Question

Express the following in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.
$0.4\overline{7}$

Answer 1

Let $x=0.4\overline{7}$ (Mixed recurring decimal)
i.e. $x=0.47777\dots$(1)
Multiplying (1) by $10$, to make it pure recurring decimal, we get
$10x=4.7777\dots$(2)
Further multiplying (2) by $10$, we get
$100x=47.777\dots .$(3)
Subtracting (2) from (3), we get
$100x–10x=(47.777\dots )–(4.7777\dots )$
$\Rightarrow 90x=43$
$\Rightarrow x=\frac{43}{90}$

Thus $0.4\overline{7}=\frac{43}{90}$