$$2) 1\frac{1}{22} \div 3-\frac{2}{3} \cdot (2\frac{5}{6}+3\frac{7}{23})=\frac{23}{22} \div 3-\frac{2}{3} \cdot (\frac{17}{6}+\frac{76}{23})=\frac{23}{22} \cdot \frac{1}{3}-\frac{2}{3} \cdot (\frac{17 \cdot 23+76 \cdot 6}{6 \cdot 23})=\frac{23}{66}-\frac{2}{3} \cdot (\frac{391+456}{138})=\frac{23}{66}-\frac{2}{3} \cdot \frac{847}{138}=\frac{23}{66}-\frac{847}{207}=\frac{23 \cdot 3}{66 \cdot 3}-\frac{847 \cdot 2}{207 \cdot 2}=\frac{69}{198}-\frac{1694}{414}=$$
$$=\frac{69 \cdot 7}{198 \cdot 7}-\frac{1694 \cdot 3}{414 \cdot 3}=\frac{483}{1386}-\frac{5082}{1242}=\frac{483 \cdot 6}{1386 \cdot 6}-\frac{5082 \cdot 7}{1242 \cdot 7}=\frac{2898}{8316}-\frac{35574}{8694}=\frac{2898-35574}{8316}=-\frac{32676}{8316}=-\frac{5446}{1386}=-\frac{2723}{693}=-3\frac{634}{693}$$
Ответ: $$-3\frac{634}{693}$$