How to Add Fractions

Adding fractions is straightforward once you remember one idea: you can only add pieces that are the same size. This guide shows every case, from matching denominators to mixed numbers.

Case 1: same denominator

When the denominators match, keep the denominator and add the numerators. Three sevenths plus two sevenths is five sevenths, because you are counting pieces of the same size.

Case 2: different denominators

  1. Find the lowest common denominator (LCD), the smallest number both denominators divide into.
  2. Rewrite each fraction with that denominator by multiplying top and bottom by the same number.
  3. Add the numerators and keep the denominator.
  4. Simplify, and convert to a mixed number if the result is improper.

Case 3: mixed numbers

Add the whole numbers and the fractions separately, then combine. If the fraction part is 1 or more, carry the extra whole number across. Alternatively convert each to an improper fraction first.

Case 4: algebraic fractions

Letters follow the same rule. To add 1/x + 1/3, use the common denominator 3x: (3 + x)/(3x). If you need to practise a mix of all four operations, use the fractions solver.

Worked examples

Example 1: Like denominators: 3/7 + 2/7

  1. The denominators match, so add the numerators: 3 + 2 = 5.

Final answer: 5/7

Example 2: Unlike denominators: 2/3 + 1/4

  1. LCD of 3 and 4 is 12.
  2. 2/3 = 8/12 and 1/4 = 3/12.
  3. Add: 8/12 + 3/12 = 11/12.

Final answer: 11/12

Example 3: Unlike denominators: 5/6 + 7/9

  1. LCD of 6 and 9 is 18.
  2. 5/6 = 15/18 and 7/9 = 14/18.
  3. Add: 29/18, which is 1 11/18.

Final answer: 29/18 = 1 11/18

Example 4: Mixed numbers: 2 1/2 + 1 3/4

  1. Fractions: 1/2 = 2/4, so 2/4 + 3/4 = 5/4 = 1 1/4.
  2. Whole numbers: 2 + 1 = 3.
  3. Combine: 3 + 1 1/4.

Final answer: 4 1/4

Example 5: Algebraic: 1/x + 1/3

  1. Common denominator: 3x.
  2. 1/x = 3/(3x) and 1/3 = x/(3x).
  3. Add the numerators.

Final answer: (3 + x) / (3x)

Common mistakes to avoid

  • Adding denominators: 1/2 + 1/3 is not 2/5, it is 5/6.
  • Changing only the denominator when finding an equivalent fraction.
  • Using the product of denominators every time, which gives answers that are hard to simplify.
  • Forgetting to convert improper fractions into mixed numbers when asked.
  • Not simplifying the final answer.

Frequently asked questions

What is the lowest common denominator?

It is the smallest number that is a multiple of every denominator. It keeps numbers small and easy to simplify.

Can I just multiply the denominators?

Yes, it always works, but the result may need extra simplifying.

How do I subtract fractions?

Use exactly the same process with subtraction at the final step.

How do I add a whole number and a fraction?

Write the whole number as a fraction over 1, or keep it separate in a mixed number.

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