Directions: Each of the following questions consists of a question followed by three statements I, II and III. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. In how many days A alone can complete a work? I. A and B can complete the work in 8 days. II. B takes twice the time taken by A in completing the work. III. A and B together take $\frac{1}{3}$ of the time taken by B alone in completing the work.

### Concept & Proportional Work Rates To find the time taken by one person alone, we need their work rate. This can be derived from the combined work rate and the ratio of their efficiencies. $$ \text{Work Rate} = \frac{\text{Total Work}}{\text{Time Taken}} $$ ### Step-by-Step Solution * **Statement I:** A and B together take 8 days. $\Rightarrow \frac{1}{A} + \frac{1}{B} = \frac{1}{8}$ * **Statement II:** B takes twice the time of A. $\Rightarrow B = 2A$ * **Statement III:** Together they take $\frac{1}{3}$ of B's time. $\Rightarrow 8 = \frac{1}{3}B \Rightarrow B = 24$. Also, substituting combined rate: $\frac{1}{A} + \frac{1}{B} = \frac{3}{B} \Rightarrow \frac{1}{A} = \frac{2}{B} \Rightarrow B = 2A$. * Notice that Statement II and Statement III provide the exact same relationship between A and B's efficiencies (B takes twice as long as A). * To find A's absolute time, we need a concrete time value (provided by I) and a ratio (provided by II or III). * Combining I and II allows us to solve for A: $\frac{1}{A} + \frac{1}{2A} = \frac{1}{8} \Rightarrow A = 12$. * Combining I and III also allows us to solve for A, as it gives $B=24$ directly, from which $A$ can be derived using statement I or the ratio. * Therefore, Statement I is essential, and either Statement II or Statement III is required. The necessary combination is "I and either II or III". * Since this specific combination is not listed in options (a), (b), (c), or (d), the answer is "None of these". ### Exam Strategy & Shortcut Recognize redundant statements quickly. Statements II and III convey the exact same efficiency ratio. Thus, if I and II are sufficient, I and III must also be sufficient. Since "I and either II or III" is not an option, you can confidently choose "None of these". ### Common Pitfall Assuming "None of these" means the question cannot be answered. It simply means the correct logical combination of statements is missing from the given standard choices. ### Final Answer Therefore, the correct answer is **None of these**.