Two trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The rtio of their speed is?

Explanation:-

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