Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. What will be the difference between the total simple interest and the total compound interest at the end of 8 years on a certain sum at the same rate of interest? I. The total simple interest on the same sum at the end of 3 years is ₹ 750. II. The total compound interest on the same sum at the end of 2 years is ₹ 512.50. III. The rate of interest is 5 p.c.p.a.

### Concept & Solving with Two Unknowns To calculate the difference between $CI$ and $SI$ for 8 years, we need both the Principal ($P$) and the Rate of interest ($R$). We have two unknowns, so we need two independent mathematical relationships (equations) involving $P$ and $R$ to solve for them. ### Step-by-Step Solution 1. **Analyze the Statements:** - **Statement I:** $SI$ for 3 years = 750. This means $SI$ for 1 year = 250. Equation: $\frac{P \times R}{100} = 250 \Rightarrow PR = 25000$. - **Statement II:** $CI$ for 2 years = 512.50. Equation: $P[(1 + \frac{R}{100})^2 - 1] = 512.50$. - **Statement III:** $R = 5\%$. 2. **Evaluate Combinations:** - **I and III:** We know $PR = 25000$ and $R = 5$. We can easily find $P$ ($P = 5000$). With $P$ and $R$ known, we can find the 8-year difference. (Sufficient) - **II and III:** We know $P[(1.05)^2 - 1] = 512.50$. We can solve for $P$. With $P$ and $R$ known, we can find the 8-year difference. (Sufficient) - **I and II:** From I, $SI$ for 1 year is 250. For 2 years, $SI$ is 500. We are given $CI$ for 2 years is 512.50. The difference for 2 years is $12.50$. Using the 2-year difference formula: $\frac{R}{100} = \frac{CI - SI \text{ for 2 years}}{SI \text{ for 1 year}} = \frac{12.50}{250} = 0.05 \Rightarrow R = 5\%$. Once $R$ is known, we can use I ($PR = 25000$) to find $P$. (Sufficient) 3. **Conclusion:** Any combination of two statements provides enough information to determine both $P$ and $R$. ### Exam Strategy & Shortcut Recognize the structural dependencies. You need $P$ and $R$. - I gives $P \times R$. - II gives a relation of $P$ and $R$ via compound growth. - III gives $R$ directly. Any pair of these distinct relationships forms a solvable system of equations. ### Common Pitfall Wasting time calculating the actual 8-year difference. Data sufficiency questions only ask *if* the data is sufficient to find the answer, not what the answer actually is. ### Final Answer Therefore, the correct answer is **Any two of I, II and III**.