Решение:
$$\begin{aligned}
5\frac{5}{6}:8\frac{9}{10} &= \frac{5 \cdot 6 + 5}{6} : \frac{8 \cdot 10 + 9}{10} = \frac{35}{6} : \frac{89}{10} = \frac{35}{6} \cdot \frac{10}{89} = \frac{35 \cdot 10}{6 \cdot 89} = \frac{350}{534} = \frac{175}{267}
\end{aligned}$$
Ответ:
$$\frac{175}{267}$$