A cistern is normally filled in 6 hours but takes 6 hours longer to fill…

Concept: For an inlet that fills a tank at a fixed rate, a leak in the bottom reduces the effective (net) filling rate. The leak's own emptying rate is simply the normal filling rate minus this net rate -- a direct application of the work-rate principle, in which the leak contributes a negative rate to the combination.

Method:

1. Normal filling rate = 1/6 of the cistern per hour (the cistern fills in 6 hours without the leak).

2. With the leak, the cistern takes 6 hours longer to fill, i.e. 6 + 6 = 12 hours; so the net filling rate with the leak present is 1/12 of the cistern per hour.

3. The leak's own emptying rate = normal rate − net rate = 1/6 − 1/12 = 1/12 of the cistern per hour.

4. At a rate of 1/12 per hour, the leak alone empties a full cistern in 1 ÷ (1/12) = 12 hours.

Cross-check: If the leak alone empties the full cistern in 12 hours, combining it with the inlet gives a net rate of 1/6 − 1/12 = 1/12, i.e. a fill time of 12 hours = 6 + 6 — exactly matching the condition stated in the question.

Answer: The leak alone empties the full cistern in 12 hours.