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?

### Concept & Relative Speed The length of the train remains constant when it overtakes two different moving objects. By setting the distance expressions (relative speed multiplied by time) equal to each other, we can solve for the unknown speed of the train. $$ \text{Distance} = (v - u) \times t $$ ### Step-by-Step Solution * Let the speed of the train be $v$ km/hr. * Relative speed with respect to the first person = $(v - 4.5)$ km/hr. Time taken $t_1 = 8.4$ seconds. * Length of train $L = (v - 4.5) \times \frac{5}{18} \times 8.4$. * Relative speed with respect to the second person = $(v - 5.4)$ km/hr. Time taken $t_2 = 8.5$ seconds. * Length of train $L = (v - 5.4) \times \frac{5}{18} \times 8.5$. * Equate the two length equations: $(v - 4.5) \times 8.4 = (v - 5.4) \times 8.5$. (The $\frac{5}{18}$ constant cancels out). * Expand the terms: $8.4v - 37.8 = 8.5v - 45.9$. * Rearrange to solve for $v$: $45.9 - 37.8 = 8.5v - 8.4v$. * $8.1 = 0.1v \Rightarrow v = \frac{8.1}{0.1} = 81$ km/hr. ### Exam Strategy & Shortcut To quickly find the speed without full algebra, use the difference formula: $v = \frac{v_2 \times t_2 - v_1 \times t_1}{t_2 - t_1}$. Calculate $(5.4 \times 8.5) - (4.5 \times 8.4)$ divided by $(8.5 - 8.4)$. $45.9 - 37.8 = 8.1$. Divide by $0.1$ to get $81$ km/hr. ### Common Pitfall Making arithmetic errors when multiplying decimals like $5.4 \times 8.5$ or $4.5 \times 8.4$. Careful decimal placement is critical here. ### Final Answer Therefore, the correct answer is **81 km/hr**.