16a. Trains problems step by step solution procedure and detailed explanation for each problem.

16a.  Trains problems - step by step solution procedure and detailed explanation for each problem.

Home page of Arithmetic Problems 

Trains:
  • Relative speed (same direction): |Speed1 - Speed2|
  • Relative speed (opposite direction): Speed1 + Speed2
  • Time to pass = (Sum of lengths) / Relative speed
  • Distance = Speed × Time

Section 1: Trains (10 Problems)
Train Problem 1: Passing a Stationary Pole
A train 150 m long travels at 72 km/h. How long does it take to pass a stationary pole?
Step-by-Step Solution:
  1. Identify given values:
    • Train length = 150 m
    • Speed = 72 km/h
  2. Convert speed to m/s: 72 × 5/18 = 20 m/s
  3. Calculate time: Time = Distance / Speed = 150 / 20 = 7.5 seconds
  4. State result: The train takes 7.5 seconds.
Detailed Explanation: The train passes a pole when its entire length (150 m) moves past it. Converting speed to m/s ensures unit consistency. Dividing the train’s length by its speed gives the time. The pole’s negligible length simplifies the problem to a basic distance-over-speed calculation.
Answer: 7.5 seconds
Train Problem 2: Passing a Platform
A train 200 m long travels at 90 km/h. How long does it take to pass a 300 m long platform?
Step-by-Step Solution:
  1. Identify given values:
    • Train length = 200 m
    • Platform length = 300 m
    • Speed = 90 km/h
  2. Calculate total distance: 200 + 300 = 500 m
  3. Convert speed to m/s: 90 × 5/18 = 25 m/s
  4. Calculate time: Time = 500 / 25 = 20 seconds
  5. State result: The train takes 20 seconds.
Detailed Explanation: To pass the platform, the train’s front must travel from the platform’s start to its end, plus the train’s length (500 m total). Speed conversion ensures matching units. The time is the total distance divided by speed, reflecting the train clearing the platform.
Answer: 20 seconds
Train Problem 3: Two Trains, Opposite Directions
Two trains, 120 m and 180 m long, approach each other at 54 km/h and 72 km/h. How long do they take to pass completely?
Step-by-Step Solution:
  1. Identify given values:
    • Train 1 length = 120 m, Speed = 54 km/h
    • Train 2 length = 180 m, Speed = 72 km/h
  2. Calculate total distance: 120 + 180 = 300 m
  3. Calculate relative speed: 54 + 72 = 126 km/h = 126 × 5/18 = 35 m/s
  4. Calculate time: Time = 300 / 35 ≈ 8.57 seconds
  5. State result: The trains take approximately 8.57 seconds.
Detailed Explanation: In opposite directions, relative speed is the sum of their speeds. The total distance (sum of lengths) is covered at this relative speed. Converting to m/s aligns units, and the time reflects when the trains’ rears clear each other after their fronts meet.
Answer: ≈ 8.57 seconds
Train Problem 4: Two Trains, Same Direction
Two trains, 100 m and 150 m long, travel in the same direction at 60 km/h and 48 km/h. How long does the faster train take to pass the slower one?
Step-by-Step Solution:
  1. Identify given values:
    • Faster train: 100 m, 60 km/h
    • Slower train: 150 m, 48 km/h
  2. Calculate total distance: 100 + 150 = 250 m
  3. Calculate relative speed: 60 - 48 = 12 km/h = 12 × 5/18 = 10/3 m/s
  4. Calculate time: Time = 250 / (10/3) = 250 × 3/10 = 75 seconds
  5. State result: The faster train takes 75 seconds.
Detailed Explanation: In the same direction, relative speed is the difference, as the faster train gains on the slower one. The total distance is the sum of their lengths, covered at the relative speed. The time (75 seconds) is longer than in opposite directions due to the smaller relative speed.
Answer: 75 seconds
Train Problem 5: Train Catching Up
Train A travels at 50 km/h. After 2 hours, train B starts at 80 km/h in the same direction. How long does train B take to catch train A?
Step-by-Step Solution:
  1. Identify given values:
    • Train A speed = 50 km/h, Head start = 2 hours
    • Train B speed = 80 km/h
  2. Calculate head start distance: 50 × 2 = 100 km
  3. Calculate relative speed: 80 - 50 = 30 km/h
  4. Calculate time: Time = 100 / 30 = 10/3 ≈ 3.33 hours
  5. State result: Train B takes approximately 3.33 hours.
Detailed Explanation: Train A’s head start creates a 100 km gap. Train B closes this at the relative speed (30 km/h). The time is the gap divided by relative speed, assuming catching up occurs when fronts align (negligible lengths). The result is from when train B starts.
Answer: ≈ 3.33 hours
Train Problem 6: Train and Man
A train 200 m long approaches a man walking at 5 km/h in the same direction at 65 km/h. How long does it take to pass him?
Step-by-Step Solution:
  1. Identify given values:
    • Train length = 200 m, Speed = 65 km/h
    • Man speed = 5 km/h
  2. Calculate relative speed: 65 - 5 = 60 km/h = 60 × 5/18 = 50/3 m/s
  3. Calculate time: Time = 200 / (50/3) = 200 × 3/50 = 12 seconds
  4. State result: The train takes 12 seconds.
Detailed Explanation: The man’s speed reduces the train’s relative speed. The train passes when its 200 m length covers the man (negligible size). Converting to m/s, the time is calculated as distance over relative speed, showing the effect of the man’s motion.
Answer: 12 seconds
Train Problem 7: Delayed Start, Opposite Directions
Two trains, each 200 m long, travel toward each other at 54 km/h. The second train starts 5 seconds late. How long after the second train starts do they pass?
Step-by-Step Solution:
  1. Identify given values:
    • Train length = 200 m each
    • Speed = 54 km/h each
    • Delay = 5 seconds
  2. Convert speed: 54 × 5/18 = 15 m/s
  3. Calculate distance during delay: 15 × 5 = 75 m
  4. Calculate remaining distance: (200 + 200) - 75 = 325 m
  5. Calculate relative speed: 15 + 15 = 30 m/s
  6. Calculate time to meet: 325 / 30 ≈ 10.83 seconds
  7. Calculate passing time: 400 / 30 ≈ 13.33 seconds
  8. Calculate total time: 5 + 10.83 + 13.33 ≈ 29.16 seconds
  9. State result: The trains pass in approximately 29.16 seconds.
Detailed Explanation: The first train covers 75 m during the delay. The remaining distance (325 m) is covered at the relative speed until the fronts meet, then the full 400 m is covered to pass completely. Total time includes delay, meeting time, and passing time.
Answer: ≈ 29.16 seconds
Train Problem 8: Train Passing a Bridge
A train 250 m long travels at 108 km/h. How long does it take to cross a 350 m bridge?
Step-by-Step Solution:
  1. Identify given values:
    • Train length = 250 m
    • Bridge length = 350 m
    • Speed = 108 km/h
  2. Calculate total distance: 250 + 350 = 600 m
  3. Convert speed: 108 × 5/18 = 30 m/s
  4. Calculate time: Time = 600 / 30 = 20 seconds
  5. State result: The train takes 20 seconds.
Detailed Explanation: Crossing a bridge requires the train to cover its length plus the bridge’s length (600 m). Speed conversion ensures unit consistency. The time is the total distance divided by speed, similar to passing a platform but with different context.
Answer: 20 seconds
Train Problem 9: Relative Speed with Stationary Train
A train 180 m long travels at 72 km/h. How long does it take to pass a stationary train 120 m long?
Step-by-Step Solution:
  1. Identify given values:
    • Moving train length = 180 m, Speed = 72 km/h
    • Stationary train length = 120 m
  2. Calculate total distance: 180 + 120 = 300 m
  3. Convert speed: 72 × 5/18 = 20 m/s
  4. Calculate time: Time = 300 / 20 = 15 seconds
  5. State result: The train takes 15 seconds.
Detailed Explanation: The stationary train’s speed is 0, so the moving train’s speed is the relative speed. The total distance is the sum of both trains’ lengths, as the moving train must clear the stationary one. Time is calculated as distance over speed.
Answer: 15 seconds
Train Problem 10: Time to Meet
Two trains start 200 km apart, traveling toward each other at 60 km/h and 40 km/h. How long until they meet?
Step-by-Step Solution:
  1. Identify given values:
    • Distance = 200 km
    • Speed 1 = 60 km/h, Speed 2 = 40 km/h
  2. Calculate relative speed: 60 + 40 = 100 km/h
  3. Calculate time: Time = 200 / 100 = 2 hours
  4. State result: The trains meet in 2 hours.
Detailed Explanation: The trains approach each other, so their relative speed is the sum of their speeds. The time to meet is the initial distance divided by relative speed, assuming they meet when their fronts align (lengths irrelevant for meeting point).
Answer: 2 hours


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