$15 \cdot 18$ in simplest form has a denominator of 30, find the sum of all such positive rational numbers less than 10.
(10th American Invitational Mathematics Examination, 1992)
[Solution] Fractions less than 10, with a denominator of 30, can be written in the form $\frac{30 n+r}{30}$, where $n$ and $r$ are integers satisfying $0 \leqslant n \leqslant 9,0 \leqslant r<30$.
The fraction $\frac{30 n+r}{30}$ is in its simplest form if and only if $r$ is coprime with 30, so we should have
$$
r \in\{1,7,11,13,17,19,23,29\} \text {. }
$$

Thus, $n$ has 10 possible choices, and $r$ has 8 possible choices, and no two pairs $(n, r)$ can make $\frac{30 n+r}{30}$ have the same value.
Therefore, the sum in question has a total of $8 \cdot 10=80$ terms.
Furthermore, if $\frac{k}{30}$ satisfies the problem's conditions, then $10-\frac{k}{30}$ also satisfies the conditions, allowing us to pair $\frac{k}{30}$ with $\left(10-\frac{k}{30}\right)$ for calculation.
Thus, the sum of all such fractions is
$$
\begin{array}{l}
\left(\frac{1}{30}+\frac{300-1}{30}\right)+\left(\frac{7}{30}+\frac{300-7}{30}\right)+\cdots+\left(\frac{149}{30}+\frac{300-149}{30}\right) \\
=40 \cdot 10=400
\end{array}
$$