A diamond broke into three pait, the ratio of weight of three part is 2:3:4. The price of diamond is directely proportional to its square weight. If the loss of 52000 after breaking diamond then what was the starting price of diamond

Options

A. 80000
B. 50000
C. 60000
D. 81000

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the weights of the diamond parts
The diamond broke into three parts with a weight ratio of 2:3:4. We can express the weights of the three parts in terms of a variable \( k \):
- Weight 1 = \( 2k \)
- Weight 2 = \( 3k \)
- Weight 3 = \( 4k \)

### Step 2: Calculate the total weight of the diamond
The total weight of the diamond before it broke is the sum of the weights of the three parts:
\[ \text{Total Weight} = 2k + 3k + 4k = 9k \]

### Step 3: Establish the relationship between price and weight
The price of the diamond is directly proportional to the square of its weight. We can express this relationship mathematically:
\[ \text{Price} \propto \text{Weight}^2 \]
Let \( x \) be the constant of proportionality. Therefore, the price can be expressed as:
\[ \text{Price} = x \cdot w^2 \]

### Step 4: Calculate the price of each part after breaking
Now, we calculate the price of each part after the diamond breaks:
- Price of Weight 1 = \( x \cdot (2k)^2 = x \cdot 4k^2 \)
- Price of Weight 2 = \( x \cdot (3k)^2 = x \cdot 9k^2 \)
- Price of Weight 3 = \( x \cdot (4k)^2 = x \cdot 16k^2 \)

### Step 5: Calculate the total price after breaking
The total price after breaking the diamond is:
\[ \text{Total Price after breaking} = x \cdot (4k^2 + 9k^2 + 16k^2) = x \cdot 29k^2 \]

### Step 6: Calculate the price of the whole diamond before breaking
The price of the whole diamond before it broke is:
\[ \text{Price before breaking} = x \cdot (9k)^2 = x \cdot 81k^2 \]

### Step 7: Set up the equation for the loss
According to the problem, the loss after breaking the diamond is \( 52,000 \). Therefore, we can set up the equation:
\[ \text{Price before breaking} - \text{Total Price after breaking} = 52,000 \]
Substituting the expressions we derived:
\[ x \cdot 81k^2 - x \cdot 29k^2 = 52,000 \]
This simplifies to:
\[ x \cdot 52k^2 = 52,000 \]

### Step 8: Solve for \( k^2 x \)
Dividing both sides by \( 52 \):
\[ k^2 x = 1,000 \]

### Step 9: Calculate the starting price of the diamond
Now we want to find the starting price of the diamond:
\[ \text{Starting Price} = x \cdot 81k^2 \]
Substituting \( k^2 x = 1,000 \):
\[ \text{Starting Price} = 81 \cdot 1,000 = 81,000 \]

### Final Answer
The starting price of the diamond was **81,000**.