G10.3The fraction $\frac{p}{q}$ is in its simplest form. If $\frac{7}{10}<\frac{p}{q}<\frac{11}{15}$ where $q$ is the smallest possible positive integer and $c=p q$. Find the value of $c$.
$$
\begin{array}{l}
\frac{7}{10}1-\frac{p}{q}>1-\frac{11}{15} \\
\frac{3}{10}>\frac{q-p}{q}>\frac{4}{15} \Rightarrow \frac{10}{3}<\frac{q}{q-p}<\frac{15}{4} \\
\frac{1}{3}<\frac{q}{q-p}-3<\frac{3}{4} \Rightarrow \frac{1}{3}<\frac{3 p-2 q}{q-p}<\frac{3}{4} \\
\frac{3 p-2 q}{q-p}=\frac{1}{2} \Rightarrow 3 p-2 q=1, q-p=2
\end{array}
$$

Solving the equations gives $p=5, q=7, \frac{7}{10}<\frac{5}{7}<\frac{11}{15}, c=35$