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Difficulty: MediumInverse Proportions

A unit of 3636 combat engineers can construct a perimeter defense system in 33 hours and 4545 minutes. If the commanding officer orders the defense system to be completed in exactly 11 hour and 4040 minutes, how many combat engineers in total must be assigned to the task, assuming all engineers work at the same constant rate?

  1. A
    16
  2. B
    32
  3. 81Answer
  4. D
    89

Answer

81
The problem describes an inverse proportion because decreasing the allotted time requires increasing the number of workers to accomplish the identical task. By converting all times to minutes, the total required work effort is calculated as 3636 engineers ×\times 225225 minutes = 8,1008,100 engineer-minutes. Dividing this constant work effort by the new strict time constraint of 100100 minutes yields the total required personnel: 8181 engineers.

Step-by-Step Solution

1
Convert both time durations entirely into minutes to establish a single, consistent unit of measure.
33 hours and 4545 minutes = (3×60)+45=225(3 \times 60) + 45 = 225 minutes. 11 hour and 4040 minutes = (1×60)+40=100(1 \times 60) + 40 = 100 minutes.
Converting to minutes prevents errors associated with mixed units and base-60 time fractions.
2
Set up an inverse proportion equation, recognizing that the total amount of work (personnel multiplied by time) remains constant.
Engineers1×Time1=Engineers2×Time2⇒36×225=x×100\text{Engineers}_1 \times \text{Time}_1 = \text{Engineers}_2 \times \text{Time}_2 \Rightarrow 36 \times 225 = x \times 100
When completing a fixed task, the number of workers and the time required are inversely proportional.
3
Solve the equation to find the total number of engineers required for the new timeframe.
8,100=100x⇒x=8,100100=818,100 = 100x \Rightarrow x = \frac{8,100}{100} = 81 engineers.
Dividing the total required engineer-minutes (8,1008,100) by the available minutes (100100) determines the necessary workforce.

Key Concept

Inverse Proportions
Estimated Time:1m 30s
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Topics

Inverse Proportions

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