Adding Fractions With Different Denominators

Adding 1/5 + 2/5 is easy: the pieces are the same size, so it's 3/5. But 1/2 + 1/3 stops many children (and quite a few adults) in their tracks, because the pieces don't match. The fix is to cut both fractions into pieces of the same size first. Once that idea is clear, the method is four steps that never change.

Why can't you just add the tops and bottoms?

This is the most common mistake: 1/2 + 1/3 = 2/5. A quick check shows it can't be right. A half plus a third is more than a half, but 2/5 is less than a half. The denominator tells you the size of the pieces, and you can only add pieces of the same size (the fraction visualiser shows how much bigger a half's slices are than a third's), just as you can't add 3 metres and 2 centimetres and call it 5 of anything.

What are the steps to add fractions with different denominators?

1. Find a common denominator: a number both denominators divide into.
2. Rewrite each fraction as an equivalent fraction with that denominator, by multiplying the top and bottom by the same number.
3. Add the numerators. Keep the denominator the same.
4. Simplify the answer, and turn an improper fraction into a mixed number if needed.

What does a worked example look like?

Let's add 2/3 + 1/4.

Step	Working
1. Common denominator	Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. Use 12.
2. Rewrite 2/3	3 × 4 = 12, so multiply the top by 4 too: 2/3 = 8/12
2. Rewrite 1/4	4 × 3 = 12, so multiply the top by 3 too: 1/4 = 3/12
3. Add the tops	8/12 + 3/12 = 11/12
4. Simplify	11 and 12 share no factors, so 11/12 is the answer

The fraction addition practice walks children through exactly these steps, one at a time, and shows the common denominator being found.

How do you find a common denominator?

There are three situations, from easiest to hardest:

• One denominator is a multiple of the other. For 1/2 + 3/8, 8 is already a multiple of 2, so use 8: 4/8 + 3/8 = 7/8. This is the Year 5 case in England.
• The denominators share no factors. For 1/3 + 1/5, multiply them: 3 × 5 = 15. So 5/15 + 3/15 = 8/15.
• The denominators share a factor. For 1/4 + 1/6, multiplying gives 24, which works, but 12 is smaller. List the multiples of the bigger number (6, 12…) and stop at the first one the smaller number goes into.

All of this is times tables in disguise. A child who knows the times tables well can spot that 12 is in both the 4s and the 6s without writing lists.

What if the answer is more than a whole?

Take 3/4 + 2/3. The common denominator is 12: 9/12 + 8/12 = 17/12. The top is bigger than the bottom, so it's more than one whole. 12 twelfths make 1, with 5 twelfths left over, so the answer is 1 5/12.

How do you add mixed numbers?

Add the whole numbers and the fractions separately, then combine. For 1 1/2 + 2 1/3: the wholes make 3, and 1/2 + 1/3 = 3/6 + 2/6 = 5/6. So the answer is 3 5/6. If the fractions add up to more than one, carry the extra whole over to the whole numbers.

Does the same method work for subtraction?

Yes. Find a common denominator, rewrite, then subtract the numerators: 5/6 − 1/4 = 10/12 − 3/12 = 7/12. You can practise it with subtracting fractions. Multiplying fractions is different: you just multiply the tops and multiply the bottoms, with no common denominator needed (try multiplying fractions).

How can you help at home?

• Make sure equivalent fractions are secure first; they are the whole method.
• Use paper strips or a chocolate bar to show why a half and a third can't just be added.
• Connect fractions to decimals and percentages, as in how they connect.
• Estimate first: "Will the answer be more or less than one?"
• Print a fractions worksheet with an answer key for extra practice, or pick another operation in the fraction workouts.