If
\[\frac{x}{y} = \frac{3}{4}, \frac{y}{z} = \frac{2}{3}, \;\text{and} \; \frac{z}{w} = \frac{5}{8}\]what is the value of $\frac{x + y + w}{w}$? Express your answer as a common fraction.
y/z * z/w = y/w = 5/12
y/w * x/y = x/w = 5/16
y/w + x/w + 1 = 166/96 = 83/48