AFCAT Numerical Ability · Time and Work

Pipes and Cisterns

A pipe is a worker whose job is filling a tank: filling pipes add their rates, and emptying pipes and leaks take theirs away.

All Time and Work notes

Why this matters

No AFCAT past-year question is here yet, and most of the page is EASY or MODERATE. The marks go on the sign of a rate: a leak or a waste pipe is taken away, not added, and when the emptying rate is the larger one the tank never fills.

Concept 1 of 4: Filling and emptying together

A pipe that fills a tank in 5 hours puts in

\tfrac 15

of it each hour; one that empties it in 10 hours takes out

\tfrac 1{10}

. Open both, and the tank gains the difference,

\tfrac 1{10}

an hour.

Definition

Net rate = sum of the filling rates minus sum of the emptying rates.
Time to fill an empty tank = 1 ÷ net rate. If the net rate is negative, the tank never fills; a full tank empties instead.
An unknown waste pipe: its rate is the filling rates minus the net rate.
One pipe k times as fast as another: give them rates $k$ and 1 units, as on Combined Work.

Net rate

\dfrac 1T = \dfrac 1a + \dfrac 1b - \dfrac 1c

Worked example

Pipes A and B fill a tank in 10 and 15 hours, and pipe C empties it in 12 hours. If all three are opened together, how long does the empty tank take to fill?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Time and Work · Pipes and Cisterns

A cistern can be filled by pipes

A

and

B

in 6 h and 4 h, respectively. When full, the cistern can be emptied by pipe

C

in 8 h. If all the pipes are turned on, then in how many hours will the cistern be filled?

Take the outlet away

Taps filling in 20 and 30 hours take 12 hours together. With a drain that empties the tank in 15 hours also open, they take 60 hours, not less than 12.

A negative net rate never fills

If the emptying pipe is faster than the filling pipes together, an empty tank stays empty. Look for an option that says the tank is never filled.

Drill 12 more on filling and emptying together

Concept 2 of 4: Finding a leak from the delay

A leak shows itself by slowing the filling. The rate with the leak is less than the rate without it, and the difference is the leak's own rate.

Definition

Leak rate = $\tfrac 1a - \tfrac 1b$ , where $a$ is the time to fill without the leak and $b$ the time with it.
So the leak alone empties a full tank in $\dfrac{ab}{b - a}$ .
A delay given in minutes: change both times to the same unit before you subtract.

Time for the leak

L = \dfrac{ab}{b - a}

Worked example

A pump fills a tank in 4 hours. Because of a leak it takes 5 hours. How long does the leak take to empty the full tank?

Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Time and Work · Pipes and Cisterns

An electric pump can fill a tank in 3 h. Because of a leak in the tank, it took

3\frac{1}{2}

h to fill the tank. If the tank is full, how much time will the leak take to empty it?

The leak is slower than both times

A pump taking 4 hours, and 5 hours with a leak, does not mean the leak empties the tank in 1 hour, or in 9. Its rate is the difference of the two rates, so it takes 20 hours.

Drill 3 more on finding a leak from the delay

Concept 3 of 4: Pipes opened or closed partway

This is the method of Partial Work, Joining and Leaving with pipes in place of people. Work out how full the tank is at the moment a pipe opens or closes, then fill the rest at the new net rate.

Definition

Part filled in a stage = net rate of the open pipes × time.
After a pipe is closed, the rest fills at the rate of the pipes still open.
An unknown pipe: compare the rate of a stage with it and a stage without it, as with one person from a joint rate.

Stages

\sum (\text{net rate} \times \text{time}) = 1

Worked example

Pipes A and B fill a tank in 10 and 20 minutes. Both are opened, and after 4 minutes A is closed. How much longer does the tank take to fill?

Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Time and Work · Pipes and Cisterns

Two pipes

P

and

Q

fill a cistern in 12 and 15 min, respectively. If both are opened together and at the end of 3 min the first is closed, how much longer will the cistern take to fill?

Leave out the closed pipe

After a pipe is shut, only the pipes still open fill the rest. Keeping the closed pipe's rate in the sum makes the answer too short.

Drill 4 more on pipes opened or closed partway

Concept 4 of 4: Capacity, pipes in turn and mixtures

A few pipe questions add a twist: a capacity in litres, pipes opened in turn, a gap between two pipes' times, or the mix of liquids the pipes bring in. Each is an earlier method in a new setting.

Definition

Capacity: when a pipe's share of the tank per minute is known and so is its flow in litres per minute, capacity = litres per minute ÷ share per minute.
Pipes in turn: run cycles as on Working on Alternate Days. With a filling and an emptying pipe in turn, the tank can fill during a filling turn, so check the level at the end of each filling turn.
Mixture: while all pipes are open, the liquid from each is in the ratio of their rates.
A gap in time with a joint time: 'A takes 5 hours more than B, and together they take 6 hours' gives a quadratic. Trying the options is often quicker.

Capacity

\text{capacity} = \dfrac{\text{litres per minute}}{\text{share of the tank per minute}}

Worked example

Pipes A and B fill a tank in 12 and 18 minutes. With a drain also open, which lets out 2 litres a minute, the tank fills in 9 minutes. Find the capacity of the tank.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Time and Work · Pipes and Cisterns

Two pipes

A

and

B

can fill a water tank in 20 and 24 min, respectively and a third pipe

C

can empty it at the rate of 3 gallons per min. If

A

B

and

C

together can fill the tank in 15 min, the capacity (in gallons) of the tank is

The tank can fill partway through the cycles

With a filling pipe and an emptying pipe in turn, the level peaks at the end of each filling turn. Dividing the whole tank by the gain per cycle overshoots, because the last filling turn finishes the job before the emptying turn comes round.

Drill 5 more on capacity, pipes in turn and mixtures

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Filling and emptying together
Net rate
$\dfrac 1T = \dfrac 1a + \dfrac 1b - \dfrac 1c$
Finding a leak from the delay
Time for the leak
$L = \dfrac{ab}{b - a}$
Pipes opened or closed partway
Stages
$\sum (\text{net rate} \times \text{time}) = 1$
Capacity, pipes in turn and mixtures
Capacity
$\text{capacity} = \dfrac{\text{litres per minute}}{\text{share of the tank per minute}}$

Watch out for (5)

Take the outlet away
→ Filling and emptying together
A negative net rate never fills
→ Filling and emptying together
The leak is slower than both times
→ Finding a leak from the delay
Leave out the closed pipe
→ Pipes opened or closed partway
The tank can fill partway through the cycles
→ Capacity, pipes in turn and mixtures

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