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PROOF.-4 × 72=12 × 24. (763, PRIN. 1.) In practice, that number which is of the same unit value as the required term, is generally made the antecedent of the second couplet or third term of the proportion, and the required term, a, the fourth term. The terms of the first couplet are so arranged as to have the same ratio to each other, as the terms of the second couplet, have to each other, which is easily determined by inspection. The product of the means 12 and 24, divided by the given extreme 4, gives the other extreme, or required term, $72. (763, PRIN. 3.)

Drill exercises like the following, will soon make the pupil familiar with the principles and operations of proportion.

2. If 4 horses eat 12 bushels of oats in a given time, how many bushels will 20 horses eat in the same time?

In this example, what two numbers have the same unit value ? What do they form? What is the denomination of the third term? Of the required term? What is the antecedent of the second couplet? From the conditions of the question, is the consequent of the second couplet or required term, greater or less than the antecedent? If greater, how must the antecedent and consequent of the first couplet compare with each other? If less, how compare? What is the ratio of the first couplet? Why not 20 to 4? Make the statement. How is the required term found?

3. If 96 cords of wood cost $240, what will 40 cords cost? 4. If 20 lb. of sugar cost $1.80, find the cost of 45 lb. 5. If 18 bu. of wheat make 4 barrels of flour, how many barrels will 200 bu. make?

RULE.-I. Make the statement so that two of the given numbers which are of the same unit value, shall form the first couplet of the proportion, and have a ratio equal to the ratio of the third given term to the required term.

II. Divide the product of the means by the given extreme, and the quotient will be the number required.

CAUSE AND EFFECT.

768. The terms of a proportion have not only the relations of magnitude, but also the relations of cause and effect.

Every problem in proportion may be considered as a comparison of two causes and two effects.

Thus, if 4 tons as a cause will bring when sold, $24 as an effect, 12 tons as a cause will bring $72 as an effect. Or, if 6 horses as a cause draw 10 tons as an effect, 9 horses as a cause will draw 15 tons as an effect.

769. Since like causes produce like effects, the ratio of two like causes equals the ratio of two like effects produced by these causes. Hence,

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

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WRITTEN EXERCISES.

70. 1. If 8 men earn $32 in one week, how much will 15 men earn at the same rate, in the same time?

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The effect of the first cause is $32 earned, the effect of the second cause is $r earned, or the required term. Since like effects have the same ratio as their causes (769), the causes may form the first couplet, and the effects the second couplet of the proportion. The required term is readily obtained by (763, 3).

2. If 20 bushels of wheat produce 6 barrels of flour, how many bushels will be required to produce 24 barrels ?

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ANALYSIS.-In this ex

ample a cause is required. The first cause is 20 bu., the second cause is x bu. or the required term.

The effect of the first

cause is 6 bbl. of flour, Since like causes

the effect of the second cause is 24 bbl. of flour. have the same ratio as their effects (769), the statement is made as in Ex. 1, and the required term found by (763, 2).

3. If 5 horses consume 10 tons of hay in 8 mo., how many horses will consume 18 tons in the same time?

Drill Exercise.-In this example, what is the first cause? The second cause? The first effect? The second effect? Is the required term a cause or an effect? A mean or an extreme? What is the first couplet? What, the second? Make the statement. How is the required term found?

4. If 8 yards of cloth cost $6, how many yards can be bought for $75?

5. How many men will be required to build 32 rods of wall in the same time that 5 men can build 10 rods?

RULE.-I. Arrange the terms in the statement so that the ratio of the causes which form the first couplet, shall equal the ratio of the effects which form the second couplet, putting x in the place of the required term.

II. If the required term be an extreme, divide the product of the means by the given extreme; if the required term be a mean, divide the product of the extremes by the given mean.

To shorten the operation, equal factors in the first and second, or in the first and third terms may be canceled.

Solve the following by either of the foregoing methods: 6. If 5 sheep can be bought for $20.75, how many sheep can be bought for $398.40?

7. When 10 barrels of flour cost $112.50, what will be the cost of 476 barrels of flour?

8. If a railroad train run 30 miles in 50 min., in what time will it run 260 miles?

9. How many bushels of peaches can be purchased for $454.40, if 8 bushels cost $10.24?

10. If a horse travel 12 miles in 1 hr. 36 min., how far, at the same rate, will he travel in 15 hours?

11. How many days will 12 men require to do a piece of work, that 95 men can do in 7 days?

12. If of an acre of land cost $60, what will 45 acres cost?

13. At the rate of 72 yards for £44 16s., how many yards of cloth can be bought for £5 12s.?

14. If of a barrel of cider cost $1, what is the cost of of a barrel?

15. If the annual rent of 35 A. 90 P. is $284.50, how much land can be rented for $374.70 ?

16. What will 87.5 yd. of cloth cost, if 14 yd. cost $1.26? 17. If by selling $5000 worth of dry goods, a merchant gains $456.25, what amount must he sell to gain $1000?

18. Bought coal at $4.48 per long ton, and sold it at $7.25 per short ton. What was the gain per ton?

19. What will be the cost of a pile of wood 80 ft. long, 4 ft. wide, 4 ft. high, if a pile 18 ft. long, 4 ft. wide, 6 ft. high cost $30.24 ?

20. If 36 bu. of wheat are bought for $44.50, and sold for $53.50, what is gained on 480 bu. at the same rate ? 21. If a business yield $700 net profits in 1 yr. 8 mo., in what time will the same business yield $1050 profits?

COMPOUND PROPORTION.

1. A Compound Proportion is an expression of equality between two ratios, one or both of which are compound.

All the terms of every problem in compound proportion appear in couplets, except one, and this is always of the same unit value as the required term.

The order of the ratios, and of the terms composing the ratios, is the same as in simple proportion.

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772. 1. If 18 men build 126 rd. of wall in 60 da., working 10 hr. a day, how many rods will 6 men build in 110 da., working 12 hr. a day?

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ANALYSIS.-All the terms in this example appear in couplets, except 126 rods, which is of the same unit value as the required term, and is made the third term of the proportion, and x rods, the fourth. The required number of rods depends upon three conditions: 1st, the number of men employed; 2d, the number of days they work; and 3d, the number of hours they work each day.

Consider each condition separately, and arrange the terms of the same unit value in couplets, and make the statement as in simple proportion (767). Then find the required term by (763, 3),

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