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RULE.-Express the ratio between the two numbers that are like numbers. Consider, from the conditions of the problem, whether the proportion is direct or indirect, and arrange the other number and the term wanted so that the two ratios will be equal.

Divide the product of the extremes or means by the single extreme or mean. The result will be the term wanted.

Problems in proportion are sometimes regarded as illustrations of cause and effect, in which two causes and their corresponding effects are compared, giving the following proportion:

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1st cause : 2d cause :: Ist effect : 2d effect.

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3. If 6 men earn $75 in one week, how much will 10 men earn in the same time?

4. If 16 yards of cloth cost $20, what will be the cost of 7 yards?

5. A man can buy 45 sheep for $112.50. · How much will 18 sheep cost at the same rate ?

6. If 8 horses consume 15 tons of hay in 6 months, how much hay will 14 horses consume in the same time?

7. If 6 men can do a piece of work in 45 days, how many days will it take 11 men to do the same work ?

8. If 10 men can do a piece of work in 6 days, in how many days can 13 men do the same work?

9. How many men will it require to build 60 rods of wall in the same time that 8 men can build 40 rods?

10. If 8 men can dig a ditch in 15 days, how many days will it take 13 men to dig it?

11. If 6 bushels of wheat can be bought for $7.32, how many bushels can be bought for $45?

12. How many barrels of apples can be bought for $2250, if 15 barrels cost $33.75 ?

13. If it requires 13 men to lay a certain number of bricks in 28 days, how many days will it take 9 men to lay the same number?

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14. If 165 bushels of potatoes can be raised on 132 acres of ground, how many bushels can be raised on 34 acres ?

15. If it requires 15 acres of ground to raise 405 bu. of carrots, how many acres will it require to raise 975 bu. ?

16. Five horses cost a man $626.25. What would be the cost of 13 horses at the same rate ?

17. It required 26 men to build an embankment in 80 days. How long would it require 32 men to do the same work?

18. It took 9 horses to move a stick of timber weighing 12590 pounds. How many pounds would a stick weigh which could be moved by 7 horses?

19. If an gcean steamer sails 1775 miles in 5 days, how many miles will she sail in 61 days ?

20. If a locomotive runs 96% miles in 31 hours, how many miles will it run in 57 hours ?

21. A dog is chasing a rabbit, which has 45 rods the start of the dog. The dog runs 19 rods while the rabbit runs 17. How far must the dog run before he catches the rabbit?

22. A cistern has 3 pipes. The first will fill it in 12 hours, the second in 16 hours, and the third in 18 hours. If all run together how long will it take them to fill it?

23. If it requires 15 compositors 15 days to set up a book of 675 pages, how many days will they need to set up a book of 900 pages?

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COMPOUND PROPORTION.

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495. A Compound Ratio is the product of two or more simple ratios.

496. A Compound Proportion is a proportion in which either ratio is compound.

497. PRINCIPLE. --The product of two or more proportions is a proportion.

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WRITTEN EXERCISES.

PROCESS.

198. 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 hours per day?

ANALYSIS.—A simple pro(1) 7: 6:: 2 :(1. days.)

portion is a proportion that

has but one condition. A com(2) 24:56:: 1 :(4 days.)

pound proportion has more (3) 12:10 :: 4 : (3} days.)

than one condition. The conOr,

ditions are introduced one at

a time, therefore examples in 7:6

compound proportion may be (4)

24:56:: 2:33 solved as several simple pro12:10)

portions.

The first condition Means 6 X 56 X 10 X 2

in this example is: If 6 men

=3} can mow 24 acres of grass in Extremes 7 x 24 x 12

2 days, how long will it take

7 men to do the same work? This, solved by simple proportion, (1), gives 13 days. The second condition is: If the men can mow. 24 acres of grass in 15 days, how long will it take them to mow 56 acres? This, solved by simple proportion, (2), gives 4 days. The third condition is: If the work can be done by the men in 4 days, by working 10 hours per day, how many days will it take to do the work if they work 12 hours per day? This, solved by simple proportion, (3), gives 31 days, the time it will take 7 men to mow 56 acres of grass, by working 12 hours per day, if six men can mow 24 acres in 2 days by working 10 hours per day. Or,

Since every simple proportion is an equality of ratios, the product of the three proportions, (1), (2), (3), will be an equality of ratios ; and, since 14 and 4 appear in both antecedent and consequent, they may be omitted, and the simple proportions will assume the form of the compound proportion, (4).

The problem may be stated, as in the second part of the process, by writing for the third term the term that is like the one sought, and by arranging the others in couplets, considering their relation to the ratio between the third term and the term sought.

RULE. --Solve by successive simple proportions, introducing the conditions one at a time. Or,

Use for the third term the number which is of the same kind as the term required.

Arrange the like numbers in couplets, as in simple proportion.

The product of the means divided by the product of the extremes will be the term required.

Problems in compound proportion are readily solved by cause and effect. Example 1, stated by cause and effect, is as follows: Cause. 2d Cause.

1st Effect.

2d Effect.
6 men

7 men
2 days
: 0 days

24 acres :

56 acres
10 hours 12 hours
Means 6X2 X 10 X 56

- 3}
Extremes 7 X 12 X 24

The

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2. If 15 men can dig a ditch in 45 days by working 10 hours a day, how many days will it take 20 men to dig it, by working 12 hours a day?

3. If a block of granite 6 feet long, 3 feet wide, and 2 feet thick, weighs 5940 pounds, what will be the weight of a block of the same kind, which is 9 feet long, 4 feet wide and 3 feet thick ?

4. If I place $1500 at interest for 18 months and receive $135 interest, what sum must I place at interest at the same rate, so that I may receive $275 interest in 8 months ?

5. If it cost $180 to support 5 grown persons and 3 children for 3 weeks, what will it cost to support 8 grown persons and 6 children for 7 weeks, allowing that it costs 1 as much to support a child as a grown person ?

6. If 20 men working 8 hours a day, can dig a trench 65 feet long, 9 feet wide and 6 feet deep, in 25 days, how many days will it take 25. men, working 10 hours a day, to dig a trench 75 feet long, 8 feet wide, and 7 feet deep?

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7. If it costs $240 to board 16 persons 5 weeks, how much will it cost to board 9 persons 22 weeks ?

8. If $800 placed at interest, amounts to $880 in 15 months, what sum must be placed at interest at the same rate, to amount to $975 in one year?

9. If it requires 275 yards 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?

10. If a bin which is 8 feet long, 6 feet wide and 8 feet deep, holds 309 bushels of wheat, how many bushels will a bin hold that is 14 feet long, 8 feet wide and 9 feet deep?

11. If 15 men, working 10 hours a day, can do a certain piece of work in 18 days, how many days will it require for 13 men to do the same work, by working 8 hours a day?

12. If 12 horses consume 40 bushels of oats in 8 days, how long will 140 bushels of oats last 16 horses?

13. If a regiment of 1025 soldiers consume 11500 pounds of bread in 15 days, how many pounds will 3 regiments of the same size, consume in 12 days ?

14. If the water that fills å vat, which is 8 feet long, 4 feet wide and 5 feet deep, weighs 10000 pounds, what will be the weight of the water required to fill a vat, which is 10 feet long, 5 feet wide and 6 feet deep?

15. If 5 horses eat as much as 6 cattle, and 8 horses and 12 cattle eat 12 tons of hay in 40 days, how much hay will be needed to keep 7 horses and 15 cattle 65 days?

16. If 15 men working 6 hours a day, can dig a cellar 80 feet long, 60 feet wide and 10 feet deep in 25 days, how many days will it require 25 men working 8 hours a day, to dig a cellar 120 feet long, 70 feet wide and 8 feet deep?

17. If 52 men can dig a trench 355 feet long, 60 feet wide and 8 feet deep in 15-days, how long will a trench be, that is 45 feet wide and 10 feet deep, which 45 men can dig in 25 days?

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