To evaluate \( 4 \frac{1}{2} - 2 - 5 \frac{2}{3} \), we can convert all the mixed numbers to improper fractions and then perform the calculations.
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Convert \( 4 \frac{1}{2} \) to an improper fraction: \[ 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \]
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Convert \( 5 \frac{2}{3} \) to an improper fraction: \[ 5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \]
Now, we rewrite the expression: \[ \frac{9}{2} - 2 - \frac{17}{3} \]
To subtract \( 2 \) (which can also be expressed as a fraction): \[ 2 = \frac{2 \times 6}{6} = \frac{12}{6} \]
The expression now looks like: \[ \frac{9}{2} - \frac{12}{6} - \frac{17}{3} \]
Next, we will find a common denominator for the fractions. The denominators are \( 2, 6, \) and \( 3 \). The least common multiple of these is \( 6 \):
Convert \( \frac{9}{2} \) to a denominator of \( 6 \): \[ \frac{9}{2} = \frac{9 \times 3}{2 \times 3} = \frac{27}{6} \]
Convert \( \frac{17}{3} \) to a denominator of \( 6 \): \[ \frac{17}{3} = \frac{17 \times 2}{3 \times 2} = \frac{34}{6} \]
Now substituting back into the expression: \[ \frac{27}{6} - \frac{12}{6} - \frac{34}{6} \]
Now we can combine the fractions: \[ \frac{27 - 12 - 34}{6} = \frac{27 - 46}{6} = \frac{-19}{6} \]
Thus, the final result is: \[
- \frac{19}{6} = -3 \frac{1}{6} \]
So, the answer is: D \(-4 \frac{1}{6}\)