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GRE Quant: Venn Diagrams for Overlapping Sets

Venn diagrams help you organize overlapping sets by placing each group into a clear visual structure. When sets share members, a Venn diagram helps you track what belongs only to one set, what belongs to both, and how the totals connect. This simple visual approach supports layered GRE Quant questions where careful counting and clean organization lead to accurate results.

The following video, part of our GRE prep online course, explains Venn diagrams for overlapping sets in a clear, structured, and beginner friendly manner. It shows how to build the diagram step by step, how to place values in the right regions, and how to keep totals consistent. The video then applies the method on GRE-style problems so you experience the application first-hand. Use this learning steadily across your GRE prep, practice exercises, GRE sectional mocks, and GRE mocks.

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Venn Diagram Involving Two Overlapping Sets

Venn diagram Involving two overlapping sets

A Venn diagram is a visual tool used to represent the relationships between two sets, labeled here as A and B. It breaks down data into distinct categories based on membership in these sets:

Total A

The entire group belonging to set A, including those that also belong to B.

Total B

The entire group belonging to set B, including those that also belong to A.

Only A

The portion of set A that does not overlap with set B.

Only B

The portion of set B that does not overlap with set A.

Both A and B

The intersection where members belong to both sets simultaneously.

Neither A nor B

The area outside of both circles, representing members that do not belong to either set.

In a typical mathematical context, the relationship between these parts is defined by the following logic: Total = (Only A) + (Only B) + (Both A and B) + (Neither A nor B).

An Explanatory Example: Venn Diagram for GRE Quant

An explanatory Example – Venn diagram for GRE quant

Correct Answers
Both: 60
Exactly one of the two: 50
Only Soccer: 10
Only Cricket: 40
At least one of the two: 110

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

For visual illustration, here is a snapshot from the video…

An explanatory example 1

Following is a concise, step-wise written explanation…

 

Question

In a group of 120, 70 play soccer, 100 play cricket, and 10 play neither. How many play…

  1. Both
  2. Exactly one of the two
  3. Only Soccer
  4. Only Cricket
  5. At least one of the two

 

Step 1: Find the total number of players

The total group is 120. If 10 people play neither sport, then the number of people who play at least one sport is 120 – 10 = 110.

Step 2: Find how many play Both

If we add the soccer players (70) and cricket players (100), we get 170. Since there are only 110 players in total, the extra people must be counted in both groups. 170 – 110 = 60. Both = 60

Step 3: Find Only Soccer and Only Cricket

  • Only Soccer = Total Soccer – Both = 70 – 60 = 10.
  • Only Cricket = Total Cricket – Both = 100 – 60 = 40.
  • Only Soccer = 10, Only Cricket = 40

Step 4: Find Exactly one and At least one

  • Exactly one of the two = Only Soccer + Only Cricket = 10 + 40 = 50.
  • At least one of the two = Only Soccer + Only Cricket + Both = 10 + 40 + 60 = 110.
  • Exactly one of the two = 50, At least one of the two = 110

 

Correct Answers
Both: 60
Exactly one of the two: 50
Only Soccer: 10
Only Cricket: 40
At least one of the two: 110

Another Explanatory Example…

Explanatory example 2

Answers

Neither: 5

Exactly one of the two: 60

Only Soccer: 15

Only Cricket: 45

At least one of the two: 115

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

For visual illustration, here is a snapshot from the video…

Explanatory example 3

Following is a concise, step-wise written explanation…

 

Question

In a group of 120, 70 play soccer, 100 play cricket, and 55 play both. How many play:

  1. Neither
  2. Exactly one of the two
  3. Only Soccer
  4. Only Cricket
  5. At least one of the two

 

Step by Step Solution

Breakdown of Data

Total people = 120

Soccer (Total) = 70

Cricket (Total) = 100

Both = 55

Calculations

Only Soccer = Total Soccer minus Both = 70 – 55 = 15

Only Cricket = Total Cricket minus Both = 100 – 55 = 45

Exactly one of the two = Only Soccer plus Only Cricket = 15 + 45 = 60

At least one of the two = Only Soccer plus Only Cricket plus Both = 15 + 45 + 55 = 115

Neither = Total people minus At least one = 120 – 115 = 5

 

Answers

Neither: 5

Exactly one of the two: 60

Only Soccer: 15

Only Cricket: 45

At least one of the two: 115

GRE-style Practice Question 1

GRE-style practice question 1

Correct Answer: C.

The two quantities are equal.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

In a group of 100 members, 80 drink tea, and 60 drink both tea and coffee.

Quantity A

Number of individuals who drink tea but not coffee

Quantity B

20

  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

 

Step-by-Step Explanation

  • Total people who drink tea = 80
  • People who drink both tea and coffee = 60
  • To find those who drink only tea, subtract the “both” group from the total tea group.
  • Calculation: 80 – 60 = 20
  • Therefore, Quantity A is 20.
  • Quantity B is also 20.

 

Correct Answer: C

The two quantities are equal.

GRE-style Practice Question 2

GRE-style practice question 2

Correct Answer: C

The two quantities are equal.

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

Following is a concise, step-wise written explanation…

 

Question

In a group of 100 members, everybody drinks at least one of the two drinks — tea and coffee. 70 members of the group drink coffee.

Quantity A

Number of individuals who drink tea but not coffee

Quantity B

30

  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given.

 

Step-by-step Solution

  • Total members : 100
  • Coffee drinkers : 70
  • Condition : Every person drinks at least one of the two drinks. This means there are no people who drink neither.
  • Calculation : To find those who drink tea but not coffee, subtract the coffee drinkers from the total group.
  • Equation : 100 – 70 = 30
  • Result : There are exactly 30 people who must drink only tea to reach the total of 100.
  • Comparison : Quantity A is 30 and Quantity B is 30.

 

Correct Answer: C

The two quantities are equal.

GRE-style Practice Question 3

GRE-style practice question 3

Correct Answer: 20

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

For visual illustration, here is a snapshot from the video…

Example 4 – Select One

Following is a concise, step-wise written explanation…

 

Question

In a group of 100 members, 20 drink neither tea nor coffee. 70 members of the group drink coffee and 30 members of the group drink tea. How many members drink both tea and coffee?

  1. 0
  2. 10
  3. 20
  4. 30
  5. 40

 

Step-by-Step Solution

1. Find the number of people who drink at least one beverage

Subtract the people who drink neither from the total group. Total members = 100 Neither = 20 People who drink tea, coffee, or both = 100 – 20 = 80

2. Add the individual beverage drinkers

Coffee drinkers = 70 Tea drinkers = 30 Sum = 70 + 30 = 100

3. Calculate the overlap (Both)

The sum (100) is higher than the actual number of drinkers (80) because the people who drink both were counted twice. Both = Sum of individuals – Total drinkers Both = 100 – 80 = 20

 

Correct Answer: 20

GRE-style Practice Question 4

GRE-style practice question 4

Correct Answer: 80

 

For a detailed explanation, please refer to the video presented earlier on this page.

 

For visual illustration, here is a snapshot from the video…

Example 5 – Numeric entry

Following is a concise, step-wise written explanation…

 

Question

Out of the students in a cohort, 50 percent signed up for the nonprofit club and 40 percent signed up for the networking club. If 30 percent of the students in the cohort signed up for the networking club but not the nonprofit club, what percent of the students who signed up for the nonprofit club did not sign up for the networking club?

 

Step by Step Solution

Identify the Groups

Assume the total number of students in the cohort is 100.

  • Total Nonprofit (NP) = 50 students
  • Total Networking (NW) = 40 students

Find the Intersection

The problem states that 30 percent of the cohort (30 students) are in Networking but NOT in Nonprofit.

  • Networking Only = 30
  • Both Clubs = Total Networking minus Networking Only
  • Both Clubs = 40 minus 30 = 10 students

Find Nonprofit Only

  • Nonprofit Only = Total Nonprofit minus Both Clubs
  • Nonprofit Only = 50 minus 10 = 40 students

Calculate the Final Percentage

The question asks for the percent of the Nonprofit club members who did not join the Networking club.

  • Calculation = (Nonprofit Only divided by Total Nonprofit) times 100
  • Calculation = (40 divided by 50) times 100
  • Calculation = 0.8 times 100 = 80 percent

 

Correct Answer: 80

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