KS3 · Maths

Speed, distance and time

50 km/h isn't just a number. It's a promise: 50 kilometres in every single hour. Read that promise properly and how far, how fast and how long all follow.

Speed is a rate

50 km/h: fifty kilometres in every hour

time travelled: 4/12. Time 1 h. Time in hours 1 h. Distance 50 km. Distance ÷ time 50 ÷ 1 = 50

Time1 hTime in hours1 hDistance50 kmDistance ÷ time50 ÷ 1 = 50

A coach drives at a steady 50 km/h for 3 hours. Drag along the journey and watch the distance. Every hour adds another 50 km, and every quarter of an hour adds a quarter of that: 12.5 km.

The steeper the line, the faster the journey

Speed 50 km/h

Speed: 50 km/h. Distance after 2 hours: 100 km

The same idea as a picture. A steady speed draws a straight line through 0, because every hour adds the same distance. Grab the point at 2 hours and lift it: a faster speed tips the whole line steeper.

One relationship, three ways to write it

Cover the quantity you want to find. What is left tells you whether to multiply or divide.

Tap the quantity you want to find. The triangle shows you the formula.

Cover distance, speed or time to reveal its rearranged formula, then plug in numbers to solve.

The slip everyone makes once

How many hours is 1 hour 30 minutes?

A family drive at a steady 60 km/h for 1 hour 30 minutes. To use distance = speed × time with a speed in km/h, the time has to be written in hours.

Which is closest to what you would write for the time?

1.3 hours. The 3 stands for the 30 minutes.1.30 hours, the way a clock shows 1:30.1.5 hours, because 30 minutes is half of 60.

How sure are you?

I'm sureI think soI'm guessing

1.3 hours. The 3 stands for the 30 minutes.

This is a common misconception

Tempting, but a decimal counts in tenths and hundredths, and minutes don't. 0.3 of an hour is 3 tenths of 60 minutes, which is only 18 minutes. Using 1.3 would give 60 × 1.3 = 78 km, when the family really drove further.

1.30 hours, the way a clock shows 1:30.

This is a common misconception

A clock writes hours and minutes side by side, but 1.30 as a decimal is exactly the same number as 1.3, which is 1 hour 18 minutes. The clock's '30' means 30 out of 60, so it has to be turned into a fraction of an hour first.

1.5 hours, because 30 minutes is half of 60.

This matches the evidence

Yes. Minutes are sixtieths of an hour, so 30 minutes = 30 ÷ 60 = 0.5 of an hour. The drive lasts 1.5 hours and covers 60 × 1.5 = 90 km.

Minutes are out of 60, not out of 100. Divide the minutes by 60 to get the decimal part of the hour: 15 minutes = 0.25 h, 30 minutes = 0.5 h, 45 minutes = 0.75 h.

Your turn to fill the gaps

Change 72 km/h into m/s

A car travels at 72 km/h. What is this speed in metres per second (m/s)? Convert each part of the unit in turn.

72 km/h means 72 km in each hour.
missing step

Which line is step 2?

= 0.072 m in each hour= 72 000 m in each hour= 7200 m in each hour

Predict, then check

Commit to an answer before you work anything out.

Sam cycles 30 km to the lake at 15 km/h, then rides the same 30 km home at 30 km/h. What is Sam's average speed for the whole trip?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

How far? How fast? How long? Three questions, one idea, and it is hiding in the word 'per'.

What you need to know

Speed is a rate: the distance travelled in each unit of time. 50 km/h means 50 km in each hour.
A compound unit is built from two units. Speed units such as m/s, km/h and mph are read with 'per': metres per second, kilometres per hour, miles per hour.
speed = distance ÷ time, distance = speed × time, time = distance ÷ speed. Use the one with the unknown as its subject.
Units must match: with km/h use km and hours; with m/s use metres and seconds.
Minutes are sixtieths of an hour: 30 min = 0.5 h, 15 min = 0.25 h, and 1 h 30 min = 1.5 h (not 1.3 h).
km/h to m/s: × 1000 then ÷ 3600. m/s to km/h: × 3600 then ÷ 1000.
Average speed for a whole journey = total distance ÷ total time. In general it is not the mean of the separate speeds.
At a steady speed, distance is directly proportional to time: the distance–time graph is a straight line through the origin, and its gradient is the speed.

The big picture

Speed is a rate: how much distance is covered in each unit of time. A speed of 50 km/h means 50 km in each hour, so speed = distance ÷ time, distance = speed × time and time = distance ÷ speed. Make the units match first (minutes become fractions of an hour, and km/h converts to m/s part by part). For a journey in parts, average speed is total distance ÷ total time. At a steady speed a distance–time graph is a straight line through the origin, and its gradient is the speed.

Key points

1Read 'per' as 'in each': km/h is kilometres in each hour, and the unit tells you the calculation, kilometres ÷ hours.

2One relationship, three arrangements: s = d ÷ t, d = s × t, t = d ÷ s.

3Match the units before you calculate, and give the answer the unit you built it from.

4Turn minutes into hours by dividing by 60, never by moving the decimal point.

5Convert a compound unit one part at a time: the distance part, then the time part.

6Average speed = total distance ÷ total time.

7Steady speed means a straight distance–time line through the origin. Steeper means faster.

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Worked example

Problem

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

⚠ Watch out

Writing a time in hours and minutes as if it were a decimal, like 2 hours 15 minutes as 2.15 hours. Minutes are out of 60, so 15 minutes is 15 ÷ 60 = 0.25 of an hour, and the time is 2.25 hours.

🧠

Memory hook

Say the unit out loud and it hands you the sum: kilometres PER hour is kilometres ÷ hours.

✓

Check yourself

A drone flies at a steady 12 m/s. How far does it fly in 1 minute? (Answer: the units must match, so 1 minute = 60 seconds, and 12 × 60 = 720 m.)

Flashcards

(14)

What does a speed of 50 km/h mean?

50 kilometres are travelled in each hour.

What is a compound unit?

A unit made from two other units, such as km/h, which combines a distance unit with a time unit.

How do you read m/s out loud?

Metres per second: the number of metres travelled in each second.

Speed formula

speed = distance ÷ time

Rearrange speed = distance ÷ time to find distance, and to find time.

distance = speed × time; time = distance ÷ speed

Which arrangement of the formula should you use?

The one that has the unknown quantity as its subject.

The speed is in km/h. What units must the distance and time be in?

Distance in kilometres, time in hours.

How do you turn a number of minutes into hours?

Divide by 60, because there are 60 minutes in an hour. For example, 15 minutes = 0.25 hours.

Is 1 hour 30 minutes equal to 1.3 hours?

No. 30 minutes is half an hour, so it is 1.5 hours.

How do you change km/h into m/s?

Multiply by 1000 (metres in a kilometre), then divide by 3600 (seconds in an hour).

How do you change m/s into km/h?

Multiply by 3600, then divide by 1000.

How do you find the average speed for a journey made of several parts?

Total distance ÷ total time.

What does a steady speed look like on a distance–time graph?

A straight line through the origin, because distance is directly proportional to time.

What does the gradient of a distance–time graph tell you?

The speed (distance ÷ time). A steeper line means a greater speed.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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The steeper the line, the faster the journey

How many hours is 1 hour 30 minutes?

Change 72 km/h into m/s

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