# How do I simplify fractions?

Reduces a fraction to lowest terms by dividing the numerator and the denominator by their greatest common factor, with the working.

- Page: https://www.acalculator.org/math/simplify-fraction-calculator
- JSON spec: https://www.acalculator.org/math/simplify-fraction-calculator.json
- Version: 734c6d307ef8

## Default answer

Example with the default inputs (Numerator (top) 42, Denominator (bottom) 56): 42/56 in simplest form is 3/4.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| n | Numerator (top) | The top number of the fraction, a whole number that may be negative. |
| d | Denominator (bottom) | The bottom number of the fraction, a whole number other than 0. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| simplified | Simplest form | The fraction in lowest terms with a positive denominator, or a whole number. |
| gcf | Greatest common factor | The largest whole number that divides both the numerator and the denominator. |
| mixed | As a mixed number | An improper fraction as a whole number and a proper fraction, such as 2 7/22. |
| decimal | As a decimal | The value of the fraction as a decimal. |
| fraction | Fraction | The fraction as typed, numerator over denominator. |
| steps | Working | Euclid's algorithm for the greatest common factor, then the division of both parts. |

## Method

a/b = (a ÷ g)/(b ÷ g), where g = GCF(a, b) is found with Euclid's algorithm: divide the larger number by the smaller, then the smaller by the remainder, until the remainder is 0; the last divisor is g.

## Assumptions

- The numerator and the denominator are whole numbers; the arithmetic is exact for numbers of any size up to 30 digits.
- A minus sign on the denominator moves to the numerator, so the simplest form has a positive denominator.
- The denominator cannot be 0.

## Worked examples

1. n = 42, d = 56 gives simplified = 3/4, gcf = 14, decimal = 0.75, steps = 56 = 1 × 42 + 14; 42 = 3 × 14 + 0; GCF(42, 56) = 14; Divide the top and the bottom by 14: 42 ÷ 14 = 3 and 56 ÷ 14 = 4; 42/56 = 3/4. Source: Method from OpenStax Prealgebra 2e, section 4.2: https://openstax.org/books/prealgebra-2e/pages/4-2-multiply-and-divide-fractions.
2. n = 1071, d = 462 gives simplified = 51/22, gcf = 21, mixed = 2 7/22, decimal = 2.318182.
3. n = 18, d = -24 gives simplified = -3/4, gcf = 6, steps = Move the minus sign to the top: 18/-24 = -18/24; 24 = 1 × 18 + 6; 18 = 3 × 6 + 0; GCF(18, 24) = 6; Divide the top and the bottom by 6: 18 ÷ 6 = 3 and 24 ÷ 6 = 4; 18/-24 = -3/4.
4. n = 100, d = 25 gives simplified = 4, gcf = 25, decimal = 4.
5. n = 7, d = 13 gives simplified = 7/13, gcf = 1.

## FAQ

### How do I simplify a fraction?

Find the greatest common factor (GCF) of the numerator and the denominator, then divide both by it. For 42/56 the GCF is 14, and 42 ÷ 14 = 3 and 56 ÷ 14 = 4, so 42/56 = 3/4.

### How do I find the greatest common factor of two numbers?

Use Euclid's algorithm. Divide the larger number by the smaller one and keep the remainder. Then divide the smaller number by that remainder, and repeat until the remainder is 0. The last number you divided by is the GCF. For 56 and 42: 56 = 1 × 42 + 14, then 42 = 3 × 14 + 0, so the GCF is 14.

### How do I know a fraction is in simplest form?

A fraction is in simplest form (lowest terms) when the only whole number that divides both the numerator and the denominator is 1. 7/13 is in simplest form; 6/8 is not, because 2 divides both.

### What happens to a negative fraction?

The minus sign goes on the top. 18/−24 and −18/24 are the same number, and both simplify to −3/4. A negative divided by a negative is positive: −6/−8 = 3/4.

### What if the simplified fraction is bigger than 1?

An improper fraction such as 51/22 is already in simplest form when its GCF is 1. The calculator also writes it as a mixed number: 51 ÷ 22 = 2 remainder 7, so 51/22 = 2 7/22.

### Can the bottom number be 0?

No. A fraction with a denominator of 0 has no value, because nothing times 0 gives a number other than 0. The calculator asks for another denominator.

### Does simplifying change the value of the fraction?

No. Dividing the top and the bottom by the same number is the same as dividing the fraction by 1, so 42/56 and 3/4 are equal: both are 0.75.

## Sources

- OpenStax, Prealgebra 2e, section 4.2 Multiply and Divide Fractions (simplify fractions, the equivalent fractions property): https://openstax.org/books/prealgebra-2e/pages/4-2-multiply-and-divide-fractions
- Euclid, Elements, Book VII, Propositions 1 and 2 (the greatest common measure of two numbers)