How do you change repeating decimals into fractions?

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Answer 1 · 2015-04-08T14:29:47

Multiply so that the decimal parts subtract to leave $0$ . Then subtract and divide. Finally simplify.

Example:

Change $0.743252525252525 . . .$ to a fraction.

Solution: Let $x = 0.743252525252525 . . .$

We want two multiples of $x$ whose decimal parts are $.25252525...$

$1000x = 10^3 x= 743.25252525...$

And $100000x = 10^5 x = 74325.252525...$ .

Subtracting the first from the second, we get:

$99000x= 73582$

So $x= 73582/99000 = 36791 / 49500$

Easier example

Change $n = 0.246246246 . . .$ to a fraction.

The repeating part is 246, so we want two multiples with that for their decimal parts.

We can use the 'multiple' $1n = 0.246246246. . .$
And $1000n = 246.246246246. . .$

Subtracting the first from the second gives us:

$999n=246$

So $n = 246/999 = 82/333$ .