Calculating Total Distance Walked A Step-by-Step Guide
In this comprehensive guide, we will dive into solving a common mathematical problem calculating the total distance walked over multiple days. This type of problem often appears in everyday situations, making it essential to grasp the underlying concepts. Whether you're a student tackling homework or simply someone looking to brush up on your math skills, this article will provide a clear and concise explanation. We will focus on a specific scenario involving walking distances on Thursday and Friday, but the principles discussed can be applied to various similar problems. So, let's embark on this mathematical journey together and unlock the secrets of distance calculation!
Distance calculation is a fundamental skill in mathematics with practical applications in real life. Before diving into the solution, let's break down the core question: A person walked 2/3 of a mile on Thursday and 5/7 of a mile on Friday. What is the total distance, in miles, the person walked? This problem requires us to add two fractions representing distances covered on different days. Understanding the question is the first step towards finding the correct answer. We need to determine the combined distance by adding the fractions 2/3 and 5/7. This involves finding a common denominator and then summing the numerators. Let's explore the step-by-step process to solve this problem efficiently.
Fraction Review Essential Concepts
To tackle this distance calculation problem, we need to revisit some fundamental concepts about fractions. A fraction represents a part of a whole, consisting of a numerator (the top number) and a denominator (the bottom number). The denominator indicates the total number of equal parts the whole is divided into, while the numerator shows how many of those parts we are considering. For instance, in the fraction 2/3, the denominator 3 tells us the whole is divided into three equal parts, and the numerator 2 indicates we are considering two of those parts. When adding fractions, it’s crucial to have a common denominator. This means that both fractions must have the same denominator before we can add their numerators. If the denominators are different, we need to find the least common multiple (LCM) to make them the same. Understanding these basics is essential for accurately solving the problem at hand.
Finding the Common Denominator The Key to Adding Fractions
The process of finding a common denominator is crucial when adding fractions with different denominators, as is the case in our distance calculation problem. The common denominator allows us to add the fractions accurately by expressing them in terms of the same whole. To find the common denominator for 2/3 and 5/7, we need to determine the least common multiple (LCM) of the denominators 3 and 7. The LCM is the smallest number that is a multiple of both denominators. In this case, the multiples of 3 are 3, 6, 9, 12, 15, 18, 21, and so on, while the multiples of 7 are 7, 14, 21, 28, and so on. The smallest number that appears in both lists is 21, making it the LCM of 3 and 7. Therefore, the common denominator for our fractions is 21. This step is essential because it allows us to rewrite the fractions with the same base, making addition straightforward.
Converting Fractions to Equivalent Forms Step-by-Step Guide
Once we've identified the common denominator, the next step in our distance calculation is to convert the original fractions into equivalent forms with the new denominator. This involves adjusting both the numerator and the denominator of each fraction while maintaining its value. For the fraction 2/3, we need to find a number that, when multiplied by 3, gives us the common denominator 21. In this case, 3 multiplied by 7 equals 21. So, we multiply both the numerator and the denominator of 2/3 by 7, resulting in (2 * 7) / (3 * 7) = 14/21. Similarly, for the fraction 5/7, we need to find a number that, when multiplied by 7, gives us 21. Here, 7 multiplied by 3 equals 21. Thus, we multiply both the numerator and the denominator of 5/7 by 3, giving us (5 * 3) / (7 * 3) = 15/21. Now that both fractions have the same denominator, we can proceed with the addition.
Adding the Fractions The Final Calculation
With the fractions now having a common denominator, we can proceed with the addition to find the total distance walked in this distance calculation. We have converted 2/3 to 14/21 and 5/7 to 15/21. To add these fractions, we simply add their numerators while keeping the denominator the same. So, we add 14 and 15, which equals 29. The denominator remains 21. Therefore, the sum of the fractions is 29/21. This result represents the total distance walked, but it's an improper fraction (the numerator is greater than the denominator). To make it more understandable, we can convert it into a mixed number.
Converting Improper Fractions to Mixed Numbers Understanding the Result
After adding the fractions, we arrived at the improper fraction 29/21. To better understand the total distance walked, we need to convert this into a mixed number. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator). To convert 29/21 to a mixed number, we divide the numerator (29) by the denominator (21). The quotient represents the whole number part, and the remainder becomes the numerator of the fractional part, with the original denominator staying the same. When we divide 29 by 21, we get a quotient of 1 and a remainder of 8. Therefore, 29/21 is equivalent to the mixed number 1 8/21. This means the person walked 1 and 8/21 miles in total. This format provides a clearer understanding of the distance covered, as it separates the whole miles from the fractional part of a mile.
Simplifying Fractions The Importance of Reduced Form
While we have found the total distance walked as 1 8/21 miles, it's always a good practice to check if the fractional part can be simplified. Simplifying fractions involves reducing them to their lowest terms, where the numerator and the denominator have no common factors other than 1. In the fraction 8/21, we need to determine if there is a common factor that divides both 8 and 21. The factors of 8 are 1, 2, 4, and 8, while the factors of 21 are 1, 3, 7, and 21. The only common factor is 1, which means the fraction 8/21 is already in its simplest form. Therefore, the final answer remains 1 8/21 miles. Simplifying fractions ensures the result is presented in the most concise and understandable manner.
Final Answer and Interpretation Putting It All Together
After meticulously following each step of the distance calculation, we have arrived at the final answer. The person walked a total distance of 1 8/21 miles on Thursday and Friday. This answer is derived from adding the distances walked on each day, 2/3 of a mile and 5/7 of a mile, respectively. We found the common denominator, converted the fractions, added them, and converted the improper fraction to a mixed number. The mixed number 1 8/21 provides a clear interpretation of the distance, indicating that the person walked one full mile and an additional 8/21 of a mile. Understanding the process and arriving at the correct answer demonstrates a strong grasp of fraction arithmetic and problem-solving skills. This final result encapsulates the solution to the initial question, providing a complete and accurate response.
In conclusion, mastering distance calculation and fraction arithmetic is crucial for solving real-world problems. We successfully determined the total distance walked by a person over two days by adding the fractions representing the distances covered each day. The process involved finding a common denominator, converting the fractions, performing the addition, and simplifying the result. This step-by-step approach not only provides the correct answer but also enhances our understanding of mathematical concepts. By breaking down the problem into manageable steps, we can confidently tackle similar challenges in the future. This guide serves as a valuable resource for anyone looking to improve their math skills and apply them in practical scenarios. Remember, consistent practice and a clear understanding of fundamental principles are key to mathematical proficiency.
