Chapter 6 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

M4.5.6 Chapter 6

Speed, distance and time

A car drives 60 km at 60 km per hour, then 40 km at 20 km per hour. What is its average speed?

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

  • Use speed = distance ÷ time with the right units
  • Change km per hour into metres per second
  • Find the average speed for a journey in two parts

Speed tells you how far something goes in one unit of time. A cyclist who goes 15 km in each hour has a speed of 15 km per hour, written 15 km/h.

Km per hour is a compound unit. It is made of two units, a unit of distance (km) and a unit of time (hours), joined by "per". Metres per second (m/s) is another.

The rule is speed = distance ÷ time. It can be turned round: distance = speed × time, and time = distance ÷ speed.

A table with the head To find, Use. Rows: speed, distance ÷ time; distance, speed × time; time, distance ÷ speed.To findUsespeeddistance ÷ timedistancespeed × timetimedistance ÷ speed
One rule, three ways to use it.

The units must fit. If the distance is in km and the time is in hours, the speed is in km/h. If the time is given in minutes, change it to hours first. There are 60 minutes in an hour.

A bar with one part. It is labelled 24 km above it and 90 minutes inside it.24 km90 minutes
A cyclist rides 24 km in 90 minutes.

Worked example

Ravi rides his bicycle 24 km in 90 minutes. What is his speed in km/h?

  1. The time is in minutes. Change it to hours: 90 ÷ 60 = 1.5 hours.
  2. Speed = distance ÷ time = 24 ÷ 1.5.
  3. 24 ÷ 1.5 = 16 km/h.

Answer: 16 km/h

Now change units. In science, speeds are often in metres per second. One km/h is 1 000 m in 3 600 seconds, which is 1 ÷ 3.6 m/s. So to change km/h into m/s, divide by 3.6. To change m/s into km/h, multiply by 3.6.

A table with the head km/h, m/s. Rows: 36, 10; 72, 20; 108, 30.km/hm/s3610722010830
Divide the number in km/h by 3.6 to get m/s.

Worked example

A train goes at 72 km/h. What is its speed in m/s?

  1. One hour is 3 600 seconds, and 1 km is 1 000 m.
  2. So 1 km/h = 1 000 ÷ 3 600 m/s, which is 1 ÷ 3.6 m/s.
  3. To change km/h to m/s, divide by 3.6.
  4. 72 ÷ 3.6 = 20 m/s.

Answer: 20 m/s

Now a journey in two parts. A car drives 60 km in 1 hour, then 40 km in the next 2 hours. The bar below shows the two parts. Each part is as long as its distance.

A bar in two parts. The first part is labelled 60 km above it, 1 hour inside it and 60 km/h below it. The second part is labelled 40 km above it, 2 hours inside it and 20 km/h below it. A bracket under both is labelled 100 km in 3 hours.60 km1 hour60 km/h40 km2 hours20 km/h100 km in 3 hours
A journey in two parts. The speeds are 60 km/h and 20 km/h.

It is tempting to say the average speed is halfway between 60 and 20, which is 40 km/h. That is wrong, because the car spends 2 hours at 20 km/h and only 1 hour at 60 km/h.

The average speed is the one steady speed that covers the same total distance in the same total time. So average speed = total distance ÷ total time.

Worked example

A car drives 60 km in 1 hour, then 40 km in 2 hours. What is its average speed, to 1 decimal place?

  1. Total distance = 60 + 40 = 100 km.
  2. Total time = 1 + 2 = 3 hours.
  3. Average speed = total distance ÷ total time = 100 ÷ 3.
  4. To 1 decimal place, 100 ÷ 3 = 33.3 km/h.
  5. This is not the mean of the two speeds, 60 and 20, which would be 40.

Answer: 33.3 km/h

A distance time graph shows the same journey. Time goes across and distance goes up. The steeper the line, the faster the speed. The dashed line is the average, a steady journey from the start to the end.

A graph with the x axis time (hours) from 0 to 3 and the y axis distance (km) from 0 to 100. A steep line named 60 km/h goes from (0, 0) to (1, 60). A flatter line named 20 km/h goes on from (1, 60) to (3, 100). A dashed straight line named average goes from (0, 0) to (3, 100).60 km/h20 km/haverage123204060801000time (hours)distance (km)
The dashed line is steady. It reaches the same end point in the same time.

Try it

Measure your walking speed

You need: a long tape or string, marked at 20 m, a clock or phone that shows seconds, a friend to time you.

  1. Lay the string flat along a straight path, 20 m long.
  2. Walk along it at your usual pace. Your friend times you in seconds.
  3. Work out speed = distance ÷ time, in m/s.
  4. Multiply by 3.6 to change it to km/h.

What you will see: Most people cover 1 to 1.5 m every second when they walk. That is about 3.6 to 5.4 km/h.

If you cannot try it, this is what happens.
A bar with one part. It is labelled 20 m above it, 15 seconds inside it and about 1.3 m/s below it.20 m15 secondsabout 1.3 m/s

A wrong idea many people have

The average speed of a journey is the mean of the two speeds.

Why it feels right: We find an average of numbers by adding them and dividing, so it looks right to do it with speeds.

The test that breaks it: Drive 60 km at 60 km/h, then 40 km at 20 km/h. The mean of 60 and 20 is 40 km/h. But the car goes 100 km in 3 hours, which is 33.3 km/h. The car spent longer at the slow speed.

What is true: Average speed is the total distance divided by the total time. Use the mean of two speeds only if the times are equal.

A wrong idea many people have

A cyclist who rides 24 km in 90 minutes has a speed of 24 ÷ 90 km/h.

Why it feels right: The rule says divide the distance by the time, and the time is the number 90.

The test that breaks it: 24 ÷ 90 = 0.27, which is far slower than anyone walks. But km/h needs the time in hours. 90 minutes is 1.5 hours, and 24 ÷ 1.5 = 16 km/h.

What is true: Make the units fit first. For km/h the time must be in hours.

Key words

compound unit
a unit made of two units, such as km per hour or metres per second.
average speed
the total distance of a journey divided by the total time.

Back to your guess

The answer is A: About 33 km per hour.

The average speed is the total distance divided by the total time. The car goes 100 km in 3 hours, which is 33.3 km per hour. It spends longer at the slow speed, so the average is pulled towards 20, not halfway.

Check yourself

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

  1. A train goes 150 km in 2.5 hours at a steady speed. What is its speed?

    A bar with one part. It is labelled 150 km above it and 2.5 hours inside it.150 km2.5 hours
    Tap your answer.

    1. Yes, that is right. Speed = distance ÷ time = 150 ÷ 2.5 = 60 km/h.

    2. Not this one. You multiplied the distance by the time. Speed is the distance divided by the time. Go to M4.5.6. Try another.

    3. Not this one. You added the numbers. Speed is how far you go in each hour, so divide 150 by 2.5. Go to M4.5.6. Try another.

    See the answer

    60 km/h

    Speed = distance ÷ time = 150 ÷ 2.5 = 60 km/h.

    • If you chose B: You multiplied the distance by the time. Speed is the distance divided by the time. Go to M4.5.6.
    • If you chose C: You added the numbers. Speed is how far you go in each hour, so divide 150 by 2.5. Go to M4.5.6.

  2. A bus goes at 45 km/h for 40 minutes. How far does it go?

    Tap your answer.

    1. Not this one. You multiplied 45 by 40 and left the time in minutes. The speed is per hour, so change 40 minutes into hours first. Go to M4.5.6. Try another.

    2. Not this one. You divided the speed by the time. To find the distance, multiply speed by time, with the time in hours. Go to M4.5.6. Try another.

    3. Yes, that is right. 40 minutes is 40 ÷ 60 = 2/3 of an hour. Distance = speed × time = 45 × 2/3 = 30 km.

    See the answer

    30 km

    40 minutes is 40 ÷ 60 = 2/3 of an hour. Distance = speed × time = 45 × 2/3 = 30 km.

    • If you chose A: You multiplied 45 by 40 and left the time in minutes. The speed is per hour, so change 40 minutes into hours first. Go to M4.5.6.
    • If you chose B: You divided the speed by the time. To find the distance, multiply speed by time, with the time in hours. Go to M4.5.6.

  3. A lorry must go 210 km at 60 km/h. How long does the journey take?

    Tap your answer.

    1. Not this one. You read 3.5 hours as 3 hours 50 minutes. The 0.5 is half an hour, which is 30 minutes. Go to M4.5.6. Try another.

    2. Not this one. You divided the speed by the distance. Time is the distance divided by the speed. Go to M4.5.6. Try another.

    3. Not this one. You multiplied the two numbers. Time = distance ÷ speed. Go to M4.5.6. Try another.

    4. Yes, that is right. Time = distance ÷ speed = 210 ÷ 60 = 3.5 hours, which is 3 hours 30 minutes.

    See the answer

    3.5 hours

    Time = distance ÷ speed = 210 ÷ 60 = 3.5 hours, which is 3 hours 30 minutes.

    • If you chose A: You read 3.5 hours as 3 hours 50 minutes. The 0.5 is half an hour, which is 30 minutes. Go to M4.5.6.
    • If you chose B: You divided the speed by the distance. Time is the distance divided by the speed. Go to M4.5.6.
    • If you chose C: You multiplied the two numbers. Time = distance ÷ speed. Go to M4.5.6.

  4. A car goes at 90 km/h. What is its speed in m/s?

    Tap your answer.

    1. Yes, that is right. To change km/h into m/s, divide by 3.6. 90 ÷ 3.6 = 25 m/s.

    2. Not this one. You multiplied by 3.6. That changes m/s into km/h. To change km/h into m/s, divide by 3.6. Go to M4.5.6. Try another.

    3. Not this one. You divided by 60, which changes km per hour into km per minute. For m/s, divide by 3.6. Go to M4.5.6. Try another.

    See the answer

    25 m/s

    To change km/h into m/s, divide by 3.6. 90 ÷ 3.6 = 25 m/s.

    • If you chose B: You multiplied by 3.6. That changes m/s into km/h. To change km/h into m/s, divide by 3.6. Go to M4.5.6.
    • If you chose C: You divided by 60, which changes km per hour into km per minute. For m/s, divide by 3.6. Go to M4.5.6.

  5. A car drives 30 km at 60 km/h. Then it drives 30 km at 30 km/h. What is its average speed for the whole journey?

    A bar in two equal parts. The first part is labelled 30 km above it and 60 km/h below it. The second part is labelled 30 km above it and 30 km/h below it.30 km60 km/h30 km30 km/h
    Tap your answer.

    1. Not this one. You found the mean of 60 and 30. The car spends longer at 30 km/h, so the average is lower than 45. Go to M4.5.6. Try another.

    2. Not this one. That is the first speed. The average uses the total distance and the total time of both parts. Go to M4.5.6. Try another.

    3. Yes, that is right. The first part takes 30 ÷ 60 = 0.5 hours. The second part takes 30 ÷ 30 = 1 hour. The total is 60 km in 1.5 hours, so the average speed is 60 ÷ 1.5 = 40 km/h.

    See the answer

    40 km/h

    The first part takes 30 ÷ 60 = 0.5 hours. The second part takes 30 ÷ 30 = 1 hour. The total is 60 km in 1.5 hours, so the average speed is 60 ÷ 1.5 = 40 km/h.

    • If you chose A: You found the mean of 60 and 30. The car spends longer at 30 km/h, so the average is lower than 45. Go to M4.5.6.
    • If you chose B: That is the first speed. The average uses the total distance and the total time of both parts. Go to M4.5.6.

  6. From an earlier chapter

    Change 4.2 m² into cm².

    See the answer

    42 000 cm²

    One square metre is 10 000 cm². So 4.2 m² is 4.2 × 10 000 = 42 000 cm².

  7. A cyclist rides 18 km in 45 minutes. What is the speed in km/h?

    See the answer

    24 km/h

    45 minutes is 45 ÷ 60 = 0.75 hours. Speed = 18 ÷ 0.75 = 24 km/h.

  8. A train goes at 54 km/h. What is its speed in m/s?

    See the answer

    15 m/s

    To change km/h into m/s, divide by 3.6. 54 ÷ 3.6 = 15 m/s.

  9. A car drives 80 km at 40 km/h, then 60 km at 60 km/h. What is its average speed, to 1 decimal place?

    See the answer

    46.7 km/h

    The first part takes 80 ÷ 40 = 2 hours. The second part takes 60 ÷ 60 = 1 hour. The total is 140 km in 3 hours, so the average speed is 140 ÷ 3 = 46.7 km/h to 1 decimal place.

Keep it

Copy this into your exercise book.

Speed = distance ÷ time. Distance = speed × time. Time = distance ÷ speed. km/h to m/s: ÷ 3.6. Average speed = total distance ÷ total time.

Draw: A bar in two parts with its distances and times, and the total under a bracket.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (741 KB). It opens on any phone.