Venn Diagram Algebra Question

Question

 

In a survey of the dining preferences of 110 dormitory students at the end of the spring semester, the following facts were discovered about Adam's Lunch (AL), Pizza Tower (PT), the Dining Hall (DH).

 

25 liked AL but not PT

15 liked AL only

58 liked AL

63 liked PT

27 liked DH

25 liked PT and AL but not DH

17 liked PT and DH

 

(a) How many liked PT or DH?

 

(b) How many liked all three?

 

(c) How many liked only DH?

 

How do I answer these questions? I believe (c) is 27, but how do I answer (a) and (b)

Answer 1

The typical Venn Diagram problem presents a sequence of values for the area in the diagram going from the outside in.
However, the solution is usually done in the opposite direction – from the inside out. So, you process the statements in the reverse order or in some other logical order.

The numbers must add up. That is very important.

In a Venn Diagram, there is a “Universe” (110 dormitory students) and there are interconnecting circles (AL, PT, and DH).

This diagram looks like this (imagine interconnecting circles AL=abce; PT=acdf; DH=abdg):

Universe = 110

 

           AL                                                                     PT

 

                       e                       c                              f

 

                                                a

 

                                          b         d

 

 

                                                g

 

 

                                           DH




25 liked AL but not PT means                         e + b = 25         [eq1]
15 liked AL only means                                  e = 15               [eq2]
58 liked AL means                                         e+a+b+c = 58   [eq3]
63 liked PT means                                         a+c+d+f = 63    [eq4]
27 liked DH means                                       a+b+d+g = 27    [eq5]
25 liked PT and AL but not DH means             c = 25                [eq6]
17 liked PT and DH means                             a+d = 17            [eq7]

Let’s find all of the values a-g:
15 liked AL only means  [eq2]                   e = 15 
Use [eq2] with [eq1] to determine that:          b=10
Consider [eq3] with [eq6] and e=15 and b=10, so:        a=8
Consider [eq7] with a=8, so:                       d=9
Consider [eq4] with a,c,d known, so:          f=21
Are all 110 students represented?
Is 8+10+25+9+15+21 = 110 ? not yet, so:         g= 22

Questions:

(a) How many liked PT or DH? “or” means add all areas;
               a+b+c+d+f+g = 8+10+25+9+15+21 = 88
(b) How many liked all three?      a = 8
(c) How many liked only DH?      g = 22