В $$10\frac{1}{2}:1\frac{1}{4} \cdot \frac{3}{14}-1\frac{23}{35}+2\frac{2}{7}$$

Выполним действия с дробями.

1. $$10\frac{1}{2}:1\frac{1}{4}=\frac{21}{2}:\frac{5}{4}=\frac{21}{2} \cdot \frac{4}{5}=\frac{21 \cdot 4}{2 \cdot 5}=\frac{84}{10}=8\frac{4}{10}=8\frac{2}{5}$$
2. $$8\frac{2}{5} \cdot \frac{3}{14}=\frac{42}{5} \cdot \frac{3}{14}=\frac{42 \cdot 3}{5 \cdot 14}=\frac{126}{70}=\frac{9}{5}=1\frac{4}{5}$$
3. $$1\frac{23}{35}=\frac{58}{35}$$
4. $$2\frac{2}{7}=\frac{16}{7}$$
5. $$-1\frac{23}{35}+2\frac{2}{7}=-\frac{58}{35}+ \frac{16}{7}=-\frac{58}{35}+ \frac{16 \cdot 5}{7 \cdot 5}=-\frac{58}{35}+\frac{80}{35}=\frac{22}{35}$$
6. $$1\frac{4}{5}+\frac{22}{35}=\frac{9}{5}+\frac{22}{35}=\frac{9 \cdot 7}{5 \cdot 7}+\frac{22}{35}=\frac{63}{35}+\frac{22}{35}=\frac{85}{35}=\frac{17}{7}=2\frac{3}{7}$$

Ответ: $$2\frac{3}{7}$$