Given:
Train A started after 2 hours of train D
Both train meet at a distance of 297 km from Delhi
Formula used:
Total distance = speed × time
Solution:
The length of all the trains (in meters)
⇒ (A + B + B + C + C + D + D + E + E + A)
⇒ 600 + 800 + 700 + 500 + 660
⇒ (A + B + C + D + E) = 3260/2 = 1630 meters
Now, (A + B + C) = (A + B + C + D + E) - (D + E)
⇒ A + B + C = 1630 - 500 = 1130
So, the length of A = (A + B + C) - (B + C) = 1130 - 800 = 330 meter
The length of C = (A + B + C) - (A + B) = 1130 - 600 = 530 meter
The length of B = 800 - 530 = 270 meter
The length of D = 700 - 530 = 170 meter
The length of E = 660 - 330 = 330 meter
While traveling in the opposite direction,
Relative speed of A + B = 600/12 = 50 m/s
Relative speed of B + C = 800/20 = 40 m/s
Relative speed of C + D = 700/25 = 28 m/s
Relative speed of D + E = 500/10 = 50 m/s
Relative speed of E + A = 660/12 = 55 m/s
Similarly, total relative speed of all trains (m/s)
⇒ (A + B + C + D + E) = 223/2 = 111.5 m/s
Now, A + B + C = (A + B + C + D + E) - (D + E)
⇒ A + B + C = 111.5 - 50 = 61.5 m/s
The speed of A = 61.5 - 40 = 21.5 m/s
The speed of B = 50 - 21.5 = 28.5 m/s
The speed of C = 61.5 - 40 = 21.5 m/s
The speed of D = 28 - 11.5 = 16.5 m/s
The speed of E = 55 - 21.5 = 33.5 m/s
| Train | Train Length(m) | Speed(m/s) |
|---|
| A | 330 | 21.5 |
| B | 270 | 28.5 |
| C | 530 | 11.5 |
| D | 170 | 16.5 |
| E | 330 | 33.5 |
Calculation:
Speed of D in kmph = 16.5 × 18/5 = 59.4 kmph
Speed of A in kmph = 21.5 × 5/18 = 77.4 kmph
Time taken by train D to travel 297 km = 297/59.4 = 5 hours
Train A started 2 hours late, so running time of train A = 5 - 2 = 3 hours
In 3 hours train A will go = 77.4 × 3 = 232.2 km
The distance between Delhi and Patna = 297 + 232.2 = 529.2 km
∴ Option (3) is the correct answer.