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COR.—It is evident that 8 men will do it in less time; that is, in one-half the time that it will take 4, which ratio is expressed by the diminishing ratio of 4 and 8, that is, $ = i, in which the given term is the numerator of the ratio and the required term the denominator.

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EXAMPLES.

1. If 5 men can dig a ditch in 20 days, how many men will dig it in 25 days? Given

5 men and 20 days. Required, ? and 25 days.

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REM.-An increase in the 25 will require less men.

2. If 6 horses eat a certain quantity of hay in 30 weeks, how many horses will consume the same quantity of hay in 9 weeks?

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3. If a man perform a journey in 12 days, when the days are 9 hours long, how many days of 12 hours will it take him?

Given
Required,

12 days and 9 hours.

? and 12 hours.

4. If 10 men can build a wall in 40 days, how many men will be requireå to build the same wall in 10 days ? Given

10 men and 40 days. Required ? and 10 days. 40 x 18

= 40 men.

An increase in the required term of the ratio demands a diminution in the unknown.

5. If 12 horses eat a certain quantity of hay in 54 weeks, how many horses will consume the same hay in 9 weeks ?

Ans. 72 horses. 6. If a man perform a journey in 24 days when the days are 9 hours long, how many days will it take him when the days are 12 hours long ? Ans. 18 days.

7. If 10 men reap 30 acres of wheat in 3 days, how long will it take 5 men to reap the same field ? Ans. 6 days.

COMPOUND RATIO.

When there are two or more ratios, it is termed Compound Ratio; thus,

If 3 men in 12 days build 40 rods of wall, how many
rods will 9 men build in 24 days ?
Given

3 men, 12 days, 40 rods.
9 men, 24 days, ?

1

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If 12 men dig a ditch 20 rods long in 18 days by work

ing 8 hours a day, how many men will dig a ditch 40 rods long in 24 days, working 6 hours a day? Given 12 men, 20 rods, 18 days, 8 hours.

? 40 rods, 24 days, 6 hours.

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12 men X

x x = 24 men.

8 EXEMPLIFICATION.—The longer the trench, the more men it will take, and the ratio is direct; but the greater the number of days and the more hours of each day, the fewer men would be required; hence these two ratios are inverse.

COR.—Each ratio must be dealt with as in the preceding article.

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1. If 4 men in 12 days build 40 rods of wall, how many rods will 6 men build in 18 days ? Given

4 men, 12 days, 40 rods. Required 6 men, 18 days, ?

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2. If 18 men dig a trench 30 rods long in 24 days by working 8 hours a day, how many men will dig a trench 50 rods long in 64 days working 6 hours a day?

Given 18 men, 30 rods, 24 days, 8 hours.
Required ? 60 rods, 64 days, 6 hours.

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Observe, the longer the trench, the more men will be required; but the more days the less men, and the more hours the less men ; the first ratio is direct, the other two inverse.

3. If 6 men in 16 days of 9 hours each, build a wall 20 feet long 6 feet high and 4 feet thick, in how many days of 8 hours each will 24 men build a wall 200 ft. long 8 ft. high and 6 ft. thick ?

Ans. 90 days. 4. If 12 men mow 24 acres of grass in 2 days of 10 hours each, how many hours a day must 16 men work to mow 80 acres in 4 days ? Given

24 acres, 2 days, 10 hours. Required 16 men, 80 acres, 4 days, ?

12 men,

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5. If $100 in 12 months gain $6, how long will it take $600 to gain $24?

Given $100 prin., 12 mo., $6 interest.
Required $600 prin., ? $24 interest.

a% * *4 = 8 months. 6. If 8 horses eat 42 bushels of oats in 24 days, how many bushels will suffice 16 horses 36 days?

Ans. 126 bushels. 7. If it cost $30 to transport 6 cwt. 2 qrs. 180 miles, what will be the cost of transportation of 19 cwt. 2 qrs. 270 miles ?

Ans. $135. 8. If 42 men in 270 days of 83 hours each can build a wall 985 yards long 71 ft. high and 27 ft. thick, in how many days of 113 hours each can 63 men build a wall 45} yds. long 61. ft. high and 31 ft. thick ? 270 x 43 x 12 X X 136 x 18 x 35 x 365 X* X = 68 da.

REM.-When all the terms are arranged in ratios, the cancella. tion is more easily performed. It is better, however, first to arrange the like terms together, and then mark the direct and indirect ratios.

9. If 14 men can reap 84 acres in 6 days, how many men must be employed to reap 44 acres in 4 days?

10. A wall 600 feet in length is to be built in 30 days; 10 men have been employed at it for 12 days, and have built 240 feet. How many more men must be employed in order to finish it in the given time?

Ans. The 10 men will finish it in the given time. 11. If 12 men make 600 pairs of shoes in 30 days, how many men will make 12000 pairs in 90 days ?

Ans. 80 men.

SIMPLE EQUATIONS.

An Equation consists of two equal members placed opposite each other with the sign of equality between them. The members are called the right and left hand members. Thus, 8 +4= 6 + 6.

8 The analysis of simple ratios may be rendered by equations, thus:

1. If 12 bushels of wheat cost $15, what will 42 bushels cost?

12 bu. = $15. Divide both members by 12. 1 bu. = $1.25. Multiply both members by 42. 42 bu. = $52.50.

That is, if 12 bushels equals or costs $15, one bushel will cost of $15; that is, $1.25 and 42 bushels will cost $1.25 x 42 = $52.50.

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