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The question asks:

**Which two expressions are equivalent to** \( 4 \frac{1}{2} \div 2 \frac{1}{4} \)? 

Let's break down the problem step-by-step.

1. **Convert the mixed numbers to improper fractions**:
   - \( 4 \frac{1}{2} = \frac{9}{2} \)
   - \( 2 \frac{1}{4} = \frac{9}{4} \)

2. **Perform the division**:
   Dividing by a fraction is the same as multiplying by its reciprocal. So,
   \[
   4 \frac{1}{2} \div 2 \frac{1}{4} = \frac{9}{2} \times \frac{4}{9}
   \]
   
3. **Multiply the fractions**:
   \[
   \frac{9}{2} \times \frac{4}{9} = \frac{9 \times 4}{2 \times 9} = \frac{36}{18} = 2
   \]
   
Now we need to find which two expressions among the options are equivalent to this calculation.

### Evaluating the options:

- **Option A**: \( 4 \frac{1}{2} \times 2 \frac{4}{1} \) 
   - Converting to improper fractions:
     \[
     4 \frac{1}{2} = \frac{9}{2}, \quad 2 \frac{4}{1} = 4
     \]
     So, \( \frac{9}{2} \times 4 = \frac{36}{2} = 18 \). This is not equal to 2, so **not correct**.

- **Option B**: \( \frac{9}{2} \div \frac{9}{4} \)
   - Dividing by \( \frac{9}{4} \) is the same as multiplying by \( \frac{4}{9} \):
     \[
     \frac{9}{2} \times \frac{4}{9} = \frac{36}{18} = 2
     \]
     This is **correct**.

- **Option C**: \( \frac{8}{2} \times \frac{8}{4} \)
   - Multiplying the fractions:
     \[
     \frac{8}{2} \times \frac{8}{4} = \frac{64}{8} = 8
     \]
     This is not equal to 2, so **not correct**.

- **Option D**: \( \frac{9}{2} \times \frac{4}{9} \)
   - This is exactly the same as the calculation we did earlier, and it equals 2. So, this is **correct**.

- **Option E**: \( \frac{8}{2} \div \frac{4}{9} \)
   - Dividing by \( \frac{4}{9} \) is the same as multiplying by \( \frac{9}{4} \):
     \[
     \frac{8}{2} \times \frac{9}{4} = \frac{72}{8} = 9
     \]
     This is not equal to 2, so **not correct**.

### Conclusion:
The correct answers are **B** and **D**.

Would you like more details or have any further questions?

Here are some related questions:
1. How do you convert mixed numbers into improper fractions?
2. What are the steps to multiply two fractions?
3. How do you divide fractions?
4. What is the reciprocal of a fraction and how is it used in division?
5. Can you explain why multiplying by a reciprocal gives the same result as division?

**Tip**: Always simplify fractions when possible to avoid errors and save time!

Which two expressions are equivalent to 4 1/2 ÷ 2 1/4?

Learn how to solve 4 1/2 ÷ 2 1/4 by converting mixed numbers to improper fractions and using reciprocal multiplication. This problem is ideal for students in Grades 6-8, covering division of fractions and fraction multiplication.

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