Question

A train leaves a station A at 7 am and reaches another station B at 11 am. Another train leaves B at 8 am and reaches A at 11.30 am. The two trains cross one another at.

• A. 8:26 am
• B. 9:26 am
• C. 9:24 am
• D. 10:00 am

Answer 1

Answer: (C)

9:24 am

Answer 2

Let the distance between station A and station B be $D$.

Train 1 leaves station A at 7 am and reaches station B at 11 am.
Time taken by Train 1 = 11 am - 7 am = 4 hours.
Speed of Train 1, $v_1 = \frac{D}{4}$ km/h.

Train 2 leaves station B at 8 am and reaches station A at 11:30 am.
Time taken by Train 2 = 11:30 am - 8 am = 3 hours 30 minutes = 3.5 hours = $\frac{7}{2}$ hours.
Speed of Train 2, $v_2 = \frac{D}{7/2} = \frac{2D}{7}$ km/h.

Train 1 starts at 7 am and Train 2 starts at 8 am. At 8 am, Train 1 has already been traveling for 1 hour.
Distance covered by Train 1 in the first hour (from 7 am to 8 am) = $v_1 \times 1 = \frac{D}{4} \times 1 = \frac{D}{4}$.

At 8 am, Train 1 is at a distance $\frac{D}{4}$ from station A towards B. Train 2 is at station B, which is at a distance $D$ from station A.
The distance between the two trains at 8 am is the total distance minus the distance covered by Train 1:
Distance between trains at 8 am = $D - \frac{D}{4} = \frac{3D}{4}$.

From 8 am onwards, Train 1 moves from its position towards B with speed $v_1$, and Train 2 moves from B towards A with speed $v_2$. Since they are moving towards each other, their relative speed is the sum of their speeds:
Relative speed = $v_1 + v_2 = \frac{D}{4} + \frac{2D}{7}$.
To add the speeds, find a common denominator (28):
Relative speed = $\frac{7D}{28} + \frac{8D}{28} = \frac{7D + 8D}{28} = \frac{15D}{28}$ km/h.

The time taken for the two trains to meet from 8 am is the distance between them at 8 am divided by their relative speed:
Time taken to meet = $\frac{\text{Distance between trains at 8 am}}{\text{Relative speed}} = \frac{3D/4}{15D/28}$.
Time taken to meet = $\frac{3D}{4} \times \frac{28}{15D} = \frac{3 \times 28}{4 \times 15}$.
Simplify the expression:
Time taken to meet = $\frac{3 \times (4 \times 7)}{4 \times (3 \times 5)} = \frac{7}{5}$ hours.

Convert the time into hours and minutes:
$\frac{7}{5}$ hours = $1 \frac{2}{5}$ hours = 1 hour + $\frac{2}{5}$ hours.
$\frac{2}{5}$ hours = $\frac{2}{5} \times 60$ minutes = $2 \times 12$ minutes = 24 minutes.
So, the time taken to meet from 8 am is 1 hour and 24 minutes.

The trains meet at 8 am + 1 hour 24 minutes = 9:24 am.