To solve the problem step by step, we will first determine the efficiencies of the pipes A, B, and C, and then use this information to find the capacity of the tank.
### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 20 minutes. Therefore, its filling rate (efficiency) is:
\[
\text{Efficiency of A} = \frac{1 \text{ tank}}{20 \text{ minutes}} = \frac{1}{20} \text{ tanks per minute}
\]
- Pipe B can fill the tank in 24 minutes. Therefore, its filling rate (efficiency) is:
\[
\text{Efficiency of B} = \frac{1 \text{ tank}}{24 \text{ minutes}} = \frac{1}{24} \text{ tanks per minute}
\]
### Step 2: Determine the emptying rate of pipe C
- Pipe C can empty the tank at the rate of 3 gallons per minute. To find its efficiency in terms of the tank capacity, we need to convert this rate into a fraction of the tank. If we denote the capacity of the tank as \( C \) gallons, the efficiency of C (in terms of tanks per minute) is:
\[
\text{Efficiency of C} = -\frac{3}{C} \text{ tanks per minute}
\]
(Note: The negative sign indicates that it is emptying the tank.)
### Step 3: Combine the efficiencies of A, B, and C
When all three pipes are opened together, the net efficiency can be expressed as:
\[
\text{Efficiency of A + B - C} = \frac{1}{20} + \frac{1}{24} - \frac{3}{C}
\]
### Step 4: Find a common denominator and simplify
To add the fractions, we find the least common multiple (LCM) of 20 and 24, which is 120. Thus:
\[
\frac{1}{20} = \frac{6}{120}, \quad \frac{1}{24} = \frac{5}{120}
\]
So,
\[
\text{Efficiency of A + B} = \frac{6}{120} + \frac{5}{120} = \frac{11}{120}
\]
Now substituting this into the equation:
\[
\frac{11}{120} - \frac{3}{C} = \text{Efficiency of A + B - C}
\]
### Step 5: Set the equation for the time taken to fill the tank
According to the problem, A, B, and C together fill the tank in 15 minutes:
\[
\text{Efficiency of A + B - C} = \frac{1}{15}
\]
### Step 6: Set up the equation
Equating the two expressions for efficiency:
\[
\frac{11}{120} - \frac{3}{C} = \frac{1}{15}
\]
### Step 7: Solve for C
To eliminate the fractions, we can multiply the entire equation by \( 120C \):
\[
11C - 3 \cdot 120 = 8C
\]
This simplifies to:
\[
11C - 360 = 8C
\]
Rearranging gives:
\[
3C = 360 \implies C = 120 \text{ gallons}
\]
### Step 8: Conclusion
The capacity of the tank is 120 gallons.
