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How to Convert Repeating Decimals to Fractions, Part 2

Learn how to convert more complicated repeating decimals from decimal to fractional form.

By
Jason Marshall, PhD
5-minute read
Episode #51

In the last article, we learned how to turn simple repeating decimal numbers into fractions. Specifically, we learned how to convert decimals in which the same number repeats over and over again starting right after the decimal point. But that’s not the only type of repeating decimal that you need to know how to convert. So today we’re going to continue where we left off last time and learn how to turn more complicated types of repeating decimals into fractions too.

Recap: How to Turn a Repeating Decimal Digit Into a Fraction

But before we get too far into today’s main topic, let’s take a minute to recap exactly what we learned last time. Our goal in that article was to understand how to convert simple repeating decimals to fractions. In particular, we looked at decimals like 0.111…, 0.444…, 0.777…, and any other decimal where the same number repeats forever starting right after the decimal point.

The quick and dirty rule we discovered is that these types of repeating decimals are equivalent to the fraction that has the number doing the repeating in its numerator and the number 9 in its denominator. So 0.111… = 1/9, 0.444… = 4/9, 0.777… = 7/9, and so on.

But what about decimals that have a whole pattern of repeating digits instead of one single repeating number—something like 0.818181…? Or what if the digits don’t start repeating right away—something like 0.7222… where there’s an extra 7 in there before 2 starts repeating forever? Well, these situations require a little extra explanation—so let’s start by looking at dealing with repeating patterns.

How to Turn Repeating Decimal Patterns Into Fractions

To figure out how to convert a decimal like 0.818181… that has a repeating pattern, let’s first recall why the quick and dirty method for converting simple repeating decimals to fractions that we talked about last time works. Namely, if we take the repeating decimal 0.777… and multiply it by 10, we get the new repeating decimal 7.777…. If we now subtract the original 0.777… from this, we’re left with 7 since the repeating decimal part subtracts away. But what we’ve really done here is to subtract 1 of something from 10 of something, which leaves us with 9 of something. And that means that we’ve found that 9 of something is equal to the number 7 in this problem. So this something, which is actually our repeating decimal 0.777…, is just equal to 7/9. If you need a more thorough reminder about how this all works, you can go back and take a look at the last article where we explain it in much more detail.

Believe it or not, this is exactly the same technique that we need to use to convert the repeating pattern of digits 0.818181… into a fraction—with one small twist. The twist is that this time we’re not going to multiply the number by 10, we’re going to multiply it by 100. When we do that, we get the new repeating decimal 81.818181…. Just like before, let’s now subtract these two numbers to get 81.818181… – 0.818181… = 81.

In this case, what we’ve really done is to subtract 1 of something from 100 of something (before it was from 10 of something), which is just equal to 99 of something. That means we’ve found that 99 of something is equal to 81 in this problem. So this something, which is actually our repeating decimal 0.818181…, must be equal to the fraction 81/99. As it turns out, you can divide both the top and bottom of this fraction by 9, which means that 0.818181… = 81/99 = 9/11.

The General Rule for Dealing with Repeating Decimals

But why did we multiply by 100 in this new problem dealing with a repeating pattern of two digits instead of by 10 as we did earlier when dealing with a single repeating digit? Well, the important thing to realize is that in order for this technique to work, we had to multiply whatever the repeating decimal was by some number so that the repeating part after the decimal point cancelled out when we subtracted the original number from the new multiplied number. And for the case of 0.818181…, that meant that we had to multiply by 100. But now that we’ve seen this, we can also see that there’s a general rule that tells us how to do this for any problem where the decimal starts repeating right after the decimal point.

The quick and dirty tip is that any repeating decimal can be converted into a fraction that has the repeating pattern in its numerator and a denominator containing the same number of nines as there are digits in the numerator. In other words, for 0.818181… the numerator of the equivalent fraction must be 81 (since that’s the pattern of repeating digits), and the denominator must be 99 (since it has to have the same number of nines as there are digits in the repeating pattern 81). Or for the repeating decimal 0.142857142857…, the numerator of the equivalent fraction is 142,857, and the denominator is 999,999. So 0.142857142857… is equal to 142,857/999,999 which, believe it or not, after dividing both the top and bottom by 142,857 is equal to the fraction 1/7!

Practice Problems

[[AdMiddle]At this point you now know everything that’s needed to convert just about any repeating decimal into an equivalent fraction…except for one thing: How do you deal with repeating decimals that don’t start repeating right away? In other words, how do you turn decimals like 0.7222… and 0.91666… into fractions? Well (big surprise here!), unfortunately, we’re out of time for today. Which means that we’ll tackle this last type of conversion next time in the third and final part of this series.

But before we finish, here are a couple of practice problems for you to work on to help you make sure you understand everything we talked about today:

  1. 0.272727… = ______

  2. 0.123123… = ______

  3. 0.454545… = ______

You can find the answers to these questions at the very end of the article. After checking your answers, feel free to leave a comment at the bottom of the page and let me know how you did.

Wrap Up

If you have questions about how to solve these practice problems or any other math questions, please email them to me at mathdude@quickanddirtytips.com, send them via Twitter, or become a fan of the Math Dude on Facebook and get help from me and the other math fans there.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading math fans!

Practice Problem Answers

  1. 0.272727… = 27/99 (since 27 is the repeating part of the decimal and it contains 2 digits). We can reduce this fraction (a process that we’ll talk more about in a future article) by noticing that we can divide both the numerator and denominator by 9 to get 0.272727… = 27/99 = 3/11.

  2. 0.123123… = 123/999 (since 123 is the repeating part of the decimal and it contains 3 digits). We can divide both the top and bottom parts by 3 to find that 0.123123… = 123/999 = 41/333.

  3. Similar to the first problem, 0.454545… = 45/99 (since 45 is the repeating part of the decimal and it contains 2 digits). We can divide both the top and bottom parts by 9 to find that 0.454545… = 45/99 = 5/11.

About the Author

Jason Marshall, PhD

Jason Marshall is the author of The Math Dude's Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.