To multiply the fractions \( \frac{3}{10} \) and \( \frac{8}{7} \), you multiply the numerators together and the denominators together:
\[ \frac{3}{10} \times \frac{8}{7} = \frac{3 \times 8}{10 \times 7} = \frac{24}{70} \]
Next, we can simplify \( \frac{24}{70} \) by finding the greatest common divisor (GCD) of 24 and 70. Both numbers can be divided by 2:
\[ \frac{24 \div 2}{70 \div 2} = \frac{12}{35} \]
So, the simplified result is:
\[ \frac{12}{35} \]