A diamond falls and breaks into pieces whose weights are in the ratio ` 2 : 3 : 5`. The value of the diamond is directly proportional to the square of its weight. Find the loss incurred, if the actual cost of the diamond is Rs. 96, 000 . (in Rs.)

Options

A. ` 36, 480`
B. ` 59, 520`
C. ` 72,960`
D. None of these

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the weights of the diamond pieces
The weights of the pieces are in the ratio 2:3:5. Let's assume the weights of the pieces are:
- Weight of the first piece = 2x
- Weight of the second piece = 3x
- Weight of the third piece = 5x

### Step 2: Calculate the total weight of the diamond
The total weight of the diamond is the sum of the weights of all pieces:
\[ \text{Total weight} = 2x + 3x + 5x = 10x \]

### Step 3: Find the original value of the diamond
We are given that the actual cost of the diamond is Rs. 96,000. The value of the diamond is directly proportional to the square of its weight. Therefore, we can express the original value \( P_1 \) as:
\[ P_1 \propto (10x)^2 = 100x^2 \]
Since the original cost is Rs. 96,000, we have:
\[ 100x^2 = 96,000 \]
From this, we can solve for \( x^2 \):
\[ x^2 = \frac{96,000}{100} = 960 \]

### Step 4: Calculate the new values of the diamond pieces
Next, we calculate the new value \( P_2 \) after the diamond breaks. The value of each piece is proportional to the square of its weight:
- Value of the first piece = \( 2x \) → \( (2x)^2 = 4x^2 \)
- Value of the second piece = \( 3x \) → \( (3x)^2 = 9x^2 \)
- Value of the third piece = \( 5x \) → \( (5x)^2 = 25x^2 \)

Now, we can find the total new value \( P_2 \):
\[ P_2 = 4x^2 + 9x^2 + 25x^2 = 38x^2 \]

### Step 5: Substitute the value of \( x^2 \)
Substituting \( x^2 = 960 \) into the equation for \( P_2 \):
\[ P_2 = 38 \times 960 = 36,480 \]

### Step 6: Calculate the loss incurred
Now, we find the loss incurred by subtracting the new value from the original value:
\[ \text{Loss} = P_1 - P_2 = 96,000 - 36,480 = 59,520 \]

### Final Answer
The loss incurred is Rs. 59,520.

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