Calculating Sandwich Costs A Math Problem For Charlie's Lunch

Introduction

In this article, we will tackle a real-world math problem involving Charlie, who wants to order lunch for his friends and little brother. This problem involves calculating how much Charlie can spend on each sandwich, given his budget and other expenses. We'll break down the problem step-by-step, focusing on forming the correct inequality and solving it to find the maximum cost per sandwich. This exercise is not only a practical application of mathematical concepts but also an excellent way to enhance problem-solving skills. So, let's delve into the details of Charlie's lunch order and figure out the cost per sandwich.

Problem Statement Charlie's Lunch Dilemma

Charlie is in charge of ordering lunch for his friends, and he's got a budget to stick to. He needs to order 6 sandwiches, and he also wants to get a $2 kid's meal for his little brother. Charlie has a total of $32 to spend. The challenge here is to determine how much Charlie can spend on each sandwich if they are all the same price. This problem is a classic example of how inequalities can be used in everyday situations to manage budgets and make purchasing decisions. We need to find the maximum amount Charlie can allocate to each sandwich while staying within his budget. This requires us to set up an inequality that represents the situation and then solve it to find the maximum possible cost per sandwich. The solution will give us a clear understanding of how to approach similar real-life budgeting problems. By the end of this article, you'll have a solid grasp of how to translate a word problem into a mathematical inequality and solve it effectively.

Setting up the Inequality for Sandwich Costs

To solve Charlie's lunch dilemma, we need to translate the word problem into a mathematical inequality. This is a critical step in problem-solving, as it allows us to use algebraic methods to find the solution. Let's break down the information given: Charlie wants to buy 6 sandwiches, each costing the same amount, and a $2 kid's meal. He has a total of $32 to spend. We can represent the cost of each sandwich with the variable x. The total cost of the 6 sandwiches will then be 6x. Adding the $2 for the kid's meal, the total expenditure becomes 6x + 2. Since Charlie cannot spend more than $32, we can set up the following inequality:

6x + 2 ≤ 32

This inequality states that the total cost of the sandwiches and the kid's meal must be less than or equal to $32. This is a crucial equation because it sets the foundation for solving the problem. The left side of the inequality, 6x + 2, represents the total amount Charlie will spend, and the right side, 32, represents his budget limit. The inequality symbol, ≤, indicates that the total spending must be within or equal to his budget. This mathematical representation of the problem allows us to use algebraic techniques to determine the maximum value of x, which is the most Charlie can spend on each sandwich. By understanding how to set up this inequality, we are well on our way to finding the solution and helping Charlie plan his lunch order effectively.

Solving the Inequality Step-by-Step

Now that we have set up the inequality 6x + 2 ≤ 32, the next step is to solve it for x. Solving this inequality will give us the maximum amount Charlie can spend on each sandwich while staying within his $32 budget. Let's go through the steps one by one. First, we need to isolate the term with the variable x. To do this, we subtract 2 from both sides of the inequality:

6x + 2 - 2 ≤ 32 - 2 6x ≤ 30

This step simplifies the inequality by removing the constant term from the left side, making it easier to isolate x. Next, we need to isolate x completely. To do this, we divide both sides of the inequality by 6:

(6x) / 6 ≤ 30 / 6 x ≤ 5

This step gives us the solution for x, which represents the maximum cost per sandwich. The inequality x ≤ 5 means that Charlie can spend at most $5 on each sandwich. This result is crucial because it provides a clear limit for Charlie's spending. Any amount greater than $5 per sandwich would exceed his budget of $32. By solving the inequality step-by-step, we have determined the maximum allowable cost for each sandwich, ensuring Charlie can order lunch for his friends and brother without overspending. This process demonstrates the practical application of algebraic inequalities in managing real-world financial constraints.

Determining the Maximum Cost per Sandwich

From solving the inequality x ≤ 5, we've determined that Charlie can spend a maximum of $5 on each sandwich. This result is the key to solving the problem, as it provides a clear upper limit on the cost per sandwich. To ensure Charlie stays within his budget, he needs to choose sandwiches that cost $5 or less. This is a practical application of mathematical problem-solving, where we use an inequality to find a real-world constraint. It's important to understand that x ≤ 5 means x can be any value less than or equal to 5. In the context of the problem, this means Charlie can choose sandwiches that cost $5 each, or he can opt for cheaper options to save money. This flexibility allows Charlie to make the best decision based on the available sandwich prices and his preferences. For instance, if the sandwiches cost $4.50 each, he would still be within budget and have some money left over. Understanding the solution x ≤ 5 provides Charlie with the necessary information to make an informed decision and manage his budget effectively. This step is crucial in translating a mathematical result into a practical solution for the problem at hand, demonstrating the real-world relevance of algebra.

Conclusion Charlie's Savvy Lunch Planning

In conclusion, by setting up and solving the inequality 6x + 2 ≤ 32, we have successfully determined that Charlie can spend a maximum of $5 on each sandwich. This problem highlights the practical application of inequalities in managing budgets and making informed purchasing decisions. The step-by-step approach we used—translating the word problem into a mathematical inequality, solving the inequality, and interpreting the result—is a valuable skill that can be applied to various real-life scenarios. Understanding these concepts not only helps in solving mathematical problems but also in making sound financial decisions. For Charlie, this means he can confidently order lunch for his friends and little brother, knowing he is staying within his budget. The ability to break down a problem, represent it mathematically, and solve it is a fundamental aspect of both mathematics and everyday life. This exercise demonstrates the importance of these skills and how they can be used to solve practical problems efficiently. By mastering these techniques, you can tackle similar challenges with confidence and make informed decisions in various aspects of your life.