$10\%$ loss on selling price is what percent loss on the cost price?

### Concept & Changing Base of Percentage In standard definitions, profit and loss are calculated as a percentage of Cost Price (CP). If a problem specifies a loss percentage based on the Selling Price (SP), we must redefine the relationship to find the CP and then calculate the standard loss percentage. $$ \text{True Loss } \% = \frac{\text{Loss Amount}}{\text{Cost Price}} \times 100 $$ ### Step-by-Step Solution * Since the loss is given on the Selling Price, let the Selling Price (SP) = $100$. * Given: Loss = $10\%$ on Selling Price. * Calculate Loss amount: $10\% \text{ of } 100 = 10$. * The formula for Cost Price when there is a loss is: $CP = SP + \text{Loss}$. * Calculate CP: $100 + 10 = 110$. * Now we have the true base (CP) which is $110$, and the Loss amount which is $10$. * Calculate standard Loss \%: $(\frac{\text{Loss}}{CP}) \times 100$. * Substitute the values: $(\frac{10}{110}) \times 100 = \frac{100}{11}\%$. * Convert to a mixed fraction: $100 \div 11 = 9$ with a remainder of $1$. * The result is $9 \frac{1}{11}\%$. ### Exam Strategy & Shortcut Loss on SP is $10\%$, which is the fraction $\frac{1}{10}$. This means Loss = 1 unit, SP = 10 units. Therefore, CP = $SP + Loss = 10 + 1 = 11$ units. True loss percentage (on CP) is $\frac{\text{Loss}}{CP} = \frac{1}{11}$. The standard percentage equivalent of $\frac{1}{11}$ is exactly $9 \frac{1}{11}\%$. ### Common Pitfall Assuming a $10\%$ loss on SP is mathematically identical to a $10\%$ loss on CP, or mistakenly calculating CP as $SP - Loss$ (yielding 90 instead of 110), which would lead to an incorrect answer of $11 \frac{1}{9}\%$. ### Final Answer Therefore, the correct answer is **$9 \frac{1}{11}\%$**.