1. **State the problem:** A gas barbecue has a series of discounts: $420 less, 33.3\%, 20\%, and 5\%$ off the original price. Then, it is marked up by 60\% of the regular selling price. Finally, during the end of season sale, it is marked down by 45\%. We need to find the end of season sale price. 2. **Define variables and understand the problem:** Let the original price be $P$. 3. **Apply the $420$ less discount:** Price after this discount: $P - 420$ 4. **Apply the 33.3\% discount:** Price after this discount: $(P - 420) \times (1 - 0.333) = (P - 420) \times 0.667$ 5. **Apply the 20\% discount:** Price after this discount: $(P - 420) \times 0.667 \times (1 - 0.20) = (P - 420) \times 0.667 \times 0.80$ 6. **Apply the 5\% discount:** Price after this discount: $(P - 420) \times 0.667 \times 0.80 \times (1 - 0.05) = (P - 420) \times 0.667 \times 0.80 \times 0.95$ 7. **Calculate the regular selling price:** The regular selling price is the price after all discounts: $$\text{Regular Price} = (P - 420) \times 0.667 \times 0.80 \times 0.95$$ 8. **Apply the 60\% markup on the regular selling price:** Markup amount: $0.60 \times \text{Regular Price}$ Selling price after markup: $$\text{Selling Price} = \text{Regular Price} + 0.60 \times \text{Regular Price} = 1.60 \times \text{Regular Price}$$ 9. **Apply the 45\% markdown during the end of season sale:** End of season sale price: $$\text{Sale Price} = \text{Selling Price} \times (1 - 0.45) = 1.60 \times \text{Regular Price} \times 0.55$$ 10. **Substitute Regular Price back:** $$\text{Sale Price} = 1.60 \times (P - 420) \times 0.667 \times 0.80 \times 0.95 \times 0.55$$ 11. **Simplify the constants:** Calculate the product of constants: $$1.60 \times 0.667 \times 0.80 \times 0.95 \times 0.55 = 0.446$$ 12. **Final formula for sale price:** $$\text{Sale Price} = 0.446 \times (P - 420)$$ Without the original price $P$, the sale price depends on $P$ as above. **If the original price $P$ is known, substitute it to find the exact sale price.**