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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.

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• 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? • 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
• okk but how do I solve the repeated beatings I get for not getting all A's • GET ME TO 100 UP VOTES AND I WILL DONATE \$100 dollars to Khan Academy • 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. • why is this so hard • whoever see this. have a good day • 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. 