Adding fractions with unlike denominators: a parent’s guide

The bottom number, the denominator, tells you what size the pieces are. Halves are big pieces. Thirds are a bit smaller. Fifths are smaller again.

So 1/2 + 1/3 is one big piece plus one smaller piece. The answer cannot be 2/5, because two fifths is less than one half on its own. You started with more than a half and ended up with less. Something went wrong.

The real problem is that you cannot count pieces of different sizes together. One apple plus one watermelon is not two of anything useful. You need to cut both fractions into pieces of the same size first. That shared size is the common denominator.

1. 1Find a common denominator: a number both denominators divide into. The smallest one is easiest to work with.
2. 2Rewrite each fraction with that denominator. Whatever you multiply the bottom by, multiply the top by the same number.
3. 3Add (or subtract) the numerators. Keep the denominator the same.
4. 4Simplify if you can, and turn an improper fraction into a mixed number if the worksheet asks for one.

Step 2 is the one to slow down on. Multiplying the top and bottom by the same number does not change the size of the fraction. It cuts the same amount into more, smaller pieces. Three sixths is exactly as much as one half.

Common denominator: list the multiples of 4 (4, 8, 12, 16) and of 6 (6, 12, 18). The first one in both lists is 12.

Rewrite: 1/4 is 3/12, because we multiply top and bottom by 3. 1/6 is 2/12, because we multiply top and bottom by 2.

Add the tops: 3/12 + 2/12 = 5/12. Five and twelve share no factor except 1, so that is already in simplest form. So 1/4 + 1/6 = 5/12.

You could also use 24 as the common denominator (4 × 6). You would get 10/24, which simplifies to 5/12. Multiplying the two denominators always works, it just means more simplifying at the end.

Try 2/3 + 1/2. The common denominator is 6. 2/3 becomes 4/6 and 1/2 becomes 3/6. Then 4/6 + 3/6 = 7/6.

Seven sixths is more than one whole, because six sixths make one. Take out the whole and you have one sixth left, so 2/3 + 1/2 = 1 1/6.

For 3/4 − 1/6, use twelfths again. 3/4 is 9/12 and 1/6 is 2/12. Then 9/12 − 2/12 = 7/12.

Only the numerators change. The denominator is the size of the pieces, and taking some pieces away does not change their size.

A note on mixed numbers

Mixed numbers, like 1 1/2 and 2 1/3, can be added in two parts. Add the whole numbers: 1 + 2 = 3. Add the fractions with a common denominator: 1/2 + 1/3 = 5/6. Put them together and the answer is 3 5/6. If the fraction part comes to more than one whole, move that whole over to the whole-number part, much like carrying in column addition.

• Adding the denominators. Go back to the paper strips. Ask whether the answer is bigger than each fraction on its own.
• Changing the bottom but not the top. Writing 1/4 as 1/12. The fraction just got three times smaller. Whatever happens to the bottom happens to the top.
• Not simplifying. 10/24 is correct but most teachers want 5/12. Check whether the top and bottom share a factor.
• Getting lost finding the common denominator. Listing multiples is slow but reliable. Encourage it until they can spot common denominators by eye.

Children usually add fractions with the same denominator in fourth grade, then move to unlike denominators in fifth grade. Before that, they spend a lot of time on what a fraction is and on equivalent fractions, which is exactly the skill step 2 uses. If your child is stuck, a little time on equivalent fractions often fixes it.

The fractions practice page has the steps and mistakes on one screen, and our 3rd and 4th grade skills checklist shows the fraction ideas that come before this.

MathPad has a fractions pad that walks through the same steps: find the common denominator, rewrite both fractions, add or subtract, then simplify. Your child’s guide explains each step in words like the ones above, and the answer can be checked as a fraction or a mixed number.