Pipes A and B can fill a cistern in 20 minutes and 30 minutes respectively, while pipe C can empty it in 15 minutes. Starting with the cistern empty, the pipes are opened one at a time, for 1 minute each, in the order A, B, C, A, B, C, and so on. After how many minutes will the cistern be full?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 167 minutes
Take the capacity as the LCM of 20, 30 and 15, that is 60 units. Each 3-minute round adds a small net amount, but the cistern becomes full in the middle of a round, as soon as a filling pipe tops it up, before C can empty any of it.
Per minute: A fills \(\frac{60}{20}=3\) units, B fills \(\frac{60}{30}=2\) units, C empties \(\frac{60}{15}=4\) units
Net gain in one round of 3 minutes \(=3+2-4=1\) unit
Within a round the level rises by 3 after A and by 5 after B, so the cistern fills during a round once the level at its start is \(60-5=55\) units
55 units are reached after 55 rounds \(=55\times3=165\) minutes
Minute 166 (A): \(55+3=58\) units; minute 167 (B): \(58+2=60\) units, full
Check one round earlier: from 54 units, A gives 57 and B gives 59, still short, so the cistern is not full before 167 minutes