To solve the expression \( \frac{3}{2} + \left( \frac{4}{5} + \frac{1}{2} \right) \times \frac{1}{6} \), 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. **Evaluate inside the parentheses first:**
\[
\frac{4}{5} + \frac{1}{2}
\]

To add these fractions, we need a common denominator. The least common denominator (LCD) of 5 and 2 is 10.

Convert each fraction:
\[
\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10}
\]
\[
\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}
\]

Now add the fractions:
\[
\frac{8}{10} + \frac{5}{10} = \frac{8 + 5}{10} = \frac{13}{10}
\]

2. **Next, multiply the result by \(\frac{1}{6}\):**
\[
\frac{13}{10} \times \frac{1}{6} = \frac{13 \times 1}{10 \times 6} = \frac{13}{60}
\]

3. **Finally, add \(\frac{3}{2}\) to the result:**
\[
\frac{3}{2} + \frac{13}{60}
\]

To add these fractions, we need a common denominator. The least common denominator of 2 and 60 is 60.

Convert each fraction:
\[
\frac{3}{2} = \frac{3 \times 30}{2 \times 30} = \frac{90}{60}
\]

Now add the fractions:
\[
\frac{90}{60} + \frac{13}{60} = \frac{90 + 13}{60} = \frac{103}{60}
\]

So, the final result is:
\[
\frac{3}{2} + \left( \frac{4}{5} + \frac{1}{2} \right) \times \frac{1}{6} = \frac{103}{60}
\]