The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone weighing 18 units. If she breaks it into four pieces with each piece having distinct integer weight, then the difference between the highest and lowest possible values of the total price of the four pieces will be 288000. Then, the price of the original precious stone is

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Accepted Answer

1. **Price Formula:**  
   The price $$P$$ is directly proportional to the square of the weight $$W$$. Thus, $$P = k \cdot W^2$$, where $$k$$ is a constant.

2. **Price of Original Stone:**  
   For the unbroken stone weighing 18 units:  
   $$
   P = k \cdot (18)^2 = 324k.
   $$

3. **Breaking into Four Pieces:**  
   - **Minimum Price Configuration:** Weights are close to each other: $$3, 4, 5, 6$$. Total price:  
     $$
     P_{	ext{min}} = k \cdot (3^2 + 4^2 + 5^2 + 6^2) = k \cdot (9 + 16 + 25 + 36) = k \cdot 86.
     $$  
   - **Maximum Price Configuration:** Weights are far apart: $$1, 2, 3, 12$$. Total price:  
     $$
     P_{	ext{max}} = k \cdot (1^2 + 2^2 + 3^2 + 12^2) = k \cdot (1 + 4 + 9 + 144) = k \cdot 158.
     $$

4. **Difference Between Prices:**  
   Given that the difference between maximum and minimum prices is Rs. 288000:  
   $$
   P_{	ext{max}} - P_{	ext{min}} = (158k - 86k) = 72k = 288000.
   $$  
   Solving for $$k$$:  
   $$
   k = \frac{288000}{72} = 4000.
   $$

5. **Price of Original Stone:**  
   Substituting $$k = 4000$$ into $$P = k \cdot (18)^2$$:  
   $$
   P = 4000 \cdot (324) = Rs.1296000.
   $$

Thus, the price of the original precious stone is **Rs.1296000**.

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