6. The above question reversed. If 20 horses consume 70 bushels of oats in 3 weeks, how many bushels will serve 6 horses the same time? Ans. 21 bushels. 7. If 365 men consume 75 barrels of provisions in 9 months, how much will 500 men consume in the same time? Ans. 1025 10254 barrels. 8. If 500 men consume 102 barrels of provisions in 9 months, how much will 365 men consume in the same time? Ans. 75 barrels. 9. A goldsmith sold a tankard for 10 £. 12 s., at the rate of 5s. 4 d. per I demand the weight of it. ounce; Ans. 39 oz. 15 pwt. 10. If the moon move 13° 10′ 35′′ in 1 day, in what time does it perform one revolution? Ans. 27 days, 7 h. 43 m. 11. If a person, whose rent is $145, pay $12'63 parish taxes, how much should a person pay whose rent is $378? 12. If I buy 7 lbs. of sugar for 75 cents, how many pounds $6? Ans. $32 925. can I buy for Ans. 56 lbs. 13. If 2 lbs. of sugar cost 25 cents, what will 100 lbs. of coffee cost, if 8 lbs. of sugar are worth 5 lbs. of coffee? Ans. $20. 14. If I give $6 for the use of $100 for 12 months, what must I give for the use of $357'82 the same time? Ans. $21'469. 15. There is a cistern which has 4 pipes; the first will fill it in 10 minutes, the second in 20 minutes, the third in 40 minutes, and the fourth in 80 minutes; in what time will all four, running together, fill it? = To +20 + 10 +15 cistern in 1 minute. Ans. 5 minutes. 16. If a family of 10 persons spend 3 bushels of malt in a month, how many bushels will serve them when there are 30 in the family? Ans. 9 bushels. Note. The rule of proportion, although of frequent use, is not of indispensable necessity; for all questions under it may be solved on general principles, without the formality of a proportion; that is, by analysis, as already shown, ¶ 65, ex. I. Thus, in the above example,-If 10 persons spend 3 hushels, 1 person, in the same time, would spend of 3 bushels, that is, of a bushel; and 30 persons would spend 30 times as much, that is, 9 bushels, as before. 17. If a staff, 5 ft. 8 in. in length, cast a shadow of 6 feet, how high is that steeple whose shadow measures 153 feet? Ans. 144 feet. 18. The same by analysis. If 6 ft. shadow require a staff of 5 ft. 8 in. = 63 in., 1 ft. shadow will require a staff of of 68 in. or 68 in.; then, 153 ft. shadow will require 153 times as much; that is, 68 x 153 = : 10층으로. 1734 in. = 144 ft., as before. = 19. If 3 £. sterling be equal to 4 £. Massachusetts, how much Massachu setts is equal to 1000 £. sterling? Ans. 1333 £. 6 s. 8 d. 20. If 1333 £. 6 s. 8 d. Massachusetts, be equal to 1000 £. sterling, how much sterling is equal to 4 £. Massachusetts? Ans. 3 £. 21. If 1000 £. sterling be equal to 1333 £. 6 s. 8 d. Massachusetts, how much Massachusetts is equal to 3 £. sterling? Ans. 4 £. 22. If 3 £. sterling be equal to 4 £. Massachusetts, how much sterling is equal to 1333 £. 6 s. 8 d. Massachusetts? Ans. 1000 £. 23. Suppose 2000 soldiers had been supplied with bread sufficient to last them 12 weeks, allowing each man 14 ounces a day; but, on examination, they find 105 barrels, containing 200 lbs. each, wholly spoiled; what must the allowance be to each man, that the remainder may last them the same time? Ans. 12 oz. a day. 24. Suppose 2000 soldiers were put to an allowance of 12 oz. of bread per day for 12 weeks, having a seventh part of their bread spoiled; what was the whole weight of their bread, good and bad, and how much was spoiled? Ans. The whole weight, 147000 lbs. 21000 lbs. Spoiled,. 25. -2000 soldiers, having lost 105 barrels of bread, weighing 200 lbs. each, were obliged to subsist on 12 oz. a day for 12 weeks; had none been lost, they might have had 14 oz. a day; what was the whole weight, including what was lost, and how much had they to subsist on? Ans. Whole weight, 147000 lbs. Left, to subsist on, 126000 lbs. 26. 2000 soldiers, after losing one seventh part of their bread, had each 12 oz. a day for 12 weeks; what was the whole weight of their bread, including that lost, and how much might they have had per day, each man, if none had been lost? Whole weight, 147000 lbs. Ans. Loss,. 21000 lbs. 14 oz. per day, had none been lost. 27. There was a certain building raised in 8 months by 120 workmen; but the same being demolished, it is required to be built in 2 months; I demand how many men must be employed about it. Ans. 480 men. 28. There is a cistern having a pipe which will empty it in 10 hours; how many pipes of the same capacity will empty it in 24 minutes? Ans. 25 pipes. 29. A garrison of 1200 men has provisions for 9 months, at the rate of 14 oz. per day; how long will the provisions last, at the same allowance, if the garrison be reinforced by 400 men? Ans. 64 months. 30. If a piece of land, 40 rods in length and 4 in breadth, make an acre, how wide must it be when it is but 25 rods long? rods. Ans. 62 31. If a man perform a journey in 15 days when the days are 12 hours long, in how many will he do it when the days are but 10 hours long? Ans. 18 days. 32. If a field will feed 6 cows 91 days, how long will it feed 21 cows? Ans. 26 days. 33. Lent a friend 292 dollars for 6 months; some time after, he lent me 806 dollars; how long may I keep it to balance the favor? Ans. 2 months 5+ days. 34. If 30 men can perform a piece of work in 11 days, how many men will accomplish another piece of work, 4 times as big, in a fifth part of the time? Ans. 600 men. 35. If lb. of sugar cost of a shilling, what of a lb. cost? Ans. 4 d. 3487 will q. Note. See T 65, ex. 1, where the above question is solved by analysis. The eleven following are the next succeeding examples in the same T. 36. If 7 lbs. of sugar cost of a dollar, what cost 12 lbs. ? 39. If oz. cost $1, what costs 1 oz. ? 40. If of 2 lbs. ? 41. If Ans. $14. Ans. $4'269 Ans. $1'283, lb. less by lb. cost 13 d., what cost 14 lbs. less by yd. cost $%, what will 40 yds. cost? 42. If 76 Ans. 4 £. 9 s. 925 43. At 3 £. per cwt., what will 93 lbs. cost? Ans. $59'062. Ans. $53'785. Ans. 6 s. 35 d. 44. A merchant, owning of a vessel, sold of his share for $957; what was the vessel worth? Ans. $1794'375. 45. If yd. cost £., what will of an ell English cost? Ans. 17 s. 1 d. 2 q. 46. A merchant bought a number of bales of velvet, each containing 12917 yds., at the rate of $7 for 5 yds., and sold them out at the rate of $11 for 7 yds., and gained $200 by the bargain; how many bales were there? Ans. 9 bales. 47. At $33 for 6 barrels of flour, what must be paid for 178 barrels ? Ans. $979. Ans. $14'195. Ans. $5'55. 48. At $2'25 for 3'17 cwt. of hay, how much is that per ton? 49. If 2'5 lbs. of tobacco cost 75 cents, how much will 185 lbs. cost? 50. What is the value of '15 of a hogshead of lime, at $2'39 per hhd. ? Ans. $0'3585. 51. If '15 of a hhd. of lime cost $0'3585, what is it per hhd.? Ans. $2′39. COMPOUND PROPORTION. 96. It frequently happens, that the relation of the quantity required, to the given quantity of the same kind, depends upon several circumstances combined together; it is then called Compound Proportion, or Double Rule of Three. 1. If a man travel 273 miles in 13 days, travelling only 7 hours in a day, how many miles will he travel in 12 days, if he travel 10 hours in a day? This question may be solved several ways. First, by analysis :— If we knew how many miles the man travelled in 1 hour, it is plain, we might take this number 10 times, which would be the number of miles he would travel in 10 hours, or in 1 of these long days, and this again, taken 12 times, would be the number of miles he would travel in 12 days, travelling 10 hours each day. 13 If he travel 273 miles in 13 days, he will travel of 273 miles, that is, 273 miles in 1 day of 7 hours; and of 273 miles is 273 miles, the distance he travels in 1 hour; then, 10 times 273-2730 miles, the distance he travels in 10 hours; and 12 times 2730=32760 distance he travels in 12 days, travelling 10 hours each day. 91 91 360 miles, the Ans. 360 miles But the object is to show how the question may be solved by proportion: First; it is to be regarded, that the number of miles travelled over depends upon two circumstances, viz. the number of days the man travels, and the number of hours he travels each day. We will not at first consider this latter circumstance, but suppose the number of hours to be the same in each case; the question then will be,-If a man travel 273 miles in 13 days, how many miles will he travel in 12 days? This will furnish the following proportion: 13 days 12 days: 273 miles : miles, which gives for the fourth term, or answer, 252 miles. Now, taking into consideration the other circumstance, or that of the hours, we must say,-If a man, travelling 7 hours a day for a certain number of days, travels 252 miles, how far will he travel in the same time, if he travel 10 hours in a day? This will lead to the following proportion: 7 hours: 10 hours: 252 miles: ...... miles. This gives for the fourth term, or answer, 360 miles. We see, then, that 273 miles has to the fourth term, or answer, the same proportion that 13 days has to 12 days, and that 7 hours has to 10 hours. Stating this in the form of a proportion, we have 13 days: 12 days 7 hours: 10 hours: 273 miles : ...... miles, by which it appears that 273 is to be multiplied by both 12 and 10; that is, 273 is to be multiplied by the product of 12 x 10, and divided by the product of 13 x 7, which, being done, gives 360 miles for the fourth term, or answer, as before. In the same manner, any question relating to compound proportion, however complicated, may be stated and solved. 2. If 248 men, in 5 days, of 11 hours each, can dig a trench 230 yards long, 3 wide, and 2 deep, in how many days, of 9 hours each, will 24 men dig a trench 420 yards long, 5 wide, and 3 deep? Here the number of days in which the proposed work can be done, depends on five circumstances, viz. the number of men employed, the number of hours they work each day, the length, breadth, and depth of the trench. We will consider the question in relation to each of these circumstances, in the order in which they have been named : 1st. The number of men employed. Were all the circumstances in the two cases alike, except the number of men and the number of days, the ques tion would consist only in finding in how many days 24 men would perform the work which 248 men had done in 5 days; we should then have 24 men: 248 men: : 5 days: days. 2d. Hours in a day. But the first laborers worked 11 hours in a day, whereas the others worked only 9; less hours will require more days, which will give 9 hours 11 hours 5 days: ...... days. 3d. Length of the ditches. The ditches being of unequal length, as many more days will be necessary as the second is longer than the first; hence we shall have 230 length: 420 length: 5 days: ...... days. 4th. Widths. Taking into consideration the widths, which are different, we have 5th. Depths. Lastly, the depths being different, we have 2 deep 3 deep 5 days: ...... days. It would seem, therefore, that 5 days has to the fourth term, or answer, the same proportion that 24 men has to 248 men, whose ratio is 248 that 9 hours has to 11 hours, the ratio of which is 1, that 230 length has to 420 length, 1238, 97. The continued product of all the second terms, 248 × 11 × 420 X5 X 3, multiplied by the third term, 5 days, and this product divided by the continued product of the first terms, 24 X 9 X 230 × 3 × 2, gives 288 849 60 days for the fourth term, or answer. 288 5.9 207 298080 But the first and second terms are the fractions 24, 1, 130, and, 흉 which express the ratios of the men, and of the hours, of the lengths, widths, and depths of the two ditches. Hence, it follows that the ratio of the number of days given to the number of days sought, is equal to the product of all the ratios which result from a comparison of the terms relating to each circumstance of the question. The product of all the ratios is found by multiplying together the fractions 248 X 11 X 420 × 5 × 3 which express them, thus, 24 X 9 X 230 × 3 × 2 17186400 fraction, 298080 17186400 298080 -' and this represents the ratio of the quantity required to the given quantity of the same kind. A ratio resulting in this manner, from the multiplication of several ratios, is called a compound ratio. From the examples and illustrations now given, we deduce the following general RULE for solving questions in compound proportion, or double rule of three, viz.-Make that number which is of the same kind with the required answer, the third term; and, of the remaining numbers, take away two that are of the same kind, and arrange them according to the directions given in simple proportion; then, any other two of the same kind, and so on, till all are used. Lastly, multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms, and the quotient will be the fourth term, or answer required. EXAMPLES FOR PRACTICE. 1. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 16 days, In what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick Ans. 80 days. 2. If the freight of 9 hhds. of sugar, each weighing 12 cwt., 20 leagues, cost 16 £., what must be paid for the freight of 50 tierces, each weighing 2 cwt., 100 leagues? Ans. 92 £. 11 s. 10 20 103 d. 3. If 56 lbs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days? Ans. 36 lbs. The same by analysis. If 7 men consume 56 lbs. of bread, 1 man, in the same time, would consume 56 lbs.; and if he consume 56 lb. in 1 day. of 56 lbs. lbs. in 14 days, he would consume of 56 would consume 21 times as much as 1 man; that is, 21 times 56 21 men 98 1776 lbs. in 1 day, and in 3 days they would consume 3 times as much; that is, 3528 98 36 lbs., as before. Ans. 36 lbs. Note. Having wrought the following examples by the rule of proportion, let the pupil be required to do the same by analysis. 4. If 4 reapers receive $11'04 for 3 days' work, how many men may be hired 16 days for $103'04 ? Ans. 7 men. 5. If 7 oz. 5 pwt. of bread be bought for 44 d., when corn is 4 s. 2 d. per bushel, what weight of it may be bought for 1 s. 2 d. when the price per bushel is 5 s. 6 d. ? Ans. 1 lb. 4 oz. 3479 pwts. 6. If $100 gain $6 in 1 year, what will $400 gain in 9 months? Note. This and the three following examples reciprocally prove each other. 7. If $100 gain $6 in 1 year, in what time will $400 gain $18? 8. If $400 gain $18 in 9 months, what is the rate per cent. per annum? 9. What principal, at 6 per cent. per annum, will gain $18 in 9 months? 10. A usurer put out $75 at interest, and at the end of 8 months received, for principal and interest, $79; I demand at what rate per cent. he received interest. Ans. 8 per cent. 11. If3 men receive 80 £. for 19 days' work, how much must 20 men receive for 100 days? Ans, 305 £. 0 s. 8 d. |