A man bought goods worth ₹ $6000$ and sold half of them at a gain of $10\%$. At what gain percent must he sell the remainder so as to get a gain of $25\%$ on the whole?

### Concept & Target Average Profit When a quantity is divided into equal halves, the overall average profit percentage is the simple average of the profit percentages of the two halves. $$\text{Overall Profit } \% = \frac{\text{Profit}_1\% + \text{Profit}_2\%}{2}$$ ### Step-by-Step Solution * **Total Cost Price:** ₹ $6000$. * **Target overall gain:** $25\%$ on the whole. * Total desired profit = $25\%$ of $6000 = \text{₹ } 1500$. * **First half:** Half of the goods cost ₹ $3000$. * Gain on first half = $10\%$ of $3000 = \text{₹ } 300$. * **Second half:** The remaining goods cost ₹ $3000$. * Required gain on remainder = Total desired profit - Gain on first half * Required gain = $1500 - 300 = \text{₹ } 1200$. * **Gain percent on remainder:** $$\frac{1200}{3000} \times 100 = 40\%$$ ### Exam Strategy & Shortcut Since the goods are split into two exactly equal halves, you can ignore the ₹ 6000 completely. It is extraneous information. Let the required profit percentage be $x$. $$\frac{10\% + x\%}{2} = 25\%$$ $$10 + x = 50$$ $$x = 40$$ The remainder must be sold at a $40\%$ gain. ### Common Pitfall A common error is to directly subtract the percentages ($25\% - 10\% = 15\%$), failing to realize that the $10\%$ only applies to half the quantity, not the whole. ### Final Answer Therefore, the correct answer is **40%**.