Calculating Total Travel Distance A Mathematical Journey To Work

In this article, we will explore a mathematical problem involving distances traveled using different modes of transportation. The problem focuses on calculating the total distance Jimmy covered on his way to work, which included biking and train travel. Understanding how to solve such problems is crucial for developing essential mathematical skills, particularly in the areas of fractions, mixed numbers, and addition. By dissecting the problem step by step, we will not only arrive at the correct answer but also gain insights into practical applications of mathematical concepts in everyday scenarios. This article aims to provide a comprehensive guide to solving this problem while emphasizing clarity and accuracy.

Jimmy's commute to work involves multiple legs, each utilizing a different mode of transportation. Initially, he rode his bike a distance of $10 \frac{4}{5}$ miles to reach the train station. Subsequently, he boarded a train and traveled into the downtown area. Upon arriving downtown, Jimmy again rode his bike, this time covering a distance of $4 \frac{3}{4}$ miles to finally reach his workplace. The primary objective is to determine the total distance Jimmy traveled on his journey to work. To solve this, we need to consider the distances covered by biking and the distance covered by train. The problem highlights the importance of understanding mixed numbers and fractions, which are fundamental concepts in mathematics.

Breaking Down the Problem

To solve this problem effectively, we must break it down into manageable steps. First, identify the known distances, which are the distances Jimmy biked: $10 \frac{4}{5}$ miles to the train station and $4 \frac{3}{4}$ miles from downtown to work. The unknown distance is the train travel, which we are not given a specific value for, but we know it's part of the total distance. The question asks for the total distance traveled, implying that we need to sum the distances of each leg of the journey. Therefore, the core task is to add the two mixed numbers representing the biking distances. This involves converting mixed numbers to improper fractions, finding a common denominator, adding the fractions, and then converting the result back to a mixed number. Each of these steps is crucial for accurate calculation, and we will delve into each one to ensure a thorough understanding.

Step-by-Step Solution

To calculate the total distance Jimmy traveled, we must sum the distances he biked. The problem states that he biked $10 \frac{4}{5}$ miles to the train station and $4 \frac{3}{4}$ miles from downtown to work. We will add these two distances together to find the total biking distance. This process involves several steps, including converting mixed numbers to improper fractions, finding a common denominator, adding the fractions, and simplifying the result. Let’s walk through each step to ensure clarity and accuracy.

Converting Mixed Numbers to Improper Fractions

The first step in adding mixed numbers is to convert them into improper fractions. A mixed number consists of a whole number and a proper fraction, while an improper fraction has a numerator greater than or equal to its denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction and add the numerator. This result becomes the new numerator, and the denominator remains the same. For example, to convert $10 \frac{4}{5}$ to an improper fraction, we multiply 10 by 5 to get 50, add 4 to get 54, and keep the denominator 5, resulting in $\frac{54}{5}$. Similarly, to convert $4 \frac{3}{4}$ to an improper fraction, we multiply 4 by 4 to get 16, add 3 to get 19, and keep the denominator 4, resulting in $\frac{19}{4}$.

Finding a Common Denominator

Before adding fractions, they must have a common denominator. The common denominator is the least common multiple (LCM) of the denominators. In this case, we need to find the LCM of 5 and 4. The multiples of 5 are 5, 10, 15, 20, 25, and so on, while the multiples of 4 are 4, 8, 12, 16, 20, 24, and so on. The least common multiple of 5 and 4 is 20. Therefore, we will convert both fractions to have a denominator of 20. To convert $\frac{54}{5}$ to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 4, resulting in $\frac{54 \times 4}{5 \times 4} = \frac{216}{20}$. Similarly, to convert $\frac{19}{4}$ to a fraction with a denominator of 20, we multiply both the numerator and the denominator by 5, resulting in $\frac{19 \times 5}{4 \times 5} = \frac{95}{20}$.

Adding the Fractions

Now that the fractions have a common denominator, we can add them by adding their numerators and keeping the denominator the same. We have $\frac{216}{20}$ and $\frac{95}{20}$, so we add the numerators 216 and 95 to get 311. The denominator remains 20, so the sum is $\frac{311}{20}$. This improper fraction represents the total biking distance in miles. To better understand this distance, we will convert the improper fraction back to a mixed number.

Converting the Improper Fraction Back to a Mixed Number

To convert the improper fraction $\frac{311}{20}$ back to a mixed number, we divide the numerator (311) by the denominator (20). The quotient becomes the whole number part of the mixed number, the remainder becomes the numerator of the fractional part, and the denominator remains the same. Dividing 311 by 20 gives us a quotient of 15 and a remainder of 11. Therefore, the mixed number is $15 \frac{11}{20}$. This means that Jimmy biked a total of $15 \frac{11}{20}$ miles.

Final Answer and Conclusion

Based on our calculations, Jimmy traveled a total of $15 \frac{11}{20}$ miles on his way to work. This distance includes the $10 \frac{4}{5}$ miles he biked to the train station and the $4 \frac{3}{4}$ miles he biked from downtown to his workplace. By breaking down the problem into smaller steps—converting mixed numbers to improper fractions, finding a common denominator, adding the fractions, and converting the result back to a mixed number—we were able to solve the problem accurately. This exercise demonstrates the importance of understanding and applying fundamental mathematical concepts in practical situations. The process of solving this problem reinforces skills in fraction manipulation and addition, which are essential in various real-life applications. Furthermore, it underscores the significance of meticulous calculation and step-by-step problem-solving strategies.

The method used to solve this problem highlights the importance of a step-by-step approach in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can tackle it more effectively and reduce the likelihood of errors. Each step builds upon the previous one, ensuring a logical and coherent solution. In the case of Jimmy’s commute, converting mixed numbers to improper fractions was a crucial first step, as it simplified the addition process. Finding a common denominator allowed us to add the fractions accurately, and converting the result back to a mixed number provided a clear and understandable answer. This methodical approach is not only beneficial in mathematics but also in various other problem-solving scenarios in life.

Application in Real-Life Scenarios

Understanding how to solve problems involving distances and fractions has practical applications in everyday life. For instance, calculating travel distances, measuring ingredients for cooking, and managing time are all situations where these mathematical skills come into play. In the context of travel, knowing how to calculate total distances can help in planning routes, estimating travel times, and budgeting for transportation costs. In cooking, accurate measurements are essential for achieving the desired results, and this often involves working with fractions. Time management also requires a good understanding of fractions, as we frequently divide our time into segments for various activities. By mastering these mathematical concepts, we can enhance our problem-solving abilities and make more informed decisions in our daily lives.

Review of Key Concepts

To ensure a thorough understanding of the solution, let's review some key mathematical concepts that were applied in this problem. Mixed numbers and improper fractions are essential components of the problem. A mixed number is a combination of a whole number and a proper fraction, while an improper fraction has a numerator greater than or equal to its denominator. Converting between these forms is crucial for performing arithmetic operations, especially addition and subtraction. Finding a common denominator is another fundamental concept. Before adding or subtracting fractions, they must have the same denominator. The least common multiple (LCM) is the smallest number that is a multiple of both denominators, and it serves as the common denominator. By mastering these concepts, we can confidently tackle a wide range of mathematical problems.

Conclusion

In conclusion, Jimmy traveled a total of $15 \frac{11}{20}$ miles on his way to work. This calculation involved adding the distances he biked, which required converting mixed numbers to improper fractions, finding a common denominator, and simplifying the result. The problem underscores the importance of a step-by-step approach in mathematical problem-solving and highlights the practical applications of fractions and mixed numbers in everyday life. By understanding these concepts and practicing problem-solving techniques, we can enhance our mathematical skills and improve our ability to tackle real-world challenges.