problem solving using venn diagram worksheet

Venn Diagrams Practice Questions

Click here for questions, click here for answers, gcse revision cards.

5-a-day Workbooks

Primary Study Cards

Privacy Policy

Terms and Conditions

Corbettmaths © 2012 – 2024

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Free ready-to-use math resources

Hundreds of free math resources created by experienced math teachers to save time, build engagement and accelerate growth

15 Venn Diagram Questions And Practice Problems (Middle & High School): Exam Style Questions Included

Beki Christian

Venn diagram questions involve visual representations of the relationship between two or more different groups of things. Venn diagrams are first covered in elementary school and their complexity and uses progress through middle and high school.

This article will look at the types of Venn diagram questions that might be encountered at middle school and high school, with a focus on exam style example questions and preparing for standardized tests. We will also cover problem-solving questions. Each question is followed by a worked solution.

How to solve Venn diagram questions

In middle school, sets and set notation are introduced when working with Venn diagrams. A set is a collection of objects. We identify a set using braces. For example, if set A contains the odd numbers between 1 and 10, then we can write this as: 

A = {1, 3, 5, 7, 9}

Venn diagrams sort objects, called elements, into two or more sets.

This diagram shows the set of elements 

{1,2,3,4,5,6,7,8,9,10} sorted into the following sets.

Set A= factors of 10 

Set B= even numbers

The numbers in the overlap (intersection) belong to both sets. Those that are not in set A or set B are shown outside of the circles.

Different sections of a Venn diagram are denoted in different ways.

ξ represents the whole set, called the universal set.

∅ represents the empty set, a set containing no elements.

Venn Diagrams Worksheet

Download this quiz to check your students' understanding of Venn diagrams. Includes 10 questions with answers!

Let’s check out some other set notation examples!

In middle school and high school, we often use Venn diagrams to establish probabilities.

We do this by reading information from the Venn diagram and applying the following formula.

For Venn diagrams we can say

Middle School Venn diagram questions

In middle school, students learn to use set notation with Venn diagrams and start to find probabilities using Venn diagrams. The questions below are examples of questions that students may encounter in 6th, 7th and 8th grade.

Venn diagram questions 6th grade

1. This Venn diagram shows information about the number of people who have brown hair and the number of people who wear glasses.

How many people have brown hair and glasses?

The intersection, where the Venn diagrams overlap, is the part of the Venn diagram which represents brown hair AND glasses. There are 4 people in the intersection.

2. Which set of objects is represented by the Venn diagram below?

We can see from the Venn diagram that there are two green triangles, one triangle that is not green, three green shapes that are not triangles and two shapes that are not green or triangles. These shapes belong to set D.

Venn diagram questions 7th grade

3. Max asks 40 people whether they own a cat or a dog. 17 people own a dog, 14 people own a cat and 7 people own a cat and a dog. Choose the correct representation of this information on a Venn diagram.

There are 7 people who own a cat and a dog. Therefore, there must be 7 more people who own a cat, to make a total of 14 who own a cat, and 10 more people who own a dog, to make a total of 17 who own a dog.

Once we put this information on the Venn diagram, we can see that there are 7+7+10=24 people who own a cat, a dog or both.

40-24=16 , so there are 16 people who own neither.

4. The following Venn diagrams each show two sets, set A and set B . On which Venn diagram has A ′ been shaded?

\mathrm{A}^{\prime} means not in \mathrm{A} . This is shown in diagram \mathrm{B.}

Venn diagram questions 8th grade

5. Place these values onto the following Venn diagram and use your diagram to find the number of elements in the set \text{S} \cup \text{O}.

\xi = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \text{S} = square numbers \text{O} = odd numbers

\text{S} \cup \text{O} is the union of \text{S} or \text{O} , so it includes any element in \text{S} , \text{O} or both. The total number of elements in \text{S} , \text{O} or both is 6.

6. The Venn diagram below shows a set of numbers that have been sorted into prime numbers and even numbers.

A number is chosen at random. Find the probability that the number is prime and not even.

The section of the Venn diagram representing prime and not even is shown below.

There are 3 numbers in the relevant section out of a possible 10 numbers altogether. The probability, as a fraction, is \frac{3}{10}.

7. Some people visit the theater. The Venn diagram shows the number of people who bought ice cream and drinks in the interval.

Ice cream is sold for $3 and drinks are sold for $ 2. A total of £262 is spent. How many people bought both a drink and an ice cream?

Money spent on drinks: 32 \times \$2 = \$64

Money spent on ice cream: 16 \times \$3 = \$48

\$64+\$48=\$112 , so the information already on the Venn diagram represents \$112 worth of sales.

\$262-\$112 = \$150 , so another \$150 has been spent.

If someone bought a drink and an ice cream, they would have spent \$2+\$3 = \$5.

\$150 \div \$5=30 , so 30 people bought a drink and an ice cream.

High school Venn diagram questions

In high school, students are expected to be able to take information from word problems and put it onto a Venn diagram involving two or three sets. The use of set notation is extended and the probabilities become more complex.

In advanced math classes, Venn diagrams are used to calculate conditional probability.

Lower ability Venn diagram questions

8. 50 people are asked whether they have been to France or Spain.

18 people have been to France. 23 people have been to Spain. 6 people have been to both.

By representing this information on a Venn diagram, find the probability that a person chosen at random has not been to Spain or France.

6 people have been to both France and Spain. This means 17 more have been to Spain to make 23  altogether, and 12 more have been to France to make 18 altogether. This makes 35 who have been to France, Spain or both and therefore 15 who have been to neither.

The probability that a person chosen at random has not been to France or Spain is \frac{15}{50}.

9. Some people were asked whether they like running, cycling or swimming. The results are shown in the Venn diagram below.

One person is chosen at random. What is the probability that the person likes running and cycling?

9 people like running and cycling (we include those who also like swimming) out of 80 people altogether. The probability that a person chosen at random likes running and cycling is \frac{9}{80}.

10. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16\}

\text{A} = \{ even numbers \}

\text{B} = \{ multiples of 3 \}

By completing the following Venn diagram, find \text{P}(\text{A} \cup \text{B}^{\prime}).

\text{A} \cup \text{B}^{\prime} means \text{A} or not \text{B} . We need to include everything that is in \text{A} or is not in \text{B} . There are 13 elements in \text{A} or not in \text{B} out of a total of 16 elements.

Therefore \text{P}(\text{A} \cup \text{B}^{\prime}) = \frac{13}{16}.

11. ξ = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}

A = \{ multiples of 2 \}

By putting this information onto the following Venn diagram, list all the elements of B.

We can start by placing the elements in \text{A} \cap \text{B} , which is the intersection.

We can then add any other multiples of 2 to set \text{A}.

Next, we can add any unused elements from \text{A} \cup \text{B} to \text{B}.

Finally, any other elements can be added to the outside of the Venn diagram.

The elements of \text{B} are \{1, 2, 3, 4, 6, 12\}.

Middle ability high school Venn diagram questions

12. Some people were asked whether they like strawberry ice cream or chocolate ice cream. 82% said they like strawberry ice cream and 70% said they like chocolate ice cream. 4% said they like neither.

By putting this information onto a Venn diagram, find the percentage of people who like both strawberry and chocolate ice cream.

Here, the percentages add up to 156\%. This is 56\% too much. In this total, those who like chocolate and strawberry have been counted twice and so 56\% is equal to the number who like both chocolate and strawberry. We can place 56\% in the intersection, \text{C} \cap \text{S}

We know that the total percentage who like chocolate is 70\%, so 70-56 = 14\%-14\% like just chocolate. Similarly, 82\% like strawberry, so 82-56 = 26\%-26\% like just strawberry.

13. The Venn diagram below shows some information about the height and gender of 40 students.

A student is chosen at random. Find the probability that the student is female given that they are over 1.2 m .

We are told the student is over 1.2m. There are 20 students who are over 1.2m and 9 of them are female. Therefore the probability that the student is female given they are over 1.2m is   \frac{9}{20}.

14. The Venn diagram below shows information about the number of students who study history and geography.

H = history

G = geography

Work out the probability that a student chosen at random studies only history.

We are told that there are 100 students in total. Therefore:

x = 12 or x = -3 (not valid) If x = 12, then the number of students who study only history is 12, and the number who study only geography is 24. The probability that a student chosen at random studies only history is \frac{12}{100}.

15. 50 people were asked whether they like camping, holiday home or hotel holidays.

18\% of people said they like all three. 7 like camping and holiday homes but not hotels. 11 like camping and hotels. \frac{13}{25} like camping.

Of the 27 who like holiday homes, all but 1 like at least one other type of holiday. 7 people do not like any of these types of holiday.

By representing this information on a Venn diagram, find the probability that a person chosen at random likes hotels given that they like holiday homes.

Put this information onto a Venn diagram.

We are told that the person likes holiday homes. There are 27 people who like holiday homes. 19 of these also like hotels. Therefore, the probability that the person likes hotels given that they like holiday homes is \frac{19}{27}.

Looking for more Venn diagram math questions for middle and high school students ?

• Probability questions
• Ratio questions
• Algebra questions
• Trigonometry questions
• Long division questions
• Pythagorean theorem questions

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

The content in this article was originally written by secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary math teacher Katie Keeton.

Related articles

28 7th Grade Math Problems With Answers And Worked Examples

15 Probability Questions And Practice Problems for Middle and High School: Harder Exam Style Questions Included

9 Algebra Questions And Practice Problems To Do With Your Middle Schoolers

15 Trigonometry Questions And Practice Problems To Do With High Schoolers

Solving Inequalities Questions [FREE]

Downloadable skills and applied questions about solving inequalities.

Includes 10 skills questions, 5 applied questions and an answer key. Print and share with your classes to support their learning.

Privacy Overview

Venn Diagram Worksheets

Venn diagrams are one of the most important concepts in the field of mathematics. It is one of the most fundamental concepts which one needs to understand in order to understand set theory in depth. Venn Diagrams are often used to represent a set of data that may or may not be mutually exclusive. It helps people to understand complex data easily. Venn diagrams become useful when you need to represent two or more parameters in a given data.

Benefits of Venn Diagram Worksheets

Cuemath experts have developed a set of Venn diagram worksheets that contain many solved examples as well as questions. Students would be able to clear their concepts by solving these questions on their own and clear their school exams as well as competitive exams like Olympiads and represent complex data easily.

Download Venn Diagram Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

Core Math Worksheets

Addition worksheets, subtraction worksheets, multiplication worksheets, division worksheets, fact family worksheets, long division worksheets, negative numbers, exponents worksheets, order of operations worksheets, fraction worksheets, fractions worksheets, graphic fractions, equivalent fractions, reducing fractions, comparing fractions, adding fractions, subtracting fractions, multiplying fractions, dividing fractions, fractions as decimals, fraction decimal percent, word problems, pre-algebra word problems, money word problems, combining like terms, properties of multiplication, exponent rules, linear equations, one step equations, two step equations, factoring polynomials, quadratic equations, other worksheets, place value, percentages, rounding numbers, ordering numbers, standard, expanded, word form, mean median mode range, ratio worksheets, probability worksheets, roman numerals, factorization, gcd, lcm, prime and composite numbers, pre-algebra, geometry worksheets, blank clocks, telling analog time, analog elapsed time, greater than and less than, arithmetic sequences, geometric sequences, venn diagram, graph worksheets, measurement & conversions, inches measurement, metric measurement, metric si unit conversions, customary unit conversions, customary and metric, patterns and puzzles, number patterns, patterns with negatives, missing operations, magic square, number grid puzzles, word search puzzles, color by number, addition color by number, subtraction color by number, multiplication color by number, division color by number, color by number, holiday & seasonal, valentine's day, st. patrick's day, thanksgiving, early learning, base ten blocks, printable flash cards, number matching, number tracing, missing numbers, picture math addition, picture math subtraction, picture math multiplication, picture math division, multiplication chart, multiplication table, prime numbers chart, hundreds chart, place value chart, roman numerals chart, handwriting paper, graph paper, coordinate plane, spaceship math check-off, square root chart, fraction chart, probability chart, measurement chart, number line, comic strip template, calculators, age calculator, factoring calculator, fraction calculator, slope calculator, degrees to radians, percentage calculator, prime factorization calculator, roman numeral converter, long division calculator, multiplication calculator, math worksheets by grade, preschool math worksheets, kindergarten math worksheets, 1st grade math worksheets, 2nd grade math worksheets, 3rd grade math worksheets, 4th grade math worksheets, 5th grade math worksheets, 6th grade math worksheets, worksheet news, venn diagram worksheets and templates.

Check out this page for Venn diagram worksheets, blank Venn diagram templates and practice for Venn diagram concepts. Venn diagrams are useful for learning set concepts such as intersection, exclusion and complements.

Blank Venn Diagram Templates

Venn Diagram Vocabulary Terms and Symbols

Venn Diagram Worksheets with Sets

Venn Diagram Shaded Worksheets

Venn Diagram Set Notation Worksheets

Venn Diagram Shaded the Regions Worksheets

Venn Diagram Word Problems

Working with Venn Diagrams

The Venn Diagram worksheets on this page start with getting students to familiarize themselves with terminologies and symbols that are essential when using Venn Diagrams.

The other set of worksheets require students to shade and name regions, solve set notation problems using set theory and illustrate sets through a Venn diagram - both using two and three sets. The last set of worksheets are fun word problems that will test the depth of understanding and knowledge of the students about Venn diagrams. All with answer keys and solutions!

Go ahead and print a Venn diagram worksheet from this page and check out our free printable blank Venn diagram templates too or what some may refer to as Venn diagram graphic organizer. A free Venn diagram template is available for you, whatever it is that you might need – a blank Venn diagram, lined Venn diagram, black and white, colored Venn diagram and more. These are perfect for practicing along with examples in solving questions or problems using Venn diagrams.

Venn Diagram as Graphic Organizer

A Venn diagram, also called a Set diagram or Logic diagram is a logical representation of relationships between different group of things. It usually uses overlapping circles or sometimes other shapes to illustrate the relationships between two or more sets of items. Each circle by itself represents a set , which may include numbers, letters, ideas, concepts, objects, items, members or similar terms.. Often, it serves to graphically organize things, highlighting how the items are similar and different, making it quite important in both quantitative and logical reasoning and therefore also sometimes called as graphic organizer.

To make it simpler, Venn diagram is a visual representations of mathematical sets—or collections of objects—that are studied using a branch of mathematical logic called set theory. Basically, Venn diagram is used to show set theories in an illustration/drawing.

Refer to the image below. This is an illustration of 2 sets - A, B in a Venn diagram.

People use this diagram or graphic organizer to describe similarities and differences between two objects, which in this case, Set A and Set B.

What about symbols and notations? We will discuss that in a few… but before we proceed with that area of our topic here, let me share some important key words that you should take note when dealing with Venn diagrams… Make sure to keep these words and their meaning in mind. Another simple tip: What I normally do when solving problems using a Venn diagram is I highlight these keywords in the given question or word problem then write the equivalent meaning so as not to mix one with the other especially “or” and “and”.

and, all three = intersection

not, neither-nor = complement

but not, only = difference

Understanding Venn Diagrams Symbols

Now that we already know the basics, let’s move on to Venn diagram symbols and what they represent. Venn diagram symbols are a collection of mathematical symbols that are used within set theory. Here are the most commonly used ones:

Let’s discuss further the symbols for union, intersection, and complement of a set. These three are the perfect foundation when you are just starting to take a grasp of Venn diagrams .

The union symbol ( ∪ ): represents the union of all sets (not to be confused with the letter “u”). Refer to the image below. This is an illustration of a two-set Venn diagram. The items that are in set A or set B are called the “union of A or B” or which we notate as “A ∪ B”. Take note that we used the word “or” , which means union .

The intersection symbol ( ∩ ): represents all elements shared or common within the selected sets or groupings. We notate this as “A ∩ B”, pronounced as: "A intersect B”. This is the intersection in the middle or the gray area where Set A and Set B overlap. In the image below, it represents shared elements within sets A and B. Take note that we used the word “and” this time which means intersection .

The complement symbol (A’): represents whatever is not represented in a particular set. In this case, everything that is “ not” in set A or purple circle. This means that given a universe, everything that is in the universe, except for A, is the absolute complement of A in U. The diagram below shows the complement of A, all of the gray area except for purple.

How to Use a Venn Diagram

Let’s move on to three-set Venn diagram. Refer to the image below. This is an illustration of 3 sets - A, B, C in a Venn diagram.

We added a third set (C), making the unions and intersection a bit complicated than what we have discussed earlier but fret not! Same symbols and rules apply to three-set diagrams (same as two-set diagrams). To help you better understand the practical application of Venn diagram, let’s try this word problem…

Example: 30 kids were asked about their favorite colors. 7 kids like only purple, 8 kids like only green, while 5 kids like only pink. 4 kids like both purple and green, 2 kids like both pink and purple, 1 kid likes both green and pink, while 3 kids like all purple, pink, and green.

Set A contains all of the kids that like purple. Set B represents all of the kids who like green and set C represents all of the kids who like pink. Then the region marked as AB represents all of the kids who like both purple and green. The region marked BC represents all of the kids who like both green and pink. Similarly, the region AC represents all of the kids who like both purple and pink. The region in the middle ABC represents all of the kids who like all purple, green, and pink.

Take note of the word “only” in the word problem. This is different from “Set A”. Set A includes all of the kids who like purple (all those marked with A, AB,AC, ABC). Meanwhile, all of the kids who like only purple is the region marked as “A” only. Same thing applies to Set B and C.

Okay! Let’s figure this out together. Draw or print a blank Venn diagram then try to illustrate as per the information given in the word problem. Let’s compare our answers after.

Here’s what I came out with my Venn diagram. I’ve written down my solutions below as well.

And here’s the summary based on what I have on my Venn diagram…

16 answered purple from regions A+AB+AC+ABC (7+4+2+3). I computed the sum from all of the regions where purple is indicated. This is Set A.

16 answered green from regions B+AB+BC+ABC (8+4+1+3). I computed the sum from all of the regions where green is indicated. This is Set B.

11 answered pink from regions C+AC+BC+ABC (5+2+1+3). I computed the sum from all of the regions where pink is indicated. This is Set C.

22 answered purple or pink. from regions A+C+AB+AC+BC+ABC (7+5+4+2+1+3). I computed the sum from all of the regions where purple or pink is indicated.

25 answered purple or green from regions A+B+AB+AC+BC+ABC (7+8+4+2+1+3). I computed the sum from all of the regions where purple or green is indicated.

23 answered green or pink from regions A+C+AB+AC+BC+ABC (8+5+4+2+1+3). I computed the sum from all of the regions where green or pink is indicated.

All of the kids answered at least one color because the total number of all of the numbers in the circles of the Venn diagram is 30, which is the total number of all of the kids asked about their favorite colors as per the given problem.

Things to Remember About Venn Diagrams:

• Venn diagrams are also called set diagrams or logic diagrams .
• A Venn diagram in math is used in logic theory and set theory to show various sets or data and their relationship with each other.
• A set is a group or collection of things, that typically have something in common.
• An element or member of a set is any one of the distinct objects that belong to a set.
• A set of all possible elements is called the Universal set .
• Venn Diagrams were invented by John Venn , an English logician, in the 1880s.
• While there are many ways to organize a Venn diagram, they most often consist of overlapping circles .
• When circles overlap or intersect, those sub-sets that are shared are known as union or intersection .
• The middle of a Venn diagram where two or more sets overlap is known as the intersection .
• The categories not represented in a set are known as the complement of a set .
• Tip: Always start filling values in the Venn diagram from the innermost value.
• Note: The curly braces “” are the customary notation for sets. Do not use parentheses or square brackets.
• While often illustrated as a pair or triplet of circles, Venn diagram can use any number of circles or any other shape to show the differences and intersections of different sets.

Copyright 2008-2024 DadsWorksheets, LLC

Child Login

• Kindergarten
• Number charts
• Skip Counting
• Place Value
• Number Lines
• Subtraction
• Multiplication
• Word Problems
• Comparing Numbers
• Ordering Numbers
• Odd and Even
• Prime and Composite
• Roman Numerals
• Ordinal Numbers
• In and Out Boxes
• Number System Conversions
• More Number Sense Worksheets
• Size Comparison
• Measuring Length
• Metric Unit Conversion
• Customary Unit Conversion
• Temperature
• More Measurement Worksheets
• Writing Checks
• Profit and Loss
• Simple Interest
• Compound Interest
• Tally Marks
• Mean, Median, Mode, Range
• Mean Absolute Deviation
• Stem-and-leaf Plot
• Box-and-whisker Plot
• Permutation and Combination
• Probability
• Venn Diagram
• More Statistics Worksheets
• Shapes - 2D
• Shapes - 3D
• Lines, Rays and Line Segments
• Points, Lines and Planes
• Transformation
• Quadrilateral
• Ordered Pairs
• Midpoint Formula
• Distance Formula
• Parallel, Perpendicular and Intersecting Lines
• Scale Factor
• Surface Area
• Pythagorean Theorem
• More Geometry Worksheets
• Converting between Fractions and Decimals
• Significant Figures
• Convert between Fractions, Decimals, and Percents
• Proportions
• Direct and Inverse Variation
• Order of Operations
• Squaring Numbers
• Square Roots
• Scientific Notations
• Speed, Distance, and Time
• Absolute Value
• More Pre-Algebra Worksheets
• Translating Algebraic Phrases
• Evaluating Algebraic Expressions
• Simplifying Algebraic Expressions
• Algebraic Identities
• Quadratic Equations
• Systems of Equations
• Polynomials
• Inequalities
• Sequence and Series
• Complex Numbers
• More Algebra Worksheets
• Trigonometry
• Math Workbooks
• English Language Arts
• Summer Review Packets
• Social Studies
• Holidays and Events
• Worksheets >
• Statistics >
• Venn Diagram >
• Word Problems: Three sets

Venn Diagram Word Problem Worksheets - Three Sets

This page contains worksheets based on Venn diagram word problems, with Venn diagram containing three circles. The worksheets are broadly classified into two skills - Reading Venn diagram and drawing Venn diagram. The problems involving a universal set are also included. Printable Venn diagram word problem worksheets can be used to evaluate the analytical skills of the students of grade 6 through high school and help them organize the data. Access some of these worksheets for free!

Reading Venn Diagram - Type 1

These 6th grade pdf worksheets consist of Venn diagrams containing three sets with the elements that are illustrated with pictures. Interpret the Venn diagram and answer the word problems given below.

• Download the set

Reading Venn Diagram - Type 2

The elements of the sets are represented as symbols on the three circles of the Venn diagram. Count the symbols and write the answers in the space provided. Each worksheet contains two real-life scenarios.

Standard Word Problems - Without Universal Set

These PDF worksheets contain 3-circle Venn diagrams without universal set. Study the Venn diagram and answer the word problems.

Standard Word Problems - With Universal Set

This section features extensive collection of Venn diagram word problems with a universal set for grade 7 and grade 8 students. Real-life scenarios like doctor appointments, attractions of Niagara Falls and more are used.

Drawing Venn Diagram - Without Universal Set

Draw a venn diagram with three intersecting circles and fill in the data from the given descriptions. Also, ask 8th grade and high school students to answer questions based on union, intersection, and complement of three sets.

Draw Venn Diagram - With Universal Set

Six exclusive printable worksheets on Venn diagram word problems are included here. Draw three overlapping circles and fill in the regions using the information provided. Based on your findings, answer the word problems.

Related Worksheets

» Venn Diagram Word Problems - 2 sets

» Pictograph

» Stem-and-leaf Plot

» Tally Marks

Become a Member

Membership Information

Printing Help

How to Use Online Worksheets

How to Use Printable Worksheets

Privacy Policy

Terms of Use

Copyright © 2024 - Math Worksheets 4 Kids

This is a members-only feature!

• Skills by Standard
• Skills by Grade
• Skills by Category

Go to profile

• Assignments
• Assessments
• Report Cards
• Our Teachers

Need some help or instruction on how to do this skill?

Share MathGames with your students, and track their progress.

Learn Math Together.

Once you're comfortable with this skill, test your speed with our games.

If you notice any problems, please let us know .

Venn Diagram Notation: Worksheets with Answers

Whether you want a homework, some cover work, or a lovely bit of extra practise, this is the place for you. And best of all they all (well, most!) come with answers.

Mathster keyboard_arrow_up Back to Top

Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers.

Corbett Maths keyboard_arrow_up Back to Top

Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. It really is one of the very best websites around.

Venn Diagram Worksheets

Here you find our free grade 6 or 7 Venn diagram worksheets that can enhance both math and logic skills. Click on the worksheets to go to the download page.

Our Venn diagram worksheets are made for primary 6 and high school math students. Venn diagrams are used to picture the relationship between different groups or things To draw a Venn diagram you start with a big rectangle (called universe) and then you draw to circles overlap each other (or not). When presented with the Venn diagrams, students will have to answer all kinds of questions about the 2 groups of data. The concept of Venn diagram can be quite challenging for grade 6 math students and we would like to advise educators and teachers to print out our Venn worksheets and go through them step by step. Once the concept is understood, students will find them challenging but engaging. Our Venn diagram math materials and exercises come as free PDF printable.

• Being able to understand Venn diagrams
• Understand the concept of sets
• Understand logic of overlapping sets
• Being able to solve word problems based on the presented information
• Being able to apply deductive logic reasoning skills

• Home   |  
• About   |  
• Contact Us   |  
• Privacy   |  
• Newsletter   |  
• Shop   |  
• 🔍 Search Site
• Easter Color By Number Sheets
• Printable Easter Dot to Dot
• Easter Worksheets for kids
• Kindergarten
• All Generated Sheets
• Place Value Generated Sheets
• Addition Generated Sheets
• Subtraction Generated Sheets
• Multiplication Generated Sheets
• Division Generated Sheets
• Money Generated Sheets
• Negative Numbers Generated Sheets
• Fraction Generated Sheets
• Place Value Zones
• Number Bonds
• Addition & Subtraction
• Times Tables
• Fraction & Percent Zones
• All Calculators
• Fraction Calculators
• Percent calculators
• Area & Volume Calculators
• Age Calculator
• Height Calculator
• Roman Numeral Calculator
• Coloring Pages
• Fun Math Sheets
• Math Puzzles
• Mental Math Sheets
• Online Times Tables
• Online Addition & Subtraction
• Math Grab Packs
• All Math Quizzes
• Kindergarten Math Quizzes
• 1st Grade Quizzes
• 2nd Grade Quizzes
• 3rd Grade Quizzes
• 4th Grade Quizzes
• 5th Grade Quizzes
• 6th Grade Math Quizzes
• Place Value
• Rounding Numbers
• Comparing Numbers
• Number Lines
• Prime Numbers
• Negative Numbers
• Roman Numerals
• Subtraction
• Add & Subtract
• Multiplication
• Fraction Worksheets
• Learning Fractions
• Fraction Printables
• Percent Worksheets & Help
• All Geometry
• 2d Shapes Worksheets
• 3d Shapes Worksheets
• Shape Properties
• Geometry Cheat Sheets
• Printable Shapes
• Coordinates
• Measurement
• Math Conversion
• Statistics Worksheets
• Bar Graph Worksheets
• Venn Diagrams
• All Word Problems
• Finding all possibilities
• Logic Problems
• Ratio Word Problems
• All UK Maths Sheets
• Year 1 Maths Worksheets
• Year 2 Maths Worksheets
• Year 3 Maths Worksheets
• Year 4 Maths Worksheets
• Year 5 Maths Worksheets
• Year 6 Maths Worksheets
• All AU Maths Sheets
• Kindergarten Maths Australia
• Year 1 Maths Australia
• Year 2 Maths Australia
• Year 3 Maths Australia
• Year 4 Maths Australia
• Year 5 Maths Australia
• Meet the Sallies
• Certificates

Venn Diagram Worksheets 3rd Grade

Welcome to our Venn Diagram Worksheets for 3rd graders. Here you will find a wide range of free venn diagram sheets which will help your child learn to classify a range of objects, shapes and numbers using venn diagrams.

For full functionality of this site it is necessary to enable JavaScript.

Here are the instructions how to enable JavaScript in your web browser .

Venn Diagram Worksheets

What is a venn diagram.

A Venn diagram is a way of classifying groups of objects with the same properties.

Typically, a venn diagram has two or three circles that intersect each other.

There is also a space outside the circles where objects that do not fit any of the properties can go.

The diagram below shows you how a venn diagram with two circles works.

Need to practice using venn diagrams?

Looking for some worksheets to help you to classify a range of different objects using venn diagrams?

Then this page should hopefully be what you are looking for!

The worksheets on this page have been split into 2 sections:

• 2 circle venn diagrams
• 3 circle venn diagrams

Each sheet consists of sorting either a set of animals, a table of facts, shapes or numbers.

Each sheet contains one set of objects and two different venn diagrams to sort the objects with.

There is a progression from easier to harder sheets within each section.

Using the sheets in this section will help your child to:

• practice using 2 and 3 circle venn diagrams;
• converting data from a table to a venn diagram;
• practice classifying a range of objects using different properties.

2 Circle Venn Digarms

• Venn Diagrams Sheet 3:1
• PDF version
• Venn Diagrams Sheet 3:2
• Venn Diagrams Sheet 3:3
• Venn Diagrams Sheet 3:4
• Venn Diagrams Sheet 3:5
• Venn Diagrams Sheet 3:6

3 Circle Venn Digrams

Below are some sheets to help you practice using 3 circle venn diagrams.

If you are looking for some support or want to see our full range of sheets, then visit our 3-circle-venn-diagram suport page.

• 3 Circle Venn Diagram Support page
• 3 CIrcle Venn Diagram Sheet 1
• 3 CIrcle Venn Diagram Sheet 2
• 3 CIrcle Venn Diagram Sheet 3
• 3 CIrcle Venn Diagram Sheet 4

Looking for help with venn diagrams?

If venn diagrams are a bit of a mystery to you and you would like some more support, then try our page all about venn diagrams.

There is a video to watch with some simple examples worked through.

• What is a venn diagram page

Looking for some easier Venn Diagram Worksheets?

If the sheets here are too tricky, then take a look at our 2nd grade venn diagram sheets.

These sheets involve sorting and classifying shapes, numbers and other objects at a 2nd grade level.

Using the link below will open up the 2nd Grade Math Salamanders website in a new browser window.

• Venn Diagram Worksheets 2nd Grade

Looking for some more challenging Venn Diagram Worksheets?

If the sheets here are too easy, then take a look at our 4th grade venn diagram sheets.

These sheets involve sorting and classifying shapes, numbers and other objects at a 4th grade level.

• Venn Diagram Worksheet 4th Grade

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Third Grade Bar Graph & Line Plot Worksheets

Take a look at some of our statistics and data worksheets for 3rd graders.

The sheets on this page involve reading and interpreting more complex bar graphs going up in a range of different scales.

We also have a selection of line plot and bar graph worksheets.

There are a selection of bar graphs, including graphs involving real-life data such as tree heights or insect speeds.

• Line Plot Worksheets 3rd grade
• Bar Graph Worksheets 3rd grade
• Tally Chart Worksheets

Here is our selection of tally chart worksheets for 1st and 2nd graders.

These sheets involve counting and recording tallies.

Rounding, Inequalities and Multiples Worksheets

The sheets in this section will help your child to learn all about inequalities and multiples.

They support the classification and sorting sheets on this page and give help on what multiples and inequalities are.

• 3rd Grade Rounding Inequalities Multiples Worksheets

How to Print or Save these sheets 🖶

Need help with printing or saving? Follow these 3 steps to get your worksheets printed perfectly!

• How to Print support

Subscribe to Math Salamanders News

Sign up for our newsletter to get free math support delivered to your inbox each month. Plus, get a seasonal math grab pack included for free!

• Newsletter Signup

Return to 3rd Grade Math Hub

Return to Statistics Worksheets

Return from Venn Diagram Worksheets to Math Salamanders Homepage

Math-Salamanders.com

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.

New! Comments

TOP OF PAGE

© 2010-2024 Math Salamanders Limited. All Rights Reserved.

• Privacy Policy
• Copyright Policy

Venn Diagram Word Problems

Related Pages Venn Diagrams Intersection Of Two Sets Intersection Of Three Sets More Lessons On Sets More GCSE/IGCSE Maths Lessons

In these lessons, we will learn how to solve word problems using Venn Diagrams that involve two sets or three sets. Examples and step-by-step solutions are included in the video lessons.

What Are Venn Diagrams?

Venn diagrams are the principal way of showing sets in a diagrammatic form. The method consists primarily of entering the elements of a set into a circle or ovals.

Before we look at word problems, see the following diagrams to recall how to use Venn Diagrams to represent Union, Intersection and Complement.

How To Solve Problems Using Venn Diagrams?

This video solves two problems using Venn Diagrams. One with two sets and one with three sets.

Problem 1: 150 college freshmen were interviewed. 85 were registered for a Math class, 70 were registered for an English class, 50 were registered for both Math and English.

a) How many signed up only for a Math Class? b) How many signed up only for an English Class? c) How many signed up for Math or English? d) How many signed up neither for Math nor English?

Problem 2: 100 students were interviewed. 28 took PE, 31 took BIO, 42 took ENG, 9 took PE and BIO, 10 took PE and ENG, 6 took BIO and ENG, 4 took all three subjects.

a) How many students took none of the three subjects? b) How many students took PE but not BIO or ENG? c) How many students took BIO and PE but not ENG?

How And When To Use Venn Diagrams To Solve Word Problems?

Problem: At a breakfast buffet, 93 people chose coffee and 47 people chose juice. 25 people chose both coffee and juice. If each person chose at least one of these beverages, how many people visited the buffet?

How To Use Venn Diagrams To Help Solve Counting Word Problems?

Problem: In a class of 30 students, 19 are studying French, 12 are studying Spanish and 7 are studying both French and Spanish. How many students are not taking any foreign languages?

Probability, Venn Diagrams And Conditional Probability

This video shows how to construct a simple Venn diagram and then calculate a simple conditional probability.

Problem: In a class, P(male)= 0.3, P(brown hair) = 0.5, P (male and brown hair) = 0.2 Find (i) P(female) (ii) P(male| brown hair) (iii) P(female| not brown hair)

Venn Diagrams With Three Categories

Example: A group of 62 students were surveyed, and it was found that each of the students surveyed liked at least one of the following three fruits: apricots, bananas, and cantaloupes.

34 liked apricots. 30 liked bananas. 33 liked cantaloupes. 11 liked apricots and bananas. 15 liked bananas and cantaloupes. 17 liked apricots and cantaloupes. 19 liked exactly two of the following fruits: apricots, bananas, and cantaloupes.

a. How many students liked apricots, but not bananas or cantaloupes? b. How many students liked cantaloupes, but not bananas or apricots? c. How many students liked all of the following three fruits: apricots, bananas, and cantaloupes? d. How many students liked apricots and cantaloupes, but not bananas?

Venn Diagram Word Problem

Here is an example on how to solve a Venn diagram word problem that involves three intersecting sets.

Problem: 90 students went to a school carnival. 3 had a hamburger, soft drink and ice-cream. 24 had hamburgers. 5 had a hamburger and a soft drink. 33 had soft drinks. 10 had a soft drink and ice-cream. 38 had ice-cream. 8 had a hamburger and ice-cream. How many had nothing? (Errata in video: 90 - (14 + 2 + 3 + 5 + 21 + 7 + 23) = 90 - 75 = 15)

Venn Diagrams With Two Categories

This video introduces 2-circle Venn diagrams, and using subtraction as a counting technique.

How To Use 3-Circle Venn Diagrams As A Counting Technique?

Learn about Venn diagrams with two subsets using regions.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

• Member login
• Pre-algebra lessons
• Pre-algebra word problems
• Algebra lessons
• Algebra word problems
• Algebra proofs
• Advanced algebra
• Geometry lessons
• Geometry word problems
• Geometry proofs
• Trigonometry lessons
• Consumer math
• Baseball math
• Math for nurses
• Statistics made easy
• High school physics
• Basic mathematics store
• SAT Math Prep
• Math skills by grade level
• Ask an expert
• Other websites
• K-12 worksheets
• Worksheets generator
• Algebra worksheets
• Geometry worksheets
• Free math problem solver
• Pre-algebra calculators
• Algebra Calculators
• Geometry Calculators
• Math puzzles
• Math tricks

Venn diagram word problems

The Venn diagram word problems in this lesson will show you how to use Venn diagrams with 2 circles to solve problems involving counting. 

Venn diagram word problems with two circles

Word problem #1

A survey was conducted in a neighborhood with 128 families. The survey revealed the following information.

• 106 of the families have a credit card
• 73 of the families are trying to pay off a car loan
• 61 of the families have both a credit card and a car loan

Answer the following questions:

1. How many families have only a credit card?

2.  How many families have only a car loan?

3. How many families have neither a credit card nor a car loan?

4. How many families do not have a credit card?

5. How many families do not have a car loan?

6. How many families have a credit card or a car loan?

• Let C be families with a credit card
• Let L be families with a car loan
• Let S be the total number of families

The Venn diagram above can be used to answer all these questions. 

Tips on how to create the Venn diagram. Always put  first , in the middle or in the intersection, the value that is in both sets. For example, since 61 families have both a credit card and a car loan, put 61 in the intersection before you do anything else. In C only, put 45 since 106 - 61 = 45

In L only, put 12 since 73 - 61 = 12

Outside C and L, put 10 since 128 - 61 - 45 - 12 = 10

The expression, " only a credit card" means that it is only in C. Any number in L cannot be included. 1.  The number of families with only a credit card is 45. Do not add 61 to 45 since 61 is in L.

2.  The number of families with only a car loan is 12. 

3. The number of families with neither a credit card nor a car loan is 10. 10 is not in C nor in L.

4. The number families without a credit card is found by adding everything that is not in C. 12 + 10 = 22

5.  The number families without a car loan is found by adding everything that is not in L. 45 + 10 = 55

6. The number of families with a credit card or a car loan is found by adding anything in C only, in L only and in the intersection of C and L?

45 + 61 + 12 = 118

Word problem #2

A survey conducted in a school with 150 students revealed the following information:

• 78 students are enrolled in swimming class
• 85 students are enrolled in basketball class
• 25 are enrolled in both swimming and basketball class

1.  How many students are enrolled only in swimming class?

2.  How many students are enrolled only in basketball class?

3.  How many students are neither enrolled in swimming class nor basketball class?

4.  How many students are not enrolled in swimming class?

5.  How many students are not enrolled in basketball class?

6. How many students are enrolled in swimming class or basketball class?

• Let S be students enrolled in swimming class
• Let B be students enrolled in basketball class
• Let E be the total number of students

Using the same technique as in problem #1 , we have the following Venn diagram

1. The number of students enrolled only in swimming class is 53 2.  The number of students enrolled only in basketball class is 60

3. The number of students who are neither enrolled in swimming class nor basketball class is 12

4. Students not enrolled in swimming class are enrolled in basketball class only or are enrolled in neither of these two activities. In other words, everything that is not in S.

60 + 12 = 72

5.  Students not enrolled in basketball class are enrolled in swimming class only or are  enrolled in neither of these two activities. In other words, everything that is not in B.

53 + 12 = 65

6.  The number of students enrolled in swimming class or basketball class is found by adding anything in S only, in B only and in the intersection of S and B?

53 + 25 + 60 = 138

A tricky Venn diagram word problem with two circles

Word problem #3

In a survey of 100 people, 28 people smoke, 65 people drink, and 30 people do neither. How many people do both?

• Let K be the number of people who smoke
• Let D be  the number of people who drink
• Let E be the total number of people
• Let x be the number of people who smoke and drink

If we make a Venn diagram, here is what we have so far.

We end up with the following equation to solve for x.

(65 - x) + x + (28 - x) + 30 = 100

65 - x + x + 28 - x + 30 - 30 = 100 - 30

65 - x + x + 28 - x  = 70

65 + 0 + 28 - x  = 70

93 - x  = 70

Since 93 - 23 = 70, x  = 23

The number of people who do both is 23.

3-circle Venn diagram

Applied math

Calculators.

100 Tough Algebra Word Problems. If you can solve these problems with no help, you must be a genius!

 Recommended

About me :: Privacy policy :: Disclaimer :: Donate   Careers in mathematics  

Copyright © 2008-2021. Basic-mathematics.com. All right reserved

Venn Diagram Worksheets

• Kindergarten

Reading & Math for K-5

• Kindergarten
• Learning numbers
• Comparing numbers
• Place Value
• Roman numerals
• Subtraction
• Multiplication
• Order of operations
• Drills & practice
• Measurement
• Factoring & prime factors
• Proportions
• Shape & geometry
• Data & graphing
• Word problems
• Children's stories
• Leveled Stories
• Sentences & passages
• Context clues
• Cause & effect
• Compare & contrast
• Fact vs. fiction
• Fact vs. opinion
• Main idea & details
• Story elements
• Conclusions & inferences
• Sounds & phonics
• Words & vocabulary
• Reading comprehension
• Early writing
• Numbers & counting
• Simple math
• Social skills
• Other activities
• Dolch sight words
• Fry sight words
• Multiple meaning words
• Prefixes & suffixes
• Vocabulary cards
• Other parts of speech
• Punctuation
• Capitalization
• Narrative writing
• Opinion writing
• Informative writing
• Cursive alphabet
• Cursive letters
• Cursive letter joins
• Cursive words
• Cursive sentences
• Cursive passages
• Grammar & Writing

Breadcrumbs

• Data & Graphing

Venn diagrams

Download & Print From only $3.10

Overlapping data sets

Venn diagrams are used to show the relationship between overlapping data sets. In these Venn diagram worksheets, students map a data set into double or triple Venn diagrams.

2 categories:

3 categories:

Blank Venn diagram:

These worksheets are available to members only.

Join K5 to save time, skip ads and access more content. Learn More

What is K5?

K5 Learning offers free worksheets , flashcards  and inexpensive  workbooks  for kids in kindergarten to grade 5. Become a member  to access additional content and skip ads.

Our members helped us give away millions of worksheets last year.

We provide free educational materials to parents and teachers in over 100 countries. If you can, please consider purchasing a membership ($24/year) to support our efforts.

Members skip ads and access exclusive features.

Learn about member benefits

This content is available to members only.

Venn Diagram Examples, Problems and Solutions

On this page:

• What is Venn diagram? Definition and meaning.
• Venn diagram formula with an explanation.
• Examples of 2 and 3 sets Venn diagrams: practice problems with solutions, questions, and answers.
• Simple 4 circles Venn diagram with word problems.
• Compare and contrast Venn diagram example.

Let’s define it:

A Venn Diagram is an illustration that shows logical relationships between two or more sets (grouping items). Venn diagram uses circles (both overlapping and nonoverlapping) or other shapes.

Commonly, Venn diagrams show how given items are similar and different.

Despite Venn diagram with 2 or 3 circles are the most common type, there are also many diagrams with a larger number of circles (5,6,7,8,10…). Theoretically, they can have unlimited circles.

Venn Diagram General Formula

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

Don’t worry, there is no need to remember this formula, once you grasp the meaning. Let’s see the explanation with an example.

This is a very simple Venn diagram example that shows the relationship between two overlapping sets X, Y.

X – the number of items that belong to set A Y – the number of items that belong to set B Z – the number of items that belong to set A and B both

From the above Venn diagram, it is quite clear that

n(A) = x + z n(B) = y + z n(A ∩ B) = z n(A ∪ B) = x +y+ z.

Now, let’s move forward and think about Venn Diagrams with 3 circles.

Following the same logic, we can write the formula for 3 circles Venn diagram :

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

Venn Diagram Examples (Problems with Solutions)

As we already know how the Venn diagram works, we are going to give some practical examples (problems with solutions) from the real life.

2 Circle Venn Diagram Examples (word problems):

Suppose that in a town, 800 people are selected by random types of sampling methods . 280 go to work by car only, 220 go to work by bicycle only and 140 use both ways – sometimes go with a car and sometimes with a bicycle.

Here are some important questions we will find the answers:

• How many people go to work by car only?
• How many people go to work by bicycle only?
• How many people go by neither car nor bicycle?
• How many people use at least one of both transportation types?
• How many people use only one of car or bicycle?

The following Venn diagram represents the data above:

Now, we are going to answer our questions:

• Number of people who go to work by car only = 280
• Number of people who go to work by bicycle only = 220
• Number of people who go by neither car nor bicycle = 160
• Number of people who use at least one of both transportation types = n(only car) + n(only bicycle) + n(both car and bicycle) = 280 + 220 + 140 = 640
• Number of people who use only one of car or bicycle = 280 + 220 = 500

Note: The number of people who go by neither car nor bicycle (160) is illustrated outside of the circles. It is a common practice the number of items that belong to none of the studied sets, to be illustrated outside of the diagram circles.

We will deep further with a more complicated triple Venn diagram example.

3 Circle Venn Diagram Examples:

For the purposes of a marketing research , a survey of 1000 women is conducted in a town. The results show that 52 % liked watching comedies, 45% liked watching fantasy movies and 60% liked watching romantic movies. In addition, 25% liked watching comedy and fantasy both, 28% liked watching romantic and fantasy both and 30% liked watching comedy and romantic movies both. 6% liked watching none of these movie genres.

Here are our questions we should find the answer:

• How many women like watching all the three movie genres?
• Find the number of women who like watching only one of the three genres.
• Find the number of women who like watching at least two of the given genres.

Let’s represent the data above in a more digestible way using the Venn diagram formula elements:

• n(C) = percentage of women who like watching comedy = 52%
• n(F ) = percentage of women who like watching fantasy = 45%
• n(R) = percentage of women who like watching romantic movies= 60%
• n(C∩F) = 25%; n(F∩R) = 28%; n(C∩R) = 30%
• Since 6% like watching none of the given genres so, n (C ∪ F ∪ R) = 94%.

Now, we are going to apply the Venn diagram formula for 3 circles. 

94% = 52% + 45% + 60% – 25% – 28% – 30% + n (C ∩ F ∩ R)

Solving this simple math equation, lead us to:

n (C ∩ F ∩ R)  = 20%

It is a great time to make our Venn diagram related to the above situation (problem):

See, the Venn diagram makes our situation much more clear!

From the Venn diagram example, we can answer our questions with ease.

• The number of women who like watching all the three genres = 20% of 1000 = 200.
• Number of women who like watching only one of the three genres = (17% + 12% + 22%) of 1000 = 510
• The number of women who like watching at least two of the given genres = (number of women who like watching only two of the genres) +(number of women who like watching all the three genres) = (10 + 5 + 8 + 20)% i.e. 43% of 1000 = 430.

As we mentioned above 2 and 3 circle diagrams are much more common for problem-solving in many areas such as business, statistics, data science and etc. However, 4 circle Venn diagram also has its place.

4 Circles Venn Diagram Example:

A set of students were asked to tell which sports they played in school.

The options are: Football, Hockey, Basketball, and Netball.

Here is the list of the results:

The next step is to draw a Venn diagram to show the data sets we have.

It is very clear who plays which sports. As you see the diagram also include the student who does not play any sports (Dorothy) by putting her name outside of the 4 circles.

From the above Venn diagram examples, it is obvious that this graphical tool can help you a lot in representing a variety of data sets. Venn diagram also is among the most popular types of graphs for identifying similarities and differences .

Compare and Contrast Venn Diagram Example:

The following compare and contrast example of Venn diagram compares the features of birds and bats:

Tools for creating Venn diagrams

It is quite easy to create Venn diagrams, especially when you have the right tool. Nowadays, one of the most popular way to create them is with the help of paid or free graphing software tools such as:

You can use Microsoft products such as:

Some free mind mapping tools are also a good solution. Finally, you can simply use a sheet of paper or a whiteboard.

Conclusion:

A Venn diagram is a simple but powerful way to represent the relationships between datasets. It makes understanding math, different types of data analysis , set theory and business information easier and more fun for you.

Besides of using Venn diagram examples for problem-solving and comparing, you can use them to present passion, talent, feelings, funny moments and etc.

Be it data science or real-world situations, Venn diagrams are a great weapon in your hand to deal with almost any kind of information.

If you need more chart examples, our posts fishbone diagram examples and what does scatter plot show might be of help.

About The Author

Silvia Valcheva

Silvia Valcheva is a digital marketer with over a decade of experience creating content for the tech industry. She has a strong passion for writing about emerging software and technologies such as big data, AI (Artificial Intelligence), IoT (Internet of Things), process automation, etc.

Well explained I hope more on this one

WELL STRUCTURED CONTENT AND ENLIGHTNING AS WELL

Leave a Reply Cancel Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed .

WORD PROBLEMS ON SETS AND VENN DIAGRAMS

Basic stuff.

To understand, how to solve Venn diagram word problems with 3 circles, we have to know the following basic stuff.  

u ----> union (or)

n ----> intersection (and)

Addition Theorem on Sets

Theorem 1 :

n(AuB) = n(A) + n(B) - n(AnB)

Theorem 2 :

=n(A) + n(B) + n(C) - n(AnB) - n(BnC) - n(AnC) + n(AnBnC)

Explanation :

Let us come to know about the following terms in details.

n(AuB) = Total number of elements related to any of the two events A & B.

n(AuBuC) = Total number of elements related to any of the three events A, B & C.

n(A) = Total number of elements related to A

n(B) = Total number of elements related to B

n(C) = Total number of elements related to C

For  three events A, B & C, we have

n(A) - [n(AnB) + n(AnC) - n(AnBnC)] :

Total number of elements related to A only

n(B) - [n(AnB) + n(BnC) - n(AnBnC)] :

Total number of elements related to B only

n(C) - [n(BnC) + n(AnC) + n(AnBnC)] :

Total number of elements related to C only

Total number of elements related to both A & B

n(AnB) - n(AnBnC) :

Total number of elements related to both (A & B) only

Total number of elements related to both B & C

n(BnC) - n(AnBnC) :

Total number of elements related to both (B & C) only

Total number of elements related to both A & C

n(AnC) - n(AnBnC) :

Total number of elements related to both (A & C) only

For  two events A & B, we have

n(A) - n(AnB) :

n(B) - n(AnB) :

Solved Problems

Problem 1 :

In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics and chemistry, 22 had taken chemistry and physics course, and 14 had taken all the three courses. Find how many had taken one course only.

Let M, C, P represent sets of students who had taken mathematics, chemistry and physics respectively.

From the given information, we have

n(M) = 64, n(C) = 94, n(P) = 58,

n(MnP) = 28, n(MnC) = 26, n(CnP) = 22

n(MnCnP) = 14

From the basic stuff, we have

Number of students who had taken only Math

= n(M) - [n(MnP) + n(MnC) - n(MnCnP)]

= 64 - [28 + 26 - 14]

Number of students who had taken only Chemistry :

= n(C) - [n(MnC) + n(CnP) - n(MnCnP)]

= 94 - [26+22-14]

Number of students who had taken only Physics :

= n(P) - [n(MnP) + n(CnP) - n(MnCnP)]

= 58 - [28 + 22 - 14]

Total n umber of students who had taken only one course :

= 24 + 60 + 22

Hence, the total number of students who had taken only one course is 106.

Alternative Method (Using venn diagram) :

Venn diagram related to the information given in the question:

From the venn diagram above, we have

Number of students who had taken only math = 24

Number of students who had taken only chemistry = 60

Number of students who had taken only physics = 22

Total  Number of students who had taken only one course :

Problem 2 :

In a group of students, 65 play foot ball, 45 play hockey, 42 play cricket, 20 play foot ball and hockey, 25 play foot ball and cricket, 15 play hockey and cricket and 8 play all the three games. Find the total number of students in the group  (Assume that each student in the group plays at least one game).

Let F, H and C represent the set of students who play foot ball, hockey and cricket respectively.

n(F) = 65, n(H) = 45, n(C) = 42,

n(FnH) = 20, n(FnC) = 25, n(HnC) = 15

n(FnHnC) = 8

Total number of students in the group is  n(FuHuC).

n(FuHuC) is equal to

= n(F) + n(H) + n(C) - n(FnH) - n(FnC) - n(HnC) + n(FnHnC)

n(FuHuC) = 65 + 45 + 42 -20 - 25 - 15 + 8

n(FuHuC) = 100

Hence, the total number of students in the group is 100.

Alternative Method (Using Venn diagram) :

Venn diagram related to the information given in the question :

Total number of students in the group :

= 28 + 12 + 18 + 7 + 10 + 17 + 8

So, the total number of students in the group is 100.

Problem 3 :

In a college, 60 students enrolled in chemistry,40 in physics, 30 in biology, 15 in chemistry and physics,10 in physics and biology, 5 in biology and chemistry. No one enrolled in all the three. Find how many are enrolled in at least one of the subjects.

Let C, P and B represents the subjects Chemistry, Physics  and Biology respectively.

Number of students enrolled in Chemistry :

Number of students enrolled in Physics :

Number of students enrolled in Biology :

Number of students enrolled in Chemistry and Physics :

n(CnP) = 15

Number of students enrolled in Physics and Biology :

n(PnB) = 10

Number of students enrolled in Biology and Chemistry :

No one enrolled in all the three.  So, we have

n(CnPnB) = 0

The above information can be put in a Venn diagram as shown below.

From, the above Venn diagram, number of students enrolled in at least one of the subjects :

= 40 + 15 + 15 + 15 + 5 + 10 + 0

So, the number of students  enrolled in at least one of the subjects is 100.

Problem 4 :

In a town 85% of the people speak Tamil, 40% speak English and 20% speak Hindi. Also 32% speak Tamil and English, 13% speak Tamil and Hindi and 10% speak English and Hindi, find the percentage of people who can speak all the three languages.

Let T, E and H represent the people who speak Tamil, English and Hindi respectively.

Percentage of people who speak Tamil :

Percentage of people who speak English :

Percentage of people who speak Hindi :

n(H)  =  20

Percentage of people who speak English and Tamil :

n(TnE) = 32

Percentage of people who speak Tamil and Hindi :

n(TnH) = 13

Percentage of people who speak English and Hindi :

n(EnH) = 10

Let x be the percentage of people who speak all the three language.

From the above Venn diagram, we can have 

100 = 40 + x + 32 – x + x + 13 – x + 10 – x – 2 + x – 3 + x

100 = 40 + 32 + 13 + 10 – 2 – 3 + x 

100 = 95 – 5 + x

100 = 90 + x

x = 100 - 90

x = 10% 

So, the percentage of people who speak all the three languages is 10%.

Problem 5 :

An advertising agency finds that, of its 170 clients, 115 use Television, 110 use Radio and 130 use Magazines. Also 85 use Television and Magazines, 75 use Television and Radio, 95 use Radio and Magazines, 70 use all the three. Draw Venn diagram to represent these data. Find 

(i) how many use only Radio?

(ii) how many use only Television?

(iii) how many use Television and Magazine but not radio?

Let T, R and M represent the people who use Television, Radio and Magazines respectively.

Number of people who use Television :

Number of people who use Radio :

Number of people who use Magazine :

Number of people who use Television and Magazines

n (TnM) = 85

Number of people who use Television and Radio :

n(TnR) = 75

Number of people who use Radio and Magazine :

n(RnM) = 95

Number of people who use all the three :

n(TnRnM) = 70

From the above Venn diagram, we have

(i) Number of people who use only Radio is 10.

(ii) Number of people who use only Television is 25.

(iii) Number of people who use Television and Magazine but not radio is 15.

Problem 6 :

In a class of 60 students, 40 students like math, 36 like science, 24 like both the subjects. Find the number of students who like

(i) Math only, (ii) Science only  (iii) Either Math or Science (iv) Neither Math nor science.

Let M and S represent the set of students who like math and science respectively.

From the information given in the question, we have

n(M) = 40, n(S) = 36, n(MnS) = 24

Answer (i) :

Number  of students who like math only :

= n(M) - n(MnS)

Answer (ii) :

Number  of students who like science only :

= n(S) - n(MnS)

=   12

Answer (iii) :

Number  of students who like either math or science :

= n(M or S) 

= n(MuS) 

= n(M) + n(S) - n(MnS)

= 40 + 36 - 24

Answer (iv) :

Total n umber  students who like Math or Science subjects :

n(MuS) = 52

Number  of students who like neither math nor science

Problem 7 :

At a certain conference of 100 people there are 29 Indian women and 23 Indian men. Out of these Indian people 4 are doctors and 24 are either men or doctors. There are no foreign doctors. Find the number of women doctors attending the conference.

Let M and D represent the set of Indian men and Doctors respectively.

n(M) = 23, n(D) = 4, n(MuD) = 24

n(MuD) = n(M) + n(D) - n(MnD)

24 = 23 + 4 - n(MnD)

n(MnD) = 3 

n(Indian Men and Doctors) = 3

So, out of the 4 Indian doctors,  there are 3 men.

And the remaining 1 is Indian women doctor.

So, the number women doctors attending the conference is 1.

Kindly mail your feedback to   [email protected]

We always appreciate your feedback.

© All rights reserved. onlinemath4all.com

• Sat Math Practice
• SAT Math Worksheets
• PEMDAS Rule
• BODMAS rule
• GEMDAS Order of Operations
• Math Calculators
• Transformations of Functions
• Order of rotational symmetry
• Lines of symmetry
• Compound Angles
• Quantitative Aptitude Tricks
• Trigonometric ratio table
• Word Problems
• Times Table Shortcuts
• 10th CBSE solution
• PSAT Math Preparation
• Privacy Policy
• Laws of Exponents

Recent Articles

Sat math resources (videos, concepts, worksheets and more).

Sep 16, 24 11:07 AM

Digital SAT Math Problems and Solutions (Part - 42)

Sep 16, 24 09:42 AM

Digital SAT Math Problems and Solutions (Part - 41)

Sep 14, 24 05:55 AM

Venn Diagram Questions

Venn diagram questions with solutions are given here for students to practice various questions based on Venn diagrams. These questions are beneficial for both school examinations and competitive exams. Practising these questions will develop a skill to solve any problem on Venn diagrams quickly.

Venn diagrams were first introduced by John Venn to represent various propositions in a diagrammatic way. Venn diagrams are used for representing relationships between given sets. For example, natural numbers and whole numbers are subsets of integers represented by the Venn diagram:

Using Venn diagrams, we can easily understand whether given sets are subsets of each other or disjoint sets or have something in common.

• Intersection of Sets
• Union of Sets
• Complement of Set
• Set Operations

Following are some set operations and their meaning useful while solving problems on the Venn diagram:

Some important formulae:

Venn Diagram Questions with Solution

Let us practice some questions based on Venn diagrams.

Question 1: If A and B are two sets such that number of elements in A is 24, number of elements in B is 22 and number of elements in both A and B is 8, find:

(i) n(A ∪ B)

(ii) n(A – B)

(ii) n(B – A)

Given, n(A) = 24, n(B) = 22 and n(A ∩ B) = 8

The Venn diagram for the given information is:

(i) n(A ∪ B) = n(A) + n(B) – n(A ∩ B) = 24 + 22 – 8 = 38.

(ii) n(A – B) = n(A) – n(A ∩ B) = 24 – 8 = 16.

(iii) n(B – A) = n(B) – n(A ∩ B) = 22 – 8 = 14.

Question 2: According to the survey made among 200 students, 140 students like cold drinks, 120 students like milkshakes and 80 like both. How many students like atleast one of the drinks?

Number of students like cold drinks = n(A) = 140

Number of students like milkshake = n(B) = 120

Number of students like both = n(A ∩ B) = 80

Number of students like atleast one of the drinks = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

= 140 + 120 – 80

Question 3: In a group of 500 people, 350 people can speak English, and 400 people can speak Hindi. Find how many people can speak both languages?

Let H be the set of people who can speak Hindi and E be the set of people who can speak English. Then,

n(H ∪ E) = 500

We have to find n(H ∩ E).

Now, n(H ∪ E) = n(H) + n(E) – n(H ∩ E)

⇒ 500 = 400 + 350 – n(H ∩ E)

⇒ n(H ∩ E) = 750 – 500 = 250.

∴ 250 people can speak both languages.

Questions 4: The following Venn diagram shows games played by the number of students in a class:

How many students like only cricket and only football?

As per the given Venn diagram,

Number of students only like cricket = 7

Number of students only like football = 14

∴ Number of students like only cricket and only football = 7 + 14 = 21.

Question 5: In a class of 40 students, 20 have chosen Mathematics, 15 have chosen mathematics but not biology. If every student has chosen either mathematics or biology or both, find the number of students who chose both mathematics and biology and the number of students chose biology but not mathematics.

Let, M ≡ Set of students who chose mathematics

B ≡ Set of students who chose biology

n(M ∪ B) = 40

n(B) = n(M ∪ B) – n(M)

⇒ n(B) = 40 – 20 = 20

n(M – B) = 15

n(M) = n(M – B) + n(M ∩ B)

⇒ 20 = 15 + n(M ∩ B)

⇒ n(M ∩ B) = 20 – 15 = 5

n(B – M) = n(B) – n(M ∩ B)

⇒ n(B – M) = 20 – 5 = 15

Question 6: Represent The following as Venn diagram:

(i) A’ ∩ (B ∪ C)

(ii) A’ ∩ (C – B)

Question 7: In a survey among 140 students, 60 likes to play videogames, 70 likes to play indoor games, 75 likes to play outdoor games, 30 play indoor and outdoor games, 18 like to play video games and outdoor games, 42 play video games and indoor games and 8 likes to play all types of games. Use the Venn diagram to find

(i) students who play only outdoor games

(ii) students who play video games and indoor games, but not outdoor games.

Let V ≡ Play video games

I ≡ Play indoor games

O ≡ Play outdoor games

n(V) = 60, n(I) = 70, n(O) = 75

n(I ∩ O) = 30, n(V ∩ O) = 18, n(V ∩ I) = 42

n(V ∩ I ∩ O) = 8

Hence, by Venn diagram

Number of students only like to play only outdoor games = 35

Number of students like to play video games and indoor games but not outdoor games = 34

Note : Always begin to fill the Venn diagram from the innermost part.

Question 8: Using the Venn diagrams, verify (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).

The shaded portion represents (P ∩ Q) ∪ R in the Venn diagram.

Comparing both the shaded portion in both the Venn diagram, we get (P ∩ Q) ∪ R = (P ∪ R) ∩ (Q ∪ R).

Question 9: Prove using the Venn diagram: (B – A) ∪ (A ∩ B) = B.

From the Venn diagram, it is clear that (B – A) ∪ (A ∩ B) = B

Question 10: In a survey, it is found that 21 people read English newspaper, 26 people read Hindi newspaper, and 29 people read regional language newspaper. If 14 people read both English and Hindi newspapers; 15 people read both Hindi and regional language newspapers; 12 people read both English and regional language newspaper and 8 read all types of newspapers, find:

(i) How many people were surveyed?

(ii) How many people read only regional language newspapers?

Let A ≡ People who read English newspapers.

B ≡ People who read Hindi newspapers.

C ≡ People who read Hindi newspapers.

n(A) = 21, n(B) = 26, n(C) = 29

n(A ∩ B) = 14, n(B ∩ C) = 15, n(A ∩ C) = 12

n(A ∩ B ∩ C) = 8

(i) Number of people surveyed = n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) = 21 + 26 + 29 – 14 – 15 – 12 + 8 = 43

(ii) By the Venn diagram, number of people who only read regional language newspapers = 10.

Video Lesson on Introduction to Sets

Practice Questions on Venn Diagrams

1. Verify using the Venn diagram:

(i) A – B = A ∩ B C

(ii) (A ∩ B) C = A C ∪ B C

2. For given two sets P and Q, n(P – Q) = 24, n(Q – P) = 19 and n(P ∩ Q) = 11, find:

(iii) n (P ∪ Q)

3. In a group of 65 people, 40 like tea and 10 like both tea and coffee. Find

(i) how many like coffee only and not tea?

(ii) how many like coffee?

4. In a sports tournament, 38 medals were awarded for 500 m sprint, 15 medals were awarded for Javelin throw, and 20 medals were awarded for a long jump. If these medals were awarded to 58 participants and among them only three medals in all three sports, how many received exactly two medals?

Keep visiting BYJU’S to get more such Maths lessons in a simple, concise and easy to understand way. Also, register at BYJU’S – The Learning App to get complete assistance for Maths preparation with video lessons, notes, tips and other study materials.

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.