First, let's round each number to the nearest whole number.
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For \(11 \frac{1}{2}\):
- \(11 \frac{1}{2} = 11.5\)
- Rounding \(11.5\) gives us \(12\).
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For \(2 \frac{5}{7}\):
- \(2 \frac{5}{7} = 2.7142857\) (approximately)
- Rounding \(2.7142857\) gives us \(3\).
Now, we subtract the rounded numbers: \[ 12 - 3 = 9 \]
Thus, the difference is approximately \(9\).