A has to pay ₹ 220 to B after 1 year. B asks A to pay ₹ 110 in cash and defer the payment of ₹ 110 for 2 years. A agrees to it. If the rate of interest be 10% per annum, in this mode of payment (a) There is no gain or loss to any one (b) A gains ₹ 7.34 (c) A loses ₹ 7.34 (d) A gains ₹ 11

### Concept & Time Value of Money Equivalence To determine a gain or loss, we must compare the equivalent future values of both payment streams at a common point in time. Let's use the end of year 2 ($t=2$). $$ \text{Amount} = \text{Principal} + \left(\frac{\text{Principal} \times R \times T}{100}\right) $$ ### Step-by-Step Solution - **Original Agreement (Value at $t=2$):** A pays ₹ 220 at the end of year 1. B can reinvest this ₹ 220 for the remaining 1 year at 10% simple interest. Future Value at $t=2$: $220 + \frac{220 \times 10 \times 1}{100} = 220 + 22 = 242$. - **New Agreement (Value at $t=2$):** A pays ₹ 110 immediately ($t=0$). B can reinvest this ₹ 110 for the full 2 years at 10% simple interest. Future Value of immediate payment at $t=2$: $110 + \frac{110 \times 10 \times 2}{100} = 110 + 22 = 132$. A also pays the deferred ₹ 110 at the end of year 2. Total Future Value at $t=2$: $132 + 110 = 242$. - **Conclusion:** Since B ends up with exactly ₹ 242 in both scenarios, and A shells out the equivalent value of ₹ 242 in both scenarios, neither party gains or loses. ### Exam Strategy & Shortcut Recognize this classic setup. If a payment is split such that the early payment's interest exactly covers the deferred payment's lack of interest over symmetric periods, the net result is always zero sum. ### Common Pitfall Trying to calculate just the Present Worth of the new payments without accounting for the opportunity cost (interest earned) of the ₹ 110 paid in cash upfront. ### Final Answer Therefore, the correct answer is **(a) There is no gain or loss to any one**.