The value of a diamond is directly proportional to the square of its weight. A diamond unfortunately breaks into three pieces with weights in the ratio of 3 : 4 : 5 thus a loss of ₹ 9.4 lakh is incurred. What is the actual value of diamond :

Options

A. 28.8 lakh
B. 13.5 lakh
C. 14.4 lakh
D. 18.8 lakh

Answer

To solve the problem, we need to determine the actual value of the diamond based on the information given about its weight and the loss incurred after it breaks into pieces.

### Step-by-Step Solution:

1. **Understanding the Proportionality**:
The value of a diamond (V) is directly proportional to the square of its weight (W). This can be expressed as:
\[ V \propto W^2 \]
This means that if the weight of the diamond changes, the value changes according to the square of that weight.

2. **Weight Ratio**:
The diamond breaks into three pieces with weights in the ratio of 3:4:5. Let's denote the weights of the pieces as:
- Weight of piece 1 = 3x
- Weight of piece 2 = 4x
- Weight of piece 3 = 5x

3. **Total Weight Calculation**:
The total weight (W_total) of the diamond before it broke is:
\[ W_{total} = 3x + 4x + 5x = 12x \]

4. **Value Calculation Before Breaking**:
The value of the diamond before breaking can be expressed in terms of its total weight:
\[ V_{total} = k \cdot (W_{total})^2 = k \cdot (12x)^2 = 144k \cdot x^2 \]
where k is the constant of proportionality.

5. **Value Calculation After Breaking**:
We need to calculate the value of each piece after breaking:
- Value of piece 1 = \( k \cdot (3x)^2 = 9k \cdot x^2 \)
- Value of piece 2 = \( k \cdot (4x)^2 = 16k \cdot x^2 \)
- Value of piece 3 = \( k \cdot (5x)^2 = 25k \cdot x^2 \)

The total value after breaking is:
\[ V_{after} = 9k \cdot x^2 + 16k \cdot x^2 + 25k \cdot x^2 = 50k \cdot x^2 \]

6. **Loss Calculation**:
The loss incurred due to breaking the diamond is given as ₹ 9.4 lakh. Therefore, we can set up the equation:
\[ V_{total} - V_{after} = 9.4 \text{ lakh} \]
Substituting the values we have:
\[ 144k \cdot x^2 - 50k \cdot x^2 = 9.4 \text{ lakh} \]
Simplifying this gives:
\[ 94k \cdot x^2 = 9.4 \text{ lakh} \]

7. **Finding the Value of kx²**:
Dividing both sides by 94:
\[ k \cdot x^2 = \frac{9.4 \text{ lakh}}{94} = 0.1 \text{ lakh} = 10,000 \]

8. **Calculating the Actual Value of the Diamond**:
Now, substituting back to find the total value of the diamond:
\[ V_{total} = 144k \cdot x^2 = 144 \cdot 10,000 = 1,440,000 \text{ (or ₹ 14.4 lakh)} \]

### Final Answer:
The actual value of the diamond is ₹ 14.4 lakh.