2. If 6 acres and 2 roods of land can be bought for $80, how much may be bought for $100?

Ans. 8 acres and 20 sq. rods.

3. How many pounds sterling are equal in value to 24 l. 15s. New-England currency, if 4s. 6d., or 54d., of the former currency be equal to 6s., or 72d., of the latter?

As 72d.: 54d. :: 24l. 15s. : the 4th term.

4)72
-18

54-the 2d term.

4)24.. 15
-6.. 3..9

L 18.. 11 3 Ans.

Here, the first term diminished by of itself is equal to the second term; and therefore the third term diminished by of itself is the answer.

4. How many pounds, &c. New-England currency, are equal in value to 8l. 16s. 4d. New-York currency; 6 shillings of the former currency being equal to 8 shillings of the latter? Ans. 67. 12s. 3d.

Note. The two last methods of contraction are very useful; for we may not only solve some particular questions by them, but also find general rules, for making, in the shortest manner possible,many numerical calculations which frequently occur in transacting business. The particular rules for the Reduction of Currencies, given in the 3d Problem in Exchange, are found by these methods of contraetion.

Solution of questions in Simple Proportion by Analysis.

Questions in Simple Proportion may sometimes be easily solved by analysis, that is, by general principles, without the formality of stating the proportions. This method of solving such questions may be illustrated by a few examples.

Ex. 1. If 2 yards of cloth cost $4.20, what would 7 yards cost?

It is evident that if 2 yards cost $4.20, one yard would cost one-half of $4.20, viz. $2.10; and 7 yards would cost 7 times as much; that is, $2.10×7=$14.70, Ans. 2. If 8 sheep cost $10, what would 3 sheep cost? Ans. $3.75 3. If a staff, 5 feet, 8 inches, in length, cast a shadow of

6 feet, how high is that steeple whose shadow measures 153 feet?

68

If 6 feet shadow require a staff of 5 ft. 8 in.=68 inches, one foot shadow will require a staff of of 68 inches, or 58 inches, and 153 feet shadow will require 153 times as much; 68×153 10404 =1734 inches=144 feet, Ans. 6.

that is,

6

4. If 4 tons of hay will keep 3 horses through the winter, how many tons will keep 15 horses the same time?

Ans. 20.

PRACTICE,

Is a contraction of the Single Rule of Three when the first term is 1; and has its name from its frequent use in business, being a concise method of finding the value of any number of articles when the price of one is known.

The method of proof is by the Rule of Three, or by varying the order of the question.

Note 1.-One number is said to be an aliquot part of another, when the former will divide the latter without a remainder: So, 2 is an aliquot part of 8, and 5 is an aliquot part of 15. Therefore, to find whether any given number is an aliquot part of a greater number, divide the greater number by the less, and, if the division terminates without a remainder, the less number is an aliquot part of the greater: If the quotient be 2, the less number is equal to of the greater; if the quotient be 3, the less number is of the greater; and so on.

Note 2.-To find 4, or 4, or 4, &c. of any given quantity or number, divide the quantity by the denominator of the fraction, viz. by 2, or 3, &c. and the quotient will be the part required: Thus, of 8 is-8÷2-4; and of 15 is =15÷3=5.

A Table of Aliquot or Proportional Parts of Money, Weight, and Measure.

Note. If 100 lb. make 1 cwt., then 1, 2, 3, &c. pounds will be the same parts of 1cwt. that 1, 2, 3, &c. cents are of $1.

CASE I.

When the given quantity is of one denomination only, and the price of one is also of one denomination.

RULE.

Multiply together the price of one, and the given number of articles, or yards, &c., and the product will be the answer in the same denomination with the price.

Or, if the price of 1 yard, &c. be an aliquot part of I of some higher denomination of that kind of money, then you may consider the given number of yards, &c. as so many units of the said higher denomination of money, and take such proportional part of this number as the price is of 1 of the said denomination, and you will have the value required.

EXAMPLES.

1. What will 734 pounds of cheese come to, at 4d. a pound?

Operation by the first method.

734 lb.
4 d.

2936 d. 121. 4s. 8d. Ans.

Operation by the second method.

3)734 s. value at 1 s. a lb.

Ans. 244s. 8d.-12l. 4s. 8d. value at 4d. a lb.

In working by the second method, I consider the given quantity, viz. 734lb., as so many shillings, and this sum is the value of the said quantity at 1 s. a lb.: Then, because 4d. is of 1s., I take of 734s., and have 244s. 8d. for the answer; which being reduced to pounds, is 127. 4s. 8d. The operation may be performed otherwise, thus:4d.-l.

6,0)73,47. value at 17. a lb.

Ans. 127. 4s. 8d. value at 4d. a lb. Here I consider 734 lb. as 7347. of money; and, because 4d. is of L1, I take of 7347., and have 127. 4s. 8d. for the answer.

Note. If after the division of the given quantity by the aliquot part of L 1, &c. there is a remainder, its value may be found in the lower denominations, as in Compound Division. Thus, in the foregoing example, after dividing 734s. by 3, 2 shillings remain, which I reduce to pence, and then divide by 3, and the quotient is 8d., - which I annex to the shillings of the answer.

2. What is the value of 75lb. at 6d. a lb.?

N

6d=s.l.: Therefore, 75s.÷2=37s. 6d.=17. 17s. 6d. the Ans.

Or, L75÷40-17. 17s. 6d. the ans. as before. 3. What is the value 92lb. of butter, at 12 cents a lb.? 12c. Hence $92÷8-$11.50, Ans.

When the given quantity is of one denomination only, and the price of one is of different denominations.

RULE I.

1. When the price is an aliquot part of 1 of some higher denomination of money; then the like part of the quantity will be the answer in the said higher denomination, as in Case I:

2. When the price is not an aliquot part of 1 of any higher denomination of that kind of money; if it can conveniently be divided into two or more parts which shall be aliquot parts, either of 1 of some higher denomination, or of each other; then take the like parts of the given quantity, and add them together for the answer. But, if the price cannot conveniently be divided into such aliquot parts; or if the highest denomination of the price be the highest denomination of that kind of money; then multiply the given number of yards, or pounds, &c., by the price of one yard, &c., and take parts for the lower denominations of the