We are asked to evaluate the sum of three rational numbers: $(-\frac{1}{2}) + (+1\frac{2}{3}) + (-\frac{5}{6})$.

1. Problem Description

We are asked to evaluate the sum of three rational numbers: $(-\frac{1}{2}) + (+1\frac{2}{3}) + (-\frac{5}{6})$.

2. Solution Steps

First, convert the mixed number $1\frac{2}{3}$ to an improper fraction.

$1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3+2}{3} = \frac{5}{3}$

The problem becomes:

$(-\frac{1}{2}) + (\frac{5}{3}) + (-\frac{5}{6})$.

To add these 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. $-\frac{1}{2} = -\frac{1 \times 3}{2 \times 3} = -\frac{3}{6}$

$\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}$

$-\frac{5}{6}$ is already in the correct form.

Now, we add the fractions:

$-\frac{3}{6} + \frac{10}{6} - \frac{5}{6} = \frac{-3+10-5}{6} = \frac{7-5}{6} = \frac{2}{6}$

Finally, simplify the fraction:

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

3. Final Answer

$\frac{1}{3}$