Example  1
Evaluate and simplify \(\frac{{15}}{{22}} \times \frac{{44}}{{63}}\).


Step 1: Simplify by eliminating equal factors found in the numerator and denominator of either fraction.

\[\begin{align}
 \frac{{15}}{{22}} \times \frac{{44}}{{63}} &= \frac{{{\color{red}\cancel {\color{#444}{3}} }\times 5}}{{{\color{red}\cancel {\color{#444}{2}}} \times {\color{red}\cancel {\color{#444}{11}}}}} \times \frac{{{\color{red}\cancel {\color{#444}{2}}} \times 2 \times {\color{red}\cancel {\color{#444}{11}}}}}{{{\color{red}\cancel {\color{#444}{3}}} \times 3 \times 7}} \\
  &= \frac{5}{1} \times \frac{2}{{3 \times 7}} \\
 \end{align}\]

Step 2: Multiply the fractions.

\[\begin{align}
 \frac{5}{1} \times \frac{2}{{3 \times 7}} &= \frac{{5 \times 2}}{{3 \times 7}} \\
  &= \frac{{10}}{{21}} \\
 \end{align}\]

Therefore, \(\frac{{15}}{{22}} \times \frac{{44}}{{63}} = \frac{{10}}{{21}}\).