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If the ratio of the length of the two trains E and F is 7:9 and these two trains cross each other while travelling in the same direction in 128 seconds, and the speed of trains E and F is 88 km/hr and 52 km/hr respectively, then find the difference between the lengths of trains E and F in meters?

a) 300

b) 100

c) 220

d) 160

e) 240

Answer :

To solve this problem, we need to find the difference between the lengths of trains E and F. Let's break this down step by step.

  1. Convert Speeds to Meters per Second:

    • Train E's speed: [tex]88 \text{ km/hr} = \frac{88 \times 1000}{3600} \text{ m/s} = \frac{220}{9} \text{ m/s}[/tex].
    • Train F's speed: [tex]52 \text{ km/hr} = \frac{52 \times 1000}{3600} \text{ m/s} = \frac{130}{9} \text{ m/s}[/tex].

  2. Find the Relative Speed:

    When two objects move in the same direction, the relative speed is the difference in their speeds.

    • Relative speed = [tex]\frac{220}{9} - \frac{130}{9} = \frac{90}{9} \text{ m/s} = 10 \text{ m/s}[/tex].

  3. Calculate the Total Length of Both Trains:

    Since the trains cross each other while traveling in the same direction, the time taken to completely cross each other is the time it takes one train to cross the full length of the other in addition to its own length. The total length of both trains together can be found using:

    • Total length = Relative speed [tex]\times[/tex] Time = [tex]10 \text{ m/s} \times 128 \text{ s} = 1280 \text{ meters}[/tex].

  4. Using the Ratio of Lengths:

    We know the ratio of the lengths of trains E and F is 7:9. Let's denote the lengths of train E and F by [tex]7x[/tex] and [tex]9x[/tex] respectively.

    • According to the ratio, [tex]7x + 9x = 1280[/tex].
    • This simplifies to [tex]16x = 1280[/tex], so [tex]x = \frac{1280}{16} = 80[/tex].

  5. Find the Difference in Lengths:

    • Length of train E = [tex]7x = 7 \times 80 = 560[/tex] meters.
    • Length of train F = [tex]9x = 9 \times 80 = 720[/tex] meters.
    • Difference in lengths = [tex]720 - 560 = 160[/tex] meters.

Thus, the difference between the lengths of trains E and F is 160 meters.

The correct answer is option d) 160.

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