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? The whole weight, 147000 lbs. Ans. Spoiled, 21000 lbs. 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 ? Whole weight, Ans. {Left, to visit on, 125000 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. 6 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? Ans. 63 rods. 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. some time 33. Lent a friend 292 dollars for 6 months; after, he lent me 806 dollars; how long may I keep it to balance the favour? 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 will of a lb. cost? Ans. 4 d. 34335 9. 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. ? Ans. $1. 37. If 64 yds. of cloth cost $3, what cost 94 yds.? 38. If 2 oz. of silver cost $2'24, what costs Ans. $4'269 43. At 3 £. per cwt., what will 93 lbs. of her worth? Ans. $53 785. cost? Ans. 6 s. 3 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. 29 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. 48. At $225 for 3'17 cwt. of hay, how much is that per Ans. 14'195. ton? 49. If 25 lbs. of tobacco cost 75 cents, how much will 185 lbs. cost? Ans. $5'55. 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 nhd.? Ans. $239, COMPOUND PROPORTION. T 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. 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 4 of 273 miles is 273 miles, the distance he travels in 1 hour: then, 10 times 272: 2330 miles, the distance he travels in 10 hours; and 12 times 2730 32760360 miles, the distance he travels in 12 days, travelling 10 hours each day. 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. but We will not at first consider this latter circumstance, 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 the form of a proportion, we have 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 × 10, and divided by the product of 13 × 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 question 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 days. 24 men: 248 men :: 5 days: 2d. Hours in a day. But the first labourers 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. Lengin 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 3 wide 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, that 230 length has to 420 length, all which stated in form of a proportion, we have Width, 3: ¶ 97. The continued product of all the second terms 248 X 11 X 420 × 5 × 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 8884960 days for the fourth term, or answer. 288,597. But the first and second terms are the fractions 248, 1, 138, and, which express the ratios of the men, and of the hours, of the lengths, widths and depths of the tw ditches. Hence it follows, that the ratio of the numbe days given to the number of days sought, is equal to the pro duct 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 which express them, thus, 24 × 9 × 230 248 X 11 X 420 17186400 and this fraction, 298030, represents the X5X3 17186400 X3X2 298080 > |