Let's break this down step by step.

The expression you provided is:

$\frac{2}{3} - \frac{1}{5} \cdot \frac{2}{4} \div \frac{2}{8} + \frac{3}{2}$

Step 1: Simplify the multiplication and division first.

$\frac{1}{5} \cdot \frac{2}{4} = \frac{2}{20} = \frac{1}{10}$

Now, divide by $\frac{2}{8}$:

$\frac{1}{10} \div \frac{2}{8} = \frac{1}{10} \cdot \frac{8}{2} = \frac{8}{20} = \frac{2}{5}$

Step 2: Substitute this into the original expression:

$\frac{2}{3} - \frac{2}{5} + \frac{3}{2}$

Step 3: Find a common denominator.

The denominators are 3, 5, and 2. The least common denominator (LCD) is 30. Rewrite each fraction with a denominator of 30:

$\frac{2}{3} = \frac{20}{30}, \quad \frac{2}{5} = \frac{12}{30}, \quad \frac{3}{2} = \frac{45}{30}$

Step 4: Perform the operations.

Now, calculate:

$\frac{20}{30} - \frac{12}{30} + \frac{45}{30} = \frac{20 - 12 + 45}{30} = \frac{53}{30}$

Final Answer:

$\frac{53}{30}$

This is the simplified result of the given expression.

Would you like more details on any of the steps? Here are 5 follow-up questions to deepen your understanding:

1. How do you find the least common denominator?
2. What is the rule for dividing fractions?
3. Can you explain how to convert improper fractions to mixed numbers?
4. Why do we perform multiplication and division before addition and subtraction?
5. How can we verify if a fraction is in its simplest form?

Tip: Always simplify fractions as much as possible at each step—it helps avoid unnecessary complications!