Converting Repeating Decimals to Fractions

Now, let's consider this never-ending, repeating decimal:

$x = 0.\overline{1} = 0.111111111111111111\cdots$

We have used the letter $x$ to give this decimal number a name that is shorter and easier to say! Now, what happens when we multiply this number by 10? (Try it yourself before reading ahead!)

We obtain the new decimal:

$10x = 1.\overline{1} = 1.1111111111111111\cdots = 1 + 0.111111111111111\cdots$

We separated the "whole number" part after the second equal sign and the "fractional part" appearing to the right of the decimal point. In fact, that separated decimal portion is exactly our original number $x$ again! This gives us a straightforward equation to manipulate (one that only involves $x$s and whole numbers):

$\begin{align*} 10x &= 1+0.111111111111111\cdots = 1 + x \\ 10x &= 1+x \\ 9x &= 1 \\ x &= \frac{1}{9} \end{align*}$

We subtracted the number $x$ from both sides of the equation in the third line and then divided both sides of the equation by $9$ in the last one.

As long as you have a repeating decimal, this process can be used to convert it into a fraction – no matter how "long" it takes the decimal to repeat. For instance, the repeating decimal:

$y = 0.\overline{123} = 0.123123123123123\cdots$

This repeats a chunk or string of three digits, so we can multiply it by $10^{3}$ to discover its secret fraction name. Proceeding just as we did above for $1/9$, we find:

$\begin{aligned} 10^{3}y &= 123 . \overline{123} = 123 + y \\ 10^{3}y &= 123 + y \\ \left(10^{3}-1\right)y &= 123 \\ 999y &= 123 \\ y &= \frac{123}{999}\end{aligned}$

A similar process can also be carried out for decimal expressions whose "repeating part" takes a while to "kick in". For example, we can manipulate the expression:

$z = 0.25\overline{1} = 0.25111111111111111\cdots$

This lets us discover:

$100z - 25 = 1000z - 251$

We can rearrange and solve this to find:

$z = \frac{226}{900} = \frac{113}{450}$.