First we need to know how much each jeweller got at the end of the exchange. To help you see a clearer picture we can find this by placing all the four jewellers in a square shown in the file attached.
Next we have to subtract 3 gems from each jeweller because there are 4 jewellers and each jeweller gives one gem to each of the other jeweller. Then we add one from each of the other 3 types of gem because we will gain one gem from each jeweller. For example the ruby jeweller had 8 rubies at the start then we subtract 3 rubies, which is 5 rubies then add one pearl, one diamond and one sapphire. Let R be ruby, P be pearl, D be diamond and S be sapphire, at the end we have the following expressions; 5R+P+D+S, 97P+D+S+R, 2D+S+R+P, 7S+R+P+D
Because it is given in the question that the values of all new combinations are the same therefore we can have the 6 following equations.
For example to solve the first one, 5R+P+D+S = 97P+D+S+R, we minus D and S from both sides, then we are left with 5R+P = 97P+R. Next we will take R and P away from both sides leaving us with 4R = 96P. Lastly we can simplify the equation by dividing both sides by 4, R = 24P, which is one of the relative values. We are going to do the same to the other equations.
5R+P+D+S = 97P+D+S+R
5R+P = 97P+R
4R = 96P
R = 24P
2D+S+R+P = 7S+R+P+D
2D+S = 7S+D
D = 6S
5R+P+D+S = 7S+R+P+D
5R+S = 7S+R
4R = 6S
2R = 3S
97P+D+S+R = 2D+S+R+P
97P+D = 2D+P
96P = D
5R+P+D+S = 2D+S+R+P
5R+D = 2D+R
4R = D
97P+D+S+R = 7S+R+P+D
97P+S = 7S+P
96P = 6S
16P = S
From the equations we can find out the relative values of each gem.
R = 24P
D = 6S
2R = 3S
96P = D
4R = D
16P = S
Once we have the relative values of each gem we can find out how much each jeweller gains or lose by expressing what each jeweller got in similar terms. For example every jeweller will have a combination of every other gem, hence there is no standardize comparison, the jewellers will have to have all of their collection expressed in terms of their type of gem.
Ruby:
First we set it out like this
5R (what the ruby jeweller has left) + P + D + S
To express their collection in terms of their type of gem we mustuse the relative values we found earlier:
For example; R = 24P we divide both sides by 24
1/24R = P
So we will replace each of the other gem in terms of R:
5R+1/24R+4R+2/3R = 9.71R (2.d.p)
9.71R (2.d.p) > 8
What the jeweller has now is greater than what he had before the exchange shows that the jeweller has gained. To work out how much we have to take the old amount and subtract it from the new amount.
9.71R (2.d.p)-8 = 1.71 (2.d.p)
Therefore the ruby jeweller gained 1.71 (2.d.p)
We will repeat this for the other three jewellers.
Pearl:
97P+24P+96P+16P = 233
233 > 100
233 - 100 = 133
The pearl jeweller gained 133
Diamond:
2D+1/4D+1/96D+1/6D = 2.43(2.d.p)
2.43(2.d.p) < 5
2.43(2.d.p) – 5 = -2.57(2.d.p)
The diamond jeweller loses 2.57
Sapphire:
7S+6S+3/2S+1/16S = 14.56(2.d.p)
14.56(2.d.p) > 10
14.56(2.d.p) – 10 = 4.56(2.d.p)
The sapphire jeweller gained 14.56(2.d.p)