Pipe A can fill an empty cistern in 6.8 hours. Pipe B can fill it up in 10.2 hours. Only Pipe B is turned on for 1.7 hours after which Pipe A is also turned on. How much time will it take in all to fill up the cistern? Options are (a) 5 hours 6 minutes (b) 5 hours 12 minutes (c) 5 hours 5 minute (d) 5 hours 10 minutes.

Options

A. 5 hours 6 minutes
B. 5 hours 12 minutes
C. 5 hours 5 minute
D. 5 hours 10 minutes

Answer

To solve the problem step by step, we will first determine the rates at which both pipes A and B can fill the cistern, then calculate the total time taken to fill the cistern based on the operations of both pipes.

### Step 1: Calculate the rates of Pipe A and Pipe B
- Pipe A can fill the cistern in 6.8 hours, so its rate is:
\[ \text{Rate of Pipe A} = \frac{1}{6.8} \text{ cisterns per hour} \]

- Pipe B can fill the cistern in 10.2 hours, so its rate is:
\[ \text{Rate of Pipe B} = \frac{1}{10.2} \text{ cisterns per hour} \]

### Step 2: Calculate the amount filled by Pipe B in 1.7 hours
- In 1.7 hours, the amount filled by Pipe B is:
\[ \text{Amount filled by Pipe B} = \text{Rate of Pipe B} \times 1.7 = \frac{1}{10.2} \times 1.7 \]
\[ = \frac{1.7}{10.2} \approx 0.1667 \text{ cisterns} \]

### Step 3: Calculate the remaining amount to be filled
- The total amount of the cistern is 1. Thus, the remaining amount to be filled after Pipe B has worked for 1.7 hours is:
\[ \text{Remaining amount} = 1 - 0.1667 \approx 0.8333 \text{ cisterns} \]

### Step 4: Calculate the combined rate of Pipe A and Pipe B
- When both pipes are turned on, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} = \frac{1}{6.8} + \frac{1}{10.2} \]

- Finding a common denominator (LCM of 6.8 and 10.2 is 20.4):
\[ \frac{1}{6.8} = \frac{3}{20.4}, \quad \frac{1}{10.2} = \frac{2}{20.4} \]
\[ \text{Combined Rate} = \frac{3}{20.4} + \frac{2}{20.4} = \frac{5}{20.4} \text{ cisterns per hour} \]

### Step 5: Calculate the time taken to fill the remaining amount
- The time taken to fill the remaining 0.8333 cisterns at the combined rate is:
\[ \text{Time} = \frac{\text{Remaining amount}}{\text{Combined Rate}} = \frac{0.8333}{\frac{5}{20.4}} = 0.8333 \times \frac{20.4}{5} \]
\[ = 3.36 \text{ hours} \]

### Step 6: Calculate the total time taken
- The total time taken to fill the cistern is:
\[ \text{Total Time} = 1.7 \text{ hours (Pipe B)} + 3.36 \text{ hours (Both pipes)} \approx 5.06 \text{ hours} \]

### Step 7: Convert the total time into hours and minutes
- Converting 0.06 hours into minutes:
\[ 0.06 \times 60 \approx 3.6 \text{ minutes} \approx 4 \text{ minutes} \]

- Thus, the total time is approximately:
\[ 5 \text{ hours and } 4 \text{ minutes} \]

### Final Answer
- The total time taken to fill the cistern is approximately **5 hours and 4 minutes**.