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 of finding 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, we can follow these steps:

### Step 1: Understand the problem
We have three equal cubes placed adjacently in a row. We need to find the total surface area of the cuboid formed and compare it to the total surface area of the three individual cubes.

### Step 2: Define the side length of the cubes
Let the side length of each cube be denoted as \( A \).

### Step 3: Calculate the total surface area of one cube
The total surface area (TSA) of one cube is given by the formula:
\[ \text{TSA of one cube} = 6A^2 \]

### Step 4: Calculate the total surface area of three cubes
Since there are three cubes, the total surface area of the three cubes is:
\[ \text{Sum of TSA of three cubes} = 3 \times \text{TSA of one cube} = 3 \times 6A^2 = 18A^2 \]

### Step 5: Determine the dimensions of the cuboid
When the three cubes are placed adjacently, they form a cuboid. The dimensions of the cuboid are:
- Length = \( 3A \) (since there are three cubes lined up)
- Width = \( A \) (same as the side of the cube)
- Height = \( A \) (same as the side of the cube)

### Step 6: Calculate the total surface area of the cuboid
The total surface area of a cuboid is given by the formula:
\[ \text{TSA of cuboid} = 2 \times (l \times w + w \times h + h \times l) \]
Substituting the dimensions:
\[ \text{TSA of cuboid} = 2 \times (3A \times A + A \times A + A \times 3A) \]
Calculating each term:
- \( 3A \times A = 3A^2 \)
- \( A \times A = A^2 \)
- \( A \times 3A = 3A^2 \)

Now, substituting back:
\[ \text{TSA of cuboid} = 2 \times (3A^2 + A^2 + 3A^2) = 2 \times (7A^2) = 14A^2 \]

### Step 7: Find the ratio of the surface areas
Now, we need to find the ratio of the total surface area of the cuboid to the sum of the total surface areas of the three cubes:
\[ \text{Ratio} = \frac{\text{TSA of cuboid}}{\text{Sum of TSA of cubes}} = \frac{14A^2}{18A^2} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{14}{18} = \frac{7}{9} \]

### 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:
\[ \text{Ratio} = 7 : 9 \]