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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 ?

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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.

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Venn Diagram Word Problems

Venn Diagram Word Problems can be very easy to make mistakes on when you are a beginner.

It is extremely important to:

Read the question carefully and note down all key information.

Know the standard parts of a Venn Diagram

Work in a step by step manner

Check at the end that all the numbers add up coorectly.

Let’s start with an easy example of a two circle diagram problem.

Venn Diagrams – Word Problem One

“A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 8 students said they had only ever had a dog. 6 students said they had only ever had a cat. 10 students said they had a dog and a cat. 4 students said they had never had a dog or a cat.”

Note that the word “only” is extremely important in Venn Diagram word problems.

Because the word “only” is in our problem text, it makes it an easy word problem. Since this question is about dogs and cats, it will require a two circle Venn Diagram.

Here is the type of diagram we will need.

Our problem is an easy one where we have been given all of the numbers for the items required on the diagram.

We do not need to work out any missing values.

All we need to do is place the numbers from the word problem onto the standard Venn Diagram and we are done.

Venn Diagrams – Word Problem Two

The answer for this question will actually be the same as the Cats and Dogs question in Example 1.

However this time we are given less information, and so we will have work out the missing information.

Here is Problem 2: “A class of 28 students were surveyed and asked if they ever had dogs or cats for pets at home. 18 students said they had a dog. 16 students said they had a cat. 4 students said they had never had a dog or a cat.”

The above question does not contain the word “only” anywhere in it, and this is an indication that we will have to do some working out.

The question states that: “18 students said they had a dog” without the word “only” in there.

This means that the total of the Dogs circle is 18.

The 18 total students for Dogs includes people that have both a cat and a dog, as well as people who only have a dog.

Some people, who do not read this question carefully, will simply take the above figures and put them straight into a Venn Diagram like this.

Always check at the end that the numbers add up to the “E” Grand total.

16 + 18 + 4 = 38 which is much bigger than the “E” total of 28.

From the given information we have been able to work out that the circles total is 24. (Eg. Everything Total – No Cats and No Dogs = 28 – 4 = 24. This is vital information we now use to work on the rest of the problem.

Let’s first work out the “Only Cats” value.

Next we work out the “Only Dogs” number of people.

All we have left to work out is the number of Cats and Dogs for the center of the diagram.

We can do this any of three possible ways: Cats and Dogs = Total Cats – Only Cats or

Cats and Dogs = Total Dogs – Only Dogs

Cats and Dogs = E Total – Only Cats – Only Dogs – (No cats and No Dogs)

Any way that we work it out, the answer is 10.

So here is the final completed Venn Diagram Answer.

When putting answers into our Mathematics Workbook, we do not have to color in the diagram.

A final answer like the following is quite acceptable.

We can summarise the steps we used to work out this problem as follows.

Word Problem Two – Summary of Steps

– Work out What Information is given, and what needs to be calculated.

– Circles Total = E everything – (No Cats and No Dogs) – Cats Only = Circles Total – Total Dogs

– Dogs Only = Circles Total – Total Cats

– Cats and Dogs = Cats Total – Cats Only

– Finally, check that all the numbers in the diagram add up to equal the “E” everything total.

Word Problem Three – Subsets

“Fifty people were surveyed and only 20 people said that they regularly eat Healthy Foods like Fruit and Vegetables. Of these 20 healthy eaters, 12 said that they ate Vegetables every day. Draw a Venn Diagram to represent these results.”

This problem is quite different to our other two circle diagrams.

Cats and Dogs are very different to each other, and so we needed two separate circles.

However Healthy Foods and Vegetables are not different to each other because Vegetables are a type of Healthy Food.

We say that vegetables are a “Subset” of Healthy Foods.

This means that we do not separate the circles. We actually need to draw our circles inside each other like this.

The total adds up to 50, and the 12 people who include vegetables in their healthy foods are shown as being fully inside the Healthy Foods circle.

Word Problem Four – Disjoint Sets “Draw a Venn Diagram which divides the twelve months of the year into the following two groups: Months whose name begins with the letter “J” and Months whose name ends in “ber”. You will need a two circle Venn Diagram for your answer.” The first step is to list the twelve months of the year:

January – named after Janus, the god of doors and gates February – named after Februalia, when sacrifices were made for sins March – named after Mars, the god of war April – from aperire, Latin for “to open” (buds) May – named after Maia, the goddess of growth of plants June – named after junius, Latin for the goddess Juno July – named after Julius Caesar in 44 B.C. August – named after Augustus Caesar in 8 B.C. September – from septem, Latin for “seven” October – from octo, Latin for “eight” November – from novem, Latin for “nine” December – from decem, Latin for “ten”

Months starting with J = { January, June, July }

Months ending in “ber” = { September, October, November, December }

The two sets do not have any items in common, and so we will not overlap them. The remaining months will need to go outside of our two circles.

There should be all twelve months in the diagram when we are finished.

The completed Venn Diagram is shown below:

Venn Word Problems – Summary We have not included three circle diagrams, as they will be covered in a separate lesson.

Remember the working out steps for harder problems are:

Work out What Information is given, and what needs to be calculated.

Check to see if the two sets are “Subsets” or “Disjoint” sets.

If they are “Intersecting Sets” then some of the following formulas may be needed.

Circles Total = E everything – (Not in A and Not in B)

In A Only = Both Circles Total – Total in B

In A Only = The A Circle Total – Total in the intersection (A and B)

In B Only = Both Circles Total – Total in A

In B Only = The B Circle Total – Total in the intersection (A and B)

In the Intersection (A and B) = Total in B – In B Only

In the Intersection (A and B) = Total in A – In A Only

Finally, check that the numbers in the diagram all add up to equal the “E” everything total.

Venn Word Problems Videos

The following video shows a typical two circles word problem.

Here is a video that covers a two circles problem, where we need to find the number of items that are ( not in “A” and not in “B”)

Here is a Video which shows how to solve Venn Diagram Survey Problems.

Related Items

Introduction to Venn Diagrams Three Circle Venn Diagrams Real World Venn Diagrams

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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.

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• Word Problems: Two sets

Venn Diagram Word Problem Worksheets: Two sets

Venn diagram word problems are based on union, intersection, complement and difference of two sets. This batch of printable word problems on Venn diagram with two circles for students of grade 5 through grade 8 is illustrated with images, numbers, words and symbols. Few word problems may contain universal set. Use the key information from the given data to solve the word problems. Click on our free worksheets and kick-start your practice!

Reading Venn Diagram - Type 1

The elements of the set are represented as pictures on the two circles of these Venn diagrams. Read each Venn diagram and answer the given questions.

• Download the set

Reading Venn Diagram - Type 2

The elements of the set are multiples, names, days, etc. Answer the word problems that follow the Venn diagram on each printable worksheet. Children should learn to differentiate the overlapping regions and the relation between the sets at the end of the practice.

Reading Venn Diagram - Type 3

These Venn diagram word problem pdfs require the 5th grade and 6th grade children to count the number of elements in the region. The cross-markings represent the customers in cafes, pizzeria, books in the library, park goers, or members of a gym.

Venn Diagram Word Problems - No Universal Set

Several practice problems on 2-circle Venn diagrams without universal set are given in these printable worksheets. Read each Venn diagram carefully and write down the answer.

Venn Diagram Word Problems - With a Universal Set

These Venn diagram word problems worksheet pdfs feature two sets representing the quantities of the data. Analyze the regions including universal set to solve these word problems.

Drawing Venn Diagram - No Universal Set

Follow the direction and create a Venn diagram with two intersecting circles for the given data. Fill in the key information to complete the Venn diagram with 2 circles. Use all the available information to answer the questions.

Drawing Venn Diagram - With a Universal Set

Task 7th grade and 8th grade students to draw a Venn diagram with a universal set and fill in each region with the information provided. Interpret the Venn diagram and answer the word problems given below.

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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.

WORD PROBLEM INVOLVING VENN DIAGRAM

WILMA B. CRISTOBAL

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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.

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Using Venn Diagrams Problems Worksheets

Students have long used Venn diagrams in English and Social Studies classes by the time we encourage them to use it math. Students sometimes will do a double take when they first see them being used in math class. Venn diagrams are great to evaluate the relationships. Especially when it comes to data. We can use them to visualize our data and see if there is something that separates it or connects it. These tools are primarily used in statistics, probability, and situations where we are evaluating the logic of something. Students learn how to create, read, and interpret Venn Diagrams imaging of logic statements with this collection of worksheets and lessons.

Aligned Standard: High School Data Modeling

• Getting to Work Step-by-Step Lesson - How do all the employees get to work? There are even some that do it multiple ways.
• Guided Lesson - Examine a voters poll, the participation at an annual school dance, and what employees drink at the office.
• Guided Lesson Explanation - I feel that it is fair to disclose that number two is very difficult for most students.
• Practice Worksheet - I give you ten scenarios that I would like to know a singular part of, but they require Venn diagrams.
• Practice Worksheet 2 - I not only double up on the practice in this area, but I also give you a fitting picnic clip art.
• Matching Worksheet - Find the answers to all the complex questions I throw at you.
• Using Venn Diagrams Five Pack of Worksheets - I really tried to find situations where Venns applied. I actual did a few hours of research to see where this skill is used in the real world.
• Answer Keys - These are for all the unlocked materials above.

Homework Sheets

Diagramming these helps out a great deal.

• Homework 1 - In an office of 40 employees, 17 travel by car, 23 travel by bike, and 10 take both cars and bikes.
• Homework 2 - How many employees in the office are not enrolled in either Car or Bike?
• Homework 3 - Now we want to find the number of students interested in participating in the different events. Let's diagram what we were told.

Practice Worksheets

I left a template of Venn diagrams for you to work with.

• Practice 1 - There are 35 people working in a bakery. 15 people are making cakes, 20 people are making chocolate cookies, and 8 people are making both cakes and cookies. How many people in the shop are not making cakes or cookies?
• Practice 2 - 18 students are in a college class. 10 students have Samsung cell phones. 8 students have Nokia cell phones. 5 students have both a Samsung and Nokia cell phone. How many students have only a Nokia cell phone?
• Practice 3 - 10 of the students in Johnson's class like to eat oranges. 7 students like to eat cherries, and 4 students like to eat both oranges and cherries. How many students like to eat cherries, but not oranges?

Math Skill Quizzes

We don't ask you to create Venn diagrams. Instead you are asked to apply them.

• Quiz 1 - Kenny takes a poll on favorite things to eat. 8 people like to eat candy. 10 people like to eat cookies. 7 people like to eat both candy and cookies. How many people did he poll?
• Quiz 2 - Sybil takes a poll of people's favorite dangerous animal. 7 like polar bears and 3 like sharks. 2 people like both polar bears and sharks. How many people like polar bears only?
• Quiz 3 - 10 of the students in Lea's class can swim the backstroke and 7 can swim the breaststroke. 4 students can swim both the strokes. How many students can swim the breaststroke, but not the backstroke?

How to Use Venn Diagrams for Math Problems

This is a schematic way of representing the elements of a group or set. In Venn diagrams, you get a number of elements or a bunch of numbers, and then all you have to do is use that information to construct a Venn diagram and then figure out the rest of the information.

They can help you understand the relationships that do and do not exist between these elements. Venn diagrams help you represent a diagram in a pictorial form. It represents the intersection and overlapping of two groups in an understandable manner. This allows you to quickly and easily compare and contrast these elements.

To understand how Venn diagrams help, consider the following example: There are a total of 40 students, out of which 14 are taking English Composition, and 26 are taking Chemistry. If there are 5 students taking English and Chemistry in both classes, how many students are in neither class? And how many of them are in either class?

To solve these, lets walk through how we would approach preparing a Venn diagram for this situation. Draw two circles, one representing English (blue in our diagram) and the other representing Chemistry (red in our diagram) with a part of circles overlapping into each other. Now write the respective numbers into the parts of the circle and use the information to solve the word problem you have at hand. 14 would be placed in the blue circle, 6 in the red circle, and 5 where the overlap is. To answer these questions we need to realize that those 5 students are accounted for already in each of those classes. So that would mean that there a total of 5 less enrolled in each class. Seeing that through the Venn diagram allows us to come to that conclusion a little easier.

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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.

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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.

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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

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Venn Diagrams: Exercises

Intro Set Not'n Sets Exercises Diag. Exercises

Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance:

Out of forty students, 14 are taking English Composition and 29 are taking Chemistry.

• If five students are in both classes, how many students are in neither class?
• How many are in either class?
• What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

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There are two classifications in this universe: English students and Chemistry students.

First I'll draw my universe for the forty students, with two overlapping circles labelled with the total in each:

(Well, okay; they're ovals, but they're always called "circles".)

Five students are taking both classes, so I'll put " 5 " in the overlap:

I've now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so I'll put " 9 " in the "English only" part of the "English" circle:

I've also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so I'll put " 24 " in the "Chemistry only" part of the "Chemistry" circle:

This tells me that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both). This gives me the answer to part (b) of this exercise. This also leaves two students unaccounted for, so they must be the ones taking neither class, which is the answer to part (a) of this exercise. I'll put " 2 " inside the box, but outside the two circles:

The last part of this exercise asks me for the probability that a agiven student is taking Chemistry but not English. Out of the forty students, 24 are taking Chemistry but not English, which gives me a probability of:

24/40 = 0.6 = 60%

• Two students are taking neither class.
• There are 38 students in at least one of the classes.
• There is a 60% probability that a randomly-chosen student in this group is taking Chemistry but not English.

Years ago, I discovered that my (now departed) cat had a taste for the adorable little geckoes that lived in the bushes and vines in my yard, back when I lived in Arizona. In one month, suppose he deposited the following on my carpet:

• six gray geckoes,
• twelve geckoes that had dropped their tails in an effort to escape capture, and
• fifteen geckoes that he'd chewed on a little

In addition:

• only one of the geckoes was gray, chewed-on, and tailless;
• two were gray and tailless but not chewed-on;
• two were gray and chewed-on but not tailless.

If there were a total of 24 geckoes left on my carpet that month, and all of the geckoes were at least one of "gray", "tailless", and "chewed-on", how many were tailless and chewed-on, but not gray?

If I work through this step-by-step, using what I've been given, I can figure out what I need in order to answer the question. This is a problem that takes some time and a few steps to solve.

They've given me that each of the geckoes had at least one of the characteristics, so each is a member of at least one of the circles. This means that there will be nothing outside of the circles; the circles will account for everything in this particular universe.

There was one gecko that was gray, tailless, and chewed on, so I'll draw my Venn diagram with three overlapping circles, and I'll put " 1 " in the center overlap:

Two of the geckoes were gray and tailless but not chewed-on, so " 2 " goes in the rest of the overlap between "gray" and "tailless".

Two of them were gray and chewed-on but not tailless, so " 2 " goes in the rest of the overlap between "gray" and "chewed-on".

Since a total of six were gray, and since 2 + 1 + 2 = 5 of these geckoes have already been accounted for, this tells me that there was only one left that was only gray.

This leaves me needing to know how many were tailless and chewed-on but not gray, which is what the problem asks for. But, because I don't know how many were only chewed on or only tailless, I cannot yet figure out the answer value for the remaining overlap section.

I need to work with a value that I don't yet know, so I need a variable. I'll let " x " stand for this unknown number of tailless, chewed-on geckoes.

I do know the total number of chewed geckoes ( 15 ) and the total number of tailless geckoes ( 12 ). After subtracting, this gives me expressions for the remaining portions of the diagram:

only chewed on:

15 − 2 − 1 − x = 12 − x

only tailless:

12 − 2 − 1 − x = 9 − x

There were a total of 24 geckoes for the month, so adding up all the sections of the diagram's circles gives me: (everything from the "gray" circle) plus (the unknown from the remaining overlap) plus (the only-chewed-on) plus (the only-tailless), or:

(1 + 2 + 1 + 2) + ( x )

+ (12 − x ) + (9 − x )

= 27 − x = 24

Solving , I get x = 3 . So:

Three geckoes were tailless and chewed on but not gray.

(No geckoes or cats were injured during the production of the above word problem.)

For more word-problem examples to work on, complete with worked solutions, try this page provided by Joe Kahlig of Texas A&M University. There is also a software package (DOS-based) available through the Math Archives which can give you lots of practice with the set-theory aspect of Venn diagrams. The program is not hard to use, but you should definitely read the instructions before using.

URL: https://www.purplemath.com/modules/venndiag4.htm

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