Find the rate percent per annum if Rs 2000 amounts to Rs 2315.25 in one and half years interest being compounded half yearly.

110%
211.5%
35%
420%

Answer:- 1

Explanation:-

Solution:

\begin{aligned} According to the question \\ compounded half yearly \\ Rate  = \frac{R}{2} \\ Time  = \frac{2T}{3} \\ Amount  =  P {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow 2315.25 = 2000 {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow \frac{2315.25}{2000} = {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow \frac{231525}{200000} = {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow \frac{9261}{8000} = {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow {(\frac{21}{20})}^{3} = {(1 + \frac{R}{2 \times 100})}^{3} \\ \Rightarrow 1 + \frac{R}{200} = \frac{21}{20} \\ \Rightarrow R =  10 % \end{aligned}

$\eqalign{ & {\text{According to the question}} \cr & {\text{compounded half yearly}} \cr & {\text{Rate = }}\frac{{\text{R}}}{2} \cr & {\text{Time = }}\frac{{{\text{2T}}}}{3} \cr & {\text{Amount = P}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow 2315.25 = 2000{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow \frac{{2315.25}}{{2000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow \frac{{231525}}{{200000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow \frac{{9261}}{{8000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow {\left( {\frac{{21}}{{20}}} \right)^3} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \Rightarrow 1 + \frac{{\text{R}}}{{200}} = \frac{{21}}{{20}} \cr & \Rightarrow {\text{R = 10}}\% \cr}$