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Two pipes can fill a tank in 20 and 24 minutes, respectively, and a waste pipe can empty 3 gallons per minute. All the three pipes working together can fill the tank in 15 minutes. The capacity of the tank is:
60 gallons
180 gallons
100 gallons
120 gallons
120 gallons
Use the LCM of filling times (20, 24, 15) to find total capacity in units. Then compare the net unit flow with the given waste rate to find the total gallon capacity.
Two pipes fill a tank in 20 min and 24 min respectively. A waste pipe empties at 3 gallons/min. All three pipes together fill the tank in 15 min.
TtotalтАЛ1тАЛ=T1тАЛ1тАЛ+T2тАЛ1тАЛтИТTwasteтАЛ1тАЛ
Use the LCM of filling times (20, 24, 15) to find total capacity in units. Then compare the net unit flow with the given waste rate to find the total gallon capacity.
Students often forget to subtract the rate of the waste pipe from the combined filling rate, or miscalculate the net efficiency of the system.
Find Total Units
Assume the tank capacity is the LCM of the times taken by the three pipes: LCM(20, 24, 15) = 120 units.
LCM(20,24,15)=120
Calculate Individual Efficiencies
Efficiency of pipe A = 120/20 = 6 units/min, pipe B = 120/24 = 5 units/min. Combined efficiency (A + B - Waste) = 120/15 = 8 units/min.
EffA+BтИТWтАЛ=8,EffAтАЛ=6,EffBтАЛ=5
Determine Waste Pipe Efficiency
From the combined rate, (6 + 5 - Waste) = 8. Thus, the waste pipe empties 3 units per minute.
11тИТW=8тЗТW=3┬аunits/min
Calculate Total Capacity
Since 3 units/min corresponds to the given 3 gallons/min, 1 unit equals 1 gallon. Therefore, 120 units equal 120 gallons.
120┬аunits=120├Ч1=120┬аgallons
D is correct because the net filling rate calculation leads to the conclusion that the tank capacity is 120 gallons.
This concept of net efficiency is identical to relative speed problems in kinematics where two objects move towards or away from each other.
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