Quick Review 7-1

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:

Reduce to lowest terms

The numerator and denominator have a common factor of 8xyz³. We can rewrite the fraction as

u7obj1-04.gif (1266 bytes)

Since 8xyz³ cancel we are left with 3x/y² which is equivalent to

u7obj1-03.gif (485 bytes) which is equal to (3x)/y^2

Reduce u7obj1-06.gif (462 bytes)

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.

u7obj1-07.gif (586 bytes)

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:

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

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).