Chapter 8 of 8

All 8 chapters
  1. 1Prisms and cylinders, readunderstood
  2. 2Pyramids and cones, readunderstood
  3. 3Spheres, readunderstood
  4. 4Joined and cut solids, readunderstood
  5. 5Changing units of area and volume, readunderstood
  6. 6Speed, distance and time, readunderstood
  7. 7Density and pressure, readunderstood
  8. 8Rates of flow and rate graphs, readunderstood

M4.5.8 Chapter 8

Rates of flow and rate graphs

A tap fills a tank at 12 litres a minute. The tank holds 120 litres. How long does it take?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Find how long a tank takes to fill at 12 litres a minute
  • Read the gradient of a straight line graph as a rate

A rate of flow says how much liquid moves in each unit of time. A tap that gives 12 litres every minute has a rate of 12 litres per minute, written 12 litres/min. It is a compound unit, like km/h.

The rule is the same shape as the rule for speed. Rate = volume ÷ time. It can be turned round: volume = rate × time, and time = volume ÷ rate.

A table with the head To find, Use. Rows: rate, volume ÷ time; volume, rate × time; time, volume ÷ rate.To findUseratevolume ÷ timevolumerate × timetimevolume ÷ rate
One rule, three ways to use it.

The picture shows a tank with a tap above it. The tank is partly full, and more water is flowing in. The tank is 60 cm long, 40 cm wide and 50 cm deep.

A tank in the shape of a box, partly full of water. The length along the front is labelled 60 cm. The width along the right is labelled 40 cm. The depth on the right is labelled 50 cm. An arrow above the tank is labelled 12 litres a minute.12 litres a minute60 cm40 cm50 cm
A tank that fills at 12 litres a minute.

Worked example

Hana has the tank in the picture. It is 60 cm long, 40 cm wide and 50 cm deep. It starts empty, and the picture shows it part way through filling. A tap fills it at 12 litres per minute. How long does it take to fill the whole tank?

  1. The volume of the tank is 60 × 40 × 50 = 120 000 cm³.
  2. Change to litres. One litre is 1 000 cm³, so 120 000 ÷ 1 000 = 120 litres.
  3. Time = volume ÷ rate = 120 ÷ 12.
  4. 120 ÷ 12 = 10 minutes.

Answer: 10 minutes

Now draw a graph. Put time in minutes across, and volume in litres up. When the tap runs steadily, each minute adds the same volume. So the points lie on a straight line through (0, 0).

See it three ways

With things

A tap fills a jug with 2 litres every minute. Pour each jugful into a tank and write down how much water the tank holds after each minute.

As a picture

A graph with the x axis time (minutes) from 0 to 5 and the y axis volume (litres) from 0 to 10. Dots at (0, 0), (1, 2), (2, 4), (3, 6), (4, 8) and (5, 10) lie on a straight line named 2 litres/min.2 litres/min123452468100time (minutes)volume (litres)

In symbols

Volume = rate × time. After 5 minutes the tank holds 2 × 5 = 10 litres.

The steepness of the line is its gradient. The gradient is how much the volume goes up for each 1 minute across. That is the rate of flow. Gradient = change in volume ÷ change in time.

A graph with the x axis time (minutes) from 0 to 12 and the y axis volume (litres) from 0 to 80. A straight line goes through (0, 0), (2, 16) and (6, 48). A dashed horizontal line from (2, 16) to (6, 16) is named 4 minutes. A dashed upright line from (6, 16) to (6, 48) is named 32 litres.4 minutes32 litres24681012204060800time (minutes)volume (litres)
Two points on the line. Across 4 minutes the volume goes up 32 litres.

Worked example

The graph shows water running steadily into a tank. The line goes through (2, 16) and (6, 48). Find the rate of flow.

  1. Choose two points on the line that are far apart: (2, 16) and (6, 48).
  2. The volume goes up by 48 − 16 = 32 litres.
  3. The time goes up by 6 − 2 = 4 minutes.
  4. Gradient = 32 ÷ 4 = 8 litres per minute.

Answer: 8 litres per minute

A graph can have more than one part. A flat part means that the volume is not changing, so the rate is 0. A line that goes down means water is leaving the tank.

Hana's tank holds 120 litres. The graph shows it filled, left, and then emptied.

A graph with the x axis time (minutes) from 0 to 14 and the y axis volume (litres) from 0 to 80. A line rises from (0, 0) to (5, 60), named filling. A flat line goes from (5, 60) to (8, 60), named tap off. A line falls from (8, 60) to (14, 0), named emptying.fillingtap offemptying2468101214204060800time (minutes)volume (litres)
A tank fills, rests, and empties.

Worked example

The graph shows Hana's tank, which holds 120 litres. It is filled, left, and then emptied. Find the rate of flow when it fills and when it empties.

  1. Filling: the tank has 60 litres after 5 minutes. The rate is 60 ÷ 5 = 12 litres per minute.
  2. The flat part, from 5 to 8 minutes, has a gradient of 0. No water flows.
  3. Emptying: the water goes from 60 litres to 0 between 8 and 14 minutes. That takes 14 − 8 = 6 minutes.
  4. Water leaves at 60 ÷ 6 = 10 litres per minute.

Answer: 12 litres per minute in, 10 litres per minute out

Try it

Time a tap

You need: a tap, a 1 litre bottle or jug, a clock that shows seconds.

  1. Open the tap fully and hold the bottle under it.
  2. Time how many seconds it takes to fill the bottle to the 1 litre mark.
  3. Work out how many litres would flow in 60 seconds: 60 ÷ your time.

What you will see: The table shows the seconds taken to fill 1 litre, and the rate in litres per minute. If the bottle fills in 10 seconds, the tap gives 6 litres in a minute. If it fills in 5 seconds, it gives 12 litres a minute.

If you cannot try it, this is what happens.
A table with the head Seconds, Litres/min. Rows: 10, 6; 5, 12.SecondsLitres/min106512

A wrong idea many people have

A flat part of a volume time graph means that the tank is full.

Why it feels right: The line stops going up, and a full tank cannot take any more water.

The test that breaks it: Look at the graph of the tank. The line is flat at 60 litres from 5 to 8 minutes, but the tank holds 120 litres. It is half full, and the tap is turned off.

What is true: A flat part means the volume is not changing. The rate is 0. It does not tell you whether the tank is full or not.

A wrong idea many people have

The time to fill a tank is the volume times the rate.

Why it feels right: When we find a distance we multiply, so multiplying looks like the right thing to do.

The test that breaks it: Volume 120 litres times rate 12 litres per minute is 1 440. That would be a whole day for a small tank. Each minute adds 12 litres, so you count how many 12s fit in 120. That is a division, 10 minutes.

What is true: Time = volume ÷ rate. Multiply only when you want a volume: volume = rate × time.

Key words

rate of flow
how much liquid moves in each unit of time, such as litres per minute.
gradient
how steep a line is: the change up divided by the change across.

Back to your guess

The answer is A: 10 minutes, because 120 ÷ 12 = 10.

Each minute adds 12 litres, so you count how many lots of 12 litres make 120. That is 120 ÷ 12 = 10 minutes. Taking one number from the other has no meaning, and 1 440 minutes would be a whole day.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. A tank is 50 cm long, 40 cm wide and 30 cm deep. It starts empty, and the picture shows it part way through filling. A tap fills it at 12 litres a minute. How long does it take to fill the whole tank?

    A tank in the shape of a box, with a little water at the bottom. The length along the front is labelled 50 cm. The width along the right is labelled 40 cm. The depth on the right is labelled 30 cm. An arrow above the tank is labelled 12 litres a minute.12 litres a minute50 cm40 cm30 cm
    Tap your answer.

    1. Not this one. You divided the volume in cm³, 60 000, by 12. The rate is in litres, so change the volume into litres first, 60 000 ÷ 1 000 = 60. Go to M4.5.5. Try another.

    2. Not this one. You divided the rate by the volume. Time = volume ÷ rate, with the volume on the top. Go to M4.5.8. Try another.

    3. Yes, that is right. The volume is 50 × 40 × 30 = 60 000 cm³, which is 60 litres. Time = 60 ÷ 12 = 5 minutes.

    See the answer

    5 minutes

    The volume is 50 × 40 × 30 = 60 000 cm³, which is 60 litres. Time = 60 ÷ 12 = 5 minutes.

    • If you chose A: You divided the volume in cm³, 60 000, by 12. The rate is in litres, so change the volume into litres first, 60 000 ÷ 1 000 = 60. Go to M4.5.5.
    • If you chose B: You divided the rate by the volume. Time = volume ÷ rate, with the volume on the top. Go to M4.5.8.

  2. Water flows from a hose at 8 litres per minute for 15 minutes. How many litres flow?

    Tap your answer.

    1. Yes, that is right. Volume = rate × time = 8 × 15 = 120 litres.

    2. Not this one. You divided the time by the rate, 15 ÷ 8. To find a volume, multiply the rate by the time. Go to M4.5.8. Try another.

    3. Not this one. You added the two numbers. In each of 15 minutes the hose gives 8 litres, so multiply 8 by 15. Go to M4.5.8. Try another.

    See the answer

    120 litres

    Volume = rate × time = 8 × 15 = 120 litres.

    • If you chose B: You divided the time by the rate, 15 ÷ 8. To find a volume, multiply the rate by the time. Go to M4.5.8.
    • If you chose C: You added the two numbers. In each of 15 minutes the hose gives 8 litres, so multiply 8 by 15. Go to M4.5.8.

  3. The graph shows water running steadily into a tank. The line goes through (0, 0) and (5, 60). What is the rate of flow?

    A graph with the x axis time (minutes) from 0 to 6 and the y axis volume (litres) from 0 to 80. A straight line goes from (0, 0) up to a dot named A (5, 60), with dashed lines down to the x axis and across to the y axis.A (5, 60)123456204060800time (minutes)volume (litres)
    Tap your answer.

    1. Not this one. You read the height of point A. That is the volume after 5 minutes. The rate is the volume divided by the time. Go to M4.5.8. Try another.

    2. Yes, that is right. The line starts at (0, 0), so the gradient is 60 ÷ 5 = 12 litres per minute.

    3. Not this one. You read the time of point A. The rate is the volume divided by the time, 60 ÷ 5. Go to M4.5.8. Try another.

    See the answer

    12 litres per minute

    The line starts at (0, 0), so the gradient is 60 ÷ 5 = 12 litres per minute.

    • If you chose A: You read the height of point A. That is the volume after 5 minutes. The rate is the volume divided by the time. Go to M4.5.8.
    • If you chose C: You read the time of point A. The rate is the volume divided by the time, 60 ÷ 5. Go to M4.5.8.

  4. Look at the graph of Hana's tank, which holds 120 litres. What does the flat part, from 5 to 8 minutes, tell you?

    A graph with the x axis time (minutes) from 0 to 14 and the y axis volume (litres) from 0 to 80. A line rises from (0, 0) to (5, 60), named filling. A flat line goes from (5, 60) to (8, 60), named tap off. A line falls from (8, 60) to (14, 0), named emptying.fillingtap offemptying2468101214204060800time (minutes)volume (litres)
    Tap your answer.

    1. Not this one. The tank holds 120 litres, and the flat part is at 60 litres. A flat part shows that the volume is not changing, not that the tank is full. Go to M4.5.8. Try another.

    2. Not this one. If water flowed in, the line would rise. The line is flat, so the volume stays the same and the rate is 0. Go to M4.5.8. Try another.

    3. Yes, that is right. The gradient of the flat part is 0. The volume stays at 60 litres, so no water goes in and none goes out.

    See the answer

    No water flows in or out.

    The gradient of the flat part is 0. The volume stays at 60 litres, so no water goes in and none goes out.

    • If you chose A: The tank holds 120 litres, and the flat part is at 60 litres. A flat part shows that the volume is not changing, not that the tank is full. Go to M4.5.8.
    • If you chose B: If water flowed in, the line would rise. The line is flat, so the volume stays the same and the rate is 0. Go to M4.5.8.

  5. A tank already holds 10 litres. Water then runs in steadily. The graph of volume against time is a straight line through (2, 26) and (6, 58). What is the rate of flow?

    A graph with the x axis time (minutes) from 0 to 8 and the y axis volume (litres) from 0 to 80. A straight line starts at the dot (0, 10) on the y axis. It goes through the dots named A (2, 26) and B (6, 58).A (2, 26)B (6, 58)12345678204060800time (minutes)volume (litres)
    Tap your answer.

    1. Yes, that is right. The volume goes up by 58 − 26 = 32 litres while the time goes up by 6 − 2 = 4 minutes. The gradient is 32 ÷ 4 = 8 litres per minute.

    2. Not this one. You divided the volume at B, 58, by its time, 6. The line does not start at 0 litres, so use the change in volume and the change in time. Go to M4.5.8. Try another.

    3. Not this one. You found the change in volume, 58 − 26 = 32. That is the rise. Divide it by the change in time, 6 − 2 = 4. Go to M4.5.8. Try another.

    See the answer

    8 litres per minute

    The volume goes up by 58 − 26 = 32 litres while the time goes up by 6 − 2 = 4 minutes. The gradient is 32 ÷ 4 = 8 litres per minute.

    • If you chose B: You divided the volume at B, 58, by its time, 6. The line does not start at 0 litres, so use the change in volume and the change in time. Go to M4.5.8.
    • If you chose C: You found the change in volume, 58 − 26 = 32. That is the rise. Divide it by the change in time, 6 − 2 = 4. Go to M4.5.8.

  6. From an earlier chapter

    A block has a mass of 360 g and a volume of 120 cm³. What is its density, in g/cm³?

    See the answer

    3 g/cm³

    Density = mass ÷ volume = 360 ÷ 120 = 3 g/cm³.

  7. A tank holds 180 litres. A tap fills it from empty at 15 litres per minute. How many minutes does it take?

    See the answer

    12 minutes

    Time = volume ÷ rate = 180 ÷ 15 = 12 minutes.

  8. Water flows from a pipe at 18 litres per minute for 25 minutes. How many litres flow?

    See the answer

    450 litres

    Volume = rate × time = 18 × 25 = 450 litres.

  9. A straight line graph of volume against time goes through (2, 14) and (6, 38). What is the rate of flow, in litres per minute?

    See the answer

    6 litres per minute

    The volume goes up by 38 − 14 = 24 litres while the time goes up by 6 − 2 = 4 minutes. The gradient is 24 ÷ 4 = 6 litres per minute.

Keep it

Copy this into your exercise book.

Rate = volume ÷ time. Volume = rate × time. Time = volume ÷ rate. On a volume time graph the gradient is the rate. Flat means no flow. Down means water leaves.

Draw: A volume time graph with a rising line, a flat part and a falling line, each part labelled.

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