1. **State the problem:** Add the fractions $\frac{1}{4}$ and $\frac{1}{7}$ and simplify the result to lowest terms. 2. **Formula used:** To add fractions, use the formula: $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$ where $a$, $b$, $c$, and $d$ are integers and $b, d \neq 0$. 3. **Apply the formula:** $$\frac{1}{4} + \frac{1}{7} = \frac{1 \times 7 + 1 \times 4}{4 \times 7} = \frac{7 + 4}{28} = \frac{11}{28}$$ 4. **Simplify the fraction:** Check if numerator and denominator have common factors. Since 11 is prime and does not divide 28, the fraction $\frac{11}{28}$ is already in lowest terms. 5. **Final answer:** The sum of $\frac{1}{4}$ and $\frac{1}{7}$ simplified is: $$\boxed{\frac{11}{28}}$$ The numerator is 11.