Question

A train overtakes two people walking along a railway track. The first one walked at 4.5 km/hr. the other walking at 5.4 km/hr. The train needs 8.4 and 8.5 seconds respectively to cover them. What is the speed of the train, if both the persons are walking in the same direction as the train?

Answer 1

Hint: If the train and person are traveled in the same direction then their relative speed will be equal to the difference in their speed.
If the speed of the train is $x$ m/sec and the speed of the person is $y$m/sec.
Thus, the relative speed will be $(x-y)$m/sec.
To convert km/hr to m/sec. just multiply that number (km/hr) to $\dfrac{5}{18}$the speed will be in the parameter of m/sec.
For example, let's say the speed of the car is $90$km/hr. To change its parameter in m/sec multiply is with $\dfrac{5}{18}$
$\Rightarrow 90\times \dfrac{5}{18}$m/sec.
$\Rightarrow 25$m/sec.

Complete step by step solution:
Let's say the speed of the train is $x$m/sec.
The speed of the first person is $4.5$km/hr.
$\Rightarrow 4.5\times \dfrac{5}{18}=\dfrac{22.5}{18}$m/sec.
Therefore the relative speed of first-person and train is $S$(say)
$\Rightarrow S=\left( x-\dfrac{22.5}{18} \right)$m/sec.
Now the speed of the second person is $5.4$km/hr
$\Rightarrow 5.4\times \dfrac{5}{18}=\dfrac{27}{18}$m/sec.
Thus the relative speed of the second person and train is $P$(assumed)
$\Rightarrow P=\left( x-\dfrac{27}{18} \right)$m/sec.
Hence the speed of the train is
$\Rightarrow \left( S\times 8.4=P\times 8.5 \right)$m/sec
$\Rightarrow \left( \left( x-\dfrac{22.5}{18} \right)\times 8.4=\left( x-\dfrac{27}{18} \right)\times 8.5 \right)$ m/sec
$\Rightarrow \left( \left( \dfrac{18\times 8.4x-22.5\times 8.4}{18} \right)=\left( \dfrac{18\times 8.5x-27\times 8.5}{18} \right) \right)$m/sec
$\Rightarrow \left( 151.2x-189=153x-229.5 \right)$m/sec
$\Rightarrow \left( 153x-151.2x=229.5-189 \right)$m/sec
$\Rightarrow \left( 1.8x=40.5 \right)$m/sec.
$\Rightarrow x=\dfrac{40.5}{1.8}$m/sec
$\Rightarrow x=22.5$m/sec
$\Rightarrow x=22.5\times \dfrac{18}{5}$km/hr
$\Rightarrow 81$km/hr.

The speed of the train will be 81 km/hr.

Note:
Again convert m/sec to km/hr. just multiply that number (m/sec) to $\dfrac{18}{5}$the speed will be in the parameter of km/hr.
For example, the speed of a vehicle is 10 m/sec then it will be in km/hr as
$\Rightarrow 10\times \dfrac{18}{5}$km/hr
$\Rightarrow 36$km/hr