SAMPLE PROBLEMS
FOR GRADE 9 MATHEMATICS
GRADE
9
DIRECTIONS: This section provides sample mathematics problems for the Grade 9 test forms. These
problems are based on material included in the New York City curriculum for Grade 8. (The Grade 8
problems on sample forms A and B cover mathematics material through Grade 7.) General directions
for how to answer math questions are located on pages 48 and 86. There is no sample answer sheet for this
section; mark your answers directly on this page or on a separate piece of paper.
1.
3.
WATER IN POOL
R
2,000
A.
B.
C.
D.
E.
1,500
1,000
500
0
1
2
3
4
5
0
V
1
R
S
T
U
V
6
Time (in hours)
7
4.
22
17
21
x
1 70 5 ___
49
What is the value of x in the equation above?
A swimming pool is being filled with water at
a constant rate. The figure above is a portion
of a graph that shows how the number of
gallons of water in the pool changes over time.
Starting with an empty pool, at the end of
hour 5 there are 2,000 gallons in the pool.
If the pool continues to fill at this rate, how
much water will be in the pool at the end of
hour 20? (Assume that the pool holds a total
of 100,000 gallons.)
F. 8
G. 9
H. 10
J. 57
K. 58
5.
A. 5,600 gal.
B. 6,000 gal.
C. 8,000 gal.
D. 40,000 gal.
E. 80,000 gal.
W
5 cm
R
S
2 cm
4 cm
T
V
In the figure above, R
wS
wT
w and V
wS
wW
w are
straight line segments and W
wT
w is parallel
WT
to R
wV
w. What is }
?
ST
Define the operation ■
as follows:
(
a■
bc = ba , where b and c are not zero.
c
3
4
If 2 ■
x = 2 , what is the value of x?
(
( (
(
U
On the number line above, which letter could
represent the location of x2 ?
2,500
2
A. }
5
(
2.
x T
–1
–2
3,000
Amount
of Water
(in gallons)
S
4
B. }
5
F. 1
G. 2
H. 3
J.
6
K. 12
C. 1
5
D. }
4
E. 20
106
If (12.6 3 1018) 2 (1.1 3 1017) 5 k 3 1019,
what is the value of k?
6.
8.
STUDENTS OWNING PETS
F. 0.016
G. 1.150
H. 1.249
J. 11.500
K. 16.000
7.
Number of
Pets Owned
Number of
Students
0
5
1
7
2
3
3
4
4
0
5
1
N
M
P
L
There are 20 students in a class. The
frequency table above shows the number of
these students that own 0, 1, 2, 3, 4, or 5 pets.
What is the mean number of pets owned
per student in this class?
L'
N'
1
F. 1__
M'
2
A geometry game awards a different score
for each geometric transformation. Each
90° rotation about a point will earn a score
of 2, a reflection over a horizontal or vertical
line will earn a score of 3, and a horizontal
or vertical translation will earn a score of 4.
Which set of transformations would earn the
highest score to transform LMNP to L9M9N9P
as shown above?
A.
B.
C.
D.
E.
G. 3
1
H. 3__
3
J. 4
K. 5
9.
two reflections
two translations
two 90° rotations
a translation, followed by a reflection
a 90° rotation, followed by a reflection
Let (x, y) → (x 1 10, y 2 10). Using that rule,
if (n, r) → (100, 100), what is (n, r)?
A.
B.
C.
D.
E.
10.
(90, 90)
(90, 110)
(100, 100)
(110, 90)
(110, 110)
Raul has two containers. One is a cylinder
with an inner radius of 4 inches and an inner
height of 8 inches. The other is a cube with
inner height, width, and length each equal to
8 inches. The cylinder is filled with water and
the cube is empty. If Raul pours the contents
of the cylinder into the cube, how deep will the
water be in the cube?
F.
2 in.
2
G. __
p in.
3
H.
4 in.
J.
2p in.
K. 4p in.
107
11.
14.
N
P
p
p2
2
2 ___
__
,
p
1
q,
p
2
q,
p
1
q
,
q
q2
1_
___
If p 5 q 5
, which one of the expressions
Ï2
above does not represent a rational number?
6 cm
M 2 cm T
p
F. __
q
R
12 cm
G. p 1 q
In the figure above, if M
wN
w is parallel to w
Tw
P,
what is the length of M
wN
w?
H. p 2 q
A. 7 cm
B. 8 cm
C. 10 cm
D. 12 cm
E. 14 cm
J. p2 1 q2
p2
K. ___
q2
15.
|x 2 1| , 3
|x 1 2| , 4
12.
How many integer values of x satisfy both
inequalities shown above?
F.
G.
H.
J.
K.
A. y = ⁻ }3} x + 3
2
0
1
3
4
5
B. y = ⁻ }2} x − 3
3
C. y = ⁻ }2} x + 2
3
D. y = ⁻ }1} x + 3
3
13.
E. y = }2} x − 2
3
y
16.
O
x
(1, 0)
Seven consecutive integers are arranged
in increasing order. Their sum is 7k.
What is the value of the second integer
in terms of k?
F.
G.
H.
J.
K.
(0, ᎑ 2)
The straight line shown above is the graph
of y 5 f (x). Which of the following points
satisfies the inequality y . f (x)?
A.
B.
C.
D.
E.
Straight line k passes through the point (23, 4)
with an x-intercept of 3. What is the equation
of line k?
17.
(22, 27)
(21, 23)
(1, 0)
(2, 1)
(3, 4)
k26
k22
k
k11
7k 2 6
A tiny robot sits on the point (1, 22) of the
coordinate plane. At each flash of a blue light,
it moves 4 units to the right and 5 units down.
At each flash of a red light, it moves 1 unit to
the left and 4 units up. If, at the end of 15 red
flashes and n blue flashes, the robot is sitting
on the line y 5 x, what is n?
A. 5
B. 8
C. 14
D. 15
E. 44
108
GRADE 9 MATHEMATICS
EXPLANATIONS OF CORRECT ANSWERS
1. (C) At the beginning (hour 0), the pool is empty.
GRADE
9
6. (H) In order to add or subtract two numbers in scientific notation, the exponent on the 10 must
be the same. Since the question asks for the
value of k 3 1019, change both terms into this
same power of 10:
After 5 hours, the pool holds 2,000 gallons.
Thus, the rate of change (or slope of the line)
2,000 − 0
2,000
is _________ 5 ______ 5 400 gallons per hour.
5−0
5
To find the number of gallons after 20 hours,
12.6 × 1018 5 (1.26 3 10) 3 1018 5 1.26 3 1019
multiply the rate by the number of hours:
1.1 × 1017 5 (0.011 3 102) 3 1017 5 0.011 3 1019
400 3 20 5 8,000 gallons.
Now, perform the subtraction:
(1.26 × 1019) 2 (0.011 3 1019)
3
2 5 __
2. (H) _____
2
4
__
x
x 5 __
3
t__
4
2
2x 5 6
5 (1.26 2 0.011) 3 1019
( )
5 1.249 3 1019
Thus, k 5 1.249
x53
7. (A) The quickest way to solve this problem may be
to test the options and see which results in the
highest score. We can immediately eliminate
options B, D, and E because those do not result
in the correct transformation. Option A results
in a score of 3 1 3 5 6. Option C results in
a score of 2 1 2 5 4. Thus, A is the correct
answer.
3. (D) Since x is a negative number between 21 and 0,
assign a value to x in that range and calculate x2.
2
4
For example, let x 5 2 __
. Then x2 5 __
, which
3
9
roughly corresponds to point U.
4. (J)
2
2
x
7 2 1 7 1 1 70 5 ___
49
x
1 1 __
1 1 1 5 ___
___
49
7
49
8. (F) First, determine the total number of pets that
the students own by multiplying the number of
pets owned by the number of students in each
row of the table. Then add that column to get
the total number of pets.
1 1 7 1 49 5 x
57 5 x
Number of
Pets Owned
____
___
___
5. (B) WT and RV are parallel, and RT is a transversal; thus ∠RVS and ∠TWS are alternate
interior angles and are congruent. Angles WST
Number
of Students
Number of Pets 3
Number of Students
0
5
0
1
7
7
2
3
6
and VSR are vertical angles, and therefore they
3
4
12
are congruent. Since there are two sets of
4
0
0
congruent angles, the third set of angles must
5
1
5
Total: 30
also be congruent. Thus, nWTS is similar to
Now, calculate the mean by dividing the total
number of pets owned by the total number of
students:
nVRS. Since the triangles are similar, a
WT .
proportion can be set up to solve for ____
ST
VR
WT = ____ 5 __
4
____
5
ST
SR
30
1
___
5 1 __
20
2
109
GRADE 9 MATHEMATICS
EXPLANATIONS OF CORRECT ANSWERS
9. (B) Using the translation equation given in the
question, set up two small equations to find
n and r:
|x – 1| , 3 can also be written as
23 , x – 1 , 3
Adding 1 to each term results in
22 , x , 4
For r:
y – 10 5 100
y 5 110
Since these are only “less than” and not “less
than or equal to,” the possible values of x for
this inequality are 21, 0, 1, 2, and 3.
So, (n, r) 5 (90, 110)
10. (J) First, calculate the volume of the cylinder:
Similarly, |x 1 2| , 4 can also be written as
24 , x 1 2 , 4
2
V = πr h 5 π(4) (8) 5 128π cubic inches
The volume of water in the cube will be the
same as the volume of water in the full cylinder.
Use the volume formula of a cube to calculate
the depth (h) of the water in the cube:
Subtracting 2 from each term results in
22 , x , 2
The possible values of x in this inequality are
21, 0, and 1.
V 5 lwh
128π 5 (8)(8)h
128π = 64h
2π = h
The possible x values in common between
the two inequalities are 21, 0, and 1, so the
answer is 3.
11. (A) Triangles MNR and TPR are similar,
so use a
____
proportion to solve for the length of MN:
MN 5 ____
TP
_____
MR
TR
6
MN 5 ___
_______
2 1 12
9
12. (H) First, determine which integer values of x
would make each inequality true:
For n:
x 1 10 5 100
x 5 90
2
GRADE
12
1 (14) 5 7 cm
MN 5 __
2
110
GRADE 9 MATHEMATICS
EXPLANATIONS OF CORRECT ANSWERS
GRADE
9
13. (B) First, calculate the equation of the given line f(x).
The slope is calculated using the difference in
y-values of two given points divided by the
difference in x-values of those points. In this
(22 − 0)
case, m 5 ________ 5 2. The y-intercept (b) is
(0 − 1)
the value of y when the line crosses the y-axis, so
b 5 22. Using the slope-intercept form of a line
(y 5 mx 1 b), the equation is f(x) 5 2x 2 2.
15. (C) An x-intercept of 3 means the point (3, 0) is on
line k. Using (3, 0) and (23, 4), calculate the
slope (m) of the line:
To determine which of the given values satisfies y . f(x), or in this case y . 2x – 2, insert
each value into the inequality to find which
one makes the inequality true. You can
immediately eliminate Option C, because the
point (1, 0) is shown in the graph as one of the
points on the line.
Next, find which of the two equations is true for
the point (3, 0). To solve, substitute 3 for x in
each equation and find the one in which y 5 0.
(420)
4
2
m 5 _______
5 __
5 ] __
3
(2323) 26
2
,
The equation of line k must contain slope ] __
3
so only Options B and C are potentially correct.
2
Option B: y 5 ] __
(3) − 3 5 22 2 3 5 25
3
2
Option C: y 5 ] __
(3) 1 2 5 22 1 2 5 0
3
For inequality y > 2x – 2:
Option C is the correct answer.
Option A
27 . 2(22) 2 2
27 . 26 is false
16. (G) The question asks for the second integer, so let
n be the second integer. Then, the sum of the
7 integers is:
(n – 1) 1 n 1 (n 1 1) 1 (n 1 2) 1 (n 1 3) 1
(n 1 4) 1 (n 1 5) 5 7k
7n 1 14 5 7k
7(n 1 2) 5 7k
n125k
n5k–2
Option B
23 . 2(21) 2 2
23 . 24 is true
Option D
1 . 2(2) – 2
1 . 2 is false
Option E
4 . 2(3) 2 2
4 . 4 is false
(Note that this point is also on the given line.)
17. (B) Since the number of red flashes is known (15),
first calculate where the robot would be after the
15 red flashes. For each red flash,
(x, y)
(x – 1, y 1 4). So, after 15 red flashes:
(1 2 [1 3 15], 22 1 [4 3 15]) 5 (214, 58)
14. (G) A rational number is a number that can be
p
written as a fraction. Since p = q, then __
q = 1,
Next, use the point (214, 58) to calculate where
the robot will be after n blue flashes. For each
blue flash, (x, y)
(x 1 4, y 2 5). So, after n
blue flashes: (214 1 4n, 58 2 5n)
p2
___
5 1, and p 2 q 5 0, all of which are
q2
rational. That leaves two expressions to test:
1__
1__
2__
p 1 q 5 ____
1 ____
5 ____
Ï2
Ï2 __Ï2
(irrational because Ï 2 is irrational)
1__
p2 1 q2 5 ____
1
1
The question states that the robot’s final
position is on the line y 5 x, which means the
x- and y-coordinates will have the same value.
To find n, set the two coordinates above as
equal and solve for n:
214 1 4n 5 58 – 5n
9n 5 72
n58
1
__ 2 5 __ 1 __ 5 1 (rational)
( Ï 2 )2 1 (____
2
2
Ï2 )
Thus, p 1 q is not a rational expression.
Answer Key for Grade 9 Mathematics
1. C
2. H
3. D
4. J
5. B
6. H
7. A
8. F
9. B
10. J
111
11. A
12. H
13. B
14. G
15. C
16. G
17. B