To add fractions with different denominators, rewrite them so the bottoms match, then add the tops. For 1/6 + 1/4, the smallest shared denominator is 12, so you get 2/12 + 3/12 = 5/12. The same trick handles subtraction. Below are the steps, a quick way to find the least common denominator, mixed numbers, and the rules for multiplying and dividing, which are actually easier.
The denominator tells you what size the pieces are. One-sixth of a pizza and one-quarter of a pizza are different-sized slices, so "1 + 1" doesn't give you two of anything useful. Before you can add them, you have to cut both into the same size of piece. Twelfths work here: 1/6 is 2/12 and 1/4 is 3/12, so together you have 5/12.
That's the whole idea. Rewrite each fraction so the bottoms match, then add or subtract the tops.
Subtraction is identical except for step 3. For 5/8 − 1/6, the LCM of 8 and 6 is 24, so you get 15/24 − 4/24 = 11/24.
For small numbers, list the multiples of the larger denominator until you hit one the smaller denominator divides into. Multiples of 18 are 18, 36, 54... and 12 goes into 36, so 36 is the LCM of 12 and 18.
For bigger numbers, prime factors are quicker. Write each denominator as primes and take the highest power of every prime that appears. 12 = 2² × 3 and 18 = 2 × 3², so the LCM is 2² × 3² = 36. The factoring calculator gives you the prime breakdown if you'd rather not do it by hand, and our guide to finding prime factorization walks through the method.
You can always skip the LCM and just multiply the two denominators together. It works every time, but the numbers get bigger than they need to be and you'll have more simplifying to do at the end:
| Problem | Multiply denominators | Least common denominator | Answer |
|---|---|---|---|
| 1/6 + 1/4 | 24 | 12 | 5/12 |
| 2/3 + 3/5 | 15 | 15 | 19/15 = 1 4/15 |
| 5/8 − 1/6 | 48 | 24 | 11/24 |
| 3/10 + 4/15 | 150 | 30 | 17/30 |
| 7/12 + 5/18 | 216 | 36 | 31/36 |
When the denominators share no factors, like 3 and 5, both approaches give the same number. When they do share factors, the LCM keeps things much smaller. 7/12 + 5/18 over 216 is 126/216 + 60/216 = 186/216, which you then have to reduce by 6. Over 36 it's just 21/36 + 10/36 = 31/36.
You've got two options. The first is to add the whole numbers and the fractions separately, then tidy up. Say you're adding two board lengths, 3 5/16 inches and 1 7/8 inches:
The second option is to convert both to improper fractions first. 3 5/16 is (3 × 16 + 5)/16 = 53/16, and 1 7/8 is 15/8 = 30/16. Add them to get 83/16, which is 5 3/16. Same answer. The improper route is easier with subtraction, because you never have to borrow from the whole number.
Good news: neither needs a common denominator.
To multiply, multiply straight across. 3/4 × 2/9 = 6/36, which simplifies to 1/6. You can save a step by cancelling first: the 3 on top and the 9 on the bottom share a 3, and the 2 and the 4 share a 2, leaving 1/2 × 1/3 = 1/6.
To divide, flip the second fraction and multiply. 3/4 ÷ 1/8 becomes 3/4 × 8/1 = 24/4 = 6. That reads naturally, too: there are six eighths in three quarters.
| Operation | Rule | Example |
|---|---|---|
| Add | Common denominator, add numerators | 1/6 + 1/4 = 2/12 + 3/12 = 5/12 |
| Subtract | Common denominator, subtract numerators | 5/8 − 1/6 = 15/24 − 4/24 = 11/24 |
| Multiply | Numerator × numerator, denominator × denominator | 3/4 × 2/9 = 6/36 = 1/6 |
| Divide | Flip the second fraction, then multiply | 3/4 ÷ 1/8 = 3/4 × 8 = 6 |
A quick sanity check catches most of these. Estimate with decimals first: 1/6 is about 0.17 and 1/4 is 0.25, so the sum should be a bit over 0.4. 5/12 is 0.417, which fits.
That decimal habit helps outside of homework, too. Fractions turn into percentages the same way. 5/12 is 41.7%, and the percentage calculator handles the conversions and "what percent of" questions that come up once you're working with real numbers.
Add, subtract, multiply or divide two fractions and get a simplified result plus the decimal.
Fraction Calculator →To add or subtract fractions with different denominators, rewrite both over the least common denominator, combine the numerators, and simplify. Multiplication goes straight across, and division flips the second fraction first. Estimate with decimals before you commit to an answer.
Find the least common denominator, rewrite each fraction over it by multiplying top and bottom by the same number, add the numerators, and simplify. For 1/6 + 1/4, use 12: 2/12 + 3/12 = 5/12.
5/6. Over a common denominator of 6, one-half is 3/6 and one-third is 2/6, and 3 + 2 = 5.
Yes. Multiplying the denominators always gives a common denominator, but it's often bigger than it needs to be, so you'll have to simplify more at the end. The least common multiple keeps the numbers small.
No. Multiply the numerators together and the denominators together, then simplify. 3/4 × 2/9 = 6/36 = 1/6.
The simplest way is to convert both to improper fractions, subtract over a common denominator, and convert back. For 5 1/4 − 2 2/3, that's 21/4 − 8/3 = 63/12 − 32/12 = 31/12, or 2 7/12.