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4th Std Scholarship · Intelligence Test

Venn Diagram

5 concepts · 40 practice questions with answers · Free to read

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Venn diagram of two groups

From everyday life

In a class some children like cricket, some like kabaddi and some like both. To show such overlapping groups we draw two circles that overlap.

Simple definition

Venn diagram : one circle for each group. The middle part of two circles holds those who are in both groups. The part outside the circles holds those who are in none.

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Parts of the figure

PartMeaning
only Ain the first group only
middlein both groups
only Bin the second group only
outsidein no group
1
Rule 1

Each part of the figure has its own number; it is the number of children in that part.

  • e.g. only A : 8.
2
Rule 2

Total in A = only A + the middle part.

  • e.g. 8 + 6 = 14.
3
Rule 3

A or B (or both) = only A + middle + only B.

  • e.g. 8 + 6 + 5 = 19.
4
Rule 4

All the children = all parts in the circles + the number outside.

  • e.g. 19 + 3 = 22.
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Example 1 · easy

circle A = children who like kabaddi, circle B = children who like drawing. How many children like only kabaddi?

  1. Look at the part of A only.
  2. Answer : 9.
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Example 2 · medium

circle A = children who like kho-kho, circle B = children who like cricket. How many children like kho-kho in all?

  1. A only + the middle : 3 + 3.
  2. Answer : 6.
11710AB7
Example 3 · exam-style

circle A = children who like kho-kho, circle B = children who like cricket. How many children like kho-kho or cricket (or both)?

  1. Add the three parts : 11 + 7 + 10.
  2. Answer : 28.
Remember

Read each part separately; the middle number is counted in both circles.

Common mistake

Forgetting the middle number when finding the total of A.

Shortcut

Memory clue : "A circle is a house; the middle is the house shared by both."

Shortcut

"or" means add; "both" means the middle number; "neither" means the outside number.

Spot the difference

WordWhich number
only Athe part of A only
A in allonly A + middle
boththe middle part

Practice questions

Easy
  1. 110410AB6 circle A = children who like cricket, circle B = children who like kho-kho. How many children like only cricket?
  2. 2787AB1 circle A = children who like kabaddi, circle B = children who like cricket. How many children like kabaddi in all?
  3. 3833AB5 circle A = children who like cricket, circle B = children who like singing. How many children like both?
Medium
  1. 4467AB7 circle A = children who like singing, circle B = children who like kabaddi. How many children like singing or kabaddi (or both)?
  2. 5845AB7 circle A = children who like kabaddi, circle B = children who like singing. How many children like neither?
  3. 6699AB5 circle A = children who like drawing, circle B = children who like singing. How many children are there in all?
Challenging
  1. 7774AB5 circle A = children who like reading, circle B = children who like kabaddi. How many children like exactly one game?
  2. 86912AB2 circle A = children who like singing, circle B = children who like kabaddi. How many children like singing in all?
Show answers (try them yourself first)
  1. 110
  2. 215
  3. 33
  4. 417
  5. 57
  6. 629
  7. 711
  8. 815

Adding, subtracting and the sums on "both"

From everyday life

A class has 30 children; 18 like tea and 15 like milk. 18 + 15 = 33 but there are only 30 children! It means some children were counted in both.

Simple definition

When you add two groups those who are in both are counted twice. So total = first group + second group − both.

13510AB2

Words and operations

WordOperation
oradd, middle number once
and / boththe middle number
onlysubtract the middle number
nonesubtract from the total
1
Rule 1

Add the two groups and subtract the number of both once.

  • e.g. 18 + 15 − 5 = 28.
2
Rule 2

The difference between the total and "those who like at least one" is those who like neither.

  • e.g. 30 − 28 = 2.
3
Rule 3

To find those in only one group subtract both from that group.

  • e.g. 18 − 5 = 13 only tea.
4
Rule 4

When you get the answer add all the parts and match with the total.

  • e.g. 13 + 5 + 10 + 2 = 30.
Example 1 · easy

A class has 22 children. 7 like kabaddi and 10 like singing. 3 like both. How many like only kabaddi?

  1. 7 − 3 = 4.
  2. Answer : 4.
Example 2 · medium

A class has 21 children. 8 like cricket and 12 like reading. 4 like both. How many like neither?

  1. Those who like at least one : 8 + 12 − 4 = 16.
  2. Those who like neither : 21 − 16 = 5.
  3. Answer : 5.
Example 3 · exam-style

A class has 31 children. 18 like kabaddi and 12 like singing. 8 like neither. How many like both?

  1. Those who like at least one : 31 − 8 = 23.
  2. Both : 18 + 12 − 23 = 7.
  3. Answer : 7.
Remember

After adding two groups subtract the number in both once.

Common mistake

Not subtracting the number in both; it is counted twice.

Shortcut

Memory clue : "Added twice, subtract once."

Shortcut

First draw the circles and fill all the parts; then read off the answer.

Spot the difference

QuestionOperation
likes bothA + B − (at least one)
only AA − both
neithertotal − (at least one)

Practice questions

Easy
  1. 1A class has 29 children. 18 like kabaddi and 16 like drawing. 9 like both. How many like only kabaddi?
  2. 2A class has 27 children. 12 like kabaddi and 21 like reading. 9 like both. How many like only reading?
  3. 3A class has 28 children. 9 like kho-kho and 18 like cricket. 6 like both. How many like neither?
Medium
  1. 4A class has 25 children. 17 like reading and 10 like drawing. 3 like neither. How many like both?
  2. 5A class has 23 children. 15 like drawing and 16 like cricket. 9 like both. How many like neither?
  3. 6A class has 20 children. 8 like reading and 11 like kabaddi. 3 like neither. How many like both?
Challenging
  1. 7A class has 21 children. 12 like reading and 12 like drawing. 5 like both. How many like only reading?
  2. 8A class has 31 children. 15 like cricket and 16 like kabaddi. 4 like both. How many like neither?
Show answers (try them yourself first)
  1. 19
  2. 212
  3. 37
  4. 45
  5. 51
  6. 62
  7. 77
  8. 84

Venn diagram of three groups

From everyday life

Cricket, kabaddi and kho-kho : some children play all three, some only two and some only one. When three circles overlap seven parts are formed.

Simple definition

The three circles make seven parts : only A, only B, only C, common to A and B, common to A and C, common to B and C, and common to all three. The eighth part is outside the circles.

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Seven parts

PartOf how many
only A, B or Cof one only
AB, AC, BC (without the third)of two
the middleof all three
1
Rule 1

Total of one circle = all four parts inside that circle.

  • e.g. A : only A + AB + AC + all three.
2
Rule 2

"Exactly one" = only A + only B + only C.

  • e.g. 5 + 4 + 6 = 15.
3
Rule 3

"Exactly two" = AB + AC + BC (do not take the middle).

  • e.g. 3 + 2 + 4 = 9.
4
Rule 4

"At least one" = the sum of all seven parts; the outside part is separate.

  • e.g. 5 + 4 + 6 + 3 + 2 + 4 + 1 = 25.
6448838ABC6
Example 1 · easy

circle A = children who like kabaddi, circle B = children who like cricket, circle C = children who like drawing. How many children like only kabaddi?

  1. The part of A only.
  2. Answer : 6.
4517732ABC9
Example 2 · medium

circle A = children who like singing, circle B = children who like kho-kho, circle C = children who like cricket. How many children like singing in all?

  1. All four parts of A : 4 + 7 + 7 + 2.
  2. Answer : 20.
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Example 3 · exam-style

circle A = children who like drawing, circle B = children who like kho-kho, circle C = children who like kabaddi. How many children like exactly two games?

  1. Leave out the middle of all three : 5 + 7 + 1.
  2. Answer : 13.
Remember

Seven parts in three circles; decide which parts to take from the words of the question.

Common mistake

Taking the middle part (all three) in "exactly two".

Shortcut

Memory clue : "Only : one, two, three — count the parts."

Shortcut

When you take a circle run your finger over its four parts and add them; none is missed.

Spot the difference

WordParts
exactly oneonly A + B + C
exactly twoAB + AC + BC
all threethe middle

Practice questions

Easy
  1. 15685585ABC7 circle A = children who like kabaddi, circle B = children who like cricket, circle C = children who like drawing. How many children like only kabaddi?
  2. 23767222ABC9 circle A = children who like kabaddi, circle B = children who like drawing, circle C = children who like singing. How many children like all three?
  3. 32655681ABC4 circle A = children who like kho-kho, circle B = children who like kabaddi, circle C = children who like cricket. How many children like none of the games?
Medium
  1. 48264456ABC7 circle A = children who like singing, circle B = children who like kabaddi, circle C = children who like reading. How many children like singing in all?
  2. 57517262ABC4 circle A = children who like kho-kho, circle B = children who like reading, circle C = children who like cricket. How many children like exactly one game?
  3. 61656354ABC6 circle A = children who like cricket, circle B = children who like singing, circle C = children who like kho-kho. How many children like at least one game?
Challenging
  1. 74583372ABC6 circle A = children who like kho-kho, circle B = children who like drawing, circle C = children who like kabaddi. How many children like exactly two games?
  2. 84818818ABC2 circle A = children who like singing, circle B = children who like kabaddi, circle C = children who like reading. How many children are there in all?
Show answers (try them yourself first)
  1. 15
  2. 22
  3. 34
  4. 422
  5. 513
  6. 630
  7. 713
  8. 840

Choosing the right figure for a relation

From everyday life

To show animal, cow and tiger in one figure the cow and the tiger sit inside the big circle of animals, but apart from each other.

Simple definition

Three groups can be related in a few ways : one inside another, two separate ones inside one big one, all three separate, or all three overlapping. Spot the relation when you choose the figure among the options.

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Kinds of relation

FigureRelation
figure 1one inside another (fruit, mango, Alphonso)
figure 2two separate ones inside one big one (animal, cow, tiger)
figure 3all three separate (cow, pen, river)
figure 4all three overlapping (doctor, teacher, player)
1
Rule 1

First compare the groups two at a time : does one fall inside the other, are they apart, or do they share something.

  • e.g. cow and animal : the cow is among animals.
2
Rule 2

If one group falls wholly inside another the small circle is drawn inside the big one.

  • e.g. a mango is among fruits.
3
Rule 3

If two groups can share something their circles overlap.

  • e.g. a doctor can also be a teacher.
4
Rule 4

When all three pairs are settled choose the figure with the same relations.

  • e.g. three separate : three separate circles.
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Example 1 · easy

Which figure shows the relation among Vehicle, Four-wheeler, Car?

  1. a car is a four-wheeler and a four-wheeler is a vehicle.
  2. Answer : 4.
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Example 2 · medium

Which figure shows the relation among Animal, Cow, Tiger?

  1. cow and tiger are both animals but different from each other.
  2. Answer : 2.
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Example 3 · exam-style

Which figure shows the relation among Men, Women, Doctor?

  1. men and women are separate; doctors can be in both.
  2. Answer : 4.
Remember

Compare two groups at a time to settle the relation; then choose the three-circle figure.

Common mistake

Showing men and women as overlapping; they are separate.

Shortcut

Memory clue : "If it falls inside draw it inside; if apart draw it apart; if it mixes let it overlap."

Shortcut

Look at the group that sits between two groups; if it mixes with both its circle spreads over both.

Spot the difference

GroupsFigure
fruit, mango, Alphonsoone inside another
animal, cow, tigertwo separate inside a big one
men, women, doctortwo separate + one over both

Practice questions

Easy
  1. 11234 Which figure shows the relation among India, Maharashtra, Pune?
  2. 21234 Which figure shows the relation among Sun, Table, Fish?
  3. 31234 Which figure shows the relation among Game, Cricket, Kabaddi?
Medium
  1. 41234 Which figure shows the relation among Student, Player, Singer?
  2. 51234 Which figure shows the relation among Men, Women, Doctor?
  3. 61234 Which figure shows the relation among Fruit, Mango, Alphonso?
Challenging
  1. 71234 Which figure shows the relation among Doctor, Teacher, Player?
  2. 81234 Which figure shows the relation among Boys, Girls, Player?
Show answers (try them yourself first)
  1. 13 (Pune is in Maharashtra and Maharashtra is in India).
  2. 23 (three different things).
  3. 33 (cricket and kabaddi are games but different).
  4. 44 (a student can also be a player and a singer).
  5. 51 (men and women are separate; doctors can be in both).
  6. 64 (Alphonso is a kind of mango and mango is a fruit).
  7. 72 (one person can be a doctor, a teacher and a player; the groups share members).
  8. 81 (boys and girls are separate; players can be in both).

Mixed questions and exam tricks

From everyday life

In the exam sometimes a figure is given and sometimes the question is in words. First draw or read the figure, then pick exactly the right parts and calculate.

Simple definition

Three steps : (1) draw or read the figure, (2) pick the parts from the words of the question, (3) add or subtract and match with the total.

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Three steps

StepJob
1draw or read the figure
2pick the parts from the words
3add or subtract and match
1
Rule 1

If there is no figure draw two circles and fill every part.

  • e.g. total 40, both 6 : the middle part is 6.
2
Rule 2

When the question says "only" subtract the middle part.

  • e.g. A = 20, both = 6 : only A = 14.
3
Rule 3

Add all parts and match with the total children.

  • e.g. 14 + 6 + only B + outside = 40.
4
Rule 4

With three groups lay out the seven parts and fill them starting from the middle.

  • e.g. all three first, then two at a time, then one at a time.
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Example 1 · easy

circle A = children who like singing, circle B = children who like kho-kho. How many children like neither?

  1. The number outside the circles.
  2. Answer : 4.
Example 2 · medium

A class has 18 children. 8 like cricket and 13 like drawing. 4 like both. How many like only drawing?

  1. 13 − 4 = 9.
  2. Answer : 9.
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Example 3 · exam-style

circle A = children who like kabaddi, circle B = children who like kho-kho, circle C = children who like reading. How many children like at least one game?

  1. Add all seven parts inside the circles.
  2. Answer : 33.
Remember

Three steps : figure, pick the parts, calculate; match the sum at the end.

Common mistake

Calculating in the head without drawing the figure.

Shortcut

Memory clue : "First the circles, then the numbers."

Shortcut

In a three-group question fill the middle first; the other parts follow from it.

Spot the difference

QuestionOperation
at least oneadd all parts
exactly oneadd the only-parts
nonethe outside number

Practice questions

Easy
  1. 19912AB2 circle A = children who like singing, circle B = children who like kho-kho. How many children are there in all?
  2. 2A class has 24 children. 14 like drawing and 13 like kho-kho. 8 like both. How many like neither?
  3. 38445857ABC6 circle A = children who like kabaddi, circle B = children who like kho-kho, circle C = children who like drawing. How many children like kabaddi and kho-kho but not drawing?
Medium
  1. 4797AB7 circle A = children who like reading, circle B = children who like singing. How many children like exactly one game?
  2. 5A class has 26 children. 15 like drawing and 8 like singing. 7 like neither. How many like both?
  3. 66258443ABC4 circle A = children who like kabaddi, circle B = children who like kho-kho, circle C = children who like cricket. How many children like exactly one game?
Challenging
  1. 76245185ABC4 circle A = children who like drawing, circle B = children who like kho-kho, circle C = children who like cricket. How many children are there in all?
  2. 81234 Which figure shows the relation among Animal, Bird, Sparrow?
Show answers (try them yourself first)
  1. 132
  2. 25
  3. 35
  4. 414
  5. 54
  6. 613
  7. 735
  8. 83 (a sparrow is a bird and a bird is an animal).

Quick revision of the whole lesson

TopicMemory clue
Two groupsor : add; both : the middle; only : subtract the middle
SumAdded twice, subtract once
Three groupsseven parts; the middle first
Relationinside, apart or overlapping

🎯 Most important points for the exam

  • Total = first group + second group − both.
  • "Only" means subtract the middle part.
  • Three circles have seven parts; the outside part is separate.
  • Do not take the middle part of all three in "exactly two".
  • To settle a relation compare the groups two at a time.
  • Add all parts and match with the total.
🧠 Study mantra

"First the circles, then the numbers; subtract both once."