Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- (Lesson 1) A bus company carries 840 passengers in 7 hours. How many passengers per hour is that?
Answer
120. 840 ÷ 7 = 120 passengers per hour. - (Year 7, Unit 2, lesson 3) A 60 km road is 35% built. How many kilometres are built?
Answer
21 km. 10% is 6 km, so 30% is 18 km and 5% is 3 km: 18 + 3 = 21. - (Year 7, Unit 3, lesson 6) Solve 12t = 90.
Answer
t = 7.5, because 90 ÷ 12 = 7.5.
Words used in this lesson
- speed
- distance travelled for one unit of time. For example: 60 km per hour.
- distance
- how far something goes. For example: 150 km.
- time
- how long something takes. For example: 3.5 hours.
- average speed
- total distance divided by total time. For example: 150 km in 3 hours is 50 km per hour.
- leg
- one part of a journey. For example: the first leg, 60 km.
Same units, or the answer is nonsense
A ferry crosses a channel at 30 km per hour, and the crossing takes 40 minutes. Predict first: how far is the crossing? A friend multiplies 30 × 40 and gets 1200 km. Why cannot that be right, and what would you change?
The figure puts a speed and a time on a road. Watch the distance, and watch the minutes beside it.
80 km per hour for 2.5 hours: 80 × 2.5 = 200 km. The time is 150 minutes.
Show me the steps
- 80 km per hour for 2.5 hours: 80 × 2.5 = 200 km. 2.5 hours is 150 minutes.
- 30 km per hour for 4 hours: 30 × 4 = 120 km.
- 100 km per hour for 0.5 hours, which is 30 minutes: 100 × 0.5 = 50 km.
- 120 km per hour for 5 hours: 120 × 5 = 600 km, the whole road.
Speed is the distance for one unit of time: speed = distance ÷ time. The same rule rearranged gives distance = speed × time and time = distance ÷ speed.
The units have to match. A speed in kilometres per hour wants a time in hours. For minutes, divide by 60 first: 40 minutes is 40 ÷ 60 of an hour.
Average speed is the total distance divided by the total time, not the mean of the speeds. When the speeds differ, the two agree only if equal times are spent at each speed.
Back to the ferry: 40 minutes is 40 ÷ 60 = 2 ÷ 3 of an hour, so the distance is 30 × 2 ÷ 3 = 20 km. Or work in minutes: 30 km per hour is 0.5 km each minute, and 0.5 × 40 = 20 km. The 1200 came from using minutes as if they were hours.
When the speed changes
A lorry drives 60 km uphill at 30 km per hour, then 60 km back down at 60 km per hour. Predict first: is its average speed for the whole 120 km the mean of the two speeds, 45 km per hour? The figure draws two legs of 60 km as bars of time: the longer the bar, the longer that leg takes.
Leg 1: 60 km at 20 km/h takes 3 h. Leg 2: 60 km at 60 km/h takes 1 h. In all, 120 km in 4 h: 120 ÷ 4 = 30 km per hour, not the mean of the speeds, 40.
Show me the steps
- 20 km/h then 60 km/h: 3 h + 1 h = 4 h, and 120 ÷ 4 = 30 km per hour. The mean of the speeds, 40, is too high.
- 30 km/h then 60 km/h: 2 h + 1 h = 3 h, and 120 ÷ 3 = 40 km per hour. The mean of the speeds is 45.
- 30 km/h then 30 km/h: 2 h + 2 h = 4 h, and 120 ÷ 4 = 30 km per hour. With equal speeds the average and the mean agree.
- 20 km/h then 30 km/h: 3 h + 2 h = 5 h, and 120 ÷ 5 = 24 km per hour. The mean of the speeds is 25.
A common mistake: “Average speed is the two speeds added and halved.” Try 20 and 60 km per hour over 60 km each. The mean is 40, but the journey takes 3 + 1 = 4 hours for 120 km, which is 30 km per hour. More of the journey is spent going slowly, so the slow speed counts for more.
Worked example: the lorry
A lorry drives 60 km at 30 km per hour, then 60 km at 60 km per hour. Find its average speed.
- Time for each leg: 60 ÷ 30 = 2 hours, and 60 ÷ 60 = 1 hour.
- Total distance 60 + 60 = 120 km, total time 2 + 1 = 3 hours.
- Average speed = 120 ÷ 3 = 40 km per hour, not 45.
Worked example: how long?
A coach travels 150 km at 60 km per hour. How long does it take, in hours and minutes?
- Time = distance ÷ speed = 150 ÷ 60 = 2.5 hours.
- 0.5 hours is half of 60 minutes, which is 30 minutes.
- 2 hours 30 minutes. Check: 60 × 2.5 = 150 km.
Your go. The first step is given. A cyclist in Hanoi rides 18 km in 45 minutes. Find her speed in kilometres per hour.
- Change the time to hours: 45 ÷ 60 = 0.75 hours.
- Now divide the distance by the time, then check.
Check the answer
Speed = 18 ÷ 0.75 = 24 km per hour. Check: 24 × 0.75 = 18 km.
Check before you go on
1. A train travels at 80 km per hour for 1 hour 15 minutes. How far does it travel?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A motorbike rider travels at 45 km per hour for 4 hours. How many kilometres does she go?
3. A boat covers 96 km in 3 hours. What is its speed in kilometres per hour?
Level 2 · Building
4. A cyclist rides 36 km at 12 km per hour. How many hours does it take?
5. A runner jogs 6 km at 12 km per hour. How many minutes does that take?
6. Zuri walks at 5 km per hour for 24 minutes. How many kilometres does she walk?
Level 3 · Stretch
7. A bus drives 90 km at 45 km per hour, then 90 km at 90 km per hour. What is its average speed for the whole journey, in kilometres per hour?
8. A cyclist rides 24 km at 12 km per hour. She wants to do the same ride in 1.5 hours. How many kilometres per hour faster must she ride?
Something to think about
A car drives the first 100 km of a 200 km trip at 50 km per hour. How fast must it drive the second 100 km to average 100 km per hour over the whole trip? Decide, then test your answer.
One way to think about it
It cannot be done. 200 km at an average of 100 km per hour allows 200 ÷ 100 = 2 hours in all, and the first 100 km at 50 km per hour has already taken 100 ÷ 50 = 2 hours. No speed, however fast, can make the second half take no time.
Exit ticket
9. Amara runs for 1 hour at 12 km per hour, then for 1 hour at 8 km per hour. Rosa says “Her average speed is (12 + 8) ÷ 2 = 10 km per hour.” Which reply is right?