The fair and back

You ride somewhere fast and come back slowly. Is your average speed halfway between the two?

About 35 minutes
Class 7
Mathematics and money

What you will learn: Use speed = distance ÷ time, change minutes into hours, and find the average speed of a whole trip from the total distance and the total time.

What should I know first?

If this feels new, try these first. It is fine to go back.

Dadi's chai for six Class 7
Last bus from the fair Class 5

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: Check yourself · For grown-ups · For teachers

Safety first

Ride only with a grown-up, on safe roads, wearing a helmet.

Keep it safe:

Any cycling or walking for this activity is with a grown-up, on safe paths, with a helmet for cycling.
Time things at home or in a park, never on a busy road.
Keep small things such as bottle caps and pen lids away from babies: nothing goes in the mouth.

You will need

Paper and a pencilFor a table of distance, speed and time.
A clock or a phone timer, used by a grown-upIf you time a walk.

Time: About 35 minutes.

What do you think?

Tanmay says: "12 km/h there and 6 km/h back, the same distance each way, so my average is 9 km/h." Is he right?

Make a prediction and write one reason for it. Then tap the guess closest to yours to see how to test it. It is fine to change your mind after checking.

Yes, 9 km/h

Let's test it. How long did each part take? Add up the whole distance and the whole time.

No, less than 9

Why less? Which part of the trip took longer?

No, more than 9

Check it: find the time for each part, then total distance ÷ total time.

Not sure

That is fine. Work out how long each part took first: 12 km at 12 km/h, then 12 km at 6 km/h.

Need a hint?

Find the time for each part first. He spends longer on the slow part.

The story

Tanmay cycles 12 km to the fair on a flat road at 12 km/h. Coming back is uphill, so he rides at 6 km/h. "Halfway between 12 and 6 is 9," he tells Dadi, "so I averaged 9 km/h." Dhruv looks at his watch. "You left at 2 o'clock and got home at 5, and the fair took no time at all in your story. Let's check."

The idea

Speed tells you how far you go in each hour: speed = distance ÷ time. 12 km in 1 hour is 12 km/h.

Turn it round to find the others: time = distance ÷ speed, and distance = speed × time.

For a whole trip with fast and slow parts, the average speed is the total distance ÷ the total time.

Do not average the two speeds. You spend more time at the slow speed, so the average is pulled towards it.

A table of Tanmay's trip. There, on the flat: 12 km at 12 km/h, 1 hour. Back, uphill: 12 km at 6 km/h, 2 hours. Whole trip: 24 km in 3 hours, average speed to be found.

Formula cards

From counting to a formula

Speed

What it finds: How far something goes in each hour (or each minute).
The formula: Speed = distance ÷ time
Why it works: If you ride 12 km in 1 hour, each hour carries you 12 km. In 2 hours at the same speed you go 24 km, and 24 ÷ 2 is still 12.
When it works: For one steady part of a trip. For a whole trip with fast and slow parts, use total distance ÷ total time.
Units: km/h when the distance is in km and the time in hours. Change minutes to hours first: 30 minutes = 0.5 hours, 15 minutes = 0.25 hours.
Use it backwards: Time = distance ÷ speed: 10 km at 5 km/h takes 10 ÷ 5 = 2 hours. Distance = speed × time: 3 hours at 4 km/h is 12 km.

Worked example

Tanmay's real average

Tanmay rides 12 km at 12 km/h, then 12 km back at 6 km/h. What is his average speed for the whole trip?

Time there: 12 km ÷ 12 km/h = 1 hour.
Time back: 12 km ÷ 6 km/h = 2 hours.
Total: 24 km in 3 hours.
Average speed = 24 ÷ 3 = 8 km/h. Check: it is nearer 6 than 12, because he rode slowly for longer.

Tanmay's average speed was 8 km/h, not 9.

Try it

Make a table.
Draw four columns: part of the trip, distance, speed, time. Fill in Tanmay's two parts.
Find each time.
Work out the time for each part with time = distance ÷ speed.
Add it all up.
Add the distances and add the times. Then divide total distance by total time.
Try a different trip.
Dadi's bus goes 30 km in 45 minutes. Change 45 minutes into hours first, then find its speed.
Need a hint?
45 minutes is three quarters of an hour: 0.75 hours. Then 30 ÷ 0.75.
Time a real walk.
With a grown-up, walk to a nearby place and back, timing each way. Work out each speed and the average.
Explain the trap.
Write one sentence on why the average of two speeds is not the average speed of the trip.

What you make (paper or offline)

Your trip planner

An empty planner table with rows for part 1, part 2 and the whole trip, and columns for distance, speed and time.

Plan a trip with two parts at different speeds, the same distance each way.

Work out the time for each part, the total time and the average speed.

Then change it so the fast part is longer. What happens to the average?

Make it on paper. Keep it with you. Nothing to upload.

What did we find out?

Read this after you have tried it. Look back at your guess and your reason first.

Speed = distance ÷ time. 12 km at 12 km/h takes 1 hour; 12 km at 6 km/h takes 2 hours.

The average speed of the whole trip is total distance ÷ total time = 24 km ÷ 3 hours = 8 km/h, not 9.

Averaging the two speeds gives the wrong answer because Tanmay spent twice as long at the slow speed.

Think back

Why is the average speed nearer the slow speed than the fast one?

When would averaging the two speeds give the right answer?

Investigate further

If Tanmay had ridden for 1 hour at 12 km/h and 1 hour at 6 km/h, what would his average speed be? Why is it different?
A runner does 400 m at 8 m/s and 400 m at 4 m/s. Find the average speed.

Can you use it here?

With a grown-up, time a walk to a nearby place and back. Work out your speed each way and your average speed.

Ask someone at home

Ask someone at home: if a bus is fast on the highway and slow in the town, is its average speed nearer the fast one or the slow one? Why?

No one free to join in? That's fine. You can do it by yourself with drawings, and ask a grown-up to help with the materials.

For grown-upsTap a heading to open it. The safety notes are at the top of the page.

Before you start

Paper and a pencil. Any walking or cycling is with you, on safe paths, with a helmet for cycling.

Questions that help

"How long did each part take?"
"Which part did you spend longer on?"
"Is your answer nearer the slow speed or the fast one?"

What's really going on

Averaging two speeds is the most common mistake, even for adults. Working out the time for each part shows why it fails.

Changing minutes into hours (45 minutes = 0.75 hours) is the second common slip.

Averaging two speeds only works when equal times are spent at each, not equal distances.

Other ways to do it

No timer? Use a bus or train timetable and the distances between stops.
Time a toy car down a ramp and back.

For teachersLearning aims, classroom ideas and what to look for.

Learning aims

Use speed = distance ÷ time and its two inverse forms, with consistent units.
Find the average speed of a journey as total distance ÷ total time.
Explain why the mean of two speeds is not the average speed when the times differ.

Subject links

Builds on ratio ("Dadi's chai for six") and averages (Class 7).
Leads to distance-time graphs (Class 9) and rates of change (Classes 11-12).

These links are general. They have not been checked against a specific board syllabus.

In the classroom

Groups invent a two-part trip and swap: the other group finds the average speed and checks whether the speed average works.
Compare equal-time and equal-distance trips side by side.

What to look for

Finds the time for each part before averaging.
Converts minutes to hours correctly.
Says why the answer is nearer the slower speed.