How to Add Fractions with Different Denominators
To add fractions with different bottom numbers, you first make the bottom numbers the same. Find the lowest common denominator (the smallest number both denominators divide into), rewrite each fraction over it, add the top numbers, and simplify. That's genuinely all there is to it. The examples below just show the moves.
Why you can't just add the tops and bottoms
It's tempting to say 1/2 + 1/3 = 2/5. But think about it with pizza. Half a pizza plus a third of a pizza is clearly more than half a pizza, and 2/5 is less than half. The pieces are different sizes, so you have to cut them into same-sized pieces before you can count them. That's what a common denominator does.
Example 1: 1/3 + 1/4
- Lowest common denominator of 3 and 4 is 12.
- 1/3 = 4/12 (multiply top and bottom by 4).
- 1/4 = 3/12 (multiply top and bottom by 3).
- 4/12 + 3/12 = 7/12.
Example 2: 2/5 + 3/10
Here one denominator already divides into the other, which makes life easy. 10 is a multiple of 5, so the common denominator is just 10.
- 2/5 = 4/10.
- 4/10 + 3/10 = 7/10.
Example 3: 5/6 + 3/8
- Multiples of 6: 6, 12, 18, 24. Multiples of 8: 8, 16, 24. So the lowest common denominator is 24.
- 5/6 = 20/24 and 3/8 = 9/24.
- 20/24 + 9/24 = 29/24.
- That's more than 1, so as a mixed number it's 1 5/24.
The shortcut: cross-multiply
If finding the lowest common denominator feels slow, there's a formula that always works:
a/b + c/d = (a × d + b × c) / (b × d)
For 5/6 + 3/8: (5 × 8 + 6 × 3) / (6 × 8) = (40 + 18) / 48 = 58/48 = 29/24. Same answer. The catch is that you'll often get bigger numbers that need simplifying at the end, but it's reliable when you're in a hurry.
Adding mixed numbers
Turn them into improper fractions first. It saves mistakes.
2 1/3 + 1 3/4
- 2 1/3 = 7/3 and 1 3/4 = 7/4.
- Common denominator 12: 28/12 + 21/12 = 49/12.
- 49 ÷ 12 = 4 remainder 1, so the answer is 4 1/12.
Subtracting works the same way
3 1/4 − 1 2/3 = 13/4 − 5/3 = 39/12 − 20/12 = 19/12 = 1 7/12. Converting to improper fractions first means you never have to "borrow" from the whole number, which is where most subtraction mistakes happen.
Where this turns up in real life
Recipes are the big one. Doubling a recipe that needs 3/4 cup plus 2/3 cup of flour? That's 17/12, or 1 5/12 cups, before doubling. Woodworking and DIY measurements in inches are the other: 5/8" plus 3/16" is 13/16". Quick check with the fraction calculator, which shows every step, and if you need to multiply or divide next, see how to multiply and divide fractions.
Quick recap
- Make the denominators match (lowest common denominator, or cross-multiply).
- Add or subtract the numerators. Keep the denominator.
- Simplify, and convert to a mixed number if the top is bigger than the bottom.
Frequently asked questions
How do you add fractions with different denominators?
Find the lowest common denominator, rewrite each fraction over it, add the numerators, and simplify. For example 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
What is the lowest common denominator?
The smallest number that both denominators divide into evenly. For 6 and 8 it's 24.
Is there a shortcut for adding fractions?
Yes: a/b + c/d = (ad + bc) / bd. It always works but may need simplifying afterwards.
How do you add mixed numbers?
Convert each to an improper fraction, add them using a common denominator, then convert the answer back to a mixed number.