Important Formulas: Problems on Trains | Quantitative Aptitude (Quant) - CAT PDF Download

Introduction

• Problems on Trains are a key part of Quantitative Aptitude. They are based on speed, time, and distance, often involving relative motion and unit conversions.
  
• A train is considered a moving body of certain length (L) that must cover a distance equal to its own length (or more) to completely cross another object.

Basic Formula

Relationship between Speed, Distance & Time

• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

Unit Conversion

Different Cases: Train Crossing

(a) Train crossing a stationary object (like a pole, tree, or man)
Distance covered = Length of train
Time = Length of train ÷ Speed

(b) Train crossing a platform or bridge
Distance covered = Length of train + Length of platform (or bridge)
Time = (Length of train + Length of platform) ÷ Speed

(c) Two trains crossing each other (moving in opposite directions)
Relative Speed = Sum of their speeds
Time = (Length of train 1 + Length of train 2) ÷ Relative Speed

(d) Two trains moving in the same direction
Relative Speed = Difference of their speeds
Time = (Length of train 1 + Length of train 2) ÷ Relative Speed

Additional/Advanced Cases

1. Train Crossing a Man Walking

If a man is walking at speed ‘x’ and the train’s speed is ‘y’:

Case 1: Walking in same direction
Relative Speed = (y – x)
Time = Length of train ÷ (y – x)

Case 2: Walking in opposite direction
Relative Speed = (y + x)
Time = Length of train ÷ (y + x)

2. Train Crossing a Signal Post / Lamp Post

In this case, the distance covered is just the length of the train.
Time taken = Length of train ÷ Speed

3. Train Crossing Another Train

Example Formula:
Time = (Length of Train A + Length of Train B) ÷ (Sum or Difference of Speeds, depending on direction)

Use Sum of Speeds if moving in opposite directions.
Use Difference of Speeds if moving in the same direction.

4. Train Crossing a Tunnel or Bridge

If the train length is L₁ and tunnel length is L₂:
Total distance = L₁ + L₂
Time taken = (L₁ + L₂) ÷ Speed

Shortcut Methods

Shortcut 1:

When two trains of equal length and equal speed cross each other,
Time = (2 × Length of train) ÷ (Sum of Speeds)

Shortcut 2:
When a train crosses a pole in ‘t₁’ seconds and a platform in ‘t₂’ seconds,
Length of Platform = Speed × (t₂ – t₁)

Shortcut 3:
If a train crosses a platform twice its own length in 45 seconds, and a man in 15 seconds,
Speed ratio = Distance ratio = (3 × train length) : (train length) = 3 : 1
So, platform length = 2 × train length.

Shortcut 4:
If two trains of lengths L₁ and L₂ and speeds S₁ and S₂ cross each other in time ‘t’,
Then,
L₁ + L₂ = t × (S₁ + S₂) (when opposite directions)
or
L₁ + L₂ = t × |S₁ – S₂| (when same direction)

Solved Examples

Example 1: A train 180 m long passes a man walking at 6 km/hr in the same direction in 30 seconds. Find the speed of the train.
Sol: Relative Speed = Distance ÷ Time
= 180 ÷ 30 = 6 m/s = (6 × 18/5) = 21.6 km/hr

Train speed = Relative speed + Man’s speed = 21.6 + 6 = 27.6 km/hr

Example 2: A train 200 m long crosses a platform 300 m long in 50 seconds. Find the speed of the train in km/hr.
Sol: Distance = 200 + 300 = 500 m
Speed = 500 ÷ 50 = 10 m/s = (10 × 18/5) = 36 km/hr

Example 3: Two trains of lengths 150 m and 250 m are moving in opposite directions with speeds 40 km/hr and 60 km/hr respectively. Find the time to cross each other.
Sol: Relative Speed = 40 + 60 = 100 km/hr = (100 × 5/18) = 27.78 m/s
Total Distance = 150 + 250 = 400 m
Time = 400 ÷ 27.78 = 14.4 seconds

Summary Table

EduRev Tip: Always convert all units (speed in m/s, distance in meters, time in seconds) before applying formulas to avoid mistakes.