Question

How do i turn reapeating decimals into fractions?

Answer 1

Easy example that shows $0.999\ldots=1$:

Let $x=0.999\ldots=0.\overline9$.

Then $10x=9.999\ldots=9.\overline9$.

So $10x-x=9.\overline9-0.\overline9$, or $9x=9$, and so $x=1$.

The basic idea is to find the period of the repeating decimal, move the $n$ digits belonging to one period over to the left of the decimal point by multiplying by $10^n$, then subtract the original repeating decimal from this new number, and finally divide by $10^n-1$.

A slightly more complicated example:

Let $x=0.142857142857142857\ldots=0.\overline{142857}$.

Then $10^6x=142857.\overline{142857}$.

Then $10^6x-x=142857.\overline{142857}-0.\overline{142857}$, or

$999999x=142857\implies x=\dfrac{142857}{999999}=\dfrac17$