Three smaller cubes with side lengths of 3 cm, 4 cm, and 5 cm are melted and reshaped into a single large cube. What is the ratio of the combined surface area of the three small cubes to the surface area of the large cube?

Explanation

First, calculate the total volume of the three smaller cubes: 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 cm³. The side length of the large cube formed by this volume is the cube root of 216, which is 6 cm. Now, find the total surface area of the smaller cubes: 6 × (3² + 4² + 5²) = 6 × (9 + 16 + 25) = 6 × 50 = 300 cm². The surface area of the large cube is 6 × 6² = 6 × 36 = 216 cm². Therefore, the ratio of the total surface areas is 300 : 216, which simplifies to 25 : 18.