Solving Apple Equations Determining Quantities Of Fuji And Golden Delicious For Childcare Center

Introduction: The Apple Dilemma

In this mathematical exploration, we'll delve into a real-world scenario involving a childcare center's purchase of apples. Apples, a nutritious and beloved fruit, are at the heart of our problem. The center aims to provide a healthy snack for the children, opting for a mix of Fuji and Golden Delicious varieties. The core challenge lies in determining the exact quantities of each type of apple purchased, given the price per pound and the total expenditure. This problem serves as a fantastic illustration of how systems of equations can be applied to solve practical, everyday situations. We'll break down the information provided, set up the necessary equations, and walk through the steps to arrive at the solution. Understanding this process will not only enhance your mathematical skills but also provide a glimpse into the decision-making processes involved in managing budgets and making purchases.

Defining the Variables: Fuji vs. Golden Delicious

To begin our journey towards solving this apple equation, we need to establish some clear definitions. Let's introduce the variables that will represent the unknown quantities we're trying to find. We'll use 'x' to denote the number of pounds of Fuji apples purchased by the childcare center. Fuji apples, known for their crisp texture and sweet flavor, are priced at $3.00 per pound. Similarly, we'll use 'y' to represent the number of pounds of Golden Delicious apples acquired. Golden Delicious apples, with their mellow sweetness and soft flesh, come at a cost of $2.00 per pound. By assigning these variables, we transform the word problem into a more manageable algebraic form. This step is crucial in translating real-world scenarios into mathematical models, allowing us to apply the power of equations to find solutions. The careful selection of variables sets the stage for the subsequent steps in the problem-solving process.

Constructing the Equations: Pounds and Dollars

Now that we have our variables defined, it's time to translate the given information into mathematical equations. This is where we bridge the gap between the narrative of the problem and the precise language of algebra. We're presented with two key pieces of information: the total weight of apples purchased and the total cost of the purchase. The first piece of information tells us that the childcare center bought a total of 30 pounds of apples. This translates directly into our first equation: x + y = 30. Here, 'x' represents the pounds of Fuji apples and 'y' represents the pounds of Golden Delicious apples, and their sum equals the total pounds purchased. The second piece of information concerns the cost. Fuji apples are priced at $3.00 per pound, so the cost of 'x' pounds of Fuji apples is 3x dollars. Similarly, Golden Delicious apples cost $2.00 per pound, making the cost of 'y' pounds 2y dollars. The total expenditure on apples was $80, leading us to our second equation: 3x + 2y = 80. These two equations form a system of linear equations, a powerful tool for solving problems involving multiple unknowns. By carefully constructing these equations, we've laid the foundation for finding the solution to our apple dilemma.

Solving the System: Substitution or Elimination

With our system of equations firmly in place, the next step is to choose a method to solve for our unknown variables, 'x' and 'y'. There are two primary techniques we can employ: substitution and elimination. Let's explore each method and its application to our specific problem. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with a single variable, which we can then solve directly. Alternatively, the elimination method focuses on manipulating the equations so that the coefficients of one variable are opposites. By adding the equations together, we eliminate that variable, again resulting in a single equation with a single unknown. For this particular problem, let's demonstrate the substitution method. From the first equation (x + y = 30), we can easily solve for 'y': y = 30 - x. Now, we'll substitute this expression for 'y' into the second equation (3x + 2y = 80). This substitution yields: 3x + 2(30 - x) = 80. By simplifying and solving this equation, we'll find the value of 'x', the pounds of Fuji apples. Once we have 'x', we can easily plug it back into either of our original equations to find 'y', the pounds of Golden Delicious apples. The choice between substitution and elimination often depends on the specific equations at hand, but both methods are valuable tools in solving systems of linear equations.

Finding the Solution: Fuji and Golden Delicious Quantities

Having chosen the substitution method, let's proceed with the algebraic manipulation to find the solution. We left off with the equation: 3x + 2(30 - x) = 80. The first step is to distribute the 2 across the parentheses: 3x + 60 - 2x = 80. Next, we combine like terms on the left side of the equation: x + 60 = 80. To isolate 'x', we subtract 60 from both sides: x = 20. This tells us that the childcare center purchased 20 pounds of Fuji apples. Now that we know the value of 'x', we can substitute it back into either of our original equations to find 'y'. Let's use the simpler equation: x + y = 30. Substituting x = 20, we get: 20 + y = 30. Subtracting 20 from both sides, we find: y = 10. Therefore, the childcare center purchased 10 pounds of Golden Delicious apples. We've successfully solved the system of equations and determined the quantities of each type of apple purchased. This methodical approach, from defining variables to algebraic manipulation, highlights the power of mathematics in solving real-world problems.

Verification: Ensuring Accuracy

Before we declare our solution final, it's crucial to verify its accuracy. This step ensures that our calculations are correct and that our answers make sense in the context of the problem. To verify, we'll plug our values for 'x' and 'y' back into both of our original equations. Our solution is x = 20 (pounds of Fuji apples) and y = 10 (pounds of Golden Delicious apples). Let's start with the first equation: x + y = 30. Substituting our values, we get: 20 + 10 = 30, which is indeed true. This confirms that the total weight of apples purchased is 30 pounds. Now, let's check the second equation: 3x + 2y = 80. Substituting our values, we get: 3(20) + 2(10) = 80. Simplifying, we have: 60 + 20 = 80, which is also true. This verifies that the total cost of the apples is $80. Since our values for 'x' and 'y' satisfy both equations, we can confidently conclude that our solution is correct. This verification step is a cornerstone of mathematical problem-solving, providing assurance in the accuracy of our results.

Conclusion: Apples and Equations

In conclusion, we've successfully navigated the apple equation, a practical problem that demonstrates the power of systems of linear equations. We began by carefully defining our variables, representing the unknown quantities of Fuji and Golden Delicious apples. We then translated the given information into two equations, one representing the total weight of apples and the other representing the total cost. Employing the substitution method, we solved the system of equations, revealing that the childcare center purchased 20 pounds of Fuji apples and 10 pounds of Golden Delicious apples. To ensure the accuracy of our solution, we performed a verification step, confirming that our values satisfied both original equations. This exercise highlights the real-world applicability of mathematics, showcasing how algebraic techniques can be used to solve everyday problems involving purchasing decisions and budgeting. The process of translating a word problem into a mathematical model, solving the equations, and verifying the results is a valuable skill that extends far beyond the realm of academics. Understanding these concepts empowers us to make informed decisions and solve problems effectively in various aspects of life.