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Converting repeating decimals to fractions (part 2 of 2)

Learn how to convert the repeating decimals 0.363636... and 0.714141414... and 3.257257257... to fractions. Created by Sal Khan.

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  • aqualine tree style avatar for user j0903544
    Why would the repeating decimal 0.714141414... which equals x be multiplied by 100 instead of 1000 or 10?
    (35 votes)
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  • blobby green style avatar for user bmfarrell
    what is 0.333333333333333 in a fraction
    (13 votes)
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    • duskpin ultimate style avatar for user Polina Vitić
      Here's a little table of repeating decimals. Notice that they all follow a pattern:

      1/9 = 0.111111111111111...
      2/9 = 0.222222222222222...
      3/9 = 0.333333333333333...
      4/9 = 0.444444444444444...
      5/9 = 0.555555555555555...
      6/9 = 0.666666666666666...
      7/9 = 0.777777777777777...
      8/9 = 0.888888888888888...

      Because 3/9 = 1/3 and 6/9 = 2/3, the following are also true:

      1/3 = 3/9
      1/3 = 0.333333333333333...

      2/3 = 6/9
      2/3 = 0.666666666666666...

      Hope this helps!
      (67 votes)
  • male robot hal style avatar for user ☾AlienGames☽
    How would you solve a number like 5.345555555...?
    (10 votes)
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    • spunky sam blue style avatar for user kans
      You can do it in this way too...
      x=5.345555555...
      10x=53.455555...
      100x=534.5555...
      1000x=5345.555...
      Since the decimal part is same, we can subtract 100x from 1000x.
      So,
      1000x=5345.555...
      - 100x=-534. 555...
      ________________________
      900x=4811
      ________________________
      x=4811/900
      Since x=5.345555...
      5.345555...=4811/900
      (9 votes)
  • aqualine tree style avatar for user Gustaf Liljegren
    In the second example, where Sal gets a decimal numerator, I found it easier to use 1000x and 10x instead. Why 1000x and 10x? You make a subtraction where the minuend (top part) is the number multiplied by 10 as many times as you need to move the decimal point the to the right side of the repeating part. For 0.714 with 14 repeating, you multiply by 1000 to get to 714.14 with the last 14 repeating.

    For the subtrahend (bottom part of a subtraction), you multiply the number by 10 as many times as you need to get the decimal point to the left side of the repeating part. For 0.714 with 14 repeating, use 10x to get to 7.14 with 14 repeating. Now the repating part is directly after the decimal point in both minuend and subtrahend, so they cancel out nicely.
    (9 votes)
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  • stelly green style avatar for user DalyThePenguin
    okk but how do I solve the repeated beatings I get for not getting all A's
    (6 votes)
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  • sneak peak blue style avatar for user 01K747DR
    How do you know when to multiply them by 1000x and 10x or 100x and 1x?
    (5 votes)
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    • blobby green style avatar for user sanpatnas29
      If there are 2 digits that are behind the decimal point, it is 100x (or whatever variable you want to use.) If there are 3 digits that are behind the decimal point, it is 1000x. You can see that depending on the number of digits that are behind the decimal point, the same number of zeros trail along. 2 digits behind the decimal point are 100x You can see that 100 has 2 zeros. So depending on the number of digits that are behind the decimal point, that's how many zeros are gonna be trailing 1. 4 digits that are behind the decimal point are going to be 10000x as there are 4 zeros in 10000. You can see what you should do now.
      (2 votes)
  • marcimus pink style avatar for user Pinkpiggy
    But, do you have to divide for every single problem? Is there a simpler way?
    (0 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Good question! Yes, there’s an alternative method. For this answer, we will consider just repeating decimals between 0 and 1 (if the repeating decimal is greater than 1 or negative, we can convert the part after the decimal point to a fraction and so make a mixed number, negative fraction, or negative mixed number.)

      To create the denominator, we use a digit 9 for every digit in the repeating group, then we add a digit 0 to the right for every digit after the decimal point not part of the repeating group (if any).

      To create the numerator, we subtract the number formed by decimal digits not part of the repeating group (if any), from the number formed by the decimal digits up to and including the last digit of the first occurrence of the repeating group.

      Then we reduce the fraction as needed.

      Example: convert 0.4136767... (where 67 repeats) to a fraction.
      Two digits (67) repeat, and three digits after the decimal point (413) are not part of the repeating group. So we use 99000 for the denominator.

      We use 41367 - 413 = 40954 for the numerator.

      So the answer is 40954/99000, which reduces to 20477/49500.
      (11 votes)
  • aqualine ultimate style avatar for user wid27hal
    why is this so hard
    (3 votes)
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  • female robot grace style avatar for user ekonadu
    what r all the numbers of pi? (write it out like 3.14.......) please i would sooo much like 2 know!!!
    (2 votes)
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    • leaf grey style avatar for user CasualJames
      Pi is irrational. It never ends. It is defined as the ratio between a circle's circumference and its diameter. pi = C/d It has been estimated to be a lot of things, including 3.14, 22/7, or 3.141592654. They're all close (to varying degrees), but none of them is exactly right, because they can't be.
      (5 votes)
  • female robot grace style avatar for user JaniceD
    do not vote for me
    (3 votes)
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Video transcript

In the last video, we did some examples where we had one digit repeating on and on forever, and we were able to convert those into fractions. In this video, we want to tackle something a little bit more interesting, which is multiple digits repeating on and on forever. So let's say I had 0.36 repeating, which is the same thing as 0 point-- since the bar's over the 3 and the 6, both of those repeat-- 363636. And it just keeps going on and on and on like that forever. Now the key to doing this type of problem is, so like we did in the last video, we set this as equal to x. And instead of just multiplying it by 10-- 10 would only shift it one over-- we want to shift it over enough so that when we line them up, the decimal parts will still line up with each other. And to do that we, want to actually shift the decimal space two to the right. And to shift it two to the right, we have to have multiplied by 100 or 10 to the second power. So 100x is going to be equal to what? We're shifting this two to the right-- one, two. So 100x is going to be equal to-- the decimal is going to be there now, so it's going to be 36.363636 on and on and on forever. And then let me rewrite x over here. We're going to subtract that from the 100x. x is equal to 0.363636 repeating on and on forever. And notice when we multiplied by 100x, the 3's and the 6's still line up with each other when we align the decimals. And you want to make sure you get the decimals lined up appropriately. And the reason why this is valuable is now that when we subtract x from 100x, the repeating parts will cancel out. So let's subtract. Let us subtract these two things. So on the left-hand side, we have 100x minus x. So that gives us 99x. And then we get, on the right-hand side, this part cancels out with that part. And we're just left with 36. We can divide both sides by 99, and we are left with x is equal to 36 over 99. And both the numerator and the denominator is divisible by 9, so we can reduce this. If we divide the numerator by 9, we get 4. The denominator by 9-- we get 11. So 0.363636 forever and forever repeating is 4/11. Now let's do another interesting one. Let's say we have the number 0.714, and the 14 is repeating. And so this is the same thing. So notice, the 714 isn't going to repeat. Just the 14 is going to repeat. So this is 0.7141414, on and on and on and on. So let's set this equal to x. Now you might be tempted to multiply this by 1,000x to get the decimal all the way clear of 714. But you actually don't want to do that. You want to shift it just enough so that the repeating pattern can be right under itself when you do the subtraction. So again in this situation, even though we have three numbers behind the decimal point, because only two of them are repeating, we only want to multiply by 10 to the second power. So once again, you want to multiply by 100. So you get 100x is equal to-- we're moving the decimal two to the right, one, two-- so it's going to be 71.4141, on and on and on and on. So it's going to be 71.4141414 and on and on and on. And then let me rewrite x right below this. We have x is equal to 0.7141414. And notice, now the 141414's, they're lined up right below each other. So it's going to work out when we subtract. So let's subtract these things. 100x minus x is 99x. And this is going to be equal to-- these 1414's are going to cancel with those 1414's. And we have 71.4 minus 0.7. And we can do this in our head, or we can borrow if you like. This could be a 14. This is a 0. So you have 14 minus 7 is 7 and then 70 minus 0. So you have 99x is equal to 70.7. And then we can divide both sides by 99. And you could see all of the sudden something strange is happening because we still have a decimal. But we can fix that up at the end. So let's divide both sides by 99. You get x is equal to 70.7 over 99. Now obviously, we haven't converted this into a pure fraction yet. We still have a decimal in the numerator. But that's pretty easy to fix. You just have to multiply the numerator and denominator by 10 to get rid of this decimal. So let's multiply the numerator by 10 and the denominator by 10. And so we get 707/990. Let's do one more example over here. So let's say we had something like-- let me write this way-- 3.257 repeating, and we want to convert this into a fraction. So once again, we set this equal to x. And notice, this is going to be 3.257257257. The 257 is going to repeat on and on and on. Since we have three digits that are repeating, we want to think about 1,000x, 10 to the third power times x. And that'll let us shift it just right so that the repeating parts can cancel out. So 1,000x is going to be equal to what? We're going to shift the decimal three to the right-- one, two, three. So it's going to be 3,257 point-- and then the 257 keeps repeating. 257257257 keeps going on and on and on forever. And then we're going to subtract x from that. So here is x. x is equal to 3. You want to make sure you have your decimals lined up. It's 3.257257257 dot, dot, dot-- keeps going on forever. And notice, when we multiplied it by 1,000, it allowed us to line up the 257's so that when we subtract, the repeating part cancels out. So let's do that subtraction. On the left-hand side, 1,000 of something minus 1 of that something-- you're left with 999 of that something. This part is going to cancel out with that part. It's going to be equal to-- let's see, 7 minus 3 is 4. And then you have the 5, the 2, and the 3. So you get 999x is equal to 3,254. And then you can divide both sides of this by 999. And you are left with x is equal to 3,254/999. And so obviously, this is an improper fraction. The numerator is larger than the denominator. You could convert this to a proper fraction if you like. One way, you could have just tried to figure out what to the 0.257 repeating forever is equal to and just had the 3 being the whole number part of a mixed fraction. Or you could just divide 999 into 3,254. Actually, we could do that pretty straightforwardly. It goes into it three times, and the remainder-- well, let me just do it, just to go through the motions. So 999 goes into 3,254. It'll go into it three times. And we know that because this is originally 3.257, so we're just going to find the remainder. So 3 times 9 is 27. But we have to add the 2, so it's 29. 3 times 9 is 27. We have a 2, so it's 29. And so we are left with, if we subtract, if we regroup or borrow or however we want to call it, this could be a 14. And then this could be a 4. Let me do this in a new color. And then the 4 is still smaller than this 9, so we need to regroup again. So then this could be a 14, and then this could be a 1. But this is smaller than this 9 right over here, so we regroup again. This would be an 11, and then this is a 2. 14 minus 7 is 7. 14 minus 9 is 5. 11 minus 9 is 2. So we are left with-- did I do that right? Yep-- so this is going to be equal to 3 and 257/999. And we're done.