Two inlet pipes can fill a tank in 20 minutes and 24 minutes respectively, and a waste pipe can empty water from the tank at the rate of 3 gallons per minute. When all three pipes are opened together, the tank is filled in 15 minutes. What is the capacity of the tank in gallons?

Introduction / Context:
This problem mixes fractional filling rates with an absolute emptying rate in gallons per minute. Two pipes fill the tank, while a waste pipe empties it at a fixed volume rate. We must find the total capacity of the tank given the net effect when all three operate together.

Given Data / Assumptions:

Pipe 1 fills the tank in 20 minutes.

Pipe 2 fills the tank in 24 minutes.

The waste pipe empties water at 3 gallons per minute.

All three pipes operating together fill the tank in 15 minutes.

Concept / Approach:
Let the capacity of the tank be V gallons. The two inlets fill at rates V / 20 and V / 24 gallons per minute. The waste pipe empties at 3 gallons per minute. The net filling rate is then V / 20 + V / 24 - 3 gallons per minute. Since the tank is filled in 15 minutes, the net rate must also equal V / 15 gallons per minute. Setting these expressions equal allows us to solve for V.

Step-by-Step Solution:
Step 1: Let tank capacity be V gallons.Step 2: Rate of pipe 1 = V / 20 gallons per minute.Step 3: Rate of pipe 2 = V / 24 gallons per minute.Step 4: Rate of waste pipe = 3 gallons per minute (emptying).Step 5: Net rate with all three pipes = V / 20 + V / 24 - 3 gallons per minute.Step 6: Because the tank is filled in 15 minutes, net rate must also equal V / 15 gallons per minute.Step 7: Set up the equation: V / 20 + V / 24 - 3 = V / 15.Step 8: Compute V / 20 + V / 24 = V * (1 / 20 + 1 / 24) = V * (6 / 120 + 5 / 120) = 11V / 120.Step 9: So we have 11V / 120 - 3 = V / 15.Step 10: Multiply through by 120 to clear denominators: 11V - 360 = 8V.Step 11: Rearrange: 11V - 8V = 360, so 3V = 360 and V = 120.Step 12: Therefore, the capacity of the tank is 120 gallons.

Verification / Alternative check:
Check the rates with V = 120. Pipe 1 fills at 120 / 20 = 6 gallons per minute. Pipe 2 fills at 120 / 24 = 5 gallons per minute. Combined inlets give 11 gallons per minute. Subtracting the waste rate of 3 gallons per minute, net rate is 8 gallons per minute. In 15 minutes, total volume filled = 8 * 15 = 120 gallons, which matches the tank capacity.

Why Other Options Are Wrong:
60 gallons and 100 gallons produce net rates that do not match the given 15 minute fill time when substituted into the equation.

180 gallons would imply larger inlet rates and would not satisfy the relation that all three together fill the tank exactly in 15 minutes.

Common Pitfalls:
Students sometimes forget that the waste pipe rate is in gallons per minute, not a fraction of the tank per minute, and treat it as 1 / t. Others add instead of subtract the waste rate, which reverses its effect. Careful unit tracking and consistent use of the tank capacity symbol V avoid these issues.

Final Answer:
The capacity of the tank is 120 gallons.