\[\boxed{\text{5.560.}}\]
\[1)\ \frac{1}{4}\ :\frac{1}{4} \cdot \frac{2}{7} = 1 \cdot \frac{2}{7} = \frac{2}{7}\]
\[2)\ \frac{1}{3} \cdot \frac{7}{8}\ :\frac{9}{14} = \frac{1}{3} \cdot \frac{7}{8} \cdot \frac{14}{9} = \frac{98}{216} =\]
\[= \frac{49}{108}\]
\[3)\ \frac{1}{3} \cdot \frac{7}{8} \cdot \frac{9}{14} = \frac{3}{16}\]
\[4)\ \frac{6}{7}\ :\frac{5}{7} \cdot \frac{4}{9} = \frac{6}{7} \cdot \frac{7}{5} \cdot \frac{4}{9} =\]
\[= \frac{2 \cdot 1 \cdot 4}{1 \cdot 5 \cdot 3} = \frac{8}{15}\]