Question

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

• A. 1620000
• B. 1296000
• C. 1944000
• D. 972000

Answer 1

The Correct Option is B

"A precious stone's price is directly correlated with the square of its weight." 
\(P = k × W^2\), where W and P are the stone's weight and price, respectively.
The total cost of the intact stone will be \(18^2 × k = 324 k. \)

“The difference between the highest and lowest possible values of the total price of the four pieces will be 288000 if she breaks it into four pieces, each with a distinct integer weight.” 
When the shattered stone weights are close to one another, that is, when the weights are 3, 4, 5, and 6 units, the minimum profit is made. 
In this instance, the four stones' combined worth =\((3^2+4^2+5^2+6^2)k=86k  \)
When the broken stone weights are separated by a large amount, i.e., 1, 2, 3, and 12 units, the largest benefit is realized. 
The combined value of the four stones in this instance is equal to \((1^2+2^2+3^2+12^2)k=158k.\)

The overall value difference is 2,88,000. 
1,88,000 – 86 000 = 72 000 = 4,000 
Thus, the original stone cost 324 k, or 12,96,000. 

The correct option is (B): 1296000