Adding and subtracting fractions: a step by step guide

How to add fractions with the same denominator

To add fractions with the same denominator, add the numerators and keep the denominator the same. The denominator tells you the size of the parts, and the numerator tells you how many parts there are. Adding fractions with the same denominator is just counting parts of the same size.

Take ³⁄₈ + ²⁄₈. Government guidance for teachers puts it like this: "3 one-eighths plus 2 one-eighths is equal to 5 one-eighths". So ³⁄₈ + ²⁄₈ = ⁵⁄₈.

It works in the same way as 3 tens and 2 tens making 5 tens. The eighths stay eighths, just as the tens stay tens.

When the answer is more than one whole

Sometimes the parts add up to more than one whole. ⁴⁄₅ + ³⁄₅ = ⁷⁄₅, which is an improper fraction: the numerator is bigger than the denominator. Five of the fifths make one whole, and 2 fifths are left over, so ⁷⁄₅ = 1 ²⁄₅.

From Year 5, children are expected to write answers like this as a mixed number. The improper fractions and mixed numbers guide shows how to change one into the other.

How to subtract fractions

To subtract fractions with the same denominator, subtract the numerators and keep the denominator. ⁵⁄₈ − ²⁄₈ = ³⁄₈, because 5 one-eighths take away 2 one-eighths leaves 3 one-eighths.

If you're wondering how to minus fractions with different denominators, it's the same first step as adding: make the denominators the same, then subtract the numerators. There's more on that below.

Why you never add the denominators

The most common mistake with adding fractions is to add the denominators as well as the numerators, so ¹⁄₂ + ¹⁄₄ becomes ²⁄₆. The denominator is the size of the parts, not a number of parts, so it doesn't get added.

A quick check catches it: ²⁄₆ is smaller than ¹⁄₂, and adding something to ¹⁄₂ can't make it smaller. Drawing the two fractions as bars of the same length, or using a fraction wall, shows the real answer is ³⁄₄.

How to add fractions with different denominators

To add fractions with different denominators, change them into equivalent fractions with the same denominator, then add the numerators. You can only add parts once they're the same size.

When one denominator is a multiple of the other

The curriculum starts here, with ¹⁄₂ + ¹⁄₈. Eight is a multiple of 2, so change the halves into eighths. Multiply the numerator and the denominator by 4: ¹⁄₂ = ⁴⁄₈.

Now both fractions are in eighths, so add the numerators: ⁴⁄₈ + ¹⁄₈ = ⁵⁄₈.

When neither denominator is a multiple of the other

For ²⁄₃ + ¹⁄₄, find a number that is in both the 3 times table and the 4 times table. The smallest is 12, so use twelfths. This is where knowing the times tables really helps.

1. Change ²⁄₃ into twelfths. 3 × 4 = 12, so multiply the top and bottom by 4: ²⁄₃ = ⁸⁄₁₂.
2. Change ¹⁄₄ into twelfths. 4 × 3 = 12, so multiply the top and bottom by 3: ¹⁄₄ = ³⁄₁₂.
3. Add the numerators: ⁸⁄₁₂ + ³⁄₁₂ = ¹¹⁄₁₂.

Subtracting works the same way. For ⁵⁄₆ − ¹⁄₄, use twelfths again: ⁵⁄₆ = ¹⁰⁄₁₂ and ¹⁄₄ = ³⁄₁₂. Then ¹⁰⁄₁₂ − ³⁄₁₂ = ⁷⁄₁₂.

If the answer can be simplified, Year 6 children are expected to do it, so ²⁄₈ would become ¹⁄₄. The simplifying fractions guide explains how.

Adding and subtracting mixed numbers

To add mixed numbers, add the whole numbers, then add the fractions. For 2 ¹⁄₃ + 1 ¹⁄₂:

1. Add the whole numbers: 2 + 1 = 3.
2. Add the fractions, using sixths: ¹⁄₃ = ²⁄₆ and ¹⁄₂ = ³⁄₆, so ²⁄₆ + ³⁄₆ = ⁵⁄₆.
3. Put them together: 2 ¹⁄₃ + 1 ¹⁄₂ = 3 ⁵⁄₆.

If the fractions add up to more than one whole, change the improper fraction into a mixed number and add the extra whole on.

Subtracting mixed numbers can need a little more thought, when the fraction you're taking away is bigger than the one you have. Take 3 ¹⁄₄ − 1 ¹⁄₂. One way, from government guidance, is to take away in pieces and go through a whole number:

1. Take away the whole number first: 3 ¹⁄₄ − 1 = 2 ¹⁄₄.
2. ¹⁄₂ is the same as ²⁄₄, so there are 2 quarters still to take away.
3. Take away 1 quarter to reach the whole number: 2 ¹⁄₄ − ¹⁄₄ = 2.
4. Take away the last quarter: 2 − ¹⁄₄ = 1 ³⁄₄.

Another way is to change both mixed numbers into improper fractions first: 3 ¹⁄₄ = ¹³⁄₄ and 1 ¹⁄₂ = ⁶⁄₄. Then ¹³⁄₄ − ⁶⁄₄ = ⁷⁄₄, which is 1 ³⁄₄. The guidance says this is sometimes a good approach, but not always the quickest, so it's worth asking which way your child's class uses.

When children learn to add and subtract fractions

Adding and subtracting fractions builds up over four years of primary school:

Year	What children learn
Year 3	Add and subtract fractions with the same denominator, within one whole
Year 4	The same, with answers beyond one whole
Year 5	Denominators that are multiples of the same number, such as halves and eighths, and answers over 1 written as mixed numbers
Year 6	Any denominators and mixed numbers, using equivalent fractions

Common mistakes to watch for

• Adding the denominators. ¹⁄₃ + ¹⁄₃ is ²⁄₃, not ²⁄₆. The parts are still thirds.
• Adding before the denominators match. With different denominators, the first step is always to make them the same.
• Changing the denominator but not the numerator. To change ²⁄₃ into twelfths, the numerator is multiplied by 4 as well, giving ⁸⁄₁₂, not ²⁄₁₂.
• Forgetting the whole numbers. With mixed numbers, it's easy to work out the fractions and leave the wholes behind.
• Leaving an improper fraction. In Years 5 and 6, an answer like ⁷⁄₅ is usually written as 1 ²⁄₅.

How to help at home

• Say the parts out loud. "Three eighths and two eighths make five eighths" helps a child hear that the eighths don't change.
• Draw bars. Two bars of the same length, split into different parts, show why halves and eighths can't be added until they match.
• Use food. A pizza or a chocolate bar cut into equal pieces makes adding and taking away fractions easy to see.
• Practise times tables. Finding a common denominator is a times tables question: which number is in both the 3s and the 4s?

Adding and subtracting fractions questions to try

1. ²⁄₉ + ⁵⁄₉
2. ⁷⁄₁₀ − ³⁄₁₀
3. ³⁄₄ + ³⁄₄ (write the answer as a mixed number)
4. ¹⁄₂ + ³⁄₈
5. ²⁄₃ − ¹⁄₅
6. 1 ¹⁄₄ + 2 ¹⁄₃

Answers

1. ²⁄₉ + ⁵⁄₉ = ⁷⁄₉
2. ⁷⁄₁₀ − ³⁄₁₀ = ⁴⁄₁₀, which simplifies to ²⁄₅
3. ³⁄₄ + ³⁄₄ = ⁶⁄₄, and ⁶⁄₄ = 1 ²⁄₄, which simplifies to 1 ¹⁄₂
4. ¹⁄₂ = ⁴⁄₈, so ⁴⁄₈ + ³⁄₈ = ⁷⁄₈
5. Use fifteenths: ²⁄₃ = ¹⁰⁄₁₅ and ¹⁄₅ = ³⁄₁₅, so ¹⁰⁄₁₅ − ³⁄₁₅ = ⁷⁄₁₅
6. 1 + 2 = 3, then ¹⁄₄ = ³⁄₁₂ and ¹⁄₃ = ⁴⁄₁₂, so the fractions make ⁷⁄₁₂. The answer is 3 ⁷⁄₁₂

Fractions work comes up again in multiplying fractions and dividing fractions in Year 6. For a start on what fractions are, see the fractions guide, and for everything else children learn in these years, see KS2 maths. Fraction calculations like these come up on the Year 6 SATs arithmetic paper.