Pipes and Cisterns
Pipes are Time and Work with signs: filling pipes add, emptying pipes (and leaks) subtract.
Why this topic matters
A pipe that fills a tank is a worker. A pipe that empties it is a worker doing negative work. That is the only difference from time and work.
Same method as time and work: total = LCM, inlets add, outlets subtract.
Inlets add, outlets subtract
Let the tank's capacity be the LCM of the times. Each filling pipe adds its units per hour, each emptying pipe takes away its units per hour.
If the net rate is negative, the tank never fills; it empties.
- Tank = LCM(12, 15, 20) = 60 units.
- A +5, B +4, C -3 per hour.Emptying counts as negative.
- Net +6 per hour.
- 60 / 6 = 10 hours.
A leak, and a pipe closed partway
A leak: a tank normally fills in 6 hours but takes 8 with a leak. Tank = 24 units: pipe +4, together +3, so the leak is -1 an hour and empties a full tank in 24 hours.
Pipe closed after some time: work out what was filled while it was open, then finish the rest with the pipes still running.
- Tank = LCM(6, 8) = 24.
- Pipe +4/h, with leak +3/h, so leak = -1/h.The difference is the leak.
- 24 hours.
- Tank 60. A 3/min, B 2/min.
- 5 minutes together: 25 units.
- Left 35, B alone: 17.5 minutes.
Formula summary
Shortcuts and tricks
Leak time
Use it when: 'Fills in 6 h, takes 8 h because of a leak'.
- Leak rate = 1/6 - 1/8 = 1/24.
- 24 hours.
Before you move on, you should be able to...
- treat emptying pipes as negative rates
- find a leak's rate from the slower filling time
- handle a pipe closed partway
Practice questions (7)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy A fills in 20 min, B empties in 30 min. Both open?
- 1/20 - 1/30 = 1/60.
Answer: 60 min
easy A fills in 12 h, B in 15 h, C empties in 20 h. All open?
- (5 + 4 - 3)/60 = 1/10.
Answer: 10 h
medium A and B fill a tank in 20 and 30 min. Both are opened; after how many minutes must B be closed so the tank fills in exactly 15 min?
- A does 15/20 = 3/4.
- B must do 1/4 → 30/4 = 7.5 min.
Answer: 7.5 min
hard A fills in 6 h, B in 8 h, opened alternately for 1 hour each, A first. Time to fill?
- 24 units. A 4, B 3 → 7 per 2 h.
- 6 h → 21. A needs 3 of 4 → 3/4 h.
Answer: 6 3/4 h
hard Two pipes fill in 10 and 12 h; an outlet empties in 20 h. All open, but the outlet is closed after 2 h. Total time? (hours)
- First 2 h: 2 × (6 + 5 - 3)/60 = 16/60.
- Left 44/60 at 11/60 per hour = 4 h.
- Total 6 h.
Answer: 6 h
hard Two inlets fill a tank in 12 and 15 min. With an outlet also open, it fills in 20 min. Outlet alone empties it in?
- 1/12 + 1/15 - 1/x = 1/20.
- 1/x = 9/60 - 3/60 = 1/10.
Answer: 10 min
hard One pipe fills a tank in 4 h. When it is half full, 3 more identical pipes are opened. Total time?
- First half: 2 h.
- Then 4 pipes: the other half takes 1/2 h.
Answer: 2 1/2 h
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up. |
| 30 min | Inlets and outlets with the tank picture. |
| 25 min | Leaks. |
| 20 min | Pipe closed after a while. |
| 30 min | Practice set. |
| 5 min | Recap. |