Aditya went to a stationery shop to buy some Parker pens. Gel pens cost ₹ 300 each while fountain pens cost ₹ 400 each. Aditya spent a total of ₹ 3600 on pens. If he had bought as many fountain pens as the number of gel pens he actually bought and vice versa, he would have saved an amount equal to half the cost of one pen of one of the two types. Find the total number of pens he bought.

### Concept & Logic The problem involves setting up simultaneous equations based on quantities and costs. The key insight is that swapping the quantities of two items changes the total cost by the difference in their individual prices multiplied by the difference in their quantities. Let the number of gel pens be $g$ and fountain pens be $f$. The total cost equation is: $$300g + 400f = \text{Total Cost}$$ ### Step-by-Step Solution * **Given:** * Cost of one gel pen = ₹ 300 * Cost of one fountain pen = ₹ 400 * Total spent = ₹ 3600 * Savings upon swapping quantities = Half the cost of one of the pen types. * **Calculation:** 1. Formulate the actual cost equation: $$300g + 400f = 3600$$ Dividing by 100, we get: $$3g + 4f = 36$$ 2. Calculate the hypothetical savings. If quantities are swapped, the new cost is $300f + 400g$. Savings = Actual Cost - Hypothetical Cost Savings = $(300g + 400f) - (300f + 400g)$ Savings = $100f - 100g = 100(f - g)$ 3. Equate savings to the given condition. Half the cost of a pen is either ₹ 150 (gel) or ₹ 200 (fountain). Possibility 1: $100(f - g) = 150$ gives $f - g = 1.5$. Since quantities must be integers, this is invalid. Possibility 2: $100(f - g) = 200$ gives $f - g = 2$. This implies $f = g + 2$. 4. Substitute $f = g + 2$ into the simplified actual cost equation: $$3g + 4(g + 2) = 36$$ $$3g + 4g + 8 = 36$$ $$7g = 28$$ $$g = 4$$ 5. Find $f$ and the total number of pens: $$f = 4 + 2 = 6$$ Total pens = $g + f = 4 + 6 = 10$ ### Exam Strategy & Shortcut **Option Elimination using the Cost Equation:** We know $3g + 4f = 36$. Since total pens $T = g + f$, we have $f = T - g$. Substitute this into the equation: $3g + 4(T - g) = 36 \Rightarrow 4T - g = 36$. Now, test the options for $T$: * If $T = 8$: $32 - g = 36 \Rightarrow g = -4$ (Invalid) * If $T = 9$: $36 - g = 36 \Rightarrow g = 0$. If $g=0$, swapping quantities gives a savings of ₹ 900, which does not match half the cost of any pen. * If $T = 10$: $40 - g = 36 \Rightarrow g = 4$. Then $f = 6$. Swap savings = $100(6-4) = 200$. This perfectly matches half of ₹ 400. ### Common Pitfall A major pitfall is getting stuck on the phrase "half the cost of one pen of one of the two types" and trying to create a complex fractional algebraic equation rather than simply testing the two possible values (150 and 200) against the integer constraint. Always remember that physical items (like pens) must result in integer quantities. ### Final Answer Therefore, the correct answer is 10.