Pipes and cisterns

The essentials

Pipes and cisterns problems are work problems in disguise. An inlet pipe fills a tank and an outlet pipe (or leak) empties it. If a pipe can fill a tank alone in n hours, it fills 1/n of the tank in one hour. If an outlet can empty a full tank in m hours, it empties 1/m per hour, which we count as −1/m.

When several pipes work together, their hourly rates add. If the net rate is r (a fraction of the tank per hour), the time to fill the tank is 1/r hours.

Key points

• Two inlets of times a and b together fill in ab/(a + b) hours.
• An inlet (a hours) and an outlet (b hours, b > a) together fill in ab/(b − a) hours.
• Example: inlets of 12, 18 and 36 hours give 3/36 + 2/36 + 1/36 = 6/36 = 1/6 per hour, so 6 hours together.
• When pipes are opened or closed at different times, find the part done in each stage and add the parts to 1 (the full tank).
• Using the LCM of the times as the tank capacity (in "units") can make arithmetic easier.

Exam tips

• Always convert to "part per hour" (or per minute) first.
• Keep track of which pipes are open in each time interval.

Common mistakes

• Adding times instead of rates.
• Forgetting the negative sign for an outlet or leak.