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Question

Difficulty: MediumRatio, Proportion, and Rate

A factory employs 1212 workers to produce 180180 identical wooden chairs in 55 days, working 88 hours per day. If 44 workers are reassigned to another department, how many days will the remaining workers take to produce 210210 such chairs if they work 77 hours per day?

  1. 10 days10\text{ days}Answer
  2. B
    20 days20\text{ days}
  3. C
    15 days15\text{ days}
  4. D
    8 days8\text{ days}

Answer

The remaining workers will take 10 days10\text{ days} to complete the task.
The correct answer is 10 days10\text{ days}. Using compound proportion, the relationship is given by W1×D1×H1C1=W2×D2×H2C2\frac{W_1 \times D_1 \times H_1}{C_1} = \frac{W_2 \times D_2 \times H_2}{C_2}. Substituting the given values 12×5×8180=8×D2×7210\frac{12 \times 5 \times 8}{180} = \frac{8 \times D_2 \times 7}{210} yields 480180=56D2210\frac{480}{180} = \frac{56 D_2}{210}, which simplifies to 83=56D2210\frac{8}{3} = \frac{56 D_2}{210}. Solving for D2D_2 gives D2=10 daysD_2 = 10\text{ days}.

Step-by-Step Solution

1
Calculate the total man-hours required for the initial production batch.
Total man-hours = 12 workers×5 days×8 hours/day=480 man-hours12\text{ workers} \times 5\text{ days} \times 8\text{ hours/day} = 480\text{ man-hours}.
Determining total labor input needed for 180180 chairs.
2
Find the man-hours required per chair.
Man-hours per chair = 480180=83 hours/chair\frac{480}{180} = \frac{8}{3}\text{ hours/chair}.
Establishing the unit rate of work.
3
Calculate the total man-hours required for the new target of 210210 chairs.
Total man-hours needed = 210×83=560 man-hours210 \times \frac{8}{3} = 560\text{ man-hours}.
Scaling the unit work rate to the new quantity.
4
Determine the new workforce size and daily labor capacity.
Remaining workers = 12−4=8 workers12 - 4 = 8\text{ workers}. Daily man-hours = 8×7=56 man-hours/day8 \times 7 = 56\text{ man-hours/day}.
Accounting for the reduced workforce and new daily hours.
5
Calculate the number of days required.
Days = 560 man-hours56 man-hours/day=10 days\frac{560\text{ man-hours}}{56\text{ man-hours/day}} = 10\text{ days}.
Dividing total required work by daily capacity.

Key Concept

Compound Proportion and Work-Rate Relationships
Estimated Time:1m 30s
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Topics

Ratio, Proportion, and Rate

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