Calculating Total Distance Walked A Step-by-Step Guide
In this article, we will delve into a mathematical problem involving the calculation of total distance walked over two days. Specifically, we will explore how to determine the total number of miles a person walked on Thursday and Friday, given that they walked $\frac{2}{3}$ of a mile on Thursday and $\frac{5}{7}$ of a mile on Friday. This problem is a classic example of fraction addition, a fundamental concept in mathematics. Understanding how to solve such problems is crucial for various real-life applications, from measuring distances to calculating quantities. Our goal is to provide a clear, step-by-step solution to this problem, ensuring that readers not only understand the answer but also grasp the underlying mathematical principles. We will also discuss common mistakes and strategies to avoid them, making this a comprehensive guide to solving fraction addition problems. This article aims to enhance your problem-solving skills and build a solid foundation in mathematical concepts. By the end of this article, you will be well-equipped to tackle similar problems with confidence and accuracy. We will also explore the importance of these calculations in everyday life and how they relate to more advanced mathematical concepts. So, let's embark on this mathematical journey and unravel the solution to this intriguing problem!
Understanding the Problem
Before diving into the solution, let's first understand the problem statement. The core of the problem lies in calculating the total distance a person walked over two days. On Thursday, the person walked $\frac{2}{3}$ of a mile, and on Friday, they walked $\frac{5}{7}$ of a mile. The question asks for the total number of miles walked, which implies that we need to add these two fractions together. This seemingly simple problem highlights the importance of understanding fraction addition, a critical skill in mathematics. Fractions represent parts of a whole, and adding them requires a different approach than adding whole numbers. We need to ensure that the fractions have a common denominator before we can add their numerators. The common denominator represents the total number of equal parts into which the whole is divided, making it possible to combine the fractions accurately. Without a common denominator, we would be adding fractions with different-sized "pieces," leading to an incorrect result. Therefore, understanding the concept of a common denominator is paramount to solving this problem. This problem also underscores the relevance of mathematics in everyday life. Whether it's calculating distances, measuring ingredients for a recipe, or managing finances, fractions are ubiquitous. Mastering fraction addition is not just an academic exercise; it's a practical skill that can be applied in various real-world scenarios. In the following sections, we will break down the steps involved in solving this problem, making sure to explain each concept thoroughly. By understanding the problem and the necessary mathematical tools, we can confidently arrive at the correct solution.
Step-by-Step Solution
To find the total distance the person walked, we need to add the distances walked on Thursday and Friday. This involves adding the fractions $\frac2}{3}$ and $\frac{5}{7}$. The first crucial step in adding fractions is to find a common denominator. A common denominator is a number that is a multiple of both denominators. In this case, we need to find a common multiple of 3 and 7. The easiest way to find a common multiple is to multiply the two denominators together3}$ to a fraction with a denominator of 21, we multiply both the numerator and the denominator by 73}$ * $\frac{7}{7}$ = $\frac{14}{21}$. Similarly, to convert $\frac{5}{7}$ to a fraction with a denominator of 21, we multiply both the numerator and the denominator by 37}$ * $\frac{3}{3}$ = $\frac{15}{21}$. Now that we have both fractions with a common denominator, we can add them together. We add the numerators and keep the denominator the same{21}$ + $\frac{15}{21}$ = $\frac{14 + 15}{21}$ = $\frac{29}{21}$. The result is $\frac{29}{21}$, which is an improper fraction (the numerator is greater than the denominator). To make it easier to understand, we can convert this improper fraction to a mixed number. To do this, we divide 29 by 21. 29 divided by 21 is 1 with a remainder of 8. So, $\frac{29}{21}$ is equal to 1 $\frac{8}{21}$. Therefore, the person walked a total of 1 $\frac{8}{21}$ miles. This step-by-step approach ensures accuracy and clarity in solving the problem. By understanding each step, you can apply this method to other fraction addition problems.
Common Mistakes to Avoid
When dealing with fraction addition, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One of the most frequent errors is adding the numerators and denominators directly without finding a common denominator. For example, adding $\frac{2}{3}$ and $\frac{5}{7}$ as $\frac{2+5}{3+7}$ = $\frac{7}{10}$ is incorrect. This method fails to recognize that fractions represent parts of different wholes unless they have the same denominator. Always remember that you must find a common denominator before adding fractions. Another common mistake is incorrectly determining the common denominator. While multiplying the denominators always yields a common denominator, it may not always be the least common denominator (LCD). Using a larger common denominator than necessary can lead to more cumbersome calculations, although it will still result in the correct answer if the fraction is simplified at the end. To find the LCD, you can list the multiples of each denominator and identify the smallest multiple they have in common. A third error occurs when converting fractions to equivalent fractions with the common denominator. It's crucial to multiply both the numerator and the denominator by the same number. For instance, if you want to convert $\frac{2}{3}$ to an equivalent fraction with a denominator of 21, you need to multiply both 2 and 3 by 7. Multiplying only the denominator or only the numerator will result in an incorrect fraction. Furthermore, students sometimes forget to simplify the final fraction. The answer should always be expressed in its simplest form. If the numerator and denominator have a common factor, divide both by that factor to reduce the fraction. In our case, $\frac{29}{21}$ is already in its simplest form, but if we had obtained a fraction like $\frac{14}{21}$, we would need to simplify it to $\frac{2}{3}$. Finally, when converting an improper fraction to a mixed number, ensure you correctly divide the numerator by the denominator and express the remainder as a fraction of the original denominator. Avoiding these common mistakes will significantly improve your accuracy and confidence in solving fraction addition problems. Practice and attention to detail are key to mastering this skill.
Real-World Applications
Understanding fraction addition isn't just an abstract mathematical concept; it has numerous real-world applications that make it a valuable skill in everyday life. From cooking and baking to construction and finance, fractions are ubiquitous, and the ability to add them accurately is essential. In the kitchen, recipes often call for fractional amounts of ingredients. For example, you might need $\frac{1}{2}$ cup of flour and $\frac{1}{4}$ cup of sugar. To determine the total amount of dry ingredients, you need to add these fractions. Similarly, in baking, precise measurements are crucial, and fraction addition is often required to adjust recipes or calculate quantities. In construction and carpentry, measurements frequently involve fractions. If you're building a bookshelf, you might need to add the lengths of several pieces of wood to determine the total length. Accurately adding these fractional measurements is vital for ensuring the structural integrity and aesthetic appeal of the finished product. Financial calculations also often involve fractions. For instance, if you're calculating the total interest earned on an investment over several periods, you might need to add fractional interest rates. Understanding how to add fractions is also crucial for budgeting and managing personal finances. In fields like engineering and architecture, fraction addition is used extensively in design and planning. Architects need to calculate fractional dimensions and proportions to create accurate blueprints, while engineers use fractions to determine stresses and strains on structures. Even in seemingly simple tasks like planning a road trip, fraction addition can be useful. If you're calculating the total distance you'll travel over several days, and each day's journey is expressed as a fraction of the total distance, you'll need to add those fractions to find the overall proportion of the trip you've completed. These examples illustrate the broad applicability of fraction addition in various aspects of life. Mastering this skill not only enhances your mathematical abilities but also equips you with a practical tool for solving real-world problems. The ability to confidently and accurately add fractions is a valuable asset in both personal and professional contexts.
Conclusion
In summary, we have explored a problem that involves calculating the total distance a person walked over two days by adding fractions. The problem stated that the person walked $\frac2}{3}$ of a mile on Thursday and $\frac{5}{7}$ of a mile on Friday. To find the total distance, we added these two fractions together. The key to adding fractions is to first find a common denominator, which in this case was 21. We converted both fractions to equivalent fractions with the common denominator3}$ became $\frac{14}{21}$, and $\frac{5}{7}$ became $\frac{15}{21}$. We then added the numerators, keeping the denominator the same, resulting in $\frac{29}{21}$. Finally, we converted this improper fraction to a mixed number, which gave us the answer{21}$ miles. Throughout this article, we emphasized the importance of understanding the underlying concepts of fraction addition and the steps involved in solving such problems. We also highlighted common mistakes to avoid, such as adding numerators and denominators directly without finding a common denominator, and the importance of simplifying the final fraction. Furthermore, we discussed the numerous real-world applications of fraction addition, demonstrating its relevance in various fields, from cooking and construction to finance and engineering. By understanding fraction addition, you gain a valuable skill that can be applied in many practical situations. Mastering this fundamental mathematical concept not only enhances your problem-solving abilities but also equips you with a tool for navigating everyday challenges that involve fractional quantities. We hope this comprehensive guide has provided you with a clear understanding of how to solve fraction addition problems and has instilled confidence in your mathematical abilities. Remember, practice is key to mastery, so continue to apply these concepts to different problems and scenarios.
