Ratio Product Evaluation

We want $\frac{abc}{def} = \left(\frac{a}{f}\right) \left(\frac{b}{e}\right) \left(\frac{c}{d}\right)$. Note that: $\frac{a}{f} = \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{e} \cdot \frac{e}{f} = \frac{1}{3} \cdot 2 \cdot \frac{1}{2} \cdot 3 \cdot \frac{1}{4} = \frac{1}{4}$. $\frac{b}{e} = \frac{b}{c} \cdot \frac{c}{d} \cdot \frac{d}{e} = 2 \cdot \frac{1}{2} \cdot 3 = 3$. $\frac{c}{d} = \frac{1}{2}$. So $\frac{abc}{def} = \frac{1}{4} \times 3 \times \frac{1}{2} = \frac{3}{8}$.