Problem 14 - Prealgebra

Question

Solve each problem by writing and solving an equation. POPULATION By 2020 , California is expected to have 2 million more senior citizens than Florida, and the sum of the number of senior citizens in the two states is expected to be 12 million. Find the expected senior citizen population of Florida in 2020 .

Step-by-Step Solution

Answer

Florida is expected to have 5 million senior citizens in 2020.

Step 1: Set Up Variables

Let \( F \) represent the number of senior citizens in Florida in 2020. Let \( C \) represent the number of senior citizens in California in 2020. We know that \( C = F + 2,000,000 \) because California has 2 million more senior citizens than Florida.

Step 2: Form the Equation

The total number of senior citizens in Florida and California combined is 12 million, so our equation will be the sum of \( C \) and \( F \), equalling 12 million: \( C + F = 12,000,000 \).

Step 3: Substitute and Simplify

Substitute the expression for \( C \) from Step 1 into the equation from Step 2: \((F + 2,000,000) + F = 12,000,000\). Simplify it into: \(2F + 2,000,000 = 12,000,000\).

Step 4: Solve for Florida's Population

Subtract 2,000,000 from both sides to get: \(2F = 10,000,000\). Then divide both sides by 2 to isolate \( F \): \(F = 5,000,000\).

Key Concepts

Problem Solving

Problem solving is a structured method that helps us tackle real-world questions and scenarios. This involves understanding a problem, gathering relevant information, and systematically working out the solution. Let's break down the approach:

• Identify the Problem: Understand what is being asked. In our scenario, it's about finding the senior citizen population in Florida by 2020.
• Gather Information: We know that California has 2 million more seniors than Florida, and together, they sum up to 12 million.
• Plan: Decide how to use this information. Use variables to represent unknowns, and create equations that represent the relationships described in the problem.
• Execute: Implement the plan by solving the equations step by step.
• Review: Check if the solution makes sense in the context of the question.

By applying these steps methodically, problems become more manageable and less intimidating.

Variables and Expressions

In math, variables are symbols that stand for unknown values. They are essential in forming equations that model real-life problems. Expressions are combinations of numbers, variables, and operations that represent a value.Consider the problem:

• We set variable F for the number of senior citizens in Florida.
• Variable C for the number of senior citizens in California.

Using variables helps us to express relationships clearly:

• Since California has 2 million more seniors than Florida, we express this as: \( C = F + 2,000,000 \).
• The total seniors in both states is represented as:\( C + F = 12,000,000 \).

Once we have expressions, these equations can then be manipulated using algebraic operations to find the unknown values.
This concept is foundational in algebra as it allows us to work with real situations symbolically before solving them.

Population Estimation

Population estimation involves predicting the number of people in a given category or area at a certain time. It's vital for planning resources, such as health care and infrastructure. Here's how it applies:

• Understanding Growth Trends: Estimations use demographic trends and prior data. In our example, California's and Florida's elderly population projections were based on known data and patterns.
• Creating Equations: These estimations help form equations that reflect expected relationships. For instance: - We know the projected increase in California's seniors relative to Florida (+2 million). - We know the joint expected total (12 million).
• Purpose of Estimations: The significance is not only academic but practical, aiding policymakers in resource allocation.

Estimations are both a prediction tool and a model from which to derive critical logistical insights, crucially affecting economic and social planning.