Problems on Trains are among the most scoring and formula-driven topics in Speed, Time and Distance. But many students find them confusing because unlike simple objects - a train has length, it moves relative to other objects, and the effective speed changes depending on direction.
If you understand these three ideas:
✔ Train length
✔ Distance to be covered
✔ Relative speed
Then you can solve ANY question effortlessly.
This Problems on Trains for Speed, Time and Distance guide covers all formulas, concepts, shortcuts, cases, diagrams, FAQs, and exam tips, making it the perfect one-stop resource for competitive exams.
Quick Overview: Problems on Trains Formulas
| Concept / Situation | Distance Considered | Speed Used | Formula (With Meaning of Symbols Inside Row) |
| Convert km/hr → m/s | – | – | Speed (m/s) = a × (5/18)(a = speed in km/hr) |
| Convert m/s → km/hr | – | – | Speed (km/hr) = a × (18/5)(a = speed in m/s) |
| Basic Speed Formula | – | – | Speed = Distance ÷ Time |
| Basic Time Formula | – | – | Time = Distance ÷ Speed |
| Basic Distance Formula | – | – | Distance = Speed × Time |
| Train passing a pole / man | l (train length only) | S (actual speed) | Time = l / S(l = length of train) (S = speed of train) |
| Train passing a platform (b) | l + b | S | Time = (l + b) / S(l = train length) (b = platform/bridge/tunnel length) |
| Two trains crossing (opposite direction) | a + b | u + v | Time = (a + b) / (u + v)(a = Train A length) (b = Train B length) (u & v = speeds of trains) |
| Two trains crossing (same direction) | a + b | u – v | Time = (a + b) / (u – v)(u & v = speeds; u > v) |
| Relative speed (opposite direction) | – | – | Relative Speed = u + v(u = speed of first train, v = speed of second train) |
| Relative speed (same direction) | – | – | Relative Speed = u – v |
| Speed ratio after meeting | – | – | Speed A : Speed B = b : a(a = time taken by Train A after meeting) (b = time taken by Train B after meeting) |
Formulas for Problems on Trains:
Speed Conversion Formulas
Speed conversion is the first step in solving any train problem. Since many exams use both km/hr and m/s, mixing units leads to wrong answers. So always convert speed to the correct unit before plugging it into a formula.
1. Convert km/hr to m/s
Because trains move large distances in short time, m/s is more accurate.
a km/hr = a×(18/5) m/s
This helps you solve questions with distances in meters and speeds originally given in km/hr.
2. Convert m/s to km/hr
When speed is given in m/s and you want results in km/hr:
a m/s = a× (5/18) km/hr
Basic Speed Time–Distance Formulas (Foundation of Train Problems)
All train questions are built on these three formulas. They tell you how speed, time, and distance relate. Before using train-specific formulas, understand these basics clearly.
- Speed = Distance ÷ Time
- Time = Distance ÷ Speed
- Distance = Speed × Time
These apply everywhere in poles, platforms, overtaking, crossing, and meeting-point problems.
Train Passing Object Formulas
In Problems on Trains, a train must cover its entire length to cross an object. The object may be a pole, man, signal post, platform, or even a tunnel. Understanding this “distance to be covered” clearly is the key to solving such questions accurately.
1. Train Passing a Pole or a Man (Zero-Length Object Formula)
When a train crosses a pole, man, or signal post, the object has no length. So the train only needs to cover its own length to completely pass the object.
Formula: Time Taken by Train to Cross a Pole
Time = l / S
Where:
- l = length of the train (in meters)
- S = actual speed of the train (converted to m/s)
Why this formula works
A train is a long object. To “cross” anything, the train must move forward until its last compartment has passed the object.
For a pole (0 meters long):
Distance = train length
Because the head of the train reaches the pole instantly, the remaining length takes time to clear.
Common mistakes students make
- Using speed in km/hr without converting to m/s
- Thinking train's speed changes, it does not
- Confusing “crossing a pole” with crossing a platform (different distance)
Key tip : Whenever the object has no measurable length → distance = l only
2. Train Passing a Platform, Bridge or Tunnel (Object With Length)
In this case, the object has a length, so the total distance the train must travel is:
Train length + platform/bridge/tunnel length
This makes the time greater than crossing a pole.
Formula: Time to Cross a Platform
Time = (l + b) / S
Where:
- l = length of train
- b = length of platform (or bridge/tunnel)
- S = speed of train (use m/s)
Why do we add (l + b)?
Because the train must:
- Completely enter the platform
- Completely exit the platform
This requires the train to travel its entire own length, plus the entire platform length.
When to use this formula
Use this when object is:
- Platform
- Bridge
- Tunnel
- Railway crossing
- Any long fixed structure
Common errors
- Forgetting to add both lengths
- Mixing meter and kilometer units
- Using wrong speed unit
3. Trains Moving in Opposite Directions (Relative Speed – Speeds Add)
When two trains move toward each other, they close the distance between them faster. Hence, the effective or relative speed becomes the sum of both speeds.
Relative Speed Formula (Opposite Direction)
Relative Speed = u + v
Where:
- u = speed of train A
- v = speed of train B
Total Distance to be Crossed
Two trains must pass both their lengths, so distance = a + b
Crossing Time Formula
Time = (a + b) / (u + v)
Where:
- a = length of train A
- b = length of train B
- u + v = combined speeds
Conceptual explanation
Imagine two trains rushing toward each other:
Each second, the gap between them decreases by the sum of their speeds. So crossing takes very little time, even for long trains.
Keywords for identifying this case
- “moving toward each other”
- “opposite direction”
- “head-on approach”
- “coming from opposite sides”
4. Trains Moving in Same Direction (Relative Speed – Speeds Subtract)
When two trains travel in the same direction, the faster train must “catch up” with the slower one, which means the effective speed is smaller.
Relative Speed Formula (Same Direction)
Relative Speed = u-v
(Only when u > v)
Meaning:
- u = speed of faster train
- v = speed of slower train
Total Distance to be Covered
Same as before:
Two trains must clear both their lengths
Distance = a + b
Crossing Time Formula
Time =(a+b)/(u-v)
Why do we subtract speeds?
Because when trains move in same direction:
- Faster train gains a small advantage every second
- Effective speed is ONLY the difference between the two
Keywords to identify this case
- “moving in the same direction”
- “overtakes”
- “travelling parallel”
5. Understanding Relative Speed in Train Problems (Detailed Explanation)
Relative speed simply tells you how quickly two moving bodies approach or separate from each other.
Case 1: Opposite Direction = Speeds Add
Relative Speed = u+v
Reason: Both are moving towards each other → gap closes faster.
Case 2: Same Direction = Speeds Subtract
Relative Speed = u-v
Reason: Faster one closes gap slowly.
Where relative speed is used
- Two trains crossing
- Two trains overtaking
- Two vehicles/train and vehicle moving
- Meeting point questions
6. Speed Ratio Formula After Meeting (Important Shortcut)
This is a common formula used in CSAT, SSC and Banking reasoning questions.
If two trains meet and then reach their destinations in times:
- a = time taken by Train A after meeting
- b = time taken by Train B after meeting
Then:
Speed Ratio Formula
Speed of A : Speed of B = b : a
Why does this formula work?
Because after meeting:
- Both trains travel remaining distances
- Longer time = slower train
- Shorter time = faster train
So speed becomes inversely proportional to remaining time.
Smart Tips and Practical Tricks for Solving Problems on Trains
Mastering Problems on Trains becomes simple when you understand how speed, distance, and length work together. Most students make mistakes not because formulas are difficult, but because they apply them without understanding the situation. This section breaks down the most important concepts into clear, actionable tips so you can solve questions faster and more accurately.
1. Convert Speed into Correct Units First
Many questions mix distances in meters and speeds in km/hr. If you don’t convert, the answer will always go wrong.
Always convert km/hr → m/s using:
Speed (m/s)=Speed (km/hr)×(18/5)
Example:
54 × (18/5) = 15 m/s
This one step improves accuracy in almost every question.
2. Identify the Exact Distance the Train Must Cover
Students often forget that distance changes depending on the obstacle.
- Crossing a pole → Distance = train length (l)
- Crossing a platform → Distance = train length + platform length (l + b)
- Two trains crossing → Distance = sum of both train lengths (a + b)
Distance for platform = l+b
Visualizing the train as a moving block makes this concept very easy to understand.
3. Use Relative Speed for Moving Trains
Whenever two trains are involved, direction decides the effective speed:
Trains in opposite directions
They move toward each other, so speeds combine:
Relative Speed = u+v
Trains in the same direction
One is catching up with the other, so speeds subtract:
Relative Speed = u−v
Recognizing the direction instantly reduces calculation time.
4. Draw a Simple Line Diagram
A very underrated but powerful trick.
- Draw both trains as blocks.
- Add their lengths.
- Mark the direction of movement.
This simple visual lets you see whether you need to add or subtract speeds and what total distance the train covers. It prevents most conceptual mistakes.
5. Focus on Patterns Instead of Memorizing Formulas
Most questions follow repeated patterns:
- Train crossing pole
- Train crossing platform
- Train crossing train
- Overtaking scenarios
- Meeting scenarios
- Relative speed cases
All of these come from the same base formula:
Time = Distance/Speed
If you learn to identify the pattern, solving becomes automatic.
6. Double-Check Lengths Before Calculating
Many students ignore length during calculation and use only speed and time. But in train questions:
Length = Distance
This is the biggest reason students get wrong answers.
Always re-check:
✔ Did you add both train lengths?
✔ Did you add platform/tunnel length?
✔ Did you consider only train length when crossing a pole?
7. Practice Typical Exam Patterns
Train questions appear frequently in:
- SSC (CGL, CHSL, GD)
- Railway RRB
- Banking (IBPS, SBI)
- UPSC CSAT
- Campus aptitude tests
Practicing these patterns increases familiarity, reducing time per question.
FAQs About Problems on Trains
Q1. Why does a train require more distance to cross a platform than to cross a pole?
A train has a physical length. When it crosses a pole, it only needs to clear its own length. But a platform also has length, so the train must pass the entire platform length plus its own. The formula becomes:
Distance = l+b
This increased distance is why time required becomes greater.
Q2. Why is m/s preferred over km/hr in train calculations?
Most train lengths and platforms are measured in meters. Using km/hr creates mismatched units, which leads to mistakes. Converting speed to m/s keeps all units compatible, making calculation more accurate.
Q3. What makes relative speed essential in train problems?
Two moving trains change the effective speed. When they move toward each other, their speeds combine, increasing the rate at which they approach:
u+v
But when moving in the same direction, only the difference matters:
u−v
This change in effective speed is what determines crossing time.
Q4. Why do long trains cross each other quickly when moving in opposite directions?
Because their combined speed is very high. Even if both trains are long, the effective speed (u + v) reduces the crossing time drastically. This is why head-on crossings take only a few seconds.
Q5. How can diagrams help in solving Problems on Trains faster?
Visualizing the trains as blocks helps identify the correct distance (sum of lengths) and direction. When you see direction clearly, you instantly know whether to use u + v or u – v.
Q6. Why does the meeting-point formula use the ratio b : a?
After meeting, the trains cover the remaining distances in different times. The one taking less time must be faster. Therefore:
Speed of A : Speed of B = b:a
This formula works because distance is proportional to speed when time is given.
Q7. Why do students often get crossing-time formulas wrong?
They forget that total distance is always the sum of both train lengths, not just one. The correct formula is:
Crossing Time = (Relative Speed)/(a+b)
Ignoring this leads to incorrect answers.
Q8. What is the simplest way to master this topic?
Recognize patterns: pole → train length, platform → train + platform, two trains → lengths added + relative speed. Once you identify the pattern, solving becomes mechanical and effortless.
Q9. How do train problems help in competitive exams?
These questions test logical application rather than memory. Mastering them boosts your conceptual clarity in all speed–distance topics, improving your overall quantitative ability.
Q10. Why is practice so important in train problems?
Most errors occur due to confusion between adding and subtracting speeds or mixing units. Regular practice helps you automatically select the correct formula and method without thinking.