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On the plant ZOG,colored coins are used formoney. Suppose 12 white, 9 red, 8 yellow, 4 blue, and 0 green can be exchanged for 2 white, 1 red, 0 yellow, 1 blue ,and 1 green. Also, suppose 1 green = N blue, 1 blue = N yellow, 1 yellow = N red, and 1 red = N white coins. What is the rate of exchange for these coins?

  • math -

    From your first condition
    12w+9r+8y+4b = 2w+1r+1b+1g
    so 10w+8r+8y+3b-g = 0

    from your last conditions, w is the smallest denomination and g is the largest, so that
    r=Nw
    y=N^2w
    b=N^3w and
    g = N^4w

    putting those back in
    10w + 8Nw + 8n^2w + 3N^3w - N^4w = 0
    divide by w
    10 + 8N + 8N^2 + 3N^3 - N^4 = 0
    Hoping that N is a whole number, I only had to try N = 1,2, or 5 (factors of 10)
    sure enough N=5 worked, so

    smallest coin W
    1 R = 5 W
    1 Y = 25 W
    1 B = 125 W and
    1 G = 625 W

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