Guests at a winter-themed party were served snowball cookies, snowman cookies, and snowflake cookies. All of the guests who ate snowball cookies also ate snowman cookies. Three-fourths of the snowball eaters did not also eat snowflake cookies. There were six guests who ate snowflakes and snowmen. Of the snowflake-eaters who did not eat snowballs, half of them ate snowmen. Eighty percent of all the guests ate the snowman cookies. One guest was on a health kick and ate none of the cookies.
If there were more than 20 guests and less than 30 guests at the party, how many guests ate snowman cookies but did not eat snowball cookies or snowflake cookies?
draw a Venn diagram and start filling in the areas. You will find that the number of people who ate only snowflakes does not matter. You will find, however, that if b ate snowballs and m ate snowmen and f ate only snowflakes, then if the total people was x,
.2x = b-1+f
so, since x is a multiple of 5, then there were 25 people, 7 of whom ate only snowmen
The answer is supposed to be eight
I just don't know how to get it
S - did you follow Steve's advice?
yes
To solve this problem, we will use a logical approach to deduce the answer step by step.
Let's represent the guests who ate snowball cookies, snowman cookies, and snowflake cookies as A, B, and C, respectively. We will use Venn diagrams to visualize the information and relationships provided.
We know that all guests who ate snowball cookies also ate snowman cookies. So, the set A is a subset of B (A ⊆ B).
It is given that three-fourths of the snowball eaters did not also eat snowflake cookies. This means that 1/4 of the snowball eaters did eat snowflake cookies. We can deduce that the set of guests who ate snowball and snowflake cookies is 1/4 of set A (1/4 * A).
There were six guests who ate snowflakes and snowmen. This information implies that the intersection between sets B and C is 6 (B ∩ C = 6).
Of the snowflake-eaters who did not eat snowballs, half of them ate snowmen. This means that 1/2 of the guests in set C are also in set B but not in set A (1/2 * C ∩ B).
Eighty percent of all the guests ate the snowman cookies. This information tells us that the union of sets A, B, and C is 80% of the total number of guests.
Now, let's break down the problem step by step:
1. From the given information, we have the following:
- A ⊆ B (All guests who ate snowballs also ate snowmen).
- 1/4 * A = "guests who ate snowball and snowflake cookies."
- B ∩ C = 6 (intersection of snowman and snowflake eaters).
- 1/2 * C ∩ B = "guests who ate snowflake cookies but not snowballs and ate snowman cookies."
2. Using the information from step 1, we can derive the following relationship:
- B = A ∪ (1/2 * C ∩ B) (Guests who ate snowman cookies are either in A or the intersection "guests who ate snowflake cookies but not snowballs and ate snowman cookies").
3. We are given that 80% of all guests ate snowman cookies. So, we can write the following equation:
- A ∪ B ∪ C = 0.8 * total number of guests
4. To find the number of guests who ate snowman cookies but did not eat snowball cookies or snowflake cookies, we need to refine our equation from step 3:
- B ∖ A = "guests who ate snowman but did not eat snowball or snowflake cookies"
- B ∖ A = B ∩ (A' ∩ C') (Guests who ate snowman but not snowball or snowflake cookies are in B but not in A or C)
5. Combining the equations from steps 2 and 4, we get:
- B ∖ A = B ∩ (A' ∩ C')
- B ∖ A = (A ∪ (1/2 * C ∩ B)) ∩ (A' ∩ C')
Now, let's summarize the steps up to this point:
1. A ⊆ B
2. B = A ∪ (1/2 * C ∩ B)
3. A ∪ B ∪ C = 0.8 * total number of guests
4. B ∖ A = (A ∪ (1/2 * C ∩ B)) ∩ (A' ∩ C')
Using these relationships, we can now solve for the number of guests who ate snowman cookies but did not eat snowball cookies or snowflake cookies.
Given that there were more than 20 guests and less than 30 guests at the party, we can substitute values within this range to find a solution that satisfies all the conditions.