What it does
Reducing a fraction (also called simplifying it, or writing it in lowest terms) means finding the same amount with the smallest possible whole numbers on top and bottom. 18/24 and 3/4 are the same number, but 3/4 is easier to read and compare. Type a fraction, an improper fraction like 45/12, or a mixed number like 2 6/8. You'll get the simplest form, the mixed number, the decimal (with repeating digits marked) and every step.
How a fraction is reduced
A fraction is in simplest form when the top and bottom have no common factor other than 1. Dividing both by the same number doesn't change the value, so divide them by their greatest common divisor (GCD, also called the greatest common factor) and you're done in one step.
Euclid: GCD(a, b) = GCD(b, a mod b), and GCD(a, 0) = a
To find the GCD, this calculator uses Euclid's algorithm: divide the larger number by the smaller, keep the remainder, and repeat with the smaller number and the remainder. The last remainder that isn't zero is the GCD. It's fast even for numbers with hundreds of digits, where listing factors would be hopeless.
Worked example: 1071/462
462 = 3 × 147 + 21
147 = 7 × 21 + 0
The GCD is 21. Then 1071 ÷ 21 = 51 and 462 ÷ 21 = 22, so 1071/462 = 51/22. As a mixed number that's 2 7/22, and as a decimal 2.318181…, where the 18 repeats forever.
Mixed numbers and improper fractions
To turn a mixed number into an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the denominator: 2 6/8 = (2 × 8 + 6)/8 = 22/8, which reduces to 11/4. To go back, divide: 11 ÷ 4 = 2 remainder 3, so 11/4 = 2 3/4.
Common fractions in lowest terms
| Fraction | GCD | Lowest terms | Decimal |
|---|---|---|---|
| 6/8 | 2 | 3/4 | 0.75 |
| 4/12 | 4 | 1/3 | 0.333… |
| 10/15 | 5 | 2/3 | 0.666… |
| 12/16 | 4 | 3/4 | 0.75 |
| 25/100 | 25 | 1/4 | 0.25 |
| 45/60 | 15 | 3/4 | 0.75 |
| 100/250 | 50 | 2/5 | 0.4 |
When it doesn't apply
- Whole numbers only on top and bottom. To turn a decimal into a fraction, use the decimal to fraction calculator.
- The denominator can't be 0, because division by zero has no value.
- A GCD of 1 isn't an error: it means the fraction is already in lowest terms, like 17/31.