Rational Numbers from Repeating Fractions

Any rational number can be written in the form $p/q$, where $p$ and $q$ are integers. All rational numbers less than $1$ (that is, those for which p < q) can be written as a decimal fraction, but this expansion may require some digits to repeat. For example, the rational number $7/22$ has the decimal representation $0.3181818\dots$ Here, the pair of digits $1$ and $8$ repeat indefinitely. Typically, these repeating digits are shown with a horizontal bar above them.

If the decimal representation of a rational fraction is given (with the repeated digits indicated, if necessary), we can determine the rational fraction (that is, express it as $p/q$ for integers $p$ and $q$) using the following algorithm.

Suppose there are $k$ non-repeating digits just after the decimal point, followed by a group of $j$ repeating digits. In the example $7/22$, we have k = 1 (for digit $3$) and j = 2 (for digits $1$ and $8$). To recover the original number $X$ ($7/22$), compute the numerator and denominator as follows:

After computing, the numerator will be:

The denominator is 1000 - 10 = 990. The calculated numerator and denominator will always be integers for the given inputs. Thus, we write the result as the fraction $315/990$. Reducing the fraction gives $7/22$ as expected.