Fiona's Shopping Spree Solve Socks And Belts Cost Problem

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#h1 Fiona's Shopping Spree Unraveling the Cost of Socks and Belts

Introduction to Fiona's Purchase

In this mathematical exploration, we delve into a real-world scenario involving Fiona's shopping trip. Fiona, with a keen eye for fashion and practicality, decided to purchase some socks and belts. The cost of each pair of socks was $4.95, while each belt was priced at $6.55. Fiona's total expenditure amounted to $27.95. Our goal is to dissect this scenario using algebraic principles, representing the number of pairs of socks purchased by the variable 'a' and the number of belts by the variable 'b.' This problem provides an excellent opportunity to apply our understanding of linear equations and problem-solving strategies.

This detailed analysis aims to unravel the number of socks and belts Fiona acquired, offering insights into how such real-life scenarios can be effectively modeled and solved using mathematical concepts. By the end of this exploration, you will not only understand the specific solution to Fiona's shopping spree but also grasp the broader applicability of algebraic methods in everyday problem-solving. So, let's embark on this mathematical journey, carefully examining the given information and constructing a clear path towards the solution. We'll break down each step, ensuring a comprehensive understanding of the process and the underlying principles. This is more than just solving an equation; it's about developing critical thinking and problem-solving skills that are valuable in various aspects of life.

Defining the Variables and Setting Up the Equation

To begin, let's clearly define the variables involved. As stated in the problem, 'a' represents the number of pairs of socks Fiona purchased, and 'b' represents the number of belts. With these variables in place, we can now translate the given information into a mathematical equation. The cost of each pair of socks is $4.95, so the total cost of the socks is 4.95a. Similarly, the cost of each belt is $6.55, making the total cost of belts 6.55b. Fiona's total spending was $27.95. Therefore, we can express this situation as a linear equation:

4.  95a + 6.55b = 27.95

This equation is the cornerstone of our solution. It encapsulates all the essential information provided in the problem statement. Now, the challenge lies in finding the values of 'a' and 'b' that satisfy this equation. It's important to recognize that this is a single equation with two unknowns, which means there might be multiple solutions or a constraint on the possible values of 'a' and 'b.'

Before we delve into solving the equation, let's pause and appreciate the power of this initial step. By translating a real-world scenario into a mathematical equation, we have laid the foundation for a systematic and logical solution. This process of translating words into symbols is a fundamental skill in mathematics and problem-solving. It allows us to move from the realm of vague descriptions to the precision of algebraic expressions. The equation 4.95a + 6.55b = 27.95 is not just a jumble of numbers and letters; it's a concise representation of Fiona's shopping spree, ready to be analyzed and解剖.

Exploring Possible Solutions and Constraints

Now that we have our equation, 4.95a + 6.55b = 27.95, the next step is to explore potential solutions. However, before we jump into algebraic manipulations, let's consider some real-world constraints. Since 'a' and 'b' represent the number of pairs of socks and belts, respectively, they must be non-negative integers. Fiona cannot buy a fraction of a sock or a belt, and she certainly cannot buy a negative number of items. This constraint significantly narrows down the possible solutions.

To further refine our search, we can rearrange the equation to express one variable in terms of the other. Let's solve for 'a':

4.  95a = 27.95 - 6.55b
a = (27.95 - 6.55b) / 4.95

This form of the equation allows us to test different integer values of 'b' and see if the resulting value of 'a' is also a non-negative integer. This is a crucial step because it incorporates the real-world constraints into our mathematical process. We're not just looking for any solution to the equation; we're looking for a solution that makes sense in the context of Fiona's shopping trip.

By systematically testing values of 'b,' we can create a table of possible solutions. This approach is methodical and helps us avoid overlooking potential answers. It also highlights the importance of logical reasoning in problem-solving. We're not blindly guessing; we're using the equation and the constraints to guide our search. This process of exploration and refinement is a hallmark of mathematical thinking. It's about trying, testing, and adjusting our approach based on the results we obtain. This iterative process is not just applicable to this specific problem; it's a valuable skill in any situation that requires problem-solving and decision-making.

Solving for Integer Solutions

As we established, 'a' and 'b' must be non-negative integers. Let's use the equation a = (27.95 - 6.55b) / 4.95 and test integer values for 'b' to see if we get an integer value for 'a'. We'll start with b = 0 and increase until the numerator becomes negative, as that would result in a negative value for 'a,' which is not possible.

  • If b = 0, a = (27.95 - 6.55 * 0) / 4.95 = 27.95 / 4.95 β‰ˆ 5.65 (Not an integer)
  • If b = 1, a = (27.95 - 6.55 * 1) / 4.95 = 21.40 / 4.95 β‰ˆ 4.32 (Not an integer)
  • If b = 2, a = (27.95 - 6.55 * 2) / 4.95 = 14.85 / 4.95 = 3 (Integer!)
  • If b = 3, a = (27.95 - 6.55 * 3) / 4.95 = 8.30 / 4.95 β‰ˆ 1.68 (Not an integer)
  • If b = 4, a = (27.95 - 6.55 * 4) / 4.95 = 1.75 / 4.95 β‰ˆ 0.35 (Not an integer)

We found an integer solution when b = 2 and a = 3. Let's check if this solution makes sense in the original equation:

4.  95 * 3 + 6.55 * 2 = 14.85 + 13.10 = 27.95

The solution checks out! This confirms that Fiona bought 3 pairs of socks and 2 belts. This methodical approach to finding integer solutions is a powerful technique in problem-solving. It combines algebraic manipulation with logical reasoning and the consideration of real-world constraints. The process of testing different values and checking the results is a cornerstone of the scientific method and is applicable in various fields beyond mathematics.

Finding integer solutions often involves a bit of trial and error, but by systematically exploring the possibilities and using the equation as a guide, we can efficiently narrow down the options. The solution we found is not just a set of numbers; it's an answer to a real-world question. It tells us something concrete about Fiona's shopping trip. This connection between abstract mathematics and tangible situations is what makes problem-solving so engaging and rewarding.

Verifying the Solution and Conclusion

Having found a potential solution, it's crucial to verify its accuracy. We determined that Fiona bought 3 pairs of socks and 2 belts. To confirm this, we substitute these values back into our original equation:

4.  95a + 6.55b = 27.95
4.  95(3) + 6.55(2) = 27.95
5.  85 + 13.10 = 27.95
6.  95 = 27.95

The equation holds true, affirming that our solution is correct. Fiona indeed purchased 3 pairs of socks and 2 belts. This final verification step is essential in any problem-solving process. It ensures that we haven't made any calculation errors or overlooked any critical details. It's a way of double-checking our work and building confidence in our answer.

In conclusion, by translating the word problem into an algebraic equation, considering real-world constraints, systematically testing values, and verifying our solution, we successfully determined the number of socks and belts Fiona bought. This exercise demonstrates the power of mathematical tools in solving everyday problems. It highlights the importance of clear problem definition, careful variable selection, and logical reasoning. The skills we've employed in this problem are transferable to a wide range of situations, both within and outside the realm of mathematics. The ability to break down a complex problem into smaller, manageable steps, to use equations as models of reality, and to verify our results is a valuable asset in any field.

This journey through Fiona's shopping spree has not only provided us with a specific answer but also reinforced the broader principles of problem-solving. It's a reminder that mathematics is not just about numbers and formulas; it's about thinking critically, reasoning logically, and applying our knowledge to the world around us. The next time you encounter a problem, whether it's a mathematical puzzle or a real-life challenge, remember the steps we've taken here: define the problem, translate it into a model, explore potential solutions, and verify your answer. With these tools in hand, you'll be well-equipped to tackle any challenge that comes your way.

#h2 Keywords and their meaning

Repair Input Keyword

Original Keyword: Let 'a' represent the number of pairs of socks purchased and 'b' the number of belts.

Repaired Keyword: How many pairs of socks (represented by 'a') and belts (represented by 'b') did Fiona purchase?

Explanation of the Repair

The original keyword was a statement defining the variables 'a' and 'b.' While this is important context, it doesn't directly represent a question or a problem to be solved. The repaired keyword transforms this statement into a question that encapsulates the core problem. It asks directly about the number of socks and belts purchased, making it clear what needs to be determined. This revised keyword is more effective for SEO because it aligns with how someone might search for help with this type of problem. It also clarifies the objective of the problem, guiding the reader towards the solution more effectively. The revised keyword uses clear and concise language, making it easily understandable and relevant to the topic. It also maintains the original intent of defining 'a' and 'b' but frames it within the context of the overarching question.

By framing the keyword as a question, it becomes more searchable and user-friendly. People often search for solutions by asking questions, so this revised keyword is more likely to attract the right audience. It also helps to focus the content on providing a clear and direct answer to the question posed. This ensures that the content is relevant and valuable to the reader, increasing engagement and satisfaction. The keyword now serves as a concise summary of the problem, allowing readers to quickly grasp the main idea and determine if the content is relevant to their needs. This is crucial for SEO and user experience, as it helps to improve click-through rates and time spent on the page.

SEO Title

Original Title: Fiona bought some socks that cost $4.954.95$ for each pair and some belts that cost $6.556.55$ each. Fiona spent $27.9527.95$ in all. Let $a$ represent the number of pairs of socks purchased and $b$ the

SEO Title: Fiona's Shopping Spree Solve Socks and Belts Cost Problem

Explanation of the SEO Title

The original title, while descriptive, is too long and doesn't effectively target keywords for search engine optimization (SEO). It also includes mathematical notation which is not ideal for a title intended for general readability and searchability. The revised SEO title, Fiona's Shopping Spree Solve Socks and Belts Cost Problem, is significantly shorter, more focused, and incorporates relevant keywords.

The phrase Fiona's Shopping Spree adds a human element and makes the title more engaging. It suggests a narrative or real-world scenario, which can attract readers who are looking for practical applications of mathematics. The keywords Solve Socks and Belts Cost Problem directly address the core topic of the problem. These keywords are likely to be used by individuals searching for help with similar math problems. The word Solve indicates that the content provides a solution, which is a strong signal to users looking for answers. The title is concise and to the point, making it easy to understand at a glance. This is crucial for SEO, as search engines prioritize titles that accurately reflect the content and are easily readable.

By using a clear and concise title with relevant keywords, we increase the likelihood that the content will rank well in search engine results and attract the right audience. The revised title also avoids the use of special characters and mathematical notation, making it more user-friendly and accessible. It focuses on the essence of the problem, which is to determine the cost and quantity of items purchased, and frames it in a way that is both informative and engaging. This balance between SEO optimization and user experience is essential for creating content that is both discoverable and valuable.