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 is the principal sum? (SNAP, 2005) I. The interest amount after 30 months is half the interest amount after 5 years. II. The sum amounts to ₹ 1050 in 5 years at simple interest rate. III. The rate of interest is 8% p.a.

### Concept & Simple Interest Principal To find a unique Principal sum ($P$), we need quantitative constraints that allow us to solve a linear equation. We typically need the Amount ($A$), Time ($T$), and Rate ($R$). $$A = P \times (1 + \frac{R \times T}{100})$$ ### Step-by-Step Solution * **From the Question:** We are tasked with finding the exact numerical value of the principal ($P$). * **Analyze Statement I:** The interest after 30 months (2.5 years) is half the interest after 5 years. * Because Simple Interest accrues linearly, this statement is a universal mathematical truth for *any* principal and *any* rate. It provides absolutely zero new mathematical information or constraints. Statement I is useless. * **Analyze Statement II:** The amount $A = 1050$ at the end of $T = 5$ years. * We have $1050 = P \times (1 + \frac{5R}{100})$. We have two variables ($P$ and $R$), so it's insufficient alone. * **Analyze Statement III:** The rate $R = 8\%$ p.a. * This gives us the rate but no monetary values. Insufficient alone. * **Combine Statements:** * Since Statement I provides no constraints, it cannot be part of the solution. * Combining II and III: We have $A = 1050$, $T = 5$, and $R = 8\%$. * Substituting these into the formula: $1050 = P \times (1 + \frac{8 \times 5}{100}) \Rightarrow 1050 = P(1.40) \Rightarrow P = 750$. * Statements II and III together are sufficient. ### Exam Strategy & Shortcut Identify tautologies (statements that are always true by definition). Statement I basically says "half the time means half the simple interest." Since this is always true, it can never help you solve for an unknown variable. Immediately eliminate any answer choice containing Statement I. ### Common Pitfall A common pitfall is assuming Statement I contains a hidden clue just because it lists specific numbers like "30 months" and "5 years". Don't be distracted by the numbers; evaluate the logical relationship they define. ### Final Answer Therefore, the correct answer is **II and III only**.