1. **State the problem:** We have a rectangular box with dimensions 8 inches, 20 inches, and 40 inches. The dimensions of the box are reduced by half, and we need to find the constant of proportionality between the volume of the new box and the volume of the original box. 2. **Formula for volume of a rectangular box:** $$V = l \times w \times h$$ where $l$, $w$, and $h$ are the length, width, and height respectively. 3. **Calculate the original volume:** $$V_{original} = 8 \times 20 \times 40 = 6400$$ 4. **Calculate the new dimensions:** Each dimension is reduced by half: $$l_{new} = \frac{8}{2} = 4$$ $$w_{new} = \frac{20}{2} = 10$$ $$h_{new} = \frac{40}{2} = 20$$ 5. **Calculate the new volume:** $$V_{new} = 4 \times 10 \times 20 = 800$$ 6. **Find the constant of proportionality:** $$\text{constant} = \frac{V_{new}}{V_{original}} = \frac{800}{6400}$$ 7. **Simplify the fraction:** $$\frac{800}{6400} = \frac{\cancel{800}}{\cancel{6400}} = \frac{1}{8}$$ **Answer:** The constant of proportionality is $\boxed{\frac{1}{8}}$, which corresponds to option C.