The difference between the compound interest compounding half yearly for 1 year and the simple interest for 1 year on a certain sum of money lent out at 8% per annum is Rs 64. What is the sum (in Rs)?

Let the given sum = Rs. $$100x$$

Rate of interest = 8% and time period = 1 year

Compound interest compounded half yearly = $$P [(1 + \frac{R}{200})^{2T} - 1]$$

= $$100x [(1 + \frac{8}{200})^2 - 1]$$

= $$100x [(\frac{26}{25})^2 - 1] = 100x (\frac{676 - 625}{625})$$

= $$100x \times \frac{51}{625} = \frac{204 x}{25}$$

Simple interest = $$\frac{P \times R \times T}{100}$$

= $$\frac{100x \times 8 \times 1}{100} = 8x$$

=> Difference between simple and compound interests = $$\frac{204 x}{25} - 8x = 64$$

=> $$\frac{204x - 200x}{25} = 64$$

=> $$4x = 64 \times 25$$

=> $$x = \frac{64 \times 25}{4} = 16 \times 25 = 400$$

$$\therefore$$ Value of given sum = $$100 \times 400 = Rs. 40,000$$

=> Ans - (A)