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.

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 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)`

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)`