Question

Show that 0.477777... can be expressed in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0.

Answer 1

Let $x=0.477777...$.
Multiplying it by 10, we get
$10x=4.77777........\left(i\right)$
Multiplying (i) by 10 again, we get
$100x=47.7777........\left(ii\right)$
Subtracting (i) from (ii), we get
$$\begin{gathered} 90x=43 \\ \Rightarrow x=\frac{43}{90} \end{gathered}$$
Hence, $\frac{p}{q}=\frac{43}{90}$.