A 31 foot board is cut in two pieces so that the longer piece is 1 foot less than three times the shorter piece
The correct answer and explanation is:
To solve the problem, we are given:
A 31-foot board is cut into two pieces so that the longer piece is 1 foot less than three times the shorter piece.
Let’s define variables and solve it step by step.
Step 1: Define Variables
Let the shorter piece be:
👉 xx feet
Then the longer piece is:
👉 3x−13x – 1 feet (because it is 1 foot less than three times the shorter)
Step 2: Write an Equation
The total length of the board is 31 feet, so: x+(3x−1)=31x + (3x – 1) = 31
Step 3: Simplify and Solve
x+3x−1=31⇒4x−1=31⇒4x=32⇒x=8x + 3x – 1 = 31 \Rightarrow 4x – 1 = 31 \Rightarrow 4x = 32 \Rightarrow x = 8
Step 4: Find the Lengths
- Shorter piece = x=8x = 8 feet
- Longer piece = 3x−1=3(8)−1=24−1=233x – 1 = 3(8) – 1 = 24 – 1 = 23 feet
✅ Answer:
- Shorter piece = 8 feet
- Longer piece = 23 feet
300-Word Explanation
This is a classic algebra word problem involving two parts that add up to a known total. We start by defining a variable, xx, to represent the shorter piece of the board. Since the longer piece is 1 foot less than three times the shorter, we express the longer piece as 3x−13x – 1.
The key insight is realizing that both pieces together must equal the total length of the board, which is 31 feet. So we create the equation x+(3x−1)=31x + (3x – 1) = 31, combining like terms to get 4x−1=314x – 1 = 31. Solving this equation by isolating xx gives us x=8x = 8, meaning the shorter piece is 8 feet long.
Substituting back, the longer piece becomes 3(8)−1=233(8) – 1 = 23 feet. When we add these two parts—8 feet and 23 feet—we get 31 feet, confirming the solution is correct.
This kind of problem teaches valuable algebra skills: variable representation, equation formation from word descriptions, and basic solving techniques. It’s often used to build foundational problem-solving skills in mathematics.