The sum of money which when given on compound interest at 18% per annum would fetch Rs 960 more when the interest is payable half-yearly then when it was payable annually for 2 years is = ?

Answer: Option D

Rate of interest = 18%
Time = 2 year
When the interest is payable half yearly
Then, rate of interest = 9%
Time = 4 half - years
Let the principal be Rs. x
$$\begin{array}{rl}& \text{C}\text{.I}\text{. = }x\left[{\left(1+\frac{R}{100}\right)}^T-1\right]\\ & =x\left[{\left(1+\frac{9}{100}\right)}^4-1\right]\\ & =x\left[{\left(\frac{109}{100}\right)}^4-1\right]\\ & =x\left[1.4116-1\right]\\ & =Rs.0.4116x\\ & \text{According to question}\\ & =x\left[{\left(1+\frac{18}{100}\right)}^2-1\right]\\ & =x\left[{\left(\frac{118}{100}\right)}^2-1\right]\\ & =x\left[{\left(1.18\right)}^2-1\right]\\ & =x\left[1.3924-1\right]\\ & =Rs.0.3924x\\ & \text{According to question,}\\ & 0.4116x-0.3924x=960\\ & \Rightarrow x=\frac{960}{0.0192}\\ & \Rightarrow x=\frac{960\times 10000}{192}\\ & \Rightarrow x=50000\\ & \text{Hence, sum of money}\\ & \text{ = Rs. 50000}\end{array}$$