Answer: Option B
$$\eqalign{
& {\text{Ratio of speeds}} \cr
& {\text{ = }}\sqrt 4 :\sqrt {6\frac{1}{4}} \cr
& = \sqrt 4 :\sqrt {\frac{{25}}{4}} \cr
& = 2:\frac{5}{2} \cr
& = 4:5 \cr }$$
Let the speeds of the two trains be 4x and 5x km/hr respectively
Then time taken by trains to meet each other
$$\eqalign{
& {\text{ = }}\left( {\frac{{270}}{{4x + 5x}}} \right){\text{hr}} \cr
& {\text{ = }}\left( {\frac{{270}}{{9x}}} \right){\text{hr = }}\left( {\frac{{30}}{x}} \right){\text{hr}} \cr
& {\text{Time taken by slower train to travel}} \cr
& {\text{ 270 km = }}\left( {\frac{{270}}{{4x}}} \right){\text{hr}} \cr
& \therefore \frac{{270}}{{4x}} = \frac{{30}}{x} + 6\frac{1}{4} \cr
& \Rightarrow \frac{{270}}{{4x}} - \frac{{30}}{x} = \frac{{25}}{4} \cr
& \Rightarrow \frac{{150}}{{4x}} = \frac{{25}}{4} \cr
& \Rightarrow 100x = 600 \cr
& \Rightarrow x = 6 \cr
& {\text{Hence speed of slower train}} \cr
& {\text{ = 4}}x \cr
& = \,24\,{\text{km/hr}} \cr} $$