Question

Neeraj lent Rs 65536 for 2 years at $12\frac{1}{2}\%$ per annum, compounded annually. How much more could he earn if the interest were compounded half-yearly?

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{principal}\mathrm{amount}\mathrm{be}P=\mathrm{Rs}.65536. \\ \mathrm{Annual}\mathrm{rate}\mathrm{of}\mathrm{interest},R=\frac{25}{2}\% \\ \mathrm{Rate}\mathrm{of}\mathrm{interest}\mathrm{for}\mathrm{a}\mathrm{half}\mathrm{year}=\frac{25}{4}\% \\ \mathrm{Time},\mathit{}n\mathit{}=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\mathit{}=P\times {\left(1+\frac{R}{100}\right)}^n \\ =\mathrm{Rs}.65536\times {\left(1+\frac{25}{100\times 4}\right)}^4 \\ =\mathrm{Rs}.65536\times {\left(\frac{400+25}{400}\right)}^4 \\ =\mathrm{Rs}.65536\times {\left(\frac{425}{400}\right)}^4 \\ =\mathrm{Rs}.65536\times {\left(\frac{17}{16}\right)}^4 \\ =\mathrm{Rs}.65536\times \left(\frac{17}{16}\right)\times \left(\frac{17}{16}\right)\times \left(\frac{17}{16}\right)\times \left(\frac{17}{16}\right) \\ =\mathrm{Rs}.\left(17\times 17\times 17\times 17\right) \\ =\mathrm{Rs}.83521 \\ \mathrm{Now},\mathrm{CI}=A-P \\ =\mathrm{Rs}.(83521-65536)=\mathrm{Rs}.17985 \\ \mathrm{Therefore},\mathrm{interest}\mathrm{earned}\mathrm{when}\mathrm{compounded}\mathrm{half}\mathrm{yearly}=\mathrm{Rs}.17985 \\ \mathrm{Amount}\mathrm{when}\mathrm{the}\mathrm{interest}\mathrm{is}\mathrm{compounded}\mathrm{yearly}\mathrm{is}\mathrm{given}\mathrm{by} \\ A=P\times {\left(1+\frac{R}{100}\right)}^n \\ =Rs.65536\times {\left(1+\frac{25}{100\times 2}\right)}^2 \\ =\mathrm{Rs}.65536\times {\left(\frac{200+25}{200}\right)}^2 \\ =\mathrm{Rs}.65536\times {\left(\frac{225}{200}\right)}^2 \\ =\mathrm{Rs}.65536\times {\left(\frac{9}{8}\right)}^2 \\ =\mathrm{Rs}.65536\times \left(\frac{9}{8}\right)\times \left(\frac{9}{8}\right) \\ =\mathrm{Rs}.82944 \\ \mathrm{Therefore},\mathrm{CI}=A-P=\mathrm{Rs}.(82944-65536)=\mathrm{Rs}.17408 \\ \therefore \mathrm{Difference}\mathrm{between}\mathrm{the}\mathrm{interests}\mathrm{compounded}\mathrm{half}\mathrm{yearly}\mathrm{and}\mathrm{yearly}=\mathrm{Rs}.(17985-17408)=\mathrm{Rs}.577 \end{gathered}$$