A man travels 600 km partly by train and partly by car. If he covers 120 km by train and the rest by car, it takes him 8 hours. But if he travels 200 km by train and the rest by car, he takes 20 minutes longer. Find the speed of the car and that of the train.

Let x km/hr be the speed of train and y km/hr be the speed of car.

$Time = \dfrac{\text{Distance}}{\text{Speed}}$

Given,

120 km by train, rest (600 - 120 = 480 km) by car takes 8 hours.

$\Rightarrow \dfrac{120}{x} + \dfrac{480}{y} = 8$ ……..(1)

Given,

200 km by train, rest (600 - 200 = 400 km) by car takes 8 hours 20 mins = $8 + \dfrac{20}{60} = 8 + \dfrac{1}{3} = \dfrac{25}{3}$ hours.

$\Rightarrow \dfrac{200}{x} + \dfrac{400}{y} = \dfrac{25}{3}$ ……….(2)

Substituting, u = $\dfrac{1}{x}$, v = $\dfrac{1}{y}$ in equation (1), we get :

⇒ 120u + 480v = 8

⇒ 120u + 480v - 8 = 0

⇒ 8(15u + 60v - 1) = 0

⇒ 15u + 60v - 1 = 0

⇒ 15u = 1 - 60v

⇒ u = $\dfrac{1 - 60v}{15}$     ……….(3)

Substituting, u = $\dfrac{1}{x}$, v = $\dfrac{1}{y}$ in equation (2), we get :

⇒ 200u + 400v = $\dfrac{25}{3}$     ………(4)

Substituting value of u from equation (3) in equation (4), we get :

$\Rightarrow 200 \Big(\dfrac{1 - 60v}{15}\Big) + 400v = \dfrac{25}{3} \\[1em] \Rightarrow \Big(\dfrac{200 - 12000v}{15}\Big) + 400v = \dfrac{25}{3} \\[1em] \Rightarrow \Big(\dfrac{200 - 12000v + 6000v}{15}\Big) = \dfrac{25}{3} \\[1em] \Rightarrow 200 - 6000v = \dfrac{25}{3} \times 15 \\[1em] \Rightarrow 200 - 6000v = 25 \times 5 \\[1em] \Rightarrow 200 - 6000v = 125 \\[1em] \Rightarrow 6000v = 200 - 125 \\[1em] \Rightarrow 6000v = 75 \\[1em] \Rightarrow v = \dfrac{75}{6000} \\[1em] \Rightarrow v = \dfrac{1}{80}.$

Substituting value of v in equation (1), we get :

$\Rightarrow u = \dfrac{1 - 60 \times \dfrac{1}{80}}{15} \\[1em] \Rightarrow u = \dfrac{1 - \dfrac{3}{4}}{15} \\[1em] \Rightarrow u = \dfrac{\dfrac{1}{4}}{15} \\[1em] \Rightarrow u = \dfrac{1}{60}.$

Substituting,

⇒ u = $\dfrac{1}{x}$

⇒ $\dfrac{1}{60} = \dfrac{1}{x}$

⇒ x = 60.

⇒ v = $\dfrac{1}{y}$

⇒ $\dfrac{1}{80} = \dfrac{1}{y}$

⇒ y = 80.

Hence, speed of train = 60 km/h, speed of car = 80 km/h.