Calculating Hanna's Shopping Spree Total Cost Of Socks And Blouses

Introduction

In this article, we will delve into a mathematical problem involving Hanna's shopping trip. Hanna is purchasing socks and blouses, and we aim to determine the total cost based on the number of items she buys. This scenario provides a practical application of algebra, allowing us to use variables and equations to represent real-world situations. Specifically, we'll be looking at how to represent the total cost of Hanna's purchases using an algebraic expression, considering the individual prices of socks and blouses and the quantities she buys of each. This exercise is not only a great way to practice algebraic principles but also to understand how math is applicable in everyday scenarios like shopping. The beauty of this problem lies in its simplicity and its ability to be easily visualized, making it a perfect example for learners to grasp the concept of algebraic expressions. By the end of this discussion, you should have a clear understanding of how to formulate an equation that represents the total cost of Hanna’s shopping spree, setting a solid foundation for more complex algebraic problems in the future. So, let’s dive in and explore the world of shopping math with Hanna!

Problem Statement: Defining the Variables and Costs

In this mathematical problem, Hanna is shopping for socks and blouses. Each pair of socks costs $2.99, while each blouse is priced at $12.99. We are given that x represents the number of pairs of socks Hanna purchases, and y represents the number of blouses she buys. The core question we need to address is: How can we represent the total cost of Hanna's purchases using an algebraic expression? This problem introduces us to the concept of variables and how they can be used to represent unknown quantities in a real-world scenario. Here, x and y are variables that stand for the number of socks and blouses, respectively. The prices of the items are constants, meaning they don't change regardless of how many items Hanna buys. Understanding this distinction between variables and constants is crucial in algebra. The problem sets the stage for us to construct an expression that combines these variables and constants to accurately depict the total cost. The challenge lies in translating the given information into a mathematical equation, which will involve multiplying the quantity of each item by its respective price and then adding those amounts together. This is a fundamental skill in algebra, and mastering it will allow you to solve a wide range of problems involving costs, quantities, and totals. So, let's break down the components of the problem and see how we can formulate the required algebraic expression.

Building the Algebraic Expression: Socks

To begin constructing the algebraic expression, let's focus on the cost of the socks. We know that each pair of socks costs $2.99, and Hanna buys x pairs of socks. To find the total cost of the socks, we need to multiply the price per pair by the number of pairs purchased. This can be represented mathematically as 2.99 * x, or simply 2.99x. This expression, 2.99x, tells us that the total amount Hanna spends on socks is directly proportional to the number of pairs she buys. For example, if Hanna buys 1 pair of socks, the cost is 2.99 * 1 = $2.99. If she buys 5 pairs, the cost is 2.99 * 5 = $14.95. This simple multiplication forms the first part of our overall expression for the total cost. It illustrates how algebra can be used to represent a direct relationship between quantity and price. Understanding this component is crucial because it lays the groundwork for the next part, where we calculate the cost of the blouses. By breaking down the problem into smaller parts like this, we can approach it more methodically and ensure we accurately capture all the information in our algebraic expression. So, with the cost of socks clearly represented, let's move on to the blouses and see how we can incorporate their cost into our equation.

Building the Algebraic Expression: Blouses

Now, let's consider the cost of the blouses. We are given that each blouse costs $12.99, and Hanna purchases y blouses. Similar to how we calculated the cost of the socks, we need to multiply the price per blouse by the number of blouses purchased to find the total cost. This can be represented as 12.99 * y, or simply 12.99y. This expression, 12.99y, signifies that the total amount Hanna spends on blouses is directly proportional to the number of blouses she buys. For instance, if Hanna buys 1 blouse, the cost is 12.99 * 1 = $12.99. If she buys 3 blouses, the cost is 12.99 * 3 = $38.97. This multiplication provides the second key component of our total cost expression. Just as with the socks, understanding this relationship between quantity and price is fundamental to formulating the correct equation. The cost of blouses, represented by 12.99y, needs to be combined with the cost of socks to give us the total expenditure of Hanna’s shopping trip. By clearly defining the cost of each item separately, we are building a comprehensive picture of the total cost. The next step is to combine these two expressions to create a single equation that represents the total amount Hanna spends on both socks and blouses. So, let's move on to the final step of piecing together our algebraic expression.

Combining the Expressions: Total Cost

Having determined the individual costs of socks and blouses, we can now combine these expressions to find the total cost of Hanna's shopping spree. We found that the cost of socks is represented by 2.99x, and the cost of blouses is represented by 12.99y. To find the total cost, we simply add these two expressions together. This gives us the algebraic expression: 2.99x + 12.99y. This expression represents the total amount Hanna spends, considering both the number of socks (x) and the number of blouses (y) she purchases. It encapsulates the entire problem in a concise mathematical form. The plus sign in the expression indicates that we are summing the costs of the two items to arrive at the total. This is a crucial step in solving the problem, as it demonstrates how individual components can be combined to represent a larger whole. The expression 2.99x + 12.99y is a linear expression, which means it represents a straight-line relationship when graphed. Each term in the expression contributes to the total cost, and the variables x and y allow us to calculate the cost for any combination of socks and blouses. This final expression is the answer to our problem, and it provides a powerful tool for calculating Hanna's expenses. By understanding how to build and interpret such expressions, we can apply these skills to various real-world scenarios involving costs, quantities, and totals. So, with the total cost now clearly represented, let's recap our findings and highlight the key takeaways from this problem.

Conclusion: The Total Cost Expression

In conclusion, we have successfully formulated an algebraic expression to represent the total cost of Hanna's shopping trip. We started by identifying the individual costs of socks and blouses, with socks priced at $2.99 per pair and blouses at $12.99 each. We then introduced the variables x and y to represent the number of pairs of socks and blouses purchased, respectively. By multiplying the price of each item by its quantity, we derived the expressions 2.99x for the cost of socks and 12.99y for the cost of blouses. Finally, we combined these expressions to obtain the total cost expression: 2.99x + 12.99y. This expression is a powerful tool that allows us to calculate Hanna's total spending for any given number of socks and blouses. It demonstrates the practical application of algebra in everyday scenarios, such as shopping and budgeting. The process of building this expression involved breaking down the problem into smaller, manageable parts, a strategy that is often useful in problem-solving. We first focused on the cost of socks, then on the cost of blouses, and finally combined these costs to find the total. This step-by-step approach not only made the problem easier to solve but also provided a clear understanding of how each component contributes to the final result. The expression 2.99x + 12.99y is a testament to the power of algebra in representing real-world situations concisely and accurately. It is a valuable skill to master, as it can be applied to a wide range of problems involving costs, quantities, and totals. With this understanding, you can confidently tackle similar problems and appreciate the relevance of algebra in everyday life. The problem not only reinforces the concepts of variables, constants, and expressions but also highlights the importance of careful analysis and logical reasoning in mathematics. By understanding the problem statement, identifying the relevant information, and applying the appropriate mathematical operations, we were able to arrive at a solution that accurately represents the total cost of Hanna's shopping trip.