Word problems with fractions

A fraction of a quantity is one of the most useful skills in real-world maths. Recipe halving, splitting a bill, working out a discount — all of them start with "what's a third / half / quarter of …".

Fraction of a whole number

The recipe: divide by the bottom, multiply by the top.

What is $\frac{2}{3}$ of 12?

• 12 ÷ 3 = 4 (one third)
• 4 × 2 = 8 (two thirds)

So $\frac{2}{3}$ of 12 = 8.

Or in one step: $\frac{2}{3}\times 12=\frac{2\times 12}{3}=\frac{24}{3}=8$.

Worked examples

Example 1 — halving and quartering

A pizza is cut into 8 slices. Jack eats a quarter of the pizza. How many slices does he eat?

A quarter is $\frac{1}{4}$, so $\frac{1}{4}$ of 8 = 8 ÷ 4 = 2 slices.

Example 2 — fraction of a length

A piece of ribbon is 60 cm long. We cut off two thirds. How long is the piece we cut off?

$\frac{2}{3}$ of 60 = 60 ÷ 3 × 2 = 20 × 2 = 40 cm.

How long is the piece we keep? 60 − 40 = 20 cm (one third).

Example 3 — combining fractions

Anna walks half of a 4 km route in the morning and a quarter of it in the afternoon. How far has she walked?

• Half of 4 km = 2 km.
• A quarter of 4 km = 1 km.
• Total = 2 + 1 = 3 km.

She has $\frac{1}{4}$ km left for tomorrow.

Example 4 — fraction of fraction of …

In a class of 30 students, two thirds play sport. A half of those who play sport play football. How many students play football?

• Two thirds of 30 = 20 students play sport.
• Half of those who play sport = 20 ÷ 2 = 10 students play football.

In a single calculation: $\frac{1}{2}\times \frac{2}{3}\times 30=10$.

Fraction equivalences in word problems

Sometimes the problem uses a fraction word and you have to recognise the equivalent.

Common pitfalls

Confusing "half" with "halve"

"Halve 12" means 12 ÷ 2 = 6. "Half of 12" also = 6. Same arithmetic, different verb form.

"Two thirds of a third"

This isn't $\frac{2}{3}\cdot 3=2$. It's $\frac{2}{3}\cdot \frac{1}{3}=\frac{2}{9}$. Read carefully — is it "of a third" (fraction of a fraction) or "of 3" (fraction of a whole)?

Forgetting to add up parts

When the problem says "she walks half then a quarter then keeps a quarter", make sure your three parts add to 1 whole. If they don't, you've misread.

Practice

• 🧁 Word problems with fractions
• 🍰 Fraction of a quantity
• 🔄 Decimal ↔ fraction

Related reading

• Multi-step word problems — intro
• Fractions of a quantity — the underlying technique