Problems on Trains
WISDOM HELPS MATHS CLASSES BY MAHESH SIR
Concepts Formulas Solved Examples 28-Minute Practice Test
Build accuracy first, then improve your speed.
1. Learning Objectives
- Understand how train length, speed and time are connected.
- Convert kilometres per hour into metres per second and back.
- Solve questions involving poles, platforms, bridges and people.
- Use relative speed when two trains move in the same or opposite directions.
- Apply shortcuts without skipping the logic behind them.
- Measure your performance using the timed test at the end.
2. Subtopics Covered
Speed-unit conversion; crossing a pole or stationary person; crossing a platform or bridge; two trains moving in opposite directions; two trains moving in the same direction; finding train length, speed or crossing time; and average-speed-free train applications based on constant speed.
3. Concept Explanation
Concept 1: The basic relationship
Distance = Speed × Time
Time = Distance ÷ Speed
For train questions, length is usually measured in metres and time in seconds. Therefore, convert the speed into metres per second before substituting values.
Concept 2: Speed conversion
To convert km/h into m/s, multiply by 5 over 18. To convert m/s into km/h, multiply by 18 over 5.
15 m/s = 15 × 18 over 5 = 54 km/h.
Shortcut: cancel before multiplying. For example, 90 × 5 over 18 = 5 × 5 = 25 m/s.
Concept 3: Train crossing a pole or a standing person
A pole has negligible length for these questions. The train must travel its own length to pass it completely.
Example: A 150 m train moves at 54 km/h. Speed = 54 × 5 over 18 = 15 m/s. Time = 150 over 15 = 10 seconds.
Concept 4: Train crossing a platform or bridge
To clear a platform fully, the train’s front must travel the train’s own length plus the platform’s length.
Time = (Train length + Platform/bridge length) ÷ Speed
Example: A 120 m train crosses a 180 m platform at 15 m/s. Total distance = 120 + 180 = 300 m. Time = 300 over 15 = 20 seconds.
Concept 5: Two trains crossing each other
When two trains pass completely, the distance covered relative to one another is the sum of their lengths.
Same direction: relative speed = faster speed − slower speed
Crossing time = (Length1 + Length2) ÷ Relative speed
Always convert both speeds into the same unit before adding or subtracting them.
Concept 6: Train passing a moving person
If the person is moving along the track, use relative speed. Opposite directions mean add speeds; same direction means subtract speeds. The distance is the train’s length if the person is treated as a point.
Common mistakes and exam tips
- Do not use only platform length when the train must clear the platform completely.
- Do not add speeds for trains moving in the same direction.
- Do not mix km/h with metres and seconds. Convert units first.
- For a pole, person or signal post, use only the train length.
- For two trains crossing, use the sum of both train lengths.
- Check whether the question says “crosses completely,” “passes,” or “just meets.”
4. Formula Sheet with Examples
Formula: speed × 5 over 18. Use when distance is in metres and time in seconds.
Example: 108 km/h = 108 × 5 over 18 = 30 m/s.
Shortcut: divide by 18, then multiply by 5.
Formula: speed × 18 over 5. Use to express a speed in km/h.
Example: 25 m/s = 25 × 18 over 5 = 90 km/h.
Shortcut: divide by 5, then multiply by 18.
Formula: time = train length ÷ speed in m/s.
Example: 240 m at 20 m/s → 240 over 20 = 12 s.
Shortcut: length and speed units should be metres and m/s.
Formula: time = (train length + platform length) ÷ speed.
Example: lengths 100 m and 200 m, speed 15 m/s → 300 over 15 = 20 s.
Shortcut: add the lengths before dividing.
Formula: time = L1 + L2 over S1 + S2, with speeds in m/s.
Example: lengths 100 m and 150 m; speeds 10 and 15 m/s → 250 over 25 = 10 s.
Shortcut: add both lengths and both speeds.
Formula: time = L1 + L2 over faster speed − slower speed.
Example: total length 300 m; speeds 25 and 15 m/s → 300 over 10 = 30 s.
Shortcut: use the speed difference, not the sum.
Formula: length = speed × time.
Example: speed 18 m/s and pole-crossing time 14 s → 18 × 14 = 252 m.
Shortcut: the distance is exactly the train length.
Formula: speed = (train length + platform length) ÷ time.
Example: total crossing distance 360 m in 24 s → 360 over 24 = 15 m/s = 54 km/h.
Shortcut: calculate the total distance first.
5. Solved Questions
Examples 1–10 are independently written for this learning module. Examples 11–18 are exam-style questions from the class practice set. Try each question before opening its solution.
Basic 1
A train 180 m long travels at 54 km/h. How long does it take to pass a pole?
Show answer and solution
Answer: B) 12 s.
Step 1: Convert speed: 54 × 5 over 18 = 15 m/s.
Step 2: Pole-crossing distance = train length = 180 m.
Step 3: Time = 180 over 15 = 12 s.
Shortcut: Convert to m/s, then divide length by speed. Do not add any extra distance for the pole.
Basic 2
A 240 m train passes a standing person in 16 seconds. Find its speed in km/h.
Show answer and solution
Answer: C) 54 km/h.
Speed = 240 over 16 = 15 m/s. Convert: 15 × 18 over 5 = 54 km/h.
Common mistake: 15 is the speed in m/s, not km/h.
Basic 3
A 150 m train runs at 72 km/h. How long will it take to cross a 250 m platform?
Show answer and solution
Answer: C) 20 s.
72 km/h = 20 m/s. Total distance = 150 + 250 = 400 m. Time = 400 over 20 = 20 s.
Shortcut: Add the train and platform lengths first.
Basic 4
A train passes a pole in 9 seconds at 20 m/s. What is its length?
Show answer and solution
Answer: C) 180 m.
Length = speed × time = 20 × 9 = 180 m. A pole contributes no meaningful length.
Intermediate 5
Two trains of lengths 120 m and 180 m travel in opposite directions at 36 km/h and 54 km/h. Find their crossing time.
Show answer and solution
Answer: B) 12 s.
Speeds: 36 km/h = 10 m/s; 54 km/h = 15 m/s. Relative speed = 10 + 15 = 25 m/s. Combined length = 300 m. Time = 300 over 25 = 12 s.
Intermediate 6
A 160 m train overtakes a 140 m train moving in the same direction. Their speeds are 72 km/h and 54 km/h. How long does the faster train take to pass the slower one completely?
Show answer and solution
Answer: C) 60 s.
Speeds = 20 m/s and 15 m/s. Relative speed = 20 − 15 = 5 m/s. Combined length = 300 m. Time = 300 over 5 = 60 s.
Tip: Same direction means subtract speeds.
Intermediate 7
A 200 m train crosses a bridge in 25 seconds at 72 km/h. Find the bridge length.
Show answer and solution
Answer: C) 300 m.
Speed = 20 m/s. Total distance = 20 × 25 = 500 m. Bridge length = 500 − 200 = 300 m.
Shortcut: Bridge length = speed × time − train length.
Intermediate 8
A 210 m train crosses a pole in 14 seconds. At the same speed, how long does it take to cross a 240 m platform?
Show answer and solution
Answer: C) 30 s.
Speed = 210 over 14 = 15 m/s. Platform crossing distance = 210 + 240 = 450 m. Time = 450 over 15 = 30 s.
Advanced 9
Two trains of equal length cross each other in opposite directions in 10 seconds. Their speeds are 45 km/h and 63 km/h. Find the length of each train.
Show answer and solution
Answer: C) 150 m.
Speeds are 12.5 m/s and 17.5 m/s, so relative speed = 30 m/s. Combined length = 30 × 10 = 300 m. Equal lengths mean each is 300 over 2 = 150 m.
Advanced 10
A train passes a 180 m platform in 18 seconds and a pole in 6 seconds. Find the train’s speed.
Show answer and solution
Answer: C) 54 km/h.
Let speed be v m/s and train length be L. Pole: L = 6v. Platform: L + 180 = 18v. Substitute: 6v + 180 = 18v, so 180 = 12v and v = 15 m/s. In km/h: 15 × 18 over 5 = 54 km/h.
Shortcut: Difference between crossing times is 18 − 6 = 12 s, which corresponds to the 180 m platform. Hence speed = 180 over 12 = 15 m/s.
Exam-Style Practice Questions
Exam-style 11
The length of a train and that of a platform are equal. If the train, moving at 90 km/h, crosses the platform in one minute, find the length of the train (in metres).
Show answer and solution
Answer: C) 750 m.
90 km/h = 90 × 5 over 18 = 25 m/s. Distance covered in 1 minute = 25 × 60 = 1500 m.
This distance is train length + platform length. The two lengths are equal, so train length = 1500 over 2 = 750 m.
Exam-style 12
A train passes two bridges of lengths 800 m and 400 m in 100 seconds and 60 seconds respectively. Find the length of the train.
Show answer and solution
Answer: C) 200 m.
Let the train length be L m and its speed v m/s. First bridge: L + 800 = 100v. Second bridge: L + 400 = 60v.
Subtract: 400 = 40v, so v = 10 m/s. Then L = 100 × 10 − 800 = 200 m.
Shortcut: Difference in bridge lengths ÷ difference in times gives the speed: 400 over 40 = 10 m/s.
Exam-style 13
A train 300 m long is running at a speed of 25 m/s. In how much time will it cross a bridge 200 m long?
Show answer and solution
Answer: C) 20 seconds.
Total distance = 300 + 200 = 500 m. Time = 500 over 25 = 20 s.
Exam-style 14
A 200 m long train is running at a speed of 72 km/h. How long will it take to cross an 800 m long bridge?
Show answer and solution
Answer: B) 50 s.
72 km/h = 72 × 5 over 18 = 20 m/s. Total distance = 200 + 800 = 1000 m. Time = 1000 over 20 = 50 s.
Exam-style 15
An 800 m long train is running at a speed of 78 km/h. If it crosses a tunnel in 1 minute, find the length of the tunnel (in metres).
Show answer and solution
Answer: B) 500 m.
78 km/h = 78 × 5 over 18 = 65 over 3 m/s. Distance covered in 60 s = 65 over 3 × 60 = 1300 m.
Tunnel length = 1300 − 800 = 500 m.
Exam-style 16
Two trains of lengths 80 m and 120 m are running at 25 km/h and 35 km/h respectively, in the same direction on parallel tracks. How many seconds will they take to pass each other?
Show answer and solution
Answer: C) 72 s.
Same direction, so subtract the speeds: 35 − 25 = 10 km/h = 10 × 5 over 18 = 25 over 9 m/s.
Combined length = 80 + 120 = 200 m. Time = 200 ÷ 25 over 9 = 200 × 9 over 25 = 72 s.
Exam-style 17
A train 150 m long takes 30 seconds to cross a bridge 500 m long. How much time will the train take to cross a platform 370 m long?
Show answer and solution
Answer: C) 24 s.
Bridge: distance = 150 + 500 = 650 m in 30 s, so speed = 650 over 30 = 65 over 3 m/s.
Platform: distance = 150 + 370 = 520 m. Time = 520 ÷ 65 over 3 = 520 × 3 over 65 = 24 s.
Exam-style 18
A man running at 15 km/h crosses a bridge in 5 minutes. Find the length of the bridge.
Show answer and solution
Answer: D) 1250 metres.
The man is treated as a point, so the bridge length equals the distance he runs.
5 minutes = 1 over 12 hour. Distance = 15 × 1 over 12 = 1.25 km = 1250 m.
6. Quick Revision Notes
- km/h → m/s: multiply by 5 over 18.
- m/s → km/h: multiply by 18 over 5.
- Pole/person: distance = train length.
- Platform/bridge: distance = train length + platform/bridge length.
- Opposite directions: relative speed = sum of speeds.
- Same direction: relative speed = difference of speeds.
- Two trains crossing: distance = sum of both lengths.
- Length from pole data: speed × time.
- Platform length: speed × time − train length.
7. Interactive 28-Question Timed Test
Default duration: 28 minutes. Select one answer per question. Navigate between questions, mark uncertain items for review, and submit when finished. Answers and explanations remain hidden until submission. The test uses a fresh question order each time you retake it.
Before you start
- Keep a rough sheet for calculations.
- Questions are worth one point each; no negative marking is applied in this practice test.
- Use “Mark for review” for questions you want to revisit.
- When time expires, your answers are submitted automatically.
Green with ✓ = answered; amber with ★ = marked for review; dark outline = current question.
Submit your test now?
8. Answer Key and Detailed Solutions
The assessment answer key and explanations appear automatically only after submission. The learning examples above are separate from the timed assessment.
9. Performance Analysis
After submission, the test calculates your score, accuracy, unanswered count, time taken and average time per question from your actual responses. It also groups incorrect and unanswered questions by concept and recommends what to revise next. No sample score is shown before you take the test.
10. Three Recommended Next Learning Activities
- Next Topic — Time, Speed and Distance: strengthen the base relationship used in every train problem.
- Practice Recommendation — Platform and Two-Train Crossing: practise identifying total distance and relative speed until the setup becomes automatic.
- Advanced Recommendation — Relative Motion Applications: extend the same-direction and opposite-direction method to moving people and more complex crossing conditions. If your accuracy is below 80%, revise the formula sheet and retake this test before moving to advanced applications.
Learn the concept. Apply the formula. Practise with a timer. Review every mistake.
Educational content prepared for student learning. Exam patterns and marking rules may differ by examination.
