on 5
1
length.
, sothan the
If feet less
which
of the following. could be a value of x ?
Explanation for
for Incorrect
Incorrect Answer
Answer E
C ::
Explanation
on 6
on 7
(A)(E)
Choice
(C) is
is not
not correct.
correct. See
Although
it is true that
some
integers
in set (B).are even,
Choice
the explanation
for Explanations
the correct
response
Test
Math
Answers
and
is it8 isfeet,
what9:
is the
areathat
of
the
rug in square
feet?
not
necessarily
true
all
integers
in
set
are
even.
(B)
on 8
on 9
on 10
4
3
(C)
Explanation
(A) 16 for Incorrect Answer D :
(D)
Some is
integers
in set X are
even. official explanations to the math
The following
the
College
(B)(D)
48is not
Choice
correct.
Since Boardʼs
some integers
in set
are even, it cannot be true
(E)in sample Test 9.
questions
that (C)
all integers
in
set
are
odd.
66
If
and
what is the value of 2r?
(D)
80
If the statement above is true, which of the following must also be true?
A NSWERS
AND E XPLANATIONS
(E) 2
96
(A)
Math (B)
Section
2
4
Explanation for Correct Answer A :
5
(C) has
6 a perimeter of 13 and one side of length 3. If the lengths of the other
A triangle
A NSWERS
E XPLANATIONS
(A) IfAND
an integer
is even, it is ,inthen
set X.
Choice (A) is correct. If
. The solutions to this
1. two
sides
(D) are
8 equal, what is the length of each of them?
(B) If an integer is odd, it is in set X..
equation
are
and
. So
could be
Explanation
(E) 12 for Correct Answer B :
6
2.
(C) All integers in set X are even.
(A) 4
Choice (B) is correct. The area of the rug is equal to the length of the rug times
(D) All integers in set X are odd.
(B) 5of the rug. Since its length is
the width
feet, and its length is
feet more than
A NSWERS
AND
XPLANATIONS
(E) Not
allEintegers
in set X are odd.
its width,
(C) 6the width of the rug is . So the area of the rug is
feet times
feet,(D)
which
square feet.
Explanation
Answer
7 is for
Explanation
for Incorrect
Correct Answer
EB
: :
(E)(B)8 AND
Choice
is notEcorrect.
The value
A NSWERS
XPLANATIONS
and of
3. Choice (E) is correct. If
could
, thennot be
, because. Since
,
.
it follows that
.
Explanation for Correct Answer E :
A NSWERS AND E XPLANATIONS
Ain:: set
Choice (E) is correct.
If someAnswer
integersC
for Incorrect
4. Explanation
are even, then those even integers
aregrid
members
ofshows
set
that
are notB
odd.
So not
all integers
in set
odd.
The
abovefor
the
number
of
hours
worked
last week
by 12are
students
of
Explanation
Correct
Answer
(A) is not
See
the explanation
the
response (B).
Choice
(C)
correct.
The
value
of:
couldfor
not
becorrect
, because
various ages at after-school jobs. Which of the following is true, according to this
. Since the
Choice
(B) is correct.
perimeter
and one side has
5. grid?
Explanation
for Incorrect
Answer
C : of the triangle is
7
length
, it follows that the sum of the lengths of the other two sides is
.
Choice (C) is not
correct.
SeeAnswer
the explanation
for the correct response (B).
Explanation
for
Incorrect
D
:
Since these
twoofsidesand
are 5
ofgives
equal the
length
andresult
the sum
their lengths
is of
Squaring
the other
product
same
as of
squaring
the sum
Explanation
for
Incorrect
Answer
A : could
and
Which
of
the
following
equations
be
used
to
find
all
possible
values of
,Choice
it5.
follows
that
length
of
each
of
them
is
.
Explanation
for
Incorrect
Answer
D
:
(D) is not correct. The value of
could not be
, because
(A) Half of the students worked more than 12 hours each.
Choice (A) is not correct. Although
it is true that some integers in set
are even,
. width
Choice (D) is not correct. If the
of the rug were
feet more than its length
One student
exactly
hours.
it is (B)
not necessarily
trueworked
that every
even15
integer
belongs to set
. So there may
of
feet, then the width would be
feet, and the area of the rug would be
be some
integers
are
not in
. 11 hours.
(C)6 even
Onefor
16
year that
old worked
more
Explanation
Incorrect
Answer
Eset
: than
(A)
square
feet. However, this answer is incorrect, because the width of the rug is
Morefor
18Incorrect
year oldsAnswer
than 17Ayear
olds worked at least 13 hours each.
Explanation
feet (D)
less
the
length.
Choice
(E)than
is not
correct.
TheAnswer
value ofB :: could not be
, because
Explanation
for
Incorrect
(B)
(E) Most of the students were
under 16 years old.
.
Choice
(A)
If
the
triangle
two sides
length in set
and one
of
Explanation
for correct.
Incorrect
Answer
E true
:had that
Choice
(B) is
is not
not
correct.
Although
it is
some of
integers
areside
even,
length
, the perimeter
of the triangle
would
. But may
the be odd
it is (C)
not necessarily
true that
contains
any be
odd integers. So there
Choice (E) is not correct. See the explanation for the correct response (B).
perimeter
is are, not
not in set
. Thus,
.
integers
that
. the two equal sides cannot have length
A
NSWERS
AND E XPLANATIONS
(D)
Explanation for Incorrect Answer C :
Explanation
for Correct Answer D :
(E)
2
3
Choice (C) is not correct. If the triangle had two sides of length
and one side of
length
rectangular
rug to
is 2the
feet
more than
its width.
of the rug
Choice
(D)ofis acorrect.
According
information
in the
grid, Ifofthe
thelength
year
6. The
The
grid
above
shows
the
number
of
hours
worked
length
, the perimeter of the triangle would be
. But last
the week by 12 students of
olds worked at various
least
hours
and
ofjobs.
the Which
yearofolds
at least
ages
at each
after-school
theworked
following
is true, according to this
If perimeter
and
the
value
2r?
isAND E
, XPLANATIONS
notwhat. is
Thus,
the
twoofequal
sides cannot have length
.
A NSWERS
hours eachgrid?
each. So more
year olds than
year olds worked at least
hours each.
Explanation
for Incorrect Answer D :
Explanation
(A) 2 for Correct Answer D :
Choice
(B)(D)
4 is not correct. If the triangle had two sides of length
and one side of
(D)
is correct. Theofsquare
of the would
product
of
and
is . But ,the
and the
7. Choice
length
, the perimeter
the of
triangle
beworked more than
(A) Half
the students
12 hours each.
(C) 6
perimeter is
, not
. Thus, the two equal sides cannot have length
.
square
is
. If squaring
and
One student
worked
exactly the
15 product
hours. of
(D)of 8the sum of (B) and
Explanation for Incorrect Answer A :
(C)
One
16 the
year
than
hours.
Explanation
for
Incorrect
Answer
Esum
:oldofworked
gives
the 12
same
result
as squaring
and more
, then
the11
equation
(E)
Choice (A) is not correct. According to the information in the grid, only
of the
18this
year
olds
than
17of
year
worked
least
is (D)
true.IfMore
Solving
equation
would
give
allolds
possible
of of 13 hours each.
Choice (E) is not correct.
the triangle
had
two
sides
length
and values
oneat
side
students worked more than
hours each. So it is not true that half of the
(E) ofMost
of the students
old.
, the perimeter
the triangle
would be were under 16 .years
But the
.length
students
worked
than
hours each.
A NSWERS
AND Emore
XPLANATIONS
perimeter is
, not
. Thus, the two equal sides cannot have length
.
Explanation for Incorrect Answer B :
Explanation for Correct Answer E :
A NSWERS AND E XPLANATIONS
Choice (B) is not correct. According to the information in the grid, none of the
Choice (E) is correct. If
and
, then
. Since
,
Page
1 of
12
students worked Explanation
exactly
hours.
So
it is
not true that one student worked
for Correct
Explanation
A : Answer D :
it follows that for Incorrect Answer
.
exactly
hours.
Choice (D) is correct. According to the information in the grid,
of the
year
Choice (A) is not correct. The square of the sum of
and
is
, but the
Explanation forolds
Incorrect
workedAnswer
at least C : hours each and
of the
year olds worked at least
inches
true. (E)
If th 5
, of
then
, so is is not necessarily
true. Neither
statement
the
term
the sequence
. The
th term
of the
nor
must
be
true.
sequence is
. So the
th term of the sequence is greater than
.
the
th term by
.
forE
Incorrect
E :B above, which of the following must be true?
If Explanation
nA NSWERS
is a member
of
both setAnswer
A and set
AND
XPLANATIONS
Test 9: Math Answers and Explanations
8.
Explanation for Incorrect Answer C :
Choice
not correct.
If inisthe
a memberabove
of both
setcoordinates
and set
, thensuch that
Which
of(E)
theislettered
points
has
Explanation
for Correct
Answer figure
A :
I . n (C)
is anis integer.
Choice
not correct. If
is directly proportional to
, then there is a
must be
,
, or
. If
, then
is not an integer, so
is not necessarily
Choice
is integer.
correct.
foot,
a piece of wire, that
constant
such
that Since . Ifinches is equal
when to
, then
so is
II . 4n (A)
is an
Explanation
for
Incorrect
Answer
A
:
feet long is
inches long. The greatest number of
-inch pieces of wire that
III. If and , then . When , which, is
true.
not
an integer;
so
is not ,necessarily
the
value
of The
cannot
be
because
can
be
cut
from
the
-inch-long
wire
is
pieces.
total
length
of
these
Choice (A) is not correct. The number
is
greater than
the
number
, but
pieces
of term
wire, is
inches.
So
the
piece of
wire
after
cutting
the
evenly.
the
of
the sequence,
,not
is
greater
than
theleftTherefore,
th term
of the
true.
If th
then
,does
sonot divideis
necessarily
true.
none
offive
the
(A) A
. long.
-inch
pieces
is
inch
three
statements
must
be
true.
sequence,
, by
.
(B) B
Explanation
for Incorrect
Incorrect Answer
Answer D
B ::
(C) C for
Explanation
9
(A)
(D)(B)None
D not correct. The
Choice
th term proportional
of the sequence
and the
Choice
(D) is
is not correct. If
is directly
to is, then ,there
is a
(B)
II
only
(E)
E
term
of
the
sequence
is
.
The
difference
between
these
numbers
is
not
constant
such
. If
, then
, so
Explanation
forthat
Incorrect Answer
B : when
I and
II only
If y is(C)
directly
proportional
to x and if
11
when
th
.
what is the value of y
Explanation
Answer
D
: the
Choice (B)
not Incorrect
correct.
The
total
length
of
wire
th term
of
sequence
is cutting
andis for
. the
When
, the
value
ofleft after
cannot
be
pieces
. The
,-inch
because
th term of the
when (D) I and III only
wire from
a
-inchsequence
piece of wire
cannot be
inches
long. If th
it were
inches
is
. So the
term of the
sequence is greater than
AofNSWERS
XPLANATIONS
Choice
(D) AND
is notEcorrect.
The
th term of the sequence is
, and the
th
(E)then
I, the
II, and
III pieces
long,
-inch
together
would
measure
inches.
But
the
th term by
.
.
term of the sequence is
. The difference between these numbers is not
Explanation
for Correct
Answer B :
does not divide
evenly.
(A)
of the lettered points in the figure above has coordinates
Explanation for IncorrectWhich
Answer
E :
are
. Since
Choice
(B) AND
is correct.
The coordinates
NSWERS
E XPLANATIONS
9. A
Explanation
for
Incorrect
Answer of
C :point
12
13
.
such that
Choice
The
th
term
of the sequence such
is
, and the
th
(B)(E) is not correct., point
has
coordinates
that
Choice (C) is for
not Correct
correct. The
total
length
of wire left after cutting
-inch pieces .
Explanation
Answer
A
:
term of the sequence is Explanation
. The difference
between
these numbers
is not
.
for Incorrect
Answer
A :
A
of wire from a
-inch piece (A)
of wire
cannot be
inches long. If it were
inches
(C)
23
(B)
B
Choice (A) is correct. If Choice
is a (A)
member
of both set
and set
, then
must than
be
10.The
is not
The number
is inches.
greater
long,
piecesis together
would
Butmuchthe number
nth then
termthe
of a -inch
sequence
defined
tocorrect.
be measure
The 50th term
is
how
(C)
C
(D) 27
the
thDis
term
sequence,
is greater than
th term of the
does
greater
than
the
term?
, not
, ordivide
. If45thevenly.
, then (D)
not of
anthe
integer,
so
is not, necessarily
true.the
If
(E) E
(E) 30
sequence,
, by
.
Explanation
Explanation for
for Incorrect
Incorrect Answer
Answer AD::
A NSWERS
AND an
E XPLANATIONS
(A), then
5
,Explanation
which
is not
not necessarily
true. If
forinteger;
Incorrect is
Answer
B :
Choice
(A)
is
not
correct.
The
coordinates
of
point
Choice
(D)
is
not
correct.
The
total
length
of
wire
after cutting . Since
-inch pieces
Explanation for Correct
Answer
B : left are
(B)
16
A NSWERS
AND E XPLANATIONS
, then
, so Choice
is of
not
necessarily
true.
Therefore,
none
the
three
(B)
not
correct.
The
thlong.
term
the
sequence
is
, and the
of wire
from a
-inch
piece
wire
cannot
be
inches
Ifof
it of
were
inches
Choice
(B)
isis
correct.
The
coordinates
of point
are
. Since
, point
does not have coordinates
such that
(C)
20
statements
must
be
true.
,
point
has
coordinates
such
that
.
term
of
the Esequence
is
. The differenceinches.
between
these numbers is not
long, then thefor Correct
-inch pieces
together
But
Explanation
Answer
: would measure
(D) 23 .
does not divide
evenly.
for Incorrect
:
Choice
correct. If Explanation
is directly
proportional
, then D
there
is a
11. Which
(E)
24isrectangular
of(E)
the
solids
shown
above hastoaAnswer
volume
closest
toconstant
the volume of a
Explanation
for
Incorrect
Answer
A
:
Explanation
for
Incorrect
Answer
C
:
Explanation
for Incorrect
Answer
E :height 4?
right
circular cylinder
with
radius
2
and
(D)when
is not correct.
The
th term
of the sequence
, and the
such that
. IfChoice
, then
, so
and is
Choice (A) is not correct. The coordinates of point
are
. Since
term
of
the
sequence
is
.
The
difference
between
these
numbers
is not
Choice (C)
(E) is
is not
not correct.
correct. The
The
total length
of
wire does
leftnotare
after
cutting
-inch
Explanation
for
Incorrect
Answer
B : of
Choice
coordinates
point
. Since
, point
have coordinates
such
that pieces of
A NSWERS AND E XPLANATIONS
12.
.
wire from a
-inch piece
ofvalue
wire cannot
be
inches long.
If it were
inches
, Ifthe
offor
is
. and
, Explanation
point
does
notIncorrect
have
coordinates
that
Choice (B). isWhen
not correct.
is a
member
of both
set
, then
Answer
E set
: such
Explanation
for Incorrect
Answer
C:
long,
then
the
-inch
pieces
together
would
measure
inches.
But
Explanation
(A) A for Correct Answer C :
.
Choice
(C)
is
not
correct.
The
coordinates
of
point
are
.
Since
Choice (E) is not correct. The
th term of the sequence is
, and the
(B)(C)B is correct. If the th term of a, point
Choice
sequence
is defined
, then
does not have
coordinates to be
such that
term of the sequence is
. The difference between these numbers is not
Explanation
for
Answer
the
ofIncorrect
the sequence
is . D :
. The
th term of the
(C) thCterm
sequence
(D) Dis
the
. So the
, but
th
.
th
.
th
.
th term of the sequence is greater than
Explanation for Incorrect Answer D :
th term by
.
(E) E
13
A NSWERS AND E XPLANATIONS
Explanation for Incorrect Answer A :
Explanation for Correct Answer E :
Choice (A) is not correct. The number
is
greater than the number
, but
volume of a right circular cylinder with radius
and
13. Choice (E) is correct. The
of the rectangular
shownth
above
a volume closest to the volume o
the
th term of the Which
sequence,
, is greater solids
than the
term has
of the
height
is
. Since
closeradius
to
, 2the
volume
of 4?
right circular
cylinderis with
and
height
sequence,
, by
.
rectangular solid
, which is
, is slightly less than
, which
Explanation
Incorrect
B :
is
about
. for
Solids
and Answer
, with respective
volumes of
and
are too
small; solids
and
, with respective volumes of
and
, are too large.
Choice (B) is not correct. The
, and the
th
(A) Ath term of the sequence is
So the volume of the cylinder is closest to the volume of solid
.
term of the sequence is
. The difference between these numbers is not
.
(B) B
C D :
Explanation for Incorrect (C)
Answer
(D)Page
Dth term
2 of of
12the sequence is
Choice (D) is not correct. The
term of the sequence is
, and the
(E)
E difference between these numbers is not
. The
Explanation for Incorrect Answer E :
th
.
(A)
16
A NSWERS AND E XPLANATIONS
Test 9: Math Answers and Explanations
(B)
Explanation for Correct Answer D :
14.
Choice
(C)(D) is correct. If
because
(D) 1
is a negative integer, then
is negative. Since
is negative
and
are positive,
Choice
(D)that
is not correct.
unshaded itregions
it follows
andThe total. area
Sinceof the
is negative,
followsisthat
,
(E) 2 ABCD above, the area of the shaded region is given by
Inwhich
rectangle
is approximately
. The answer of
is too large.
, so
If the
. Therefore,
Choice
(D)
the to
explanation
for thewhole
correct
area
of the shaded
region is Answer
what
is not
thecorrect.
total See
area,
the nearest
Explanation
for Incorrect
E :is
. Putting
the inequalities
and
A
NSWERS
AND
E
XPLANATIONS
number, of the unshaded regions
of rectangle
ABCD?Answer E :
Explanation
for Incorrect
Choice
(E)
is not correct. The
total area of the unshaded regions is
together
gives
.
response (B).
,
Explanation
for Correct. Answer
B(E)
: of
which is approximately
TheChoice
answer
is too As
large.
is not correct.
explained for the correct response (B),
water,
mixed
by weight
in the
ratio
of and
watersalt
are are
required
to make
pounds
of the
dough. But the question
15. Choice (B) is correct. The flour,pounds
(A) , 4
respectively. So if it requires
pounds
salt to
make
pounds
of this
asks for the number
of of
pounds
of salt,
not of water,
needed
to
(B) 6
dough, then
; so
, and
make the dough.
.
(C) 8
The city
donated some children’s books to Mr. Clark’s first-grade class. If
(D)library
9
each student takes 4 books, there will be 20 books left. If 3 students do not take a
16
10 rest of the
book (E)
and the
students take 5 books each, there will be no books left. How
many books were donated to the class?
17
Explanation for Incorrect Answer A :
A NSWERS
AND E XPLANATIONS
(A) 120
16.
Choice (A) is not correct. See the explanation for the correct response (B).
(B) 140 for Correct Answer B :
Explanation
Explanation for Incorrect Answer C :
(C) 160
Choice (B) is correct. The total
area
of the
unshaded
regions
of the
rectangle
rectangle
ABCD
above, the
area of
shaded region is given by
Choice
is not correct. SeeInthe
explanation
for the correct
response
(B).
(D)(C)
is equal
to175
the area of the entire rectangle minus the area of the shaded region.
area of the shaded region is
If the
what is the total area, to the nearest whole
(E) 185 for Incorrect Answer D :
Explanation
Since the area of the shaded number,
region isof the unshaded
, it
follows
. So
regions
of that
rectangle ABCD?
the area of the rectangle is
, and the area of the shaded region is
A NSWERS
AND E XPLANATIONS
the
total area
of the unshaded regions is
. Thus,
, which is approximately
(A) 4
.
Explanation for Correct Answer(B)
C :6
8
in the class and if
17. Choice (C) is correct. If there are (C) students
books will be
(D) 9
left after each student takes
books,
then the number of books is
. If
Explanation
for
Incorrect
Answer
(E) A10:
students do not take a book and the rest take
books each, there will be no
books left.
Sonot
thecorrect.
numberThe
of books
is also
given
by
. Soisthe number
Choice
(A) is
total area
of the
unshaded
regions
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XPLANATIONS
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.too
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Explanation for Incorrect Answer C :
.
Choice (B) is correct. The total area of the unshaded regions of rectangle
Choice (C) is not correct. The total
area of the unshaded regions is
,
is equal to the area of the entire rectangle minus the area of the shaded region.
which is approximately
. The answer of
is too large.
Since the area of the shaded region is
, it follows that
. So
Explanation for Incorrect Answer
the area D
of :the rectangle is
, and the area of the shaded region is
. Thus,
the
total
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, which is approximately
.
Explanation for Incorrect Answer A
: of the unshaded regions is
Choice (A) is not correct. See the explanation for the correct response (C).
Explanation for Incorrect Answer B :
Explanation for Incorrect Answer A :
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Explanation for Incorrect Answer
:
which is D
approximately
. The answer of
,
is too small.
Explanation
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C:
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explanation
for the correct
(C).
Choice (C) is not correct. The total area of the unshaded regions is
Explanation for Incorrect Answer
E :
which is approximately
. The answer of
is too large.
Choice (E) is not correct. See the
explanation
Page
3 of 12 for the correct response (C).
Explanation for Incorrect Answer D :
,
In the figure above, if line
has a slope of
what is the y-intercept of
A NSWERS AND E XPLANATIONS
Explanation for Correct Answer E :
Test 9: Math Answers and Explanations
Choice (E) is correct. If
(A) 7
19
8
For all(B)
positive
integers 18
w and y, where
(C) 9
(D) 10
is defined for all positive integers
let the operation
, to be
(E) 12
positive integers
(A) None
positive integers
equal to 4 ?
. So
greater than
, where
be defined by
, then
For how many positive integers w is
and
for all
. Therefore,
, which is more than
for infinitely many
.
(B) One
A NSWERS
AND E XPLANATIONS
(C) Two
Explanation for Correct Answer D :
(D) Four
the figure
above,
if line Answer
slope
for
Incorrect
A :of
is correct.
FromExplanation
theInfigure,
you
can
see
that has
the apoint
18. Choice
(E)(D)
More
than four
line
what is the y-intercept of
lies on
. The slope of line Choice
is
, soisthe
line
is
, correct response (E).
(A)
notequation
correct. of
See
the explanation
for the
where
-intercept of . Since
lies on , it follows that
(A) is the
Explanation
(A) 7for Incorrect Answer B :
A NSWERS AND E XPLANATIONS
. Solving this equation for
gives
.
(B) 8
Choice (B)
is not correct. See the explanation for the correct response (E).
Explanation
for Correct Answer (C)
E : 9
(B)
19. Choice (E) is correct. If
Explanation
for Incorrect Answer C :
(D) 10
is defined
for all positive integers
and
(E) 12
, where
(C)
Choice (C) is not correct. See the explanation for the correct response (E).
, to befor Incorrect Answer
, thenA :
Explanation
Explanation for Incorrect Answer D :
A NSWERS AND E XPLANATIONS
Choice
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; if the
-intercept of
were
,
(D)(A) is not correct. Line
. So See the explanation
for for
all the correct response (E).
ChoiceExplanation
(D) is notfor
correct.
Correct
Answer
D
:
then
would have the equation
. But then the point
would
positive integers
greater than . Therefore,
for infinitely many
not (E)
be on .
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lies on
positive integers
, which is more than
.
line
. The slope of line
Explanation for Incorrect Answer
B is
: the
where
Choice (B) is not correct. Line
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A NSWERS AND 20
E XPLANATIONS
then
would have the equation
is
, so the equation of line
-intercept of
. Since
. Solving this equation for
; if the
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gives
-intercept of
. But then the point
is
,
, it follows that
were
.
,
would
Explanation
for Correct
Incorrect
Answer
not be on . for
Explanation
Answer
C :A :
Explanation
for for
Incorrect
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A :
Choice
(A) is
is correct.
not
See
the
explanation
the Tina
correct
(E).
Explanation
for correct.
Incorrect
Answer
C : Karen
(C)
TheThe
probability
that
and
each
will be
assigned
figure
above
represents
four
offices
that
will bean
assigned randomly to four
20. Choice
(A) of
isemployee
not
correct.
has slope
; if the
-intercept
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were
,
office indicated with an Xemployees,
is the Choice
number
ways
this Line
can
by
the
one
per happen
office.
Ifdivided
Karen
and
Tina are
of the
four
Explanation
for correct.
Incorrect
Answer
B :
Choice (C) is not
Line
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;the
if the
-intercept of . Butwere
,point
then
would
have
equation
then
the
would
total number of ways of employees,
assigning thewhat
four offices
the four employees.
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probability
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thenthe(B)
would
have
theand
equation
. But
then
the response
point
bethe
on rightmost
.
has
leftmost
office
Tina
has
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there
are
ways
to
an
X the
? not
Choice
is not
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explanation
for
the
correct
(E).would
not be the
on other
.
assign
two offices. Similarly,
if Tina
the leftmost
and Karen
Explanation
for has
Incorrect
Answer Boffice
:
Explanation
for Incorrect
Answer
C
:
has
the rightmost
office, there
are
ways
to assign the other two offices. Thus,
Choice E
(B): is not correct. Line
has slope
; if the
-intercept of
Explanation for Incorrect
there
waysSee
to Answer
assign
Karen
andfor
Tina
offices
marked
with(E).
an X. Now
Choiceare
(C) is not correct.
the
explanation
the
correct
response
then
would have
the
equation
. But then the point
count
number
of Line
ways ofhas
assigning the; four
there are
ways to ,
Choicethe
(E)total
is not
correct.
if theoffices:
-intercept
of
were
not be slope
on .
Explanation
Incorrect
assign
the firstfor
office,
leavingAnswer
waysDto: assign the second, leaving
ways to
then
would have the equationExplanation for Incorrect
. But then
the point
would
Answer C :
assign
the
third,
leaving
waythe
to explanation
assign the fourth;
this
is response (E). ways.
Choice
(D)
is
not
correct.
See
for
the
correct
not be on .
Choice
is not
correct.
Line
hasan
slope
if thean -intercept
of
So the probability that Karen and
Tina(C)
will
each
be assigned
office ;with
X is
then
.
would have the equation
not be on
. But then the point
were
,
would
were
,
would
.
Explanation for Incorrect Answer E :
20
Choice (E) is not correct. Line
then
would have the equation
not be on
has slope
; if the
-intercept of
. But then the point
.
Explanation for Incorrect Answer A :
The figure above represents four offices that will be assigned randomly to four
Choice (A) is not correct. See the explanation for the correct response (C).
employees, one employee per office. If Karen and Tina are two of the four
employees,
what
the probability
that
Explanation
for is
Incorrect
Answer
B : each will be assigned an office indicated with
an X ?
Choice (B) is not correct. This is the probability that Karen will be assigned the
leftmost office and Tina will be assigned the rightmost office. But Karen could be
assigned the rightmost office and Tina the leftmost office. See the explanation for
Page 4 of 12
the correct response (C).
Explanation for Incorrect Answer D :
Choice (D) is not correct. See the explanation for the correct response (C).
were
would
,
on 9
A NSWERS AND E XPLANATIONS
Choice (D) is correct. Since
Explanation for Incorrect Answer E :
Explanation
Correct
D :
is notfor
divisible
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.
on 10
The figure above shows five lines. If
,
is divisible by
and
which of the following is NOT equal to 90
Choice
(E) is not
Since the angle with
measure
is supplementary to a
Test
9: correct.
Math
?
Choice Answers
(D) is correct. and
Since Explanations
is a linear function, it has the form
right angle, it follows that
.
Answers and Explanations
where
and
are constants. From the table, you have
, and thus
Test Sections
Online. - Practice
Testis #6
Choice (A)
not correct. Since
However, since
,
(B) s
,
, so
, the equationBack to Score Repor
. Since
Math Section
6
Explanation for Incorrect Answer A :
View Answers
andholds,
Explanations
so
, and thus
Section 1
(A) r
Section 2
. However,
. Therefore,
,
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and
.
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Which(C)
of the
t following is divisible by 3 and by 5 but is not divisible by 10 ?
Section24
Section 5
(D) u
(A)
(E) 30
v
(B) 35
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1
Explanation for Incorrect Answer B :
Choice (B) is not correct. Since
,
is divisible by
but not divisible
Explanation for Incorrect Answer A :
by
.
(C) 40
Choice (A) is not correct. From the table, you have
. If
were defined
A NSWERS AND E XPLANATIONS
Explanation
for
Incorrect
Answer
C
:
(D) 45
, then
. But then
would be equal to
, which is false. So
Section 7
Section 8
Section 9
Section 10
Explanation
Answer
A not
: define
(E) 60 for Correct
Choice
(C)
is
correct. Since
cannot
.
1.
,
The figure above shows five lines. If
by
.
Choice (A) is correct. Since
, and since the angle with measure
Explanation
for Incorrect Answer B :
?
angle with measure
are corresponding angles,
. Therefore,
A NSWERS AND E XPLANATIONS
Explanation for Incorrect Answer E :
equal
to
.
Choice (B) is not correct. From the table, you have
(A)
Explanation for Correct
Answer
: correct. Since
(E) isD not
as Choice
, then
. But then
(B)
2.
is divisible by
but not divisible
which of the following is NOT equal to 90
and the
is not
. If
were defined
is divisible
by isand
would ,be equal
to , which
false. .
(A) r since
.
So However,
cannot define , . is also divisible by
Choice
,
is divisible by
and
. However,
(C) (D) is correct. Since
(B) s
is not divisible by
.
Explanation
for Incorrect Answer C :
(D)
Explanation
for Incorrect
Answer
B :
(C)
t
(E) (B) is not correct.
ChoiceSince
(C) is
correct.
the table,
have
(D)
u notangle
Choice
the
withFrom
measure
is you
supplementary
to .aIf
were defined
.as
(E), then
v
3
false. So
Explanation
C:
for
Incorrect Answer A
A
NSWERS AND
E XPLANATIONS
, which is
. But then
right angle,
3.
cannot define
would be equal to
.
Explanation
for Incorrect
E with
: bymeasure
A NSWERS
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Segment
(A)(C)
Choice
is not correct.
Since AND
,Eand
since
the
(A)
, Answer
is angle
divisible
and
. and
Explanation
for Correct Answer D :
the angle with
angles,
. Since the angle with
However,
sincemeasure
,areiscorresponding
is also
divisible
bythe table,
.
(B) Segment
Choice
(E)
not
correct.
From
. If
were
Explanation
for
Correct
Answer
A : you have
Choice
(D) is is
correct.
Since
istoaabove
function,
it has
, . values of t. If the graph
The table
of the
the
function.f So
for several
measure
supplementary
alinear
rightgives
angle,values
it follows
thatform
defined
as
,Bthen
. But then
would be equal to
,
Explanation
Incorrect
Answer
: the
(C) Ray
is
a line,
of
following
defines
where
andfor are
constants.
From
the
table,
you have
, ?soangle with measure
Choice
(A)Which
is correct.
Since
, and since the
and the
Explanation for Incorrect
Answer
D
:
which is false. So
cannot define
.
with measure
are
corresponding
angles,
is not
(D)(B)
Ray
,angle
and
thus
. Since
, the
Choice
is not correct.
Since
,
is divisible
byequation
but
not divisible . Therefore,
Choice
is not correct.
Since
angle with measure
and a right
equal
. , , the
by (E)
. (D)
holds,
so to
and thus
. Therefore,
Line
angle are corresponding angles. So
.
.
Explanation for Incorrect Answer C :
A
NSWERS
XPLANATIONS
Choice
(C) AND
is notEcorrect.
Since
4.
3
,
is divisible by
but not divisible
4
Explanation for Incorrect Answer B :
by
.
Explanation for Correct Answer B :
Explanation for Incorrect
A :correct. Since the angle with measure
ChoiceAnswer
(B) is not
is supplementary to a
Explanation
for Incorrect
Answer
E :the
Choice
(B) is correct.
Rayfigure
isabove,
the set
whose
members of
areray
point
andray
all
In the
intersection
and
is
right
angle,
.
Choice
(A)the
is not
From
. If members
wereare
defined
points to
rightcorrect.
of
on
thethe
line.table,
Ray you have
is the set whose
point
Choice (E) is not correct. Since
,
is divisible by
and
.
and
all points to theExplanation
left .ofBut then
on for
the Incorrect
line. The
these
sets
, then
would
be intersection
equal
to C, :of
which
is two
false.
So is
Answer
However,
sincemembers are ,pointsis also
divisible
.
the
set whose
and
andby
all points
on the line between
cannot define
.
Choice
(C) is not correct.
Since
, and since the angle with measure
and
and
. This set of points
is segment
.
the angle with measure
are corresponding angles,
. Since the angle with
Explanation for Incorrect Answer B :
measure
is supplementary to a right angle, it follows that
. So
.
Choice (B) is not correct. From the table, you have
. If
were defined
Explanation for Incorrect Answer D :
as
, then
. But then
would be equal to , which is false.
Explanation for Incorrect Answer A :
Choice
(D)
is
not
correct.
Since
, the angle with measure
and a right
So
cannot define
.
angle
are corresponding
angles.
Somembers. are points
Choice (A) is not correct.
Segment
is the set
whose
Explanation
for
Incorrect
C:
and
and all
points
on theAnswer
line between
and
. Ray
is the set whose
members are point
and all points to the left of
on the line. Any point
Choice
(C)
is notgives
correct.
Fromofthe
table,
you have
If
were
The
table
above
values
the
function
f for several. values
of
t. Ifdefined
the graph of f
between
and
on the line is a member of segment
but not a member of
isas
a line, Which
, thenof the following defines
. But then ? would be equal to , which is
ray
. Therefore, segment
cannot be the intersection of ray
Page 5 of.12
false.. So
cannot define
and ray
Explanation
Explanation for
for Incorrect
Incorrect Answer
Answer E
C ::
Choice (E)
have members .are
If pointwereand all
(C) is
is not
not correct.
correct. From
Ray the table,
is the you
set whose
According to the graph above, if there are 6,000 registered voters aged 60 and over
in Washington County, how many registered voters are under the age of 30?
Your Response:
Test
9: Math3 Answers
Correct
Response(s):
or 6 or 9 or 12 and
Explanations
Explanation:
(A) 1,000
2,000
The (B)
correct
answer is
5
,
,
, or
. The sum of the measures of the three
(C)of 3,000
angles
a triangle is equal to
so
If
then(D) 4,000 so
Since
and
are positive integers and
Choice (B) is not correct. The values of
for which
or
are
(E) 5,000 it follows that
must be a multiple of
the values of
for which
is negative.
. So the only possible values for
are
,
,
, or
.
Explanation for Incorrect Answer D :
A NSWERS AND E XPLANATIONS
Choice (D) is not correct. For some values of
in the interval
Explanation
for
Correct
Answer
E
:
value of
is positive. However, this is not true for all values of
13
5.
the
in that
According to the graph above, if there are 6,000 registered voters aged 60 and over
interval.
For
if in Washington
, to
then
is inhow
themany
interval
,under
but you
Choice (E)
is example,
correct. According
the
graph,
of
the registered
voters
in
County,
registered
voters are
the age of 30?
Washington
are that
age
can
see fromCounty
the graph
or over.
. If
is the total number of registered
voters in Washington
County,
then
of
Choice (D)Answer
is
not correct.
Explanation
for Incorrect
E :From
(A) 1,000
have that there
are a total of
is
. the
So correct response (E), you
.
the explanation
for
registered voters, and from the graph, you
have that (B) 2,000
these
voters are of
ages
Solving (E)
for is not
gives
.ofOf
allregistered
the
voters,
Choice
correct.
For some
values
of registered
in the interval
to
of
to
. are
Sinceof
of
theages
value
3,000
is(C)
, theall
number
of registered
voters
of ages
of
, and
are under
. Thus,
of
the
voters
are
is positive.
However,
thisage
is not true
for
values
of registered
in
that
interval.
(D) 4,000
For example, if
under the age of
the graph that
to
is
you were to find the number
registered
votersfrom
, then
is .inHowever,
the interval
but ofyou
can see
;under
thisthe
is age
of
which
is the age of
.
(E) of5,000
, which includes, those
under
.
.
A NSWERS AND E XPLANATIONS
The incomplete table above
is an expense
sheet
for ECarmen’s
business trip. If her
Explanation
for Correct
Answer
:
hotel expenses 6were the same each day, what were her total expenses for Friday, in
Choice (E) is correct. According to the graph,
of the registered voters in
dollars?
(Disregard
the $ sign
when gridding
your answer.)
Explanation
for Incorrect
Answer
A :
Washington County are age
or over. If
is the total number of registered
Bernardo drives to work at an average speed of 50 miles per hour and returns along
Choice (A) is not correct. From
the
explanation
for
the correct
response
(E), you
voters in
Washington
County,
then
of
is
. So
.
the same route at an average speed of 25 miles per hour. If his total travel time is 3
have that there are a total of
registered voters, and from the graph, you
Solving for
gives in the round-trip?
. Of all the registered voters,
are of ages
hours, what is the total number
of miles
7
have that
of these registered
voters are
under the age
. Since
of
to
, and
are under age
. Thus,
of the registered voters
Your Response:
are
under
(A) Response(s):
225
is
Correct
96
.
the age
of
, the
; this isof registered
of
, whichunder
is
number
voters
the age
(B) 112.5
6.
of
is
. However, you were to find the number of registered voters under
Explanation:
(C) of100 .
the age
Based on the graph of the function f above, what are the values of x for which f(x)
The correct answer is
. If Carmen’s hotel expenses were
dollars each day,
positive?
(D) 62.5
Incorrect Answer
then her totalfor
expenses
inExplanation
dollars
forfor
the
Explanation
Incorrect
Answer
B : trip were
A :
is
. The table
(E) 50
Choice (A) is not correct. From the explanation for the correct response (E), you
gives her total expenses as
, so
. Solve this
Choice (B) is not correct. See
explanation
response
(E).from the graph, you
have the
that there
are a totalfor
of the correct
registered
voters, and
(A)
or
equation for
:
,
so
,
and
.
Carmen’s
have that
of these registered voters are under the age total
. Since
of
(B) Answer or
Explanation for Incorrect
C:
expenses
Friday,
in dollars, were
A
NSWERSfor
AND
E XPLANATIONS
is or
(C)
7.
.
, the number of registered voters under the age
Choice (C) is not correct.(D)
See the explanation for the correct response (E).
is
Explanation for Correct ofAnswer
C :. However, you were to find the number of registered voters
(E)
the age of
.
Explanation for Incorrect Answer D :
Choice (C) is correct. If Bernardo’s route to work is
miles long, and if he takes
Explanation for Incorrect Answer B :
hours to drive to work
at an
average
speed of
miles per hour, then
A NSWERS
AND
E XPLANATIONS
Choice (B) is not correct. See the explanation for the correct response (E).
. His total travel
timeforgoing
toAnswer
and from
work is
Explanation
Correct
C:
hours, so he takes
Explanation for Incorrect Answer C :
hours to drive back.
Since
he returns
alongof the
same
route,
Choice (C)
is correct.
From the graph
, you
can see
that for
Therefore,
.
any value of
(C) is
not correct.
Seeofthe explanation
for the. correct
response
strictlyChoice
between
and
value
is greater than
of (E).
., the
Solving
this equation
gives For any .value
Since
, the value of
thus
greater
than
and
than or equal
to D, :the value of
Explanation
forless
Incorrect
Answer
is
.
less than or equal to
for which
is greater than
. The total number of miles in the round-trip is
. For any other value of
between
and
. So the values of
or
inclusive, the value of
for which
is
is positive are those
.
Explanation
for Incorrect
Explanation for Incorrect
Answer
A : Answer A :
Choice (A) is not correct. For the values of
such that
, the value of
Choice (A) is not correct. See
the However,
explanation
for the
(C).
is positive.
if
, thecorrect
value of response
is negative.
Page 6 of 12
Explanation
for Incorrect
Explanation for Incorrect
Answer
B : Answer B
:
Choice (B) is not correct. See the explanation for the correct response (C). (Keep
in mind that Bernardo’s average speed for the round-trip was not the average of
under
(E) I, II, and III
Explanation:
Your Response:
The
correct
answer is
Correct
Response(s):
9 or
. If
is the number, then when
is
A NSWERS AND E XPLANATIONS
Test
Math
Answers
and Explanations
multiplied
by 9: and
the product
is then multiplied
by
, the result is
Explanation:
9
When a certain number is multiplied by
. So
and the product is then multiplied by 32,
Explanation for Correct Answer C :
The correct answer is
. If
, then
, and so
. The
the result is 60. .What
is the number?, so the number is
Therefore,
.
greatest
integer
valueIfof isfor
is
.
(C)
is correct.
an which
integer such that
, then
or
8. Choice
Choice (D) is not correct. From the explanation for the correct response (E), you
. So
, which means
couldvoters,
be true.
If from
is the
an integer
have that
there could
are a be
total of
registered
and
graph, such
you
that
, then
be
, voters
so
cannot
be true. to could
be true,
have that
of thesemust
registered
are of ages
. Since
of
Your Response:
because
could be
and
is
, in which case
. So
is
, the number of registered voters of ages of
Correct
7.5,be
15/2
could
be Response(s):
true and
could
true, but
cannot be true.
An object thrown upward from a height of h feet with an initial velocity of v feet per
What
integer you
value
of x
to is the
is greatest
. However,
were
tofor
findwhich
the number of registered voters
11
10
with
an initial velocity of
Explanation:
feet per second, then
and
in the
second
reach
height those
of
If the
is thrown upward
under will
the age
of a maximum
, which includes
under feet.
the age
of object
.
9.
6
10.
12
10
The correct answer
is formula
or gives the
. If maximum
is the number,
when
is object
formula
. The
height, then
in feet,
that the
from a height of 6 feet with an initial velocity of 32 feet per second, what will be its
multipliedheight,
by for and
the product
is then
, the result is
. So
maximum
in feet?
Explanation
Incorrect
Answer
A : multiplied by
Your Response:
will reach, so this height, in feet, is
.
Choice
is not correct. 9From the explanation for the correct answer,
could be
Correct(A)
Response(s):
. Therefore,
, so the number is
.
true, but
could also be true.
with an initial velocity of
feet per second, then
and
in the
Explanation:
Your Response:
formula
Explanation for Incorrect
Answer .BThe: formula gives the maximum height, in feet, that the object
The correct answer is
. If
, then
, and so
. The
Correct Response(s): 22
Choice
(B)
is notvalue
correct.
the
explanation
for
the
correct
answer,
cannot
greatest
integer
of From
for
which
is
.
will reach, so this height, in feet, is
.
be
true. The value of
Explanation:
must be
. Also,
could be true and
could be true.
The three angles of a triangle have measures of
and
, where
The is
correct
answer
isinteger
. IfAnswer
the object
feet,
What
the greatest
value
ofDx :is
forthrown
whichupward from a height of
11. and
Explanation
for Incorrect
y are integers,
what is one
possible
value of
with an(D)
initial
velocity
of From
feet
second, then
and answer, in the
Choice
is not
correct.
theper
explanation
for the correct
cannot
be true. Also,
12
The three angles of a triangle have measures of
and
could be true.
, where
If x
If x
and
y are a
integers,
possible
Anformula
object thrown. upward
from
height
ofishone
feet
withvalue
aninofinitial
velocity
of v feet per
The formula
gives
thewhat
maximum
height,
feet, that
the object
11
Your Response:
Explanation
for Incorrect Answer E :
Your Response:
second
will reach a maximum height of
feet. If the object is thrown upward
Correct Response(s): 9
will
reach,
so this height,3Your
inorfeet,
Response:
Correct
Response(s):
6 or is9 or 12
.
from a height of 6 feet with
an Response(s):
initial velocity
of 32 feet per second, what will be its
3 or 6 or 9 or 12
Explanation:
Based
on the
graph
theCorrect
function
f above,
what are the values of x for which f(x) is
maximum
height,
inof
feet?
Explanation:
Explanation:
positive?
12.
The correct answer is
. If
, then
, and so
. The
The correct answer is
, The ,correct
, or
of theof three
answer is. The
,
, sum
, or of .the
The measures
sum of the measures
the three
greatest integer value of angles
forofwhich
is
.
a triangle is equal to
so
If
angles of a triangle is equal to
so
If
then Response:so
Your
then
so
Since
and
it follows
that
Since
(A) angles of a or
The three
triangle
. So have
the only measures
possible values of
for
12
it follows 22
that
Correct Response(s):
or is one possible value of
and y(B)
are integers, what
. So the only possible values for
Explanation:
(C)
or
are
,
and
are positive integers and
are positive integers
must and
be a multiple of
,
are
, or
,
and
, where
,
, or
must
be .a multiple of
If x
.
An object thrown upward from a height of h feet with an initial velocity of v feet per
The (D)
correct answer is
. If the object is thrown upward from a height of
feet,
13
second
will
reach
a
maximum
height
of
feet.
If
the
object
is
thrown
upward
(E)
11
Your Response:
from a height of 6 feet with an initial velocity of 32 feet per second, what will be its
Correct Response(s): 3 or 6 or 9 or 12
maximum height, in feet?
A NSWERS AND E XPLANATIONS
13
Explanation:
Explanation
for Correct
The correct answer
is
, Answer
,
, orC : . The sum of the measures of the three
13.
The incomplete
is an expense sheet for Carmen’s business
angles
of
triangle
equal
to graphtable
soabove
If of trip. If her
Choice
(C)a is
correct.is From
of
you each
can day,
seewhat
thatwere
for her
any
value
Your Response:
hotelthe
expenses were
the, same
total
expenses
for Friday, in
$ sign when
gridding
your integers
answer.) and
then
Since
are
positive
strictly
between soand dollars?
, the (Disregard
value
of the and
is greater
than
. For any value of
Correct Response(s): 22
that
be ais multiple
of
greater than it follows
and less
than or equal to
, the value must
of
greater than
Explanation:
. For
So the
possible
values
for
are and
,
, inclusive,
, or
. the value of
Your Response:
anyonly
other
value of
between
is
The correct
less
than or answer
equal tois
Correct
Response(s):
96
. Ifthe
the
objectof
is thrown
upward from
a heightare
of those
feet,
. So
values
for which
is positive
Explanation:
The
table or
above
is an expense
sheet for Carmen’s business trip. If her
for incomplete
which
.
hotel expenses were the same
each
day,
what
were
her
totalwere
expenses
for day,
Friday, in
The correct answer is
. If Carmen’s
hotel
expenses
dollars each
then her
total expenses
in dollars
for the
trip were
. The table
dollars? (Disregard the $ sign
when
gridding
your
answer.)
gives her total expenses as
13
equation for
:
, so
, so
expenses for Friday, in dollars, were
. Solve this
, and
. Carmen’s total
.
Page 7Aof
Explanation for Incorrect Answer
: 12
Your Response:
Choice (A) is not correct. For the values of
such that
Correct Response(s): 96
is positive. However, if
, the value of
, the value of
is negative.
Explanation:
is the correct answer. From the information in the figure, you know that
is a right triangle. Since
and
, by the Pythagorean
Test 9: Math Answers and Explanations
theorem, the length of
In
angle,
is
. Since
is a right
is also a right angle, and
is a right triangle. Since
14
and
What is the length of
, by the Pythagorean theorem, the length of
is
above,
and
.
Your Response:
Correct
SaturdayResponse(s):
morning, then 8
. So
.
Explanation:
15
morning
Mr. From
Smiththe
had
a certain in
amount
of money
thatthat
he planned to
is the correct
answer.
information
the figure,
you know
14. On Monday
In subsequent
above,
What is the length of
spend during the week. On each
morning, he had one and
fourth the amount
is a right triangle. Since
16
angle,
the list?
and
Your Response:
15.
16
, by the Pythagorean
is also a right angle, and
is a right triangle. Since
Your theorem,
Response:
, by the Pythagorean
the length of
is
Your Response:
Correct Response(s): 1024
Saturday morning, then
Correct Response(s): 109
Explanation:
15
and
of the previous morning. On Saturday morning, 5 days later, he had $1. How many
theorem,
is start with on Monday
. Since
is a right the $ sign
dollars
did the
Mr.length
Smithoforiginally
morning? (Disregard
when
gridding
your
answer.)
The median of a list of 99 consecutive integers is 60. What is the greatest integer in
Explanation:
The correct answer is
Correct Response(s): 8
.
Explanation:
is the correct answer.
From the information
in the figure, you know that
. So
.
is a right triangle. Since
and
, by the Pythagorean
length
of
. If theorem,
Mr. Smiththe
had
dollars
on is
Monday morning and
. Since
is a right
correctmorning
answer isMr. Smith
. If the
median
of a amount
list
of aofright
consecutive
integers
is is a
angle,
is also
angle,
and
OnThe
Monday
had
a certain
money
that
he planned
toright triangle. Since
had one fourth that amount on Tuesday morning, then he had
dollars on
spend, during
the are
week. On
eachand
subsequent
he
had one
thethe
amount
then there
integers
in
the list thatmorning,
are
andfourth
integers
, byless
the than
Pythagorean
theorem,
length of
is
of in
the
previous
morning.
On
Saturday
morning,
5
days
later,
he
had
$1.
How
many
the
list
that
are
greater
than
.
Since
the
integers
are
consecutive,
the
Tuesday morning. On Wednesday morning, he had one fourth the amount he had
dollars
did Mr.
originally
start with
on Monday
(Disregard
the
$ sign
The
median
of Smith
a in
listthe
of
99 is
consecutive
integers
is 60.morning?
What is the
greatest
integer
in
.
greatest
integer
list
.
onlist?
Tuesday
or
dollars. Continuing in this way, you see that he
when
griddingmorning,
your answer.)
the
had
dollars on Saturday morning. If he had
on
Your
Your Response:
Response:
17
16
When
the positive integer
m is divided by 5, the remainder is 3. What is the
Correct
Saturday
morning, then
So
.
Correct Response(s):
Response(s):
109 On Monday morning Mr.. Smith
151024
had a certain
amount of money that he pla
remainder when 20m is
divided
by 25?
spend during the week. On each subsequent morning, he had one fourth t
Explanation:
Explanation:
of the previous morning. On Saturday morning, 5 days later, he had $1. H
did Mr.
startmorning
with
onand
Monday
morning? (Disregard
The
correct
answer
is
.
If
Mr.median
Smith
hadaSmith
on
Monday
. If dollars
the
of
listdollars
of originally
consecutive
integers
is
16. The correct answer is
when gridding your answer.)
, then
there are
integers in the list that are less than
and
integers
Your
Response:
had
one
fourth that amount on Tuesday morning, then he had
dollars on
in the list that are greater than
. Since the integers are consecutive, the
Correct Response(s): 10
greatest morning.
integer inOn
theWednesday
list is
Tuesday
morning,
he. hadisone
amount
he had
The
median of a list
of 99 consecutive
integers
60.fourth
What the
is the
greatest
integer in
Explanation:
the
list?
Your
Response:
on Tuesday morning, or
dollars. Continuing in this way, you see that he
17. The correct answer is
17
18
. WhenCorrect
the positive
integer
is
divided by
, the
Response(s):
1024
remainder is
. So there is a positive integer
such that
. Then
had
dollars on Saturday morning. If he had
on
Explanation:
. Therefore, the remainder is
Your Response:
Whenwhen
the positive
integer m is divided
by 5,
the remainder
the dollars on Monday morning
The
correct
answer
is
. is
If 3.
Mr.What
Smithishad
is divided by
.
remainder
when 20m is divided
by 25?
Correct Response(s):
109
had one fourth that amount on Tuesday morning, then he had
dollars on
Explanation:
The correct answer is
Your Response:
, then there are
Tuesday morning. On Wednesday morning, he had one fourth the amount he h
. If the median of a list of
consecutive integers is
morning,
or than
dollars. Continuing
integerson
in Tuesday
the list that
are less
and
integers in this way, you see that
Correct
in
the listResponse(s):
that are greater10than
greatest integer in the list is
Explanation:
The correct answer is
remainder is
. Since the integers are consecutive, the
had
.
dollars on Saturday morning. If he had
Page
of 12 integer
. When
the 8
positive
. So there is a positive integer
is divided by
such that
, the
. Then
. Therefore, the remainder is
when
is divided by
.
So
on 4
on 5
on 6
on 7
on 8
on 9
on 10
and Explanations
Explanation:
Explanation for Incorrect Answer A :
Choice (A) is not correct. Since the average of
and
is
and
,
. Solving for
gives
, so Ali made
sandwiches.
must be greater than
. See the explanation for the correct response (D).
Back to Score Report
Explanation for Incorrect Answer B :
ections
on 2
.
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Test
9: Math49/5,
Answers
Correct
Response(s):
9.8
s and Explanations
on 1
, and
View Answers
and Explanations
Choice (B) is not correct. See the explanation
for the correct response (D).
Online -Explanation
Practice Test for
#6 Incorrect Answer B :
Explanation for Incorrect Answer
:
or
.CFrom
the figure and the information in the
18. The correct answer is
(B) iscost
not correct.
See
the
explanation
for$32
the each,
correctwhich
response
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1
If Choice
notebooks
$2
each
and
backpacks
cost
of
question,
cancorrect.
draw the
figure
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is similar
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triangle
Choice
(C)you
is not
See
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for the correct
response
(D).
represents
the
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n
notebooks
and
b
backpacks?
, because
are right
triangles
. To find
,
Explanation
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Incorrect
Answer
C :with
Explanation for Incorrect Answer E :
you (A)
can(C)
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solve See
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You have
and
Choice
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the explanation for the. correct
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Choice (E) is not correct. The sum of
and
is equal to
, but
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(B)
Explanation
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Incorrect
Answer
D
:
equal to
. See the explanation for the correct response (D).
, so
. Therefore,
, so
, and
(C)
2
The
average
6, 19, andfor
x is
19.
Whatresponse
is the value
of x?
Choice
(D) is(arithmetic
not correct.mean)
See theofexplanation
the
correct
(A). Ben
.
made
as Ali, and Ali made
sandwiches, so
is
(D) times as many sandwiches
the number
(A) 19of sandwiches Ben made. However, the question asked how many
(E)
sandwiches Ali made.
(B) 25
Ali,
Ben,
and Carla made a total of 20 sandwiches. Ben made 3 times as many as
Explanation
(C) 31 for Incorrect Answer E :
Ali,
and Carla
made
twice as many as Ben. How many sandwiches did Ali make?
A NSWERS
AND
E XPLANATIONS
(D)(E)32
Choice
is not correct. See the explanation for the correct response (A).
3
Math Section 8
Explanation
(E)
(A) 57
Twofor Correct Answer D :
(B)(D)Four
is correct. If notebooks cost
1. Choice
dollars.
If backpacks
cost
(C)
Five
A NSWERS
AND
E XPLANATIONS
each, then the cost of
each, then the cost of
notebooks is
Back to Score Report
backpacks is
dollars.
So,
cost,
in dollars,
of
notebooks
and Policy
notebooks
ght © 2005 College Entrance Examination
Board.
All
rights
reserved.
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Termsisof Use Contact Us
(D)
Sixtheof
4
If 0.03
percent
n is 3, what is 3 percent of n?
Explanation for Correct Answer D :
(E) Ten
2.
(A) 900
Choice (D) is correct. If the average of
(B) 600
A NSWERS AND E XPLANATIONS
So
, and Answer A :
Explanation
for Incorrect
(C) 300
and
, then
.
.
Explanation
for correct.
CorrectSee
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A :
Choice
is not
the explanation
for the correct response (D).
(D)(A)0.006
(E)(A)
0.003
Choice
is correct.
If Ali made
and Ben made three times as
for Incorrect
Answersandwiches,
B :
3. Explanation
many as Ali, then Ben made
sandwiches. Carla made twice as many
Choice
(B) is not
SeeAnswer
the explanation
for the correct response (D).
Explanation
for correct.
Incorrect
A :
sandwiches as Ben, so Carla made
sandwiches. The total number of
A
NSWERS AND
XPLANATIONS
sandwiches
was
equal to
,and
so
Explanation
forEmade
Incorrect
Answer
C : . This
Choice
(A) isthey
not
correct.
Since
the average
of is alsoand
is
,
.
Solving
for
gives
,
so
Ali
made
sandwiches.
must be greater than
. See the explanation for the correct response (D).
Explanation
for correct.
CorrectSince
Answer C :
Choice
(C) is not
this would be the cost,
4.
Explanation
Incorrect
Answer
B :
in
dollars, of for
notebooks
and
backpacks
if each notebook cost
and each
Choice (C) is correct. If
percent of
is
, then
. So
backpack
cost
.
Choice (B) is not correct. See the explanation for the correct response (D).
Explanation for
Incorrect Answer B
, and
C ::
Explanation for
Incorrect Answer E
. Thus,
percent of
is
Choice (B) is not correct. See the explanation for the correct response (A).
Choice (E)
(C) is
is not
not correct.
correct. Since
See the explanation for the correct
Choice
thisresponse
would be(D).
the cost,
.
Explanation for Incorrect Answer C :
in
dollars, of for
notebooks
and
backpacks
if each notebook cost
and each
Explanation
Incorrect
Answer
E :
Choice
(C)cost
is not correct.
See the explanation for the correct response (A).
backpack
.
Choice (E) is not correct. The sum of
and
is equal to
, but
is not
equal to
. for
SeeIncorrect
the explanation
forDthe
Explanation
Answer
: correct response (D).
Choice (D) is not correct. See the explanation for the correct response (A). Ben
Explanation for Incorrect Answer A :
made
times as many sandwiches as Ali, and Ali made
sandwiches, so
is
the
number
sandwiches
Ben made.
However, for
thethe
question
how (C),
many
Choice
(A) isofnot
correct. From
the explanation
correctasked
response
you
sandwiches
have
is Ali made.
. So
percent of
. However, the question asked for
3
Page 9 of 12
percent
of Carla
.
Ali,
Ben, and
made a total
of 20
Explanation
for Incorrect
Answer
E :sandwiches. Ben made 3 times as many as
Ali, and Carla made twice as many as Ben. How many sandwiches did Ali make?
Explanation
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Incorrect
B :
Choice
(E) is not
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Choice
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(A)(B)Two
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Explanation for Incorrect Answer
D : (E) is not correct. From the explanation for the correct response (C), you
have
is
. So
is
percent of
asked for
percent of
.
Choice
(C)(D) is not correct. If the portion of the number line between
and
divided into eighths, as in the figure above, you can see that point
. However, the question
is
is greater
Test 9: Math Answers and Explanations
6
than(D).
5
foron
Incorrect
Answer
:
If Explanation
the (E)
tick marks
the number
line Eabove
are equally spaced, which of the lettered
Explanation for Incorrect Answer A :
points
A (E)
through
is between
and of the number line between
Choice
is not Ecorrect.
If the portion
and
is
Choice (A) is not correct. See the explanation for the correct response (D).
divided into eighths, as in the figure above, you can see that point
is greater
Explanation
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Answer B :
A NSWERS AND
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than
.
Choice (B) is not correct. See the explanation for the correct response (D).
Explanation for Correct Answer B :
(A) A
Explanation for Incorrect Answer C :
What is an is
equation of line , where
in the figureisabove?
the slope
5. Choice
(B)(B)
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of
and
the
-intercept
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. From the
you can
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Choice
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7
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. So
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,
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. Since
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for Incorrect Answer A :
(D)
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6. Choice
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is number
not correct.
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the correct response (D).
(C)
into eighths, the result is
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(E) for Incorrect Answer B :
Explanation
Choice (B) is not correct. See the explanation for the correct response (D).
(E)
A NSWERS
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In the figure above, what is the value of y?
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ChoiceExplanation
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and
is
.
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is
, where
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Explanation for Incorrect Answer E :
of
and
is the
-intercept of
. From the figure, you can see that the points
Explanation for Correct Answer D :
Choice (E) isand
not correct.
See the explanation for the correct
response
(D).
(A) 40
are on , so the slope is
. Since
intersects the
The
only lettered
point between
A
NSWERS
AND E XPLANATIONS
9
If
, then
, so
7. If Choice
(B)(D)
45is correct.
which of the following is a possible value of
(C) 50
-axis at the point
Explanation for Incorrect Answer A :
.
, the
-intercept of
(D) 60
and
has the equation
(A)
(E)
72
(B) into
10 eighths,8as in the figure above, you can see that point
divided
Choice (A) is not correct. If the portion of the number line between
.
.
and
is
. So
and
,
is
is less than
(C) 20
(D) 30AND E XPLANATIONS
A NSWERS
(E) 870
Explanation for Incorrect Answer C :
Explanation for Correct Answer D :
Choice (C) is not correct. If the portion of the number line between
is
what that
is the value ofand
y?
, it follows
and
8. Choice (D) is correct. Since In the figure above,
A
NSWERS
E XPLANATIONS
divided
intoAND
eighths,
as in the figure above, you can see that point
. Vertical angles are equal, so
. Thus,
.
is greater
than
.
Explanation
for Correct Answer C :
(A) , 40
then
. So
Explanation for Incorrect Answer D :
(B)
45
, and either
or
. If
, then
Explanation
for correct.
Incorrect
Answer
A of
: the number line between
Choice (D) is not
If the
portion
and
is
(C) 50
.
divided
into
eighths,
as
in
the
figure
above,
you
can
see
that
point
is
greater
Choice (A) is not correct. See the (D)
explanation
for the correct response (D).
60
9. Choice (C) is correct. If
(E) B72
than
.
Explanation
for Incorrect Answer
:
Choice (B) is not correct. See the explanation for the correct response (D).
Explanation for Incorrect Answer E :
Explanation for Incorrect Answer
A :AND E XPLANATIONS
A NSWERS
Explanation for Incorrect Answer C :
Choice (E) is not correct. If the portion of the number line between
and
is
Choice
(A) is
not
If Explanation
, then
. So
for
Correct
Answer
D
: (D).
10
of
12
Choice
not correct.
correct.
the
explanation
for
thesee
correct
divided (C)
intoiseighths,
as inSee
the Page
figure
above,
you
can
that response
point
is greater
, and either
or
. If
, then
Choice (D)
, it follows that
Explanation
for Incorrect Answer
E :is correct. Since
than
.
. If
, then
So
. Vertical angles are equal, so
. .Thus,
Choice (E) is not correct. See the explanation for the correct response (D).
and
.
Choice (E)
(C) is
is not
not correct.
correct. This
If
boxesnot
arerepresent
selected, Mark’s
there could
be red
scarves
in it
Choice
could
trip. One
reason
is that
(A)
of them, brown
scarves
in
of them,during
yellow the
scarves
of them,
a gray
indicates
a negative
distance
traveled
last in minutes
of and
the trip.
This
scarf
of them.
is
not in
possible.
Test 9: Math Answers and Explanations
Explanation for Incorrect Answer D :
(E)
Choice (D) is not correct. If
of them, brown scarves in
boxes are selected, there could be red scarves in
of them, yellow scarves in
of them, and gray
scarves
of them.
There
arein6 red,
6 brown, 6 yellow, and 6 gray scarves packaged in 24 identical,
11
unmarked boxes, 1 (B)
scarf per box. What is the least number of boxes that must be
selected in order to be sure that among the boxes selected 3 or more contain
scarves of the same color?
A NSWERS AND E XPLANATIONS
(A) 3
Explanation
for Correct Answer B :
12
10.
(B) 6
Choice (B) is correct. This graph could show Mark’s distance traveled as a function
of time.
(C) The
7 first portion of the graph indicates that Mark rode at a constant rate
of
mile in
minutes (
miles per hour) and covered a distance of
mile.
(D) 8
The next portion of(C)
the graph is horizontal, which indicates that he rested for
(E) 9
minutes and did not cover any distance during that period. The graph indicates
In the figure above, ABCDEF is a regular hexagon, and its center is point O. What is
that during the final
minutes Mark rode at a constant rate of
miles in
the value of x ?
minutes, or
miles per hour, which is faster than he rode in the first
A NSWERS AND E XPLANATIONS
minutes.
Explanation for Correct Answer E :
11.
(A) 80
Choice (E) is correct. If
boxes are selected, there could be red scarves in
them, brown scarves in
of them, yellow scarves in
(B) 60
in
(D)
(C)
40 in this case, there would not be
of them;
color(D)
among
30 these
of
of them, and gray scarves
or more scarves of the same
boxes. However, if one more box is selected, the color of the
scarf(E)
in that
20 box must be the same as the color of the scarves in
boxes, and there will then be
of the same color. So,
of the original
is the least number
of boxes that must be selected in order to be sure that among the boxes selected
or more AND
contain
scarves of the same color.
A NSWERS
E XPLANATIONS
Explanation for Correct Answer B :
(E)
12.
Choice (B) is correct. Since
is the center of the hexagon, it follows that
. Drawing some additional lines and labeling some angles yields
additional information from the figure:
A NSWERS AND E XPLANATIONS
Explanation for Correct Answer B :
Choice (B) is correct. This graph could show Mark’s distance traveled as a function
of time. The first portion of the graph indicates that Mark rode at a constant rate
of
mile in
minutes (
miles per hour) and covered a distance of
mile.
The next
portion
of hexagon,
the graph itis follows
horizontal,
From the symmetry
of the
regular
that which indicates that he rested for
minutes and did not cover any distance during that period.. The
graph indicates
Since
. Since
that during the final
minutes Mark rode at a constant rate of
miles in
the sum of these angles is
, each of these angles has a measure of
. So
minutes, or
miles per hour, which is faster than he rode in the first
. It is also true that
, so
. Since
, and
minutes.
, it follows that
. Therefore,
, and
.
Explanation for Incorrect Answer A :
Choice (A) is not correct. See the explanation for the correct response (B).
Page 11 of 12
Explanation for Incorrect Answer C :
Choice (C) is not correct. See the explanation for the correct response (B).
Explanation for Incorrect Answer D :
(E) 25pr
Choice (E)
(E) is
is not
not correct.
correct. See
If the explanation
, then
system
of equations
Choice
(D) isthe
not for
correct.
Since
for all is
numbers
Choice
the
correct
response
(C).
, you have
. This expression is not equivalent to
and(A)
. The solutionFor
toexample,
this system
is, then
if
A NSWERS
AND E XPLANATIONS
Test
9: Math Answers
and,
.
but
and Explanations
.
.
Explanation for Incorrect Answer E :
(B)
Explanation for Correct AnswerChoice
C : (E) is not correct. Since
for all numbers
, you have
. This expression is not equivalent to
15
13. Choice (C) is correct. Since
(C)
forif
example,
all numbers
, then
, you , have
but
. For
.
.
16
RESULTS
(D) OF BEANBAG GAME
For which
(E) of the following values of k will the system of equations above have no
14
solution?
Explanation for Incorrect Answer A :
A
NSWERS AND E XPLANATIONS
Choice (A) is not correct. Since
Explanation
for Correct Answer C :
(A)
for all numbers
, you have
The circle
above has an
ofequivalent
and is divided
. This
is area
notcongruent
to intois8 congruent
. For regions. What is
(B)(C) is correct. The perimeter
ofexpression
each of the
regions
14. Choice
the perimeter of one of these regions?
(C) 0
example, if, where
is
the radius of the circle
and
, then
, but
is the.circumference of the
(D) 5
(A) a beanbag at a target until the person missed
In a certain game, each person threw
circle.
area of the circle is
, so
. The circumference of the
(E)The10
the
target.
The
table
shows
the
results
Explanation for Incorrect Answer (B)
B : for the 25 people who played the game. For
circle is 4 people
, so hit the target
. The perimeter
of one
of the regions,
then,on
is their 4th
example,
on their first
3 throws
and missed
throw. Based on the information in
the
(C) table, which of the following must be true?
Choice (D) is not correct. If
, then the system of equations is
A NSWERS AND E XPLANATIONS .
(D)
I . More than half the people hitand
the
target on .their
first throw.
The solution
to this system
(E)
is
and
.
Explanation for Correct Answer A :
II . For all of the throws attempted, more hit the target than missed the target.
Explanation for Incorrect Answer E :
NSWERS
ANDsystem
E XPLANATIONS
is hit
correct.
If
, Athen
the
of equations is
15. Choice
III. No(A)
one
the target
5 times.
Choice (E) is not correct. If
, then the system of equations is
and
Explanation
for Correct Answer
. The first equation
is equivalent
to C :
are no values
and
for which
and
. The solution to this system is
system of equations has no solution if
(A) I only
, where
.is the radius of the circle and
circle. The area of the circle is
(B) II only
circle is
(C) I and III only
. There
and
.
Choice (C) is correct.
perimeter
of each of the
canTheequal
both
and congruent
. Soregions
the is
, so
is the circumference of the
, so
. The circumference of the
. The perimeter of one of the regions, then, is
RESULTS OF BEANBAG GAME .
16
(D) II and III only
Explanation for Incorrect Answer B :
(E) I, II, and III
Choice (B) is not correct. If
and
, then the system of equations is
. The solution to this system is
and
A NSWERS AND E XPLANATIONS
.
Explanation for
for Incorrect
Correct Answer
EC
: :
Explanation
Answer
In a certain game, each person threw a beanbag at a target until the person missed
the target. The table shows the results for the 25 people who played the game. For
Choice (C)
(E) is
To determine
to
question,
you
decide
is correct.
not correct.
If
, the
thenanswer
system
equations
is must
16. Choice
example,
4the
people
hit the
theof
target
on their
first
3 throws
and missed on their 4th
which of the three statements are
true.
throw.
Based on the information in the table, which of the following must be true?
and
. The solution to this Isystem
is half the people
andhit the target.on their first throw.
. More than
Statement : From the table, you can see that only
of the
people
II . For all of the throws attempted, more hit the target than missed the target.
missed the target on their first throw. So
III. D
No :one
Explanation for Incorrect Answer
the target on their first throw.
is true.
Statement
, which is more than half of
, hit
hit the target 5 times.
: The total number of throws attempted was
. The total
(A) I only
number of throws that missed the target
. So
(B) IIwas
only
throws hit the target,(C)
which
isIII
more
I and
only than the number of throws
that missed the target.
Statement
: Of the
is true.
(D) II and III only
(E) I, II, and III
people,
missed before their fifth throw and
missed their fifth throw. So no one hit the target
times.
is true.
A NSWERS AND E XPLANATIONS
Explanation for Correct Answer E :
Choice (E) is correct. To determine the answer to the question, you must decide
which of the three statements are true.
Explanation for Incorrect Answer
A :of 12
Page 12
Statement
: From the table, you can see that only
missed the target on their first throw. So
are all true. So
the target on their first throw.
is true.
is not the only true
statement.
Statement
: The total number of
of the
people
, which is more than half of
Choice (A) is not correct. From the explanation for the correct answer,
,
, and
throws attempted was
. The total
, hit