1. Content Covered by the ACT Mathematics Test
In the Mathematics Test, three subscores are based on six content areas:
pre-algebra, elementary algebra, intermediate algebra, coordinate geometry, plane geometry, and trigonometry.

2. Pre-Algebra
Pre-Algebra (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.

3. Elementary Algebra
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.

4. Intermediate Algebra
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.

5. Coordinate Geometry
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.

6. Plane Geometry
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.

7. Trigonometry
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.

8. Read each question carefully to make sure you understand the type of answer required.
If you use a calculator, be sure it is working on test day and has reliable batteries.
Solve the problem.
Locate your solution among the answer choices.
Make sure you answer the question asked.
Make sure your answer is reasonable.
Check your work.

9. Pre-Algebra 1. Number Problems 2. Multiples, Factors, & Primes
3. Divisibility and Remainders
4. Percentages
5. Ratios and Proportions
6. Mean, Median, & Mode
7. Probability
8. Absolute Value
9. Exponents and Roots
10. Series

10. Elementary Algebra 1. Substitution
2. Simplifying Algebraic Expressions
3. Writing Expressions & Equations
4. Solving Linear Equations
5. Multiplying Binomials
6. Inequalities

11. Intermediate Algebra 1. Solving & Factoring Quadratic Equations
2. Solving Systems of Equations
3. Relationship between Sides of an Equation
4. Functions
5. Matrices
6. Logarithms

12. Plane Geometry 1. Angles 2. Triangles 3. Polygons 4. Circles
5. Simple 3-D Geometry

13. Coordinate Geometry 1. Number Lines & Inequalities
2. The (x,y) Coordinate Plane
3. Distance and Midpoints
4. Slope
5. Parallel & Perpendicular Lines
6. Graphing Equations
7. Conic Sections

14. Trigonometry 1. SOHCAHTOA 2. Solving Triangles
3. Trigonometric Identities
4. Trigonometric Graphs

15. Whole Numbers

16. Terms and Operations Basic Terms sum (total) difference product
quotient
remainder

17. Terms and Operations 2. Symbols of inclusion parentheses brackets
braces
(2 + 3) •4 = 20
2 + (3 • 4) = 14

18. Terms and Operations 3. Order of Operation Parentheses
Exponents, radicals
Multiplication/division
Addition/subtraction

19. Terms and Operations
Factoring and Canceling
5( ) = 5(9) = 45

20. Factors, Multiples, and Primes
Factor - A number that divides another number evenly.
Multiple - A number that is evenly divisible by another number.
Prime - A number that is only divisible by 1 and itself.

21. Odd and Even Numbers
Odd number - A number that is not evenly divisible by 2.
Even number - A number that is divisible by 2.
1. Even + Even = Even Even • Even = Even
2. Even + Odd = Odd Even • Odd = Even
3. Odd + Even = Odd Odd • Even = Even
4. Odd + Odd = Even Odd • Odd = Odd

22. Consecutive Integers
Consecutive integers immediately follow one another
1st nd rd
n n n + 2
Consecutive even integers and consecutive odd integers
1st nd rd
n n n + 4

23. Fractions Convert Mixed Numbers to Improper Fractions
Convert Improper Fractions to Mixed Numbers
Reducing Fractions to Lowest Terms
Common Denominators
Operations of Fractions
Comparing Fractions

24. Signed Numbers Absolute Value Adding Negative Numbers
Subtracting Negative Numbers
Multiplying Negative Numbers
Dividing Negative Numbers
Subtracting a negative is the same as adding a positive number.
Adding a negative number is the same as subtracting a positive number.
Multiplying or dividing an odd number of negative signs will always result in a negative number.
Multiplying or dividing an even number of negative signs will always result in a positive number.

25. Decimals
A decimal is a special way of writing fractions using a denominator that is a power of 10.
Millionths
Hundred Thousandths
Ten Thousandths
Thousandths
Hundredths
Tenths

26. Decimals Converting Fractions to Decimals
Converting Mixed Numbers to Decimals
Converting Improper Fractions to Decimals
Converting Decimals and Fractions to Mixed Numbers
Operations of Decimals

27. Percents Converting to and from Percents Operations of Percents
Percent Story Problems

28. Mean, Median, and Mode Mean - The average of a set data values Median
The median of an odd number of data values is the middle value
The median of an even number of data values is the average of the middle two
Mode - The value that occurs most often

29. Ratios and Proportions
Ratio - A relationship between two quantities
Proportion - An equation relating two proportions
1:2 = 2:4 1*4 = 2*2
The product of the means equals the product of the extremes.

30. Exponents Rules For Exponents
If a > 0 and b > 0, the following hold true for all real
numbers x and y.

31. Roots and Radicals
Square Root
Cube Root
Rational Exponents

32. Algebraic Operations Elements of Algebra Algebraic Terms Variables
Coefficient
Variable
Exponent
Variables
Coefficients
Exponent

33. Operations of Algebraic Terms
Adding and Subtracting
Multiplying and Dividing
Algebraic Fractions
Factoring Algebraic Expressions
Absolute Value in Algebraic Expressions
Radicals in Algebraic Expressions

34. Algebraic Equations and Inequalities
Solving Algebraic Formulas
Solving Linear Equations
Solving Simultaneous Equations
Solving Quadratic Equations
Algebraic Inequalities
Exponents in Equations and Inequalities
Rational Equations and Inequalities
Radical Equations

35. Geometry Notation
Line Segment PQ
P Q B
Angle BAC <BAC
Vertex A C

36. Angle Properties
90°
180° 0°
270°

37. Angle Properties Right Angle Acute Angle Obtuse Angle Vertical Angle
x = z
w = y x y w z

38. Angle Properties Transversal - A line that intersects parallel line
5 6
3 4
Alternate Interior
Alternate Exterior
Vertical Angles
Same Side Interior Angles

39. Polygon Properties
Polygon - A closed figure created by three or more lines.
Triangle - A polygon with exactly three sides
Quadrilateral - A polygon with exactly four sides
Pentagon - A polygon with exactly five sides
Hexagon - A polygon with exactly six sides

40. Polygon Properties
Regular Polygon - A polygon with equal sides and equal angles.
The sum of the exterior angles of a polygon is 360°
The sum of the interior angles of a polygon is 180(n - 2) where n is the number of sides in the polygon

41. Triangle Properties Scalene Triangle - A triangle with no equal sides
Isosceles Triangle - A triangle with exactly two sides equal, If the two sides are equal than the angles opposite the two sides are equal and vice versa.
Equilateral Triangle - A triangle with three sides equal, If the three sides are equal than each angle equal 60° and vice versa

44. Perimeter and Area

45. Area and Volume

46. Circle Properties
chord
diameter
secant
radius
tangent

47. Circle Properties Central Angle - An angle with
the vertex at the center and
with sides that are radii.
Inscribed Angle - An angle with
a vertex on the circle and with
sides that are chords. B A O D C E

48. Coordinate Geometry

49. Coordinate Geometry
Slope
Equations of Lines

50. Coordinate Geometry Parallel lines m1 = m2
Perpendicular lines m1 = -1/m2

51. Coordinate Geometry Transformations
h(x) = af(x) vertical stretch or shrink
h(x) = f(ax) horizontal stretch or shrink
h(x) = f(x) + k vertical shift
h(x) = f(x+h) horizontal shift
h(x) = -f(x) reflection in the y-axis
h(x) = f(-x) reflection in the x-axis

52. Logarithms log a (ax) = x for all x   a log ax = x for all x > 0
log a (xy) = log a x + log a y
log a (x / y) = log a x – log a y
log a xn = n log a x
Common Logarithm: log 10 x = log x
Natural Logarithm: log e x = ln x
All the above properties hold.

53. Trigonometric Functions
60° 1 2
30°
45° 1 S A T C

54. Trigonometric Functions

55. Trigonometric Functions

56. Trigonometric Functions

57. Trigonometric Functions

58. Trigonometric Functions

59. Trigonometric Functions

60. Coordinate Geometry Quadratic functions f(x) = ax2+bx+c
Complete the square f(x) = a(x-h)2+ k
Discriminant b2-4ac
Vertex (h,k)
Quadratic Formula

61. Conic Sections Circles (x – h)2 + (y – k)2
Parabolas (x – h)2 = 4a(y – k) (Up/Down) (y – k)2 = 4a(x – h) (Right/Left)
Ellipses (x – h)2 /a2 + (y – k)2/ b2 = (y – k)2 /a2 + (x – h)2/ b2 = 1
c 2 = a 2 - b 2
Hyperbolas (x – h)2 /a2 - (y – k)2/ b2 = 1
(y – k)2 /a2 - (x – h)2/ b2 = 1
c 2 = a 2 + b 2

62. Circles Standard Equation of a Circle [Center at (h,k)]
The standard form of the equation of a circle
centered at (h,k) is y
Radius r
Point on
circle (x,y)
Center (h,k) x

63. Parabolas (x – h)2 = 4a(y – k) Vertex (h, k) Focus (h, k + a)
a > 0 opens up
a < 0 opens down
(y – k)2 = 4a(x – h)
Vertex (h, k)
Focus (h + a, k)
a < 0 opens left
a > 0 opens right

64. Ellipses

65. Hyperbolas -

66. DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document. Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test. You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator. Note: Unless otherwise stated, all of the following should be assumed.
1 Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane.
3. The word line indicates a straight line.
4. The word average indicates arithmetic mean.

67. 1. Which of the following is equivalent to (x)(x)(x)(x), for all x ?
a. 4x b. x c. x d. 4 + x e. 2x2

68. 2. A rectangle is twice as long as it is wide
2. A rectangle is twice as long as it is wide. If the width of the rectangle is 3 inches, what is the rectangle's area, in square inches?
F G H J K. 18

69. 3. A vendor has 14 helium balloons for sale: 9 are yellow, 3 are red, and 2 are green. A balloon is selected at random and sold. If the balloon sold is yellow, what is the probability that the next balloon, selected at random, is also yellow?

70. 4. 3 x = ? F. -30, G H J K. 0

71. 5. For all x > 0,
simplifies to:
A. x + 3
B. x + 4
C. 2(x + 3)
D. 2(x + 4)
E. 2(x + 3)(x + 4)

72. If one leg of a right triangle is 8 inches
long, and the other leg is 12 inches long,
how many inches long is the triangle's
hypotenuse? F. G. H. J.
K. 4

73. 7. In the standard (x,y) coordinate plane,
the graph of (x - 2)2 + (y + 4)2 = 9 is a circle.
What is the area enclosed by this circle,
expressed in square coordinate units?
A. 3π
B. 4π
C. 6π
D. 9π
E. 16π

74. 8. How many solutions are there to
the equation x = 0 ?
F. 0
G. 1
H. 2
J. 4
K. 15

75. 9. A circle with center (-3,4) is tangent to
the x-axis in the standard (x,y)
coordinate plane.
What is the radius of this circle?
A. 3
B. 4
C. 5
D. 9
E. 16

76. sin A = what is the value of cos A ?
In ∆ABC, if <A and <B are acute angles, and
sin A = what is the value of cos A ? F. G. H. J. K

77. What are the values of a and b, if any,
where a|b - 2| > 0 ?
A. a < 0 and b < 2
B. a < 0 and b = 2
C. a < 0 and b > 2
D. a > 0 and b < 2
E. There are no such values of a and b.

78. In a shipment of 1,000 light bulbs,
of the bulbs were defective. What is the
ratio of defective bulbs to nondefective bulbs? F. G. H. J. K.

79. 13. Which of the following is divisible by 3 (with no remainder)?
2,725
4,210
4,482
6,203
8,105

80. 14. A particle travels 1 x 108 centimeters per
second in a straight line for 4 x 10 -6
seconds. How many centimeters has it
traveled?
F x 102
G x 10 13
H. 4 x 102
J x
K x

81. 15. The triangle below is isosceles with
<M = < N. What is the measure of <N ?
A. 22°
B. 68°
C. 78°
D. 79°
E. 89°

82. 16. In the figure, AB = AC and BC is 10
units long. What is the area, in square
inches, of  ABC ?
F. 12.5
G. 25 H.
J. 50
K. Cannot be determined from the given
information

83. 17. Which of the following statements
completely describes the solution set for
3(x - 4) = 3x - 12 ?
F. x = 3 only
G. x = 0 only
H. x = -12 only
J. There are no solutions for x.
K. All real numbers are solutions for x.

84. 18. When graphed in the (x,y) coordinate
plane, at what point do the lines
x + y = 5 and y = 7 intersect?
A. (-2,0)
B. (-2,7)
C. (0,7)
D. (2,5)
E. (5,7)

85. where h is the altitude, and b1 and b2 are the
19. The area of a trapezoid is h/2 (b1 + b2),
where h is the altitude, and b1 and b2 are the
lengths of the parallel bases. If a trapezoid
has an altitude of 5 inches, an area of 55
square inches, and one base 13 inches long,
what is the length, in inches, of its other
base?
F.  9.0
G. 16.8
H. 19.4 J
K.  97.0

86. 20. If you have gone 4.8 miles in 24 minutes, what was your average speed, in miles per hour?B
C.  12.0 D E

87. 21. If a and b are any real numbers such
that 0 < a < 1 < b, which of the
following must be true of the value of
ab ?
F. 0 < ab < a
G. 0 < ab < 1
H. a < ab < 1
J. a < ab < b
K. b < ab