Problems on Train – MBA CUET PG – Notes & Practice Questions

Train problems = Time–Speed–Distance + Relative Speed + Length of moving objects

Trains always cover their own length, plus any extra length of the object they cross.

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

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Distance covered includes train length + object length.

Speed Conversions

$$1\text{ km/hr} = \frac{5}{18}\text{ m/s}$$

$$1\text{ m/s} = \frac{18}{5}\text{ km/hr}$$

Train questions often require m/s.

Passing a Standing Object

Objects: man, pole, tree, signboard → length = 0

Distance covered = length of train

$$t = \frac{\text{Length of Train}}{\text{Speed}}$$

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Crossing a Platform

Platform has length.
Distance covered:

$$\text{Distance} = \text{Train Length} + \text{Platform Length}$$

$$t = \frac{L_{\text{train}} + L_{\text{platform}}}{\text{Speed}}$$

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Two Trains Crossing – Opposite Directions

Relative speed increases:

$$S_{\text{rel}} = S_1 + S_2$$

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Distance = sum of lengths.

$$t = \frac{L_1 + L_2}{S_1 + S_2}$$

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Two Trains Overtaking – Same Direction

Relative speed decreases:

$$S_{\text{rel}} = |S_1 – S_2|$$

Distance = sum of lengths.

$$t = \frac{L_1 + L_2}{|S_1 – S_2|}$$

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Train Passing a Moving Man / Car / Bike

Treat man/car as moving object.

Same Direction:

$$S_{\text{rel}} = S_{\text{train}} – S_{\text{man or car}}$$

Opposite Direction:

$$S_{\text{rel}} = S_{\text{train}} + S_{\text{man or car}}$$

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Time:

$$t = \frac{L_{\text{train}}}{S_{\text{rel}}}$$

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Train Meeting Problem

If two trains meet:

$$\text{Distance} = S_{\text{rel}} \times t$$

Typical structure:

• Starts from opposite ends

• Meet after t hours

Used to find unknown speeds or distances.

Trains with Percent Change in Speed

If speed increases by x%:

$$T_{\text{new}} = T_{\text{old}} \times \frac{100}{100+x}$$

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If speed decreases by x%:

$$T_{\text{new}} = T_{\text{old}} \times \frac{100}{100-x}$$

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Used when crossing time changes.

Train Length from Two Situations

If train takes

$t_1$

t1​ sec to cross pole → gives train length
If same train takes

$t_2$

t2​ sec to cross platform:

$$L_{\text{platform}} = S t_2 – S t_1$$

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Average Speed in Train Problems

If crossing two objects of different distances at different speeds → use:

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

Average Speed=Total TimeTotal Distance​

But NOT for equal distances with different speeds → then use:

$$\text{Avg Speed} = \frac{2uv}{u+v}$$

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Train Starting & Stopping, Delays

If train slows due to temporary speed drop:

Steps:

1. Compute time at normal speed

2. Compute time at reduced speed

3. Compare

4. Adjust distance accordingly

Shortcuts

1: Overtaking Time

If Train A overtakes Train B:

$$t = \frac{L_A + L_B}{S_A – S_B}$$

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2: Opposite Direction Crossing

$$t = \frac{L_A + L_B}{S_A + S_B}$$

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3: Platform Length Formula

$$\text{Platform Length} = S(t_{\text{platform}} – t_{\text{pole}})$$

4: Train Length from Speed & Time

$$L = S \times t$$

5: Convert km/hr → m/s instantly

Divide by 3.6
(Equivalent to × 5/18)

Races Involving Trains

If track length = D
Train speed = S

Time to complete =

$$\frac{D}{S}$$

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If two trains compete:

• Faster beats slower by difference in times

• Or faster gains distance per second = relative speed

Used for “train A beats train B by ___ seconds”.

Observations for Exams

• Any situation where both objects move → use relative speed

• If crossing happens → distance = sum of lengths

• If overtaking → same direction relative speed

• If object is stationary → distance = only train’s length

• Change in crossing time → find speed change or length change

Fractional Tricks

Some standard speed conversions:

• 54 km/h = 15 m/s

• 36 km/h = 10 m/s

• 72 km/h = 20 m/s

• 90 km/h = 25 m/s

• 108 km/h = 30 m/s

Most Common Patterns

Expect questions like:

• Train crosses pole, then platform → find lengths

• Train crossing another moving train in same/opposite direction

• Train crosses a person/car running at x m/s

• Train reduces speed; crossing time changes → find % change

• Relative speed based on delayed arrival times

• Finding train speed from two different crossing times

• Train meets/starts/just crosses at signals

• Trains starting at different times

Practice Questions

1. A train 180 m long passes a pole in 12 s. Time to cross a platform 420 m long?
A) 28 s
B) 30 s
C) 32 s
D) 36 s

2. Two trains 200 m & 300 m long run in opposite directions at 54 & 36 km/h. Time to cross?
A) 18 s
B) 20 s
C) 24 s
D) 30 s

3. A train passes a pole in 10 s and a 300 m platform in 25 s. Find train length.
A) 120 m
B) 150 m
C) 180 m
D) 200 m

4. A train increases speed by 20% and now crosses a pole in 12 s instead of 15 s. Find original speed.
A) 36 km/h
B) 45 km/h
C) 54 km/h
D) 60 km/h

5. A train 240 m long crosses another train 360 m long running in same direction at 12 m/s & 8 m/s. Time to cross?
A) 90 s
B) 120 s
C) 150 s
D) 180 s