Pipes and Cisterns: Time-Work Problems Solved Easily

Pipes and cisterns problems are time-and-work problems wearing a different costume. A pipe that fills a tank in n hours does 1/n of the filling in one hour, exactly like a worker finishing a job. The twist is the outlet pipe: it does negative work, emptying the tank instead of filling it.

This model matters wherever tanks are involved — water supply systems, fuel storage, swimming pools and chemical plants all balance inflow against outflow. Knowing the net rate tells an operator whether the tank is filling, draining, or holding steady.

Key Concepts

• Inlet pipe: fills the tank; its hourly share is +1/n for a pipe that fills it alone in n hours.
• Outlet pipe (leak): empties the tank; its hourly share is −1/m for a pipe that empties it alone in m hours.
• Net rate: add all hourly shares, keeping the minus sign on outlets. Positive means filling, negative means draining.
• Cistern: just an old word for a water tank.
• Time to fill or empty: 1 ÷ |net hourly share|.

Formulas

• One hour's work of a pipe = 1 ÷ (hours it takes alone). Inlets positive, outlets negative.
• Net hourly rate = (1/a + 1/b + …) − (1/c + 1/d + …) for inlets a, b and outlets c, d.
• Time to fill/empty = 1 ÷ net rate (fill if positive, empty if negative). Assumes constant flow and all stated pipes open the whole time.
• Leak problems: if a tank fills in F hours with the leak and would fill in f hours without it, the leak alone empties a full tank in 1 ÷ (1/f − 1/F) hours.

Worked Example

Pipe A fills a tank in 6 hours. Pipe B empties the full tank in 8 hours. With both open, how long to fill an empty tank?

1. A's hourly share = +1/6; B's hourly share = −1/8.
2. Net rate = 1/6 − 1/8 = 4/24 − 3/24 = 1/24 of the tank per hour.
3. Time to fill = 1 ÷ (1/24) = 24 hours.

The tank fills in 24 hours. Sense-check: the net inflow (1/24) is exactly one quarter of A's inflow (1/6), so filling takes 4 times A's solo time: 6 × 4 = 24 hours. The arithmetic and the logic agree.

Common Misconceptions

• "Two pipes filling in 6 and 8 hours take 14 hours together." Same trap as time-and-work: add the hourly fractions (1/6 + 1/8 = 7/24), then flip: 24/7 ≈ 3.43 hours, not 14.
• "An outlet pipe just slows filling; the answer can't be negative." If the outlet is faster than the inlet, the net rate is negative and the tank can never fill — the correct answer is that it drains, not a positive time.
• "The leak formula needs the tank's size in litres." No. Since every share is a fraction of one full tank, the litres cancel out. The tank's capacity is irrelevant unless a question asks about volumes.

Real-World Applications

• Municipal water tanks balance pump inflow against consumer outflow.
• Fuel depots schedule loading while trucks draw fuel out simultaneously.
• Swimming pools run filters (outlet) and refill hoses (inlet) together.
• Chemical plants control reactor levels with matched feed and drain rates.

Pipes and cisterns is time-and-work with a minus sign: inlets add, outlets subtract, and the net hourly fraction decides everything. Flip the net rate to get the time, and watch the sign — a negative net rate means the tank drains.