To subtract \( 4 \frac{2}{3} - 2 \frac{1}{2} \), we'll first convert both mixed numbers to improper fractions.
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Convert \( 4 \frac{2}{3} \): \[ 4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3} \]
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Convert \( 2 \frac{1}{2} \): \[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]
Now we have the subtraction: \[ \frac{14}{3} - \frac{5}{2} \]
To subtract these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.
- Convert both fractions to have a denominator of 6: \[ \frac{14}{3} = \frac{14 \times 2}{3 \times 2} = \frac{28}{6} \] \[ \frac{5}{2} = \frac{5 \times 3}{2 \times 3} = \frac{15}{6} \]
Now we can subtract the fractions: \[ \frac{28}{6} - \frac{15}{6} = \frac{28 - 15}{6} = \frac{13}{6} \]
Finally, we convert \( \frac{13}{6} \) back to a mixed number: \[ \frac{13}{6} = 2 \frac{1}{6} \]
So, the answer is: \[ \boxed{2 \frac{1}{6}} \]