PSAT Math : Data Analysis
Study concepts, example questions & explanations for PSAT Math
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Example Questions
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?
A∩B = { }
A∩B = {6, 12}
A∩B = {4, 6, 8}
A∩B = {6, 12, 18}
A∩B = {6}
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}
Example Question #2 : How To Find The Intersection Of A Venn Diagram
Students at a local high school are given the option to take one gym class, one music class or one of each. Out of 100 students, 60 say that they are currently taking a gym class and 70 say that they are taking a music class. How many students are taking both?
This problem can be solved two ways, with a formula or with reason.
Using the formula, the intersection of the Venn diagram for which classes students take is:
By using reason, it is clear that 60 + 70 is greater than 100 by 30. It is assumed that this extra 30 students come from students who were counted twice because they took both classes.
Example Question #1 : Intersection
There are 75 juniors at a high school. 15 of the students are enrolled in Physics and 40 students are enrolled in Chemistry. 30 students are not enrolled in either Physics or Chemistry. How many students are enrolled in both Physics and Chemistry?
5
10
25
15
30
10
First, subtract the students that are in neither class; 75 – 30 = 45 students.
Thus, 45 students are enrolled in Chemistry, Physics, or both. Of these 45 students, we know 40 are in Chemistry, so that leaves 5 students who are enrolled in Physics only; with 15 total students in Physics, that means 10 must be in Chemistry as well. So 10 students are in both Physics and Chemistry.
Example Question #1 : Venn Diagrams
In a group of 83 gym members, 51 are taking kickboxing and 25 are taking yoga. Of the students taking kickboxing or yoga, 11 are taking both classes. How many members are not taking either course?
18
24
39
7
18
If 11 people are taking both courses, this means 51-11 or 40 are taking kickboxing only and 25-11 or 14 are taking yoga only. The number of people taking at least one course, therefore, is 40 + 14 + 11 = 65. The 83 members minus the 65 that are taking courses leaves 18 who are not taking any courses.
Example Question #1 : Data Analysis
Doug has a cow farm. Some of Doug's cows are used for milk, some are used for reproduction and some are used for both. If he has a total of 40 cows and 10 are used only for milk and 3 are used for both milk and reproduction, then how many cows are used for reproduction?
40
33
13
30
27
30
Since we know that only 10 cows are for milk only we must subtract this number from the total amount of cows to get our answer: 40 – 10 = 30 cows. The cows that do both are still used for reproduction, so the correct answer is 30 cows.
Example Question #1 : Venn Diagrams
All students have to take at least one math class and one language class. Twenty students take calculus, and thirty students take statistics. Fifteen students take Spanish and twenty-five take French. If there are thirty-five students total, what is the maximum number of students taking both two math classes and two language classes.
7
9
5
18
15
5
Totalling the number in math there are 50 students on the rosters of all the math classes. With 35 total students this means that there are 15 students taking 2 math classes. For the language classes there are 40 students on the roster, showing that 5 students are taking 2 language classes. The maximum number of students taking two math classes and two language classes is only as great as the smallest number taking a double math or language class, which is 5 students (limited by the language doubles).
Example Question #1 : Data Analysis
The class of 2034 at Make Believe High School graduated 50 students. 13 students studied only math. 35 students studied English. 30 students studied only 2 subjects. Only 4 students studied writing, it was the third subject for all of them. How many students did not study anything?
10
2
11
0
5
2
The answer is 2. Taking away the 4 writing students, the 13 math-only students, and the remaining 31 English students, we have 2 students remaining.
Example Question #1 : Data Analysis
In a school of 1250 students, 50% of the students take an art class and 50% of the students take a gym class. If 450 students take neither art nor gym class, then how many students take both art and gym?
You can construct a Venn diagran in which one circle represents art (A), the other represents gym (B), the region of overlap is designated (C), and the number of students not present in either circle is designated (S).
First, it is given that S=450.
50% of students take art and 50% take gym and the total number of students is 1250. 50% of 1250 is 625. Thus, A+C=625 and B+C=625.
Setting them equal to each other we get A+C=B+C.
Subtract C from both sides to get A=B; so the same number of students take art only and gym only.
Now 1250-450= A+B+C=800.
Since A=B we can use substitution to get 2A+C=800.
Finally you can solve the system of equations using the method of your choice (substitution or elimination) to solve the system with A+C=625 and 2A+C=800.
For substitution, we solve for C in the first equation to get C=625-A. Then we substitute this value into the second equation to get 2A+ (625-A)=800. Solve for A to get A=175, so B=175 also. Since A+B+C = 800, C=450.
Example Question #1 : Data Analysis
The universal set is positive counting numbers less than 11. Set A = {1, 3, 5} and Set B = { 2, 4, 6}.
What is 
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {1, 3, 5}
A' = {2, 4, 6, 7, 8, 9, 10}
B = {2, 4, 6}
B' = {1, 3, 5, 7, 8, 9, 10}
Example Question #2 : Data Analysis
Sets P, Q, and R consist of the positive factors of 48, 90, and 56, respectively. If set T = P U (Q ∩ R), which of the following does NOT belong to T?
28
48
24
3
6
28
First, let's find the factors of 48, which will give us all of the elements in P. In order to find the factors of 48, list the pairs of numbers whose product is 48.
The pairs are as follows:
1 and 48; 2 and 24; 3 and 16; 4 and 12; 6 and 8
Therefore the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Now we can write P = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.
Next, we need to find the factors of 90.
Again list the pairs:
1 and 90; 2 and 45; 3 and 30; 5 and 18; 6 and 15; 9 and 10
Then the factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Thus, Q = {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}.
Next find the factors of 56:
1 and 56; 2 and 28; 4 and 14; 7 and 8
Set R = {1, 2, 4, 7, 8, 14, 28, 56}
Now, we need to find set T, which is P U (Q ∩ R).
We have to start inside the parantheses with Q ∩ R. The intersection of two sets consists of all of the elements that the two sets have in common. The only elements that Q and R have in common are 1 and 2.
Q ∩ R = {1, 2}
Lastly, we must find P U (Q ∩ R).
The union of two sets consists of any element that is in either of the two sets. Thus, the union of P and Q ∩ R will consist of the elements that are either in P or in Q ∩ R. The following elements are in either P or Q ∩ R:
{1, 2, 3, 4, 6, 8, 12, 16, 24, 48}
Therefore, T = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}.
The problem asks us to determine which choice does NOT belong to T. The number 28 doesn't belong to T.
The answer is 28.
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