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?

Answer 1

Correct Answer: 15

Step 1: Assign variables

Let the speed of train \( T \) be \( t \) km/h. Speed of train \( S \) is: \[ 0.75t \] Train \( T \) starts at **3 PM**, train \( S \) starts at **4 PM**. When they meet, train \( T \) has traveled **1 hour more** than train \( S \).

Step 2: Define meeting time

Let the trains meet \( x \) hours after 3 PM. Then: - Time for train \( T \) = \( x \) hours - Time for train \( S \) = \( x - 1 \) hours Distances: \[ \text{Train T: } xt \] \[ \text{Train S: } (x - 1) \times 0.75t = 0.75xt - 0.75t \]

Step 3: Ratio of distances

It is given that train \( T \) has traveled \(\frac{3}{5}\) of the total distance, train \( S \) has traveled \(\frac{2}{5}\) of the total distance. Thus: \[ \frac{xt}{0.75xt - 0.75t} = \frac{3}{2} \]

Step 4: Solve for \( x \)

Cross-multiplying: \[ 2xt = 2.25xt - 2.25t \] \[ 0.25xt = 2.25t \] Dividing through by \( t \) (assuming \( t > 0 \)): \[ 0.25x = 2.25 \] \[ x = 9 \ \text{hours} \]

Step 5: Total travel time for train \( T \)

Train \( T \) covers \(\frac{3}{5}\) of the distance in 9 hours. Time for the entire distance: \[ \frac{9 \times 5}{3} = 15 \ \text{hours} \]

✅ Final Answer: Train \( T \) takes 15 hours to cover the entire distance.