Maximising Resource Allocation

📌 CONCEPT: A linear inequality is a mathematical statement that describes a relationship between two or more variables, where one expression is greater than, less than, or equal to another expression. In this case, we need to maximize the total profit by allocating resources between two projects, subject to a budget constraint. The goal is to find the optimal investment in the first project that maximizes the total profit.

📐 RULE / FORMULA: The rule for solving this problem is to use the concept of linear inequalities and the budget constraint to set up an equation. We can represent the investment in the first project as x and the investment in the second project as 100,000 - x. The total profit is then given by the sum of the profits from each project, which can be expressed as an inequality: x + 1.1(100,000 - x) ≤ 100,000.

💡 WORKED EXAMPLE: Let's say the first project yields an annual profit of 10% of the investment, and the second project yields 10% more profit. If the total investment is Rs 80,000, we can set up the inequality 0.1x + 1.1(80,000 - x) ≤ 80,000 and solve for x. Simplifying the inequality, we get 0.1x + 88,000 - 1.1x ≤ 80,000, which leads to -0.1x ≤ -8,000. Dividing by -0.1, we get x ≥ 80,000.

⚠️ COMMON MISTAKE: Students often make the mistake of not considering the budget constraint and investing too much in one project, resulting in a loss. They may also forget to account for the 10% increase in profit for the second project.