Unit 7 - Objective 1 - Equivalent Fractions

Multiplying or dividing both the numerator and denominator of a fraction by the same numbers, except zero, results in a fraction that is equivalent to the original fraction.  Equivalent means they represent the same number.  We want to reduce the fraction to lowest terms or simplest terms.

Note:

 because   and if we have -x/y the negative can be with the numerator as in -x/y or in the denominator as in x/-y but the negative cannot be with both. General procedure for reducing a fraction is to factor numerator and denominator then divide out factors they have in common.

Examples:

 1. Reduce    to lowest terms The numerator and denominator have a common   factor of 8xyz3.  We can rewrite the fraction as Since 8xyz3 cancel we are left with 3x/y2 which is equivalent to which is equal to (3x)/y^2 2. Reduce  Notice that x2 is a factor in the numerator but it is not a factor for both terms in the denominator.   We need to factor the denominator which is the difference of two squares.  The numerator cannot be factored farther. We can now cancel the x - 4 term and we are left with -2x2/(x+4) as an equivalent fraction. (answer: (-2x^2)/(x+y)

Note:

 (factor out a (-1) changes the signs)

Examples:

 1. Reduce    to lowest terms We do not have a common factor for the numerator and denominator so we have to factor them separately. Notice that (5 - b) and (b - 5) differ by a negative 1 so we can rewrite either one. We can now cancel the quantity (b - 5). = -a/3 2. Reduce    to lowest terms. Notice that (2x - 1) and (1 - 2x) differ by a negative 1 so we can rewrite either one.. We can now cancel the quantity (1 - 2x).   or