Correct Answer: $8$

Solution : Given: $\left(5 \div 2\right.$ of $\left.\frac{1}{2}\right)+\left(5 \frac{1}{4} \div \frac{3}{7}\right.$ of $\left.\frac{1}{2}\right) \div\left(5 \frac{1}{9}-7 \frac{7}{8} \div 9 \frac{9}{20}\right) \times \frac{11}{21}$
$=\left(5 \div (2\right.\times\left.\frac{1}{2})\right)+\left( \frac{21}{4} \div (\frac{3}{7}\right.\times\left.\frac{1}{2})\right) \div\left( \frac{46}{9}-\frac{63}{8} \div \frac{189}{20}\right) \times \frac{11}{21}$
$=5+\left( \frac{21}{4} \div \frac{3}{14}\right.) \div\left( \frac{46}{9}-\frac{63}{8} \times \frac{20}{189}\right) \times \frac{11}{21}$
$=5+\left( \frac{21}{4} \times \frac{14}{3}\right.) \div\left( \frac{46}{9}-\frac{5}{6} \right) \times \frac{11}{21}$
$=5+\frac{49}{2} \div \frac{92-15}{18} \times \frac{11}{21}$
$=5+\frac{49}{2} \times \frac{18}{77} \times \frac{11}{21}$
$=5+3$
$=8$
Hence, the correct answer is $8$.