Train $A:$ Speed $ = 50\;km/hr,$ Time $ = 3\;hr$
Distance from $A$ to $C = 50 \times 3 = 150\;km$
Train $B:$ Speed $ = 60\;km/hr,$ Time $ = 2\;hr$
Distance from $B$ to $C = 60 \times 2 = 120\;km$
$\therefore$ The required ratio $ = \dfrac{AC}{BC} = \dfrac{150}{120} = \dfrac{5}{4}$
Hence, the ratio of the distances $AC$ to $BC$ is $ = 5:4.$
So, the correct answer is $(B).$