Srinivas sold an article for ₹ 6800 and incurred a loss. Had he sold the article for ₹ 7850, his gain would have been equal to half of the amount of loss that he incurred. At what price should he sell the article to have 20% profit?

### Concept & Equations Let the Cost Price be $CP$. We are given two scenarios linking loss and gain based on different selling prices. We can formulate a linear equation to find the exact Cost Price. ### Step-by-Step Solution 1. Let the Cost Price of the article be $x$. 2. **First Scenario (Loss):** Selling Price = ₹ $6800$. Loss Amount = $CP - SP = x - 6800$. 3. **Second Scenario (Gain):** Selling Price = ₹ $7850$. Gain Amount = $SP - CP = 7850 - x$. 4. **Link the scenarios:** The problem states: Gain = $\frac{1}{2} \times$ Loss $$7850 - x = \frac{1}{2} \times (x - 6800)$$ Multiply the entire equation by 2 to clear the fraction: $$2 \times (7850 - x) = x - 6800$$ $$15700 - 2x = x - 6800$$ $$15700 + 6800 = x + 2x$$ $$22500 = 3x$$ $$x = 7500$$ So, the Cost Price is ₹ $7500$. 5. **Calculate desired Selling Price:** To gain a $20\%$ profit: Required SP = $CP \times 1.20 = 7500 \times 1.20 = 9000$. ### Exam Strategy & Shortcut The difference between the two selling prices ($7850 - 6800 = 1050$) bridges the gap from a "full loss" amount to a "half gain" amount. Total gap = Loss $+ 0.5 \times$ Loss = $1.5 \times$ Loss. $1.5 \times \text{Loss} = 1050 \implies \text{Loss} = 700$. CP = $6800 + 700 = 7500$. Desired SP for $20\%$ profit = $7500 + 1500 = 9000$. ### Common Pitfall A common error is confusing the wording and setting the Loss equal to half the Gain, which reverses the multiplier and leads to an incorrect Cost Price. ### Final Answer Therefore, the correct answer is **₹ 9000**.