Solving A Population Puzzle Determining Residents Of Town A And Town B
Unveiling the Population Mystery of Towns A and B
In this intricate population puzzle, we embark on a journey to unravel the demographic composition of two towns, A and B. The initial clue sets the stage: Town A boasts a population exceeding Town B by a margin of 400 residents. This establishes a foundational relationship between the two towns, hinting at a disparity in their sizes. However, the plot thickens with a significant population shift. A migration of 800 individuals from Town B to Town A occurs, triggering a transformation in their relative populations. This movement acts as a catalyst, altering the demographic landscape and setting the stage for a compelling mathematical challenge.
As the dust settles from this migration, a new equilibrium emerges. The population of Town A swells, becoming three times the size of Town B's population. This pivotal piece of information serves as the linchpin in our quest to decipher the exact populations of the two towns. To solve this puzzle, we must delve into the realm of algebraic equations, translating the narrative into mathematical expressions. We will employ the power of variables to represent the unknown populations, and through a series of logical deductions and algebraic manipulations, we will strive to unveil the true demographic composition of Towns A and B. This endeavor will not only test our mathematical prowess but also our ability to dissect complex scenarios and transform them into solvable equations. Our journey begins with a careful examination of the given information, laying the groundwork for a methodical and insightful solution.
Setting Up the Equations for Population Dynamics
To effectively solve this population dynamics problem, we must translate the word problem into a system of algebraic equations. Let's denote the initial number of people in Town A as x and the initial number of people in Town B as y. Our first clue states that Town A has 400 more people than Town B. This translates directly into our first equation: x = y + 400. This equation establishes a clear relationship between the populations of the two towns before any migration occurs. It tells us that the population of Town A is directly dependent on the population of Town B, with an additional 400 residents. This is a crucial foundation upon which we will build our solution.
Next, we consider the migration of 800 people from Town B to Town A. This event alters the populations of both towns. After the migration, Town A's population becomes x + 800, while Town B's population decreases to y - 800. The problem states that after this migration, Town A's population becomes three times the size of Town B's population. This crucial piece of information forms the basis for our second equation: x + 800 = 3(y - 800). This equation captures the state of the populations after the migration and establishes another key relationship between x and y. With these two equations, we have a system that we can solve to find the values of x and y. The first equation provides a direct link between the initial populations, while the second equation describes the relationship after the migration. By solving this system, we can unravel the mystery of the populations of Towns A and B.
Solving the System of Equations to Find the Populations
With our equations firmly in place, we can now embark on the process of solving the system of equations. Our system consists of two equations:
- x = y + 400
- x + 800 = 3(y - 800)
To solve this system, we can employ the method of substitution. Since the first equation already expresses x in terms of y, we can substitute this expression into the second equation. This substitution eliminates x from the second equation, leaving us with an equation involving only y. By substituting y + 400 for x in the second equation, we get:
(y + 400) + 800 = 3(y - 800)
Now, we simplify and solve for y:
y + 1200 = 3y - 2400
Combining like terms, we get:
3600 = 2y
Dividing both sides by 2, we find:
y = 1800
This tells us that the initial population of Town B was 1800 people. Now that we have found y, we can substitute this value back into the first equation to find x:
x = 1800 + 400
x = 2200
Thus, the initial population of Town A was 2200 people. We have successfully solved the system of equations, revealing the populations of the two towns. This process highlights the power of algebraic manipulation in solving real-world problems. By carefully translating the word problem into equations and employing techniques like substitution, we can unravel complex scenarios and arrive at definitive solutions.
Verifying the Solution and the Final Answer
With our potential solutions for the populations of Town A and Town B in hand, it's crucial to verify their accuracy. To do this, we'll substitute the values we found, x = 2200 and y = 1800, back into our original equations. This step ensures that our solutions not only satisfy the individual equations but also accurately reflect the relationships described in the problem statement. Let's begin by revisiting our first equation, x = y + 400. Substituting our values, we get:
2200 = 1800 + 400
2200 = 2200
This confirms that our values satisfy the first condition, where Town A initially had 400 more people than Town B. Now, let's move on to our second equation, which describes the population dynamics after the migration: x + 800 = 3(y - 800). Substituting our values, we get:
2200 + 800 = 3(1800 - 800)
3000 = 3(1000)
3000 = 3000
This confirms that our values also satisfy the second condition, where after the migration, Town A's population became three times that of Town B. Having verified our solutions against both equations, we can confidently state that our answers are correct. Therefore, the initial population of Town A was 2200 people, and the initial population of Town B was 1800 people. This verification step is a critical component of problem-solving, ensuring that our solutions are not just mathematically sound but also logically consistent with the problem's narrative. It reinforces the accuracy of our analysis and provides a sense of closure to our investigation.
Conclusion: Unraveling Population Mysteries with Math
In conclusion, we have successfully navigated the intricate population dynamics of Towns A and B, employing the power of algebraic equations to decipher their demographic composition. By carefully translating the word problem into a system of equations, we were able to unravel the relationships between the town populations and solve for the unknown values. Our journey began with establishing the initial conditions, where Town A had 400 more residents than Town B. This led to our first equation, x = y + 400, which served as a foundational link between the two towns. The narrative then introduced a migration event, where 800 people moved from Town B to Town A, altering their respective populations. This critical event paved the way for our second equation, x + 800 = 3(y - 800), which captured the new population equilibrium after the migration.
With our system of equations in place, we skillfully employed the method of substitution to eliminate variables and solve for the unknown populations. This process revealed that Town B initially had 1800 residents (y = 1800), and Town A had 2200 residents (x = 2200). To ensure the accuracy of our solution, we meticulously verified our values against both equations, confirming their consistency with the problem's conditions. This rigorous verification step underscored the reliability of our mathematical approach and provided a definitive resolution to the puzzle. The successful solution of this problem highlights the versatility and effectiveness of algebra in modeling and solving real-world scenarios. By transforming complex narratives into mathematical expressions, we can unlock hidden relationships and arrive at precise answers, further solidifying the power of mathematics as a tool for understanding the world around us.
