13.167. A motorboat and a sailboat, being 30 km apart on a lake, move towards each other and meet after 1 hour. If the motorboat were 20 km behind the sailboat and was catching up to it, it would take 3 hours and 20 minutes. Determine the speeds of the boat and the sailboat, assuming they are constant and unchanged in both cases.
## Solution.

Let $x$ km/h be the speed of the boat, and $y$ km/h be the speed of the sailboat. According to the problem, $x \cdot 1 + y \cdot 1 = 30$. In the second case, in 3 hours and 20 minutes, the boat would have traveled
$3 \frac{1}{3} y + 20$ km. Then $\frac{3 \frac{1}{3} y + 20}{x} = 3 \frac{1}{3}$. Solving the system $\left\{\begin{array}{l}x + y = 30, \\ \frac{3 \frac{1}{3} y + 20}{x} = 3 \frac{1}{3},\end{array}\right.$

we find $x = 18 \text{ km/h}, y = 12 \text{ km/h}$.

Answer: 18 and 12 km/h.