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Difficulty: HardRatios, Rates, and Percentages

A manufacturing facility uses two processing lines, Line A and Line B, to produce liquid chemical compounds. Line A operates at a constant rate of rAr_A liters per hour and yields a mixture containing 15%15\% active solute by volume. Line B operates at a constant rate of rBr_B liters per hour and yields a mixture containing 40%40\% active solute by volume.

When Line A operates for 4 hours and Line B operates for 6 hours, the combined output is 1,6001,600 liters of a solution containing 30%30\% active solute by volume.

If Line A and Line B are run simultaneously for 5 hours, how many liters of active solute will be produced in total?

Answer: 440 liters

Answer

440 liters of active solute will be produced in total.
By setting up a system of two linear equations relating the individual line flow rates (rAr_A and rBr_B) to total volume and total solute, we find rA=160r_A = 160 L/hr and rB=160r_B = 160 L/hr. Running Line A and Line B together yields 0.15(160)+0.40(160)=880.15(160) + 0.40(160) = 88 liters of active solute per hour. Over 5 hours, this produces 88×5=44088 \times 5 = 440 liters of active solute.

Step-by-Step Solution

1
Formulate total volume equation
4rA+6rB=1600  ⟹  2rA+3rB=8004r_A + 6r_B = 1600 \implies 2r_A + 3r_B = 800
Line A runs for 4 hours and Line B runs for 6 hours to produce a total volume of 1,600 liters.
2
Formulate active solute equation
0.15(4rA)+0.40(6rB)=0.30(1600)  ⟹  0.6rA+2.4rB=480  ⟹  rA+4rB=8000.15(4r_A) + 0.40(6r_B) = 0.30(1600) \implies 0.6r_A + 2.4r_B = 480 \implies r_A + 4r_B = 800
The total solute produced is 30%30\% of the 1,600-liter mixture, which equals 480 liters.
3
Solve the system of linear equations
rA=160r_A = 160 liters/hour and rB=160r_B = 160 liters/hour
Substituting rA=800−4rBr_A = 800 - 4r_B into 2rA+3rB=8002r_A + 3r_B = 800 yields 5rB=800  ⟹  rB=1605r_B = 800 \implies r_B = 160 and rA=160r_A = 160.
4
Calculate solute rate per hour when both run simultaneously
0.15(160)+0.40(160)=24+64=880.15(160) + 0.40(160) = 24 + 64 = 88 liters of solute per hour
Each line produces solute at its respective concentration percentage of its hourly fluid rate.
5
Compute total solute produced in 5 hours
88×5=44088 \times 5 = 440 liters
Multiplying the combined solute production rate of 88 liters per hour by 5 hours gives the final quantity.

Key Concept

Simultaneous equations combining volumetric rates and mixture percentage concentrations.
Estimated Time:2m 30s
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Topics

Ratios, Rates, and Percentages

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