Simplify the following expression.
$2\frac{1}{4}\times 3\frac{1}{5}+5\frac{1}{6}\text{ of }2\frac{4}{31}÷3\frac{2}{3}+4\frac{1}{7}\text{ of }1\frac{6}{29}$

The correct option is $15\frac{1}{5}$.

Given expression:

$2\frac{1}{4}\times 3\frac{1}{5}+5\frac{1}{6}\text{ of }2\frac{4}{31}÷3\frac{2}{3}+4\frac{1}{7}\text{ of }1\frac{6}{29}$

Step 1: Convert all mixed fractions into improper fractions:

$2\frac{1}{4}=\frac{9}{4}$

$3\frac{1}{5}=\frac{16}{5}$

$5\frac{1}{6}=\frac{31}{6}$

$2\frac{4}{31}=\frac{66}{31}$

$3\frac{2}{3}=\frac{11}{3}$

$4\frac{1}{7}=\frac{29}{7}$

$1\frac{6}{29}=\frac{35}{29}$

Substitute these improper fractions into the expression:

$\frac{9}{4}\times \frac{16}{5}+\left(\frac{31}{6}\text{ of }\frac{66}{31}\right)÷\frac{11}{3}+\left(\frac{29}{7}\text{ of }\frac{35}{29}\right)$

Step 2: According to the BODMAS rule, evaluate the "of" operations first:

First "of" operation:

$\frac{31}{6}\times \frac{66}{31}=\frac{66}{6}=11$

Second "of" operation:

$\frac{29}{7}\times \frac{35}{29}=\frac{35}{7}=5$

Now, the expression becomes:

$\frac{9}{4}\times \frac{16}{5}+11÷\frac{11}{3}+5$

Step 3: Perform division and multiplication:

Multiplication:

$\frac{9}{4}\times \frac{16}{5}=\frac{9\times 4}{5}=\frac{36}{5}$

Division:

$11÷\frac{11}{3}=11\times \frac{3}{11}=3$

Now, the expression becomes:

$\frac{36}{5}+3+5$

Step 4: Add the terms together:

$\frac{36}{5}+8=\frac{36+40}{5}=\frac{76}{5}$

Convert the improper fraction back into a mixed fraction:

$\frac{76}{5}=15\frac{1}{5}$

Thus, the simplified value of the expression is $15\frac{1}{5}$.