work problem

Question

A survey of 105 college students was taken to determine the musical styles they liked. Of, course 37 students listened to rock, 36 to classical, and 25 to jazz. Also, 13 students listened to rock and jazz, 23 listened to rock and classical, and 11 to classical and jazz. Finally 9 students listened to all three musical styles.

Answer 1

a survey of 59 customers was taken at a bookstore regarding the types of books purchased. the survey found that 34 customers purchased mysteries, 28 purchased science fiction,21 purchased romance novels, 14 purchased mysteries and science fiction, 10 purchased mysteries and romance novels, 7 purchased science fiction and romance novels, and 2 purchase all three types of books

 

Let’s solve this using a Venn Diagram:

 

There are 59 customers in the Universe.

    34 Mysteries

    28 Science Fiction

    21 Romance

    14 Mysteries and Science Fiction

    10 Mysteries and Romance

     7 Science Fiction and Romance

     2  All Three

 

There are three circles:  Mysteries, Science Fiction, Romance

 

Solve (fill in the areas) in reverse order:

   Put 2 in the center

   7 Science Fiction and Romance = (all three) +x, so this is 2+x, x=5

   10 Mysteries and Romance = (all three) +x, so this is 2+x, x=8

   14 Mysteries and Science Fiction =(all three)+x, so this is 2+x, x=12

    21 Romance = (all three)+(M&R)+(S&R)+x, so this is 2+8+5+x, x=6

    28 Science Fiction = (all three)+(S&M)+(S&R)+x, so this is 2+12+5+x, x=9

    34 Mysteries = (all three)+(M&S)+(M&R)+x, so this is 2+12+8+x, x=12

 

Imagine 3 inter-locking circles with 4 numbers each:

 

Universe = 59 customers

 

                               No books = 5

 

      Mysteries                                        Science Fiction

 

               12                                                      9

                                          12

 

 

                                          2

 

                            8                          5

 

 

                                        6

 

                                  Romance

 

Check:

     12 + 9 + 12 + 2 + 8 + 5 + 6  = 54   There must be 5 people who purchased 0 books.

 

 

A survey of 105 college students was taken to determine the musical styles they liked. Of, course 37 students listened to rock, 36 to classical, and 25 to jazz. Also, 13 students listened to rock and jazz, 23 listened to rock and classical, and 11 to classical and jazz. Finally 9 students listened to all three musical styles.

 

There are 105 students in the Universe.

    37 Rock

    36 Classical

    25 Jazz

    13  Rock and Jazz

    23 Rock and Classical

    11 Classical and Jazz

     9  All Three

 

There are three circles:  Rock, Classical, and Jazz

 

Solve (fill in the areas) in reverse order:

   Put 9 in the center

   11 Classical and Jazz = (all three) + x, so this is 9+x, x=2

   23 Rock and Classical = (all three) + x, so this is 9+x, x=14

   13 Rock and Jazz = (all three) + x, so this is 9+x, x=4

   25 Jazz = (all three)+(R&J)+(C&J)+x, so this is 9+4+2+x, x=10

   36 Classical = (all three)+(C&R)+(C&J)+x, so this is 9+14+2+x, x=11

    37 Rock = (all three)+(R&C)+(R&J)+x, so this is 9+14+4+x, x=10

 

Imagine 3 inter-locking circles with 4 numbers each:

 

Universe = 105 students

 

                No Music = 45

 

      Rock                                                  Classical

 

             10                                                      11

                                         14

 

 

                                          9

 

                             4                        2

 

 

                                       10

 

                                     Jazz

 

Check:

  45+ 10 + 11 + 14 + 9 + 4 + 2 + 10 = 105

 

how many listened to only rock music?     10

how many listened to classical and jazz, but not rock?     2

how many listened to classical or jazz, but not rock?      11+2+10=23

how many listened to music in exactly one of the musical styles?     10+11+10=31

how many listened  to music in exactly tow of the musical styles?     14+4+2=20

how many did not listen to any of the musical styles?    45