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### Course: 8th grade (Illustrative Mathematics)>Unit 8

Lesson 16: Lesson 15: Infinite decimal expansions

# 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.

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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
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• 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!
• 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?
• i u have 10x is more easy to suptrac but u can us other nunmbers if u want
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(1 vote)
• 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!
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• 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.
(1 vote)

## Video transcript

In this video, I want to talk about how we can convert repeating decimals into fractions. So let's give ourselves a repeating decimal. So let's say I had the repeating decimal 0.7. And sometimes it'll be written like that, which just means that the 7 keeps on repeating. So this is the same thing as 0.7777 and I could just keep going on and on and on forever with those 7s. So the trick to converting these things into fractions is to essentially set this equal to a variable. And we'll just show it, do it step-by-step. So let me set this equal to a variable. Let me call this x. So x is equal to 0.7, and then the 7 repeats on and on forever. Now what would 10x be? Well, let's think about this. 10x. 10x would just be 10 times this. And we could even think of it right over here. It would be, if we multiplied this times 10, you'd be moving the decimal 1 over to the right, it would be 7.777, on and on and on and on forever. Or you could say it is 7.7 repeating. Now this is the trick here. So let me make these equal to each other. So we know what x is. x is this, just 0.777 repeating forever. 10x is this. And this is another repeating thing. Now the way that we can get rid of the repeating decimals is if we subtract x from 10x. Right? Because x has all these 0.7777. If you subtract that from 7.77777, then you're just going to be left with 7. So let's do that. So let me rewrite it here just so it's a little bit neater. 10x is equal to 7.7 repeating, which is equal to 7.777 on and on forever. And we established earlier that x is equal to 0.7 repeating, which is equal to 0.777 on and on and on forever. Now what happens if you subtract x from 10x? So we're going to subtract the yellow from the green. Well, 10 of something minus 1 of something is just going to be 9 of that something. And then that's going to be equal to, what's 7.7777 repeating minus 0.77777 going on and on forever repeating? Well it's just going to be 7. These parts are going to cancel out. You're just left with 7. Or you could say these two parts cancel out. You're just left with 7. And so you get 9x is equal to 7. To solve for x, you just divide both sides by 9. Let's divide both sides by nine. I could do all three sides, although these are really saying the same thing. And you get x is equal to 7/9. Let's do another one. I'll leave this one here so you can refer to it. So let's say I have the number 1.2 repeating. So this is the same thing as 1.2222 on and on and on. Whatever the bar is on top of, that's the part that repeats on and on forever. So just like we did over here, let's set this equal to x. And then let's say 10x. Let's multiply this by 10. So 10x is equal to, it would be 12.2 repeating, which is the same thing as 12.222 on and on and on and on. And then we can subtract x from 10x. And you don't have to rewrite it, but I'll rewrite it here just so we don't get confused. So we have x is equal to 1.2 repeating. And so if we subtract x from 10x, what do we get? On the left-hand side, we get 10x minus x is 9x. And this is going to be equal to, well, the 2 repeating parts cancel out. This cancels with that. If 2 repeating minus 2 repeating, that's just a bunch of 0. So it's 12 minus 1 is 11. And you have 9x is equal to 11. Divide both sides by 9. You get x is equal to 11/9.