1. (10 points) Calculate: $39 \frac{18}{19} \times 18 \frac{19}{20}=757 \frac{1}{380}$ -
【Analysis】Analysis: Convert the two mixed numbers into improper fractions first, there's no need to calculate the numerators, then write them in the form of "integer - fractional unit", and calculate accordingly.

【Solution】Solution: $39 \frac{18}{19} \times 18 \frac{19}{20}$,
$$
\begin{array}{l}
=\frac{18+39 \times 19}{19} \times \frac{19+18 \times 20}{20}, \\
=\left(40-\frac{1}{19}\right) \times\left(19-\frac{1}{20}\right), \\
=40 \times 19-40 \times \frac{1}{20}-19 \times \frac{1}{19}+\frac{1}{19} \times \frac{1}{20}, \\
=760-2-1+\frac{1}{380}, \\
=757 \frac{1}{380} ;
\end{array}
$$

Therefore, the answer is: $757 \frac{1}{380}$.