Three equal cubes are placed adjacently in a row. Find the ratio of the total surface area of the resulting cuboid to that of the sum of the total surface areas of the three cubes.

Answer

To solve the problem, we need to find the ratio of the total surface area of the resulting cuboid formed by three equal cubes placed in a row to the total surface area of the three individual cubes.

### Step-by-Step Solution:

1. **Define the side length of the cube**:
Let the side length of each cube be \( a \).

2. **Determine the dimensions of the resulting cuboid**:
When three equal cubes are placed adjacently, the dimensions of the resulting cuboid will be:
- Length = \( 3a \) (since there are three cubes in a row)
- Breadth = \( a \) (the same as the side of the cube)
- Height = \( a \) (the same as the side of the cube)

3. **Calculate the total surface area of the cuboid**:
The formula for the total surface area (TSA) of a cuboid is given by:
\[ \text{TSA} = 2 \times (l \times b + b \times h + h \times l) \]
Substituting the dimensions of the cuboid:
\[ \text{TSA}_{\text{cuboid}} = 2 \times (3a \times a + a \times a + a \times 3a) \]
Simplifying this:
\[ = 2 \times (3a^2 + a^2 + 3a^2) = 2 \times (7a^2) = 14a^2 \]

4. **Calculate the total surface area of one cube**:
The total surface area of one cube is given by:
\[ \text{TSA}_{\text{cube}} = 6a^2 \]

5. **Calculate the total surface area of three cubes**:
Since there are three cubes, the total surface area will be:
\[ \text{TSA}_{\text{3 cubes}} = 3 \times \text{TSA}_{\text{cube}} = 3 \times 6a^2 = 18a^2 \]

6. **Calculate the ratio of the total surface area of the cuboid to the total surface area of the three cubes**:
The ratio is given by:
\[ \text{Ratio} = \frac{\text{TSA}_{\text{cuboid}}}{\text{TSA}_{\text{3 cubes}}} = \frac{14a^2}{18a^2} \]
Simplifying this:
\[ = \frac{14}{18} = \frac{7}{9} \]

### Final Answer:
The ratio of the total surface area of the resulting cuboid to that of the sum of the total surface areas of the three cubes is \( 7:9 \).