1
If
is an even integer, then
could be which of the following?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer B :
Choice (B) is correct. The sum of any even number and
since
options is
is even, it follows that
so
must be odd. Therefore,
must be odd. The only odd integer among the
could be
Explanation for Incorrect Answer A :
Choice (A) is not correct. If
even. Thus,
then
would be
which is odd, not
could not be
Explanation for Incorrect Answer C :
Choice (C) is not correct. If
even. Thus,
then
would be
which is odd, not
would be
which is odd, not
could not be
Explanation for Incorrect Answer D :
Choice (D) is not correct. If
even. Thus,
then
could not be
Explanation for Incorrect Answer E :
Choice (E) is not correct. If
even. Thus,
2
then
would be
which is odd, not
could not be
A certain building has
square feet of surface that needs to be painted. If
gallon of paint will cover
square feet, what is the least whole number of gallons
that must be purchased in order to have enough paint to apply one coat to the
surface? (Assume that only whole gallons can be purchased.)
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer C :
Choice (C) is correct. This problem can be solved by dividing the total number of
square feet to be painted by the number of square feet that can be painted with
gallon:
Since you are asked to find the least whole number of gallons
that must be purchased, you would have to buy the next highest whole number of
gallons, which is
gallons.
Explanation for Incorrect Answer A :
Choice (A) is not correct. If
gallon of paint covers
paint cover only
to cover
square feet. Thus,
square feet, then
gallons of
gallons is not enough paint
square feet.
Explanation for Incorrect Answer B :
Choice (B) is not correct. If
gallon of paint covers
of paint cover only
paint to cover
square feet, then
square feet. Thus,
gallons
gallons is not enough
square feet.
Explanation for Incorrect Answer D :
Choice (D) is not correct. It is true that
feet, but only
gallons are needed. Thus,
of gallons that must be purchased.
gallons of paint could cover
square
gallons is not the least whole number
Explanation for Incorrect Answer E :
Choice (E) is not correct. It is true that
feet, but only
gallons are needed. Thus,
of gallons that must be purchased.
gallons of paint could cover
square
gallons is not the least whole number
3
On a report, the typing begins
inch from the left edge of the paper and ends
inches from the right edge. If the width of the paper is
per line are used for typing?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
inches, how many inches
Explanation for Correct Answer E :
Choice (E) is correct. The width devoted to typing is the width of the paper,
inches, minus the widths of the left and the right margins, or
inches.
Explanation for Incorrect Answer A :
Choice (A) is not correct.
inches is the width of the paper minus the width of the
left margin only. The number of inches per line used for typing is the width of the
paper minus the widths of both the left and the right margins.
Explanation for Incorrect Answer B :
Choice (B) is not correct.
inches is too much by
inches.
Explanation for Incorrect Answer C :
Choice (C) is not correct. inches is the width of the paper minus the width of the
right margin only. The number of inches per line used for typing is the width of the
paper minus the widths of both the left and the right margins.
Explanation for Incorrect Answer D :
Choice (D) is not correct.
4
Brenda received pledges from
inches is too much by
people for a
inch.
mile bike-a-thon. If Brenda rode
miles and each person gave
for each mile she rode, which of the following
expressions gives the total dollar amount of money Brenda collected?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer A :
Choice (A) is correct. Since Brenda rode
miles and each person gave
each mile she rode, it follows that each person gave
were
for
dollars. Since there
people, the total number of dollars that were given was
Explanation for Incorrect Answer B :
Choice (B) is not correct. The total amount of money that Brenda collected was
dollars. The value of
is
Brenda did
not collect
dollars; she collected
dollars.
Explanation for Incorrect Answer C :
Choice (C) is not correct. The total amount of money that Brenda collected was
dollars. The value of
collect
dollars; she collected
is
Brenda did not
dollars.
Explanation for Incorrect Answer D :
Choice (D) is not correct. The total amount of money that Brenda collected was
dollars. The value of
collect
dollars; she collected
is
Brenda did not
dollars.
Explanation for Incorrect Answer E :
Choice (E) is not correct. The total amount of money that Brenda collected was
dollars. The value of
collect
5
dollars; she collected
is
Brenda did not
dollars.
During a sale at a music store, if a customer buys one tape at full price, the customer
is given a
percent discount on a second tape of equal or lesser value. If Linda buys
two tapes that have full prices of
two tapes reduced during the sale?
and
by what percent is the total cost of the
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer B :
Choice (B) is correct. This question involves calculating the percent decrease, which is
given by
discount is
The full price for both tapes before the
The
tape is discounted, because the discounted tape must be of
equal or lesser value. With the
reduction of
discount, the
tape costs
This is a
Therefore, the percent decrease for the total purchase is
Explanation for Incorrect Answer A :
Choice (A) is not correct. The reduction in price is equal to
decrease.
Explanation for Incorrect Answer C :
but this is not a
Choice (C) is not correct. The reduction in price would be
had the same value.
only if the two tapes
Explanation for Incorrect Answer D :
Choice (D) is not correct. This answer corresponds to a
discount for the more
expensive tape, but only the tape of lesser value will be discounted.
Explanation for Incorrect Answer E :
Choice (E) is not correct. This answer corresponds to a
but only the tape of lesser value will be discounted.
6
If
microns equal
centimeter and
discount for both tapes,
angstrom units equal
centimeter, how many angstrom units equal
micron?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer C :
Choice (C) is correct. Since
centimeter equals both
angstrom units, it follows that
angstrom units. Therefore,
microns and
microns are equal to
micron is equal to
angstrom units.
Explanation for Incorrect Answer A :
Choice (A) is not correct. A micron is larger than an angstrom unit, so the answer
must be greater than
Explanation for Incorrect Answer B :
Choice (B) is not correct. This is the number of microns that are equal to
unit.
angstrom
Explanation for Incorrect Answer D :
Choice (D) is not correct.
is too large by a factor of
Explanation for Incorrect Answer E :
Choice (E) is not correct.
correct answer is
is
but the
7
In a hotel, all
rooms are occupied by either
occupied by more than
person are occupied by
person and
or
people. If
of the rooms are
of the rooms that are occupied by more than
people, how many rooms are occupied by
people?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer D :
Choice (D) is correct. Since
follows that
rooms are occupied by more than
rooms are occupied by
be occupied by
of the rooms are occupied by more than
people, it follows that
person, it
person. Since
of these
of these
rooms, or
must
people.
Explanation for Incorrect Answer A :
Choice (A) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer B :
Choice (B) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer C :
Choice (C) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer E :
Choice (E) is not correct. This is the number of rooms that are occupied by more than
person. According to the problem, more than
person could mean either
people. The problem asks for the number occupied by
8
or
people.
If the product of three consecutive integers written in increasing order equals the
middle integer, what is the least of the three integers?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer D :
Choice (D) is correct. The product of the three consecutive integers starting with
Since the product equals the middle integer,
Another way to solve the problem is to set up an equation. Let
middle number; then
divide both sides by
to
If
represent the
is not equal to
The resulting equation is
Because there are no integers
is
this is correct.
you can
which simplifies
that satisfy this equation,
is the
only integer solution to
Explanation for Incorrect Answer A :
Choice (A) is not correct. The product of the three consecutive integers starting with
is
which does not equal the middle integer,
the correct choice.
Therefore,
is not
Explanation for Incorrect Answer B :
Choice (B) is not correct. The product of the three consecutive integers starting with
is
which does not equal the middle integer,
correct choice.
Therefore,
is not the
Explanation for Incorrect Answer C :
Choice (C) is not correct. The product of the three consecutive integers starting with
is
correct choice.
which does not equal the middle integer,
Therefore,
is not the
Explanation for Incorrect Answer E :
Choice (E) is not correct. The product of the three consecutive integers starting with
is
which does not equal the middle integer,
not the correct choice.
9
If
is divided by
the remainder is
What is the remainder if
Therefore,
is
is divided by
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer C :
Choice (C) is correct. Since the remainder is
quotient
written as
such that
Thus,
when
is divided by
there is some
which can be
Therefore, the remainder when
is
divided by
is
Explanation for Incorrect Answer A :
Choice (A) is not correct. If
value of
is
is divided by
the remainder is
The remainder when
so one possible
is divided by
is
not
Explanation for Incorrect Answer B :
Choice (B) is not correct. If
value of
is
is divided by
the remainder is
The remainder when
so one possible
is divided by
is
not
Explanation for Incorrect Answer D :
Choice (D) is not correct. If
value of
is
is divided by
the remainder is
The remainder when
so one possible
is divided by
is
not
Explanation for Incorrect Answer E :
Choice (E) is not correct. If
value of
is
is divided by
the remainder is
The remainder when
so one possible
is divided by
is
not
10
Which of the following has the same effect as dividing by
and then multiplying by
(A)
Multiplying by
(B)
Multiplying by
(C)
Dividing by
(D)
Dividing by
(E)
Dividing by
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer A :
Choice (A) is correct. If a number
is divided by
this number is then multiplied by
the result is
Therefore, dividing
by
the result is
and then multiplying the result by
If
has the same effect
as multiplying
by
Explanation for Incorrect Answer B :
Choice (B) is not correct. Multiplying
by
has the same effect as multiplying
by
and then dividing the result by
Explanation for Incorrect Answer C :
Choice (C) is not correct. Dividing
by
has the same effect as multiplying
by
and then dividing the result by
Explanation for Incorrect Answer D :
Choice (D) is not correct. See the explanation for the correct response (A).
Explanation for Incorrect Answer E :
Choice (E) is not correct. See the explanation for the correct response (A).
11
A certain computer chip processes
million instructions per second. Which of the
following is another way of writing the number of instructions per second that the
computer chip can process?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer C :
Choice (C) is correct. One million is
Therefore,
million is
or, in scientific notation,
or
Explanation for Incorrect Answer A :
Choice (A) is not correct.
or
million.
Explanation for Incorrect Answer B :
Choice (B) is not correct.
Explanation for Incorrect Answer D :
or
million.
Choice (D) is not correct.
or
billion.
Explanation for Incorrect Answer E :
Choice (E) is not correct.
or
billion.
12
In the sequence above, the first term is
and each term after the first is
previous term. Which of the following expresses the
is a positive integer?
times the
term of this sequence, where
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer B :
Choice (B) is correct. This sequence is an example of a geometric sequence. The ratio
of adjacent terms is constant. The
as
where
term of any geometric sequence can be written
is the constant ratio of adjacent terms, and
the sequence. Since we know the first term of the sequence (it’s
constant ratio (it’s
), the
is the first term of
) and we know the
term can be written as
Explanation for Incorrect Answer A :
Choice (A) is not correct. If the
first term would be
term of the sequence were
then the
but the first term is
Explanation for Incorrect Answer C :
Choice (C) is not correct. If the
term would be
term of the sequence were
but the second term is
then the second
Explanation for Incorrect Answer D :
Choice (D) is not correct. If the
term would be
term of the sequence were
then the first
but the first term is
Explanation for Incorrect Answer E :
Choice (E) is not correct. If the
first term would be
term of the sequence were
then the
but the first term is
13
In the rectangular box above, how many different paths are there of total length
from
to
along the edges?
(A)
Three
(B)
Four
(C)
Five
(D)
Six
(E)
Eight
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer D :
Choice (D) is correct. The box has edges of lengths
and
Since the box is
rectangular, the edges parallel to the edges with lengths marked
same lengths. Also, each path of length
length
one of length
(it is true that
to
from
and one of length
have the
must contain one edge of
to have total length
because
but there are clearly no paths from
consisting of three edges of length
the first edge of the path beginning at
to
and
). There are
possible edges to choose for
Then, from any of the three vertices
reached, there are possible edges for the second edge of the path (namely, the
edges that have a different length from that of the first edge used). Finally, from each
of the vertices reached, there is only
edge possible to complete the path to
Therefore, the total number of different paths of length
from
to
is
Explanation for Incorrect Answer A :
Choice (A) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer B :
Choice (B) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer C :
Choice (C) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer E :
Choice (E) is not correct. See the explanation for the correct response (D).
14
If
and
are integers,
, and
which of the following must be true?
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer E :
Choice (E) is correct. Since
and
it follows that
and
signs: one positive, the other negative. Therefore, the product of
have opposite
and
is negative:
Explanation for Incorrect Answer A :
Choice (A) is not correct. The statement
could be false. For example, if
and
is not true.
then
and
but
Explanation for Incorrect Answer B :
Choice (B) is not correct. The statement
and
then
and
but
could be false. For instance, if
is not true.
Explanation for Incorrect Answer C :
Choice (C) is not correct. The statement
implies that
Therefore,
is equal to, not less than,
Explanation for Incorrect Answer D :
Choice (D) is not correct. The statement
and
, then
and
could be false. For instance, if
but
is not true, because
15
Set
is the set of positive even integers less than
squares of the members of set
Set
is the set of the
How many integers are members of both set
and
set
(A)
(B)
(C)
(D)
(E)
ANSWERS AND EXPLANATIONS
Explanation for Correct Answer A :
Choice (A) is correct. Set
is
. Set
is
which is
The
question asks for the number of integers that are members of both set
that is, the number of members in the intersection of set
have to be less than
these criteria:
set
and set
perfect squares, and even. There are
and
Thus, there are
and set
These integers
integers that meet
members in the intersection of
and set
Explanation for Incorrect Answer B :
Choice (B) is not correct. See the explanation for the correct response (A).
Explanation for Incorrect Answer C :
Choice (C) is not correct. See the explanation for the correct response (A).
Explanation for Incorrect Answer D :
Choice (D) is not correct. See the explanation for the correct response (A).
Explanation for Incorrect Answer E :
Choice (E) is not correct. See the explanation for the correct response (A).
16
In the sequence of integers above, each term after the first term,
integer that is greater than
value of
Your Response:
is the least
of the immediately preceding term. What is one possible
Correct Response(s): 114 or 115 or 116 or 117
Explanation:
The possible correct answers are
values of
than
or
By the rule for generating the sequence,
it follows that
is the least integer greater
Therefore,
From these values for
so
you can find the possible values for
could be
If
is
so
could be
or
If
Therefore,
so
could be
or
Therefore,
On a certain map, a distance of
miles are represented by
or
then
Therefore,
could be any of the four integers
17
. First, find the possible
is
then
and
miles is represented by
centimeter. How many
centimeters on this map?
Your Response:
Correct Response(s): 82.5
Explanation:
18
The correct answer is
. Each centimeter represents
represents
miles.
miles, so
centimeters
How many different assignments of campers, Alice, Barbara, and Connie, to
different tents are possible if each tent must contain at least one of these campers?
Your Response:
Correct Response(s): 6
Explanation:
The correct answer is . Since each tent must contain at least one camper, one
camper is assigned to one tent and two campers are assigned to the other tent. Let
and
stand for Alice, Barbara, and Connie, respectively. If there is only one
camper in the first tent, then
assignments are possible.
With two campers in the first tent, there are likewise
Therefore, a total of
19
different assignments are possible.
In an election, a total of
Candidate
received
and candidate
assignments possible.
votes were cast for the three candidates,
votes. If candidate
received more votes than candidate
received more votes than candidate
of votes that candidate
and
what is the least number
could have received?
Your Response:
Correct Response(s): 2051
Explanation:
The correct answer is
together candidate
votes than
. Since candidate
and candidate
it follows that
If
votes. Since
the least number of votes candidate
received is
20
received
of the
votes,
received more
must have received more than half of those
votes. Since
than
received
(In this case,
received
could have
votes, which is more
received.)
percent of
is equal to
percent of
, then
is equal to what number?
Your Response:
Correct Response(s): 4
Explanation:
The correct answer is
. To find the value of
equation. Since percent means "per hundred,"
and
translates into the equation
percent of
translate the given sentence into an
percent of
is
and solving for
is
Therefore, the sentence
gives