A bathtub can be filled by the cold water pipe in 20 minutes and by the hot water pipe in 30 minutes. A person leaves the bathroom after turning on both the pipes simultaneously and returns at the moment when the bath should be full. Finding, however, that the waste pipe has been open, he now closes it. In 6 minutes more the bathtub is full. In what time would the waste pipe empty it?

Let waste pipe empty in x minutes (rate = 1/x per minute). Cold fills at 1/20 per minute, hot at 1/30 per minute. Combined fill rate = 1/20 + 1/30 - 1/x = 5/60 - 1/x. The tub should be full in 1/(1/20+1/30) = 12 minutes with all three pipes working normally. But when waste is open, it takes 12 + 6 = 18 minutes to fill. In those 18 minutes, actual fill occurred at rate (1/20 + 1/30) for full time, minus waste effect for 12 minutes. Actually: (1/20 + 1/30) × 18 - (1/x) × 12 = 1 (full tub). So 3/60 × 18 - 12/x = 1, giving 9/10 - 12/x = 1, so -12/x = 1/10, x = -120. That's wrong. Let me reconsider: The person expected tub to be full in 12 min (1/20+1/30=5/60=1/12). But when he returns, waste pipe has been open the whole time. In those 12 minutes, some water drained. Then he closes waste and it takes 6 more minutes to fill. In first 12 min: (1/20+1/30-1/x) × 12 = some fraction filled. Then in next 6 min: (1/20+1/30) × 6 = 6/12 = 1/2 fills remaining half. So first 12 min filled 1/2: (5/60-1/x) × 12 = 1/2. So 1 - 12/x = 1/2, giving 12/x = 1/2, x = 24 minutes. ✓