A fair price shopkeeper takes $10\%$ profit on his goods. He lost $20\%$ goods during theft. His loss percent is

### Concept & Effective Profit and Loss When a portion of goods is lost or destroyed, the profit must be calculated based on the revenue generated solely from the remaining goods compared to the total initial investment. ### Step-by-Step Solution * Let the initial number of goods be $100$. * Let the cost price (CP) of $1$ good be Re. $1$. * Total initial Cost Price (CP) for all goods = Rs. $100$. * The shopkeeper sets his price to take a $10\%$ profit on individual goods. * Expected Selling Price (SP) of $1$ good = Re. $1 + 10\%$ of $1 = \text{Rs. } 1.10$. * A theft occurs, and he loses $20\%$ of his goods. * Remaining goods available for sale = $100 - 20\% \text{ of } 100 = 80$ goods. * Total Selling Price (SP) realized from the remaining goods = $80 \times 1.10 = \text{Rs. } 88$. * Now, compare the actual total SP to the total initial CP. * Total CP = $100$, Total SP = $88$. * Since SP < CP, he incurs a loss. * Loss amount = $100 - 88 = \text{Rs. } 12$. * Loss percentage = $\frac{\text{Loss}}{\text{Total CP}} \times 100 = \frac{12}{100} \times 100 = 12\%$. ### Exam Strategy & Shortcut Use the multiplier method for successive changes. Volume multiplier (due to theft) = $0.80$ (since only $80\%$ is sold). Price multiplier (due to profit margin) = $1.10$. Overall multiplier = $0.80 \times 1.10 = 0.88$. A final multiplier of $0.88$ represents a $12\%$ drop from the original value of $1.00$. Thus, a $12\%$ loss. ### Common Pitfall A common mistake is simply subtracting the two percentages ($-20\% + 10\% = -10\%$). This fails because the price markup and the physical volume loss are multiplicative factors affecting total revenue, not additive ones. ### Final Answer Therefore, the correct answer is **$12$**.