Simplify Fractions Calculator

What this calculator does

So, you have a clunky fraction like 18/24 and you want it in its cleanest form. This tool reduces a fraction to lowest terms, and it does not just hand you the answer, it shows the common factor it pulled out, so you can see why 18/24 becomes 3/4.

There are two boxes: a numerator on top and a denominator on the bottom. It is built for tidying up a real, positive fraction, so it expects positive whole numbers in both.

How to use it

1. Type the numerator, the top number.
2. Type the denominator, the bottom number.
3. Press Calculate.

Both need to be positive whole numbers, since the job here is to simplify an existing fraction rather than to evaluate a negative or a decimal.

What "simplest form" actually means

A fraction is in simplest form, also called lowest terms, when the top and the bottom share no common factor except 1. 2/3 is already there, because nothing but 1 divides both 2 and 3. 12/18 is not, because both are divisible by 6.

The important thing is that reducing never changes the value. 12/18 and 2/3 sit on the exact same point of the number line; you have simply described that point with smaller, cleaner numbers. Simpler digits, same amount.

How it finds the number to divide by

To reduce a fraction you divide the top and the bottom by the same number, and to get all the way to lowest terms in one move you divide by the largest number that goes into both. That is the greatest common divisor. The tool finds it with the Euclidean algorithm, a method old enough that Euclid wrote it down around 300 BC and still the one computers reach for: replace the pair with the smaller number and the remainder of dividing the larger by it, and repeat until the remainder is zero. The last number standing is the greatest common divisor.

For 12 and 18 that runs as gcd(18, 12), then gcd(12, 6), then gcd(6, 0), which stops at 6. Divide both parts of 12/18 by 6 and you land on 2/3. The tool shows this as the step (2 × 6) / (3 × 6), so the shared factor of 6 is right there in front of you rather than hidden.

A worked example, with the step shown

Take 18/24.

• The greatest common divisor of 18 and 24 is 6.
• Divide both by 6: 18 becomes 3, and 24 becomes 4.
• So the tool shows 18/24 = (3 × 6) / (4 × 6) = 3/4.

3 and 4 share no factor but 1, so 3/4 is as simple as it gets. That factored middle step is the part most calculators skip, and it is the part that makes the reduction make sense.

When the answer is a whole number or a mixed one

Reducing does not always leave you with a neat proper fraction, and the tool handles the other endings too. If the denominator reduces all the way to 1, the fraction was really a whole number in disguise: 12/4 reduces to 3/1, so the answer is just 3. And if the top is still larger than the bottom after reducing, you get a mixed number as well. 20/8 reduces to 5/2, which the tool also reports as 2 1/2, so you see both the reduced fraction and its tidiest reading.

The opposite trip: scaling a fraction up

Simplifying moves a fraction down to the smallest numbers that carry its value. There is an opposite move, multiplying the top and bottom by the same number to make a bigger fraction of equal value: 2/3 becomes 4/6, then 6/9, and so on. Those are equivalent fractions, and if you want to list them rather than reduce, that is what the equivalent fractions calculator is for. Reducing and scaling up are simply the same idea walked in opposite directions.

Questions people ask

How do I know a fraction is in simplest form?

When the top and bottom have no common factor except 1. If any number bigger than 1 divides both, there is still reducing to do.

What if the fraction is already in lowest terms?

Then it comes back unchanged. The greatest common divisor is 1, dividing by 1 does nothing, and you get the same fraction you entered.

Can I simplify a negative fraction or a decimal?

This tool is built for positive whole-number fractions, so it expects a positive numerator and denominator. It is about tidying a fraction, not evaluating one.

What happens if the top is bigger than the bottom?

It still reduces the fraction, and it additionally gives you the mixed number, so 20/8 comes back as 5/2 and as 2 1/2.

How does it find the common factor?

With the Euclidean algorithm: it repeatedly replaces the larger number with the remainder of dividing it by the smaller, until nothing is left over. The final value is the greatest common divisor, and it divides both top and bottom by it.

References

On the reducing step. Cutting a fraction to lowest terms means dividing through by the greatest common divisor, and the procedure for finding it is Euclid's, set out in the Elements around 300 BC. Book VII, Proposition 2 is the greatest common divisor construction itself, and the same book goes on to treat ratios written in lowest terms.

1. Euclid, Elements, Book VII, Proposition 2 (c. 300 BC), the Euclidean algorithm for the greatest common divisor, in David E. Joyce's online edition: Euclid's Elements, Book VII, Proposition 2.
2. Eric W. Weisstein, "Reduced Fraction," from MathWorld, a Wolfram resource, on a fraction written in lowest terms.