Evaluate the expression in the numerator.

First, simplify the expression inside the parenthesis:

$$4\frac{1}{3} - 3\frac{5}{6} = \frac{13}{3} - \frac{23}{6}$$

Find a common denominator, which is 6:

$$\frac{13 \times 2}{3 \times 2} - \frac{23}{6} = \frac{26}{6} - \frac{23}{6} = \frac{26 - 23}{6} = \frac{3}{6} = \frac{1}{2}$$

Now, multiply this result by $\frac{5}{6}$ ("of" means multiplication):

$$\frac{5}{6} \times \frac{1}{2} = \frac{5}{12}$$

So, the numerator is $\frac{5}{12}$.

First, perform the multiplication:

$$\frac{5}{12} \times \frac{3}{25} = \frac{5 \times 3}{12 \times 25} = \frac{15}{300}$$

Simplify the fraction by dividing both numerator and denominator by 15:

$$\frac{15 \div 15}{300 \div 15} = \frac{1}{20}$$

Next, perform the division. Convert mixed numbers to improper fractions:

$$1\frac{5}{9} = \frac{9 \times 1 + 5}{9} = \frac{14}{9}$$ $$2\frac{1}{3} = \frac{3 \times 2 + 1}{3} = \frac{7}{3}$$

Now divide:

$$\frac{14}{9} \div \frac{7}{3} = \frac{14}{9} \times \frac{3}{7}$$

Simplify by canceling common factors (14 and 7, 3 and 9):

$$\frac{14 \div 7}{9 \div 3} \times \frac{3 \div 3}{7 \div 7} = \frac{2}{3} \times \frac{1}{1} = \frac{2}{3}$$

Now, add the results of the multiplication and division:

$$\frac{1}{20} + \frac{2}{3}$$

Find a common denominator, which is 60:

$$\frac{1 \times 3}{20 \times 3} + \frac{2 \times 20}{3 \times 20} = \frac{3}{60} + \frac{40}{60} = \frac{3 + 40}{60} = \frac{43}{60}$$

So, the denominator is $\frac{43}{60}$.

$$\frac{\frac{5}{12}}{\frac{43}{60}} = \frac{5}{12} \div \frac{43}{60}$$

To divide fractions, multiply the first fraction by the reciprocal of the second fraction:

$$\frac{5}{12} \times \frac{60}{43}$$

Simplify by canceling common factors (12 and 60):

$$\frac{5}{1} \times \frac{60 \div 12}{43} = \frac{5}{1} \times \frac{5}{43} = \frac{25}{43}$$

The final answer is $\frac{25}{43}$.