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

### Concept & Formula Compound interest for 2 years compounded annually is given by: $$CI = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]$$ Simple interest is calculated as: $$SI = \frac{P \times R \times T}{100}$$ ### Step-by-Step Solution 1. **Find the Principal ($P$):** Given $CI = \text{₹ } 525$, $T = 2\text{ years}$, and $R = 10\%$. $$CI = P \left[\left(1 + \frac{10}{100}\right)^2 - 1\right] = P \left[\left(\frac{11}{10}\right)^2 - 1\right]$$ $$525 = P \left[\frac{121}{100} - 1\right] = P \left(\frac{21}{100}\right)$$ $$P = \frac{525 \times 100}{21} = 25 \times 100 = \text{₹ } 2500$$ 2. **Calculate the Simple Interest ($SI$):** * Double the time: $T' = 2 \times 2 = 4\text{ years}$ * Half the rate: $R' = \frac{10\%}{2} = 5\%$ $$SI = \frac{2500 \times 5 \times 4}{100} = 25 \times 20 = \text{₹ } 500$$ ### Exam Strategy & Shortcut * Effective $CI$ rate for 2 years at $10\% = 10 + 10 + \frac{10 \times 10}{100} = 21\%$. * Effective $SI$ rate for double the time ($4\text{ years}$) at half the rate ($5\%$) is $4 \times 5\% = 20\%$. * Therefore, $SI = 525 \times \frac{20}{21} = 25 \times 20 = \text{₹ } 500$. ### Common Pitfall Students often forget to scale the rate and time for the simple interest part, mistakenly using the original time and rate values. ### Final Answer Therefore, the correct answer is **₹ 500**.