Question

Express each of the following decimal numbers in the rational form.
(i) $0.\overline{4}$
(ii) $0.\overline{54}$

Answer 1

(i) $0.\overline{4}$
Let $x=$$0.\overline{4}$
$\Rightarrow x=\mathrm{0.444...}$---(i)
Here the periodicity of the decimal is one.
So, we multiply both sides of (i) by $10$ and we get
$10x=\mathrm{4.44...}$---(ii)
Subtracting (i) from (ii), we get
$10x-x=\mathrm{4.444...}-\mathrm{0.444...}$
$\Rightarrow 9x=4$
Hence, $x=\frac{4}{9}$
(ii)$0.\overline{54}$
Let $x=$$0.\overline{54}$
$\Rightarrow x=\mathrm{0.5454...}$---(i)
Here the periodicity of the decimal is two.
So, we multiply both sides of (i) by $100$ and we get
$100x=\mathrm{54.5454...}$---(ii)
Subtracting (i) from (ii), we get
$100x-x=\mathrm{54.5454...}-\mathrm{0.5454...}$
$\Rightarrow 99x=54$
Hence, $x=\frac{54}{99}$