154385

1 We will denote the value of 1 ruby as r, of 1 sapphire as s, of 1 pearl as p and
2 of one diamond as d
3 Jeweller 1- J1, Jeweller 2-J2, Jeweller 3- J3, Jeweller 4-J4
4 J1 J2 J3 J4
5 8r 10s 100p 5d
6 Each jeweller gives one of his gems to to each of the rest, meaning each 7jeweller lost 3 of his gems and the others gained those gems in a 1:1:1
8 ratio. We now have
9 J1 J2 J3 J4
10 5r 7s 97p 2d
11 s r r r
12 p p s s
13 d d d p
14 J1 Value of gems= J2 Value of gems= J3 Value of gems= J4 Value of gems
15 Using J1=J2, we have
16 5r+s+p+d=7s+r+p+d
17 5r+s=7s+r
18 4r=6s
19 r=(3/2)s
20 Using J2=J3, we have
21 7s+r+p+d=97p+r+s+d
22 7s+p=97p+s
23 6s=96p
24 s=16p (or p=(1/16)s)
25 Using J3=J4, and substituting p=(1/16)s and r=(3/2)s, we have
26 (97/16)s+(3/2)s+s+d=(3/2)s+s+(1/16)s+2d
27 (137/16)s+d=(41/16)s+2d
28 d=6s
29 Substituting s=16p into line 28 and line 19, we get
30 d=96p and r=24p
31 So p<s<r<d , where the relative values of the gems in terms of p are
32 p=p s=16p r=24p d=96p
33 J1 lost 8r-(5r+s+p+d)=
34 3r-s-p-d=
35 72p-16p-p-96p=-41p
36 Losing a negative value is gaining a positive value, so J1 gained the value of
37 41 pearls
38 J2 lost 10s-(7s+r+p+d)=
39 3s-r-p-d=
40 48p-24p-p-96p= -73p
41 J2 gained the value of 73 pearls
42 J3 lost 100p-(97p+r+s+d)=
43 3p-r-s-d=
44 3p-24p-16p-96p= -133p
45 J3 gained the value of 133 pearls
46 J4 lost 5d-(2d+r+s+p)=
47 3d-r-s-p=
48 288p-24p-16p-p= 247p
49 J4 lost the value of 247 pearls
50 Adding the net loss between the four jewellers, we get
51 41p+73p+133p-247p=0p
52 This is a valid result- no gems were added or taken away to any of the
53 jewellers, only rearranged between them