The value of

$\frac{3}{4}÷2\frac{1}{4}$ of $\frac{2}{3}-\frac{\left(\frac{1}{2}\right)-\left(\frac{1}{3}\right)}{\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)}\times 3\frac{1}{3}+\frac{5}{6}$ is

A. $\frac{29}{12}$

B. $\frac{49}{54}$

C. $\frac{2}{3}$

D. $\frac{1}{6}$

The correct option is C: $\frac{2}{3}$

$\frac{3}{4}÷2\frac{1}{4}$ of $\frac{2}{3}-\frac{\left(\frac{1}{2}\right)-\left(\frac{1}{3}\right)}{\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)}\times 3\frac{1}{3}+\frac{5}{6}$

$=\frac{3}{4}÷[2\frac{1}{4}\times \frac{2}{3}]-\frac{\left(\frac{1}{2}\right)-\left(\frac{1}{3}\right)}{\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)}\times 3\frac{1}{3}+\frac{5}{6}$

$=\frac{3}{4}÷\frac{3}{2}-\frac{\left(\frac{1}{2}\right)-\left(\frac{1}{3}\right)}{\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)}\times 3\frac{1}{3}+\frac{5}{6}$

$=\frac{3}{4}\times \frac{2}{3}-\frac{\left(\frac{3}{6}\right)-\left(\frac{2}{6}\right)}{\left(\frac{3}{6}\right)+\left(\frac{2}{6}\right)}\times 3\frac{1}{3}+\frac{5}{6}$

$=\frac{9}{4}-\frac{\left(\frac{1}{6}\right)}{\left(\frac{5}{6}\right)}\times \frac{10}{3}+\frac{5}{6}$

$=\frac{1}{2}-[\left(\frac{1}{6}\right)\times \left(\frac{6}{5}\right)\times \frac{10}{3}]+\frac{5}{6}$

$=\frac{1}{2}-[\frac{1}{5}\times \frac{10}{3}]+\frac{5}{6}$

$=\frac{1}{2}-\frac{2}{3}+\frac{5}{6}[\text{L.C.M of}2,3,6=6]$

$=\frac{3-4+5}{6}$

$=\frac{4}{6}$

$=\frac{2}{3}$