Find and simplify the quotient \( \frac{2}{3} \div \frac{8}{9} \)

To simplify the quotient \( \frac{2}{3} \div \frac{8}{9} \), we first convert the division of fractions into multiplication by the reciprocal. This gives us: \[ \frac{2}{3} \div \frac{8}{9} = \frac{2}{3} \times \frac{9}{8} \] Next, we multiply the numerators together and the denominators together: \[ \frac{2 \times 9}{3 \times 8} = \frac{18}{24} \] Now, we simplify \( \frac{18}{24} \) by finding the greatest common divisor (GCD) of 18 and 24, which is 6. We then divide both the numerator and the denominator by 6: \[ \frac{18 \div 6}{24 \div 6} = \frac{3}{4} \] Thus, the simplified quotient is: \[ \boxed{\frac{3}{4}} \]