A contractor employed 30 men to do a piece of work in 38 days. After 25 days, he employed 5 men more and the work was finished one day earlier. How many days he would have been behind, if he had not employed additional men?

### Concept & Man-Days Equivalence The total amount of work in a project can be quantified as the total sum of "man-days" (the number of workers multiplied by the number of days they worked). $$ \text{Total Work} = (M_1 \times D_1) + (M_2 \times D_2) $$ ### Step-by-Step Solution 1. **Calculate the actual total man-days required:** - Initial phase: 30 men worked for 25 days. ($30 \times 25 = 750$ man-days) - Final phase: 5 more men joined ($30 + 5 = 35$ men). The work finished "one day earlier" than the planned 38 days, meaning the total project took 37 days. - The final phase lasted $37 - 25 = 12$ days. - Work completed in final phase: $35 \times 12 = 420$ man-days. - Total work = $750 + 420 = 1170$ man-days. 2. **Calculate the hypothetical time for the original crew:** - If only the original 30 men worked the whole time, the total days required would be: - $\text{Days} = \frac{\text{Total Work}}{\text{Men}} = \frac{1170}{30} = 39$ days. 3. **Determine the delay:** - The planned deadline was 38 days. - Actual time taken by the original 30 men would be 39 days. - Delay = $39 - 38 = 1$ day. ### Exam Strategy & Shortcut Focus entirely on the remaining work after 25 days. The remaining work was successfully completed by 35 men in 12 days ($35 \times 12 = 420$ units). If the original 30 men had done this remaining work, they would have taken $\frac{420}{30} = 14$ days. The total time would have been $25 \text{ (initial)} + 14 \text{ (remaining)} = 39$ days. Since the deadline was 38 days, the contractor would simply be $39 - 38 = 1$ day behind. ### Common Pitfall Miscalculating the duration of the second phase. "Finished one day earlier" refers to the original 38-day estimate, making the total project duration 37 days, not meaning 37 days for the second phase alone. ### Final Answer Therefore, the correct answer is **1**.