A Cord Of Length 143/2 Units Is Cut Into 26 Pieces Of Equal Length. How Long Is Each Piece?
Introduction
In this article, we will delve into a mathematical problem involving the division of a cord into equal pieces. Specifically, we are given a cord with a length of 143/2 units and tasked with cutting it into 26 pieces of equal length. Our goal is to determine the length of each of these pieces. This problem involves basic arithmetic operations, particularly division, and provides a practical application of fractions in real-world scenarios. Understanding how to solve such problems is crucial for developing a strong foundation in mathematics and enhancing problem-solving skills. So, let's embark on this mathematical journey and unravel the solution step by step.
Understanding the Problem
To effectively solve this problem, it's crucial to first grasp the core concept it presents. We have a cord, the total length of which is 143/2 units. This cord needs to be divided into 26 equal parts. The question we aim to answer is: what will be the length of each of these 26 parts? This is essentially a division problem, where we are dividing the total length of the cord by the number of pieces we want to create. The total length, 143/2, is a fraction, and we will be dividing it by a whole number, 26. Understanding this basic framework is the first step towards finding the solution. Let's break down the components involved: the total length, the number of pieces, and the operation required to find the length of each piece. By clearly defining these elements, we set the stage for a systematic approach to solving the problem. Now, let's move on to the next step: converting the total length into a more workable form.
Converting the Total Length
Before we can proceed with the division, it's often helpful to convert the total length, which is given as an improper fraction (143/2), into a mixed number. This conversion can make the value more intuitive and easier to work with, especially for those who are more comfortable with mixed numbers. To convert 143/2 into a mixed number, we divide the numerator (143) by the denominator (2). The quotient will be the whole number part of the mixed number, and the remainder will be the numerator of the fractional part, with the denominator remaining the same. When we divide 143 by 2, we get a quotient of 71 and a remainder of 1. This means that 143/2 is equivalent to 71 and 1/2. So, the total length of the cord can be expressed as 71 and a half units. This conversion doesn't change the value of the length; it simply represents it in a different form. Now that we have the total length expressed as a mixed number, we can move on to the next step: setting up the division problem. This involves translating the word problem into a mathematical expression that we can solve. Let's see how we can do that.
Setting Up the Division
Now that we understand the problem and have the total length in a usable form, we can set up the division problem. The core of the problem is to divide the total length of the cord (143/2 or 71 1/2) by the number of pieces (26). This can be expressed mathematically as: (143/2) ÷ 26. This expression tells us exactly what operation we need to perform to find the length of each piece. However, dividing by a whole number can sometimes be tricky when dealing with fractions. To simplify this process, we can rewrite the division as multiplication by the reciprocal of the divisor. In other words, dividing by 26 is the same as multiplying by 1/26. This transformation allows us to work with multiplication, which is often easier to handle than division, especially with fractions. So, our division problem (143/2) ÷ 26 becomes a multiplication problem: (143/2) × (1/26). This new expression sets us up for the next step: performing the multiplication to find the length of each piece. Let's see how we can carry out this multiplication and arrive at the solution.
Performing the Division
With our problem set up as a multiplication ((143/2) × (1/26)), we can now proceed with the calculation. To multiply fractions, we multiply the numerators together and the denominators together. So, (143/2) × (1/26) becomes (143 × 1) / (2 × 26), which simplifies to 143/52. Now we have a fraction that represents the length of each piece. However, this fraction is in its improper form, and it's not yet in its simplest form. To make it more understandable and easier to interpret, we need to simplify it. The first step in simplification is to convert the improper fraction (143/52) into a mixed number. To do this, we divide the numerator (143) by the denominator (52). The quotient will be the whole number part, and the remainder will be the numerator of the fractional part, with the denominator remaining the same. When we divide 143 by 52, we get a quotient of 2 and a remainder of 39. This means that 143/52 is equivalent to 2 and 39/52. We're not quite done yet, as the fractional part (39/52) can be further simplified. To simplify a fraction, we look for the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. In this case, the GCD of 39 and 52 is 13. Dividing both 39 and 52 by 13 gives us 3/4. So, the simplified form of 39/52 is 3/4. Putting it all together, the length of each piece is 2 and 3/4 units. This is the final answer to our problem. Let's summarize our steps and the solution we've found.
Solution
After working through the problem step-by-step, we have arrived at the solution. We started with a cord of length 143/2 units, which we converted to 71 1/2 units for easier understanding. We then set up the division problem to find the length of each piece when the cord is cut into 26 equal parts. This division was expressed as (143/2) ÷ 26, which we transformed into a multiplication problem: (143/2) × (1/26). Performing the multiplication, we got 143/52, which we then simplified. The simplification process involved converting the improper fraction to a mixed number, resulting in 2 and 39/52. Finally, we simplified the fractional part to its lowest terms, giving us 2 and 3/4. Therefore, the length of each piece of the cord is 2 and 3/4 units. This means that each of the 26 pieces cut from the original cord will be 2.75 units long. This solution demonstrates the practical application of fractions and division in everyday situations. By breaking down the problem into manageable steps and applying the appropriate mathematical operations, we were able to find the answer. Now, let's summarize the entire process in a concise conclusion.
Conclusion
In this article, we tackled the problem of dividing a cord of length 143/2 units into 26 equal pieces. We successfully determined that the length of each piece is 2 and 3/4 units, or 2.75 units. This involved several key steps: understanding the problem, converting the total length into a more workable form, setting up the division as a multiplication problem, performing the multiplication, and simplifying the resulting fraction. This exercise highlights the importance of understanding fractions and their operations, as well as the ability to apply these concepts to real-world scenarios. By breaking down complex problems into smaller, manageable steps, we can effectively find solutions and gain a deeper understanding of the underlying mathematical principles. The problem-solving skills demonstrated here are not only valuable in mathematics but also in various aspects of life. We hope this detailed explanation has provided clarity and enhanced your understanding of division and fractions. Remember, practice is key to mastering these concepts, so keep exploring and solving similar problems to further strengthen your skills.