In this guide
Every question in this topic comes from one formula: distance = speed × time. Trains, boats and "reached 10 minutes late" questions are the same formula with one extra idea each. Clerk papers commonly include a question or two from this family in the arithmetic block, usually a train or boat question with clean numbers.
The fastest way through the topic is to learn each extra idea with its reason. Then you are not memorising eight formulas, only applying one.
The basics
- Speed = distance ÷ time, and time = distance ÷ speed.
- km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
Why 5/18: 1 km/h is 1,000 metres in 3,600 seconds, which is 1,000/3,600 = 5/18 m/s. Multiples of 18 km/h convert cleanly: 18 → 5, 36 → 10, 54 → 15, 72 → 20, 90 → 25 m/s.
Proportion shortcuts
- Same distance: speed and time are inversely proportional. If speed becomes 3/4 of usual, time becomes 4/3 of usual.
- Same time: distance is proportional to speed.
- Same speed: distance is proportional to time.
These let you skip equations in many questions.
Average speed
Average speed = total distance ÷ total time. It is not the mean of the speeds.
For two equal distances at speeds a and b, average speed = 2ab ÷ (a + b).
Why: take each distance as d. The time is d/a + d/b, and the total distance is 2d. Dividing gives 2d ÷ (d/a + d/b) = 2ab ÷ (a + b). The average is pulled towards the slower speed because you spend more time at it.
Relative speed and trains
- Opposite directions: relative speed = sum of speeds.
- Same direction: relative speed = difference of speeds.
- Train passing a pole, a person or a signal: distance = the train's length.
- Train passing a platform, bridge or another train: distance = sum of the two lengths.
Why lengths add: the front of the train must travel until its tail clears the far end of the platform, which is its own length plus the platform's.
Boats and streams
- Downstream speed d = boat + stream. Upstream speed u = boat − stream.
- So boat speed = (d + u) ÷ 2 and stream speed = (d − u) ÷ 2.
The last two follow from adding and subtracting the first two.
Worked examples
Example 1. A person travels equal distances at 40 km/h and 60 km/h. Find the average speed.
- 2 × 40 × 60 ÷ (40 + 60) = 4,800 ÷ 100 = 48 km/h.
- Check: take 120 km each way. Times are 3 h and 2 h. Total 240 km in 5 h = 48 km/h.
Example 2. A 200 m train runs at 72 km/h. How long does it take to pass a pole, and a 300 m platform?
- 72 km/h = 20 m/s.
- Pole: 200 ÷ 20 = 10 seconds.
- Platform: (200 + 300) ÷ 20 = 25 seconds.
Example 3. Trains of 150 m and 100 m run at 50 km/h and 40 km/h. How long do they take to cross each other in opposite directions, and in the same direction at 60 km/h and 42 km/h?
- Opposite: relative speed 90 km/h = 25 m/s. Distance 250 m. Time 10 seconds.
- Same direction at 60 and 42: relative speed 18 km/h = 5 m/s. Time 250 ÷ 5 = 50 seconds.
Example 4. A train passes a pole in 12 seconds and a 240 m platform in 24 seconds. Find its length and speed.
- The extra 12 seconds is the time to cover the platform's 240 m.
- Speed = 240 ÷ 12 = 20 m/s = 72 km/h.
- Length = 20 × 12 = 240 m.
Example 5. A boat goes 36 km downstream in 3 hours and 24 km upstream in 3 hours. Find the speeds of the boat and the stream.
- d = 12 km/h, u = 8 km/h.
- Boat = (12 + 8) ÷ 2 = 10 km/h. Stream = (12 − 8) ÷ 2 = 2 km/h.
Example 6. Walking at 3/4 of the usual speed, a person reaches the office 20 minutes late. Find the usual time.
- Same distance, so time becomes 4/3 of usual.
- The extra 1/3 of usual time is 20 minutes, so the usual time is 60 minutes.
Common mistakes
- Averaging speeds directly for equal distances: 40 and 60 give 48, not 50.
- Forgetting to convert km/h to m/s when lengths are in metres.
- Using only the train's length when it crosses a platform or another train.
- Adding speeds for trains moving in the same direction.
- Mixing up boat speed with downstream speed.
| Situation | Distance | Speed to use |
|---|---|---|
| Train passes pole | Train length | Train speed |
| Train passes platform | Train + platform | Train speed |
| Two trains, opposite directions | Sum of lengths | Sum of speeds |
| Two trains, same direction | Sum of lengths | Difference of speeds |
| Boat downstream | Given | Boat + stream |
| Boat upstream | Given | Boat − stream |
Practice
Set a 6-minute timer.
- Convert 90 km/h to m/s.
- A 300 m train at 54 km/h passes a pole. How long does it take?
- Equal distances are covered at 30 km/h and 60 km/h. Find the average speed.
- A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. How long to go 36 km upstream?
- A 180 m train at 54 km/h crosses a 120 m bridge. How long does it take?
- Trains of 110 m and 90 m run in the same direction at 65 km/h and 47 km/h. How long does the faster one take to pass the slower one?
- Walking at 5 km/h, a person is 10 minutes late; at 6 km/h, 5 minutes early. Find the distance.
- A boat goes 20 km downstream in 2 hours and 20 km upstream in 4 hours. Find the speeds of the boat and the stream.
Answers:
- 25 m/s. 90 × 5/18.
- 20 seconds. 54 km/h = 15 m/s; 300 ÷ 15.
- 40 km/h. 2 × 30 × 60 ÷ 90.
- 3 hours. Upstream speed 12 km/h; 36 ÷ 12.
- 20 seconds. 300 m at 15 m/s.
- 40 seconds. Relative speed 18 km/h = 5 m/s; 200 ÷ 5.
- 7.5 km. The time difference is 15 minutes = 1/4 hour. d/5 − d/6 = d/30 = 1/4, so d = 7.5.
- Boat 7.5 km/h, stream 2.5 km/h. d = 10, u = 5; (10 + 5) ÷ 2 and (10 − 5) ÷ 2.
What to do next
- Memorise the 18 km/h conversion ladder up to 108 km/h = 30 m/s.
- Solve five train and five boat questions a day for a week, drawing a quick sketch for each train question.
- Revisit time and work, which uses the same rate thinking.
- Practise mixed arithmetic in word problems once each topic feels comfortable.
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 Institute of Banking Personnel Selection website .
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