Content Covered by the ACT Mathematics Test

Content Covered by the ACT Mathematics Test
The ACT Mathematics Test is a 60­question, 60­minute test designed to measure the mathematical skills students have typically acquired in courses taken by the end of 11th grade.
Content Covered by the ACT Mathematics Test
In the Mathematics Test, three subscores are based on six content areas: pre­algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry.
Pre­Algebra/Elementary Algebra
• Pre­Algebra (23%). Questions in this content area are based on basic operations using whole numbers, decimals, fractions, and integers; place value; square roots and approximations; the concept of exponents; scientific notation; factors; ratio, proportion, and percent; linear equations in one variable; absolute value and ordering numbers by value; elementary counting techniques and simple probability; data collection, representation, and interpretation; and understanding simple descriptive statistics. • Elementary Algebra (17%). Questions in this content area are based on properties of exponents and square roots, evaluation of algebraic expressions through substitution, using variables to express functional relationships, understanding algebraic operations, and the solution of quadratic equations by factoring.
Intermediate Algebra/Coordinate Geometry
• Intermediate Algebra (15%). Questions in this content area are based on an understanding of the quadratic formula, rational and radical expressions, absolute value equations and inequalities, sequences and patterns, systems of equations, quadratic inequalities, functions, modeling, matrices, roots of polynomials, and complex numbers. • Coordinate Geometry (15%). Questions in this content area are based on graphing and the relations between equations and graphs, including points, lines, polynomials, circles, and other curves; graphing inequalities; slope; parallel and perpendicular lines; distance; midpoints; and conics.
Plane Geometry/Trigonometry
• Plane Geometry (23%). Questions in this content area are based on the properties and relations of plane figures, including angles and relations among perpendicular and parallel lines; properties of circles, triangles, rectangles, parallelograms, and trapezoids; transformations; the concept of proof and proof techniques; volume; and applications of geometry to three dimensions. • Trigonometry (7%). Questions in this content area are based on understanding trigonometric relations in right triangles; values and properties of trigonometric functions; graphing trigonometric functions; modeling using trigonometric functions; use of trigonometric identities; and solving trigonometric equations.
1
ACT Prep: Pre-Algebra / Elementary Algebra
Practice Problems
miles on Monday and
miles on Tuesday. What was the
1. Kaya ran
total distance, in miles, Kaya ran during those 2 days?
E. 3 11/15
D. 3 7/15
C. 3 2/5
B. 3 3/8
A. 3 2/15
3. Mr. Dietz is a teacher whose salary is $22,570 for this school year,
which has 185 days. In Mr. Dietz's school district, substitute teachers are
paid $80 per day. If Mr. Dietz takes a day off without pay and a
substitute teacher is paid to teach Mr. Dietz's classes, how much less does
the school district pay in salary by paying a substitute teacher instead of
paying Mr. Dietz for that day?
E. $122
C. $97 D. $105
B. $80
A. $42
2
4. So far, a student has earned the following scores on four 100-point
tests this grading period: 65, 73, 81, and 82. What score must the
student earn on the fifth and last 100-point test of the grading period to
earn an average test grade of 80 for the 5 tests?
F. 75
G. 76 H. 78
J. 99
K. The student can't earn an avg. of 80
5. The oxygen saturation level of a river is found by dividing the amount
of dissolved oxygen the river water currently has per liter by the
dissolved oxygen capacity per liter of the water and then converting to a
percent. If the river currently has 7.3 milligrams of dissolved oxygen per
liter of water and the dissolved oxygen capacity is 9.8 milligrams per liter,
what is the oxygen saturation level, to the nearest percent?
A. 34% B. 70% C. 73% D. 74% E. 98%
3
7. The expression a[b + (c - d)] is equivalent to:
C. ab + ac - d
B. ab + ac + ad
A. ab + ac - ad
E. ab + c - d
D. ab + c + d
8. If 4x + 3 = 9x - 4, then x = ?
F. 7/5
G. 5/7 H. 7/13
J. 1/5
K. -1/5
4
11. A formula for the volume V of a sphere with radius r is
If the radius of a spherical rubber ball is
to the nearest cubic inch?
A. 5
B. 7
C. 8
D. 16
E. 65
.
inches, what is its volume
12. If a marble is randomly chosen from a bag that contains exactly 8
red marbles, 6 blue marbles, and 6 white marble, what is the
probability that the marble will NOT be white?
K. 7/10
J. 3/10
H. 4/5
G. 3/5
F. 3/4
5
The following chart shows the current enrollment in all the
mathematics classes offered by Eastside High School.
Course title
Pre-Algebra
Algebra 1
Geometry
Algebra 2
Pre-Calculus
Section
A
A
B
C
A
B
A
A
Period
3
2
3
4
1
2
4
6
Enrollment
23
24
25
29
21
22
28
19
* For use in problems #14 & #15
6
14. What is the average number of students enrolled per section in
Algebra 1?
F. 24
G. 25
H. 26
J. 27
K. 29
15. The school owns 2 classroom sets of 30 calculators each, which
students are required to have during their mathematics class. There are
2 calculators from one set and 6 calculators from the other set that are
not available for use by the students because these calculators are being
repaired. For which of the following class periods, if any, are there NOT
enough calculators available for each student to use a school-owned
calculator without having to share?
C. Period 4 only
A. Period 2 only B. Period 3 only
D. Periods 3 & 4 only E. There are enough calculators for each class
period
7
16. What expression must the center cell of the table below contain so
that the sums of each row, each column, and each diagonal are equivalent?
F. 6x
H. 2x
K. -4x
G. 4x
J. -2x
x 8x -3x
6x
-2x ?
7x -4x 3x
18. Kareem has 4 sweaters, 6 shirts, and 3 pairs of slacks. How many
distinct outfits, each consisting of a sweater, a shirt, and a pair of slacks,
can Kareem select?
K. 216
J. 72
H. 42
G. 36
F. 13
8
19. At a refinery, 100,000 tons of sand are required to produce each
60,000 barrels of a tarry material. How many tons of sand are required to
produce 3,000 barrels of this tarry material?
A. 5000
B. 18000
C. 20000
D. 40000
E. 50000
23. Which of the following is a solution to the equation x2 - 36x = 0?
A. 72
B. 36
C. 18
D. 6
E. -6
9
26. The length, L, in meters, of a spring is given by the equation
L = 2/3 F + 0.03, where F is the applied force in newtons. What force, in
newtons, must be applied for the spring's length to be 0.18 meters?
F. 0.13
G. 0.15
H. 0.225
J. 0.255
K. 0.27
34. If a = b + 2, then (b - a)4 = ?
H. 1
G. -8
F. -16
J. 8
K. 16
10
29. The graph below shows the number of cars assembled last year in 4
cities, to the nearest 5000 cars. According to the graph, what fraction of
the cars assembled in all 4 cities were assembled in Coupeville?
key
City
CarTown
Coupeville
Truck City
Sedan Falls
Cars assembled
= 10000 cars
A. 1/5
C. 3/11
B. 1/4
D. 3/10
E. 1/3
11
32. For all pairs of real numbers M and V where M = 3V + 6, V = ?
K. M + 6
J. M - 6
H. 3M - 6
G. M + 6
F. M - 6
3
3
3
3
36. The larger of two numbers exceeds twice the smaller number by 8.
The sum of twice the larger and 3 times the smaller number is 65. If x is
the smaller number, which equation below determines the correct value of
x?
F. 3(2x + 8) + 2x = 65
G. 3(2x - 8) + 2x = 65
H. (4x + 8) + 3x = 65
J. 2(2x + 8) + 3x = 65 K. 2(2x - 8) + 3x = 65
12
42. Of the 804 graduating seniors in a certain high school, approximately 2/5
are going to college and approximately 1/4 of those going to college are going
to a state university. Which of the following is the closest estimate for how
many of the graduating seniors are going to a state university?
F. 80
G. 90
H. 160 J. 200 K. 320
43. If x and y are positive integers such that the greatest common
factor of x2y2 and xy3 is 45, then which of the following could y equal?
E. 3
D. 5
C. 9
B. 15
A. 45
13
44. If 115% of a number is 460, what is 75% of the number?
K. 400
J. 345
H. 320
G. 300
F. 280
59. If x = 2t - 9 and y = 5 - t, which of the following expresses y in terms
of x?
A. y = 1 - x
B. y = 19 - x C. y = 14 - 2x
D. y = 5 - x E. y = 1 - x
2
2
14
49. Which of the following statements describes the total number of dots
in the first n rows of the triangular arrangement illustrated below?
1st row
2nd row
3rd row
4th row
5th row
A. This total is always equal to 25 regardless of the number of rows.
B. This total is equal to twice the number of rows.
C. This total is equal to 5 times the number of rows.
D. This total is equal to the square of the number of rows.
E. There is no consistent relationship between this total and the number of
rows.
15
ACT Prep: Intermediate Algebra / Coordinate Geometry
Practice Problems
F.
G.
H.
J.
K.
9. What 2 numbers should be placed in the blanks so that the difference
between consecutive numbers is the same? 17, ___, ___, 41
A. 23, 29
B. 24, 34
C. 25, 33
D. 26, 35
E. 27, 31
16
10. If x is a real number such that x3 = 64, then x2 + √x = ?
K. 47
J. 20
H. 18
G. 10
F. 4
22. What is the slope-intercept form of 8x - y - 6 = 0?
F. y = -8x - 6
G. y = -8 + 6
H. y = 8x - 6
J. y = 8x + 6
K. y = 6x - 8
17
13. The number of students participating in fall sports at a certain high
school can be shown by the following matrix:
Football
Cross-Country
Soccer
Tennis
80 ]
80
60
[ 40
The athletic director estimates the ratio of the number of sports awards
that will be earned to the number of students participating with the
0.3
Tennis
following matrix.
0.4
Soccer
0.2
Cross-country
0.5
Football
Given these matrices, what is the athletic director's estimate for the
number of sports awards that will be earned for these fall sports?
A. 80
B. 88
C. 91
D. 92
E. 99
18
17. Point A is to be graphed in a quadrant, not on an axis, of the standard
y
(x, y) coordinate plane below.
quadrants of
the standard
(x,y) coordinate
plane
I
II
x
o
III
IV
If the x-coordinate and the y-coordinate of point A are to have opposite
signs, then point A must be located in:
A. Quadrant II only
B. Quadrant IV only
C. Quadrant I or III only
D. Quadrant I or IV only
E. Quadrant II or IV only
19
21. For all positive integers x, y, and z, which of the following expressions is
equivalent to x/y?
A.
B.
C.
D.
E.
31. What is the x-coordinate of the point in the standard (x,y) coordinate
plane at which the 2 lines y = 2x + 6 and y = 3x + 4 intersect?
A. 1
B. 2
C. 4
D. 6
E. 10
20
35. A park has the shape and dimensions in blocks given below. A water
fountain is located halfway between point B and point D. Which of the
following is the location of the water fountain from point A? (Note: The
park's borders run east-west or north-south.)
A. 3.5 blocks east and 6 blocks north
B. 5 blocks east and 4.5 blocks north
C. 5 blocks east and 6 blocks north
D. 8.5 blocks east and 4.5 blocks north
E. 9 blocks east and 7.5 blocks north
F
7
E
3
D
C
N
12
A
10
B
21
45. What is the distance in the standard (x,y) coordinate plane between
the points (1,0) and (0,5)?
A. 4
B. 6
C. 16
D. 36
E. √26
47. A circle in the standard (x,y) coordinate plane is tangent to the x-axis
at 5 and tangent to the y-axis at 5. Which of the following is an equation
of the circle?
A. x2 + y2 = 5
B. x2 + y2 = 25
C. (x - 5)2 + (y - 5)2 = 5
D. (x - 5)2 + (y - 5)2 = 25
E. (x + 5)2 + (y + 5)2 = 25
22
48. In the complex numbers, where i 2 = -1,
F. i - 1
G. 1 + i
H. 1 - i
J. 1 - i
2
K. 1 + i
2
50. After polling a class of 20 music students by a show of hands, you find
that 8 students play the guitar and 9 students play the piano. Given that
information, what is the minimum number of students in this music class
who play both the guitar and the piano?
K. 17
J. 9
H. 8
G. 1
F. 0
23
51. Which of the following is the set of all real numbers x such that
x + 3 > x + 5?
A. The empty set
B. The set containing all real numbers
C. The set containing all negative real numbers
D. The set containing all nonnegative real numbers
E. The set containing only zero
24
55. Which of the following systems of inequalities is represented by the
shaded region of the graph below?
A. y < -2x and x > 3
C. y > -2x and x > 3
E. y > -2x and x < 3
B. y < -2x or x > 3
D. y > -2x or x > 3
25
56. If f(x) = x2 - 2, then f(x+h) = ?
H. x2 + h2 - 2
G. x2 - 2 + h
F. x2 + h2
K. x2 + 2xh + h2 - 2
J. x2 + 2xh + h2
58. A triangle, ΔABC, is reflected across the x-axis to have the image
ΔA'B'C' in the standard (x,y) coordinate plane; thus, A reflects to A'. The
coordinates of point A are (c,d). What are the coordinates of point A'?
F. (c, -d)
G. (-c, d)
H. (-c, -d)
J. (d, c)
K. Cannot be determined from the given information.
26
57. Which of the following is the graph, in the standard (x,y) coordinate
?
plane, of
A.
D.
B.
C.
E.
27
ACT Prep: Plane Geometry/Trigonometry
Practice Problems
6. A rectangular lot that measures 150 ft by 200 ft is completely fenced.
What is the approximate length, in feet, of the fence?
F. 300 G. 350 H. 400 J. 700 K. 1400
20. If a rectangle measures 54 meters by 72 meters, what is the length,
in meters, of the diagonal of the rectangle?
J. 126 K. 252
H. 90
G. 63
F. 48
28
24. For right triangle ΔRST shown below, what is tan R?
R
F. r/s
J. t/s
G. r/t
K. s/t
H. t/r
t
S
s
r
T
25. A chord 24 inches long is 5 inches from the center of a circle, as
shown below. What is the radius of the circle, to the nearest tenth of
an inch?
A. 29.0 B. 24.5 C. 16.9 D. 13.0 E. 10.9
5
24
r
29
27. After a snowstorm, city workers removed an estimated 10000 cubic
yards of snow from the downtown area. If this snow were spread in an
even layer over the entire rectangular football field shown below, about
how many yards deep would the layer of snow be?
.5
53
yd
A. Less than 1
B. Between 1 and 2
C. Between 2 and 3
D. Between 3 and 4
E. More than 4
10 yd
100 yd
10 yd
30
28. The hypotenuse of the right triangle ΔPQR shown below is 16 feet long.
The sine of < P is 3/5. About how many feet long is QR? P
F. 8.0
G. 9.6
H. 12.4
J. 14.3
16
K. 15.4
R
Q
30. Points B and C lie on AD as shown below. The length of AD is 30 units;
AC is 16 units long; and BD is 20 units long. How many units long, if it can
be determined, is BC?
A
F. 4
G. 6
H. 10
B
J. 14
C
D
K. Cannot be determined from the
given information
31
33. Parallelogram ABCD, with dimensions in inches, is shown in the diagram
below. What is the area of the parallelogram, in square inches?
B
A. 18
B. 36
C. 39
D. 45
C
E. 72
5
A
4
3
6
D
37. Members of the fire department lean a 30-foot ladder against a
building. The side of the building is perpendicular to the level ground so
that the base of the ladder is 10 feet away from the base of the building.
To the nearest foot, how far up the building does the ladder reach?
A. 10
B. 20
C. 28
D. 31
E. 40
32
38. A square is circumscribed about a circle of 7-foot radius as shown
below. What is the area of the square, in square feet?
J. 49π K. 196
H. 98
G. 56
F. 49
7
39.The ratio of the side lengths for a triangle is exactly 12:14:15. In a
second triangle similar to the first, the shortest side is 8 inches long. To
the nearest tenth of an inch, what is the length of the longest side of the
second triangle?
A. 11.0 B. 10.0 C. 9.3
D. 6.4
E. cannot be determined from the
given information
33
40. In the figure below, ABCD is a trapezoid, E lies on AD, and
<
angle measures are as marked. What is the measure of BDC?
H. 30
J. 35
K. 45
B
C
) 30
?
)
G. 25
) 60
A
105
)
F. 15
D
E
41. In the figure shown below, each pair of intersecting line segments
meets at a right angle, and all the lengths given are in inches. What is the
perimeter, in inches, of the figure?
A. 40
B. 52
C. 56
D. 66
E. 80
10
4
6
6
4
26
34
46. The ratio of the radii of two circles is 4:9. What is the ratio of their
circumferences?
K. 9:18π
J. 4:8π
H. 16:81
G. 4:9
F. 2:3
35
52. Pentagons have 5 diagonals, as illustrated below.
How many diagonals does the octagon below have?
F. 8
H. 20
K. 40
G. 16
J. 30
36
53. Douglas wants to draw a circle graph showing the favorite colors of his
friends. When he polled his friends asking each their favorite color, 25%
of his friends said red; 30% of his friends said blue; 20% of his friends
said green; 10% of his friends said purple; and the remaining friends said
colors other than red, blue, green, and purple. The colors other than red,
blue, green, and purple will be grouped together in an Other sector. What
will be the degree measure of the Other sector?
E. 10
D. 15
C. 27
B. 54
A. 108
37
54. If sin θ = -3/5 and π < θ < 3π/2, then tan θ + ?
F. -5/4 G. -3/4 H. -3/5 J. 3/4 K. 4/5
60. What is sin π/12 given that π/12 = π/3 - π/4 and that
sin (α - β) = (sin α)(cos β) - (cos α)(sin β)?
(Note: You may use the following table of values.)
θ
sin θ
cos θ
π/6
1/2
√3/2
π/4
√2/2
√2/2
π/3
√3/2
1/2
F. 1/4
G. 1/2
H. √3 - 2
J. √3 - √2
K. √6 - √2
4
2
4
38
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39
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