A piece of string is 40 centimeters long. It is cut into three pieces. The longest piece is 3 times as long as the middle- sized piece and the shortest piece is 23 centimeters shorter than the longest piece. Find the length of the shortest piece.

Options

A. 27
B. 5
C. 4
D. 9

Answer

To solve the problem step by step, we can define the lengths of the three pieces of string based on the information given:

1. **Define Variables**:
- Let the length of the middle-sized piece be \( x \) centimeters.
- The longest piece is 3 times as long as the middle-sized piece, so its length will be \( 3x \) centimeters.
- The shortest piece is 23 centimeters shorter than the longest piece, so its length will be \( 3x - 23 \) centimeters.

2. **Set Up the Equation**:
- According to the problem, the total length of the string is 40 centimeters. Therefore, we can write the equation:
\[ x + 3x + (3x - 23) = 40 \]

3. **Combine Like Terms**:
- Now, combine the terms on the left side of the equation:
\[ x + 3x + 3x - 23 = 40 \]
This simplifies to:
\[ 7x - 23 = 40 \]

4. **Isolate the Variable**:
- Next, we will isolate \( x \) by adding 23 to both sides:
\[ 7x = 40 + 23 \]
\[ 7x = 63 \]

5. **Solve for \( x \)**:
- Now, divide both sides by 7 to find \( x \):
\[ x = \frac{63}{7} \]
\[ x = 9 \]

6. **Find the Lengths of Each Piece**:
- Now that we have \( x \), we can find the lengths of each piece:
- Middle-sized piece: \( x = 9 \) cm
- Longest piece: \( 3x = 3 \times 9 = 27 \) cm
- Shortest piece: \( 3x - 23 = 27 - 23 = 4 \) cm

7. **Conclusion**:
- The length of the shortest piece is \( 4 \) centimeters.