The cost of a piece of diamond varies with the square of its weight. A diamond of Rs. 5,184 value is cut into 3 pieces whose weights are in the ratio 1 : 2 : 3. Find the loss involved in the cutting.

Price of diamond = Rs.5184. As price is proportional to square of weight of diamond, weight of diamond = √5184 = 72.
This weight of 72 units is divided in the ratio 1 : 2 : 3.
Hence their weights are $$(\frac{1}{6})\times72, (\frac{2}{6})\times72, (\frac{3}{6})\times72$$ i.e. 12, 24, 36 units by weight.
Hence their price are $$12^{2}, 24^{2}, 36^{2}$$ , i.e., 144, 576, 1296.
Hence total price of three pieces = 144+576+1296 = 2016.
Therefore loss involved in cutting = 5184 - 2016 = 3168