Question

Bob is planning a birthday party for his son and has a maximum budget of \(\$ 475\) to spend. He intends to rent a party room that costs \(\$ 125\) for the day. He expects a total of 35 people to attend the party and plans to provide them all with lunch. Which of the following inequalities shows how to calculate the amount \(x\) that Bob can spend on lunch for each person? (A) \(125 x+35 \geq 475\) (B) \(125 x+35 \leq 475\) (C) \(125+35 x \geq 475\) (D) \(125+35 x \leq 475\)

Short answer

\(125 + 35x \leq 475\)

Step-by-step solution

Identify the Maximum Budget and Renting Fee

Bob has a maximum budget of \(\$475\) to spend on the entire party and the renting fee is \(\$125\). So, he has \(475-125\) dollars left for the lunch.

Calculate the Remaining Budget

Subtract the renting fee from the total budget: \[475 - 125 = 350\] Bob has \(\$350\) left to spend on lunch for the guests.

Relate the Remaining Budget to the Lunch Expenses

The remaining budget must be divided among 35 people. So, the cost of lunch per person can be represented as \(35x\) where x is the amount Bob can spend on lunch for each person. The inequality would then indicate that the cost of lunch for all people is less than or equal to the remaining budget: \[35x \leq 350.\]

Observe the Given Options and Select the Correct Inequality

The inequality obtained in Step 3 matches with option (D): \[125 + 35x \leq 475.\] So, the answer is (D).

Key concepts

Understanding Inequalities

Inequalities are a fundamental concept in mathematics that allow us to compare two values, expressions, or quantities. In the context of budget constraints, inequalities help determine limits on spending. Instead of stating exact values, an inequality provides a range or a boundary. For example, in Bob's party planning, we used the inequality \(125 + 35x \leq 475\) to show that the total cost of the room and lunch should not exceed \(475.

This inequality tells us that the sum of the room rental fee (\)125) plus the cost of lunch for 35 people must be less than or equal to Bob's budget of $475. When dealing with real-world constraints like budget limits, inequalities help ensure decisions stay within allowed boundaries. They are crucial for effective and responsible problem-solving.

Problem-Solving with Constraints

Problem-solving often involves dealing with constraints or limits, and budget constraints are a common type of constraint. This requires a systematic approach to ensure all elements fit within the available limits. In Bob's party planning exercise, we faced a constraint that his total expenses must not exceed $475. This meant we had to calculate and allocate resources effectively.

Here's how to approach similar problems:

• Identify all given constraints, such as total budget and fixed costs (room rental).
• Subtract fixed costs from the total budget to find remaining resources (money for lunch).
• Use inequalities to express how variables (cost per person) relate to constraints.
• Choose the solution that best fits all constraints, ensuring the plan stays viable.

This approach simplifies problem-solving, making it easier to handle multiple variables and limits.

The Role of Mathematical Reasoning

Mathematical reasoning is the logical thought process used to solve problems and make decisions based on mathematical concepts. In our exercise, we applied mathematical reasoning to determine the most Bob could spend per person on lunch without exceeding his budget.

Here's the reasoning process:

• First, identify known values and fixed variables, like the total budget and rental fees.
• Calculate what remains after accounting for fixed costs, which in this case is $350 for lunches.
• Express the variable costs (lunch expenses) with an inequality to ensure all guests are fed within budget.
• Compare the inequality with given options to select the most accurate representation of the scenario.

Developing strong mathematical reasoning skills aids in breaking down problems into manageable parts, ensuring logical and accurate solutions.