The difference between simple interest and compound interest on ₹ $P$ at $R$% p.a. in 2 years is

### Concept & 2-Year Interest Difference The mathematical difference between Compound Interest (CI) and Simple Interest (SI) for exactly 2 years on a principal $P$ at rate $R\%$ is simply the interest earned on the first year's interest. $$ \text{Difference}_{2 years} = P(\frac{R}{100})^2 $$ ### Step-by-Step Solution * **Step 1:** Define the formula for Simple Interest for 2 years. $SI = \frac{P \times R \times 2}{100} = \frac{2PR}{100}$ * **Step 2:** Define the formula for Compound Interest for 2 years. $CI = P(1 + \frac{R}{100})^2 - P$ $CI = P(1 + \frac{2R}{100} + \frac{R^2}{100^2}) - P$ $CI = P + \frac{2PR}{100} + \frac{PR^2}{100^2} - P$ $CI = \frac{2PR}{100} + \frac{PR^2}{100^2}$ * **Step 3:** Subtract SI from CI to find the algebraic difference. $CI - SI = (\frac{2PR}{100} + \frac{PR^2}{100^2}) - \frac{2PR}{100}$ $\text{Difference} = \frac{PR^2}{100^2}$ or $\frac{PR^2}{(100)^2}$ ### Exam Strategy & Shortcut This is a standard identity formula in competitive mathematics. Memorizing $D = P(\frac{R}{100})^2$ for 2 years and $D = P(\frac{R}{100})^2(\frac{R}{100} + 3)$ for 3 years is highly recommended to save time. ### Common Pitfall Confusing the denominator as just $100$ instead of $100^2$ (or $10000$). Option (c) is a trap for this exact mistake. ### Final Answer Therefore, the correct answer is **₹ $\frac{PR^2}{(100)^2}$**.