How to add fractions

A fraction has two numbers. The bottom one, the denominator, says what size the pieces are: quarters, sixths, tenths. The top one, the numerator, says how many of those pieces you have. Adding fractions means counting the pieces together, and that only works when they are all the same size.

The rule

When the bottom numbers are the same, add the top numbers and keep the bottom number. When they are different, change the fractions into equal fractions with the same bottom number first.

How to do it

Look at the bottom numbers. If they are the same, go straight to adding.
If they are different, find a number that both bottom numbers divide into. The smallest one is their lowest common multiple, and using it keeps the numbers small.
Change each fraction to have that bottom number. Whatever you multiply the bottom by, multiply the top by the same: 1/3 becomes 2/6, because both numbers are multiplied by 2.
Add the top numbers and write the answer over the bottom number. The bottom number stays as it is.
Simplify if you can. If the top and the bottom both divide by the same number, divide them, so 3/6 becomes 1/2. If the top is bigger than the bottom, the answer is more than 1, and you can write it as a mixed number.

Worked examples

What is 1/4 + 2/4?

1/4 + 2/4 = 3/4

The bottom numbers match, so every piece is a quarter. 1 quarter and 2 quarters make 3 quarters, and the bottom stays 4.

What is 1/3 + 1/6?

1/3 = 2/6
2/6 + 1/6 = 3/6 = 1/2

Thirds and sixths are different sizes, but every third is two sixths, so 1/3 is 2/6. Then 2/6 + 1/6 = 3/6, and since 3 and 6 both divide by 3, the answer simplifies to 1/2.

What is 2/3 + 1/4?

The LCM of 3 and 4 is 12
2/3 = 8/12 and 1/4 = 3/12
8/12 + 3/12 = 11/12

Neither bottom number goes into the other, so use 12, the first number in both the 3 and the 4 times tables. Multiply 2/3 top and bottom by 4, and 1/4 top and bottom by 3. 11 and 12 share no factor but 1, so 11/12 is already as simple as it gets.

What is 3/4 + 2/4?

3/4 + 2/4 = 5/4 = 1 1/4

5 quarters is more than the 4 quarters in a whole. Fill one whole with 4 of them and 1 quarter is left over, so the answer is one and a quarter. 5/4 is an improper fraction and 1 1/4 is a mixed number, and they are the same amount.

The quick way, multiply the bottoms

If the lowest common multiple is hard to spot, multiply the two bottom numbers together. That always works as a shared bottom number, though it may not be the smallest.

For 1/4 + 1/6: 4 × 6 = 24, so 1/4 = 6/24 and 1/6 = 4/24, and the sum is 10/24. That simplifies to 5/12, the same answer the lowest common multiple, 12, gives straight away: 3/12 + 2/12 = 5/12.

Adding mixed numbers

Add the whole numbers, then add the fractions. For 1 1/2 + 2 1/4: 1 + 2 = 3, and 1/2 + 1/4 = 2/4 + 1/4 = 3/4, so the answer is 3 3/4.

If the fractions add up to more than 1, carry the whole across, the way a carry works in long addition. 2 3/4 + 1 2/4 is 3 and 5/4, and 5/4 is 1 1/4, so the answer is 4 1/4.

Why the bottom number stays the same

The bottom number is a name, not an amount. It says what kind of piece you have, the way the word apples does in 3 apples. 1 quarter and 2 quarters make 3 quarters, just as 1 apple and 2 apples make 3 apples. The pieces are still quarters, so the bottom stays 4.

That is also why the bottoms must match first. 1/3 + 1/4 is one big piece and one smaller piece, and there is no one word for what you have. Cut them both into twelfths and you can count: 4 twelfths and 3 twelfths make 7 twelfths, 7/12.

Changing 1/3 into 4/12 does not change the amount. Cutting every third into 4 smaller pieces gives you 4 times as many pieces, each 4 times smaller, so you have exactly as much as before.

Where mistakes happen

Adding the bottom numbers as well as the tops. 1/2 + 1/2 would come out as 2/4, which is still only a half. Two halves make a whole, 2/2, so the bottom has to stay the same.
Adding before the bottoms match. 1/3 + 1/6 is not 2/9. Change 1/3 into 2/6 first, and the answer is 3/6.
Changing the bottom number but not the top. When 1/3 becomes sixths, the bottom is multiplied by 2, so the top must be too: 2/6, not 1/6.
Giving an improper fraction when the question asks for a mixed number, or the other way round. 5/4 and 1 1/4 are the same amount, but check which one the question wants.

Questions people ask

Do I have to simplify the answer?

3/6 and 1/2 are the same amount, so 3/6 is not wrong, just not finished. Most school tests want the simplest form, so it is a good habit. In the Math Basics practice the answer is left as it comes out, 3/6 rather than 1/2, so that the adding is what gets practised.

Do I have to use the lowest common multiple?

No. Any number that both bottoms divide into will work. The lowest one, called the lowest common denominator, keeps the numbers small, so there is less to simplify at the end.

What are improper fractions and mixed numbers?

An improper fraction has a top bigger than its bottom, such as 5/4, and is worth more than 1. British teachers sometimes call it top-heavy. A mixed number writes the same amount as a whole number and a fraction, 1 1/4. To turn 5/4 into one, divide: 5 ÷ 4 = 1 remainder 1, so 1 whole and 1 quarter.

Is it done the same way in America?

Yes, the method is the same. American books often say reduce instead of simplify, and least common denominator instead of lowest common denominator.