Выполним вычисления по порядку:
1. $$3 - 1\frac{8}{15} = 3 - \frac{23}{15} = \frac{45}{15} - \frac{23}{15} = \frac{22}{15}$$.
2. $$\frac{4}{9} : \frac{22}{15} = \frac{4}{9} \cdot \frac{15}{22} = \frac{2}{3} \cdot \frac{5}{11} = \frac{10}{33}$$.
3. $$\frac{9}{22} - \frac{10}{33} = \frac{27}{66} - \frac{20}{66} = \frac{7}{66}$$.
4. $$\frac{7}{66} + \frac{8}{11} = \frac{7}{66} + \frac{48}{66} = \frac{55}{66} = \frac{5}{6}$$.
Ответ: $$\frac{5}{6}$$