Question

How do you convert $0.83$ ($3$ repeating) to a fraction?

Answer 1

Convert $0.8\overline{3}$ to a fraction.

Let $x=0.8\overline{3}$.

That is, $x=0.83333...\dots \left(1\right)$.

Since, only one term is repeating, multiply equation $\left(1\right)$ with $10$.

$$\begin{gathered} \Rightarrow 10x=10\left(0.83333\overline{3}\right) \\ \Rightarrow 10x=8.33333...\dots \left(2\right) \end{gathered}$$

Subtract equation $\left(1\right)$ form equation $\left(2\right)$.

$$\begin{gathered} \Rightarrow 10x-x=8.33333...-0.83333... \\ \Rightarrow 9x=7.5 \\ \Rightarrow x=\frac{7.5}{9}\left[\mathrm{Dividing}\mathrm{both}\mathrm{sides}\mathrm{by}\left(9\right)\right] \\ \Rightarrow x=\frac{75}{9\times 10}\left[\mathrm{Remove}\mathrm{decimal}\mathrm{by}\mathrm{multiplying}\mathrm{denomnator}\mathrm{by}\left(10\right)\right] \\ \therefore x=\frac{5}{6} \end{gathered}$$

Hence, $0.8\overline{3}$ as fraction is $\frac{5}{6}$.