Write the ratio \( 2 \frac{8}{9} \) to \( 5 \frac{1}{3} \) as a simplified fraction.

To express the ratio \( 2 \frac{8}{9} \) to \( 5 \frac{1}{3} \) as a simplified fraction, we first convert the mixed numbers into improper fractions. For \( 2 \frac{8}{9} \): \[ 2 \frac{8}{9} = 2 \times 9 + 8 = 18 + 8 = 26 \quad \text{(improper fraction: } \frac{26}{9}\text{)} \] For \( 5 \frac{1}{3} \): \[ 5 \frac{1}{3} = 5 \times 3 + 1 = 15 + 1 = 16 \quad \text{(improper fraction: } \frac{16}{3}\text{)} \] Now we write the ratio: \[ \frac{2 \frac{8}{9}}{5 \frac{1}{3}} = \frac{\frac{26}{9}}{\frac{16}{3}} \] To divide by a fraction, we multiply by its reciprocal: \[ \frac{26}{9} \div \frac{16}{3} = \frac{26}{9} \times \frac{3}{16} = \frac{26 \times 3}{9 \times 16} \] Calculating the numerator and the denominator: \[ = \frac{78}{144} \] Now we simplify \( \frac{78}{144} \). First, we find the greatest common divisor (GCD) of 78 and 144, which is 6. We divide both by 6: \[ \frac{78 \div 6}{144 \div 6} = \frac{13}{24} \] Therefore, the ratio \( 2 \frac{8}{9} \) to \( 5 \frac{1}{3} \) as a simplified fraction is: \[ \frac{13}{24} \]