Question

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

Short answer

The maximum number of magazines Marie can buy in a week is 6.

Step-by-step solution

Defining the variables

Let's begin by assigning the variables: M = the number of magazines Marie can buy P = the price of one magazine, which is $4 dollars B = the total budget, which is $24 dollars The problem asks for the maximum value of M.

Setting up the equation

Marie's total spending on magazines is the number of magazines multiplied by the price of each magazine (M times P). The total spending should not exceed the total budget she has. Therefore, we can set up the following equation: \(M \times P \leq B\)

Plugging in known values

We know that the price of a magazine (P) is \(4 dollars and Marie's total budget (B) is \)24 dollars. Let's plug these values into the equation: \(M \times 4 \leq 24\)

Solving for the maximum number of magazines (M)

Now, let's solve for M by dividing both sides of the inequality by 4: \(M \leq 24 \div 4\) \(M \leq 6\) The maximum number of magazines Marie can buy in a week is 6. So the answer is 6 magazines.

Key concepts

Consumer Choice and Budget Constraint

Understanding consumer choice is essential in everyday life decisions, like how Marie manages her weekly spending. She has to decide how best to allocate her budget of $24 to maximize her satisfaction. This involves selecting the right combination of products, in Marie's case, magazines and pies.

The budget constraint represents the boundary of choices available given a limited income. It's depicted by the equation \(M \times P \leq B\), where \(M\) represents the number of magazines, \(P\) is the price of each magazine, and \(B\) is the total budget. Marie's task is to identify the combination of goods that provides the most utility or satisfaction without breaching this limit.

• The consumer must always consider the cost of a choice in terms of other opportunities they forfeit (opportunity cost).
• Informed decisions reflect personal preference and the utility derived from different goods.

By carefully weighing her options, Marie can make the best possible use of her limited budget.

Inequality Constraint

An inequality constraint provides a way to express limitations in consumer choices. When dealing with money and spending, it's important to remember that expenditures should not exceed available funds.

Marie faces an inequality constraint in the form of \(M \times 4 \leq 24\). This simple inequality ensures that the total amount spent on magazines does not exceed her $24 budget. She's allowed to buy up to, but not exceeding, this limit.

• Inequalities help simplify complex decisions by setting clear boundaries.
• They automatically adjust permissible choices when any input changes, like price or total available funds.

Understanding how to resolve inequalities is a fundamental skill to manage personal finances effectively.

Mathematical Modeling in Consumer Decisions

Mathematical modeling transforms a real-world scenario into a mathematical formula to simplify complex decision-making. The budget constraint model uses algebraic equations to illustrate potential choices within a defined budget.

When Marie calculates how many magazines she can buy, she uses the model \(M \times 4 \leq 24\) to find the maximum value of \(M\). The steps involved include:

• Defining Variables: Determine what each symbol represents in the problem.
• Setting up Equations: Use these variables to formulate an equation that captures the essence of the problem. This equation reflects the balance between spending and income.
• Solving: Solve the equation to find the maximum or best solution.

Such mathematical models empower consumers like Marie to make data-driven decisions, visualize potential outcomes, and effectively plan their finances.