In this guide
Time and distance questions ask how fast something moves, how far it goes, or how long it takes. Buses, cyclists, people walking to school and trains passing platforms all turn up. It looks like a big chapter, but every question comes from one formula and three or four ideas built on it.
This topic uses the same "rate" thinking as time and work. If you are comfortable with work per day, speed (distance per hour) will feel familiar.
The one formula
Distance = Speed × Time
Cover the part you want and read the rest:
- Speed = Distance ÷ Time
- Time = Distance ÷ Speed
Before you calculate, make the units match. If speed is in km/h, time must be in hours and distance in km. If speed is in m/s, use metres and seconds.
Changing units
- km/h to m/s: multiply by 5/18.
- m/s to km/h: multiply by 18/5.
Why 5/18? One km is 1,000 m and one hour is 3,600 seconds, so 1 km/h = 1,000/3,600 m/s = 5/18 m/s.
Learn these pairs by heart. They come up in almost every train question.
| km/h | m/s |
|---|---|
| 18 | 5 |
| 36 | 10 |
| 54 | 15 |
| 72 | 20 |
| 90 | 25 |
| 108 | 30 |
Average speed
Average speed = total distance ÷ total time. It is not the simple average of the two speeds.
When the same distance is covered at two speeds x and y (for example going and coming back), there is a shortcut:
Average speed = 2xy ÷ (x + y)
The answer is always a little below the simple average, because you spend more time at the slower speed.
Trains
A train has length, so the distance it covers depends on what it passes.
| Train passes | Distance covered |
|---|---|
| A pole, a signal or a standing person | Its own length |
| A platform, bridge or tunnel | Its length + the platform's length |
| Another train | Sum of both lengths |
Relative speed
When two things move at once, use their speed compared with each other.
| Direction | Relative speed |
|---|---|
| Towards each other or opposite directions | Add the speeds |
| Same direction (one chasing the other) | Subtract: faster − slower |
Solved examples
Example 1. A motorbike runs at 54 km/h. How far does it go in 20 minutes?
- Time = 20/60 = ⅓ hour.
- Distance = 54 × ⅓ = 18 km.
Example 2. A person drives 60 km to a town at 30 km/h and returns at 20 km/h. Find the average speed for the whole trip.
- Time going = 60 ÷ 30 = 2 hours. Time returning = 60 ÷ 20 = 3 hours.
- Total distance = 120 km. Total time = 5 hours.
- Average speed = 120 ÷ 5 = 24 km/h.
- Shortcut check: 2 × 30 × 20 ÷ 50 = 1,200 ÷ 50 = 24. Note that the simple average, 25, is wrong.
Example 3. A 240 m train runs at 54 km/h. How long does it take to pass a 360 m platform?
- Speed = 54 × 5/18 = 15 m/s.
- Distance = 240 + 360 = 600 m.
- Time = 600 ÷ 15 = 40 seconds.
Example 4. Two trains, 150 m and 100 m long, run at 50 km/h and 40 km/h. How long do they take to cross each other (a) in opposite directions (b) in the same direction?
- Distance in both cases = 150 + 100 = 250 m.
- (a) Relative speed = 50 + 40 = 90 km/h = 25 m/s. Time = 250 ÷ 25 = 10 seconds.
- (b) Relative speed = 50 − 40 = 10 km/h = 10 × 5/18 = 25/9 m/s. Time = 250 ÷ 25/9 = 250 × 9/25 = 90 seconds.
Example 5. A cyclist leaves a town at 8 a.m. at 10 km/h. A second cyclist leaves the same place at 10 a.m. at 15 km/h on the same road. When and where does the second catch the first?
- By 10 a.m. the first is 2 × 10 = 20 km ahead.
- The gap closes at 15 − 10 = 5 km/h.
- Time to catch up = 20 ÷ 5 = 4 hours, so at 2 p.m.
- Distance from town = 15 × 4 = 60 km. Check: the first rode 6 hours × 10 = 60 km.
Example 6. A student walking at 4 km/h reaches school 10 minutes late. Walking at 5 km/h, the student is 5 minutes early. How far is the school?
- The two times differ by 10 + 5 = 15 minutes = ¼ hour.
- Distance/4 − Distance/5 = ¼, so Distance/20 = ¼.
- Distance = 5 km. Check: 5 ÷ 4 = 75 minutes and 5 ÷ 5 = 60 minutes, which differ by 15 minutes.
Common mistakes
| Mistake | Correct way |
|---|---|
| Dividing metres by km/h | Convert km/h to m/s first |
| Taking 30 minutes as 0.30 hours | 30 minutes = 0.5 hours |
| Averaging two speeds directly | Total distance ÷ total time |
| Forgetting the platform length | Train + platform |
| Adding speeds for a chase | Same direction: subtract |
Practice set
- A car covers 225 km in 4½ hours. Find its speed.
- Convert 90 km/h into m/s.
- Convert 20 m/s into km/h.
- A 180 m train at 36 km/h passes a pole. How long does it take?
- A 300 m train at 72 km/h crosses a 200 m bridge. How long does it take?
- A person goes to a place at 40 km/h and returns at 60 km/h. Find the average speed.
- Two people 72 km apart walk towards each other at 5 km/h and 7 km/h. When do they meet?
- A cyclist leaves at 8 km/h. Half an hour later, a second cyclist leaves the same point at 10 km/h. How long does the second take to catch up?
Answers:
- 225 ÷ 4.5 = 50 km/h.
- 90 × 5/18 = 25 m/s.
- 20 × 18/5 = 72 km/h.
- 36 km/h = 10 m/s. 180 ÷ 10 = 18 seconds.
- 72 km/h = 20 m/s. (300 + 200) ÷ 20 = 25 seconds.
- 2 × 40 × 60 ÷ 100 = 48 km/h.
- Combined speed 12 km/h. 72 ÷ 12 = 6 hours.
- Head start = 8 × ½ = 4 km. Gap closes at 2 km/h. 4 ÷ 2 = 2 hours.
What to do next
- Learn the km/h and m/s table until you can say it without looking.
- Solve 10 train questions: five with a pole, five with a platform.
- Practise five average-speed questions using both the long way and 2xy ÷ (x + y).
- Try five meeting and chasing questions, and write "add" or "subtract" before you start.
- Read common maths mistakes to see how unit errors cost marks.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Staff Selection Commission website .
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