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

To solve this problem, we need to use the concept of relative speed.

When the boat is going upstream, it is going against the current, so the effective speed of the boat is (speed of boat - speed of stream).

When the boat is going downstream, it is going with the current, so the effective speed of the boat is (speed of boat + speed of stream).

Let's denote the speed of the boat as 'b' and the speed of the stream as 's'.

From the first condition, we have the equation: 12/(b-s) + 40/(b+s) = 8

From the second condition, we have the equation: 16/(b-s) + 32/(b+s) = 8

Now we have a system of two equations, and we can solve it to find the values of 'b' and 's'.

To simplify the equations, we can multiply all terms by (b-s)(b+s), which gives us:

12(b+s) + 40(b-s) = 8(b² - s²) 16(b+s) + 32(b-s) = 8(b² - s²)

Solving these equations will give us the speed of the boat and the speed of the stream.

This is a complex system of equations that might require the use of algebraic methods or numerical methods to solve.

Please note that the solution to this system of equations might not be unique, meaning that there might be several pairs of (b, s) that satisfy both equations.