Stacey's Cloth Square Puzzle How To Find The Original Side Length

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Introduction

This article delves into a classic geometry problem involving a square piece of cloth and its transformation. The problem presents a scenario where a square is reduced in size, and we are tasked with determining the dimensions of the original square. This kind of problem is not just a mathematical exercise; it sharpens our problem-solving skills and our ability to translate word problems into mathematical equations. Geometry problems, like this one, are foundational in mathematics, appearing in various standardized tests and real-world applications. Understanding how to approach and solve them is a valuable asset. So, let's unravel the mystery of Stacey's cloth square and discover the side length of the original piece.

Problem Statement

Stacey has a square piece of cloth. She cuts 3 inches off the length and 3 inches off the width, effectively creating a smaller square. The crucial piece of information is that the area of the smaller square is 14\frac{1}{4} the area of the original square. The question we aim to answer is: What was the side length of the original square?

This problem can be solved using algebraic principles, specifically by setting up equations that represent the areas of the squares. By defining variables and expressing the given relationships mathematically, we can methodically work towards the solution. The problem emphasizes the relationship between the areas of two squares, one derived from the other, which adds an intriguing layer to the challenge. Before we dive into the solution, let's break down the components and plan our approach.

Breaking Down the Problem

To effectively solve this problem, we need to identify the key pieces of information and how they relate to each other.

  1. Original Square: We start with a square piece of cloth. Let's denote the side length of this original square as 'x' inches. The area of this square would then be x². Understanding the concept of area for a square is the first step. The area of a square is simply the side length multiplied by itself. This forms the basis of our initial equation.

  2. Cutting the Cloth: Stacey cuts 3 inches from both the length and the width. This action reduces the dimensions of the square, resulting in a smaller square. If the original side length was 'x', the new side length becomes 'x - 3'. This step is crucial as it defines the dimensions of the second square in relation to the original. The act of cutting the cloth creates a direct link between the two squares, which is vital for setting up the equations.

  3. Smaller Square: The smaller square has a side length of (x - 3) inches. Consequently, its area is (x - 3)². This is a direct application of the area formula for a square. The side length of the smaller square is a result of the modification done to the original square. Accurately expressing the area of this smaller square is critical for establishing the core equation of the problem.

  4. Area Relationship: The problem states that the area of the smaller square is 14\frac{1}{4} the area of the original square. This statement is the cornerstone of our solution strategy. It provides the direct mathematical link between the two squares, allowing us to set up an equation. This relationship is the key to solving for the unknown variable 'x'. Without this link, it would be impossible to relate the dimensions of the two squares.

By carefully dissecting the problem, we have identified the essential elements and their interconnections. Next, we will translate these elements into a mathematical equation, which will pave the way for solving the problem.

Setting up the Equation

Now that we have a clear understanding of the problem, let's translate the information into a mathematical equation. This is a critical step in solving mathematical problems, especially those involving geometry and word problems. The equation will serve as a roadmap, guiding us towards the solution. The process of setting up an equation involves expressing the relationships described in the problem using mathematical symbols and operations.

We know the following:

  • The area of the original square is x².
  • The area of the smaller square is (x - 3)².
  • The area of the smaller square is 14\frac{1}{4} the area of the original square.

Using this information, we can set up the equation as follows:

(x - 3)² = 14\frac{1}{4}x²

This equation represents the core of the problem. It mathematically states that the area of the smaller square is one-fourth the area of the original square. Solving this equation will give us the value of 'x', which represents the side length of the original square. The equation encapsulates all the key information from the problem statement, making it the central tool for finding the solution.

This equation is a quadratic equation, meaning it involves a term with x². Solving quadratic equations often involves algebraic manipulation, such as expanding terms, simplifying, and possibly factoring or using the quadratic formula. Before we delve into solving the equation, let's take a moment to appreciate the power of mathematical equations to represent real-world scenarios. This equation is not just an abstract concept; it is a representation of a physical situation involving squares and areas.

In the next section, we will proceed to solve this equation step by step, revealing the side length of Stacey's original square piece of cloth. The process of solving the equation is a blend of algebraic techniques and logical deduction, ultimately leading us to the answer.

Solving the Equation

With our equation set up, the next step is to solve it. This involves using algebraic techniques to isolate the variable 'x' and find its value. Solving the equation will provide us with the side length of the original square, which is our ultimate goal. This section will walk through the steps required to solve the equation (x - 3)² = 14\frac{1}{4}x².

  1. Expand the left side: The first step is to expand the term (x - 3)². This can be done using the formula (a - b)² = a² - 2ab + b². Applying this formula, we get:

x² - 6x + 9 = 14\frac{1}{4}x²

Expanding the term helps to remove the parentheses and allows us to combine like terms in the subsequent steps. This is a standard algebraic technique used to simplify equations. By expanding, we are essentially rewriting the equation in a different form that is easier to manipulate.

  1. Multiply both sides by 4: To eliminate the fraction, we can multiply both sides of the equation by 4. This gives us:

4(x² - 6x + 9) = 4 * 14\frac{1}{4}x²

Simplifying this, we get:

4x² - 24x + 36 = x²

Eliminating fractions often makes equations easier to work with, especially when dealing with more complex algebraic manipulations. This step clears the way for rearranging the equation into a standard form for a quadratic equation.

  1. Rearrange the equation: Now, we need to rearrange the equation to bring all the terms to one side, setting the equation equal to zero. This is a crucial step in solving quadratic equations, as it allows us to use methods like factoring or the quadratic formula. Subtracting x² from both sides, we get:

3x² - 24x + 36 = 0

Rearranging the equation into this form is a key step in solving quadratic equations. It sets the stage for applying standard solution methods.

  1. Simplify by dividing by 3: We can simplify the equation further by dividing all terms by 3. This makes the coefficients smaller and the equation easier to handle. Dividing both sides by 3, we get:

x² - 8x + 12 = 0

Simplifying the equation helps to reduce the complexity of the numbers involved, making it easier to factor or apply the quadratic formula.

  1. Factor the quadratic: The next step is to factor the quadratic equation. We are looking for two numbers that multiply to 12 and add up to -8. These numbers are -6 and -2. So, we can factor the equation as follows:

(x - 6)(x - 2) = 0

Factoring is a powerful technique for solving quadratic equations. It allows us to break down the equation into simpler factors, each of which can be set to zero to find the solutions.

  1. Solve for x: Setting each factor equal to zero, we get:

x - 6 = 0 or x - 2 = 0

Solving these equations, we find two possible solutions for x:

x = 6 or x = 2

These are the potential values for the side length of the original square. However, we need to consider the context of the problem to determine which solution is valid.

In the next section, we will analyze these solutions and determine the correct answer to the problem.

Analyzing the Solutions

In the previous section, we found two possible solutions for the side length of the original square: x = 6 inches and x = 2 inches. However, in the context of the problem, not all solutions may be valid. It is crucial to analyze each solution and determine if it makes sense within the given scenario. Analyzing the solutions is a critical step in problem-solving, ensuring that the answer is not only mathematically correct but also logically sound.

Let's consider each solution:

  1. x = 6 inches: If the original side length was 6 inches, Stacey cut 3 inches off each side, resulting in a smaller square with a side length of 6 - 3 = 3 inches. The area of the original square would be 6² = 36 square inches, and the area of the smaller square would be 3² = 9 square inches. Is the area of the smaller square 14\frac{1}{4} the area of the original square? Yes, 9 is indeed 14\frac{1}{4} of 36. So, this solution is valid.

  2. x = 2 inches: If the original side length was 2 inches, Stacey cut 3 inches off each side. This would result in a side length of 2 - 3 = -1 inches, which is not physically possible. A square cannot have a negative side length. Therefore, this solution is not valid in the context of the problem. This highlights the importance of checking solutions against the real-world constraints of the problem.

By analyzing the solutions, we can see that only x = 6 inches makes sense in the given context. The solution x = 2 inches is an extraneous solution, meaning it is a solution to the equation but not to the original problem. Extraneous solutions often arise when dealing with equations derived from real-world scenarios, emphasizing the need for careful analysis.

Therefore, the side length of the original square piece of cloth was 6 inches. We have successfully solved the problem by setting up an equation, solving it, and analyzing the solutions within the context of the problem. In the final section, we will summarize the steps we took to solve the problem and highlight the key concepts involved.

Conclusion

In this article, we tackled the problem of finding the original side length of Stacey's square piece of cloth. The problem involved a reduction in the size of a square and a relationship between the areas of the original and smaller squares. We successfully navigated through the problem by breaking it down into manageable steps and applying algebraic principles. Let's recap the journey we took to arrive at the solution.

  1. Understanding the Problem: We began by carefully reading and understanding the problem statement. We identified the key pieces of information, such as the cutting of the cloth and the relationship between the areas of the squares. This initial step is crucial for any problem-solving endeavor, as it sets the foundation for the subsequent steps.

  2. Setting up the Equation: We translated the information into a mathematical equation. By denoting the original side length as 'x', we expressed the areas of both squares in terms of 'x'. The crucial equation we derived was (x - 3)² = 14\frac{1}{4}x², which mathematically represented the given relationship between the areas. This step is where we transitioned from a word problem to a mathematical problem.

  3. Solving the Equation: We employed algebraic techniques to solve the equation. This involved expanding terms, simplifying, and factoring the quadratic equation. We arrived at two possible solutions for 'x': 6 inches and 2 inches. The process of solving the equation demonstrated the power of algebraic manipulation in finding unknown values.

  4. Analyzing the Solutions: We critically analyzed the solutions within the context of the problem. We realized that the solution x = 2 inches was not valid because it would result in a negative side length for the smaller square. The solution x = 6 inches, on the other hand, made logical sense and satisfied the conditions of the problem. This step underscored the importance of checking solutions against the real-world constraints of the problem.

Through this methodical approach, we determined that the side length of the original square piece of cloth was 6 inches. This problem serves as an excellent example of how mathematical concepts can be applied to solve real-world scenarios. It also highlights the importance of careful problem-solving techniques, including understanding the problem, setting up equations, solving equations, and analyzing solutions.

Geometry problems like this one are not just academic exercises; they hone our analytical and logical reasoning skills. These skills are valuable not only in mathematics but also in various aspects of life. By mastering such problems, we equip ourselves with the tools to approach and solve a wide range of challenges.

In conclusion, Stacey's cloth square puzzle is a testament to the power and elegance of mathematical problem-solving. It illustrates how a seemingly complex problem can be unraveled through a systematic approach and a solid understanding of mathematical principles.