To simplify \(7\frac{1}{12} - 4\frac{3}{4} + 2\frac{1}{2}\), we need to convert all the mixed numbers to improper fractions and then perform the arithmetic operations.

1. \(7\frac{1}{12} = \frac{7 \cdot 12 + 1}{12} = \frac{85}{12}\)

2. \(4\frac{3}{4} = \frac{4 \cdot 4 + 3}{4} = \frac{19}{4}\)

3. \(2\frac{1}{2} = \frac{2 \cdot 2 + 1}{2} = \frac{5}{2}\)

Now, the expression becomes:

\(\frac{85}{12} - \frac{19}{4} + \frac{5}{2}\)

To add and subtract fractions, we need a common denominator. The least common multiple of 12, 4, and 2 is 12. So, we'll rewrite the fractions with the common denominator of 12:

\(\frac{85}{12} - \frac{57}{12} + \frac{30}{12}\)

Now we can combine the numerators:

\(\frac{85 - 57 + 30}{12} = \frac{29}{6} = 4\frac{5}{6}\)