How to use the fraction calculator
- Enter the numerator and denominator for the first fraction.
- Select the operation: add, subtract, multiply or divide.
- Enter the numerator and denominator for the second fraction.
- Read the simplified result, mixed number and decimal value below.
How fraction arithmetic works
To add or subtract fractions you need a common denominator. The standard approach is to cross-multiply: for a/b + c/d, the result is (a x d + c x b) / (b x d). For example, adding 1/2 and 1/3 gives (1 x 3 + 1 x 2) / (2 x 3) = 5/6.
Multiplication is simpler: multiply the numerators together and the denominators together. So 2/3 x 3/8 = 6/24, which reduces to 1/4 after dividing both parts by their greatest common divisor (GCD), which is 6.
Division uses the reciprocal. To divide by a fraction, flip the second fraction and multiply: a/b / c/d = a/b x d/c. Dividing 1/2 by 3/4 becomes 1/2 x 4/3 = 4/6, simplifying to 2/3.
Common fraction operations reference
| Operation | Example | Unsimplified | Simplified |
|---|---|---|---|
| Addition | 1/4 + 1/6 | 10/24 | 5/12 |
| Subtraction | 3/4 - 1/3 | 5/12 | 5/12 |
| Multiplication | 2/5 x 5/8 | 10/40 | 1/4 |
| Division | 3/4 / 1/2 | 6/4 | 3/2 |
Every result above is reduced by dividing the numerator and denominator by their GCD. A fraction like 10/40 has a GCD of 10, so it becomes 1/4.
Improper fractions and mixed numbers
An improper fraction has a numerator larger than its denominator, such as 7/3. You can rewrite it as a mixed number by dividing: 7 divided by 3 is 2 with a remainder of 1, giving 2 1/3.
This fraction calculator displays both forms when the result is improper. Mixed numbers are common in cooking, carpentry and everyday measurements where whole units and partial units are easier to read together. If you need to go the other direction and convert a terminating decimal into a fraction, use the decimal to fraction tool.
Simplifying with the greatest common divisor
Simplification means reducing a fraction to its lowest terms. The key is the greatest common divisor (GCD), the largest integer that divides both the numerator and denominator evenly. The Euclidean algorithm finds the GCD efficiently: repeatedly replace the larger number with the remainder of dividing the larger by the smaller until the remainder is zero.
For example, to simplify 24/36: the GCD of 24 and 36 is 12. Dividing both by 12 gives 2/3. Without simplification, fraction arithmetic quickly produces unwieldy numbers, especially after multiple operations.
Handling negative fractions and zero
A negative sign belongs on the numerator. If you enter a negative denominator, the calculator moves the sign to the top so the math stays consistent: -3/4 and 3/-4 both become -3/4.
Zero as a numerator gives a result of zero regardless of the denominator. Zero as a denominator is undefined and will produce an error. When dividing fractions, if the second numerator is zero, the operation requires dividing by zero, which is also invalid.
Worked examples
Each result below was produced by the same code the tool runs, so you can check your own numbers against them.
| Example | Input | Result |
|---|---|---|
| Add 1/2 and 1/3 | Numerator 1: 1; Denominator 1: 2; Operation: +; Numerator 2: 1; Denominator 2: 3 | Result: 5/6; Decimal: 0.8333333333; Steps: (1 x 3 + 1 x 2) / (2 x 3) = 5/6 |
| Subtract 3/4 from 5/6 | Numerator 1: 5; Denominator 1: 6; Operation: -; Numerator 2: 3; Denominator 2: 4 | Result: 1/12; Decimal: 0.0833333333; Simplified by: 2; Steps: (5 x 4 - 3 x 6) / (6 x 4) = 2/24 |
| Multiply 2/3 by 3/8 | Numerator 1: 2; Denominator 1: 3; Operation: *; Numerator 2: 3; Denominator 2: 8 | Result: 1/4; Decimal: 0.25; Simplified by: 6; Steps: (2 x 3) / (3 x 8) = 6/24 |
| Divide 1/2 by 3/4 | Numerator 1: 1; Denominator 1: 2; Operation: /; Numerator 2: 3; Denominator 2: 4 | Result: 2/3; Decimal: 0.6666666667; Simplified by: 2; Steps: (1 x 4) / (2 x 3) = 4/6 |
FAQ
How do I add fractions with different denominators?
Find a common denominator by multiplying the two denominators together, then adjust each numerator accordingly. Alternatively, use the cross-multiply method: (a x d + c x b) over (b x d). Simplify the result afterward.
Why does my fraction need to be simplified?
Simplified fractions are easier to read, compare and use in further calculations. A fraction like 4/8 represents the same value as 1/2, but 1/2 is the standard lowest-terms form.
Can this fraction calculator handle negative numbers?
Yes. Enter a negative sign in either the numerator or denominator field. The calculator normalizes the sign to the numerator before computing the result.
What happens if I enter zero as a denominator?
Division by zero is undefined in mathematics. The calculator will display an error message asking you to enter a non-zero denominator.
How do I divide one fraction by another?
Multiply the first fraction by the reciprocal of the second. Flip the numerator and denominator of the second fraction, then multiply straight across and simplify.
Is the result always exact?
Yes. The calculator uses integer arithmetic internally, so there is no floating-point rounding error in the fraction itself. The decimal equivalent shown may be rounded for display.