The price of a diamond is directly proportional to the square of its weight. A man accidentally breaks a diamond into three pieces in the ratio 3 : 5 : 7 and thereby incurs a loss of Rs 42600. What was the original price of the diamond (in rupees)?

Introduction / Context:
This question tests your understanding of proportionality and how value can depend on the square of a physical quantity, in this case the weight of a diamond. Diamonds and many other precious items often follow non linear pricing models where larger pieces are disproportionately more valuable than smaller pieces. When such an item is broken into parts, there is a loss because total value no longer follows the same square law as the original single piece.

Given Data / Assumptions:

• Price of a diamond is directly proportional to the square of its weight.
• The diamond is broken into three parts with weights in the ratio 3 : 5 : 7.
• The resulting loss due to breaking is Rs 42600.
• We need to find the original price of the intact diamond.
• We assume the proportionality constant remains the same before and after breaking.

Concept / Approach:
If the weight of the full diamond is W, its price is proportional to W^2. When broken into three parts with weights 3k, 5k and 7k, the sum of weights is still W = 3k + 5k + 7k = 15k. The price of each piece is proportional to the square of its weight, so total price after breaking is proportional to 3k squared + 5k squared + 7k squared. The difference between original price and total price after breaking is equal to the loss given in the question.

Step-by-Step Solution:
Let common factor of weights be k, so the three pieces have weights 3k, 5k and 7k.Total original weight W = 3k + 5k + 7k = 15k.Price is proportional to square of weight, so original price is proportional to (15k)^2 = 225k^2.After breaking, total price is proportional to 3k^2 + 5k^2 + 7k^2 = 9k^2 + 25k^2 + 49k^2 = 83k^2.Loss in terms of proportional units is 225k^2 - 83k^2 = 142k^2.Given actual loss is Rs 42600, so 142k^2 corresponds to Rs 42600.Therefore k^2 = 42600 / 142 = 300.Original price corresponds to 225k^2 = 225 * 300 = Rs 67500.

Verification / Alternative check:
We can use a simple proportional constant, say 1 rupee per k^2 unit, to visualise the calculation. Original price = 225 units, price after breaking = 83 units, loss = 142 units. If 142 units equal Rs 42600, 1 unit equals Rs 300. Thus original price = 225 * 300 = Rs 67500, exactly as calculated. This confirms the correctness of our result.

Why Other Options Are Wrong:
Values like 60000 or 75000 do not satisfy the proportional relationship between loss and original price when computed using the ratio based square weights. 11786 is far too small given a loss of Rs 42600. Only Rs 67500 yields a loss equal to Rs 42600 when we recalculate using the square law of pricing.

Common Pitfalls:
A frequent mistake is to assume that price is directly proportional to weight and not to the square of weight as given. Another error is to use the ratio 3 : 5 : 7 directly on price without squaring the terms. Always pay attention to whether the relationship is linear or involves squares, cubes or higher powers, since this completely changes the computation.

Final Answer:
The original price of the diamond was Rs 67500.