Given the ratio of side lengths of two similar cubes is \( \frac{2}{5} \), we can derive the ratios for perimeter, surface area, and volume.
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Perimeter of one face:
- The perimeter of a face of a cube is \( 4 \times \text{side length} \).
- Using the ratio of side lengths \( \frac{\text{smaller}}{\text{larger}} = \frac{2}{5} \), the ratio of perimeters is: \[ \frac{\text{perimeter of smaller}}{\text{perimeter of larger}} = \frac{4 \times \text{side length of smaller}}{4 \times \text{side length of larger}} = \frac{\text{side length of smaller}}{\text{side length of larger}} = \frac{2}{5} \]
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Total surface area:
- The total surface area of a cube is \( 6 \times \text{side length}^2 \).
- The ratio of surface areas is: \[ \frac{\text{surface area of smaller}}{\text{surface area of larger}} = \frac{6 \times (\text{side length of smaller})^2}{6 \times (\text{side length of larger})^2} = \frac{(\text{side length of smaller})^2}{(\text{side length of larger})^2} = \left(\frac{2}{5}\right)^2 = \frac{4}{25} \]
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Volume:
- The volume of a cube is \( \text{side length}^3 \).
- The ratio of volumes is: \[ \frac{\text{volume of smaller}}{\text{volume of larger}} = \frac{(\text{side length of smaller})^3}{(\text{side length of larger})^3} = \left(\frac{2}{5}\right)^3 = \frac{8}{125} \]
Final Ratios:
- Perimeter of smaller to larger: \( \frac{2}{5} \)
- Surface area of smaller to larger: \( \frac{4}{25} \)
- Volume of smaller to larger: \( \frac{8}{125} \)