Simplifying Fractions: Reducing to Lowest Terms

What simplest form means

Look at these two bars.

Same shading. Same amount. But $\tfrac{3}{4}$ uses smaller numbers, so it is quicker to read and quicker to work with.

A fraction is in simplest form when no whole number bigger than $1$ divides both the top and the bottom.

• $\tfrac{6}{8}$ is not in simplest form, because $2$ divides both $6$ and $8$.
• $\tfrac{3}{4}$ is in simplest form, because nothing but $1$ divides both $3$ and $4$.

Why dividing both parts is allowed

You already know the rule for equivalent fractions: multiply or divide the top and the bottom by the same number, and the value stays put.

Simplifying is that rule run backwards.

$$\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$$

On the bar, you took the eight small parts and taped them together in pairs. Four bigger parts, three of them shaded, same shading. Nothing left the bar.

Method 1 — chip away

Divide by any factor you spot, then look again. Repeat until nothing is left to divide by.

$$\frac{24}{36} = \frac{12}{18} = \frac{6}{9} = \frac{2}{3}$$

Here we divided by $2$, then $2$, then $3$. Four short steps, no big numbers to think about.

This method is forgiving. You never have to spot the best factor first, only a factor.

Method 2 — divide by the GCF

The greatest common factor of two numbers is the largest number that divides both. Divide by it once and you land in simplest form immediately.

For $24$ and $36$:

Number	Factors
$24$	$1, 2, 3, 4, 6, 8, 12, 24$
$36$	$1, 2, 3, 4, 6, 9, 12, 18, 36$

The common factors are $1, 2, 3, 4, 6$ and $12$. The greatest is $12$.

$$\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}$$

Same answer as before, in one step. Both methods are correct — pick the one that suits the numbers in front of you.

Quick tests for a common factor

If both numbers	They share
are even	$2$
end in $0$ or $5$	$5$
have digits that add to a multiple of $3$	$3$
end in $0$	$10$

Try $\tfrac{45}{60}$. Both end in $0$ or $5$, so divide by $5$ to get $\tfrac{9}{12}$. Now $9$ and $12$ both have digits adding to a multiple of $3$, so divide by $3$ to get $\tfrac{3}{4}$.

Worked examples

Common mistakes

Should every answer be simplified?

For schoolwork, yes — most teachers and tests expect simplest form, and it is the version other people can read at a glance.

There is one useful exception. When you are about to add or subtract fractions, you often want a bigger bottom number rather than a smaller one, so simplify at the end rather than in the middle.

Practice problems