DIVISION BY FRACTIONS. $207. Suppose we wish to know how many times is contained in . We would divide in the same manner that we follow in dividing 6 pieces by 3 pieces; and say is contained in, two times. In the same manner, is contained in, five times. The principle is general that when the divisor and the dividend have a common denominator, the division is performed by dividing the numerator of the dividend by the numerator of the divisor. Explanation. In this example, the dividend and the divisor has not a common denominator. Our first object then, is to bring them to a common denominator. done by §205, ab bax-by ху by abxy abx-by This is = abs ab2: ux Ans. 9. Operation. §208. It will be seen that in sums of this kind, after we have brought the terms to a common denominator, the division is performed by putting the numerator of the dividend for the numerator of the answer, and the numerator of the divisor for the denominator of the answer, and make no use at all of the denominators. Let us see then how we obtain these two terms. We multiply the numerator of the dividend by the denominator of the divisor; and this becomes the numerator of the answer. And we multiply the numerator of the divisor by the denominator of the dividend; and this becomes the denominator of the answer. By looking at the last two sums, it will be seen that this is the true operation. §209. Hence we obtain the general rule for dividing by a fraction. Multiply the numerator of the dividend by the denominator of the divisor, for a new numerator; and multiply the denominator of the dividend by the numerator of the divisor for a new denominator. §210. When the dividend is a whole number, it is changed into a fraction, by putting 1 under it for a denominator. REDUCTION OF COMPLEX FRACTIONS TO SIMPLE ONES. §211. We have shown, that when we multiply a fraction by its denominator, we obtain for the answer, the same quantity as the numerator. We have also shown that where both terms of a fraction are multiplied by the same quantity, the value of the fraction is not altered. By these two principles, we obtain the following rule for reducing a complex fraction to a simple one. §212. Multiply both terms of the fraction by the denominator that is found either in the entire numerator or denominator. If the fraction is still complex, multiply the result in both terms by the remaining denominator that is found in the entire term. Thus, |