If \(y\) is proportional to \(z\), and \(y=3\) when \(z=8\):

Find \(y\) when \(z=10\) \begin{align}y & \propto z\\ So \quad y & =kz \\ By \; substitution \quad k & =\frac{3}{8}\\ \therefore \quad y & =\frac{3}{8}z\\ And \; when \; z & =10\\ y & =\frac{3}{8}(10)\\ \therefore \quad y & =\frac{30}{8} \end{align}

Find \(z\) when \(y=7\) \begin{align}y & \propto z\\ So \quad y & =kz \\ By \; substitution \quad k & =\frac{3}{8}\\ \therefore \quad y & =\frac{3}{8}z\\ And \; when \; y & =7\\ 7 & =\frac{3}{8}(z)\\ z & =\frac{8}{3}(7)\\ \therefore \quad z & =\frac{56}{3} \end{align}