Two pipes A and B can fill a cistern in 10 minutes and 15 minutes respectively. A person opens both pipes together but accidentally leaves a waste pipe open as well. He expects the cistern to be full when only A and B would have filled it, but at that time it is not full. He then closes the waste pipe, and 3 minutes later the cistern becomes full. In how many minutes can the waste pipe alone empty a full cistern?

Introduction / Context:
This question involves two inlet pipes and one waste pipe. The person expects the cistern to be full at a certain time based on the filling rates of the two inlets, but discovers a waste pipe is also open. After closing the waste pipe, the cistern becomes full 3 minutes later. From this information we must deduce the emptying rate of the waste pipe and hence how long it would take to empty a full cistern on its own.

Given Data / Assumptions:

Pipe A alone fills the cistern in 10 minutes.

Pipe B alone fills the cistern in 15 minutes.

The person initially opens both A and B together, but the waste pipe is also open.

If only A and B were open, the cistern would be full in 6 minutes.

At that 6 minute mark, he discovers the waste pipe is open, closes it, and the cistern becomes full 3 minutes later.

Concept / Approach:
We first find the combined filling rate of A and B. Since the person discovers the problem at the time when A and B alone should have filled the cistern (6 minutes), we know that up to that time all three pipes have been operating. We compute the fraction filled after 6 minutes with all three pipes, then deduce the waste pipe rate from the fact that the remaining volume is filled in 3 minutes by A and B only.

Step-by-Step Solution:
Step 1: Rate of A = 1 / 10 cistern per minute.Step 2: Rate of B = 1 / 15 cistern per minute.Step 3: Combined rate of A and B = 1 / 10 + 1 / 15 = (3 + 2) / 30 = 1 / 6 cistern per minute.Step 4: If only A and B were working, they would fill the cistern in 6 minutes.Step 5: Let waste pipe rate be W cistern per minute (emptying), so net rate with all three open is 1 / 6 - W.Step 6: In the first 6 minutes, volume filled = 6 * (1 / 6 - W) = 1 - 6W.Step 7: After the waste pipe is closed, only A and B fill the cistern. In the next 3 minutes, volume added = 3 * (1 / 6) = 1 / 2.Step 8: This 1 / 2 must be exactly the remaining fraction of the cistern, so remaining fraction after 6 minutes = 1 / 2.Step 9: Therefore, 1 - 6W = 1 / 2, so 6W = 1 / 2 and W = 1 / 12 cistern per minute.Step 10: Time taken by waste pipe alone to empty a full cistern = 1 / W = 12 minutes.

Verification / Alternative check:
Check the partial filling: with W = 1 / 12, net rate during the first 6 minutes = 1 / 6 - 1 / 12 = 1 / 12 cistern per minute. In 6 minutes that fills 6 * (1 / 12) = 1 / 2 of the cistern. Then A and B alone fill the remaining half in 3 minutes at rate 1 / 6, since 3 * (1 / 6) = 1 / 2. The total time is 9 minutes, consistent with the story, and the calculation of W is confirmed.

Why Other Options Are Wrong:
10 minutes, 8 minutes and 15 minutes correspond to waste pipe rates that do not result in exactly half the tank remaining after 6 minutes and thus do not satisfy the information about the last 3 minutes of filling.

Common Pitfalls:
Some students incorrectly assume the first phase lasts an unknown time rather than the 6 minutes determined by the A and B filling time. Others mix up net rates, adding instead of subtracting the waste pipe rate. Keeping track of volumes and writing equations carefully for each phase prevents such errors.

Final Answer:
The waste pipe alone can empty a full cistern in 12 minutes.