Melting Cubes into a Large Cube

Volume of new cube $V = 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216\text{ cm}^3$. Side of the new cube $A = \sqrt[3]{216} = 6\text{ cm}$. Total surface area of the three smaller cubes $= 6(3^2 + 4^2 + 5^2) = 6(9 + 16 + 25) = 6(50) = 300\text{ cm}^2$. Total surface area of the large cube $= 6(6^2) = 216\text{ cm}^2$. Ratio $= \frac{300}{216} = \frac{25}{18}$.