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Pipes and Cisterns

Competitive Mathematics · Time, Work and Distance

8 Part2 worked examples18 practice questions

Think of the water tank on your roof. A tap fills it in 6 hours, while a small leak at the bottom would empty it in 12. If both run together, will the tank fill, and when? That is the whole chapter. It is the same as work-and-time, except the work is filling or emptying a tank. An inlet adds a part of the tank and an outlet (or a leak) removes a part. When they run together, the net rate is simply the sum of what they add minus what they remove each hour.

The basic rule
1

Inlets, outlets and parts

How much in one hour

A pipe’s one-hour part, the whole time from a part, and a pipe’s diameter.

Part and time

and its reciprocal

A pipe that fills in hours fills of the tank in one hour.

  • Inlet: fills an empty tank. Outlet: empties a full one. Both work on — only the sign differs: filling , emptying .
  • A pipe that fills in one hour fills the whole tank in hours. The reciprocal of the part is the time.
  • For a fraction of the tank the time is (time for the whole).
  • Effect of diameter: the flow is proportional to the area of the opening, and area . Double the diameter ⟹ four times the flow ⟹ a quarter of the time.
Example 1 · Basic
An inlet fills of a tank in one hour. How long will it take to fill the whole empty tank?
Solution, step by step
  1. 1The reciprocal of the part is the time.
  2. 2Time hours
Answer 9 hours
Example 2 · Standard
An outlet empties a full tank in 10 hours. In how many hours will it empty of the tank?
Solution, step by step
  1. 1For of the tank the time is .
  2. 2 hours
Answer 6 hours
BasicTry it yourself first
A tap fills a tank in 6 hours. What part of the tank does it fill in 1 hour?
BasicTry it yourself first
A tap fills a tank in 12 minutes. What part of the tank does it fill in 1 minute?
StandardTry it yourself first
A tap fills a tank in 15 minutes. What part of the tank does it fill in 1 minute?
StandardTry it yourself first
A tap fills a tank in 25 hours. What part of the tank does it fill in 1 hour?
BasicTry it yourself first
A pipe fills of a tank in 1 hour. How long will it take to fill the whole empty tank?
BasicTry it yourself first
A pipe fills of a tank in 1 minute. How long will it take to fill the whole empty tank?
StandardTry it yourself first
A pipe fills of a tank in 1 minute. How long will it take to fill the whole empty tank?
StandardTry it yourself first
A pipe fills of a tank in 1 hour. How long will it take to fill the whole empty tank?
AdvancedTry it yourself first
A pipe fills of a tank in 1 minute. How long will it take to fill the whole empty tank?
BasicTry it yourself first
An outlet empties a full tank in 6 hours. How long will it take to empty of the tank?
StandardTry it yourself first
An outlet empties a full tank in 20 minutes. How long will it take to empty of the tank?
StandardTry it yourself first
An outlet empties a full tank in 15 hours. How long will it take to empty of the tank?
AdvancedTry it yourself first
An outlet empties a full tank in 24 minutes. How long will it take to empty of the tank?
AdvancedTry it yourself first
An outlet empties a full tank in 18 hours. How long will it take to empty of the tank?
BasicTry it yourself first
A tap of diameter empties a tank in 36 minutes. How long will a tap of diameter 2 take for the same tank? (Take the flow as proportional to the area.)
StandardTry it yourself first
A tap of diameter empties a tank in 48 minutes. How long will a tap of diameter 3 take for the same tank? (Take the flow as proportional to the area.)
StandardTry it yourself first
A tap of diameter empties a tank in 80 minutes. How long will a tap of diameter 2 take for the same tank? (Take the flow as proportional to the area.)
AdvancedTry it yourself first
A tap of diameter empties a tank in 90 minutes. How long will a tap of diameter 3 take for the same tank? (Take the flow as proportional to the area.)
2

Opening together: the net rate

Add and subtract

Time to fill or empty the tank when two or three pipes run together.

Two pipes

Both fill, or one fills and one empties

Two inlets: add the rates. One inlet, one outlet: subtract.

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Three pipes

Three rates in one line

Two inlets and an outlet, or three inlets.

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3

Leaks and the hidden pipe

The unknown rate from a difference

The leak’s time, the third pipe’s time, and “how much faster”.

Leaks

The gap between two rates

The leak’s rate from the filling time with and without the leak.

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One pipe unknown

Total rate − known rates

When the total and two pipes are known find the third, plus and “ times faster” questions.

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4

Staged questions

Add the filling of each stage

A pipe closes or opens midway, pipes open in turns or at different times, or only a percent of the tank is to be filled.

Closing or opening midway

Make stages

Add net rate time stage by stage until the sum reaches .

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In turns and at different times

Cycles and the clock

Pipes open for one minute each in turns, or are opened at different hours.

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5

Capacity, percents and counts

Litres and the number of pipes

Capacity from a rate in litres, the number of inlets and outlets, and a percent difference in tap capacity.

Capacity in litres

Rate difference = litres per minute

When a pipe’s rate is given in litres, equate the difference in the tank’s rates with it.

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6

Traps and fast tricks

Slips · tricks

Where marks go in pipes-and-cisterns questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

LCM units and checks

Habits for speed.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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