SSC CGL Reasoning · Venn Diagrams

Counting from a Venn Diagram

A drawn diagram puts a number in each region of overlapping circles, and the question asks how many belong to a described part: only one class, both of two, exactly two, or one class but not another.

All Venn Diagrams notes

Why this matters

Fourteen questions in the bank, all from the 2024 shifts, and 13 of them EASY. The arithmetic is a sum of two or three numbers; the marks are lost by reading the wrong regions, so naming each region before adding is the whole skill.

Concept 1 of 2: Reading one region at a time

Three overlapping circles make seven regions: three where a circle stands alone, three where two circles overlap without the third, and one middle region inside all three. Every question names some of these regions, so name them first and only then add.

Definition

Only A: the part of circle A outside both other circles.
A and B only (but not C): the overlap of A and B outside C.
All three: the middle region.
Exactly one: the three "only" regions added. Exactly two: the three "two only" regions added.
More than one: exactly two plus all three.
Fast route: point at each region the question names before you write any number; most wrong options add one region too many.

Worked example

In a diagram of three circles, Tea only holds 20, Coffee only 15, Juice only 10, Tea and Coffee only 6, Coffee and Juice only 4, Tea and Juice only 5, and all three 3. How many people like only tea or only juice?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 23 Sep 2024, Shift 3 · Q9Easy

Example 1 · Venn Diagrams · Counting from a Venn Diagram

How many people like either only spicy or only sweet as per the given Venn diagram?

Adding an overlap when the question says only

"Only tea" excludes every region shared with another circle. Use the part of the Tea circle that touches no other circle.

Counting the middle region as exactly two

The middle region is in all three circles, so it is not part of "exactly two". Add it only when the question asks for two or more.

Drill 6 more on reading one region at a time

Concept 2 of 2: Adding the regions of a described group

"People who like tea" means every region inside the Tea circle, whatever else they like. "Tea and coffee both" means every region inside both circles, including the middle. Build the group region by region and add.

Definition

All of A: only A + each two-circle region inside A + the middle.
A and B both: the A and B only region + the middle.
A but not C: the regions inside A that are outside C: only A and the A and B only region.
A and B but not C: the A and B only region alone.
Difference questions: work out each group first, then subtract.
Fast route: for "both" questions, the answer is always one two-circle region plus the middle region.

The whole of one circle in a three-circle diagram

n(A) = \text{only } A + (A \cap B \text{ only}) + (A \cap C \text{ only}) + (A \cap B \cap C)

Worked example

In a diagram of three circles, Tea only holds 20, Coffee only 15, Juice only 10, Tea and Coffee only 6, Coffee and Juice only 4, Tea and Juice only 5, and all three 3. How many people like tea?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 10 Sep 2024, Shift 3 · Q6Easy

Example 2 · Venn Diagrams · Counting from a Venn Diagram

Study the given Venn diagram and answer the question that follows. How many smart people are polite?

Leaving out the middle region from "both"

People who like tea and coffee include those who like all three. Add the Tea and Coffee only region and the middle region.

Keeping regions inside the excluded circle

"Women entrepreneurs who are not Indian" must leave out every region inside the Indian circle, including the middle one. Strike those regions before you add.

Drill 6 more on adding the regions of a described group

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Adding the regions of a described group
The whole of one circle in a three-circle diagram
$n(A) = \text{only } A + (A \cap B \text{ only}) + (A \cap C \text{ only}) + (A \cap B \cap C)$

Watch out for (4)

Adding an overlap when the question says only
→ Reading one region at a time
Counting the middle region as exactly two
→ Reading one region at a time
Leaving out the middle region from "both"
→ Adding the regions of a described group
Keeping regions inside the excluded circle
→ Adding the regions of a described group

Test yourself on Venn Diagrams

15 past SSC CGL questions from this chapter, timed at 9 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Venn Diagrams chapter test Drill all 14 questions