Thank you for the clarification! Now let's solve the expression $\frac{2}{\frac{2}{3}} \div \frac{\frac{2}{3}}{2}$.

Step 1: Simplify each part of the expression

1. First part: $\frac{2}{\frac{2}{3}}$

   • Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{2}{\frac{2}{3}} = 2 \times \frac{3}{2} = \frac{6}{2} = 3$

2. Second part: $\frac{\frac{2}{3}}{2}$

   • Simplify this fraction: $\frac{\frac{2}{3}}{2} = \frac{2}{3} \times \frac{1}{2} = \frac{2 \times 1}{3 \times 2} = \frac{2}{6} = \frac{1}{3}$

Step 2: Combine the simplified parts

Now, we have: $3 \div \frac{1}{3}$

Step 3: Divide by the fraction

Dividing by a fraction is the same as multiplying by its reciprocal: $3 \times 3 = 9$

So, the value of the expression $\frac{2}{\frac{2}{3}} \div \frac{\frac{2}{3}}{2}$ is 9.

Would you like further details or have any questions?

Here are five related questions:

1. What is the result of $\frac{4}{\frac{1}{2}} \div \frac{\frac{3}{4}}{2}$?
2. How do you simplify complex fractions?
3. Why is dividing by a fraction the same as multiplying by its reciprocal?
4. Can you simplify $\frac{\frac{3}{5}}{4}$ before performing operations?
5. What happens when you multiply a whole number by a fraction?

Tip: When simplifying complex fractions, always remember to flip the divisor and multiply when dividing by another fraction.