Fraction operations

In Year 4 you added and subtracted fractions with the same denominator — easy, you just add or subtract the numerators. Year 5 takes you two steps further: doing the same thing when the denominators are different, and multiplying a fraction by a whole number.

Adding and subtracting fractions with different denominators

If the denominators don't match, you can't add the numerators directly: a third and a half aren't the same size of slice. The trick is to rename both fractions with a common denominator.

The recipe

1. Find a denominator that both denominators divide into. The easiest reliable choice is to multiply the two denominators together.
2. Convert each fraction to that denominator — multiply numerator and denominator by the right number so the value doesn't change.
3. Now both fractions have the same denominator. Add (or subtract) the numerators. The denominator stays.
4. If you can, simplify the answer.

Worked example — adding

$\frac{1}{2}+\frac{1}{3}$

Common denominator: 2 × 3 = 6.

$\frac{1}{2}=\frac{3}{6}$, $\frac{1}{3}=\frac{2}{6}$

$\frac{3}{6}+\frac{2}{6}=\frac{5}{6}$

Worked example — subtracting

$\frac{3}{4}-\frac{1}{6}$

Common denominator: 4 × 6 = 24 (or 12 if you spot the LCM).

$\frac{3}{4}=\frac{9}{12}$, $\frac{1}{6}=\frac{2}{12}$

$\frac{9}{12}-\frac{2}{12}=\frac{7}{12}$

Multiplying a fraction by a whole number

This one is simpler than it sounds. Multiplying $\frac{2}{5}$ by 3 means "three lots of two fifths" — six fifths.

Numerator × whole. Denominator stays the same. Simplify if needed.

$\frac{2}{5}\times 3=\frac{6}{5}$ (an improper fraction — that's fine, it just means more than one whole)

$\frac{3}{4}\times 2=\frac{6}{4}=\frac{3}{2}$ (simplified by dividing top and bottom by 2)

$\frac{1}{8}\times 6=\frac{6}{8}=\frac{3}{4}$

Common mistakes to avoid

• Adding denominators. $\frac{1}{2}+\frac{1}{3}\ne \frac{2}{5}$. The denominators describe the size of the pieces — they don't change when you add.
• Forgetting to change both top and bottom. When you rename $\frac{1}{2}$ as $\frac{3}{6}$, multiply both the 1 and the 2 by 3. Just changing the bottom turns it into a different fraction.
• Skipping simplification. $\frac{6}{8}$ is correct, but $\frac{3}{4}$ is the same thing in its simplest form. Either is accepted in the practice exercises here.

Practice

• ➕ Add fractions with different denominators
• ➖ Subtract fractions with different denominators
• ✖️ Multiply a fraction by a whole number
• ⚖️ Compare fractions — useful for checking your answer is sensible
• 🟰 Equivalent fractions — practise the renaming step

Related reading

• Year 4 — introduction to fractions
• Equivalent fractions
• Comparing fractions