mary and tom jointly buy the vanilla-chocolate cake for $20. Suppose that Mary values vanilla cake four times more than she values chocolate cake. Find the dollar value to Mary of each of the following pieces of the cake:

a. the vanilla half of the cake

b. the chocolate half of the cake

c. the 75 degree slice of vanilla cake

To find the dollar value to Mary of each piece of the cake, we first need to determine the ratio of Mary's value for vanilla cake to chocolate cake.

Let's assign the value of vanilla cake to be V and the value of chocolate cake to be C. We are given that Mary values vanilla cake four times more than chocolate cake, so we can setup the equation:

V = 4C

Now let's solve for the values of V and C.

Since Mary and Tom jointly bought the cake for $20, we know that the total value of the cake is $20. Since Mary values the vanilla cake four times more than the chocolate cake, we can set up the equation:

V + C = 20

Substituting the value of V from the previous equation into this equation, we get:

4C + C = 20
5C = 20
C = 4

Now we know the value of C, which represents the dollar value to Mary of the chocolate half of the cake.

a. To find the dollar value to Mary of the vanilla half of the cake, we can substitute the value of C into the equation V = 4C:

V = 4 * 4
V = 16

Therefore, the dollar value to Mary of the vanilla half of the cake is $16.

b. The dollar value to Mary of the chocolate half of the cake is $4.

c. To find the dollar value to Mary of the 75 degree slice of vanilla cake, we need to consider the proportion of the vanilla half of the cake that the 75 degree slice represents.

If the vanilla half of the cake is valued at $16, then the 75 degree slice represents (75/180) * $16 = $6.67.

Therefore, the dollar value to Mary of the 75 degree slice of vanilla cake is approximately $6.67.