Wei's Garland Creation How Many Balloons Can He Buy

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Introduction

In this article, we delve into a mathematical problem involving Wei, who has $150.00 to create a garland using 60-cent balloons. The heart of the problem lies in determining the number of white balloons Wei can purchase, given that he intends to buy 100 blue balloons and the white balloons are on sale for half price. This scenario provides an excellent opportunity to explore equation formulation and problem-solving techniques in a real-world context. We will break down the problem step-by-step, analyzing the information provided and constructing an equation that accurately represents the situation. By solving this equation, we can determine the precise number of white balloons Wei can afford while staying within his budget. This exercise highlights the practical application of mathematical concepts in everyday decision-making, demonstrating how algebraic equations can be used to model and solve financial constraints. Furthermore, this exploration can enhance understanding of cost analysis, budgeting, and the importance of considering sale prices when making purchasing decisions. Through this analysis, readers will gain insights into the mathematical principles underlying everyday transactions and the value of strategic planning in financial matters. This introduction sets the stage for a detailed examination of Wei's balloon-buying dilemma, emphasizing the relevance of mathematical skills in addressing real-world challenges. We will now proceed to dissect the problem, identifying the key variables and formulating the equation necessary to find the solution.

Problem Statement

Wei's Balloon Budget: Wei has a budget of $150.00 to create a garland using balloons that cost 60 cents each. He plans to buy 100 blue balloons and an unknown number of white balloons. The white balloons are on sale for half price. The main goal is to write and solve an equation to determine the number of white balloons Wei can purchase within his budget. This problem requires a careful analysis of the costs involved and the constraints imposed by Wei's budget. We must consider the cost of the blue balloons, the sale price of the white balloons, and the total amount Wei can spend. Formulating the correct equation is crucial, as it will accurately represent the relationship between the number of balloons, their prices, and the total budget. The solution to the equation will provide the answer to our question: How many white balloons can Wei buy? This problem-solving process involves several steps, including defining variables, translating the word problem into a mathematical expression, and using algebraic techniques to solve for the unknown. The challenge lies in correctly interpreting the information provided and representing it in a way that allows us to find the solution efficiently. By breaking down the problem into smaller parts, we can address each component systematically and arrive at the final answer. This exercise not only tests our mathematical skills but also our ability to apply these skills in a practical setting. The problem highlights the importance of budgeting and making informed decisions when faced with financial constraints. We will now proceed to identify the variables and formulate the equation that will guide us to the solution, ensuring that Wei's garland creation remains within his financial reach. This methodical approach will help us understand the underlying principles and apply them to similar scenarios in the future.

Defining Variables

To tackle this problem effectively, we need to define our variables clearly. Let's use the following:

  • Let x represent the number of white balloons Wei can purchase.
  • The cost of each balloon is $0.60.
  • The cost of each white balloon on sale is $0.60 / 2 = $0.30.
  • Wei wants to purchase 100 blue balloons.
  • Wei's total budget is $150.00.

Defining these variables is a crucial step in translating the word problem into a mathematical equation. Each variable represents a specific quantity or piece of information that we need to consider. By assigning symbols to these quantities, we can manipulate them algebraically and establish relationships between them. The variable x is particularly important as it represents the unknown quantity we are trying to find: the number of white balloons. The cost per balloon and the sale price are also key elements, as they determine the overall expense. The number of blue balloons Wei plans to buy is a fixed quantity that contributes to the total cost, and the total budget of $150.00 acts as a constraint on the number of balloons Wei can purchase. With these variables clearly defined, we can now proceed to formulate an equation that accurately represents the problem. The equation will incorporate these variables and their relationships, allowing us to solve for x and determine the number of white balloons Wei can afford. This structured approach ensures that we address all the essential aspects of the problem and move towards a logical solution. The clear definition of variables sets the foundation for the subsequent steps in the problem-solving process, ensuring clarity and accuracy in our calculations and analysis.

Equation Formulation

Now, let's formulate the equation based on the information and variables we've defined:

The cost of 100 blue balloons is 100 * $0.60 = $60.00. The cost of x white balloons at half price is x * $0.30 = $0.30x. The total cost of the balloons should not exceed Wei's budget of $150.00.

Therefore, the equation is:

$60.00 + $0.30x = $150.00

This equation encapsulates the essence of the problem, expressing the relationship between the cost of the blue balloons, the cost of the white balloons, and Wei's total budget. The left-hand side of the equation represents the total cost of the balloons, which is the sum of the cost of the blue balloons and the cost of the white balloons. The right-hand side represents Wei's budget, which acts as the upper limit on the total cost. The equation states that the total cost of the balloons must be equal to Wei's budget. This is a linear equation in one variable, x, which makes it relatively straightforward to solve. The formulation of this equation is a critical step in the problem-solving process. It requires a clear understanding of the problem's constraints and the relationships between the variables. By expressing the problem in mathematical terms, we can apply algebraic techniques to find the solution. The equation serves as a model of the real-world situation, allowing us to make quantitative predictions and decisions. The accuracy of the solution depends on the correct formulation of the equation. Any errors or omissions in the equation can lead to incorrect results. Therefore, it is essential to carefully consider all the relevant information and ensure that the equation accurately represents the problem. With the equation now formulated, we can proceed to solve for x and determine the number of white balloons Wei can purchase within his budget. This will provide us with a concrete answer to the problem and demonstrate the practical application of algebraic principles.

Solving the Equation

To solve the equation $60.00 + $0.30x = $150.00, we will follow these steps:

  1. Subtract $60.00 from both sides of the equation:

    $0. 30x = $150.00 - $60.00

    $0. 30x = $90.00

  2. Divide both sides by $0.30:

    x = $90.00 / $0.30

    x = 300

Therefore, Wei can purchase 300 white balloons.

Solving the equation involves isolating the variable x on one side of the equation to determine its value. The steps we took are based on the fundamental principles of algebra, which allow us to manipulate equations while maintaining their balance. Subtracting $60.00 from both sides simplifies the equation by isolating the term containing x. This step eliminates the constant term on the left-hand side, making it easier to solve for x. Dividing both sides by $0.30 then isolates x, giving us the solution. The result, x = 300, indicates that Wei can purchase 300 white balloons within his budget. This solution is a numerical answer to our original question, providing a concrete quantity that Wei can use to plan his garland creation. The process of solving the equation demonstrates the power of algebraic techniques in addressing real-world problems. By following these steps, we can systematically find the value of the unknown variable and gain insights into the situation being modeled. The solution not only provides an answer but also validates the equation we formulated earlier. If the solution does not make sense in the context of the problem, it may indicate an error in the equation or in the solving process. In this case, the solution of 300 white balloons seems reasonable, given Wei's budget and the cost of the balloons. With the solution in hand, we can now move on to interpret the result and consider its implications for Wei's garland project. This comprehensive approach ensures that we not only find the answer but also understand its significance and application.

Interpretation of the Solution

The solution x = 300 means that Wei can purchase 300 white balloons in addition to the 100 blue balloons he initially planned to buy. This outcome is crucial for Wei, as it provides a clear understanding of how many white balloons he can afford while staying within his $150.00 budget. The interpretation of the solution is as important as the solution itself. It involves understanding what the numerical answer means in the context of the original problem. In this case, the number 300 represents a physical quantity: the number of white balloons. Wei can use this information to plan his garland project, ensuring that he has enough balloons to create the desired effect. The solution also highlights the impact of the sale price on the white balloons. The reduced cost allows Wei to purchase a significantly larger number of white balloons than he would have been able to afford at the regular price. This demonstrates the importance of considering sales and discounts when making purchasing decisions. The interpretation of the solution can also involve further analysis and planning. For example, Wei might want to consider the color balance in his garland. He now knows he can buy 300 white balloons, but he might decide that a smaller number of white balloons would create a more visually appealing garland. He could then adjust his purchase accordingly, perhaps buying additional blue balloons or balloons of other colors. This type of analysis goes beyond the simple numerical solution and considers the broader context of the problem. The interpretation of the solution is a crucial step in the problem-solving process. It connects the mathematical result to the real-world situation, allowing us to make informed decisions and take appropriate action. In Wei's case, the solution provides him with the information he needs to move forward with his garland project, knowing that he can afford 300 white balloons within his budget. This comprehensive understanding of the solution enhances the value of the mathematical exercise and its practical application.

Conclusion

In conclusion, by formulating and solving the equation $60.00 + $0.30x = $150.00, we determined that Wei can purchase 300 white balloons. This mathematical exercise demonstrates how algebraic equations can be used to model and solve real-world problems involving budgeting and purchasing decisions. The process involved defining variables, formulating an equation, solving the equation, and interpreting the solution, all of which are essential steps in mathematical problem-solving. The problem also highlighted the importance of considering discounts and sale prices when making purchasing decisions, as the half-price sale on white balloons significantly increased the number of balloons Wei could afford. This exercise not only provided a solution to a specific problem but also reinforced the broader applicability of mathematical skills in everyday life. The ability to formulate and solve equations is a valuable tool for making informed decisions in various financial and practical situations. By breaking down the problem into smaller parts and addressing each component systematically, we were able to arrive at a clear and accurate solution. The solution, 300 white balloons, provides Wei with the information he needs to plan his garland project effectively. The conclusion summarizes the key steps and findings of the problem-solving process, emphasizing the practical relevance of the mathematical concepts involved. It reinforces the idea that mathematics is not just an abstract subject but a powerful tool for addressing real-world challenges. The exercise also serves as a reminder of the importance of careful planning and budgeting when making purchases, ensuring that expenses stay within the available resources. Overall, this exploration of Wei's balloon-buying dilemma provides valuable insights into the application of mathematics in everyday decision-making and the benefits of a systematic problem-solving approach.

Keywords

Wei's balloon purchase, Garland creation budget, Equation for balloon cost, White balloon sale price, Number of balloons affordable