A cistern stands one-third full. Its inlet alone fills the whole cistern in 24 minutes and its outlet alone empties a full cistern in 40 minutes. Both are opened together. How long before the cistern is full?

Take the cistern as 120 units: the inlet gives 5 a minute, the outlet takes 3, a net gain of 2. One-third is already there, so 80 units are wanted: 80 ÷ 2 = 40 minutes. 60 minutes fills all 120 units at the net rate, forgetting that the cistern was not empty to begin with. 16 minutes uses the 80 units but ignores the outlet (80 ÷ 5). 20 minutes takes the 40 units already in the cistern as the amount still to come (40 ÷ 2) — the one-third is what is there, not what is missing.