1. **State the problem:** We know 9 men produce 30 pans in 6 days. We want to find how long 12 men will take to produce 42 pans at the same work rate. 2. **Formula and concept:** Work done = Number of men × Number of days × Rate per man Since rate per man is constant, work is proportional to men × days. 3. **Calculate the total work done by 9 men:** Work = 9 men × 6 days = 54 man-days to produce 30 pans. 4. **Find work per pan:** Work per pan = 54 man-days / 30 pans = 1.8 man-days per pan. 5. **Calculate total work needed for 42 pans:** Total work = 42 pans × 1.8 man-days per pan = 75.6 man-days. 6. **Find how many days 12 men need to do 75.6 man-days:** Days = Total work / Number of men = 75.6 / 12 = 6.3 days. **Final answer:** It will take 12 men approximately $6.3$ days to produce 42 pans at the same rate.