Hence, to multiply one fraction by another, multiply the numerators together for a new numerator, and the denominators together

for a new denominator.

Arith. Art. XVII.

We have seen that a fraction may be divided by multiplying its denominator, because the parts are made smaller; on the contrary, a fraction may be multiplied by dividing its denominator, because the parts are made larger. Arith. Art. XVIII. If the denominator be divided by 2, the unit is divided into only one half as many parts; consequently the parts must be twice as large as before. If the denominator be divided by 5, the unit is divided into only one fifth as many parts; hence the parts must be five times as large as before, and if the same number of parts be used as at first, the value of the fraction will be five times as great, and so on.

If we divide the denominator by b, the fraction becomes

part as many parts; hence

the parts, and consequently the fraction is b times as large as

Dividing the denominator by 5 it becomes †, or 3.

by

5.

Dividing the denominator by b it becomes 4, or a.

α

= 1, and being a times

must give a product a times as large, or a , must give a product

times 1, which is a.

Hence, if a fraction be multiplied by its denominator, the product will be the numerator.

Two ways have been shown to multiply fractions, and two

ways to divide them.

Reducing Fractions to Lower Terms.

XVII. If both numerator and denominator be multiplied by the same number, the value of the fraction will not be altered.

Arith. Art. XIX.

For multiplying the numerator multiplies the fraction, and multiplying the denominator divides it; hence it will be multiplied and the product divided by the multiplier, which reproduces the multiplicand.

α

In other words, signifies that a contains b a certain num

b

ber of times, if a is as large or larger than b; or a part of ontime, if b is larger than a. Now it is evident that 2 a will contain 2 b just as often, since both numbers are twice as large as before.

So dividing both numerator and denominator, both divides and multiplies by the same number.

Hence, if a fraction contain the same factor both in the numerator and denominator, it may be rejected in both, that is, both may be divided by it. This is called reducing fractions to lower terms.

8. Divide 35 a2 b m3 x3 by 7 a3n m3 x.

Write the divisor under the dividend in the form of a frac

tion, and reduce it to its lowest terms.