### All ACT Math Resources

## Example Questions

### Example Question #1 : Venn Diagrams

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

**Possible Answers:**

**Correct answer:**

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 Find The Union Of A Venn Diagram

Sixty high school seniors were polled to see if they were taking history and calculus. A total of 29 students said they were taking calculus, and a total of 50 students said they were taking history. What is the minimum number of students who take both history and calculus?

**Possible Answers:**

**Correct answer:**

We can draw a Venn diagram to see these two sets of students.

We need to find the overlap between these two sets. To find that, add up the total number of students who are taking history and the total number of students who are taking calculus.

Notice that we have more students this way than the total number who were polled. That is because the students who are taking history AND calculus have been double counted. Subtract the total number of students polled to find out how many students were counted twice.

### Example Question #1 : Venn Diagrams

A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set ?

**Possible Answers:**

Bob

Steph

Molly

Patrick

Andy

**Correct answer:**

Patrick

The notation stands for "union," which refers to everything that is in either set. refers to the group of students taking either Calculus or Spanish (everyone on this diagram except those taking only Biology). From the diagram, Patrick and Ashley are the only students taking neither Calculus nor Spanish, so Patrick is the correct answer.

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

Forty students play soccer and/or basketball after school. Twenty-four students play soccer and twenty-nine play basketball. How many students play both soccer and basketball?

**Possible Answers:**

**Correct answer:**

We can draw a Venn diagram of these students.

Drawn this way, there are more students on the Venn diagram than we have.

This is because some of the students play both sports and should be in the overlap on the Venn diagram. To find the number of students in the overlap, subtract the total number of students given from the number on the diagram.

This represents the number of students who were counted twice, or the number in the overlap.

We can redraw the correct Venn diagram with this number.

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

Giving the Venn diagram above, what is the sum of the numbers in the set ?

**Possible Answers:**

**Correct answer:**

The notation stands for "A union C," which refers to everything that is in either set or set .

When we add the numbers together, we get:

### Example Question #11 : Venn Diagrams

In a class of senior high-school students, have pet cats, have pet dogs, have both cats and dogs, and have neither cats nor dogs. How many total students are in the class?

**Possible Answers:**

**Correct answer:**

A Venn diagram can help us determine the total number of students in the class.

First, we must calculate the number of students who have ONLY cats or ONLY dogs. First, for cats, 15 students have cats, and 5 students have both cats and dogs.

Ten students have only cats.

For dogs, 12 students have dogs, and 5 students have both cats and dogs.

Seven students have only dogs.

Using this information, we can fill in the Venn diagram.

This diagram shows the 10 students with only cats, the 7 students with only dogs, the 5 students with both, and the 8 students with neither. Adding up the numbers will give us the total number of students.

### Example Question #1 : Venn Diagrams

In the venn diagram above, let the set and let , what is the set Use set notation to enumerate your answer.

**Possible Answers:**

**Correct answer:**

means the intersection of the sets and , the only things in the intersection are elementes that are in BOTH of the sets. Since there is nothing shared by the two sets (no elements are in both), the interesction is the emptyset, or:

### Example Question #1 : Venn Diagrams

In a venn diagram, let and let

What is

**Possible Answers:**

**Correct answer:**

is the union of the sets and which is the set that contains anything that is in either set. Thus, the total of is every element that is in one of the two sets, and so

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

Fifty 6^{th} graders were asked what their favorite school subjects were. Three students like math, science and English. Five students liked math and science. Seven students liked math and English. Eight people liked science and English. Twenty students liked science. Twenty-eight students liked English. Fourteen students liked math. How many students didn’t like any of these classes?

**Possible Answers:**

3

5

10

7

None of the answers are correct

**Correct answer:**

5

Draw a Venn diagram with three subsets: Math, Science, and English. Start in the center with students that like all three subjects. Next, look at students that liked two subjects. Be sure to subtract out the ones already counted in the middle. Then, look at the students that only like one subject. Be sure to subtract out the students already accounted for. Once all of the subsets are filled, look at those students who don’t like any of these subjects. To find the students who don’t like any of these subjects add all of the students who like at least one subject from the total number of students surveyed, which is 50.

M = math

S = science

E = English

M∩S∩E = 3

M∩S = 5 (but 3 are already accounted for) so 2 for M and S ONLY

M∩E = 7 (but 3 are already accounted for) so 4 for M and E ONLY

S∩E = 8 (but 3 are already accounted for) so 5 for S and E ONLY

M = 14 (but 3 + 2 + 4 are already accounted for) so 5 for M ONLY

S = 20 (but 3 + 2 + 5 are already accounted for) so 10 for S ONLY

E = 28 (but 3 + 4 + 5 are already accounted for) so 16 for E ONLY

Therefore, the students already accounted for is 3 + 2 +4 + 5 + 5 + 10 + 16 = 45 students

So, those students who don’t like any of these subjects are 50 – 45 = 5 students

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

Set A contains the positive even integers less than 14. Set B contains the positive multiples of three less than 20. What is the intersection of the two sets?

**Possible Answers:**

A∩B = { }

A∩B = {6, 12}

A∩B = {6, 12, 18}

A∩B = {6}

A∩B = {4, 6, 8}

**Correct answer:**

A∩B = {6, 12}

A = {2, 4, 6, 8, 10, 12}

B = {3, 6, 9, 12, 15, 18}

The intersection of a set means that the elements are in both sets: A∩B = {6, 12}

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