problem: If \[\frac{x}{y} = \frac{4}{5}, \; \frac{y}{z} = \frac{3}{10}, \;\text{and} \; \frac{z}{w} = \frac{6}{7},\] what is the value of $\dfrac{x + y + w}{z}$? Express your answer as a common fraction.
solution: If we multiply the first two fractions, we can find the value of $x/z$: $$\frac{x}{y}\cdot\frac{y}{z}=\frac{x}{z}=\frac{4}{5}\cdot\frac{3}{10}=\frac{12}{50}.$$
  
  Inverting the given $\dfrac{z}{w} = \dfrac{6}{7}$ gives $$\frac{w}{z}=\frac{7}{6}.$$
  
  Adding these results to the given value of $y/z$ gives the value we are looking for: \begin{align*}
  \frac{x}{z}+\frac{y}{z}+\frac{w}{z}&=\frac{x+y+w}{z} \\&= \frac{12}{50}+\frac{7}{6}+\frac{3}{10}\\
  & = \frac{36}{150}+\frac{175}{150}+\frac{45}{150}\\
  & = \frac{256}{150} \\
  &= \boxed{\frac{128}{75}}.\end{align*}