# 5. Equivalent Fractions

Recall the following fraction properties:

3/4=15/20

This is true because we have multiplied both the top (numerator) and the bottom (denominator) by 5. We say 3/4 and 15/20 are equivalent fractions.

7/21=1/3

This is true because we have divided both the numerator and the denominator by 7. We say 7/21 and 1/3 are equivalent fractions.

We now apply these ideas to fractions involving algebraic expressions.

### Example 1

Divide the numerator and the denominator of (6a^3b^2)/(9a^5b^4) by 3ab2.

(6a^3b^2-:3ab^2)/(9a^5b^4-:3ab^2)=(2a^2)/(3a^4b^2)

NOTE: This answer is not in simplest form. We could divide top and bottom again by a2, to give 2/(3a^2b^2)

## Know when to stop!

The following expression cannot be simplified further because there is an addition sign in the numerator and a subtraction in the denominator:

(3+x^2)/(2-x)

We cannot cancel the x and the x2.

However, if the terms in the numerator and denominator are multiplied, then we can do further simplifying like this:

(3x^2)/(2x)=(3xxx^2)/(2xxx)=(3x)/(2)

### Example 2

Reduce to simplest form:

(10x^2+15x)/(2x+3)

We start by factoring the numerator and then observe we can divide top and bottom by one of the factors:

(10x^2+15x)/(2x+3)=(5x(2x+3))/(2x+3)=5x

### Example 3

Reduce to simplest form:

(2x^2-8)/(4x+8)

### Exercises

Simplify:

(1) (2a^2xy)/(6axyz^2)

(2) (t-a)/(t^2-a^2)

(3) (x^2-y^2)/(x^2+y^2)

(4) (x^2-y^2)/(y-x)