A boat goes 16 km upstream and 24 km downstream in 6 hrs. Also it covers 12 km upstream and 36 km downstream in the same time. Find the speed of the boat upstream and downstream

Explanation

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr. Then,
speed upstream = ` (x - y )` km/hr
and speed downstream = = ` (x + y ) `km /hr.
Time taken to cover 16 km upstream = `(16)/((x- y)) `hours .
Time taken to cover 24 km downstream = ` ( 24 )/(( x+ y ))` hours
Total time taken = 6 hours.
` therefore (16)/(x -y ) + ( 24)/( x + y ) = 6 " " `... (i)
Again, time taken to cover 12 km upstream = `( 12)/(( x - y ))` hours.
Time taken to cover to 36 km downstream = ` ( 36)/(( x + y )) `hours.
Total time taken = 6hours.
` therefore ( 12)/(x- y ) + ( 36)/( x+ y ) = 6 " " `... (ii)
Putting ` ( 1)/((x - y )) = u and (1)/((x + y )) = v ` in (i) and (ii), we get
` 16 u+ 24v = 6 rArr 8 u + 12 v = 3," " ` ... (iii)
`12 u + 36 v = 6 rArr 2u + 6v = 1 " " `... (iv)
On multiplying (iv) by 4 and subtracting (iii) from the result, we get
` 12 v = 1 rArr v = (1)/(12)`
` rArr (1)/(x + y ) = (1)/(12) rArr x + y = 12 " "` ... (v)
On multiplying (iv) by 2 and subtracting the result from (iii), we get
` 4u = 1 rArr u = (1)/(4) `
` rArr (1)/(x - y ) = (1)/(4) rArr x - y =4. " " `... (vi)
On adding (v) and (vi), we get ` 2x = 16 rArr x = 8`
On subtracting (vi) from (v), we get `2y = 8 rArr y = 4 `.
` therefore ` speed of the boat in still water = 8 km/hr ,
and speed of the stream = 4 km/hr.