A motorboat takes 6 hours to cover 100 km downstream and 30 km upstream. If the motorboat goes 75 km downstream and returns back to its starting point in 8 hours, find the speed of the motorboat in still water and the rate of the stream.

Let x km/hr be the speed of motorboat in still water and y km/hr be the speed of stream.

Downstream speed = x + y km/h

Upstream speed = x - y km/h

Given,

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

Motorboat takes 6 hours to cover 100 km downstream and 30 km upstream.

$\Rightarrow \dfrac{100}{x + y} + \dfrac{30}{x - y} = 6$     ………(1)

Given,

Motorboat goes 75 km downstream and returns back to its starting point in 8 hours.

$\Rightarrow \dfrac{75}{x + y} + \dfrac{75}{x - y} = 8$     ….(2)

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

⇒ 100u + 30v = 6

⇒ 10(10u + 3v) = 6

⇒ (10u + 3v) = $\dfrac{6}{10}$     ……..(3)

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

⇒ 75u + 75v = 8

⇒ 75(u + v) = 8

⇒ u + v = $\dfrac{8}{75}$

⇒ u = $\dfrac{8}{75} - v$     ……..(4)

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

$\Rightarrow 10 \Big(\dfrac{8}{75} - v\Big) + 3v = \dfrac{6}{10} \\[1em] \Rightarrow \dfrac{80}{75} - 10v + 3v = \dfrac{6}{10} \\[1em] \Rightarrow \dfrac{80}{75} - 7v = \dfrac{6}{10} \\[1em] \Rightarrow 7v = \dfrac{80}{75} - \dfrac{6}{10} \\[1em] \Rightarrow 7v = \dfrac{160 - 90}{150} \\[1em] \Rightarrow 7v = \dfrac{70}{150} \\[1em] \Rightarrow v = \dfrac{7}{15 \times 7} \\[1em] \Rightarrow v = \dfrac{1}{15}.$

Substituting value of v in equation (4), we get :

$\Rightarrow u = \dfrac{8}{75} - v \\[1em] \Rightarrow u = \dfrac{8}{75} - \dfrac{1}{15} \\[1em] \Rightarrow u = \dfrac{8 - 5}{75} \\[1em] \Rightarrow u = \dfrac{3}{75} \\[1em] \Rightarrow u = \dfrac{1}{25}.$

Since,

$\Rightarrow u = \dfrac{1}{x + y} \\[1em] \Rightarrow \dfrac{1}{25} = \dfrac{1}{x + y}$

⇒ x + y = 25     …….(5)

$\Rightarrow v = \dfrac{1}{x - y} \\[1em] \Rightarrow \dfrac{1}{15} = \dfrac{1}{x - y}$

⇒ x - y = 15     ……..(6)

Adding equations (5) and (6),

⇒ x + y + x - y = 25 + 15

⇒ 2x = 40

⇒ x = $\dfrac{40}{2} = 20$.

Substituting value of x in equation (6), we get :

⇒ x - y = 15

⇒ 20 - y = 15

⇒ 20 - 15 = y

⇒ y = 5.

Hence, the speed of the motorboat in still water = 20 km/hr and the speed of the stream = 5 km/hr.