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 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. 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. Two ways have been shown to multiply fractions, and two ways to divide them. 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. |