Averages , Ratio and Proportion Question 1 - CAT 2023
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 a 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:
Step-by-Step Explanation
Let the price of the stone be C and its weight be W. Since price is directly proportional to the square of the weight:
C = kW²
For the original uncut stone of weight 18:
Cₒᵣᵢgᵢₙₐₗ = k(18)² = 324k
- Find the maximum possible price of the broken pieces:
To maximize the sum of squares of a fixed sum (18), we must make one piece as large as possible and the others as small as possible. Since weights must be distinct integers:
Smallest distinct weights = 1, 2, 3
Largest remaining piece = 18 − (1 + 2 + 3) = 12
Maximum Price = k(1² + 2² + 3² + 12²)
= k(1 + 4 + 9 + 144) = 158k
- Find the minimum possible price of the broken pieces:
To minimize the sum of squares, the pieces must be as close to each other in weight as possible (around the average of 18/4 = 4.5).
Closest distinct integer weights = 3, 4, 5, 6 (Sum = 18)
Minimum Price = k(3² + 4² + 5² + 6²)
= k(9 + 16 + 25 + 36)
= 86k
- Calculate the original price:
The difference between the maximum and minimum price is 288,000.
158k − 86k = 288,000
72k = 288,000
⟹ k = 288,000/72 = 4,000
Original Price = 324k = 324 × 4,000 = 1,296,000
Correct Option: D
💡 FundaMakers Tip:
Optimization of sums of squares is a standard CAT trick.
- To maximize: Push values to the extremes (e.g., 1, 2, 3, 12).
- To minimize: Squeeze values as close to the arithmetic mean as possible (e.g., 3, 4, 5, 6).
Finding these combinations takes just a few seconds if you know this rule!