Write the ratio \( 4 \frac{1}{3} \) to \( 3 \frac{2}{3} \) as a simplified fraction.

To express the ratio \( 4 \frac{1}{3} \) to \( 3 \frac{2}{3} \) as a simplified fraction, we first convert the mixed numbers to improper fractions. For \( 4 \frac{1}{3} \): \[ 4 \frac{1}{3} = 4 \times 3 + 1 = 12 + 1 = 13 \] So, \( 4 \frac{1}{3} = \frac{13}{3} \). For \( 3 \frac{2}{3} \): \[ 3 \frac{2}{3} = 3 \times 3 + 2 = 9 + 2 = 11 \] Thus, \( 3 \frac{2}{3} = \frac{11}{3} \). Now, we can write the ratio: \[ \frac{4 \frac{1}{3}}{3 \frac{2}{3}} = \frac{\frac{13}{3}}{\frac{11}{3}}. \] Dividing fractions is the same as multiplying by the reciprocal: \[ \frac{\frac{13}{3}}{\frac{11}{3}} = \frac{13}{3} \times \frac{3}{11} = \frac{13 \times 3}{3 \times 11}. \] The \( 3 \) in the numerator and denominator cancels out: \[ = \frac{13}{11}. \] Thus, the simplified fraction for the ratio \( 4 \frac{1}{3} \) to \( 3 \frac{2}{3} \) is \( \frac{13}{11} \).