5. Equivalent Fractions

Recall the following fraction properties:


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.


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.



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:


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:


Example 2

Reduce to simplest form:



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


Example 3

Reduce to simplest form:




(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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