Question

A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. find the speed of the board in still water and the speed of the stream.

Answer 1

let us take the speed of boat in still water as v1
speed of  stream is v2
in up stream the total relative speed will be v1-v2
in down stream the total relative speed will be v1+v2
According to condition 1
time taken up stream+ time taken down stream = 8hrs
we know distance/speed
(12/v1-v2)+(40/v1+v2)=8@ Equation 1
On solving (52v1-28v2)=8((v1*v1)-(v2*v2))
Keep that aside
now according to condition two 
(12/v1-v2)+(40/v1+v2)=(16/v1-v2)+(32/v1+v2)=8
on solving we get
v1-3v2=0@ Equation 2
Substitute Equation 2 in 1 to get v2 and then v2 in Equation 2 to get v1
final result will be
v1=7.31 Km/hr
v2=2.43 Km/hr
Note: you may notice v2=0 as one of the answers but if v2=0 there is no up stream and down stream

Answer 2

let  the speed of boat in still water be x
speed of  stream be y
in up stream the speed of boat will be x - y
in down stream the speed of boat will be x + y
so given,
time taken to go up stream+ time taken to go down stream = 8hrs
we know distance/speed

⇒(12/(x - y))+(40/(x +y))=8
On solving (
52x- 28y)=8(x² - y²)

now according to condition two 
(12/(x - y))+(40/(x +y))=(16/(x - y))+(32/(x + y))=8

on solving we get

x - 3y = 0
Substitute Equation 2 in 1 to get x and then y in Equation 2
to get x
final result will be
x = 7.31 Km/hr
y = 2.43 Km/hr