Two pipes can fill a cistern in 6 minutes and 7 minutes respectively. Both the pipes are opened alternatively for 1 minute each. In what time will they fill the cistern.
A)6 minutes
B)`6 (2)/(3)` minutes
C)`6 (3)/(7)` minutes
D)`3 (1)/(2)` minutes

Options

A. 6 minutes
B. `6 (2)/(3)` minutes
C. `6 (3)/(7)` minutes
D. `3 (1)/(2)` minutes

Answer

To solve the problem, we need to determine how long it will take for two pipes, which fill a cistern in 6 minutes and 7 minutes respectively, to fill the cistern when they are opened alternately for 1 minute each.

### Step-by-Step Solution:

1. **Determine the Filling Rates of Each Pipe:**
- Pipe A fills the cistern in 6 minutes. Therefore, its rate of filling is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ cistern}}{6 \text{ minutes}} = \frac{1}{6} \text{ cistern per minute} \]
- Pipe B fills the cistern in 7 minutes. Therefore, its rate of filling is:
\[ \text{Rate of Pipe B} = \frac{1 \text{ cistern}}{7 \text{ minutes}} = \frac{1}{7} \text{ cistern per minute} \]

2. **Calculate the Amount Filled in 2 Minutes:**
- In the first minute, Pipe A is open, and it fills:
\[ \text{Amount filled by A in 1 minute} = \frac{1}{6} \]
- In the second minute, Pipe B is open, and it fills:
\[ \text{Amount filled by B in 1 minute} = \frac{1}{7} \]
- Therefore, in 2 minutes, the total amount filled is:
\[ \text{Total amount filled in 2 minutes} = \frac{1}{6} + \frac{1}{7} \]
- To add these fractions, find a common denominator (which is 42):
\[ \frac{1}{6} = \frac{7}{42}, \quad \frac{1}{7} = \frac{6}{42} \]
\[ \text{Total amount filled in 2 minutes} = \frac{7}{42} + \frac{6}{42} = \frac{13}{42} \]

3. **Determine How Many Full Cycles Are Needed:**
- Each cycle (2 minutes) fills \(\frac{13}{42}\) of the cistern.
- To find out how many cycles are needed to fill the cistern, we can set up the equation:
\[ n \cdot \frac{13}{42} \geq 1 \]
- Solving for \(n\):
\[ n \geq \frac{42}{13} \approx 3.23 \]
- This means we need 3 full cycles (6 minutes) and a bit more.

4. **Calculate the Amount Filled in 6 Minutes:**
- After 3 cycles (6 minutes), the amount filled is:
\[ 3 \cdot \frac{13}{42} = \frac{39}{42} \text{ cistern} \]

5. **Determine Remaining Volume to Fill:**
- The remaining volume to fill is:
\[ 1 - \frac{39}{42} = \frac{3}{42} = \frac{1}{14} \text{ cistern} \]

6. **Calculate Time Needed to Fill Remaining Volume:**
- Now, Pipe A will fill the remaining volume. The time taken by Pipe A to fill \(\frac{1}{14}\) of the cistern is:
\[ \text{Time} = \text{Volume} \div \text{Rate of A} = \frac{1/14}{1/6} = \frac{6}{14} = \frac{3}{7} \text{ minutes} \]

7. **Total Time Taken:**
- The total time taken to fill the cistern is:
\[ \text{Total time} = 6 \text{ minutes} + \frac{3}{7} \text{ minutes} = \frac{42}{7} + \frac{3}{7} = \frac{45}{7} \text{ minutes} = 6 \frac{3}{7} \text{ minutes} \]

### Final Answer:
The time taken to fill the cistern is \(6 \frac{3}{7}\) minutes.