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Difficulty: MediumWork Rate and Combined Work

A commercial coffee roasting facility operates two industrial roasters, Roaster XX and Roaster YY. Working alone at its constant rate, Roaster XX can process a standard batch of coffee beans in 1212 hours. Working alone at its constant rate, Roaster YY can process the exact same batch in 88 hours.

Roaster XX begins processing a standard batch alone. After 33 hours, Roaster YY is turned on, and both roasters work together at their respective constant rates until the batch is completely processed. What is the total time, in hours, required to process the entire batch of coffee beans?

  1. A
    3.63.6 hours
  2. 6.66.6 hoursAnswer
  3. C
    7.87.8 hours
  4. D
    9.09.0 hours
  5. E
    10.510.5 hours

Answer

6.6 hours
In the first 33 hours, Roaster XX completes 3×112=143 \times \frac{1}{12} = \frac{1}{4} of the total batch, leaving 34\frac{3}{4} of the batch remaining. When both roasters work together, their combined rate is 112+18=524\frac{1}{12} + \frac{1}{8} = \frac{5}{24} batch per hour. The time required for both roasters to finish the remaining 34\frac{3}{4} batch is 3/45/24=3.6\frac{3/4}{5/24} = 3.6 hours. Adding the initial 33 hours yields a total elapsed time of 6.66.6 hours.

Step-by-Step Solution

1
Calculate individual work rates per hour.
Roaster XX rate = 112\frac{1}{12} batch/hr; Roaster YY rate = 18\frac{1}{8} batch/hr.
Work rate is the reciprocal of the time required to complete one full job.
2
Determine the amount of work completed by Roaster XX in the first 33 hours and the remaining work.
Work done = 3×112=143 \times \frac{1}{12} = \frac{1}{4} batch. Remaining work = 1−14=341 - \frac{1}{4} = \frac{3}{4} batch.
Roaster XX worked alone for the first 33 hours.
3
Calculate the combined work rate of both roasters working together.
Combined rate = 112+18=224+324=524\frac{1}{12} + \frac{1}{8} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} batch/hr.
Simultaneous work rates are additive.
4
Find the time taken to complete the remaining work together.
Combined time = 3/45/24=34×245=185=3.6\frac{3/4}{5/24} = \frac{3}{4} \times \frac{24}{5} = \frac{18}{5} = 3.6 hours.
Time equals remaining work divided by the combined work rate.
5
Add the initial single-roaster phase to the combined work phase to find total time.
Total time = 3+3.6=6.63 + 3.6 = 6.6 hours.
The question asks for the total time to process the entire batch.

Key Concept

Work Rate and Combined Work
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Topics

Work Rate and Combined Work

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