Solution to the Problem: Here is the solution:
Two solutions are:
(1) Diamond = $16,000,
Sapphire = $12,000,
and Ruby = $8,000
(2) Diamond = $8,000,
Sapphire = $18,000,
and Ruby = $20,000
Let R = cost of the ring with the Ruby
Let S = cost of the ring with the Sapphire
Let D = cost of the ring with the Diamond
Then we can write the following algebraic statements:
R = 2 | D - S |
D = 4 | S - R |
S = 2000n where n is an integer
Let's suppose that the diamond ring costs the most and the ruby ring costs the least.
Then we can remove the absolute value signs above and write:
R = 2 (D - S)
D = 4 (S - R)
Distribution and Substitution leads to:
R = 2D - 2S
D = 4S - 4R
R = 2(4S - 4R) - 2S
R = 8S - 8R - 2S
R = 6S - 8R
9R = 6S
R = (2/3)S
D = 4 (S - 2S/3)
D = (4/3)S
So S must be a multiple of 3 and since it is also a multiple of $2,000, then it must also be a multiple of $6,000.
If S = $6,000, D = $8,000 and R = $4,000.
S + R + D = $18,000, which is not between $20,000 and $50,000.
So, try the next multiple of $6,000 which is $12,000.
If S = $12,000, D = $16,000 and R = $8,000.
S + R + D = $36,000, which is between $20,000 and $50,000, so that is one of our solutions.
So, try the next multiple of $6,000 which is $18,000.
If S = $18,000, D = $24,000 and R = $12,000.
S + R + D = $54,000, which is not between $20,000 and $50,000.
To find a second solution, instead of assuming that D > S > R, reverse the inequality signs and assume that D < S < R.
Then we can write the following:
R = 2 (S - D)
D = 4 (R - S)
Distribution and Substitution leads to:
R = 2S - 2D
D = 4R - 4S
R = 2S - 2(4R - 4S)
R = 2S - 8R + 8S
9R = 10S
R = (10/9)S
D = (40/9)S - 4S
D = (4/9)S
So S must be a multiple of 9 and since it is also a multiple of $2,000, then it must also be a multiple of $18,000.
If S = $18,000, D = $8,000 and R = $20,000.
S + R + D = $46,000, which is between $20,000 and $50,000, so that is our second solution.
If you switched the variables in the difference part of the equation, you obtain a third answer:
Diamond = $8,000, Sapphire = $14,000, and Ruby = $12,000
Dr. Kishan found several more answers:
(Ruby, Diamond, Sapphire) =
(16000/3, 32000/3, 8000),
(20000/3, 40000/3, 10000),
(28000/3, 56000/3, 14000)
and (32000/3, 64000/3, 16000)
Click here to see his work
Davit Banana found these five answers that work:
Ivy Joseph also found these five answers that work:
There was nothing in the problem that prohibited non-integer solutions!