To find the speed of the stream, we first calculate the downstream and upstream speeds.

Downstream Speed Calculation

The man rows 12 km downstream in 3 hours. The formula for speed is distance divided by time.

Speed downstream = $\frac{Distance}{Time}$ = $\frac{12 \text{ km}}{3 \text{ h}} = 4 \text{ km/h}$

Let $v_m$ be the speed of the man in still water and $v_s$ be the speed of the stream. The downstream speed is the sum of these two speeds:

$v_m + v_s = 4$ km/h (Equation 1)

Upstream Speed Calculation

The man rows 6 km upstream in 2 hours.

Speed upstream = $\frac{Distance}{Time}$ = $\frac{6 \text{ km}}{2 \text{ h}} = 3 \text{ km/h}$

The upstream speed is the difference between the man's speed and the stream's speed:

$v_m - v_s = 3$ km/h (Equation 2)

Stream Speed Determination

To find the speed of the stream ($v_s$), we can subtract Equation 2 from Equation 1:

$(v_m + v_s) - (v_m - v_s) = 4 \text{ km/h} - 3 \text{ km/h}$

$v_m + v_s - v_m + v_s = 1 \text{ km/h}$

$2v_s = 1 \text{ km/h}$

$v_s = \frac{1}{2} \text{ km/h}$

$v_s = 0.5 \text{ km/h}$

The speed of the stream is 0.5 km/h.