Travelling at full speed agaist the current of a river, a 
motorboat moves at the rate of 12 kilometers per hour relative
to the land.  Travelling at half speed with the current, it 
moves 8 kilometers per hour. find the maximum speed of the boat 
in still water.

Let x = maximum speed in still water in k/h
Let y = speed of the river current in k/h.

The river current slows the boat down from speed x k/h 
to x-y k/h, so

>>...travelling at full speed agaist the current of a river, 
a motorboat moves at the rate of 12 kilometers per hour relative 
to the land...<<

tells us  

x - y = 12

When the boat travels at half speed with the current, the current
speeds up the boat from half speed (1/2)x k/h to (1/2)x+y k/h, so

>>...travelling at half speed with the current, it moves 8 
kilometers per hour...<<

tells us

(1/2)x + y = 8

so we have this system of 2 equations in 2 unknowns:

     x - y = 12
(1/2)x + y = 8

Can you solve that?

x = 40/3 k/h, y = 4/3 k/h

So the boat goes 13 1/3 k/h in still water, and the 
current's speed is 1 1/3 k/h

Edwin