Shelly Vs Lynne Who Ran Farther And By How Much
In this article, we'll delve into a common type of mathematical problem: comparing distances run by two individuals. Specifically, we'll analyze the distances run by Shelly and Lynne, who ran 2 1/4 miles and 3 1/8 miles, respectively. Our goal is to determine who ran farther and by how much. This kind of problem not only tests our understanding of fractions and mixed numbers but also our ability to apply these concepts in real-world scenarios. Understanding these types of problems is crucial for developing problem-solving skills that extend beyond the classroom. So, let’s embark on this mathematical journey to unravel the solution step by step, ensuring a clear understanding of each process involved. By the end of this article, you'll be equipped with the knowledge to tackle similar distance comparison problems with confidence.
Before diving into calculations, the first step in solving any mathematical problem is to clearly understand what is being asked. In this scenario, we need to compare the distances Shelly and Lynne ran. Shelly ran 2 1/4 miles, and Lynne ran 3 1/8 miles. The problem has two parts: first, we must determine who ran the greater distance; second, we need to calculate the difference in miles between their distances. To accurately compare these distances, we need to work with fractions effectively. Recognizing that we're dealing with mixed numbers (a whole number and a fraction combined) is crucial, as we'll need to convert them into a common format to perform comparisons and subtractions. The ability to break down a problem into smaller, manageable steps, like identifying the need for comparison and subtraction, is a key skill in mathematics. This foundational understanding sets the stage for the subsequent steps in solving the problem. This initial analysis ensures we're on the right track and prevents errors arising from misinterpreting the question. Understanding the core of the problem is fundamental to navigating the solution process efficiently and accurately. By identifying the key elements – the distances run by Shelly and Lynne, and the need for comparison and difference calculation – we establish a solid foundation for solving the problem.
To effectively compare and subtract the distances, the next critical step involves converting the mixed numbers into improper fractions. A mixed number, as we have with Shelly's and Lynne's distances (2 1/4 and 3 1/8 miles), combines a whole number and a fraction. Improper fractions, on the other hand, have a numerator (the top number) that is greater than or equal to the denominator (the bottom number). This conversion is necessary because it simplifies the process of performing arithmetic operations like subtraction. For Shelly's distance, 2 1/4 miles, we multiply the whole number (2) by the denominator (4) and add the numerator (1), which gives us 9. We then place this result over the original denominator (4), resulting in the improper fraction 9/4. Similarly, for Lynne's distance, 3 1/8 miles, we multiply 3 by 8 and add 1, yielding 25. This is placed over the denominator 8, giving us 25/8. These improper fractions, 9/4 and 25/8, now represent Shelly's and Lynne's distances, respectively, in a format that is easier to work with for comparison and subtraction. Understanding the mechanics of this conversion is vital, as it transforms the mixed numbers into a form suitable for direct mathematical operations. This step is not just a mathematical procedure; it's a bridge that allows us to apply the rules of fraction arithmetic effectively.
Now that we have the distances in improper fraction form (9/4 miles for Shelly and 25/8 miles for Lynne), the next crucial step is to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions we are working with. In this case, the denominators are 4 and 8. Finding a common denominator is essential because we cannot directly compare or subtract fractions unless they have the same denominator. This is because the denominator represents the number of equal parts a whole is divided into, and to compare fractions, we need these parts to be of the same size. The least common multiple (LCM) of 4 and 8 is 8, which makes 8 the least common denominator (LCD). To convert 9/4 to an equivalent fraction with a denominator of 8, we multiply both the numerator and the denominator by 2, resulting in 18/8. The fraction 25/8 already has the desired denominator, so it remains unchanged. Now, with both distances expressed as fractions with the same denominator (18/8 miles and 25/8 miles), we can easily compare and subtract them. This step of finding a common denominator is a fundamental concept in fraction arithmetic, enabling us to perform operations across fractions with different denominators. It’s a key to unlocking the ability to accurately compare and manipulate fractions in a variety of mathematical contexts.
With the distances now expressed as fractions with a common denominator (18/8 miles for Shelly and 25/8 miles for Lynne), comparing the distances becomes straightforward. Since the denominators are the same, we can directly compare the numerators. The fraction with the larger numerator represents the greater distance. In this case, 25/8 has a larger numerator than 18/8, indicating that Lynne ran farther than Shelly. This step highlights the importance of having a common denominator, as it allows for a direct comparison of the fractional parts. Without a common denominator, it would be difficult to accurately determine which fraction represents a larger quantity. The comparison is now reduced to a simple numerical assessment: 25 is greater than 18, therefore 25/8 is greater than 18/8. This direct comparison is a powerful demonstration of how mathematical manipulations, such as finding a common denominator, can simplify complex problems. It transforms the problem from a comparison of unlike fractions to a simple numerical comparison. The ability to make such comparisons is not just a mathematical skill; it’s a valuable tool in everyday life, where we often need to compare quantities, sizes, or values.
Having determined that Lynne ran farther, the next step is to calculate the difference in distances. This will tell us by how much Lynne's distance exceeded Shelly's. With the distances expressed as fractions with a common denominator (25/8 miles for Lynne and 18/8 miles for Shelly), subtraction is a simple matter of subtracting the numerators while keeping the denominator the same. So, we subtract 18 from 25, which gives us 7. This result is then placed over the common denominator of 8, giving us a difference of 7/8 miles. This fraction represents the additional distance Lynne ran compared to Shelly. The process of subtracting fractions with a common denominator is a fundamental arithmetic operation, and it's the key to answering the second part of our problem: by how much did Lynne outrun Shelly? The result, 7/8 miles, provides a clear and concise answer to this question. This calculation not only gives us the numerical difference but also reinforces the practical application of fraction subtraction. It demonstrates how mathematical operations can be used to quantify differences and make comparisons in real-world scenarios. Understanding and performing this subtraction is a crucial aspect of mastering fraction arithmetic and its applications.
In our case, the difference in distances is 7/8 miles. Since 7 is less than 8, this is already a proper fraction, meaning the numerator is less than the denominator. Therefore, no further conversion is needed. However, it's important to understand when and how to convert an improper fraction (where the numerator is greater than or equal to the denominator) back into a mixed number. This conversion is often necessary to present the answer in its simplest and most understandable form. For instance, if our difference had been 11/8 miles, we would divide 11 by 8 to get 1 with a remainder of 3. This would be expressed as the mixed number 1 3/8 miles. Understanding how to convert between improper fractions and mixed numbers is a crucial skill in fraction arithmetic, as it allows us to express quantities in the most appropriate form depending on the context. In this particular problem, the difference of 7/8 miles is already in its simplest form, but the knowledge of conversion is invaluable for other scenarios where the result may be an improper fraction. This skill ensures that we can always present our answers in the clearest and most practical way.
In conclusion, by carefully comparing the distances run by Shelly and Lynne, we've determined that Lynne ran farther than Shelly by 7/8 miles. This problem demonstrates the importance of several key mathematical concepts: converting mixed numbers to improper fractions, finding a common denominator, comparing fractions, and subtracting fractions. Each step played a crucial role in arriving at the correct solution. Understanding these concepts is not only essential for solving mathematical problems but also for applying mathematical thinking to real-world situations. The ability to work with fractions is a fundamental skill, and this problem provides a practical example of its application. By breaking down the problem into smaller, manageable steps, we were able to navigate the solution process effectively. This approach, of systematically analyzing each component of a problem, is a valuable strategy for tackling more complex mathematical challenges. The final answer, Lynne ran 7/8 miles farther than Shelly, is a testament to the power of these mathematical tools and techniques. This exercise not only reinforces the mechanics of fraction arithmetic but also cultivates a deeper appreciation for the role of mathematics in everyday problem-solving. Furthermore, this kind of problem-solving skill is highly beneficial in various fields that require analytical thinking and precise calculations.
The final answer is (C) Lynne by 7/8 miles.
