1. \(\left(6-2\frac{1}{10}:2\frac{1}{7}\right)\cdot\frac{4}{15}\)

\(2\frac{1}{10}:2\frac{1}{7}=\frac{21}{10}:\frac{15}{7}=\frac{21}{10}\cdot\frac{7}{15}=\frac{49}{50}\).

\(6-\frac{49}{50}=\frac{300}{50}-\frac{49}{50}=\frac{251}{50}\).

\(\frac{251}{50}\cdot\frac{4}{15}=\frac{1004}{750}=\frac{502}{375}\).

Ответ: \(\frac{502}{375}\).