Directions: Each of the questions given below consists of a statement and/or a question followed by two statements labelled I and II. 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. In how many days 10 men will finish the work while working together? (Bank P.O., 2008) I. Only 12 women can finish the work in 16 days. II. 4 men and 6 women can finish the work in 16 days.

### Concept & Efficiency Ratio To find the time taken by a specific group (10 men), we need to establish the Total Work and the individual rate of that group. When different groups are involved, establishing an efficiency ratio between them (e.g., $1m = x \cdot w$) is essential. $$ \text{Total Work} = \text{Workers} \times \text{Days} $$ ### Step-by-Step Solution We need to find $D$ in the equation: $10 \text{ men} \times D = \text{Total Work}$. From Statement I: * 12 women can finish the work in 16 days. * Total Work $= 12W \times 16 = 192W$ (where W is the work done by 1 woman in 1 day). * This does not give us information about men. Statement I is insufficient alone. From Statement II: * 4 men and 6 women can finish the work in 16 days. * Total Work $= (4M + 6W) \times 16 = 64M + 96W$. * This does not give us the Total Work in terms of just Men, nor does it give an efficiency ratio. Statement II is insufficient alone. Combining Statement I and Statement II: * We have two expressions for the same Total Work. * Equate them: $192W = 64M + 96W$. * $192W - 96W = 64M$ * $96W = 64M \Rightarrow \frac{W}{M} = \frac{64}{96} = \frac{2}{3}$. * This means 3 women do the work of 2 men ($2M = 3W$, or $M = 1.5W$). * We can now substitute this back to find Total Work in terms of Men: $Total Work = 192 \times (\frac{2}{3}M) = 128M$. * We need to find days for 10 men: $10M \times D = 128M \Rightarrow D = 12.8$ days. * Since we need both statements to solve it, both are necessary. ### Exam Strategy & Shortcut Recognize the pattern: Statement I gives "Total Work" in terms of one variable. Statement II gives "Total Work" as a mixture of two variables. Setting them equal to each other will *always* yield the efficiency ratio between the two variables. Once you have the ratio and the total work, any specific sub-question can be answered. Both are required. ### Common Pitfall A common error is stopping the calculation at finding the efficiency ratio ($2M=3W$) and forgetting to check if the total work is actually calculable. Always ensure you have a path to the final requested metric (days for 10 men). ### Final Answer Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question.**