Let's solve the expression step by step. The expression given is:

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

Step 1: Simplify Inside the Brackets

First, we need to handle the subtraction inside the brackets:

$\frac{5}{6} - \frac{1}{3}$

To subtract these fractions, convert them to have a common denominator. The least common denominator (LCD) of 6 and 3 is 6.

$\frac{1}{3} = \frac{2}{6}$

So,

$\frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$

Step 2: Division Inside the Brackets

Next, we divide $\frac{1}{2}$ by $\frac{1}{3}$:

$\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$

Step 3: Multiplication Inside the Brackets

Now, multiply the result by $\frac{1}{3}$:

$\frac{3}{2} \times \frac{1}{3} = \frac{3 \times 1}{2 \times 3} = \frac{3}{6} = \frac{1}{2}$

Step 4: Division Outside the Brackets

Finally, divide the result by $-\frac{2}{3}$:

$\frac{1}{2} \div -\frac{2}{3} = \frac{1}{2} \times -\frac{3}{2} = -\frac{3}{4}$

Final Answer

The final answer is:

$-\frac{3}{4}$

Would you like more details on any part of the solution, or do you have any other questions?

Related Questions:

1. How do you simplify complex fractions?
2. What are the rules for dividing fractions?
3. How do you find the least common denominator?
4. Can you multiply and divide fractions with different denominators?
5. What is the process for adding or subtracting fractions with unlike denominators?
6. How do negative numbers affect division in fractions?
7. What is the importance of simplifying fractions in mathematical expressions?
8. How do you handle mixed operations with fractions?

Tip: When dividing fractions, remember that dividing by a fraction is the same as multiplying by its reciprocal.