7. Multiply by 3.

A X3, by Rule 1; or,

2

1 x 3 = 332

by Rule 2.

NOTE 2. -The 1st rule is preferable in this and all similar examples, because the 2d gives a complex fraction.

(a) The correctness of the first rule is also apparent from the following reasoning: It is just as evident that 3 times & are as that 3 times 2 cents are 6 cents; or that 3 times 2 are 6; i. e. when the numerator is multiplied by 3, the fraction represents 3 times as many parts as before, and each part remains of the same size; .. the fraction is multiplied by 3.

(b) The 2d rule may be explained thus: 2 ninepences, in New England, are 25 cents; i. e. 2 times of a dollar are † of a dollar, and, as evidently, 2 times g are ; i. e. if the denominator is divided by 2, the fraction represents just as many parts as before, but each part is twice as great, and, .., the whole fraction is twice as great.

14. Multiply by 15. 15 = 5 × 3.

NOTE. We may here, as in whole numbers (44), use the component parts of the multiplier, and, in using these component parts, we may apply the 1st or the 2d rule, or both.

19. Multiply 18,569 by 63.

99

20. Multiply 278 by 1008.

21. Multiply by 5.

X 5 == 4, by Rule 2; ..,

(c) If we multiply a fraction by its denominator, the product

126. To divide a fraction by a whole number,

RULE 1.-Divide the numerator by the whole number (59, b, and 118); or,

RULE 2.-Multiply the denominator by the whole number

(59, c).

Ex. 1. Divide by 4.

골을 응

42, by Rule 1; or,

f÷42, by Rule 2.

NOTE 1.-The 1st rule is preferable in this example.

Why?

Ans.

A÷2, by Rule 2.

NOTE 2. The 2d rule is preferable in this example.

(a) It is just as evident that of 4 is 4, as that of 6 cents is 2 } cents, or that of 6 is 2; i. e. the 1st rule may be explained by saying that, if the numerator is divided by 3, the fraction will express only as many parts, and each part remains of the same size; hence the value of the fraction is divided by 3.

(b) By the 2d rule each part expressed by the fraction is made smaller, while the number of parts taken remains the same; .. the value of the fraction is divided when we multiply the denominator.

14. Divide by 20. 20= 4 × 5.

X

&,

127. To multiply by 3, 1st, × 3 = (125, Rule 1); but the multiplier, 3, is 5 times,.. the product, , is 5 times the product sought; hence, 2d, ÷ 5 = (126, Rule 2) is the product sought; i. e.

To multiply a fraction by a fraction,

RULE.-Multiply the numerators together for a new numerator, and the denominators for a new denominator.

(a) To multiply by a fraction is only to multiply by the numerator, and then divide the product by the denominator. In Ex. 6 we multiply by 5 and obtain 42 (125, Rule 2), and then 2 divided by 6 gives (126, Rule 1), the result sought.

NOTE 1.—In this simple operation is involved the whole principle of can

The 7th example is solved on the same principle as the 6th. Since the product of the numerators is a dividend, and that of the denominators a divisor (118), and since the quotient is not affected by dividing both dividend and divisor by the same number (61, Cor.), we may cancel (strike out, or reject) the factors 3 and 7 from both numerator and denominator; i. e. we may divide both numerator and denominator by 3 and 7, and thus obtain, the product sought.

NOTE 2.-There can be no difficulty in canceling so long as we remember the simple principle, that it rests upon rejecting equal factors from dividend and divisor. The process is only to strike out or cancel the same factors from numerator and denominator, and it often saves much labor.

(b) In canceling 3 and 5 in Example 13, we obtain the quotients 1 and 1 in the numerators, and whenever an entire term cancels we obtain 1 to place instead of the term canceled; but since 1, as a multiplier or divisor, is valueless, there is no need of retaining it under any circumstances except where all the numerators are canceled; in such a case, 1 is the true numerator and must be retained.

NOTE. The 14th and similar examples may be more conveniently writ ten as follows: