A fruitseller has $24$ kg of apples. He sells a part of these at a gain of $20\%$ and the balance at a loss of $5\%$. If on the whole he earns a profit of $10\%$, the amount of apples sold at a loss is

### Concept & Rule of Alligation with Loss The Rule of Alligation applies even when one part is sold at a loss. The crucial step is to represent the loss as a negative percentage. $$ \text{Ratio} = \frac{\text{Profit}_1 - \text{Mean}}{\text{Mean} - \text{Loss}_2} $$ ### Step-by-Step Solution * **Part 1 (Gain):** $+20\%$ * **Part 2 (Loss):** $-5\%$ * **Mean Result (Gain):** $+10\%$ * Apply Alligation cross-subtraction: * Left side (Quantity at $20\%$ gain) ratio part: $10 - (-5) = 10 + 5 = 15$. * Right side (Quantity at $5\%$ loss) ratio part: $20 - 10 = 10$. * **Ratio of quantities:** (Quantity at $20\%$ gain) : (Quantity at $5\%$ loss) = $15 : 10 = 3 : 2$. * The total quantity is divided into $3 + 2 = 5$ parts. * **Total apples:** $24$ kg. * The amount sold at a loss corresponds to the 2 parts. $$\text{Amount sold at loss} = \frac{2}{5} \times 24 = \frac{48}{5} = 9.6 \text{ kg}$$ ### Exam Strategy & Shortcut Write down the values in the alligation format, ensuring the loss has a minus sign: $20 \quad \quad \quad -5$ $\quad \searrow \quad \swarrow$ $\quad \quad 10$ $\quad \swarrow \quad \searrow$ $15 \quad : \quad 10$ Ratio is $3 : 2$. The loss portion is $\frac{2}{5}$ of the total. $\frac{2}{5} \times 24 = 9.6$. ### Common Pitfall The most common and fatal mistake is ignoring the negative sign for the loss. If you use $+5$ instead of $-5$, the ratio becomes $(10 - 5) : (20 - 10) = 5 : 10 = 1 : 2$, leading to an incorrect calculation of $16$ kg for the loss portion. ### Final Answer Therefore, the correct answer is **9.6 kg**.