Data Analysis - ACT Math
Card 0 of 1845
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?
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?
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.
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.
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Given the Venn diagram below, which of the following does not belong to 
?
Given the Venn diagram below, which of the following does not belong to
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.
The symbol
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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?
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?
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.
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.
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Giving the Venn diagram above, what is the sum of the numbers in the set 
?
Giving the Venn diagram above, what is the sum of the numbers in the set
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:
The notation
When we add the numbers together, we get:
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A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set 
?
A group of high school juniors are taking Biology, Calculus, and Spanish as shown above. Which student is not in the set
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.
The notation
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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?
In a class of senior high-school students,
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.
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.
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In the venn diagram above, let the set 
and let 
, what is the set 
Use set notation to enumerate your answer.
In the venn diagram above, let the set
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: 
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In a venn diagram, let 
and let 
What is 
In a venn diagram, let
What is
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 
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Fifty 6th 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?
Fifty 6th 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?
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
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
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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?
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 = {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}
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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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?
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?
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.
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.
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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?
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.
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.
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In the Venn Diagram shown above, let the set 
be the set of all prime numbers less than 
. Let the set 
be the set of all odd numbers less than 
. What is the set 
, given by 
? Use set notation to enumerate.
In the Venn Diagram shown above, let the set
To find the intersection of a set of numbers, we need to find all numbers that are in both sets (definition of intersection). We can see that all the primes less than 
that are odd are also odd numbers less than 
, therefore they are in both sets and in the intersection. Thus the answer:
To find the intersection of a set of numbers, we need to find all numbers that are in both sets (definition of intersection). We can see that all the primes less than
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In the Venn Diagram shown above, let 
, let 
and let 
. If the Universe is all integers less than or equal to 10, what is 
? Use set notation to enumerate your answer.
In the Venn Diagram shown above, let
means the complement of 
union 
- this is everything in the universe that is not in 
or 
. First we need to find 
:
- we find this by knowing anything in either set is in the union. Next we need to find things in the universe that are NOT in 
:
Since the universe is integers less than or equal to 10, we see that the only ones that fit that description that are not in 
are 
Since the universe is integers less than or equal to 10, we see that the only ones that fit that description that are not in
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Angela scores 17, 19, 13, 24, and 14 points in the first five games of a seven-game basketball season. If the scoring leader in Angela’s league averages 18 points per game, how many points must Angela score in the final two games combined to end the season with the highest scoring average in the league AND have a higher scoring average than any other player?
Angela scores 17, 19, 13, 24, and 14 points in the first five games of a seven-game basketball season. If the scoring leader in Angela’s league averages 18 points per game, how many points must Angela score in the final two games combined to end the season with the highest scoring average in the league AND have a higher scoring average than any other player?
Since a given player’s scoring average can be determined by dividing the sum total of points scored by the number of games, we can determine the total points of the scoring leader by multiplying the average points per game by the total number of games. 18 x 7 = 126. Angela would have to score 1 more point than the current scoring leader. Angela’s current total is 87 points; therefore, she must score 40 (87 + 40 = 127) over the course of the final two games to have the highest average points per game in the league.
Since a given player’s scoring average can be determined by dividing the sum total of points scored by the number of games, we can determine the total points of the scoring leader by multiplying the average points per game by the total number of games. 18 x 7 = 126. Angela would have to score 1 more point than the current scoring leader. Angela’s current total is 87 points; therefore, she must score 40 (87 + 40 = 127) over the course of the final two games to have the highest average points per game in the league.
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What is the mean of the following set of numbers: 13, 15, 100, 54, 345, rounded to the nearest 1’s place?
What is the mean of the following set of numbers: 13, 15, 100, 54, 345, rounded to the nearest 1’s place?
mean = sum of all values divided by the number of values.
mean = sum of all values divided by the number of values.
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If the average of A, B, and C is 50, which of the following expressions represents the average of A, B, C, and D?
If the average of A, B, and C is 50, which of the following expressions represents the average of A, B, C, and D?
We take the average 50 and multiply it by 3 (the number of terms in the set) to get the total sum of the initial set. Then you take the total sum and add D and divide by the numbers of terms in the new set.
We take the average 50 and multiply it by 3 (the number of terms in the set) to get the total sum of the initial set. Then you take the total sum and add D and divide by the numbers of terms in the new set.
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Calculate the arithmetic mean of the following set of numbers: 54, 35, 50, 49, 54, 65, 82, 54, and 25.
Calculate the arithmetic mean of the following set of numbers: 54, 35, 50, 49, 54, 65, 82, 54, and 25.
To solve this question, you must know that the mean is the sum of the values divided by the number of values, in this case: 468/9=52. It is not 54, which is the median (the most commonly occurring number of the set).
To solve this question, you must know that the mean is the sum of the values divided by the number of values, in this case: 468/9=52. It is not 54, which is the median (the most commonly occurring number of the set).
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The mean of five numbers is 40. The mean of the smallest two numbers is 25. What is the mean of the other three numbers?
The mean of five numbers is 40. The mean of the smallest two numbers is 25. What is the mean of the other three numbers?
The equation for the mean of a group of numbers is to find the sum of all of the numbers and then divide by how many numbers are in the group. This means that if we know the mean and how many numbers are in the group, we can find the sum of those numbers.
(sum of all five numbers) / 5 = 40 --> sum of all five numbers = 200
(sum of two smallest numbers) / 2 = 25 --> sum of two smallest numbers = 50
Subtracting the sum of the two smallest numbers from the sum of all five gives us the sum of the remaining three. We can then divide by three to find the mean of those three remaining numbers.
200 – 50 = 150
150 / 3 = 50
The equation for the mean of a group of numbers is to find the sum of all of the numbers and then divide by how many numbers are in the group. This means that if we know the mean and how many numbers are in the group, we can find the sum of those numbers.
(sum of all five numbers) / 5 = 40 --> sum of all five numbers = 200
(sum of two smallest numbers) / 2 = 25 --> sum of two smallest numbers = 50
Subtracting the sum of the two smallest numbers from the sum of all five gives us the sum of the remaining three. We can then divide by three to find the mean of those three remaining numbers.
200 – 50 = 150
150 / 3 = 50
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In a theatre there are 600 people. Out of this, 100 males have an average age of 50 and 500 females have an average age of 30. To the nearest whole year, what is the average age of the town’s entire population?
In a theatre there are 600 people. Out of this, 100 males have an average age of 50 and 500 females have an average age of 30. To the nearest whole year, what is the average age of the town’s entire population?
To find the average age, compute:
100 * 50 + 500 * 30= 5000+ 15000= 20000. Then divide by the total amount of people that are at the theatre: 20000/600= 33.3=33.
To find the average age, compute:
100 * 50 + 500 * 30= 5000+ 15000= 20000. Then divide by the total amount of people that are at the theatre: 20000/600= 33.3=33.
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