Question

How do you change repeating decimals into fractions?

Answer 1

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`.