A boat covers 24 km upstream and 72 km downstream in 8 hours, while it covers 48 km upstream and 108 km downstream in 14 hours. Find the speed of the boat in still water and the speed of the stream respectively.

Let the speed of the boat in still water be V and speed of the stream be V'.

Relative speed of boat in upstream = V - V', as water stream flows against the direction of boat.

whereas Relative speed of boat in downstream = V + V', as water stream flows in the direction of boat.

Case (1)

Given Total time taken = Time taken during upstream + Time taken during downstream = 8 hours.

$$\Rightarrow \frac{24}{V - V'} + \frac{72}{V + V'} = 8$$

$$\Rightarrow 3 [\frac{1}{V - V'} + \frac{3}{V + V'}] =1$$.............(1)

$$\Rightarrow 3 [4V - 2V'] = V^2 - V'^2$$......................(2)

Case (2)

Given Total time taken = Time taken during upstream + Time taken during downstream = 14 hours

$$\Rightarrow \frac{48}{V - V'} + \frac{108}{V + V'} = 14$$

$$\Rightarrow 6 [\frac{4}{V - V'} + \frac{9}{V + V'}] = 7$$

$$\Rightarrow 6[13V - 5V'] = 7 [V^2 - V'^2]$$...................(3)

Dividing equation (3) by (2), we get

$$\frac{2 [13V - 5V']}{4V - 2V'} = 7$$

$$\Rightarrow 26V - 10V' = 28V - 14V'$$

$$\Rightarrow V = 2V'$$

Substituting this value in equation (1) we get,

$$\Rightarrow 3 [\frac{1}{V'} + \frac{1}{V'}] =1$$

$$\Rightarrow V' = 6$$

$$\Rightarrow V = 12$$

Hence, Speed of the boat in still water = 12 km/h.

and Speed of the stream = 6 km/h.