CHAPTER VIII FRACTIONS REDUCTION OF FRACTIONS α b 138. A fraction is an indicated quotient; thus is identical with a÷b. The dividend a is called the numerator and the divisor b the denominator. The numerator and the denominator are the terms of the fraction. 139. Since a fraction represents an indicated division, the proofs of all fundamental properties may be based upon the definition of division; viz.: 140. If both of the terms of a fraction be multiplied by the same expression, the value of the fraction is not thereby altered. Expressed in symbols, But in this form the correctness of the conclusion is obvious. 141. If both the terms of a fraction be divided by the same expression, the value of the fraction is not thereby altered; or ma α mb b This follows from the preceding article. 142. A fraction is in its lowest terms when its numerator and its denominator have no common factors. Ex. 1. Reduce 6 xy2z1 Remove successively all common divisors of numerator and denominator, as 3, x, y2, and 23 (or divide the terms by their H. C. F. 3xy2z3). 143. To reduce a fraction to its lowest terms, resolve numerator and denominator into their factors, and cancel all factors that are common to both. Never cancel terms of the numerator or the denominator; cancel factors only. That is, a fraction is not altered if the signs of both its terms are changed; but the sign of the fraction is changed if the sign of either term is changed. 145. If only one of the terms of the fraction can be factored by inspection, find, if possible, the H. C. F. according to § 129, and divide numerator and denominator by it. (Art. 131.) But 3 a 5 cannot be a factor of the numerator. Hence only 3 a 7 need to be tried. Actual division shows that 3 a3 13 a2 + 23 a by 3 a7, and that the quotient is a2 2a + 3. 21 is exactly divisible (3a-7)(a2 - 2a + 3) a2 - 2a +3 146. Since a factor of each term of a fraction is also a factor of their difference or sum, we can frequently find the H. C. F. of these terms without factoring them, and thereby reduce the fraction. Ex. 2. Reduce 2x34x2+5x-33 to its lowest terms. The difference between numerator and denominator = 3x24 x 15 3x + 5 is not contained in either term. (Art. 131.) Hence if there is a H. C. F., it is x 3. =(3x+5)(x-3). 3 is found to be the H. C. F., and the fraction reduces by 2 x2 + 2 x + 11. By trial x division to 147. Reduction of fractions to equal fractions of lowest common denominator. Since the terms of a fraction may be multiplied by any quantity without altering the value of the fraction, we may use the same process as in arithmetic for reducing fractions to the lowest common denominator. |