There are six faces in a cube . Rajeev fix one cube on each of the faces . The dimensions of all the cubes are same. What is the ratio of total surface area of the newly formed solid to the area of a single cube ?

Options

A. `7:1`
B. `6:1`
C. `5:1`
D. `41:9`

Answer

To solve the problem, we need to find the ratio of the total surface area of the newly formed solid (which consists of one cube fixed on each face of the original cube) to the surface area of a single cube.

### Step-by-Step Solution:

1. **Understand the Dimensions of the Cube**:
Let's assume the side length of the original cube is \( a \). The surface area of a single cube is given by the formula:
\[ \text{Surface Area of a Cube} = 6a^2 \]

2. **Calculate the Surface Area of One Cube**:
Since we are assuming the side length \( a = 1 \) for simplicity, the surface area of one cube becomes:
\[ \text{Surface Area of One Cube} = 6 \times 1^2 = 6 \]

3. **Determine the Surface Area of the Newly Formed Solid**:
When a cube is placed on each face of the original cube, each cube has 5 of its faces exposed (since one face is attached to the original cube).

- The surface area of one of these smaller cubes is:
\[ \text{Surface Area of One Smaller Cube} = 5 \times 1^2 = 5 \]

- Since there are 6 smaller cubes (one on each face), the total surface area contributed by all 6 cubes is:
\[ \text{Total Surface Area of 6 Cubes} = 6 \times 5 = 30 \]

4. **Calculate the Total Surface Area of the Newly Formed Solid**:
The original cube has a surface area of 6, but since the cubes are attached to it, we need to account for the fact that the faces of the original cube that are connected to the smaller cubes do not contribute to the total surface area.

- The total surface area of the newly formed solid is:
\[ \text{Total Surface Area of Newly Formed Solid} = \text{Surface Area of Original Cube} + \text{Surface Area of 6 Smaller Cubes} - \text{Area of 6 Faces of Original Cube} \]
\[ = 6 + 30 - 6 = 30 \]

5. **Find the Ratio of Total Surface Area of Newly Formed Solid to the Area of a Single Cube**:
Now, we can find the ratio:
\[ \text{Ratio} = \frac{\text{Total Surface Area of Newly Formed Solid}}{\text{Surface Area of One Cube}} = \frac{30}{6} = 5 \]

### Final Answer:
The ratio of the total surface area of the newly formed solid to the area of a single cube is \( 5:1 \).