Trains, Boats and Streams Formula Guide: Speed, Crossing Time and Upstream Downstream Formulas for SSC Maths
Rahul Kumar
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PDF Notes: Trains, Boats and Streams Formula Guide: Speed, Crossing Time and Upstream Downstream Formulas for SSC Maths
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Problems on trains and on boats and streams use the relative speed concept in a fixed set of formulas. This trains, boats and streams formula guide for SSC gives all the formulas in tables, with the unit conversion and the common tricks, along with solved examples, practice questions and FAQs.
Unit Conversion
| Conversion | Rule |
|---|---|
| km/h to m/s | Multiply by 5/18 |
| m/s to km/h | Multiply by 18/5 |
| 1 km/h | 5/18 m/s (about 0.278 m/s) |
| 36 km/h | 10 m/s |
| 54 km/h | 15 m/s |
| 72 km/h | 20 m/s |
Problems on Trains
| Case | Time or Formula |
|---|---|
| Train crosses a pole or a man (standing) | Length of the train / speed of the train |
| Train crosses a platform or a bridge | (Length of the train + length of the platform) / speed |
| Train crosses a man moving in the same direction | Length / (speed of the train − speed of the man) |
| Train crosses a man moving in the opposite direction | Length / (speed of the train + speed of the man) |
| Two trains crossing in opposite directions | (L1 + L2) / (v1 + v2) |
| Two trains crossing in the same direction (overtaking) | (L1 + L2) / (v1 − v2) |
| Two trains start at the same time from two stations towards each other and after meeting take t1 and t2 hours | v1 : v2 = √t2 : √t1 |
| Distance covered to cross | The length of the train (a pole) or the sum of the lengths |
Boats and Streams
| Term | Formula |
|---|---|
| Speed in still water | u |
| Speed of the stream (current) | v |
| Downstream speed | u + v |
| Upstream speed | u − v |
| Speed in still water | (Downstream + upstream)/2 |
| Speed of the stream | (Downstream − upstream)/2 |
| Average speed for a round trip (same distance) | (Downstream × upstream) × 2 / (downstream + upstream) = (u² − v²)/u |
| Time for a round trip of D km each way | D/(u + v) + D/(u − v) = 2uD/(u² − v²) |
| Ratio of the time upstream and downstream | (u + v) : (u − v) |
- If the speed of a boat in still water is x times the speed of the stream, the ratio of the times for the same distance upstream to downstream is (x + 1) : (x − 1).
- A swimmer follows the same rules; the speed in still water is the swimmer’s own speed.
- If the time upstream is twice the time downstream for the same distance, then u = 3v.
Solved Examples
Example 1. A 150 m train crosses a pole in 10 seconds. Find its speed.
150/10 = 15 m/s = 54 km/h.
Example 2. The same train crosses a platform of 250 m. Find the time.
(150 + 250)/15 = 26.67 seconds.
Example 3. Two trains of lengths 100 m and 200 m run in opposite directions at 20 m/s and 25 m/s. Find the time of crossing.
300/45 = 6.67 seconds.
Example 4. Two trains of lengths 150 m and 100 m run in the same direction at 72 km/h and 54 km/h. Find the time for the faster to overtake the slower.
Relative speed = 18 km/h = 5 m/s; time = 250/5 = 50 seconds.
Example 5. A boat goes at 12 km/h in still water and the stream is 3 km/h. Find the time for a 30 km trip each way (down and up).
Downstream 30/15 = 2 hours; upstream 30/9 = 3.33 hours; total = 5 hours 20 minutes.
Example 6. A man rows downstream at 10 km/h and upstream at 4 km/h. Find the speed in still water and the speed of the stream.
Still water = (10 + 4)/2 = 7 km/h; stream = (10 − 4)/2 = 3 km/h.
Quick Tips
- Convert the speed to m/s when the length is in metres.
- Add the lengths when a train crosses a train or a platform.
- Add the speeds for opposite directions and subtract them for the same direction.
- Use (down + up)/2 and (down − up)/2 for boats.
Quick Revision Points
- km/h to m/s: × 5/18.
- Crossing a platform: (L + P)/v.
- Opposite directions: add speeds; same direction: subtract.
- Downstream = u + v; upstream = u − v.
- Still water speed = (D + U)/2.
Practice Questions With Answers
- Convert 72 km/h to m/s.
Answer: 20 m/s - A 200 m train crosses a pole in 8 s. Find its speed.
Answer: 25 m/s (90 km/h) - A 100 m train at 36 km/h crosses a 100 m bridge. Find the time.
Answer: 20 s - Two trains of 120 m and 180 m run towards each other at 15 m/s and 15 m/s. Find the crossing time.
Answer: 10 s - A boat goes 18 km downstream in 2 hours and 18 km upstream in 3 hours. Find the speed in still water.
Answer: 7.5 km/h - Find the speed of the stream in the above question.
Answer: 1.5 km/h - A boat with a speed of 8 km/h in still water goes 20 km downstream in 2 hours. Find the speed of the stream.
Answer: 2 km/h - The speed of a man in still water is 6 km/h and the stream is 2 km/h. Find the upstream speed.
Answer: 4 km/h - Convert 25 m/s to km/h.
Answer: 90 km/h - A train 250 m long passes a man running at 6 km/h in the opposite direction in 10 s. Find the speed of the train.
Answer: 84 km/h
How to Practise This Chapter
Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Formula Guide notes page, read the previous topic Average, Ages and Partnership Formula Guide and the next topic Permutation, Combination and Probability Formula Guide.
Frequently Asked Questions
How do I convert km/h to m/s?
Multiply by 5/18.
What is the time for a train to cross a pole?
The length of the train divided by its speed.
What is the time to cross a platform?
(Length of the train + length of the platform) divided by the speed.
What is the relative speed in opposite directions?
The sum of the speeds.
What is the relative speed in the same direction?
The difference of the speeds.
What is the downstream speed?
The speed of the boat in still water plus the speed of the stream.
What is the upstream speed?
The speed of the boat in still water minus the speed of the stream.
How do I find the speed of the stream?
Half the difference of the downstream and the upstream speeds.
Is this topic asked in SSC?
Yes. Trains and boats are asked in every SSC maths paper.
How much time should a question take?
About 40 seconds.
What is the length covered when two trains cross?
The sum of their lengths.
What is the average speed of a round trip in a stream?
(u² − v²)/u.
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Average, Ages and Partnership Formula Guide: Important Formulas, Tricks and Solved Examples for SSC Maths
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Permutation, Combination and Probability Formula Guide: nPr, nCr, Circular Arrangements and Probability Rules for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.