To solve the problem, we need to understand the relationship between the number of men employed and the time taken to complete a piece of work. The key concept here is that the time taken to do a piece of work is inversely proportional to the number of men employed. This means that as the number of men increases, the time taken decreases, and vice versa. ### Step-by-Step Explanation 1. **Understanding Inverse Proportionality**: - If we denote the number of men as \( M \) and the time taken as \( T \), the relationship can be expressed mathematically as: \[ T \propto \frac{1}{M} \] - This implies that \( T \times M = k \), where \( k \) is a constant. 2. **Finding the Constant \( k \)**: - From the problem, we know that 45 men can complete the work in 5 days. We can use this information to find the constant \( k \): \[ T = 5 \text{ days}, \quad M = 45 \text{ men} \] \[ k = T \times M = 5 \times 45 = 225 \] 3. **Using the Constant to Find Time for 25 Men**: - Now, we need to find out how long it will take for 25 men to complete the same piece of work. We can use the constant \( k \) we just calculated: \[ T \times M = k \] \[ T \times 25 = 225 \] - To find \( T \), we rearrange the equation: \[ T = \frac{225}{25} = 9 \text{ days} \] ### Conclusion Thus, it will take 25 men **9 days** to complete the work. ### Explanation of Other Options - **Option A: 15 days**: This option suggests that it would take longer than the original scenario with 45 men. Since fewer men are employed (25 instead of 45), the time should decrease, making this option incorrect. - **Option B: 12 days**: Similar to option A, this option also suggests a longer time than the original scenario. It does not align with the inverse relationship, making it incorrect. - **Option C: 5 days**: This option suggests that 25 men can complete the work in the same time as 45 men, which contradicts the principle of inverse proportionality. Fewer men would require more time, so this option is also incorrect. ### Revision Summary - The time taken to complete work is inversely proportional to the number of workers. - Use the formula \( T \times M = k \) to find the constant \( k \). - Substitute the number of men into the equation to find the new time required. - Fewer men will always result in a longer time to complete the same work. The correct answer is **D. 9 days**.