Hemant sold $10$ sarees for a total profit of ₹ $460$ and $12$ sarees for a total profit of ₹ $144$. At what profit per saree should he sell the remaining $20$ sarees so that he gets an average profit of ₹ $18$ per saree?

### Concept & Target Average To achieve a specific average profit across multiple items, the sum of all profits must equal the target average multiplied by the total number of items. $$\text{Total Target Profit} = \text{Total Quantity} \times \text{Target Average Profit}$$ ### Step-by-Step Solution * **Total number of sarees:** $10 + 12 + 20 = 42 \text{ sarees}$. * **Target overall profit:** $42 \text{ sarees} \times \text{₹ } 18/\text{saree} = \text{₹ } 756$. * **Profit already earned:** Profit from first 10 + Profit from next 12 = $460 + 144 = \text{₹ } 604$. * **Required profit from remaining 20 sarees:** $\text{Target Profit} - \text{Profit already earned} = 756 - 604 = \text{₹ } 152$. * **Required profit per saree for the remaining batch:** $\frac{\text{₹ } 152}{20 \text{ sarees}} = \text{₹ } 7.60$. ### Exam Strategy & Shortcut You can also use the method of deviations from the mean. Target mean = $18$. Batch 1 (10 items): average profit = $460/10 = 46$ (deviation = $+28$). Total deviation = $+280$. Batch 2 (12 items): average profit = $144/12 = 12$ (deviation = $-6$). Total deviation = $-72$. Net deviation so far = $280 - 72 = +208$. To balance to 0, the remaining 20 items must have a deviation of $-208$. Deviation per item = $-208 / 20 = -10.4$. Required profit per item = $18 - 10.4 = 7.60$. ### Common Pitfall Misreading "total profit" as "profit per saree" for the first two batches. Ensure you note whether a value represents an aggregate or a per-unit metric. ### Final Answer Therefore, the correct answer is **₹ 7.60**.