4 pounds; and the tare, 1 hundred weight, 3 quarters, 12 pounds. Subtracting the tare from the gross weight, the result is the suttleweight; then by arranging the quantities, as in the last example, and proceeding by compound division, we find the neat weight to be 38 hundredweight, 1 quarter, 16 pounds.

Exercises.-92. In 25 barrels of figs, each 2cwt. 1qr. gross, tare 16lbs. per cwt., how much neat?

and

Ans. 48cwt. Oqr. 24lbs. 93. In 152cwt. 1qr. 3lbs. gross, tare 10lbs. per cwt., tret as usual; how much neat?

Ans. 133cwt. 1qr. 10lbs. 133 oz. 94. What is the value of 13hhds. of tobacco, at $7.50 per cut. each weighing 4cwt. 3qrs. 17lbs. gross, tare 13lbs. Ans. $422 45 per cut.? 95. Find the neat weight of 22 bags, weighing gross 45cwt. 1qr. 19lbs., tare 5lbs. per bag, tret as usual.

485

Ans. 42cwt. 2qrs. 2515 lbs. 96. Find the neat weight of 35 barrels of anchovies, each weighing 1qr. 12lbs., tare 14lbs. per cat.

Ans. 10cwt. 3qr. 21lb. 97. What is, the value of 26hhds. of tobacco, at $7.75 per cut. neat weight, each weighing 4cwt. 2qrs., tare 13lbs... Ans. $801.50-21. per cut.? 98. What is the price of 50 casks of butter, gross weight 45cwt. 1qr. 25lbs., tare 15lbs. per cask, at $21 per cwt. neatAns. $814.50. weight?

99. I want to know the height of a tree, by means of the length of its shadow; I set up a straight stick that measures above the ground 3 feet, 4 inches; the shadow of this is 5 feet, 2 inches, and the shadow of the tree, at the same moAns. 123ft. 8 in. ment, I find to be 79 feet, 10 inches.

Questions.

What is simple proportion?

What is its object?

What is compound proportion?

Repeat the rule for finding a fourth proportional to three given numbers.

What is direct proportion?

What is inverse proportion?

If there be a remainder after dividing the product of the second and third by the first term, how do you proceed?

When the first term is an unit, how is the answer found? When the second or third term is an unit, how is the answer found?

When the first and second terms are not of the same denomination, how are you to proceed with the operation ? What is to be done with the third term, when it is of more denominations than one?

When the first and second, or the first and third terms, admit of a common measure, how is the process to be performed?

Compound Proportion.

138. RULE I.-By the rule for simple proportion, find a fourth proportional to two given terms of the same kind with one another, and to the term which is of the same kind as the answer. To two other terms of the same kind, and to the last obtained, find a fourth proportional; and thus proceed if there be more terms: the final result will be the answer.

139. RULE II.-Place the term which is of the same kind as the required term, in the last place. Comparing the other given terms by pairs, place each as antecedent or consequent, according to the rule for simple proportion. Divide the continual product of all the consequents, and the last term, by the continual product of all the antecedents; the quotient will be the

answer.

As a contraction in the use of the latter rule, divide an antecedent and any consequent by any number that will measure them, and employ the results instead of those terms or if an antecedent and any consequent, or an antecedent and the last term be the same, reject them.

:

Example 1. If 40 gallons of ale serve 17 persons for 5 days, how many gallons will serve 9 persons for a year, at . the same rate?

I. As 5 days

: 365 days :: 40 gal. 2920 gal.

As 17 persons 9 persons: 2920 gal.: 15451 gal. In this operation, we first proceed as if the number of persons in both cases were 17, and on this supposition we find that 2920 gallons would be used by those persons in a

year. But the number of persons being 9, instead of 17, we find by the second analogy, that if 17 persons would use 2920 gallons in a year, 9 persons would use 154515 gallons in the same time.

17 persons : 9 persons

:: 40gals. 15451 gals.

In the second method that term is placed last which is of the same kind as the required term. Thus, were the num ber of persons the same, it is evident that the answer would be greater than 40 gallons; and therefore we put 365 days in the second and 5 days in the first place; but were the number of days the same, it is obvious that the required term would be less than 40 gallons; and therefore we put 9 persons in the second place and 17 in the first. We then multiply the product of the consequents by the common third term, 40, and divide the result by the product of the antecedents, and the same answer is found as before. This operation is in effect the same as that in the first method; for the result of the first single analogy, without the actual 365 X 40 multiplication and division, is

5

the second analogy becomes 17: 9 ::

and, consequently,

365x40 9X365 X 40

5

:

5X17 →whence it appears that in both methods the same multiplications and divisions are in reality performed, and consequently the one is only a modification of the other. this method, 5 and 365, or 5 and 40, might be divided by 5 as a contraction.*

In

•Exam. 2. If a family of 13 persons spend $64 on butcher's meat in 8 months when the meat is 6 cents per pound, how much, at the same rate, should a family of 12 persons spend in 9 months, when the meat is 6 cents per pound?

As 13 persons : 12 persons

8 months : 9 months :: $64: $72

6 cents.

: 61 cents

Here the last ratio being the same with that of 12 : 13,

*The principles on which the operations in Compound and in Simple Proportion depend are the same.

It may be remarked that one very considerable advantage which the second rule possesses, is, that by it the operation is kept entirely free from fractions till the conclusion; while in the other mode fractions often arise from the first analogy, and render the remaining part of the work more intricate and difficult.

the terms of the first and last ratios may be erased, and therefore as 89: $64 to the answer, which is known by inspection to be $72.

Exam. 3. If the carriage of 66cwt. for 42 American miles be $30, what is the carriage of 36 hundreds, long weight, (see § 37, 40,) for 90 Irish miles, at the same rate?

As 66cwt. 36 long cwt.

In this example, since the answer is to be in money, $30 is placed as the common third term. Then it is evident that were all things alike except the number of hundreds, the answer would be less than $30; and therefore 36 is put as consequent and 66 as antecedent. But had the for

mer number of hundreds been equal, the answer must have been greater than $30,.on account of the difference in their magnitudes; and therefore 112 is made antecedent and 120 consequent. The arrangement of the remaining terms proceeds on similar principles, and the answer is found by dividing the continual product of $30 and the consequents by the continual product of the antecedents.

The operation is 11 : 6 (14): 15

much abbreviated,

as in the margin, by

dividing the terms

7: 15

11 : (14)

$30 $46.63439.

of the first and third ratios by 6, and those of the second by 8, and neglecting in the work 14, which occurs as an antecedent in one ratio, and a consequent in another.*

Exercises.

1. If 3 masons, working 7 hours a day, build a wall in 6 days, how many hours a day must 4 masons work in order to build it in 5 days? Ans. 6 hours, 18 minutes.

* This question might also have been wrought by four analogies in Simple Proportion. Every one of these, however, would have given origin to fractions, the neglecting of which would have prevented the precise result from being obtained. The work might also have been modified by reducing the hundreds and the miles, respectively, to the same kind. The using 112, indeed, as an antecedent and 120 as a consequent, serves this purpose, as the hundreds are thus reduced to pounds and the multiplication by 11 and 14 serves a similar purpose in relation to the miles.

2. If 9 bushels of corn serve 7 horses 10 days, how many bushels at the same rate will serve 20 horses 21 Ans. 54 bushels. days?

3. If a family of 19 persons expend $235 in 8 months, how much at the same rate will a family of 12 persons exAns. $92.76.3. pend in 5 months?

4. If 96 men, working 9 hours for 10 days, can dig a trench 400 yards long, 3 wide, and 2 deep, in how many days at the same rate can 108 men, working 7 hours a day, dig a trench of 175 yards long, 4 wide, and 3 deep?

Ans. 10 days.

5. If the carriage of 59cwt. 19 miles cost £ 2 16s. how how far may 43cwt. be carried at the same rate for £2 48.? Ans. 2081 miles. 6. If the carriage of 13cwt. 65 miles cost $9, how many hundreds may be carried 40 miles at the same rate for $15? Ans. 35 cwt.

7. If 12 oxen in 5 days plough 11 acres, would plough 33 acres in 18 days?

how

many oxen Ans. 10.

8. If a person walking 12 hours each day, perform a journey of 250 miles in 9 days, in how many days, walking 10 hours each day at the same rate, would he complete a journey of 400 miles?

Ans. 17

days.

9. If the expenses of a family of 8 persons amount to $42 in 16 weeks, how long will $100 support a family of Ans. 5050 weeks. 6 persons at the same rate? 10 If 29 men, in 5 days of 12 hours each, reap 32 American acres of wheat, in how many days of 13 hours each, will 20 men, working equally, reap 40 Irish acres?

1573

Ans. 13,86 days. 11. If $312 pay 16 labourers for 18 days, how many labourers, at the same rate, will $702 pay for 24 days? Ans. 27. 12. If 36 yards of cloth, 7 quarters wide, cost $504, what cost 120 yards of the same quality, but only 5 quarAns. $1200. ters wide?

13. If a tradesman earn 16 guineas in 108 days, how many sovereigns would he earn, at the same rate, in 270 days; 20 guineas being equivalent to 21 sovereigns? Ans. 42.

15. If 3000 copies of a history of the United States, of paper, how each containing 11 sheets, require 66 reams much paper will 5000 take if the work be extended to 121 Ans. 125 reams. sheets?