Calculate Total Surface Area of the Larger Cube

The total surface area of a cube of side $L = 5\text{ cm}$ is given by $6L^2$.

$TSA_{\text{larger}} = 6 \times 5^2 = 150\text{ cm}^2$

Find the Number of Smaller Cubes

Dividing the volume of the larger cube by the volume of a single smaller cube gives the total number of smaller cubes $n$.

$n = \frac{5^3}{1^3} = 125$

Calculate Sum of Total Surface Areas of All Smaller Cubes

The surface area of one smaller cube is $6 \times 1^2 = 6\text{ cm}^2$. For $125$ cubes, multiply by $125$.

$TSA_{\text{smaller sum}} = 125 \times (6 \times 1^2) = 750\text{ cm}^2$

Determine the Required Ratio

Divide the total surface area of the larger cube by the sum of surface areas of all smaller cubes.

$\text{Ratio} = \frac{150}{750} = \frac{1}{5} = 1: 5$