An amount of Rs. 10000 becomes Rs. 14641 in 2 years if the interest is compounded half - yearly. What is the rate of compound interest p.c.p.a. ?

110%
212%
316%
420%

Answer:- 1

Explanation:-

Solution:

\begin{aligned} Let the rate be R%  p .a . \\ Then, \\ 10000 {(1 + \frac{R}{2 \times 100})}^{4} = 14641 \\ \Rightarrow {(1 + \frac{R}{200})}^{4} = \frac{14641}{10000} = {(\frac{11}{10})}^{4} \\ \Rightarrow 1 + \frac{R}{200} = \frac{11}{10} \\ \Rightarrow \frac{R}{200} = \frac{1}{10} \\ \Rightarrow R = 20% \end{aligned}

$\eqalign{ & {\text{Let the rate be R% p}}{\text{.a}}{\text{. }} \cr & {\text{Then,}} \cr & {\text{10000}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^4} = 14641 \cr & \Rightarrow {\left( {1 + \frac{{\text{R}}}{{200}}} \right)^4} = \frac{{14641}}{{10000}} = {\left( {\frac{{11}}{{10}}} \right)^4} \cr & \Rightarrow 1 + \frac{{\text{R}}}{{200}} = \frac{{11}}{{10}} \cr & \Rightarrow \frac{{\text{R}}}{{200}} = \frac{1}{{10}} \cr & \Rightarrow {\text{R}} = {\text{20% }} \cr}$