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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 inust 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

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If we divide the denominator by b, the fraction be

comes ,

1 in which a is divided into part as many ō

parts; hence the parts, and consequently the fraction is times as large as before.

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Dividing the denominator by 5 it becomes, or 3,

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times as much as 10 must give a product a times as

large, or a times 1, which is a.

Hence, if a fraction be multiplied by its denominator,

the product will be the numerator.

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Two ways have been shown to multiply fractions,

and two ways to divide them.

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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.

a

In other words, signifies that a contains 6 a cerb

tain number of times, if a is as large or larger than b; or a part of one time, if 3 is larger than a. Now it is evident that 2 a will contain 2b 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.

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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.

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Write the divisor under the dividend in the form of

a fraction, and reduce it to its lowest terms.

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