# Decimal to Fraction Calculator

By Wojciech Sas, PhD candidate
Last updated: Jul 02, 2020

Welcome to our decimal to fraction calculator - a smart tool that helps you convert any decimal to a fraction in the blink of an eye. You'll find how to turn a decimal into a fraction, or even how to change repeating decimals to fractions. The basic idea of this fraction converter is to rewrite any decimal as a fraction - a ratio of two integer numbers.

## Why is it helpful to convert a decimal to a fraction?

We use numbers in everyday life, both decimals and fractions. Although decimals may feel more natural for writing, there are a few problems that could arise, sooner or later.

For example, by using rational numbers, we are sometimes forced to round values at some point, depending on how many significant figures we need to use. Writing the same number in its fractional form gives us the exact value.

Working out fractional exponents is more straightforward with fractions than with decimals. What's `42.5`? Well, it's not so obvious at first glance. But what about `45/2`? It's easier to visualize this as `(√4)⁵ = 2⁵ = 32`. This issue gets even more problematic when we deal with repeating digits, so it's worth knowing how to turn a repeating decimal to fraction.

There are also more practical ways of using fractions instead of decimals. Imagine you're at a party, and you want to divide a cake or a pizza into even parts. If there is a group of six people, how much of the total would everybody get? It's either around `0.166` or exactly `1/6` - that's your choice.

On the other hand, you may also want to work things out the other way around, by changing any fraction to a decimal, or even a fraction into a percent. It all depends on the context.

## How to turn a decimal into a fraction?

Our goal is to find two integer numbers, a numerator and a denominator, which divided by themselves make the initial value. Let's say that we want to evaluate what `0.125` is as a fraction:

1. Set your initial numerator to be the same as the starting number (`0.125`), and set the denominator as `1`.

2. Move the decimal dot to the end of numerator `0.125 → 1.25 → 12.5 → 125`. Each leap corresponds to multiplying the numerator by 10.

3. As we've moved the dot by three digits, it means that we should multiply the denominator by 1000, which is 10 to the power 3.

4. Work out the greatest common factor of `125` and `1000`, which is 125.

5. Divide both values by 125, the numerator is `1`, and the denominator is `8`.

6. As a result of this fraction converter, we found that `0.125` as a fraction is `1/8`.

## How to convert a repeating decimal to a fraction?

Converting a repeating decimal to a fraction is a bit more challenging. Let's see how our decimal to fraction calculator deals with the task. Take into account `0.6252525…`, which is `0.625` with two repeating digits. We can also write it as `0.625`, or `0.6(25)`:

1. Let our number be `x`, so that `x = 0.6252525…`.

2. Multiply it by 100, which is 10 to the power 2 (the number of repeating digits). `100x = 62.5252525…`.

3. Subtract these two values: `100x - x = 99x = 62.5252525… - 0.6252525… = 61.9`. As you can see, by using this trick the trailing digits just cancel out!

4. Move the decimal dot until the value is an integer: `61.9 → 619`. It's equivalent to multiplying the number by 10, so `990x = 619`.

5. Divide both sides by 990, so we have `x = 619/990`.

6. Estimate the greatest common divisor of 619 and 990 to find if we can simplify the fraction. The gcd is, in fact, 1. It means 619 and 990 are coprime numbers, so our fraction is already in simplest form.

7. `0.625` as a fraction is then `619/990`.

## How to use decimal to fraction calculator? - Fraction converter in practice

Now that we know how to change a decimal to a fraction, let's take a look at a problem. Is there a difference between `a = 1.83` and `b = 1.833`? In other words, is the number of repeating digits relevant to fraction conversion, and, if so, in what way? Begin with `a`:

• A single repeating figure means we need to find `9a`, which is `16.5`.
• Multiplying by 10 results in `90a = 165`, so that `a = 165/90`.
• The gcf of 165 and 90 is 15, so we can write the fraction in its simplest form, `a = 11/6`.

So how about turning the other decimal into a fraction by following the same procedure?

• Two repeating digits mean we need to find `99b`, which is `181.5`.
• Multiplying by 10 results in `990b = 1815`, so that `b = 1815/990`.
• The gcf of 1815 and 990 is 165, so we can write the fraction in its simplest form, `b = 11/6`.

Great! `a` and `b` are precisely the same. You can check it with this decimal to fraction calculator if you aren't convinced yet! You can write your decimal in several different ways, and it doesn't change the outcome of transforming it to a fraction.

Additionally, you can convert the outcomes from an improper fraction to a mixed number. There are two ways of doing so:

• Divide the numerator and the denominator, and take the integer part of the result. For the fractional part, use the modulo operator over the numerator with the denominator as a divisor.

• … or rewrite the integer part at the beginning and simply convert the decimal part to a fraction. In the end, combine both pieces.

Wojciech Sas, PhD candidate
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