When the tens and units of the divisor are 25 or 75, it is sometimes convenient to multiply both the divisor and dividend by 4, and use the products in their places.

Example: To divide £ 1486 17 2 by 175

Or, using the Rule just given for dividing by 100, we may make the division thus ;

We may here observe, that this method is frequently of use, when the two lower figures of the divisor are other parts than fourths of 100;—as 12, multiply by 8; with 314, multiply by 16, or 32; but to know when these can be employed, requires some proficiency in fractions.

Other contractions of divisions may be made by employing decimal valuations, and by dropping the dividing figures, for which the student may refer to the contractions of division, at the end of that part of this work which treats on Decimals.

In Ex. 1. we may also multiply by 8, and divide by 1000. In Ex. 3. we may multiply 16, and divide by 10000.

MULTIPLICATION

BY A FRACTIONAL NUMBER.

Case 1. When the multiplier is a simple fraction.

Rule. Multiply the given quantity by the numerator of the fraction, and divide the product by the denominator.

Or, when the fraction can conveniently be separated into parts, find the product for each part, and take the amount.

N. B. The numerator is the upper number of the fraction, and the denominator is the lower number.

It must be observed, that when the result is found by the deduction of a part, if there is any remainder, the lowest part or quantity of the amount to be subtracted must be reckoned as 1 more—thus, in the third method, if the 4 d. had been 42 d., instead of 1, we should have reckoned 14 d.

EXERCISES.

Ex. 1. Multiply £ 8 17 6 by Product £ 6 13 14

2.

3.

4.

5.

6.

6 14 4

7

0 10/1/2

4 07

5 17 41 6651 5 11 5

12 19 0%

Case 2. When the multiplier is a mixed number

Rule. Multiply separately by the whole number and by the fraction, and add the products together.

As the following form of multiplication sometimes occurs, it is here given to show how it may be performed.

DIVISION

BY A FRACTIONAL NUMBER.

Rule. Multiply both the divisor and the dividend by the denominator of the fraction, and divide the latter product by the former.

N. B. When a whole number with a fraction, that is a mixed number, is multiplied by the denominator of the fraction, it requires only the whole number to be multiplied, and the numerator of the fraction to be added in.

We multiply the dividend and the divisor by 4 the denominator of this fraction 2, saying, in working the divisor, 4 times 3 are 12 and 3 (the numerator) are 15.

If the divisor is only a fraction (as) we have to multiply by the denominator (as 4) and divide the product by the numerator (as 3).

DIRECT PROPORTION.

Four quantities are said to be in direct proportion, when the third term is the same product of the first, that the fourth is of the second, or, when the fourth term is the same product of the third, that the second is of the first.

If the first three terms of a direct proportion are given, the fourth may be found by the following Rule.

THE RULE OF THREE DIRECT.

State the three given terms in the form of a question, making the second term of the same kind as the answer required; then, of the two remaining terms, make that the first term of which the second term is the result, and make that the third term of which the answer required is to be the result. Bring the first and third terms into the same denomination; then multiply the second term by the number of the third, and divide the product by the number of the first, and the quotient will be the answer in the same denomination as the second term.

N. B. 1. If the three terms are of the same kind, as all in money, either the first and second, or the first and third, may be reduced into the same denomination.

2. When the first and third terms are simple quantities, if they are in the same denomination, they of course require no reduction; if they are not in the same denomination, the higher must be reduced into the denomination of the lower.-When either the first or the third term is a compound quantity, reduction must be made into the lowest denomination that either term contains.

3. When the second term is a compound quantity, it may be reduced into its lowest denomination for convenience of multiplication, when the third term is expressed by a large number; otherwise, the second term may be multiplied, without reducing it, according to any form of Compound Multiplication.

4. When the first and third terms are of the same denomination, if the first term is an integer, only a multiplication is required; if the third term is an integer, only a division is required.

5. If the first and second terms, or the first and third terms, can be exactly divided by any number, they may be divided, and their quotients be used in their places.