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?

(Train)

Let, the length of train is x meter and the velocity of the train is v km/hr.

According to first condition,$\frac{x}{(v-4.5){\displaystyle \frac{5}{18}}}=8.4.............\left(i\right) [\because \frac{Dis\tan ce}{Velocity}=Time]$

According to second condition,$\frac{x}{(v-5.4){\displaystyle \frac{5}{18}}}=8.5............\left(ii\right)$

From equation (i) $\frac{x}{(v-4.5){\displaystyle \frac{5}{18}}}=8.4$

$$\begin{gathered} =x=(v-4.5)\times \frac{5}{18}\times 8.4 \\ =x=(v-4.5)\times \frac{42}{18} \\ =x=\frac{7(v-4.5)}{3} \\ =3x=7v-314.5 \\ =18x=42v-182............\left(iii\right) \\ From Equation \left(ii\right)\frac{x}{(v-5.4){\displaystyle \frac{5}{18}}}=8.5 \\ =x=8.5\times \frac{5}{18}(v-5.4) \\ =18x=42.5v-229.5..........\left(iv\right) \\ Subtracting equation \left(iii\right) from equation \left(iv\right) \\ \frac{18x=42.5v-229.5 \\ 18x=42v-189}{0=0.5v-40.5} \\ =0.5v=40.5 \\ \therefore v=\frac{40.5}{0.5}=81 \\ \therefore The velocity of the train is 81kmph. \left(answer\right) \end{gathered}$$