Let's define the length of the shorter section as \( x \) inches. According to the problem, the longer section is then \( x + 5 \) inches.
We know from the problem statement that: \[ 2 \times (\text{length of the longer section}) = 3 \times (\text{length of the shorter section}) \]
Substituting the expressions for the sections into the equation, we get: \[ 2(x + 5) = 3x \]
Now, we can solve for \( x \):
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Distribute the 2 on the left side: \[ 2x + 10 = 3x \]
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Subtract \( 2x \) from both sides: \[ 10 = 3x - 2x \] \[ 10 = x \]
So, the length of the shorter section \( x \) is 10 inches.
To verify, if \( x = 10 \) inches, then the longer section would be: \[ x + 5 = 10 + 5 = 15 \text{ inches} \]
Now checking the original condition:
- Twice the length of the longer section: \[ 2 \times 15 = 30 \text{ inches} \]
- Three times the length of the shorter section: \[ 3 \times 10 = 30 \text{ inches} \]
Both sides are equal, confirming that our solution is correct.
Therefore, the length of the shorter section is 10 inches, not 12 inches.
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