Fractions (Years 6–11) — Maths

A fraction is a way of writing part of a whole. Imagine a pizza cut into 4 equal slices and you eat 3 of them — you've eaten 3/4 ("three quarters") of the pizza.

Why this matters: fractions are everywhere once you look — half an hour, a quarter of a pizza, three-fifths off in a sale, ingredients in a recipe. They're also the gateway to decimals, percentages and algebra, so the skills here pay off again and again.

3 of the 4 equal parts shaded = 3/4

The two numbers in a fraction

Every fraction has a top number and a bottom number, and they mean different things:

The bottom number is the denominator — it tells you how many equal pieces the whole has been split into.
The top number is the numerator — it tells you how many of those pieces you have.

In 3/4: the 4 means the whole is split into 4 equal pieces, and the 3 means we have 3 of them.

Why the pieces must be equal: "3 out of 4" only makes sense if all 4 pieces are the same size. Three big slices and one tiny one wouldn't be three quarters.

Equivalent fractions (the same amount, written differently)

If you cut each slice in half, you now have 8 smaller pieces and your 3 slices become 6 pieces — but you still have the same amount of pizza. So 3/4 = 6/8. These are equivalent fractions.

You make equivalent fractions by multiplying (or dividing) the top and bottom by the same number:

1/2	2/4	3/6	5/10
×1	×2	×3	×5

Split each quarter in two — the shaded length doesn't move, so 3/4 = 6/8.

Simplifying (making a fraction as small as possible)

To simplify, divide top and bottom by the biggest number that goes into both (their highest common factor).

Simplify 6/8: 2 divides into both → 6÷2 = 3 and 8÷2 = 4, so 6/8 = 3/4. Nothing divides into both 3 and 4, so it's fully simplified.

Adding & subtracting fractions

You can only add or subtract pieces that are the same size. Quarters and thirds are different-sized pieces, so first you rewrite both fractions so they have the same denominator (a "common denominator"), then you just add the top numbers.

2/3 + 1/4:
• A denominator that both 3 and 4 go into is 12.
• 2/3 = 8/12 (×4 top and bottom) and 1/4 = 3/12 (×3 top and bottom).
• Now the pieces match: 8/12 + 3/12 = 11/12.

Once both are twelfths, the 8 purple and 3 amber pieces just add: 8/12 + 3/12 = 11/12.

Common mistake: never add the bottoms! 1/2 + 1/2 = 1 (a whole), not 2/4. The denominator stays the same once the pieces match.

Multiplying fractions

This one's friendly: multiply the tops together, and the bottoms together — no common denominator needed.

3/4 × 2/5 = (3×2)/(4×5) = 6/20 = 3/10 (simplified).

3 columns of 4, times 2 rows of 5 → the darker overlap is 6 cells out of 20 = 6/20.

What "×" means here: the word "of" is multiplication. "1/2 of 1/2 a pizza" is a quarter — and 1/2 × 1/2 = 1/4. ✓

Dividing fractions

Dividing asks "how many of this fit into that?" The quick method is Keep, Change, Flip:

Keep the first fraction · Change ÷ into × · Flip the second fraction upside-down. Then multiply as normal.

3/4 ÷ 1/2 → keep 3/4, change to ×, flip 1/2 to 2/1 → 3/4 × 2/1 = 6/4 = 3/2 (= 1½).
Check: how many halves fit into three-quarters? One and a half — ✓.

Finding a fraction of an amount

Divide by the bottom, then multiply by the top.

3/4 of 20: 20 ÷ 4 = 5 (that's one quarter), then 5 × 3 = 15.

In a nutshell: + and − need the same denominator first; × multiply straight across; ÷ keep-change-flip; and always simplify your answer at the end.

Drag the slider until it shows exactly two fifths (2 out of 5):