Adding 1/2 + 1/3 is not as simple as adding the tops and the bottoms — if it were, the answer would be 2/5, which is actually smaller than where you started. The real rule makes sense once you picture what a fraction means. Here is how to add and subtract any fractions, step by step.
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Definition: Fraction
Case 1: Same Denominator (the Easy Case)
When the denominators already match, the pieces are the same size. Just add (or subtract) the numerators and keep the denominator the same.
Work out 3/8 + 2/8
Check the denominators
Add the numerators only
Can it be simplified?
Case 2: Different Denominators (Find a Common Denominator)
When the denominators are different, the pieces are different sizes — so you cannot add them yet. First rewrite both fractions so they share a common denominator. The easiest common denominator to use is the lowest common multiple (LCM) of the two bottom numbers.
The 3-Step Recipe
- 1. Find a common denominator (the LCM of the two denominators).
- 2. Rewrite each fraction with that denominator (multiply top and bottom by the same number).
- 3. Add or subtract the numerators, then simplify.
Work out 1/2 + 1/3
Find the common denominator
Rewrite 1/2 with denominator 6
Rewrite 1/3 with denominator 6
Now the pieces match — add the numerators
Work out 5/6 − 1/4
Find the common denominator
Rewrite both fractions over 12
Subtract the numerators
Simplify if possible
Mixed Numbers (a Whole Number plus a Fraction)
A mixed number like 2 1/3 means “2 wholes and one third.” The safest way to add or subtract them is to turn each into an improper fraction first, then follow the same rules.
Work out 2 1/3 + 1 1/2
Convert to improper fractions
Common denominator
Add
Convert back to a mixed number
Quick Reference
| If the denominators are… | Do this |
|---|---|
| The same | Add/subtract the numerators, keep the denominator |
| Different | Find the LCM, rewrite both, then add/subtract numerators |
| Mixed numbers | Convert to improper fractions first, then proceed |
- A fraction is parts of a whole: numerator over denominator.
- You can only add/subtract fractions with the same denominator.
- Same denominator: add/subtract the tops, keep the bottom.
- Different denominators: find the LCM and rewrite both fractions.
- Never add the denominators together.
- Convert mixed numbers to improper fractions before calculating.
- Always simplify the final answer.
Practice Problems
- 1
Work out 2/7 + 3/7.
Hint: Same denominator — just add the numerators.
- 2
Work out 3/4 + 1/6.
Hint: LCM of 4 and 6 is 12. Rewrite both over 12.
- 3
Work out 7/10 − 2/5.
Hint: Rewrite 2/5 with denominator 10.
- 4
Work out 5/8 − 1/3.
Hint: LCM of 8 and 3 is 24.
- 5
Work out 1 3/4 + 2 1/2.
Hint: Convert both to improper fractions first.
Next: Multiplying and Dividing Fractions
Once adding and subtracting feels natural, multiplying is actually easier — you multiply straight across, no common denominator needed. Dividing has a neat “flip and multiply” trick. If fractions keep tripping you up, start with a consultation and we will work through them together until they click.
