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The problem of diamonds and rubies

The problem of diamonds and rubies

It is good to know that the value of diamonds increases according to the square of their weight just as the value of rubies does according to the cube of their weight.

Thus, for example, if a carat diamond is worth $ 100 one of two carats of the same quality will cost $ 400 while one of three carats of equal purity will cost $ 900. On the other hand, if an oriental carat ruby ​​is worth $ 200, a two-carat stone will cost $ 6,400.

A well-known merchant familiar with the Brazilian diamond mines of Cape Colony and other areas of the globe, showed me a pair of earrings with a diamond set each one that had exchanged for two diamonds of different sizes on the basis already explained that a Carat costs $ 100.

Can you deduce the size of the two different diamonds that changed by a pair of earrings with a diamond of the same size each?

Of course there are many possible answers, so we ask you to discover the smallest possible size of the two equal stones of the earrings, equivalent to the value of the two diamonds of different sizes without using carat fractions.

Solution

The diamond of each earring was 5 carats so each of them was worth $ 2,500 or $ 5,000 a couple. Diamonds of different sizes were 1 carat ($ 100) and 7 carats ($ 4,900) so that their added values ​​also gave $ 5,000.