TARE AND TRET.

The whole weight of any commodity, together with the weight of the box, barrel, &c., which contains it, is called its GROSS WEIGHT.

The weight of the box, barrel, &c., which contains any commodity, is called its TARE.

The SUTTLE WEIGHT is what remains after the tare is deducted.

TRET is an allowance of 4 lbs. for each 104 tbs. of the suttle, in goods that are liable to waste.

CLOFF is an old-fashioned name for an allowance of 2 tbs. which used to be made in every 3 cwt. of what remains after the other deductions are made. This allowance was for the turning of the beam in weighing the goods. Nowadays balances are so good that no such allowance is needed.

The REAL or NET WEIGHT is what remains after all deductions are made.

The arithmetical working of allowances and deductions such as those described in Tare and Tret is so simple that neither rules nor examples are needed.

COMPOUND PROPORTION.

RULE I. (1.) 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. (2.) To two other terms of the same kind, and to the proportional last obtained find a fourth proportional; and thus proceed if there be more terms: the final result will be the

answer.

RULE II. (1.) Place the term which is of the same kind as the required term, in the last place. (2.) Comparing the other given terms by pairs, place each as antecedent or consequent, according to the general rule for simple proportion. (3.) Divide the continual product of all the consequents and the last term, by the

To find the price at 17s. 4d.: Find the price at 16s., by the method given in page 206; and for the price at 1s. 4d. take a twelfth of the result: and to find the price at 14s. 8d., from the price at 16s., take a twelfth of itself. In like manner, to find the price at 15s. 2d., to the price at 14s., add a twelfth of itself; and to find the price at 12s. 10d., from the price at 14s., take a twelfth of itself: and, lastly, to find the price at 19s. 6d., to the price at 188., add a twelfth of itself; and to find the price at 16s. 6d., from the price at 18s., take a twelfth of itself.

continual product of all the antecedents; the quotient will be the answer.

As a contraction in the use of this latter rule, divide an antecedent, and either the last term, or any consequent, by any number that will divide them without remainders, 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.

Exam. 1. If 40 gallons of ale serve 17 persons 5 days, how many gallons will 9 persons use in a year, at the same rate? I. As 5 days: 365 days::40 gallons: 2920 gallons.

As 17 persons 9 persons:: 2920 gallons: 154515 gallons. 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 serve those persons for 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 only 154515 gallons in the same time.

II. As 5 days: 365 days 17 persons 9 persons

}

::40 gallons: 1545 gallons.

In this method, that term is placed last which is of the same kind as the required term. Then were the number of persons the same, the answer would evidently 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, the required term would be less than 40 gallons; wherefore, we put 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 365 x 40 the actual multiplication and division, is 5

quently, the second analogy becomes 17: 9::

and, conse

365 x 40, 9 × 365 × 40

5

5 x 17

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. In this method 5 and 365, or 5 and 40, might be divided by 5 as a contraction.

One very considerable advantage belonging to the second rule is, that by it the operation is kept 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.

Exam. 2. If 15 men, working 12 hours daily, reap 60 acres in 16 days, in what time would 20 women, working 10 hours daily,

reap 98 acres, 7 men being able to reap as much as 8 women, in the same time?

As 20 persons: 15 persons

10 hours

60 acres
7

: 12 hours
: 98 acres
: 8

::16 days: 262 days.

In this example, since the answer is to be in time, we place 16 days as the common third term. Then, it is evident, that, were all things alike, except the number of workers, the time required would be less than 16 days, 20 persons requiring less time than 15; and therefore 15 is put as consequent, and 20 as antecedent. As, however, the women work only 10 hours daily, while the men work 12, the required time would be longer on this account; therefore 12 is put in the second column, and 10 in the first. The other terms are arranged on similar principles; and the answer is found by dividing the continual product of 16 and the consequents, by the continual product of the antecedents.

(4): 3 5: (6) 5(10)(60): 14

::16: 262

The work may be shortened, as in the margin, by dividing 20 and 15 by 5, and 10 and 12 by 2; then, by omitting 7, and writing 14 instead of 98, as 98 is equal to 14 times 7. For similar reasons, we omit 6, and write 10 instead of 60: we omit also 4, and write 2 in place of 8; and, lastly, this 2 is omitted, and the antecedent 10 is changed into 5. We then divide 672, the continual product of 3, 14, and 16, by 25, the product of 5 and 5, and we obtain the same answer, as before,

(8)(2)

This question might also have been wrought by means of four operations in simple proportion. All these, however, except one, would give origin to fractions, the neglecting of which would prevent the true answer from being found. In every respect, therefore, the second rule is greatly preferable.

Exer. 1. If the carriage of 59 cwt., 19 miles, cost £2 - 16, how far may 43 cwt. be carried, at the same rate, for £2-4? Answ. 20291 miles.

602

2. If the carriage of 13 cwt., 65 miles, cost £2 - 5, how many hundreds may be carried 40 miles, at the same rate, for £3 - 15? Answ. 354 cwt.

3. If 12 horses plough 11 acres in 5 days, how many horses would plough 33 acres in 18 days? Answ. 10.

4. If a man walking 12 hours each day, travel 250 miles in 9 days, in how many days, walking 10 hours each, at the same rate, would he travel 400 miles? Answ. 17 days.

5. If the expenses of a family of 8 persons, amount to £42 in 16 weeks, how long will £100 support a family of 6 persons, at the same rate? Answ. 50 weeks.

6. If 29 men, in 5 days of 12 hours each, reap 32 acres, in how many days of 13 hours each, will 20 men, working equally, reap Answ. 812 days.

40 acres?