Train and Speed

In GMAT preparation, the concepts of Train and Speed are foundational in understanding and solving questions related to time, speed, and distance. Let’s break it down with detailed explanations and examples.

Key Concepts

1. Speed: The rate at which an object (e.g., a train) travels. It is usually expressed in units like kilometers per hour (km/h) or miles per hour (mph).
   • Formula: Speed=DistanceTimetext{Speed} = frac{text{Distance}}{text{Time}}Speed=TimeDistance​
2. Distance: The length of the path traveled by the object.
   • Formula: Distance=Speed×Time
3. Time: The duration for which the object travels.
   • Formula: Time=Distance ÷ Speed​
4. Relative Speed: When two objects are moving, their effective speed relative to each other depends on their direction:
   • Same Direction: Relative Speed=Speed1−Speed2​
   • Opposite Direction: Relative Speed=Speed1+Speed2​
5. Train Length: Often included in problems to account for how long it takes a train to cross an object or another train.

Common Types of GMAT Problems

1. Train Crossing a Stationary Object

The time a train takes to pass a stationary object (like a pole, tree, or signal) depends on its length and speed.

Example:
A train 200 meters long is moving at 60 km/h. How much time will it take to pass a pole?

Solution:
Convert speed to meters per second:

60 km/h=(60×1000) ÷ 3600=16.67 m/s

Time = Distance ÷ Speed = 200 ÷ 16.67 =12 seconds

2. Train Crossing a Platform

When a train passes a platform, the distance to be covered is the sum of the train’s length and the platform’s length.

Example:
A train 150 meters long crosses a platform 350 meters long in 25 seconds. Find the train’s speed.

Solution:
Total Distance = Length of Train + Length of Platform = 150+350=500 meters
Speed = Distance ÷ Time = 500 ÷ 25=20 m/s
Convert to km/h: 20×3.6=72 km/h

3. Two Trains Passing Each Other

When two trains pass each other, their relative speed determines the time taken.

Example:
Two trains, 120 meters and 150 meters long, travel at speeds of 45 km/h and 30 km/h, respectively, in opposite directions. How long will they take to cross each other?

Solution:
Relative Speed = 45+30=75 km/h

Convert to m/s:

75 km/h=(75×1000) ÷ 3600=20.83 m/s

Total Distance = 120+150=270 meters
Time = Distance ÷ Speed = 27 ÷ 020.83≈13 seconds

4. Two Trains Moving in the Same Direction

Here, the relative speed is the difference in their speeds.

Example:
Two trains, each 180 meters long, travel at speeds of 60 km/h and 40 km/h in the same direction. How long will the faster train take to pass the slower train?

Solution:
Relative Speed = 60−40=20 km/h

Convert to m/s:

20 km/h=(20×1000) ÷ 3600=5.56 m/s

Total Distance = 180+180=360 meters
Time = Distance ÷ Speed = 360 ÷ 5.56 ≈ 65 seconds

5. Train Catching Up

A faster train overtaking a slower one involves using relative speed in the same direction.

Example:
A train traveling at 90 km/h catches up with another train moving at 60 km/h. The faster train is 100 meters long. How much time will it take to completely overtake the slower train?

Solution:
Relative Speed = $90-60=30\text{km/h}$. Convert to m/s:

30 km/h=30×10003600=8.33 m/s30 , text{km/h} = frac{30 times 1000}{3600} = 8.33 , text{m/s}30km/h=360030×1000​=8.33m/s

Distance = Length of Faster Train = $100\text{meters}$.
Time = Distance ÷ Speed = 1008.33≈12 secondsfrac{100}{8.33} approx 12 , text{seconds}8.33100​≈12seconds.

Tips for GMAT Success

1. Convert Units: Always ensure consistency in units (e.g., km/h to m/s). Use $\times 5/18$ for km/h to m/s and $\times 18/5$ for m/s to km/h.
2. Practice Relative Speed: These problems are frequent on the GMAT, so mastering them will boost your performance.
3. Memorize Formulas: Know the formulas and when to apply them.
4. Draw Diagrams: Visualizing the problem can help simplify complex scenarios.
5. Solve Strategically: Skip lengthy calculations when approximate answers are acceptable.

Would you like additional practice problems or further clarification?