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

Repeated decimals can be converted into fractions by shifting the decimal to the right and subtracting the decimals. To do this, multiply the number by 10 to the second power, then subtract. For example, 0.363636 repeating is 4/11 and 0.7141414 repeating is 707/990. Another example is 3.257257257 repeating, which is 3257/999. This calculation can be done in the head or by borrowing. After the subtraction, the numerator and denominator can be reduced and the fraction can be simplified. Created by Sal Khan.

## Want to join the conversation?

• what is 0.333333333333333 in a fraction
• 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!
• Why would the repeating decimal 0.714141414... which equals x be multiplied by 100 instead of 1000 or 10?
• so you will have 71 left over and you will get rid of the other numbers that are after the decimal.
• How would you solve a number like 5.345555555...?
• 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
• 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.
• Even I was confused as to why was he complicating it. I found using 1000x and 10x easier as well.
• 2 Minutes into studying and my brain abandoned skull
• when will i use any of this in the future?
• I don't know they just make school students learn the bases of math some you will use a lot some you might never use and some that make absolutely no sense whatsoever
(1 vote)
• do not vote for me
• At least you are original, trying to use reverse psychology to try and get votes without breaking the rules of not soliciting votes.
• But, do you have to divide for every single problem? Is there a simpler way?
• 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.
• why sometimes we do 100-1 and sometimes 100-10
• What about if you have a number like 0.67 with the sevens repeating?
• You do it the same way, this time though you have to times it by 10. Because there is only one number recurring.

So, it will be:
10x = 6.77777... recurring.
-x = 0.67777 (By the way the decimal point should be lined up don't know why it's not.)
9x = 6.10000
Then you divide both sides by 9:
9x/9 = 6.1/9
And you can cancel out the 9x/9, leaving 6.1/9.
Finally, you times 6.1/9 both by ten, leaving 61/90.