Problems Based on Trains

Competitive Mathematics · Time, Work and Distance

7 Part5 worked examples46 practice questions

For a car we ignore its length — it is a point. But a 200 m train passing a pole raises a question: when has it passed? When its last coach has cleared the pole. So from the engine reaching the pole to the last coach leaving it, the train has covered its whole length. All train questions come from this one idea: distance covered while passing = length of the train + length of whatever it passes. The rest is plain speed-time-distance.

दूरी = L_{ट्रेन} + L_{वस्तु}, समय = \frac{दूरी}{( सापेक्ष ) चाल}

The formula for every passing question

Pole, platform and bridge

Passing something that stands still

Length for a pole; length + platform for a platform or a bridge.

Pole, man and signal

An object of negligible length

Passing a pole or a standing man covers exactly the train’s length.

See what the train passes — for a pole / standing man / signal the distance is the train’s length $L$ .
If the speed is in km/h and lengths in metres, convert the speed to m/s ( $\times \frac{5}{18}$ ).
Length $=$ speed $\times$ time; speed $=$ length $\div$ time.
If the speed is asked, convert back to km/h ( $\times \frac{18}{5}$ ).

Example 1 · Basic

A 180 m train passes a pole in 9 seconds. Find its speed in km/h.

Solution, step by step

1Speed $= \frac{180}{9} = 20$ m/s.
2 $20 \times \frac{18}{5} = 72$ km/h

Answer 72 km/h

Example 2 · Basic

A train at 54 km/h passes a pole in 8 seconds. Find its length.

Solution, step by step

1 $54 \times \frac{5}{18} = 15$ m/s.
2Length $= 15 \times 8 = 120$ m

Answer 120 m

BasicTry it yourself first

A 150 m long train passes a signal post in 10 seconds. What is its speed in km/h?

BasicTry it yourself first

A 120 m long train passes a signal post in 8 seconds. What is its speed in km/h?

StandardTry it yourself first

A 300 m long train passes a signal post in 18 seconds. What is its speed in km/h?

StandardTry it yourself first

A 250 m long train passes a signal post in 15 seconds. What is its speed in km/h?

AdvancedTry it yourself first

A 360 m long train passes a signal post in 12 seconds. What is its speed in km/h?

AdvancedTry it yourself first

A 480 m long train passes a signal post in 16 seconds. What is its speed in km/h?

BasicTry it yourself first

A train running at 72 km/h passes a pole in 9 seconds. What is its length?

BasicTry it yourself first

A train running at 36 km/h passes a pole in 15 seconds. What is its length?

StandardTry it yourself first

A train running at 90 km/h passes a pole in 12 seconds. What is its length?

StandardTry it yourself first

A train running at 63 km/h passes a pole in 8 seconds. What is its length?

AdvancedTry it yourself first

A train running at 108 km/h passes a pole in 10 seconds. What is its length?

AdvancedTry it yourself first

A train running at 45 km/h passes a pole in 16 seconds. What is its length?

Platforms and bridges

Train + platform

Passing a platform covers $L + P$ ; solving the unknown from two situations.

Platform / bridge: distance $=$ train $+$ platform. From the engine reaching the start to the last coach leaving the end, that is the distance.
Pole and bridge both given: the speed is equal, so $\frac{L}{t _{2}} = \frac{L + B}{t _{1}}$ gives $L = \frac{B t _{2}}{t _{1} - t _{2}}$ .
Two bridges given: the train’s length cancels; speed $= \frac{B _{1} - B _{2}}{t _{1} - t _{2}}$ .
A platform as long as the train: total distance $= 2 L$ .
Time for one platform known: get the speed, then the time for another.

Example 3 · Standard

A 200 m train at 72 km/h crosses a platform in 25 seconds. How long is the platform?

Solution, step by step

172 km/h $= 20$ m/s; distance in 25 s $= 500$ m.
2Platform $= 500 - 200 = 300$ m

Answer 300 m

Example 4 · Standard

A train crosses a 200 m bridge in 15 seconds and a man standing on it in 10 seconds. Find its length and speed.

Solution, step by step

1The extra $200$ m takes $15 - 10 = 5$ seconds ⟹ speed $= 40$ m/s.
2Length $= 40 \times 10 = 400$ m; speed $= 144$ km/h

Answer 400 m; 144 km/h

Example 5 · Standard

A 150 m train crosses a 250 m platform in 20 seconds. How long will it take to cross a 350 m platform?

Solution, step by step

1Speed $= \frac{150 + 250}{20} = 20$ m/s.
2Time $= \frac{150 + 350}{20} = 25$ seconds

Answer 25 seconds

BasicTry it yourself first

A 150 m long train running at 72 km/h crosses a platform in 25 seconds. How long is the platform?

BasicTry it yourself first

A 200 m long train running at 54 km/h crosses a platform in 30 seconds. How long is the platform?

StandardTry it yourself first

A 180 m long train running at 36 km/h crosses a platform in 40 seconds. How long is the platform?

StandardTry it yourself first

A 240 m long train running at 90 km/h crosses a platform in 20 seconds. How long is the platform?

AdvancedTry it yourself first

A 120 m long train running at 63 km/h crosses a platform in 24 seconds. How long is the platform?

AdvancedTry it yourself first

A 300 m long train running at 108 km/h crosses a platform in 30 seconds. How long is the platform?

BasicTry it yourself first

A 200 m long train crosses a 300 m long platform in 25 seconds. Find its speed in km/h.

BasicTry it yourself first

A 160 m long train crosses a 240 m long platform in 20 seconds. Find its speed in km/h.

StandardTry it yourself first

A 180 m long train crosses a 270 m long platform in 30 seconds. Find its speed in km/h.

StandardTry it yourself first

A 170 m long train crosses a 280 m long platform in 25 seconds. Find its speed in km/h.

AdvancedTry it yourself first

A 250 m long train crosses a 350 m long platform in 24 seconds. Find its speed in km/h.

AdvancedTry it yourself first

A 300 m long train crosses a 500 m long platform in 40 seconds. Find its speed in km/h.

StandardTry it yourself first

A train crosses a 240 m bridge in 18 seconds and a man standing on it in 10 seconds (steady speed). How long is the train?

StandardTry it yourself first

A train crosses a 180 m bridge in 18 seconds and a man standing on it in 12 seconds (steady speed). How long is the train?

StandardTry it yourself first

A train crosses a 240 m bridge in 20 seconds and a man standing on it in 12 seconds (steady speed). How long is the train?

AdvancedTry it yourself first

A train crosses a 300 m bridge in 26 seconds and a man standing on it in 16 seconds (steady speed). How long is the train?

AdvancedTry it yourself first

A train crosses a 160 m bridge in 14 seconds and a man standing on it in 10 seconds (steady speed). How long is the train?

AdvancedTry it yourself first

A train crosses a 250 m bridge in 22 seconds and a man standing on it in 12 seconds (steady speed). How long is the train?

StandardTry it yourself first

A train crosses bridges of 400 m and 250 m in 30 s and 20 s at a steady speed. Find the speed in km/h.

StandardTry it yourself first

A train crosses bridges of 360 m and 240 m in 28 s and 20 s at a steady speed. Find the speed in km/h.

AdvancedTry it yourself first

A train crosses bridges of 500 m and 300 m in 35 s and 25 s at a steady speed. Find the speed in km/h.

AdvancedTry it yourself first

A train crosses bridges of 600 m and 350 m in 45 s and 30 s at a steady speed. Find the speed in km/h.

BasicTry it yourself first

A train crosses a platform of its own length in 30 seconds at 72 km/h. What is the length of the train?

StandardTry it yourself first

A train crosses a platform of its own length in 40 seconds at 54 km/h. What is the length of the train?

StandardTry it yourself first

A train crosses a platform of its own length in 24 seconds at 63 km/h. What is the length of the train?

AdvancedTry it yourself first

A train crosses a platform of its own length in 36 seconds at 90 km/h. What is the length of the train?

AdvancedTry it yourself first

A train crosses a platform of its own length in 32 seconds at 45 km/h. What is the length of the train?

StandardTry it yourself first

A 200 m long train crosses a 300 m platform in 25 seconds. How long will it take to cross a 400 m platform?

StandardTry it yourself first

A 180 m long train crosses a 320 m platform in 25 seconds. How long will it take to cross a 420 m platform?

AdvancedTry it yourself first

A 240 m long train crosses a 360 m platform in 30 seconds. How long will it take to cross a 480 m platform?

AdvancedTry it yourself first

A 100 m long train crosses a 200 m platform in 15 seconds. How long will it take to cross a 500 m platform?

AdvancedTry it yourself first

A 120 m long train passes a tree in 4 seconds. How long will it take to cross a 180 m platform and a 140 m bridge, 40 m apart?

AdvancedTry it yourself first

A 100 m long train passes a tree in 5 seconds. How long will it take to cross a 200 m platform and a 140 m bridge, 60 m apart?

AdvancedTry it yourself first

A 80 m long train passes a tree in 4 seconds. How long will it take to cross a 120 m platform and a 70 m bridge, 30 m apart?

Passing each other and passing a man

Through the relative speed

Two trains crossing each other; a train passing a moving man; trains of equal length.

Two trains crossing each other

Add the lengths; add or subtract the speeds

Distance $= x + y$ ; relative speed: $u + v$ in opposite directions, $u - v$ in the same direction.

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Passing a moving man

Add or subtract the man’s speed

The man’s length is negligible, so the distance is the train’s length; the speed is relative.

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Meeting between stations

Distance, head start and the clock

Meeting after starting from two stations, a late start, one covering more, and speed ratios.

Place and time of meeting

Lead first, then the joint speed

When one starts late, first find the other’s lead by the time he starts.

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Times after meeting and speed ratios

The square-root ratio

If two trains start together and the times to finish after meeting are given, find the ratio of speeds.

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Mixed questions

Reduced speed, an accident and stoppages

More time when the speed drops, an accident on the way, and minutes lost to stoppages.

Reduced speed, accident, stoppages

Same distance, different speeds

The distance is the same in every case; that gives the equation.

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Traps and fast tricks

Slips · tricks

Where marks go in train questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

\frac{5}{18}

and relative speed

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Problems Based on Trains

Pole, platform and bridge

Pole, man and signal

Platforms and bridges

Passing each other and passing a man

Two trains crossing each other

Passing a moving man

Meeting between stations

Place and time of meeting

Times after meeting and speed ratios

Mixed questions

Reduced speed, accident, stoppages

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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