The compound interest on a certain sum of money for 2 years at 10% per annum is ₹ 525. The simple interest on the same sum of money for double the time at half the rate percent per annum is?

### Concept & Relating Compound and Simple Interest First, find the principal using the effective compound interest rate for 2 years. Then apply the simple interest formula with the newly adjusted time and rate constraints. $$Effective\ CI\% = 2R + \frac{R^2}{100}$$ $$SI = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution * **Given:** CI for 2 years at 10% = 525. * Effective CI rate for 2 years at 10% = $10 + 10 + \frac{10 \times 10}{100} = 21\%$. * So, $21\%$ of Principal = 525. * $P = \frac{525}{0.21} = \frac{52500}{21} = 2500$. * **New Conditions:** Double the time $T = 2 \times 2 = 4$ years. Half the rate $R = 10 / 2 = 5\%$. * $SI = \frac{2500 \times 5 \times 4}{100}$ * $SI = 25 \times 20 = 500$. ### Exam Strategy & Shortcut The effective CI rate is 21%. The effective SI rate for the new conditions ($4 \text{ years} \times 5\%$) is 20%. Therefore, the SI is just $\frac{20}{21}$ of the CI. $525 \times \frac{20}{21} = 25 \times 20 = 500$. This skips finding the principal entirely! ### Common Pitfall Calculating the principal correctly but forgetting to adjust both the time (double) and the rate (half) before calculating the simple interest. ### Final Answer Therefore, the correct answer is **₹ 500**.