Question

How do you convert $0.123123123\dots$ to a fraction?

Answer 1

Express $\mathbf{0}\mathbf{.}\mathbf{123123123}\mathbf{\dots }$ as a fraction:

Let the given decimal expansion be represented by $x$, $x=0.123123123\dots$.

Multiply both sides of the equation $x=0.123123123\dots$ by $1000$.

$\begin{array}{rcl}x& =& 0.123123123\dots \left(i\right)\\ & \Rightarrow & 1000x=1000\times 0.123123123\dots \\ & \Rightarrow & 1000x=123.123123123\dots \left(ii\right)\end{array}$

Thus, $1000x=123.123123123\dots$.

Subtract the equation $x=0.123123123\dots$ from $1000x=123.123123123\dots$:

$\begin{array}{rcl}1000x-x& =& 123.123123123\dots -0.123123123\dots \\ & \Rightarrow & 999x=123\\ & \Rightarrow & x=\frac{123}{999}\\ & \Rightarrow & x=\frac{41}{333}\end{array}$

Thus, the decimal expansion $0.123123123\dots$ is equal to the fraction $\frac{\mathbf{41}}{\mathbf{333}}$.