word problem

Question

A survey of 67 customers was taken to a bookstore regarding the types of books purchased. The survey found  that 42 customers purchased mysteries, 34 purchased  science fiction, 26 purchased  romance novels, 19 purchased mysteries and  science fiction,15 purchased  mysteries and romance novels, 12 purchased science fiction and romance novels, and 6 purchase all three types of books.

 

 How many of the customers surveyed purchased only romance novels?
How many purchased mysteries and science fiction, but not romance novels?
How many purchased mysteries or science fiction?
How many purchased mysteries or science fiction, but not romance novels?
How many purchased exactly two types of books?

Answer 1

The simplest way to solve this is by using venn diagrams.

There will be 3 intersecting circles M, SF and RN for the 3 genres.

Draw 2 at the top, 1 at the bottom.

The part common to all 3 circles has 6 corresponding to those who buy all 3 genres

19 buy M & SF out of which 6 also buy RN - so, 19-6=13 buy only M & SF

12 buy SF and RN out of which 6 also buy M - so, 12-6 = 6 buy only SF and RN

15 buy RN and M out of which 6 also buy SF - so, 15-6=9 buy only RN and M.

Now, find out how many get only 1 genre.

Only M = 42-(13+6+9)  = 14

Only SF = 34 -(13+6+6)=9

Only RN = 26-(6+6+9)=5

 

How many of the customers surveyed purchased only romance novels?

Only RN =5
How many purchased mysteries and science fiction, but not romance novels?

Only M + Only SF + Both M & SF but not RN

 = 14 + 9 + 13 = 36
How many purchased mysteries or science fiction?

Total M + SF excluding M & SF (to avoid double counting) = 42+9+6=57
How many purchased mysteries or science fiction, but not romance novels?

Only M + Only SF + Both M & SF (excl. RN) = 14+9+13=36
How many purchased exactly two types of books?

M & SF but not RN =13

M & RN but not SF = 9

SF & RN but not M = 6

So, exactly 2 genres = 13+9+6=28