The profit earned by selling an article for ₹ 900 is double the loss incurred when the same article is sold for ₹ 450. At what price should the article be sold to make 25% profit?

### Concept & Proportional Profit and Loss This problem involves a relationship where the profit is a multiple of the loss. By setting up a linear equation based on the Cost Price (C.P.), we can find the unknown base price. $$ \text{Profit} = \text{Multiplier} \times \text{Loss} $$ $$ (\text{S.P.}_1 - \text{C.P.}) = n \times (\text{C.P.} - \text{S.P.}_2) $$ ### Step-by-Step Solution * Let the Cost Price of the article be $x$. * Loss incurred when sold for ₹ $450 = x - 450$. * Profit earned when sold for ₹ $900 = 900 - x$. * The problem states that Profit = $2 \times$ Loss. Therefore: * $900 - x = 2(x - 450)$ * $900 - x = 2x - 900$ * Add $x$ to both sides and add $900$ to both sides: * $3x = 1800$ * $x = 600$. The Cost Price is ₹ 600. * To make a $25\%$ profit, calculate the new Selling Price: * $\text{New S.P.} = 600 \times \left(1 + \frac{25}{100}\right) = 600 \times 1.25$ * $\text{New S.P.} = 600 + 150 = 750$. ### Exam Strategy & Shortcut Understand the "gap" between the two prices. The difference between ₹ 900 and ₹ 450 is ₹ 450. This ₹ 450 difference is split into the loss amount and the profit amount, which is twice the loss (Ratio of Loss:Profit = 1:2). So, divide the ₹ 450 gap by 3 parts = ₹ 150 per part. Loss = ₹ 150. $\text{C.P.} = 450 + 150 = 600$. $25\%$ of $600 = 150$. $\text{Final S.P.} = 600 + 150 = 750$. ### Common Pitfall Writing the loss equation incorrectly as $(450 - x)$ instead of $(x - 450)$ will yield negative or incorrect results. Always ensure Loss is $\text{C.P.} - \text{S.P.}$ ### Final Answer Therefore, the correct answer is **₹ 750**.