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?
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
Im srry i still dont get it.
To determine the rate of exchange for the colored coins on the planet ZOG, we need to establish the value of each coin in relation to another color. Let's break down the given information step by step:
1. Given: 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.
This implies that the value of 2 white coins is equivalent to 1 red coin, 1 blue coin, and 1 green coin, as white coins are the baseline.
2. Given: 1 green = N blue
Since we already established that 1 white coin is worth 1 blue coin, we can deduce that N must be 1. Therefore, 1 green coin is equal to 1 blue coin.
3. Given: 1 blue = N yellow
Since we established that 1 green coin is equal to 1 blue coin, and 1 blue coin is equal to N yellow coins, we can conclude that N is also 1. Therefore, 1 blue coin is equal to 1 yellow coin.
4. Given: 1 yellow = N red
Using a similar logic to the previous steps, we find that N must be 1, making 1 yellow coin equal to 1 red coin.
5. Given: 1 red = N white
Again, following the pattern established above, N must be 1, meaning that 1 red coin is equal to 1 white coin.
To summarize the exchange rates for the colored coins on the planet ZOG:
- 1 white coin = 1 red coin
- 1 red coin = 1 yellow coin
- 1 yellow coin = 1 blue coin
- 1 blue coin = 1 green coin
Thus, each colored coin is valued equally to the other, with no specific exchange rate other than 1:1 across all colors.