A cube of side 4 cm cut into small cubes of each side 1 cm. The ratio of the surface area of all smaller cubes to that of large one is

Options

A. `1:2`
B. `1:4`
C. `4:1`
D. `2:1`

Answer

To solve the problem, we need to find the ratio of the surface area of all the smaller cubes to that of the larger cube.

### Step-by-Step Solution:

1. **Calculate the volume of the large cube:**
The side of the large cube is given as 4 cm.
\[ \text{Volume of the large cube} = \text{side}^3 = 4^3 = 64 \text{ cm}^3 \]

2. **Calculate the volume of a small cube:**
The side of each small cube is given as 1 cm.
\[ \text{Volume of the small cube} = \text{side}^3 = 1^3 = 1 \text{ cm}^3 \]

3. **Determine the total number of small cubes:**
To find the total number of small cubes, divide the volume of the large cube by the volume of one small cube.
\[ n = \frac{\text{Volume of the large cube}}{\text{Volume of the small cube}} = \frac{64}{1} = 64 \]

4. **Calculate the surface area of the large cube:**
The surface area \( A \) of a cube is given by the formula:
\[ A = 6 \times \text{side}^2 \]
For the large cube:
\[ \text{Surface area of the large cube} = 6 \times 4^2 = 6 \times 16 = 96 \text{ cm}^2 \]

5. **Calculate the surface area of one small cube:**
Using the same formula for surface area:
\[ \text{Surface area of the small cube} = 6 \times 1^2 = 6 \text{ cm}^2 \]

6. **Calculate the total surface area of all small cubes:**
Multiply the surface area of one small cube by the total number of small cubes:
\[ \text{Total surface area of small cubes} = n \times \text{Surface area of one small cube} = 64 \times 6 = 384 \text{ cm}^2 \]

7. **Find the ratio of the surface area of all smaller cubes to that of the large cube:**
\[ \text{Ratio} = \frac{\text{Total surface area of small cubes}}{\text{Surface area of large cube}} = \frac{384}{96} = 4 \]

8. **Express the ratio in the form of a:b:**
The ratio can be expressed as:
\[ 4:1 \]

### Final Answer:
The ratio of the surface area of all smaller cubes to that of the large cube is \( 4:1 \).