Let's simplify the given expression step by step:

\(\frac{\frac{7}{9} - \frac{2}{3}}{\frac{1}{3} + \frac{\frac{2}{5}}{\frac{4}{5}}}\)

First, simplify the fractions in the numerator:

\(\frac{\frac{7}{9} - \frac{2}{3}}{\frac{1}{3} + \frac{2}{5} \cdot \frac{5}{4}}\)

\(= \frac{\frac{7}{9} - \frac{6}{9}}{\frac{1}{3} + \frac{10}{20}}\)

\(= \frac{\frac{1}{9}}{\frac{1}{3} + \frac{1}{2}}\)

Now, find a common denominator for the fractions in the denominator:

\(= \frac{\frac{1}{9}}{\frac{2}{6} + \frac{3}{6}}\)

\(= \frac{\frac{1}{9}}{\frac{5}{6}}\)

Invert and multiply to divide by a fraction:

\(= \frac{1}{9} \cdot \frac{6}{5}\)

\(= \frac{6}{45}\)

Simplify the fraction further:

\(= \frac{2}{15}\)