Calculating Remaining Distance In A Race A Runner's Challenge
Introduction
In the realm of mathematics, we often encounter problems that require us to apply basic arithmetic principles to solve real-world scenarios. One such scenario involves calculating distances, particularly in the context of running or other forms of endurance exercise. In this article, we will delve into a problem where a runner is participating in a 5-mile race and has already covered a portion of the distance. Our goal is to determine the remaining miles the runner needs to complete to finish the race. This problem provides an excellent opportunity to reinforce our understanding of fractions, mixed numbers, and subtraction, all of which are fundamental concepts in mathematics.
This article aims to break down the problem step-by-step, providing a clear and concise explanation of the solution process. We will begin by restating the problem and identifying the key information needed to solve it. Then, we will convert the mixed number representing the distance already run into an improper fraction, making it easier to perform the subtraction. Next, we will subtract the distance run from the total distance of the race to find the remaining distance. Finally, we will express the answer in both improper fraction and mixed number forms, ensuring a comprehensive understanding of the solution. By working through this problem, we will not only enhance our mathematical skills but also develop our problem-solving abilities, which are crucial in various aspects of life. So, let's embark on this mathematical journey and uncover the remaining miles the runner needs to conquer.
Understanding the Problem
To effectively solve this problem, a clear understanding of the given information is crucial. Our central question revolves around calculating the remaining distance a runner needs to cover in a 5-mile race, having already completed a portion of it. The key pieces of information we have at our disposal are: the total distance of the race, which is 5 miles, and the distance the runner has already run, which is 2 7/10 miles. These two figures form the foundation upon which our solution will be built.
The challenge here lies in determining the difference between these two distances. In mathematical terms, this translates to a subtraction problem. We need to subtract the distance already covered (2 7/10 miles) from the total race distance (5 miles). However, before we can perform this subtraction, it's important to recognize that we are dealing with a mixed number (2 7/10), which combines a whole number and a fraction. To simplify the subtraction process, we will need to convert this mixed number into an improper fraction. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This conversion is a crucial step in making the subtraction more manageable. By carefully extracting and understanding the given information, we set the stage for a successful solution to this problem.
Converting Mixed Numbers to Improper Fractions
Before we can subtract the distances, we need to convert the mixed number 2 7/10 into an improper fraction. This conversion is essential because it allows us to perform the subtraction more easily. A mixed number, as the name suggests, combines a whole number (in this case, 2) and a fraction (7/10). To convert it into an improper fraction, we follow a specific procedure that involves both multiplication and addition.
The first step in this conversion process is to multiply the whole number (2) by the denominator of the fraction (10). This gives us 2 * 10 = 20. Next, we add the result of this multiplication (20) to the numerator of the fraction (7). This yields 20 + 7 = 27. This sum, 27, will become the new numerator of our improper fraction. The denominator of the improper fraction remains the same as the denominator of the original fraction, which is 10. Therefore, the improper fraction equivalent of the mixed number 2 7/10 is 27/10. Now that we have successfully converted the mixed number into an improper fraction, we are one step closer to solving the problem. This conversion allows us to express both distances (the total distance and the distance already run) in a consistent fractional form, which is necessary for performing the subtraction. With this conversion complete, we are now well-prepared to calculate the remaining distance the runner needs to cover.
Subtracting Fractions
Now that we've converted the mixed number to an improper fraction, the next step is to subtract the distance the runner has already run (27/10 miles) from the total race distance (5 miles). However, before we can perform this subtraction, we need to express the whole number 5 as a fraction with the same denominator as 27/10, which is 10. This is a crucial step because we can only subtract fractions that have the same denominator. To convert 5 into a fraction with a denominator of 10, we simply multiply it by 10/10, which is equivalent to 1. This gives us 5 * (10/10) = 50/10. Now we have both distances expressed as fractions with the same denominator: 50/10 miles (total distance) and 27/10 miles (distance already run).
With both distances expressed as fractions with a common denominator, we can now proceed with the subtraction. To subtract fractions with the same denominator, we subtract the numerators and keep the denominator the same. In this case, we subtract 27 from 50, which gives us 50 - 27 = 23. The denominator remains 10. Therefore, the result of the subtraction is 23/10 miles. This fraction represents the remaining distance the runner needs to cover. However, to provide a more intuitive understanding of the distance, we can convert this improper fraction back into a mixed number. This conversion will give us a whole number and a fraction, making it easier to visualize the remaining distance.
Converting Improper Fractions to Mixed Numbers
Having obtained the remaining distance as an improper fraction (23/10 miles), it's beneficial to convert it back into a mixed number. This conversion provides a more intuitive understanding of the distance, as it separates the whole number portion from the fractional part. An improper fraction, as we recall, is one where the numerator (23) is greater than the denominator (10).
The process of converting an improper fraction to a mixed number involves division. We divide the numerator (23) by the denominator (10). In this division, 10 goes into 23 two times (2 * 10 = 20), with a remainder of 3. The quotient, which is 2, becomes the whole number part of the mixed number. The remainder, which is 3, becomes the numerator of the fractional part. The denominator of the fractional part remains the same as the original denominator, which is 10. Therefore, the mixed number equivalent of the improper fraction 23/10 is 2 3/10. This means that the runner has 2 whole miles and 3/10 of a mile left to run. By converting the improper fraction back to a mixed number, we have provided a more easily understandable representation of the remaining distance. This conversion completes the process of solving the problem, giving us a clear and concise answer.
Solution
After carefully working through the problem, we have arrived at the solution. The runner has 2 3/10 miles left to run. This result provides a clear and concise answer to the question posed in the problem. We began by understanding the problem, identifying the key information, and recognizing that we needed to subtract the distance already run from the total distance. We then converted the mixed number representing the distance already run (2 7/10 miles) into an improper fraction (27/10 miles) to facilitate the subtraction process.
Next, we expressed the total race distance (5 miles) as a fraction with the same denominator as the distance already run, resulting in 50/10 miles. We then subtracted the two fractions, (50/10) - (27/10), which gave us 23/10 miles. This improper fraction represented the remaining distance. To provide a more intuitive understanding of the distance, we converted the improper fraction 23/10 into a mixed number. This conversion involved dividing 23 by 10, which resulted in a quotient of 2 and a remainder of 3. Thus, the mixed number equivalent of 23/10 is 2 3/10. Therefore, the runner has 2 3/10 miles remaining in the race. This final answer encapsulates the entire solution process and provides a clear and easily understandable answer to the problem.
Conclusion
In conclusion, we have successfully determined the remaining distance a runner needs to cover in a 5-mile race, given that she has already run 2 7/10 miles. The solution, which is 2 3/10 miles, was obtained through a step-by-step process that involved converting mixed numbers to improper fractions, subtracting fractions with common denominators, and converting improper fractions back to mixed numbers. This problem serves as a valuable exercise in applying fundamental arithmetic principles to solve real-world scenarios. It reinforces our understanding of fractions, mixed numbers, and subtraction, all of which are essential concepts in mathematics.
Moreover, this problem highlights the importance of breaking down complex problems into smaller, more manageable steps. By systematically addressing each step, we were able to arrive at the solution with confidence. This problem-solving approach is not only applicable to mathematics but also to various other areas of life. The ability to analyze a problem, identify the key information, and develop a logical solution strategy is a valuable skill that can be applied in numerous contexts. Therefore, the lessons learned from solving this problem extend beyond the realm of mathematics and contribute to the development of critical thinking and problem-solving abilities. As we continue our mathematical journey, we can apply these skills to tackle more challenging problems and deepen our understanding of the world around us.
