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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 $\frac{z}{w}=\frac{6}{7}$ gives $$\frac{w}{z}=\frac{7}{6}.$$

Adding these results to the given value of $y/z$ gives the value we are looking for: $$\begin{array}{rl}\frac{x}{z}+\frac{y}{z}+\frac{w}{z}& =\frac{x+y+w}{z}\\ & =\frac{12}{50}+\frac{7}{6}+\frac{3}{10}\\ & =\frac{36}{150}+\frac{175}{150}+\frac{45}{150}\\ & =\frac{256}{150}\\ & =\boxed{\frac{128}{75}}.\end{array}$$