To multiply the fractions \( \frac{4}{5} \) and \( \frac{5}{8} \), you multiply the numerators together and the denominators together:
\[ \frac{4 \times 5}{5 \times 8} = \frac{20}{40} \]
Next, simplify \( \frac{20}{40} \) by dividing both the numerator and denominator by their greatest common divisor, which is 20:
\[ \frac{20 \div 20}{40 \div 20} = \frac{1}{2} \]
So, the answer in simplest form is \( \frac{1}{2} \).