Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in $10$ minutes while train B takes $9$ minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train $B$ to travel from station $Y$ to station $X$ is

Two trains start simultaneously from opposite stations and travel towards each other. We need to find how long train B takes to complete its entire journey.

Given Information:

Train A starts from X, reaches Y in 10 minutes total

Train B starts from Y, takes 9 minutes to reach X after meeting A

We need: Total time for train B's journey from Y to X

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Let's say the trains meet after t minutes from the start.

At this meeting point:

Train A has been traveling for t minutes

Train B has also been traveling for t minutes

After meeting, A needs (10 - t) minutes to reach Y

After meeting, B needs 9 minutes to reach X

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Here's the key insight: After meeting, each train covers the distance that the other train covered before meeting.

Think about it:

Train A covers distance X → meeting point in t minutes

Train B must cover this same distance (meeting point → X) in 9 minutes

Similarly:

Train B covers distance Y → meeting point in t minutes

Train A must cover this same distance (meeting point → Y) in (10 - t) minutes

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Since both trains travel at constant speeds:

Distance covered by A before meeting = Distance covered by B after meeting

Using the relationship: Distance = Speed × Time

If train A covers a certain distance in t minutes, then train B covers the same distance in 9 minutes.

The speeds are inversely proportional to time, so:

$\tfrac{\text{Speed of A}}{\text{Speed of B}} = \tfrac{9}{t}$

Similarly, for the other part of the journey:

$\tfrac{\text{Speed of A}}{\text{Speed of B}} = \tfrac{t}{10-t}$

Since both ratios are equal:

$\tfrac{9}{t} = \tfrac{t}{10-t}$

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$9(10-t) = t^2$

$90 - 9t = t^2$

$t^2 + 9t - 90 = 0$

$(t + 15)(t - 6) = 0$

Since time cannot be negative: t = 6 minutes

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Train B's total journey time = Time to meeting + Time after meeting

$= 6 + 9 = 15$ minutes

Therefore, train B takes 15 minutes to travel from station Y to station X.

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This problem uses a fundamental principle: when two objects start simultaneously towards each other, the time each takes to complete the other's pre-meeting journey depends on their relative speeds. The mathematical relationship we derived captures this beautifully and works for any similar meeting problem.