1. **Problem statement:** An article is sold by X to Y at a loss of 20%, Y to Z at a gain of 15%, Z to W at a loss of 5%, and W to V at a profit of 10%. V pays 500. We need to find how much X originally paid for the article. 2. **Understanding the problem:** Let the cost price for X be $x$. Each transaction changes the price by a percentage gain or loss. 3. **Formulas:** - Loss of 20% means selling price = $0.8 \times$ cost price. - Gain of 15% means selling price = $1.15 \times$ cost price. - Loss of 5% means selling price = $0.95 \times$ cost price. - Profit of 10% means selling price = $1.10 \times$ cost price. 4. **Calculate the final price step-by-step:** - Price after X to Y: $0.8x$ - Price after Y to Z: $1.15 \times 0.8x = 0.92x$ - Price after Z to W: $0.95 \times 0.92x = 0.874x$ - Price after W to V: $1.10 \times 0.874x = 0.9614x$ 5. **Given:** V pays 500, so $$0.9614x = 500$$ 6. **Solve for $x$:** $$x = \frac{500}{0.9614} \approx 520.23$$ **Answer:** X originally paid approximately 520.23.