COMPOUND PROPORTION.

96. It frequently happens, that the relation of the quantity required, to the given quantity of the same kind, depends upon several circumstances combined together; it is then called Compound Proportion, or Double Rule of Three.

1. If a man travel 273 miles in 13 days, travelling only 7 hours in a day, how many miles will he travel in 12 days, if he travel 10 hours in a day?

This question may be solved several ways. First, by analy

sis:

If we knew how many miles the man travelled in 1 hour, it is plain, we might take this number 10 times, which would be the number of miles he would travel in 10 hours, or in 1 of these long days, and this again, taken 12 times, would be the number of miles he would travel in 12 days, travelling 10 hours each day.

If he travel 273 miles in 13 days, he will travel of 273 miles; that is, 273 miles in 1 day of 7 hours; and 4 of 27 miles is 273 miles, the distance he travels in 1 hour: then, 10 times = 2330 miles, the distance he travels in 10 hours; and 12 times 2730 22760 360 miles, the distance he travels in 12 days, travelling 10 hours each day. Ans. 360 miles.

But the object is to show how the question may be solved by proportion:

First; it is to be regarded, that the number of miles travelled over depends upon two circumstances, viz. the num ber of days the man travels, and the number of hours he travels each day.

We will not at first consider this latter circumstance, but suppose the number of hours to be the same in each case: the question then will be,-If a man travel 273 miles in 13 days, how many miles will he travel in 12 days? This wil furnish the following proportion :

13 days 12 days: 273 miles :

miles

which gives for the fourth term, or answer, 252 miles.

Now, taking into consideration the other circumstance, or that of the hours, we must say,-If a man, travelling 7 houre a day for a certain number of days, travels 252 miles, how fur

will he travel in the same time, if he travel 10 hours in a day? This will lead to the following proportion :

7 hours 10 hours :: 252 miles :

miles.

This gives for the fourth term, or answer, 360 miles.

We see, then, that 273 miles has to the fourth term, or answer, the same proportion that 13 days has to 12 days, and that 7 hours has to 10 hours. Stating this in the form of a proportion, we have

days 12 days

13 hours: 10 hours }

:: 273 miles :

miles

by which it appears, that 273 is to be multiplied by both 12 and 10; that is, 273 is to be multiplied by the product of 12 X 10, and divided by the product of 13 × 7, which, being done, gives 360 miles for the fourth term, or answer, as before..

In the same manner, any question relating to compound proportion, however complicated, may be stated and solved.

2. If 248 men, in 5 days, of 11 hours each, can dig a trench 230 yards long, 3 wide, and 2 deep, in how many days, of 9 hours each, will 24 men dig a trench 420 yards long, 5 wide, and 3 deep?

Here the number of days, in which the proposed work can be done, depends on five circumstances, viz. the number of men employed, the number of hours they work each day, the length, breadth, and depth of the trench. We will consider the question in relation to each of these circumstances, in the order in which they have been named :

1st. The number of men employed. Were all the circumstances in the two cases alike, except the number of men and the number of days, the question would consist only in find ing in how many days 24 men would perform the work which 248 men had done in 5 days; we should then have

days.

24 men: 248 men :: 5 days: 2d. Hours in a day. But the first labourers worked 11 hours in a day, whereas the others worked only 9; less hours will require more days, which will give

9 hours 11 hours

5 days :

days.

3d. Length of the ditches. The ditches being of unequal

length, as many more days will be necessary as the second is longer than the first; hence we shall have

230 length: 420 length: 5 days: ....... days.

4th. Widths. Taking into consideration the widths, which are different, we have

3 wide

5th. Depths. 2 deep

Lastly, the depths being different, we have 3 deep 5 days:

days.

It would seem, therefore, that 5 days has to the fourth term, or answer, the same proportion

that 24 men has to 248 men, whose ratio is

that 9 hours has to 11 hours, the ratio of which is that 230 length has to 420 length,

3 width has to

5 width,

3 depth,

that
that 2 depth has to

................

all which stated in form of a proportion, we have

248

$38,

97. The continued product of all the second terms 248 X 11 X 420 × 5 × 3, multiplied by the third term, 5 days, and this product divided by the continued product of the first terms, 24 X 9 X 230 × 3 × 2, gives 288. 288,84960 days for the fourth term, or answer.

But the first and second terms are the fractions 24, 130, and, which express the ratios of the men, and of the hours, of the lengths, widths and depths of the two ditches. Hence it follows, that the ratio of the number of days given to the number of days sought, is equal to the product of all the ratios, which result from a comparison of the terms relating to each circumstance of the question. The product of all the ratios is found by multiplying to248 X 11 X 420 gether the fractions which express them, thus, 24 X 9 X 230

X5X3

X3X2

17186400

298080

17186400

and this fraction,

298080 represents the

ratio of the quantity required to the given quantity of the same kind. A ratio resulting in this manner, from the multiplice tion of several ratios, is called a compound ratio.

From the examples and illustrations now given we deduce the following general

RULE

for solving questions in compound proportion, or double rule of three, viz.-Make that number which is of the same kind with the required answer, the third term; and, of the remaining numbers, take away two that are of the same kind, and arrange them according to the directions given in simple proportion; then, any other two of the same kind, and so on till all are used.

Lastly, multiply the third term by the continued product of the second terms, and divide the result by the continued product of the first terms, and the quotient will be the fourth term, or answer required.

EXAMPLES FOR PRACTICE.

1. If 6 men build a wall 20 ft. long, 6 ft. high, and 4 ft. thick, in 16 days, in what time will 24 men build one 200 ft. long, 8 ft. high, and 6 ft. thick? Ans. 80 days.

2. If the freight of 9 hhds. of sugar, each weighing 12 cwt., 20 leagues, cost 16 £., what must be paid for the freight of 50 tierces, each weighing 24 cwt., 100 leagues? Ans. 92 £. 11 s. 10 d.

3. If 56 lbs. of bread be sufficient for 7 men 14 days, how much bread will serve 21 men 3 days? Ans. 36 lbs.

The same by analysis. If 7 men consume 56 lbs. of bread, 1 man, in the same time, would consume of 56 lbs. = 6 lbs.; and if he consume 56 lbs. in 14 days, he would consume of 56 = 8 lb. in 1 day. 21 men would consume 21 times so much as 1 man; that is, 21 times = 7 lbs. in 1 day, and in 3 days they would consume 3 times as much; that is, 353 = 36 lbs., as before.

Ans. 36 lbs.

Note. Having wrought the following examples by the rule of proportion, let the pupil be required to do the same by analysis.

4. If 4 reapers receive $11'04 for 3 days' work, how many men may be hired 16 days for $103'04 ?

Ans. 7 men

5. If 7 oz. 5 pwt. of bread be bought for 44 d. when corn is 4 s. 2 d. per bushel, what weight of it may be bought for 1. 2 d. when the price per bushel is 5 s. 6 d. ?

Ans. 1 lb. 4 oz. 3474 pwts. 6. If $100 gain $6 in 1 year, what will $400 gain in 9 months?

Note. This and the three following examples reciprocally prove each other.

7. If $100 gain $6 in 1 year, in what time will $400 gain $18?

8. If $400 gain $18 in 9 months, what is the rate per cent. per annum ?

9. What principal, at 6 per cent. per. ann., will gain $18

in 9 months?

10. A usurer put out $75 at interest, and, at the end of 8 months, received, for principal and interest, $79; I demand at what rate per cent. he received interest.

Ans. 8 per cent. 11. If 3 men receive 8 £. for 19 days' work, how much must 20 men receive for 100 days'?

Ans. 305. 0 s. 8 d.

SUPPLEMENT TO THE SINGLE RULE OF THREE.

QUESTIONS.

1. What is proportion? 2. How many numbers are required to form a ra? 3. How many to form a proportion? 4. What is the firerm of a ratio called? 5. the second

term? 6. Which is taken for the numerator, and which for the denominator of the fraction expressing the ratio? 7. How may it be known when four numbers are in proportion? 8. Having three terms in a proportion given, how may the fourth term be found? 9. What is the operation, by which the fourth term is found, called? 10. How does a ratio become inverted ? 11. What is the rule in proportion? In what denomination will the fourth term, or answer, be found? 13. If the first and second terms contain different denominations, what is to be done? 14. What is compound proportion, or double rule of three ? 15. Rule?

12.