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Question

Train \(X\) crosses a man standing on the platform in 24 seconds and train \(Y\) crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of \(X\) to speed of \(Y\)?

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
1:2

Solving Train Speed Ratio Problems

This problem involves calculating the ratio of the speeds of two trains, Train X and Train Y, based on how long they take to cross a stationary man and how long they take to cross each other.

Understanding the Concepts

Key concepts used here are:

  • Speed, Distance, Time Relationship: The fundamental formula is \(Speed = Distance / Time\).
  • Train Crossing a Man: When a train crosses a stationary man, the distance covered is equal to the length of the train itself. The man is considered a point object.
  • Trains Crossing Each Other (Opposite Directions): When two trains move in opposite directions, their relative speed is the sum of their individual speeds (\(v_X + v_Y\)). The total distance they need to cover to completely cross each other is the sum of their lengths (\(L_X + L_Y\)).

Setting up the Equations

Let:

  • \(v_X\) be the speed of Train X.
  • \(v_Y\) be the speed of Train Y.
  • \(L_X\) be the length of Train X.
  • \(L_Y\) be the length of Train Y.

We are given the following information:

  • Train X crosses a man in 24 seconds.
  • Train Y crosses a man in 18 seconds.
  • Train X and Train Y cross each other in 20 seconds while running in opposite directions.

Step-by-Step Solution

  1. Train X crossing a man:

    Using the speed-distance-time formula, where distance is the length of the train:

    \(v_X = L_X / 24\)

    From this, we can express the length of Train X in terms of its speed:

    \(L_X = 24 * v_X\)

  2. Train Y crossing a man:

    Similarly, for Train Y:

    \(v_Y = L_Y / 18\)

    Expressing the length of Train Y in terms of its speed:

    \(L_Y = 18 * v_Y\)

  3. Trains X and Y crossing each other:

    Since they are moving in opposite directions, their relative speed is \(v_X + v_Y\).

    The total distance to cover is the sum of their lengths, \(L_X + L_Y\).

    The time taken is 20 seconds.

    Therefore, the equation is:

    \(v_X + v_Y = (L_X + L_Y) / 20\)

  4. Substituting lengths into the crossing equation:

    Now, substitute the expressions for \(L_X\) and \(L_Y\) from steps 1 and 2 into the equation from step 3:

    \(v_X + v_Y = ( (24 * v_X) + (18 * v_Y) ) / 20\)

  5. Solving for the ratio of speeds:

    Multiply both sides by 20:

    \(20 * (v_X + v_Y) = 24 * v_X + 18 * v_Y\)

    Distribute the 20:

    \(20 * v_X + 20 * v_Y = 24 * v_X + 18 * v_Y\)

    Rearrange the terms to group speeds of X on one side and speeds of Y on the other:

    \(20 * v_Y - 18 * v_Y = 24 * v_X - 20 * v_X\)

    Simplify the terms:

    \(2 * v_Y = 4 * v_X\)

    Now, find the ratio of the speed of X to the speed of Y (\(v_X / v_Y\)):

    \(v_X / v_Y = 2 / 4\)

    Simplify the fraction:

    \(v_X / v_Y = 1 / 2\)

Conclusion

The ratio of the speed of Train X to the speed of Train Y is 1:2.

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  3. If the lengths of the two trains A and B are 400 m and 500 m respectively, then what is the time taken by them to cross each other?
     


Important Questions from Problem on Trains

  1. Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.

    If the route between G and C is closed, which one of the following stations need not be passed through while travelling from H to C?

  2. A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the Shuttle Service?

  3. A train with a uniform speed passes a 122 meters long platform in 17 seconds and a 210 meters long bridge in 25 seconds. The speed of the train is:

  4. How long does a train 153 meters long running at the rate of 90 kmph take to cross a bridge 622 meters in length?

  5. A train passes a 360 metre long platform in 40 seconds and a man standing on the platform in 16 seconds. The speed of the train is:

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