The cost of a diamond varies directly as the square of Its weight. A diamond broke into four pieces with their weights in the ratio of `1 : 2 : 3 : 4`. If the loss in total value of the diamond was Rs. 70,000. what was the price of the original diamond?

Options

A. Rs. 1.00,000
B. Rs. 1,40,000
C. Rs. 1,50,000
D. Rs. 1,75,000

Answer

To solve the problem step-by-step, we will follow the logic presented in the video transcript.

### Step 1: Understand the relationship between weight and cost
The cost of a diamond varies directly as the square of its weight. This means if the weight of the diamond is \( w \), then the cost \( C \) can be expressed as:
\[ C = k \cdot w^2 \]
where \( k \) is a constant of proportionality.

### Step 2: Determine the weights of the diamond pieces
The diamond broke into four pieces with weights in the ratio \( 1:2:3:4 \). Let's denote the weights of the pieces as:
- Weight of piece 1 = \( 1x \)
- Weight of piece 2 = \( 2x \)
- Weight of piece 3 = \( 3x \)
- Weight of piece 4 = \( 4x \)

### Step 3: Calculate the total weight
The total weight \( W \) of the diamond is:
\[ W = 1x + 2x + 3x + 4x = 10x \]

### Step 4: Calculate the cost of each piece
Using the relationship \( C = k \cdot w^2 \), we can find the cost of each piece:
- Cost of piece 1 = \( k \cdot (1x)^2 = k \cdot 1x^2 \)
- Cost of piece 2 = \( k \cdot (2x)^2 = k \cdot 4x^2 \)
- Cost of piece 3 = \( k \cdot (3x)^2 = k \cdot 9x^2 \)
- Cost of piece 4 = \( k \cdot (4x)^2 = k \cdot 16x^2 \)

### Step 5: Calculate the total cost of the diamond
The total cost \( C_{total} \) of the original diamond is:
\[ C_{total} = k \cdot 1x^2 + k \cdot 4x^2 + k \cdot 9x^2 + k \cdot 16x^2 = k(1 + 4 + 9 + 16)x^2 = k \cdot 30x^2 \]

### Step 6: Determine the loss in value
The problem states that the loss in total value of the diamond was Rs. 70,000. The original value of the diamond was \( C_{total} \) and the value after breaking was Rs. 70,000 less:
\[ C_{total} - C_{broken} = 70,000 \]

### Step 7: Calculate the cost of the broken pieces
The cost of the broken pieces is:
\[ C_{broken} = k \cdot (1 + 4 + 9 + 16)x^2 = k \cdot 30x^2 \]

### Step 8: Set up the equation
Since the loss is Rs. 70,000:
\[ C_{total} - C_{broken} = 70,000 \]
This implies:
\[ k \cdot 30x^2 - k \cdot 30x^2 = 70,000 \]
This shows that the total cost of the original diamond was:
\[ C_{total} = k \cdot 30x^2 \]

### Step 9: Relate the loss to the total cost
From the loss of Rs. 70,000, we can relate it to the cost:
If the loss corresponds to 70 parts (since the total cost is 100 parts), we can set up the ratio:
\[ \frac{70,000}{70} = 1,000 \text{ (value of one part)} \]

### Step 10: Calculate the original price of the diamond
To find the original price (100 parts):
\[ \text{Original Price} = 1,000 \times 100 = 100,000 \]

### Final Answer
The price of the original diamond was Rs. 100,000.

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