problem: If $3x=8y$ and $5y=15z$, what is the value of $\frac{x}{z}$? Express your answer in simplest form.
solution: Rewrite the given equations as $\frac{x}{y}=\frac{8}{3}$ and $\frac{y}{z}=\frac{15}{5}$.  Multiply these equations to find $\frac{x}{z}=\frac{8}{3}\cdot\frac{15}{5}=\boxed{8}$.