Outre Ornaments, Inc. sells baubles, gewgaws, and trinkets. While there, a customer spoke to three different salespeople.

1. The first salesperson the customer talked to told him that any 7 baubles together with any 5 gewgaws have the same value as any 6 trinkets.
2. The second salesperson he talked to told him that any 4 baubles together with any 9 trinkets have the same value as any 5 gewgaws.
3. The third salesperson he talked to told him that that any 6 trinkets together with any 3 gewgaws have the same value as any 4 baubles.
4. When the customer bought some of each kind of ornament, he found out exactly one of these salespersons was lying, while the other two told the truth.

Which salesperson was the liar?

Algebraically the three statements are:

1. 7 B + 5 G = 6 T
2. 4 B + 9 T = 5 G
3. 6 T + 3 G = 4 B

We can restate 2 and 3:

2.  9 T = 5 G - 4 B
3.  6 T = 4 B - 3 G

Each of these clearly shows that T has a lesser value than statement 1 allows, and thus statement 1 is incompatible with both and must be the lie: The first salesperson is the liar.

Solving 2 and 3 we see that G, B and T are in the ratio:

G:B:T = 30 : 28.50 : 4

As these are proportions, the actual values of G, B and T could be any multiple of this set of numbers.

Posted by Charlie on 2013-03-12 14:26:20