Adding and Subtracting Fractions

Fractions are pieces of a whole

Before we add or subtract, remember what a fraction means. A fraction shows a part of a whole. The bottom number, the denominator, tells you how many equal pieces the whole is cut into. The top number, the numerator, tells you how many of those pieces you have.

So in 3/4, the whole is cut into 4 equal pieces and you have 3 of them. If you need a refresher, visit our Introduction to Fractions lesson first.

Adding and subtracting fractions is really just counting pieces — but only when the pieces are the same size.

Same denominator: the easy case

When two fractions have the same denominator, their pieces are already the same size. You just count them.

Rule: add (or subtract) the numerators. Keep the denominator the same.

Example 1 — Adding. You eat 2/8 of a pizza, then 3/8 more.

1. The pieces are all eighths, so they are the same size.
2. Add the tops: 2 + 3 = 5.
3. Keep the bottom: 8.
4. So 2/8 + 3/8 = 5/8.

Example 2 — Subtracting. A jug holds 7/10 of a litre. You pour out 4/10.

1. Same denominator (tenths), so subtract the tops: 7 − 4 = 3.
2. Keep the bottom: 10.
3. So 7/10 − 4/10 = 3/10.

Why you never add the denominators

This is the most common mistake, so let's understand the why. Imagine cutting a cake into 4 pieces. Each piece is one quarter. If you add 1/4 + 1/4 and changed the bottom to get 2/8, you would be saying the pieces suddenly became eighths — smaller! But they didn't change size at all. You simply have two quarters: 1/4 + 1/4 = 2/4. The denominator stays the same because the size of each piece stays the same.

Different denominators: make them match

What about 1/2 + 1/3? Here the pieces are different sizes — halves and thirds. You cannot count them together until they are the same size. We fix this by making equivalent fractions with a common denominator.

A common denominator is a number that both denominators divide into evenly. For 2 and 3, a common denominator is 6.

Example 3 — Adding 1/2 + 1/3.

1. Find a common denominator for 2 and 3. The smallest is 6.
2. Rewrite each fraction with denominator 6:
3. 1/2 = (1×3)/(2×3) = 3/6
4. 1/3 = (1×2)/(3×2) = 2/6
5. Now the pieces match. Add the numerators: 3 + 2 = 5.
6. So 1/2 + 1/3 = 5/6.

Example 4 — Subtracting 3/4 − 1/6.

1. A common denominator for 4 and 6 is 12.
2. Rewrite each fraction:
3. 3/4 = (3×3)/(4×3) = 9/12
4. 1/6 = (1×2)/(6×2) = 2/12
5. Subtract: 9 − 2 = 7.
6. So 3/4 − 1/6 = 7/12.

Here is a quick reference table showing how to convert to a common denominator:

Problem	Common denominator	Rewritten	Answer
1/2 + 1/4	4	2/4 + 1/4	3/4
1/3 + 1/6	6	2/6 + 1/6	3/6 = 1/2
5/6 − 1/3	6	5/6 − 2/6	3/6 = 1/2
2/5 + 1/2	10	4/10 + 5/10	9/10

Simplify your answer

A fraction is in its simplest form when the top and bottom share no common factor except 1. After adding or subtracting, always check if you can simplify.

Example 5. Add 1/6 + 1/6.

1. Same denominator: 1 + 1 = 2, giving 2/6.
2. The greatest common factor of 2 and 6 is 2.
3. Divide both: 2 ÷ 2 = 1 and 6 ÷ 2 = 3.
4. So 2/6 = 1/3.

A practice activity

Grab a sheet of paper and fold it. Try this with real folds:

1. Fold a strip into 4 equal parts. Colour 1 part. That is 1/4.
2. Colour 1 more part. Now you have coloured 1/4 + 1/4 = 2/4. Notice the strip is still in quarters!
3. Now take a new strip, fold it into 2, and colour 1 part (1/2). Lay it next to two quarters. They cover the same length, showing 2/4 = 1/2.

Then try these on paper (answers below): (a) 3/5 + 1/5, (b) 7/8 − 3/8, (c) 1/2 + 1/4, (d) 2/3 − 1/6.

Answers: (a) 4/5, (b) 4/8 = 1/2, (c) 3/4, (d) 4/6 − 1/6 = 3/6 = 1/2.

Where this leads

Once you can add and subtract fractions, you are ready to work with Equivalent Fractions more deeply, and later with decimals and percentages, which are just fractions in disguise. Practise by sharing real food and combining the parts — it makes the maths feel natural.