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CAT 2023 Slot 2 — QA Question 12

VariationEasy

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

Answer & solution

  • A

    972000

  • 1296000

  • C

    1944000

  • D

    1620000

Solution

Easy

Price ∝\propto weight2^2, so P=k w2P=k\,w^2. Breaking one stone into pieces changes the total price because squaring is not additive. The total is largest when the weights are as spread out as possible and smallest when they are as bunched as possible (with distinct integer weights summing to 1818). The gap between the two fixes kk, which then prices the original stone.

1

Maximise the total: make three pieces tiny and one huge — 1,2,3,121,2,3,12 (distinct, sum 1818):

Pmax⁡=k(12+22+32+122)=k(1+4+9+144)=158k\begin{aligned} &P_{\max}=k\left(1^2+2^2+3^2+12^2\right)=k(1+4+9+144)=158k \end{aligned}
2

Minimise the total: make the pieces as equal as possible — 3,4,5,63,4,5,6 (distinct, sum 1818):

Pmin⁡=k(32+42+52+62)=k(9+16+25+36)=86k\begin{aligned} &P_{\min}=k\left(3^2+4^2+5^2+6^2\right)=k(9+16+25+36)=86k \end{aligned}
3

Use the given difference to solve for kk:

158k−86k=28800072k=288000⇒ k=4000\begin{aligned} &158k-86k=288000\\ &72k=288000\\ &\Rightarrow\ k=4000 \end{aligned}
4

Price the original stone of weight 1818:

P=k×182=4000×324=1296000\begin{aligned} &P=k\times18^2=4000\times324=1296000 \end{aligned}

The price of the original stone is 1296000\mathbf{1296000}.

Since P=k w2P=k\,w^2 and w=18w=18 is fixed, the original price is just k×324k\times324. So you only need kk: spread vs. bunched gives 158k−86k=72k=288000158k-86k=72k=288000, hence k=4000k=4000 and price =324k=1296000=324k=1296000.

CAT 2023 Slot 2 QA Q12: The price of a precious stone is directly proportional to the square of its weight. Sita has a precious stone — Solution | TheCATExam