Find the amount and the compound interest on Rs 160000 for 2 years at 10% per annum, compounded half-yearly.

$$\begin{gathered} \mathrm{Principal},P=\mathrm{Rs}.160000 \\ \mathrm{Annual}\mathrm{rate}\mathrm{of}\mathrm{interest},R\mathit{}=10\% \\ \mathrm{Rate}\mathrm{of}\mathrm{interest}\mathrm{for}\mathrm{a}\mathrm{half}\mathrm{year}=\frac{10}{2}\%=5\% \\ \mathrm{Time},n=2\mathrm{years}=4\mathrm{half}\mathrm{years} \\ \mathrm{Then}\mathrm{the}\mathrm{amount}\mathrm{with}\mathrm{the}\mathrm{compound}\mathrm{interest}\mathrm{is}\mathrm{given}\mathrm{by} \\ A=P\times {\left(1+\frac{R}{100}\right)}^n \\ =160000\times {\left(1+\frac{5}{100}\right)}^4 \\ =160000\times {\left(\frac{100+5}{100}\right)}^4 \\ =160000\times {\left(\frac{105}{100}\right)}^4 \\ =160000\times \left(\frac{21}{20}\right)\times \left(\frac{21}{20}\right)\times \left(\frac{21}{20}\right)\times \left(\frac{21}{20}\right) \\ =\left(21\times 21\times 21\times 21\right) \\ =\mathrm{Rs}194481 \\ \mathrm{Therefore},\mathrm{compound}\mathrm{interest}=\mathrm{amount}-\mathrm{principal}=\mathrm{Rs}(194481-160000)=\mathrm{Rs}34481 \end{gathered}$$