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

1. $2:3$
2. $2:1$
3. $3:2$
4. $5:3$

Let the length of train $\text{A}$ be $x \; \text{m},$ and the length of train $\text{B}$ be $y \; \text{m}.$

Let the speed of train $\text{A}$ be $5s \; \text{m/sec},$ and the speed of train $\text{B}$ be $3s \; \text{m/sec}.$

Let’s draw the diagram for a better understanding.

The front end of $\text{A}$ crossed the rear end of $\text{B 46}$ seconds after the front ends of the trains had crossed each other.

Here, $y = 46 \times 8s \quad \longrightarrow (1)$

It took another $69$ seconds for the rear ends of the trains to cross each other.

Here, $x = 69 \times 8s \quad \longrightarrow (2)$

Divide equation $(2)$ by equation $(1).$

$\Rightarrow \frac{x}{y} = \frac{ 69 \times 8s}{46 \times 8s}$

$\Rightarrow \frac{x}{y} = \frac{3}{2}$

$\therefore$ The ratio of length of train $\text{A}$ to that of train $\text{B} = x : y = 3 : 2.$

Correct Answer $: \text{C}$

$\textbf{PS:}\;\text{Important points:}$

• When two trains are going in the same direction, then their relative speed is the difference between the two speeds.
• When two trains are moving in the opposite direction, then their relative speed is the sum of the two speeds.
• When a train crosses a stationary man/ pole/ lamp post/ sign post- in all these cases, the object which the train crosses is stationary, and the distance traveled is the length of the train.
• When it crosses a platform/ bridge- in these cases, the object which the train crosses are stationary and the distance traveled is the length of the train and the length of the object.
• When two trains are moving in the same direction, then their speed will be subtracted.
• When two trains are moving in opposite directions, then their speed will be added.
• In both the above cases, the total distance is the sum of the length of both the trains.
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