1. The problem is to divide the mixed number $1 \frac{1}{2}$ by the fraction $\frac{3}{4}$. 2. First, convert the mixed number to an improper fraction. $1 \frac{1}{2} = \frac{3}{2}$ because $1 \times 2 + 1 = 3$. 3. Division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. 4. So, $\frac{3}{2} \div \frac{3}{4} = \frac{3}{2} \times \frac{4}{3}$. 5. Multiply the numerators and denominators: $\frac{3 \times 4}{2 \times 3} = \frac{12}{6}$. 6. Simplify the fraction $\frac{12}{6}$ by dividing numerator and denominator by 6: $\frac{12 \div 6}{6 \div 6} = \frac{2}{1} = 2$. 7. Therefore, $1 \frac{1}{2} \div \frac{3}{4} = 2$.