fractions, and multiply the numerator and denominator of that fraction by the quotient; and deal in a similar manner with all the other fractions: we thus obtain fractions equal to the given fractions, all of which have the L. C. M. for denominator. The L. C. M. of the denominators is a2b2 (x2 - a2). Dividing the L. C. M. by a b (x +a), ab2 (x − a), and ab (x2 — a2), we have the quotients b (x- a), a (x + a), and ab respectively. Hence NOTE. In the above process it is not absolutely necessary to take the lowest common multiple of the denominators; any common multiple will do equally well, but by using the L. C. M. there is a saving of labor. 165. Addition of Fractions. The sum (or difference) of two fractions which have the same denominator is a fraction whose numerator is the sum (or difference) of their numerators, and which has the common denominator. This follows from Art. 80. When two fractions have not the same denominator, they must first be reduced, by the method explained in the last article, to equivalent fractions which have the same denominator: their sum or difference will then be found by taking the sum or difference of their numerators, retaining the common denominator. When more than two fractions are to be added, or when there are several fractions, some of which are to be added and the others subtracted, the process is precisely the same. The fractions are first reduced to a common denominator, and then the numerators of the reduced fractions are added or subtracted as may be required. NOTE. 818 + b y C x xyz y X xz - It may be necessary to remind the student that when there is no sign between a fraction and a letter or number, the sign of multiplication is understood. Thus, 2 means 2 × b × 2 and x + a The L. C. M. of the denominators is (x − a) (x + a); and Beginners should always see that the denominators of the fractions which are to be added are all arranged according to descending powers, or all according to ascending powers, of some particular letter. This is not the case in the present example; but It is sometimes best not to add all the fractions at once; this is particularly the case when the denominators are not all of the same dimensions. Here again it is best not to reduce all the fractions to a common denominator at once: the work is simplified by proceeding as hence the L. C. M. of the denominator is (x 2) (x − 3) (x − 4). Hence we have х 4 x-2 + 1 (a - b)(ac) (b − a) (b − c)' (c − a) (c — b) In examples of this kind it is best for beginners to arrange all the factors in the denominators of the fractions so that a precedes bor c, and that b precedes c. We therefore change b a into - (a - b), c a into (ac), and c − b into − (b − c). |