Question

How do you convert $0.1\overline{6}$ to a fraction?

Answer 1

Converting decimal to fraction:

Consider the given decimal as follows:

$x=0.1\overline{6}$

Multiplying $x$ by $10$; and $10x=1.\overline{6}$

Multiplying $x$ by $100$; and $100x=16.\stackrel{\rightharpoonup }{6}$

Subtracting $10x$ from $100x$ obtained above:

$\begin{array}{rcl}100x-10x& =& 16.\overline{6}-1.\overline{6}\\ & \Rightarrow & 90x=15\\ & \Rightarrow & x=\frac{15}{90}\\ & \Rightarrow & x=\frac{1}{6}\end{array}$

Therefore, the fraction form of the given repeating decimal $\mathbf{0}\mathbf{.}\mathbf{1}\overline{\mathbf{6}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}}$.