Question

How do you convert $0.\overline{7}$ to a fraction?

Answer 1

Converting decimal to fraction:

Consider the given decimal as: $x=0.\overline{7}$

Multiply $x$ by $10$ and find the difference between $10x$ and $x$.

That is,

$\begin{array}{rcl}10x-x& =& 7.\overline{7}-0.\overline{7}\\ & \Rightarrow & 9x=7\\ & \Rightarrow & x=\frac{7}{9}\end{array}$

Therefore, the fraction form of the given repeating decimal is $\mathbf{0}\mathbf{.}\overline{\mathbf{7}}\mathbf{=}\frac{\mathbf{7}}{\mathbf{9}}$.