Arithmetic

Time, Speed & Distance

Medium

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

$3 : 2$

$2 : 1$

$5 : 3$

$2 : 3$

✅ Correct Option: 1

We define our variables clearly:

Speed of train A = 5x (where x is some unit of speed)

Speed of train B = 3x (given that speeds are in ratio 5:3)

Length of train A = L₁

Length of train B = L₂

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds = 5x + 3x = 8x

We visualize what happens step by step:

At t = 0: Front ends of both trains cross each other (this is our reference point)

At t = 46 seconds: Front end of train A crosses the rear end of train B

At t = 46 + 69 = 115 seconds: Rear ends of both trains cross each other

From t = 0 to t = 46 seconds:

The front of train A travels from the front of train B to the rear of train B

This distance covered = Length of train B = L₂

Using the fundamental equation: Distance = Speed × Time

Distance covered by both trains combined = Relative speed × Time

L₂ = 8x × 46

L₂ = 368x

From t = 46 to t = 115 seconds (a duration of 69 seconds):

The trains continue moving, and now the rear end of A crosses the rear end of B

In these 69 seconds, the distance covered = Length of train A = L₁

Using the same principle:

L₁ = 8x × 69

L₁ = 552x

The ratio of length of train A to train B:

$\frac{L₁}{L₂} = \frac{552x}{368x} = \frac{552}{368}$

To simplify this fraction, we find the GCD of 552 and 368:

552 = 368 × 1 + 184

368 = 184 × 2 + 0

So GCD = 184

$\frac{552}{368} = \frac{552 ÷ 184}{368 ÷ 184} = \frac{3}{2}$

In relative motion problems involving two objects moving in opposite directions, we always use their combined speed (relative speed) to calculate distances. This is because from the perspective of one train, the other train appears to be moving at the relative speed.

When trains cross each other completely, the total distance covered equals the sum of their lengths. But in this problem, we are tracking specific crossing points, so we get individual train lengths.

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

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