To solve the expression \( \frac{25}{12} : 5 + \frac{5}{6} + \frac{1}{2} \times 3 \), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

1. **Division and Multiplication (from left to right):**
\[
\frac{25}{12} : 5 = \frac{25}{12} \times \frac{1}{5} = \frac{25}{60} = \frac{5}{12}
\]
\[
\frac{1}{2} \times 3 = \frac{3}{2}
\]

2. **Substitute these results back into the expression:**
\[
\frac{5}{12} + \frac{5}{6} + \frac{3}{2}
\]

3. **Find a common denominator for the fractions:**
The least common denominator for 12, 6, and 2 is 12.

\[
\frac{5}{12} = \frac{5}{12}
\]
\[
\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}
\]
\[
\frac{3}{2} = \frac{3 \times 6}{2 \times 6} = \frac{18}{12}
\]

4. **Add the fractions:**
\[
\frac{5}{12} + \frac{10}{12} + \frac{18}{12} = \frac{5 + 10 + 18}{12} = \frac{33}{12}
\]

5. **Simplify the fraction:**
\[
\frac{33}{12} = \frac{11}{4} = 2.75
\]

So, the value of the expression \( \frac{25}{12} : 5 + \frac{5}{6} + \frac{1}{2} \times 3 \) is \( 2.75 \).