Cancellation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cancellation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Cancellation is the process of removing a common factor from the numerator and denominator of a fraction, or from both sides of an equation, to simplify. It works because dividing both parts by the same nonzero number leaves an equivalent but simpler form.

$\frac{6}{8} = \frac{3}{4}$ because we can cancel the common factor 2 from top and bottom.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Cancellation removes a shared nonzero factor from numerator and denominator to leave an equivalent, simpler form.

Common stuck point: The procedure for cancellation is the easy part; the trap is canceling across addition. Asking "Is there a factor that divides the entire numerator and the entire denominator?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there a factor that divides the entire numerator and the entire denominator?

Worked Examples

Example 1

easy

Simplify

\dfrac{4 \times 6}{4 \times 3}

using cancellation.

Answer

First step

Write as:

\dfrac{4 \times 6}{4 \times 3}

Full solution

2
The 4 appears in both top and bottom (common factor).
3
Cancel: $\dfrac{\cancel{4} \times 6}{\cancel{4} \times 3} = \dfrac{6}{3}$ .
4
Simplify: $\dfrac{6}{3} = 2$ .

Cancellation uses

\frac{ac}{bc} = \frac{a}{b}

. The common factor

c

cancels from top and bottom.

Example 2

medium

Simplify the fraction

\dfrac{18}{24}

using cancellation (find the GCF first).

Example 3

medium

Simplify

\frac{6x+12}{6}

Example 4

hard

Simplify

\frac{x^2-9}{x^2-x-6}

for

x\ne 3,-2

Example 5

challenge

Simplify

\frac{x^3-1}{x-1}

for

x\ne 1

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Simplify

\dfrac{3 \times 5}{3 \times 7}

Example 2

medium

Simplify

\dfrac{30}{42}

completely.

Example 3

easy

Simplify

\frac{6}{8}

by cancelling the common factor.

Example 4

easy

Simplify

\frac{10}{15}

Example 5

easy

Simplify

\frac{4\times 7}{4}

Example 6

easy

Simplify

\frac{12}{18}

Example 7

easy

Can you cancel the

x

\frac{x+3}{x+5}

Example 8

easy

Simplify

\frac{9}{9}

Example 9

easy

Simplify

\frac{5a}{5}

Example 10

easy

Simplify

\frac{8}{12}

Example 11

medium

Simplify

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

Example 12

medium

Simplify

\frac{x(x+1)}{x}

for

x\ne 0

Example 13

medium

Solve

4x = 20

using cancellation.

Example 14

medium

Simplify

\frac{2}{3}\times\frac{9}{4}

by cancelling before multiplying.

Example 15

medium

A student simplifies

\frac{16}{64}

by 'crossing out the 6' to get

\frac{1}{4}

. Is the method valid?

Example 16

medium

Simplify

\frac{6x+9}{3}

Example 17

medium

Solve

\frac{x}{5}=3

using cancellation.

Example 18

medium

Simplify

\frac{14}{21}+\frac{1}{3}

by reducing first.

Example 19

challenge

Simplify

\frac{x^2-9}{x+3}

for

x\ne -3

Example 20

challenge

Why is cancelling the 2 in

\frac{2+x}{2}

to get

x

wrong? Give the correct simplification.

Example 21

challenge

For what nonzero values can you cancel to simplify

\frac{a^2 b}{ab^2}

, and what is the result?

Example 22

medium

Simplify

\frac{15}{25}

to lowest terms.

Example 23

easy

Simplify

\frac{4}{10}

by cancelling.

Example 24

easy

Simplify

\frac{20}{25}

Example 25

easy

Simplify

\frac{16}{24}

Example 26

easy

Simplify

\frac{7x}{14}

for

x\ne 0

Example 27

easy

Simplify

\frac{2(x-1)}{4}

Example 28

easy

Simplify

\frac{0}{5}

Example 29

medium

Simplify

\frac{12ab}{18b}

for

b\ne 0

Example 30

medium

Simplify

\frac{4}{9}\times\frac{27}{8}

by cancelling before multiplying.

Example 31

medium

Simplify

\frac{(x+2)^2}{x+2}

for

x\ne -2

Example 32

medium

Solve

\frac{3x}{4}=6

using cancellation.

Example 33

medium

Simplify

\frac{15}{45}

in one step.

Example 34

medium

Simplify

\frac{10}{6}\div\frac{5}{3}

Example 35

medium

Simplify

\frac{8m^2}{12m}

for

m\ne 0

Example 36

hard

Simplify

\frac{x^2-4}{x-2}

for

x\ne 2

Example 37

hard

Simplify

\frac{x^2+5x+6}{x+2}

for

x\ne -2

Example 38

hard

Simplify

\frac{2x^2-8}{x^2-4}

for

x\ne \pm 2

Example 39

hard

Simplify

\frac{a+b}{a^2-b^2}

for

a\ne \pm b

Example 40

hard

Why is

\frac{x+2}{x+3} \ne \frac{2}{3}

Example 41

challenge

Simplify

\frac{3x^2-12}{6x-12}

for

x\ne 2

Example 42

challenge

Evaluate

\frac{19!}{17!}

using cancellation.

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Related Concepts

Fractions Factors Simplification

Background Knowledge

These ideas may be useful before you work through the harder examples.

fractionsfactors