Question

how do you convert $0.16(6\mathrm{repeating})$ to a fraction?

Answer 1

Converting $0.16$ (repeating) into a fraction:

Let, $y=0.16666...$

Multiplying both sides by $10$:

$$\begin{gathered} 10\times y=10\times 0.16666... \\ \Rightarrow 10y=1.6666... \\ \Rightarrow 10y=1+0.6666...\left(\because \frac{2}{3}=0.6666...\right) \\ \Rightarrow 10y=1+\frac{2}{3} \\ \Rightarrow 10y=\frac{5}{3} \end{gathered}$$

Dividing both sides by $10$:

$$\begin{gathered} \frac{10y}{10}=\frac{5}{3}\times \frac{1}{10} \\ \Rightarrow y=\frac{5}{30} \\ \Rightarrow y=\frac{1}{6} \end{gathered}$$

Hence, the fraction form of $0.16$ (repeating) is $\frac{1}{6}$.