A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/hr. The train needs 8.4 and 8.5 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?

Step 1: Convert walking speeds to m/s

• Speed of first person = 4.5 km/hr = (4.5 × 5) / 18 = 1.25 m/s
• Speed of second person = 5.4 km/hr = (5.4 × 5) / 18 = 1.5 m/s

Step 2: Let the length of the train be L meters

We know:

Time = Distance / Relative Speed
So, L / (Train Speed - Person Speed) = Time

Use the first case (Person walking at 1.25 m/s):

L / (V - 1.25) = 8.4  → (1)

Use the second case (Person walking at 1.5 m/s):

L / (V - 1.5) = 8.5  → (2)

Step 3: Solve the two equations

From (1): L = 8.4 × (V - 1.25)

From (2): L = 8.5 × (V - 1.5)

Equating both expressions for L:

8.4(V - 1.25) = 8.5(V - 1.5)

Expand both sides:

8.4V - 10.5 = 8.5V - 12.75

Bring all terms to one side:

8.5V - 8.4V = 12.75 - 10.5
0.1V = 2.25
⇒ V = 2.25 / 0.1 = 22.5 m/s

Step 4: Convert train speed to km/h

22.5 m/s = 22.5 × 18 / 5 = 81 km/h

Answer: The speed of the train is 81 km/h.

This type of problem uses the concept of relative speed and distance = speed × time. It’s a common question in aptitude tests, and mastering it builds speed and accuracy for competitive exams.