Finding The Gym Location A Mathematical Journey
Embark on a mathematical journey to pinpoint the location of a gym situated two-thirds of the way between Ping's and Ari's residences. This exploration delves into the practical application of fractions and distances in a real-world scenario, providing a clear, step-by-step approach to solving this problem. This article will guide you through the process of understanding the problem, visualizing the scenario, and applying the necessary mathematical concepts to arrive at the solution. Whether you're a student honing your math skills or simply curious about practical applications of mathematical principles, this article offers a comprehensive and engaging explanation. The concepts discussed here are fundamental to understanding spatial relationships and can be applied to various real-life situations, from planning routes to understanding maps.
Understanding the Problem
To begin, let's clearly define the problem at hand. Ping resides at the intersection of 3rd Street and 6th Avenue, while Ari's home is located at the intersection of 21st Street and 18th Avenue. Our objective is to determine the precise location of a gym positioned $\frac{2}{3}$ of the distance from Ping's residence to Ari's. This problem inherently involves understanding spatial relationships and applying fractional distances within a coordinate-like system. We need to treat the street and avenue numbers as coordinates to calculate the distance and then find the point that is two-thirds along that distance. This requires a combination of spatial reasoning and mathematical calculation, making it a practical exercise in applying mathematical concepts to real-world scenarios. Before diving into the calculations, it's crucial to visualize the problem. Imagine a grid where streets and avenues form the axes, and Ping and Ari's homes are points on this grid. The gym lies somewhere on the line connecting these two points, closer to Ping's house since it's located two-thirds of the way from Ping to Ari. This visualization helps in understanding the magnitude and direction of the distances involved.
Visualizing the Scenario
Visualizing the scenario is crucial for grasping the spatial relationships involved. Picture a city grid where streets run horizontally and avenues run vertically. Ping's location at 3rd Street and 6th Avenue can be represented as a point on this grid, and similarly, Ari's location at 21st Street and 18th Avenue forms another point. The gym is situated along the imaginary line connecting these two points. Mentally mapping out these locations helps to understand the relative positions and the distances involved. This mental image provides a framework for understanding the numerical calculations that follow. By visualizing the scenario, we can better appreciate the directional aspects of the problem. We're not just dealing with abstract numbers; we're dealing with locations in a two-dimensional space. This spatial understanding makes the mathematical calculations more intuitive and easier to follow. Moreover, visualizing the scenario helps in identifying potential errors in the calculations. If the calculated location of the gym seems out of place in the mental map, it's a sign that there might be a mistake in the calculations. Therefore, developing a strong visual representation of the problem is an essential first step in solving it effectively. It bridges the gap between abstract mathematical concepts and concrete spatial realities.
Calculating the Distance
To accurately locate the gym, we must first calculate the distance between Ping's and Ari's homes. Since we're dealing with a grid-like system of streets and avenues, we can treat this as a problem involving two-dimensional coordinates. Ping's location is (3, 6), representing 3rd Street and 6th Avenue, while Ari's location is (21, 18). To find the distance, we need to consider the difference in street numbers and the difference in avenue numbers separately. The difference in street numbers is 21 - 3 = 18 streets, and the difference in avenue numbers is 18 - 6 = 12 avenues. These differences represent the legs of a right-angled triangle, where the direct distance between the two homes is the hypotenuse. To calculate the direct distance, we can apply the Pythagorean theorem: distance = √((difference in streets)^2 + (difference in avenues)^2). Plugging in the values, we get distance = √(18^2 + 12^2) = √(324 + 144) = √468. This gives us the total direct distance between Ping's and Ari's homes, which is a crucial piece of information for locating the gym. It's important to remember that this distance is not measured in standard units like meters or miles, but rather in terms of the number of street and avenue blocks. This approach simplifies the problem by treating the grid as a uniform coordinate system.
Applying the Pythagorean Theorem
Applying the Pythagorean Theorem is the key to finding the straight-line distance between Ping's and Ari's homes. This theorem, a cornerstone of Euclidean geometry, states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our scenario, the difference in street numbers and the difference in avenue numbers form the two shorter sides of the right-angled triangle, and the direct distance between the homes is the hypotenuse. The formula derived from the Pythagorean Theorem is: c = √(a^2 + b^2), where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. As calculated earlier, the difference in street numbers (a) is 18, and the difference in avenue numbers (b) is 12. Squaring these values gives us 18^2 = 324 and 12^2 = 144. Summing these squares yields 324 + 144 = 468. Finally, taking the square root of 468 gives us the direct distance, which is approximately 21.63 blocks. This result represents the shortest distance between Ping's and Ari's residences, a crucial value in determining the gym's location. Understanding and applying the Pythagorean Theorem in this context demonstrates its practical utility in solving real-world spatial problems. It allows us to translate the grid-like street and avenue system into a quantifiable distance, which is essential for further calculations.
Locating the Gym
Now that we've determined the distance between Ping's and Ari's homes, the next step is locating the gym. The problem states that the gym is located $\frac{2}{3}$ of the distance from Ping's home to Ari's. This means we need to find a point along the line connecting Ping's and Ari's residences that is two-thirds of the way from Ping's location. To do this, we'll calculate the fractional change in street and avenue numbers separately. First, consider the change in street numbers. The total difference in streets is 18 (21 - 3). To find $\frac{2}{3}$ of this difference, we multiply 18 by $\frac{2}{3}$, which gives us 12 streets. This means the gym is 12 streets away from Ping's street number. Adding this to Ping's street number (3) gives us 3 + 12 = 15th Street. Next, we repeat the process for the avenues. The total difference in avenues is 12 (18 - 6). Multiplying this by $\frac{2}{3}$ gives us 8 avenues. This means the gym is 8 avenues away from Ping's avenue number. Adding this to Ping's avenue number (6) gives us 6 + 8 = 14th Avenue. Therefore, the gym is located at the corner of 15th Street and 14th Avenue. This method of calculating fractional distances by considering each coordinate separately is a practical way to solve problems involving distances in a grid system.
Calculating Fractional Distances
Calculating fractional distances involves determining the position of a point that lies a certain fraction of the way along a line segment. In this case, we need to find the location of the gym, which is $ rac2}{3}$ of the distance from Ping's home to Ari's. The key to solving this is to consider the changes in street and avenue numbers separately. For the streets, we calculated the difference between Ari's street number and Ping's street number (21 - 3 = 18 streets). To find $ rac{2}{3}$ of this distance, we multiply 18 by $ rac{2}{3}$, which equals 12 streets. This value represents the number of streets we need to move from Ping's location towards Ari's. Adding this to Ping's street number (3) gives us the gym's street number{3}$ gives us 8 avenues. Adding this to Ping's avenue number (6) gives us the gym's avenue number: 6 + 8 = 14th Avenue. This method effectively breaks down the problem into smaller, more manageable steps. By calculating the fractional changes in each dimension (streets and avenues) separately, we can pinpoint the exact location of the gym. This approach is widely applicable in various scenarios, such as map reading, navigation, and spatial planning. Understanding how to calculate fractional distances is a valuable skill in both mathematical problem-solving and real-world applications.
Conclusion
In conclusion, by meticulously following a step-by-step approach, we successfully pinpointed the location of the gym. Starting with a clear understanding of the problem, we calculated the distance between Ping's and Ari's homes using the Pythagorean Theorem. This gave us a quantifiable measure of the total distance. Next, we applied the concept of fractional distances to determine the point that is $\frac{2}{3}$ of the way from Ping's to Ari's. By considering the changes in street and avenue numbers separately, we accurately calculated the gym's location at the corner of 15th Street and 14th Avenue. This exercise demonstrates the practical application of mathematical concepts, such as the Pythagorean Theorem and fractions, in solving real-world spatial problems. The process of visualizing the scenario, calculating distances, and applying fractional proportions is applicable in a wide range of contexts, from urban planning to navigation. Moreover, this problem highlights the importance of breaking down complex problems into smaller, more manageable steps. By addressing each component individually and then synthesizing the results, we can effectively solve intricate challenges. The ability to apply mathematical principles to everyday situations is a valuable skill, and this exercise provides a clear example of how mathematics can be used to solve practical problems.
