CAT 2018 Slot 1 QA Question & Solution
Question
Train T leaves station X for station Y at 3 pm. Train S, traveling at three quarters of the speed of T, leaves Y for X at 4 pm. The two trains pass each other at a station Z, where the distance between X and Z is three-fifths of that between X and Y. How many hours does train T take for its journey from X to Y?
Solution
Train $T$ starts at 3 PM and train $S$ starts at 4 PM.
Let the speed of train $T$ be $t$.
Therefore, the speed of train $S$ is $0.75t$.
When the trains meet, train $T$ would have traveled for one more hour than train $S$.
Let us assume that the two trains meet $x$ hours after 3 PM. Train $S$ would have traveled for $x-1$ hours.
The distance traveled by train $T$ is:
$$
\text{Distance traveled by } T = x \cdot t
$$
The distance traveled by train $S$ is:
$$
\text{Distance traveled by } S = (x - 1) \cdot 0.75t = 0.75xt - 0.75t
$$
We know that train $T$ has traveled three-fifths of the distance. Therefore, train $S$ should have traveled two-fifths of the distance between the two cities.
Thus, we have the equation:
$$
\frac{x \cdot t}{0.75xt - 0.75t} = \frac{3}{2}
$$
Simplifying this:
$$
2xt = 2.25xt - 2.25t
$$
$$
0.25x = 2.25
$$
$$
x = 9 \text{ hours}
$$
Train $T$ takes 9 hours to cover three-fifths of the distance. Therefore, to cover the entire distance, train $T$ will take:
$$
9 \cdot \frac{5}{3} = 15 \text{ hours}
$$
Hence, the correct answer is $15$ hours.