Problem 335 - Elementary Algebra

Question

Wedding Budget Adele and Walter found the perfect venue for their wedding reception. The cost is \(\$ 9,850\) for up to 100 guests, plus \(\$ 38\) for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than \(\$ 12,500 ?\)

Step-by-Step Solution

Answer

169 guests can attend.

Step 1: Understand the Cost Structure

The total cost for the wedding reception includes a fixed cost of \( \$ 9,850 \) for up to 100 guests, plus an additional \( \$ 38 \) for each guest beyond the first 100.

Step 2: Define Variables

Let \( x \) represent the number of additional guests beyond the first 100 guests. So, the total number of guests is \( 100 + x \).

Step 3: Set Up the Inequality

The total cost should not exceed \( \$ 12,500 \). Therefore, we set up the inequality: \[ 9850 + 38x \leq 12500 \]

Step 4: Solve for x

First, isolate the term with \( x \): \[ 9850 + 38x \leq 12500 \]Subtract \( 9850 \) from both sides: \[ 38x \leq 12500 - 9850 \]Simplify the right-hand side: \[ 38x \leq 2650 \]Finally, divide by \( 38 \) to solve for \( x \): \[ x \leq \frac{2650}{38} \]\[ x \leq 69.7368 \]

Step 5: Interpret the Result

Since \( x \) represents the number of additional guests, round down to the nearest whole number if necessary: \[ x = 69 \]Therefore, the total number of guests is \( 100 + x = 100 + 69 = 169 \).

Key Concepts

solving inequalities

Solving inequalities is a fundamental skill in algebra. In this exercise, we solved an inequality to determine the maximum number of guests Adele and Walter could invite to their wedding reception without exceeding their budget. Here’s a quick overview of the steps we took:

• First, we identified the various costs involved: a fixed cost for up to 100 guests and a variable cost for additional guests.
• We then set up an inequality to represent the scenario. Inequalities in algebra use symbols like \(\leq\) and \(\geq\) to show a range of possible values.
• We isolated the variable by performing algebraic operations like subtraction and division on both sides of the inequality.
• Finally, we interpreted the result by rounding down to ensure we did not exceed the budget.

These steps help you understand the fundamental principles of solving inequalities, which can be applied to various real-world problems such as budgeting, as demonstrated here.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the context of this problem, we used algebraic expressions to represent the total cost of the wedding reception.

• We defined a variable, \(x\), to represent the number of additional guests beyond the initial 100 guests.
• The total cost was then described as the sum of a fixed cost (\$9850) and a variable cost (\$38 per additional guest), leading to the expression \(9850 + 38x\).

By manipulating this expression, we could set up an inequality to ensure the total cost did not exceed a specified amount. Understanding how to create and manipulate algebraic expressions is crucial for solving various mathematical problems. It's like building a mathematical sentence to describe a real-world situation.

budgeting problems

Budgeting problems often involve making sure costs do not exceed a certain limit. In this exercise, Adele and Walter wanted to know the maximum number of guests they could invite without spending more than \(\$12,500\).

• We began by understanding the cost structure, including a fixed cost and a variable cost.
• We then used algebra to represent and analyze the total cost. This helped us see how different numbers of guests would affect the overall expense.
• Finally, we solved an inequality to ensure the budget constraint was met. This step-by-step approach can be applied to various budgeting problems in real life, such as planning events, managing expenses, or even personal finance.

Budgeting is an essential life skill that involves careful planning and analysis. By applying algebraic techniques, you can make informed decisions and ensure you stay within your financial limits.