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Difficulty: HardLinear Equations in One Variable

Two water pumps, Pump AA and Pump BB, operate at constant rates to fill a 910910-gallon storage tank. The rate at which Pump AA fills the tank is 1515 gallons per hour more than twice the rate of Pump BB. Both pumps begin working together to fill the empty tank. After 44 hours, Pump AA shuts down, and Pump BB continues working alone for another 55 hours to finish filling the tank completely. What is the pumping rate of Pump AA, in gallons per hour?

  1. A
    50
  2. B
    85
  3. C
    100
  4. 115Answer
  5. E
    130

Answer

115 gallons per hour
Letting rr represent the rate of Pump B in gallons per hour, the rate of Pump A is 2r+152r + 15. During the first 4 hours, both pumps operate together at a combined rate of (3r+15)(3r + 15) gallons per hour, contributing 4(3r+15)=12r+604(3r + 15) = 12r + 60 gallons. In the remaining 5 hours, Pump B alone contributes 5r5r gallons. Setting the total amount equal to the tank capacity gives 12r+60+5r=91012r + 60 + 5r = 910, which simplifies to 17r=85017r = 850, so r=50r = 50. Substituting r=50r = 50 into 2r+152r + 15 yields 115115 gallons per hour.

Step-by-Step Solution

1
Define the variable for the unknown rate.
Let rr be the pumping rate of Pump BB in gallons per hour. Then the rate of Pump AA is 2r+152r + 15 gallons per hour.
Expressing both rates in terms of a single variable rr allows us to formulate a linear equation in one variable.
2
Determine the combined pumping rate and the work done during the first phase.
Combined rate =r+(2r+15)=3r+15= r + (2r + 15) = 3r + 15 gallons per hour. Work completed in 4 hours =4(3r+15)=12r+60= 4(3r + 15) = 12r + 60 gallons.
When both pumps work together, their individual rates add together.
3
Formulate the total work equation for filling the 910-gallon tank.
(12r+60)+5r=910  ⟹  17r+60=910(12r + 60) + 5r = 910 \implies 17r + 60 = 910
The sum of the work done in the first 4 hours by both pumps and the work done in the next 5 hours by Pump B alone equals the total capacity.
4
Solve the linear equation for rr.
17r=850  ⟹  r=5017r = 850 \implies r = 50 gallons per hour.
Subtracting 60 from both sides and dividing by 17 gives the rate of Pump B.
5
Calculate the pumping rate of Pump A.
Rate of Pump A=2(50)+15=115A = 2(50) + 15 = 115 gallons per hour.
Substitute r=50r = 50 into the expression for Pump A's rate (2r+152r + 15).

Key Concept

Linear equation formulation from work and rate relationships
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Topics

Linear Equations in One Variable

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