Problem 50 - Elementary Algebra

Question

In the following exercises, solve each number word problem. Find three consecutive even integers whose sum is 222 .

Step-by-Step Solution

Answer

The three consecutive even integers are 72, 74, and 76.

Step 1: Define the Variables

Let the first even integer be denoted as x.Since the integers are consecutive even numbers, the second integer will be x + 2 and the third will be x + 4.

Step 2: Set Up the Equation

The sum of the three consecutive even integers is given as 222. Therefore, the equation can be set up as: x + (x + 2) + (x + 4) = 222.

Step 3: Simplify the Equation

Combine like terms in the equation: 3x + 6 = 222.

Step 4: Solve for x

Isolate x by subtracting 6 from both sides of the equation: 3x = 216. Then divide both sides by 3: x = 72.

Step 5: Find the Three Integers

Now that x is found, the three consecutive even integers are: x = 72, x + 2 = 74, and x + 4 = 76.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order. For example, 2, 4, 6, and 8 are consecutive even integers because each number is exactly 2 more than the previous one.
Consecutive integers can also be odd or even. In this exercise, we are working with consecutive even integers.
Here, we start with the first even integer, denote it as \( x \), and the subsequent even integers become \( x + 2 \) and \( x + 4 \). These are terms commonly used in problems involving consecutive numbers, making it easier to set up and solve the equations.

Algebraic Equations

Algebraic equations represent relationships between different values. In this exercise, we express the sum of three consecutive even integers as an algebraic equation.
By defining the variables and setting up the equation based on the given sum, we have: \( x + (x + 2) + (x + 4) = 222 \).
This step is crucial because it translates the word problem into a mathematical form that we can solve using algebraic methods. Combining like terms, as shown in the next steps, is part of solving these types of equations.

Solving Word Problems

Solving word problems involves several steps, including carefully reading the problem, defining variables, setting up equations, and solving them.
In this exercise, the word problem asks us to find three consecutive even integers whose sum is 222. We begin by defining \( x \) as the first integer. Next, we set up our equation based on the given information.
Then, we broke down the original equation by combining like terms to get a more straightforward form: \( 3x + 6 = 222 \). Once simplified, we solve for \( x \) to find the specific integers.

Basic Algebra Concepts

Understanding basic algebra concepts is essential for solving problems like the one in this exercise.
One of the basic concepts is combining like terms. For example, in the equation \( x + (x + 2) + (x + 4) = 222 \), combining like terms gives us \( 3x + 6 \).
Another important concept is isolating the variable. We did this by subtracting 6 from both sides of the equation to get \( 3x = 216 \), and then dividing by 3 to find \( x = 72 \). Finally, substituting back to find the consecutive integers completes the process.