A 160-m long train is running at 45 km/hr. In what time will it pass Meera, who is running at 9 km/hr in the same direction as the train?

16 seconds. Let train length L = 160 m, train speed = 45 km/hr, and Meera's speed = 9 km/hr. Both move the same way, so relative speed = difference of speeds. Relative speed = 45 - 9 = 36 km/hr. Convert to m/sec: 36 x 5/18 = 10 m/sec. To fully pass Meera, the train covers its own length L relative to Meera. Time = L / relative speed = 160/10 = 16 seconds. Check: 10 m/sec x 16 sec = 160 m, which matches L.

A 540-m long train is running at 99 km/hr. In what time will it pass Karan, who is running at 9 km/hr in the direction opposite to that of the train?

18 seconds. Let train length L = 540 m, train speed = 99 km/hr, and Karan's speed = 9 km/hr. They move towards each other, so relative speed = sum of speeds. Relative speed = 99 + 9 = 108 km/hr. Convert to m/sec: 108 x 5/18 = 30 m/sec. To fully pass Karan, the train covers its own length L relative to Karan. Time = L / relative speed = 540/30 = 18 seconds. Check: 30 m/sec x 18 sec = 540 m, which matches L.

Two trains, 240 metres and 160 metres long, are running towards each other on parallel tracks at 54 km/hr and 90 km/hr. How much time will they take to cross each other?

10 seconds. Let the train lengths be 240 m and 160 m, with speeds 54 km/hr and 90 km/hr. They move towards each other, so relative speed = sum of speeds. Relative speed = 54 + 90 = 144 km/hr. Convert to m/sec: 144 x 5/18 = 40 m/sec. To fully cross, the trains together cover the sum of their lengths. Distance = 240 + 160 = 400 m. Time = distance / relative speed = 400/40 = 10 seconds. Check: 40 m/sec x 10 sec = 400 m, which matches.

Two trains, 220 metres and 240 metres long, are running towards each other on parallel tracks at 36 km/hr and 36 km/hr. How much time will they take to cross each other?

23 seconds. Let the train lengths be 220 m and 240 m, with speeds 36 km/hr and 36 km/hr. They move towards each other, so relative speed = sum of speeds. Relative speed = 36 + 36 = 72 km/hr. Convert to m/sec: 72 x 5/18 = 20 m/sec. To fully cross, the trains together cover the sum of their lengths. Distance = 220 + 240 = 460 m. Time = distance / relative speed = 460/20 = 23 seconds. Check: 20 m/sec x 23 sec = 460 m, which matches.

Two trains, 170 metres and 290 metres long, are running on parallel tracks in the same direction, the faster one at 108 km/hr and the slower one at 36 km/hr. How long will the faster train take to completely cross the slower one?

23 seconds. Let the faster train be 170 m long at 108 km/hr. Let the slower train be 290 m long at 36 km/hr. They move the same way, so relative speed = difference of speeds. Relative speed = 108 - 36 = 72 km/hr. Convert to m/sec: 72 x 5/18 = 20 m/sec. To fully overtake, the faster train covers the sum of both lengths relative to the slower one. Distance = 170 + 290 = 460 m. Time = distance / relative speed = 460/20 = 23 seconds. Check: 20 m/sec x 23 sec = 460 m, which matches.

What distance must a train of length l metres cover to completely pass a stationary pole or signal post?

l metres (its own length). A pole or signal post is a single point, with no length of its own. The train has passed it only once the train's rear also clears that point. So the front must travel a distance equal to the train's own length. That distance is l metres.

Two trains of lengths a metres and b metres move towards each other at u m/sec and v m/sec. What is the correct formula for the time they take to cross each other?

(a + b) / (u + v). Let the train lengths be a metres and b metres. To fully cross, together they must cover the sum of their lengths, a + b metres. Since they move towards each other, relative speed = u + v m/sec. Time = distance / relative speed = (a + b) / (u + v).

Two trains start at the same time from stations A and B towards each other. After crossing, the train from A takes 9 seconds to reach B, and the train from B takes 36 seconds to reach A. What is the ratio of the speed of the train from A to the speed of the train from B?

6 : 3. Let speed of the train from A = Sa, and speed of the train from B = Sb. Let t = time in seconds for the two trains to meet. Before meeting, the A-train covers Sa x t and the B-train covers Sb x t. After meeting, each train still has to cover the other train's pre-meeting distance. So Sb x t = Sa x 9, since the A-train covers that gap in 9 seconds. Also Sa x t = Sb x 36, since the B-train covers that gap in 36 seconds. Dividing these two equations gives (Sa/Sb)^2 = 36/9. So Sa : Sb = sqrt(36) : sqrt(9) = 6 : 3.

Problems on Trains: Formulas, Shortcuts and Solved Examples

Train problems are speed, distance and time with one twist.

This lesson gives each rule, the reason it works and seven worked examples. It also lists the traps that cost marks.

The idea in brief

Convert: km/hr × 5/18 = m/sec.
Pole: distance = l.
Platform: distance = l + b.
Head-on: add speeds.
Chase: subtract speeds.

What Train Problems Ask

What are these questions about?

A train crosses a pole, a platform, a person or another train. You find how long it takes.

Every one is a speed, distance and time sum.

Which unit trap comes first?

Speeds are often in km/hr but lengths are in metres. Convert before you divide.

To convert km/hr to m/sec, multiply by 5/18.

Why it works: 1 km is 1000 m and 1 hr is 3600 sec, and 1000/3600 = 5/18.

Check 1

To change km/hr into m/sec, multiply by 18/5.

What distance does the train cover?

It covers its own length to pass a point. It covers train plus object length to pass a long object.

A pole has no length. A platform adds its own length to the distance.

Core Rules and Why They Work

How long to pass a pole or signal post?

Time = l ÷ speed.

Why it works: the train moves its own length to clear a point.

How long to pass a platform of length b?

Time = (l + b) ÷ speed.

Why it works: the front travels l + b before the tail clears the far end.

How long for trains moving towards each other?

Time = (a + b) ÷ (u + v).

Why it works: they close the gap at the sum of their speeds.

How long to overtake in the same direction?

Overtaking time = (a + b) ÷ (u − v).

Why it works: the faster train gains only the difference in speeds each second.

Head-on, speeds add. In a chase, speeds subtract.

How do I get a speed ratio after trains cross?

If they take t₁ and t₂ seconds to finish the journey, the speed ratio is √t₂ : √t₁.

Why it works: each train covers the other’s distance in that time, so speeds follow the square roots.

Why do the lengths add when trains cross?

Crossing starts when the two fronts meet. It ends when the two tails clear each other.

In that time the trains must slide past by a full a plus a full b.

So the gap to close is a + b, whatever the speeds are.

How can I convert quickly?

Divide by 3.6 to go from km/hr to m/sec.

For example, 90 km/hr is 25 m/sec.

Why it works: dividing by 3.6 is the same as multiplying by 5/18.

Check 2

Two trains in the same direction have a relative speed of u + v.

Seven Solved Examples

Easy, step by step: a 120 m train runs at 54 km/hr. How long does it take to pass a pole?

Step 1: 54 × 5/18 = 15 m/sec.

Step 2: distance = 120 m.

Step 3: time = 120 ÷ 15 = 8.

The answer is 8 seconds.

Easy, step by step: a 150 m train at 72 km/hr passes a 250 m platform. Find the time.

Step 1: 72 × 5/18 = 20 m/sec.

Step 2: distance = 150 + 250 = 400 m.

Step 3: time = 400 ÷ 20 = 20.

The answer is 20 seconds.

Medium, step by step: a 160 m train at 45 km/hr passes Meera running at 9 km/hr in the same direction.

Step 1: relative speed = 45 − 9 = 36 km/hr = 10 m/sec.

Step 2: distance = 160 m, since a person has no length.

Step 3: time = 160 ÷ 10 = 16.

The answer is 16 seconds.

Medium, step by step: a 540 m train at 99 km/hr passes Karan running at 9 km/hr in the opposite direction.

Step 1: relative speed = 99 + 9 = 108 km/hr = 30 m/sec.

Step 2: distance = 540 m.

Step 3: time = 540 ÷ 30 = 18.

The answer is 18 seconds.

Medium, step by step: trains of 240 m and 160 m run towards each other at 54 and 90 km/hr. How long do they take to cross?

Step 1: relative speed = 54 + 90 = 144 km/hr = 40 m/sec.

Step 2: distance = 240 + 160 = 400 m.

Step 3: time = 400 ÷ 40 = 10.

The answer is 10 seconds.

Exam level, step by step: trains of 170 m and 290 m run in the same direction at 108 and 36 km/hr. How long does the faster one take to overtake?

Step 1: relative speed = 108 − 36 = 72 km/hr = 20 m/sec.

Step 2: distance = 170 + 290 = 460 m.

Step 3: time = 460 ÷ 20 = 23.

The answer is 23 seconds.

Exam level, step by step: after they meet, train A takes 9 seconds to reach B and train B takes 36 seconds to reach A. Find the speed ratio.

Step 1: the ratio is √36 : √9.

Step 2: that is 6 : 3.

Step 3: the ratio is the same as 2 : 1.

The speed ratio A : B is 6 : 3.

Check 3

In example 3, Meera adds nothing to the distance because a person has no length.

Common Traps and Shortcuts

Which traps cost marks?

Forgetting to convert km/hr into m/sec.
Adding the speeds in a same-direction chase.
Subtracting the speeds in a head-on meeting.
Leaving out the platform or second-train length.
Using the second train’s length when passing a person.

Which shortcuts save time?

Learn the common pairs: 36 km/hr is 10 m/sec and 72 km/hr is 20 m/sec.
Keep speeds in km/hr until the end, then convert once.
Check that the time looks sensible, usually a few seconds to a minute.

What should I do before I submit?

Re-read the question. Note whether the object is a pole, a person, a platform or a train.

Then pick the distance rule first, and the speed rule second.

Check 4

Keeping speeds in km/hr while distances are in metres gives the right time directly.

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