Fraction Calculator
Add, subtract, multiply, and divide fractions.
About the Fraction Calculator
Add, subtract, multiply or divide two fractions and get the answer reduced to lowest terms, without converting to decimals and losing exactness.
How to use it
- Enter the first fraction as 1/2.
- Choose the operation.
- Enter the second fraction.
The formula
Fractions follow four rules, and only one of them needs any real work.
Add / subtract: a/b ± c/d = (ad ± cb) ÷ bd — needs a common denominator
Multiply: a/b × c/d = ac ÷ bd — straight across
Divide: a/b ÷ c/d = a/b × d/c — flip and multiply
Multiplication and division are the easy ones. Addition is the awkward one, because you cannot add thirds to quarters any more than you can add feet to kilograms — the pieces are different sizes. Converting both to twelfths makes them comparable, and that is all a common denominator is.
Simplifying at the end means dividing the top and bottom by their greatest common divisor. 8/12 becomes 2/3 because both divide by 4.
Worked examples
| Problem | Common denominator | Working | Simplified |
|---|---|---|---|
| 1/2 + 1/3 | 6 | 3/6 + 2/6 = 5/6 | 5/6 |
| 1/2 + 3/4 | 4 | 2/4 + 3/4 = 5/4 | 5/4, or 1¼ |
| 3/4 − 1/6 | 12 | 9/12 − 2/12 = 7/12 | 7/12 |
| 2/3 × 3/4 | not needed | 6/12 | 1/2 |
| 1/2 ÷ 1/4 | not needed | 1/2 × 4/1 = 4/2 | 2 |
The second row shows what this calculator gives you: 1/2 + 3/4 comes back as 5/4, not 1.25. An improper fraction is an exact answer and a decimal often is not — 1/3 has no finite decimal form at all — which is why fraction work stays in fractions until the very last step. The last row is the one that trips people up. Dividing by a quarter gives a bigger answer, not a smaller one, because the question is really 'how many quarters fit into a half?' — and the answer is two. Dividing by any number below one always increases the result, which is exactly why flipping the second fraction works.
Common mistakes
- Adding the denominators. 1/2 + 1/3 is not 2/5. You must convert to a common denominator first, giving 3/6 + 2/6 = 5/6. Adding across is the single most common fraction error there is.
- Finding a common denominator when multiplying. Multiplication does not need one. Multiply the tops together and the bottoms together, then simplify.
- Flipping the wrong fraction when dividing. Invert the second one only — the divisor. 1/2 ÷ 1/4 is 1/2 × 4/1, not 2/1 × 1/4.
- Leaving the answer unsimplified. 6/12 is correct and 1/2 is finished. Most marking schemes require the simplified form, and it is far easier to read.
Terms explained
- Numerator
- The top number: how many parts you have.
- Denominator
- The bottom number: how many parts make a whole. It sets the size of each piece.
- Common denominator
- A shared bottom number that lets two fractions be added or compared. Any multiple works; the lowest is tidiest.
- Improper fraction
- One where the top exceeds the bottom, such as 7/4. Perfectly valid, and often easier to compute with than a mixed number.
- Mixed number
- A whole number with a fraction attached, such as 1¾. Easier to picture, harder to multiply.
- Reciprocal
- A fraction turned upside down. Dividing by a fraction is multiplying by its reciprocal.
Common questions
- How do I add fractions with different denominators?
- Convert both to the same denominator, add the numerators, then simplify. For 1/2 + 1/3, use sixths: 3/6 + 2/6 = 5/6.
- What is the fastest way to find a common denominator?
- Multiplying the two denominators always works and sometimes gives a larger number than necessary. The lowest common multiple keeps things tidy — for 4 and 6 it is 12, not 24.
- Why do I flip the second fraction when dividing?
- Because dividing by a number is the same as multiplying by its reciprocal. Asking how many quarters fit in a half is the same as multiplying a half by four.
- How do I convert a mixed number to an improper fraction?
- Multiply the whole number by the denominator and add the numerator. 2⅓ becomes (2 × 3 + 1)/3 = 7/3.
- How do I simplify a fraction?
- Divide top and bottom by their greatest common divisor. For 8/12 that is 4, giving 2/3.
- How do I turn a fraction into a decimal?
- Divide the numerator by the denominator. 3/8 is 3 ÷ 8 = 0.375. Some, like 1/3, repeat forever.
- Which fraction is bigger?
- Convert both to the same denominator and compare the tops, or cross-multiply: for 3/5 against 4/7, compare 3 × 7 = 21 with 4 × 5 = 20, so 3/5 is larger.
- Why can a denominator never be zero?
- It says how many pieces make a whole, and dividing something into zero pieces has no meaning. Division by zero is undefined for the same reason.