Pipes & Cisterns • Topic 3 of 3

Leak Problems

A leak is just an outlet you did not plan for — give it a negative rate and the signed-sum machinery handles it. The signature CAT setup: a pipe that should fill the tank in x hours actually takes x + d hours because of a leak; find how long the leak alone would empty a full tank. Work in rates: fill rate = 1/x, observed net = 1/(x+d), so leak rate = 1/x − 1/(x+d) = d / [x(x+d)], giving leak-empties-in = x(x+d)/d hours. Memorise that product form. The other staple: a tank is filled, then a leak is discovered that empties the full tank in L hours — combine +inlet and −leak for the real fill time, or compute how long the stored water lasts. Beware the direction of the wording: "leak empties the full tank in L hours" sets the leak rate; "with the leak the tank fills in t hours" gives the net rate. Mixing these up is the number-one error here.

✅ Solved examples

1. A pipe fills a tank in 6 hours, but with a leak it takes 8 hours. How long would the leak alone take to empty a full tank?
Leak-empties-in = x(x+d)/d with x = 6, d = 2 ⇒ 6×8/2 = 24 hours. (Check: 1/6 − 1/8 = 1/24.)
2. An inlet fills in 4 h; a leak empties the full tank in 12 h. With both, fill time?
Capacity = LCM(4,12) = 12. Inlet +3, leak −1, net +2. Time = 12/2 = 6 hours.
3. A cistern fills in 10 h normally; a leak makes it 15 h. Leak alone empties the full cistern in?
x = 10, x+d = 15, d = 5 ⇒ 10×15/5 = 30 hours. (1/10 − 1/15 = 1/30.)
4. A tank is full. Two inlets (each fill in 8 h) and a leak (empties full tank in 6 h) are all open. Net rate and time to fill from empty?
Capacity = 24. Inlets +3 each = +6, leak −4, net +2. From empty, time = 24/2 = 12 hours.

✏️ Practice — try these, take hints as needed

1. A pipe fills in 12 h, but with a leak it takes 16 h. Leak alone empties a full tank in?
Use x(x+d)/d.
x = 12, d = 4.
12×16/4.
48 hours
2. Inlet fills in 5 h; leak empties full tank in 20 h. Both open — fill time?
Capacity = 20.
Inlet +4, leak −1.
Net +3, 20/3.
20/3 hours (6 h 40 min)
3. A pipe fills in 8 h; with a leak it takes 10 h. Leak alone empties the tank in?
x = 8, d = 2.
8×10/2.
Or 1/8 − 1/10.
40 hours
4. A full tank’s leak empties it in 9 h; an inlet fills (empty tank) in 6 h. Both open — fill time?
Capacity = 18. Inlet +3, leak −2.
Net +1.
18/1.
18 hours
5. Two pipes fill in 10 h and 15 h; a leak empties the full tank in 30 h. All open — fill time?
Capacity = 30. Rates +3, +2, −1.
Net = 3 + 2 − 1 = +4.
30/4.
7.5 hours

📝 Topic test — 8 questions

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