To multiply the fractions \( \frac{5}{3} \) and \( \frac{5}{7} \), you multiply the numerators together and the denominators together:
\[ \frac{5}{3} \times \frac{5}{7} = \frac{5 \times 5}{3 \times 7} = \frac{25}{21} \]
Now, to convert \( \frac{25}{21} \) to a mixed number, we divide the numerator by the denominator:
\[ 25 \div 21 = 1 \quad \text{(with a remainder of 4)} \]
This gives us:
\[ \frac{25}{21} = 1 \frac{4}{21} \]
Thus, the final answer is:
\[ 1 \frac{4}{21} \]