1. **State the problem:** Simplify the expression $$\left(\frac{p^2 q^3}{4}\right) \times \left(\frac{8}{2p^2 q}\right) \div pq$$. 2. **Rewrite the expression clearly:** $$\frac{p^2 q^3}{4} \times \frac{8}{2 p^2 q} \times \frac{1}{p q}$$ 3. **Multiply the numerators and denominators:** $$\frac{p^2 q^3 \times 8 \times 1}{4 \times 2 p^2 q \times p q}$$ 4. **Simplify constants:** $$\frac{p^2 q^3 \times \cancel{8}}{\cancel{4} \times 2 p^2 q \times p q} = \frac{p^2 q^3 \times 2}{2 p^2 q \times p q}$$ 5. **Cancel the 2 in numerator and denominator:** $$\frac{p^2 q^3 \times \cancel{2}}{\cancel{2} p^2 q \times p q} = \frac{p^2 q^3}{p^2 q \times p q}$$ 6. **Simplify powers of $p$ and $q$ in numerator and denominator:** $$\frac{\cancel{p^2} q^3}{\cancel{p^2} q \times p q} = \frac{q^3}{q \times p q}$$ 7. **Multiply denominator terms:** $$q \times p q = p q^2$$ 8. **Rewrite expression:** $$\frac{q^3}{p q^2}$$ 9. **Simplify powers of $q$:** $$\frac{q^{3}}{q^{2}} = q^{3-2} = q$$ 10. **Final simplified expression:** $$\frac{q}{p}$$ **Answer:** $$\frac{q}{p}$$