The problem is to evaluate the expression $\frac{1}{2} - 1\frac{1}{3} + 2\frac{1}{6}$.

1. Problem Description

The problem is to evaluate the expression $\frac{1}{2} - 1\frac{1}{3} + 2\frac{1}{6}$.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$1\frac{1}{3} = \frac{1 \cdot 3 + 1}{3} = \frac{3+1}{3} = \frac{4}{3}$

$2\frac{1}{6} = \frac{2 \cdot 6 + 1}{6} = \frac{12+1}{6} = \frac{13}{6}$

Now, the expression becomes $\frac{1}{2} - \frac{4}{3} + \frac{13}{6}$.

To add or subtract fractions, we need to find a common denominator. The least common multiple of 2, 3, and 6 is

6. So we convert each fraction to have a denominator of

6. $\frac{1}{2} = \frac{1 \cdot 3}{2 \cdot 3} = \frac{3}{6}$

$\frac{4}{3} = \frac{4 \cdot 2}{3 \cdot 2} = \frac{8}{6}$

$\frac{13}{6}$ remains the same.

Now the expression is $\frac{3}{6} - \frac{8}{6} + \frac{13}{6}$.

Combine the fractions: $\frac{3-8+13}{6} = \frac{-5+13}{6} = \frac{8}{6}$.

Simplify the fraction: $\frac{8}{6} = \frac{4}{3}$.

Convert the improper fraction to a mixed number: $\frac{4}{3} = 1\frac{1}{3}$.

3. Final Answer

$1\frac{1}{3}$