19.2.1 * Find the sum of all positive rational numbers less than 10 that, when written as reduced fractions, have a denominator of 30.
Proper fractions less than 10 with a denominator of 30 can be written as
$$
\frac{30 n+r}{30}, 0 \leqslant n \leqslant 9,0 \leqslant r<30,(r, 30)-1 .
$$

Therefore, the values of $r$ form the set $\{1,7,11,13,17,19,23,29\}$. Hence, the required sum is
$$
\begin{aligned}
& 8 \times(1+2+\cdots+9)+10 \times\left(\frac{1}{30}+\frac{7}{30}+\frac{11}{30}+\frac{13}{30}+\frac{17}{30}+\frac{19}{30}+\frac{23}{30}+\frac{29}{30}\right) . \\
= & 360+40=400 .
\end{aligned}
$$