A cloth merchant sold half of his cloth at $20\%$ profit, half of the remaining at $20\%$ loss and the rest was sold at the cost price. In the total transaction, his gain or loss will be

### Concept & Net Effect via Fractions To find the total gain or loss percentage, apply the given profit and loss rates to their respective fractions of the total quantity. Losses are treated as negative values, and selling at cost price means a 0% profit. $$\text{Net Effect } \% = \sum (\text{Fraction} \times \text{Rate})$$ ### Step-by-Step Solution * **Part 1:** "Half of his cloth" = $\frac{1}{2}$. * Rate = $+20\%$ (Profit). * **Part 2:** "Half of the remaining". * Remaining cloth after first sale = $1 - \frac{1}{2} = \frac{1}{2}$. * Half of this remainder = $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$. * Rate = $-20\%$ (Loss). * **Part 3:** "The rest". * Cloth accounted for so far = $\frac{1}{2} + \frac{1}{4} = \frac{3}{4}$. * The rest = $1 - \frac{3}{4} = \frac{1}{4}$. * Rate = $0\%$ (Sold at cost price). * **Calculate the Overall Net Effect:** $$\text{Net } \% = \left(\frac{1}{2} \times 20\right) + \left(\frac{1}{4} \times -20\right) + \left(\frac{1}{4} \times 0\right)$$ $$\text{Net } \% = 10 - 5 + 0$$ $$\text{Net } \% = +5\%$$ * Since the result is positive, it is a $5\%$ gain. ### Exam Strategy & Shortcut Assume the total cloth is 100 meters, costing ₹ 1 per meter (Total CP = ₹ 100). - Sell 50 meters at 20% profit: SP = $50 \times 1.2 = \text{₹ } 60$. - Sell 25 meters at 20% loss: SP = $25 \times 0.8 = \text{₹ } 20$. - Sell remaining 25 meters at CP: SP = $\text{₹ } 25$. Total SP = $60 + 20 + 25 = 105$. Gain = $105 - 100 = 5$. Since base is 100, gain is exactly 5%. ### Common Pitfall A common misinterpretation is reading "half of the remaining" simply as "the other half". It is crucial to mathematically compute fractions sequentially based on what is left over. ### Final Answer Therefore, the correct answer is **5% gain**.