Gift of Gems

Two solutions were received to this problem, both using the same method, and both from Yorkshire. Thank you to Vassil Vassilev of Lawnswood High School in Leeds and Catherine Harrison of The Mount School in York.

First, we need to see what type of jewels each jeweller has:

• 1st - 5 rubies, 1 sapphire, 1 pearl, 1 diamond
• 2nd - 1 ruby, 7 sapphires, 1 pearl, 1 diamond
• 3rd - 1 ruby, 1 sapphire, 97 pearls, 1 diamond
• 4th - 1 ruby, 1 sapphire, 1 pearl, 2 diamonds

Then, we convert this to equations:
( r = ruby, s = sapphire, p = pearl, d = diamond )

5r + s + p + d = r + 7s + p + d => 4r = 6s
r + 7s + p + d = r + s + 97p + d => 6s = 96p
r + s + 97p + d = r + s + p + 2d => 96p = d

From here, we can see that:

1 diamond = 96 pearls
1 sapphire = 16 pearls
1 ruby = 24 pearls

If we say that a pearl has a price of 1, then:

• 1st jeweller with 8 rubies has a stock worth 192.
• 2nd jeweller with 5 sapphires has a stock worth 160.
• 3rd jeweller with 100 pearls has a stock worth 100.
• 4th jeweller with 5 diamonds has a stock worth 480.

After the sharing of jewels, each jeweller has a stock worth 233.