A departmental store receives a shipment of $1000$ pens, for which it pays ₹ $9000$. The store sells the pens at a price $80$ percent above cost for one month, after which it reduces the price of the pens to $20$ percent above cost. The store sells $75$ percent of the pens during the first month and $50$ percent of the remaining pens afterwards. How much gross income did the sales of the pens generate?

### Concept & Gross Income Calculation Gross income is the total revenue generated from sales before any expenses are deducted. To find it, calculate the Cost Price (CP) per unit, apply the respective markups to find the Selling Prices (SP) for different periods, and multiply by the quantities sold during those periods. $$\text{Gross Income} = (Q_1 \times SP_1) + (Q_2 \times SP_2)$$ ### Step-by-Step Solution * **Cost Price per pen:** The store pays ₹ 9000 for 1000 pens. $$\text{CP per pen} = \frac{9000}{1000} = \text{₹ } 9$$ * **First Month Sales ($Q_1$ & $SP_1$):** * Price is 80% above cost: $SP_1 = 9 + (0.80 \times 9) = 9 + 7.20 = \text{₹ } 16.20$. * Quantity sold ($75\%$ of 1000) = $750 \text{ pens}$. * Revenue 1 = $750 \times 16.20 = \text{₹ } 12150$. * **Afterwards Sales ($Q_2$ & $SP_2$):** * Remaining pens after first month = $1000 - 750 = 250 \text{ pens}$. * Quantity sold ($50\%$ of remaining) = $0.50 \times 250 = 125 \text{ pens}$. * New Price is 20% above cost: $SP_2 = 9 + (0.20 \times 9) = 9 + 1.80 = \text{₹ } 10.80$. * Revenue 2 = $125 \times 10.80 = \text{₹ } 1350$. * **Total Gross Income:** $$\text{Total Revenue} = 12150 + 1350 = \text{₹ } 13500$$ ### Exam Strategy & Shortcut Work with markups fractionally to speed up multiplication. CP = 9. SP1 markup is 80% (4/5). SP1 = $9 \times \frac{9}{5}$. Rev 1 = $750 \times \frac{81}{5} = 150 \times 81 = 12150$. SP2 markup is 20% (1/5). SP2 = $9 \times \frac{6}{5}$. Rev 2 = $125 \times \frac{54}{5} = 25 \times 54 = 1350$. Sum = 13500. ### Common Pitfall Many students mistakenly assume all remaining 250 pens were sold in the second phase. The problem specifically states "50 percent of the remaining pens", which is only 125 pens. Reading carefully prevents this trap. ### Final Answer Therefore, the correct answer is **₹ 13500**.