The difference between compound interest and simple interest on a sum for 2 years at 10% per annum, when the interest is compounded annually is ₹ 16. If the interest were compounded half-yearly, the difference in two interests would be

### Concept & Logic First, use the annual difference to find the Principal ($P$). $$D_2 = P \left(\frac{R}{100}\right)^2$$ Then, find the new difference using the half-yearly compounding rate over the same duration. ### Step-by-Step Solution 1. **Find the Principal ($P$):** Given $D_2 = \text{₹ } 16$, $R = 10\%$, $T = 2\text{ years}$. $$16 = P \left(\frac{10}{100}\right)^2$$ $$16 = P \left(\frac{1}{100}\right) \implies P = \text{₹ } 1600$$ 2. **Calculate SI for 2 years at 10%:** $$SI = \frac{1600 \times 10 \times 2}{100} = \text{₹ } 320$$ 3. **Calculate CI compounded half-yearly:** Rate per half-year = $\frac{10\%}{2} = 5\%$. Time periods = $2 \times 2 = 4$ half-years. $$A = 1600 \left(1 + \frac{5}{100}\right)^4 = 1600 \times (1.05)^4$$ $$(1.05)^4 = 1.21550625$$ $$A = 1600 \times 1.21550625 = 1944.81$$ $$CI = 1944.81 - 1600 = \text{₹ } 344.81$$ 4. **Find the New Difference:** $$\text{New Difference} = CI - SI = 344.81 - 320 = \text{₹ } 24.81$$ ### Exam Strategy & Shortcut From $10\%$ annually for 2 years, the difference is exactly $1\%$ of $P$. Thus, $P = 1600$. For half-yearly compounding, effective CI rate is $21.55\%$. SI rate is $20\%$. Difference in effective rates = $1.55\%$. $1.55\%$ of $1600 = 16 \times 1.55 = 24.8$. Using exact decimal yields $24.81$. ### Common Pitfall Forgetting that the new difference requested is still between the new CI (half-yearly) and the original SI, not between the two different CIs. ### Final Answer Therefore, the correct answer is **₹ 24.81**.