Answer:

Speed of boat in still water = 8 km/hr

Step-by-step explanation:

Given that, speed of a stream is 3 km/hr.

Let assume that speed of the boat in still water be x km/hr

So,

Speed of boat in upstream = x - 3 km/hr

Speed of boat in downstream = x + 3 km/hr

Case :- 1

Distance covered in upstream = 15 km

Speed of boat in upstream = x - 3 km/hr

So, time taken by boat to covered 15 km in upstream,

[tex]\sf \: t_1 = \dfrac{15}{x - 3} \: hr - - - (1) \\ \\ [/tex]

Case :- 2

Distance covered in downstream = 22 km

Speed of boat in downstream = x + 3 km/hr

So, time taken by boat to covered 22 km in downstream,

[tex]\sf \: t_2 = \dfrac{22}{x + 3} \: hr - - - (2) \\ \\ [/tex]

Now, According to statement

[tex]\sf \: t_1 + t_2 = 5 \\ \\ [/tex]

[tex]\sf \: \dfrac{15}{x - 3} + \dfrac{22}{x + 3} = 5 \\ \\ [/tex]

[tex]\sf \: \dfrac{15(x + 3) + 22(x - 3)}{(x - 3)(x + 3)} = 5 \\ \\ [/tex]

[tex]\sf \: \dfrac{15x + 45 + 22x - 66}{{x}^{2} - 9} = 5 \\ \\ [/tex]

[tex]\sf \: \dfrac{37x - 21}{{x}^{2} - 9} = 5 \\ \\ [/tex]

[tex]\sf \: 5( {x}^{2} - 9) = 37x - 21 \\ \\ [/tex]

[tex]\sf \: 5{x}^{2} - 45 = 37x - 21 \\ \\ [/tex]

[tex]\sf \: 5{x}^{2} - 45 - 37x + 21 = 0 \\ \\ [/tex]

[tex]\sf \: 5{x}^{2} - 37x - 24 = 0 \\ \\ [/tex]

[tex]\sf \: 5{x}^{2} - 40x + 3x - 24 = 0 \\ \\ [/tex]

[tex]\sf \: 5x(x - 8) + 3(x - 8) = 0 \\ \\ [/tex]

[tex]\sf \: (x - 8) \: (5x + 3) = 0 \\ \\ [/tex]

[tex]\sf \: x = 8 \: \: or \: \: x = - \dfrac{3}{5} \\ \\ [/tex]

[tex]\sf\implies x = 8 \: \: \: \{as \: x \: can \: never \: be \: negative \}\\ \\ [/tex]

Hence,

Speed of boat in still water = 8 km/hr