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.

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 12 tons of hay cost $180.25, what will 81 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 96 miles in 3 hours, how many miles will it run in 5 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 $5. Find the cost of & yd.

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?

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

21. If a train, at the rate of 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 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. How many yd. will he gain while B. is running 971⁄2 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 14 yd. of cloth cost $192, how much will 197 yd.

cost?

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

29. If 21 gal. of molasses cost 65 cents, what will 31⁄2 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 15 yd. of silk that is yd. wide will make a dress, how many yards of muslin that is 13 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?

34. A butcher in selling meat sells 14 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 12 hhd. in 2 hr. 15 min. ?

can pump

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

37. If of of 64 bbl. of beef cost $78, how much will of 1⁄2 of 31 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 1 in., what distance is represented on the same map by 73 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?

COMPOUND PROPORTION.

A Compound ratio indicates the product of two or more simple ratios; for instance, ק is a compound ratio, being the product of the simple ratios and §, written 3:7

5: 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.

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

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 19 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 & 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