A cube of edge 5 cm is cut into cubes each of edge of 1 cm. The ratio of the total surface area of one of the small cubes to that of the large cube is equal to :

Options

A. `1:125`
B. `1:5`
C. `1:625`
D. `1:25`

Answer

To solve the problem, we need to find the ratio of the total surface area of one of the small cubes (1 cm edge) to that of the large cube (5 cm edge).

### Step 1: Calculate the total surface area of the large cube.
The formula for the total surface area (TSA) of a cube is given by:
\[ \text{TSA} = 6 \times \text{side}^2 \]
For the large cube with an edge of 5 cm:
\[ \text{TSA}_{\text{large}} = 6 \times (5 \, \text{cm})^2 = 6 \times 25 \, \text{cm}^2 = 150 \, \text{cm}^2 \]

### Step 2: Calculate the total surface area of one small cube.
Using the same formula for the small cube with an edge of 1 cm:
\[ \text{TSA}_{\text{small}} = 6 \times (1 \, \text{cm})^2 = 6 \times 1 \, \text{cm}^2 = 6 \, \text{cm}^2 \]

### Step 3: Find the ratio of the total surface area of one small cube to that of the large cube.
Now, we can find the ratio:
\[ \text{Ratio} = \frac{\text{TSA}_{\text{small}}}{\text{TSA}_{\text{large}}} = \frac{6 \, \text{cm}^2}{150 \, \text{cm}^2} \]
Simplifying this ratio:
\[ \text{Ratio} = \frac{6}{150} = \frac{1}{25} \]

### Final Answer:
The ratio of the total surface area of one of the small cubes to that of the large cube is:
\[ \frac{1}{25} \]

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