To simplify the expression, we will first simplify the numerator and the denominator separately, then divide the simplified numerator by the simplified denominator.

$$5\frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{23}{4}$$ $$1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}$$ $$4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}$$

The numerator is $\frac{23}{4} \div \frac{4}{3} - \frac{9}{2} \times \frac{2}{15}$.
First, perform the division:

$$\frac{23}{4} \div \frac{4}{3} = \frac{23}{4} \times \frac{3}{4} = \frac{23 \times 3}{4 \times 4} = \frac{69}{16}$$

Next, perform the multiplication:

$$\frac{9}{2} \times \frac{2}{15} = \frac{9 \times 2}{2 \times 15} = \frac{18}{30}$$

Simplify the fraction $\frac{18}{30}$ by dividing both numerator and denominator by their greatest common divisor, which is 6:

$$\frac{18 \div 6}{30 \div 6} = \frac{3}{5}$$

Now, perform the subtraction in the numerator:

$$\frac{69}{16} - \frac{3}{5}$$

Find a common denominator for 16 and 5, which is 80.

$$\frac{69 \times 5}{16 \times 5} - \frac{3 \times 16}{5 \times 16} = \frac{345}{80} - \frac{48}{80} = \frac{345 - 48}{80} = \frac{297}{80}$$

So, the numerator is $\frac{297}{80}$.

The denominator is $\frac{1}{2} - \frac{7}{20}$.
Find a common denominator for 2 and 20, which is 20.

$$\frac{1 \times 10}{2 \times 10} - \frac{7}{20} = \frac{10}{20} - \frac{7}{20} = \frac{10 - 7}{20} = \frac{3}{20}$$

So, the denominator is $\frac{3}{20}$.

$$\frac{\frac{297}{80}}{\frac{3}{20}} = \frac{297}{80} \div \frac{3}{20}$$

To divide by a fraction, multiply by its reciprocal:

$$\frac{297}{80} \times \frac{20}{3}$$

Simplify by canceling common factors. 20 divides into 80 four times, and 3 divides into 297 ninety-nine times.

$$\frac{297 \div 3}{80 \div 20} = \frac{99}{4}$$

Convert the improper fraction to a mixed number:

$$\frac{99}{4} = 24\frac{3}{4}$$

The simplified expression is \boxed{\text{24\frac{3}{4}}}.