\[\boxed{\mathbf{521.}\mathbf{\ }}\]
\[\frac{24}{25}\ :\frac{2}{5} = \frac{24}{25} \cdot \frac{5}{2} = \frac{12}{5}\]
\[\frac{12}{5}\ :6 = \frac{12}{5} \cdot \frac{1}{6} = \frac{2}{5}\]
\[\frac{2^{\backslash 3}}{5} - \frac{1^{\backslash 5}}{3} = \frac{6}{15} - \frac{5}{15} = \frac{1}{15}\]
\[\frac{1}{15}\ :\frac{1}{5} = \frac{1}{15} \cdot 5 = \frac{1}{3}\]
\[\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}\]