Abdul travelled 300 km by train …

Suppose, speed of the train be x km/hr and the speed of taxi be y km/h.
time taken to cover 300 km by the train = $\frac { 300 } { x }$hours
time taken to cover 200 km by the taxi = $\frac { 200 } { y }$ hours
Total time taken = $5 \frac { 30 } { 60 } \text { hours } = 5 \frac { 1 } { 2 } \text { hours } = \frac { 11 } { 2 }$ hours
$\therefore \frac { 300 } { x } + \frac { 200 } { y } = \frac { 11 } { 2 }$
$\Rightarrow \frac { 600 } { x } + \frac { 400 } { y } = 11$
Put $\frac 1x$ = u and $\frac 1y$ = v
$\Rightarrow$$600u + 400v = 11$..........(i)

time taken  to cover $260\ km$ by the train = $\frac { 260 } { x }$hours
time taken to cover $240\ km$ by the taxi = $\frac { 240 } { y }$ hours
Total time taken = $5 \frac { 36 } { 60 } \text { hours } = 5 \frac { 1 } { 2 } \text { hours } = \frac { 11 } { 2 }$hours
$\Rightarrow$$1300u + 1200v = 28$..........(ii)
Multiplying (i) by 3 and subtracting (ii) from it, 
$\Rightarrow500 u = 5 \Rightarrow u = \frac { 5 } { 500 } \Rightarrow u = \frac { 1 } { 100 }$
Substituting u = $\frac{1}{100}$ in (i), $\Rightarrow$ v = $\frac{1}{80}$
$\therefore u = \frac { 1 } { 100 } \Rightarrow \frac { 1 } { x } = \frac { 1 } { 100 } \Rightarrow x = 100$
$v = \frac { 1 } { 80 } \Rightarrow \frac { 1 } { y } = \frac { 1 } { 80 } \Rightarrow y = 80$
$\therefore$ the speed of the train = $100\ km/hr$
the speed of the taxi = $80\ km/hr$