15. How many yards of cloth, 11⁄2 yd. in width, are equal in measure to 30 yds. 1 ell English in width?

16. How long must a board, 4 contain a square foot

Answer, 25 yards. inches in breadth, be to Ans., 32 inches.

17. A certain building was raised in 8 months by 120 workmen; how many workmen could have done the same amount of labor in 2 months? Ans., 480. men.

18. How much in length that is 16 rods in width will it take to make an acre?

Answer, 10.

19. There is a cistern having a pipe, which will empty it in 6 hours. How many pipes of the same capacity will empty it in 20 minutes ?

Answer, 18 pipes.

20. If 30 men can perform a piece of work in 11 days, how many men will accomplish another piece of work 4 times as large in a fifth part of the time?

Answer, 600 men.

Thus far, Simple Proportion has been considered as Direct and Inverse, that the scholar might have a better understanding of the doctrine of ratio; but when the nature of proportion and the question proposed are well understood, the statement may be made by the following general RULE, and solved either by Proportion or by Cancelling.

General Rule by Proportion. General Rule by Cancelling.

Make that number which Draw a perpendicular line, is of the same kind with the and place the sign of the ananswer required, the third swer on the left, at the top of term, and the two remaining the line, and that number ones the first and second, put- which is of the same kind ting the greater, or less, first, with the answer, on the right; according as the 3d is greater and as this number is greater or less than the term sought; or less than the answer sought, then the fourth term will be place the greater or less of found by multiplying the sec- the two remaining numbers ond and third, and dividing on the right and the other on by the first.

the left, and proceed in all reIspects as before instructed.

Examples by Proportion. Examples by Cancelling. 1. If 500 men consume 1. If 500 men consume 1024 barrels of flour in 9102 barrels of flour in 9 months, how many barrels will months, how many barrels will 365 men consume in the same 365 men consume in the same

500)27375|00

bar.

Having reduced the mixed

73)5475(75 Answer. number, 10234,to an improp

511

365

365

er fraction, as the answer is in barrels, we place its numerator on the right of the line, (see p. 122,) and as the

Having reduced 1024 bar-answer is to be less than this rels to an improper fraction, number, we place 365, the we make it the third term, less of the other two numbers, because it is the same kind also on the right, and 500, the with the answer required; and remaining number, on the left. as 365 men would consume Then by cancelling-the two a less number of barrels than ciphers on the right cancel the 500 in a given time, we make two ciphers on the left; (this 365 the second term and 500 is the same as dividing both the first; we then multiply sides of the line by 100) also 73 is contained in 365—5 the second term by the nuThen 5 on the right merator of the fraction, and times. divide the product by 500, the cancels 5 on the left, and 75, first term, and the quotient the remaining number on the divided by the denominator right, is the answer. of the fraction is the answer.

2. A merchant bartered 58 2. A merchant bartered 58 cwt. of sugar, at 63d. per lb. cwt. of sugar at 63d. per lb., -for tea, at 8ğs. per lb., how for tea at Sgs. per lb.-how much tea did he receive? much tea did he receive?

4. If 2 lbs. of sugar cost 4. If 2 lbs. of sugar cost of a dollar, what will 100 lbs. of a dollar, what will 100 lbs. of coffee cost, if 8 lbs. of su- of coffee cost, if 8 lbs. of sugar are worth 5 lbs. of coffee? gar are worth 5 lbs. of coffee? Operation.

coff. sug. coff

58 100 1st STAT.

5)800

Operation. Whatwill$100 coffee 20 coffee 58 sugar.

sugar 21$
4

5. If

2)4000

$20,00 Answer.

lb. less by cost 134d., what will 14 lbs. less by of 2 lbs. cost? Ans., £4 9s. 9d.. 6. A merchant failing in trade owes $6000-his property amounts to $2400. What does his creditor receive to whom he owes $500, and what does he pay on the dollar? (Creditor, $200 on dollar,

Ans.,

7. Bought yd. of cloth for $3, sold of

Did I make or lose?

40 cts.
of

for gs. Ans., made 87 cts. 8. If 40 yds. of cloth cost $32, what will 1 ell English

cost?

Ans., $1.

Ans., $240.

9. If of a yd. cost £ sterling, what will 60 yds. cost in dollars? 10. A mercer bought 3 pieces of silk, each piece containing 24 yds. at 6s. 6d. per yd. What does the whole cost him? Ans., £27 13s. 7d.

11. If § of a gallon cost §£, what will of tun cost?

Answer, £140.

12. If 24 yds. of cloth cost 60 cts., what will 125 yds. cost? Answer, $30,18. 13. If g of a cord of wood is worth $6, what is 40 cords worth? Ans., $470.

COMPOUND PROPORTION.

WHEN a proportion is formed by the combination of two or more simple proportions, it is called Compound Proportion, or Double Rule of Three.

1. If 8 men consume 24 bushels of wheat in 5 months, how many bushels will 4 men consume in 15 months?

In this question, the number of bushels consumed depends on two circumstances-the number of men, and the time. We may consider the circumstances separately, and solve the question by two statements in the Single Rule of Three. First, the number of men. If 8 men consume 24 bushels in 5 months, how many bushels will 4 men consume in the same time?

Operation.

How many bushels. 4 men.

2 men 824 bushels 12

12 bushels, Answer.

Secondly, the time. If 4 men consume 12 bushels in 5 months, how many bushels will the same number of men consume in 15 months?

Operation.

How many bushels. 15 mo. 3

mo. 512 bu.

36 bu. Answer.

All questions in Compound Proportion may be solved by two or more statements in Simple Proportion, or they may be analyzed-thus: If 8 men consume 24 bushels in 5 months, 1 man would consume of 24-3 bushels, and in 1 month of 3=3 of 1 bushel. Then 4 men would consume 4 times 3-12, in 1 month, and in 15 months, 15 times 12-36 bushels, the answer.

5

NOTE.-The foregoing is, perhaps, the most natural method of solving proportional questions, but it is not always the shortest; other methods are adopted rather for the sake of brevity than cor

rectness.

Compound Proportion teaches to solve by one state

QUESTIONS. 1. How is Compound Proportion formed? 2. What does Compound Proportion teach? 3. What is a compound ratio?