48. There are two flocks of sheep which are to each other as 15 to 20, and the greater contains 500: how many does the less contain?

49. An express traveling 60 miles a day had been dispatched 5 days, when a second was sent after him traveling 75 miles a day: how long will it take the latter to overtake the former ?

50. A fox has 150 rods the start of a hound, but the hound runs 8 rods while the fox runs 5 rods: how far must the hound run before he catches the fox?

51. A stack of hay will keep a cow 20 weeks, and a horse 15 weeks: how long will it keep them both?

52. A traveler divided 80s. among 4 beggars in such a manner, that as often as the first received 10s., the second received 9s., the third 8s., and the fourth 7s.: what did each receive?

53. Pure water is composed of oxygen and hydrogen in the ratio of 8 to 1 by weight: what is the weight of each in a cubic foot of water, or 1000 ounces avoirdupois?

COMPOUND PROPORTION.

507. COMPOUND PROPORTION is an equality between a compound ratio and a simple one. (Arts. 479, 480.)

Thus, 6:3

Into 4:2

123, is a compound proportion.

That is, 6X4: 3×2::12:3; for, 6×4X3=3×2×12.

OBS. Compound proportion is chiefly applied to the solution of examples which would require two or more statements in simple proportion. It is sometimes called Double Rule of Three.

Ex. 1. If 8 men can reap 32 acres in 6 days, how many acres can 12 men reap in 15 days?

Suggestion. When stated in the form of a compound proportion, the question will stand thus:

That is, the product of the antecedents 8×6, has the same

QUEST.-507. What is compound proportion? Obs. To what is it chiefly applied 1 What is it sometimes called?

ratio to the product of the consequents 12X15, as 32 has to the answer; or simply, 8 into 6: 12 into 15: 32 to the answer.

And

Operation. 32X12X15=5760, 8X 6=48.

Now 5760 48 120.

Ans. 120 acres.

The product of the numbers standing in the 2d and 3d places divided by the product of those standing in the first place, will give the answer.

Note.-The learner will observe that it is not the ratio of 8 to 12 alone, nor that of 6 to 15, which is equal to the ratio of 32 to the answer, as it is sometimes stated; but it is the ratio compounded of 8 to 12, and 6 to 15, which is equal to the ratio of 32 to the answer. Thus, 8×6: 12×15:: 32: 120, the answer. A compound proportion when stated as above, is read, "the ratio of 3X6 is to 12X15 as 32 is to the answer."

2. If 6 men can earn £42 in 60 days working 8 hours per day, how much can 10 men earn in 84 days working 12 hours a day?

Operation.

6m.: 10m. 60d. 84d.

8hrs. 12hrs.

Now

}

£42 to Ans.

State the question, then multiply and divide, as before.

10×84×12×42=423360; and 6×60×8=2880.
4233602880=147. Ans. £147.

508. From the foregoing illustrations we derive the following general

RULE FOR COMPOUND PROPORTION.

I. Place that number which is of the same kind as the unswer equired for the third term.

II. Then take the other numbers in pairs, or two of a kind, and arrange them as in simple proportion. (Art. 503.)

III. Finally, multiply together all the second and third terms, divide the result by the product of the first terms, and the quotient will be the fourth term or answer required.

QUEST.-508. In stating a question in compound proportion, which number do you put for the third term? How arrange the other numbers? Having stated the question, how is the answer found?

PROOF.-Multiply the answer into all of the first terms or ante cedents of the first couplets, and if the product is equal to the con tinued product of all the second and third terms, the work is right. (Art. 498.)

OBS. 1. Among the given numbers there is but one which is of the same kind as the answer. This is sometimes called the odd term, and is always to be placed for the third term.

2. If the antecedent and consequent of any couplet are compound numbers, they must be reduced to the lowest denomination mentioned in either, before the multiplication is performed. When the third term contains different denominations, it must also be reduced to the lowest mentioned in it.

3. Questions in Compound Proportion may be solved by Analysis; also by Simple Proportion, by making two or more separate statements.

3. If 12 horses can plough 11 acres in 5 days, how many horses can plough 33 acres in 18 days?

4. If a man walking 12 hours a day, can travel 250 miles in 10 days, how long will it take him to travel 400 miles, if he walks but 10 hours a day?

5. If 40 gallons of water will last 20 persons 5 days, how many gallons will 9 persons drink in a year?

6. If 16 laborers can earn £15, 12s. in 18 days, how many laborers will it take to earn £35, 2s. in 24 days?

COMPOUND PROPORTION BY CANCELATION.

7. If a person can make 60 rods of wall in 45 days, working 12 hours a day, how many rods can he make in 72 days, working 8 hours a day?

QUEST.-How are questions in compound proportion proved? Obs. Among the given numbers, how many are of the same kind as the answer? Cin questions in compound proportion be solved in any other way?

509. When the first terms have factors common to the second or third terms.

Cancel the factors which are common, then divide the product of those remaining in the second and third terms by the product of those remaining in the first, and the quotient will be the answer.

PROOF. Place the answer in the denominator, or on the left of the perpendicular line, and if the factors of the divisor and dividend exactly cancel each other, the work is right.

OBS. 1. Instead of placing points between the antecedents and consequents of the left hand couplets of the proportion, it is sometimes more convenient to put a perpendicular line between them, as in division of fractions. (Art. 232.) This will bring all the terms whose product is to be divided on the right of the line, and those whose product is to form the divisor, on the left. In this case the third term should be placed below the second terms, with the sign of proportion (::) before it, to show its origin, and its relation to the answer.

2. It will be observed that Cancelation can be applied in Compound Proportion to all those examples whose first terms have factors common to the second terms, or to the third term.

8. If 24 men can saw 90 cords of wood in 6 days, when the days are 9 hours long, how many cords can 8 men saw in 36 days, when they are 12 hours long?

9. If 6 men can make 120 pair of boots in 20 days, working 8 hours a day, how long will it take 12 men to make 360 pair, working 10 hours a day?

10. If 12 men can build a wall 30 ft. long, 6 ft. high, and 4 ft. thick, in 18 days, how long will it take 36 men to build one 360 ft. long, 8 ft. high, and 6 ft. thick.

11. If a horse can travel 120 miles in 4 days when the days are 8 hours long, how far can he travel in 30 days when the days are 10 hours long?

QUEST.-509. When the first terms have factors common to the second or third terms hcw proceed?

12. If $250 gain $30 in 2 years, what will be the interest of

$750 for 5 years?

13. What will be the interest of $500 for 4 years, if $600 will gain $42 in 1 year?

14. If $360 gain $14.40 in 8 months, what will $4800 gain in 32 months?

15. If a family of 8 persons spend $200 in 9 months, how much will 18 persons spend in 12 months?

16. If 15 men, working 12 hours a day, can hoe 60 acres in 20 days, how long will it take 30 boys, working 10 hours a day, to hoe 96 acres, 6 men being equal to 10 boys?

CONJOINED PROPORTION.

510. When each antecedent of a compound ratio is equal in value to its consequent, the proportion is called Conjoined Proportion.

OBS. Conjoined Proportion is often called the chain rule. It is chiefly used in comparing the coins, weights and measures of two countries, through the medium of those of other countries, and in the higher operations of exchange. The odd term is sometimes called the demand.

17. If 20 lbs. United States make 12 lbs. in Spain; and 15 lbs. Spain 20 lbs. in Denmark; and 40 lbs. Denmark 60 lbs. in Russia: how many pounds in Russia are equal to 100 lbs. U. S.?

Arrange the given terms in pairs, making the first term the antecedent, and its equal the consequent; then since it is required to find how many of the last kind are equal to a

given number (100 lbs.) of the first, place the odd term or de

mand under the consequents.

Then, 20X15X40:12 × 20×60 :: 100: Ans.

That is 201 12

1$ 20 10,40

60, A

:: 100, 10

Ans. 12x10=120 lbs.

Cancel the factors common to both sides, and the product of those remaining on the right divided by the product of those on the left, is the answer.