Problem 32 - College Algebra

Question

In 2000 , the population of Greece was \(10,600,000,\) with projections of a population decrease of \(28,000\) people per year. In the same year, the population of Belgium was \(10,200,000,\) with projections of a population decrease of \(12,000\) people per year. (Source: United Nations) According to these projections, when will the two countries have the same population? What will be the population at that time?

Step-by-Step Solution

Answer

The countries Greece and Belgium will have the same population in the year 2025, and the projection for the population at that time is 9,700,000.

Step 1: Set up linear equations for Greece and Belgium

Let \(t\) represent the number of years since 2000. The population of Greece can be represented as \(P_G = 10,600,000 - 28,000t\). The population of Belgium can be represented as \(P_B = 10,200,000 - 12,000t\). These equations simply state that each year the population decreases by a certain amount from the starting population.

Step 2: Find when the population of the two countries will be equal

We set the two equations equal to each other and solve for \(t\): \(10,600,000 - 28000t = 10,200,000 - 12000t\). Solving this equation gives us the time when the populations will be equal.

Step 3: Solve for t

Distributing and solving for \(t\) yields \(16,000t = 400,000\), so \(t = 25\) years. Therefore, according to the projections, Greece and Belgium will have the same population 25 years after 2000, which is in the year 2025.

Step 4: Find the population at that time

We substitute \(t = 25\) into either of the equations. Let's choose Greece's equation: \(P_G = 10,600,000 - 28,000 \times 25 = 9,700,000\). Thus, the population of both countries in 2025 would be 9,700,000 according to the projections.

Key Concepts

Population Projection

Population projection is a crucial tool used by governments and organizations to forecast future population changes. It helps in planning for future resources and infrastructure, and understanding demographic shifts. These projections often involve assumptions about birth rates, death rates, and migration patterns. In this particular exercise, we consider population decrease based on historical trends.
For Greece and Belgium, the projections show a decrease in population each year. This indicates potential challenges like economic implications or social infrastructure adjustments if the trend continues.

• For Greece, the decrease is at a rate of 28,000 people per year.
• For Belgium, the decrease is at a rate of 12,000 people per year.

This difference in rate highlights how different demographic factors can impact population changes. By calculating when the populations converge, planners can anticipate and prepare for specific years that might require targeted policy interventions.

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality between two expressions. In our exercise, we use two separate algebraic equations to model the population projections for Greece and Belgium over time.
These equations are linear, meaning they represent a constant rate of change:

• Greece: \(P_G = 10,600,000 - 28,000t\)
• Belgium: \(P_B = 10,200,000 - 12,000t\)

Here, \(t\) is the variable representing years since the year 2000. For each year, the population decreases by a fixed number, so the equation becomes a simple linear representation of this decline.
Understanding algebraic equations in real-world contexts, like population change, allows us to make predictions and solve for unknowns such as the year when two populations become equal.

Problem Solving with Equations

Problem solving with equations involves setting up and manipulating mathematical expressions to find solutions to real-world questions. The exercise illustrates a classic problem-solving process:
First, we establish the equations for each country's population decrease over time. Then, to find when the two populations are the same, we equate and solve for \(t\):

• Set up: \(10,600,000 - 28,000t = 10,200,000 - 12,000t\)
• Solve: Simplifying gives \(16,000t = 400,000\)
• Result: \(t = 25\)

This means 25 years after 2000, i.e., in 2025, both populations will be equal. Substituting \(t = 25\) into one of the original equations shows the population at that time will be 9,700,000.
Thus, through logical steps and solving equations, we conclude not only the timing but also the future predicted population size, demonstrating the utility of algebra in problem-solving.