Problem 50 - Algebra A Combined Function

Question

A student is building a bookcase with stepped shelves for her dorm room. She buys a 48 -inch board and wants to cut the board into three pieces with lengths equal to three consecutive even integers. Find the three board lengths.

Step-by-Step Solution

Answer

The board lengths are 14 inches, 16 inches, and 18 inches.

Step 1: Define Consecutive Even Integers

Consecutive even integers can be represented as \( x \), \( x + 2 \), and \( x + 4 \). These are three even numbers that follow each other in sequence.

Step 2: Set Up the Equation

We know that the sum of the lengths of the three pieces is 48 inches. Therefore, we can set up the equation: \( x + (x + 2) + (x + 4) = 48 \).

Step 3: Combine Like Terms

Simplify the equation by combining like terms: \( x + x + x + 2 + 4 = 48 \). This simplifies to \( 3x + 6 = 48 \).

Step 4: Solve for x

Subtract 6 from both sides of the equation to isolate the term with \( x \): \( 3x + 6 - 6 = 48 - 6 \), giving \( 3x = 42 \). Divide both sides by 3 to solve for \( x \): \( x = 14 \).

Step 5: Find the Lengths of the Pieces

Substitute \( x = 14 \) back into the expressions for the three lengths: \( x = 14 \), \( x + 2 = 16 \), and \( x + 4 = 18 \). Therefore, the three pieces are 14 inches, 16 inches, and 18 inches long.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. For even numbers, this means the difference between them is consistently 2. For example, if one integer is 2, the next consecutive even integer would be 4, and the next would be 6.

In the problem about cutting a board into pieces, the integers involved are consecutive even integers. This is why they are represented as \(x, x+2, x+4\).

Understanding the concept of consecutive integers helps recognize patterns in sequences and can be essential for solving many mathematical problems. If you imagine a series of numbers lined up, consecutive even integers are those even numbers that would stand right next to each other.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations like addition and multiplication. In our exercise, expressions such as \(x, x+2, x+4\) represent the lengths of the board pieces.

Algebraic expressions help translate real-world situations, such as finding lengths, into mathematical terms that make them easier to solve.

• Variables: These are symbols used to represent unknown numbers, commonly \(x\) in our example.
• Operations: The expressions involve algebraic operations (+2, +4) to represent the sequential addition needed for consecutive numbers.

Writing and solving algebraic expressions is a core skill not only in algebra but in almost all areas of mathematics.
They allow us to create equations that can represent and solve real-life problems, making them indispensable for any budding mathematician.

Integer Operations

Integer operations are basic arithmetic operations that are applied to whole numbers, including addition, subtraction, multiplication, and division.

In the given problem, multiple integer operations are used to solve for \(x\):

• **Addition**: Used to combine lengths of the board pieces \(x, x+2, x+4\) to equal 48.
• **Subtraction**: This operation helps isolate \(x\) by removing constants from the equation \(3x + 6 = 48\).
• **Division**: Used to solve for \(x\) by dividing both sides of the equation by 3, resulting in \(x = 14\).

These integer operations are foundational in mathematics and are used for simplifying and solving equations. They allow us to manipulate algebraic expressions so that we can find unknown values, like the lengths of the board pieces in our exercise. Understanding and accurately performing integer operations is essential for solving not only algebraic equations but also broader mathematical challenges.