Question

Radha took a loan for Rs 1,00,000 at 12 % per annum compounded half yearly.She paid Rs 1,12,360 if (1.06)² =1.1236 , then the period for which she took the loan is?

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{formula}\mathrm{for}\mathrm{the}\mathrm{amount}\text{'}\mathrm{A}\text{'}\mathrm{on}\mathrm{the}\mathrm{principal}\mathrm{amount}\text{'}\mathrm{P}\text{'}\mathrm{charged}\mathrm{at}\text{'}\mathrm{r}\text{'}\mathrm{percent}\mathrm{per}\mathrm{annum} \\ \mathrm{for}\text{'}\mathrm{n}\text{'}\mathrm{years}\mathrm{is}\mathrm{given}\mathrm{by}, \\ \mathrm{A}=\mathrm{P}{\left(1+\frac{\mathrm{r}}{100}\right)}^{\mathrm{n}} \\ \mathrm{Here}\mathrm{P}=1,00,000,\mathrm{A}=1,12,360,\mathrm{r}=12\%\mathrm{p}\mathrm{a},\mathrm{t}=? \\ \mathrm{Since}\mathrm{the}\mathrm{interest}\mathrm{is}\mathrm{compounded}\mathrm{half}\mathrm{yearly}. \\ \mathrm{r}=6\%\mathrm{per}\mathrm{half}\mathrm{year} \\ \mathrm{t}=2\mathrm{n},\mathrm{where}\mathrm{n}\mathrm{is}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{years} \\ \mathrm{So}\mathrm{the}\mathrm{amount}\mathrm{is}\mathrm{given}\mathrm{by}, \\ 1,12,360=1,00,000{\left(1+\frac{6}{100}\right)}^{2\mathrm{n}} \\ \frac{1,12,360}{1,00,000}={\left(1+0.06\right)}^{2\mathrm{n}} \\ 1.12360={\left(1.06\right)}^{2\mathrm{n}} \\ 1.1236={\left[{\left(1.06\right)}^2\right]}^{\mathrm{n}} \\ {\left(1.1236\right)}^1={\left(1.1236\right)}^{\mathrm{n}}\left(\mathrm{Since}{\left(1.06\right)}^2=1.1236\mathrm{given}\right) \\ \mathrm{Compare}\mathrm{exponent}\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{to}\mathrm{get}, \\ \mathrm{n}=1 \\ \mathbf{Thus}\mathbf{}\mathbf{Radha}\mathbf{}\mathbf{took}\mathbf{}\mathbf{loan}\mathbf{}\mathbf{for}\mathbf{}\mathbf{1}\mathbf{}\mathbf{year}\mathbf{.}\mathbf{} \end{gathered}$$