E. E. 4 qrs. 1 na. by 5. Ans. 19 yds. 2 qrs. 3 na.-i- J'2. Divide 4J gallons of brandy equally among 14 i soldiers. Ans. 1 gill aviece. 13. Bought a dozen of silver spoons, which together weighed 3 Ib. 2 oz. 13 pwt. 12 grs. how much silver did each spoon contain 1 An$, 3 oz. -ipwt. 11 gr. 14. Bought 17 cwt. C qrs. 19 Ib. of sugar, and sold out one third of it ; how much remains unsold 1 Ans. 11 cwt. 3 qrs. 2:3 Ib. 15. From a piece of cloth containing 64 yards 2 na. a tailor was ordered to make 9 soldiers' coats, which took ono third of the whole piece ; how many yards did each cort Ans. 2 irrh. 1 nr. v 30 -COMPOUND PRACTICAL QUESTIONS. 1. If 9 yards of cloth cost 4/. ,3's. *i/J. wliat is thai per yard 7 s. d. qr. 9)4 372 932 Answer. 2. If 11 tons of hay cost 23J. Os. 2er pound ? Ans. 5}d. 5. Bought 48 pairs of stockings for 1 II. 2s. how much a pair do they stand me in ? Ans. 4s. 7^d. 6. If a reckoning of 51. Ss. W%d. be paid equally among 1 3 persons, what do they pay apiece ? Ans. Ss. d. 7. A piece of cloth containing 24 yards, cost 18J. 13s. what did it cost per yard? . Ans. 15s. 3d. S. If a hogshead of wine cost 331. 12s. what is it a gal- lon? Ans. 10s. Sd. 9. If 1 cwt. of sugar cost 3/. 10s. what is it per pound Ans. 7%d. 10. If a man spend 71/. 14s. 6d. a year, what is that per calendar month ? Ans. 5 19s. Q\d. 11. The Prince of Wales' salary is 150,000?. a year, what is that a day ? AJIS. 410 19s. 2d. 12. A privateer takes a prize worth rx M65 dollars, of which the owner takes one half, the officers one fourth, and the re- mainder is equally divided among the sailors, who ore 125 in number ; how much is each sailor's part 1 Ans. $24 93 cts. 13. Three merchants A, B, and C, have a ship in com- pany. A hath |, B , and C 1, and they receive for freight 228?. 16s. Sd. It is required to divide it among the own- ers according to their respective shares. Ans. As share 143 Os. 5d. B's share 57 4s. 2J, f "* share 28 12s. Id. ] I, A privateer haying taken a prize worth $6850 : , it livided into one hundred shares ; of which the captain is to lave 11; 2 lieutenants, each 5; 12 midsipmen, each 2; ind the remainder is to be divided equally among the sailors, who are 105 in number. Ans. Captain's share $753 50 cts. ; lieutenant's, $342 50 cts.; a midshipman's, $137, and a sailor's, $35 88. REDUCTION, TEACHES to bring or change numbers from one name o another, without altering their value. ' Reduction is either Descending or Ascending. Descending, is when great names are brought into small, s pounds into shillings, days into hours, &c. This is done >y Multiplication. Ascending, is when small names are brought into great, shillings into pounds, hours into days, &c. This is per- formed by Division. REDUCTION DESCENDING. RULE. Multiply the highest denomination given by so many of tfte next less as make one of that greater, and thus continue till you have brought it down as low as your question requires. PROOF. Change the order of the question, and divide your last product by the last multiplier, and so on. EXAMPLES. 1. In 25Z. 15s. 9(7. %qrs. how many farthings! 5. d. qrs. 25 15 9 2 Proof. 20 4)24758 Ans. 24758. 515 shillings. '12)6189 2 qrs. 12 6189 pence. 210)51)5 -t 25 15 9 2 24758 farthings. NOTE. In multiplying by 20, 1 added in tbe 15s. by 1% the 9d. and by 4 the 2qrs. which must always be dope in like csrses. In 31Z. Us. I0s. 8d. 1 Ans.lZl. Reduction Ascending and Descending. 1. MONEY. 1. In 12U Os, 9id. how many half-pence? A ns. 58099. 2. In 58099 half-pence, how many pounds ? Ans. 121/. Os. 9.^. 3. Bring 23760 half- pence into pounds. Ans. 19 10^. 4. In2l4/; Is. 3d. how many shillings, six-ponces, three- pences, and farthings'? Ans. 4281 s. 8562 sir^pc.nces^ 17125 fhrec-pences, and 205500 farthings. 5. In 1377. how many pence, and English or French crowns, at 6: 6. Is! 249 English half-crowns, how many pence and pom 7. li 346 guineas, fit 21 s. each, how many shillings, groats, and pence ? Ans.TSGGs. 21798 gr'fs - ' "7192^ 8. In 48 guineas, at 28s. ' ich, how many 4^d. pieces ? Am. 358. 81 guineas, at 27s. 4d. each, how many pounds I Ans. 110 l t>2 REDUCTION. 10. la 24396 pence, how many shillings, pounds, and pistoles 1 A?is. 2033^. 101 13s. and 92 pistoles. 9s. over. 11. In 252 moidores, at 36s. each, how many guineas at 28s. each 1 Ans. 324. 12. In 1680 Dutch guilders, at 2s. 4d. each, how many pistoles at 22s. each] Ans 178 pistoles, 45. 13. Borrowed 1248 English crowns, at 6s. 8d. each, how many pistareens, at 14|d. each, will pay the debt? ^ws/6885 pistareens, and 7^d. 14. In 50J. how many shillings, nine-pences, six-pences, four-pences, and pence, and of each, an equal number? iM.+M.+6d.+4d.+ l Answer 64 KEDUC/iiuA. 3. AVOIRDUPOIS WEIGHT. In 89 cwt. 3'qrs. 14 Ib. 12 oz. how many ounces ? 4 359 quarter* Proof. 28 16)161068 2876 28)10066 12 oz. 719 10066 pounds 4)359 14 Ib. 16 Ans. 89 cwt. 3 qrs. 14 Ib. 12 oz. 6039S 10067 * 161068 ounces. Answer. 2. In 19 Ib. 14 oz. 11 dr. how many drains? Ans. 5099. 3. In 1 tun, how many drams? Ans. 573440. 4. In 24 tuns. 17 cwt. 3 qrs. 17 Ibs. 5 oz. how many ounces ? .4715. 892245. * 5. Bring 5099 drams into pounds. Ans. I9lb. \4toz. II dr. 0. Bring 573440 drams into tuns. Ans. 1. 7. Bring 892245 ounces into tuns. Ans. 24 tuns, 17 cwt. 3 qrs. 17 Ib. 5oz. 8. In 12 hhds. of sugar, each 11 cwt. 25 Ib. how many pounds ? Ans. 15084. 9. I: 42 pigs of lead, each weighing 4 cwt. 3 qrs. how many foihcr, at 19 cwt. 2 qrs. ? Ans. 10 / 'other, 4 cwt. 10. A. gentleman lias 20 hhd^. of tobacco, each 8 cwt. 3 qrs. 14 Ib. and wishes to put it into boxes containing 70 Ib. each, I demand the number of boxes he must get? Ans. 284. 4. APOTHECARIES' WEIGHT. 1. In 9fc 8 3 1 3 2 D J9 grs. how many grains? Ans. 55799. . In 55799 grains. ho\v many pounds? Ans. 9 ft 83 13 29 1 KEDl i;!iiK\. 66 5. CLOTH MEASURE. 1. In 95 yards, how many quarters and nails'? Ans. 3SOqrs. I520w. 2. In 341 yards, 3 qrs. 1 na. how many nails ? Ans. 5460. 3. In 3783 nails, how many yard., 1 Ans. 236 y ds. I qr. 3 na. 4. In 61 Ells English, how iminy (quarters and nails '? Ans. 3Q5 qrs. 1220 na. 5. In 56 Ells Flemish, how many quarters and nails ? Ans, 168 qrs. 672 na. 6. In 148 Ells English, how many Ells Flemish ? Ans; 246 E. F. 2 qrs. 7. In 1920 nails, how many yards, Ells Flemish, and Ells English 1 Ans. 120 yds*. 169 E. F. and 96 E. E. 8. How many coats can bo made out of S6J yards of broadcloth, allowing If yards to a coat ? Ans. 21. 1. In 136 bushels, ho\v many pecks, quarts and pints 'I Ans. 544pfo. 4352 qts. 87Q4pts. 2. In 49 bush. 3pks. 5 qts. how many quarts? Ans. 1597. 3. In 8704 pints, how many bushels ? Ans. 136. 'n 1597 quarts, how many busilota 1 Ans. 49 bush . 3 pks. 5 qts. 5. A man would ship 720 bushels of corn in barrels, which hold 3 bushels 3 pecks each, how many barrels must he get 1 Ans. 192. 7. \VINE MEASURE. 1. In 9 tuns of wine, how many hogsheads; gallons aird quarts? Ans.SGhhds. 2Z68gals. 90720*;. 2. In 24 hhds. 18 gals. 2 qts. how many pints ? Ans. 12244. 3. In 9072 quarts how many tuns? Ans. 9. 4. In 1906 pints of wine, how many hogsheads ? Ans. 3 hhds. 49 gals. Ipt. &* In 1789 quarts of cider, how many barrels? Ans. 14 Ms. 25-2*5. lib iiELLi/no:v. 6. What number of bottles, containing a pint and a halt each, can be filled with a barrel of cider ? Ans. 168. 7. Hovv many piais, quarts, and two quarts, each an equal number, may be filled from a pipe of wine? Ans. 144. 8. LONG MEASURE. 1. Ill 51 miles, hew many furlongs and poles? Ans. 40H/wr. 10320 poles. 2. In 49 yards, how many feet, inches, and barley-corns ? Ana. U7ft. 1764 inch.- 5292 b. c. 3. How many inches from Boston to New-York, it being 248 miles? Ans. 15713280 inch. 4. In 4352 inches, how many yards ? Ans. 120 yds. 2ft. 8 in. 5. In 682 yards, how many rod- Ans. 632 x 2 -rl 1=124 rods. 6. In 15840 yards, how many miles and leagues ? - Ans. 9 m. 3 lea. 7. How many times will a carrk.^e wheel, Hi feet and 9 inches in circumference, turn ro^.d in going from New- York to Philadelphia; it beinjr 9G tnilcs? Ans. 30261 tihics, and S\fect over. 8. How many barley-corns will reach round the globe, it being 360 degrees ? Ans. 4755S01600. 9. LAND OR SQUARE MEASURE. 1. In 241 acres, 3 roods, and 25 poles, how many square rods or perches? Ans. 38705 perches. 2. In 20692 square poles, how many acres ? Ans. 129 a. 1 r. \2poS. 3. If a piece of land contain 24 acres, and au enclosure of 17 acres, 3 roods, and 20 rods, be taken out of it, how many perches are there in the remainder? Ans. 980 perches. 4. Three fields contain, the first 7 acres, the second 10 acres, the third 12 acres, 1 rood ; how many shares can they be divided into, each share to contain 76 rods ? Ans. 61 shores and 44 rods over. 10. SOLID MEASURE. 1. In 14 tons of hewn timber, bow many solid inches ? Ans. 14 50 X 17*28:=: 1 209600. 2. In 19 tons of round timber, how many inches'? Ans. 1313280. 3. In 21 cords of wood, how many solid feet ? Ans. 21 128=2688. 4. In 12 cords of wood, how many solid feet and inches ? Ans. 153n/i. and 2054208 inch. 5. In 4608 solid feet of wood, how many cords] Ans, 36 cds. 11. TIME. 1. In 41 weeks, how many days, hours, minutes, and seconds? Ans. 287 d. 6888 h. 413280 min. and 24796800 sec. 2. In 214 d. 15 h. 31 m. 25 sec. how many seconds? Ans. 18545485 sec. 3. In 24796800 seconds, how many weeks? 4>?s. 41 wks. 4. In 184009 minutes, how many days? Ans. 137 '/. 18 h. 49 min. 5. How many days from the birth of Christ, to Christ- mas, 1797, allowing the year to contain 365 days, 6 hours? Ans. 656354 d. 6 7t. 6. Suppose your age to be 16 years and 20 days, how many seconds old ai-f you, allowing 365 days and 6 hours to the year? Aus. 506649600 sec. 7. From March 2d, to November 19th following, inclu- sive, how many days ? Ans. 262. 12. CIRCULAR MOTION. 1. In 7 signs, 15 24' 40", how many degrees, minutes, and seconds? Ans. 225 13524' and 811480". 2, Bring 1020300 seconds into signs. Ans. 9 signs, 13 '25'. Questions to exercise Reduction. 1. In 1259 groats, how many farthings, pence, shillings. and guineas, at 28s. ? Ans. 20144#rs. 50368 UEDUCTiOA 2. Borrowed 10 English guineas at 28s. each, ai)d 24 English crowns at (is. and 8d. each; how in a jay p< stoles at 22s. each, will pay the debt? * Ana. 20. 3. Four UK!.' brought each 171. 10s. sterling value in gold into the mint, how many *um< .as at 2ls. eaci) must tliey receive in return t /jy/s. (S^ git in. 14s. 4. A biivcrsinith received three ingots of silver, each weighing 27 ounces, with directions to make them into spoons of 2 oz., cups of 5 oz., salts of 1 oz., and snuiF-boxes of 2 oz., and deliver an equ,-l number of each ; what was the number? /' each, v(;u! i n cannon ball be, at that rate in ;'rom lit re to the sun ? Ans. 22 yr. 216 d. 12 h. 40 m. ^ IK The sun .travels ihrouab 6 signs of the zodiac in tialf a Year ; bdMmany decrees, minutes, and seconds ? Aus. ISO deg. 10800 min. 648000 sec. low many strokes does a regular clock strike in 36ft lays, or a year ? * Ans. 56940. 13. How long will it take to count a million, at the rate of 50 a minute ? An*. 333 7i. -20 m. or 13 d. 21 //. 20 m. 14. The national debt of England amounts 'to about 279 millions of pouhds sterling; how long would it take to count this debt in dollars (4s. 6d. sterling) reckoning without in- termission twelve hours a day at the i^ite of 50 dols. a mi- nute, and 365 days to the year "? Ans. 94 years, 134 days, 5 'hours, 20 min. FRACTIONS. FRACTIONS, or broken numbers, are expressions for any assignable part of a unit or whole number, and (in general) are of two kinds, viz. VULGAR AND DECIMAL. A Vulgar Fraction, is represented by two numbers placed one above another, with a line drawn between them, thus, J, f, &c. signifies three fourths, five eighths, &c. The figure above the line, is called the numerator, and that below it, the denominator ; Thus { 5 Numerator - ) 8 Denominator. The denominator (which is the divisor in division) shows liow many parts the integer is divided into ; and the nume.r rator (which is the remainder after division) shows how ma- ny of those parts are meant by the fraction. A fraction is said to be in its least or lowest terms, when t is expressed by the least numbers possible, as when re- duced to its lowest terms will be J, and T q is equal tof , 26 3 2=4400 Num. a mile =5280 Denom. Ans. fJfS=| 7. Reduce 7 oz. 4 pvvt. to the fraction of a pound troy. Ans. J 8. What part of an acre is 2 roods, 20 poles ] Ans. | 9. Reduce 54 gallons to the fractioivof a hogshead of vine. 10. What part of a hogshead ; s 9 gallons ? Ans. \ 11. What part of a pound troy is 10 oz. 10 pwt. 10 grs, Ans. DECIMAL FRACTIONS. \DecimalFraction is that whose denominator is a unit, ith a cipher, or ciphers annexed to it. Thus, /2 FRACTIONS. The integer is always divided either into 10, 100, 1000, &c. equal parts; consequently the denominator of the frac- tion will always be either 10, 100, 1000, or 10000,&c. which being understood, need not be expressed ; for the true value of the fraction may be expressed by writing the numerator only with a point, before it on the left hand thus, T \ is writ- ten ,5 ;^ T VV ,45 ; -^ Tt ,725, g;;ificar,t figure conti- nues to possess the same place : so ,5 ,50 and ,500 are all the same value, n-u! equal to / or J. But ciphers placed at the left hand of decimals, decrease their value ir a tenfold proportion, by ren.ov ng them fur- ther from the decimal point. Thus, ,5 ,05 ,005, &e. are five tenth parts, five hundredth parts, five thousandth parts, ,098 by 2,23 Product, ,041883 6,74352 3. Multiply 25,238 by 12,1.7. Ayisieers. 307,14646 4. MiiUiniy 2461 by ,0529. 130,1869 5. Multiply 7853 by 3.5. 27485,6 i). Multiply ,007853 by ,0:1.5. ,000274855 7. Multiply 004 by ,004. ,000016 /hut* cost 6,21 yards of cloth, at 2 dols. 32 cents, 5 iijs, per vard ? Ans. 14, 4d. 8e. S^\m. 1>. j^ultiply 7,0;> dollars by 5.27 dollar?. Ans. 35,9954 dob. or $36 9D ets. 5, li). Multiply 41dols. 25cts. by 120 dollars. Ans. 4950 11. Multiply 3 dols. 45 els. by 16 cts. yl/w, "$0,5520=55 els. 12. Multiply 65 cents, by ,09 or 9 cents. Ans. $0,05855 cts. $ mills. 13. Multiply 10 dels, by 10 cts. Ans. $1 1 4:"ktultiply 341 ,45 dols. by ,007 or 7 mills. Ans. $2,39 To multiply by 10, 100, 1000, &c. remove the separating. point so many places to the right hand, as the multiplier has ciphers. ( Multiplied by 10, makes 4.25 So ,425 I by 100, makes 42,5 ( by 1000, is ,425 For ,425X10 Is 4,250, &o. DIVISION OF DECIM \ llui.E. 1. The places of the decimal parts oi JJ.KJJIMA.L FRACTIONS. divide as in whole numbers, and from the right hand of the quotient, point off so many places for decimals, as the decimal places ill the dividend exceed those in the divisor. 2. If the places in the quotient be not so many as the rule requires, supply the defect by prefixing ciphers to the left hand of said quotient. NOTE. If the decimal places in the divisor be more than Chose in the dividend, annex as many ciphers to the divi- dend as you please, so as to make it equal, (at least,) to the divisor. Or, if there be a remainder, you may annex ciphers -to i,t, and carry on the quotient to any degree of exactness. EXAMPLE-. 9,51)77,4114(3.11 7(3,08 3.8),21316(,0561 190 1,331 951 3804 3804 328 38 38 00 Answers. 32,12 ,23068+ ,00758 ,00150+ ,038356 + ,40736+ 611,9+ 3. Divide 780,517 by 24,3. 4. Divide 4,1 8 by ,1812. 3. Divide 7,25406 by 957. (>. Divide ,00078759 by ,525. 7. Divide 14 by 365. 8. Divide $246,1476 by $604,25. 9. Divide $186513,239 by $304,81. U>. Divide $1,28 by $8,31 11. Divide 56 cts. by 1 dol. 12 cts. 12. Divide 1 dollar by 12 cents. 13. If 213 or 21,75 yajds of cloth cost 34,317 dollars, what will one yard cost 1 $ 1 ,577 + NOTE.- When decimals, or whole numbers, are to be di- ^ided by 10, 100, 1000, &e. (viz. unity witli ciphers,) it i* performed by removing the separatrix in the dividend, sc many places towards the left hand as there one ciphers in the divisor. 8,333 + r 10, the quotient, is 57, ~ 57-2 divided by J 100, - - - - 5,7:2 f 1000, - - - - ,57:2 REDUCTION OF DECIMALS. CASE I. To reduce a Vulgar Fraction to its equivalent Decimal. RULE. Annex ciphers to the numerator, and divide by the deno- ninator; and the quotient will be the decimal required. NOTE. So many ciphers as you annex to the given ny- ucrator, so many places must be pointed in the quotient : irul if there be not so many places of figures in the quotient, nakc up the deficiency by placing ciphers to the left hand .if the said quotient. EXAMPLES. 1. Reduce -J- to a decimal. 8)1,000 Ans. ,1.25 ^. What decimal is equal to J ? Ansivers. ,5 3. What decimal is equal to J ? ,75 4. Reduce 4- to a decimal. ------ .2 5. Reduce -j -J- to a decimal. ----- />75 6. Reduce ~-J to a decimal. ---...., 5 g> 7. Bring % to a decimal. ----- ,00375 8. What decimal is equal to ^ T 1 - - - ,0370*37+ 1). Reduce ^ to a decimal. - ' * - - ,333333 -j- 10. Reduce T^J to ^ s equivalent decimal. - - ,008 11. Reduce 2 *Vto a decimal. - - - . 1923076 -f- CASE II. To reduce quantities of several denominations to a Decimal* RULE. 1. Bring the given denominations first to a vulgar fraction by Problem HI. page 71 ; and reduce said vulgar fraction to its equi- valent decimal ; or, 2. Place tfto several denominations above each other, lotting U;e highest denomination stand at the bottom ; then divide each denomi- nation (beginning at the top) by its value in the next dcaoinh' the last ouotie'nt, vrjll give the decimal reoui;- 3DECIAI AL jbUiACiTO.N S . EXAMPLES. 1. Keduce 1'2 . 6d. '3qrs. to the decimal of a pound. 150 4 5760 2700 1920 7800 768.0 Answer. By Rule 4 3, *400 1920 4800 4800 2. Ileduce los. 9d. 3qrs. to the decimal of a pound. Ans. ,700625 3. Reduce Dd. 3 qrs. to the decimal of a shilling. -4ns. ,8125 4. Jleduce TJ farthings to the decimal of a shilling. Ans. ,0625 5. Reduce 3s. 4d. New-England currency, to the deci- mal of a dollar. Ans. ,555555+ 6. Reduce 12s. to the decimal of a pound. Ans. ,6 NOTE. When the shillings are even, half the number, with a point prefixed, is their decimal expression ; but if the number be odd, annex a cipher to the shillings, and then by halving* them, you will have their decimal expression-. 7. Reduce 1, 2, 4, 9, 16 and 19 shillings to decimals. Shillings 1 2 4 9 16 19 ,'0.5 ,1 .2 ,45 ,8 ,95 Dfc : JMAL FRACTIONS. . Write down 47 18 10] in a decimal expression. Ans. 47,943 >. Reduce \ 8s. 2d. to an equivalent decimal. Ans. 1 ,40 PROBLEM II. .1 short and easy method to find the value of any decimal of a pound by inspection. RULE. Double the first figure, or place of tenths, for shillings, and if the second figure be 5, or more than 5, reckon another shilling ; then, after this 5 is deducted, call the figures in the second and third places o many farthings, abating 1 when they are above 12, and 2 when above 36, and the result will be the answer. NOTE. When the decimal has but 2 figures, if any thing remains after the shillings are taken out, a cipher must be annexed to the left hand, or supposed to be so. EXAMPLES. 1. Find the value of . ,679 by inspection. 12s=double of 6 . 1 for the 5 in the second place which is to be [deducted out of 7 Add 7Jd. 29 farthings remain to be added. Deduct Jd. for the excess of 12. Ans. 13s. ~d. '2. Find the value of . ,870 by inspection. Ans. 17s.Gjf7. IK Find the value of . ,842 by inspection. Ans. 16$. Wd. V Find the value of . ,.097 by inspect! on. Aits, }?. 11 W. REDUCTION OF CURRENCIES. RULES for reducing the Currencies of the several United States* into Federal Money. CASE I. To reduce the currencies of the different states, where a crollaris an even number of shillings, to Federal Money. They are f New-England, Nero- York, and ) J Virginia, North Carolina. ( *j Kentucky, and Tennessee. RULE. 1. When the sum consists of pounds only, annex a ciphn to the pounds, and divide by half the number of shillings in a dollar; the quotient will bo dollars. t ^ 2. But if tiie eiim 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 whole by the number of shillings con- tained in a dollar, and the quotient will be dollars, cents, mills. &:c. EXAMPLES. 1. Reduce 737. New-England and Virginia currency, to Federal money. 3)730 | cts. $243j-243 83 1 2. Reduce 45/. 155. 1\d. New-England currency, to fedc- 20 [ral monev. d. A dollar=6)91 5,625 12)7,500 $152,6044- Ans. ,625 decimal. * 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 de- preciated in their value, in some states more, in others less, &. Thus a dollar is reckoned in Jfeic-England. Virginia, Kentucky, and Tennessee, New-Jersey, Pennsylvania, Delaware, and Jilaryland, South- Georgia, ^few-York, and fi jforth Carolina, ) Cl t Adding a cipher to the pounds, multiplies the whole ny 10, them into tenths of a pound ; then because a dollar is just three tenth? of:' pound, N. Li. currency, dividing 1 throse ifntbs byS ? brih^the!fl inio(Muar. 1 farthing is ,25 J which annex to the pence, and 2 ,50 > divide by 12, you will have the 3 = ,75 j decimal required. 3. Reduce 345/. 10s. ll\d. New-Hampshire, &c. curren- ^v to Spanish milled dollars, or federal monev. 345 10 III 20 cL 12)11,2500 6)6910,937^ ,9375 decimal $1151,8229+ Ans. 4. Reduce 105Z. 14s. 3*d. New-York and Nortb-Caroli- la currency, to federal money. ,105 14 3J- d 20 12)3,7500 A doliar=8)21 14,3125 ,3125 decimal $264,28906 Ans. Or $ dcm. T W 5. Reduce 4317. New-York currency to federal money. r his being' pounds only,* 4)4310 Ans. C. Reduce 2SZ. lls. 6d. New-England and Virginia cur- ency, to federal money. Ans. $95, 25 cts. 7. Change 463/f. 10^. Sd. New-England, &c. currency, o federal money. Ans. $1515, llefa. lm.-r 8. Reduce 35/. 19s. Virginia, &c. currency, to federal noncy. -4ns. $119, 83 cts. 3 m. -f 9. Reduce 214?. 10s. 7d. New- York, &c, currency, to edoral money. Ans. $536, 32 cts. 8 w^.+ 10. Reduce 304?. 11s. 5tf. North-Carolina, &c. currency, o federal money. Ans. 761 42 cts. 7 m.+ 11. Change 219Z. 11s. 7$d. New-England and Virginia currency, to federal money. Ans. $731 94 cts.-{- * A dollar is 8s. in this currency ,4=4-10 of a pound ; therefore, multi- ly by 10. and divide by 4, brings the pounds into dollars, &c. 84 REDUcfio:; OF CUIUIESLILS. 12. Change 241Z. New-England, &c. currency, into le* deral money. Ans. $803, 33 ct*. 13. Bring 20/. 185. 5f<7. New-England currency, into dollars. Ans. $69, 74 cts. 6| m.+ 14. Reduce 468?. New-York currency to federal money. Ans. $1170* 15. Reduce 17s. 9j0-^3==$653j=$658, 33-cfc NOTE. When there are shillings, pence, &c. in the eivqa sum, reduce them to the decimal of a pound, then multiply and divide as above, &c. 2. Reduce 36J. 11s. 8d. New-Jersey, declined expression. 30 7)1645,2180 AKS. 285,0311 s, d. 3. Reduce 94 14 8 to federal A Rpflnpo 1Q 17 ft* ANSWERS. S cts. m. money, 405 99 8+ cx ix 74- ^ l?n/]iifn 417 1-1 ft" 170H 9^ G~p4>f1ii"r> 1,1ft 10 f> 1\\?> IJOi. 7 R<>/]npj> Iftfl O ftQ.X 71 J. > SRf.'Iiipn f> 11 ft O 1^ /fJ- 0. RP.IM^P 41 17 I7O .^1 4 8 CASE IV. To reduce the currency of Canada and Nova-Scotia to Federal Moitey. RULE. Multiply the given sum by 4, the product will be dollars. NOTE. Five shillings of this currency are equal to ri dollar ; consequently 4 dollars make one pound. EXAMPLES. 1. Reduce 1257. Canada and Nova-Scotia currency, to federal money. 125 4 Ans. $500 4s. Sd. or 56f7. to the dollar -x 5 ^^ nf a pound ; 00-?-7, 80 KKDUCTION OF COltf. 2. Reduce 55?. 10s. 6d. Nova-Scotia currency, to dollar?, 55,525 decimal value. 4 -- I cts. Ans. $222, 100=r222 10 ANSWERS'. 3. Reduce 241/. 18s. 9J. to federal money, 007 75 4. Reduce 58 13 6J 234 fO 5. Reduce 528 17 8 2115 53 G. Reduce 120 4 50 7. Reduce 224 19 - 899 80 8. Reduce 13 Hi 2 79 REDUCTION OF COIN. RULES for reducing the Federal Money to the currencies of the several United States. To reduce Federal Money to the currency of RuLE ._ Mu ltiply the given sum by ,3, and tho I. < j-?^ 1 , ' ,> product will be pounds, and decimals of a ) RULE. Multiply the given sum by ,4, and thfj \ product will be pounds-, and decimals of a LK. Multiply the given sum by ,3, and di~ Pennsylvania^ I vide the p rot ] uct by 8, and the quotient will Delawarejondi bc poundSi and decimals of a pound. Maryland. j South r< firn1i n f , ) RULE. Multiply the given sum by ,7 and and T divide by 3, the quotient will be the Georgia C answer in pounds, and decimals of a 3 pound. Examples in the foregoing Rules. . Reduce $152, 60 cts. to New-England currency* ,3 45, 780 ^ws.=45 15s. 20 But the value of any decimal of -- a pound, may be found by inspec- 15, 600 tion. See Problem IJ. page 81. 200 REDUCTION OF COIN*. 67 2. In $196, how many pounds, N. England currency t __ ._fl 58,8 Ans.=5S 16 3. Reduce $629 into New- York, &c. currency. ,4 _ -251,6 ^4ns.~251 12 4. Bring SI 10, 51cts. 1m. into New- Jersey, &c. currency. #110,511 ,3 Double 4 makes 8s. Then 39 farthings 8)331,533 are 9d. 3qrs. See Problem II. page 81. " "11,441 ^4ns.=:41 8s. 9J<7. % Inspection. . Brin* $65, 36cts. into South-Carolina, &c. currency. 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|4- 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 Nova- Scotia currency. RULE. Divide the dollars, c. by 4, the quotient will be pounds, ind decimals of a pound. EXAMPLES. 1. Reduce $741 into Canada and Nova-Scotia currency. S cts. ~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. 4. Reduce $2114, 50 cts. into Canada currency. Ans. 528 12$, Orf, RULES FOR REDUCING CURRENCIES. RULES far reducing the Currencies of the several United States, aiao Canada, JVova Scotia, and Sterling, to the par of all the others. tCjP* See the given currenc) in the left hand column, and then cast your eye to the right hand, till you come under the required currency, and you will have the rule. MagMmcoBw Jfew-Ene- and, yir- ffinia, Km- ucky, and renncsuce. NewJersey, Pennsylva- nia, Dcla- irnre, and Maryland. JVw York, and Jforth- Carolina. South- Ca- rol in a, and Georgia. Canada, and NovaScotia, Sterling. *\*t to- Eng- land, I'ir- pinia, Ken- tucky, and yjfnnessce. Add one fourth to the given sum. Add one third to the given sum. Multiply the i^iven sum tjy 7, and di- vide the pro- duct by , Multiply the given sum by 5, and di- vide the pro- duct by 6. Deduct one ouilh from he givtn sum. New Jersey, Pennsylva- nia, Dela- jrnre, and Maryland. Deduct one fifth from he given aum. Add one fifteenth to the given sum. Multiply the Riven sum by 28, and livido the product by Deduct one third from the given sum. Vuli'ply the iv n sum y 3, iuid di- vide the pro- duct by 5. Mic- York, and Jfortft- Caiolina. Deduct one fourth from the Ncw- York, &c. Deduct one sixteenth from the N. York. Multiply the sjivcn sum by 7, and di- vide the pro- duct by 12. Multiply the ?iven sum By 5, and di- vide the pro- duct by 8. Multiply the ;iven sum jy 9, and di- vide the pro- duct by 16. South- Ca rstliiia, and Georgia. Multiply thg iven sum y 9, and di- vide the pro- duct by 7. Multiply the given sum by 45, and divide the product by 28. Multiply the given sum by 12, and divide the product by 7. Multiply the given Btiir by 15, and divide tht product by 14. From the given sum deduct one twenty- eighth. Canada, and WovaScotia Add one fifth lo the Canada.&c Add one half to the Canada sum.*' Multiply the riven sum by 8, and di- vide the pro- duct by 5. Deduct one fifteenth from the gi- ven sum. Deduct one tenth from the eiven yum. Sterling. To the En- glish sum add one third. Multiply the English mo- ney by 5, and divide the product by3. Multiply the English sum by 10, and divide the 1 product by9. To the Eng ish .money add one twenty-se- venth. Add one ninth to the given sum. REDUCTION OF CulX. % Implication of the Rules contained in the foregoing Table. EXAMPLES. 1. Reduce 46/. 10s. 6d. of the currency of New-Hamp- hire, into that of New-Jersey, Pennsylvania, &c. . s. d. See the rule 4)46 10 6 in the table. +11 12 7J Ans. 58 3 H 2. Reduce 25/. 13s. 9d. Connecticut "currency, to New- fork currency. . s. d. 3)25 13 9 By the table, +, &c. -f-8 11 3 Ans. 34 5 3. Reduce 1257. 10g. 4d. New-York, &c. currency, to South-Carolina currency. . s. d. Rule by the table, 125 10 4 x7,-rby 12, &c. 7 12)878 12 4 Ans. 73 4 4 4. Reduce 467. lls. 8d. New-York and N. Carolina cur- ency, to sterling or English money. s. d. 4G 11 8 9 See the table. \ IG^X 4)419 5 X given sum by > 4)104 16 3 9,-:-byl6,&c. ) Ans. 26 4 OJ To reduce any of the different currencies of the several States into eacli other, at par ; you may consult the prece- ding 1 table, which will give you the rules. MORE EXAMPLES FOR EXERCISE. 5. Reduce 84/. 10s. 8d. New-Hampshire, &c. currency, into New- Jersey currency. Ans. 105 13s. 4d. 6. Reduce 120?. 8s. 3d. Connecticut currency, into New- York currency. Ans. 160 lls. M. H 2 4 30 RULE OF THREE DIRECT. 7. Reduce 120Z. 10s. Massachusetts currency, into South- Carolina and Georgia currency. Ans. 93 14s. 5]s. 1344 "672 672 o. 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 51Z. 6s. what will one pair cost ? Ans. 4s. 6 d. 7. At I OU1. per pound, what is the value of a flr'dn of butter, weight 50 pounds? Ans. 2 9s. 8. How much sugar can you buy for 23/. 2s. at 9d. per pound I Ans. 5 C. 2 qrs. 9. Bought 8 chests ^f sugar, each 9 cwt. 2 qrs. what do they come to at 2/. 5s. per cwt. 1 Ans. 171. 10. If a man's wages be 75/. 10s. a year, what is that a calendar month? Ans. 6 5s. Wd. 11. If 4; tuns of hay will keep 3 cattle over the winter; how many tuns will it take to keep 25 cattle the same time? Ans. 37 J tuns. 12. If a man's yearly income be 2087. Is. what is that a day? Ans. Us. 4u:>t to ? Ans. $255 99 cts. 34. Suppose a gentleman's income is $1835 a year, and : ^ 19 cts. a day, one day with another, how muck r.ll !,v. !i;ivc saved at the year's end? ,-i?;9.$562, 15 cts. 35 T r >iy horse stond me in 20 cts. per day keeping, rhat will be the charge of 11 horses for the vvW. at that Ans. 96 RULE OF THiXEE 36. A merchant bought 14 pipes of wine, and is allowe 6 months credit, but for ready money gets U 8 cts. a gallo cheaper ; how much did he save by paying ready money ? Ans. $141, }%cts. Examples promiscuously placed. 37. Sold a ship for 537?. and I owned f of her ; wha was my purt of the money 1 Ans. 201 7s. 6d. 38. If , y of a ship cost 781 dollars 25 cents, what is the whole v ->rth? As 5 : 781,25 : : 16 : $2500 Ans. 39. V I buy 54 yards of cloth for 3U 10s. what did it cost p< : Ell English 1 Ans. 14s. 7d. 40. knight of Mr. Grocer, 11 cwt. 3qrs. of sugar*, at 8 dollai i2 cents per cwt. and gave him James Paywell's note for 19Z. 7s. (New-England ciirrtvicy) the rest I pay in cash , tell me how many dols. will make up the balance 1 Ans. 30, 91 cts. > 41. If a staff 5 feet long cast a shade or. level ground 8 feet, what is the height of that steeple whose shade at th2, 46 cts. 5m. 43. Bouo-ht 51) pieces of Lerseys, each 34 Eils Flemish, at ' 8s. 4d. per Ell English; what did the whol^ cost? Ans. 425. 44. Bought 200 yards of camhriok for 90/. but being da* mnged, I am willing to lose 7L 10?. by the sale of it ; what must I demand per EH English? Ans. 10s. 3fd. 45. How many pieces of Holland, each 20 Ells Flemish, may 1 have for23/.8s. at 6s^6d. per Ell English? .4ns. 6; 46. A merchant bought a bale of cloth containing 240 yds. ' at the rate of 7^ for 5 ycb. and sold it again at the rate oi $1H for 7 yards ; did he gain or lose by the bargain, and how much ? Ans. He gained $25, 71 cts. 4 :. 47. Bought a pipe of wine for 84 dollars, and found it hat leaked out 12 gals. : I sold the remainder at 12^ cts. a pint what did I gain or lose ? Ans. I gained $30. 48. A gentleman bought 18 pipes of wine at 12s. Gd (New-Jersey currency) per gallon ; how many dollars wil pay the purchase ? An*- $3760. 49. Bought a quantity of plate, weighing 15 Ib. 11 02?. 13 >\vt. 17 gr. how many dols. will pay for it, at the rate of 12s* Td. New-York currency, per oz.? Ans. 301, 50, cts.Sfym. 50. A factor bought n certain quantity of broadcloth and trugget, which together cost 817. the quantity of broadcloth ivas 50 yds., at 18s. per yd., and for every 5 yds. of broad- cloth he had 9 yards of drugget ; I demand how many yds. }f drugget he had, and what it cost him per yard ? Ans. 90 yds. at 8s. per yd. 51. If I give 1 eagle, 2 dols. 8 dimes,2ets. and 5m.for675 lops, how many tops will 19 mills buy ? Ans. 1 top. 52 Whereas an eagle and a cent just threescore yards did buy, How many yards of that same cloth for 15 dimes had 1 1 Ans'. 8 yds. 3 qrs. 3 na.-\- 53. If the legislature of a state grant a tax of 8 mills on tire dollar, how niuoh must that man pay who is 319 dols, 75 ceuts on the list 1 Ans. $2, 55 cts. S 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 gallons of water, in one hour, fall into a cistern containing 300 gallons, and by a pipe in the cistern 35 gal^ Ions run out in an hour; 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 clays before B, at the rate of 15 miles a day ; B follows at the rate of 20 mile a day ; what distance must he travel to overtake A? Ans. 300 miles. RULE OF THREE INVERSE. THE Rule of Three Inverse, teaches by having three numbers given to find a fourth, which shall have the same proportion to the second, as the first has to the third. If more requires more, or less requires less, the question belongs to the Rule of Three Direct. But if more requires less, or less requires more, the ques- tion belongs to the Rule of Three Inverse ; which may al- ways be known from the nature and tenor of the question For example r UF THREE If 2 men can mow a field in 4 dnys> how many day 3 vvifl it require 4 men to mow it? men days men 1. If 2 require 4 how much time will 4 require? Answer, 2 days. Here more requires less, viz. 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, vizr. the less tho number of men arc, the more days are required therefore the question belongs to Inverse Proportion, RULE. 1. State anil reduce the terms as in the Rule of Three Di- rect. 2. Multiply the first and second terms together, and divide the pro- tiuct by the third ; the quotient will he the answer in the same deno- mination as the middle term was reduced into. EXAMPLES. 1. If 12 men con 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 dnys, when the day is 12 hours long', in how many days will he perform it wlnm the day is but 10 hours long? Ana. 6 days. 3. What length of board 7 J inches wide, will make a square foot?. Ans. 19.1 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 money? Ans. 20 mifas. 5. If when wheat is 7s. 6d. the bushel, the penny Ion f will Weigh 9 oz. what ought it to weigh when wheat is O's. per bushel ? Ans. 11 oz. 5pwt. 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 bushels. 7. If 100Z. in 12 months gain 6L 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 worfc only 8 hours per day ? Aiis. 7 months. 9. What number of men must be employed to finish ia , -what 15 aaen^ould bo 20 days about? 10. buppose 050 men art; in a garrison, and their provi- sions calculated to last but 2 months, how many men must leave the garrison that the same provisions may be suffi- cient lor those who remain 5 months 'I 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 ? 1 J yds. wide, and lined with shalloon yd. wide ; how ma- ny yards of shalloon will complete the lining? Ans. 6911 yds.'S qrs. 2J na. PRACTICE. PRACTICE is a contraction of the Rule of Three Direct, Jpvhen the first term happens to be a unit or one, and is h concise method of resolving most questions that occur in trade or business where money is reckoned in pounds, shil- lings and pence ; but reckoning in federal money will ren- der this rule almost useless : for which reason 1 shall not tmlarge so much on the subject as many oilier writers have dune. Tables of Aliquot, or Even Pa/'tis. Parts of a shilling d. s. is -J- 4 = | 5 ? Parts of 2 shillings. i i Is. is 8d. = 1 6d. -> 4*1- * Parts of a pound, s. d. 10 G 8 5 4 3 4 2 (> 1 S ' Parts of a cwt. Ib. cwt. r>G is -.>- 2S r^= - 1 10 14 3d. The aliquot part of any number is such a part of it, as being taken a cer- tain number of times, exactly mnkcb that number. Ta CASE I. When the price of one yard, pound, &. is an oven part of one shilling Find the value of the given quantity at Is. a yard, pound, &c, and divicl* it by that even part, and ?,ho rpmfjPT't will ]. t-'.c n*r?Vv'.-r in' sVf 100 Or find the value of the given quantity at 2s. per yd. &e. and divide said value by the even part which the given price is of 2s. and the quotient will be the answe'r in shiT- lmtvs,&c. which reduce to pounds. N. B. To find the value of any quantity at Ss. you need only double the unit figure for shillings ; tire other figures will be pounds. EXAMPLES. 1. What will 46] i yds. of tape come to at ld. per yd. ? s. d. l|d. | | | 461 6 value of 461^ yds. at Is. per yd. 5,7 8| 2 17s. 8J7. value at 1-Jd. 2. What cost 256 Ib. of cheese at 8d. pe*r pound? ScL j -J- | 25 12s. value of 256 Ib. at 2s. perlh. 8 10s. 3d. value at 8d. per pound. Yards, per i/ard> . s. d. 486 at Id. Answers. 2 6j 8G2 at 2d. 7 3 8 Oil at 3d. II 7 9 . 749 at 4d. 12 9 8 113 at 6d. 2 16 6 899 at' 8d. 29 19 4 CASE If. When the price is an even part of a pound Find the value of the given quantity at one pound per yard, fcc. and divide it by that even part, and the quotient will be tlte an- swer in pounds. EXAMPLES. What will 129i- yards cost at 2s. 6d. per yard ? 5v d. . 9. . 2 6 ] -J- | 129 10 value at I per yard. Ans. 16 2s. &d. value at 2s. 6d. per yank Y(fs. 5. ?. . &. d. T23 at 10 per van}. An SWA. 61 10 () 6874 at 5 , I 171 17 ft I'KA- . ^. \tif. & d.. . s. a, 21 1J at 4 per yard. 42 5-0 543 at 6 8 l$l 127 at 3 4 - - 21 3 4 461 at 1 8 38 8 4 NOTE. When the price is pounds ouly, the given quan- ity multiplied thereby, will be the answer. EXAMPLE. 11 tuns of hay at 4Z. per tun. Thus, 11 4 Ans. 44 CASE III. When the given price is any number of shillings un- der 20. 1. When the shillings are an even number, multiply the quantity by half the number of shillings, and double the first figure of the product for shillings; and the jest 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. 1st. 124 yds. at Sg. 3d. 132 yds. at 7s. per vd. 4* 7 49 12s. Ans. 2,0)92,4 46,4 Ans. Yds. . s. Yds. . s . 562 at 4s. Ans.112 8 j 372 at 11*. An*. 204 12 378 at 2s. 37 16 1 264 at Ps. 118 16 913 at 14s. 630 2 | 250 at 16s. 200 00 CASE IV. When the given price is pence, or pence and farthings, and not an even part of a shilling Find the value of the given quantity at Is. per yd. &c. which divide by the great- est even part of a shilling contained in the given price, and take parts of the quotient for the remainder of the price, and the sum of these several quotients will be the answer s. Are, which reduce to pound?. , i> . ; - EXAMPLES. Wha! will 24511). of raisins come fo at 9|d. per Ik s. d. 6d. % ' 245 value of 245 Ib. at Is. per ; 3d. 122 6 value of do. at 6d. per Ib. fd. J 61 3 value of do. at 3d. per Ib. 15 3 value of do. at Jd. per Ib. Ib. 2,0)19,9 Of Ans. 9 19 Oj value of the whole at 9|d. per Ib. d. . a. d. Ib. d. . s. d. 372 at 1 Ans. 2 14 3 325 at 2} 30 11} 27 at 4} 15 10 H 18 20 17 0{ 32 18 576 at 7 541 at 9} 672 at 11 J CASE V. When the price is shillings, pence and farthings, and pot the aliquot part of a pound Multiply the given quantity f>y the 'shillings, and take parts for the pence and farthings, as in the foregoing cases, and add them together ; the sum will be the answer in shillings. EXAMPLES. 1. What will 246 yds. of velvet come to, at 7s. 3d. per yd.'? s. d. 3d. | -J- | 246 value of 346 yiards at Is. per yd. 1722 value of do. at 7s. per yard. 61 6 value of do. at 3d. per yard. 2,0)178, 3 6 An. 89 3 6 Value of g particulars : 1. Gross Weight, which is the whole weight of any sort f goods, together with the box, cask, or bag, &c. which ontains them. . Tare, which is an allowance made to the buyer, for be weight of the box, cask, or bag, &c. which contains the ;oods bought, and is either at so much per box, &c. or at much per cw.t. or at. sr> much in the whole gross weight. 3. Tret, which is an allowance of 4 lh. on every lOlll, for waste, dust, &c. 4. Cloff, which is an allowance made of 2 Ib. upon ever,y 3 cwt. 5. Buttle, is what reflnains 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 the total tare from the whole gross, and the remainder will he the neat weight. EXAMPLES. 1. What is the neat weight of 4 hogsheads of Tobacco, marked with the gross weight as follows : Ib. Tare 100 95 83 81 359 total tare. neat. 2. What is the neat weight of 4 barrels of Indigo, No,- and weight as follows : C. qr. Ib. Ib. jVo. 1 4 1 10 Tare 36} o _ 3 3 02 39 ( 3 _ 4 19 32 f cwt.gr. Ib 4_4 o 35J4^.1501I CASE II. When the tare is at so much per box, cask, bag, &c. Multiply .the tare of 1 by the number of bags, bales, &c the product is the whole tare, which subtract from the gros? and the remainder will be the neat weight. EXAMPLES. 1. In 4 hhds. of sugar, each weighing 10 cwt. 1 qr. 15 Ib gross; tare 75 Ib. per hhd. how much neatl Cwt. qrs. Ibs. 10 115 gross weight of one hhd. 4 [Carried up.l C. qr. Ib. No. 1 9 12 2 8 3 4 3 7 1 4 6 3 25 Whole gross 32 13 Tare 359 lb.= 3 3 23 Ans. 28 3 18 TARE AXD TUKT. 105 41 2 4 gross weight of the whole: ~>5x4=2 2 20 whole tare. 8 3 12 neat. 2. What is the neat weight of 7 tierces of ricre, each ^veiirhinjr 4 cwt. 1 qr. 9 Ib. gross, tare per tierce 34 Ib. ? Ans. 28 C. qr. 21 Ib. 3. In 9 firkins of butter, each weighing 2 qrs. 12 Ib. gross<, tare 1 lib. per firkin, how much neat? Ans. 4 C. 2 qrs. $ Ib. 4. If 241 bis. of figs, each 3 qrs. 19 Ib. gross, tare 10 Ib, per barrel ; how many pounds neat 1 Ans. 22413. 5. In 16 bags of pepper, each 85 Ib. 4 oz. gross, tare per bag, 3 Ib. 5 oz. ; how many pounds neat ? Ans. 1311. 0. In 75 barrels of figs, each 2 qrs. 27 Ib. gross, tare in the whole 597 Ib. ; how much neat weight ? Ans. 50 C. 1 qr. 7. What is the neat weight of 15 hhds. of Tobacco, each wfiighinjr 7 cwt. 1 qr. 13 Ib. tare 100 Ib. per hhd. ? Ans. 97 C. (tor. 11 Ib. CASE III. When the tare is at so much per ewt. Divide the gross weight by the aliquot part of a cwt. for the tare, which sub- tract from the gross, and the remainder will be neat weight. EXAMPLES. 1. What is the neat weight of 44 cwt. 3 qrs. 16 Ib. grosa, tare 14 Ib. per cwt.? C. qrs. Ib. [ 141b. | | ] 44 3 16 gross. 5 2 12 tare. Ans. 39 1 3J neat. 2. What is the neat weight of 9 hhds. of Tobacco, each weighing gross 8 cwt. 3 qrs. 14 Ib. tare 16 Ib. per cwt. ? Ans. 68 C.I qr. 24 Ib. 3. What is the neat weight of 7 bis. of potash, each weighing 201 Ib. gross, tare 10 Ib. pel- cwt. ? Ans. 1281 Ib. 6 oz. 4. In 25 bis. of figs, each 2 cwt. 1 qr. gross, tare per. cwt. 16 Ib. ; how much neat weight? Ans. 48 cwt. 24 Ib. 5. In 83 cwt. 3 qrs. gross, tare 20 Ib. per cwt. what neat weight? Ans. 68 cwt. 3 qrs. 5 Ib. 6. In 45 cwt. 3 qrs. 21 Ib. gross, tare 8 Ib. per cwt, how flinch neat weight ? Ans. 42 cwt. 2 qrs. 17 Ib. ".'. AVhnt is fire vaTiro of tire n-rnt weight of 8 hhds. of FU-* 10t> TARE AND TRET. gar, at $9, 54 cts. per cwt. each weighing 10 cwt. 1 qr. 14 Ib. gross, tare 14 Ib. per cwt. Ans. $692, 84 cts. 2w. CASE IV. When Tret is allowed with the Tare. 1. Find the tare, which subtract from the gross, and call the remainder suttle. 2. Divide the suttle by 26, and the quotient will bo the tret, which subtract from the suttle, and the remainder will IA> the neat weight. EXAMPLES. 1. In a hogshead of sugar, weighing 10 cwt. 1 qr. 12 Ib. gross, tare 14 Ib. per cwt., tret 4 Ib. per 104 Ib.,* how much neat weight 1 Or thus, cwt. qr. Ib. cwt. qr. Ib. 10 1 12 14=^=J)10 1 12 gross. 4 115 tare. 41 26)9 7 suttle, 28 I 11^ tret. 330 Ans. 8~~2~~24 neat. 83_ 14=1)1160 gross. 145 tare. 26)10l5 suttle. 39 tret. Ans. 976 Ib. neat. 2. In 9 cwt. 2 qrs. 17 Ib. gro^, tare 41 Ib., tret 4 Ib. per 104 Ib., how much neat 1 Ans. 8 cwt. 3 qrs. 20 Ib. 3. In 15 chests of sugar, weighing 117 cwt. 21 Ib. gross, tare 173 Ib., tret 4 Ib. per 104, how many cwt. neat ? Ans. Ill cwt.Zllb. 4. What is the neat weight of 3 tierces of rice, each weigh- ing 4 cwt. 3 qrs. 14 Ib gross, tare 16 Ib. per cwt., and allow- ing tret as usual ? Ans. 12 cwt. qrs. 6 Ib. 5. In 25 bis. of figs, each 84 Ib. gross, tare 12 Ib. per cwt. tret41b. per 104 Ib. ; how many pounds neat? Ans. 1803 -f- * This is the tret allowed in London. The reason of divividing by 2t> ? because 4 Ib. is 1-26 of *04lb. but if the tret is at any other rrtf, other ra " mu*t he tnkrr tucc.oTiiing to ftp) rate proposed, &-<\ IAKIS A:,D TUE'I. 107 0. What is the value of the neat weight of 4 barrels of Spanish tobacco; numbers, weights, and allowances as fbl- ows, at 9jd. per pound 1 cwt. qrs. No. 1 Gross 1 l Tret 4 Ib. per 104 Ib. Jns. 17 16^. _ CASE V. When Tare, Tret, and ClolT, arc allowed : Deduct the tare and tret as before, and divide the suttlo >y 168 (because 2 Ib. is the T -Jg- of 3 cwt.) the quotient will e the clofi', which subtract from the suttlc, and the remain* r will be the neat weight. EXAMPLES. 1. In 8 hogsheads of tobacco, each weighing 13 cwt. 3qre. 31b. gross, tare 1071b. per hlid., tret 4 Ib. per 104 Ib., and Ib. per 3 cwt., as usual ; how much neat? cwtf . grs. Ib. 13 3 23 4 55 28 443 nan i"5^ Ib. gross, of 1 hhiU 3 4689 whole gross. 107X3=^321 tare. 26)4368 suttle, 168 tret. 168)^00 suttle. 25 clofF. Ans. 4175 neat weight. & What is the neat weight of 26 cwt. 3 qrs. 20 ib. gross, are 62 Ib., the allowance of tret and cloff as usual ? . neat 25 ci&, 1 qr, 5Z, I oz, nearly ; omitting fur- ther frac* ?>??5.. 103 INTEREST. INTEREST is of two kinds; Simple and Compound* SIMPLE INTEREST. Simple Interest is the sum paid by the borrower to the lender for the use of money lent ; and is generally ut a cer- tain rate per cent, per annum, which in several of the Uni- ted States is fixed by law at G per cent, per annum ; that is, 6/. for the use of 1007. or 6 dollars for the use of 100 dol- lars for one year, &c. Principal, is the sum lent. Rate, is the sum per cent, agreed on. Amount, is the principal and interest added together. CASE I. To find the interest of any given sum for one year. RULE. :Multiply the principal by the rate per cent, and divide tfio product by 100 ; the quotient will be the answer. EXAMPLES. 1. What is the interest of 397. 11s. Sid. for one year aj 67. per cent, per annum 1 . s. d. 39 11 8-!- 2J37 10 3 20 7(50 0|12 Ans. 2 7s. (k/. T {fe. 2. What is tlfc interest of 23Gi 10s. 4d- for a year, at 5 per cerft ? A nr. 1 1 16* &! . . 109 3. What is the interest of 571 1. 13s. 9d. for* one year, at G/. per cent. ? Ans. 34 6s. Q\d. 4. What is the interest of 2J. 12s. 9-J-d. for a year, at 67. per cent. 1 Ans. 3s. %J. FEDERAL MONEY. 5. What is the interest of 468 dols. 45 cts. for one year, at 6 per cent. 1 $ cts. 468, 45 6 Ans. 28J10, 70=$28, Here I cut off the two right hand integers, which divide by 100 : but to divide federal money by 100, you need only call the dollars so many cents, and the inferior denomina- tions decimals of*a cent, and it is done. Therefore you may multiply the principal by the rate, and place the separatrix in the product, as in multiplication of federal money, and all the figures at the left of the sepa- ratrix, will be the interest in cents, arid the first figure on the right will be mills, and the others decimals of a mil!, as in the following EXAMPLES. 6. Required the interest of 135 dols. 25 cts. for a year at 6 per cent ? $ cts. 135, 25 6 Ans. 811, 50=88, 11 cts. 3m. 7. What is the interest of 19 dols. 51 cts. for one year, at 5 per cent. ? cts. 19, 51 Ans. 97, 55=97 cts. fym. 8. What is the interest of 436 dols. for one year, at 6 per cent.? 6 Ans. 2616 cfe.=$26, 16 cte 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. What is the interest of 73, 65 cents for a year, at C per cent. ? Principal 7365 cent?. (j Ans. 441,90=441-^- cts. or $4, 41 eta. 9m. 10. Required the interest of $85, 45 cts. for a year, at 7 per cent. I Cents. Principal 8515 7 Jbi*.89S; 15 cents f=$o CASE II To find the simple interest of any sum of money, for any number of years, and parts of a year. GENERAL RULE. 1st. Find the interest of the given sum for one year. 2d. Multiply the interest of one yearj>y the given number of years, and the product will be the answer for that time. 3d. If there be parts of a year, us months and days, work for tho months by the aliquot parts of a year, and for the days by the Rule of 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 you will have the interest of any ordinary sum sufficiently exact for common use ; but if thr sum be very large, you may say, As 365 days : is to the "interest of one year : : so is the given number of days : to the interest required, .SIMPLE INTEREST. ill EXAMPLES. I. What is the interest of 751. 8s. 4d. for 5 years and 2 months, at 61. per cent, per annum ? . s. d. 75 8 4 . s. d. 6 2 w0.=)4 10 6 Interest for 1 year. 5 4152 10 20 10J50~ 12 22 126 do. 5 years. 15 1 do. for two months. 6|00 23 7 7 Ans. 2. What is the interest of 64 dollars 58 cents for 3 years, 5 months, and 10 days, at 5 per cent. ? $ 64,58 5 j 322,90 nterest for 1 year in cents, per 3 [Case I. 968,70 do. for 3 years. 4 mo. i 107,63 do. for 4 months. 1 mo. V j 26,90 do. for 1 month. 10 days, ^ J 8,96 do. for 10 days. Ans. 1112,19=1112cfe. or $ 11, 12c. l^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, per 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. Of 3257. 12s. 3d. for five years, at 6 per cent. ? Ans. 97 135. Sd. 7. Of 174/. 10s. 6d. for 3 and a half years, at 6 per cent.? Ans. 36 13*. 8. Of 150/. 16s. 8d. for 4 vears and 7 months, at 6 per ? Aw. 4!9*.7ff 1J2 COMMISSION. 9. Of 1 dollar for 12 years, at 5 per cent.? Ans. 60 cte. 10. Of 215 dollars 34 cts. for 4 arid a half years, at 3 and a half per cent. Ans. $33, 91 cts. 6m. 11. What is the amount of 3*24 dollars 61 cents for 5 years and 5 months, at 6 per cent. ? Ans. $430, 10 cts. S^m. 12. What will 30007. amount to in 12 years and 10 months, at 6 per cent. ? Ans. 5310. 13. What is the interest of 2577. 5s. Id. for 1 year and 3 quarters, at 4 per cent. ? Ans. 18 Os. Id. 3qrs. 14. What is the interest of 279 dollars 87 cents for 2 vears and a half, at 7 per cent, per annum ? Ans. $48, 97c*s. 1\m. 15. What will 279Z. 13s. 8d. amount to in 3 years and a half, at 5 per cent, per annum? Ans. 331 Is. 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, 9l cts. 17. What will 730 dols. amount to at 6 per cent, in 5 years, 7 months, and 12 days, or ---/j of a year ? Ans. $975, 99 cts. 18. What is the interest of 1825Z. 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 cal- culate Commission, Brokerage, Ensurance, or any thing else estimated at a rate per cent. COMMISSION, IS an allowance of so much per cent., to a factor or cor- respondent abroad, for buying and selling goods for his em- ployer. EXAMPLES. 1. What will the commission of 843/. 10s, come to at 5 per rf^nt. ? EAGE. . f. Or thus, 843 10 * 5 5 is aV)843 10 42| 17 10 -4ns. 42 3 6 20 3150 12 6[00 42 3*. 6cL 2. Required the commission on 964 dols. 90 cts. at 2} per cent. 1 Ans. $21 , 71 cts. 3. What may a factor demand on 1^ per cent, commis-* sion for laying out 3568 dollars 1 Ans. 62, 44c?s. BROKERAGE, IS an allowance of so much per cent, to persons assist- ing merchants, or factors, in purchasing or selling goods. EXAMPLES. 1. What is the brokerage of 750J. 8s. 4d. at 6s. 8d. pel- cent. ? s. d. 750 8 4 Here I first find the brokerage at 1 pound I per cent, and then for the given rate* which is of a pound. 7,50 8 4 20 ; 6-. d. . s. d. qrs. 6 8=i)7 10 1 10,08 12 Ans. 2 10 1-J- 1,00 2. What is the brokerage upon 4125 dols. at or 75 cents per cent. 1 Ans. 30, 93 cts. 7 m. 3. If a broker sell goods to the amount of 5000 dollars, what is his demand at 65 cts. per cent. 1 Ans. 32. 50 ct*. K rt : yl4 X, L 4. What may a broker demand, when he sells goods to the value of 508/. 17s. lOd. and I allow him U per cent. ? Ans. 1 12s. Sd. 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, 2 6 9 3. Required the interest of 94?. 7s. 6d. for one year, five months and a half, at 6 per cent, per annum ? Ans. 8 55. Id. 3,5grs. 4. What is the interest of 12/. 18s. for one third of a month, at 6 per cent. ? H. FOR FEDERAL MONEY. RULE. 1. Divide the principal by 2, placing the separatrix as usual, ami the quotient will be the interest for one month in cents, and deci- mals of a cent ; that is, the figures at the left of the separatrix will cents, and those on the right, decimals of a cent. 2. Multiply the interest of on month by the given number of months, or months and decimal parts thereof, or for tho days take the even parts of a momth. &c. 1 1(3 SHORT I'RACTICL RULES EXAMPLES. 1. What is the interest of 341 dols. 52 cts. for 7 months 2)341,52 Or thus, 170,76 Int. for 1 month. 170,76 Int. for 1 month. X 7,5 months. 85380 1 195,32 do. for 7 mo. 1 19532 85,38 do. for ^ mo. $ cts. m. 1280,700cfc. = 12,80 7 1280,70 Ans. 12SO,7c>s.-=$12, SQcts. 7m. 2. Required the interest of 10 dols. 44 cts. for 3 years, 5 months, and 10 days. 2)}0,44 10 days ) 5,22 interest for 1 month. 41 months. 5,22 208,8 214,02 ditto for 41 months. 1,74 ditto for 10 days. 215,76 cts. Ans. =$2, 15 cts. 7 m.+ 3. What is the interest of 342 dollars for 11 months'? The is 171 interest for one month. 11 Ans. 1881 cte.=$18, 81 cts. NOTE. To find the interest of any sum for two months, at 6 per cent, you need only call the dollars so many cents, and the inferior denominations decimals of a cent, arid 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. 4 m. &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* in cent* and per cent, and there were pay- ments endorsed upon it as follows, viz. First payment, 148 dollars, May 7, 1794. Second payment, 341 dols. August 17, 179(5. Third payment, 99 dols. Jan. 2, 1798. I demand how much 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 1 184, 50 amount 341, 00 second payment, Aug. 17, 1796. Yr. mo. 37, 51 interest to June 17, 1798. 1 10 378, 51 amount. "vied over."j SHORT raACriCAL $ cts. 99, 00 third payment, January 2, 17SB. 2, 72 interest to June 17, 1798.= 5J 101, 72 amount. 184, 50 several amounts. 184, 50 J 378, 51 > 101, 72) 664, 73 total amount of payments. 675, 00 note, dated April 17, 1793. Yr. mo. 209, 25 interest to June 17, 1798. 5 2 884, 25 amount of the note. 6154, 73 amount of payments. $219, 52 remains due on the note, June 17, 1798. 2. On the 16th January, 1795, 1 lent James Paywell 500 dollars, on interest at 6 per cent, which I received back in the following partial payments, as under, viz. 1st of April, 1796 * - $ 50 16th of July, 1797 - 400 1st of Sept'. 1798 60 How stands the balance between us, on the 16th Novem- ber, 1800 1 Ans. due to me, $63, 18 cts. 3. A PROMISSORY NOTE, VIZ. 62105. New-London, April 4, 1797. On demand, I promise to pay Timothy Careful, sixty-two pounds, ten shillings, and interest at 6 per cent, per annum, till paid ; value received. JOHN STANBY, PETER PAYWELL, RICHARD TESTIS. Endorsements. . s. 1st. Received in part of the above note, . September 4, 1799, 50 And payment June 4, 1800, 12 10 How much remains due on said note, the 4th day of De- cember, 180CX . s. d. 4ns. 9 12 G FoR oALuL-LATKvu IXTERJjST. 1:21 % . f NOTEU T^hfi preceding Rule r by- c"us.to.th;, is r$ hcferefl so* p*qpular and sx> much: practised and esteemed by* irra'ny QA account of its being simple and concise, that I have- given it a place : it may answer for short periods of time, hut in. a long course of years., it will be found to- be very errone- ous. Although this method seems at first view to be upon the ground of simple interest, yet upon a little attention tire following objection will be found most clearly to lie against it viz. that the interest will, in a course of years, complete- ly expunge, or as it may be said, eat up the de.hr. For an explanation of this^ take the following A lends B lOi) dollars-, at C per cht. interest, and takes his note of hand ; B does no more tlraa pay A at eXery };eitr*s end 6 dollars, (which is then Justly 1 due 1 toi B -fop tlie use 6f his money) and has it endorsed an his noev At the and of 10 year-s B takes up his note, and the sum he lias to jtny is reckoned thus : The principal 100 dollars, art itite- rei 10 year's amounts to 160 dollars ; there are nine* en- dorjaements of G dollars ench, upon which the debtor claims fteresj: ; one for nine years, the second for S years, the third for 7 years, and so down to the time of settletnejtt^ the whole amount of the several endorsements and theifin- tei*est, (as any one can stse by casting it) is $70, 20 cts. thir 4 years, at 6 per cent, per annum ? Ans. $131,238+ 6. What will 1000 dollars amount to in 4 years, at 7 per cent, per annum, compound interest? Ans. $1310, 79 cfe6.m -f 7. What is tile amount of 750 dollars for 4 years-, at 6 per cent, per annum, compound interest? Ans. 1946, 85 ct$. 7,72 #h 8. What is the compound interest of 876 dols. 90 cents for'3J- years, at 6 per cent, per annum? t BS. V-:.; {'& EXAMPLES. 1. It* an annuity of 707. be forborne o y.ears, what will be due for the principal and interest at the end of said term, simple interest being computed at 5 per cent, per annum? Yr. . s. ^Hfr- Interest of 70/. at 5 per ce'n't. for 1 3 10 at the giveji rau* 2 7 deb*. 310 LO 4 14 Q RULE. As the amount of 1007. or 1.00 dollars, at ..~_ ft ami time : is to the interest of 100, at the sam'e rjate and time : . :? tfte given sum : to the discount- Subtract the discount from the given sum., and {lie -remainder is the present worth. Or as the amount of 100 : is to. IjOO : : so is tluj given sum or debt : to the present worth. PROOF. Find the amount of the present worth, at tire given rate and time, and if the work is right, tfra't will be equal t the ^fiven sum. KXA&PLES. 1.. What must be discounted for the fendy payment of 100 d'attars, due a year hence at 6 per cent, a year ? $ S $ cts. Afe 1QG : 6 : : 100 : 5 66 the answer, 100,00 given sum* 5jG6 discount. $94,34 the present worth*. & "What sum in ready money \vill discharge a debi of . due 1 year and 8 months b<5npe 4 at 6 TKT cent. ? 100 * 10 interest for 20 months.. 110 Am't ^. .. . v s. ,n+ RULE. FTnd the interest TQF a year *- T - inent of trre whole debt ? 100 x 6 = 600 120 x 7-840 160 x 10 = 1GOO 380 }3040(8 months; An*. 2. A merchant hath owing him 800J. to be paid as &!- laws : 507. at 2 months, 100/. at 5 months, and the rest at 8 months ; and it is agreed to make one payment of the vvtiole : I demand the equated time ? Ans. 6 months-. 3. P owBsH 1000 dollars, whereof 200 dollars is to be jpaid present, 400 dollars at 5 months, and the rest at 1*5 months, but they agree to make one payment of the whole; I demand when that time must be 1 Ans. S months-. 4. A i&erclaant has due to him a certain sum ofmorrey, to he paid one sixth at % months, one third at 3 months, and the rest at 6 months ; "what is the equated time for the jgrayment of the whole ? Ails. 4| months. BARTER, JS the exfchaaging of one commodity fop another, and iftrjeets merchants and traders how to make tUe exfchaorge without loss to either party. RtiLE. Find the value of the Commodity whose tfuattUty is given ; then find what quantity of the fcthep at the g^posed -rate can, be bought for the same money, and it gives the,answer>. 127 EXAMPLES, 1. \Vl*at quantity of flax at 9 cts. per Ib. must be giveji in barter for 12 Ib. of indigo, at 2 dols. 19 cents per Ib. ? 12 Ib. of indigo at 2 dols. 19 Cts. per Ib. comes to 28 rfols. 28 cts. therefore, As 9 cts. : 1 Ib. : : 26,28 cts. '. 292 the answer. 2. How much wheat at 1 dol. 25 cts. a bushel, must he given in barter for 50 bushels of lye, at 70 cts. a bushel 1 Ans. 28 bushels. 3. How much rice at 28s. per cwt. must be bartered fq'r TS the crtffil on ea'ch pound 1 1 An*. itUrf. 5. B'oirgla a hhd. of molasses containing 119 gallons,, at 52 cents per gallon ; paid for carting the same 1 dollar 23 cents, and by accident 9 gallons leaked out ; at what rate mfrst I sell the remainder per gallon, to gain 13 in the whole ? Ans. 69 cts. IL To know what is gained or lost per cent. RULE. First see what the gain or loss is by subtraction ; then, Aa tB"p price it coat : is to the gain or loss^: : so is 100/. or $100, to this gain or loss per cent. EXAMPLES. I. If I buy Irish linen at 2s. per yard, and sell it again at #s. 8d. per yard ; what do I gain per cent, or in layin-g t}UJt 100/. : As : 2s. 8d. : : IOO/. : 33 65. Qd. Ans. 2-. If I buy broadcloth at 3 dols. 44 cts. per yard, and sell ft again at 4 dols. 30 cts. per yard : what do I gain per ct. or in laying out 100 dollars 1 $ cts.^} Sold for 4, 30 $ cts. cts. -$ $ Cost 3, 44 > As 3 44 : 86 : : 100 ; 25 Am. 25 per cent. {Jainerd per yd. 86 j 3. If I buy a cvvt. of cotton for 34 dols. 86 cts. and sell it again at 41 cts. per Ib. what do I gain or lose, and what j>er cent. ? $ cts. 1 cwt. at 41^ cts. per Ib. comes to 46,48 Prime co,st 34,86 Gained in the gross, $11,61 As 34,86 : 11,62 : : 100 : 33J An*. 33J per ccjit. 4. Bought sugar at 8{d. per Ib. and sold it again at 4?. 17s. per cwt. what did I gain per cent, t Ans. 25 19s. 5?, At Hd. prtffft in a sliflfing, how much per'cenk ? Ate. 12 JO.?. 7- At 25 ets. profit in a dollar, how muesli p'er cent t NOTE. When goods are bought or sold on credit, you must calculate (by discount) the present worth of fhfifr prire, in order to find your true gain or loss, &c. EXAMPLES. 1. Bought 164 yards of broadcloth, at 14s. Gd. per yard ready money, and sold the same again for 154/. IQs. on G months credit ; what did I gain by the whole ; allowing discount at G per cent, a year ] . . . s. . s. As 103 : 100 : : 154 10 : 150 present worth. 118 18 prime cqsK Gained 31 2 Answer. & If I buy cloth at 4 dols. 1G cts. per yard, on eigljt months credit, and sell it again at 3 dols. 90 cts. per y.d. ready money, what do I lose per cent, allowing 6 per cent. iTiscount on the purchase price '( -4ns. 2^ per ccnf f III. To know how a commodity must be sjold, to gain Or lose so much per cent. RULE. As 100 : is to the purchase price : : so is 100J. or 100 dollars, with the profit, added, or loss subtracted : to Ove selling prite. EXAMPLES. 1. If I buy Irish linen at 2s. 3d. per yard ; how must I stjl it per yard to gain 25 per cent. ? As 100Z. : 2s. 3d. : : 125/. to 2s. 9rf. 3 qrs. Any. 2. If I buy rum at 1 dol. 5 cts. per gallon ; how must I &} it per gallon to gain 30 per cent. ? As 100 : $1,05 : : $130 : $l,36c&. A~s 8. If tea cost 54 cts. per Ib. ; how must it be sold per Ibt to lose 12^ per cent. 1 As $100 : 54 cts. : : $87, 50 cts. : 47 cts. 2 m. An$. 4. Bought cloth at 17s. Gd. per yard, which not proving so good as I expected, I am obliged to lose 15 percent. bV it ; how mirst I s^ll it er yard ? AT*. 14*. 5. Il'l'l cjy*. 1 qr. 25 Ib. of sugar co'&f Ii6 dok 50 c; how must it he sold per Ib. to aiu 30 per cent, t Ans. 12 cte. &#. G. Bought 90 gallons of wine at 1 dol. 20 cts. per gnll. hut hy accident 10 gallons leaked out ; at what rate mast I sell the remainder per gallon to gain upon the whole prime co,st> at the rate of 12 J per cent. I Ans. 1, 51 cts. S^m^ IV. When there is gained or lost pel* cent, to wfmt the commodity cost. RULE. As 100J. or 100 dole, with the gain per cent, added, r los? per cent, subtracted, is lo the price, so is 100 to the prime cost. EXAMPLES. li 'IF a yard of cloth be sold at 14s. 7d. and there is gaiir- eti 1W . 13,s. 4d. per cent. ; what did the yard cost I . s. d. s. d. . As 116 13 4 : 14 7: : 100 to 12*. 6d. Aos. 2. By selling broadcloth at 3 dols. 25 cts. per yard, I Ipse at the rate of 20 per cent. ; what is the prime cost of sakl cloth per yard ? Ans. $4, 06 cts. 2jz?/. 3. If 40 Ib. of chocolate be sold at 25 cts. per Ib. and I gain 9 per cent. ; what did the whole cost me ? Ans. $9, 17 cts. 4w.-f 4. Bought 5 cwt. of sugar, and sold it again at 12 cents pfcr Ib. by which I gained at the rate of 25 per ceufr. J wllat did the sugar cost me pc# cwt. 1 Ans. $10, TOcte. 9m<+ V. If by wares sold at a given rate, there is so much grained or lost per cent, to know what would be gained ol- lost per cent, if sold at another rate. RULE. As the first price : is to 100/. or 100 dols. with the profit- per cent, added, or loss per cent, subtracted : : so is the other price to the gain or loss per cent, at the other rate. $. B. If your answer exceed 1007. or 100 dols. tlra excess is your gain per cent. ; but if it be less tlran IOD, that (Jkfici;en<5y is tire loss per cent. 13J. x . . EXAMPLES'. 1. If I seJl cloth at is. per yd. and thereby 1*4111 15 per t. what shall I gain per cent, if I ssll it at (Js, ppr yd, S $. s. . As 5 : 115 : : 6 : 138 Ans. gamed 38ptr ccjit, 2. If I retail rum at 1 dollar 50 cents per gallon, a*n As 288 : 19* : : Sum 288 3 I Proof 19 . Two merchants traded in company; A put in 215 dols. for 6 months, and B 390 dols. for 9 months, but by misfortune they lose 200 dols. ; how must they share tht? loss ? Ans. A's lass #53, 75 cts. JB's 1146, 25 cts. 3. Three persons had received 665 dols. interest: A had pat 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 110/. 12s. : A's stock 120/. 105. for 4 months, and B's 200Z. for 61 months ; \vhat is each man's part of the gain 1 Ans. A's part 29 18s.3id.}^ff- B's 80 iSs.Sld.-fffe 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 do.l- lajs ; what is each man's share of the gain 1 A i #324, 07 4 A's share. ' \ #375,92 5 -B's do. 0. 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 wirt\ A at the year's end I Mo. g Mo. $ As 12 : 1090 : J 8 ; 1000 x 12m 1500 Am* JJ-OUBLE RULE OT T DOUBLE RULE OF THREE, THE Double Rule of Three teaches to resolve at onc swch questions as require two or more statings in simpha 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 hrst two 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, &jc k in the second place ; and the remaining term in the third place. Place the terms of demand, under those of the same kind iu the supposition. If the blank place, or term sought, fall un- der the third term, the proportion is direct ; then multiply the i'frst and second terms together for a divisor, and the other three for a dividend : hut 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 vrdcnd, and the quotient will be the answer. EXAMPLES. 1. If 7 men can build 36 rods of wall in 3 days ; how many rods can 20 men build in 14 days 1 7 : 3 : : 36 Terms of supposition 20 : 14 Terms of demand> 36 * 42 .// c 504 20 7 X 3=21)10080(480 rods. Ans, 2. If 100Z. principal will gain 6Z. interest in what will 400/. gain in 7 months ? Principal 1007. : 12 ma. : : 6/-. interest. 400 : 7 Jtos, 3. If 100?. will gain 61 a year ; in what time will 4W. gain 14/. mo. 100 : 1-2 : : 6 400 : : : 14 Ans. 7 months. 4. If 400/. gain 14/. 111 7 months : what is the rate pef Cent, per annum 1 . mo. Int. 400 : 7 : : 14 100 : 12 Ans. 6. 3. What principal at 6/. per cent, per annum, will give 147. in 7 months 1 . mo. Int. 100 : 1*2 : : 6 7 : : 14 Ans. 400. 6. An usurer put out 86?. to receive interest for the same ; and when it had continued 8 months, he received principal and interest, SSL 17s. 4d. ; I demand at what rate per ct. per ami. he received interest? Ans. 5 per cent. 7. If 20 bushels of wheat are sufficient for a family at* 8 persons 5 months, how much will be sufficient for 4 per* 'sons 12 months 7 Ans. 24 bushels. 8. If 30 men perform a piece of work in 20 days ; how many men will accomplish another piece of work 4 time's ;vs large in a fifth part of the time ? 80 : 20 : : 1 4 : : 4 Ans. COO. 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 xjwt. 2 qrs. 25 Ib. 64 miles, at the same rate 1 Ans.$U,QScts. 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 1 8 : 12 : : 20x6x4 24 : 200 x 8 x 6 80 days. Ans. CONJOINED PROPORTION, IS when the coins, weights or measures of several coun- tries are compared in the same question ; or it is joining ntany proportions together, and by the relation wh&h C OX JOINED PROPOft T JO X'. several antecedents have to their consequents, the prop^iv i'ion between the first antecedent and the last consequent is discovered, as well as the proportion between the others in their several respects. NOTE. This rule may generally he abridged by can- celling equal quantities, or terms that happen to be the same in both columns : and it may be proved by as many statings in the Single Rule of* Three as the nature of the question may require. CASE I. When it is required to find how many of the first st)rt of coin, weight or measure, mentioned in the question, are equal tq a given quantity of the last. RULE. Place the numbers alternately, beginning at the left hand, and let the last number stand on the left hand column ; then multi- ply 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 Ib. Flemish, and 19 Ih. Flemish 25 lb. at Bologna ; how many pounds English are equal to 50 lb. at Bologna? * Ib. lb. 100 Eng.=95 Flemish. 19 Fie. =25 Bologna. 50 Bologna. Then 95 X 25==2375 the divistm S5000 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 Ih. at New- York are equal to 144 lb. at Leghorn ? Ans. 100/6. 3. If 70 braces at Venice be equal to 75 braces at.Lteg- Iiorn, and 7 braces at Leghorn be equal to 4 American yards ; how many braces at Venice are equal to 64 Ameri- "cao yards? Ans. 104^. CASE II. When it is required to find how many of the last sort of ct)in, weight or measure, mentioned in the question-, are equal to a given quantity of the uLv Place the numbers alternately, beginning at the left hantfc, Und let the last number stand on the right hand ; then multiply Cre first row for a divisor, and the second for a dividend. EXAMPLES. 1. If 24 Ib. at New-London make 20 Ib. at Amsterdam-, and 50 Ib. at. Amsterdam 60 Ib. at Paris ; how many at Paris are equal to 40 at New-London ? Left. Right. 24 = 20 20 x 60 x 40 = 48000 50 = 60 = 40 Ans. 40 24 x 50 = 1200 &. If 50 Ib. at New-York make 45 at Amsterdam, and 80 Ib. at Amsterdam make 103 at Dantzic ; how many Ib* at Dantzic are equal to 240 at N. York ? Ans. 278 T V 3. If 20 braces at Leghorn be equal to 1 1 vares at Lis- bon, and 40 vares at Lisbon to 80 braces at Lucca ; how jriany braces at Lucca are equal to 100 braces at Leghorn '{ Ans. 110. EXCHANGE. BY this rule merchants know what sum of money onght to be received in one country, for any sum of different spe- cie paid in another, according to the given course of ex- change. To reduce the moneys of foreign nations to that of tire United States, you may consult the following TABLE : Showing 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 lavre of France, 18 1 Guilder or Florin of the U. Netherlands, 39 * Mark Banco of Hamburgh, 3 tfrately after reduction of whole numbers, and given somd general definitions, and a, few such problems therein as \yere necessary to prepare and lead the scholar immediate- ly to decimals ; the learner is therefore requested to nrotl those general definitions in page 69. Vulgar Fractions are either proper, improper, single* Compound, or mixed. 1. A single, simple, or proper fraction, is when the nu> tnerator is less than the denominator, as , J, f , --, &.- 2. An Improper Fraction, \s when the numerator ex-*- tieexls tlie denominator, as 3, J, ^ , &c. 3. A Compound Fraction, is the fraction of a fraction, Coupled by the word of, thus, -*- of T l jt | of of J, &c. 4. A fifixed Number, is composed of a whole number nntf a fraction, thus, 8|, 14 ^, &c. 5. Aii3 r whole number may be expressed like a fractrorc by dravving a line under it, and putting 1 for denominator,, thus, 8==f, and 12 thus, y, &c. 6. The common measure of two or more numbers, is that number which will divide each of them without a re- mainder ; thus, 3 is the common measure of 12, 24, and 3Q.J and the greatest number which will do this is called tire greatest common measure. 7. A' number, which can be measured by two or more numbers, is called their common multiple : and if it be the least number that can be so measured, it is called the least common multiple : thus 24 is the common multiple 2, 3 and 4 ; but their least common multiple is 12. To fmdjihe least common multiple of two or more num- bers. RUT E. 1. Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients, tog.e^ ther with the undivided numbers, in a line beneath. 2. Divide the second liner-' as bsforo, and so on till there are no two- numbers that can be divided ; then the continued product of tire di-. vistfrs and quotients, will give the multiple remii: 144 Bj^uroiro'ft OP vo- 1. What is the least common multiple of 4, S, 6 and 10 1 X5)4 5 6 10 X2)4 1 6 Q X2 1x3 1 5x2x2x3 m An A S. What is the common multiple of 6 and SI Ans. 24. 3. What is the least number that 3, 5, 8 and 12 will measure ? ^ 7W . 130. 4. What is the least number that can be divided by tht? 9 digits separately, without a remainder 2 Ans. 252I>. DEDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form into another, in o*iv cfer to prepare them for the operation of Addition, Sub- traction, &c. CASE I. To abbreviate or reduce fractions to their lowest-teTms*. RULE. 1. Find a common measure, by dividing the greater terjrt by the les, and this divisor by the remainder, and so on, always di-f viding the last divisor by the last remainder, till nothing remains^; the laft divisor is the common measure.* 2. Divide both of the terms of the fraction by the common nroa- sure, and the quotients will make the fraction required. * To find the greatest common measure of more than two numbers, you, must find the greatest corr.mon measure of two of them as per rule above j 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 ccimiroir mea- sure last tfwiM he the GN OF VULGAR FRACT.IQN3. 145 OK, it* 3^1 chogp^, yo,u may take th&t etiy mtho,d in Brobfgth I. age 62.) EXAMPLES; 1. Reduce ff to its lowest terms. 48)f|(i " Operation. sT\48/6 common measure, 8)f$=v$ Ans. /tfv Tlcpi OfiT **y/- %. Reduce $f to its lowest terms. Ans. & 3. Reduce || to its lowest terms. Ans. i| 4. Reduce -f Jg-f to its lowest terms. Ans. % CASE II. To reduce a mixed number to its equivalent improper fraction. RULE. Multiply the whole number by the denominator o'f thfe gi- ven fraction, and to the product add the numerator, this surti written above the denominator will form the fraction 1. Reduce 45 J to its equivalent improper fraction. 2. Reduce 19}| to its equivalent improper fraction. Ans. \*/ 3. Reduce l^yW to an improper fraction. Ans. t-V^ 8 4. Reduce 6lf| to its equivalent improper fraction. Ans. ff|5 CASE III. To find the value of an improper fraction. RULE. Divide the numerator by the denominator, and the qur? tient will be the value sought. EXAMPLES. ANSWERS, 1. Find the value of y 5)48(9f * 2. Find the value of 3 T y !9}| 3. Find the value of 9 T 3 T 3 4. Fifid the value of 2 |S 5 5. Find the value of V N 146 REDUCTION OF VULGAR CASE IV. To reduce a whole number to an equivalent fraction^ Irav - 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, 7x9=63, and V the Ans. 2. Reduce 18 to a fraction whose denominator shall be 12. Ans. a -f-f 3. Reduce 100 to its equivalent fraction, having 90 fbr a .denominator. Ans. 9 fJ= 9 J= 1 J CASE V. To reduce a compound fraction to a simple one of equal value. RULE. 1. Reduce all whole and mixed numbers to their equiva- lent fractions. 2. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator ; and they will f fraction required. EXAMPLES. 1. Reduce J- off of of T Vto a simple fraction. 1x2x3x4 2x3x4x10 2. Reduce f of of f to a single fraction. Ans. 3. Reduce f of ji of ^f to a single fraction. Ans. 4. Reduce f of | of 8 to a simple fraction. Ans. =3 5. Reduce | of Jf of 42 to a simple fraction. Ans. 'fjr NOTE. If the denominator of any member of a com- pound fracti'on be equal to the numerator of another mem- REDUCTION Or VULGAR FRACTIONS. 147 ber thereof, they may both be expunged, and the other members continually multiplied (as by the rule) will pro- duce the fraction required in lower terms. (3. Reduce f off off to a simple fraction. Thus 2x5 =*=& An*. 4X7 7. Reduce -J off off of {4 to a simple fraction. Ans.^=-ll CASE VI. To 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 seve- ral ire\v numerators will give the fractions required. EXAMPLES. 1. Reduce |-, f , J, to equivalent fractions, having a coni,- irron denominator. -J + -| -f -4 24 common denominator. 1 2 3 % x3 2 3 3 4 :<4 4 2 12 !&: 18 new numerators. 24 24 24 denominators. 2. Reduce f, -,%, and |4, to a common denominator. 3. Reduce |, , f , and |, to 'a common denominator. Ans, f f . 4ff , Iff, flu* 4'8 REDUCTION F V U LG AR FR At T iUX b.. 4. Reduce -, -? , and T \, to a common denominator^ 800 300" 400 1000 1000 1000 & Reduce J, > and 12^ 3 tp a common denominator. ^, , W- 6. Reduce f, f, and of |i, to a common denominator. ' The fore^in^ is a gemeral rule for reducing fractions to a comroou denominator ; but as it will save much labour to keep the fractions in the lowest terms possible, the follow- ing Rule is much preferable. RULE II. For reducing fractions to the least common denominator. (By Rule, page 143) find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required, in which divide each particular denominator, and multiply the quotient by its own numerator, for a new numerator, and the new nume- rators being placed over the common denominator, will ex- press the fractions required in their lowest terms. EXAMPLES. 1. Reduce ^ f ,and | , to their iftsf common denominator, 4)2 4 8 111 4x=r8 the least com. denominator. S 2x1=4 the 1st numerator. .84x3=6 the 2d numerator. 8 $X5=5 the 3d numerator, numbers placed over the denominator, give (he tmswer f- , f , f , equal in value, and in much lower terms than the general Rule would produce ff, f f , f . 2i ll-editee f , f , and T V, to their least common denomina- tor. An*. U, 4f , 4ft. REDLTCtLQ.V OF VULGAIi FKA.C.Tl' 149 tf. Reduce i f f and -^ to their least common denomi- nator. Ans. -J3- -/ T if J-f 4. Reduce I J |- and ^ to their least common denomi- nator. Ans. & If }|- W CASE VII. To Reduce the fraction of one denomination to the frac- tion of another, retaining the same value. Reduce the given fraction to such a compound one, as will express the value of the given fraction, by comparing it with all the denominations between it and that denomi- nation you would reduce it to ; lastly, reduce this com- pound fraction to a single one, by Case Y. EXAMPLES. 1. Reduce | of a penny to the fraction of a pound* By comparing it, it becomes f of V V of ^- - of a poun'd. 5 x 1 x 1 5 6x 12x20 1440 2. Reduce TT 5 T ^ of a pound to the fraction of a penny. Compared thus T / T77 of 2 T of yd. Then 5 x 20 x 12 1440 1 1 3. Reduce ^ of a farthing to the fraction of a shilling. Ans. ij 4. Reduce f of a shilling to the fraction of a pound. Ans. 7 f -^y Q 5. Reduce 4 of a pwt. to the fraction of a pound troy. AllS. T/Jo- 3"3F 6. Reduce f of a pound avoirdupois to the fraction of a cwt. 7. What part of a pound avoirdupois is T J F of a cwt. Compounded thus T | off of V=iW =# 8. What part of an hour is ^ of a week. 60 RKDl'i.TIuN OF VULGAR i;HACT 1O 9. Reduce f of a pint to the fraction of a hhd. Ans. ^- 2 - 10. Rediice J of a pound to the fraction of a guinea. Compounded thus, J of 2 T of ^V S>==: T 11. Express 5J- furlongs in the fraction of a mile. Thus 5J=V of i=H 12. Reduce f of ail English crown, at 6s. 8d. to the fjjac- t?0n of a guinea at 28s. Ans. - T o/ rt guinea. CASE Vllt. T!o find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &e. RULE. Multiply the numerator hy the parts in the next inferior denomination, and divide the product by the denominator ; $nd if any trthig remains, multiply it by the next inferior de* nomination, and divide by the denominator as before, and so on as far as necessary, and the quotient will be the answer. NOTE. This and the following Case are the same with ^Problems II. and III. pages TO and 71 ; but for the scho- lar^s exercise, I shall give a few more examples in each. EXAMPLES. 1. What is the value of f 11 of a p6und ? Ans. 8s. 9d. . Ifind tlie value of of a c\vt. Ans. 3 qrs. 3 Ib. 1 oz.Vty dr. &. Bind the value off of 3s. 6d. Ans. &s. 4. B^w imrch is y 6 ^- of a pound avoirdupois ? Ans. 7 ov. 10 d'r, &. How mucfe is f of a hhd. of wine ? Ans. 6. What is the value of |f of a dollar ? Ans. 5*5. 7). 'What is the value of T \ of a guinea ? Arts. tSs\ ADDITION OF VULGAR F.RACTIONS. 151 8. Required the value of J-J of a pound apothecaries. Ans. 2 oz. 3 grs. 9. How much is | of 57. 9s. 1 Ans. 4 13s. 5|rf. 10. How much is -- of f of J of a hhd. of wine t -4^5. 15 gals. 3 ??fc. CASE IX. To reduce any given quantity to the fraction of any greater denomination of the same kind. [See the Rule in Problem III. page 71.] EXAMPLES FOR EXERCISE. 1. Reduce 12 Ib. 3 oz. to the fraction of a cwt. 4 K *lWjf 2. Reduce 13 cwt. 3 qrs. 20 Ib. to the fraction of a ton. Ans. || 3. Reduce 16s. to the fraction of a guinea. Ans. -| 4. Reduce 1 hhd. 4$ gals, of wine to the fraction of a tun. t Ans. % 5. What part of 4 cwt. 1 qr. 24 Ib. is 3 cwt. 3 qrs. 17 Ib. 8 oz. ? Ans. I ADDITION OF VULGAR FRACTIONS. RULE. Reduce compound fractions to single ones ; mixed num- bers to improper fractions ; and all of them to their least common denominator, (by Case VI. Rule II.) then the sum of the numerators written over the common denominator ;II1 be the sum f>f the fractions required. EXAMPLES. 1. Aild Bj- J and f off together. 5=y and f of f J| Then y , J, 4J reduced to their least common denominator by Ca'se VI. Rule If. will become W, f* H Then 13^-f- I8-V 14r= W =0,14 or G% Ans\ 152 ADDITION OF VULGAR 2. Add |, , and $ together. ANSWERS. 1J 3. Add i, J, and f together. 1 1 4- Add l2i 3f and 4 together. 20 J4 5. Add J of 95 and $ of 14 J- together. 44j| NOTE 1. In adding mixed numbers that are not com- pounded with other fractions, you may first find the sum of the fractions, to which add the whole numbers of the given mixed numbers. 6- Find the sum of 5J, 7f and 15. I find the sain of J and to be Then lll+5+7 7. Add f and 17^ together. ANSWERS. 17-^- 8. Add 25, 8} and -J- of | of -/ 33^ NOTE 2. To add fractions of money, weight, &c. reduce fractions of different integers to those of the same. Qr, if you please, you may find the value of each fraction e V T II1. in Reduction, and then add tliem in their yroper terms. 9. Add -\ of a shilling to f of a pound, 1st method 2d method. 4 of *V=T* T - ^=7s. Cd. Oqrs. Then T H+f- T 4 f^. 1 ^-=0 6 3f Whole value by Case VIII. is 8s. Od. 3f qrs. Arts. Ans. 8 3 J By Case VIII. Reduction'. 10. Add | Ib. Ti?oy, to | of a pwt. Ans. 7o 11. Add 4 of a ton, to T ^ of a cwt. Ans. 12 cz^. 1 gr. 8 12. Add f of a mile to T \ of a furlong. Ans. 13. Add ^ of a yard, of a foot, and J- of atiib together. ^is. 1540 y^s. g ^f. 9 in. 14. Add J of a week, of a day, of An hour, and of fe. 2 ^/, 2 7w* 3f) ??F^ 45 5ert SUBTRACTION OF VUL.GAR FRACTIONS. 163 SUBTRACTION OF VULGAR FRACTIONS. RULE.* Ptepare the fraction as in Addition, and the difference of the numerators written above the eommon denominator, will give Ihe difference of the fraction required. EXAMPLES 1. From $ take f of | of J=M=T 7 - Then a an d T \=rV T V Therefore 9 7=f f =-J */< -4ns. 2. From |f take 4 /Insiders. Ji 3. From }-J take T % ^VV 4. From i 4 take ] f 13 r^ 5. What is the difference of T \ and |-f ? ir'H 6. What differs T V from -J 1 T W 7. From 14^ take | of 19 1 T V 8. From f i take i]i remains. 9. From |4 of a pound, take \ of a shilling. of i_= r j 7r . Then from }i. take T |o^. ^W5. f|. NOTE. In fractious of money, weight, &c. you may, if you please, find the value of the given fractions (by Case VIII. in Reduction) and then subtract them in their proper terms. 10. From T V&. take 3J shillings. -4ns. 5s. (yd. 2| qrs. 11. From f of an oz. take J of a pwt. A/i5. 11^?^^^. 3^v. 13w From ^ of a cwt. take T 7 ^ of a Ib. Ans. 1 $r. 27 Ib. 6 oz. lO, 3 ^ ^- 13. From 3| weeks, take ^ of a day, and -} of f of j- of an hour. Ans.Qw. &da. 12 ho. \9min. 17^ sec. * In subtracting mixed numbers, when the lower fraction is greater than the upper one, you may, without reducing them to improper fractions, sub- tract the numerator of the lower fraction from the common denominator, and to that difference add the upper numerator, carrying one to the unit's glace 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 over tile (Denominator, then taking oiw from the whole number. 154 MULTIPLICATION, DIVISION, &Q. MULTIPLICATION OF VULGAR FRACTIONS. RULE. -\C v Reduce whole and mixed numbers to the improper frao tions, mixed fractions to simple ones, 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. EXAMPLE?. 1. Multiply | by -*- Answers. 44=-J- 2. Multiply f by ^ - 3. Multiply 5} by 4. Multiply | of 7 by 4- 3|i 5. Multiply }Jf by Vv M G. Multiply | of 8 by J of 5 7. Multiply 7 by 9} 8. Multiply f of J by of 3} f 9. What is the continued product of J of f , 7, 5J aricL J of ? u4/zs. 4 pV DIVISION OF VULGAR FRACTIONS- RULE. Prepare the fractions as before ; then, invert the divisor and proceed exactly as in Multiplication : The products will be the quotient required. EXAMPLES; 4x5 1. Divide -f by J Thus, =|f Ans. 3x7 3v Divide ^ by J ^?w?^r5. 1 T % 3. Divide of | by 4 What is the quotient of 17 by f ? 59^ 5. Divide 5 by & 6. Divide i of?, of f by | of J 3^ 7. Divide 4f r by ' of 4 2.V 8. Divide 71 by 127 &V 9. Divide 52054- bv 4 of 91 RULE OF THREE DIKECT, INVERSE, &,d. 155 RULE OP THREE DIRECT IN VULGAR FRACTIONS. RULE. Prepare the fractions as before, then state your questkw 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. Iff of a yard cost ~ of a pound, what will ^ of an Ell English cost ? yd.=f off of =! or Ell English. Ell . EU. s. d. yrs. As : -f : : -r And \ x % x rV^fVi^-^0 3 H An *- 2. If f of a yard cost J of a pound, what will 40 yards come to 1 Ans. 59 8s. 6d. 3. If 50 bushels of wheat cost 17f Z. what is it per bush- el? Ans. 7s. Qd. Iff qrs. 4. If a pistareen be worth 14 J pence, what are 100 pistar reens worth ? Ans. 6 5. A merchant sold 51 pieces of cloth, each containing 24 yards at 9s. ?id. per yard ; what did the whole amount to? Ans. 60 105. Od. 3% qrs. 6. A person having f of a vessel, sells f of his share for 312Z. ; what is the whole vessel worth ? Ans. 780 7. If | of a ship be worth f of her cargo, valued at 8000?. what is the whole ship and cargo worth ? Ans. 10031 14s. INVERSE PROPORTION. RULE. Prepare the fractions and state the question as before^ then invert the third term, and multiply all the three terms .ogether. the product will be tht? answer; fcU-LE OF TH&EE DMIBCC IN DECIMALS. EXAMPLE'S. 1. How much shalloon that is J yard wid^ t will line 5J,- yards of cloth which is ] J yard wide ? Fic/5. yr/5. yds. Yds. As 1J : 5i : : f And J x y *f =W *<>& ^ W5 - 2. If a man perform a journey in 3} days, when the day is 12J hours long ; in how many days will he do it when the day is but 9| hours ? Ans. 4 7 \ 4 j days. 3. If 13 men in 11| days, mow 21 acres, in how many days will 8 men do the same? Ans. 18ff days. 4. How much in length that is 7 inches broad, will make a square foot 1 Ans. 20 inches. 5. If 25f s. will pay for the carriage of a cwt. 145} mites ; how far may 6 cwt. he carried for the same money 1 Ans. 22^- miles. 6. How many yards of baize which is 1} yards wide, will line 18 J- yards of camblet J yard wide? Ans. 11 yds. 1 qr. 1J net. RULE OF THREE DIRECT IN DECIMALS. RULE. Reduce your fractions to decimals, and state ytfur ques- tion as in whole numbers; multiply the second and third tfl* gether ; divide by the first, and the quotient will b the an- swer, &4N EXAMPLES* 1. If I of a yard cost T T of a pound ; what will 15 yards come to ? i=,875 T \=,583+ and =,75 Yds. . Yds. . . s. d. qrs. As ,875 : ,583 : : 15,75: 10,494=10 9 10 2,24 Ans. 2. If 1 pint of wine cost 1,2s. what cost 12,5 hhds? Ans. 378 0. If 4] rafds cost 3s. 44,d. what will 30| yards cost t SIMPLE INTEREST BY DECIMALS. 1*57 4. If 1,4 cwt. of sugar cost 10 dols. 9 cts., what will 9 Civt. 3 qrs. cost at the same rate ? not. $ 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 4.35 J yards come to 1 Ans. $590, 21 cts. 7f$ m. 6. If 345 yards of tape cost 5. dols. 17 cents, 5 m., what will one yard cost 1 Ans. ,015=1^ cts. 7. If a man lay out 121 dollars 23 cts. in merchandise, and thereby gains 139,51 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 can I buy for 25J dols. if 29J yards cost 4] 'dollars 1 Ans. 178 yards. 9. If 178J yds. cost 25 J- dollars, what cost 29f yards 1 Ans. $4 10. If 1.6 cwt. of sugar cost 12 dols. 12 cts., what cost 3 hhds., each 11 cwt. 3 qrs. 10,12 Ib. ? Ans. 269,072 rfofc.=$2G9, 7 cts. 2m.+ SIMPLE INTEREST BY DECIMALS. A TABLE OF RATIOS. Rate per cent. Ratio. \Rate per cent. Ratio. 3 4 4 5 ,03 ,04 ,045 ,05 K\ ? 6i 7 ,055 ,06 ,065 ,07 Ratio is the simple interest of II. for one year ; or in fe- deral money, of $1 for one year, at the rate percent, 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 c.ent. per annum ? 1'08 SIMPLE" flSTTERlST' fc BECIMALV. $ cts. 211,45 principal. ,05 ratio. 10,5725 interest for one yeah 5 multiply by the time. 52,8625 ylws.=g52, 86 cts. 2J m. 2. What is the interest of 645?. 10s. for 3 years, at & pi' cent, per annum 1 <645,5x06x3=116,190=116 3s. 9d. 2,4 qrs. Am. 3. What is the interest of 121/. 8s. 6d. for 4 years, at 6 per cent, per annum 1 Ans. 32 15s. &d. 1,36 per cent, per annum 1 Ans. 9 years. 3. In what time will 340 dols. 25 cts. amount to 62)5 dols. 6 cts. at 7 per cent per annum? Ans. 12 years. 4. In what time will 6457. 15s. amount to 9567. 10s, 4 ? 125d. at 5| per ct. per annum ? Ans. 8,75=8J 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 num- ber of days in a year) and it will give the interest required. EXAMPLES. J . What is the interest of 3007. 10s. for 146 days, at 6 rfr. ct.? 160 SIMPLE INTEREST BY DECIMALS, 360,5 xl 46 x, 06 s. d. qrs. -=8652=8 13 1,9 365 2. What is the interest of 640 dols. 60 cts. for 100 days, at 6 per cent, per annum ? Ans. $10, 53.cfo.-f 3. Required the interest of 250Z. 17$. for 120 days, at 5 per cent, per annum ? Ans. 4,1235=47. 2s. 5J + 4. Required the interest of 481 dollars 75 cents, for 25 days, at 7 per cent, per annum ? Ans. $2, 30 cts. 9m. + c? S 2 K f* y5 Ob s b a > .a 1 ^ 1 CO iO H- -* to o g 2; 00 Oi CG o O CO to OD CO CO O CO tS H-* h- H-A to o o i 00 Oi .h^ J, ^J Ci CO CO Oi CO o 1 to to S3 S CO tO tO h- ! 58 CO Ci CO OJ H- ^ C^ to to 135, ;n per cent. 1st payment February 19, 1798, $200 2d payment Juie '29, 1799, *500 3d payment November 14, 1799, 260 How much rcu'.a^'.s due o, fcal-J iiOte th* j , ? 5< J, ^ .Decem- ber, \- r.ts. Fnncii>ti5, Jaiii!;:;y -i, I 1000,00 Interest to February ki-", 1798. (U>- i?z?.) 67,60 Amount, 1007,50 ary 19, 17 ^00,00 Kfr,,- .|>al, 867,50 Interest to Ju ;;-y, (itfj wr.) 70,84 Amount, 938,34 Paid Jiuic-iO, 17 500,00 Reirii,. : U November 14, 17 f ), .-il, 188,20 rc-rt ;- $ Balance due on said note, Dec. 24, 1800, 200,90 $ cts. The balance by Rule I. 200,579 ' Rule II. 200,990 DrflViwce, 0,411 Aiv le MI RuVj'U. A hor.f? or i;ary 1, "1800, v,-jv pven for --. ^ * ;: -^- ^ '** sj. ; Way T '-^00. 40,00 2d payment November 14, 1800 8,00 COMPOUND INTEREST BY DECIMALS. 165 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- tember, 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, (4J mo.) 10,92 Balance due on the note, Sept. 16, 1801, $455,57 OJr^The payments being applied according to this Rule, keep down the interest, and no part of the interest ever forms apart of the principal carrying interest. COMPOUND INTEREST BY DECIMALS. . RULE. Multiply the given principal continually by the amount of one pound, or one dollar, for one year, at the rate per cent, given, until the number of multiplications are equal to the given number of years, and the product will be the amount required. Or, In Table I, Appendix, find the amount of one dollar, or one pound, for the given number of years, which multiply by the given principal, and it will a'ive the amount as before* J6,G INVOLUTION. EXAMPLES. 1. What will 400Z. amount to in 4 years, at 6 per cent, par annum, compound interest ? 400x1,06 1,00 x 1,06 x l,06=504,99+oj [504 19s. 9d. 2,75 ^.-f AM. The same by Table I. Tabular amount of 1 = 1,26-247 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 J cts. -f 3. What is the compound interest of 555 dols. for 14 years at 5 per cent.? By Table I. Ans. 543,86 cts.+ 4. What will 50 dollars amount to in 20 years, at 6 per cent, compound interest? Ans. 160, 35 cts. 6ra. INVOLUTION, IS the multiplying any number with itself, and that pro- duct by the former multiplier ; and so on ; and the several products which arise are called powers. The number denoting the height of the power, is called the index or exponent of that power. . EXAMPLES. What is the 5th power of 8 ? 8 the root or 1st power, 8 64 =2d power, or square,. 8 512 =3d power, or cube. 8 <**r . 4096 =4th power, or biquadrate. 8 2768 5th power, or sitrsolkl. EVOLUTION, Oft E'XTRA6Tlfc>lV Or lldOl'S. 16? What is the square of 17,1 ? Ans. 292,41 What is the square of ,085 ? -4ns. ,007225 What is the cube of 25,4 1 Ans. 16387,004 What is the biquadrate of 12 1 4ns. 20730 What is the square of 7J ? -4ns. EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the business of finding it is called the Extraction of the Root. The root is that number, which by a continued multipli- cation into itself, produces the given power. Although there is no number but what will produce a perfect power by involution, yet there are many numbers of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root to any assigned degree of exactness. The roota which approximate are called surd roots, and those which are perfectly accurate are called rational roots, A Table of the Squares and Cubes of the nine digits. Roots. M |2 1 3 4 16 | 5 1 6 7 | Squares. u L 4 9 25 | 36 49 1 Cubes. 11 18 27 64 | 125 |216| 343 L EXTRACTION OF THE SQUARE ROOT. Any number multiplied into itself produces a square. To extract the square root, is only to find a number, which being multiplied into itself shall produce the given number. RULE. 1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on ; and if there are decimals, point them in the same manner, from units towards the right band ; which points show the num- ber of figures the root will consist of. 2. Find the greatest square number in the first, or left hand period, place the root of it at the right hand of th^ 168 TiVOLUT'lOtt, Oil EXTRACTION OF given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number un- der 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 multiplied into the whole (increased divisor) the product shall be equal to, or the next less than the dividend, and it will be the- second figure in the root. 5. Subtract the product from the dividend, and to the remainder join the next period for a new dividend. 6. Double the figures already found in the root, for a new divisor, and from these iind 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 dividend (omitting the right hand figure thereof) by double the root already found, and the quotient will commonly be the 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 belonging to any given number are exhausted, and still 67)512 leave a remainder, the operation may 469 be continued at pleasure, by annexing periods of ciphers, &c. 745)4325 3725 7508)60064 60064 remains, ^. What is the square root of 1296 '? ANSWERS. 36 3. Of 5 ? J 4.. What is the square root of 20} ? 4< 5. What is the square root of 243,^ ' I.5J- SURDS. 6. What is the square root of Jf ? 9128 -f 7. What is the square root of 41 ? ,7745 -f 8-. Required the square root of 36 ? 6,0207-^- APPLICATION AND~USE OE 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 o. square ? 170 EVOLVTiON, OR EXTRACTION OF KOD'i>. RULE. Extract the square root of the given number. V5184=7 Am. PROB. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained in one of its sides? \/20736 144 Ans. PROB. III. To find a mean proportional between two numbers. RULE. Multiply the given numbers together and extract the square root .of the product. What is the mean proportional between 18 and 7:2 'I 72 x 18=1296, and V 1296=36 Ans. PROF.. IV. To form any body of soldiers so that they may be double, triple &c. as many in rank as in file. RULE. Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men in file, which double, triple, &c. and the product will be tife number in rank. EXAMPLES. Let 1312*2 men be so formed, as that the number in rank may be double the number in file. 13122-^2=6561, and ^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 cer- tain time ; I demand what the diameter of another pipe must be to discharge four times as much water in the same time. RULE. Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer. 2=2,5, and 2,5x2,5=6,25 square. 4 given proportion. ^ 25,00=5 inch, diam, Ans. E VO.L U TIQX , OR E XT R A ( T 1 X Q F II Q OT .. 171 PRQB. VI. The sum of any two numbers, and their pro- ducts being given, to find each number. RULE-. From the square of their sum, subtract 4 times their pro- duct, 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 ha!i' difference subtracted from the half sum., gives the lessor imr-ibor. 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 the pro, Then to the -? r sum of 21 ,5 [.numb. f and 4,5 A /81=9 diff. of the Greatest n anbcr, '20,0 ) 4 the % dift'. > Answers^ Least n limber, 17,0 J 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, be- ing multiplied into its square, shall produce the given num- ber. RULE. 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and c very- third figure from the place of units to the left, and if there be decimals, to the right. 2. Find the greatest cube in the left hand period, and place its root in the quotient. 3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, calling this the dividend. 4. Multiply the square of the quotient by 300, calling it the divisor. f. \ OLUTHJX. OK KXTRAi PlOJi 6F ROC T^ 5. rJeek how often the divisor may be had in the divi- dend, and place the result in the quotient ; then multiply the divisor by this last quotient figure, placing the product under the dividend. G. 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; A "equently 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 subtract the subtrahend from the dividend. EXAMPLES. I. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans*. 8 2x2r,,4x 300=1200)10399 first dividend, . 7200 6 x 6^36 X 2=72 x 30=2160 6x6x6= 216 9576 1st subtrahend, 26 X 26=676 x 300=202800)823744 2d dividend. 811200 4x4=-16x26rr=416x30r=: 12480 4X4X4= 64 "-W44 2d .suM V <4L L'.T ID r \ , OR E X T R A ( . 1" 1 '. 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 cqn tin ue the operation as far as you think it necessary. 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 ? 4,32-1. 8. Of - ,162771336? ,546* 9. Of ,000684134? ,088+ 10. Of --- 32261n327232? 4968 RULE. 1. Find by trial, a cube near to the givon number, and call it the supposed cube. 2. Then, as twice the supposed cube, added to the given number, is lo twice the given number added to tlis 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 degrou 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 1.3 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 : 1,2599 root, which is true to the last place of decimals ; but might by re^ peating the operation, be brought to greater exactness. 2. Wliat is the cube root of 584,277056 ? Ans. 8,36. p 2 174 K-VOLUTI.O:N, OK EXTRACTION OF 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 solid inches. I demand the side of a cubic box, which shall con- tain that quantity ? Z/ 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 Ib. what will a bullet of the same metal weigh, whose diameter is 6 in- ches ? 3x3x3=27 6x6x6=216. As 27 : 4 Ib. : : 216: 32 Ib. Ans. 3. If a solid globe of silver, of 3 inches diameter, be worth 150 dollars ; what is the value of another globe of silver, whose diameter is six inches? 3x3x3=27 6x6x6=216, As 27 : 150 : : 216 : $1200. Ans. The side of a cube being given, to find the side of that cube which shall be double, triple, &c. in quantity to the given cube. RULE. Cube your given side, and multiply by the given propor- tion between the given and required cube, and the cube root of the product will be the side sought. EXAMPLES. 4. If a cube of silver, whose side is two inches, be worth 20 dollars ; I demand the side of a cube of like silver whose value shall be 8 times as much ? 2 x 2 x 28, and 8 X 8=64 ^/64=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 times as much ? 4 x 4 x 4=64, and 64 x 4=256 v/ 256=6, 349 +/*. Am. 6\ A cooper fiaving a ca^sk 40 inches long, and 32 in- EVOLUTION, OR EXTRACTION OF ROOTS. 175 ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much ; what will be the bung diameter and length of the new cask ? 40 X 40 x 40 X 2=128000 'then V 1 2800050,3 -f kng? \ . 32 X 32 x 32 x 2=65536 and -^6553640,3+ bung diajfi. A General Rule for extracting the Roots nf 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 next period, and call it the dividend. 4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor. 5. Find how many times the divisor may be had in the dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, an^sub- tract 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 ho finished. NOTE. When the number to be subtracted is greater than those periods from which it is to be taken, the last quotient 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 ROOTS. 135796744(51,4 the root. 125=lst 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. 514x514x514=135796744 3d subtrahend. 2. Required the sursolid or 5th root of 6436343. 0436343(23 root. 32 2x2x2x2x 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 the square and cube roots only ; therefore, when any even power is given, the easiest method will be (especially in a very high power) to extract the square root of it, which re- duces it to half the given power, then the square root of that power reduces it to half the same power; and so an, till you come to a square or a cube. For example : suppose a 12th power be given ; the square root of that reduces it to a 6th power : and the square root of a 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 12S30590 464. Ans. 4& 5. Extract the square, biquadrate, or 8th root of 72138 95789338336. Arts. 96. ALLIGATION. 177 V ALLIGATION, IS the method of mixing several simples of different qua- lities, 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 arid prices of several things afe given, to find the mean price of the mixture composed of those materials. RULE. As the whole composition : is to the whole value : : so is any part of the composition : to its mean price. EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bush- el, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel ; I demand what a bushel of this mixture is worth 1 bit. cts. $cts. hn. $ cts. bit. 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 I 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 1 3. A tobacconist mixed 30 Ib. of .tobacco, at Is. (jd. per Ib. 12 Ib. at 2s. a pound, with 12 Ih. at Is, 10d. per Ib. ; what is the price of a pound of this mixture 1 Ans. Is. S(L 4. A grocer mixed 2 C. of sugar at 56s?. per C. and 1 C. at 43s. per C. and 2 C. at 50s. per C. together ; I de- mand the price of 3 cwt. of this mixture ? Ans. 7 13s. 5. A wine merchant mixes 15 Dillons of wine at 4s. 2d. per gallon, with 24 gallons at 6s. 8d. and 20 gallons at 6?. 3d. ; what is a gallon of this composition worth ? Ans. 5.s\ 1(V/.. 24-3. ^r^ 178 ALLIGATION' ALTERNATE. *f>. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt. nnd he would mix an equal quantity of each together; 1 demand the price of 3^ cwt. of this mixture ? Am. 34 \Zcts.5m. 7. A. goldsmith melted together 5 Ib. of silver bullion, of 8 oz. fine, 10 Ib. of 7 oz. fine, and 15 Ib. of 6 oz. fine ; pray what is the quality or fineness of this composition 1 Ans. 6 oz. I3pwt. 8 gr. fine. 8. Suppose 5 Ib. of gold of 22 carats fine, 2 Ib. of 21 carats fine, and 1 Ib. of alloy be melted together; what is the quality or fineness of this mass ? Ans. 19 carats fine. ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate ; so that it is the reverse of Alligation Mg^ dial, 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 quan- 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 ingredi- ent, which is less than that of the mean rate, with one or any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num- ber of the less. 3-. Place the difference, between the mean price (or mix- ture rate) and that of each of the simples, opposite to thql rflfp* with which thev are rovme*trl. ALLIGATION ALTERNATE. 179 4. TJien, it* 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 Ib. some at Is. some at 2s. and some at 2s. 6d. per Ib. how much of each sort must he mix, that he may seil the mixture at Is. 8"d. per pound 1 Ib. d. cL Ib. 10at9^ f 9. 4 12 I Gives the d. \ 12dL^ 10 8 W[ Answer} or 20]243 1 11 11 303 [30 83 T 2. A grocer would mix the following qualities of sugar ; viz. at 10 cents, 13 cents, and 16 cents per Ib. ; what quan- tity of each sort must be taken to make a mixture worth 12 cents per pound 1 Arts. 5 Ib. at 10 cts. 2 Ib. at 13 cts. and 2 Ib. at 10 cts. per Ib. 3. A grocer has two sorts of tea, viz. at 9s. and at 15s. jr Ib. how must he mix them so as to afford the composi- for 12s. per Ib. 1 -4ns. He must mix an equal quantity of each sort. 4. A goldsmith would mix gold of 17 carats fine, with some of 19, 21, and 24 carats fine, so that the compound ntay be 22 carats fine ; what quantity of each must he take? Ans.% 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. 7s. and 9e. per gallon, with water at Q per gallon, toge- ther, so that the mixture may be worth 6s. per gallon ; how much of each sort must the mixture consist of? J Ant. 1 gal. of rum at 5s., 1 do. at 7s., 6 do. at 9s. and 3 gals, wafer. Or, 3 gals, rum at 5s., 6 do. at 7s., I do. at 9s. and 1 gal. water. 1 0. A grocer hath several sorts of sugar, viz. one sort at 12 |cts. per Ib. another at 11 cts. a third at 9 cts. and a fourth at 8 cts. per Ib. ; I demand how much of each sort he must ijmix together, that the whole quantity may be afforded at IjlO cerfts per pound ? J80 ALTERNATION PARTIAL. lb. cts, Ib. cts. lb. ct. " at 12 Cl at 12 f 3 at 12 2 at 8 [l at 8 3at 8 4/* ^iws. 3 /&. of each 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, hit proportion to the quantity given. RULE. Take the differences between each price, and the mean rate, and place them alternately as in CASE I. Then, as the difference standing against that simple whose quantity is] 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 cents per bushel, with rye at 48 cts. corn at 36 cts. and barley at 30 cts. per bushel, so that a bushel of the composition may be sold for 38 cts*; what quantity of each must be taken ? {70 ^ 8 stands against the given quan- 3(0 J10 30' 32 ( 2 : 21 bushels of rye. As 8 : 10 : : \ 10 : 12 bushels of corn. ( 32 : 40" bushels of barley. * These four answers arise From as marir various ways of linking the rates of the ingredients together. Questions in this rule adfmitof an infinite variety of answers : for after the quantities are found from different methods of linking; ; any other numbers in the same proportion between themselves, as the numbers which compose thn answer, 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. per gallon 1 Ans. 20 gallons. 3. A farmer 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 1 Ans. 20 bushels of 'barley, and 61 T ? r bushels of oats. 4. With 95 gallons of rum at 8s. per gallon, I mixed other rum at 6s. 8d. per gallon, and some water; then I found it stood me in 6s. 4d. per gallon ; I demand how much rum and how much water I took 1 Ans. 95 gals, rum at 6s. Sd. and 30 gals, water. CASE III. When the whole composition is limited to a given quantity. RULE. Place the difference between the mean rate, and the se- veral prices alternately, as in CASE I. ; then, As the sum of the quantities, or difference thus determined, is to the given quantity, or whole composition : so is the difference of each rate, to the required quantity of each rate, EXAMPLES. 1. A grocer had four sorts of tea, at Is. 3s. 6s. and 10s, per Ib. the worst would not sell, and the best were too dear; he therefore mixed 120 Ib. and so much of each sort, as to sell it at 4s. per Ib. ; how much of each sort did he take ? i . 6 re : 60 at n 3^ I 2 Ib. Ib. ) 2 : 20 3 I ^ lh 6j 1 As 12 : 120 : : ] 1 : 10 6 \ p 10 -^ 3 ^3:30 10 J Sum, 12 120 182 ARITHMETICAL PROGRESSION. 2. How much water at per gallon, must be mixed with wiive at 90 cents per gallon, so as to fill a vessel of 100 gal- lons, which may be afforded at 60 cents per gallon 1 Am. 33i gals, water, and 66| gals. wine. 3. A grocer having sugars at 8 cts. 16 cts. and 24 cts. per pound, would make a composition of 240 Ib. worth 20 cts. per Ib. without gain or loss ; what quantity of each must be taken ? Ans. 40 Ib. at 8 cts. 40 Ib. at 16 cts. and 160 Ib. 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. 4J of 5 oz.fme, and 7-J- of 10 oz. fine. 5. Brandy at J3s. 6d. and 5s. 9d. per gallon, is to be mixed, so that a hhd. of 63 gallons may be sold for 12Z. 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. g ( 2,4,6,8, &e. 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 are called the extremes.* PROBLEM I. 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 under- stood by an algebraic process, and are here omitted. ARITHMETICAL PROGRESSION. 183 RULE. Multiply the sum of the extremes by the number of terms, and half the product will be the answer. EXAMPLES. 1. The first term of an arithmetical series is 3, the last term 23, and the number of terms 1 1 ; required the sum of the series. 234-326 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. $5Q., 4. A man bought 19 yards of linen in arithmetical progression, for the first yard he gave Is. and for the last yd. I/. 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 1 Ans. Amount $252^, and the average price is $2, 52 cts* 5 mitts 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 1 Ans. 23 miles, 5 furlongs, 180 yds. PROBLEM II. The 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 of terms less 1. and the quotient will be the common difference. 184 ARITHMETICAL PROGRESSION. EXAMPLES. 1. The extremes are 3 and 29, and the number of terms 14, what is the common difference 1 29) T7 * _g > Extremes. Number of terms less 1 1 3)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 ? 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, increa- sing 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 ? Arts. The daily increase is 5, and the whole journey 207 miles. 4. A debt is to be discharged at 16 different payments (in arithmetical progression,) the first payment is to be 147. the last 100?. ; What is the common difference, and the sum of the whole debt ? Ans. 51. 14s. Sd. common difference, and 912/. the whole debt. PROBLEM III. Given the first term, last term, and common difference, to find the number of terms. RULE. Divide the difference of the extremes by the common difference, and the quotient increased by 1 is the number of terms. EXAMPLES. 1. If the extremes be 3 and 45, and the common differ- ence 2 ; what is the number of terms ? Ans. 22. 2. A man going a journey, travelled the first day five miles, the last day 45 miles, and each day increased his journey by 4 miles ; how many days did he travel, and how far ? Arts. 11 days, and the whole distance travelled 275 mil* $. UEOMETRIOAL PROGRESSION". 185 GEOMETRICAL PROGRESSION, IS when any rank or series of numbers increase by one ^common 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 ra- tio given, to find the sum of the series. RULE. Multiply the last term by the ratio, and from the pro- duct subtract the first term ; then divide the remainder by the ratio, less by 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 1 3x14582 =2186 the Answer. 31 2. The extremes of a geometrical series are 1 and 65536, and the ratio 4 ; what is the sum of the series 1 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. f common ainerencc 13 i. t When the first term of the series and the ratio are equal, the indices must begin with the unit, and in this .case, the product of any two terms ia equal to that term, signified by the sum of their indices : 186 GEOMETRICAL PROGRESSION. 1. Write down a few of the leading terms of the series, and place their indices over them, beginning the indices with a unit or 1. 2. Add together such indices, whose sum shall make up 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 term sought. 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. 32x32x8=8192 Ans. 2. A draper sold 20 yards of superfine cloth, the first yard for 3d., the second for 9d., the third for 27d., &c. in triple proportion geometrical ; what did the cloth come to at that ratel The ,'M)th, or last :term,is 3^66784401^. Then 3+34867844013 =5230176600^. the sum of all 31 the terms (by Prob. I.) equal to 21792402, 10s. 3. A rich miser thought 20 guineas a price too much for 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 1 Ans. The 12 horses came to $223696, 20 cts., and the average price was $18641, 35 cts. per head. 123 4 5, &c. indices or arithmetical series. 2 4 8 16 32, w 4x8 = 32 ~ the fifth term. GEOMETRICAL PROGRESSION. 187 CASE II. 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 ndex.less by 1 than the number expressing the place of the erms sought. 3. Multiply the terms of the geometrical series together >elongingto those indices, and make the product a dividend. 4. Raise the first term to a power whose index is one ess than the number of the terms multiplied, and make the esult 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 2, what is the 7th term 1 0, I, 2, 3, Indices, 4, 12, '36, 108, leading terms. 3 + 2 -{-1=6, the index of the 7th term. 108x36x1246656 2916 the 7th term required. 16 Here the number of terms multiplied are three; there- fore 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 indices nust begin with a cipher, and the sum of the indices made choice of must 36 one less than the number of terms given in the question : because 1 in Jie indices stands over the second term, and 2 in the indices over the third ,erm, &c. and in this case, the product of any two terms, divided by the first s equal to that term beyond the first, signified by the sum of their indices. Th ( 0, 1, 2, 3, fc 4, &c. Indices. \ 1, 3, 9, 27, 81, &c. Geometrical series. Here 4+3=7 the index of the 8th term. 81x27=2187 the 8th term, or the 7th beyond the 1st. 188 POSITION. 5. A Goldsmith sold 1 Ib. 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 1 farthing the first month, 10 farthings, or (2^d) the second, and so on, each month in a tenfold proportion ? Ans. 115740740 14s. 9d. 3 qrs. 7. 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 v/hat 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, four for the third, &c. There were four shoes, and eight nails in each shoe ; what did the horse come to at ihat rate ? Ans. 4473924 5s. 3jd. 9. Suppose a certain body, put in motion, should move the length of 1 barley-corn the first second of time, one 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 in the term of half a minute. Ans. 953199885623 yds. 1 ft. I in. Ib. which is no less than five hundred and forty- one millions of miles. POSITION. POSITION is a rule which, by false or supposed 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 of which are given in the question. POSITION, 189 RULE. 1. Take any number and perform the 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 I 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 An*. as many = 12 48 * as many =6 24 ^ as many =4 16 J as many 3 12 Result, 37 Proof, 148 2. What number is that which being increased by |, , and I of itself, the sum will be 125 1 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 $15, 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 re- ceive interest for the same at 61. per cent, per annum, sim- ple interest, and at the end of twelve years received 7317. principal and interest together ; what v/as the sum deliver- ed 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 they all fill it together? Ans. 34w??w. 17-V.w. 190 DOUBLE POSITION. DOUBLE POSITION, . TEACHES to resolve questions by making two swppo sitions of false numbers.* RULE. 1. Take any two convenient numbers, and proceed with each according to the conditions of the question. 2. Find how much the results are different from the re- sults in the question. 3. Multiply the first position by the last error, and the last position by the first error. 4. If the errors are alike, divide the difference of the pro- ? ducts by the difference of the errors, and the quotient will be the answer. 5. If the errors are unlike, divide the sum of the pro- j ducts by the sum of the errors, and the quotient will be i the answer. NOTE. The errors are said to be alike when they are I both too great, or both too small ; and unlike, when one I 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 than A, and C 8 dollars more than B, and D twice as many as C ; what is each one's share of the money 1 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 posi- tions, belong to this rule ; such as those in which the number sought is in- I creased or diminished by some given number, which is no known part of the number required. DOUBLE POSITION. 191 The errors being alike, are both too small, therefore, Pos. Err. 6 30 X 8 20 . Proof 100 1240 120 120 10)120(12 A's part. 2. A, B, and C, built a house which cost 500 dollars, of vhich A paid a certain sum ; B paid 10 dollars more than L, and C paid as much as A and B both ; how much did ach man pay 1 Ans. A paid $120, B $130, and C$250. 3. A man bequeathed 100Z. to three of his friends, after this manner ; the first must have a certain portion, the se- cond must have twice as much as the first, wanting SL and the third must have three times as much as the first, want- ing 15Z. ; I demand how much each man must have ? Ans. The first 20 10s. second 33, third 46 10s. 4. A labourer was hired for 6g days upon this condition ; that for every day he wrought he should receive 4s. and for every day he was idle should forfeit 2s. ; at the expiration of the time he received 71. 10s. ; how many days did he work, and how many was he idle 1 Ans. He wrought 45 days, and was idle 15 days. 5. "What number is that which being increased by its \ , its J, and 18 more, will be doubled 1 Ans. 72. 6. A man gave to his three* sons all his estate in money, viz. to F half, wanting 50?., to G one-third, and to H the rest, which was 10Z. less than the share of G ; I demand the sum given, and each man's part ? Ans. the sum given was 360, whereof F had 130, G 120, and H 110. 192 PERMUTATION OF QUANTITIES. 7. Two men, A and B, lay out equal sums of money in trade ; A gains 1267. and B loses 877. and A's money is now double to B's ; what did each lay out ? Ans. 300. 8. A farmer having driven his cattle to market, received for them all 130/. 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 127|, B 142J, C 54. PERMUTATION OF QUANTITIES, IS the showing how many different ways any given num- ber of things may be changed. To find the, number of Permutations, or changes, that/ can be made of any given number of things all different from each other. RULE. Multiply all the terms of the natural series of numbers from one up to the given number, continually together, and the hist product will be the answer required. EXAMPLES. 1. How many changes can be made of the first three letters of the alphabet? Proofj a b c a c b b a c b c a c b a cab . ^~ How many changes may be rung on 9 bells ? Ans. 362880, 3. Scvcii gentlemen met at an inri; aiid Wcte sp well pleaded with their host, and with eacli other, that they agreed to tarry so long as they, together with their host,*. could sit every day in a different position at dinner ; how Jong must they have staid at said inn to have fulfilled their' agreement ? An$. HO^-f years. ANNUITIES OB, PENSIO3S$i COMPUTED AT COMPOUND INTEREST, CASE t To find the auiocrat of an Annuity, or iPc'issiqi'i, iii nt Compound Interest, RUL& 1. Make i the first term of a gedttfetrlcal and the amount of $1 or 1 for one year, nt the give'a rate' pjer cent, the ratio. 2. Cf*rry on the scries irp to rig many tcritf s as tit} giyefl number of years, and find its sum. 3. Multiply the snrn thus found, by the given a*mluity, and the product will be the amount EXAMPLES. 1. If* 125 dols. yearly rent, or annuity he fo'rbjome (qr unpaid) 4 years ; what will it amount to at 6 per cent. j*e.r annum, compound interest! 1 + 1,06+1,12364-1,191016^4,37461(3, sitm of the s/> 6s.* - -Then, 4,374616 x 125=-$546,827, t!i amount sought* OR BY TABLE IT. Multiply the Tabular number under the rate, Uud ojjpb- site to the time, by the annuity, and the product will bo the amount sought, * The sum of the series thus found* is the amount of ll. or 1 Hollar ai\? nuity, for the given time, which may bo found in Table II. ready calcula- ted, Hignei, either the amount or present wSrtn of annuities miy PC readily fotmcl by tables for th*. purroso. R 2. If a salary at 60 dollars per annum tp he paid yearh;, be forborne twenty years, at 6 per cent, compound interest, what is tire amount ? Under 6 pel- ceril>. and opposite SD-, in Table !, you \vill find, Tabular mi niber -3 6,76559 60 AnnuTtv. 13 eft. 5m.+ 3. Suppose an annuity of 1007. be l&ycarB in arrears, it is required to. find what is. now due, compound interest being -allowed at 5/. per cent, per annum 1 ' ^5. 1.591 145, 0,024^ (by TaHe II.) 4. \Vhat will, a pennon of 120?. per annum, payable yearly t ampunt. to in 3 year*, at 5L per cent, compound in- ferest 1 Ms. 378 6^-. II. To find tlie present w'ortli of annuities at Compmtnd In- terest. tlULE. l^vidc llue annuity, &c. by that power of tbe r/itio sij>> nified by the number of years, and subtract tbe quotient from the annuity : This remainder being divided bj the ra- tio less 1, the quotient will be the present value of tire an- nuity swght. 1. What ready nxoney will purchase an annuity of 507, to continue 4 years, at 5?. per cent, compound interest ? From 50 Subtract 41,13513' Divis. 1,05-1^05)86487 . Arts QR t BY TABLE III. Under 5 per cent, and even with 4 years., We have 3,54595~present worth qf IL for 4 years. Multiply by 50=Annuity. Ans. 177,2975Q=present Worth of die annuity. & What is the present worth of an annuity of 60 do'ls-. per annum, to continue years, at & per cent, compound interest 1 Ans. $688, 10J tts. + 3. What is 30Z. per annum, to continue 7 years]! worjth in feady money, at 6 per cent, compound interest 1 Ans* 167 $5. 5d,+ III. To find the present worth of Annuities, Lease's, &c, ta* ken 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 tire ratio demo- ted by the time of Reversion, (or the time to come be.fore the annuity commences) and the quotient will be lire pre- sent worth of the annuity in .Reversion-, EXAMPLES. 1. What ready moey will purchase .an annuity of 50(* payable yearly, for 4 years ; but not to commen.ce till two vears, at 5 per cent/? 4th power of 1,05= 1,2 15506) 50,00000(41 ,1& 13 "Subtract the quotient=41, 13513 Divide by 1,05 1 -,05)8,86487 2d power of 1,05-1, 1025) 177,297( 160,8 136^= 100 lijTs. 3d. 1 qr. present worth of the annuity in reversion. OR BY TABLE III. Find the present value of IL at the given .rate far the sum of the time of continuance, and time in reversion added to- gtyher; from which value subtract the present worth of IL i'drthe time in reversion, and multiply the remainder by the anrmitv ; the nrfydnot will he tire answer. A\Ai;mi^- OB r.;: Thus in Example 1-. Time Of continuance, 4 years. Ditto of reversion, 2 The sum, = years, gives 5,0756^ fh A\eve^lQiT 4 =S years, - - 1,859410 Remainder, 3,210282 X 50 An*. 160,8141. : \VJiat is the present vrorth of ?5/. yearly rent, which is not to commence until 10 years hence, and then to cftn.- iinue 7 years after that time at C per cent. ? Ans. 233 15*. 9 is the yearly rent to J". Whajt is the worth of a freehold estate of 407. per fii nujn, aHo\vin-5 pep cent, to the purchaser ? As^5 : 100 : : 10 : 800 Ans. 2. An estate fcrings in yearly 1507. what would it sell foiY allowing ttio purchaser G per cent, for his money? Ans. 2500. V. To find fhc present worth of a Freehold Estate, in Re*- versien, at Compound Interest. Kur.n. 1. Find the present value of the c&tote (by the foregoing .rule) as though it \vcre to be entered on immediately, and divide tlio said vttlae by that power 6f the ratio dsnotod by the lime of rcver** sion, und the q6fiefit \vill be the present \vorth of tlic estate in rc- ver^fen. EXAMPLES. 1. "Suppose a freehold estate of 407. per annum fo com*- xilenec two years hene"e,.be put on snk- : Svhnt IF iYs allowing tl/n pn'rT'Tiris^r T>/? rffr on ft-. ? ciuES.;rio v X5 IAOLI EXE uy. $;:, 1 1 J7 As & : 100 ; : 4Q : SOJQ=pre^nt wqrth it' entered on immediately. Then, l,0.5=-l,1025)800,00(7-25,6235S-?-i5/. 12s. 5^7.=r>resent worth of 800 in two years reversion. Ans. OR BY TABLE III. Find the present worth of the annuity, or rent, for the tiiiie of reversion, which subtract from the value of the im- mediate possession, and you will {lave the value of the es- tate in reversion. Thus in the foregoing example, k859410i=present worth of l/ for 2 years. 40-= annuity or renj. 74 5 37G40Q=pregent worth of the aimuily or rent, for [the time of reversion. From 8(JO,00'00=value of immediate possession. Take 74,3764==present worth pf rcn.t f 2. Suppose an estate of 90 dollars per annum, to com* mence 10 years hence, were to be seld, gcliowing thp pur- .cltaser 6 per cent ; what is the worth \ Ans. $837, SOcfc. 2w, 3. Which is the most advantageous, a term of .15 yeaTs, iu an estate of 100Z. per annum ; or the reversion of such. an estate forever after the said 15 years, computing at tire rate of 5 per cent, per annum, compound interest? Ans. The first term of 15 years is better thtfn tire rever* son forever afterwards, t by 75 18s. 7|*c/. A COLLECTION OF QUESTIONS TO EXERCISE THE FOREGOING RULES. 1. I demand th sum of 1748^ ailed to it&el'n 'Ans. 34'97. 2. Whai i's the difference lietweejii 41 eagles, and 4U99 dihies 1 ' Ans. 10 els. 3. What number is that which btirig multiplied by 21, the 7f.tTdfict will be I3BS ? An** 65. 4rKSTio.N> F<: -1. What number is that which being divided by 19, the quotient will be 72 ? An*. 1368. 5. What number is that which being multiplied by 15, the product will he J ? Ans. -J F (>. There are 7 chests of drawers, in each of winch there are 18 drawers, and in each of these there are six divisions, in each of which is I6/. 6^. 8d. ; how much money is there in the whole 1 Ans. 12348. * 7. Bought 3$ pipes of* wine for 4530 dollars ; how mirst I fell it a pipe to save one for my OUT. use, and sell the rest % "what the whole cost ? Ana. gl.2f>. 6.0 /;ta S. Just 16 yards of German serge, For DO dimes bad I ; ITow many yards of that same cloth Will 14 eagles buy 1 Ans. 246 yds. 3 grs. 2f nu. 9. A certain quantity of pasture will last 963 sheep 7 vfceks, how many must be turned out that it will last the remainder 9 weeks ? Ans. 214. 10. A grocer bought an equal quantity of su'gmr, tea, and coffee, for 740 dollars ; he gave 10 cents per Ib. for the su- gar, 60 cts. per Ib. for the tea, and 20 cts. per Ib. for the -collee ; required the quantity of each ? Ans. 822 75. 3 02. 8| dr. 11. Bought cloth at $l a yard, and lost 25 per cent., w was it sold n yard 1 Ans. 93 cts. 12. The third part of an army was killed, the fourth pai't taken prisoners, and 1000 fled ; how many were in this ar- nry, how many killed, and bow many captives ? Ans. 2400 in the army. 800 kitted, ctnd * 600 taken prisoners. 13. Thfcrnas sold loO pine apples at 33^ cents apiece, and received as much money us Hnrry 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 had Harry? Ans. Each rcc'd $50, and Harry sold 200 melons. 14. Said John to Dick, rny purse and money are worth 9J. 2s. , but the money is twenty-five times as much as tile purse ; I deifiaml TiOw Tmreli money was in it ? 8 tXf which A advanced f , B -f , and C 140/. How much paid A and B, and what part of the vessel Ans. A paid 267 T 3 T , B 305 T $ T , and C's part of the vessel was . 23. What is the purchase of 12007. bank stock, at !(& percent. 1 Ans. 1243 16s. 24. UotrghfST pieces t)f Nankeens, each 11^ yard:*, at QUESTIONS 200 14s. 4Jd. a piece, which were sold at 18d. a yard ; required' the prime cost, what it sold for, and the gain. . s. d. c Prime, cost, 19 8 1J- Ans. { Sold for, 23 5 9 ( Gain, 3 17 7 25. Three partners, A, B and C, join their stock, anil 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 24/. per cent. : A's part of the gain is -, B's 4, and C's the rest. Required each man's particular stock. A's stock was 512,7.5 ( As stack was 512,7o Aits. { &s 205,1 I C"s 307,65 26. What is that number which being divided by , the quotient will be 21 ? Ans. 15 j. 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 1 An.s. 66. 28. A gentleman divided his fortune among his three ,sons, giving A 97. as often as B 51. and to C but 3/. as often as B 71. and yet C's dividend was 2584Z. ; what did the whole estate amount to? Ans. 19466 2s. Sd. 29. A gentleman left his son a fortune, | of which he spent in three months ; % of the remainder lasted him 10 months longer, when he had only 2524 dollars left ; pray what did his father bequeath him ? Ans. $5889, 33e^. -f * 30. In an orchard of fruit trees, -J cf them bear apples, pears, plums, 40 of them peaches, and 10 cherries :, how many trees does the orchard contain 1 AUK. 600. 31. There is a certain number which being diridet! by 7, the quotient resulting multiplied by 3, that product divided by 5, from the quotient 20 being subtracted, and 30 added to the remainder, the half mi in shall make 65 ; can you teil number t An* 1400. 3& What part of 25 is | of a unit ? Ans. ^V- 33. If A can do a piece of work alone in 10 days, B in ^20 days, C in 40 days, and D in 80 days; set all four about it together, in whut time will they finish it 1 Am. 5 days. 34. A farmer being asked how many bhcep he had, an- swered, that he had them in live fields ; in the first he had J of his fleck, in the second , in the third {, in the fourth T ^, and in the fifth 450 ; how many had he \ Ans. 120U. 35. A and B together can huild a hoat in 18 days, and with the assistance of C they can do it in 1! days ; in what time would C do it alone 1 Ans. 281- days. 36. There are three numbers, 23, 25, and 42; what is the difference between the sum of the squares of the first and last, and the cube of the middlemost I Ans. 13332. 37. Part 1200 acres of land among" A, B, and C, so that B may have 100 more than A, and C 64 more than B. Ans. A 312, B 412, C 476. 38. If 3 dozen pairs of gloves be equal in value to 2 pieces of Holland, 3 pieces of Holland to 7 yards of satin, 6 yards of satin to 2 pieces of Flanders lace, and 3 pieces of Flan- ders lace to 81 shillings; how manv dozen pairs of gloves may be bought for 28s. ? Ans. 2 dozen pairs. "39. A lets B have a hogshead of sugar of 18 cwt., worth 5 dollars, for 7 dollars the cwt. Tf of which he is to pay in cash. B hath paper worth 2 dollars per ream, which he gives A for the rest of his sugar, at 2| dollars per ream ; which gained most by the bargain 1 Ans. A % $19 20 cts. 40. A father left his two sons (the one 11 and the other 16 years old) 10,000 dollars, to be divided so that each share being put to interest at 5 per cent, might amount to equal sums when they would be respectively 21 years of age. Required the shares? Ans. 545-1 ~v and 4545/ T dollars. IK Bought n (Vrtain rwantify of broadcloth for ' QU E S T 1 0$ S FOil K XJiK c I s> t '. 5s. and if the number of shillings which it coit p#r yard were added to the number of yards bought, the sum would he 386 ; 1 demand the number of yards bought, and at what price per yard! Ans. 365 yds. at 21s. per yard. Solved by PROBLEM VI. page 171* 42. Two partners Peter and John, bought goods to the amount 'of 1000 dollars ; in the purchase of which, Peter paid more than John, and John paid I know not how much : They then sold their goods for^rcady 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, PJy part of the gain is really a handsome sum of money ; I wish I had as many such sums as your part contains dollars, I should then have $960,000. I demand each man's particular stock in purchasing tire goods. Ans. Peter paid $600 and John paid 4(M). THE FOLLOWING QUESTIONS ARE EROPOSED TO SURVEYORS I 1. Required to lay out a lot of land in form of a long square, containing 3 acres, 2 roods and 29 rods, that shati take just 100 rods of wall to enclose, or fence it round ; 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 . "* 2L A tract of land is to he 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 > 320 -=- 160-640 acres: and 320 v 4x10=12,800 rails. As 640 : 12,800 : : 12,800 : 256,000, rails, which will enclose 50,000 acres= 20 nrilcrs square >v , APPENDIX, . / CONTAINING 'SHORT RULES, CASTING INTEREST AND REBATE TOGETHER. WITH SOME USEFUL RULES, ' F.'OR FINDINCi TirE CONTENTS OF SUPERFICES, SOLIT>S, &C. SHORT RULES, FOR CASTING INTEREST AT SIX PER CEN ? P. I. To find the interest of any sum of shillings for number of days less than a month, at 6 per cent. RULE. 1. Multiply the shillings of the principal by the number of days, and that product by 2, and cut off three figures to the right hand, and all above three figures will be the interest in ponce. 2. Multiply "the figures cut off by 4, still striking off three figures to the right hand, and you will have the far- thing's, very nearly. EXAMPLES. 1. Required the interest of 51. 8s. for 25 days. . s. 5,8=108x25x2=5,400, and 400x4=1,600. Ans. 5d. IfiqrS. 2. "Wftat is tlie interest of 217. 3s. for 29 days ? 204 . FEDERAL MONEY. II. To find the interest of any number of cents for any number of days less than a month, at G per cent. RULE. Multiply the cents by the number of days, divide the ptm~ duct by 6, and point off two figures to the right, and all tho figures at the left hand of the dash, will be the interest ia mills, nearly. EXAMPLES. Required the interest of 85 dollars, for 20 days. $ cts. mitls< 85=8500x20 ^-6^283,33 Am. 283 whicli i* 28 cts. 3 mills. 2. What is the interest of 73 dollars 41 cents, or 7241 cents, for 27 days, at 6 per cent. ? Ans. 330 miffs, or 0.3' cf.* r III. When the principal is given in pounds, shilling jt, fcc. New-England currency, to find the interest for any num- ber of days, less than a month, in Federal Money. RULE. Multiply the shillings in the principal by the number of days, and divide the product by 36, the quotient will be the interest in mills, for the given time, nearly, omitting fractions. EXAMPLE. Required the interest in Federal Money, of 277. 15s. for I 27 days, at 6 per cent. . s. s. Ans. 27 1 5=555x27-^36=416 milt$.=41 cte. Gm. IV. When the principal is given in Fecferal Money, ahtl you want the interest in shillings, pence, &c. NW-' land currency* for anv number tff d^vs !c?2 thsri a Hv APPENDIX. 205 RULE. Multiply the principal, in cents, by the number of days, and 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 7 ? $ 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 cent, which admits of a variety of short methods of cast- ing : and when the rate of interest is 7 per cent, as esta- blished in New-York, &c. you may first cast the interest at 6 per cent, and add thereto one sixth of itself, and the sum will be the interest at 7 per ct., which perhaps, many times, will be found more convenient than the general rule of cast- ing interest. EXAMPLE. Required the interest of 751 for 5 months, at 7 percent. .. s. d. Interest of 1507. for 7 months, is 476 Interest of 47. 7s. 6d. for 7 months, is 2 6 Ans.0) tiv.-n divide the whole by .5 in New-England, and by ,4 in New-York currency, and the quotient will be dollars, cents, &c. EXAMPLES. 1. Reduce 547. 8s. 3^d. New-England currency, to fo deral money. ,8)5-1,415 decimally expressed. Ans. $181,38 cts. 2. Reduce 7s. ll|d. New-England currency, to federal money. 7s. lljd.=0,399 then, ,3),399 3. Reduce 5137. 16s. lOd. New-York, &c. currency, to federal rnonev. ,4)513,842 decimal. Ans. $1284,604 APPENDIX. 07 4. Reduce 19s. 5jd. New- York, &c. currency, to Fede* ral Money. ,4)0,974 decimal of 19s. 5Jd." $2,431 Ans. 5. Reduce 647. New-England currency, to Federal Money. ,3)64000 decimal expression. $213,331 Ans. NOTE. By the foregoing rule you may carry on the de-' cimal to any degree of exactness ; but in ordinary practice, the following Contraction fnay be useful. RULE II. To the shillings contained in the given sum, annex 6 times the given pence, increasing the product by 2 ; then divide the whole by the number of shillings contained in a dollar, and the quotient will be cents. EXAMPLES. 1. Reduce 45s. 6d. New-England currency, to Federal Money. 6 x 8-f 2 == 50 "to be annexed. 6)45,50 or 6)4550 . ________ __ # p[$ t $7,58-1 Ans. 758 cents. 7,58 2. Reduce 2/. 10s. 9d. New-York, &c. currency, to Federal Money. 9x84-2=74 to be annexed. Then 8)5074 Or thus, 8)50,74 Ans. 634 cents.=.6 34 $6,34 N. B. When there are no pence in the .given sum, you must annex two ciphers to the shillings ; then divide as be- fore, &c. 3. Reduce 3Z. 5s. New-England currency, to Federal Money, M. 5&=65s. Then 6)6500 Ans. 1083 ceitf*. 203 APPENDIX. SOME USEFUL RULES, FOR FINDING THE CONTENTS OF SUPERFICES AND SOLIDS.. SECTION I. OF SUPERFICES. The superfices or area of any plane surface, is comrw> sed or made up of squares, either greater or less, according to the different measures by which the dimensions of the figure are taken or measured: and because 12 inches in length make 1 foot of long measure, therefore, 12 X 12== 144 fhe square inches in a superficial foot, &e. ART. I. To find the area of a square having equal sides, RULE. Multiply the side of the square into itself and the pro* duct will be the area, or content. EXAMPLES. 1 . How many square feet of boards are contained in the floor of a room which is 20 feet square ? 20 20=400 feet, the Answer. 2. Suppose a square lot of land measures 26 rods on each side, how many acres doth it contain ? NOTE. 160 square rods make an acre. Therefore, 26x26=676 sq. rods, and 676-^160=4^ 36 r. the Answer. ART. 2. To measure a parallelogram, or long square. RULE. Multiply the length by the breadth, arid the product will be the area, or superficial content. EXAMPLES. 1. A certain garden, in form of a long square, is 96 feet long, and 54 wide ; how many square feet of ground are contained in it ? Ans. 96 X 545184 square feet. 2. A lot of land, in form of a long square, is 120 rods in length, and 60 rods wide ; how many acres are in it ? 120 ^ :' 60=7200 sq. rods, then ^^^ acres. Ans. 9. If a board or plank be 21 feet long, and 18 inches brcrad ; how many square feet are contained in it ? 18 mckes*=l,5feet, then, 21 X 1.5-31,5. Am, Ai'l'liNDIX. 2GU Or, in measuring boards, you may multiply the length iu Feet by the breadth in inches, arid divide by 12, the quo- tient will give the answer iu square feet, &c. Thus, in the foregoing example, 21 X 18 -i- 12=3 J ,5 as before. 4. If a board be 8 inches wide, how much in length will make a square foot 1 RULE. Divide 144 by the breadth, thus, ' 8)144 Ans. 1*3 in. 5. If a piece of land be 5 rods wide, how many rods in length will make an acre? RULE. Divide 160 by t^c breadth, and the quotient will bo the length required, thus, 5)160 Ans. 3*2 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 per- pendicular height, or half the base into the whole perpen- dicular, and the product will be the area. EXAMPLES. 1. Required the area of a triangle whose base or longest side is 32 inches, and the pcipen^ igki I -i i fches, ;V2 7 2ui swart inches the Answer. 2. There is a triangular ;-jnrn:-red lot of b :d whose base or longest side is 5! be perpendicular from die corner opposite the base measures 44 rods : how many acres doth it contain 1 51,5-221133 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 perpcnriicu- .far* By .marking it down on the slate, paper, &c~. 21 TO MEASURE A CIRCLE. ART. 4. The diameter of a circle being given, to find 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. The diameter is a right line drawn across the circle through its centre. EXAMPLES. 1. What is the circumference of a wheel whose diameter is 4 feet 1 as 7 : 22 : : 4 : 12,57 the circumference. 2. What is the circumference of a circle whose diameter 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. RULE. Multiply half the diameter by half the circumference, and the product is the area; or if the diameter is given without the cir- cumference, multiply the square of the diameter by ,7854, and the product will be the area. EXAMPLES. 1. Required the area of a circle whose diameter is 12 inches, and circumference 37,7 inches. ^ 18,85^=half the circumference. 6 half the diameter. 113,10 area in square inches. S. Required the area of a circular garden whose diame- ter is 11 rods? ,7854 By the second method, 11x11 =^ 121 Ans. 95,0334 rods-. SECTION 2. OF SOLIDS. Solids are estimated by the solid inch, solid foot, &c. 1728 of these inches, that is, 12 X 12 X 12 make 1 cubic or fb.ot. _ 211 AUT. 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 side, 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 solid feet. Ans. Or, 18 X 18 x 18=5832 solid inches, and f 111=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 ouMo complete the same ? 12 * 12 12=1728 sold feet, the Ans. ART. 7. To find the content of any regular solid of three dimensions, length, breadth and thickness, as a piece of timber squared, whose length is more than the breadth and depth. RULE. Multiply the breadth by the depth, or thickness, and that product by the length, which gives the solid content. EXAMPLES. I. A square piece of timber, being one foot 6 inches, or 18 inches broad, 9 inches thick, and 9 feet or 108 inches long ; how many solid feet doth it cgntain ? 1 ft. 6 ki.=l,5 foot 9 inches ---. ,75 foot. Prod. 1,125 9=10,125 solid feet, the Ans. in. in. in. solid in. Or 18x9x108=17496 1728=1 0,125 /eef. But, in measuring timber, you may multiply the breadth in inches, arid the dopth in inches, and th*it product by the length in feet, and divide the last product by 144, which will give the solid content in feet, &c. 2. A piece of timber being 16 inches broad, 11 inches thick, and 20 feet long, to find the content 1 Breadth 1G inches. Depth 11 Prod. 176 x 20=3520 then, 3520- 144=&4,4/ee*. Ans. 3. A piece of timber 1.5 inches broad, 8 inches thick, and 25 feet long ; how many solid feet doth it contain? Ans. 20,8-h/ccf. 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. BJULE. Divide 1728 by the product of the breadth and depth, anil the quotient will be the length making a solid foot. EXAMPLES. 1. If a piece of timber be 11 inches broad and 8 inches deep, how many inches in length will make a solid foot? 11x8=88)1728(19,6 inches. Ans. 2. If a piece of timber be 18 inches broad and 14 inches deep, how many inches in length will make a solid foot? 18 X 14=252 divisor, then, 252)1728(6,8 inches. Ans. ART. 0. To measure a Cylinder. Definition. A Cylinder is a round body whose bases oca -circles, like a round column or suck, of \ .-, a) f>er, of equal big- ness from end to end. RULE, Multiply tfee square of t^e clwriiuter of the end by ,7854 which gives the area of the bab-c i-law .s-.i-tiply the area of the base by the length, and the product wii- be the 'solid etm tent. % EXAMPLE. What is the solid content of a round stick of tiir-^ / of equal bigness frouj end to end, whose diameter is 18 inches*, length 20 PPENDiN. 213 IS in 1,5 ft, xl,5 Square 2,25 x ,7854=1,76715 area of the base. i 20 length. Ans. 35,34300 solid content. Or, 18 inches. 18 inches. 324x,7854=-254,4696 inches, area of the base- 20 length in feet. 144)5089,3920(35,343 solid feet. Ans. ART. 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 quotient will be the answer. EXAMPLE. If the diameter of a round stick of timber be 22 inches and its length 20 feet, how many solid feet will it contain when hewn square ? 11X11X2 20-M 44=33,6 4- feet, the solidity when hewn square. ART. 11. 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 the last article Then say, As the thickness of the board including the saw calf : is to the solid feet : : so is 12 (in- ches) to the number of feet of boards. KX. \MPLE. How many feet of square edged boards, \\ inch thick, including the saw calf, can be sawn from a log 20 feet long and 24 inches diameter 1 12 X 12 2 X 20 144=40 feet, solid content. As 1 4- : 40 : :^M : 3$4 feet, thfl Atfs-. ART. 12. The length, breadth and depth of any square box being given, to find how nmny bushels it will contain. Multiply the length by the breadth, and that product by the depth, divide the last product bv 2150,425 the solid inches in a statute bushel, and the quotient will be the an- swer. EXAMPLE. There is a square box, the length of its bottom is 50 inches, breadth of ditto 40 inches, and its depth is 60 inches ; how many bushels of corn will it hold ? 50 x 40 x 60-:-2150,425==55,84 r or 55 bushels three pecks. Am. .AiiT. 13. The dimensions of the walls of a brick building being given, to find how many bricks are necessary to build it. RULE. From the whole circumference of the wall measured round on the outside, subtract four times its thickness, then multiply the remainder by th<^ height, and that product by the thickness of t!u wall, gives the solid content of the whole wall ; which multiplied by the number of bricks contained in a solid foot gives the answer. EXAMPLE. How many bricks 8 inches long, 4 inches wide, and 2i irichcs thick, will it take to build a house 44 feet long-, 40 feet wide, and 20 feet high, and the walls to he 1 foot thick ? 8x4x2,5=80 solid inches in a brick, then 1728^80 1,6 bricks in a solid foot. 44 f* 40 f 44 4-40=108 feet, whole length of wall, 4 times the thickness, 104 remains. Multiply by 20 height. 3280 solid feet in the whole wall-. Multiply by 21,6 bricks in a solid foot. Product. 70848 bricks. 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 the beam, and 12 feet deep ; what is the tonnage] 72x24 x 12 -r 95=218,2 + tons. Ans. RULE II. Multiply the length of the keel by the breadth of the beam, anc| that 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 the tonnage ? 84 : 28 * 14-95=350,29 tons. Ans. ART. 15. From the proof of any cable, to find the strength of another. RULE. The strength of cables, and consequently the weights of Iheir anchors, are as the cube of their peripheries. Therefore ; As the cube of the periphery of any cable, Is to the weight of its anchor; So is the '-.'libe 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 2 cwt. of what weight must an anchor bt for a 12 inch cable? As 6x6 6 : 2^ cwt. : : 12 12 v 12 : IS cwt. 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| cwt.? cwt. cwt. n in. As 18 : 12 12*12 : : 2,25 : 216 v >2l6r=6 Ans. ART. 16. Having the dimensions of two similar built ships of a different capacity, with the burthen of one of them-, to find the burthen of the other. 216 APPENDIX. RULE. The burthens of similar built ships are to each other j as the cubes of their like dimensions. EXAMPLE. If a ship of 300 tons burthen be 75 feet long in the keel, I demand the burthen of another ship, whose keel is 100 feet long 1 T. cwt. qrs. Ib. As/Tox 75x75: 300 :: 100x100x100:711 2 24+ DUODEGTMALS, OR CROSS MULTIPLICATION, IS a rule made use of by workmen and artificers in cast- ing up the contents of their work. RULE. 1. Under the mulplicand write the corresponding deno-- minations of the multiplier. 2. Multiply each term into the multiplicand, beginning at the lowest, by the highest denomination in the multiplier r and write the result of each under its respective term ; ob- serving to carry an unit for every 12, from each lower de- nomination to its next superior. 8. In the same manner multiply all the multiplicand by the inches, or second denomination, in the multiplier, and set the result of each term one place removed to the right - hand of those ir*the multiplicand. 4. Do the same with the seconds in the multiplier, set- ling the result of each term two places to the right hand of those in the multiplicand, &c. EXAMPLES. F. I. F. I. F. L Multiply 73 75 46 By 47 39 58 F.L 9 7 9 7 29 " 27 9 9 25 6 \ o o . 91 10 1 Product, 33 2 9 /'. /. Multiply 4 7 By 5 10 F. J. 3 8 7 6 F. / 9 7 3 Product, 26 8 10 27 6 32 6 F. L Multiply 3 11 By 95 F. /. 6 5 . 7 6 F. t 7 Itf 8 11 Product, 36 10 7 48 1 6 69 10 2 FEET, INCHES AND SECONDS. F. L " Multiply 986 , . . By 793 [tiplier. =prod. by the feet in the mul- 7 46" =ditto by the inches. 2516 =ditto by the seconds. 75 5 3 7 6 Ans. F. L ' F. I. Multiply 719 567 By 789 8 9 10 Product, 55 2 9 3 9 48 11 2 8 10 How many square feet in a boar4 16 f eet 9 inches long, and 2 feet 3 inches wide 1 By Duodecimals. By Decimals. F. L F. L 16 9 16 9=16,75 feet. 23 2 3=2,25 33 6 8375 4 2 3 3350 3350 W! 8 3 F. L IN*. 37,6875=87 8 218 APPENDIX. TO MEASURE LOADS OF WOOD. RULE. Multiply the length by the breadth, and the product by the -depth or height, which will give the content in solid feet ; of which 64 inake half a cord, and ]'28 a cord, EXAMrhK:. 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 ? 7ft. 6 i.= 7,5 /zwd 4 ft. 2 *. =4,167 0/^ 2//. 3 w.= 2,25 ; then, 7,5 x 4,167^1^25 x 2,25=70,318125 jo&W /<;e#, /bis. But loads of \vood 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, which is still called 4 feet of wood. By measuring the breadth and height of the load, tire content may be found by the following RuLrE. Multiply the breadth by the height, and half the product will be the content in feet and inches. EXAMPLE. Required the content of a load of wood which is 3 feet 9 inches wide and 2 feet 6 inches high. Sy Duodecimals. .B?/ Decimals. F. in. F. 3 9 3,75 2 6 2,5 7 6 1875 % 10 6 750 9 4 6 9,375 F. in. Ans. -4 8 3 4,6875=4 8j- or kalfa cord and 8 incjits over. The foregoing method is concise and easy to those who are well acquainted with Duodecimals, but the following table will give the <-onttnt of any load of wood, by inspection only, sufficiently exact fo| Common practice : which \vill fie found very convenient. AH'KNDIX. 219 TABLE of Breadth, Height, and (Content, [ Breadth. Height infect -. [ft. in. I 2 3 4 1 l i 910 n" 2- 6 15 3C iH 60 I 1 21 4! 5 6i 7 7) l(j 1 t ni2 14 7 16 3147 62 1 3'i 4i 5! { 8 9 10 12 J13 14 8 16 32 4864 1 3| 4i 5i 7 8 9 11 12 13 15; 9 17 33 4966 I 4 C> 7 8 9 11 12 14 15! 10 17 345168, 2 S 4 6 7 10 11 13 114 11 18 3553 70 2 3, 4,' 6 7 9 >w 13 13115 3 18 3654 72 !2 8 5! i) 8| 9 11112 14 15 if] 1 ' 19 37(56 74 2 3 5j 6 8 911 12 14 16 17 2 19, *8J57 76 o 3 5i 6 8 ion 13 14 1617' 3 1939:59 78 2 f> 5; 7 8 10111 1315 16 JH 4 20 40 60 80 o g 5 7 8 10:1213 1517 I:- 5 21 41 62 82 2 Q ft* * iDlial 14 16117 l!> ; "IT 21 4263 Si 2 4 5 7 ^ i 11 12 u! I9i 7 22143 64 6 2 4 "q 7 9; 11 13114 L6 1820' 22 14 66,88| AW 4 61 7 9 u 13J15 17 I82QJ 9 |J23< 45 081901 2 | 11 13]] 15' 17 10 Ii23< 16 69192 4 4J (3 7j o; 12 l;j!ir> j; r 11 jfegfc 17 r094 2.1 4i ei Slid! 13 I4|l6 L820J 4 !|24|48 72*96 2' 4l (>! 8 ; ! > i4il 6 TO USE THE FOREGOING TAS.U^. First measure the breadth and heisht of voui-Ioad to (he ii! in?,h ; then find the breadth in the left hand column pi' the table, then mov 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 which #ld ih^coaieir. of tlie inches on the riffht and divide the sum by 12, and you will . true conteni of the load in feet and inches. /VoJ. The contents answering the inches btMiig nhvays smfQi^may t added by inspection. EXAMPLES, 1. Admit a load of wood is 3 feet 4 inches wide, and 2 feet 10 inches liigh, required the content. Thus, against 3 fet 4 inches, and under 2 feet, stands 40 inches ; and tin- der 10 inches at top, stands 17 inches: then 40-H?=57, true content in inches, which divide by 12, ffivos 4 feet 9 inches, the answer. 55. The breadth being 3 feel, and height -2 fret 8 inches; required the con tent. Thti, with brr-ad'li^ H-' ' ''P- sjftnas 36 220 APPENDIX. inches ; and under 8 inches, stands 12 inches : now 36 and 12 make 48, the answer in inches ; and 48-^12=4 feet, or just half a cord. 3. 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 S8-M2=7 feet 4 inches, or 7 ft. 1 qr. 2 inches, the answer. TABLE I. Showing the amount of 1, or $1, at 5 and 6 pe\ annum, Compound Interest, for 20 year. 5 and 6 per cent, per Yrs. 5 per cent.'C) per cent. \\ Yrs. 5 per cent. 6 per cent. I 1 ,05000 1,05000 11 1,71034 1,89829 1,10250 1,12360 12 1 .79585 2,01219 '3 1,1571)2 1,19101 i 13 1,88565 2,13292 4 1,21550 ,26247 i 14 1,97993 2,26090 ;> 1,27628 ,33322 15 2,07893 2,39655 <> 1,34009 ,41851 16 2,18287 2,54727 7 1,40710 ,50363 17 2,29201 2,69277 8 1,47745 ,59384 18 2,40661 2,85433 9 1,55132 1,68947 19 2,52695 3,02559 10 1,62889 1,79084 20 2,65329 3,20713 VII. The weights of the coins of the United States. Eagles, Half-Eagles, Quarter- Eagles, Dollars, Half-Dollars, Quarter-Doll ars , Dimes, Half-Dimes, Cents, Half-Cents, pwt. grs. 11 6 2 17 8 4 1 8 4 16 8 171 20i 16 8 Standard Gold. Standard Silver. Copper. 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 part's fine to 179 parts alloy -the nllov to bo \vhollv copper. ANNUITIES. TABLE il. || TABLE iil. Slioioing the amount of \ anmn-\ Showing the present worth] /y, forborne, fcr 31 year." or H:i-\\ of 1 cnnrdty^ to conli\ aer, at 5 and 6 per cent, cwft- nvc for $1 years, at !> afld pound interest. \\ 8 per cent. compoundifU.\ Yrs. 5 6 -3 6 j 1 2 3 4 5 1,000000 2,050000| 3,152500 4,310125 5,525681! 1,000000: 0.952381 2,(^0000| 1,859410 3,183fm ^,723248 4,3740 JG 8,'545950 5,60710:) 4 ,329477 0,943396 1,833398 2,673012; 3,165106] 4,212361] G z 9 Ji. 11 10 LA 13 14 15 0,801913 8,142009 9,549109 11,026564 12,577892 6,975319 5,07509^ 8.^933"^ 5,786278 9,974(>S 6,463213 11,491310 7,10 13,180? '0! 7,7^17:35 4,917324; 5,582381 6,2097941 6,801692 1 ^087; 14,206787 15,917126 17,712982 19,598C32 21,578564 14,971643] 16,86994% 18,882138 21,015 23,275%t) 8,306414 8,G63252 9,393573 9,898641 10,379668 7JS& 8,:^3844 8,852683' 9 3 2919S4' 9,712249; 16 17 18 19 20 23,<>57492 25,67#>23 25,840366 28,2123801 28,13238530,905653 30,539004 33,'/o99^ 33,065954 36,785592] 10,837769 1I.2740C6 11,689587 13,085321 1:2,462210 10,105895. 10,477-260! 10,827tJ03i 11,158116. 11,469921! 21 22 23 24 25 35,719252 38,505'2l4 41,430475 44,501999 47,727099 39,992727 43,39-2291 46,995828 50,815578! 54,864512 12,821153 13,163003 13,488574 13,798642 14,093944 11,764077} 12,041582; 12,303380; 12,550357] 12,783356 < 26 27 28 29 30 31 51,113454 54,669126 58,402583 62,322712 66,438847 70,760790 59,1,5638-2 63,705765 68,52811-2 73,639798 179,058186 184,801677 14,375185 14,613034 14,898127 15,141073 15,372451 15,50-2810 13,003166^ 13,210534 13,10(5164 13,590721 ! 13,764831 13,929086 TABLES. THE three following tables are calculated agreeable to an Act of Congress passed in November, 171^2, making foreign Gold and Silver coins a legal tender for the pay* ment of all debts and demands, at the several and respec- - live rates following, viz. The Gold Coins of Great Bri- tain and Portugal, of their present standard, at the rate of 100 cents for every 27 grains of the actual weight there- of. Those of France and Spain 27f 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 Corns. | Pwt. | Gr. \Dols. Cts. M. Johannes, - - - - - 18 16 Single ditto, - - - - 9 800 English Guinea, - 5 G 4 66| Half ditto, 2 15 2 33i French Guinea, - - - 5 6 4 59 8 Half ditto. 2 15 2 29 9 4 Pistoles, 16 12 14 45 2 2 Pistoles, 8 6 7 22 6 1 Pistole, 4 3 3 61 3 oo *& 6 14 8 j APPENDIX. -5 1 ^D^CO CO ^J Oi O W QO O r-H QO rj< rH CO rj O ts. TO CC JO Oi a" -42 SCO*?! 4 O r-i 12 9 11 9 8 11 1 8 3 8-8 14 2 13 3 14 8 13 6 9 12 5 9 3 10 16 15 16 6 15 3 10] 13 8 10 4 ^ll "l 17 8 16 6 18 5 17 11 15 2 11 1 12 2 19 18 3 20 3 18 8. i 1 10 6 12 5 13 3j 21 4 20 22 2 20 k 2 33 3 25 26 42 8 40 44 4 41 3 50 37 5 40 61 2 60 66 6 61 5 4| GO 6 50 53 3 85 7 80 88 8 82 5 83 3 62 5 66 6 107 1 100 111 1 102 5 6 100 75 60 128 5 120 133 3 123 ly / 116 6 87 5 93 3 150 140 155 5 143 5 8133 3 100 106 6 171 4 160 177 7 164 1 9J160 112 5 120 193.8180 200 184 6 10 166 6 125 133 3 214 2 200 222 2 205 1 11 183 3 137 5 146 6235 7220 |244 4 225 6 12 200 150 160 257 1 240 266 6 246 1 13 216 6 162 5 173 3 278 5 260 288 8 266 6 14 233 3 175 186 6 300 2SO 311 1 287 1 15 250 187 5 200 321 4 300 333 3 307 6 16 266 61200 213 3 342 8 320 355 5 328 2 17 283 3 212 5 226 6 364 2 340 377 7 348 7 18 300 225 240 385 6 360 400 369 2 19 316 6 237 5 253 3 407 1 380 422 2 389 T 20 333 3 250 266 6 428 5 400 444 4 410 2 APPENDIX. TABLE IX. Shewing the value of Federal Money in other Currencies. Federal Money. New Eng- land i Vir- ginia , and Kentuky currency. New York and North Carolina currency. New Jersey, Pennsylva- nia , Dela- ware, and Maryland currency. South-Car-] olina, and | Georgia currency. Cents. s. d. s. d. s. d. s. d. 1 OJ 1 1 Oi 2 o 11 2 1J- o r 3 2i 3 2J If 4 3 3J 3L 21 5 3i 4f 4i 2J 6 4i 5 5i 3| 7 5 - 6 61 4 8 5? 7| 7J 4i 9 61 8J 8 5 10 7i 9i 9 5} 11 8 101 10 6} 12 8J Hi 10J 6? 13 9i 1 Oi 11 J 7* 14 10 1 H 1 0V 7f 15 10J 1 2i i H -8i 16 Hi 3i 1 9 17 1 Oi 4 1 ?i 9i 18 1 1 51 1 4i 10 - 19 1 If 6i 1 5} 10 J 20 1 2i W 1 6 11} , 30 1 9i 2 4J 2 3 1 4J 40 2 4J 3 2>V 3 1 10i 50 3 4 0~ ' 3 9 2 4" 60 3 7i 4 9^ 4 6 2 9 70 4 2i 5 7 i 5 3 3 3} 80 4 9i i\ 4^ 6 3 8f 90 5 4f 7 2i 6 9 4 2.V 100 6 > 8 7 6 4 8" 226 APPENDIX. A FEW USEFUL FORMS IN TRANSACTING BUSINESS. AN OBLIGATORY BOND. KNOW all men by these presents, that I, C. D. of in the county of am held and firmly bound to II. W. of in the penal sum of to be paid II. 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, and sealed with my seal. Dated at this day of A. D. The condition of this obligation is 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 arid titles whatsoever. In witness whereof, I the said A. B. have hereunto set my hand and seal, this day of Signed, sealed and delivered > in presence of $ A. 15. A NOTE PAYABLE AT A BANK. #500,60] HARTFORD, May 30, 1815. FOR value received, I promise to pay to John Merchant, or order, Five Hundred Dollars and Sixty Cents, at Hartford Bank, in sixty days from the date. WILLIAM DISCOUNT. AN INLAND BILL OF EXCHANGE. [$83,34] BOSTON, June 1, 1815. TWENTY days after date, please to pay to Thomas Good- win or order, Eighty-Three Dollars and Thirty-Four Cents, and place it to my account, as per advice from your humhle servant, Mr. T. W. Merchant, \ SIMON PURSE. New- York. 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. NKW-YORK, June 10, Mr. Charles Careful, Please to deliver Mr. George Speedwell, the amount of Twenty-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 MECHAN1GKS' BEST METHOD OF BOOK-KEEPING; FOR THE EASY INSTRUCTION OF YOUTH. DESIGNED AS A COMPANION DABOLL'S ARITHMETICS BY SAMUEL GREEN. M1DDLETOWN, (Con.) PUBLISHED BY WILLIAM H. NILES. Stereotyped by A. Chandler, New- York. 18-28. 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 instructed in the practice of Book Keeping. By this it is not meant to recom- mend that the son or daughter of every farmer, mechanic, or shop keeper, should enter deeply into the science as practised by the mer- chant engaged in extensive business, for such studynvould engross a grt 6,30 84 cents a 4*y? S : ^6 ---i Anthony Billings, . Cr. By 2 galls, molasses, at 36 cts. per gall. 0,72 4 yds. of India Cotton, at ISA cents, 0,74 2 flannel shirts to Joseph Hastings, 2,16 Joseph Hastings, Dr. To 2 shirts of A. Billings, .... There put the name of the owner of the book, and first date, FORM OF A DAY BOOK. Albany, February 12, 1822. Entered. t Thomas Grosvenor, Cr. By my order in favour of Joseph Hastings, Entered. 1 Joseph Hastings, Dr. To my order on T. Grosvenor, 16 Entered. 1 Thomas Grosvenor, Dr. 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 18 Entered. 1 Edward Jones, Cr. By 4 months' hire of his son William at $ 10 a month, OA Entered. 1 Edward Jones, Dr. To my draft on Thomas Grosvenor, Entered. 1 Thomas Grosvenor, _ Cr. By my draft in favour of E. Jones, 00 Entered. 1 Thomas Grosvenor, Dr. To the frame of a barn, .... Entered. 1 Anthony Billings, Cr; For the following 1 articles, 14 Ibs. muscovado sugar at $12 pr cwt. 1,50 1 large dish, ..... 0,23 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 Wafers, 4 ; ink, 6 ; 1 bottle, 8 ; . . 0,18 Entered. 1 Peter Daboll, Dr. 1 To a cotton Coverlet delivered Sarah Bradford, byfj your written order, dated 14 Jan, 355 151 FORM OF A DAY BOOK. Albany, March 1, Entered. 1 Entered. 1 Entered. 1 Entered. 1 Entered. 1 Thomas Grosvenor, By cash paid me this date, Cr. Anthony Billings, Dr. To one barrel of Cider, . . . .$11 1 barrel containing- the same, (from Tho- mas Grosvenor,) . . . . 58 7_ Thomas Grosvenor, Cr. By 1 barrel containing Cider sold and delivered to Anthony Billings, . 10 Anthony Billings, To cash per his order to George Gilbert, 1 5 Dr. Peter Daboll, Cr. By amount of his Shoe account, . . $448 Yarn received from him for the balance of his account, ..... 1 Entered. Samuel Green, Cr. 2 By amount due for 12 months New-London Gazette, $2 00 4 Spelling Books, at 20 cents, for children, 80 1 Daboll's Arithmetic, for my son Samuel, 42 2 blank Writing Books, at 124 cents, . 25 1 quire of Letter Paper, . . . 034 Entered, Entered Notes Payable, Dr. 2 By my note of this date, endorsed by Ephraim Dodge, at 6 months, fbr a yoke of Oxen bought of Daniel Mason, at Lebanon, .28 -24- Jonathan Curtis, Dr. 2 To an old bay Horse, .... #23 00 A four wheeled Wagon, and half worn Harness, . . . . . 42 00 Entered. Samuel Green, 2 To cash in full, Dr. FORM OF A DAY BOOK. Albany, April 6, Entered. Anthony Billings, To 2 tons of Hay, at $1 1 25, Entered. 1 Entered. 1 D,r. . . $2250 Amount of order dated March 26, 1822, ) inTavour of Fanny White, paid in 1 > 54 pair yarn stockings, . . ) Hire of my wagon and horse to bring sundry articles from Providence, 3d 3 00 of this month, . . . . Thomas Grosvenor, -12- Cr. By his order on Theodore Barrel], New-London, for 68 dollars, . . ..... Ajithony Billings, Dr. To 1 hogshead Rum from Theodore Barrel!, 100 gals, at 50 cents, . . . $50 00 Cash received from said Barrell for balance due on Thomas Grosvenor's order, 18 00 Entered. Entered. i. Entered. Jonathan Curtis, -18- Cr. 2 By a coat $14,75, pantaloons $5,00, -22 Thomas Grosvenor, Dr. To mending your cart by my man William, $1 00 Paid Hunt for blacksmith's work en j r our cart, . . . . . . 58JJ Setting 6 panes of glass, and 6nding glass, 66 2 To a yoke of Oxen, at 60 days' credit Entered. Entered. -25- John Rogers, Dr. Anthony Billings, Cr. By Garden Seeds of various kinds, . . $0 56 1 pair Boots, myself, $4,00, and 1 pair for John, f 3,50 7 50 1 pair of thick Shoes for Joseph Hastings, 1 25 Tea, Sugar, and Lamp Oil, per bill, . 68 Notes Payable, Cr. By my note to Isaac Thompson, at. & months, FORM OF A DAY BOOK. Albany, May 3, Entered. 2 Entered. I Entered. 1 Entered. 1 Entered. 2 Entered: 1 _ Entered. 1 Entered. 2 Theodore Barrel!, New-London," To 16, cheese, 308 Ibs. at 5 cents, . 217 Ibs. of butter, at 15 2-3 cts. . 24 Ibs. of honey, at 12$ cents, <] Dr. 1 $15 40! 34 00, 3 00 $ 52 1 43 31 52 54 54 48 ct. 40 25 60 50 40 00 00 00 Joseph Hastings, To 1 pair shoes, 29lh April, from Anthony 1 Dr. Bilikigs, Anthony Billings, Dr. To 84 bushels of seed potatoes, at 33 1-3 cents, . . . ,".;-. . $28 00 8 pair mittens, at 20 cents, . . . 1 60 Cash, . . .' . 14 00 15 Joseph Hasting^, By 4 months wages, at 7 dollars, 20 Cr. Theodore Barrel!, By cash in full of all demands, 5 Cr. Thomas Grosvenor, By his acceptance of my order in favour .of Billings, ..... Cr. Anthony Anthony Billings, Dr. To amount of my order on Thomas Grosvenor, C~p r> Notes Payable, To cash paid for my note to D. Mason, Dr. Tho foregoing example of a Day Book, may suffice to give a good idea of the way in which it is proper to make the original eu'.nVs of all dent and credit articles. Ano- ther small book should mx? bo ur;*pa;v.d. according to the following form, termed tiu 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 ; the debt a-.nomUs to b- - ;r---^3 l.\ th*- left, aiul ( . right hand p::gr. i'hus, should it at any time be required to know the state of nn account, it will only be neces- 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 tiie Leger, it will i>e proper, op. poaite the article, to not ' the same in the margin of the Day Book, by writing th# word Entered, or making two parallel strokes with the- pen : to which should be added the figure denoting the pa,^e m the Leger where the account is. On a bi^nk pag* at the beginning or cad of the Leger, an alpnabetical index should be written, containing the names of every person with whom you haveacc.ount^in the Leger, with the number of the pace where the accounts are. FORM Or A LEGER. Dr. Joseph Hastings. 1822. Jan'y Feb'y May 5 26 12 8 To my order on Anthony Billings for goods, 2 shirts of Anthony Billings, My order on Thomas Grosvenor, 1 pair shoes. 29th ^pril, from A. Billings, - fa I 1 Ct. 50 16 50 25 Dr. Samuel Stacy. 1822. Jan'y 5 To 2 weeks' wages of my daughter, at 75 cents a week, ___-_-- * 1 ct. 50 Dr. Anthony Billings. 1822. March April May 4 10 6 12 12 25 To 1 barrel of cider, and barrel, - Cash paid your order in favour of G. Gilbert, Sundries, ------- ditto, ------- i 26 68 43 54 ct. 75 32 04 00 60 00 ditto, My order on Thomas Gros.venor, Dr. Thomas Grosvenor. 1822. 1 ct. Jan'y 15 To the frame of a house, 10000 25 Sundries, ------- 74|30 Feb'y 16 Sundries, - - 7 11 28 The frame of a barn, ----- 75 00 April 22 Sundries, - - - -'- 2|24 Dr. Edward Jones. 1822. Feb'y 1 24]To my draft on Thomas Grosvenor, 38 ct. 00 Dr. Peter Daboll. 18**. I nnaries, \ $ \ct. 5l51 FORM OF A LEGEK. A hired lad, Cr. 1822. Jan'y May l,i 15 By 3 months' wages due this day, at $6, - 4i months' wages, at $7, - 18 31 ct> 00 50 Farmer, Cr. Merchant, Cr. 1822. Jan'y Feb'v April 5 26 28 29 By my order in favour of Joseph Hastings, Sundries, ------- * ditto, _--_..- ditto, ------- 11 3 3 9 50 62 55 99 Judge of County Court, Cr. Feb'y March April May " 12 25 t^ f Border in favour of Joseph Hastings, E'flMl ra ft in favour of Edward Jones, - " /fi paid me this day, - 1 empty cider barrel, Amount of your order on Theodore Barrel], My order in favour of Anthony Billings, $3 38 75 68 54 50 00 00 58 00 00 Labourer, Cr. 1822. Jan'y Feb'y 18 By team hire at sundry times, 18 1 4 months' hire of his son William, at $10, - 5 40 ct. 64 00 Farmer, Cr. 1822. 1 March! 15 By sundries in full, 5 51 Dr. FORM OF A LEGER. Samuel Green. 1822. March 28|To cash in full of his account, $ let 3|81 Dr. Notes Payable. 1822. Sept. 24J To cash paid for my note to D. Mason, - $ ct. 4800 Dr. Jonathan Curtis. 1822. 1 March 28 To a bay horse, - A wagon and harness, 1 | ct. 2300 - 1 4200 John Rogers. 25 To 1 yoke of oxen at 60 days' credit, Theodore Barrell. ,May| 3|To 16 cheese, weight 308 Ibs. at 5 cents, - #15 40 1 217 Ibs. butter at 15 2-3 cents, - 34 00 24 Ibs. honey at 12^ cents, 3 00 52 40 INDEX TO THE LEGER. B. Barrell, Theodore, - Billings, Anthony, PAGE 2 1 H. Hastings, Joseph, PAGE 1 J. Jones, Edward, 1 C. Curtis, Jonathan, 2 N. Notes Payable, 2 D. Daboll, Peter, - 1 R. Rogers, John, - 2 G. Grosvenor, Thomas, Oreon. Samuel. 1 - S. Stacy. Samuel, I f 3~23^ f^./^'/ /> K^nPJQ^H^H P;flB|aflB -^ ' iiKflHHH9H^ ^ aSr