By selling an article at $\frac{2}{3}$ of the marked price, there is a loss of $10\%$. The profit percent, when the article is sold at the marked price, is (C.P.O., 2006)

### Concept & Marked Price and Discount The relationship between Cost Price (CP), Selling Price (SP), and Marked Price (MP) can be established through profit/loss percentages and fractional sales conditions. $$ \text{SP} = \text{MP} \times \text{Fraction} $$ $$ \text{SP} = \text{CP} \times \left(1 - \frac{\text{Loss \%}}{100}\right) $$ ### Step-by-Step Solution * Given: * New SP = $\frac{2}{3} \times \text{MP}$ * Loss at this SP = $10\%$ * Let the Marked Price (MP) be $x$. * Current $\text{SP} = \frac{2x}{3}$ * Express CP in terms of $x$ using the loss condition: * With a $10\%$ loss, $\text{SP} = 90\%$ of CP = $0.9 \times \text{CP}$. * $0.9 \times \text{CP} = \frac{2x}{3}$ * $\text{CP} = \frac{2x}{3 \times 0.9} = \frac{2x}{2.7} = \frac{20x}{27}$ * Calculate profit when sold at Marked Price: * New SP = MP = $x$ * $\text{Profit} = \text{SP} - \text{CP} = x - \frac{20x}{27} = \frac{7x}{27}$ * Calculate profit percentage: * $\text{Profit \%} = \left(\frac{\text{Profit}}{\text{CP}}\right) \times 100$ * $\text{Profit \%} = \frac{\frac{7x}{27}}{\frac{20x}{27}} \times 100$ * $\text{Profit \%} = \frac{7}{20} \times 100 = 7 \times 5 = 35\%$ ### Exam Strategy & Shortcut Assume a convenient number for MP that is divisible by 3. Let MP = 300. SP = $\frac{2}{3} \times 300 = 200$. Loss is $10\%$, so $200 = 90\%$ of CP $\Rightarrow \text{CP} = \frac{200}{0.9} = \frac{2000}{9}$. If sold at MP (300), Profit = $300 - \frac{2000}{9} = \frac{2700 - 2000}{9} = \frac{700}{9}$. Profit \% = $\frac{\frac{700}{9}}{\frac{2000}{9}} \times 100 = \frac{700}{2000} \times 100 = 35\%$. ### Common Pitfall A frequent error is calculating the final profit percentage based on the Marked Price instead of the Cost Price. Profit percentage is fundamentally always calculated over the CP unless stated otherwise. ### Final Answer Therefore, the correct answer is **$35\%$**.