Question

Find the fraction equivalent to $0.5436$ and $0.\overline{5436}$.

Answer 1

(i) $0.5436=\frac{5436}{10000}=\frac{1359}{2500}$

(ii) Let $r=0.\overline{5436}$, then ${10}^{4}r$ will be as follows:

${10}^{4}r={10}^4\times 0.\overline{5436}=5436.\overline{5436}$

Now let us find ${10}^{4}r-r$ as shown below:

${10}^{4}r-r=5436.\overline{5436}-0.\overline{5436}\Rightarrow r({10}^4-1)=5436\Rightarrow r(10000-1)=5436\Rightarrow 9999r=5436\Rightarrow r=\frac{5436}{9999}\Rightarrow r=\frac{604}{1111}$

Hence, the fractions equivalent to $0.5436$ and $0.\overline{5436}$ are $\frac{1359}{2500}$ and $\frac{604}{1111}$ respectively.