Dividing A Rod Calculating The Number Of Pieces

In this article, we will explore a practical problem involving the division of a rod into equal pieces. Specifically, we'll tackle the question: If a rod of length 208 4/5 is cut into equal pieces of length 23 1/5, then how many total rods will be obtained? This problem involves understanding fractions, mixed numbers, and the concept of division. By working through the steps, we will arrive at the correct answer and reinforce our understanding of these mathematical principles. This article aims to provide a comprehensive explanation, making it easy for anyone to follow along and grasp the solution.

To effectively solve this problem, we must first thoroughly understand the given information. We have a rod with a total length of 208 4/5 units, and we want to cut it into smaller pieces, each measuring 23 1/5 units long. The core question we need to answer is: how many of these smaller pieces can we obtain from the original rod? This is a division problem at its heart, where we are dividing the total length of the rod by the length of each piece. Before we dive into the calculations, it's crucial to recognize that both lengths are given as mixed numbers. Mixed numbers combine a whole number and a fraction, like 208 4/5, which means 208 plus 4/5. Similarly, 23 1/5 means 23 plus 1/5. To perform the division, we'll need to convert these mixed numbers into improper fractions, which are fractions where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion will simplify the division process and allow us to find the accurate number of pieces. Understanding the problem thoroughly is the foundation for a successful solution, and taking the time to break down the information ensures we approach the problem with clarity and confidence.

Before we can divide the lengths, we need to convert the mixed numbers into improper fractions. This involves a straightforward process that will make the division much easier to handle. Let's start with the total length of the rod, which is 208 4/5. To convert this into an improper fraction, we multiply the whole number (208) by the denominator of the fraction (5), and then add the numerator (4). This gives us the new numerator, and we keep the same denominator. So, the calculation is (208 * 5) + 4 = 1040 + 4 = 1044. Therefore, 208 4/5 becomes the improper fraction 1044/5. Next, we need to convert the length of each piece, which is 23 1/5, into an improper fraction. We follow the same process: multiply the whole number (23) by the denominator (5) and add the numerator (1). This gives us (23 * 5) + 1 = 115 + 1 = 116. So, 23 1/5 becomes the improper fraction 116/5. Now that we have both lengths expressed as improper fractions, 1044/5 and 116/5, we are ready to perform the division. Converting mixed numbers to improper fractions is a fundamental skill in fraction arithmetic, and it's essential for accurately solving problems like this one. By mastering this conversion, we set ourselves up for success in the next step of dividing the total length by the length of each piece.

Now that we have converted the mixed numbers into improper fractions, we can proceed with dividing the total length of the rod by the length of each piece. The total length of the rod is 1044/5, and the length of each piece is 116/5. To find out how many pieces we can obtain, we need to divide 1044/5 by 116/5. Dividing fractions involves a simple yet crucial step: we multiply by the reciprocal of the divisor. In other words, we flip the second fraction (116/5) and change the division to multiplication. The reciprocal of 116/5 is 5/116. So, the division problem becomes a multiplication problem: (1044/5) ÷ (116/5) = (1044/5) * (5/116). Now we can multiply the numerators together and the denominators together: (1044 * 5) / (5 * 116). This gives us 5220 / 580. Before we perform the final division, we can simplify the fraction by canceling out common factors. Notice that both the numerator and the denominator have a factor of 5. So, we can divide both by 5: 5220 ÷ 5 = 1044 and 580 ÷ 5 = 116. This simplifies our fraction to 1044/116. Now we need to divide 1044 by 116 to find the number of pieces. Dividing fractions by multiplying by the reciprocal is a fundamental technique, and simplifying fractions before dividing makes the calculation easier and reduces the risk of errors. This step-by-step approach ensures we arrive at the correct result.

Having simplified our fraction to 1044/116, we now need to perform the division to find out how many pieces of rod we can obtain. We divide 1044 by 116. This can be done through long division or by recognizing that 1044 is a multiple of 116. If we divide 1044 by 116, we get 9. This means that 1044/116 equals 9. Therefore, if a rod of length 208 4/5 is cut into equal pieces of length 23 1/5, we will obtain 9 pieces. This completes our calculation and answers the original question. The result indicates that the long rod can be divided into exactly 9 smaller rods of the specified length. Performing the division accurately is the final step in solving the problem, and it's crucial to ensure that the result makes sense in the context of the question. In this case, obtaining a whole number (9) as the answer confirms that the division was exact and that we have correctly determined the number of pieces. This methodical approach to solving the problem, from converting mixed numbers to dividing fractions, highlights the importance of each step in arriving at the correct solution.

In conclusion, we successfully determined the number of pieces that can be obtained by cutting a rod of length 208 4/5 into equal pieces of length 23 1/5. By converting the mixed numbers into improper fractions, dividing the fractions, and simplifying the result, we found that 9 pieces can be obtained. This problem demonstrates the practical application of fractions and division in real-world scenarios. Understanding how to work with mixed numbers and fractions is a crucial skill in mathematics, and this exercise reinforces that understanding. The step-by-step approach we followed ensures accuracy and clarity in the solution. From converting mixed numbers to improper fractions, to dividing fractions by multiplying by the reciprocal, each step played a vital role in arriving at the correct answer. This methodical approach can be applied to similar problems, making it easier to solve complex mathematical questions. By mastering these skills, we can confidently tackle a wide range of problems involving fractions and division. The solution not only answers the specific question but also enhances our overall mathematical proficiency.