124. It is often convenient to change the signs of the numerator or denominator of a fraction, or both. By the rule for the signs, in Division (Art. 69), we have, -ab If we change the sign of the numerator, we have -b. +a If we change the sign of the denominator, we have 1. The signs of both terms of a fraction may be changed, without altering its value or changing its sign, as a whole. 2. If the sign of either term be changed, the sign of the fraction will be changed. Hence, also, 3. The signs of either term of a fraction may be changed, without altering its value, if the sign of the fraction be changed at the same time. Applying the above principles, the sign of the fraction may be made plus, in all cases, if desired. Case IV.-TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO EQUIVALENT FRACTIONS HAVING A COMMON DENOMINATOR. 125.-1. Let it be required to reduce and to m' n a common denominator. If we multiply both terms of the first fraction by nr, of the second by mr, and of the third by mn, we have As the terms of each fraction have thus been multiplied by the same quantity, the value of the fractions has not been changed. (Art. 118.) Hence, TO REDUCE FRACTIONS TO A COMMON DENOMINATOR, Rule.-Multiply both terms of each fraction by the product of all the denominators, except its own. Or, 1. Multiply each numerator by the product of all the denominators except its own, for the new numerators. 2. Multiply all the denominators together for the common denominator. Reduce the fractions in each of the following to a common denominator: 126. It frequently happens, that the denominators of the fractions to be reduced contain a common factor. In such cases the preceding rule does not give the least common denominator. 1. Let it be required to reduce least common denominator. α b m'mn' and to their с nr Since the denominators of these fractions contain only three prime factors, m, n, and r, it is evident that the least common denominator will contain these three factors, and no others; that is, it will be mnr, the L.C.M. of m, mn, and nr. It now remains to reduce each fraction, without altering its value, to another whose denominator shall be mnr. To effect this, we must multiply both terms, of the first fraction by nr, of the second by r, and of the third by m. But these multipliers will evidently be obtained by dividing mnr by m, mn, and nr; that is, by dividing the L.C.M. of the given denominators by the several denominators. Hence, TO REDUCE FRACTIONS OF DIFFERENT DENOMINATORS TO EQUIVALENT FRACTIONS HAVING THE LEAST COMMON DENOMINATOR, Rule.-1. Find the L.C.M. of all the denominators; this will be the common denominator. 2. Divide the L.C.M. by the first of the given denominators, and multiply the quotient by the first of the given numerators; the product will be the first of the required numerators. 3. Proceed thus to find each of the other numerators. Reduce the fractions, in each of the following, to equivalent fractions having the least common denominator : Other exercises will be found in Addition of Fractions. NOTE. The two following Articles may be of frequent use. 127. To reduce an entire quantity to the form of a fraction having a given denominator, Rule.-Multiply the entire quantity by the given denominator, and write the product over it. 1. Reduce x to a fraction whose denominator is a. 2. Reduce 2az to a fraction whose denominator is z2. 3. Reduce x+y to a fraction whose denominator is x -y. 128. To convert a fraction to an equivalent one having a given denominator, Rule.-Divide the given denominator by the denominator of the given fraction, and multiply both terms by the quo tient. 1. Convert to an equivalent fraction, having 49 for its denominator. Ans. 21 49. Case V.-ADDITION AND SUBTRACTION OF FRACTIONS. 129.-1. Required to find the value of Since in each of these fractions the unit is supposed to be divided into d parts, it is evident that their sum will be expressed by the Rule for the Addition of Fractions.-1. Reduce the fractions, if necessary, to a common denominator. 2. Add the numerators, and write their sum over the common denominator. b 130.—2. Let it be required to subtract from d The unit being, in each case, divided into the same parts, the difference will evidently be expressed by a-b Hence, Rule for the Subtraction of Fractions.-1. Reduce the fractions, if necessary, to a common denominator. 2. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the remainder over the common denominator. |