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 given question. A, B and C together start a business with a total investment of ₹ 15,000. At the end of the year, the total profit is ₹ 3000. What is A's share in the profit? I. A's contribution is $\frac{3}{2}$ times B's. II. B's contribution is twice that of C. III. A's contribution is thrice that of C.

### Concept & Logic Since the total profit (₹ $3000$) is known, and the investments are for the same duration (one year), we only need the ratio of their investments ($A : B : C$) to find A's share. To find the ratio between three variables, we need two independent relational equations between them. ### Step-by-Step Solution * **Statement I:** $A = \frac{3}{2}B \Rightarrow \frac{A}{B} = \frac{3}{2}$ * **Statement II:** $B = 2C \Rightarrow \frac{B}{C} = \frac{2}{1}$ * **Statement III:** $A = 3C \Rightarrow \frac{A}{C} = \frac{3}{1}$ * **Checking Pair I and II:** From $A:B = 3:2$ and $B:C = 2:1$, we directly get $A:B:C = 3:2:1$. Sufficient. * **Checking Pair II and III:** From $B:C = 2:1$ and $A:C = 3:1$, we get $A:B:C = 3:2:1$. Sufficient. * **Checking Pair I and III:** From $A:B = 3:2$ and $A:C = 3:1$ (which means $A$ must be a common multiple, so $A=3, B=2, C=1$), we get $A:B:C = 3:2:1$. Sufficient. * Because any two statements can provide the full ratio needed to distribute the known profit, any two statements are sufficient. ### Exam Strategy & Shortcut Recognize that Statements I, II, and III are just equations linking variables $A$, $B$, and $C$. If you have three unknowns, you need two relative equations to establish a continuous ratio. Therefore, any combination of two statements will do the job. ### Common Pitfall Assuming all three statements are needed just to be "safe", or actually calculating A's share (which is $3000 \times \frac{3}{6} = 1500$) instead of stopping once sufficiency is proven. ### Final Answer Therefore, the correct answer is **Any two of the three**.