1. **State the problem:** We need to divide the fraction $\frac{1}{11}$ by the fraction $\frac{4}{55}$. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{1}{11} \div \frac{4}{55} = \frac{1}{11} \times \frac{55}{4}$$ 3. **Multiply the numerators and denominators:** $$\frac{1 \times 55}{11 \times 4} = \frac{55}{44}$$ 4. **Simplify the fraction:** Both numerator and denominator can be divided by 11. $$\frac{\cancel{55}^{5}}{\cancel{44}^{4}} = \frac{5}{4}$$ 5. **Final answer:** The simplified result of the division is $\frac{5}{4}$, which is an improper fraction and can also be written as the mixed number $1 \frac{1}{4}$.