9514 1404 393
Answer:
- hat = 10
- pumpkin = 4
- ghost = 2
- cauldron = 8
Step-by-step explanation:
Let the symbols hat, pumpkin, ghost, and cauldron be represented by the letters h, p, g, c, respectively. Then the 5 relations given are ...
- pgh = 80
- c^2+g = 66
- 2h +c = 28
- p^2 +g = 18
- p +g +h = 16
We start by assuming a positive integer solution. The divisors of 80 are ...
1, 2, 4, 5, 8, 10, 16, 20, 40, 80
We can choose 3 of these that have a total of 16 and a product of 80 to satisfy the first and last equations. None of the values can be 16 or larger if the last equation is to be satisfied. Those three values must be 2, 4, 10.
The fourth equation tells us p=4 and g=2, because no other pair of numbers from the set {2, 4, 10} will satisfy this equation. This means h=10.
When the value h = 10 is put into the third equation, it reduces to ...
2·10 +c = 28 ⇒ c = 8
Now, we know the values of the symbols:
- hat = 10
- pumpkin = 4
- ghost = 2
- cauldron = 8
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The second equation serves as a check on our answer: 8·8 +2 = 66.
