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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 out 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.

Prefer watching rather than reading? Learn all you need in 90 seconds with this video we made for you:

Do you already know what's the difference between ratios and fractions? Head to our ratio calculator to find out!

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, a few problems 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 42.54^{2.5}? Well, it's not so obvious at first glance. But what about 45/24^{^{5}/_{2}}? It's easier to visualize this as (4)5=25=32(\sqrt{4})^5=2^5=32. If still in doubt, compare these results using fraction exponent calculator.

This issue gets even more problematic when we deal with repeating digits, so it's worth knowing how to turn a repeating decimal into a fraction.

There are also more practical ways of using fractions instead of decimals. Imagine you're at a party and 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 0.1660.166 or exactly 1/6^1/_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 into a percent. It all depends on the context. In the latter case, Omni's fraction to percent calculator may be helpful!

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 0.1250.125 is as a fraction:

  1. Set your initial numerator to be the same as the starting number (0.1250.125), and set the denominator as 11.

  2. Move the decimal dot to the end of numerator 0.1251.2512.51250.125 \rightarrow 1.25 \rightarrow 12.5 \rightarrow 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 power 3.

  4. Work out the greatest common factor of 125125 and 10001000, which is 125. Feel free to use our GCF calculator whenever the number is less straightforward.

  5. Divide both values by 125, the numerator is 11, and the denominator is 88.

  6. As a result of this fraction converter, we found that 0.1250.125 as a fraction is 1/8^1/_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 0.62525250.6252525\ldots, which is 0.6250.625 with two repeating digits. We can also write it as 0.6250.6\overline{25}, or 0.6(25)0.6(25):

  1. Let our number be xx, so that x=0.6252525x = 0.6252525\ldots.

  2. Multiply it by 100, which is 10 to the power 2 (the number of repeating digits). 100x=62.5252525100x = 62.5252525\ldots.

  3. Subtract these two values: 100xx=99x=62.52525250.6252525=61.9100x - x = 99x = 62.5252525\ldots - 0.6252525\ldots = 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: 61.961961.9 \rightarrow 619. It's equivalent to multiplying the number by 10, so 990x=619990x = 619.

  5. Divide both sides by 990, so we have x=619/990x=\,^{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. 0.6250.6\overline{25} as a fraction is then 619/990^{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=1.83a=1.8\overline{3} and b=1.833b=1.8\overline{33}? In other words, is the number of repeating digits relevant to fraction conversion, and if so, in what way? Begin with aa:

  • A single repeating figure means we need to find 9a9a, which is 16.516.5.
  • Multiplying by 10 results in 90a=16590a = 165, so that a=165/90a=\,^{165}/_{90}.
  • The gcf of 165 and 90 is 15, so we can write the fraction in its simplest form, a=11/6a=\,^{11}/_{6}.

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

  • Two repeating digits mean we need to find 99b99b, which is 181.5181.5.
  • Multiplying by 10 results in 990b=1815990b = 1815, so that b=1815/990b=\,^{1815}/_{990}.
  • The gcf of 1815 and 990 is 165, so we can write the fraction in its simplest form, b=11/6b=\,^{11}/_{6}.

Great! aa and bb 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 into 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.

FAQ

How do I convert a decimal to a fraction?

To convert a decimal to a fraction:

  1. Count the decimal places in your number. Call this number n.
  2. Compute 10 to the power of n. That is, write 1 trailed by n zeroes. This is your denominator.
  3. Remove the decimal dot and any leading zeroes from your decimal number. This is your numerator.
  4. If possible, simplify the fraction.

What is 1.8 as a fraction?

1.8 as a fraction is 9/5. To arrive at this answer, first write 1.8 = 18/10, and then simplify by dividing both the numerator and the denominator by their greatest common divisor, i.e. by 2.

What is 0.9999… as a fraction?

0.9999… is 1. To get this astounding result, we suppose that x = 0.9999…, then multiply both sides by 10. We now have 10x = 9.9999…. Now, we subtract from the latter equation the former: 10x − x = 9.999… − 0.999… and so 9x = 9. Therefore, x=1.

Can every decimal be written as a fraction?

Yes, every decimal can be written as a fraction. A decimal is in fact a fraction whose denominator is a power of 10, so 10 or 100 or 1000. However, simplifying this fraction (by dividing both its numerator and denominator by their greatest common factor), we may end up with a fraction whose denominator is no longer a power of 10. For instance, 0.5 written as a fraction (after simplification) is ½.

Wojciech Sas, PhD
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