1. **State the problem:** We need to find the result of dividing $\frac{1}{4}$ by $\frac{2}{5}$. 2. **Recall the formula:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 3. **Apply the formula:** $$\frac{1}{4} \div \frac{2}{5} = \frac{1}{4} \times \frac{5}{2}$$ 4. **Multiply the numerators and denominators:** $$= \frac{1 \times 5}{4 \times 2} = \frac{5}{8}$$ 5. **Final answer:** $$\boxed{\frac{5}{8}}$$ This means $\frac{1}{4}$ divided by $\frac{2}{5}$ equals $\frac{5}{8}$.