How to find the missing part of a list - ISEE Upper Level Quantitative Reasoning
Card 1 of 208
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The nine hundred ninety-ninth term of the sequence
(b) The one-thousandth term of the sequence
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The nine hundred ninety-ninth term of the sequence
(b) The one-thousandth term of the sequence
Tap to reveal answer
The common ratio is 
, which means that the terms will alternate in sign. The first term is negative; therefore, all of the odd-numbered terms will be negative and all of the even-numbered terms will be positive. This makes the one thousandth term greater than the nine hundred ninety-ninth.
The common ratio is
← Didn't Know|Knew It →
A class has 25 students. If 60% of them are boys, how many students are girls?
A class has 25 students. If 60% of them are boys, how many students are girls?
Tap to reveal answer
If 60% of the students are boys, 40% are girls (100 – 60 = 40). Multiply 25 by 40% (25 * 0.4 = 10); therefore, 10 students are girls.
If 60% of the students are boys, 40% are girls (100 – 60 = 40). Multiply 25 by 40% (25 * 0.4 = 10); therefore, 10 students are girls.
← Didn't Know|Knew It →
A pair of fair dice are tossed. Which is the greater quantity?
(a) The probability that the product of the numbers will be at least 
.
(b) 
A pair of fair dice are tossed. Which is the greater quantity?
(a) The probability that the product of the numbers will be at least
(b)
Tap to reveal answer
Out of a possible thirty-six rolls, the following result in a product of 
or greater:
This is ten equally probable rolls out of thirty-six, resulting in a probability of
.
Since 
, (b) is the greater quantity.
Out of a possible thirty-six rolls, the following result in a product of
This is ten equally probable rolls out of thirty-six, resulting in a probability of
Since
← Didn't Know|Knew It →
Which is the greater quantity?
(a) The number of odd integers 
such that 
(b) The number of even integers 
such that 
Which is the greater quantity?
(a) The number of odd integers
(b) The number of even integers
Tap to reveal answer
This question can be most easily answered by matching each element in the set in (a) with the next consecutive integer, which is in the set in (b):
...
Every element in the second set has a match, but there is an unmatched element in the first set. Therefore (a) is the greater quantity.
This question can be most easily answered by matching each element in the set in (a) with the next consecutive integer, which is in the set in (b):
...
Every element in the second set has a match, but there is an unmatched element in the first set. Therefore (a) is the greater quantity.
← Didn't Know|Knew It →
An arithmetic sequence begins as follows:
Which is the greater quantity?
(a) The fourth term of the sequence
(b) 200
An arithmetic sequence begins as follows:
Which is the greater quantity?
(a) The fourth term of the sequence
(b) 200
Tap to reveal answer
The common difference of the sequence is 
, so the next two terms of the sequence are:
215 is the fourth term. This makes (a) greater.
The common difference of the sequence is
215 is the fourth term. This makes (a) greater.
← Didn't Know|Knew It →
Which quantity is greater?
(a) The number of even integers 
that satisfy the inequality 
(b) The number of multiples of 4 
that satisfy the inequality 
Which quantity is greater?
(a) The number of even integers
(b) The number of multiples of 4
Tap to reveal answer
The easiest way to answer this is to try to match each element in the first set to one the second set as follows:
...
In other words, each element in the set in (a) is paired with the element in the set in (b) that is its double. Since there is a one-to-one correspondence, the two sets are of equal aize, and (a) and (b) are equal quantities.
The easiest way to answer this is to try to match each element in the first set to one the second set as follows:
...
In other words, each element in the set in (a) is paired with the element in the set in (b) that is its double. Since there is a one-to-one correspondence, the two sets are of equal aize, and (a) and (b) are equal quantities.
← Didn't Know|Knew It →
Which quantity is greater?
(a) The number of even integers that satisfy the inequality 
(b) The number of odd integers that satisfy the inequality 
Which quantity is greater?
(a) The number of even integers
(b) The number of odd integers
Tap to reveal answer
The easiest way to answer this is to try to match each element in the first set to one in the second set as follows:
...
Since there is a one-to-one correspondence between the elements of the two sets, (a) and (b) are equal.
The easiest way to answer this is to try to match each element in the first set to one in the second set as follows:
...
Since there is a one-to-one correspondence between the elements of the two sets, (a) and (b) are equal.
← Didn't Know|Knew It →
A pair of fair dice are rolled. Which is the greater quantity?
(a) The probability that at least one die comes up 5 or 6
(b) 
A pair of fair dice are rolled. Which is the greater quantity?
(a) The probability that at least one die comes up 5 or 6
(b)
Tap to reveal answer
For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are 
ways out of 36 that this can happen, so there are 
ways for either or both of the two dice to be 5 or 6. Since half of 36 is 18, the probability of this event is greater than 
.
For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are
← Didn't Know|Knew It →
Which of the following is the greater quantity?
(a) The sum of the even integers from 
to 
(b) Twice the sum of the integers from 
to 
Which of the following is the greater quantity?
(a) The sum of the even integers from to
(b) Twice the sum of the integers from to
Tap to reveal answer
The quantities are equal. This can be proved as follows:
The sum of the integers from to is
.
(b) is twice this:
This is the same value as (a), the sum of the even integers from to .
The quantities are equal. This can be proved as follows:
The sum of the integers from to
is
(b) is twice this:
This is the same value as (a), the sum of the even integers from to
.
← Didn't Know|Knew It →
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fourth term of the sequence
(b) The sixth term of the sequence
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fourth term of the sequence
(b) The sixth term of the sequence
Tap to reveal answer
The common ratio of the geometric sequence is
The next four terms of the sequence are:
- the fourth term
- the sixth term
, so the fourth term, which is (a), is greater
The common ratio of the geometric sequence is
The next four terms of the sequence are:
← Didn't Know|Knew It →
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fourth element of the sequence
(b) 30
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fourth element of the sequence
(b) 30
Tap to reveal answer
The common ratio of the sequence is
The next two terms of the sequence can be found as follows:
This is the fourth term, which is greater than 30.
The common ratio of the sequence is
The next two terms of the sequence can be found as follows:
This is the fourth term, which is greater than 30.
← Didn't Know|Knew It →
An arithmetic sequence begins as follows:
Which of the following is the greater quantity?
(a) The tenth element of the sequence
(b) 70
An arithmetic sequence begins as follows:
Which of the following is the greater quantity?
(a) The tenth element of the sequence
(b) 70
Tap to reveal answer
The common difference of the sequence is 
; then tenth element is
,
which is greater than 70.
The common difference of the sequence is
which is greater than 70.
← Didn't Know|Knew It →
A standard deck of cards is altered by removing the red queens and the three of the jacks. A card is drawn at random from this altered deck.
Which is the greater quantity?
(a) The probability of drawing a red card
(b) The probability of drawing a black card
A standard deck of cards is altered by removing the red queens and the three of the jacks. A card is drawn at random from this altered deck.
Which is the greater quantity?
(a) The probability of drawing a red card
(b) The probability of drawing a black card
Tap to reveal answer
The three removed jacks must either comprise two red cards and one black card, or one red card and two black cards. Since the two red queens are also removed, the net result is either the removal of four red cards and one black card, or three red cards and two black cards. Either way, more black cards than red cards are left in the altered deck, and (b) must be greater.
The three removed jacks must either comprise two red cards and one black card, or one red card and two black cards. Since the two red queens are also removed, the net result is either the removal of four red cards and one black card, or three red cards and two black cards. Either way, more black cards than red cards are left in the altered deck, and (b) must be greater.
← Didn't Know|Knew It →
An arithmetic sequence begins as follows:
'
Which is the greater quantity?
(a) The fifth number in the sequence
(b) 
An arithmetic sequence begins as follows:
Which is the greater quantity?
(a) The fifth number in the sequence
(b)
Tap to reveal answer
The common difference of the sequence is
.
The fifth number in the sequence is
.
This makes (b) greater.
The common difference of the sequence is
The fifth number in the sequence is
This makes (b) greater.
← Didn't Know|Knew It →
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fifth term of the sequence
(b) 
A geometric sequence begins as follows:
Which is the greater quantity?
(a) The fifth term of the sequence
(b)
Tap to reveal answer
The common ratio of the sequence is
.
The next three terms of the sequence are:
, the fifth term.
, making (a) greater.
The common ratio of the sequence is
The next three terms of the sequence are:
← Didn't Know|Knew It →
An arithmetic sequence begins as follows:
The Fibonacci sequence begins
,
with each subsequent term being the sum of the previous two.
Which is the greater quantity?
(a) The tenth term of the arithmetic sequence
(b) The tenth term of the Fibonacci sequence
An arithmetic sequence begins as follows:
The Fibonacci sequence begins
with each subsequent term being the sum of the previous two.
Which is the greater quantity?
(a) The tenth term of the arithmetic sequence
(b) The tenth term of the Fibonacci sequence
Tap to reveal answer
(a) The arithmetic sequence has common difference 
; its tenth term is 
.
(b) By beginning with 
and taking the sum of the previous two terms to get each successive term, we can generate the Fibonacci sequence:
Therefore the two are equal.
(a) The arithmetic sequence has common difference
(b) By beginning with
Therefore the two are equal.
← Didn't Know|Knew It →
The Fibonacci sequence begins
,
with each subsequent term being the sum of the previous two.
Which is the greater quantity?
(a) The sum of the seventh, eighth, ninth, and tenth terms of the Fibonacci sequence
(b) The twelfth term of the Fibonacci sequence
The Fibonacci sequence begins
with each subsequent term being the sum of the previous two.
Which is the greater quantity?
(a) The sum of the seventh, eighth, ninth, and tenth terms of the Fibonacci sequence
(b) The twelfth term of the Fibonacci sequence
Tap to reveal answer
By beginning with 
and taking the sum of the previous two terms to get each successive term, we can generate the Fibonacci sequence:
The seventh through tenth terms, added together, yield a sum of
,
which is less than the twelfth term, 
.
By beginning with
The seventh through tenth terms, added together, yield a sum of
which is less than the twelfth term,
← Didn't Know|Knew It →
Twelve candidates are running for four vacancies on the school board. Six of the candidates are Democrats, and six are Republicans. Samantha wants to vote for two Democrats and two Republicans.
Which is the greater quantity?
(a) The number of ways Samantha can cast her ballot so that she votes for two Democrats and two Republicans
(b) 
Twelve candidates are running for four vacancies on the school board. Six of the candidates are Democrats, and six are Republicans. Samantha wants to vote for two Democrats and two Republicans.
Which is the greater quantity?
(a) The number of ways Samantha can cast her ballot so that she votes for two Democrats and two Republicans
(b)
Tap to reveal answer
The number of ways Samantha can choose two Democrats out of the six is the number of combinations of 2 chosen from a set of 6:
This is also the number of ways Samantha can choose two Republicans out of the six.
These are two independent events, so we multiply them to get the number of total ways Samantha can vote with these specifications:
Therefore (b) is greater.
The number of ways Samantha can choose two Democrats out of the six is the number of combinations of 2 chosen from a set of 6:
This is also the number of ways Samantha can choose two Republicans out of the six.
These are two independent events, so we multiply them to get the number of total ways Samantha can vote with these specifications:
Therefore (b) is greater.
← Didn't Know|Knew It →
The first six terms of a sequence are:
Which is the greater quantity?
(a) The tenth term of the sequence
(b) 
The first six terms of a sequence are:
Which is the greater quantity?
(a) The tenth term of the sequence
(b)
Tap to reveal answer
The terms are generated by alternately subtracting two and adding two. The next four terms are:
Therefore (b) is greater.
The terms are generated by alternately subtracting two and adding two. The next four terms are:
Therefore (b) is greater.
← Didn't Know|Knew It →
The Student Council comprises thirty students, from which four are chosen to fill the offices of President, Vice-President, Secretary, and Treasurer. Which is the greater quantity?
(a) The number of ways the four offices can be filled
(b) 
The Student Council comprises thirty students, from which four are chosen to fill the offices of President, Vice-President, Secretary, and Treasurer. Which is the greater quantity?
(a) The number of ways the four offices can be filled
(b)
Tap to reveal answer
(a) There are 30 ways to fill the office of President. Afterward, there will be 29 ways to fill the office of Vice-President, 28 ways to fill the office of Secretary, and 27 ways to fill the office of Treasurer. Multiply:
(b) 
This makes (b) greater.
(a) There are 30 ways to fill the office of President. Afterward, there will be 29 ways to fill the office of Vice-President, 28 ways to fill the office of Secretary, and 27 ways to fill the office of Treasurer. Multiply:
(b)
This makes (b) greater.
← Didn't Know|Knew It →