1. **State the problem:** A project requires 20 men working 8 hours a day for 36 days to complete. After 12 days, only one fourth of the work is done. We need to find how many additional men should be employed if the working hours increase to 9 hours per day to finish on time. 2. **Formula and concepts:** Total work = Number of men $ \times $ hours per day $ \times $ number of days. 3. **Calculate total work:** $$\text{Total work} = 20 \times 8 \times 36 = 5760 \text{ man-hours}$$ 4. **Work done in 12 days:** Given only one fourth of the work is done, $$\text{Work done} = \frac{1}{4} \times 5760 = 1440 \text{ man-hours}$$ 5. **Work done by 20 men in 12 days at 8 hours/day:** $$20 \times 8 \times 12 = 1920 \text{ man-hours}$$ Since only 1440 man-hours of work is done, this implies actual work done is less than expected, but we proceed with given data. 6. **Remaining work:** $$\text{Remaining work} = 5760 - 1440 = 4320 \text{ man-hours}$$ 7. **Remaining days:** $$36 - 12 = 24 \text{ days}$$ 8. **Let additional men be } x. Total men now = 20 + x. Working hours per day = 9. We want to complete remaining work in 24 days:** $$ (20 + x) \times 9 \times 24 = 4320 $$ 9. **Solve for } x:** $$ (20 + x) \times 216 = 4320 $$ $$ 20 + x = \frac{4320}{216} $$ $$ 20 + x = 20 $$ $$ x = 0 $$ 10. **Interpretation:** The calculation shows no additional men are needed if hours increase to 9 per day. But since only one fourth work was done in 12 days instead of one third, the actual work rate is less. We must adjust for actual work done. 11. **Adjusting for actual work rate:** Actual work done in 12 days = 1440 man-hours instead of expected 1920 man-hours. Work rate factor = $ \frac{1440}{1920} = \frac{3}{4} $ So effective work rate is 75% of expected. 12. **Effective total work in man-hours:** $$ 5760 \times \frac{3}{4} = 4320 \text{ man-hours} $$ 13. **Remaining work:** $$ 4320 - 1440 = 2880 \text{ man-hours} $$ 14. **New equation for remaining work:** $$ (20 + x) \times 9 \times 24 = 2880 $$ 15. **Solve for } x:** $$ (20 + x) \times 216 = 2880 $$ $$ 20 + x = \frac{2880}{216} $$ $$ 20 + x = \frac{2880}{216} = 13.333... $$ $$ x = 13.333... - 20 = -6.666... $$ Negative additional men is impossible, so this means even with 20 men working 9 hours, the work can be completed on time. 16. **Conclusion:** No additional men are needed if working hours increase to 9 per day. **Final answer:** 0 additional men need to be employed.