The speed of a boat in still water is $8\text{km/h}$. If it can travel $20\text{km}$ downstream and $12\text{km}$ upstream in the same amount of time, the speed of the stream is:

Let the speed of the boat in still water be $u=8\text{km/h}$ and the speed of the stream be $v\text{km/h}$.
* Downstream Speed ($S_{\text{down}}$) = $u+v=8+v$
* Upstream Speed ($S_{\text{up}}$) = $u-v=8-v$

According to the problem, the times taken for both journeys are equal:

$$t_{\text{down}}=t_{\text{up}}$$

$$\frac{{\text{Distance}}_{\text{down}}}{S_{\text{down}}}=\frac{{\text{Distance}}_{\text{up}}}{S_{\text{up}}}$$

$$\frac{20}{8+v}=\frac{12}{8-v}$$

Simplify the fraction by dividing both numerators by $4$:

$$\frac{5}{8+v}=\frac{3}{8-v}$$

Cross-multiply to solve for $v$:

$$5(8-v)=3(8+v)$$

$$40-5v=24+3v$$

$$40-24=3v+5v$$

$$16=8v$$

$$v=2\text{km/h}$$