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Difficulty: MediumRatios, Rates, and Percentages

A pharmaceutical laboratory uses two automated centrifuge units, Unit Alpha and Unit Beta, to process liquid compound solutions into a central storage tank.

- Unit Alpha processes a solution containing 15%15\% active compound at a constant rate of 4040 liters per hour.
- Unit Beta processes a solution containing 35%35\% active compound at a constant rate of 6060 liters per hour.

If both units operate simultaneously for 55 hours, what is the percentage of active compound in the total combined solution collected in the tank?

Answer: 27 %

Answer

The active compound concentration in the combined solution is 27%27\%.
The correct answer of 27%27\% is obtained by determining the overall mass balance: Unit Alpha contributes 3030 liters of active compound from 200200 liters of total liquid, while Unit Beta contributes 105105 liters of active compound from 300300 liters of total liquid. Dividing the total active compound (135135 liters) by the total liquid (500500 liters) gives 0.270.27, or 27%27\%.

Step-by-Step Solution

1
Determine the total volume and active compound volume produced by Unit Alpha.
Unit Alpha produces 200200 liters of total liquid (40×540 \times 5) containing 3030 liters of active compound (200×0.15200 \times 0.15).
Volume equals rate multiplied by time, and active compound volume equals total volume multiplied by its concentration percentage.
2
Determine the total volume and active compound volume produced by Unit Beta.
Unit Beta produces 300300 liters of total liquid (60×560 \times 5) containing 105105 liters of active compound (300×0.35300 \times 0.35).
Applied the same volume and percentage relationship to Unit Beta.
3
Compute the combined total volume and total active compound volume.
Total combined liquid volume = 200+300=500200 + 300 = 500 liters. Total active compound = 30+105=13530 + 105 = 135 liters.
Combining the output of both units sums their respective liquid and active compound volumes.
4
Calculate the combined percentage concentration.
135500×100%=27%.\frac{135}{500} \times 100\% = 27\%.
The final concentration is the ratio of total active compound to total liquid volume expressed as a percentage.

Key Concept

Weighted Average Percentage Concentration
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Topics

Ratios, Rates, and Percentages

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