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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
13 days : 12 days 7 hours 10 hours
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 X 7, which, being done, gives 360 miles for the fourth term, or answer, as before.
:: 273 miles :
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
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 3d. Length of the ditches.
5 days: days.
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 : 5 wide :: 5 days : days. 5th. Depths. Lastly, the depths being different, we have 2 deep 3 deep 5 days: days. It would seem, therefore, that 5 days has to the fourth term, or answer, the same proportion
Men, 24: 248
that 24 men has to 248 men,
:: 5 days:
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 × 9 × 230 × 3 × 2, gives 288,84960 days for the fourth term, or answer. 288,59. But the first and second terms are the fractions 248, 48, 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 × 9 × 230 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
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. 5 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 I man; that is, 21 times 7 lbs. in 1 day, and in 3 days they would consume 3 times as much; that is, 5 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 4reapers 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 4 d. when corn is 4 s. 2 d. per bushel, what weight of it may be bought for 1 s. 2 d. when the price per bushel is 5 s. 6 d. ?
Ans. 1 lb. 4 oz. 342 pwts. 6. If $100 gain $6 in 1 year, what will $400 gain in 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 194 days' work, how much must 20 men receive for 100
Ans. 305£. 0 s. 8 d.
SUPPLEMENT TO THE SINGLE RULE OF
1. What is proportion? 2. How many numbers are required to form a ratio? 3. How many to form a proportion?
4. What is the first term 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? 12. 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?
1. If I buy 76 yds. of cloth for $113'17, what does it cost per ell English? Ans. $1'861. 2. Bought 4 pieces of Holland, each containing 24 ells English, for $96; how much was that per yard?
Ans. $0'80. 3. A garrison had provision for 8 months, at the rate of 15 ounces to each person per day; how much must be al lowed per day, in order that the provision may last 91 months? Ans. 1218 oz. 4. How much land, at $2'50 per acre, must be given in exchange for 360 acres, at $375 per acre? Ans. 540 acres.
5. Borrowed 185 quarters of corn when the price was 19 s.; how much must I pay when the price is
6. A person, owning of a coal mine, sells for 171 £.; what is the whole mine worth? of a gallon cost § of a dollar, what
17 s. 4 d. ? Ans. 2024
of his share Ans. 380£.
of a Ans. $140. Ans. 10 d.
8. At 14 £. per cwt., what cost 31 lbs. ? 9. If 4 cwt. can be carried 36 miles for 35 shillings, how many pounds can be carried 20 miles for the same money? Ans. 9074 lbs. 10. If the sun appears to move from east to west 360 degrees in 24 hours, how much is that in each hour? in each minute?
in each second?
Ans. to last, 15" of a deg. 11. If a family of 9 persons spend $450 in 5 months, how much would be sufficient to maintain them 8 months if 5 persons more were added to the family? Ans. $1120.
Note. Exercises 14th, 15th, 16th, 17th, 18th, 19th, and 20th, "Supplement to Fractions," afford additional examples in single and double proportion, should more examples be thought necessary.
¶ 98. 1. Two men own a ticket; the first owns, and the second owns of it; the ticket draws a prize of 40 dol lars; what is each man's share of the money?