Gain Loss Articles E0B2E5

1. **Problem 1:** X sells two articles for Rs. 4000 each with no loss and no gain overall. One article is sold at a gain of 25%. Find the loss percentage on the other article. 2. **Formula and rules:** - Selling Price (SP) = Cost Price (CP) + Profit or - Loss - Gain% = $\frac{\text{Profit}}{\text{CP}} \times 100$ - Loss% = $\frac{\text{Loss}}{\text{CP}} \times 100$ 3. **Step 1:** Let the cost price of the first article be $x$. - Since it is sold at 25% gain, its selling price is $SP_1 = x + 0.25x = 1.25x$ 4. **Step 2:** Let the cost price of the second article be $y$. - Let the loss percentage on the second article be $L$%. - Then, $SP_2 = y - \frac{L}{100}y = y(1 - \frac{L}{100})$ 5. **Step 3:** Both articles are sold for Rs. 4000 each, so: $$1.25x = 4000 \quad \Rightarrow \quad x = \frac{4000}{1.25} = 3200$$ $$y(1 - \frac{L}{100}) = 4000$$ 6. **Step 4:** Since there is no loss and no gain overall, total cost price equals total selling price: $$x + y = SP_1 + SP_2 = 4000 + 4000 = 8000$$ Substitute $x=3200$: $$3200 + y = 8000 \quad \Rightarrow \quad y = 4800$$ 7. **Step 5:** Substitute $y=4800$ into the selling price equation: $$4800(1 - \frac{L}{100}) = 4000$$ Divide both sides by 4800: $$1 - \frac{L}{100} = \frac{4000}{4800} = \frac{5}{6}$$ Use cancellation notation: $$1 - \frac{L}{100} = \frac{\cancel{4000}}{\cancel{4800}} = \frac{5}{6}$$ 8. **Step 6:** Solve for $L$: $$1 - \frac{L}{100} = \frac{5}{6} \quad \Rightarrow \quad \frac{L}{100} = 1 - \frac{5}{6} = \frac{1}{6}$$ $$L = \frac{1}{6} \times 100 = 16.67\%$$ --- 9. **Problem 2:** A pair of shoes is sold at a loss of 10%. If it had been sold for Rs. 9 more, there would have been a gain of 12.5% (which is $12 \frac{1}{2}\%$) on it. Find the cost price. 10. **Step 1:** Let the cost price be $C$. - Selling price at 10% loss: $SP_1 = C - 0.10C = 0.90C$ - Selling price at 12.5% gain: $SP_2 = C + 0.125C = 1.125C$ 11. **Step 2:** According to the problem: $$SP_2 = SP_1 + 9$$ Substitute: $$1.125C = 0.90C + 9$$ 12. **Step 3:** Solve for $C$: $$1.125C - 0.90C = 9$$ $$0.225C = 9$$ $$C = \frac{9}{0.225} = 40$$ **Final answers:** - Loss percentage on the second article = $16.67\%$ - Cost price of the shoes = Rs. $40$