Equations To Find The Price Of Tire Patches For Rhyan's Mountain Biking Trip
Introduction
In anticipation of his mountain biking adventure, Rhyan has taken a proactive step by purchasing four tire patches. This demonstrates a practical approach to potential mechanical issues that may arise during his rides. To facilitate this purchase, Rhyan utilized a gift card that held an initial balance of $22.20. After completing the transaction, a remaining balance of $1.90 was left on the gift card. This scenario presents an opportunity to explore the mathematical principles involved in determining the cost of each tire patch. By carefully analyzing the information provided, we can construct equations that accurately reflect the relationship between the initial gift card balance, the remaining balance, and the total cost of the tire patches. This exercise not only helps in solving a real-world problem but also reinforces fundamental concepts in algebra and problem-solving. Understanding how to formulate and solve these equations is a valuable skill that can be applied to various financial and everyday situations. The process involves identifying the known variables, defining the unknown variable (the cost of one tire patch), and then creating an equation that represents the given scenario. By solving this equation, we can precisely determine the price Rhyan paid for each tire patch, thereby completing our understanding of this financial transaction within the context of his mountain biking preparations. This article will delve into the step-by-step process of formulating these equations and arriving at the solution, highlighting the importance of mathematical reasoning in everyday life.
Determining the Equations to Calculate the Price of One Tire Patch
To effectively determine the price of a single tire patch in Rhyan's scenario, it is crucial to establish the correct mathematical framework. This involves translating the given information into an equation that accurately represents the relationships between the variables. The key variables we need to consider are the initial balance on the gift card ($22.20), the remaining balance after the purchase ($1.90), and the number of tire patches bought (4). The unknown variable we aim to find is the price of one tire patch, which we can denote as 'x'. The fundamental principle here is that the total amount spent on the tire patches is the difference between the initial balance and the remaining balance on the gift card. Mathematically, this can be expressed as: Total amount spent = Initial balance - Remaining balance. Substituting the given values, we get: Total amount spent = $22.20 - $1.90 = $20.30. This calculation reveals that Rhyan spent a total of $20.30 on the four tire patches. Now, to find the price of one tire patch, we need to divide the total amount spent by the number of patches purchased. This leads us to the equation: Price of one tire patch (x) = Total amount spent / Number of tire patches. Substituting the values, we get: x = $20.30 / 4. This equation is a direct representation of the problem and can be used to solve for the unknown variable, 'x'. Another way to represent this situation mathematically is to express the total amount spent as the product of the number of patches and the price of each patch. This can be written as: 4x = $20.30. This equation is equivalent to the previous one and can be solved using algebraic methods to find the value of 'x'. Both equations effectively capture the essence of the problem and provide a pathway to calculate the price of one tire patch. Understanding how to formulate these equations is crucial for solving not only this specific problem but also similar mathematical scenarios in various contexts.
Equation 1: Subtracting the Remaining Balance
One effective approach to calculating the price of a tire patch involves focusing on the difference between the initial gift card balance and the remaining balance after the purchase. This method directly addresses the total amount Rhyan spent on the tire patches, which is a crucial step in determining the individual price. The initial balance on Rhyan's gift card was $22.20, and after buying the tire patches, he had $1.90 remaining. To find the total amount spent, we subtract the remaining balance from the initial balance. This can be represented mathematically as: Total amount spent = Initial balance - Remaining balance. Substituting the given values, we have: Total amount spent = $22.20 - $1.90. This subtraction yields $20.30, which signifies the total expenditure on the four tire patches. Now that we know the total amount spent, we can proceed to calculate the price of a single tire patch. Since Rhyan bought four patches, we need to divide the total amount spent by the number of patches. This can be expressed as: Price of one tire patch = Total amount spent / Number of tire patches. Substituting the values, we get: Price of one tire patch = $20.30 / 4. This equation directly calculates the cost of one tire patch. The equation encapsulates the entire process, where 'x' represents the unknown price of a single tire patch. Solving this equation will give us the exact cost Rhyan paid for each tire patch. This method is straightforward and intuitive, as it directly reflects the financial transaction that occurred. By first determining the total amount spent and then dividing it by the number of items purchased, we arrive at the individual price. This approach is applicable in various scenarios where one needs to calculate the unit price of items bought in bulk. Understanding this method is a valuable skill in both academic and real-life situations, fostering a practical understanding of financial calculations.
Equation 2: Representing the Total Cost
Another effective way to formulate an equation for this scenario is to directly represent the total cost of the tire patches in terms of the unknown price of a single patch. This approach involves using algebraic notation to express the relationship between the number of patches, the price of each patch, and the total amount spent. Let's denote the price of one tire patch as 'x'. Since Rhyan bought four tire patches, the total cost of the patches can be represented as 4 multiplied by the price of one patch, which is 4x. We already determined that the total amount Rhyan spent was the difference between his initial gift card balance and the remaining balance. This difference is $22.20 - $1.90 = $20.30. Therefore, we can set up the equation: 4x = $20.30. This equation directly links the total cost of the patches (4x) to the total amount Rhyan spent ($20.30). This equation is a concise and direct representation of the problem. It states that four times the price of one tire patch equals the total amount spent. Solving this equation for 'x' will give us the price of a single tire patch. The equation 4x = $20.30 is a linear equation, which is a fundamental concept in algebra. Solving linear equations is a crucial skill in mathematics and has numerous applications in various fields. By understanding how to set up and solve equations like this, individuals can effectively tackle real-world problems involving costs, quantities, and prices. This method not only provides a solution to the specific problem at hand but also reinforces algebraic thinking and problem-solving skills. The ability to translate a word problem into a mathematical equation is a key step in mathematical literacy and is essential for success in various academic and practical contexts. This approach offers a clear and structured way to analyze the situation and arrive at a solution, making it a valuable tool in mathematical problem-solving.
Solving the Equations and Finding the Price
Having established two different equations to represent the cost of the tire patches, we can now proceed to solve them and determine the actual price Rhyan paid for each patch. The first equation we derived was based on the total amount spent divided by the number of patches: Price of one tire patch = $20.30 / 4. To solve this, we simply perform the division: $20.30 ÷ 4 = $5.075. This result indicates that each tire patch costs $5.075. However, in practical terms, prices are usually rounded to the nearest cent, so we would round $5.075 to $5.08. Therefore, according to the first equation, the price of one tire patch is approximately $5.08. The second equation we formulated directly related the total cost to the price of a single patch: 4x = $20.30. To solve for 'x', we need to isolate 'x' by dividing both sides of the equation by 4. This gives us: x = $20.30 / 4. Again, performing the division, we get: x = $5.075. As with the first equation, we round this result to the nearest cent, giving us $5.08. This confirms that both equations lead to the same solution. Regardless of the method used, the price of one tire patch is approximately $5.08. This consistency in the results reinforces the validity of both equations and demonstrates the flexibility in problem-solving approaches. The process of solving these equations highlights the importance of basic arithmetic operations such as division and the application of algebraic principles. Understanding how to manipulate equations to isolate the unknown variable is a fundamental skill in mathematics and is crucial for solving various types of problems. In this case, both methods provided a clear pathway to the solution, showcasing the power of mathematical reasoning in everyday scenarios. By accurately calculating the price of one tire patch, we have successfully addressed the core question posed in the problem.
Conclusion
In conclusion, the problem of determining the price of one tire patch in Rhyan's mountain biking trip preparation provides a valuable exercise in mathematical problem-solving. By carefully analyzing the given information and translating it into mathematical equations, we have successfully calculated the cost of each tire patch. We explored two primary methods to formulate the equations. The first method involved subtracting the remaining balance on the gift card from the initial balance to find the total amount spent and then dividing this amount by the number of tire patches. This approach directly reflects the financial transaction and provides an intuitive way to calculate the unit price. The second method involved setting up an algebraic equation where the total cost of the patches (4x) is equated to the total amount spent ($20.30). This method highlights the power of algebraic representation and demonstrates how equations can concisely capture the relationships between variables. Both methods led to the same result, reinforcing the validity of the approaches and showcasing the flexibility in mathematical problem-solving. By solving these equations, we determined that the price of one tire patch is approximately $5.08. This result not only answers the specific question posed but also demonstrates the practical application of mathematical skills in everyday scenarios. The ability to formulate and solve equations is a crucial skill in mathematics and has numerous applications in various fields, from personal finance to scientific research. This exercise underscores the importance of mathematical literacy and the value of understanding how to translate real-world problems into mathematical models. By mastering these skills, individuals can effectively tackle a wide range of challenges and make informed decisions in various aspects of life. The process of problem-solving, from identifying the knowns and unknowns to formulating equations and arriving at a solution, is a valuable skill that extends beyond the realm of mathematics and into critical thinking and decision-making in general.
