Directions: Each of the questions given below consists of a statement and/or a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statement(s) is/are sufficient to answer the question. Read both the statements and: Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. Rahul, Anurag and Vivek started a business together. In what proportion would the annual profit be distributed among them? I. Rahul got one-fourth of the profit. II. Rahul and Vivek contributed 75% of the total investment.

### Concept & Logic To determine the exact proportion of profit distribution among three partners (Rahul, Anurag, Vivek), we need the individual profit shares or investment ratios for all three relative to the whole. ### Step-by-Step Solution * **From Statement I:** We know Rahul's share is $25\%$ ($\frac{1}{4}$) of the total profit. This leaves $75\%$ for Anurag and Vivek combined, but we don't know their individual breakup. Statement I alone is insufficient. * **From Statement II:** We know Rahul and Vivek combined contributed $75\%$ of the total investment. Assuming the investment time period is the same for all (annual profit), this means their combined profit share is $75\%$. Consequently, Anurag's profit share must be the remaining $25\%$ ($\frac{1}{4}$). However, we don't know the individual breakup between Rahul and Vivek. Statement II alone is insufficient. * **Combining I and II:** * From I, Rahul's share = $\frac{1}{4}$ ($25\%$). * From II, Anurag's share = $\frac{1}{4}$ ($25\%$). * Since total profit is $100\%$, Vivek's share = $100\% - (25\% + 25\%) = 50\%$. * The ratio of their shares (Rahul : Anurag : Vivek) is $25 : 25 : 50$, which simplifies to $1 : 1 : 2$. * Since we can find the exact proportion by using both statements together, both are necessary. ### Exam Strategy & Shortcut Map out the missing pieces. Statement I gives you one piece (Rahul), Statement II gives you another piece indirectly (Anurag). You need three pieces, and since the total is $100\%$, having two individual pieces is enough to find the third. Therefore, combining them is the key. ### Common Pitfall Misinterpreting Statement II to mean Rahul and Vivek invested equally, or forgetting that total investment/profit always sums to $100\%$ which allows you to deduce Anurag's share from Statement II. ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question.**