Concept
Pipes and cisterns is time and work with a twist: an inlet does positive work and an outlet or leak does negative work. Use the LCM (capacity units) method exactly as in time and work. Target: 45 seconds.
Key formulas
| Situation | Time to fill |
|---|---|
| Two inlets a and b hours | ab/(a + b) |
| Inlet a, outlet b (b > a) | ab/(b - a) |
| Pipe fills in x, with leak in y (y > x) | leak alone empties in xy/(y - x) |
| Three pipes | capacity / (sum of rates with signs) |
LCM method
A fills in 12 h, B in 15 h, C empties in 20 h. Capacity = 60 units. A = +5, B = +4, C = -3. Net = 6 units/h, time = 10 hours.
Typical twists
- Pipe closed after t minutes: the other pipe works the full time; subtract its work to find what the closed pipe did.
- Alternate opening: work in cycles (e.g. A for 1 hour, then B for 1 hour). Stop the cycles just before the tank is full and finish with the pipe whose turn it is.
- Capacity in litres: find the outlet's time to empty, then capacity = time x litres per minute.
- Leak discovered later: split the filling into before-leak and after-leak parts.
Worked example
A pipe fills a tank in 6 hours and an outlet empties it in 9 hours. Both are opened together. Time to fill?
Capacity = 18 units. Inlet = +3, outlet = -2, net = 1 unit/h. Time = 18 hours.
Worked example 2
A fills in 20 min, B fills in 30 min, C empties in 15 min. All opened together. Time to fill?
Capacity = 60 units: +3 +2 -4 = 1 unit/min, so 60 minutes.
Worked example 3 (leak)
A pipe fills a tank in 10 hours; with a leak it takes 12 hours. Leak alone empties the full tank in 10 x 12/(12 - 10) = 60 hours.
Common traps
- Forgetting the minus sign for an outlet.
- If the outlet is faster than the inlet, the tank never fills (it empties instead).
- Mixing minutes and hours in the same problem.
- "Tank is full and both are opened" means the question is about emptying.