Fractions Problem Solving A Piece Of Red 2m Long Cut Into Pieces

Delving into the world of fractions, this problem presents a scenario where a 2-meter long piece is subjected to multiple divisions, challenging our understanding of how fractions represent parts of a whole. To dissect this problem effectively, we will approach it step-by-step, ensuring clarity and precision in our calculations. Our primary goal is to determine the fractional representation of one of the smallest pieces in relation to the original 2-meter length.

Let's embark on this mathematical journey by first addressing the initial division. The original piece, measuring 2 meters in length, is cut into three equal pieces. This foundational step lays the groundwork for subsequent divisions and is crucial in determining the ultimate fraction. The concept of fractions revolves around dividing a whole into equal parts, and understanding this initial division is paramount to grasping the entire problem. When we cut the 2-meter piece into three equal parts, we are essentially dividing 2 by 3, resulting in each piece being 2/3 meters long. This 2/3 meter length becomes the new 'whole' that we will further divide in the next step.

The beauty of fractions lies in their ability to represent parts of a whole in a precise and quantifiable manner. In this context, each of the three pieces represents a fraction of the original 2-meter length, specifically 2/3. This understanding is pivotal as we move forward, as it allows us to track how the divisions affect the size of the pieces relative to the original length. This concept of fractions representing parts of a whole is not just a mathematical abstraction; it has practical applications in various real-world scenarios, from measuring ingredients in a recipe to calculating distances on a map.

Thus, with each piece measuring 2/3 meters, we have successfully navigated the first step of this fractional puzzle. Now, we are equipped to address the next division, where one of these 2/3 meter pieces is further divided into four equal parts. This step will require us to apply our understanding of fractions once more, building upon the foundation we have established.

Having established that each of the three initial pieces measures 2/3 meters, we now focus on one of these pieces, which undergoes further division. This second division is a critical step in our problem, as it introduces an additional layer of fractional representation. The problem states that one of the 2/3 meter pieces is cut into four equal pieces. This division implies that we are now dealing with fractions of a fraction, a concept that requires careful consideration. To accurately determine the size of these smaller pieces, we must divide the 2/3 meter length by 4. This calculation is fundamental to understanding the final fraction in relation to the original 2-meter length. Dividing a fraction by a whole number involves multiplying the denominator of the fraction by the whole number. In this case, we are dividing 2/3 by 4, which translates to multiplying the denominator 3 by 4. This process results in a new fraction, which represents the length of each of the four smaller pieces. The calculation yields 2/(3*4) = 2/12 meters.

This 2/12 meter length represents the size of each of the four pieces resulting from the second division. However, it's important to recognize that this fraction is not in its simplest form. Simplifying fractions is a crucial skill in mathematics, as it allows us to express fractions in their most concise and easily understandable form. To simplify 2/12, we look for the greatest common divisor (GCD) of the numerator (2) and the denominator (12). The GCD of 2 and 12 is 2. Dividing both the numerator and the denominator by 2, we arrive at the simplified fraction 1/6. Thus, each of the four smaller pieces measures 1/6 of a meter. This 1/6 meter length is a crucial milestone in our problem-solving journey.

However, it's important to remember that this 1/6 meter length represents the size of the piece in relation to the 2/3 meter piece that was divided. Our ultimate goal is to determine the fraction of the original 2-meter length that each of these pieces represents. To achieve this, we need to relate the 1/6 meter length back to the original 2-meter piece. This step requires us to consider the initial division, where the 2-meter piece was divided into three equal parts. With each piece now measuring 1/6 of a meter, we are one step closer to solving the puzzle. The next step involves expressing this 1/6 meter length as a fraction of the original 2-meter length.

Having meticulously navigated the divisions, we arrive at the final stage of our fractional journey. We've established that each of the smallest pieces measures 1/6 of a meter. However, the crux of the problem lies in determining what fraction of the total length, which is 2 meters, this 1/6 meter piece represents. This final calculation requires us to relate the length of the small piece (1/6 meter) to the original length (2 meters). The concept of fractions is inherently about expressing a part in relation to a whole. In this case, the 'part' is the 1/6 meter piece, and the 'whole' is the original 2-meter length. To express this relationship as a fraction, we need to divide the length of the small piece by the total length. This division will yield the fraction that represents the proportion of the small piece to the original length. Dividing 1/6 by 2 can be a bit tricky if not approached systematically.

Remember, dividing by a number is the same as multiplying by its reciprocal. The reciprocal of 2 is 1/2. Therefore, dividing 1/6 by 2 is equivalent to multiplying 1/6 by 1/2. This transformation simplifies the calculation, making it easier to arrive at the correct answer. Multiplying fractions involves multiplying the numerators together and the denominators together. So, (1/6) * (1/2) = (11) / (62) = 1/12. This result, 1/12, is the fraction we've been seeking. It represents the proportion of the smallest piece to the original 2-meter length. In other words, each of the four smallest pieces, resulting from the second division, constitutes 1/12 of the original 2-meter piece. This final answer encapsulates the entire problem-solving process, demonstrating our understanding of fractions and their application in real-world scenarios.

The journey through this problem has not only provided us with a numerical answer but has also reinforced the fundamental principles of fractions. We've seen how fractions represent parts of a whole, how they can be divided, and how they relate to each other in complex scenarios. This understanding is invaluable, as fractions are ubiquitous in mathematics and in everyday life. From measuring ingredients in a recipe to calculating proportions in financial analysis, fractions are an essential tool for quantitative reasoning. Thus, by successfully navigating this problem, we've not only solved a specific mathematical puzzle but have also strengthened our grasp of a fundamental mathematical concept. The answer is 1/12.

Therefore, one of the pieces will be 1/12 of the total length.

Options A, B, and C are not fractions and have dimensions attached to them.