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Difficulty: Very hardRatio, Proportion, and Rate

A contractor estimates that 2020 workers, working 8 hours8\text{ hours} per day, can complete a road construction project in 30 days30\text{ days}. All workers work at the same constant rate. After working for 10 days10\text{ days}, 44 workers leave the site. To make up for the loss, the daily working duration for each remaining worker is increased to 10 hours10\text{ hours} per day. However, due to fatigue, the work efficiency of each remaining worker drops by 20%20\%. How many additional days will be required to complete the remaining work?

Answer: 25 days

Answer

It will take 25 additional days to complete the remaining work.
The total project requires 4,8004,800 worker-hours (20×8×3020 \times 8 \times 30). In the first 1010 days, 1,6001,600 worker-hours are completed, leaving 3,2003,200 worker-hours. After 44 workers leave, 1616 workers remain. Working 1010 hours a day at 80%80\% efficiency, the 1616 workers produce 16×10×0.8=12816 \times 10 \times 0.8 = 128 effective worker-hours each day. Dividing the remaining 3,2003,200 worker-hours by 128128 gives exactly 2525 additional days.

Step-by-Step Solution

1
Calculate total work units required for the entire project
Total work = 4800 worker-hours4800\text{ worker-hours}
Work rate is proportional to workers multiplied by total hours worked.
2
Calculate completed work and remaining work
Work done = 1600 worker-hours1600\text{ worker-hours}, Remaining work = 3200 worker-hours3200\text{ worker-hours}
Subtracting completed work from total work gives the remaining required effort.
3
Determine the effective rate per day after conditions change
Effective daily output = 128 worker-hours/day128\text{ worker-hours/day}
16 remaining workers working 10 hours daily at 80% efficiency yield 16×10×0.8=12816 \times 10 \times 0.8 = 128 worker-hours per day.
4
Divide remaining work by the new effective daily rate
Number of additional days = 25 days25\text{ days}
Dividing 3200 worker-hours3200\text{ worker-hours} by 128 worker-hours/day128\text{ worker-hours/day} gives 25 days25\text{ days}.

Key Concept

Compound Proportion and Work Efficiency
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Topics

Ratio, Proportion, and Rate

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