What is the simplified value of expression $9\frac{5}{6}-4\frac{3}{8}+2\frac{7}{12}?$

The given expression is $9\frac{5}{6}-4\frac{3}{8}+2\frac{7}{12}$.

The given expression can be simplified as,

$9\frac{5}{6}-4\frac{3}{8}+2\frac{7}{12}$

$=\frac{9\times 6+5}{6}-\frac{4\times 8+3}{8}+\frac{2\times 12+7}{12}$

$=\frac{59}{6}-\frac{35}{8}+\frac{31}{12}$

$=\frac{236-105+62}{24}$  [L.C.M. (6,8,12) = 24]

$=\frac{298-105}{24}$

$=\frac{193}{24}$

$=8\frac{1}{24}$

Therefore, the simplified value of expression $9\frac{5}{6}-4\frac{3}{8}+2\frac{7}{12} \text{is }\mathbf{8}\frac{\mathbf{1}}{\mathbf{24}}.$