Problem 14 - Intermediate Algebra

Question

Woodworking. \(\quad\) A carpenter saws a board that is 22 feet long into two pieces. One piece is to be 1 foot longer than twice the length of the shorter piece. Find the length of each piece. (TABLE CANNOT COPY)

Step-by-Step Solution

Answer

The shorter piece is 7 feet, and the longer piece is 15 feet.

Step 1: Define the Variables

Let's denote the length of the shorter piece as \( x \). Since one piece is 1 foot longer than twice the length of the shorter piece, we can express the length of the longer piece as \( 2x + 1 \).

Step 2: Set Up the Equation

According to the problem, the total length of the board is 22 feet. Therefore, we can write the equation: \[ x + (2x + 1) = 22 \] This equation sums up the lengths of the two pieces to equal the original board length.

Step 3: Simplify the Equation

Simplify the equation by combining like terms: \[ x + 2x + 1 = 22 \] \[ 3x + 1 = 22 \]

Step 4: Solve for \( x \)

To find \( x \), subtract 1 from both sides of the equation: \[ 3x = 21 \] Then, divide by 3: \[ x = 7 \] The length of the shorter piece is 7 feet.

Step 5: Find the Length of the Longer Piece

Substitute \( x = 7 \) back into the expression for the longer piece: \[ 2x + 1 = 2(7) + 1 = 14 + 1 = 15 \] The length of the longer piece is 15 feet.

Step 6: Verify the Solution

Check that the sum of the two pieces equals the original board length: \( 7 + 15 = 22 \). Since 22 feet matches the given board length, the solution is verified.

Key Concepts

Variable Definition

When solving any mathematical problem, the first essential step is defining the variables. Variables are symbols that represent unknown or changeable values. In this woodworking exercise, our task is to find the lengths of two pieces of wood from a single board.

To easily manage this, we define the length of the shorter piece as a variable. We'll use the symbol \( x \) for this. This choice helps us express one unknown quantity in a tangible way. Once we have one variable in place, we can also express the other quantity—in this case, the longer piece—in terms of \( x \).

• The shorter piece is \( x \).
• The longer piece is given as 1 foot longer than twice the shorter piece, making it \( 2x + 1 \).

Understanding variable definition is crucial. It breaks down complex situations into familiar algebraic forms, making the entire problem approachable.

Equation Setup

Equation setup is the next step, and it's about translating the problem into a mathematical sentence. With our variables in place, the task is to write an equation representing the situation described in the problem.

For the woodworking example, the total length of the board—22 feet—helps us form the equation. The concept is straightforward: the sum of the lengths of the two pieces must equal the total length of the uncut board.

• The shorter piece's length is \( x \).
• The longer piece's length is \( 2x + 1 \).
• Thus, the equation becomes \( x + (2x + 1) = 22 \).

This step requires careful attention to the relationships between the pieces of the problem. Setting up the equation correctly is vital because the rest of the solving process builds upon it.

Problem-Solving Strategy

Having the equation is just the beginning. The core of problem-solving strategy is to manipulate and solve the equation to find unknown values. With our equation set up as \( x + (2x + 1) = 22 \), we dive into solving by following these steps:

Simplify the equation by combining like terms:

• Combine \( x \) and \( 2x \) to get \( 3x \).
• The equation then becomes \( 3x + 1 = 22 \).

To isolate \( x \), we need to perform inverse operations:

• Subtract 1 from both sides: \( 3x = 21 \).
• Then divide by 3: \( x = 7 \).

This tells us the shorter piece is 7 feet long. To find the longer piece, substitute \( x = 7 \) back into \( 2x + 1 \) to get 15 feet.

Finally, always verify your solution. Check if the sum \( 7 + 15 \) matches the original length of 22 feet. It does, so you've confirmed that the problem-solving strategy works correctly. Embracing this strategy ensures you can tackle similar challenges with confidence.