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How to Simplify a Fraction by Canceling

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Canceling, simplifying or reducing a fraction is the opposite of expanding a fraction. You will cancel fractions throughout your years in school, so you should rehearse this until it’s second nature to you!

When you cancel, the value of the fraction stays the same. For that reason, you can set an equal sign between the two expressions, even though they look different.

Rule

To simplify by canceling

1.: Factorize the numerator.
2.: Factorize the denominator.
3.: Order the factors by value (optional).
4.: Cancel one common factor in the numerator and the denominator at a time until there are no more common factors.
5.: Multiply the remaining factors in the numerator.
6.: Multiply the remaining factors in the denominator.

Example 1

Simplify the fraction $\frac{3}{6}$ by canceling.

\frac{3}{6} = \frac{3 \cdot 1}{2 \cdot 3} = \frac{1}{2}

because

\frac{3}{6} = \frac{3 \cdot 1}{2 \cdot 3} = \frac{3}{3} \cdot \frac{1}{2} = 1 \cdot \frac{1}{2} = \frac{1}{2} .

$Two equivalent fractions that both have a numerical decimal value of 0.5$

Example 2

Simplify the fraction $\frac{4}{10}$ by canceling.

\frac{4}{10} = \frac{2 \cdot 2}{5 \cdot 2} = \frac{2}{5}

because

\frac{4}{10} = \frac{2 \cdot 2}{2 \cdot 5} = \frac{2}{2} \cdot \frac{2}{5} = 1 \cdot \frac{2}{5} = \frac{2}{5} .

Example 3

Simplify the fraction $\frac{27}{21}$ by canceling.

\frac{27}{21} = \frac{3 \cdot 9}{7 \cdot 3} = \frac{9}{7}

because

\frac{27}{21} = \frac{3 \cdot 9}{3 \cdot 7} = \frac{3}{3} \cdot \frac{9}{7} = 1 \cdot \frac{9}{7} = \frac{9}{7} .

Example 4

Simplify the fraction $\frac{48}{18}$ by canceling.

\frac{48}{18} = \frac{6 \cdot 8}{3 \cdot 6} = \frac{8}{3}

because

\frac{48}{18} = \frac{6 \cdot 8}{6 \cdot 3} = \frac{6}{6} \cdot \frac{8}{3} = 1 \cdot \frac{8}{3} = \frac{8}{3} .

Example 5

Simplify the fraction $\frac{20}{70}$ by canceling.

\frac{20}{70} = \frac{10 \cdot 2}{7 \cdot 10} = \frac{2}{7}

because

\frac{20}{70} = \frac{2 \cdot 10}{7 \cdot 10} = \frac{10}{10} \cdot \frac{2}{7} = 1 \cdot \frac{2}{7} = \frac{2}{7} .

Example 6

Simplify the fraction $\frac{48}{40}$ by canceling.

\frac{48}{40} = \frac{8 \cdot 6}{5 \cdot 8} = \frac{6}{5}

because

\frac{48}{40} = \frac{6 \cdot 8}{5 \cdot 8} = \frac{8}{8} \cdot \frac{6}{5} = 1 \cdot \frac{6}{5} = \frac{6}{5} .

Example 7

Simplify the fraction $\frac{32}{36}$ by canceling.

\frac{32}{36} = \frac{4 \cdot 8}{9 \cdot 4} = \frac{8}{9}

because

\frac{32}{36} = \frac{4 \cdot 8}{4 \cdot 9} = \frac{4}{4} \cdot \frac{8}{9} = 1 \cdot \frac{8}{9} = \frac{8}{9} .

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