Simplify: `1/2 ÷ 4/3 xx 2/3 ÷ 1/2 + 3/4 - 5/2 + 2/3 xx 9/4 ÷ 2/3`

Explanation

To simplify the expression `1/2 ÷ 4/3 × 2/3 ÷ 1/2 + 3/4 - 5/2 + 2/3 × 9/4 ÷ 2/3`, we will follow the order of operations (PEMDAS/BODMAS) and handle the fractions step by step.

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ \frac{1}{2} \div \frac{4}{3} \times \frac{2}{3} \div \frac{1}{2} + \frac{3}{4} - \frac{5}{2} + \frac{2}{3} \times \frac{9}{4} \div \frac{2}{3} \]

2. **Convert Division to Multiplication**:
Remember that dividing by a fraction is the same as multiplying by its reciprocal. Thus, we can rewrite the expression:
\[ \frac{1}{2} \times \frac{3}{4} \times \frac{2}{3} \times 2 + \frac{3}{4} - \frac{5}{2} + \frac{2}{3} \times \frac{9}{4} \times \frac{3}{2} \]

3. **Simplify Each Part**:
- For the first part:
\[ \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \]
Then multiply by \(\frac{2}{3}\):
\[ \frac{3}{8} \times \frac{2}{3} = \frac{3 \times 2}{8 \times 3} = \frac{6}{24} = \frac{1}{4} \]
Now multiply by \(2\):
\[ \frac{1}{4} \times 2 = \frac{1 \times 2}{4} = \frac{2}{4} = \frac{1}{2} \]

- For the second part:
\[ \frac{2}{3} \times \frac{9}{4} = \frac{2 \times 9}{3 \times 4} = \frac{18}{12} = \frac{3}{2} \]
Then multiply by \(\frac{3}{2}\):
\[ \frac{3}{2} \times \frac{3}{2} = \frac{9}{4} \]

4. **Combine All Parts**:
Now we have:
\[ \frac{1}{2} + \frac{3}{4} - \frac{5}{2} + \frac{9}{4} \]

5. **Find a Common Denominator**:
The common denominator for \(2\) and \(4\) is \(4\):
- Convert \(\frac{1}{2}\) to \(\frac{2}{4}\)
- Convert \(-\frac{5}{2}\) to \(-\frac{10}{4}\)

Now the expression is:
\[ \frac{2}{4} + \frac{3}{4} - \frac{10}{4} + \frac{9}{4} \]

6. **Combine the Fractions**:
Combine the numerators:
\[ \frac{2 + 3 - 10 + 9}{4} = \frac{4}{4} = 1 \]

### Final Answer:
The simplified expression is:
\[ \boxed{1} \]