Three pipes A, B and C together can fill a tank in 6 hours. After they work together for 2 hours, pipe C is closed and pipes A and B together fill the remaining part of the tank in 7 hours. How many hours would pipe C alone take to fill the tank?

Introduction / Context:
This is a standard pipes and cisterns problem involving three pipes, where one pipe is later closed. We must use the information about their combined work and the work done after closing one pipe to deduce the individual rate of pipe C and then find how long C alone would need to fill the tank.

Given Data / Assumptions:

Pipes A, B and C together fill the tank in 6 hours.

All three pipes run together for 2 hours.

After 2 hours, pipe C is closed.

Pipes A and B together fill the remaining part of the tank in 7 hours.

Tank capacity is taken as 1 unit.

Concept / Approach:
We convert the given times into rates. First we obtain the combined rate of A + B + C. Then using the information about how long A and B take to fill the remaining fraction, we find the combined rate of A + B alone. Subtracting these gives the rate of C alone. Finally, we invert that rate to find the time required by C working by itself.

Step-by-Step Solution:
Step 1: Combined rate of A + B + C = 1 / 6 tank per hour.Step 2: In the first 2 hours, amount filled by all three pipes = 2 * (1 / 6) = 1 / 3 tank.Step 3: Remaining fraction of the tank after 2 hours = 1 - 1 / 3 = 2 / 3.Step 4: A and B together fill this remaining 2 / 3 in 7 hours, so rate of A + B = (2 / 3) / 7 = 2 / 21 tank per hour.Step 5: Therefore, rate of C alone = (A + B + C rate) - (A + B rate) = 1 / 6 - 2 / 21.Step 6: Compute 1 / 6 - 2 / 21 = (7 / 42) - (4 / 42) = 3 / 42 = 1 / 14 tank per hour.Step 7: Time taken by C alone to fill the full tank = 1 / (1 / 14) = 14 hours.

Verification / Alternative check:
We can check the arithmetic by recomputing combined rates. If C fills at 1 / 14, then A + B rate must be 1 / 6 - 1 / 14 = (7 - 3) / 42 = 4 / 42 = 2 / 21, which matches our earlier value. With A + B working at 2 / 21, they need 7 hours to fill 2 / 3 of the tank, since 7 * (2 / 21) = 14 / 21 = 2 / 3. This confirms the consistency of our solution.

Why Other Options Are Wrong:
10 hours and 12 hours correspond to faster rates than allowed by the given conditions and do not satisfy the combined time relationships.

16 hours is too long, implying a slower C that would require a different pattern of filling times for A and B than the one described.

Common Pitfalls:
Many students wrongly assume that the shares of work of A, B and C can be read directly from the times, without computing actual rates. Others try to combine fractions in their heads and miscalculate 1 / 6 - 2 / 21. It is safer to write fractions with a common denominator step by step and then simplify carefully.

Final Answer:
Pipe C alone would fill the tank in 14 hours.