Decimal To Fraction Converter

Converting decimals to fractions is hard. You have to multiply by 10, 100, 1000, or some multiple of 10, and subtract the original value. (for repeating decimals) Then you divide it back. Lots and lots of work. (at least for me) But fear no more! This is my guide on how to convert Decimals, to Fractions. It also comes with a built-in calculator at the end :D

Terminology

First - What does Terminating mean?

It means that it comes to an end - it isn't infinite Sooo, Non-terminating means that it, you guessed it, doesn't terminate Terminating is the opposite - it terminates If it's non-terminating, that means it goes on forever and ever and ever. For example, 0.235823582358... And if it's terminating, it'll end eventually. For example, 0.25839538

Second - What does Repeating mean?

It means that the decimals repeat at some point - for example, 0.123412341234... The 1234 part goes on forever You can also have non-repeating decimals, e.g. 0.1234 You'll notice it's also terminating

Third - What's a non-repeating prefix?

A non-repeating prefix is a non-repeating part at the start of the decimal. Not all decimals have them. For example, take 0.0505050... and 0.33505050... They both have a repeating part of 05, but the 2nd has a 33 at the start. That's the non-repeating prefix - It's a prefix, so at the start, and it's non-repeating, so it's just there and nowhere else. This creates a 'mixed' decimal - it has both non-repeating and repeating part.

Putting this all together

So, what if we have a non-repeating, non-terminating decimal? Well, you've already seen some! Take pi, π, which goes on forever, but never repeats (3.141529...) All non-repeating, non-terminating decimals are irrational

Conversion: Terminating Decimals

First, count its length. For example, 0.53835 is 5 digits long (after the decimal point) So, you take a 1, and put (length) 0's after it - getting 100,000 That's the denominator (bottom number) You can also multiply 10 * length Now, take the original decimal, and make the numerator (top number) And we get . . . 53,835 / 100,000! Which, in fact, is 0.53835 as a fraction. That's how you convert non-repeating decimals to fractions. Pretty easy, right?

Conversion: Non-Terminating Decimals

I'll show you two different ways - first, the 'official' way, And second, my personal favorite, a very convenient shortcut

Official Way

Disclaimer! This is different for whether your decimal has a non-repeating prefix or not

Repeating decimal with non-repeating prefix

First, count the length of the repeating part, as well as the length of the non-repeating part Next, add the two lengths together Once you've got that, do 10 ^ length, or just add (length) 0's after a 1 - let's call it the 10# For example, if you had 0.34787878..., you would get length(non-repeating) = 2 and length(repeating) = 2 Do note that the non-repeating part is the 34 before the 78's So your total length would be 4, and you would get 10,000 Then, set your original decimal to X in an equation - for example, x = 0.34787878... Next, multiply the equation by the 10# (in our case, 10,000) - So, we get 10,000x = 3478.787878... Now, take the original length of the non-repeating part (in our case, 2) And get the new 10# - remember, 10 ^ length Now, multiply the original equation by that new 10# So now, we have 2 equations 10,000x = 3478.787878... 100x = 34.787878... Now, subtract the 2nd, smaller equation from the first. We would get 10,000x = 3478.787878... - 100x = 34.787878... --------------------------- 9900x = 3444.0 That's because the 2 repeating parts (1 from the first, 1 from the second) cancel each other out So, our final equation is 9900x = 3444, which simplifies to x = 3444/9900 And that's how to convert a repeating decimal with a non-repeating prefix to a fraction!

Pure repeating decimals without non-repeating prefix

First, count the length of the repeating part Once you've got that, do 10 ^ length, or just add (length) 0's after a 1 - let's call it the 10# For example, if you had 0.121212..., you would get length(repeating) = 2 So our total length would be 2, and we would get 100 - let's call this the 10# Then, set your original decimal to X in an equation - for example, x = 0.121212... Next, multiply the equation by the 10# (in our case, 100) - So, we get 100x = 12.121212... So now, we have 2 equations The multiplied one, and the original one 100x = 12.121212... x = 0.121212... Now, subtract the 2nd, smaller equation from the first. We would get 100x = 12.121212... - x = 0.121212... --------------------------- 99x = 12.0 That's because the 2 repeating parts (1 from the first, 1 from the second) cancel each other out So, our final equation is 99 = 12, which simplifies to x = 12/99 And that's how to convert a repeating decimal without a non-repeating prefix to a fraction!

Recommended - Shortcut

Disclaimer! This is different for whether your decimal has a non-repeating prefix or not

Mixed Repeating decimal with non-repeating prefix

First, for each digit of the repeating part, add a 9 to the denominator Then, for each digit of the non-repeating prefix, add a 0 to the denominator (after all the 9's) For example, if we had 0.618939393... (618 is the prefix, 93 the repeating part) We'd add 2 9's, and 3 0's, to get 99000 That's our denominator! Next, we take the prefix as well as the repeating part, and combine them So we would get 61893 That's our (almost) numerator! Finally, we have to subtract the prefix from the numerator This leaves us with 61893 - 618 = 61275 So, 0.618939393... is equal to 61275/99000 And that's the shortcut for converting a decimal with a non-repeating prefix to a fraction!

Pure repeating decimals without non-repeating prefix

First, for each digit of the repeating part, add a 9 to the denominator For example, if we had 0.294294294... (293 is the repeating part) We'd add 3 9's, to get 999 That's our denominator! Next, we take the repeating part So we would get 294 That's our numerator! So, 0.293293293... is equal to 293/999 And that's the shortcut for converting a decimal without a non-repeating prefix to a fraction!

Calculator

Enter any decimal, and see it automatically converted to a fraction

Help With Calculator

Enter your decimal, and if it's repeating, check the Is Repeating checkbox For example, to convert 0.34686868 (not repeating), you would enter 0.34686868 into the decimal box, and leave the checkbox unchecked However, for 0.34686868... (repeating), you would enter 0.34686868 into the decimal box, and check the checkbox Note that for very long decimals, e.g. over thousands of digits, it can be slow(er). Also, for optimal results, 1) type in your repeating part twice - this can help you make sure that you get an accurate result and 2) don't cut off your string. Instead of typing 0.1231231231, write 0.123123123123 - make it complete