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Fraction Calculator
Enter two fractions or mixed numbers, pick an operation, and get the answer as a simplified fraction, a mixed number and a decimal, with the working shown step by step.
The four operations on fractions
Every operation here first converts mixed numbers into improper fractions. A mixed number w n/d becomes (w × d + n) / d, so 2 1/2 is 5/2. The result is then reduced by dividing the numerator and denominator by their greatest common factor.
| Operation | Rule |
|---|---|
| Add | a/b + c/d = (a×d + c×b) / (b×d), using the lowest common denominator in practice |
| Subtract | a/b − c/d = (a×d − c×b) / (b×d) |
| Multiply | a/b × c/d = (a×c) / (b×d) |
| Divide | a/b ÷ c/d = a/b × d/c, valid only when c is not zero |
This calculator uses the least common multiple of the two denominators for addition and subtraction, which keeps the intermediate numbers small. The steps appear in the result list so you can compare them with your own working.
Reading the three answer forms
The simplified fraction is in lowest terms: numerator and denominator share no factor other than 1. The mixed number splits an improper fraction into a whole part and a proper remainder, which is how recipes and measurements are usually written. The decimal is exact when the simplified denominator contains only the prime factors 2 and 5, such as 3/8 = 0.375. Otherwise it repeats forever, as 1/3 does, and the calculator marks the decimal with an approximation sign and rounds it to ten significant digits.
Entering negatives and mixed numbers
Put the whole part in the first box, or leave it at 0 for a plain fraction. A negative whole number makes the entire mixed number negative, so -1 and 1/2 means negative one and a half, not negative one plus a half. To enter a negative proper fraction such as −3/4, keep the whole number at 0 and give the numerator a minus sign. The denominator must be a positive whole number, since a denominator of zero is undefined.
Inputs are limited to six digits per box. Internally the arithmetic uses exact integers, so large denominators never produce the rounding errors that decimal approximations would.
Common mistakes with fractions
- Adding denominators. 1/2 + 1/3 is not 2/5. The pieces must be the same size first, so rewrite both over 6 to get 3/6 + 2/6 = 5/6.
- Forgetting to flip when dividing. Dividing by a fraction means multiplying by its reciprocal, so 3/4 ÷ 3/8 is 3/4 × 8/3 = 2.
- Cross-cancelling in addition. Cancelling common factors across fractions is allowed in multiplication only.
- Leaving the answer unsimplified. 6/8 is correct but 3/4 is the expected form.
Worked example
Multiply 2 1/2 by 1 1/3. As improper fractions that is 5/2 × 4/3 = 20/6, which simplifies to 10/3 or 3 1/3.
Inputs
| Fraction 1: whole number | 2 |
|---|---|
| Fraction 1: numerator | 1 |
| Fraction 1: denominator | 2 |
| Operation | Multiply (×) |
| Fraction 2: whole number | 1 |
| Fraction 2: numerator | 1 |
| Fraction 2: denominator | 3 |
Result
Mixed number: 3 1/3. Decimal ≈ 3.333333333.
| Step 1: fraction 1 as an improper fraction | 5/2 |
|---|---|
| Step 2: fraction 2 as an improper fraction | 4/3 |
| Step 3: multiply numerators and denominators | 5 × 4 over 2 × 3 = 20/6 |
| Step 4: divide top and bottom by the GCF (2) | 10/3 |
| Mixed number | 3 1/3 |
| Decimal | ≈ 3.333333333 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I add fractions with different denominators?
Find a common denominator, usually the least common multiple, rewrite each fraction over it, add the numerators and keep the denominator. Then simplify. For example 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. So 3 2/5 becomes (3×5 + 2)/5 = 17/5.
Why does the decimal sometimes show an approximation sign?
When the simplified denominator has a prime factor other than 2 or 5, the decimal never ends. Writing 1/3 as 0.3333333333 is therefore rounded, and the sign tells you so.
Can I divide by a fraction that equals zero?
No. Dividing by zero is undefined, so the calculator reports an error when the second fraction is zero and the operation is division.
What is the difference between a proper and an improper fraction?
A proper fraction has a numerator smaller than its denominator, such as 3/4. An improper fraction has a numerator equal to or larger than its denominator, such as 7/4, and can be rewritten as a mixed number.
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References and sources
- Standard school arithmetic: Euclid, Elements Book VII, for the greatest common divisor method used to simplify fractions.