Data Analysis - ISEE Upper Level Quantitative Reasoning
Card 0 of 1356
What is the greatest common factor of 
and 
?
What is the greatest common factor of
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
: 
: 
: 
Taking these together, you get:
To solve for the greatest common factor, it is necessary to get your numbers into prime factor form. For each of your numbers, this is:
Next, for each of your sets of prime factors, you need to choose the exponent for which you have the smallest value; therefore, for your values, you choose:
Taking these together, you get:
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For the Venn Diagram pictured above, what is the value 
for the overlap of the two sets drawn as circles?
For the Venn Diagram pictured above, what is the value
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal 
or 
. This is the total amount in the "universe" (
) minus the amount that is found outside of the two circles (
).
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means: 
. The overlap is the whole of circle 
!
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means:
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For the Venn Diagram pictured above, what is the value 
for the overlap of the two sets drawn as circles?
For the Venn Diagram pictured above, what is the value
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal 
or 
. This is the total amount in the "universe" (
) minus the amount that is found outside of the two circles (
).
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means:
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means:
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For the Venn Diagram pictured above, what is the value 
for the overlap of the two sets drawn as circles?
For the Venn Diagram pictured above, what is the value
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal 
or 
. This is the total amount in the "universe" (
) minus the amount that is found outside of the two circles (
).
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means:
Solving for the overlap of two sets is easy when you have all of your data. You know that the two circles added up will have to equal
Because the overlap happens once in each circle, you know that:
Given your data, you know:
Solving, this means:
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For the Venn Diagram pictured above, what is the value 
for the overlap of the two sets drawn as circles?
For the Venn Diagram pictured above, what is the value
Do not be tricked by this question! In order to solve for the overlap, you need to know the amount that is in the area outside of the circles but still inside the universal box area! You cannot figure out the answer without knowing this fact; therefore, you must select "No answer is possible." We know that the two circles do not exhaust the universe because 
. This is not large enough to fill the complete 
. If it were, you would know that the overlap is 
.
Do not be tricked by this question! In order to solve for the overlap, you need to know the amount that is in the area outside of the circles but still inside the universal box area! You cannot figure out the answer without knowing this fact; therefore, you must select "No answer is possible." We know that the two circles do not exhaust the universe because
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Let the universal set 
be the set of all positive integers. Define:
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
Let the universal set
Examine the above Venn diagram. If each integer was to be placed in its correct region, which of the following would be placed in the gray area?
The grayed portion of the Venn diagram corresponds to those integers which are not in any of 
, 
, or 
. Therefore, we eliminate any choices that are in any of the three sets.
is the set of integers which end in 1 or 6; we can eliminate 166 and 176 immediately.
is the set of perfect square integers; we can eliminate 144, since 
.
is the set of integers which, when divided by 4, yields remainder 2. Since 
, we can eliminate 154.
All four choices have been eliminated.
The grayed portion of the Venn diagram corresponds to those integers which are not in any of
All four choices have been eliminated.
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In the above Venn diagram, the universal set is defined as 
. Each of the eight letters is placed in its correct region.
What is 
?
In the above Venn diagram, the universal set is defined as
What is
is the union of sets 
and 
- that is, the set of all elements of 
that are elements of either 
or 
. We want all of the letters that fall in either circle, which from the diagram can be seen to be all of the letters except 
. Therefore,
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In a school of 
students, 
students take Greek, 
take Old English, and 
take neither. How many take both?
In a school of
Based on the information given, you can construct the following Venn Diagram:
In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract: 
. Now, since the overlap represents a duplication, you need to subtract out one of those duplicate values. Let's call that 
; therefore, we know that:
Solving for 
, you get:
Based on the information given, you can construct the following Venn Diagram:
In order to find the overlap, you need to find out how many are in the circles together. This is easy. Subtract:
Solving for
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In a group of 
people, 
have a laptop and 
have a tablet. Of those people who have a laptop or a tablet, 
have both. How many people in the total group have neither a laptop nor a tablet?
In a group of
Based on the information given, you can draw the following Venn Diagram:
To solve this, remember that the total number of values in the two circles is:
(We must do this because of the overlap. You need to subtract out one instance of that overlap.)
If we assign the value 
for the unknown region, we know:
Based on the information given, you can draw the following Venn Diagram:
To solve this, remember that the total number of values in the two circles is:
(We must do this because of the overlap. You need to subtract out one instance of that overlap.)
If we assign the value
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In a group of plants, 
are green, 
have large leaves, and 
are both green and have large leaves. How many plants are green without large leaves?
In a group of plants,
Based on the information, you can draw the following Venn Diagram:
It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely 
. We find this by eliminating the large-leaved plants from the green ones (by subtracting the overlap from the green ones).
Based on the information, you can draw the following Venn Diagram:
It is very easy to solve for the number of plants that have green leaves but not large ones. This is merely
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In a group of 
people, 
have books, 
have pens, and 
have neither books nor pens. How many people in the group have only books?
In a group of
Based on the information given, you can draw the following Venn Diagram:
Now, you must begin by solving for 
. You know that the two circles together will have 
in them. This is arrived at by subtracting the people who have neither books nor pens (
) from the "universe" of people in the sample space (
). Now, we know that 
. This is because of the overlap of 
in both groups. We have to get rid of one instance of that. Thus we can solve for 
:
Now, we can find the number of people with only books by subtracting 
from the 
to get 
.
Based on the information given, you can draw the following Venn Diagram:
Now, you must begin by solving for
Now, we can find the number of people with only books by subtracting
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Examine the above Venn diagram. Let 
be the universal set of the Presidents of the United States. 
is the set of all Presidents born in Virginia; 
is the set of all Presidents born after 1850; 
is the set of all Presidents whose first name was or is James.
James Abram Garfield was born in Ohio in 1831. In which region would he fall?
Examine the above Venn diagram. Let
James Abram Garfield was born in Ohio in 1831. In which region would he fall?
Carter would not fall in set A, since he was not a President born in Virginia.
He would not fall in B, since he was born before 1850.
He would fall in C, since his first name is James.
He would fall in the region included in set C, but not A or B - this is Region V.
Carter would not fall in set A, since he was not a President born in Virginia.
He would not fall in B, since he was born before 1850.
He would fall in C, since his first name is James.
He would fall in the region included in set C, but not A or B - this is Region V.
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Examine the above Venn diagram. Let 
be the universal set of the Presidents of the United States. 
is the set of all Presidents born in Virginia; 
is the set of all Presidents born after 1850; 
is the set of all Presidents whose first name was or is James.
James Earl Carter was born in Georgia in 1924. In which region would he fall?
Examine the above Venn diagram. Let
James Earl Carter was born in Georgia in 1924. In which region would he fall?
Carter would not fall in set A, since he was not a President born in Virginia.
He would fall in B, since he was born after 1850.
He would fall in C, since his first name is James.
He would fall in the region included in sets B and C, but not A - this is Region III.
Carter would not fall in set A, since he was not a President born in Virginia.
He would fall in B, since he was born after 1850.
He would fall in C, since his first name is James.
He would fall in the region included in sets B and C, but not A - this is Region III.
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Examine the above Venn diagram. Let universal set 
represent the set of all words in the English language.
Let 
be the set of all words whose last letter is a vowel. Let 
be the set of all words whose first letter is a consonant. Let 
be the set of all words exactly six letters in length.
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
Examine the above Venn diagram. Let universal set
Let
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
The subset must comprise words that fall inside sets 
and 
, but not 
. Therefore, all of the words in the subset must begin with a consonant, end with a vowel, and not have six letters.
Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.
The subset must comprise words that fall inside sets
Of the given choices, the only set whose elements fit this description is {plateau, portmanteau, calliope, marionette, taco}.
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Examine the above Venn diagram. Let universal set 
represent the set of all words in the English language.
Let 
be the set of all words whose last letter is a consonant. Let 
be the set of all words whose first letter is a vowel. Let 
be the set of all words exactly five letters in length.
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
Examine the above Venn diagram. Let universal set
Let
Which of the following would be a subset of the set represented by the shaded region in the diagram?
Note: for purposes of this question, "Y" is considered a consonant.
The subset must comprise words that fall inside set 
, but neither
nor 
.
Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.
The only set among the five choices that matches this description is the set
{price, value, pinna, trove, three}.
The subset must comprise words that fall inside set
Therefore, all of the words in the subset must have exactly five letters, but cannot begin with a vowel or end with a consonant - that is, we are looking for a set of five-letter words that begin with a consonant and end with a vowel.
The only set among the five choices that matches this description is the set
{price, value, pinna, trove, three}.
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The beverage menu from a coffeehouse reads as follows:
Eight friends go out for drinks. Three order cappucino, two order espresso, one orders Americano, one orders Turkish coffee, and one orders cafe latte. One of the eight agrees to leave the entire amount of the tip, which is to be 20% of the check. How much will be the amount of the tip (round to the nearest dime)?
The beverage menu from a coffeehouse reads as follows:
Eight friends go out for drinks. Three order cappucino, two order espresso, one orders Americano, one orders Turkish coffee, and one orders cafe latte. One of the eight agrees to leave the entire amount of the tip, which is to be 20% of the check. How much will be the amount of the tip (round to the nearest dime)?
The costs of the beverages will be as follows:
Add these amounts:
20% of this is
, which rounds to \$5.30.
The costs of the beverages will be as follows:
Add these amounts:
20% of this is
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A pair of fair dice are tossed. Which is the greater quantity?
(a) The probability that the sum will be at least eight
(b) The probability that the sum will be at most six
A pair of fair dice are tossed. Which is the greater quantity?
(a) The probability that the sum will be at least eight
(b) The probability that the sum will be at most six
The following table illustrates the ways that the dice can come up. The orange cells indicate a roll of 6 or less; the pink cells indicate a roll of eight or more.
Since each of these outcomes are equally likely, and there are fifteen outcomes that are favorable to each of the two events, the probabilities are the same.
The following table illustrates the ways that the dice can come up. The orange cells indicate a roll of 6 or less; the pink cells indicate a roll of eight or more.
Since each of these outcomes are equally likely, and there are fifteen outcomes that are favorable to each of the two events, the probabilities are the same.
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Refer to the above circle graph. Which of these quantities is greater?
(a) The number of votes won by Hawley
(b) The number of votes won by Wells
Refer to the above circle graph. Which of these quantities is greater?
(a) The number of votes won by Hawley
(b) The number of votes won by Wells
The gray portion of the circle graph represents Wells; the yellow portion represents Hawley. The gray portion is larger, so Wells won more votes, making (b) greater.
The gray portion of the circle graph represents Wells; the yellow portion represents Hawley. The gray portion is larger, so Wells won more votes, making (b) greater.
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Refer to the above circle graph. Which of these quantities is greater?
(a) The number of students who voted for Phelps
(b) The number of students who voted for candidates other than Phelps
Refer to the above circle graph. Which of these quantities is greater?
(a) The number of students who voted for Phelps
(b) The number of students who voted for candidates other than Phelps
The portion of the circle graph representing Phelps is light blue; it takes up less than half of the graph, so fewer than half of the votes were won by Phelps. (b) is correct.
The portion of the circle graph representing Phelps is light blue; it takes up less than half of the graph, so fewer than half of the votes were won by Phelps. (b) is correct.
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The following is a portion of the menu at Moonbucks Coffee:
Which is the greater quantity?
(a) The price of two lattes and a cappucino
(b) The price of two Americanos and an espresso
The following is a portion of the menu at Moonbucks Coffee:
Which is the greater quantity?
(a) The price of two lattes and a cappucino
(b) The price of two Americanos and an espresso
(a) Add: 
(b) Add: 
(a) is the greater quantity.
(a) Add:
(b) Add:
(a) is the greater quantity.
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