If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Converting repeating decimals to fractions (part 1 of 2)

The process of converting a repeating decimal to a fraction can be broken down into a few easy steps. To start, set the decimal equal to a variable. Multiply the decimal by 10 and subtract the original decimal from it. Finally, divide both sides by 9 to obtain the fractional form of the decimal. For example, 0.7 repeating would be 7/9, and 1.2 repeating would be 11/9.

.
Created by Sal Khan.

## Want to join the conversation?

• where did he get 10x from? Im confused
• The reason he is 10x is because he wanted to eliminate the recurring part. The recurring part is somewhat as a nuisance in your mission to convert recurring to a fraction. The only way to remove is by subracting 10x by x which means 7.7 recurring minus 0.7 recurring. This causes the recurring parts to cancel out each other. This leaves you with 9x is equal to 7. To find x divide 7 by 9. The answer you get is 7/9
• One of the practice problems is 1.83 with the 3 repeating.

The hints tell you to set up two equations and subtract them, but this is never covered in the videos. Where should I have learned this?

Also, even with the hints I continue to struggle with this problem. If I have trouble simplifying fractions like this answer to this question 165/90, where in Khan Academy can I go for extra practice?

I was able to do a few of these problems, but I did not get enough correct to be considered "familiar" with it.

Should I just move forward? This is the very first section of the 8t grade math.
• Since the repeating digit isn't in front of the decimal place, you've got to move it to the left of the decimal point with 100x. So the first step is to write it like this:
100x=183.3
But since you also moved 8, you've got to subtract 10x=18.3 from our first step:
100x=183.3
-10x= 18.3
------------
90x=165

I hope this helps. Have a great rest of your day!
• Hi! Here is a step-by-step explanation that might help you understand if you are confused.

Let's say I have the repeating decimal 0.7, as used in the video.
_
Step 1: First we put a variable to represent 0.7.
_
X=0.7

Step 2: We multiply 10 to both sides of the equation.
_
10x=7.7
_ _
Step 3: We take x=0.7 and subtract that by 10x=7.7
_
10x = 7.7
_
- x = 0.7
________

9x = 7

Step 4: We divide 9 from x which is x. Then we divide 9 from seven, which is 7/9.

x=7/9

Steps:

X=0.7
______

10x=7.7
________

10x = 7.7
_
- x = 0.7
________

9x = 7
_______
x=7/9
________________

I hope this helps you!
• Thank you so much! This helped a lot!! 😅😁
• so using this strategy, wouldnt it always be x/9?
• Good question!

If the repeating decimal is of the form 0.xxxx... where x is a digit, then yes the decimal always converts to x/9. But this is not true for other forms of repeating decimals (those with more than 1 digit that repeats, or with some digits before the repeating pattern).
• Why do you have to use 10x to subtract it from x? Can it be any other number? 20x, 56x, IDK?
• Because 10x is moving the decimal point to the right once. If it's 56x, it's not going to be moving the decimal point.
• it seems as though all repeating decimals are fractions with 9 as the denominator. Is that true?
• Good question!

This is true of all repeating decimals with just one digit that repeats, and no digits after the decimal point before the repetition begins. However, this is not true of all repeating decimals in general.
• I'm sorry, but this is really confusing! I haven't learned yet about x and all that, but I need to be able to convert repeating decimals into fractions for finding percents in my next quiz that I'm studing for. But this dosn't look at ALL like what I'm learning. Can you please just give me a more simple way to convert a repeating decimal to a fraction? Thanks!
• real though
(1 vote)
• how do you do decimals like 0.3111111111.....
• Think of it. Which part of the decimal makes this decimal look terrifying? Probably the one recurring. So we know that we have to eliminate the one recurring. How do we do that? Imagine this decimal number is equal to x. Multiply x by ten to get 3.1111111111.....

Then again multiply x by 100 to get 31.11111111....

Now you have the chance to eliminate the recurring part. How? Just subtract the answers of the previous two operations. When you subtract previous 2 operations, one get 90x = 28. Divide 28 by 90 and there you get your answer 28/90.