Two pipes P and Q alone can fill a tank in 24 min and 30 min respectively and a pipe R alone can empty the tank by throwing out 8 gallons in 2 minutes. All the three pipes working together can fill the tank in 20 minutes. Find the capacity of the tank?

Options

A. A)80 gallons
B. B)160 gallons
C. C)320 gallons
D. D)240 gallons

Answer

To solve the problem, we need to find the capacity of the tank given the rates at which pipes P, Q, and R fill or empty the tank.

### Step-by-Step Solution:

1. **Determine the filling rates of pipes P and Q:**
- Pipe P can fill the tank in 24 minutes.
- Therefore, the rate of filling by pipe P = \( \frac{120 \text{ liters}}{24 \text{ minutes}} = 5 \text{ liters/minute} \).

- Pipe Q can fill the tank in 30 minutes.
- Therefore, the rate of filling by pipe Q = \( \frac{120 \text{ liters}}{30 \text{ minutes}} = 4 \text{ liters/minute} \).

2. **Determine the combined filling rate of pipes P and Q:**
- Combined rate of P and Q = \( 5 + 4 = 9 \text{ liters/minute} \).

3. **Determine the combined filling rate of all three pipes (P, Q, and R):**
- All three pipes together can fill the tank in 20 minutes.
- Therefore, the rate of filling by P, Q, and R = \( \frac{120 \text{ liters}}{20 \text{ minutes}} = 6 \text{ liters/minute} \).

4. **Determine the emptying rate of pipe R:**
- Since the combined filling rate of P and Q is 9 liters/minute and the combined rate of all three pipes is 6 liters/minute, we can find the emptying rate of pipe R:
- Rate of R = Rate of P + Q - Rate of (P + Q + R) = \( 9 - 6 = 3 \text{ liters/minute} \).
- Therefore, pipe R empties the tank at a rate of 3 liters/minute.

5. **Determine the time taken by pipe R to empty the tank:**
- The capacity of the tank is 120 liters, and R empties it at 3 liters/minute.
- Time taken by R to empty the tank = \( \frac{120 \text{ liters}}{3 \text{ liters/minute}} = 40 \text{ minutes} \).

6. **Convert the emptying rate of pipe R from liters to gallons:**
- Pipe R can empty 8 gallons in 2 minutes, which means it empties at a rate of \( \frac{8 \text{ gallons}}{2 \text{ minutes}} = 4 \text{ gallons/minute} \).

7. **Determine the total capacity of the tank in gallons:**
- Since R empties at 4 gallons/minute and it takes 40 minutes to empty the tank, the total capacity of the tank in gallons is:
- Capacity of the tank = \( 4 \text{ gallons/minute} \times 40 \text{ minutes} = 160 \text{ gallons} \).

### Conclusion:
The capacity of the tank is **160 gallons**.