A school has to buy at least 15 chairs within a budgetary ceiling of ₹ 2000. A chair with arms costs ₹ 160 and one without arms costs ₹ 100. What is the maximum number of chairs with arms that the school can buy?

### Concept & Logic Use linear inequalities to maximize one variable while satisfying a minimum quantity constraint and a maximum budget limit. ### Step-by-Step Solution * Let $x$ be the number of chairs with arms (costing ₹ 160). * Let $y$ be the number of chairs without arms (costing ₹ 100). * Condition 1: Total chairs must be at least 15. $$x + y \geq 15 \implies y \geq 15 - x$$ * Condition 2: Total cost must be within ₹ 2000. $$160x + 100y \leq 2000$$ * Divide the budget inequality by 20 for easier calculation: $$8x + 5y \leq 100$$ * To maximize $x$ (the expensive chairs), we must minimize the budget spent on $y$. This happens when we buy exactly 15 total chairs, so we substitute $y = 15 - x$: $$8x + 5(15 - x) \leq 100$$ $$8x + 75 - 5x \leq 100$$ $$3x \leq 25$$ $$x \leq \frac{25}{3} \approx 8.33$$ * Since the number of chairs must be an integer, the maximum possible value for $x$ is 8. * Verification: If $x = 8$, then $y = 7$. Total cost = $8(160) + 7(100) = 1280 + 700 = 1980$, which perfectly satisfies the ₹ 2000 budget and the 15 chair minimum. ### Exam Strategy & Shortcut Since the goal is to maximize the number of chairs with arms, test the options starting from the highest to the lowest. Try Option (d) 12: $12 \times 160 = 1920$. Remaining budget = 80. You can't buy even 1 armless chair. Total chairs = 12. (Fails the $\geq 15$ rule). Try Option (c) 9: $9 \times 160 = 1440$. Remaining budget = 560. You can buy 5 armless chairs. Total chairs = $9 + 5 = 14$. (Fails the $\geq 15$ rule). Try Option (b) 8: $8 \times 160 = 1280$. Remaining budget = 720. You can buy 7 armless chairs. Total chairs = $8 + 7 = 15$. (Satisfies all conditions). ### Common Pitfall Students often ignore the "at least 15 chairs" constraint and simply divide the total budget by the cost of the expensive chairs ($2000 / 160 = 12.5$), incorrectly answering 12. ### Final Answer Therefore, the correct answer is 8.