Write the ratio \( 6 \frac{2}{3} \) to \( 3 \frac{1}{7} \) as a simplified fraction.

To write the ratio \( 6 \frac{2}{3} \) to \( 3 \frac{1}{7} \) as a simplified fraction, we first convert the mixed numbers to improper fractions. For \( 6 \frac{2}{3} \): \[ 6 \frac{2}{3} = 6 + \frac{2}{3} = \frac{6 \cdot 3}{3} + \frac{2}{3} = \frac{18}{3} + \frac{2}{3} = \frac{20}{3} \] For \( 3 \frac{1}{7} \): \[ 3 \frac{1}{7} = 3 + \frac{1}{7} = \frac{3 \cdot 7}{7} + \frac{1}{7} = \frac{21}{7} + \frac{1}{7} = \frac{22}{7} \] Now we can express the ratio as a fraction: \[ \frac{6 \frac{2}{3}}{3 \frac{1}{7}} = \frac{\frac{20}{3}}{\frac{22}{7}} = \frac{20}{3} \times \frac{7}{22} = \frac{20 \cdot 7}{3 \cdot 22} = \frac{140}{66} \] Next, we simplify \( \frac{140}{66} \) by finding the greatest common divisor (GCD) of 140 and 66. The GCD is 2. Dividing both the numerator and denominator by 2 gives: \[ \frac{140 \div 2}{66 \div 2} = \frac{70}{33} \] Thus, the simplified fraction representing the ratio \( 6 \frac{2}{3} \) to \( 3 \frac{1}{7} \) is \[ \frac{70}{33}. \]