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115 Rh Thus in each case we divide the unit into bd equal parts, and we take first ad of these parts, and then bc of them ; that is, we take ad + bc of the hd parts of the unit ; and this is expressed by the fraction {ad + bc}{bd} Similarly,

145. Here the fractions have been both expressed with a common denominator bd. But if b and d have a common factor, the product bd is not the lowest common denominator, and the fraction will not be in its lowest terms. To avoid working with fractions which are not in their lowest terms, we take the lowest common denominator, which is the lowest common multiple of the denominators of the given fractions.

Rule I. To reduce fractions to their lowest common denominator. Find the L. C. M. of the given denominators, and take it for the common denominator ; divide it by the denominator of the first fraction, and multiply the numerator of this fraction by the quotient so obtained; and do the same with all the other given fractions.

Ex. 1. Express with lowest common denominator a 3xyz, b 6xyz,  c 2yz The lowest common multiple of the denominators is 6 xyz. Dividing this by each of the denominators in turn, and multiplying the corresponding numerators by the respective quotients, we have the equivalent fractions {2az}{ 6 xyz}, {b} {6 xyz}, {3cx} {6 xyz }

Ex. 2. Express with lowest common denominator {5x}{ 2a(x — a) } and {4a}{3x{x2- a2) }