Pipe A can fill a cistern in 4 minutes and pipe B can fill it in 5 minutes. Pipe C can empty it in 2 minutes. First A and B are opened. After 2 minutes C is also opened. In how much time, the tank is emptied?

Options

A. 12 min.
B. 18 min.
C. 15 min.
D. 20 min.

Answer

To solve the problem step by step, we will calculate the rates of filling and emptying the cistern, determine how much water is filled in the first 2 minutes, and then find out how long it takes to empty the tank after pipe C is opened.

### Step-by-Step Solution:

1. **Determine the rates of pipes A, B, and C:**
- Pipe A fills the cistern in 4 minutes, so its rate is:
\[ \text{Rate of A} = \frac{20 \text{ (volume)}}{4 \text{ (time)}} = 5 \text{ units/min} \]
- Pipe B fills the cistern in 5 minutes, so its rate is:
\[ \text{Rate of B} = \frac{20 \text{ (volume)}}{5 \text{ (time)}} = 4 \text{ units/min} \]
- Pipe C empties the cistern in 2 minutes, so its rate is:
\[ \text{Rate of C} = \frac{20 \text{ (volume)}}{2 \text{ (time)}} = 10 \text{ units/min} \]

2. **Calculate the combined rate of pipes A and B when they are opened together:**
- Combined rate of A and B:
\[ \text{Rate of A + B} = 5 + 4 = 9 \text{ units/min} \]

3. **Calculate the volume filled in the first 2 minutes:**
- Volume filled in 2 minutes:
\[ \text{Volume filled} = \text{Rate of A + B} \times \text{Time} = 9 \text{ units/min} \times 2 \text{ min} = 18 \text{ units} \]

4. **Determine the net rate when pipe C is opened:**
- After 2 minutes, pipe C is also opened. The net rate becomes:
\[ \text{Net rate} = \text{Rate of A + B} - \text{Rate of C} = 9 - 10 = -1 \text{ unit/min} \]
- The negative sign indicates that the tank is emptying at a rate of 1 unit per minute.

5. **Calculate the time taken to empty the tank:**
- Since 18 units are already filled and the net rate of emptying is 1 unit/min, the time taken to empty the tank is:
\[ \text{Time to empty} = \frac{\text{Volume to empty}}{\text{Net rate}} = \frac{18 \text{ units}}{1 \text{ unit/min}} = 18 \text{ minutes} \]

### Final Answer:
The tank will be emptied in **18 minutes**.