To solve the problem $5 \frac{1}{2} \times 2 \frac{2}{3}$, we need to convert each mixed number into an improper fraction and then perform the multiplication.

1. Convert $5 \frac{1}{2}$ to an improper fraction: $5 \frac{1}{2} = \frac{5 \times 2 + 1}{2} = \frac{10 + 1}{2} = \frac{11}{2}$

2. Convert $2 \frac{2}{3}$ to an improper fraction: $2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}$

3. Now multiply the two improper fractions: $\frac{11}{2} \times \frac{8}{3} = \frac{11 \times 8}{2 \times 3} = \frac{88}{6}$

4. Simplify $\frac{88}{6}$: $\frac{88}{6} = \frac{44}{3} = 14 \frac{2}{3}$

After simplifying, the correct answer is: $14 \frac{2}{3}$

However, none of the options provided in the image match this answer, suggesting a possible error in the answer choices. Let me know if you have any questions or need further clarification!

Would you like more details on:

1. Steps to convert mixed numbers to improper fractions?
2. Why the multiplication of fractions involves multiplying numerators and denominators?
3. Additional examples of simplifying improper fractions to mixed numbers?
4. How to verify the result by reconverting to a mixed number?
5. Further clarification on identifying errors in multiple-choice questions?

Tip: When working with fractions, always check if the result can be simplified to its lowest terms.