Let the salary be \( S \).

Amount spent on shirts:
\(\text{Shirts} = \frac{1}{5} S\)

Remainder after shirts:
\(\text{Remainder} = S - \frac{1}{5} S = \frac{4}{5} S\)

Amount spent on transport:
\(\text{Transport} = \frac{1}{3} \left( \frac{4}{5} S \right) = \frac{4}{15} S\)

Remainder after transport:
\(\text{Remainder} = \frac{4}{5} S - \frac{4}{15} S = \frac{12}{15} S - \frac{4}{15} S = \frac{8}{15} S\)

Thus, the fraction left is: \(\frac{8}{15}\)