\begin{align} & {\text{Speed}} \cr & {\text{ = }}\left( {60 \times \frac{5}{{18}}} \right){\text{m/sec}} \cr & {\text{ = }}\frac{{50}}{3}{\text{m/sec}} \cr & {\text{Time = 27 sec}}{\text{.}} \cr & {\text{Let the length of the train be }}x{\text{ metres}}{\text{.}} \cr & {\text{Then, }}\frac{{x + 200}}{{27}}{\text{ = }}\frac{{50}}{3}{\text{ }} \cr & \Leftrightarrow x + 200 = \left( {\frac{{50}}{3} \times 27} \right) = 450 \cr & \Leftrightarrow x = 450 - 200 = 250{\text{ metres}} \cr\end{align}

\begin{align} & {\text{Length of the train}} \cr & {\text{ = Speed }} \times {\text{time}} \cr & {\text{ = 36 km/hr}} \times {\text{10 sec}} \cr & {\text{ = 36}} \times \frac{5}{{18}}{\text{m/s}} \times 10\sec \cr & = 100{\text{ metres}} \cr & {\text{Therefore, }} \cr & {\text{Time taken by train to cross a plateform}} \cr & {\text{ of 55 metre long in time}} \cr & {\text{ = }}\frac{{\left( {100 + 55} \right)}}{{36 \times \frac{5}{{18}}}} \cr & = \frac{{155}}{{10}} \cr & {\text{Time}} = 15\frac{1}{2}\, \sec \cr\end{align}

\begin{align} & {\text{Let the speed of first train be }} \cr & {{\text{S}}_1}{\text{ km/hr and speed of second train}} \cr & {\text{is }}{{\text{S}}_2}{\text{km/hr }} \cr & {\text{As we know, }} \cr & {\text{Time }} \cr & {\text{ = }}\frac{{{\text{total distance}}}}{{{\text{relative speed in same/opposite direction}}}} \cr & {\text{In the same direction}} \cr & \Rightarrow {\text{27 sec = }}\frac{{\left( {100 + 95} \right)}}{{\left( {{\text{ }}{{\text{S}}_1} - {\text{ }}{{\text{S}}_2}} \right) \times \frac{5}{{18}}}} \cr & \Rightarrow 27 = \frac{{195 \times 18}}{{\left( {{\text{ }}{{\text{S}}_1} - {\text{ }}{{\text{S}}_2}} \right) \times 5}} \cr & \Rightarrow {\text{ }}{{\text{S}}_1} - {\text{ }}{{\text{S}}_2} = 26_..(i) \cr & {\text{In the opposite direction, }} \cr & \Rightarrow 9 = \frac{{\left( {100 + 95} \right)}}{{\left( {{\text{ }}{{\text{S}}_1}{\text{ + }}{{\text{S}}_2}} \right) \times \frac{5}{{18}}}} \cr & \Rightarrow 9 = \frac{{195 \times 18}}{{\left( {{\text{ }}{{\text{S}}_1}{\text{ + }}{{\text{S}}_2}} \right) \times 5}} \cr & \Rightarrow {\text{ }}{{\text{S}}_1}{\text{ + }}{{\text{S}}_2} = 39 + 2 \cr & \Rightarrow {\text{ }}{{\text{S}}_1}{\text{ + }}{{\text{S}}_2} = 78 \cr & {\text{From equation (i) and (ii)}} \cr & \Rightarrow {\text{ }}{{\text{S}}_1} - {\text{ }}{{\text{S}}_2} = 26 \cr & \Rightarrow {\text{ }}{{\text{S}}_1}{\text{ + }}{{\text{S}}_2} = 78 \cr & \Rightarrow {\text{ }}{{\text{S}}_1} = \frac{{26 - 78}}{2} \cr & \Rightarrow {\text{ }}{{\text{S}}_1} = \frac{{104}}{2} \cr & \Rightarrow {\text{ }}{{\text{S}}_1}{\text{ = 52 km/hr and }} \cr & \, \, \, \, \, \, \, \, \, \, \, {{\text{S}}_2}{\text{ = 26 km/hr}} \cr\end{align}