Whenever we divide a fraction by another fraction, we are basically multiplying the first fraction by the reciprocal of the second fraction.
Let's use it to divide:
[tex]\begin{gathered} \frac{\frac{1}{6}}{\frac{5}{9}} \\ =\frac{1}{6}\times\frac{9}{5} \\ =\frac{1\times9}{6\times5} \\ =\frac{9}{30} \\ =\frac{3}{10} \end{gathered}[/tex]The answer (quotient) is 3/10