Divide and simplify \( 2 \frac{2}{7} \div \frac{4}{5} \)

To divide \( 2 \frac{2}{7} \) by \( \frac{4}{5} \), first convert the mixed number to an improper fraction. \( 2 \frac{2}{7} = \frac{2 \times 7 + 2}{7} = \frac{14 + 2}{7} = \frac{16}{7} \). Now, dividing by a fraction is the same as multiplying by its reciprocal. So, \[ \frac{16}{7} \div \frac{4}{5} = \frac{16}{7} \times \frac{5}{4}. \] Multiply the numerators and the denominators: \[ \frac{16 \times 5}{7 \times 4} = \frac{80}{28}. \] Now, simplify \( \frac{80}{28} \) by finding the greatest common divisor (GCD) of 80 and 28, which is 4. Dividing both the numerator and the denominator by 4 gives: \[ \frac{80 \div 4}{28 \div 4} = \frac{20}{7}. \] The simplified result is \( \frac{20}{7} \) or \( 2 \frac{6}{7} \) if you prefer to express it as a mixed number.