A boat takes 10 hours to go 44 km downstream and 30 km upstream. Again, the same boat takes 13 hours to go 55 km downstream and 40 km upstream. The speed of the boat and the current will be :

1. speed of boat = 3 kmph, speed of current = 2 kmph

2. speed of boat = 6 kmph, speed of current = 4 kmph

3. speed of boat = 9 kmph, speed of current = 2 kmph

4. speed of boat = 8 kmph, speed of current = 3 kmph

Let x be the speed of the boat in still water and y be the speed of current,

Downstream speed = (x + y) km/hr

Upstream speed = (x - y) km/hr

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

Given,

It takes 10 hours to go 44 km downstream and 30 km upstream.

⇒ $\dfrac{44}{x + y} + \dfrac{30}{x - y} = 10$ ………(1)

Given,

It takes 13 hours to go 55 km downstream and 40 km upstream.

⇒ $\dfrac{55}{x + y} + \dfrac{40}{x - y} = 13$ ……..(2)

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

⇒ 44u + 30v = 10     ….(3)

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

⇒ 55u + 40v = 13     ….(4)

Multiply equation (3) by 4, we get :

⇒ 4(44u + 30v = 10)

⇒ 176u + 120v = 40     ….(5)

Multiply equation (4) by 3, we get :

⇒ 3(55u + 40v = 13)

⇒ 165u + 120v = 39     ….(6)

Subtracting equation (5) from equation (6), we get :

⇒ (165u + 120v) - (176u + 120v) = 39 - 40

⇒ (165u + 120v - 176u - 120v) = -1

⇒ -11u = -1

⇒ u = $\dfrac{-1}{-11} = \dfrac{1}{11}$.

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

⇒ 44u + 30v = 10

⇒ $44 \times \dfrac{1}{11}$ + 30v = 10

⇒ 4 + 30v = 10

⇒ 30v = 10 - 4

⇒ 30v = 6

⇒ v = $\dfrac{6}{30} = \dfrac{1}{5}$.

$\Rightarrow \dfrac{1}{x + y} = u \\[1em] \Rightarrow \dfrac{1}{x + y} = \dfrac{1}{11} \\[1em] \Rightarrow x + y = 11 \text{ ……..(7) } \\[1em] \Rightarrow \dfrac{1}{x - y} = v \\[1em] \Rightarrow \dfrac{1}{x - y} = \dfrac{1}{5} \\[1em] \Rightarrow x - y = 5 \text{ ………(8) }$

Adding equations (7) and (8) we get,

⇒ x + y + x - y = 11 + 5

⇒ 2x = 16

⇒ x = $\dfrac{16}{2} = 8$.

Substituting value of x in equation (8),

⇒ x - y = 5

⇒ 8 - y = 5

⇒ 8 - 5 = y

⇒ y = 3.

The speed of the boat in still water is 8 km/hr and the speed of the current is 3 km/hr.

Hence, option 4 is the correct option.