Venn Diagrams

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ISEE Lower Level Quantitative Reasoning › Venn Diagrams

Questions 1 - 10

Students were asked if they prefer TV or radio. The following Venn Diagram depicts the number of students who said TV, radio, or both. How many students like both TV and radio?

Explanation

The blue circle of the Venn diagram depicts the number of students who prefer TV, the orange circle depicts the number of students who prefer radio, and the region of overlap indicates the number of students who like both. Therefore, 7 students like both TV and radio.

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The Venn diagram above represents the results from a recent survey given to middle school students. Category $\small A$ represents the amount of students that only like pizza. Category $\small B$ represents the amount of students that only chicken nuggets.

What percentage of students like both chicken nuggets and pizza?

Explanation

Since we are asked to find the percent of students that like both chicken nuggets and pizza we need to first add the percentages together.

Thus we get,

$\small 13+35=48$ .

The amount of students that like both pizza and chicken nuggets is going to be all of the students minus those that like only chicken nuggets and only pizza.

$\small 100-48=52$ .

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What percentage of students like both chicken nuggets and pizza?

Explanation

Since we are asked to find the percent of students that like both chicken nuggets and pizza we need to first add the percentages together.

Thus we get,

$\small 13+35=48$ .

The amount of students that like both pizza and chicken nuggets is going to be all of the students minus those that like only chicken nuggets and only pizza.

$\small 100-48=52$ .

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What fraction of the students only like chicken nuggets?

$\small \frac{7}{20}$

$\small \frac{30}{100}$

$\small \frac{5}{100}$

$\small \frac{100}{35}$

Explanation

Since exactly $\small 35$ of a total $\small 100$ percent of the students only like chicken nuggets, the solution is:
$\small \frac{35}{100}=\frac{7}{20}$

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The Venn diagram shown above has three categories that represent information about the Wildcats varsity baseball team.

Category $\small L$ represents the number of players on the team that are left-handed.
Category $\small P$ represents the numebr of players on the team that are pitchers.
And, the overlapping portion of the Venn diagram represents the number of players that are left-handed pitchers.

Given that $\small L=6, P=9$ and that there are $\small 3$ players in the overlapping region of the diagram.

What fraction of the players are only left-handed?

$\small \frac{1}{5}$

$\small \frac{3}{6}$

$\small \frac{1}{6}$

$\small \frac{4}{5}$

Explanation

Since, there are $\small 6$ left-handed players total and $\small 3$ left-handed pitchers there must be $\small 3$ players that are left-handed but do not pitch, because $\small 6-3=3$ .

To find what fraction this represents we need to do:

$\frac{\text{left handed}}{\text{total players}}=\frac{3}{15}=\frac{1}{5}$

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In the above Venn diagram category $\small A$ represents the number of Kevin's friends that play the flute, while category $\small B$ represents the number of Kevin's friends that play the bass.

If $\small \frac{2}{8}$ of Kevin's friends only play the flute and $\small \frac{2}{8}$ of his friends only play the bass, then what fraction of Kevin's friends play both instruments?

$\small \frac{1}{2}$

$\small \frac{2}{8}$

$\small \frac{6}{8}$

$\small \frac{2}{3}$

Explanation

To find the fraction of Kevin's friends that play both the bass and the flute, consider that the fraction $\small \frac{8}{8}$ represents all of Kevin's friends in this senario.

Thus, the solution can be found by adding those that only play flute with those that only play bass and subtracting that final answer from the fraqction that represents all of Kevin's friends:

$\small \small \frac{2}{8}+\frac{2}{8}=\frac{4}{8}$

$\frac{8}{8}-\frac{4}{8}=\frac{8-4}{8}=\frac{4}{8}=\frac{1}{2}$

If Jill likes blue, yellow, tan and green, and Doug likes red, tan, black and green, which Venn diagram is correct?

Venn1

Venn2

Venn5

Venn4

Venn3

Explanation

The middle portion of the diagram is the area that both circles share, so the color name that belongs in both circles should go in the middle area. Doug and Jill both like green and tan, so those colors should go in the middle. Only Jill likes blue and yellow, so these go on Jill's side. Only Doug likes red and black, so these go on Doug's side.

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What fraction of the students only like pizza?

$\small \frac{13}{100}$

$\small \frac{3}{5}$

$\small \frac{35}{100}$

$\small \frac{6}{50}$

Explanation

Since category $\small A$ represents the total number of students that only like pizza, the correct answer is that $\small 13$ percent is equivalent to $\small \frac{13}{100}$ .

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The Venn diagram shown above has three categories that represent information about the Wildcats varsity baseball team.

Given that $\small L=6, P=9$ and that there are $\small 3$ players in the overlapping region of the diagram.

How many players are right-handed pitchers?

Explanation

Since the question provides the information that there are $\small 3$ left-handed pitchers and $\small 9$ total pitchers, one can infer that the number of right-handed pitchers is equal to the difference between the total number of pitchers and the number of left-handed pitchers.

Thus, the solution is:

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In the above Venn diagram category $\small A$ represents the number of Kevin's friends that play the flute, while category $\small B$ represents the number of Kevin's friends that play the bass.

If $\small \frac{2}{8}$ of Kevin's friends only play the flute and $\small \frac{2}{8}$ of his friends only play the bass, then what percentage of Kevin's friends only only play the bass?

Explanation

$\small \frac{2}{8}$ of Kevin's friends only play the bass.

Thus, to find the fractional equivalent we need to multiply the fraction by 100.

The solution becomes:

$\frac{2}{8}\times100 = \frac{200}{8}=\frac{25}{1}= 25$

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