1. **State the problem:** Multiply the fractions $\frac{1}{2}$ and $8 \frac{1}{8}$.\n\n2. **Convert the mixed number to an improper fraction:**\n$8 \frac{1}{8} = \frac{8 \times 8 + 1}{8} = \frac{64 + 1}{8} = \frac{65}{8}$.\n\n3. **Write the multiplication of fractions:**\n$\frac{1}{2} \times \frac{65}{8}$.\n\n4. **Multiply the numerators and denominators:**\n$\frac{1 \times 65}{2 \times 8} = \frac{65}{16}$.\n\n5. **Simplify the fraction if possible:**\n$65$ and $16$ have no common factors other than $1$, so $\frac{65}{16}$ is in simplest form.\n\n6. **Convert the improper fraction back to a mixed number:**\nDivide $65$ by $16$: $65 \div 16 = 4$ remainder $1$.\nSo, $\frac{65}{16} = 4 \frac{1}{16}$.\n\n**Final answer:** $4 \frac{1}{16}$.