In a particular country the value of diamond is directly proportional to the surface area ( exposed ) of the diamond. Four thieves steel a cubical diamond piece and then divide equally in four parts . What is the maximum percentage increase in the value of diamond after cutting it ?

Options

A. `50%`
B. `66.66%`
C. `100%`
D. none of these

Answer

To solve the problem, we need to determine the maximum percentage increase in the value of the diamond after it is cut into four equal parts. The value of the diamond is directly proportional to its surface area.

### Step-by-Step Solution:

1. **Determine the surface area of the original cube:**
- Let the side length of the original cube be \( s \).
- The surface area \( A \) of a cube is given by the formula:
\[ A = 6s^2 \]
- For our example, let's assume \( s = 2 \) (as mentioned in the transcript).
- Thus, the surface area of the original cube is:
\[ A = 6 \times (2^2) = 6 \times 4 = 24 \text{ square units} \]

2. **Calculate the side length of each smaller cube after cutting:**
- When the cube is divided into four equal parts, the side length of each smaller cube becomes:
\[ s' = \frac{s}{2} = \frac{2}{2} = 1 \]

3. **Determine the surface area of one smaller cube:**
- The surface area \( A' \) of one smaller cube is:
\[ A' = 6(s')^2 = 6 \times (1^2) = 6 \text{ square units} \]

4. **Calculate the total surface area of all four smaller cubes:**
- Since there are four smaller cubes, the total surface area \( A_{\text{total}} \) of the four smaller cubes is:
\[ A_{\text{total}} = 4 \times A' = 4 \times 6 = 24 \text{ square units} \]

5. **Compare the surface areas:**
- The surface area of the original cube is 24 square units.
- The total surface area of the four smaller cubes is also 24 square units.
- Therefore, there is no increase in surface area after cutting.

6. **Calculate the percentage increase in value:**
- Since the value of the diamond is directly proportional to the surface area, and the surface area remains the same, the percentage increase in value is:
\[ \text{Percentage Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 = \frac{0}{\text{Original Value}} \times 100 = 0\% \]

### Conclusion:
The maximum percentage increase in the value of the diamond after cutting it into four equal parts is **0%**.