What was the rate of interest on a sum of money? I. The sum fetched a total of ₹ 2522 as compound interest at the end of 3 years. II. The difference between the simple interest and the compound interest at the end of 2 years at the same rate was ₹ 40.

### Concept & Compound Interest Data Sufficiency To find the rate of interest ($R$), we need enough information to form a solvable equation. The key formulas are Compound Interest $CI = P((1 + \frac{R}{100})^n - 1)$ and the difference for 2 years $CI - SI = P(\frac{R}{100})^2$. ### Step-by-Step Solution 1. **Statement I alone:** Gives $CI$ for 3 years = ₹ 2522. We have $P((1 + \frac{R}{100})^3 - 1) = 2522$. Two variables ($P$ and $R$), one equation. Not sufficient. 2. **Statement II alone:** Gives $CI - SI$ for 2 years = ₹ 40. We have $P(\frac{R}{100})^2 = 40$. Two variables, one equation. Not sufficient. 3. **Combining I and II:** We have two distinct equations with two variables. Substitute $P = \frac{40}{(\frac{R}{100})^2}$ into the first equation to solve for $R$. Since this yields a unique positive root for practical interest rates ($R = 5\%$), both statements together are sufficient. ### Exam Strategy & Shortcut In data sufficiency, you do not need to calculate the exact final value. Once you establish that you have $n$ independent equations for $n$ variables (and confirm they are not degenerate or multi-root in the practical domain), you can confidently deduce sufficiency. ### Common Pitfall Assuming two equations are always sufficient without checking for independence or multiple roots. Always mentally verify that a unique solution exists. ### Final Answer Therefore, the correct answer is **Both I and II are necessary to answer**.