To solve the expression $3 \frac{3}{4} \div 2 \frac{1}{2} \cdot 4 \frac{1}{2} + 1 \frac{3}{4}$, we'll first convert all the mixed numbers into improper fractions, then follow the order of operations: division and multiplication from left to right, followed by addition.

Step 1: Convert Mixed Numbers to Improper Fractions

1. $3 \frac{3}{4}$: $3 \frac{3}{4} = \frac{3 \cdot 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$

2. $2 \frac{1}{2}$: $2 \frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

3. $4 \frac{1}{2}$: $4 \frac{1}{2} = \frac{4 \cdot 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2}$

4. $1 \frac{3}{4}$: $1 \frac{3}{4} = \frac{1 \cdot 4 + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$

Step 2: Substitute and Perform Operations

The expression now becomes: $\frac{15}{4} \div \frac{5}{2} \cdot \frac{9}{2} + \frac{7}{4}$

Division

Divide $\frac{15}{4}$ by $\frac{5}{2}$: $\frac{15}{4} \div \frac{5}{2} = \frac{15}{4} \times \frac{2}{5} = \frac{15 \times 2}{4 \times 5} = \frac{30}{20} = \frac{3}{2}$

Multiplication

Multiply $\frac{3}{2}$ by $\frac{9}{2}$: $\frac{3}{2} \times \frac{9}{2} = \frac{3 \times 9}{2 \times 2} = \frac{27}{4}$

Addition

Add $\frac{27}{4}$ and $\frac{7}{4}$: $\frac{27}{4} + \frac{7}{4} = \frac{27 + 7}{4} = \frac{34}{4} = \frac{17}{2} = 8 \frac{1}{2}$

Final Result

Thus, the result of the expression is $8 \frac{1}{2}$.

If you have any questions or need more details, feel free to ask!

Here are some related questions for further exploration:

1. How do you convert a mixed number to an improper fraction?
2. What is the order of operations in mathematics?
3. How do you simplify fractions?
4. What is the difference between division and multiplication of fractions?
5. How do you add fractions with different denominators?
6. What are some real-world applications of fractions?
7. How do you convert an improper fraction back to a mixed number?
8. How do you check your work when simplifying expressions with fractions?

Tip: When performing operations with fractions, always remember to simplify your results where possible for easier interpretation and understanding.