Question

Use this information to answer the following 4 questions: Marie has a weekly budget of \(\$ 24,\) which she likes to spend on magazines and pies. If the price of a pie is \(\$ 12,\) what is the maximum number of pies she could buy in a week?

Short answer

Marie can buy a maximum of \(2\) pies in a week without exceeding her weekly budget.

Step-by-step solution

Identify the given information.

We are given Marie's weekly budget, which is \(\$24\). Additionally, we know the cost of a pie, which is \(\$12\).

Set up an inequality.

We want to find out the maximum number of pies, say x, that she can buy so that her weekly expenditures don't exceed her budget. We can set up an inequality as follows: \(12x \le 24\)

Solve the inequality.

To find the maximum number of pies Marie can buy, we need to solve the inequality: Divide both sides by 12: \(x \le \frac{24}{12}\)

Simplify the inequality.

Simplify the inequality by dividing: \(x \le 2\) In conclusion, Marie can buy a maximum of 2 pies in a week without exceeding her weekly budget.

Key concepts

Consumer Choice and Budget Constraints

When making buying decisions, consumers like Marie must consider budget constraints. This refers to the financial limits imposed by their available money. In this scenario, Marie has a weekly budget of \(\\(24\) to spend across different goods, namely magazines and pies.

The fundamental idea behind consumer choice is that individuals strive to maximize their satisfaction or utility within the confines of their budget. Marie's challenge is to optimize the purchase of pies without overspending. Each pie costs \(\\)12\), so she must calculate how many she can realistically afford. This decision-making process involves examining preferences, prices, and budget limitations.

Budget constraints often force consumers to make trade-offs. Marie must decide not only how many pies to buy but also how that might limit other potential purchases, such as magazines. Understanding these constraints forms the backbone of effective consumer choice, ensuring spending aligns with financial capacity.

Solving Inequality to Determine Maximum Purchase

To determine how many pies Marie can purchase while staying within her budget, we use inequality. Inequality helps us calculate boundaries in scenarios involving limitations, such as budget limits.

In our case, the inequality \(12x \le 24\) helps clarify how many pies Marie can afford. Here, \(x\) represents the number of pies she wishes to buy, and the expression \(12x\) indicates the total cost of these pies.

• Total Cost of Pies: Cost per pie \(\times x\) number of pies
• Budget Limitation: Ensure total cost doesn't exceed the budget

By solving this inequality, we determine the maximum value of \(x\) that satisfies the budget constraint. Through simple division, we find \(x \le 2\), meaning Marie can purchase a maximum of 2 pies without exceeding her budget. Thus, inequality serves as an essential tool in making prudent financial decisions and optimizing consumer choice.

Mathematical Reasoning in Budget Analysis

Mathematical reasoning involves logical steps to interpret given data and reach conclusions. In budget analysis, like Marie's situation, it provides clarity and structure to otherwise complex decisions.

To determine the number of pies within budget, each step of the solution uses mathematical principles:

• Identify Information: Recognizing given data—budget and pie cost.
• Set Up an Inequality: Formulate \(12x \le 24\) to represent the spending decision.
• Simplify: Calculate \(x \le 2\) through division to find a workable solution.

Mathematical reasoning transforms abstract financial situations into solvable equations. By applying these logical steps, problems become approachable, aiding decision-making. This structured approach ensures consumers understand not only their limits but also potential alterations in choice dynamics due to budget shifts.