Two inlet pipes can fill a cistern in 3 hours and 4 hours respectively, while a third outlet pipe can empty the full cistern in 8 hours. All three pipes are opened together when the cistern is already 1/12 full. How long will it take from that moment for the cistern to become completely full?

Introduction / Context:
This problem is a combination of work and time with an initial nonzero water level. Two pipes fill while a third pipe empties, and the cistern already contains some water at the start. The question tests the ability to compute net rate and to understand that only the remaining fraction of the tank needs to be filled from the initial condition to reach full capacity.

Given Data / Assumptions:

Pipe 1 fills the cistern in 3 hours.

Pipe 2 fills the cistern in 4 hours.

Pipe 3 empties the cistern in 8 hours.

The cistern is initially 1/12 full.

All three pipes are opened together at that moment.

Flow rates are constant throughout.

Concept / Approach:
We convert all times into rates in terms of cisterns per hour. Filling pipes contribute positive rates, while the outlet contributes a negative rate. The net rate when all three are open is the sum of these three rates. Since the cistern starts at 1/12 full, we only need to fill an additional 11/12 of its capacity. Dividing that remaining volume by the net rate gives the required time.

Step-by-Step Solution:
Let the capacity of the cistern be 1 unit.Rate of pipe 1 (inlet) = 1/3 cistern per hour.Rate of pipe 2 (inlet) = 1/4 cistern per hour.Rate of pipe 3 (outlet) = 1/8 cistern per hour.Net rate when all three are open = 1/3 + 1/4 - 1/8.Find a common denominator of 24: 1/3 = 8/24, 1/4 = 6/24, 1/8 = 3/24.Net rate = (8/24 + 6/24 - 3/24) = 11/24 cistern per hour.Initial volume in the cistern = 1/12 of capacity.Remaining volume to be filled = 1 - 1/12 = 11/12 of the cistern.Time required = remaining volume / net rate = (11/12) / (11/24).This simplifies to (11/12) * (24/11) = 24/12 = 2 hours.Therefore, the cistern will be completely full 2 hours after all three pipes are opened.
Verification / Alternative check:
In 2 hours, the net volume added by all three pipes is 2 * (11/24) = 22/24 = 11/12 of the cistern.Since the cistern starts at 1/12 full, final volume = 1/12 + 11/12 = 1 full cistern.This confirms the correctness of the time computed.
Why Other Options Are Wrong:
Times shorter than 2 hours, such as 1 hour 45 minutes, would not be enough to add the required 11/12 of the cistern at the given net rate. Longer times like 2 hours 10 minutes or 2 hours 11 minutes would overfill the cistern if the net rate remained unchanged. The option 2.5 hours is also inconsistent with the simple fraction calculation and gives too much total work done.

Common Pitfalls:
Some students forget to subtract the outlet rate and instead add all three rates, which would produce an unrealistically high net rate. Others mistakenly compute the time needed to fill the entire cistern from empty rather than only the remaining 11/12 volume. Also, confusion can arise when converting fractions with different denominators; using a common denominator like 24 keeps arithmetic clean and reduces errors.

Final Answer:
The cistern will be completely full in 2 hours.