# Fraction Simplifier

> Reduce a fraction to its lowest terms, and read off the mixed number, the exact decimal and the percentage that go with it. Every figure is produced by whole-number arithmetic, so a repeating decimal is shown as the block that repeats rather than as a rounded approximation of it.

Use it: https://tessalor.com/en/maths/fraction-simplifier

This tool runs entirely in the browser. Nothing entered into it is uploaded.

## How it is done

1. Read each box as the exact decimal that was typed rather than as the binary number a browser stores it in, and count how many decimal places the longer of the two has.
2. Multiply both the top and the bottom by ten to the power of that count. Scaling both halves by the same amount leaves the value of the fraction untouched while turning both into whole numbers.
3. Run Euclid's algorithm on the two whole numbers with their signs removed — divide the larger by the smaller, then the smaller by the remainder, and keep going until a remainder of zero. The last divisor used is the greatest common divisor.
4. Divide the top and the bottom by that divisor. What is left is the fraction in its lowest terms, with any minus sign moved onto the numerator so that -3/4 and 3/-4 come out the same way.
5. Divide the numerator by the denominator once more for the mixed number — the quotient is the whole part and the remainder sits over the same denominator.
6. Produce the decimal by long division, recording every remainder as it appears. A remainder of zero ends the expansion. A remainder that has already been seen means the digits from that point repeat forever, and they are shown in brackets.

## Assumptions

- A minus sign always ends up on the numerator, so -3/4 and 3/-4 both come out as -3/4. They are the same number, and one canonical form is what makes two answers comparable.
- A fraction that divides exactly is written as a plain whole number rather than as something over one, so 10/5 comes out as 2 and never as 2/1.
- Decimal inputs are multiplied by a power of ten before anything else happens, so 2.5/0.75 is treated as 250/75. The answer is exact, but a value with many decimal places produces very large lowest terms.
- The repeating block of a decimal is only bracketed if it both starts and finishes within the number of places set under More options. Beyond that the expansion is cut off with an ellipsis rather than guessed at.
- Whole numbers above 9,007,199,254,740,991 cannot be shown in the numeric boxes without losing digits, so those boxes show a dash. The fraction itself is text and stays exact whatever its size.
- The rounded decimal and the percentage are rounded, to eight and four places. A value smaller than that rounds to zero in those two boxes while the exact form beside them is unchanged.

## Inputs

| Name | Label | Type | Default | Range |
| --- | --- | --- | --- | --- |
| `numerator` | Top number | number | 1071 | -1000000000 to 1000000000 |
| `denominator` | Bottom number | number | 462 | -1000000000 to 1000000000 |
| `places` | Decimal places to work out | integer | 30 | 4 to 400 |

## Outputs

- `simplified` — In its lowest terms (text), primary
- `mixed` — As a mixed number (text)
- `decimalForm` — As a decimal (text)
- `inWords` — In words (text)
- `lowestNumerator` — Numerator (integer)
- `lowestDenominator` — Denominator (integer)
- `commonFactor` — Cancelled by (integer)
- `decimalValue` — Rounded decimal (number)
- `percent` — As a percentage (percent)

## Questions

### What is 45/60 in its simplest form?

Three quarters. The greatest common divisor of 45 and 60 is 15, and dividing both by it leaves 3/4, which is 0.75 or 75%. Cancelling by 5 first gives 9/12, which is correct but not yet finished — 9 and 12 still share a factor of 3.

### What do the brackets in the decimal mean?

The digits inside them repeat forever. 2.3(18) means 2.3181818... with the 18 continuing without end, and 0.(3) means 0.3333... The notation is exact where writing out a few digits and stopping is not.

### Is my calculation sent anywhere?

No. The arithmetic runs in this browser tab and there is no server to send it to. The page works with the network disconnected, which is the quickest way to check the claim rather than take it on trust.

### Can I type decimals into the boxes?

Yes. Both numbers are multiplied by the same power of ten until they are whole, so 2.5/0.75 becomes 250/75 and then 10/3. What you cannot type is a recurring value: 0.333 is three hundred and thirty-three thousandths, and it reduces to 333/1000 rather than to a third.

### What happens if the bottom number is zero?

Nothing is calculated, and the boxes show a dash with an explanation beside them. Division by zero has no value at all, so there is no simplest form to give and no decimal to expand.

### Why does one of the number boxes show a dash?

Because the whole number in it is larger than 9,007,199,254,740,991, above which a JavaScript number cannot count in ones. Showing a figure that is out by one or two would be worse than showing nothing, so the numeric boxes stand down and the exact fraction is given as text instead.

### Why does the tool show 2 rather than 2/1?

Because a denominator of one carries no information. Once the common factor is cancelled, 10/5 is the whole number 2, and writing it over a one would suggest there is still something to divide.

### How do I know the answer is fully simplified?

The step-by-step table below the result is the proof. The last row always ends in a remainder of zero, and the divisor on that row is the greatest common divisor — the largest number that goes into both halves. Once it has been cancelled there is nothing left to cancel.

## Sources

- [Euclid's Elements, Book VII, Proposition 2: to find the greatest common measure of two given numbers not relatively prime](https://mathcs.clarku.edu/~djoyce/elements/bookVII/propVII2.html) — Clark University, Department of Mathematics and Computer Science. Retrieved 2026-07-30.
- [National curriculum in England: mathematics programmes of study](https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study) — Department for Education, applies to 2014 programmes of study, key stages 1 to 4. Retrieved 2026-07-30.
- [ECMAScript Language Specification, 6.1.6.1: the Number type](https://tc39.es/ecma262/#sec-ecmascript-language-types-number-type) — Ecma International, applies to ECMAScript 2027 draft. Retrieved 2026-07-30.

## Variants

- [How do you turn an improper fraction into a mixed number?](https://tessalor.com/en/maths/fraction-simplifier/improper-fraction-to-mixed-number)
- [How do you simplify a fraction that has decimals in it?](https://tessalor.com/en/maths/fraction-simplifier/simplify-a-fraction-with-decimals)

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