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Work & motion

Pipes & cisterns: the net flow

Model filling and emptying as signed rates, and decide whether a tank will fill at all.

IntermediateAbout 12 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Fractions and time-and-work rates.

  • Represent inlets and outlets with signed rates
  • Calculate net filling time
  • Recognise zero or negative net flow
02 / BUILD THE CONCEPT

Understand the idea

A tank is another one-unit job

A pipe that fills an empty tank in a hours supplies 1/a tank per hour. An outlet that empties a full tank in b hours removes 1/b tank per hour.

Direction determines the sign

Add inlet rates and subtract outlet rates. Positive net flow fills a tank; zero net flow leaves its level unchanged; negative net flow lowers an existing water level.

Initial level matters

For a partially full tank, divide the unfilled fraction by the positive net rate. The standard exam model assumes constant flows while water is available; real pressure-dependent flow is outside this model.

03 / A DIFFERENT WAY TO SEE IT

Balance what enters and leaves

  1. Inlet +3/12 per hour
  2. Outlet −2/12 per hour
  3. Net +1/12 per hour
An inlet fills in 4 hours and an outlet empties in 6 hours. Together they fill an empty tank in 12 hours.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

Net rate = sum of inlet rates − sum of outlet rates

Time to fill = unfilled fraction ÷ positive net rate.
  1. Convert every stated filling or emptying time into a rate.
  2. Assign positive signs to inlets and negative signs to outlets.
  3. Check the net sign, then divide the required volume fraction by the net rate.
A useful insight

With one inlet of filling time a and one outlet of emptying time b, filling time is ab/(b − a), but only when b > a.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Inlet and outlet

An inlet fills in 4 hours and an outlet empties in 6. Find filling time with both open.

  1. Net rate = 1/4 − 1/6.
  2. Use twelfths: 3/12 − 2/12 = 1/12.
  3. Time = 1 ÷ (1/12).
Answer12 hours
EXAMPLE 02

Two inlets

Pipes fill a tank in 6 and 3 hours separately. Find combined time.

  1. Net rate = 1/6 + 1/3 = 1/2.
  2. The tank is initially empty.
  3. Time = 1/(1/2).
Answer2 hours
EXAMPLE 03

A tank that cannot fill

An inlet fills in 8 hours but an outlet empties in 4. Can they fill an empty tank together?

  1. Net rate = 1/8 − 1/4 = −1/8.
  2. The outlet's capacity exceeds the inlet's.
  3. Under this model, no positive filling time exists.
AnswerThe tank cannot fill
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1An inlet fills in 5 hours. How much fills in 2 hours?

See the explanation

2/5. Rate × time = 1/5 × 2 = 2/5 of a tank.

QUESTION 2Equal inlet and outlet rates are open. What happens to an existing level?

See the explanation

It stays unchanged. Net rate is zero.

QUESTION 3A half-full tank has net filling rate 1/6 tank/hour. Time to fill?

See the explanation

3 hours. The remaining half takes (1/2)/(1/6) = 3 hours.

07 / TAKE THE NEXT STEP

Put your understanding to work

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Pipes and Cisterns Calculator

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Pipes and Cisterns Practice Quiz 1

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Pipes and Cisterns Practice Quiz 2

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Pipes and Cisterns Practice Quiz 3

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