Problem 22 - Elementary and Intermediate Algebra

Question

World Population Growth. The world population is projected to be 9.4 billion in \(2050 .\) At that time, there is expected to be approximately 3.5 billion more people in Asia than in Africa. The population for the rest of the world will be approximately 0.3 billion less than two-fifths the population of Asia. Find the projected populations of Asia, Africa, and the rest of the world in \(2050 .\)

Step-by-Step Solution

Answer

Asia: 5.5 billion, Africa: 2 billion, Rest of the world: 1.9 billion

Step 1 Title - Define Variables

Let the projected population of Asia in 2050 be denoted by \( A \), Africa by \( B \), and the rest of the world by \( C \).

Step 2 Title - Set Up Equations

Using the given information, establish the following equations:\ 1. The total projected world population is \(A + B + C = 9.4\) billion.\ 2. The population of Asia is expected to be 3.5 billion more than Africa, so \(A = B + 3.5\).\ 3. The population for the rest of the world will be approximately 0.3 billion less than two-fifths the population of Asia, so \(C = \frac{2A}{5} - 0.3\).

Step 3 Title - Substitute and Simplify Equations

Substitute \(C\) and the value of \(A\) from equations 2 and 3 into equation 1:\ \( (B + 3.5) + B + \left( \frac{2(B + 3.5)}{5} - 0.3 \right) = 9.4 \)

Step 4 Title - Solve for Africa's Population (B)

Solve the equation from step 3 for \(B\):\ \( 2B + 3.5 + \left( \frac{2B + 7}{5} \right) - 0.3 = 9.4 \).\ Combine like terms and clear the fraction:\ \( 2B + 3.5 + 0.4B + 1.4 - 0.3 = 9.4 \), which simplifies to \( 2.4B + 4.6 = 9.4 \).\ Solve for \(B\): \( 2.4B = 4.8 \), \( B = 2 \).

Step 5 Title - Solve for Asia's Population (A)

Using \(B = 2\), substitute it into \(A = B + 3.5\): \( A = 2 + 3.5 = 5.5 \).

Step 6 Title - Solve for Rest of the World Population (C)

Using \(A = 5.5\), substitute it into \( C = \frac{2A}{5} - 0.3 \): \( C = \frac{2(5.5)}{5} - 0.3 = 2.2 - 0.3 = 1.9 \).

Key Concepts

algebraic equations

Algebraic equations form the backbone of solving word problems in algebra. In this exercise, we translate the word problem about the world population into mathematical equations. This allows for a structured and methodical approach to finding the solution.
The equations represent different relationships expressed in the problem. For example:
- The sum of populations of Asia (A), Africa (B), and the rest of the world (C) equals 9.4 billion: \ A + B + C = 9.4
- The population of Asia is 3.5 billion more than Africa: \( A = B + 3.5 \)
- The population of the rest of the world is 0.3 billion less than two-fifths of Asia's: \( C = \frac{2A}{5} - 0.3 \)
These equations help us encapsulate all given relationships in the problem, setting the stage for solving the variables step by step.

variable substitution

Variable substitution is a powerful technique in algebra that simplifies the process of solving equations. It involves replacing one variable with an expression involving another variable.
In our problem, we first define variables for the populations of Asia (A), Africa (B), and the rest of the world (C). We then use the given relationships to express these variables in terms of one another. For instance, we substitute the expression for A and C into the main equation. This reduces the number of variables we need to solve simultaneously.
After substitution, the equation looks like:
\ \((B + 3.5) + B + \bigg( \frac{2(B + 3.5)}{5} - 0.3 \bigg) = 9.4\)
This step helps convert a complex system into a simpler form that can be easily solved.

population projections

Population projections involve estimating the future population based on specific data and assumptions. Understanding how to set up such projections through algebra can provide us with crucial insights for planning and resource management.
In this exercise, the projection data for the year 2050 includes:
- Total world population: 9.4 billion
- Asia's population being 3.5 billion more than Africa's
- The rest of the world's population being 0.3 billion less than two-fifths of Asia's Therefore, projections give us a way to visualize the future demographics based on current trends and expected changes. By solving for the populations, we are effectively converting mathematical predictions into meaningful numbers representing future scenarios.

problem-solving steps

Structured problem-solving is essential in tackling algebraic word problems. Breaking down a problem into smaller, manageable steps ensures a logical progression towards the solution. In this exercise, the steps are as follows:
1. Define Variables: Name the populations for Asia (A), Africa (B), and the rest of the world (C).
2. Set Up Equations: Translate the given data into algebraic equations.
3. Substitute and Simplify: Replace variables to reduce the number of unknowns and simplify the equations.
4. Solve for Each Variable: Isolate and find the value of one variable at a time by solving the simplified equations.
This methodical approach not only helps in solving the problem but also ensures that each part of the problem is understood thoroughly. The last step involves verifying the consistency of the solutions with the original problem context to ensure accuracy.