1. **Problem statement:** A company reduces the price of a product by 20% and sales income increases by 20%. Costs per product remain the same, no fixed costs, and total profit remains constant. We need to find the percentage drop in profit per product. 2. **Define variables:** Let the original price be $P$, original quantity sold be $Q$, and cost per product be $C$. 3. **Original profit per product:** $\text{Profit per product} = P - C$ 4. **Original total profit:** $\text{Total profit} = (P - C) \times Q$ 5. **New price after 20% reduction:** $P_{new} = P \times (1 - 0.20) = 0.8P$ 6. **New total sales income after 20% increase:** $\text{New sales income} = P_{new} \times Q_{new} = 1.2 \times P \times Q$ 7. **Find new quantity sold $Q_{new}$:** $$ 0.8P \times Q_{new} = 1.2 P Q \implies Q_{new} = \frac{1.2 P Q}{0.8 P} = 1.5 Q $$ 8. **New profit per product:** $\text{Profit per product}_{new} = P_{new} - C = 0.8P - C$ 9. **New total profit:** $$ \text{Total profit}_{new} = (0.8P - C) \times 1.5 Q = 1.5 Q (0.8P - C) $$ 10. **Since total profit remains constant:** $$ (P - C) Q = 1.5 Q (0.8P - C) $$ 11. **Divide both sides by $Q$:** $$ P - C = 1.5 (0.8P - C) $$ 12. **Expand right side:** $$ P - C = 1.2 P - 1.5 C $$ 13. **Rearrange terms:** $$ P - C - 1.2 P + 1.5 C = 0 \\ -0.2 P + 0.5 C = 0 \\ 0.5 C = 0.2 P \\ C = \frac{0.2}{0.5} P = 0.4 P $$ 14. **Calculate original profit per product:** $$ P - C = P - 0.4 P = 0.6 P $$ 15. **Calculate new profit per product:** $$ 0.8 P - C = 0.8 P - 0.4 P = 0.4 P $$ 16. **Calculate percentage drop in profit per product:** $$ \frac{0.6 P - 0.4 P}{0.6 P} \times 100 = \frac{0.2 P}{0.6 P} \times 100 = \frac{1}{3} \times 100 = 33.3\%\text{ drop} $$ **Final answer:** The profit per product dropped by 33.3%.