The value of 

$$1\frac{2}{3} \div \left\{\frac{3}{7}  of  \frac{14}{5} \times 1\frac{2}{3} - \left(3\frac{1}{2} - 2\frac{1}{6} \right)\right\} + \frac{1}{2} \div \frac{3}{2}  of  \frac{1}{2}$$ is:

Given Equation : 

$$1\frac{2}{3} \div \left\{\frac{3}{7} of \frac{14}{5} \times 1\frac{2}{3} - \left(3\frac{1}{2} - 2\frac{1}{6} \right)\right\} + \frac{1}{2} \div \frac{3}{2} of \frac{1}{2}$$ is

Lets solve it as per the rule of BODMAS : 

= $$\frac{5}{3}\div\left\{\frac{3}{7}of\frac{14}{5}\times\frac{5}{3}-\left(\frac{5}{2}-\frac{13}{6}\right)\right\}+\frac{1}{2}\div\frac{3}{2}of\frac{1}{2}$$

= $$\frac{5}{3}\div\left\{\frac{3}{7}of\frac{14}{5}\times\frac{5}{3}-\left(\frac{4}{3}\right)\right\}+\frac{1}{2}\div\frac{3}{2}of\frac{1}{2}$$

= $$\frac{5}{3}\div\left\{2-\frac{4}{3}\right\}+\frac{1}{2}\div\frac{3}{2}of\frac{1}{2}$$

= $$\frac{5}{3}\div\frac{2}{3}+\frac{1}{2}\div\frac{3}{2}of\frac{1}{2}$$

= $$\frac{5}{3}\times\ \frac{3}{2}+\frac{1}{2}\div\frac{3}{4}$$

= $$\frac{5}{3}\times\ \frac{3}{2}+\frac{1}{2}\times\frac{4}{3}$$

= $$\frac{5}{2}+\frac{2}{3}$$

= $$\frac{19}{6}$$

= $$3\frac{1}{6}$$

Hence, Option A is correct.