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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What a Fraction Actually Means

Definition: Fraction

A fraction shows part of a whole. The denominator (bottom number) tells you how many equal pieces the whole is split into. The numerator (top number) tells you how many of those pieces you have.

A fraction has two parts: the numerator (top) counts the pieces you have, the denominator (bottom) tells you how many equal pieces make a whole.

You can only add or subtract pieces that are the same size. Three quarters plus two quarters is easy — they are both “quarters,” so you just count them: five quarters. The whole trick to adding fractions is first making the pieces the same size.

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.

a/c + b/c = (a + b)/c

Work out 3/8 + 2/8

Check the denominators

Both are 8 — the pieces are the same size, so we can add directly.

Add the numerators only

3 + 2 = 5. Keep the denominator: 5/8.

Can it be simplified?

5 and 8 share no common factor, so 5/8 is the final answer.

Never add the denominators. 3/8 + 2/8 is 5/8, not 5/16. The denominator names the size of the piece — it does not change just because you have more pieces.

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 least common multiple (LCM) of the two bottom numbers — see our GCF and LCM guide if that term is new (GCF is greatest common factor; HCF is the same idea used internationally).

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

Multiples of 2: 2, 4, 6… Multiples of 3: 3, 6… The LCM is 6.

Rewrite 1/2 with denominator 6

Multiply top and bottom by 3: 1/2 = 3/6.

Rewrite 1/3 with denominator 6

Multiply top and bottom by 2: 1/3 = 2/6.

Now the pieces match — add the numerators

3/6 + 2/6 = 5/6. It cannot be simplified, so the answer is 5/6.

Work out 5/6 − 1/4

Find the common denominator

Multiples of 6: 6, 12… Multiples of 4: 4, 8, 12… The LCM is 12.

Rewrite both fractions over 12

5/6 = 10/12 (×2) and 1/4 = 3/12 (×3).

Subtract the numerators

10/12 − 3/12 = 7/12.

Simplify if possible

7 and 12 share no common factor, so 7/12 is the answer.

Stuck finding the LCM? You can always use the two denominators multiplied together as a common denominator. For 1/2 + 1/3 that gives 6anyway, but for trickier pairs it may give a bigger number that you simplify at the end. It always works — it is just not always the smallest. Our GCF and LCM guide walks through faster methods.

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

2 1/3 = 7/3 (2 × 3 + 1 = 7) and 1 1/2 = 3/2 (1 × 2 + 1 = 3).

Common denominator

LCM of 3 and 2 is 6: 7/3 = 14/6 and 3/2 = 9/6.

Add

14/6 + 9/6 = 23/6.

Convert back to a mixed number

23 ÷ 6 = 3 remainder 5, so the answer is 3 5/6.

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 our factors and multiples guide or GCF and LCM guide, then book a consultation and we will work through them together until they click.

How to Add and Subtract Fractions (Step by Step)

What a Fraction Actually Means

Case 1: Same Denominator (the Easy Case)

Case 2: Different Denominators (Find a Common Denominator)

Mixed Numbers (a Whole Number plus a Fraction)

Quick Reference

Next: Multiplying and Dividing Fractions

How this topic connects

Related topics

What Does x Really Mean? Understanding Variables

What Are Factors and Multiples? A Visual Guide for Grades 5–8

Prime and Composite Numbers Explained (Plus Co-prime)

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