Example  5
Evaluate and simplify \(\frac{6}{{35}} \div \frac{9}{{25}}\).


Step 1: Rewrite the division statement as the product of the first expression and the reciprocal of the second.

\[\frac{6}{{35}} \div \frac{9}{{25}} = \frac{6}{{35}} \times \frac{{25}}{9}\]

Step 2: Complete the operation by following the rules for multiplication of fractions.

\[\begin{align}
 \frac{6}{{35}} \times \frac{{25}}{9} &= \frac{{{\color{red}\cancel {\color{#444}{6}}^2 }}}{{{\color{red}\cancel {\color{#444}{35}}^7 }}} \times \frac{{{\color{red}\cancel {\color{#444}{25}}^5 }}}{{{\color{red}\cancel {\color{#444}{9}}^3 }}} \\
  &= \frac{2}{7} \times \frac{5}{3} \\
  &= \frac{{10}}{{21}} \\
 \end{align}\]

Therefore, \(\frac{6}{{35}} \div \frac{9}{{25}} = \frac{{10}}{{21}}\).