Alexandra's Coins Equation For Nickels

Mathematics often presents us with intriguing puzzles, and one such puzzle involves Alexandra's coins. Alexandra possesses a collection of 36 coins, totaling a value of $6.00. This collection consists solely of nickels and quarters, adding a layer of complexity to the problem. The challenge lies in determining the equation that can be used to solve for the number of nickels, denoted as n. This seemingly simple problem delves into the realm of algebraic equations and problem-solving strategies. To effectively tackle this coin conundrum, we need to carefully analyze the given information and translate it into mathematical expressions. The number of coins, their individual values, and the total value of the collection are all crucial pieces of the puzzle. By establishing relationships between these elements, we can construct an equation that accurately represents the scenario and allows us to determine the unknown quantity, the number of nickels. This process not only enhances our understanding of algebraic concepts but also sharpens our problem-solving skills in real-world contexts. As we embark on this mathematical journey, we will explore different approaches, unravel the intricacies of the problem, and ultimately arrive at the correct equation that unlocks the solution to Alexandra's coin conundrum. The beauty of mathematics lies in its ability to transform seemingly complex situations into elegant equations, providing a clear path to understanding and resolution. So, let us delve into the world of coins, values, and equations, and discover the equation that will reveal the number of nickels in Alexandra's collection. This endeavor will not only provide us with a numerical answer but also offer valuable insights into the power of mathematical modeling and its applications in everyday scenarios. The process of formulating the equation involves careful consideration of the relationships between the variables and constants involved. We must accurately represent the given information in a mathematical form that captures the essence of the problem. This requires a clear understanding of the values of nickels and quarters, as well as the total number of coins and their combined value. By systematically analyzing these elements, we can construct an equation that precisely reflects the situation and allows us to solve for the unknown, the number of nickels.

Setting up the Equations: A Step-by-Step Guide

To begin, let's break down the information provided. We know that Alexandra has a total of 36 coins, which are a combination of nickels and quarters. We can represent this information with the following equation:

n + q = 36

Where:

• n represents the number of nickels
• q represents the number of quarters

This equation establishes the relationship between the number of nickels and quarters, indicating that their sum must equal 36. However, this is just the first piece of the puzzle. We also know that the total value of the coins is $6.00. To incorporate this information into our equation, we need to consider the value of each coin. A nickel is worth $0.05, and a quarter is worth $0.25. Therefore, the total value of the nickels can be represented as 0.05n, and the total value of the quarters can be represented as 0.25q. Combining these values, we get the following equation:

0. 05n + 0.25q = 6.00

This equation represents the total value of the coins in terms of the number of nickels and quarters. Now we have two equations with two variables, which means we can solve for the unknowns. However, the question asks for the equation that can be used to solve for the number of nickels, n. To achieve this, we need to eliminate the variable q from the equations. We can do this by using the first equation (n + q = 36) to express q in terms of n:

q = 36 - n

Now, we can substitute this expression for q into the second equation:

0. 05n + 0.25(36 - n) = 6.00

This equation contains only one variable, n, which represents the number of nickels. Therefore, this is the equation that can be used to solve for the number of nickels. By simplifying and solving this equation, we can determine the exact number of nickels in Alexandra's collection. This process demonstrates the power of algebraic manipulation in solving real-world problems. By carefully translating the given information into mathematical expressions and then applying algebraic techniques, we can arrive at a solution that provides valuable insights into the problem at hand. The ability to set up and solve equations is a fundamental skill in mathematics and has wide-ranging applications in various fields.

Identifying the Correct Equation: A Matter of Representation

The key to solving this problem lies in correctly representing the given information in the form of an equation. We have two crucial pieces of information: the total number of coins and the total value of the coins. Let's revisit the equations we derived earlier:

1. n + q = 36 (Total number of coins)
2. 0.05n + 0.25q = 6.00 (Total value of coins)

As we established, we need an equation that allows us to solve for n, the number of nickels. By substituting q = 36 - n into the second equation, we arrived at:

0. 05n + 0.25(36 - n) = 6.00

This equation is the correct representation of the problem and can be used to solve for n. It incorporates both the number of coins and their values, providing a comprehensive mathematical model of the situation. Let's consider why other potential equations might be incorrect. An equation that only considers the number of coins (n + q = 36) does not account for the different values of nickels and quarters. Similarly, an equation that only focuses on the total value (0.05n + 0.25q = 6.00) does not incorporate the constraint on the total number of coins. The correct equation must capture both aspects of the problem to accurately reflect the relationships between the variables. The process of identifying the correct equation highlights the importance of careful analysis and attention to detail in mathematical problem-solving. We must ensure that our equations accurately represent the given information and capture the essential relationships between the variables. This requires a clear understanding of the problem context and the ability to translate verbal descriptions into mathematical expressions. By mastering these skills, we can confidently tackle a wide range of mathematical challenges and apply our knowledge to real-world situations.

Solving for the Number of Nickels: Bringing it Home

While the question specifically asks for the equation that can be used to solve for the number of nickels, let's take it a step further and actually solve for n. This will not only reinforce our understanding of the problem but also demonstrate the practical application of the equation we derived. Starting with the equation:

0. 05n + 0.25(36 - n) = 6.00

First, we distribute the 0.25:

0. 05n + 9 - 0.25n = 6.00

Next, we combine the n terms:

-0.20n + 9 = 6.00

Now, we subtract 9 from both sides:

-0.20n = -3.00

Finally, we divide both sides by -0.20:

n = 15

Therefore, Alexandra has 15 nickels. We can verify this result by plugging it back into our equations. If n = 15, then q = 36 - 15 = 21. The value of 15 nickels is 15 * $0.05 = $0.75, and the value of 21 quarters is 21 * $0.25 = $5.25. Adding these values together, we get $0.75 + $5.25 = $6.00, which confirms our solution. This exercise demonstrates the power of algebraic equations in solving real-world problems. By translating a word problem into a mathematical equation, we can systematically solve for the unknown quantities. The process of solving the equation involves applying algebraic principles and techniques to isolate the variable of interest. This requires a solid understanding of mathematical operations and the ability to manipulate equations effectively. By mastering these skills, we can confidently tackle a wide range of mathematical challenges and apply our knowledge to practical situations.

Key Takeaways: Mastering the Art of Equation Formulation

This problem highlights several key concepts in mathematical problem-solving. Firstly, it emphasizes the importance of carefully reading and understanding the problem statement. We must identify the given information, the unknown quantities, and the relationships between them. Secondly, it demonstrates the power of translating verbal descriptions into mathematical expressions. By representing the problem in the form of equations, we can leverage the tools of algebra to solve for the unknowns. Thirdly, it underscores the importance of verifying our solutions. By plugging our results back into the original equations, we can ensure that they are consistent with the given information. In summary, Alexandra's coin conundrum provides a valuable lesson in equation formulation and problem-solving. By carefully analyzing the problem, translating it into mathematical expressions, and applying algebraic techniques, we can confidently tackle similar challenges in the future. The ability to set up and solve equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering this skill, we can enhance our problem-solving abilities and gain a deeper understanding of the world around us. This exercise also reinforces the importance of attention to detail and accuracy in mathematical calculations. A small error in the setup or solution process can lead to an incorrect answer. Therefore, it is crucial to carefully check our work and ensure that our results are consistent with the given information. By developing these habits, we can become more confident and effective problem-solvers.

In conclusion, this exploration of Alexandra's coin conundrum has provided us with valuable insights into the process of equation formulation and problem-solving. By carefully analyzing the problem, translating it into mathematical expressions, and applying algebraic techniques, we have successfully identified the equation that can be used to solve for the number of nickels. This exercise has not only enhanced our mathematical skills but also reinforced the importance of critical thinking and attention to detail in problem-solving. As we continue our mathematical journey, let us remember the lessons learned from Alexandra's coins and apply them to new and challenging problems. The world of mathematics is full of intriguing puzzles, and by mastering the art of equation formulation, we can unlock the solutions to these puzzles and gain a deeper understanding of the world around us.

Table: Coin Denominations