House of Math

How to Simplify Fractions with Variables

Here you will learn to simplify fractions that include variables. The most important part is factorizing the numerator and denominator, and then cancelling common factors.

Rule

Simplifying Fractions with Variables

1.: Factorize the numerator.
2.: Factorize the denominator.
3.: Cancel common factors.
4.: Multiply the remaining factors.

Example 1

Simplify the expression $\frac{4 x + 16}{4}$

Find the common factors and cancel:

\begin{array}{l} \frac{4 x + 16}{4} & = \frac{4 (x + 4)}{4} \\ = \frac{4 (x + 4)}{4} \\ = x + 4 \end{array}

\frac{4 x + 16}{4} = \frac{4 (x + 4)}{4} = \frac{4 (x + 4)}{4} = x + 4

Example 2

Simplify the expression $\frac{3 x + 6}{2 x + 4}$

Find the common factors and cancel:

\begin{array}{l} \frac{3 x + 6}{2 x + 4} & = \frac{3 (x + 2)}{2 (x + 2)} \\ = \frac{3 (x+2)}{2 (x+2)} \\ = \frac{3}{2} \end{array}

\frac{3 x + 6}{2 x + 4} = \frac{3 (x + 2)}{2 (x + 2)} = \frac{3 (x+2)}{2 (x+2)} = \frac{3}{2}

Example 3

Simplify the expression $\frac{3 a b + 6 a}{2 a b + 4 a}$

Find the common factors and cancel:

\begin{array}{l} \frac{3 a b + 6 a}{2 a b + 4 a} & = \frac{3 a (b + 2)}{2 a (b + 2)} \\ = \frac{3 a (b+2)}{2 a (b+2)} \\ = \frac{3}{2} \end{array}

\frac{3 a b + 6 a}{2 a b + 4 a} = \frac{3 a (b + 2)}{2 a (b + 2)} = \frac{3 a (b+2)}{2 a (b+2)} = \frac{3}{2}

Example 4

Simplify the expression $\frac{4 x^{2} - 3 x}{4 x - 3}$

Find the common factors and cancel:

\begin{array}{l} \frac{4 x^{2} - 3 x}{4 x - 3} & = \frac{x (4 x - 3)}{(4 x - 3)} \\ = \frac{x (4x-3)}{(4x-3)} \\ = x \end{array}

\frac{4 x^{2} - 3 x}{4 x - 3} = \frac{x (4 x - 3)}{(4 x - 3)} = \frac{x (4x-3)}{(4x-3)} = x

Example 5

Write $\frac{x^{2} - 81}{x - 9}$ as simply as possible

Using the third algebraic identity of quadratic expressions, you get

\begin{array}{l} \frac{x^{2} - 81}{x - 9} & = \frac{(x - 9) (x + 9)}{(x - 9)} \\ = \frac{(x-9) (x + 9)}{(x-9)} \\ = x + 9 \end{array}

\begin{array}{l} \frac{x^{2} - 81}{x - 9} = \frac{(x - 9) (x + 9)}{(x - 9)} = \frac{(x-9) (x + 9)}{(x-9)} = x + 9 . \end{array}

Think About This

Being able to factorize and cancel expressions is one of the main skills you need in mathematics. I recommend that you spend plenty of time practicing these skills.

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