To multiply the fractions \( \frac{3}{9} \) and \( \frac{4}{7} \), you multiply the numerators together and the denominators together:
\[ \frac{3 \times 4}{9 \times 7} = \frac{12}{63} \]
Next, simplify the fraction \( \frac{12}{63} \). The greatest common divisor (GCD) of 12 and 63 is 3. Dividing both the numerator and the denominator by 3:
\[ \frac{12 \div 3}{63 \div 3} = \frac{4}{21} \]
So,
\[ \frac{3}{9} \times \frac{4}{7} = \frac{4}{21} \]