A BOAT GOES 12KM UPSTREAM AND …

Let the speed of the boat in still water be x km/hr and speed of the stream be y km/hr.
Then,
Speed of the boat while going upstream     = (x - y)km/hr
Speed of the boat while going downstream = (x + y) km/hr

Also we know that, time taken to cover ' d ' Km with speed ' s ' Km/hr is {tex} \frac ds{/tex}

Hence,Time taken by the boat to cover 12 km upstream = {tex}\frac{12}{x-y}{/tex}hrs
And,Time taken by the boat to cover 40 km downstream = {tex}\frac{40}{x+y}{/tex}hrs
According to the question, Total time taken = 8 hrs
{tex}\therefore \frac { 12 } { x - y } + \frac { 40 } { x + y } = 8{/tex}.........(1)
Again, time taken by the boat to cover 16 km upstream = {tex}\frac{16}{x-y}{/tex}
And,Time taken by the boat to cover 32 km downstream = {tex}\frac{32}{x+y}{/tex}
According to the question,Total time taken = 8 hrs
{tex}\therefore\frac { 16 } { ( x - y ) } + \frac { 32 } { ( x + y ) } = 8{/tex}.........(2)
Putting {tex}\frac { 1 } { ( x - y ) } = u{/tex} and {tex}\frac { 1 } {( x + y ) } = v{/tex} in equation (1) & equation (2), so that we may get linear equations in the variables u & v as following :-
     12u + 40v = 8
{tex}\Rightarrow{/tex} 3u + 10v = 2........(3)
and
16u + 32v = 8
{tex}\Rightarrow{/tex}2u + 4v = 1.........(4)
Multiplying  equation (3) by 4 and equation (4) by 10, we get ;
12u + 40v = 8..........(5)
20u + 40v = 10........(6)
Subtracting equation (5) from equation (6), we get
{tex}8u = 2 \Rightarrow u = \frac { 1 } { 4 }{/tex}
Putting u = {tex}\frac 14{/tex} in equation (3), we get
{tex}3 \times \frac { 1 } { 4 } + 10 v = 2 \Rightarrow 10 v = \frac { 5 } { 4 } \Rightarrow v = \frac { 1 } { 8 }{/tex}
{tex}u = \frac { 1 } { 4 } \Rightarrow \frac { 1 } { x - y } = \frac { 1 } { 4 } \Rightarrow x - y = 4{/tex}.....(7)
{tex}v = \frac { 1 } { 8 } \Rightarrow \frac { 1 } { x + y } = \frac { 1 } { 8 } \Rightarrow x + y = 8{/tex}......(8)
On adding (7) and (8), we get
2x = 12
{tex}\Rightarrow{/tex}x = 6
Putting x = 6 in (8), we get
6 + y = 8
{tex}\Rightarrow{/tex}y = 8 - 6 = 2
{tex}\therefore{/tex} x = 6, y = 2
Hence, the speed of the boat in still water = 6 km/hr and speed of the stream = 2 km/hr