Problem 49 - Algebra A Combined Function

Question

The number of counties in California and the number of counties in Montana are consecutive even integers whose sum is 114 . If California has more counties than Montana, how many counties does each state have? (Source: The World Almanac and Book of Facts)

Step-by-Step Solution

Answer

Montana has 56 counties; California has 58 counties.

Step 1: Setting Up the Equation

Let's represent the number of counties in Montana as \( x \), the smaller integer. Then, since California's counties are consecutive even integers, the number of counties in California will be \( x + 2 \). The problem states that their sum is 114, so we can write the equation: \( x + (x + 2) = 114 \).

Step 2: Simplifying the Equation

Combine like terms in the equation. Adding \( x \) and \( x + 2 \) gives \( 2x + 2 = 114 \).

Step 3: Solving for x

To isolate \( x \), first subtract 2 from both sides: \( 2x = 112 \). Then, divide both sides by 2 to solve for \( x \): \( x = 56 \).

Step 4: Finding the Number of Counties

Since \( x \) represents the number of counties in Montana, Montana has 56 counties. The number of counties in California is \( x + 2 = 56 + 2 = 58 \).

Key Concepts

Understanding Consecutive Even Integers

Consecutive even integers are numbers that follow each other in order where each number is an even number. This means every subsequent number differs by 2. For example, 2, 4, and 6 are consecutive even integers. In many algebra word problems, we deal with these types of numbers because they provide a straightforward way to apply algebraic principles. To identify a consecutive even number from a given starting point, simply add 2 to the last even number you have. Thus, if you know the smallest number is represented as \( x \), the next consecutive even integer is \( x + 2 \). Such problems often appear in exercises where you're asked to find sums or differences of these numbers, making them a great method to showcase foundational algebra skills.

Setting Up Equations from Word Problems

The first step in tackling an algebra word problem is understanding how to translate words into equations. This process begins with identifying what you're solving for and assigning variables to unknown quantities. For example, in the given problem, if California and Montana have consecutive even integers for counties, label Montana's number of counties as \( x \). Then, California's counties are represented by \( x + 2 \), as they must be the next even integer.

• Clearly define what each variable represents.
• Construct an equation based on the relationship described in the problem.

In this scenario, the equation becomes \( x + (x + 2) = 114 \), reflecting the sum of counties from both states. Correctly setting up the equation is crucial since it serves as the basis for finding a solution.

Solving Equations Accurately

Solving equations requires a strategic approach to isolating the variable you're interested in. Start by simplifying the given equation as much as possible. Combine like terms to make the equation easier to manage. In the exercise, combining \( x \) and \( x + 2 \) gives \( 2x + 2 = 114 \).Next, perform operations to isolate \( x \). Subtract any constants from both sides of the equation first. Here, subtract 2, resulting in \( 2x = 112 \). Finally, to solve for \( x \), divide by 2: \( x = 56 \). This process is systematic and relies on balancing the equation step by step.

• Simplify the equation by combining like terms.
• Use inverse operations to isolate the variable.
• Check your work by substituting back into the original equation.

Once solved, use the value of \( x \) to determine other related quantities, such as the counties in California, which is \( x + 2 = 58 \). Remember, consistency and precision in solving equations are key to successful algebra problems.