loss therefore is in proportion to both his stock and the time it is continued in trade. 2. Two men hired a pasture for $9; A put in 2 oxen for six months, and B 3 oxen for 5 months; what ought each to pay for the pasture? 3. Two oxen for 6 months is the same as (2×6=) 12 oxen for 1 month, and 3 oxen for 5 months is the same as (3 × 5=) 15 oxen for 1 month: thus, 2 × 6= 12 ) 27: 12 : $9 : $4 A's Ans. 3 × 5=15 ( 27; 15: $9; $5 B's Ans. RULE. 4. Having multiplied each man's stock by the time it was in trade. then say as the sum of these products is to each man's product, so is the whole gain or loss, to each man's gain or loss. 5. Three merchants, A, B, and C, enter into partnership; A puts in $60 for 4 mo.; B $50 for 10 mo., and C $80 for 12mo.; but by misfortune they lose $50; how much loss must each man sustain? A. A's $7.058+; B's $14.705+; C's $28.235+. 6. Three butchers hire a pasture for $48; A puts in 80 sheep for 4mo.; B 60 sheep for 2mo, and C 72 sheep for 5mo. ; what share of the rent must each man pay? A. A's $19.20; B's $7.20 ; C's $21.60. 7. Two merchants entered into partnership for 16mo.; A at first put in stock to the amount of $600, and, at the end of 9 months, put in $100 more; B put in at first $750, and, at the expiration of 6 months, took out $250; with this stock they gained $386: what was each man's part? A. A's, $200.797; B's, $185.202. 8. On the first of January, A began to trade with $760, and, on the first of February following, he took in B with $540; on the first of June following, they took in C with $800; at the end of the year, they found they had gained $872; what was each man's share of the gain? A. A's, $384.929; B's, $250.71; C's, $236.36. XCIV. Q. What is Compound Fellowship? 1. In example 2, what num ber of oxen for 1 month is equal to the given number for the given time? What is the Rule? 4. APPENDIX. PART THIRD. PRACTICE. XCV. 1. PRACTICE is a concise method of answering questions in the Rule of Three, when the first term happens to be unity. 2. Operations in Practice are conducted principally by supposing a price, and taking aliquot or even parts of the same for the true price. [XLII. 1.] 3. What will 50 bushels of rye cost at 5s. a bushel? Suppose the price were £1 per bushel, then the 50 bushels would cost 5s. =£1)£50 £50; but at 5s. per bushel only as much for 5s. =£1. £12.10s. 4. What will be the cost of 8640 yards of cloth at the following prices. At 10 shillings per yard? =£1/ ་ At 6s. 8 pence per yard? =£13. 1 20 6. What is the cost of the following quantities at the prices annexed? A. £393. 15s. A. £705. 16s. 8d. A. £202.16s.. 8d. 3,150 gallons of oil at 2s. 6d. per gallon? 4,235 yards of cloth at 3s. 4d. per yard? 2,434 bushels of oats at 1s. 8d. per bushel? 2,678 dozen of oranges at 5 pence for each? 4,595 quarts of strawberries at 3d. per quart? 7. When the price is not an aliquot part, we may take the one nearest to it first, then take an aliquot part of that part, and so on. A. £669. 10s. A. £57. 8s. 9d. XCV. Q. What is Practice? How is it performed? How is example 3 performed? 3. What are the divisor and dividend when the quantity is 8640 and the price 5s. ?—is 10s. ?-6s. 8d.—4s. ?—3s. 4d. ?—2s. 6d. ?--1s. 3d. ?—5d.? When the price is not an aliquot part of the given quantity, what is the diree on ? 7. 8. What will 51 barrels of cider cost at 7s. 6d. per barrel? (2s. 6d. =£}} or 1⁄2 of 5s.) 1 2 %, ) £ 5 1 at £1 per bl. £ 12.15 s. at 5 s. Or) £5 1 £ 6. 7 s. 6 d. at 2 s. 6 d. ) £ 1 2 . 1 5 s 10. What will 11cwt. 3qr. 13lb. of rice cost at $9.60 per cwt.? $ 9.6 0=cost of 1 cwt. 2 qr.= 11. What would be the cost of the following quantities ? A. $61.20 A. $67.50. A. $24.36. A. $12.564. 25 yards 2 quarters at $2.40 per yard? 18 bushels 3 pecks at $3.60 per bushel ? 5cwt. 3qr. 5lb. at $4.20 per cwt.? 3cwt. 1qr. 24lb. at $3.60 per cwt. ? 4T. 15cwt. 3qr. at $10.50 per ton? 5 e. E. 2qr. 3na. at $2.75 per ell? 12. Suppose a merchant buys 7hhd. 7gal. 2qt. $10.62 per hhd., and sells of it for $114 per hhd.; of it for $12 2 per hhd. and the balance for $15 per hhd.; how much profit does he make on the whole? A. $15.2625. of molasses at 3 A. $12.457.+ XCVI. DUODECIMALS. 1. DUODECIMALS are so called from duodecim, the Latin for twelve, because they decrease by twelves from the left hand towards the right. 2. In Duodecimals, the foot is divided first into twelve equal parts, Q. How is example 8 performed? When the quantity is 11cwt. 3qr. 13lb., what are the several divisors? [See 10.] XCVI. Q What are Duodecimals? 1. What is the integer and its divi sions ? ? called inches or primes; each prime into 12 equal parts, called seconds; each second into 12 equal parts, called thirds, and so on. 3. That is, 1 inch or prime is of a foot. 12 1 second is of, that is of a foot. 12 1 third is of of of a foot. 12 12 1 fourth is of of2 of 1220436 of a foot. 1 1 2 12 T 4. These fractions are distinguished usually by marks called accents; thus 8/8 inches or primes; 8" or 8 seconds; 8/23 or 8 thirds, &c., each additional mark denoting an inferior denomination. 5. Since feet stand in the place of units, feet multiplied by feet must give feet; feet multiplied by 12ths must give 12ths, that is inches or primes, and so on as follows: 6. Feet multiplied by feet give feet. Feet multiplied by primes give primes. 7. That is, the product will always be of that denomination which is indicated by the sum of the accents; thus, 7×5/35/////// or 35 sevenths. 8. The operations of addition, subtraction, multiplication, division and reduction of duodecimals are the same as of other compound numbers, 12 of an inferior denomination invariably making one of the next higher denomination, as in the foregoing Table. 9. How many feet are there in 1,685/? 10. How many primes are there in 140ft. 5/? 11. How many feet are there in 31,049/? 12. How many thirds in 23ft. 4/7//8///? 13. How many feet in 2,985,984////// ? 14. How many feet in 1,504,935,936////////? 15. How many ninths in 7 feet? A. 140ft. 5'. A. 1685'. A. 215ft. 7/ 5. A. 40,412′′. A. 1 foot. A. 3 feet. A. 36,118,462,464///////// ̧ Q. What parts of a foot are these sub-divisions? 3. How are these denomi. nations distinguished? 4. What do feet, primes, &c., multiplied by each other, form? 6. How can the denomination of the product be determined? Repeat the Table. 16. Add together 425ft. 4/8/7/// 5//// 11///// 9//////: 125ft. 3′ 4′′/ 9/// 2//// 3///// /// and 43ft. 2' 5" 11 3 6 10/////. A. 593ft. 10/7// 3/// 11//// 10///// 2////// ̧ 17. If a stick of timber which contains 39ft. 2′ 3′′ 9′′ be divided into two parts, one of which shall contain 23ft. 8′ 1′′ 10", what will the other part contain? A. 15ft. 6′ 1′′ 11. 18. Suppose that a person agreed to furnish at a certain price, 15 sticks of round timber, each to contain in solid measure 90ft. 3′56′′; also 30 other sticks, each measuring 101ft. 2′ 6′′ 9/"; but on its delivery of the whole was rejected on the ground that it did not answer the description in the contract; what was the quantity received? A. 52T. 34ft. 5' 3". CROSS MULTIPLICATION OF DUODECIMALS. 13. Duodecimals are principally used by workmen and artificers in ascertaining the square or solid contents of their work. 20. The square content, we have seen, [vII. 46.] is the product cf the length by the breadth; and the solid content, the square contents, multiplied by the depth or thickness. [VII. 60.] 22. The principle illustrated in (6.) which see, forms the basis of the following rule. RULE. 22. Having written feet under feet, primes under primes, &c., multiply by each denomination separately, beginning with the highest of the multiplier and the lowest of the multiplicand. 23. Place those products that are of the same denomination under each other, which will carry the first denomination in each successive product after the first, one place farther toward the right than the former; then the sum of these partial products will form the required product. 24. In a stick of timber 20ft. 9' long, 2ft. 5' wide and 2ft. 3' thick how many solid feet does it contain? * For the value of each product see 8; recollecting always to carry by 12; thus, 2ft. 9/18/÷121ft. 6 primes; then 20ft. x 2ft.=40ft.+1ft. (to carry)=41ft. Next 5'x9'45"-3′ and 9′′; 20ft × 5′= 100+3' (to carry) 103-8ft. 7 and add the products together. To multiply by 2ft. 3', say 2ft. x 9" 18" 1′ 6′′ : 2ft. x 1' =2+1=3′ : 2ft x 50ft. 100ft. : 3'x 9/ 27///=2′′/ 3/// : 3′× 1'=3′′+2′′=5 : 3′× 50ft.=150/= 12ft. 6'. = We begin on the left of the multiplier instead of the right, because it is more convenient, as may be seen by comparing the adjacent operation with the one above, with which it cor responds, except that we begin to multiply as usual.t + Lacroix's method of illustration. |