Problem 20 - Prealgebra

Question

Georgia's Native-American population is \(10,000\) greater than Mississippi's. Mississippi's Native-American population is \(106,000\) less than Texas'. If the total population of all three is \(149,000,\) find each state's Native-American population.

Step-by-Step Solution

Answer

Texas: 113,667; Mississippi: 7,667; Georgia: 17,667.

Step 1: Understanding Given Relationships

Let's denote the Native-American population in Texas as \( T \), in Mississippi as \( M \), and in Georgia as \( G \). We have two key relationships: 1. Georgia's population is 10,000 more than Mississippi's: \( G = M + 10,000 \).2. Mississippi's population is 106,000 less than Texas': \( M = T - 106,000 \).

Step 2: Setting Up the Equation for Total Population

According to the problem statement, the total of the three states' Native-American populations is 149,000. We can express this as: \( T + M + G = 149,000 \).

Step 3: Solving for Mississippi's Population

Substitute the relational expressions of \( G \) and \( M \) into the total population equation:1. Replace \( G \) with \( M + 10,000 \).2. Replace \( M \) with \( T - 106,000 \).This gives us: \[ T + (T - 106,000) + ((T - 106,000) + 10,000) = 149,000 \].

Step 4: Simplifying the Equation

Combine like terms in the equation:\[ 3T - 202,000 + 10,000 = 149,000 \].Which simplifies to: \[ 3T - 192,000 = 149,000 \].

Step 5: Solving for Texas' Population

Add 192,000 to both sides of the equation to isolate the \( 3T \) term: \[ 3T = 341,000 \].Then divide by 3 to find \( T \): \[ T = 113,667 \].

Step 6: Finding Mississippi's Population

Using the relationship \( M = T - 106,000 \): \[ M = 113,667 - 106,000 = 7,667 \].

Step 7: Finding Georgia's Population

Using the relationship \( G = M + 10,000 \): \[ G = 7,667 + 10,000 = 17,667 \].

Step 8: Summary of the Populations

We have found the populations as: - Texas: 113,667 - Mississippi: 7,667 - Georgia: 17,667.

Key Concepts

Algebraic Expressions

Algebraic expressions are vital in representing relationships between different quantities. They use variables and constants to express such relationships clearly and concisely.
In this problem, we use algebraic expressions to describe the Native-American populations in the states of Texas, Mississippi, and Georgia.
An algebraic expression consists of:

• Variables: These are symbols, typically alphabetic, like \( T, M, \) and \( G \), used to represent unknown quantities.
• Constants: These are fixed numerical values, such as 10,000 and 106,000, specifying the relationships between populations in different states.
• Operators: Common mathematical operators like addition (+), subtraction (-), and equality (=) are used to link variables and constants to form meaningful expressions.

In the given exercise, the expressions \( G = M + 10,000 \) and \( M = T - 106,000 \) are used to depict relationships among the populations. Here, the constants represent how much more or less one state's population is compared to another. Crafting these expressions correctly is the first step in solving complex problems involving multiple unknowns.

Problem Solving

Problem solving in mathematics involves breaking down complex scenarios into manageable steps. The exercise here showcases a typical problem-solving approach. When faced with a problem involving relationships among different quantities, it's essential to:

• Understand what is given: Identify all the provided information. Here, we know that Georgia's population is 10,000 more than Mississippi's, and Mississippi's is 106,000 less than Texas'. Additionally, the total population figures are given.
• Set up equations: Translate relationships and conditions into algebraic equations. In this exercise, three equations describe the populations of Georgia, Mississippi, and Texas.
• Substitute and simplify: Insert algebraic expressions into total equations to reduce complexity. Doing so makes it easier to solve for each state's unknown population.
• Calculate and verify: Perform mathematical operations to find the results and substitute them back to ensure accuracy.

By proceeding step by step, problem-solving becomes systematic, helping to avoid mistakes and finding accurate solutions to problems involving interrelated quantities.

Linear Equations

Linear equations are equations that represent straight-line relationships between variables. In this exercise, they play a central role. Every relevant relationship in this exercise is represented as a linear equation.Key features of linear equations include:

• They have variables raised to the power of one, which means that their graph is a straight line.
• They involve constants and coefficients, making them appear as expressions like \( ax + b = c \).
• They are solvable using algebraic manipulation, such as addition, subtraction, and division.

In our problem, we used linear equations such as \( 3T - 192,000 = 149,000 \) to represent combined populations. The equation simplified and solved provides the population for Texas. Once Texas' population was determined, solving for Mississippi and Georgia became straightforward by referencing their respective relationships as linear equations. Through understanding and manipulating linear equations, students learn to see connections between variables and constant relationships.