Time and Distance
Distance = speed × time, and you can hold any one of the three fixed. When distance is fixed, speed and time trade in inverse ratio.
Why this topic matters
Speed is distance per hour. If you drive at 60 km/h for 2 hours, you cover 120 km. Every question is that one line: distance = speed × time.
Time and distance and its two cousins (trains, boats) make up 2 to 4 questions per paper. Units and relative speed cause most mistakes.
The formula and units
Distance = speed × time. Speed = distance / time. Time = distance / speed.
km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5. 72 km/h = 20 m/s. 36 km/h = 10 m/s.
When distance is fixed, speed and time are inversely proportional: go 1.5 times faster and take 2/3 the time.
- Speed = 180 / 2.5 = 72 km/h.2 h 30 min = 2.5 h.
- 72 × 5/18 = 20 m/s.Convert units last.
- Speed × 4/5 means time × 5/4.Same distance: time is inversely proportional to speed.
- The extra 1/4 of the usual time = 10 minutes.
- Usual time = 40 minutes.
Average speed
Average speed = total distance / total time. It is NOT the average of the speeds, unless the TIMES are equal.
For equal distances at speeds a and b: average = 2ab/(a + b).
- Same distance both ways, so use 2ab/(a + b).He spends longer at the slower speed, so the average is below 50.
- 2 × 40 × 60 / 100 = 48 km/h.
Relative speed
Two objects moving towards each other close the gap at the SUM of their speeds. Moving in the same direction, at the DIFFERENCE.
Meeting time = gap / relative speed.
- Relative speed = 70 + 50 = 120 km/h.Towards each other: add.
- 300 / 120 = 2.5 hours.
- Head start: 8 × 10/60 = 4/3 km.Distance the thief covers before the chase begins.
- Closing speed = 10 - 8 = 2 km/h.Same direction: subtract.
- Time = (4/3) / 2 = 2/3 h = 40 minutes.
Formula summary
Shortcuts and tricks
Usual time from 'fraction of speed'
Use it when: 'At 3/4 of usual speed he is 20 min late'.
- Time becomes 4/3 of usual; the extra 1/3 = 20 min.
- 1/3 of usual = 20 → 60.
Common mistakes
Before you move on, you should be able to...
- convert between km/h and m/s
- use average speed = total distance / total time
- add speeds for opposite directions, subtract for the same direction
Practice questions (7)
Try each one before opening the solution. Or practise them one by one so your score is saved.
easy A car goes at 40 km/h one way and 60 km/h back. Average speed?
- 2 × 40 × 60/100.
Answer: 48 km/h
medium At 5 km/h a man is 10 min late; at 6 km/h he is 5 min early. Distance?
- d/5 - d/6 = 15/60 → d = 7.5.
Answer: 7.5 km
medium A thief runs at 8 km/h. A policeman starts 10 min later at 10 km/h. When does he catch him (minutes after he starts)?
- Head start 8 × 1/6 = 4/3 km.
- Closing 2 km/h → 2/3 h.
Answer: 40 min
hard A covers a distance at 3 speeds 10, 20, 30 km/h over equal thirds. Average speed? (2 d.p.)
- 3/(1/10 + 1/20 + 1/30) = 180/11.
Answer: 16.36 km/h
hard A walks at 4 km/h. 4 hours later B cycles after him at 10 km/h. How far from the start does B catch A?
- Head start 16 km, closing speed 6 km/h → 8/3 h.
- 10 × 8/3.
Answer: 26 2/3 km
hard A man covers half a journey at 30 km/h. What speed on the second half gives an average of 40 km/h?
- 2 × 30 × v / (30 + v) = 40 → v = 60.
Answer: 60 km/h
hard Going 20% faster, a car arrives 10 min early. Usual time?
- Time becomes 5/6, the saved 1/6 = 10 min.
Answer: 60 min
Class plan (2 hours, for trainers)
A tested order for teaching this topic in one 2-hour session. Present mode follows the same order.
| 10 min | Warm-up: km/h to m/s for 18, 36, 54, 72, 90. |
| 25 min | D = S × T, units, inverse proportion (late/early). |
| 20 min | Average speed and 2ab/(a+b). |
| 30 min | Relative speed: meeting and chasing, with the diagrams. |
| 30 min | Practice set. |
| 5 min | Recap. |