An investigator interviewed 100 students to determine their preferences for the three drinks milk coffee and tea. He reported the following.

Question

An investigator interviewed 100 students to determine their preferences for the three drinks milk coffee and tea. He reported the following,  10 students had all the three drinks, 20 had milk and coffee,  30 had coffee and tea , 25 had milk and tea,  12 had milk only. 5 had coffee only, 8 had tea only.  The number of students that did not take any of the three drinks is. 

Answer 1

Here is the Venn Diagram:
(sorry, you must imagine the three interconnecting circles, but draw them so that Milk includes the overlapping sections with an M, Coffee includes the overlapping sections with C, and Tea includes the overlapping sections with T; None is in the Universe, but outside all the circles):

The Universe = 100
- - - - - - - - - - - - - - - - - - - - - - - -
    None

                    MilkOnly                                   CoffeeOnly

                                             M&C


                                           M&C&T

                              M&T                     C&T

 

 

                                          TeaOnly


Most problems like this give the values of the “Only” items first, but require you to solve the problem in the reverse order, so get used to that approach.

“100 students” means The Universe (the grand total)
“10 students had all the three drinks” means M&C&T [note: start at the middle]

Realize that M&C includes M&C&T, so thatt has to be subtracted.
“20 had milk and coffee” means M&C = 20-M&C&T = 20-10 = 10
“30 had coffee and tea” means C&T=30-M&C&T = 30-10 = 20
“25 had milk and tea” means M&T=25-M&C&T = 25-10 = 15

That fills in all of the overlapping sections of the diagram. Now, the “only” sections are given.

“12 had milk only” means MilkOnly = 20
“5 had coffee only” means CoffeeOInly = 5
“8 had tea only” means TeaOnly=8

 

Now, you have to add up all of the little segments along with the None that is outside of the circles to get the total of 100 in the Universe (the entire group)
!0+10+20+15+12+5+8+None=100

 

O.K., solve for None:
80+None=100
None=20

 

The number of students that did not take any of the three drinks is 20.

Answer 2

If we assume that those who answered that they like two drinks - i.e. milk and coffee, coffee and tea, etc. like atleast two beverages - i.e. they could either like the third beverage or or not, we can solve this as follows:

 

Those who like all the drinks - 10

Those who like milk and coffee only - 20 - 10 (those who like all of them) = 10

Those who like coffee and tea only = 30-10 = 20

Those who like milk and tea only - 25-10 - 15

Those who like milk alone = 12

Those who like coffee alone = 5

Those who like tea alone = 8

 

The sum of these numbers is 80. Therefore the students who drank none of the beverages is 100-80 = 20.