Two pipes A and B can fill a tank in 1 h and 75 min, respectively. There is also an outlet C. If all the three pipes are opened together. The tank is full in 50 min. How much time will be taken by C to empty the full tank?

Options

A. 100 min
B. 150 min
C. 200 min
D. 125 min

Answer

To solve the problem, we need to determine how long it will take for pipe C to empty the full tank, given the filling rates of pipes A and B and the combined effect of all three pipes.

### Step-by-Step Solution:

1. **Convert Time to Minutes:**
- Pipe A can fill the tank in 1 hour, which is 60 minutes.
- Pipe B can fill the tank in 75 minutes.
- The outlet pipe C's effect will be determined later.

2. **Find the Capacity of the Tank:**
- Let’s assume the capacity of the tank is 300 units (this is a convenient number since it is a common multiple of 60, 75, and 50).

3. **Calculate the Filling Rate of Pipe A:**
- Pipe A fills the tank in 60 minutes, so its rate is:
\[ \text{Rate of A} = \frac{300 \text{ units}}{60 \text{ minutes}} = 5 \text{ units/minute} \]

4. **Calculate the Filling Rate of Pipe B:**
- Pipe B fills the tank in 75 minutes, so its rate is:
\[ \text{Rate of B} = \frac{300 \text{ units}}{75 \text{ minutes}} = 4 \text{ units/minute} \]

5. **Calculate the Combined Rate of A and B:**
- The combined rate of pipes A and B is:
\[ \text{Rate of A + B} = 5 + 4 = 9 \text{ units/minute} \]

6. **Calculate the Combined Rate of A, B, and C:**
- When all three pipes are opened together, they fill the tank in 50 minutes:
\[ \text{Combined Rate (A + B + C)} = \frac{300 \text{ units}}{50 \text{ minutes}} = 6 \text{ units/minute} \]

7. **Determine the Rate of Pipe C:**
- From the combined rates, we can find the rate of pipe C:
\[ \text{Rate of C} = \text{Rate of A + B + C} - \text{Rate of A + B} \]
\[ \text{Rate of C} = 6 - 9 = -3 \text{ units/minute} \]
- The negative sign indicates that pipe C is emptying the tank at a rate of 3 units per minute.

8. **Calculate the Time Taken by Pipe C to Empty the Tank:**
- To find the time taken by pipe C to empty the full tank, we use the formula:
\[ \text{Time} = \frac{\text{Capacity of the tank}}{\text{Rate of C}} \]
\[ \text{Time} = \frac{300 \text{ units}}{3 \text{ units/minute}} = 100 \text{ minutes} \]

### Final Answer:
Pipe C will take **100 minutes** to empty the full tank.

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