Question 1201924: survey of 70 customers was taken at a bookstore regarding the types of books purchased. The survey found that 41 customers purchased mysteries, 35 purchased science fiction, 29 purchased romance novels, 17 purchased mysteries and science fiction, 14 purchased mysteries and romance novels, 11 purchased science fiction and romance novels, and 2 purchased all three types of books.
a) How many of the customers surveyed purchased only romance novels?
b) How many purchased mysteries and science fiction, but not romance novels?
c) How many purchased mysteries or science fiction?
d) How many purchased mysteries or science fiction, but not romance novels? e How many purchased exactly two types of books?
Answer by math_tutor2020(3817)   (Show Source):
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Template of a Venn Diagram with 3 circles:
Circle M = Mysteries
Circle S = Science Fiction
Circle R = Romance
There are 8 distinct regions labeled "a" through "h".- a = mystery novels only
- b = mystery and science fiction novels, but not romance novels
- c = science fiction novels only
- d = mystery and romance novels, but not science fiction novels
- e = all three genres
- f = science fiction and romance novels, but not mystery novels
- g = romance novels only
- h = none of the three genres mentioned
For example, if a person bought all three types of books, then they will be placed in region e.
Here are the given facts from the instructions
| Number | Statement | | Fact 1 | Survey of 70 customers | | Fact 2 | 41 customers purchased mysteries | | Fact 3 | 35 purchased science fiction | | Fact 4 | 29 purchased romance novels | | Fact 5 | 17 purchased mysteries and science fiction | | Fact 6 | 14 purchased mysteries and romance novels | | Fact 7 | 11 purchased science fiction and romance novels | | Fact 8 | 2 purchased all three types of books. |
Fact 8 tells us to write "2" without quotes into region e.
Fact 7 and Fact 8 combine to tell us 11-2 = 9 people bought science fiction and romance novels, but not mystery novels. The value "9" will go in region f.
Fact 6 and Fact 8 combine to tell us that 14-2 = 12 people bought a mystery and romance novel, but not science fiction.
The value 12 goes in region d.
Fact 5 and Fact 8 combine to tell us 17-2 = 15 people bought mysteries and science fiction, but not romance. This value goes in region b.
Summary so far:
b = 15
d = 12
e = 2
f = 9
Updated Venn Diagram
Highlight circle M to focus on for now.
The values in this circle must add to 41 (see Fact 2)
a+15+12+2 = 41
a+29 = 41
a = 41-29
a = 12
For circle S, Fact 3 means we add up the values and set the sum equal to 35
15+c+2+9 = 35
c+26 = 35
c = 35-26
c = 9
Use the same idea for circle M.
We use Fact 4
12+2+9+g = 29
23+g = 29
g = 29-23
g = 6
Further update:
a = 12, b = 15, c = 9
d = 12, e = 2, f = 9
g = 6
Add up the values in regions "a" through "g"
a+b+c+d+e+f+g = 12+15+9+12+2+9+6 = 65
There are 65 people who bought at least one (i.e. one or more) type of genre.
Fact 1 says 70 people were surveyed.
Therefore, 70-65 = 5 people didn't buy any of the three genres mentioned.
h = 5 goes outside all three circles.
This is the completed Venn diagram
The 8 regions are:
a = 12, b = 15, c = 9
d = 12, e = 2, f = 9
g = 6, h = 5
At this point, the questions for parts (a) through (e) should be fairly straight forward.
Use the Venn diagram to answer each.
Please let me know if you have questions about this part, or if you have a question on the Venn diagram construction.
Answers:
(a) 6 ........ see region g
(b) 15 ........ see region b
(c) 59 ........ compute a+b+c+d+e+f = 12+15+9+12+2+9 = 59
(d) 36 ........ compute a+b+c = 12+15+9 = 36
(e) 36 ........ compute b+d+f = 15+12+9 = 36
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