The value $$\frac{2}{3}\div\frac{3}{10} of \frac{4}{9}-\frac{4}{5}\times1\frac{1}{9}\div\frac{8}{15}-\frac{3}{4}+\frac{3}{4}\div\frac{1}{2} $$ is:

According to the BODMAS rule, the order is: Brackets > Orders (powers/roots) > “of” > Division/Multiplication (left to right) > Addition/Subtraction (left to right).
Hence we must evaluate the “of” operation $$\frac{3}{10}\;\text{of}\;\frac{4}{9}$$ before any division.

Step 1 - Evaluate the “of” term:
$$\frac{3}{10}\;\text{of}\;\frac{4}{9}= \frac{3}{10}\times\frac{4}{9}= \frac{12}{90}= \frac{2}{15}$$

After substituting this value the expression becomes
$$\frac{2}{3}\div\frac{2}{15}-\frac{4}{5}\times1\frac{1}{9}\div\frac{8}{15}-\frac{3}{4}+\frac{3}{4}\div\frac{1}{2}$$

Step 2 - Perform the divisions and multiplications from left to right.

Case 1: $$\frac{2}{3}\div\frac{2}{15}= \frac{2}{3}\times\frac{15}{2}=5$$

Case 2: Mixed number to improper fraction:
$$1\frac{1}{9}= \frac{10}{9}$$
Then
$$\frac{4}{5}\times\frac{10}{9}= \frac{40}{45}= \frac{8}{9}$$
and
$$\frac{8}{9}\div\frac{8}{15}= \frac{8}{9}\times\frac{15}{8}= \frac{15}{9}= \frac{5}{3}$$

Case 3: $$\frac{3}{4}\div\frac{1}{2}= \frac{3}{4}\times\frac{2}{1}= \frac{6}{4}= \frac{3}{2}$$

After completing all × and ÷ operations the expression simplifies to
$$5-\frac{5}{3}-\frac{3}{4}+\frac{3}{2}$$

Step 3 - Bring every term to a common denominator, say $$12$$:
$$5 = \frac{60}{12},\quad -\frac{5}{3}= -\frac{20}{12},\quad -\frac{3}{4}= -\frac{9}{12},\quad \frac{3}{2}= \frac{18}{12}$$

Step 4 - Add and subtract in sequence:
$$\frac{60}{12}-\frac{20}{12}-\frac{9}{12}+\frac{18}{12}= \frac{49}{12}$$

Therefore, the required value is $$\frac{49}{12}$$.
Option D which is: $$\frac{49}{12}$$