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2. If 6 men can do a piece of work in 5 days, in how many days can 10 men do the work ?

Process. We find that the answer, or x, will be less than the third quantity, for 10 men will not require so long a time as 6 men. We have, therefore :

Analysis. 10 : 6 Or, 6 : 10 = x :5.

If 6 men = 5 days, 10x

30 days; x = 3 da.

10 men = 3 days.

5 : X.

30.

1 man =

THE RULE OF THREE.

1. Let x represent the required term or answer.

2. With x and the quantity of like denomination form a ratio.

3. Compare the two terms of the ratio, and determine from the conditions of the problem whether x is greater or less than the other term.

4. With the two given like quantities form a ratio equal to the first.

5. Express the equality of the ratios, and apply formula A or B, as the case may require.

PROBLEMS.

NOTE.—Solve the following problems both by analysis and proportion.

Suggestion : Let x represent the fourth term.
1. If 31 yd. of cloth cost $62, what will 21 yd. cost ?

2. How long will it take 24 men to do a piece of work that 8 men can do in 12 days ?

3. How far can a certain load be carried for $34, if $64 will carry it 100 miles ?

4. If 231 men have provisions for 8 mo., how long will the same provisions last 308 men ?

5. If 95 cents will buy one bushel of wheat, how many bushels will $11.75 buy ?

6. A man owes $2500, and can pay only $1000. How much does he pay on a dollar ?

7. If 28 yd. of oil-cloth, .875 yd. wide, cover a certain floor, how many yards 1.25 yd. wide will cover the same floor?

8. If 22 bu. 3 pk. of corn be produced on one acre, how many acres will produce 546 bu. ?

9. If two men earn $72 in 6 da., how much will 30 men earn in the same time?

10. If 121 tons of hay cost $180.25, what will 811 tons cost? Suggestion : Let x represent the third term in the following problems.

11. A regiment of 960 men has provisions for 40 days. How long will it last if the regiment is reinforced by 240 men ?

12. A field can be mowed in 4 days of 11 hours each ; how many days of 9 hours each will it take?

13. At the time when a man 5 ft. 9 in. in height casts a shadow 4 ft. 6 in. long, what is the height of a tree that casts a shadow 52 ft. 6 in. long ?

14. If a locomotive runs 964 miles in 31 hours, how many miles will it run in 57 hours ?

15. A wheel makes 75 revolutions in 5 min. How many does it make in an hour?

16. A. can do a piece of work in 6 days, B. can do it in 7 days. If B.'s wages are $2.10 per day, how much should A. receive per day ?

17. If a 5-cent loaf of bread weighs 8 ounces when flour is worth $5, what should such a loaf weigh when flour is at $6 ? 18. / yd. cost $. Find the cost of yd. 5

i 19. If a cistern containing 3000 gal. leak 1 gal. 2 qt. a min., how long will it take to empty it?

20. If 42 yd. of carpet 2 ft. 3 in. wide are required for a room, how many yd. of carpet 2 ft. 4 in, wide will be required ?

a

Suggestion: Let a represent the second term in the following problems.

a

How many

21. If a train, at the rate of is of a mile per min., requires 31 hr. to make a certain distance, how long will it require at the rate of % of a mile a min. ?

22. If a train travels t of a mile in 18 sec., how many miles an hour does it travel ?

23. A. gains 4 yd. on B. in running 30 yd. yd. will he gain while B. is running 971 yd. ?

24. If a man spends $276 in the three summer months, how much will he spend in a year at the same rate per day? 25. If 28 men mow a field of

grass

in 12 days, how many men will be required to mow it in 8 days?

26. If 17 men can mow a field in 9 days, how long would it take to mow half of it if 5 men refuse to labor ?

27. If 141 yd. of cloth cost $191, how much will 197 yd. cost?

28. If of a ship costs £273 2s. 6d., what will be of her cost?

29. If 2; gal. of molasses cost 65 cents, what will 3} hhd. cost?

30. If a steeple 150 ft. high casts a shadow 210 ft., what is the length of the shadow cast, at the same time, by a staff 5 ft. high?

Suggestion : Let x represent the first term in the following problems.

31. If the interest of $600 for 6 mo. is $15, what principal will gain $64 in the same time?

32. If 151 yd. of silk that is tyd. wide will make a dress, how many yards of muslin that is 18 yd. wide will be required to line it? 33. If I borrow $500 and keep it 1 yr. 4 mo., for how long

. a time should I lend $240 as an equivalent for the favor ?

32

34. A butcher in selling meat sells 1411 oz. for a pound. How much does he cheat a customer who buys of him to the amount of $30?

35. In what time can a man pump 64 hhd. of water if he can pump 12 hhd. in 2 hr. 15 min. ?

36. How many men can do in 24 days a piece of work which would employ 40 men 6 days?

37. If 1 of 1 of 64 bbl. of beef cost $78, how much will { of ã of 37 bbl. cost?

38. If 450 tiles, each 12 in. square, will pave a cellar, how many tiles that are 9 in. by 8 in. will pave the same?

39. If a distance of 48 miles is represented on a map by 14 in., what distance is represented on the same map by 7} in.?

40. Twenty-four men in 30 days can finish a piece of work. After 16 days 11 men quit work. In how many days can the rest finish the work ?

a

5 3

COMPOUND PROPORTION. A Compound ratio indicates the product of two or more simple ratios ; for instance, * x is a compound ratio, being the product of the simple ratios and }, written 3 : A Compound proportion has one of its ratios compound.

ILLUSTRATION. If 5 men build a wall 6 ft. high in 7 days of 8 hr., in how many days of 9 hr. can 10 men build a wall 11 ft. high ?

Process.

:

The second ratio is simply 7 da. : x da.
The first ratio is compound, and we construct it as follows:
We write 10: 5, for 10 men require less time than 5 men.
We write 6:11, for 11 ft. require more time than 6 ft.

We write 9: 8, for 9 hr. per day require fewer days than 8 hr.

}

10: 5
Hence the proportion is : 6:11 7: X.

9: 8
2

4

5 x 11 x 8 x 7 By formula A, x =

154 = 529 days. 10 X 6 X 9

2

7 x 5 x 11 x 8 By Analysis :

1 X 10 X 6 X 9 Since 5 men require 7 da., 10 men require a shorter time, i. e., 1% of 7 da. ; since 6 ft., etc.

527 da.

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PROBLEMS.

1. If 6 men can mow 24 acres of grass in 2 days, by working 10 hours per day, how many days will it take 7 men to mow 56 acres by working 12 hrs. per day?

2. If 4 men mow 15 A. in 5 da. of 14 hr., in how many da. of 13 hr. can 7 men mow 191 A. ?

3. If 810 bricks, 8 in. long and 4 in. wide, are required for a walk 36 ft. long and 5 ft. wide, how many bricks will be required for a walk 66 ft. long and 4 ft. wide ?

4. If the interest on $640 for 4 yr. 6 mo. is $172.8, what is the interest on $820 for 2 yr. 8 mo. at the same rate ?

5. If it requires 275 yd. of cloth i yd. wide to make 75 garments, how many yards of cloth 14 yd. wide will it require to make 215 such garments ?

6. If it costs $2.40 to carry 20 cwt. 50 miles, what will it cost to carry 40 cwt. 40 miles at the same rate ?

7. If 12 candles, 8 weighing a pound, last from 5 o'clock to 11, how many candles, 6 weighing a pound, will last from 7 o'clock to 11 ?

8. A farmer owning 25 horses traded them for sheep. If 3 horses are worth 12 cows, 6 cows are worth 42 pigs, and 25

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