1. **State the problem:** Find the product when the quotient of $5 \frac{1}{4}$ and $3 \frac{1}{2}$ is multiplied by $1 \frac{1}{3}$. 2. **Convert mixed numbers to improper fractions:** $5 \frac{1}{4} = \frac{21}{4}$ $3 \frac{1}{2} = \frac{7}{2}$ $1 \frac{1}{3} = \frac{4}{3}$ 3. **Find the quotient of $\frac{21}{4}$ and $\frac{7}{2}$:** $$\frac{21}{4} \div \frac{7}{2} = \frac{21}{4} \times \frac{2}{7}$$ 4. **Multiply the fractions:** $$= \frac{21 \times 2}{4 \times 7} = \frac{42}{28}$$ 5. **Simplify the fraction by canceling common factors:** $$= \frac{\cancel{42}^{6} \times 7}{\cancel{28}^{4} \times 7} = \frac{6}{4}$$ 6. **Simplify $\frac{6}{4}$ further:** $$= \frac{3}{2}$$ 7. **Multiply the quotient by $\frac{4}{3}$:** $$\frac{3}{2} \times \frac{4}{3} = \frac{3 \times 4}{2 \times 3}$$ 8. **Simplify by canceling common factors:** $$= \frac{\cancel{3}^1 \times 4}{2 \times \cancel{3}^1} = \frac{4}{2}$$ 9. **Simplify $\frac{4}{2}$:** $$= 2$$ **Final answer:** The product is $2$.