How to solve it
- Find the least common multiple (LCM) of the denominators.
- Rewrite each fraction with that common denominator.
- Add or subtract the numerators and keep the common denominator.
- Simplify the result.
1/4 + 2/3 = 3/12 + 8/12 = 11/12
Result: 11/12
Common mistake:Wrong: 1/2 + 1/3 = 2/5.Right: 1/2 + 1/3 = 5/6.Adding numerators and denominators separately mixes parts of different sizes. The parts must be the same size before you add.
Full guide: Why denominators are not added
To add fractions, first check whether the denominators match. If they do, add the numerators. If not, rewrite each fraction over a common denominator and then add. Here is the method with a complete example, the five cases that change the procedure, and one very common mistake. For the reason behind the common denominator, with pictures, see the guide why you don’t add denominators.
Worked example: 1/6 + 1/4
You cut 1/6 of a rope and then 1/4 of the same rope. What part of the rope have you cut in total?
- Find the common denominator. The least common multiple of 6 and 4 is 12.
- Convert each fraction. Multiply top and bottom: the first by 2 and the second by 3, which gives 2/12 and 3/12.
- Add the numerators and simplify. 2/12 + 3/12 = 5/12, and 5 and 12 share no divisors.
Result: 5/12 of the rope.
Check: 1/6 ≈ 0.1667 and 1/4 = 0.25; their sum ≈ 0.4167 (error under 0.0001) matches 5/12 ≈ 0.4167.
What changes with the denominators
Fractions with the same denominator are called like fractions, and those with different denominators are unlike fractions. Like fractions you add directly; with unlike fractions you have to make the denominators match first. The table lists the most common cases and the shortest way to handle each.
| Case | What you do | Calculation |
|---|---|---|
| Same denominator | Add the numerators | 2/7 + 3/7 = 5/7 |
| One denominator divides the other | Use the larger as the common one | 3/4 + 1/8 = 6/8 + 1/8 = 7/8 |
| Relatively prime denominators | Use their product | 3/5 + 2/7 = 21/35 + 10/35 = 31/35 |
| Mixed numbers | Convert to improper fractions | 2 1/3 + 1 3/4 = 28/12 + 21/12 = 49/12 = 4 1/12 |
| A negative fraction | Add with its sign | 3/4 + (−5/6) = 9/12 − 10/12 = −1/12 |
In most cases the common denominator is the LCM, not the product of the denominators: with 4 and 8 the product would be 32, but 8 is enough. Only when the denominators share no factors, like 5 and 7, does the LCM equal the product. If you need help finding it, see the least common denominator calculator; and to check that each conversion keeps the value, see equivalent fractions.
When to use another page
The result of an addition comes out reduced, but if you only want to reduce a fraction you already have, use simplify fractions. For the inverse operation, where the order of the terms does matter, there is subtract fractions. If your problem mixes sums with products or parentheses, the calculator on the home page has an expression mode. And to check a result by hand before you trust it, the guide to verifying your answer offers three quick tests.
When you want to practice, the fraction worksheets come with exercises and answers. OpenStax covers adding with like denominators and with unlike denominators. The calculator accepts up to 100 digits per number and rejects any denominator equal to zero.
Frequently asked questions
Do I need the LCM, or will any common denominator work?
Any common multiple of the denominators works, because the conversions are equivalent. With 5/6 + 7/9 you can use 36: 30/36 + 28/36 = 58/36, which reduces to 29/18. With the LCM, which is 18, you get 15/18 + 14/18 = 29/18 with nothing left to reduce. The LCM saves that last step.
How do you add a whole number and a fraction?
Write the whole number as a fraction with denominator 1 and add over a common denominator. So 4 + 2/7 = 28/7 + 2/7 = 30/7, which is the same as the mixed number 4 2/7. In the calculator's advanced mode, just type "4" in its field.
Can you add a decimal and a fraction?
Yes, if the decimal terminates: write it as a fraction and add as usual. The decimal 0.25 is 1/4, so 0.25 + 2/3 = 3/12 + 8/12 = 11/12. In advanced mode you can type the decimal directly, with a point or a comma, and the answer is still an exact fraction.
Why can the result be greater than 1, like 19/12?
Because the two amounts together are more than one whole. With 5/6 + 3/4 = 10/12 + 9/12 = 19/12, and 19/12 reads as 1 whole and 7/12 more: 1 7/12. It is an improper fraction, valid and already in lowest terms. If you need the mixed form, the mixed numbers calculator does the conversion.
How do you add three or more fractions?
Find the LCM of all the denominators, convert each fraction, and add the numerators in one go. For 1/2 + 1/3 + 1/7 the LCM is 42, and 21/42 + 14/42 + 6/42 = 41/42. You do not need to add two at a time: the result is the same in any order.