Calculating Nathan's Food Items In Terms Of N - A Step-by-Step Solution

#h1 Tonya, Beth, and Nathan's Food Pantry Contributions

In a recent school food pantry drive, students Tonya, Beth, and Nathan made significant contributions. Tonya, taking the initiative, collected n food items. Beth, demonstrating remarkable generosity, gathered a considerably larger amount, specifically 9 more than 6 times the number of items Tonya collected. Nathan, not to be outdone, also contributed a substantial number of items, collecting 4 fewer than 13 as many as Beth collected. The central question we aim to address is: In terms of n, how many food items did Nathan contribute to the school’s food pantry drive?

This problem is an excellent example of how algebraic expressions can be used to model real-world scenarios. By representing the unknown quantities with variables and establishing relationships between them, we can systematically determine the solution. In this case, we will use n to represent the number of food items Tonya collected, and we will express the number of items collected by Beth and Nathan in terms of n. This approach allows us to understand the relative contributions of each student and to gain a deeper appreciation for the power of mathematical modeling.

To begin, we need to translate the given information into mathematical expressions. Tonya's contribution is simply n. Beth's contribution is described as 9 more than 6 times the number of food items Tonya collected. This can be expressed as 6n + 9. Nathan's contribution is slightly more complex, as it involves a fraction and a subtraction. He collected 4 fewer than 13 as many as Beth collected. To represent this mathematically, we first need to determine 13 of Beth's collection, which is (1/3)(6n + 9). Then, we subtract 4 from this result to find Nathan's total contribution. This gives us the expression (1/3)(6n + 9) - 4. By simplifying this expression, we can determine the number of food items Nathan collected in terms of n.

#h2 Unraveling Beth's Contribution: A Detailed Analysis

To accurately determine Nathan's contribution, we must first meticulously calculate the number of food items Beth collected. The problem states that Beth collected 9 more than 6 times the number of food items Tonya collected. Since Tonya collected n items, we can represent Beth's collection as 6n + 9. This expression is a fundamental building block for solving the overall problem, as it directly links Beth's contribution to Tonya's. Understanding this relationship is crucial for moving forward and calculating Nathan's contribution.

Let's break down the expression 6n + 9 to fully grasp its meaning. The term 6n represents 6 times the number of food items Tonya collected. This means that for every food item Tonya collected, Beth collected 6 times that amount. The addition of 9 indicates that Beth collected an additional 9 food items on top of the 6 times Tonya's contribution. This extra 9 highlights Beth's significant effort in contributing to the food pantry drive. The expression 6n + 9 is a linear expression, where 6 is the coefficient of n, and 9 is the constant term. The coefficient 6 indicates the rate at which Beth's contribution increases with respect to Tonya's contribution, while the constant term 9 represents the fixed amount Beth contributed regardless of Tonya's contribution.

To illustrate this with an example, let's assume Tonya collected 10 food items, so n = 10. In this case, Beth would have collected 6 * 10 + 9 = 60 + 9 = 69 food items. This demonstrates how the expression 6n + 9 accurately reflects the relationship between Tonya's and Beth's contributions. As Tonya's collection increases, Beth's collection also increases proportionally, with an additional 9 items added. This proportionality and the constant addition are key elements of the problem and must be carefully considered when calculating Nathan's contribution.

The expression 6n + 9 is not only a mathematical representation but also a reflection of Beth's generosity and dedication to the food pantry drive. By collecting 9 more than 6 times the number of items Tonya collected, Beth made a substantial contribution that significantly benefited the cause. This underscores the importance of each individual's effort in contributing to community initiatives. Understanding the components of this expression and their implications is essential for solving the problem and appreciating the real-world context of the mathematical concepts involved.

#h2 Deciphering Nathan's Collection: A Step-by-Step Calculation

With a clear understanding of Beth's contribution, we can now focus on determining the number of food items Nathan collected. The problem states that Nathan collected 4 fewer than 13 as many as Beth collected. This is a slightly more complex relationship, involving a fraction and a subtraction, but by breaking it down into smaller steps, we can arrive at the solution. First, we need to calculate 13 of Beth's collection, which is represented by the expression (1/3)(6n + 9). Then, we subtract 4 from this result to find Nathan's total contribution. This process requires careful attention to order of operations and algebraic simplification.

The expression (1/3)(6n + 9) represents one-third of Beth's collection. To simplify this, we can distribute the (1/3) across the terms inside the parentheses. This gives us (1/3) * 6n + (1/3) * 9, which simplifies to 2n + 3. This simplified expression represents the number of food items Nathan collected before the subtraction of 4. The distribution of the fraction is a key step in simplifying the expression and making it easier to work with. By understanding this step, we can avoid potential errors in the calculation.

Now, we need to subtract 4 from the simplified expression 2n + 3 to find Nathan's total contribution. This gives us the final expression 2n + 3 - 4. By combining the constant terms, we arrive at the final expression for Nathan's collection: 2n - 1. This expression tells us that Nathan collected 2 times the number of food items Tonya collected, minus 1. This is a concise and accurate representation of Nathan's contribution in terms of n. The subtraction of 1 indicates that Nathan's contribution is slightly less than twice Tonya's contribution.

To illustrate this with an example, let's assume Tonya collected 10 food items, so n = 10. In this case, Nathan would have collected 2 * 10 - 1 = 20 - 1 = 19 food items. This demonstrates how the expression 2n - 1 accurately reflects the relationship between Tonya's and Nathan's contributions. As Tonya's collection increases, Nathan's collection also increases proportionally, with 1 item subtracted. This proportionality and the constant subtraction are key elements of the problem and must be carefully considered when interpreting the solution.

The final expression, 2n - 1, is not only a mathematical representation but also a reflection of Nathan's contribution to the food pantry drive. By collecting 4 fewer than 13 as many as Beth collected, Nathan made a significant contribution that complements the efforts of Tonya and Beth. This underscores the importance of each individual's effort in contributing to community initiatives. Understanding the components of this expression and their implications is essential for appreciating the real-world context of the mathematical concepts involved.

#h2 The Final Solution: Nathan's Contribution Expressed in Terms of n

After a detailed step-by-step analysis, we have arrived at the solution. Nathan collected 2n - 1 food items. This expression accurately represents Nathan's contribution in terms of n, where n is the number of food items Tonya collected. This solution encapsulates the relationships between the contributions of Tonya, Beth, and Nathan, providing a clear and concise answer to the problem. The expression 2n - 1 is a linear expression, indicating that Nathan's contribution increases linearly with Tonya's contribution.

The expression 2n - 1 provides valuable insights into the relationship between Nathan's contribution and Tonya's contribution. The coefficient 2 indicates that Nathan collected approximately twice the number of food items Tonya collected. The subtraction of 1 suggests that Nathan's contribution is slightly less than twice Tonya's. This expression allows us to easily calculate Nathan's contribution for any given value of n. For example, if Tonya collected 20 food items, then Nathan collected 2 * 20 - 1 = 39 food items. This demonstrates the utility of the expression in quickly determining Nathan's contribution.

Moreover, the solution 2n - 1 highlights the power of algebraic expressions in modeling real-world scenarios. By translating the word problem into mathematical expressions, we were able to systematically determine the number of food items Nathan collected. This approach underscores the importance of algebraic reasoning in problem-solving. The ability to represent unknown quantities with variables and establish relationships between them is a fundamental skill in mathematics and has wide-ranging applications in various fields.

In conclusion, the solution 2n - 1 accurately represents the number of food items Nathan collected in terms of n. This solution is the result of a careful and systematic analysis of the problem, involving the translation of word descriptions into mathematical expressions, simplification of expressions, and application of algebraic principles. This problem serves as an excellent example of how mathematical concepts can be used to model and solve real-world problems, fostering a deeper understanding and appreciation for the power of mathematics.

#h2 Practical Applications and Implications of the Solution

The solution we derived, 2n - 1, is not just a mathematical answer; it has practical applications and implications in understanding the dynamics of contributions to the food pantry drive. This expression allows us to analyze how Nathan's contribution changes with respect to Tonya's, providing valuable insights into the overall impact of the students' efforts. Furthermore, it underscores the importance of mathematical modeling in understanding real-world scenarios and making informed decisions.

One of the key practical applications of the solution is the ability to predict Nathan's contribution for any given number of food items collected by Tonya. For instance, if the school aims to collect a certain number of food items and knows Tonya's contribution, they can use the expression 2n - 1 to estimate Nathan's contribution. This information can be valuable in planning and resource allocation. The ability to predict outcomes based on mathematical models is a powerful tool in various fields, including logistics, finance, and engineering.

Moreover, the solution highlights the relative contributions of Tonya and Nathan. Nathan's contribution is approximately twice Tonya's, minus one item. This information can be used to compare the efforts of the students and to identify areas where additional support may be needed. For example, if Tonya's contribution is significantly lower than expected, it may be necessary to encourage more students to contribute to ensure the food pantry's needs are met. The comparison of contributions allows for a more nuanced understanding of the situation and facilitates informed decision-making.

The process of solving this problem also demonstrates the importance of mathematical modeling in real-world scenarios. By translating the word problem into mathematical expressions, we were able to systematically determine the solution. This process highlights the power of mathematics in representing and analyzing complex relationships. Mathematical modeling is a crucial skill in various fields, allowing professionals to make predictions, optimize processes, and solve problems effectively. The ability to apply mathematical concepts to real-world situations is a valuable asset in today's data-driven world.

In conclusion, the solution 2n - 1 has practical applications and implications in understanding the contributions to the food pantry drive. It allows for the prediction of Nathan's contribution, comparison of contributions, and highlights the importance of mathematical modeling in real-world scenarios. This problem serves as an excellent example of how mathematical concepts can be used to address practical issues and make informed decisions.

#h2 Summarizing the Contributions: A Comprehensive Overview

To gain a comprehensive understanding of the food pantry drive, let's summarize the contributions of each student in terms of n. Tonya collected n food items, Beth collected 6n + 9 food items, and Nathan collected 2n - 1 food items. By combining these expressions, we can determine the total number of food items collected by the three students. This total provides a valuable overview of the overall success of the food pantry drive and the collective impact of the students' efforts.

To calculate the total number of food items collected, we simply add the expressions for each student's contribution: n + (6n + 9) + (2n - 1). By combining like terms, we can simplify this expression. The terms with n are n, 6n, and 2n, which add up to 9n. The constant terms are 9 and -1, which add up to 8. Therefore, the total number of food items collected is 9n + 8. This expression provides a concise representation of the total contribution in terms of n.

The expression 9n + 8 allows us to quickly calculate the total number of food items collected for any given value of n. For example, if Tonya collected 15 food items, then the total number of food items collected would be 9 * 15 + 8 = 135 + 8 = 143. This demonstrates the utility of the expression in quickly determining the overall impact of the food pantry drive. The expression 9n + 8 is a linear expression, indicating that the total number of food items collected increases linearly with Tonya's contribution.

Moreover, the summarized contributions provide a clear picture of the relative efforts of each student. Beth's contribution, represented by 6n + 9, is significantly larger than Tonya's and Nathan's contributions. This highlights Beth's exceptional generosity and dedication to the food pantry drive. Nathan's contribution, represented by 2n - 1, is also substantial, but slightly less than twice Tonya's contribution. This comparison allows for a nuanced understanding of the individual contributions and their impact on the overall success of the drive.

In conclusion, summarizing the contributions of each student provides a comprehensive overview of the food pantry drive. The total number of food items collected is represented by the expression 9n + 8, which allows for quick calculation and analysis. The summarized contributions also highlight the relative efforts of each student, providing valuable insights into the dynamics of the drive. This comprehensive overview underscores the importance of collaboration and individual effort in achieving a common goal.

#h1 Keywords and Problem Restatement

Repair Input Keyword: How many food items did Nathan collect in terms of n?

SEO Title: Calculating Nathan's Food Items in Terms of n - A Step-by-Step Solution