Time, speed and distance questions appear regularly in the Lok Sewa aptitude (Maths and IQ) paper, including problems on trains and on boats and streams. They all come from one simple formula, Distance = Speed × Time. This note explains unit conversion, average speed, relative speed, trains, boats and streams and a few useful shortcuts, each with a worked example.
1. The basic formula
- Distance = Speed × Time
- Speed = Distance ÷ Time
- Time = Distance ÷ Speed
Speed is usually given in kilometres per hour (km/h) or metres per second (m/s). Before using the formula, make sure all units match.
Example: A car covers 240 km in 4 hours. Speed = 240 ÷ 4 = 60 km/h.
Example: A bus runs at 45 km/h for 3 hours 20 minutes. Time = 3⅓ hours = 10/3 hours. Distance = 45 × 10/3 = 150 km.
2. Converting units
| Conversion | Multiply by | Example |
|---|---|---|
| km/h to m/s | 5/18 | 72 km/h = 72 × 5/18 = 20 m/s |
| m/s to km/h | 18/5 | 15 m/s = 15 × 18/5 = 54 km/h |
| minutes to hours | 1/60 | 45 minutes = 45/60 = 0.75 hour |
Reason: 1 km/h = 1,000 m ÷ 3,600 s = 5/18 m/s. Remember: to m/s multiply by 5/18; to km/h multiply by 18/5.
3. Speed, time and ratios
- For a fixed distance, time is inversely proportional to speed. If speeds are in the ratio a : b, times are in the ratio b : a.
- For a fixed time, distance is directly proportional to speed.
- Example: The speeds of A and B are in the ratio 3 : 4. For the same distance, their times are in the ratio 4 : 3.
- Example: A man covers a distance in 6 hours at 40 km/h. To cover it in 4 hours, distance = 240 km, so speed needed = 240 ÷ 4 = 60 km/h.
Worked example 1 (late and early): If a boy walks at 4 km/h he is 10 minutes late, and at 5 km/h he is 5 minutes early. Find the distance.
- Difference in time = 10 + 5 = 15 minutes = 1/4 hour.
- D/4 − D/5 = 1/4, so D/20 = 1/4
- D = 5 km.
4. Average speed
Average speed = Total distance ÷ Total time (it is not the simple average of the speeds).
Shortcut: If equal distances are covered at speeds x and y, average speed = 2xy ÷ (x + y).
Worked example 2: A man goes to a town at 40 km/h and returns at 60 km/h. Average speed = 2 × 40 × 60 ÷ (40 + 60) = 4,800 ÷ 100 = 48 km/h (not 50).
Worked example 3: A car travels 100 km at 50 km/h and then 150 km at 75 km/h. Time = 2 h + 2 h = 4 h. Average speed = 250 ÷ 4 = 62.5 km/h.
5. Relative speed
| Objects move … | Relative speed |
|---|---|
| in the same direction | difference of speeds (S₁ − S₂) |
| in opposite directions (towards or away from each other) | sum of speeds (S₁ + S₂) |
Worked example 4: Two places are 150 km apart. Two people start at the same time towards each other at 30 km/h and 20 km/h. Relative speed = 50 km/h. They meet after 150 ÷ 50 = 3 hours.
6. Problems on trains
| Train crosses … | Distance to cover |
|---|---|
| a pole, a man standing or a signal post | length of the train |
| a platform, bridge or tunnel | length of train + length of platform |
| another train | sum of the two train lengths (use relative speed) |
Worked example 5: A train 150 m long runs at 54 km/h. How long does it take to pass a pole? Speed = 54 × 5/18 = 15 m/s. Time = 150 ÷ 15 = 10 seconds.
Worked example 6: A train 200 m long at 72 km/h crosses a platform 300 m long. Speed = 20 m/s. Distance = 200 + 300 = 500 m. Time = 500 ÷ 20 = 25 seconds.
Worked example 7: Two trains 120 m and 180 m long run at 50 km/h and 40 km/h.
- Opposite directions: relative speed = 90 km/h = 25 m/s. Time to cross = 300 ÷ 25 = 12 seconds.
- Same direction: relative speed = 10 km/h = 25/9 m/s. Time = 300 × 9/25 = 108 seconds.
Worked example 8 (train passing a moving man): A train 125 m long passes a man running at 5 km/h in the same direction in 10 seconds. Relative speed = 125 ÷ 10 = 12.5 m/s = 12.5 × 18/5 = 45 km/h. Train speed = 45 + 5 = 50 km/h.
7. Boats and streams
Let the speed of the boat in still water be u and the speed of the stream (current) be v.
| Term | Formula |
|---|---|
| Downstream speed (with the current) | D = u + v |
| Upstream speed (against the current) | U = u − v |
| Speed of boat in still water | u = (D + U) ÷ 2 |
| Speed of stream | v = (D − U) ÷ 2 |
Worked example 9: A boat goes downstream at 16 km/h and upstream at 10 km/h. Speed in still water = (16 + 10) ÷ 2 = 13 km/h; speed of the stream = (16 − 10) ÷ 2 = 3 km/h.
Worked example 10: A boat’s speed in still water is 15 km/h and the stream flows at 3 km/h. It goes 36 km downstream and comes back. Downstream speed = 18 km/h, time = 36 ÷ 18 = 2 h. Upstream speed = 12 km/h, time = 36 ÷ 12 = 3 h. Total time = 5 hours.
8. Quick exam tips
- Convert all speeds to the same unit before calculating; for trains and seconds, use m/s.
- For a train, add the platform or other train’s length to the distance; for a pole or a standing man, use only the train’s length.
- Same direction: subtract speeds. Opposite direction: add speeds.
- Average speed for equal distances is 2xy ÷ (x + y), never (x + y) ÷ 2.
- Downstream is always faster than upstream; still water speed is the average of the two.
📌 Key points to remember
- Distance = Speed × Time.
- km/h to m/s: multiply by 5/18; m/s to km/h: multiply by 18/5.
- Average speed = total distance ÷ total time; for equal distances use 2xy ÷ (x + y).
- For the same distance, the ratio of times is the inverse of the ratio of speeds.
- Relative speed: same direction = difference; opposite directions = sum.
- A train passing a pole covers its own length; passing a platform covers train + platform length.
- Downstream = u + v; upstream = u − v.
- Boat speed = (D + U) ÷ 2; stream speed = (D − U) ÷ 2.
Test yourself: Time, Speed and Distance
Practise The basic formula, Average speed and Boats and streams with 50 questions. Each attempt picks 25 at random.