Pathik travels $600 \, km$ to his home partly by train and partly by car. He takes $8$ hours if he travels $120 \, km$ by train and the rest by car. He takes $20$ minutes longer if he travels $200 \, km$ by train and the rest by car. Find the speed of the train and the car.

(A) Let the speed of the train be $x \, km/hr$ and the speed of the car be $y \, km/hr$.
Case $1$: Time taken = $\frac{120}{x} + \frac{480}{y} = 8$ hours. Dividing by $8$,we get $\frac{15}{x} + \frac{60}{y} = 1$.
Case $2$: Time taken = $8$ hours $20$ minutes = $8 + \frac{20}{60} = 8 + \frac{1}{3} = \frac{25}{3}$ hours. The equation is $\frac{200}{x} + \frac{400}{y} = \frac{25}{3}$. Dividing by $25$,we get $\frac{8}{x} + \frac{16}{y} = \frac{1}{3}$.
Let $u = \frac{1}{x}$ and $v = \frac{1}{y}$. The equations become $15u + 60v = 1$ and $8u + 16v = \frac{1}{3}$.
Solving these,we get $u = \frac{1}{60}$ and $v = \frac{1}{80}$.
Thus,$x = 60 \, km/hr$ and $y = 80 \, km/hr$.