If a person makes a profit of $10\%$ on one-fourth of the quantity sold and a loss of $20\%$ on the rest, then what is the average percent profit or loss? (J.M.E.T., 2004)

### Concept & Fractional Weighted Average To find the total percentage profit or loss across an entire quantity, apply the given profit and loss percentages to their respective fractions of the whole. Losses are represented as negative values. $$\text{Net } \% = (\text{Fraction}_1 \times \text{Profit}_1) - (\text{Fraction}_2 \times \text{Loss}_2)$$ ### Step-by-Step Solution * **First Part:** One-fourth ($\frac{1}{4}$) of the quantity. * **Profit on First Part:** $10\%$. * **Second Part:** The rest, which is $1 - \frac{1}{4} = \frac{3}{4}$ of the quantity. * **Loss on Second Part:** $20\%$. * **Calculating the net effect:** $$\text{Net } \% = \left(\frac{1}{4} \times 10\%\right) + \left(\frac{3}{4} \times -20\%\right)$$ $$\text{Net } \% = 2.5\% - 15\%$$ $$\text{Net } \% = -12.5\%$$ * The negative sign indicates an overall loss. ### Exam Strategy & Shortcut Assume the total cost price is 100 units consisting of 4 equal parts costing 25 each. Part 1 (Cost = 25): $+10\%$ of 25 = $+2.5$ profit. Remaining 3 parts (Cost = 75): $-20\%$ of 75 = $-15$ loss. Net effect = $+2.5 - 15 = -12.5$. Since the base is 100, the net effect is exactly $-12.5\%$, meaning a 12.5% loss. ### Common Pitfall A common error is misinterpreting "the rest". If a quarter is used, the remainder is three-quarters ($\frac{3}{4}$), not a half or some other fraction. ### Final Answer Therefore, the correct answer is **12.5% loss**.