In these lessons, examples, and solutions, we will learn how to solve word problems using Venn Diagrams that involve two sets or three sets.

Venn diagrams are the principal way of showing sets diagrammatically. The method consists primarily of entering the elements of a set into a circle or circles.

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?

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?

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?

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)

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?

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: 90 - (14 + 2 + 3 + 5 + 21 + 7 + 23) = 90 - 75 = 15

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

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