Question:hard

Two trains A and B were moving in opposite directions, their speeds being in the ratio 5 : 3. The front end of A crossed the rear end of B 46 seconds after the front ends of the trains had crossed each other. It took another 69 seconds for the rear ends of the trains to cross each other. The ratio of length of train A to that of train B is

Updated On: Jan 15, 2026
  • 2 : 3
  • 2 : 1
  • 5 : 3
  • 3:2
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The Correct Option is D

Solution and Explanation

Trains A and B are moving in opposite directions. Their speeds are in a ratio of 5:3, meaning Train A's speed is five times a constant factor (5x) and Train B's speed is three times the same constant factor (3x).

Let \( L_a \) denote the length of Train A, and \( L_b \) denote the length of Train B.

Step 1: Calculate relative speed

When the front ends of the trains meet, their relative speed is the sum of their individual speeds. Given the speed ratio of 5:3, the relative speed is:

\[ \text{Relative speed} = 5x + 3x = 8x \] where \( x \) is the constant factor.

Step 2: Determine the length of Train B

The front of Train A passes the rear of Train B 46 seconds after their fronts met. The distance Train A's front travels relative to Train B during this period is the length of Train B, \( L_b \). This takes 46 seconds:

\[ L_b = \text{Relative speed} \times \text{Time} = (8x) \times 46 = 368x \]

Step 3: Determine the length of Train A

An additional 69 seconds elapse from the moment the front of Train A passes the rear of Train B until their rears pass each other. The distance covered by the front of Train A relative to the rear of Train B during this time is the length of Train A, \( L_a \). This takes 69 seconds:

\[ L_a = \text{Relative speed} \times \text{Time} = (8x) \times 69 = 552x \]

Step 4: Establish the ratio of train lengths

The ratio of the lengths can be found by dividing the distance covered in Step 3 by the distance covered in Step 2:

\[ \frac{L_a}{L_b} = \frac{552x}{368x} \] Simplifying the expression yields:

\[ \frac{L_a}{L_b} = \frac{552}{368} = \frac{3}{2} \]

Step 5: Final Result

The ratio of the length of Train A to the length of Train B is 3:2.

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