The percentage profit earned by selling an article for ₹ 1920 is equal to the percentage loss incurred by selling the same article for ₹ 1280. At what price should the article be sold to make 25% profit?

### Concept & Equating Percentage Profit and Loss When the percentage profit equals the percentage loss on the same item, the absolute profit equals the absolute loss (because both are calculated on the same Cost Price base). $$ \text{Profit } \% = \left(\frac{\text{Profit}}{\text{C.P.}}\right) \times 100 $$ $$ \text{Loss } \% = \left(\frac{\text{Loss}}{\text{C.P.}}\right) \times 100 $$ ### Step-by-Step Solution * Let the Cost Price (C.P.) be $x$. * Profit percentage = $\left(\frac{1920 - x}{x}\right) \times 100$. * Loss percentage = $\left(\frac{x - 1280}{x}\right) \times 100$. * Since Profit % = Loss %, we can equate them: * $\left(\frac{1920 - x}{x}\right) \times 100 = \left(\frac{x - 1280}{x}\right) \times 100$ * $1920 - x = x - 1280$ * $2x = 1920 + 1280$ * $2x = 3200$ * $x = 1600$. So, the Cost Price is ₹ 1600. * To make a $25\%$ profit, we find the new Selling Price: * $\text{New S.P.} = 1600 + (25\% \text{ of } 1600)$ * $\text{New S.P.} = 1600 + \left(\frac{1}{4} \times 1600\right) = 1600 + 400 = 2000$. ### Exam Strategy & Shortcut Recognize that "equal percentage profit and loss on the same item" is mathematically identical to "equal absolute profit and loss". Instantly find the Cost Price by averaging the two selling prices: $\frac{1920 + 1280}{2} = 1600$. Then add $25\%$ ($\frac{1}{4}$ of $1600 = 400$) to get $2000$. ### Common Pitfall A student might overcomplicate the algebra by strictly using the percentage formulas and struggling to isolate $x$, not realizing that dividing both sides by $\left(\frac{100}{x}\right)$ simplifies it immediately to just equating the absolute profit and loss. ### Final Answer Therefore, the correct answer is **₹ 2000**.