Вычислите: \(2\frac{1}{3} : \left(\frac{5}{6} + \frac{1}{2}\right) + 2 \cdot 1\frac{3}{4}\).

\[2\frac{1}{3}=\frac{7}{3},\qquad \frac{5}{6}+\frac{1}{2}=\frac{5}{6}+\frac{3}{6}=\frac{8}{6}=\frac{4}{3},\qquad 1\frac{3}{4}=\frac{7}{4}.\]

\[\frac{7}{3}:\frac{4}{3}+2\cdot\frac{7}{4}=\frac{7}{3}\cdot\frac{3}{4}+\frac{7}{2}=\frac{7}{4}+\frac{14}{4}=\frac{21}{4}=5\frac{1}{4}.\]

Ответ: \(5\frac{1}{4}\).