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!