Two inlet pipes can separately fill a cistern completely in 8 hours and 10 hours respectively. They are operated for 2 hours, after which the second pipe is closed, and an outlet pipe which can drain out water from the full cistern in 20 hours is opened. How much time will it take to fill the cistern completely from the instant of opening the outlet pipe?

Options

A. 7 hours
B. 8 hours
C. 7 hours 20 minutes
D. 8 hours 40 minutes

Answer

To solve the problem step by step, we will first determine the rates at which the pipes fill or drain the cistern, then calculate how much of the cistern is filled in the first 2 hours, and finally determine how long it will take to fill the remaining part of the cistern after the outlet pipe is opened.

### Step 1: Determine the filling rates of the pipes
- Pipe A fills the cistern in 8 hours, so its rate is:
\[ \text{Rate of A} = \frac{1}{8} \text{ cisterns per hour} \]
- Pipe B fills the cistern in 10 hours, so its rate is:
\[ \text{Rate of B} = \frac{1}{10} \text{ cisterns per hour} \]
- Pipe C drains the cistern in 20 hours, so its rate is:
\[ \text{Rate of C} = -\frac{1}{20} \text{ cisterns per hour} \]

### Step 2: Calculate the combined filling rate of pipes A and B
The combined rate of pipes A and B when both are open is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{8} + \frac{1}{10} \]
To add these fractions, we find a common denominator, which is 40:
\[ \text{Combined Rate} = \frac{5}{40} + \frac{4}{40} = \frac{9}{40} \text{ cisterns per hour} \]

### Step 3: Calculate the amount filled in 2 hours
In 2 hours, the amount filled by both pipes A and B is:
\[ \text{Amount filled in 2 hours} = \text{Combined Rate} \times 2 = \frac{9}{40} \times 2 = \frac{18}{40} = \frac{9}{20} \text{ cisterns} \]

### Step 4: Calculate the remaining capacity of the cistern
The total capacity of the cistern is 1 (full cistern), so the remaining capacity after 2 hours is:
\[ \text{Remaining Capacity} = 1 - \frac{9}{20} = \frac{20}{20} - \frac{9}{20} = \frac{11}{20} \text{ cisterns} \]

### Step 5: Determine the effective rate after closing pipe B and opening pipe C
After 2 hours, pipe B is closed, and pipe C (the outlet pipe) is opened. The effective rate now is:
\[ \text{Effective Rate} = \text{Rate of A} + \text{Rate of C} = \frac{1}{8} - \frac{1}{20} \]
Finding a common denominator (40):
\[ \text{Effective Rate} = \frac{5}{40} - \frac{2}{40} = \frac{3}{40} \text{ cisterns per hour} \]

### Step 6: Calculate the time to fill the remaining capacity
To find the time required to fill the remaining \(\frac{11}{20}\) cisterns at the effective rate of \(\frac{3}{40}\) cisterns per hour:
\[ \text{Time} = \frac{\text{Remaining Capacity}}{\text{Effective Rate}} = \frac{\frac{11}{20}}{\frac{3}{40}} = \frac{11}{20} \times \frac{40}{3} = \frac{11 \times 2}{3} = \frac{22}{3} \text{ hours} \]

### Step 7: Convert the time into hours and minutes
\(\frac{22}{3}\) hours is equal to:
\[ 7 \text{ hours and } \frac{1}{3} \text{ of an hour} = 7 \text{ hours and } 20 \text{ minutes} \]

### Final Answer
Thus, the time it will take to fill the cistern completely from the instant of opening the outlet pipe is **7 hours and 20 minutes**.