SETS PROBLEM

Question

The 130 consumers were asked about their buying preferences concerning another product that is sold in the market under three labels. The results were
?

12 buy only those sold under label A.

?24 buy only those sold under label B.

?27 buy only those sold under label C.

?14 buy only those sold under labels A and B.

?9 buy only those sold under labels A and C.

?11 buy only those sold under labels B and C.

?7 buy the product sold under all three labels.

?
How many of the consumers surveyed buy the product sold under none of these labels?

?

a. 106
b. 89
c. 41
d. 24
e. 19

This is confusing. Please explain. Thank you

Answer 1

To create the diagram so that you can see how the numbers relate:

For looking at 3 sets that intersect, draw 3 circles that overlap in a 3 leaf clover shape so that there is a small are in the middle where all 3 circles overlap.  Then label your circles as A B and C and fill in each section of the diagram with the appropriate count you have listed above.  So in the small sliver where A and B overlap but it is outside of C (not overlapping with C), you would put the count of 14 (14 buy only those sold under labels A and B.).

 

From the way your groups are phrased, it sounds as if ALL of your counts are mutually exclusive (none of the buyers fit into more than one of your counts).  Since your counts cover all of the possible options of people using one or more of A, B, and C, alll you have to do is add all of your counts together to get the total number of people that use some combination of A, B, and/or C.  the difference between your total of 130 consumers and this value is the total number of people who bought something OTHER THAN A B or C.

 

so your equation would be

<total number of consumers> minus <total number of people who bought something from the A B C options>

130 - (12 + 24 + 27 + 14 + 9 + 11 + 7) = ?