I purchased $120$ exercise books at the rate of ₹ $3$ each and sold $\frac{1}{3}$ of them at the rate of ₹ $4$ each, $\frac{1}{2}$ of them at the rate of ₹ $5$ each and the rest at the cost price. My profit percent is (C.P.O., 2006)

### Concept & Profit Calculation To find the overall profit percentage, we need to determine the total Cost Price (CP) of all items and the total Selling Price (SP) of all items after dividing them into their respective sales batches. $$\text{Profit } \% = \frac{\text{Total SP} - \text{Total CP}}{\text{Total CP}} \times 100$$ ### Step-by-Step Solution * **Total Cost Price:** $120 \text{ books} \times \text{₹ } 3/\text{book} = \text{₹ } 360$. * **Batch 1 (1/3 of total):** $\frac{1}{3} \times 120 = 40 \text{ books}$. * Selling Price for Batch 1 = $40 \times \text{₹ } 4 = \text{₹ } 160$. * **Batch 2 (1/2 of total):** $\frac{1}{2} \times 120 = 60 \text{ books}$. * Selling Price for Batch 2 = $60 \times \text{₹ } 5 = \text{₹ } 300$. * **Batch 3 (The rest):** Total books - (Batch 1 + Batch 2) = $120 - (40 + 60) = 120 - 100 = 20 \text{ books}$. * These are sold at cost price (₹ 3 each). * Selling Price for Batch 3 = $20 \times \text{₹ } 3 = \text{₹ } 60$. * **Total Selling Price:** $160 + 300 + 60 = \text{₹ } 520$. * **Total Profit:** $\text{Total SP} - \text{Total CP} = 520 - 360 = \text{₹ } 160$. * **Profit Percentage:** $$\frac{160}{360} \times 100 = \frac{16}{36} \times 100 = \frac{4}{9} \times 100 = \frac{400}{9}\%$$ $$\frac{400}{9}\% = 44 \frac{4}{9}\%$$ ### Exam Strategy & Shortcut You can solve this using only ratios and per-item profits without calculating the total large values. Assume total items = 6 (LCM of 3 and 2). Cost per item = 3. Total CP for 6 items = 18. - 1/3 of 6 = 2 items sold at 4 (Profit = +1 each, total profit = +2) - 1/2 of 6 = 3 items sold at 5 (Profit = +2 each, total profit = +6) - Remaining = 1 item sold at CP (Profit = 0) Total Profit on 6 items = 8. Profit \% = $\frac{8}{18} \times 100 = \frac{4}{9} \times 100 = 44 \frac{4}{9}\%$. ### Common Pitfall A common error is miscalculating the "remaining" fraction by subtracting from 1 incorrectly, or forgetting to multiply the remaining quantity by the original cost price. ### Final Answer Therefore, the correct answer is **$44 \frac{4}{9}\%$**.