A train overtakes two persons walking along a railway track. The first one walks at 3 km/hr. The other one walks at 4 km/hr. The train needs 7.6 and 7.8 seconds respectively to overtake them. What is the speed of the train if both the persons are walking in the same direction as the train?

Given:

The speed of the first person (S₁) = 3 km/hr

The speed of the second person (S₂) = 4 km/hr

Time taken to overtake the first person (T₁) = 7.6 seconds

Time taken to overtake the second person (T₂) = 7.8 seconds

Formula used:

Speed of train (S_t) - Speed of person (S_p) = Distance / Time

Calculations:

Let the speed of the train be S_t km/hr.

For the first person:

S_t - 3 = (S_t - 3) km/hr = Distance / 7.6 s

For the second person:

S_t - 4 = (S_t - 4) km/hr = Distance / 7.8 s

Converting the speeds to m/s:

1 km/hr = 5/18 m/s

S_t - 3 = (S_t × 5/18 - 3 × 5/18) m/s

S_t - 4 = (S_t × 5/18 - 4 × 5/18) m/s

Using the formula:

Distance covered in 7.6 s = 7.6 × (S_t - 3) × 5/18

Distance covered in 7.8 s = 7.8 × (S_t - 4) × 5/18

We know that the distances are the same, so:

7.6 × (S_t - 3) × 5/18 = 7.8 × (S_t - 4) × 5/18

⇒ 7.6 × (S_t - 3) = 7.8 × (S_t - 4)

⇒ 7.6S_t - 22.8 = 7.8S_t - 31.2

⇒ 31.2 - 22.8 = 7.8S_t - 7.6S_t

⇒ 8.4 = 0.2S_t

⇒ S_t = 8.4 / 0.2

⇒ S_t = 42 km/hr

∴ The correct answer is option (3).