Pipe A and B can fill a tank in 10 hours and 20 hours respectively, if an outlet pipe C is left open which can empty the tank in 40 hours then in how much time will the tank be filled if they are opened together?

Options

A. 10 hours
B. 40/7 hours
C. 8 hours
D. 7 hours

Answer

To solve the problem, we need to determine how much of the tank is filled by pipes A and B when they are opened together, while also considering the effect of the outlet pipe C that empties the tank.

### Step-by-Step Solution:

1. **Determine the capacity of the tank**:
- Let's assume the capacity of the tank is 40 liters (this is a convenient number for calculations).

2. **Calculate the filling rate of Pipe A**:
- Pipe A can fill the tank in 10 hours.
- Therefore, in 1 hour, Pipe A fills:
\[ \text{Filling rate of A} = \frac{40 \text{ liters}}{10 \text{ hours}} = 4 \text{ liters/hour} \]

3. **Calculate the filling rate of Pipe B**:
- Pipe B can fill the tank in 20 hours.
- Therefore, in 1 hour, Pipe B fills:
\[ \text{Filling rate of B} = \frac{40 \text{ liters}}{20 \text{ hours}} = 2 \text{ liters/hour} \]

4. **Calculate the emptying rate of Pipe C**:
- Pipe C can empty the tank in 40 hours.
- Therefore, in 1 hour, Pipe C empties:
\[ \text{Emptying rate of C} = \frac{40 \text{ liters}}{40 \text{ hours}} = 1 \text{ liter/hour} \]

5. **Calculate the net filling rate when all pipes are open**:
- When all three pipes are open, the net filling rate is:
\[ \text{Net filling rate} = \text{Filling rate of A} + \text{Filling rate of B} - \text{Emptying rate of C} \]
\[ \text{Net filling rate} = 4 \text{ liters/hour} + 2 \text{ liters/hour} - 1 \text{ liter/hour} = 5 \text{ liters/hour} \]

6. **Calculate the time taken to fill the tank**:
- To find the time taken to fill the tank, we use the formula:
\[ \text{Time} = \frac{\text{Capacity of the tank}}{\text{Net filling rate}} \]
\[ \text{Time} = \frac{40 \text{ liters}}{5 \text{ liters/hour}} = 8 \text{ hours} \]

### Final Answer:
The tank will be filled in **8 hours** when all three pipes are opened together.