The denominator doesn’t count pieces. It tells you how big each piece is. A half and a third are pieces of different sizes, so you can’t join them as they are: first write both in sixths, and then 3/6 + 2/6 = 5/6. If you added the denominators too, you would get 2/5, which is less than a half. The full method for any pair of fractions is in the add fractions calculator; this page explains the reason.
The picture that proves it
Imagine two equal bars, one split into 2 parts and the other into 3. One part of the first bar is bigger than one part of the second, and that is the problem: counting “one and one” mixes pieces that aren’t the same size. The fix is to split both bars into 6 parts, the smallest common size. Each of the two fractions can then be counted in sixths.
Worked example: 1/2 + 1/3
One part of a bar split into 2 is shaded, and so is one part of an equal bar split into 3. What portion of one whole bar is shaded?
- Compare the size of the parts. One part of the bar split into 2 is bigger than one part of the bar split into 3, so you can’t count them together.
- Pick a common size. The least common multiple (LCM) of 2 and 3 is 6, and 6 is divisible by both. For other pairs, the link explains how to find it.
- Convert each fraction. 1/2 = 3/6 because 1 × 3 = 3 and 2 × 3 = 6; 1/3 = 2/6 because you multiply the top and bottom by 2.
- Add the numerators. 3 + 2 = 5, and the denominator stays 6.
Result: 5/6 of a bar.
Check: five sixths is more than a half, which is three sixths, and less than six sixths, which is the whole bar.
When the denominator does stay the same
If two fractions already count parts of the same size, the denominator is just the name of the piece, and it doesn’t change when you join them. In 2/7 + 3/7 there are two sevenths and three sevenths, which make five sevenths of the same whole. It’s the same as adding 2 apples and 3 apples. Fractions with the same denominator are called like fractions, and those with different denominators are unlike fractions. Only like fractions can be added directly.
| Type | What you do with the denominator | Example |
|---|---|---|
| Like fractions | Keep it | 2/7 + 3/7 = 5/7 |
| Unlike fractions | Make them equal before adding | 3/5 + 1/10 = 6/10 + 1/10 = 7/10 |
What matters is the size of the parts, not whether the numbers look alike. In the second row, 3/5 is rewritten in tenths because 10 is a multiple of 5, and from there the parts can be compared. Units of length work the same way. 1 m + 1 cm isn’t 2 of anything, because you have to convert both to centimeters and add 100 + 1 = 101 cm.
Practice with three cases
A few well-chosen cases make the rule stick. In each one, decide whether the denominator stays or has to be made equal. To compare with other common slips, the guide to common fraction mistakes collects five. If the problem is that the two fractions aren’t about the same whole, the guide on the same whole explains why matching denominators isn’t enough.
Three exercises
- Calculate 2/9 + 4/9 and say whether the denominator changes.
- Calculate 3/10 + 1/4 with a common denominator.
- Explain why 3/10 + 1/4 can’t be 4/14.
Show the answers
- 6/9 = 2/3. The denominator stays because both addends count ninths.
- The common denominator is 20, which gives 6/20 + 5/20 = 11/20.
- 3/10 is 0.3, and the sum has to be greater than that addend. But 4/14 is almost 0.29, which is less than 3/10.
For the rule in words, the English Wikipedia article Fraction, in its section “Arithmetic with fractions,” says that only like quantities can be added, so unlike fractions must first be converted to a common denominator, which can be found by multiplying the two denominators, and it gives a/b + c/d = (ad + cb)/bd (accessed October 17, 2026). To subtract with the same logic, use the subtract fractions calculator.