# 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
3*ab*^{2}.

**Answer:**

`(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 *a*^{2}, 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
*x*^{2}.

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)`

**Answer:**

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)`

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