# GRE Math : How to find the union of a Venn Diagram

## Example Questions

### Example Question #1 : Venn Diagrams

There are 15,000 students at college X.  Of those students, 1,700 are taking both ethics and metaphysics this semester.  There are 2,200 total students taking ethics.  9,500 students are taking neither of these classes.  How many students are taking metaphysics this term?

Possible Answers:

None of the other answers.

5,000

3800

3300

1600

Correct answer:

5,000

Explanation:

The easiest way to understand this problem is to draw a Venn Diagram:

S = Total number of students

A = Total students taking ethics

B = Total students taking metaphysics

A - B = Students taking only ethics

B - A = Students taking only metaphysics

A ∪ B = The total students taking either ethics or metaphysics

A ∩ B = The total students taking both ethics and metaphysics

We know there are 15,000 total and that 9,500 are taking neither class.  Therefore, we know that 15,000 - 9,500 = 5,500 are taking at least one of the classes.  Based on our prompt, we know that there are 1,700 taking both and that 2,200 are taking ethics.  To fiind out how many are taking ONLY ethics, we have to subtract off the amount that are taking ethics and metaphysics.  Hence, 2,200 - 1,700 = 500.  Finally, if we know that there are 5,500 taking at at least one of these classes, we want to get rid of that portion taking ethics.  This will leave us with those who are taking at least metaphysics (regardless of whether or not they are taking ethics): 5,500 - 500 = 5,000.

### Example Question #2 : Venn Diagrams

In a class of 100 students, 43 play basketball and 37 play baseball. 9 students play both. How many students do not play either sport?

Possible Answers:

29

38

20

not enough information to answer the question

71

Correct answer:

29

Explanation:

In order to determine how many students are not enrolled in a sport, we must first determine how many students are. The simplest way to do this is to begin by adding the students of both sports together.

43 + 37 = 80

But wait! 9 of those students are play both baseball and basketball. To avoid double counting these students, subtract 9 from the total.

80 – 9 = 71

Now we know 71 students play sports. If there are 100 students, all that's left to do is subtract.

100 – 71 = 29

29 students do not play basketball or baseball.

### Example Question #2 : Venn Diagrams

There are  students in a class.   of them take German and  take Latin.  Some students take two languages.  There are  students who take no language whatsoever.  How many students are there who take at least one language?

Possible Answers:

Correct answer:

Explanation:

You could represent this question in the following Venn Diagram:

We know that the two circles must contain a total of  students.  Now, since there is an extra case of the overlap area when you add together the German and the Latin students, you can say:

This is the simple answer to this question!  You do not need to compute the overlap at all, as you are merely looking for the contents of the two circles.

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