In the following fractions, transfer the factors having negative Reduce each of the following fractions to the form of an entire 128. To reduce one or more fractions to a common denominator. We have seen (124) that a fraction may be reduced to lower terms by division. Conversely, a fraction must be reduced to higher terms by multiplication, and each of the higher denominators it may have, must be some multiple of its lowest denominator. Hence, 1.—A common denominator to which two or more fractions may be reduced, must be a common multiple of their lowest denominators; and 2. The least common denominator of two or more fractions, must be the least common multiple of their denominators. 1. Reduce and с d to their least common denominator. We find by inspection that the least common multiple of the given denominators is ab3. And a2l3÷a2b=l a'l3÷al3=a If, therefore, we multiply both numerator and denominator of the first fraction by l3, and of the second by a, we shall reduce the two fractions to their least common denominator, al'. Thus, c>b3=b3c, new numerator of first fraction; From these principles and illustrations we deduce the following RULE. I. Find the least common multiple of all the denominators, for the least common denominator. II. Divide this common denominator by each of the given denominators, and multiply each numerator by the corresponding quotient. The products will be the new numerators. NOTE.-Mixed numbers should first be reduced to fractions, and all fractions to their lowest terms. EXAMPLES FOR PRACTICE. In each of the following examples, reduce the fractions and mixed quantities to their least common denominator: 5. a2+ y and a3y' a су Ans. ay—1 ay'-y ay-y 129. We have seen (115) that a fraction is equal to the reciprocal of its denominator multiplied by the numerator. Hence, if two or more fractions have a common denominator, they will have a common fractional unit, which may be made the unit of addition. Thus, The intermediate steps may be omitted; hence the following RULE. I. Reduce the fractions to their least common denomi nator. II. Add the numerators, and write the result over the common denominator. NOTES. 1. If there are mixed quantities, we may add the entire and fractional parts separately. 2. Any fractional result should be reduced to its lowest terms. SUBTRACTION. 130. If two fractions have a common denominator, they will have the same fractional unit; and the one may be subtracted from the other, by taking the difference of the numerators. 1 Thus, -b с с RULE. I. Reduce the fractions to their least common denominator II. Subtract the numerator of the subtrahend from the numerator of the minuend, and write the result over the common denominator. |