Pipes & Cisterns • Topic 1 of 3

Inlet & Outlet Pipes

Start by labelling each pipe with a signed rate: an inlet that fills the tank in x hours does +1/x of the tank per hour, an outlet that empties it in y hours does −1/y per hour. The fractions are ugly, so use the LCM trick — set the tank capacity to the LCM of all the times and every rate becomes a clean integer. If an inlet fills in 6 h and an outlet empties in 8 h, take capacity = 24: the inlet does +4 units/h, the outlet −3 units/h, net +1, so the tank fills in 24 hours. The most common CAT twist is an outlet quietly left open while you fill: subtract its rate. If the outlet is faster than the inlet (net negative), the tank can never fill from empty — a favourite trap. Always check the sign of the net rate before computing time.

✅ Solved examples

1. Pipe A fills a tank in 6 hours; outlet B empties it in 8 hours. If both are open, how long to fill?
Capacity = LCM(6,8) = 24. A = +4/h, B = −3/h, net = +1/h. Time = 24/1 = 24 hours.
2. An inlet fills a cistern in 4 hours; an outlet empties the full cistern in 6 hours. Both open — fill time?
Capacity = LCM(4,6) = 12. Inlet +3, outlet −2, net +1. Time = 12/1 = 12 hours.
3. Pipe A fills in 10 h, outlet B empties in 15 h. Tank is empty and both run. Fill time?
Capacity = LCM(10,15) = 30. A = +3, B = −2, net +1. Time = 30 hours.
4. An inlet fills in 5 h; an outlet empties in 4 h. Both open on an empty tank — what happens?
Capacity = 20. Inlet +4, outlet −5, net −1. Net is negative, so the tank never fills; it stays empty (and would drain if it had water).

✏️ Practice — try these, take hints as needed

1. Inlet fills in 12 h, outlet empties in 18 h. Both open from empty — fill time?
Capacity = LCM(12,18) = 36.
Rates +3 and −2.
Net +1, time = 36/1.
36 hours
2. Inlet fills in 3 h, outlet empties in 5 h. Both open — fill time?
Capacity = 15.
Inlet +5, outlet −3.
Net +2, 15/2.
7.5 hours
3. Pipe A fills in 8 h, outlet B empties in 12 h. Both open — fill time?
Capacity = 24.
+3 and −2.
Net +1.
24 hours
4. An inlet fills in 9 h, an outlet empties in 9 h. Both open — what happens?
Equal and opposite rates.
+? and −? are the same.
Net = 0.
Tank level never changes (never fills)
5. A full tank has an inlet (fills in 6 h) and outlet (empties in 4 h) both open. Time to empty?
Capacity = 12. Inlet +2, outlet −3.
Net = −1 (emptying).
Time = 12/1.
12 hours

📝 Topic test — 8 questions

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