Problem 598

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
$$\begin{aligned} & 8 \times(1+2+\cdots+9)+10 \times\left(\frac{1}{30}+\frac{7}{30}+\frac{11}{30}+\frac{13}{30}+\frac{17}{30}+\frac{19}{30}+\frac{23}{30}+\frac{29}{30}\right) . \\ = & 360+40=400 . \end{aligned}$$