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?
Let's start by using S to represent the number of guests who ate snowball cookies, F to represent the number of guests who ate snowflake cookies, and M to represent the number of guests who ate snowman cookies. We know that:
All of the guests who ate snowball cookies also ate snowman cookies, so S is a subset of M.
Three-fourths of the snowball eaters did not also eat snowflake cookies, so 1/4 of S is equal to the number of guests who ate all three types of cookies (X), and 3/4 of S is equal to the number of guests who ate only snowball and snowman cookies (which includes Y).
There were six guests who ate snowflakes and snowmen, so F intersect M equals 6.
Of the snowflake-eaters who did not eat snowballs, half of them ate snowmen, so F minus S equals half of (F intersect M - X).
Eighty percent of all the guests ate the snowman cookies, so M equals 0.8 times the total number of guests minus the one guest who didn't eat any cookies.
We want to find the number of guests who ate snowman cookies but not snowball or snowflake cookies, which is Y. To find Y, we can use the information we have and solve for it. Here's one way to do it:
From the first bullet point, we know that S is a subset of M, so M minus S is the number of guests who ate snowman cookies but not snowball cookies.
From the second bullet point, we know that Y is included in the group of guests who ate only snowball and snowman cookies, which is 3/4 of S. So Y is less than or equal to 3/4 of S.
From the third bullet point, we know that F intersect M equals 6, so F minus (F intersect M) is the number of guests who ate snowflake cookies but not snowman cookies. Since half of this group also ate snowball cookies (according to the fourth bullet point), we can say that (F minus F intersect M)/2 is the number of guests who ate snowflake and snowball cookies but not snowman cookies.
From the fifth bullet point, we know that M equals 0.8 times the total number of guests minus one. We can substitute this expression into the equation from the first bullet point: M minus S equals the number of guests who ate snowman cookies but not snowball cookies. This gives us:
0.8T - 1 - S = M - S
0.8T - 1 = M
M = 0.8T - 1
Now we can substitute all of the expressions we have into the equation for Y:
Y <= 3/4S
Y <= S - X
X = 1/4S
F - F intersect M = (F - S intersect F)/2
M - S = 0.8T - 1 - S
M = 0.8T - 1
Simplifying and combining terms, we get:
Y <= 3/4S
Y <= S - 1/4S
X = 1/4S
F - 6 = (F - S)/2
0.8T - 2S + 1 = 0
Solving the last equation for T, we get:
T = (2S - 1)/0.8
Substituting this expression into the other equations, we get:
Y <= 3/4S
Y <= S
Combining the two inequalities for Y, we get Y <= min(3/4S, S), which simplifies to Y <= 3/4S.
Substituting X = 1/4S into the equation F - 6 = (F - S)/2 and solving for F, we get F = 2S - 12.
Substituting this expression for F into the equation for Y, we get Y <= 3/4S.
We know that there were more than 20 and less than 30 guests at the party, so we can try different values of S in this range and see which ones satisfy the conditions we've derived. If S is 24, then X is 6, F is 30, and M is 18. This satisfies all the conditions, and we can calculate Y as follows:
Y <= 3/4S
Y <= 3/4(24)
Y <= 18
Therefore, we know that there are at most 18 guests who ate snowman cookies but not snowball or snowflake cookies. To find the exact number, we need to check whether any of the guests who ate snowman cookies also ate snowball or snowflake cookies. We know that 18 guests ate snowman cookies, and 6 guests ate all three types of cookies, so there are 12 guests who ate only snowman and snowflake cookies. We also know that 80% of all the guests ate snowman cookies, which means that the total number of guests is 25. Subtracting the guests who didn't eat any cookies, we get 24 guests who ate cookies. Therefore, the number of guests who ate snowman cookies but not snowball or snowflake cookies is 18 - 12 = 6.
To find the number of guests who ate snowman cookies but did not eat snowball cookies or snowflake cookies, we need to use the given information and solve step by step.
Let's assign variables to the unknowns:
- Let A represent the number of guests who ate snowball cookies.
- Let B represent the number of guests who ate snowman cookies.
- Let C represent the number of guests who ate snowflake cookies.
Now let's break down the given information:
1. All of the guests who ate snowball cookies also ate snowman cookies. This means that the number of guests who ate snowman cookies (B) is at least equal to the number of guests who ate snowball cookies (A).
2. Three-fourths of the snowball eaters did not also eat snowflake cookies. So, (3/4)A guests who ate snowball cookies did not eat snowflake cookies. This implies that (1/4)A guests who ate snowball cookies also ate snowflake cookies.
3. There were six guests who ate snowflakes and snowmen. This means C = 6.
4. Of the snowflake-eaters who did not eat snowballs, half of them ate snowmen. This implies that (1/2)(C - A) guests who ate snowflake cookies also ate snowman cookies.
5. Eighty percent of all the guests ate the snowman cookies. So, B = 0.8(total number of guests).
6. One guest was on a health kick and ate none of the cookies. Therefore, (A + B + C) + 1 = total number of guests.
Let's summarize the information we have so far:
- B ≥ A
- (3/4)A + (1/4)A = A
- C = 6
- (1/2)(C - A) = (1/2)(6 - A)
- B = 0.8(total number of guests)
- (A + B + C) + 1 = total number of guests
Now we can use this information to solve for the number of guests (B) who ate snowman cookies but did not eat snowball cookies or snowflake cookies.
First, let's eliminate unnecessary variables and rewrite the equations:
B ≥ A --> Equation 1
(3/4)A + (1/4)A = A --> Equation 2
C = 6 --> Equation 3
(1/2)(6 - A) --> Equation 4
B = 0.8(total number of guests) --> Equation 5
(A + B + C) + 1 = total number of guests --> Equation 6
From equation 2, we simplify:
(3/4)A + (1/4)A = A
(4/4)A = A
A = 4
Now let's substitute A = 4 into equations 1 and 4:
From equation 1:
B ≥ A
B ≥ 4
From equation 4:
(1/2)(6 - A) = (1/2)(6 - 4)
(1/2)(2) = 1
(1/2)(2) = 1
1 = 1
Now let's substitute A = 4 into equation 5:
B = 0.8(total number of guests)
B = 0.8(total number of guests)
B = 0.8(total number of guests)
B = 0.8(total number of guests)
Finally, let's substitute A = 4, B ≥ 4, and C = 6 into equation 6:
(A + B + C) + 1 = total number of guests
(4 + B + 6) + 1 = total number of guests
(10 + B) + 1 = total number of guests
10 + B + 1 = total number of guests
11 + B = total number of guests
From the given information, we know that the total number of guests is between 20 and 30. So we can express it as 20 < total number of guests < 30.
Now we can plug the new expression for the total number of guests into the previous equation:
11 + B = 20
B = 20 - 11
B = 9
Given that B ≥ A (from equation 1), we have B = 9.
Therefore, the number of guests who ate snowman cookies but did not eat snowball cookies or snowflake cookies is 9.