Three pipes A, B and C together can fill a tank in 6 hours. C is closed after 2 hours of opening. A and B can fill the remaining part of the tank in 7 hours. In how much time C alone can fill it?

Options

A. 10 hours
B. 12 hours
C. 8 hours
D. 14 hours

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Determine the combined rate of pipes A, B, and C.
Given that pipes A, B, and C together can fill the tank in 6 hours, we can find their combined rate of work.

\[ \text{Rate of A + B + C} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tanks/hour} \]

### Step 2: Calculate the amount of work done by A, B, and C in the first 2 hours.
Since C is closed after 2 hours, we can calculate how much of the tank is filled in that time.

\[ \text{Work done in 2 hours} = \text{Rate} \times \text{Time} = \frac{1}{6} \times 2 = \frac{1}{3} \text{ of the tank} \]

### Step 3: Determine the remaining work after C is closed.
After 2 hours, the remaining part of the tank to be filled is:

\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \text{ of the tank} \]

### Step 4: Find the rate of A and B together.
We are told that A and B can fill the remaining part of the tank in 7 hours. Thus, their combined rate is:

\[ \text{Rate of A + B} = \frac{2/3 \text{ tank}}{7 \text{ hours}} = \frac{2}{21} \text{ tanks/hour} \]

### Step 5: Relate the rates of A, B, and C.
We know the rate of A + B + C is \(\frac{1}{6}\) tanks/hour and the rate of A + B is \(\frac{2}{21}\) tanks/hour. We can find the rate of C by subtracting the rate of A + B from the rate of A + B + C:

\[ \text{Rate of C} = \text{Rate of A + B + C} - \text{Rate of A + B} = \frac{1}{6} - \frac{2}{21} \]

### Step 6: Find a common denominator and calculate the rate of C.
The common denominator of 6 and 21 is 42. Thus, we convert the rates:

\[ \frac{1}{6} = \frac{7}{42}, \quad \frac{2}{21} = \frac{4}{42} \]

Now we can subtract:

\[ \text{Rate of C} = \frac{7}{42} - \frac{4}{42} = \frac{3}{42} = \frac{1}{14} \text{ tanks/hour} \]

### Step 7: Calculate the time taken by C to fill the tank alone.
To find the time taken by C to fill the tank alone, we use the formula:

\[ \text{Time} = \frac{\text{Volume}}{\text{Rate}} = \frac{1 \text{ tank}}{\frac{1}{14} \text{ tanks/hour}} = 14 \text{ hours} \]

### Final Answer:
C alone can fill the tank in **14 hours**.

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