132. A fraction is expressed in its lowest terms if its numerator and denominator have no common factor; and therefore any fraction can be reduced to its lowest terms by dividing both numerator and denominator by their G. C. D., because it contains all the factors common to both terms of the fraction. Since in example 3 no common factor can be determined by inspection, it is necessary to determine the G. C. D. of the numerator and the denominator by the method of division. Omit the factory from the denominator and divide. Now and (6x5xy-6y) y (2x-3y) y (3x+2y) = 6x11xy3xy ÷ (2x-3y) = x (3 x − y). 6x11xy+3xy2 x(3x-y) (2x-3y) = 6xy-5xy 6 y3 y(3x+2y)(2x-3y) = x (3 x y) y (3x+2y) 133. When the terms of the fraction can not be readily factored, then the G. C. D. must be found by division and the terms of the fraction divided by it. 2 an+rbm−le — 4 arb2m-1c2d+2ar+1bmc+6 ar¬1hm-1.n 7 ab. 10 ac - 5 be 45 a3b+c+27 a3b3cd — 9 aab2d3 6ac9be-5 c2 12 adf18bdf-10cdf 30 q3n-1brcr+2 —6q2n−4b3ç3ďr−1. 20 abr-1c2d-4a-3b2 dr+1 n2-2n+1 9. n2-1 134. To Reduce a Fraction to an Integral or Mixed Quantity. If the degree of the numerator be equal to or greater than the degree of the denominator, the fraction may be changed to the form of a mixed or integral expression by dividing the numerator by the denominator. The quotient will be the integral part, and the remainder, if any, will be the numerator, and the divisor the denominator of the fractional part of the mixed quantity. 135. To Reduce a Mixed Expression to the Form of a Fraction. We have learned in Arithmetic that In Arithmetic, the sign connecting the fraction and the integral part of a mixed number is always +, but in Algebra, it may be + or; so that a mixed expression may have either one of the following forms: To reduce a mixed quantity to a fraction, multiply the integral part by the denominator, to the product annex the numerator, and under the result write the denominator. 136. The sign before the fraction shows that the number of things of the group b indicated by the numerator must be added or subtracted according as the sign is or from the number of things in the integral part of the kind in the b group, i. e., from Ab. If the sign precedes the fraction, when the numerator is annexed, the sign of every term in the numerator must be changed. 137. Reduction of Fractions to a Lowest Common Denominator.--Some propositions concerning fractions in Arithmetic will now be recalled, and be proved to hold universally in Algebra. In the following paragraphs the letters represent positive integers, unless it is otherwise stated. 138. 1. Rule for multiplying a fraction by an integer. Either multiply the numerator by that integer, or divide the denominator by it. 2. Rule for dividing a fraction by an integer. Either multiply the denominator by that integer, or divide the numerator by it. For, Let be any fraction, and e any integer; then will = c = be b a be [1; 131, Th. I] be any fraction and c any integer; then prove that By the first part of the theorem |
