A precious stone worth Rs 6800 is dropped accidently and breaks into three pieces: The weight of three pieces are in the ratio 5: 7: 8. The value of the stone is proportional to the square of its weight. Find the loss.

Options

A. Rs 4260
B. Rs 4273
C. Rs 4454
D. Rs 3250

Answer

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

### Step 1: Understand the given information
The precious stone is worth Rs. 6800 and breaks into three pieces with weights in the ratio of 5:7:8.

### Step 2: Assign weights to the pieces
Let the weight of the pieces be represented as:
- First piece = 5x
- Second piece = 7x
- Third piece = 8x

### Step 3: Calculate the total weight
The total weight of the pieces can be calculated as:
\[ \text{Total weight} = 5x + 7x + 8x = 20x \]

### Step 4: Establish the relationship between value and weight
According to the problem, the value of the stone is proportional to the square of its weight. Thus, we can express the value of the entire stone as:
\[ \text{Value} = k \cdot (20x)^2 \]
Where \( k \) is a constant of proportionality.

### Step 5: Set up the equation for the total value
We know that the total value of the stone is Rs. 6800:
\[ k \cdot (20x)^2 = 6800 \]

### Step 6: Calculate the value of each piece
The value of each piece can be calculated based on their weights:
- Value of first piece = \( k \cdot (5x)^2 = 25kx^2 \)
- Value of second piece = \( k \cdot (7x)^2 = 49kx^2 \)
- Value of third piece = \( k \cdot (8x)^2 = 64kx^2 \)

### Step 7: Calculate the total value of the pieces
The total value of the pieces can be calculated as:
\[ \text{Total value of pieces} = 25kx^2 + 49kx^2 + 64kx^2 = 138kx^2 \]

### Step 8: Relate the two equations
From Step 5, we have:
\[ k \cdot (20x)^2 = 6800 \]
This simplifies to:
\[ k \cdot 400x^2 = 6800 \]
Thus:
\[ k \cdot x^2 = \frac{6800}{400} = 17 \]

### Step 9: Substitute back to find the total value of the pieces
Now substituting \( kx^2 = 17 \) into the total value of the pieces:
\[ \text{Total value of pieces} = 138 \cdot 17 = 2346 \]

### Step 10: Calculate the loss
The loss incurred when the stone broke can be calculated as:
\[ \text{Loss} = \text{Original Value} - \text{Total Value of Pieces} \]
\[ \text{Loss} = 6800 - 2346 = 4454 \]

### Final Answer
The loss incurred by the owner due to the breakage of the stone is Rs. 4454.

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