1. **State the problem:** Multiply the fractions $\frac{1}{4}$ and $\frac{4}{5}$.\n\n2. **Formula:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$$\n\n3. **Apply the formula:** $$\frac{1}{4} \times \frac{4}{5} = \frac{1 \times 4}{4 \times 5} = \frac{4}{20}.$$\n\n4. **Simplify the fraction:** Both numerator and denominator share a common factor of 4. Cancel it out: $$\frac{\cancel{4}}{\cancel{20}} = \frac{1}{5}.$$\n\n5. **Final answer:** The product of $\frac{1}{4}$ and $\frac{4}{5}$ is $\frac{1}{5}$.