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Difficulty: MediumRatios, Rates, and Percentages

A pharmaceutical facility uses automated Line Alpha and Line Beta to package liquid medicine vials.

- Line Alpha packages vials at a constant rate of 200200 vials per hour.
- Line Beta packages vials at a constant rate that is 50%50\% greater than the rate of Line Alpha.

During a production shift, Line Alpha operates for 99 hours. Line Beta operates for a duration such that Line Alpha's output accounts for 30%30\% of the total combined vial output of both lines during the shift.

Which of the following choices correctly identifies the total number of hours Line Beta operated and the total number of vials packaged by Line Beta during the shift?

  1. A
    Line Beta Operating Time = 9 hours; Line Beta Total Vials = 1,800 vials
  2. Line Beta Operating Time = 14 hours; Line Beta Total Vials = 4,200 vialsAnswer
  3. C
    Line Beta Operating Time = 14 hours; Line Beta Total Vials = 6,000 vials
  4. D
    Line Beta Operating Time = 21 hours; Line Beta Total Vials = 4,200 vials
  5. E
    Line Beta Operating Time = 6 hours; Line Beta Total Vials = 1,800 vials

Answer

Line Beta operated for 14 hours and packaged 4,200 vials during the shift.
The correct answer accurately determines both required quantities. Line Alpha packages 200×9=1,800200 \times 9 = 1,800 vials. Since 1,8001,800 represents 30%30\% of the shift's total output, the total output is 1,800/0.30=6,0001,800 / 0.30 = 6,000 vials. Line Beta accounts for the remaining 6,000−1,800=4,2006,000 - 1,800 = 4,200 vials. At Line Beta's rate of 200×1.5=300200 \times 1.5 = 300 vials per hour, its operating time is 4,200/300=144,200 / 300 = 14 hours.

Step-by-Step Solution

1
Calculate Line Beta's hourly packaging rate.
Line Beta Rate = 200×1.50=300200 \times 1.50 = 300 vials per hour.
Line Beta operates at a rate 50%50\% greater than Line Alpha's rate of 200200 vials/hr.
2
Calculate Line Alpha's total vial output during the shift.
Line Alpha Output = 200 vials/hr×9 hrs=1,800200 \text{ vials/hr} \times 9 \text{ hrs} = 1,800 vials.
Line Alpha runs for 99 hours at a constant rate of 200200 vials/hr.
3
Determine the total combined vial output of both lines.
Total Combined Output = 1,8000.30=6,000\frac{1,800}{0.30} = 6,000 vials.
Line Alpha's output of 1,8001,800 vials represents 30%30\% of the total combined output.
4
Calculate Line Beta's individual vial output.
Line Beta Output = 6,000−1,800=4,2006,000 - 1,800 = 4,200 vials.
Line Beta produces the remaining 70%70\% of the total combined output.
5
Calculate Line Beta's total operating time.
Line Beta Operating Time = 4,200 vials300 vials/hr=14\frac{4,200 \text{ vials}}{300 \text{ vials/hr}} = 14 hours.
Dividing Line Beta's total output by its hourly rate gives the operating time.

Key Concept

Two-Part Analysis combining relative work rates and percentage allocation
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Topics

Ratios, Rates, and Percentages

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