Unlocking Jose's Journey Calculating Distance Traveled By Foot

Introduction: The Kilometric Quest

In this mathematical journey, we embark on a quest to unravel the distance Jose traveled on foot. Jose covered a total of 150 kilometers, traversing the landscape via three modes of transport: car, bicycle, and foot. The challenge lies in deciphering the distances covered by each mode, particularly the distance covered on foot. The problem states that Jose traveled three times the distance by car compared to the bicycle, and he walked 20 kilometers less than he cycled. Our mission is to dissect this information, formulate a mathematical equation, and ultimately determine the kilometers Jose walked. This involves careful analysis of the given ratios and differences in distances, ensuring we accurately represent the relationships between the car, bicycle, and walking distances. The initial step is to define variables for the unknown distances, allowing us to translate the word problem into an algebraic equation. We will then solve this equation, systematically isolating the variable representing the bicycle distance. Once we determine the bicycle distance, calculating the walking distance becomes a straightforward subtraction. This problem exemplifies how mathematical reasoning can be applied to real-world scenarios, breaking down complex situations into manageable components. The key is to maintain precision and clarity throughout the calculation process, ensuring we arrive at the correct solution for Jose's walking distance. This exploration will not only reveal the answer but also demonstrate the power of mathematical problem-solving in understanding and quantifying everyday experiences. So, let us delve into the calculations and uncover the distance Jose traversed on foot, piecing together the puzzle of his 150-kilometer journey. By carefully applying algebraic principles and logical deduction, we will arrive at the final answer, shedding light on the pedestrian portion of Jose's extensive travel.

Defining the Variables: A Symbolic Representation

To begin, let's define our variables. We'll represent the distance traveled by bicycle as x. This is our foundational unknown, the key to unlocking the other distances. Since Jose traveled three times the distance by car compared to the bicycle, we can represent the car distance as 3x. This establishes a direct relationship between the bicycle and car distances, allowing us to express one in terms of the other. Finally, Jose walked 20 kilometers less than he cycled, so the walking distance can be represented as x - 20. Now, we have a symbolic representation of all three distances, each expressed in terms of the variable x. This is a crucial step in translating the word problem into a mathematical equation, providing a clear and concise way to represent the given information. The use of variables allows us to manipulate the unknowns algebraically, ultimately solving for the value of x and subsequently determining the other distances. This approach highlights the power of algebra in simplifying complex scenarios, transforming verbal statements into mathematical expressions. By carefully defining these variables, we lay the groundwork for a systematic solution, ensuring that each distance is accurately accounted for in our calculations. The next step involves formulating the equation that represents the total distance traveled, bringing together these individual components into a unified expression. This process of variable definition is not just about assigning symbols; it's about creating a mathematical language that mirrors the relationships described in the problem, paving the way for a clear and logical solution.

Formulating the Equation: A Mathematical Model

Now, let's formulate the equation. We know the total distance Jose traveled is 150 kilometers. This total distance is the sum of the distances traveled by car, bicycle, and foot. Using the variables we defined earlier, we can express this as an equation: x (bicycle) + 3x (car) + (x - 20) (foot) = 150. This equation is the heart of our problem, a mathematical model that encapsulates the relationships between the different modes of transport. It combines our defined variables into a single, coherent statement, allowing us to solve for the unknown value of x. The equation represents a balance, where the sum of the individual distances equals the total distance. By carefully constructing this equation, we have transformed the word problem into a solvable mathematical expression. The next step involves simplifying this equation, combining like terms, and isolating the variable x. This process of equation formulation is a critical skill in mathematical problem-solving, allowing us to translate real-world scenarios into abstract mathematical representations. The accuracy of our solution hinges on the correctness of this equation, so it's essential to ensure that all the given information is accurately incorporated. The equation serves as a roadmap, guiding us through the steps needed to unravel the unknown distances and ultimately determine how far Jose walked. This equation is not just a collection of symbols; it's a powerful tool for understanding and solving the problem at hand, bridging the gap between the narrative and the numerical solution. By carefully crafting this equation, we set the stage for a systematic and logical approach to finding the answer.

Solving for x: Unveiling the Bicycle Distance

Next, we solve for x. Our equation is: x + 3x + (x - 20) = 150. First, let's simplify by combining like terms. We have x + 3x + x, which equals 5x. So, our equation becomes: 5x - 20 = 150. Now, we want to isolate the term with x. To do this, we add 20 to both sides of the equation: 5x - 20 + 20 = 150 + 20. This simplifies to 5x = 170. Finally, to solve for x, we divide both sides of the equation by 5: 5x / 5 = 170 / 5. This gives us x = 34. Therefore, the distance Jose traveled by bicycle is 34 kilometers. This is a significant milestone in our journey, as we've now determined the value of our primary unknown. The systematic approach of simplifying the equation and isolating the variable has allowed us to unveil the bicycle distance. This value of x serves as a foundation for calculating the other distances, bringing us closer to our final goal of determining the walking distance. The process of solving for x demonstrates the power of algebraic manipulation, transforming a complex equation into a clear and concise solution. Each step, from combining like terms to isolating the variable, is a testament to the logical progression of mathematical reasoning. Now that we know the bicycle distance, we can use this information to calculate the distance Jose walked, completing the final leg of our kilometric quest. The value of x = 34 is not just a number; it's a key piece of the puzzle, unlocking the solution to the entire problem.

Calculating the Walking Distance: The Final Stretch

Now, we calculate the walking distance. We know that Jose walked x - 20 kilometers, and we've determined that x = 34 kilometers. So, the walking distance is 34 - 20 = 14 kilometers. This is our final answer! Jose walked 14 kilometers. By systematically following the steps of defining variables, formulating an equation, and solving for the unknown, we have successfully determined the distance Jose traveled on foot. This journey through the problem highlights the power of mathematical reasoning and algebraic manipulation in solving real-world scenarios. The final calculation is a simple subtraction, but it's the culmination of a carefully orchestrated process, each step building upon the previous one. The 14 kilometers represent the pedestrian portion of Jose's 150-kilometer journey, a testament to his endurance and the variety of transportation modes he employed. This solution not only provides a numerical answer but also demonstrates the importance of breaking down complex problems into manageable components. By carefully analyzing the given information and applying mathematical principles, we have successfully navigated the challenge and arrived at the correct solution. The 14 kilometers stand as a clear and concise answer, a testament to the effectiveness of our problem-solving approach. This final calculation marks the end of our kilometric quest, a journey through variables, equations, and ultimately, the distance Jose walked.

Conclusion: Jose's Footsteps Revealed

In conclusion, Jose walked 14 kilometers. This problem demonstrates the application of algebraic principles to solve real-world scenarios. By defining variables, formulating an equation, and systematically solving for the unknown, we successfully determined the distance Jose traveled on foot. This exercise highlights the importance of mathematical reasoning in understanding and quantifying everyday experiences. The journey through this problem serves as a reminder of the power of mathematics in unraveling complex situations and providing clear, concise solutions. The 14 kilometers represent not just a number, but the culmination of a logical and systematic approach to problem-solving. This exploration has not only revealed the answer but also reinforced the value of mathematical skills in navigating the challenges of the world around us. The ability to translate word problems into mathematical equations and solve them effectively is a valuable asset, empowering us to make informed decisions and understand the quantitative aspects of our lives. This final answer stands as a testament to the effectiveness of our approach and the enduring relevance of mathematical principles. The steps we took, from variable definition to final calculation, represent a clear and logical path to understanding and solving the problem at hand. So, the next time you encounter a mathematical challenge, remember the power of systematic reasoning and the ability of algebra to unlock the solution, just as we did in determining the distance Jose walked.