What are the values of each Halloween icon? (Math Logic Puzzles) (100 POINTS)

Hat: 10 Jack-O-Lantern: 4 Ghost: 2 Cauldron: 8

Step-by-step explanation:

I just did trial-and-error

9514 1404 393

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 ...

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:

The second equation serves as a check on our 8·8 +2 = 66.

(b) 240

step-by-step explanation:

first, we use the pythagorean formula using the given side lengths to find x. then we use x to find the side lengths. then we add the side lengths to find the perimeter.

pythagorean theorem formula:

a^2 + b^2 = c^2

(x - 20)^2 + (x - 40)^2 = x^2

x^2 - 40x + 400 + x^2 - 80x + 1600 = x^2

x^2 -120x + 2000 = 0

(x - 100)(x - 20) = 0

x - 100 = 0 or x - 20 = 0

x = 100 or x = 20

we see that the solution x = 20 must be discarded because it will give a negative side length and a side length of 0. the only valid solution is x = 100.

perimeter = sum of the lengths of the sides

perimeter = x + x - 20 + x - 40

perimeter = 3x - 60

replace x with 100.

perimeter = 3(100) - 60

perimeter = 300 - 60

perimeter = 240

m

step-by-step explanation: