To solve the problem, we need to understand the relationship between the number of men, the time taken, and the amount of work done. The key concept here is that the time taken to complete a piece of work is inversely proportional to the number of men employed. This means that if you increase the number of men, the time taken decreases, and vice versa. ### Step-by-Step Explanation 1. **Understanding Inverse Proportionality**: - If \( T \) is the time taken to complete the work and \( M \) is the number of men, we can express the relationship as: \[ T \propto \frac{1}{M} \] - This can be rewritten as: \[ T \times M = k \] - where \( k \) is a constant. 2. **Finding the Constant \( k \)**: - From the problem, we know that 30 men can complete the work in 6 days. We can use this information to find \( k \): \[ T = 6 \text{ days}, \quad M = 30 \text{ men} \] \[ k = T \times M = 6 \times 30 = 180 \] 3. **Setting Up the Equation for New Conditions**: - Now, we want to find out how many men (\( M' \)) are required to complete the same work in 4 days (\( T' = 4 \) days). - Using the relationship we established: \[ T' \times M' = k \] - Substituting the known values: \[ 4 \times M' = 180 \] 4. **Solving for \( M' \)**: - To find \( M' \), we rearrange the equation: \[ M' = \frac{180}{4} = 45 \] ### Conclusion The number of men required to complete the work in 4 days is **45**. Therefore, the correct option is **C**. ### Explanation of Other Options - **Option A (20)**: This option suggests that only 20 men can complete the work in 4 days. This is incorrect because fewer men would mean more time is needed to complete the same amount of work. - **Option B (35)**: This option also suggests a lower number of men than required. If only 35 men were employed, they would not be able to finish the work in the required 4 days, as calculated. - **Option D (60)**: This option suggests that 60 men are needed. While more men would indeed reduce the time, 60 men would complete the work in less than 4 days, which is not what the question asks for. ### Revision Summary - The time taken to complete work is inversely proportional to the number of men employed. - Use the formula \( T \times M = k \) to find the constant. - Substitute the known values to find the required number of men for a different time frame. - Always check if the number of men calculated makes sense in the context of the problem. This thorough understanding of the relationship between time, men, and work will help you tackle similar problems in the future!