Solving Annikas Pie Sales A Mathematical Problem

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In this article, we will delve into a mathematical problem involving Annika's pie sales. Annika, a talented baker, sells two sizes of pies: 6-inch pies for $5.00 each and 8-inch pies for $9.00 each. Last week, she had a successful week, selling twice as many 6-inch pies as 8-inch pies and earning a total of $133.00. To analyze her sales, Annika defined x as the number of 6-inch pies sold and y as the number of 8-inch pies sold. Our goal is to explore this scenario, understand the relationships between the variables, and ultimately determine the number of each type of pie Annika sold. This problem provides a practical application of algebraic principles and problem-solving techniques, which are fundamental concepts in mathematics. Throughout this exploration, we will focus on formulating equations, solving systems of equations, and interpreting the results in the context of Annika's pie sales. We will also emphasize the importance of clearly defining variables and carefully setting up the equations to accurately model the given situation. The beauty of mathematics lies in its ability to provide tools for understanding and solving real-world problems. By analyzing Annika's pie sales, we gain insights into the power of mathematical modeling and its relevance in everyday scenarios. This article aims to provide a comprehensive understanding of the problem, guiding readers through the step-by-step process of solving it and highlighting the underlying mathematical concepts involved. So, let's embark on this mathematical journey and unravel the solution to Annika's pie sales puzzle.

Setting up the Equations

To solve this problem effectively, we need to translate the given information into mathematical equations. This involves identifying the key relationships between the variables and expressing them in a concise and symbolic form. Remember, Annika sells 6-inch pies for $5.00 each and 8-inch pies for $9.00 each. She sold twice as many 6-inch pies as 8-inch pies, and her total sales amounted to $133.00. We've defined x as the number of 6-inch pies sold and y as the number of 8-inch pies sold. Let's break down the information and construct the equations step by step. First, consider the relationship between the number of 6-inch pies and 8-inch pies sold. The problem states that Annika sold twice as many 6-inch pies as 8-inch pies. This can be expressed as an equation: x = 2y. This equation establishes a direct relationship between the variables x and y, indicating that the value of x is always twice the value of y. Next, let's focus on the total sales revenue. Annika made $133.00 from her sales. The revenue from 6-inch pies is the price per pie ($5.00) multiplied by the number of 6-inch pies sold (x), which is 5x. Similarly, the revenue from 8-inch pies is the price per pie ($9.00) multiplied by the number of 8-inch pies sold (y), which is 9y. The total revenue is the sum of the revenue from both types of pies, so we can write the equation: 5x + 9y = 133. This equation represents the total sales revenue in terms of the variables x and y. Now, we have a system of two equations with two variables:

  1. x = 2y
  2. 5x + 9y = 133

This system of equations provides a mathematical representation of the given scenario. Solving this system will give us the values of x and y, which represent the number of 6-inch and 8-inch pies sold, respectively. The process of setting up equations is crucial in problem-solving. It allows us to translate real-world situations into a mathematical framework, making it easier to analyze and find solutions. In this case, we have successfully formulated two equations that capture the essential information about Annika's pie sales. In the following sections, we will explore different methods for solving this system of equations and interpreting the results.

Solving the System of Equations

Now that we have established the system of equations, the next step is to solve for the unknowns, x and y. There are several methods for solving systems of equations, including substitution, elimination, and graphical methods. In this case, the substitution method is particularly convenient due to the form of the first equation (x = 2y). This equation expresses x directly in terms of y, making it easy to substitute into the second equation. Let's proceed with the substitution method. We will substitute the expression for x from the first equation into the second equation. The first equation is x = 2y. The second equation is 5x + 9y = 133. Substituting x = 2y into the second equation, we get: 5(2y) + 9y = 133. Now, we have an equation with only one variable, y. Simplifying the equation, we get: 10y + 9y = 133, which further simplifies to 19y = 133. To solve for y, we divide both sides of the equation by 19: y = 133 / 19, which gives us y = 7. So, Annika sold 7 eight-inch pies. Now that we have the value of y, we can find the value of x using the first equation, x = 2y. Substituting y = 7, we get: x = 2 * 7, which gives us x = 14. Therefore, Annika sold 14 six-inch pies. We have successfully solved the system of equations and found the values of x and y. To ensure our solution is correct, we can verify it by plugging the values of x and y back into the original equations. For the first equation, x = 2y, we have 14 = 2 * 7, which is true. For the second equation, 5x + 9y = 133, we have 5(14) + 9(7) = 70 + 63 = 133, which is also true. Since both equations are satisfied, our solution is correct. In summary, we used the substitution method to solve the system of equations. This method involves expressing one variable in terms of the other and substituting it into the other equation. This reduces the system to a single equation with one variable, which can be easily solved. Once we find the value of one variable, we can substitute it back into one of the original equations to find the value of the other variable. The substitution method is a powerful tool for solving systems of equations and is particularly useful when one of the equations is already in a form that expresses one variable in terms of the other. In the next section, we will interpret the solution in the context of the problem and draw conclusions about Annika's pie sales.

Interpreting the Solution

Now that we have solved the system of equations and found the values of x and y, it is crucial to interpret the solution in the context of the original problem. Remember, x represents the number of 6-inch pies Annika sold, and y represents the number of 8-inch pies she sold. Our solution is x = 14 and y = 7. Therefore, Annika sold 14 six-inch pies and 7 eight-inch pies last week. This answers the main question of the problem: how many of each type of pie did Annika sell? But we can delve deeper into the interpretation of the solution. We can analyze the relationship between the number of pies sold and Annika's total revenue. Annika sold twice as many 6-inch pies as 8-inch pies, which is consistent with the information given in the problem statement. The revenue from the 6-inch pies is 14 pies * $5.00/pie = $70.00. The revenue from the 8-inch pies is 7 pies * $9.00/pie = $63.00. The total revenue is $70.00 + $63.00 = $133.00, which matches the given total sales amount. This confirms that our solution is not only mathematically correct but also makes sense in the real-world context of the problem. Interpreting the solution is an essential step in problem-solving. It involves connecting the mathematical results back to the original problem and ensuring that the answer is meaningful and reasonable. In this case, we have not only found the number of each type of pie sold but also verified that the total revenue matches the given information. This thorough interpretation strengthens our confidence in the correctness of the solution. Furthermore, we can use this information to gain insights into Annika's business. For instance, we can observe that although she sold twice as many 6-inch pies, the revenue from the 8-inch pies is almost as high due to their higher price. This might suggest that Annika could consider strategies to increase the sales of 8-inch pies, as they contribute significantly to her total revenue. In conclusion, interpreting the solution is a vital step that goes beyond simply finding the numerical answer. It involves understanding the meaning of the solution in the context of the problem and drawing meaningful conclusions. By interpreting our solution in the context of Annika's pie sales, we have gained a comprehensive understanding of her sales performance and identified potential areas for improvement.

Conclusion

In this article, we have explored a mathematical problem involving Annika's pie sales. We successfully translated the problem into a system of equations, solved the system using the substitution method, and interpreted the solution in the context of the problem. This exercise has demonstrated the power of mathematical modeling in solving real-world problems. We began by carefully reading the problem statement and identifying the key information. We defined variables to represent the unknown quantities, which is a crucial step in setting up the mathematical framework. We then translated the given information into two equations: one representing the relationship between the number of 6-inch and 8-inch pies sold, and the other representing the total sales revenue. This process of setting up equations is fundamental in mathematical problem-solving, as it allows us to express the problem in a concise and symbolic form. Once we had the system of equations, we chose the substitution method to solve for the unknowns. This method proved to be particularly efficient due to the form of one of the equations. We systematically substituted one variable in terms of the other, reducing the system to a single equation with one variable. Solving this equation gave us the value of one variable, which we then substituted back into one of the original equations to find the value of the other variable. The solution we obtained was x = 14 and y = 7, meaning Annika sold 14 six-inch pies and 7 eight-inch pies. However, our analysis did not end there. We emphasized the importance of interpreting the solution in the context of the problem. We verified that the solution made sense by checking if the total revenue matched the given information. We also analyzed the relationship between the number of pies sold and the revenue generated, gaining insights into Annika's business. This step of interpretation is crucial because it ensures that the mathematical solution is meaningful and relevant to the real-world situation. In conclusion, this article has provided a comprehensive exploration of Annika's pie sales problem. We have demonstrated the process of translating a real-world scenario into a mathematical model, solving the model using appropriate techniques, and interpreting the results in a meaningful way. This approach is applicable to a wide range of problem-solving situations and highlights the importance of mathematical thinking in everyday life. By understanding the underlying mathematical principles and developing problem-solving skills, we can effectively tackle complex challenges and make informed decisions.