If an article is sold for ₹ $x$, there is a loss of $15\%$. If the same article is sold for ₹ $y$, there is a profit of $15\%$. The ratio of $(y - x)$ to $(y + x)$ is (Hotel Mgmt, 2010)

### Concept & Proportional Relations of Selling Price When an article is sold at a loss or a profit, the selling price can be expressed as a percentage of the Cost Price (CP). A $15\%$ loss means the selling price is $85\%$ of the CP, while a $15\%$ profit means it is $115\%$ of the CP. The ratio of any two expressions involving these selling prices will be independent of the actual Cost Price. $$ \text{Selling Price} = \text{CP} \times (1 \pm \frac{\text{Profit/Loss } \%}{100}) $$ ### Step-by-Step Solution * Let the Cost Price (CP) of the article be $100$ units. * Given: Selling for ₹ $x$ results in a $15\%$ loss. * Therefore, $x = 100 - 15 = 85$ units. * Given: Selling for ₹ $y$ results in a $15\%$ profit. * Therefore, $y = 100 + 15 = 115$ units. * We need to find the ratio of $(y - x)$ to $(y + x)$. * First, calculate $(y - x)$: $115 - 85 = 30$. * Second, calculate $(y + x)$: $115 + 85 = 200$. * Form the ratio: $\frac{y - x}{y + x} = \frac{30}{200}$. * Simplify the ratio by dividing both numerator and denominator by $10$: $\frac{3}{20}$. * The final ratio is $3 : 20$. ### Exam Strategy & Shortcut Instead of writing out $100$ or using a variable like $C$, directly use the percentages. $x = 85\%$ of CP $y = 115\%$ of CP Ratio = $\frac{115 - 85}{115 + 85} = \frac{30}{200} = \frac{3}{20} = 3 : 20$. This avoids defining explicit variables and jumps straight to the solution. ### Common Pitfall A common error is to set up a complex algebraic equation for CP involving $x$ and $y$ and trying to solve for variables independently. Since the question asks for a ratio, the absolute value of CP will always cancel out. ### Final Answer Therefore, the correct answer is **3 : 20**.