To multiply the fractions \( \frac{3}{8} \) and \( \frac{5}{6} \), you multiply the numerators together and the denominators together:
\[ \frac{3}{8} \times \frac{5}{6} = \frac{3 \times 5}{8 \times 6} = \frac{15}{48} \]
Next, we can simplify the fraction \( \frac{15}{48} \). Both the numerator and the denominator can be divided by 3:
\[ \frac{15 \div 3}{48 \div 3} = \frac{5}{16} \]
So, the final simplified result is:
\[ \frac{5}{16} \]