Venn Diagrams:

Venn diagrams use circles to represent two or more sets of people or objects that have members in common. A Venn diagram and logical reasoning can help you to solve counting problems involving overlapping sets of people or objects.

Let’s look at a few examples:

1)A school picnic was attended by 100 students, 25 students ate hotdogs, 50 students ate hamburgers, and 15 students ate both hotdogs and hamburgers. How many students did not eat either hotdogs or hamburgers?

Solution: Since there are 100 total students, we will make that what we call the universal set. The 25 students and the 50 students will then become subsets. The 15 students will become the intersection of the two sets.

We can now answer the question using what we have drawn above.

Subset 1 = 25

+Subset 2 = 50

- intersection = 15

=union = 60

We then use:universal set-union =answer

Or

100 - 60 = 40

2)In Chapel Hill there are 50 students who attend DeanSmithHigh School. 20 of these students are members of the soccer team and 28 are members of the track team. How many students do not belong to one of the two teams?

Solution: First draw out the diagram below!

Now, use the diagram to come up with the solution.

Subset 1 =

+Subset 2 =

-Intersection =

=union =

Finally, universal set – union = answer

Try this one on your own:

3)In a class of 30 students, 19 are studying French, 12 are studying Spanish, and 7 are studying both French and Spanish. If no other foreign language is being studied what percent of the students in the class are not studying any foreign language?

Examples:

a)Of 25 city council members, 12 voted for proposition A, 19 voted for proposition B, and 10 voted for both propositions. If every council member voted on both propositions, what percent of the council members voted against both propositions?

b)In a school of 500 students, 65 students are in the school chorus, 255 students are on sports teams, and 40 students participate in both activities. What percent of the number of students in the school participate in either the school chorus or on sports teams but not both teams?

c)Of 76 retired people, 34 stay fit by swimming, 12 stay fit by playing golf, and 10 stay fit by swimming and playing golf. How many of the retired people do not stay fit by swimming or by playing golf?

d)In a High School of 1200 students, 900 students passed Math A, 700 students passed Math B, and 500 passed both Math A and Math B. How many students did not either pass Math A or Math B?

e)In a house made up of 9 rooms, 6 rooms have painted walls, 2 rooms have ceramic tiled walls, and 1 room has both painted and ceramic walls. How many rooms do not have either painted or ceramic tiled walls?

f)There are 1200 people working in an office building. 750 people get to work by using an automobile, 900 people get to work using public transportation, and 650 people get to work by using both an automobile and public transportation. How many people get to work without using an automobile or public transportation?

g)An automobile dealer has 200 new cars on his lot. 90 cars have air conditioning, 60 cars have automatic, and 40 cars have both. How many cars do not have either automatic transmissions or air conditioning?

h)A store selling major appliances has 550 sales this week, 300 sales are for appliances only, 10 sales are for extended service agreements only, and 100 sales are for appliances and extended service agreements. How many sales are not for appliances nor extended service agreements?

i)On Election Day, 2700 voters went to the polls. 2000 voted for the school budget, 2400 voted for the library budget, and 1800 voted for both the school and the library budgets. How many voters at the polls did not vote for either the school budget or the library budget?

j)There are 24 families living on Winding Road. 16 families have children with boys, 12 families have children with girls, and 10 families have both boys and girls. How many families do not have children?

k)There are 500 houses in a rural area, 300 houses get a bill for electric, 60 houses get a bill for gas, and 20 houses get a bill for both electricity and gas usage. How many houses do not get a bill for either electricity or gas?