To Buy A TV, Fan And Juicer, Pedro Noticed That In The Store Where He Would Make The Purchase, The Fan Cost A Quarter Of What The TV Cost And The TV Cost $180,000 More Than The Juicer. If Pedro Bought All Three
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In this article, we will delve into a mathematical problem involving Pedro's purchase of a television, a fan, and a juicer. We will break down the problem step-by-step to understand the relationships between the prices of the appliances and ultimately determine the total cost of Pedro's purchase. This exercise will not only help in solving the specific problem but also enhance our understanding of applying mathematical concepts in real-life scenarios. Let's embark on this journey of problem-solving and uncover the total expenditure Pedro incurred.
Breaking Down the Problem Statement
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The problem statement presents us with a scenario where Pedro intends to purchase three appliances: a television, a fan, and a juicer. We are given crucial information about the relative costs of these items. Specifically, the fan's cost is one-fourth of the television's cost, and the television's cost exceeds the juicer's cost by $180,000. Our ultimate goal is to determine the total amount Pedro spent on all three appliances. To achieve this, we need to systematically dissect the information provided and translate it into mathematical expressions. Let's begin by identifying the knowns and unknowns in this problem. The knowns are the relationship between the fan's and television's costs, the difference in price between the television and the juicer, and the fact that Pedro bought all three items. The unknowns are the individual prices of each appliance and the total cost. By carefully analyzing these knowns and unknowns, we can formulate a plan to solve the problem effectively. The first step is to represent the unknowns using variables, which will allow us to create equations that capture the relationships described in the problem statement. This approach will pave the way for a clear and concise solution.
Defining Variables and Formulating Equations
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To effectively solve this problem, we must first define variables to represent the unknown prices of the appliances. Let's denote the cost of the television as T, the cost of the fan as F, and the cost of the juicer as J. This is a crucial step in translating the word problem into a mathematical framework. Now, we can express the given information as equations using these variables. The problem states that the fan's cost is one-fourth of the television's cost. This can be written as: F = (1/4)T. This equation establishes a direct relationship between the cost of the fan and the cost of the television. Next, we are told that the television costs $180,000 more than the juicer. This can be expressed as: T = J + 180,000. This equation links the cost of the television to the cost of the juicer. With these two equations, we have successfully captured the essential information from the problem statement in a mathematical form. The next step is to use these equations to find the individual costs of the appliances. This can be done by employing techniques such as substitution or elimination, which are fundamental tools in algebra. By carefully manipulating these equations, we can unravel the values of T, F, and J, bringing us closer to the final solution.
Solving for the Cost of the Television
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The problem implicitly provides us with the cost of the television, which is a crucial piece of information for solving the rest of the problem. The statement mentions that the television cost $180,000 more than the juicer. While this doesn't directly give us the television's cost, it sets the stage for us to determine it through subsequent steps. However, upon closer inspection of the initial problem statement, there seems to be a missing piece of information. To accurately determine the cost of the television, we need a specific value or another equation that directly relates to its price. Without this, we can only express the television's cost in terms of the juicer's cost, as we have already done: T = J + 180,000. To proceed further and find a numerical value for T, we would typically need an additional piece of information, such as the cost of the juicer or the total cost of all three items. In a typical problem-solving scenario, we would either revisit the problem statement to look for any overlooked details or consider making a reasonable assumption if the missing information can be logically inferred. However, in this case, without any further context, we must acknowledge that the problem as stated lacks the necessary information to directly compute the cost of the television. Despite this hurdle, we can still explore how we would approach the problem if we had the television's cost. This will allow us to understand the subsequent steps in the solution process and appreciate the importance of having complete information in mathematical problem-solving.
Determining the Cost of the Fan
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Assuming we have determined the cost of the television (T), the next step is to calculate the cost of the fan (F). The problem states that the fan costs one-fourth of the price of the television. This relationship is mathematically expressed as F = (1/4)T. This equation is straightforward and allows us to directly compute the fan's cost once we know the television's cost. For example, if the television costs $400,000, then the fan would cost (1/4) * $400,000 = $100,000. This simple calculation demonstrates how the equation works in practice. The relationship between the fan's and television's costs highlights the importance of understanding fractions and proportions in real-world applications. By expressing the fan's cost as a fraction of the television's cost, we can easily determine its value. This concept is not only useful in this specific problem but also in various other scenarios where proportional relationships are involved. Furthermore, this step underscores the sequential nature of problem-solving in mathematics. We often need to solve for one unknown before we can move on to the next. In this case, we need the television's cost to find the fan's cost. This step-by-step approach is crucial for tackling complex problems and ensuring accuracy in our calculations. By breaking down the problem into smaller, manageable steps, we can systematically arrive at the solution. The ability to apply mathematical concepts like fractions and proportions is a valuable skill that extends beyond the classroom and into everyday life.
Calculating the Cost of the Juicer
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Having (hypothetically) established the cost of the television (T), we can now proceed to calculate the cost of the juicer (J). The problem states that the television costs $180,000 more than the juicer. This relationship can be expressed mathematically as T = J + 180,000. To find the cost of the juicer, we need to rearrange this equation to solve for J. This can be done by subtracting $180,000 from both sides of the equation, resulting in J = T - 180,000. This equation now allows us to directly calculate the juicer's cost if we know the television's cost. For instance, if the television costs $500,000, then the juicer would cost $500,000 - $180,000 = $320,000. This example illustrates how the rearranged equation is used to find the juicer's cost. The process of rearranging equations is a fundamental skill in algebra and is essential for solving a wide range of problems. By isolating the variable we want to solve for, we can determine its value based on the other known quantities. This technique is not only applicable in mathematical contexts but also in various fields such as physics, engineering, and economics. The relationship between the television's and juicer's costs highlights the concept of differences and how they can be expressed mathematically. By understanding this relationship, we can accurately determine the juicer's cost based on the television's cost. This step further emphasizes the importance of algebraic manipulation in problem-solving and how it allows us to uncover hidden information within a problem statement.
Determining the Total Cost
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Once we have determined the individual costs of the television (T), the fan (F), and the juicer (J), the final step is to calculate the total cost of Pedro's purchase. This is a straightforward process that involves adding the costs of the three appliances together. The total cost can be expressed as: Total Cost = T + F + J. This equation simply sums up the individual costs to give us the overall expenditure. For example, let's assume that the television costs $400,000, the fan costs $100,000, and the juicer costs $220,000. In this case, the total cost would be $400,000 + $100,000 + $220,000 = $720,000. This calculation demonstrates how the total cost is obtained by adding the individual costs. This final step underscores the importance of careful and accurate calculations in problem-solving. By ensuring that we have correctly determined the individual costs, we can confidently arrive at the total cost. This process also highlights the practical application of addition in everyday scenarios. Calculating the total cost of a purchase is a common task that we encounter in our daily lives, and understanding the underlying mathematical principles allows us to do so effectively. In conclusion, by systematically breaking down the problem, defining variables, formulating equations, and performing the necessary calculations, we can successfully determine the total cost of Pedro's appliance purchase. This exercise not only provides a solution to the specific problem but also reinforces our understanding of mathematical concepts and their application in real-world situations.
Conclusion: Problem-Solving and Real-World Applications
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In conclusion, the problem of determining the total cost of Pedro's appliance purchase serves as a valuable exercise in applying mathematical concepts to real-world scenarios. By breaking down the problem into smaller, manageable steps, we can systematically analyze the given information and arrive at a solution. The process involves defining variables, formulating equations, and performing algebraic manipulations to find the unknown quantities. While the initial problem statement appears to be missing a crucial piece of information – the actual cost of the television – we can still appreciate the steps involved in solving the problem if we were given that value. We explored how to calculate the cost of the fan and the juicer based on the television's cost, and how to ultimately determine the total cost by summing the individual prices. This exercise highlights the importance of having complete information in problem-solving and the challenges that arise when data is missing. However, it also demonstrates the power of mathematical reasoning and how we can use equations and variables to represent relationships and solve for unknowns. The skills learned in this exercise are not only applicable to mathematical problems but also to various aspects of our daily lives. From budgeting and personal finance to making informed purchasing decisions, the ability to analyze information, identify relationships, and perform calculations is essential. By engaging in problem-solving activities like this, we can enhance our critical thinking skills and develop a deeper appreciation for the role of mathematics in the world around us. The key takeaways from this exercise include the importance of clear problem definition, the power of algebraic representation, and the value of systematic problem-solving techniques. By mastering these skills, we can confidently tackle a wide range of challenges and make informed decisions in both academic and real-world contexts.