To solve the problem, let's break down the steps clearly:
1. Define the variables:
Let [tex]\( x \)[/tex] be the length of the shorter piece of rope. Since one piece is half the length of the other, the longer piece of rope would be [tex]\( 2x \)[/tex].
2. Total length of the rope:
The sum of the lengths of the two pieces should equal the total length of the rope, which is 24 feet.
3. Formulate the equation:
Given that the shorter piece is [tex]\( x \)[/tex] and the longer piece is [tex]\( 2x \)[/tex], their sum should equal the total length:
[tex]\[
x + 2x = 24
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Combine the terms on the left side:
[tex]\[
3x = 24
\][/tex]
Divide both sides by 3 to isolate [tex]\( x \)[/tex]:
[tex]\[
x = \frac{24}{3} = 8
\][/tex]
So, the length of the shorter piece of rope is [tex]\( 8 \)[/tex] feet.
5. Determine the length of the longer piece:
The longer piece is twice the length of the shorter piece:
[tex]\[
2x = 2 \times 8 = 16
\][/tex]
So, the length of the longer piece of rope is [tex]\( 16 \)[/tex] feet.
6. Conclusion:
The lengths of the two pieces of rope are:
- Shorter piece: [tex]\( 8 \)[/tex] feet
- Longer piece: [tex]\( 16 \)[/tex] feet
Let’s check the given multiple-choice statements to see which ones are correct:
- The stated solution is correct (lengths of 16 feet and 8 feet). ✔️
- Based on the way the variable is defined, the equation should be [tex]\( x + 2x = 24 \)[/tex]. ✔️
- The equation is written correctly, but there is an error in solving the equation. ❌
- The way the variable is defined, and because [tex]\( x=16 \)[/tex], the longer piece of rope would be 32 feet, which is not possible. ❌
Based on the correct solution, the only true statements are:
- The stated solution is correct (lengths of 16 feet and 8 feet).
- Based on the way the variable is defined, the equation should be [tex]\( x + 2x = 24 \)[/tex].
