Problem 80 - Thinking Mathematically

Question

One apartment is directly above a second apartment. The resident living downstairs calls his neighbor living above him and states, "If one of you is willing to come downstairs, we'll have the same number of people in both apartments." The upstairs resident responds, "We're all too tired to move. Why don't one of you come up here? Then we will have twice as many people up here as you've got down there." How many people are in each apartment?

Step-by-Step Solution

Answer

There are 2 people in the downstairs apartment and 3 people in the upstairs apartment.

Step 1: Define Variables

Let \(x\) represent the number of people in the downstairs apartment, and let \(y\) represent the number of people in the upstairs apartment. These are the two unknowns to be solved.

Step 2: Set Up Equations

Two separate scenarios have been given which can be interpreted into equations. The first scenario states that if one person from upstairs goes downstairs, the number of people will be equal in both apartments. So, \(y-1 = x\). Under the second scenario, if one person from downstairs goes upstairs, the number of people upstairs will be twice as many as downstairs. That can be represented as \(y+1=2(x-1)\). These are the two equations to be solved.

Step 3: Solve the System of Equations

First, simplify the second equation to \(y+1=2x-2\), which can be rearranged to \(y=2x-3\). Now we have two equations left: \(y = x + 1 \) and \(y = 2x - 3\). Solving these equation we get the value \(x = 2\) and \(y = 3\).

Step 4: Interpret the Solution

The solution indicates that there are 2 people in the downstairs apartment and 3 people in the upstairs apartment.

Key Concepts

Equation Solving

At the heart of algebra lies the concept of equation solving. Equations are statements that assert two expressions are equal, often involving unknown variables that we aim to determine.
For instance, in the given problem, the variables represent the number of people in two different apartments. To solve these equations means to find the values of these variables that make the equation true.

Here are some key concepts to keep in mind when solving equations:

• Identify the Unknowns: Clearly defining your variables is crucial. In this problem, we designated the number of people downstairs as \(x\) and upstairs as \(y\).
• Translate Words into Equations: The word problem gives conditions about the variables. Learning to translate verbal descriptions into algebraic expressions is essential.
• Manipulating Equations: To find the variable values, algebraic operations such as addition, subtraction, multiplication, or division are used systematically to simplify the equations or isolate the unknowns.

Mastering these concepts will make solving algebraic equations much smoother, allowing you to understand relationships between quantities and how they can be balanced.

System of Equations

A system of equations involves solving for multiple variables and typically includes two or more equations that are considered simultaneously. In our problem, we are dealing with a system of two linear equations derived from conditions mentioned.
For example, both scenarios described in the exercise provided us with two distinct relationships between \(x\) and \(y\), which resulted in:

• Equation 1: \(y - 1 = x\)
• Equation 2: \(y + 1 = 2(x - 1)\)

The goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously.

Here are crucial methods to tackle a system of equations:

• Substitution Method: Solve one of the equations for one variable, then substitute this expression into the other equation. This method simplifies the number of variables so you can solve for each.
• Elimination Method: This technique involves adding or subtracting the equations to eliminate one of the variables, making it possible to solve for the remaining one.

You can choose any method based on what feels more comfortable or appropriate for the problem at hand. Solving systems of equations is all about finding points where both equations intersect, representing the solution.

Word Problems

Word problems can be a challenging aspect of algebra because they require more than just solving equations; they demand understanding a real-world scenario and translating it into a mathematical form. Our example problem involves residents in two apartments, and understanding their conversation helped us form the mathematical equations needed to find a solution.

To effectively solve word problems, consider these strategies:

• Read Carefully: Ensure you understand the scenario fully. What information is being given? What are you being asked to find?
• Define the Variables: Clearly specify what each variable represents in the context of the problem.
• Set Up the Equations: Based on the problem, create equations that reflect the relationships described.
• Check Your Work: After finding a solution, plug the values back into the original word problem to ensure they make logical sense and solve the problem as described.

With practice, solving word problems becomes an advantageous skill, allowing you to tackle a wide array of real-life issues mathematically.