Problem 336 - Elementary Algebra

Question

Shower Budget Penny is planning a baby shower for her daughter-in-law. The restaurant charges \(\$ 950\) for up to 25 guests, plus \(\$ 31.95\) for each additional guest. How many guests can attend if Penny wants the total cost to be no more than \(\$ 1,500 ?\)

Step-by-Step Solution

Answer

42 guests.

Step 1: Identify the Fixed and Variable Costs

Determine the amount Penny needs to pay regardless of how many guests are added. The restaurant charges a fixed cost of \(\$ 950\) for up to 25 guests. The variable cost is \(\$ 31.95\) for each additional guest beyond 25.

Step 2: Set Up the Inequality

Let \( x \) be the number of additional guests Penny can invite. The total cost Penny can afford is \( \$ 1,500 \). Set up the inequality based on the given costs: \( 950 + 31.95x \leq 1,500 \)

Step 3: Solve the Inequality for Additional Guests

Subtract the fixed cost from the total budget to isolate the variable term: \( 31.95x \leq 1,500 - 950 \) Simplify the inequality: \( 31.95x \leq 550 \)

Step 4: Find the Maximum Value of x

Divide both sides of the inequality by \( 31.95 \) to solve for \( x \): \( x \leq \frac{550}{31.95} \) Calculate the value: \( x \leq 17.21 \)

Step 5: Determine the Total Number of Guests

Since \( x \) represents additional guests and must be a whole number, round down to the nearest whole number: \( x = 17 \) Add the 25 initial guests: \( 25 + 17 = 42 \)

Key Concepts

linear inequalities //the value of 'concept' will be the headline later on so DO NOT restate the headline in text

Linear inequalities are a way to represent a range of values that satisfy a given condition. In this baby shower problem, we use a linear inequality to decide how many guests can be invited without exceeding a budget.

The inequality is set up as: \(950 + 31.95x \leq 1,500\).
This inequality accounts for a fixed cost (\(950\)) plus variable costs (\(31.95\) per additional guest) that must not go over \(1,500\).
To solve it, we first isolate the variable term, which gives us \(31.95x \leq 550\).
Finally, solving for \(x\) helps us find the maximum number of additional guests.
When dealing with inequalities, remember to:

• Isolate the variable terms.
• Keep the inequality sign while performing operations.
• Deal with whole numbers if they are required by the context, like in this exercise.

budgeting //the value of 'concept' will be the headline later on so DO NOT restate the headline in text

Budgeting is the process of planning how to spend your money. In this scenario, Penny needs to ensure the total cost of the baby shower does not exceed her budget of \(\$1,500\).
Proper budgeting involves:

• Identifying fixed costs: Expenses that do not change regardless of the number of guests, like the \(\$950\) restaurant fee.
• Calculating variable costs: Expenses that change based on the number of additional guests. Here, it's \(\$31.95\) per extra guest.

Penny uses these details to set up and solve an inequality, ensuring she stays within her budget while maximizing the number of guests. Budgeting skills help in monitoring and controlling expenditures, and allocating resources effectively.

variable and fixed costs //the value of 'concept' will be the headline later on so DO NOT restate the headline in text

Understanding variable and fixed costs is crucial for effective financial planning. Fixed costs are constant, no matter how many units (like guests) are involved. In this scenario, the fixed cost is \(\$950\) for the restaurant.

Variable costs change with the number of units. Here, it's \(\$31.95\) for each additional guest beyond 25.
These types of costs are important in determining overall expenses and creating equations or inequalities for budgeting. By identifying and separating these costs, it becomes easier to:

• Set up accurate financial models.
• Predict changes in total cost with varying numbers of guests.
• Ensure the plan stays within budget by adjusting the number of units (guests, in this case).

whole numbers //the value of 'concept' will be the headline later on so DO NOT restate the headline in text

Whole numbers are integers without fractions or decimals, only complete units such as \(0, 1, 2, \) and so on. In the context of our problem, whole numbers are essential since the number of guests can’t be fractional or negative.

For the inequality \(x \leq 17.21\), we round \(x\) down to the nearest whole number: \(x = 17\).
Using whole numbers ensures that the count of additional guests is realistic and practical.
Key points to remember:

• Whole numbers are used in real-life scenarios where fractional units don't make sense, such as counting people.
• In inequalities and equations, ensure solutions are rounded appropriately to whole numbers if required by the problem context.