What are Venn diagrams and how do I use them to solve GRE Quant problems?

Venn diagrams are sometimes used in GRE quantitative reasoning section to represent sets. In addition, Venn Diagrams are excellent visualization tools to solve problems that require set manipulation. We will look at both types of examples in this lesson

What are Venn Diagrams?

Venn diagrams or set diagrams are diagrams that show all logical relationships between sets. Venn diagrams normally comprise overlapping circles. The interior of the circle symbolically represents the elements of the set, while the exterior represents elements which are not members of the set. For instance, the following figure shows a two-set Venn diagram, one circle represent the group of all student who own a PC, while another circle may represent the set of all students who own a Mac. The overlapping area or intersection represents the students who own both PC and a Mac. The area outside of the circle represents the students who do not own either a PC or a Mac.

Venn diagram can also be used to solve set based questions that involve unions and intersections. Let’s see a typical example:

Example:

In a summer camp there are a total of 28 students out of which 7 play hockey, 32 students play tennis and 5 students play neither tennis nor hockey. How many student plays both hockey and tennis?

A) 7

B) 9

C) 16

D) 21

E) 28

Solution:

To solve this problem we first need to find the number of students who play both hockey and tennis.

We can divide the entire school into 4 groups of students:

1. Group 1: Students who play hockey only
2. Group 2: Students who play tennis only
3. Group 3: Students who play hockey and tennis
4. Group 4: Students who play neither hockey nor tennis. We know there are 5 students who play neither tennis nor hockey.

The sum of the number of students in each group should be equal to the total number of students in the school = 28.

While we know that 7 students play hockey, this number also includes students who play both hockey and tennis. Let  be the number of students that play both hockey and tennis. Therefore we can conclude that:

Number of Students who play only hockey (Group 1) = 

Number of Students who play only tennis (Group 2) = 

Number of Students who play hockey and tennis (Group 3) = 

Number of Students who play neither hockey nor tennis (Group 4) = 5

Isolating the variable we get:

The correct answer is C