Shelly And Lynne's Run Comparing Distances
In this mathematical problem, we are presented with a scenario involving two individuals, Shelly and Lynne, who have each run a certain distance. Shelly's run is measured at 2 1/4 miles, while Lynne's run is recorded as 3 1/8 miles. The core question we need to address is twofold: First, we need to determine who ran the farther distance. Second, we need to quantify the difference in the distances they ran. This involves comparing the two mixed numbers representing their respective distances and then calculating the difference between them. This problem falls under the category of basic arithmetic operations with fractions, requiring us to convert mixed numbers into improper fractions, find a common denominator, and then perform subtraction. Understanding these steps is crucial not only for solving this particular problem but also for building a solid foundation in mathematical problem-solving in general. Moreover, this type of question often appears in standardized tests and real-life scenarios, emphasizing the importance of mastering these skills. The ability to accurately compare and subtract fractions is a fundamental skill in mathematics with applications extending beyond the classroom. It is essential for various calculations in fields like engineering, finance, and even everyday activities like cooking and home improvement. Therefore, a thorough understanding of these concepts is vital for both academic success and practical application in real-world contexts. By carefully analyzing the distances run by Shelly and Lynne and applying the appropriate mathematical techniques, we can confidently determine who ran farther and by exactly how much. This will not only provide the answer to the specific question at hand but also reinforce our understanding of fraction operations and problem-solving strategies.
Converting Mixed Numbers to Improper Fractions
The initial step in solving this problem is to convert the mixed numbers representing the distances run by Shelly and Lynne into improper fractions. This conversion is essential because it allows us to perform arithmetic operations, such as subtraction, more easily. A mixed number consists of a whole number and a fraction, making it slightly cumbersome to work with directly in calculations. An improper fraction, on the other hand, is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This format simplifies the process of adding, subtracting, multiplying, and dividing fractions. To convert a mixed number to an improper fraction, we follow a specific procedure. First, we multiply the whole number part of the mixed number by the denominator of the fractional part. Then, we add the numerator of the fractional part to the result. This sum becomes the new numerator of the improper fraction. The denominator of the improper fraction remains the same as the denominator of the original fractional part. Let's apply this process to the distances run by Shelly and Lynne. Shelly ran 2 1/4 miles. To convert this to an improper fraction, we multiply the whole number 2 by the denominator 4, which gives us 8. Then, we add the numerator 1 to this result, yielding 9. So, the improper fraction representing Shelly's distance is 9/4 miles. Lynne ran 3 1/8 miles. Following the same procedure, we multiply the whole number 3 by the denominator 8, resulting in 24. Adding the numerator 1 to this gives us 25. Therefore, the improper fraction representing Lynne's distance is 25/8 miles. Now that we have converted both distances into improper fractions (9/4 miles and 25/8 miles), we are better equipped to compare them and calculate the difference in their distances. This conversion is a crucial step in simplifying the problem and paving the way for accurate calculations.
Finding a Common Denominator
Before we can compare the distances run by Shelly and Lynne, which are now represented as improper fractions (9/4 miles and 25/8 miles, respectively), we need to find a common denominator. A common denominator is a shared multiple of the denominators of the fractions being compared or subtracted. It's essential because fractions can only be directly compared or subtracted if they have the same denominator. Think of it like comparing apples and oranges – you need to convert them to a common unit, like "pieces of fruit," before you can meaningfully compare them. In our case, the denominators are 4 and 8. To find a common denominator, we need to identify the least common multiple (LCM) of 4 and 8. The multiples of 4 are 4, 8, 12, 16, and so on. The multiples of 8 are 8, 16, 24, 32, and so on. The smallest number that appears in both lists is 8, making it the least common multiple of 4 and 8. Therefore, 8 will be our common denominator. Now, we need to convert both fractions to equivalent fractions with a denominator of 8. The fraction representing Lynne's distance, 25/8 miles, already has a denominator of 8, so we don't need to change it. For Shelly's distance, 9/4 miles, we need to find an equivalent fraction with a denominator of 8. To do this, we ask ourselves: what do we need to multiply 4 by to get 8? The answer is 2. So, we multiply both the numerator and the denominator of 9/4 by 2. This gives us (9 * 2) / (4 * 2) = 18/8 miles. Now, both distances are expressed as fractions with the same denominator: Shelly ran 18/8 miles, and Lynne ran 25/8 miles. This common denominator allows us to directly compare the numerators and determine who ran farther. Furthermore, it sets the stage for accurately calculating the difference in their distances. Finding a common denominator is a critical step in working with fractions, ensuring that our comparisons and calculations are meaningful and accurate.
Comparing the Distances
Now that we have expressed the distances run by Shelly and Lynne as fractions with a common denominator, we can easily compare the distances. Shelly ran 18/8 miles, and Lynne ran 25/8 miles. With a common denominator, comparing fractions becomes as simple as comparing their numerators. The fraction with the larger numerator represents the greater distance. In this case, 25 is greater than 18, which means that 25/8 miles is greater than 18/8 miles. Therefore, Lynne ran farther than Shelly. This direct comparison is only possible because we took the crucial step of finding a common denominator. Without it, we would be trying to compare fractions with different "units," making the comparison difficult and prone to error. The common denominator effectively converts the fractions to the same unit, allowing for a straightforward comparison of the numerators. This principle is fundamental to working with fractions and is essential for various mathematical operations and real-world applications. Having established that Lynne ran farther, we now need to determine exactly how much farther she ran. This requires us to calculate the difference between the two distances, which we can do by subtracting the smaller fraction from the larger fraction. The comparison step has not only answered the first part of our problem – who ran farther – but also set us up for the next step: quantifying the difference. This systematic approach to problem-solving, where we break down a complex question into smaller, manageable steps, is a valuable strategy in mathematics and beyond. By comparing the distances in this clear and concise manner, we have gained a solid understanding of the relative distances run by Shelly and Lynne, paving the way for the final calculation.
Calculating the Difference
Having determined that Lynne ran farther than Shelly, the next step is to calculate the difference in the distances they ran. This will tell us by how much Lynne's run exceeded Shelly's. To find the difference, we subtract the distance Shelly ran from the distance Lynne ran. Both distances are now expressed as fractions with a common denominator of 8: Lynne ran 25/8 miles, and Shelly ran 18/8 miles. Subtracting fractions with a common denominator is straightforward: we simply subtract the numerators and keep the denominator the same. So, the difference is (25/8) - (18/8) = (25 - 18) / 8 = 7/8 miles. This means Lynne ran 7/8 of a mile farther than Shelly. The result, 7/8 miles, is already in its simplest form because 7 and 8 have no common factors other than 1. This fraction represents the precise difference in the distances run by Lynne and Shelly. It's a clear and concise answer that directly addresses the second part of our problem. The process of subtracting fractions with a common denominator is a fundamental arithmetic operation. It's a skill that builds upon our understanding of fractions and their properties. By accurately calculating the difference in distances, we have completed the problem-solving process, arriving at a definitive answer. This calculation not only provides the numerical difference but also reinforces our ability to perform fraction subtraction with confidence. The final answer, 7/8 miles, provides a clear and quantifiable measure of the difference in the distances run by Lynne and Shelly, completing our analysis of the problem.
Final Answer
After carefully analyzing the distances run by Shelly and Lynne, we have arrived at the final answer. Lynne ran farther than Shelly, and the difference in their distances is 7/8 of a mile. This conclusion is based on a step-by-step process that involved converting mixed numbers to improper fractions, finding a common denominator, comparing the distances, and calculating the difference. Each step was crucial in ensuring the accuracy and clarity of our result. We began by converting Shelly's distance of 2 1/4 miles to the improper fraction 9/4 miles and Lynne's distance of 3 1/8 miles to 25/8 miles. This conversion allowed us to work with the distances more easily. Next, we found a common denominator for the fractions, which was 8. This enabled us to directly compare the distances by converting 9/4 to 18/8. Comparing the fractions 18/8 and 25/8, we determined that Lynne ran farther because 25 is greater than 18. Finally, we calculated the difference in their distances by subtracting 18/8 from 25/8, which resulted in 7/8 miles. Therefore, Lynne ran 7/8 of a mile farther than Shelly. This final answer is a clear and concise solution to the problem. It not only identifies who ran farther but also quantifies the difference in their distances. The process we followed demonstrates a systematic approach to problem-solving in mathematics, emphasizing the importance of understanding each step and its contribution to the overall solution. By accurately performing these calculations, we have not only answered the question but also reinforced our understanding of fraction operations and problem-solving strategies.
Therefore, the correct answer is C) Lynne by 7/8 miles.
