The value of 

$$\frac{1}{4} \times \frac{3}{4} \div 1\frac{1}{4}  of  \frac{2}{5} - \left[\frac{1}{6} \div \left\{\frac{3}{7}  of  \frac{14}{5} \times 1\frac{2}{3} - \left(3\frac{1}{2} - 2\frac{1}{6}\right) \right\}\right]$$ is:

= $$\frac{1}{4} \times \frac{3}{4} \div 1\frac{1}{4} of \frac{2}{5} - \left[\frac{1}{6} \div \left\{\frac{3}{7} of \frac{14}{5} \times 1\frac{2}{3} - \left(3\frac{1}{2} - 2\frac{1}{6}\right) \right\}\right]$$

= $$\frac{1}{4}\times\frac{3}{4}\div\frac{5}{4}of\frac{2}{5}-\left[\frac{1}{6}\div\left\{\frac{3}{7}of\frac{14}{5}\times\frac{5}{3}-\left(\frac{7}{2}-\frac{13}{6}\right)\right\}\right]$$

= $$\frac{1}{4}\times\frac{3}{4}\div\frac{10}{20}-\left[\frac{1}{6}\div\left\{2-\left(\frac{21}{6}-\frac{13}{6}\right)\right\}\right]$$

= $$\frac{1}{4}\times\frac{3}{4}\times\frac{2}{1}-\left[\frac{1}{6}\div\left\{2-\left(\frac{8}{6}\right)\right\}\right]$$

= $$\frac{3}{8}-\left[\frac{1}{6}\div\left\{2-\left(\frac{4}{3}\right)\right\}\right]$$

= $$\frac{3}{8}-\left[\frac{1}{6}\div\left\{\frac{6-4}{3}\right\}\right]$$

= $$\frac{3}{8}-\left[\frac{1}{6}\div\left\{\frac{2}{3}\right\}\right]$$

= $$\frac{3}{8}-\left[\frac{1}{6}\times\frac{3}{2}\right]$$

= $$\frac{3}{8}-\frac{1}{4}$$

= $$\frac{3}{8}-\frac{2}{8}$$

= $$\frac{1}{8}$$