To add or subtract fractions with unlike denominators, rewrite them so they share a common denominator, then add or subtract the numerators and keep the denominator. For 5/6 + 3/8, the least common denominator is 24, so the sum becomes 20/24 + 9/24 = 29/24, which is 1 5/24 as a mixed number.
The reason is simple: the denominator tells you the size of the pieces. You can add 3 eighths and 5 eighths to get 8 eighths, but you cannot directly add 5 sixths and 3 eighths any more than you can add 5 feet and 3 inches without converting first. A common denominator converts both fractions into the same size pieces.
The four-step method
- Find the least common denominator (LCD), the smallest number both denominators divide into evenly. It is the least common multiple of the denominators.
- Rewrite each fraction with the LCD. Multiply the numerator and denominator by whatever factor turns the old denominator into the LCD.
- Add or subtract the numerators. Keep the denominator unchanged.
- Simplify the result, and convert to a mixed number if that is the form you need.
Worked example: 5/6 + 3/8
Step 1. Multiples of 8 are 8, 16, 24, 32… The first one divisible by 6 is 24, so the LCD is 24.
Step 2. 6 × 4 = 24, so multiply the first fraction by 4/4: 5/6 = 20/24. Then 8 × 3 = 24, so multiply the second by 3/3: 3/8 = 9/24.
Step 3. 20/24 + 9/24 = 29/24.
Step 4. 29 and 24 share no common factor, so 29/24 is already simplified. As a mixed number, 29 ÷ 24 = 1 remainder 5, giving 1 5/24.
Subtracting fractions
Subtraction follows the identical steps; only the operation on the numerators changes.
7/10 − 1/4: the LCD of 10 and 4 is 20
7/10 = 14/20 and 1/4 = 5/20
14/20 − 5/20 = 9/20
If the second fraction is larger, the answer is negative: 3/4 − 5/6 = 9/12 − 10/12 = −1/12.
Ways to find the least common denominator
| Method | How it works | Example: LCD of 12 and 18 |
|---|---|---|
| List multiples | Count up by the larger denominator until the smaller one divides it | 18, 36 (36 ÷ 12 = 3) |
| Prime factors | Take each prime the highest number of times it appears | 12 = 2² × 3, 18 = 2 × 3², LCD = 2² × 3² = 36 |
| Formula | LCD = b × d ÷ GCF(b, d) | 12 × 18 ÷ 6 = 36 |
| Multiply | Use b × d (a common denominator, not always the least) | 216 |
Some common pairs:
| Denominators | LCD | Product b × d |
|---|---|---|
| 2 and 3 | 6 | 6 |
| 3 and 4 | 12 | 12 |
| 4 and 6 | 12 | 24 |
| 6 and 8 | 24 | 48 |
| 8 and 12 | 24 | 96 |
| 10 and 15 | 30 | 150 |
| 12 and 18 | 36 | 216 |
When the denominators share no factors, such as 5 and 7, the LCD is simply their product, 35. The LCD calculator finds the least common denominator for any set of fractions, and how to find the GCF and LCM explains the underlying methods.
The cross-multiplication shortcut
For two fractions you can skip the LCD and use the product of the denominators:
For 5/6 + 3/8 this gives (40 + 18)/48 = 58/48, which simplifies to 29/24. The answer is the same, but you have to reduce a larger fraction at the end. The shortcut is great for quick mental work with small denominators and clumsy for large ones.
Adding three or more fractions
Find one LCD that works for every denominator, convert each fraction, and add all the numerators together.
1/2 + 2/3 + 3/4: the LCD of 2, 3 and 4 is 12
6/12 + 8/12 + 9/12 = 23/12 = 1 11/12
The adding and subtracting fractions calculator accepts up to six fractions, each with its own plus or minus sign, and shows the LCD steps.
Adding and subtracting mixed numbers
Adding. Add the whole numbers and the fractions separately, then carry any extra whole.
3 1/4 + 2 5/6: whole numbers 3 + 2 = 5
Fractions: 3/12 + 10/12 = 13/12 = 1 1/12
Total: 5 + 1 1/12 = 6 1/12
Subtracting with borrowing. When the fraction you subtract is larger, borrow 1 from the whole number.
5 1/3 − 2 3/4 → 5 4/12 − 2 9/12
4/12 is smaller than 9/12, so borrow: 5 4/12 = 4 16/12
4 16/12 − 2 9/12 = 2 7/12
An alternative that avoids borrowing is to convert both mixed numbers to improper fractions first: 16/3 − 11/4 = 64/12 − 33/12 = 31/12 = 2 7/12. The mixed numbers calculator works either way.
Everyday examples
Woodworking. A 3/8-inch board on top of a 5/16-inch board: 6/16 + 5/16 = 11/16 inch total. Tape-measure fractions use denominators of 2, 4, 8 and 16, so the LCD is always the largest denominator in the problem. For longer measurements that mix feet, inches and fractions, use the feet and inches calculator.
Cooking. 2/3 cup of milk plus 3/4 cup of water is 8/12 + 9/12 = 17/12, or 1 5/12 cups, a little less than 1 1/2 cups.
Common mistakes
- Adding denominators. 1/2 + 1/3 is not 2/5. Since 2/5 is smaller than 1/2 on its own, that answer cannot be right. The correct sum is 5/6.
- Changing only the denominator. When you rewrite 5/6 as something over 24, the numerator must be multiplied by the same 4.
- Forgetting to simplify. 3/10 + 1/5 = 3/10 + 2/10 = 5/10, which should be reported as 1/2. See how to simplify fractions for fast ways to reduce.
- Losing the borrowed whole. When you borrow for a mixed-number subtraction, reduce the whole number by 1.
For any other operation, the general fraction calculator multiplies and divides fractions as well.
Frequently asked questions
How do you add fractions with different denominators?
Find the least common denominator (LCD), rewrite each fraction with that denominator, add the numerators, and simplify. For 5/6 + 3/8, the LCD is 24, so 20/24 + 9/24 = 29/24, or 1 5/24.
Why can't I just add the denominators?
Because the denominator names the size of the pieces, not how many there are. Adding 1/2 + 1/3 by adding tops and bottoms gives 2/5, which is smaller than 1/2 alone. The correct sum is 3/6 + 2/6 = 5/6.
Do I have to use the least common denominator?
No. Any common denominator works, such as the product of the two denominators. The LCD just keeps the numbers smaller and means less simplifying at the end.
How do I subtract a larger fraction from a smaller one in mixed numbers?
Borrow 1 from the whole number and add it to the fraction as a full set of pieces. In 5 1/3 − 2 3/4, rewrite as 5 4/12 − 2 9/12, borrow to get 4 16/12 − 2 9/12, and subtract to get 2 7/12.