Question

How do you convert $0.8\dot{3}$ to a fraction?

Answer 1

Converting $0.8\dot{3}$ to a fraction:

Let $x=0.8\dot{3}$ $\text{…(i)}$

Then $x=0.833\dot{3}$ and

$$\begin{gathered} 10x=10\times \left(0.8333\dot{3}\right) \\ 10x=8.\dot{3}...\text{(ii)} \end{gathered}$$

Again multiply by $10$ in equation (ii)

$100x=83.\dot{3}...\text{(iii)}$

Subtract equation (ii) from (iii)

$$\begin{gathered} 100x-10x=83.\dot{3}-8.\dot{3} \\ \Rightarrow 90x=75 \\ \Rightarrow x=\frac{75}{90} \\ \therefore x=\frac{5}{6} \end{gathered}$$

Therefore, $0.8\dot{3}=\frac{5}{6}$.