Four pipe A, B, C and D can fill a cistern in 20, 25, 40 and 50 hours respectively.
If A and B are opened as inlet pipe into the cistern and C and D are opened as outlet pipes from the cistern and all the four pipes are opened simultaneously, how many hours will it take to fill the cistern completely?

Options

A. a. 20 hours
B. b)11 hours
C. c)22 hours
D. d)45 hours

Answer

To solve the problem, we need to determine how long it will take to fill the cistern when two pipes are filling it (A and B) and two pipes are emptying it (C and D).

### Step-by-Step Solution:

1. **Determine the rates of each pipe:**
- Pipe A fills the cistern in 20 hours. Therefore, the rate of A = 1/20 cisterns per hour.
- Pipe B fills the cistern in 25 hours. Therefore, the rate of B = 1/25 cisterns per hour.
- Pipe C empties the cistern in 40 hours. Therefore, the rate of C = 1/40 cisterns per hour.
- Pipe D empties the cistern in 50 hours. Therefore, the rate of D = 1/50 cisterns per hour.

2. **Calculate the combined rate of A and B (filling pipes):**
\[ \text{Rate of A and B} = \frac{1}{20} + \frac{1}{25} \]
To add these fractions, find a common denominator. The least common multiple (LCM) of 20 and 25 is 100.
\[ \frac{1}{20} = \frac{5}{100}, \quad \frac{1}{25} = \frac{4}{100} \]
Therefore,
\[ \text{Rate of A and B} = \frac{5}{100} + \frac{4}{100} = \frac{9}{100} \text{ cisterns per hour} \]

3. **Calculate the combined rate of C and D (emptying pipes):**
\[ \text{Rate of C and D} = \frac{1}{40} + \frac{1}{50} \]
The LCM of 40 and 50 is 200.
\[ \frac{1}{40} = \frac{5}{200}, \quad \frac{1}{50} = \frac{4}{200} \]
Therefore,
\[ \text{Rate of C and D} = \frac{5}{200} + \frac{4}{200} = \frac{9}{200} \text{ cisterns per hour} \]

4. **Calculate the net rate of filling the cistern:**
\[ \text{Net rate} = \text{Rate of A and B} - \text{Rate of C and D} \]
\[ \text{Net rate} = \frac{9}{100} - \frac{9}{200} \]
To subtract these fractions, find a common denominator. The LCM of 100 and 200 is 200.
\[ \frac{9}{100} = \frac{18}{200} \]
Therefore,
\[ \text{Net rate} = \frac{18}{200} - \frac{9}{200} = \frac{9}{200} \text{ cisterns per hour} \]

5. **Calculate the time taken to fill the cistern:**
Since the net rate of filling is \(\frac{9}{200}\) cisterns per hour, the time taken to fill 1 cistern is the reciprocal of the net rate:
\[ \text{Time} = \frac{1 \text{ cistern}}{\frac{9}{200} \text{ cisterns per hour}} = \frac{200}{9} \text{ hours} \]
This can be approximated as:
\[ \frac{200}{9} \approx 22.22 \text{ hours} \]

### Final Answer:
It will take approximately **22.22 hours** to fill the cistern completely.