Two pipes X and Y can fill a cistern in 6 and 7 min, respectively. Starting with pipe X, both the pipes are opened alternately, each for 1 min. In what time will they fill the cistern?

Options

A. `6 (2)/(7) ` min
B. `6 (3)/(7) ` min
C. `6(5)/(7) ` min
D. `6 (1)/(7 )` min

Answer

To solve the problem step by step, we will first determine how much each pipe can fill in one minute, then calculate the total time taken to fill the cistern by alternating the pipes.

### Step 1: Determine the filling rates of the pipes
- **Pipe X** can fill the cistern in 6 minutes. Therefore, in 1 minute, it fills:
\[ \text{Filling rate of X} = \frac{1 \text{ cistern}}{6 \text{ minutes}} = \frac{1}{6} \text{ cistern per minute} \]

- **Pipe Y** can fill the cistern in 7 minutes. Therefore, in 1 minute, it fills:
\[ \text{Filling rate of Y} = \frac{1 \text{ cistern}}{7 \text{ minutes}} = \frac{1}{7} \text{ cistern per minute} \]

### Step 2: Calculate the total filling in 2 minutes
- In the first minute, Pipe X is opened:
\[ \text{Amount filled by X in 1 minute} = \frac{1}{6} \]

- In the second minute, Pipe Y is opened:
\[ \text{Amount filled by Y in 1 minute} = \frac{1}{7} \]

- Therefore, the total amount filled in 2 minutes is:
\[ \text{Total filled in 2 minutes} = \frac{1}{6} + \frac{1}{7} \]
To add these fractions, we need a common denominator, which is 42:
\[ \frac{1}{6} = \frac{7}{42}, \quad \frac{1}{7} = \frac{6}{42} \]
Thus,
\[ \text{Total filled in 2 minutes} = \frac{7}{42} + \frac{6}{42} = \frac{13}{42} \]

### Step 3: Determine how many complete cycles are needed
- The cistern has a total capacity of 1 (or 42 units). We need to find out how many 2-minute cycles are required to fill the cistern:
\[ \text{Let } n \text{ be the number of complete cycles.} \]
After \( n \) cycles (which take \( 2n \) minutes), the total amount filled is:
\[ \text{Total filled} = n \times \frac{13}{42} \]
We want this to equal 1:
\[ n \times \frac{13}{42} = 1 \implies n = \frac{42}{13} \approx 3.23 \]
This means we can complete 3 full cycles (6 minutes) and will need some additional time to fill the remaining part.

### Step 4: Calculate the amount filled after 3 cycles
- After 3 cycles (6 minutes):
\[ \text{Total filled} = 3 \times \frac{13}{42} = \frac{39}{42} \]

### Step 5: Determine the remaining amount to fill
- The remaining amount to fill is:
\[ 1 - \frac{39}{42} = \frac{3}{42} = \frac{1}{14} \]

### Step 6: Calculate the time taken to fill the remaining amount
- Now, Pipe X will be opened again for 1 minute. In 1 minute, Pipe X fills \( \frac{1}{6} \) of the cistern. We need to find out how long it takes to fill \( \frac{1}{14} \):
\[ \text{Time} = \frac{\text{Amount to fill}}{\text{Filling rate of X}} = \frac{\frac{1}{14}}{\frac{1}{6}} = \frac{6}{14} = \frac{3}{7} \text{ minutes} \]

### Step 7: Calculate the total time taken
- The total time taken to fill the cistern is:
\[ \text{Total time} = 6 \text{ minutes} + \frac{3}{7} \text{ minutes} = 6 + \frac{3}{7} = \frac{42}{7} + \frac{3}{7} = \frac{45}{7} \text{ minutes} \]
Converting this to mixed number:
\[ \frac{45}{7} = 6 \frac{3}{7} \text{ minutes} \]

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