SSAT Middle Level Quantitative - How to use a Venn Diagram

Venn_4

Let set $U = \mathbb{N}$ , 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.

$4,572 \div 6 = 762$

$3,438 \div 6 = 573$

$8,544 \div 6 =1,424$

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

$9,349 \div 9 = 1,038 ;R; 7$ , so 9,349 can be eliminated because it isn't a multiple of 9.

$4,077 \div 6 = 679 ; R ; 3$ and $4,077 \div 9 = 453$ , so, as both a nonmultiple of 6 and a multiple of 9, 4.077 is the correct choice.

Venn_diagram

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

$\overline{A} \cap B$

$A \cap \overline{B}$

$A \cup \overline{B}$

$\overline{A} \cup B$

$\overline{A} \cup \overline{B}$

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 $\overline{A} \cap B$ .

Vt custom venn series ssat middle

Kayla used a popular social media website to survey her friends' hobbies. All of her friends either play sports or enjoy playing video games, and some of her friends do both.

What fraction of her friends only play sports?

$\frac{1}{2}$

$\frac{3}{5}$

$\frac{3}{10}$

$\frac{1}{5}$

$\frac{2}{10}$

Explanation

In order to calculate the fraction of Kayla's friends who only play sport, first find the sum of the common portion and video game portion of the Venn diagram. Then subtract that sum from whole.

The solution is:

$\frac{3}{10}+\frac{2}{10}=\frac{5}{10}$

Note: $\frac{10}{10}=1$

Thus, $\frac{10}{10}-\frac{5}{10}=\frac{5}{10}=\frac{5\div 5}{10\div 5}=\frac{1}{2}$

This means that half of Kayla's friends only play sports.

Vt custom venn series ssat middle

Kayla used a popular social media website to survey her friends' hobbies. All of her friends either play sports or enjoy playing video games, and some of her friends do both.

What percentage of Kayla's friends play sports and video games?

Explanation

In order to solve this problem, identify that the common portion of this Venn diagram represents Kayla's friends who play sports and video games. Since $\frac{3}{10}$ of her friends play sports and video games, convert this fraction to a percent.

The solution is:

$\frac{3}{10}=\frac{3\times 10}{10\times 10}=\frac{30}{100}=30%$

Note: the most efficient way to convert this fraction to a percent is to find an equivalent fraction to $\frac{3}{10}$ with a denominator of .

Vt custom venn series ssat middle

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 ?

$\frac{9}{100}$

$\frac{3}{50}$

$\frac{4}{50}$

$\frac{9}{10}$

$\frac{90}{10}$

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:
$B=9\div100=\frac{9}{100}$

Vt custom venn series ssat middle

Ms. Dunn gave her class a survey to find out which states her student's have visited. Ms. Dunn was surprised to find that all of her student's had visited either New York City or Texas, and some of her student's had visited both locations.

The above Venn diagram represents the percentage of students who have visited only NYC, only Texas, and those who have visited both locations.

What ratio represents the number of students that have gone only to NYC, in comparison to the rest of the class?

Explanation

Since of Ms. Dunn's class have visited only NYC, out of every students must have only visited NYC. This can be represented by the ratio ; however, this ratio does not appear as an answer choice, so we must reduce this ratio by dividing each part by their greatest common divisor.

The solution is:

$46:100=46\div2:100\div2=23:50$

Vt custom venn series ssat middle

The above Venn diagram represents the total number of respondents from a survey administered in . Respondents were categorized into only group , only group or both of the groups.

What fraction of the respondents were categorized into both groups?

$\frac{3}{25}$

$\frac{3}{50}$

$\frac{6}{100}$

$\frac{12}{50}$

$\frac{12}{10}$

Explanation

Since the information provided in the Venn diagram represents percentages, convert the quantity in the common portion of the diagram from a percentage to a fraction. To convert a percentage to a fraction, divide the percentage by a divisor of , then simplify the fraction if possible.

Common portion is equal to . Therefore, the solution is:

$12\div100=\frac{12}{100}=\frac{12\div 2}{100\div 2}=\frac{6}{50}=\frac{6\div 2}{50\div 2}=\frac{3}{25}$

Vt custom venn series ssat middle

The above Venn diagram represents the total number of respondents from a survey administered in . Respondents were categorized into only group , only group or both of the groups.

What percentage of respondents were categorized into only group

Explanation

To find the missing quantity for category , first calculate the sum from the common portion of the Venn diagram and category . Then, subtract that sum from , because the total percentage of respondents must equal .

The algebraic solution is:

Vt custom venn series ssat middle

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: $\frac{60}{100}=60:100=6:10=3:5$

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

Venn_diagram_2

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

$A \cup \overline{B}$

$\overline{A }\cup B$

$\overline{A }\cap B$

$\overline{A }\cap \overline{B}$

$\overline{A }\cup \overline{B}$

Explanation

The gray region represents all elements that either are in , are not in - that is, are in $\overline{B}$ - or both. This is the union of and $\overline{B}$ , or $A \cup \overline{B}$ .

How to use a Venn Diagram

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