Problem on Trains

Detailed Explanation of Train Problems in Arithmetic Aptitude

Problems involving trains require a solid grasp of the concepts of speed, distance, and time. These problems often test your ability to apply these concepts in various contexts, such as trains passing each other, overtaking a person, or crossing a bridge. Here’s a detailed explanation of each type of problem:

1. Basic Concepts

1.1 Speed, Distance, and Time

• Speed (S): The rate at which an object covers distance. It is given by:

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

  where Speed is typically measured in meters per second (m/s) or kilometers per hour (km/h).

• Distance (D): The total length of the path traveled by the object. It can be calculated using:

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

• Time (T): The duration taken to cover a certain distance. It can be calculated using:

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

1.2 Relative Speed

• When Moving in the Same Direction: The relative speed of two trains moving in the same direction is the difference between their speeds.

  $$\text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train}$$

• When Moving in Opposite Directions: The relative speed of two trains moving in opposite directions is the sum of their speeds.

  $$\text{Relative Speed} = \text{Speed of First Train} + \text{Speed of Second Train}$$

2. Types of Train Problems

2.1 Train Passing a Point or Object

• Problem: A train passes a stationary object (like a pole or a bridge) in a given time.

• Formula to Find Speed:

  $$\text{Speed} = \frac{\text{Length of the Train}}{\text{Time Taken to Pass the Object}}$$

• Example:

  • Question: A train of length 120 meters passes a pole in 10 seconds. Find the speed of the train.
  • Solution: $$\text{Speed} = \frac{120 \, \text{meters}}{10 \, \text{seconds}} = 12 \, \text{m/s}$$ Convert to km/h: $$12 \, \text{m/s} \times \frac{3600}{1000} = 43.2 \, \text{km/h}$$

2.2 Train Passing Another Train

• Problem: Two trains moving in the same or opposite directions pass each other.

• When Moving in the Same Direction:

  • Formula for Relative Speed:

    $$\text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train}$$

  • Formula for Time Taken to Pass Each Other:

    $$\text{Time} = \frac{\text{Sum of Lengths of Both Trains}}{\text{Relative Speed}}$$

• When Moving in Opposite Directions:

  • Formula for Relative Speed:

    $$\text{Relative Speed} = \text{Speed of First Train} + \text{Speed of Second Train}$$

  • Formula for Time Taken to Pass Each Other:

    $$\text{Time} = \frac{\text{Sum of Lengths of Both Trains}}{\text{Relative Speed}}$$

• Example:

  • Question: Train A (length 200 meters) and Train B (length 150 meters) are moving towards each other at speeds of 60 km/h and 40 km/h, respectively. Find the time taken to pass each other.
  • Solution: Convert speeds to meters per second: $$60 \, \text{km/h} = 60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}$$ $$40 \, \text{km/h} = 40 \times \frac{1000}{3600} = 11.11 \, \text{m/s}$$ Relative Speed: $$16.67 + 11.11 = 27.78 \, \text{m/s}$$ Total Length of Both Trains: $$200 + 150 = 350 \, \text{meters}$$ Time Taken: $$\text{Time} = \frac{350 \, \text{meters}}{27.78 \, \text{m/s}} \approx 12.6 \, \text{seconds}$$

2.3 Train Passing a Platform or Bridge

• Problem: A train passes a platform or bridge of a given length.

• Formula to Find Speed:

  $$\text{Speed} = \frac{\text{Length of Train} + \text{Length of Platform/Bridge}}{\text{Time}}$$

• Example:

  • Question: A train of length 300 meters passes a bridge of length 500 meters in 50 seconds. Find the speed of the train.
  • Solution: Total Distance Traveled: $$300 + 500 = 800 \, \text{meters}$$ Speed: $$\text{Speed} = \frac{800 \, \text{meters}}{50 \, \text{seconds}} = 16 \, \text{m/s}$$ Convert to km/h: $$16 \, \text{m/s} \times \frac{3600}{1000} = 57.6 \, \text{km/h}$$

2.4 Train Overtaking a Person

• Problem: A train overtakes a person walking alongside the track.

• Formula to Find Speed:

  $$\text{Speed of Train} = \frac{\text{Length of Train} + \text{Distance Covered by Person}}{\text{Time Taken to Overtake}}$$

• Example:

  • Question: A train of length 240 meters overtakes a person walking at 4 km/h in 30 seconds. Find the speed of the train.
  • Solution: Convert the person’s speed to m/s: $$4 \, \text{km/h} = 4 \times \frac{1000}{3600} \approx 1.11 \, \text{m/s}$$ Distance Covered by Person in 30 seconds: $$1.11 \times 30 = 33.3 \, \text{meters}$$ Total Distance Traveled by Train: $$240 + 33.3 = 273.3 \, \text{meters}$$ Speed of Train: $$\text{Speed} = \frac{273.3 \, \text{meters}}{30 \, \text{seconds}} \approx 9.11 \, \text{m/s}$$ Convert to km/h: $$9.11 \, \text{m/s} \times \frac{3600}{1000} \approx 32.8 \, \text{km/h}$$

Summary

Understanding and solving train problems require familiarity with:

• Basic equations of speed, distance, and time.
• Converting between units (m/s and km/h).
• Calculating relative speed for trains moving in the same or opposite directions.
• Applying these concepts to real-world scenarios like passing objects, overtaking, or crossing platforms and bridges.

By practicing these problems, BBA students can improve their problem-solving skills and enhance their aptitude for handling various mathematical and analytical tasks.