A survey of 59 customers was taken at a bookstore regarding the types of books purchased. The survey found 36 customers purchased mysteries, 28 purchased science fiction, 21 purchased romance novels, 15 purchased mysteries and science fiction, 12 purchased mysteries and romance novels, 8 purchased science fiction and romance novels, and 4 purchased all three types of books
N = N(M) + N(S) + N(R) - N(M and S) - N(M and R) - N(S and R) + N(M and S and R)
= 36+28+12 - 15 - 12 - 8 + 4
= 45
If the question is, How many did not buy any type of book, then it would be 59-45 or 14
You did not state a question.
You could draw a Venn diagram and all mysteries would be revealed.
32
To solve this problem, we can use a Venn diagram to represent the different types of books purchased. Let's start by assigning variables to the number of customers who purchased each type of book:
Let M = number of customers who purchased mysteries
Let S = number of customers who purchased science fiction
Let R = number of customers who purchased romance novels
We are given the following information:
The survey found 36 customers purchased mysteries: M = 36
28 customers purchased science fiction: S = 28
21 customers purchased romance novels: R = 21
15 customers purchased mysteries and science fiction
12 customers purchased mysteries and romance novels
8 customers purchased science fiction and romance novels
4 customers purchased all three types of books
Now, let's fill in the Venn diagram using this information. We will start by placing the value of 4 in the intersection of all three sets (M ∩ S ∩ R).
M ∩ R: 12 - 4 = 8 R: 21
/ \
M: 36 S ∩ R: 8 S: 28
\ /
M ∩ S: 15 - 4 = 11
Now, we need to find the missing values (the number of customers who purchased only one type of book).
To find M ∖ (S ∪ R) (the number of customers who purchased only mysteries), we subtract the numbers we have already counted from the total number of customers (59):
M ∖ (S ∪ R) = M - (M ∩ S) - (M ∩ R) - (M ∩ S ∩ R) = 36 - 15 - 8 - 4 = 9
Similarly, we can find S ∖ (M ∪ R) (the number of customers who purchased only science fiction) and R ∖ (M ∪ S) (the number of customers who purchased only romance novels):
S ∖ (M ∪ R) = S - (M ∩ S) - (S ∩ R) - (M ∩ S ∩ R) = 28 - 15 - 8 - 4 = 1
R ∖ (M ∪ S) = R - (M ∩ R) - (S ∩ R) - (M ∩ S ∩ R) = 21 - 8 - 8 - 4 = 1
Now we can summarize our findings:
Number of customers who purchased only mysteries (M ∖ (S ∪ R)): 9
Number of customers who purchased only science fiction (S ∖ (M ∪ R)): 1
Number of customers who purchased only romance novels (R ∖ (M ∪ S)): 1
Number of customers who purchased both mysteries and science fiction (M ∩ S): 11
Number of customers who purchased both mysteries and romance novels (M ∩ R): 8
Number of customers who purchased both science fiction and romance novels (S ∩ R): 8
Number of customers who purchased all three types of books (M ∩ S ∩ R): 4
I hope this helps! Let me know if you have any further questions.
To solve this problem, we can use a method called Venn diagrams. Venn diagrams are a graphical representation of the relationships between different sets or groups of things. In this case, we will use a Venn diagram to represent the three types of books: mysteries, science fiction, and romance novels.
Step 1: Draw three overlapping circles to represent the three types of books: mysteries, science fiction, and romance novels. Label each circle accordingly.
Step 2: Fill in the information given in the problem. We know that 36 customers purchased mysteries, 28 purchased science fiction, and 21 purchased romance novels. Place these numbers in the respective circles.
Step 3: Next, place the additional information provided. 15 customers purchased both mysteries and science fiction, 12 purchased both mysteries and romance novels, and 8 purchased both science fiction and romance novels. Additionally, 4 customers purchased all three types of books. Fill in these numbers accordingly.
Step 4: Add up the numbers in each section of the Venn diagram to get the total number of customers in each category.
According to the information given, the Venn diagram would look like this:
Mysteries (36)
_______
| |
| 4 | Science Fiction (28)
| |
|_______|
Romance Novels (21)
Step 5: Analyze the Venn diagram to get the answer to the question.
a) To find the number of customers who purchased only mysteries, add up the numbers in the region that belongs only to the mysteries circle. In this case, it is the 36 customers who purchased mysteries minus the 15 who also purchased science fiction and the 12 who also purchased romance novels. So, the answer is 36 - 15 - 12 = 9.
b) To find the number of customers who purchased only science fiction, do the same in the science fiction circle. It is the 28 customers who purchased science fiction minus the 15 who also purchased mysteries and the 8 who also purchased romance novels. So, the answer is 28 - 15 - 8 = 5.
c) To find the number of customers who purchased only romance novels, do the same in the romance novels circle. It is the 21 customers who purchased romance novels minus the 12 who also purchased mysteries and the 8 who also purchased science fiction. So, the answer is 21 - 12 - 8 = 1.
d) To find the number of customers who did not purchase any of the three types of books, add up the numbers in the region outside all three circles. In this case, it is the total number of customers surveyed (59) minus the sum of all the numbers inside the Venn diagram (36 + 28 + 21 - 15 - 12 - 8 + 4). So, the answer is 59 - (36 + 28 + 21 - 15 - 12 - 8 + 4) = 59 - 74 = -15. Since having a negative number of customers doesn't make sense, we can conclude that there might be a mistake in the provided information or the Venn diagram.
The Venn diagram method allows us to visualize and solve problems involving overlapping sets, providing a clear understanding of the relationships between different groups.