Dividing A Board A Detailed Solution To A Length Problem

Introduction

In this article, we will delve into a practical mathematical problem involving the division of a board into equal pieces. Understanding fractions and division is crucial in various real-life situations, from home improvement projects to culinary endeavors. This particular problem focuses on calculating the length of each piece when a board of a given length is cut into a specific number of segments. We will explore the steps involved in solving this problem, ensuring a clear and comprehensive understanding of the underlying concepts.

The core of the problem lies in the intersection of fractions and division, two fundamental concepts in mathematics. Fractions represent parts of a whole, while division involves splitting a quantity into equal groups. The ability to seamlessly navigate these concepts is not just academically beneficial but also practically advantageous. Whether you're a student tackling math problems or a DIY enthusiast working on a project, a solid grasp of fractions and division will undoubtedly prove invaluable. So, let's embark on this mathematical journey and dissect the problem at hand, unraveling the solution step by step. We'll make sure to clarify each stage, so you can confidently apply these skills to similar challenges in the future. Remember, the goal isn't just to find the answer but to understand the process and the reasoning behind it.

Problem Statement

The problem we aim to solve is this: A board measuring $\frac{3}{4}$ feet in length is cut into 6 equal pieces. Our task is to determine the length of each of these pieces. The given options are:

A. $1 \frac{1}{2}$ inches B. 3 inches

Before we jump into calculations, it's essential to understand the problem thoroughly. We're dealing with a board that has a fractional length, and we're dividing it into a whole number of pieces. This means we'll need to apply our knowledge of fractions and division to arrive at the correct answer. The units also matter here; the initial length is given in feet, while the options are in inches. This hints at a necessary unit conversion step somewhere in our solution. Misunderstanding the problem's requirements can easily lead to incorrect calculations, so let's make sure we're clear on what's being asked. We want the length of a single piece after the board has been cut into six equal sections. Now that we've dissected the problem statement, we can move on to strategizing our approach to finding the solution. Remember, a well-defined strategy is often the key to solving mathematical problems efficiently and accurately.

Solution

To solve this problem, we will follow these steps:

1. Convert the length of the board from feet to inches.
2. Divide the length in inches by the number of pieces.

Step 1: Convert Feet to Inches

We know that 1 foot is equal to 12 inches. Therefore, to convert $\frac{3}{4}$ feet to inches, we multiply $\frac{3}{4}$ by 12:

$$\frac{3}{4} \text{ feet} = \frac{3}{4} \times 12 \text{ inches} = \frac{3 \times 12}{4} \text{ inches} = \frac{36}{4} \text{ inches} = 9 \text{ inches}$$

This conversion is a crucial first step because it brings our units into alignment. We were given the board's length in feet, but the answer options are presented in inches. Working with consistent units is paramount in mathematical problem-solving. Without this conversion, we would be comparing apples and oranges, so to speak. The multiplication we performed, $\frac{3}{4} \times 12$, effectively scales the fractional length of the board into the inch-based measurement system. Simplifying the fraction $\frac{36}{4}$ to 9 gives us a clear understanding of the board's total length in inches, which is the necessary foundation for the next step. This conversion highlights the importance of paying attention to units in mathematical problems and underscores the need for unit conversions when different units are involved. Now that we have the board's length in inches, we can proceed to divide it into the specified number of pieces.

Step 2: Divide the Length by the Number of Pieces

Now that we know the board is 9 inches long, we divide this length by the number of pieces, which is 6:

$$\frac{9 \text{ inches}}{6 \text{ pieces}} = \frac{9}{6} \text{ inches per piece}$$

We can simplify the fraction $\frac{9}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$$\frac{9}{6} = \frac{9 \div 3}{6 \div 3} = \frac{3}{2}$$

So, each piece is $\frac{3}{2}$ inches long. We can convert this improper fraction to a mixed number:

$$\frac{3}{2} = 1 \frac{1}{2}$$

Therefore, each piece is $1 \frac{1}{2}$ inches long.

This division step is the heart of the problem, as it directly addresses the question of how long each piece will be. Dividing the total length by the number of pieces gives us the length of a single piece, assuming they are all equal in size, as the problem states. The resulting fraction, $\frac{9}{6}$, represents the length per piece, but it's essential to simplify this fraction to its simplest form for clarity and ease of comparison with the answer options. Simplifying fractions is a key skill in mathematics, and in this case, it allows us to express the length in a more concise and understandable way. Converting the improper fraction $\frac{3}{2}$ to the mixed number $1 \frac{1}{2}$ makes the answer even clearer, as it expresses the length as a whole number plus a fraction, which is often easier to visualize and comprehend. This final result directly answers the problem's question, telling us the length of each individual piece of the board.

Final Answer

Comparing our result with the given options, we find that the length of each piece is $1 \frac{1}{2}$ inches, which corresponds to option A.

Therefore, the correct answer is A. $1 \frac{1}{2}$ inches.

This final step is where we connect our calculated result to the choices provided and select the one that matches. It's a crucial validation step, ensuring that we haven't made any errors in our calculations or interpretations. By comparing $1 \frac{1}{2}$ inches with the options given, we confidently identify option A as the correct answer. This process of checking and confirming our solution is vital in problem-solving, as it minimizes the chances of selecting an incorrect answer due to a simple oversight. It reinforces the importance of accuracy and attention to detail in mathematical problem-solving. Now that we've successfully navigated the problem and arrived at the solution, it's worthwhile to reflect on the steps we took and the underlying mathematical concepts.

Conclusion

In this article, we successfully solved a problem involving the division of a board into equal pieces. We applied the concepts of fractions, division, and unit conversion to arrive at the correct answer. This problem demonstrates the practical application of mathematical skills in everyday scenarios. By understanding the problem, devising a clear strategy, and executing the steps meticulously, we can confidently tackle similar challenges in the future.

This exercise not only reinforced our understanding of fractions and division but also highlighted the importance of unit conversions in mathematical problem-solving. The ability to convert between different units, such as feet and inches, is a valuable skill that extends beyond the classroom and into real-world applications. Furthermore, this problem-solving process underscores the significance of breaking down complex problems into smaller, manageable steps. By dividing the problem into conversion and division stages, we were able to approach it systematically and avoid getting overwhelmed. This step-by-step approach is a valuable strategy for tackling any complex problem, mathematical or otherwise. So, remember the key takeaways from this exercise: pay attention to units, simplify fractions, and break down problems into manageable steps. With these skills in your toolkit, you'll be well-equipped to conquer a wide range of mathematical challenges.