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Note 1. If the first and second terms contain different denominations, they must both be reduced to the same denomination; and if the third term be a compound number, it either must be reduced to integers of the lowest denomination, or the low denominations must be reduced to a fraction of the highest denomination contained in it.
Note 2. The same rule is applicable, whether the given quantities be integral, fractional, or decimal.
EXAMPLES FOR PRACTICE.
5. If 6 horses consume 21 bushels of oats in 3 weeks, how many bushels will serve 20 horses the same time? Ans. 70 bushels.
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. 1024 barrels. barrels of provisions in 9 consume in the same Ans. 75 barrels.
8. If 500 men consume 1024 months, how much will 365 men time?
9. A goldsmith sold a tankard for 10 £. 12 s., at the rate of 5 s. 4 d. per ounce; I demand the weight of it.
Ans. 39 oz. 15 fwt.
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? Ans. $32'925.
12. If I buy 7 lbs. of sugar for 75 cents, how many pounds can I buy for $6? 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?
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?
20 +20 + 0 +
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. 1. Thus, in the above example,-If 10 persons spend 3 bushels, 1 person, in the same time, would spend of 8 bushels, that is, of a bushel; and 30 persons would spend 30 times as much, that is, 18 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. - 68 in., 1 ft. shadow will require a staff of of 68 in. or 8 in.; then, 153 ft. shadow will require 153 times as much; that is, 68 X 153 = 10404 — 1734 in. ⇒ 144 ft., as before.
19. If 3 £. sterling be equal to 4 £. Massachusetts, how much Massachusetts 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?
RULE OF THREE.
The whole weight, 147000 lbs. 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?
Ans. Left, to subsist on, 126000 lbs.
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. 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 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?
31. If a man perform a journey in 15 days are 12 hours long, in how many will he do it are but 10 hours long?
Ans. 62 rods. when the days when the days 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 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 1 lb. of sugar cost of a shilling, what will of a lb. cost? Ans. 4 d. 3481 4
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 TT.
36. If 7 lbs. of sugar cost ₫ of a dollar, what
37. If 64 yds. of cloth cost $3, what cost 9 38. If 2 oz. of silver cost $2'24, what costs
39. If 1⁄2 oz. cost $11, what costs 40. If lb. less by lb. cost 13 less by of 2 lbs. ? 41. If yd. cost $7, what will 40
d., what cost 14 lbs. Ans. 4 £. 9 s. 9 d.
42. If of a ship cost $251, what is
Ans. $59'062. of her worth? Ans. $53 785.
43. At 3 £. per cwt.what will 93 lbs.
Ans. 6 s. 3. d.
45. If yd. cost £., what will
44. A merchant, owning of a vessel, sold of his share for $957; what was the vessel worth? Ans. 1794'375. of an ell English cost? Ans. 17 s. 1 d. 24 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 $2'25 for 3'17 cwt. of hay, how much is that per ton? Ans. 14'195. how much will Ans. $5'55.
49. If 2'5 lbs. of tobacco cost 75 cents, 185 lbs. cost?
cost 12 lbs. ? Ans. $12
yds.? Ans. $4'269. oz. ?
Ans. $0'84. Ans. $1283.
50. What is the value of '15 of a hogshead of lime, at $2'39 per hhd. ?
51. If '15 of a hhd. of lime cost $0'3585, what is it per
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 analy
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 27 miles, the distance he travels in 1 hour: then, 10 times 273 270 miles, the distance he travels in 10 hours; and 12 times 2730 = 32760 360 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.
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 day for a certain number of days, travels 252 miles, how for