Learn the final trick that will enable you to convert any type of repeating decimal into a fraction.
In the last two episodes, we learned how to turn simple repeating decimals like 0.444… and more complex repeating patterns of decimals like 0.818181… into fractions. But there’s one type of repeating decimal that we haven’t talked about yet—and that’s a decimal that doesn’t start repeating until some number of digits after the decimal point. Once we’ve learned to convert those into fractions as well, we’ll be able to turn any repeating decimal into a fraction…and that’s precisely our goal for today.
Recap: How to Turn Repeating Decimals Into Fractions
Before we get into the details of this last type of repeating decimal, let’s spend a little time reviewing what we’ve learned so far about dealing with the simpler types of repeating decimals.
First, the quick and dirty method for creating a fraction from a single repeating decimal digit like 0.444…—that is, a decimal number made from a single digit that starts to repeat immediately after the decimal point—is to put the digit doing the repeating in the numerator of the resulting fraction and the number 9 in its denominator. For example, using this method allows us to find that 0.444… = 4/9, 0.888… = 8/9, and 0.999… = 9/9—which means that it’s equal to 1. Yes, that’s right—believe it or not, 0.999… = 1! If you’re having trouble believing this, check out the video explanation put together by James Tanton.
Second, the quick and dirty method for converting a more complex repeating pattern of decimal digits like 0.818181…—that is, a decimal number in which a group of digits begins to repeat immediately after the decimal point—is to create a fraction whose numerator contains the digits doing the repeating and whose denominator contains the number made up of the same number of nines as there are digits in the numerator. For example, using this allows us to find that 0.818181… = 81/99 (which can be reduced to 9/11), 0.123123… = 123/999 (which can be reduced to 41/333), and so on.
What’s the Big Idea for Turning Repeating Decimals Into Fractions?
Okay, it’s now time to turn our attention to decimals like 0.7222… and 0.91666…—that is, decimals that don’t begin to repeat until some number of digits after the decimal point. To understand how this works, it’ll help to understand the big picture behind the methods we’ve been using to convert the simpler types of repeating decimals into fractions.
Whether or not you’ve realized it, the two methods we’ve learned so far are actually based on the same idea. Namely, to convert a repeating decimal into a fraction, multiply the repeating decimal by some power of 10—that is, by 10, 100, 1000, or whatever is needed—so that we can then subtract the repeating decimal part off of this new number. For example, to turn 0.444… into 4/9 we just multiply 0.444… by 10 to get 4.444…, then subtract 0.444… from this to get 4—a number without a decimal part. Afterwards, we use the fact that 1 of something from 10 of something is 9 of something—giving us the result that 0.444… = 4/9 (see the earlier article on converting simple repeating decimals for details).
How to Convert Decimals that Don’t Repeat Right Away Into Fractions
Now that we understand the main idea behind the methods we’ve been using, we can figure out how to convert a decimal like 0.7222… where the repeating doesn’t start right away into a fraction. Remember, the goal is to multiply our decimal by some number of 10s so that we can then subtract away the decimal part. We can multiply 0.7222… by 10 to get 7.222…, but subtracting the original 0.7222… from this doesn’t get rid of the decimal part. And multiplying by 100, 1000, or any other higher power of 10 doesn’t work any better since that 7 in 0.7222… always messes things up.
But what if we instead first multiply 0.7222… by 100 to get 72.222…, then multiply the original decimal 0.7222… by 10 to get 7.222…, and finally subtract these two numbers? Does that solve our problem and allow us to subtract away the decimal part? Yes, it absolutely does since 72.222… – 7.222… = 65—with no decimal part left over! Now, what we’ve actually done here is to subtract 10 of something from 100 of something, which leaves us with 90 of something. That means we’ve found that 90 of something is equal to 65 in this problem. So this something, which is actually our repeating decimal 0.7222…, must be equal to the fraction 65/90. As it turns out, you can divide both the top and bottom of this fraction by 5, which means that 0.7222… = 65/90 = 13/18.
How to Convert Any Repeating Decimal Into a Fraction
So what exactly have we done here that’s different from what we’ve done before? Well, the only difference is that instead of multiplying the repeating decimal by some power of 10 and then subtracting the original number, we’ve now allowed for the possibility that we can multiply the repeating decimal by some power of 10, and then once again multiply the original repeating decimal by some other power of 10, and finally subtract these two numbers.
In other words, the quick and dirty tip is this: To turn a decimal that doesn’t start repeating right away into a fraction, multiply the repeating decimal by two different powers of 10 so that you can then subtract the decimal part away and use the methods we’ve developed over the last few articles to finish the conversion.
[[ AdMiddle2 ]]And the great news is that once you’ve mastered converting this final type of repeating decimal into a fraction, you know all that’s necessary to convert any repeating decimal into a fraction. Which means that you’re now free and able to go forth and convert repeating decimals to your heart’s content.
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:
0.16666666… = ______
0.91666666… = ______
0.55767676… = ______
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.
If you have questions about how to solve these practice problems or any other math questions, please email them to me at email@example.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
0.16666666… = 1/6 — First multiply 0.1666… by 100 to get 16.666…, then multiply 0.1666… by 10 to get 1.666…, and finally subtract these two numbers to get 16.666… – 1.666… = 15. Since we subtracted 10 of something from 100 of something, this says that 100 – 10 = 90 of something is equal to 15. Therefore, 0.1666… = 15/90. After dividing top and bottom by 15, we get 1/6.
0.91666666… = 11/12 — First multiply 0.91666… by 1000 to get 916.666…, then multiply 0.91666… by 100 to get 91.666…, and finally subtract these two numbers to get 916.666… – 91.666… = 825. This says that 1000 of something minus 100 of something, which is 1000 – 100 = 900 of something, is equal to 825. Therefore, 0.91666… = 825/900. After dividing top and bottom by 75, we get 0.91666… = 11/12.
0.55767676… = 5521/9900 — First multiply 0.55767676… by 10,000 to get 5576.767676…, then multiply 0.55767676… by 100 to get 55.767676…, and finally subtract these two numbers to get 5576.767676… – 55.767676 = 5521. This says that 10,000 of something minus 100 of something, which is 10,000 – 100 = 9,900 of something, is equal to 5521. Therefore, 0.55767676… = 5521 / 9900.