1. **State the problem:** Simplify the expression $\frac{5\frac{1}{4}}{1\frac{2}{3} - \frac{1}{2}}$. 2. **Convert mixed numbers to improper fractions:** - $5\frac{1}{4} = \frac{21}{4}$ because $5 \times 4 + 1 = 21$. - $1\frac{2}{3} = \frac{5}{3}$ because $1 \times 3 + 2 = 5$. 3. **Rewrite the expression:** $$\frac{\frac{21}{4}}{\frac{5}{3} - \frac{1}{2}}$$ 4. **Find common denominator and subtract in the denominator:** - Common denominator of $3$ and $2$ is $6$. - Convert fractions: $\frac{5}{3} = \frac{10}{6}$ and $\frac{1}{2} = \frac{3}{6}$. - Subtract: $\frac{10}{6} - \frac{3}{6} = \frac{7}{6}$. 5. **Rewrite the expression:** $$\frac{\frac{21}{4}}{\frac{7}{6}}$$ 6. **Divide by a fraction by multiplying by its reciprocal:** $$\frac{21}{4} \times \frac{6}{7}$$ 7. **Simplify:** - $21$ and $7$ share a factor of $7$: $21 \div 7 = 3$, $7 \div 7 = 1$. - $6$ and $4$ share a factor of $2$: $6 \div 2 = 3$, $4 \div 2 = 2$. 8. **Multiply simplified fractions:** $$\frac{3}{2} \times \frac{3}{1} = \frac{9}{2}$$ 9. **Convert improper fraction to mixed number:** $$\frac{9}{2} = 4\frac{1}{2}$$ **Final answer:** $4\frac{1}{2}$