51. A farm was sold at $25.50 per acre, amounting to $1925.25; how many acres did the farm contain?

Ans. 75a. 2r. 52. Bought a horse for $75, and a pair of oxen for a price which was to the price of the horse in the duplicate ratio of 5 to 7; what was the price of the oxen?

53. A garrison of 300 men has provisions to last 60 days; how long will the same provisions last if the garrison is reenforced by 100 men?

54. A garrison of 1000 men have 14oz. of bread each per day for 120 days; how long will the same bread last them if each man is allowed but 12oz. per day?

55. If of a ship cost $25000, what is 1 of her worth? 56. At $27 per cwt., what is the cost of 374lb.?

57. If .25 of a piece of land are worth $750, what are .376 of it worth?

58. A's property is to B's in the triplicate ratio of 3 to 4; B's estate is worth $12800; what is the value of A's.?

59. The earth moves 19 miles per second in her orbit; how far does she go in 3m. 27sec.?

259. COMPOUND PROPORTION is an equality of two ratios one of which is compound and the other simple; thus,

3:12 16: 25

189, is a compound proportion;

and 48: 24: 18: 9, is the same reduced to a simple form. NOTE.-The compound ratio may consist of any number of couplets.

260. Every compound proportion may be reduced to a simple form, and, moreover, every example in compound proportion may be solved by means of two or more simple proportions.

Ex. 1. If 6 men in 8 hours thresh 30 bushels of wheat, in how many hours will 2 men thresh 5 bushels?

BY SIMPLE PROPORTION.

2:6 8:24, and 30:5: 24: 4, Ans.

In solving this question by simple proportion, we, in the first place, disregard the amount of labor, and inquire how long it will take 2 men to do as much as 6 men in 8 hours. Having found 24 hours to be the answer to this question, we next disregard the number of men, and inquire how long it will take to thresh 5 bushels of wheat if 30 bushels are threshed in 24 hours, and thus obtain 4 hours, the true answer to the question.

To shorten the work, we may consider both conditions at

once.

2:6

It will be seen that, of the first two couplets, {38},

30: 5

one is a ratio of less and the other of greater inequality (243 and 242); but there is no impropriety in this, for one condition of the question requires the answer to be greater than the 3d term, and the other condition requires it to be less.

261. There is no new principle in Compound Proportion. Hence,

To solve questions in Compound Proportion,

RULE.— Write that given number which is of the same kind as the required answer for the 3d term; take any two of the remaining terms THAT ARE ALIKE, and, considering the question as DEPENDING ON THESE ALONE, arrange them as in simple proportion; arrange each pair of LIKE NUMBERS by the same principles; and then multiply the continued product of the 2d terms by the 3d term, and divide this result by the continued product of the 1st terms; the quotient will be the term sought.

NOTE.-The work may frequently be much abridged by canceling any factor in the 2d and 3d terms, with a like factor in the 1st terms (127, a, Note 2).

Ex. 2. If 6 men in 15 days earn $135, how many dollars will 9 men earn in 18 days?

3. If 3 men, in 16 days of 12 hours each, build a wall 30ft. long, 8ft. high and 3ft. thick, in how many days of 9 hours each can 9 men build a wall 45ft. long, 9ft. high and 6ft. thick?

4. A wall, which was to be built 32 feet high, was raised 8 feet by 6 men in 12 days; how many men must be employed to finish the wall in 6 days? Ans. 36.

5. If 3 men, in 16 days of 12 hours each, build a wall 30ft. long, 8ft. high and 3ft. thick, how many men will be required to build a wall 45ft. long, 9ft. high and 6ft. thick, in 24 days of 9 hours each?

6. If a family of 6 persons spend $600 in 8 months, how many dollars will be required for a family of 10 persons in 14 months? Ans. $1750.

7. If the transportation of 9hhds. of sugar, each weighing 12 cwt., 20 leagues, cost $50, what must be paid for the transportation of 50 tierces, each weighing 2cwt., 300 miles?

8. If $100 gain $8 in 1 year, what will $300 gain in 9 months?

9. If $300 gain $18 in 9 months, what will $100 gain in 1 year?

10. If $100 gain $8 in 1 year, in what time will $300 gain $18?

11. If $100 gain $8 in 1 year, what principal will gain $18 in 9 months?

12. If $300 gain $18 in 9 months, what is the rate per cent.? 13. If a 2 penny loaf weighs 8oz. when wheat is 6s. 9d. per bushel, how much bread may be bought for 3s. 4d. when wheat is worth 13s. 6d. per bushel? Ans. 5lbs.

14. If a bar of silver 2ft. 1 inch long, 6in. wide and 3in. thick, be worth $2725, what is the value of a bar of gold 1ft. 91ĝin. long, 8in. wide and 4in. thick, the specific gravity of silver to that of gold being as 10.47 to 19.26, and the value per oz. of silver being to that of gold as 2 to 33? Ans. $128293.

15. If 496 men, in 5 days of 12h. 6m. each, dig a trench of 9 degrees of hardness 465 feet long, 33 feet wide and 42 feet deep, how many men will be required to dig a trench of 2 degrees of hardness 1683 feet long, 73 feet wide and 2 deep, in 22 days of 9 hours each? Ans. 15.

§ 28. CONJOINED PROPORTION.

262. CONJOINED PROPORTION (frequently called the Chain Rule and also Arbitration of Exchange) is a species of Compound Proportion, in which the antecedent and consequent of each couplet are in different denominations, but equivalent in value, and each antecedent is in the same denomination as the consequent in the following couplet.

263. The rule is principally employed in the operations of exchange in the currencies of different countries; but, to unfold its principles, we will apply it to one or two simple examples in reduction.

Ex. 1. If 4qr. = 1 d., 12d. = 1s. and 20s. = farthings are equal to 3£?

4qr.

: 1s.

- 1d.

3£.

1£, how many

Here, evidently, the continued product of the numbers on the left of the signs of equality, will be the answer; but, had the question read thus: - If

= 1d., 24d. = = 2s. and 20s.=1£, how many farthings are equal to 3£?, we would have arranged the numbers as in the

3£= how many qr.?

If 20s.=1£,
24d. - 2s.

and 4qr. = 1d.

5760qr.2 X 3£,

and 5760qr.2=2880qr.=3£.

margin, and then, evidently, the continued product of the numbers on the left of the signs of equality divided by 2, (i. e. by the continued product of the numbers on the right of the signs of equality,) will be the number of farthings in 3£, as before.

(a) This principle is equally applicable to examples in reduc

tion ascending.

Ex. 2. If 1£

20s., 1s= 12d and 1d. pounds are equal to 14400qr.?

= 4qr., how many

Here, as in the preceding example, the continued product of the left hand members of the equations, divided by the continued

product of the right hand members, gives the correct result.

This, and all other examples, may also be modified in the same manner as example 1; thus, If 1£= 20s., 3s. 36d. and 5d.20qr., how many pounds are equal to 14400qr.?