To find the quotient of \( \frac{5}{6} \div (1 + \frac{1}{2}) \), we first need to simplify the expression inside the parentheses:
\[ 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]
Now we can rewrite the division:
\[ \frac{5}{6} \div \frac{3}{2} \]
Dividing by a fraction is the same as multiplying by its reciprocal, so we have:
\[ \frac{5}{6} \times \frac{2}{3} \]
Next, we multiply the fractions:
\[ \frac{5 \times 2}{6 \times 3} = \frac{10}{18} \]
Now we simplify \( \frac{10}{18} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[ \frac{10 \div 2}{18 \div 2} = \frac{5}{9} \]
Thus, the simplified quotient is:
\[ \boxed{\frac{5}{9}} \]
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