A passenger train runs at the rate of $80$ km/h. It starts from the station, $6$ hours after a goods train leaves the station. The passenger train overtakes the goods train after $4$ hours. The speed of goods train is

The correct option is A $32$ km/h
Let the speed of the goods train be $x$ km/h.
Now, when the goods train is overtaken, it has travelled for a total time of $10$ hours, as it started $6$ hours early and took $4$ hours to be overtaken.
So, the distance travelled $=$ speed $\times$ time $=10x$ km

Also, by the time the train is overtaken, the passenger train should have travelled the same distance as the goods train. It is given that the speed of the passenger train is $80$ km/h and time taken is $4$ hours.
$\therefore 80\times 4=10x$
$\Rightarrow x=32$ km/h