On the planet ZOG, colored coins are used for money. Suppose 12 white, 9 red, 8 yellow, 4 blue and 0 green coins can be exchanged for 2 white, 1 red, 0 yellow, 1 blue and 1 green coin. Also suppose 1 green = n blue, 1 blue = n yellow, 1 yellow = n red, and 1 red = n white. What is the whole number rate of exchange for these coins? Is only one exchange rate possible?
To determine the exchange rates, we can start by setting up a system of equations based on the given information:
Let n be the rate of exchange between coins of different colors.
1 green = n blue
1 blue = n yellow
1 yellow = n red
1 red = n white
Using these relationships, we can express the exchange rate between the white coins and the green coins:
1 white = (1/n)^4 green
Now, we know that 12 white coins can be exchanged for 2 white coins:
12 white = 2 white
Thus, we can set up the equation:
12 * 1 = 2 * (1/n)^4
Simplifying, we get:
12 = 2/n^4
n^4 = 2/12
n^4 = 1/6
n = (1/6)^(1/4)
n = (1/2)^(1/4)
n = 1/√2
n = 1/1.414
n ≈ 0.7071
Therefore, the exchange rate between coins of different colors is approximately 0.7071.