the required term or answer, thus: 18 x = 24, as shown in the operation. Or, finding the product of the means and dividing by the ex28 x 16 x 18 tremes, we have 24. 42 × 8 7. If 5. compositors, in 16 days, of 14 hours each, can compose 20 sheets of 24 pages in each sheet, 50 lines in a page, and 40 letters in a line, in how many days, of 7 hours each, will 10 compositors compose a volume to be printed in the same letter, containing 40 sheets, 16 pages in a sheet, 60 lines in a page, and 50 letters in a line? OPERATION. Days. Comp. Hours. Sheets. Pages. Lines. Letters. 16÷10 × 14 × 20 × 24 × 58 × 48 32 days. = BY CANCELLATION. 16 210 5 50 603 40 50 SOLUTION. The required term or answer is to be in days; and we see that all the terms appear in pairs or couplets, except the 16 days, which is of the same kind as the answer sought. We will proceed to compare the terms of each couplet with the 16 days. First, if 5 compositors require 16 days, how many days will 10 compositors require? Less days; hence, the divisor is the improper fraction 10, and we have 16 ÷ 10. Next, if 14 hours a day require 16 days, how many days will 7 hours a day require? More days; hence the divisor is the proper fraction 14, and we have 160 × 7. Next, if 20 sheets require 16 days, how many days will 40 sheets require ? more days; hence the divisor is the proper fraction 28, and we have 160 ×× 28. Pursuing the same method with the other couplets, we obtain the result as shown in the operation. 32 days, Ans. RULE.-I. Of the terms composing each couplet form a ratio greater or less than 1, in the same manner as if the answer depended on those two and the third or odd term. II. Divide the third or odd term by these ratios successively and the quotient will be the answer sought. By the odd term is meant the one that is of the same kind as the answer. 8. If 16 horses consume 128 bushels of oats in 50 days, how many bushels will 5 horses consume in 90 days? 9. If a man travels 120 miles in 3 days when the days are 12 hours long, how many days of 10 hours each will he require to travel 360 miles? Ans. 10 days. 10. If 6 laborers dig a ditch 34 yards long in 10 days, how many yards can 20 laborers dig in 15 days? Ans. 170 yd. 11. If 450 tiles, each 12 inches square, will pave a cellar, how many tiles that are 9 inches long and 8 inches wide will pave the same? Ans. 900. 12. If it requires 1200 yards of cloth yd. wide to clothe 500 men, how many yards which are yd. wide will it take to clothe 960 men? Ans. 32914 yd. 13. If 8 men mow 36 acres of grass in 9 days, of 9 hours each, how many men will be required to mow 48 acres in 12 days, working 12 hours each day? Ans. 6 men. 14. If 4 men, in 21 days, mow 6 acres of grass by working 8 hours a day, how many acres will 15 men mow in 3 days by working 9 hours a day? Ans. 4010 acres. 15. If, by traveling 6 hours a day at the rate of 41 miles an hour, a man performs a journey of 540 miles in 20 days, in how many days, traveling 9 hours a day at the rate of 4 miles an hour, will he travel 600 miles? Ans. 144 days. 16. If 2 yards of cloth, 13 yards wide, cost $3.371⁄2, what will be the cost of 36 yards, 11 yards wide? Ans. $52.79+. 17. If 5 men reap 52.2 acres in 6 days, how many men will reap 417.6 acres in 12 days? Ans. 20 men. PARTITIVE PROPORTION. 401. Partitive Proportion is the process by which a number is divided into proportional parts. EXAMPLES. 1. Divide $1305 into parts proportional to the numbers 2, 3, and 4. 2+3+49 of $1305=$290 of $1305-$435 SOLUTION. The problem is to take three fractions of $1305 which shall be in the proportion of 2, 3, 4. Any fractions with a common denominator, and with 2, 3, and of $1305=$580 4 as respective numerators, would be in the proper proportion; but since the sum of the three parts of $1305, when found, must equal the whole of $1305, the sum of the three fractions must equal a whole unit, and since the sum of the numerators is 2+ 3+ 4 9, the common denominator must be 9, since = 1. Therefore the fractions must be 3, §, 1. of $1305 = $290; } of $1305 = $ 435; ‡ of $1305 = $580. 2 3 2. Divide 5200 into parts proportional to,, and . SOLUTION. When fractions have a common denominator, they have the ratios of their numerators (378). Reducing 1,., and to common denominators, we have 15,3%, and 5. The problem now is to divide 5200 into parts proportional to 15, 6, 5. 6 309 Proceeding, as in Example 1, we have 15 + 6 + 5 = 26. 15 of 5200 = 3000; 2% of 5200 = 1200; = 1000. 6 of 5200 26 15 6 1 5 蛋=疆;言=; =路 30 30 15+ 6+5=26 5 18 of 5200 = 3000 26 6 of 5200 = 1200 > Ans. 26 RULE. I. Find the sum of the numbers in proportion for a common denominator. Form fractions with this sum as denominator, and the proportional numbers as respective numerators. Take these fractional parts of the number given. II. When the numbers in proportion are fractions, reduce them to a common denominator. The numerators may then be regarded as whole numbers in the given proportion, with which proceed as before. 5, 6. 1 3. Divide 105 into parts proportional to 5, 7, 9. Ans. 25, 35, 45. 4. Divide 126 into six parts proportional to 1, 2, 3, 4, Ans. 6, 12, 18, 24, 30, 36. 5. Divide 666 into three parts proportional to 4, 4, and. Ans. 333, 222, 111. 6. Three men A, B, and C, enter into partnership. For every $3 of capital A puts in, B puts in $4, and C $1. Their whole capital amounts to $20480. How much money does each put in? Ans. A, $7680; B, $10240; C, $2560. 7. A man bequeathed $63000 worth of property to three heirs as follows: To his son as much as to his wife, and to his daughter as much as to his son. How much did each receive? Ans. Wife, $42000; Son, $14000; Dau., $7000. 8. A man bought three houses for $112500. The first cost twice as much as the second, and the third three times as much as the first. How much did he pay for each? Ans. 1st, $25000; 2d, $12500; 3d, $75000. 9. Three men, A, B, and C, owned a vessel worth $52000. B owned as much as A, and C as much as A. How much did each own? Ans. A, $24000; B, $16000; C, $12000. 10. A farmer bought a farm consisting of house, barn, implements, horses, and cows for $5300. The cows cost as much as the barn; the horses as much as the barn; the implements as much; and the house 11 times as much. How much did he pay for each? 11. Divide 115200 into parts proportional to 6, 12, 18, 24, 30. PARTNERSHIP. 402. Partnership is the association of two or more persons under a certain name, for the transaction of business. 403. The Partners are the individuals associated. 404. A Firm, Company, or House is any particular partnership association. 405. Capital, or Stock, is the money or property invested by the partners, called also Investment, or Joint Stock. 406. The Resources of a firm are the amounts due the firm, together with the property of all kinds belonging to it, called also Assets, or Effects. 407. The Liabilities of a firm are its debts. The Net Capital is the excess of resources over liabilities. 408. A Dividend is the profit to be divided. An Assessment is a tax to meet losses sustained. 409. Gains and Losses are shared in proportion to the sums invested, and the periods of investment. 410. Partnership may be simple or compound. In simple partnership the capital of each partner is invested for the same time. In compound partnership the time for which the capital of each partner is invested must be taken into account. |