1. The problem is to calculate $4 \div \frac{5}{7}$ and express the answer as a fraction in its lowest terms. 2. Recall the rule for dividing fractions: dividing by a fraction is the same as multiplying by its reciprocal. 3. So, $4 \div \frac{5}{7} = 4 \times \frac{7}{5}$. 4. Write 4 as a fraction: $\frac{4}{1}$. 5. Multiply the numerators and denominators: $$\frac{4}{1} \times \frac{7}{5} = \frac{4 \times 7}{1 \times 5} = \frac{28}{5}.$$ 6. Check if the fraction $\frac{28}{5}$ can be simplified. Since 28 and 5 have no common factors other than 1, it is already in lowest terms. 7. Therefore, the final answer is $\frac{28}{5}$.