How to simplify fractions (with examples)

Simplify $\tfrac{36}{48}$

• Both are even: $\tfrac{36}{48} = \tfrac{18}{24}$.
• Still even: $\tfrac{18}{24} = \tfrac{9}{12}$.
• Both divide by 3: $\tfrac{9}{12} = \tfrac{3}{4}$.
• 3 and 4 share no factor but 1, so $\tfrac{3}{4}$ is fully simplified.

In one step: the HCF of 36 and 48 is 12, so $\tfrac{36 \div 12}{48 \div 12} = \tfrac{3}{4}$. Find HCFs with the HCF calculator.

Dividing the top and bottom by the same number is like dividing by $\tfrac{6}{6} = 1$, which doesn't change the value. Picture a pizza cut into 24 slices with 18 eaten: that's the same amount as one cut into 4 slices with 3 eaten.

Simplify first, then convert if needed: $\tfrac{42}{12} = \tfrac{7}{2} = 3\tfrac{1}{2}$. For a mixed number, simplify the fraction part: $2\tfrac{6}{8} = 2\tfrac{3}{4}$.

When multiplying fractions, you can cancel common factors before multiplying, diagonally or vertically: $\tfrac{4}{9} \times \tfrac{3}{8} = \tfrac{1}{3} \times \tfrac{1}{2} = \tfrac{1}{6}$. It keeps numbers small. See how to multiply fractions.

• Dividing only the top or only the bottom.
• Subtracting instead of dividing: $\tfrac{9}{12} \neq \tfrac{6}{9}$.
• Stopping too early: $\tfrac{18}{24} = \tfrac{9}{12}$ isn't fully simplified.
• Cancelling digits: $\tfrac{16}{64} \neq \tfrac{1}{4}$ "because the 6s cancel" (it happens to be true here, but it's not a method).

Simplify: (a) $\tfrac{15}{25}$, (b) $\tfrac{28}{42}$, (c) $\tfrac{45}{60}$, (d) $\tfrac{56}{16}$. Answers: $\tfrac{3}{5}$, $\tfrac{2}{3}$, $\tfrac{3}{4}$, $\tfrac{7}{2}$ (or $3\tfrac{1}{2}$). More in the fractions topic guide.

Simplifying is one direction of equivalent fractions: fractions that look different but have the same value. Multiply or divide the top and bottom by the same number to move between them:

$\tfrac{3}{4}$ is the simplest form because 3 and 4 have no common factor other than 1. Writing fractions with the same denominator is also how you compare them: $\tfrac{2}{3} = \tfrac{8}{12}$ and $\tfrac{3}{4} = \tfrac{9}{12}$, so $\tfrac{3}{4}$ is bigger.

The same rule applies with letters: divide top and bottom by a common factor.

• $\dfrac{6x}{9} = \dfrac{2x}{3}$
• $\dfrac{x^2}{3x} = \dfrac{x}{3}$
• $\dfrac{2x + 4}{6} = \dfrac{2(x + 2)}{6} = \dfrac{x + 2}{3}$

You can only cancel factors, not terms that are added: $\tfrac{x + 2}{2}$ does not simplify to $x + 1$. Factorise first, then cancel. See factorising.

A ratio like $12 : 18$ simplifies exactly like a fraction: divide both parts by the HCF, 6, to get $2 : 3$. More in ratio and proportion.