# SSAT Middle Level Math : How to use a Venn Diagram

## Example Questions

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### Example Question #1 : How To Use A Venn Diagram

See the above Venn diagram. Which of the following sets is represented by the gray region?

Explanation:

The shaded area represents the set of all elements that are both in  and not in . This the intersection of  and the complement of , or .

### Example Question #1 : How To Use A Venn Diagram

See the above Venn diagram. Which of the following sets is represented by the gray region?

Explanation:

The gray region represents all elements that either are in , are not in  - that is, are in  - or both. This is the union of  and , or .

### Example Question #1 : How To Find The Union Of A Venn Diagram

Given the Venn diagram below, which of the following does not belong to ?

Explanation:

The symbol  stands for the union between two sets.  Therefore,  means the set of all numbers that are in either A or B.  Looking at our choices, the only number that isn't in either A, B, or both is 23.

### Example Question #1 : How To Use A Venn Diagram

Let set , the set of all natural numbers.

= { |  is a multiple of 6 }

= { |  is a multiple of 9 }

Which of the following numbers would appear in the gray region of the Venn diagram?

Explanation:

The gray area represents the portion of  that is not in  - in other words, all multiples of 9 that are not also multiples of 6.

Therefore, 4,572, 3,438, and 8,544 can be eliminated.

, so 9,349 can be eliminated because it isn't a multiple of 9.

and , so, as both a nonmultiple of 6 and a multiple of 9, 4.077 is the correct choice.

### Example Question #3 : How To Use A Venn Diagram

The above Venn diagram represents survey respondents from a recent political poll. Based on the respondent's political affiliation, they were classified into group  only,  only or both groups.

What percentage of the respondents were classified into both groups?

Explanation:

The common portion of this Venn diagram represents the percentage of respondents that were classified into both groups. In order to calculate the percentage that represents the amount of respondents classified into both groups, first find the sum of group  and group . Then subtract that quantity from  percent.

The solution is:

### Example Question #1 : How To Use A Venn Diagram

Mr. Robinson surveyed his class to find out what his students planned on doing during their summer vacation. Every student in his class stated that they planned on swimming with their friends and/or travel with their family.

What percentage of Mr. Robinson’s class planned on both swimming with their friends and traveling with their family?

Explanation:

The common portion of this Venn diagram represents the percentage of Mr. Robinson's students that stated they planned on both swimming and traveling during the summer. In order to calculate the percentage that represents the amount of students classified into both groups, first find the sum of the swimming group and traveling group. Then subtract that sum from

The solution is:

### Example Question #5 : How To Use A Venn Diagram

Kelly and Antonio are in a group together on a popular social media website. They realized that within the group they have mutual friendships as well as friendships exclusive of one another.

What ratio accurately represents the amount of friendships within the group that they have in common to those that they do not have in common?

Explanation:

In order to find the ratio of friends that Kelly and Antonio have in common compared to the group members who are only friends with one or the other first find the percentage value of the common portion of the Venn diagram. To calculate this value, find the sum of the two exclusive groups and then subtract that sum from

This means that  of the group members are common friends of Kelly and Antonio. To convert this percentage to a ratio, first write  as a fraction, and then simplify as a ratio.

is equal to:

This means that  out of every  group members are mutual friends of Kelly and Antonio.

### Example Question #1 : How To Use A Venn Diagram

Kelly and Antonio are in a group together on a popular social media website. They realized that within the group they have mutual friendships as well as friendships exclusive of one another.

What ratio accurately represents the amount of friendships within the group that they do not have in common to those that they do have in common.

Explanation:

To find the ratio of Kelly and Antonio's exclusive friendships to their mutual friendships--find the sum of the two exclusive groups in the Venn diagram:

This means that  of the members in the group are not friends with both Kelly and Antonio. To convert this percent to a ratio, first write  as a fraction, and then simplify as a ratio.

is equal to:

This means that  out of every  group members are not mutual friends of Kelly and Antonio.

### Example Question #7 : How To Use A Venn Diagram

Results from a recent political poll are represented by the Venn diagram above. The results indicate the percentage of voters who have only voted for Democratic presidential candidates, only Republican presidential candidates, and those that have voted for both Democratic and Republican candidates in the past.

What percentage of respondents have voted for both Democratic and Republican presidential candidates?

Not enough information is provided.

Explanation:

To solve this problem first find the sum of the two exclusive groups shown in the Venn diagram:

This means that  of respondents have exclusively voted for presidential candidates from only one political party.

To find the value of the common portion of the Venn diagram, calculate the difference between  and :

This means that  of the respondents have voted for both Democratic and Republican presidential candidates.

### Example Question #1 : How To Use A Venn Diagram

The above Venn diagram represents survey respondents from a recent political poll. Based on the respondent's political affiliation, they were classified into group  only,  only or both groups.

What fraction of respondents were classified into only group

Explanation:

Since the information provided in the Venn diagram represents percentages, convert the quantity in category  from a percentage to a fraction. To convert a percentage to a fraction, divide the percent by a divisor of , then simplify the fraction if applicable; however, in this case the fraction can't be reduced.

The solution is:

Group

Thus:

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