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Difficulty: HardRatios, Rates, and Proportions

A water desalination facility uses two types of filtration systems, System X and System Y. System X processes saltwater at a constant rate of 1515 liters per minute, producing purified water and brine in a ratio of 4:14:1. System Y processes saltwater at a constant rate of 2121 liters per minute, producing purified water and brine in a ratio of 5:25:2. If both systems operate simultaneously, how many liters of brine will be produced in total during the time it takes the two systems combined to produce 324324 liters of purified water?

  1. A
    81
  2. 108Answer
  3. C
    126
  4. D
    432

Answer

108
The correct value of 108 is found by calculating the rate of brine production for each system. System X produces brine at a rate of 15×15=315 \times \frac{1}{5} = 3 liters per minute, and System Y produces brine at 21×27=621 \times \frac{2}{7} = 6 liters per minute, giving a combined brine rate of 99 liters per minute. Meanwhile, the combined purified water rate is (15×45)+(21×57)=12+15=27(15 \times \frac{4}{5}) + (21 \times \frac{5}{7}) = 12 + 15 = 27 liters per minute. It takes the systems 32427=12\frac{324}{27} = 12 minutes to produce the purified water. During these 12 minutes, the total brine produced is 12×9=10812 \times 9 = 108 liters.

Step-by-Step Solution

1
Find the purified water and brine production rates for System X.
Purified water rate = 1212 liters per minute; Brine rate = 33 liters per minute.
Since the ratio of purified water to brine is 4:14:1, there are 4+1=54 + 1 = 5 parts in total. System X processes 1515 liters of saltwater per minute, so the purified water rate is 15×45=1215 \times \frac{4}{5} = 12 liters per minute and the brine rate is 15×15=315 \times \frac{1}{5} = 3 liters per minute.
2
Find the purified water and brine production rates for System Y.
Purified water rate = 1515 liters per minute; Brine rate = 66 liters per minute.
Since the ratio of purified water to brine is 5:25:2, there are 5+2=75 + 2 = 7 parts in total. System Y processes 2121 liters of saltwater per minute, so the purified water rate is 21×57=1521 \times \frac{5}{7} = 15 liters per minute and the brine rate is 21×27=621 \times \frac{2}{7} = 6 liters per minute.
3
Calculate the combined purified water and brine rates.
Combined purified rate = 2727 liters per minute; Combined brine rate = 99 liters per minute.
Adding the individual rates of System X and System Y gives the combined rates of 12+15=2712 + 15 = 27 liters per minute for purified water and 3+6=93 + 6 = 9 liters per minute for brine.
4
Determine the time required to produce the target amount of purified water.
Time = 1212 minutes.
Divide the target purified water volume of 324324 liters by the combined purified water rate of 2727 liters per minute to find the operating time: 32427=12\frac{324}{27} = 12 minutes.
5
Calculate the total amount of brine produced during this time.
Total brine = 108108 liters.
Multiply the operating time of 1212 minutes by the combined brine production rate of 99 liters per minute: 12×9=10812 \times 9 = 108 liters.

Key Concept

Combining rates derived from part-to-whole ratios to solve multi-step rate problems.
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Topics

Ratios, Rates, and Proportions

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