How do you change a repeating decimal into a fraction?

Answer 1

This is made quite easy with the following observations:

5/9 = 0.5555...

7/9 = 0.7777...

12/99 = 0.121212...

23/99 = 0.232323...

456/999 = 0.456456456...

78/999 = 078/999 = 0.078078078...

So we can see that fractions with a denominator of 9, 99, 999, 9999, ... are pretty useful for making repeating decimals. All we have to do is to reduce the fraction to its lowest terms. For example, let's take the repeating decimal 0.027027027...

Clearly this is 27/999 = 1/37 (having divided top and bottom of the fraction by 27)

Now let's try something more tricky. Take a look at 0.4588888888...

This isn't simply something divided by 99.. since the 45 bit doesn't repeat. What we need to do is move the decimal point over to the start of the repeating bit. In this case we multiply by 100 to get 45.888888...

Now we know the fraction part is 8/9. In total we have 45 + 8/9 = 413/9 (changing into a top-heavy fraction will make things easier for us).

So 45.88888... = 413/9

Now just divide both sides by 100 to get:

0.458888... = 413/900

Answer 2

The pattern is divide the repeating number by the amount of nines equivalent tot he amount of digits in the repeating pattern. For example: 0.44444... = 4/9 and .123123123123... = 123/999.