A boat moves downstream at the rate of 8 km per hour and upstream at 4 km per hour. The speed of the boat in still waters is:

The correct option is 6 km per hour.

To find the speed of the boat in still water, we can use the standard formulas for downstream and upstream speeds.

Let the speed of the boat in still water be $u$ km/h, and the speed of the stream be $v$ km/h.

When the boat moves downstream (in the direction of the water flow), the effective speed is the sum of the speed of the boat in still water and the speed of the stream:
Downstream speed = $u+v=8$ km/h (Equation 1)

When the boat moves upstream (against the water flow), the effective speed is the difference between the speed of the boat in still water and the speed of the stream:
Upstream speed = $u-v=4$ km/h (Equation 2)

To find the speed of the boat in still water ($u$), we can add Equation 1 and Equation 2 together:
$(u+v)+(u-v)=8+4$
$2u=12$
$u=\frac{12}{2}$
$u=6$ km/h

Alternatively, we can use the direct formula for the speed of a boat in still water:
Speed in still water = $\frac{1}{2}\times (\text{Downstream Speed}+\text{Upstream Speed})$

Substituting the given values:
Speed in still water = $\frac{8+4}{2}=\frac{12}{2}=6$ km/h

Thus, the speed of the boat in still water is 6 km per hour.