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ACT MATHEMATICS Improving College Admission Test Scores
Contributing Writers Marie Haisan L. Ramadeen Matthew Miktus David Hoffman  ACT is a registered trademark of ACT Inc.  Cop...
TABLE OF CONTENTS Introduction  iv  Glossary of Terms  vi  Summary of Formulas, Properties, and Laws  xvi  Practice Test A...
INTRODUCTION The American College Testing Program  ACT  is a comprehensive system of data collection, processing, and repo...
5. If the item asks for an equation, check to see if your equation can be transformed into one of the choices. 6. Always w...
GLOSSARY OF TERMS ABSCISSA An ordered pair  x, y  specifying the distance of points from two perpendicular number lines  x...
BASE of a power The number to which an exponent is attached. In the expression x3, x is the base, 3 is the exponent. BASE ...
CONSTANT A symbol representing a single number during a particular discussion  e.g., x2   x   5 has  5 as the constant tha...
DISTANCE The length of the line joining two points or the length of a perpendicular line joining two lines. Distance may b...
FRACTIONS  A fraction is part of a whole. It is written A . B is B  the denominator and tells how many parts the whole was...
LIKE TERMS  Terms whose variables  letters  are the same  e.g., 3x and 12x. LINE SEGMENT A part of a line that consists of...
PENTAGON A polygon that has five sides. PERCENT AGE  Hundredths  symbol     e.g., 5  of a quantity is  PRIME NUMBER A natu...
REFLEXIVE The reflexive property of equality  any number is equal to itself  e.g., 5   5.  QUADRILATERAL A polygon that ha...
SIDES A side of a polygon is any one of the line segments forming the polygon. SIMILAR terms  like terms  e.g., 5x2 and 8x...
VARIABLE A symbol representing any one of a given set of numbers. Most common are x and y. VERTEX of an isosceles triangle...
Summary of Formulas, Properties, and Laws I. Proper ties of Integer s Commutative Laws of addition and multiplication a b ...
VII. Some Commonly Used Percent, Decimal, and Fraction Equivalents   .167   1 6  62 1     .625   5 8 2  33 1     .333   1 ...
XIII. Formulas for Triangles  V   1. Perimeter of any triangle is a   b   c  sum of lengths of sides . 2. Perimeter of equ...
PRACTICE TEST A 60 minutes     60 questions  Directions  Answer each question. Choose the correct answer from the 5 choice...
4.  In the figure, what is the value of x  60    40    F. G. H. J. K. 5.  3x 3y 10 20 30  If cos      3 in the first quadr...
9.  It takes Mr. Smith H hours to mow his lawn. After three hours it begins to rain. How much of the lawn is not mowed  A....
13. In the sketch below, the area of each circle is 4  . What is the perimeter of WXZY   W X  Y  A. B. C. D. E.  Z  8 32 1...
17. Which symbol below makes this expression true  24 ___ 42 A.   B.   C.   D.    E. A relationship cannot be determined. ...
21. The function  M O  N P  is defined as  MP     NO. What is the value of A. B. C. D. E.  2 4   6 8  -8 -6 -4 -2 4  22. I...
26. What is the sum of angles a   b   c   d in terms of x  a  x    b  c d  x 2x 180     x 180     2x 360 - x  F. G. H. J. ...
31. A 24-inch diameter pizza is cut into eight slices. What is the area of one slice  A. 3   B. 6   C. 12   D. 18   E.    ...
36. The formula to convert degrees Fahrenheit to  5 Celsius is  C   F     32  . What temperature Celsius 9  is 86   Fahren...
40. What is the slope of the line perpendicular to a line with the equation ax   by   c   F.  b a  G.  -b  H.  c a  J.  -a...
44. If the radius of a circle is reduced by 50 percent, by what percent is its area reduced  F.  33 1    G.  50   H.  66 2...
47. Given circle O with minor arc     AB   60   and OA   12. What is the area of sector AOB    A B O  A. B. C. D. E.  12  ...
51. Which of the following has the greatest number of integer factors other than itself and one  A. B. C. D. E.  12 16 24 ...
55. Given rectangle ABDC, which of the following statements must be true  A  B  C  D  A. B. C. D. E.  AB   BD   AD AB   BD...
59. If  x   8    12, then x     A. B. C. D. E.  4 only 20 only either -20 or 4 either -4 or 20 either 0 or 12  60. Which o...
PRACTICE TEST B 60 minutes     60 questions  Directions  Answer each question. Choose the correct answer from the 5 choice...
3.  In the figure l1    l2 and l3 is a transversal. What is the value of q     p    DO YOUR FIGURING HERE.  l3 135    p   ...
6.  If tan x   3 , what is the value of 4  cos x   sin x   F.  4 3  G.  9 16  H.  7 5  J.  25 12  K.  1  7. A square sheet...
9. If x   ut   1 at2, what is t when x   16, 2  u   0, and a   4  A. B. C. D. E.  2 2 4 2 2 2 4  10. If 18  of the senior ...
13.  2a     3 5a   3         2 5  A.  -21  B.  -9  C.  - 21  D.  -9  E.  9 10  10  10  14. A plumber charges  35 flat fee ...
16. If the area of the triangle is 8, what is the value of x     x     2    x   2   F. G.  5 2 2 5  H.  4 3  J.  2 3  K.  ...
20.     1 2       4        2 17 34  F.  2  G.  1 2  H.  0  J.  -1  K.  -2  2  21. If  x   2 2    22 3 and x   0, what is t...
24. What is the equation of the line, in standard form, connecting points  2, -3  and  4, 4   F. G. H. J. K.  7x     2y   ...
28. If 5x   3y   23 and x and y are positive integers, which of the following can be equal to y   F. G. H. J. K.  3 4 5 6 ...
32. What is the slope of the line connecting the points  2, -2  and  3, -2   F. G. H. J. K.  undefined 1 0 -1 -4  33. Whic...
35. If    ABE is similar to    ACD, what is the value of AB   A x B  2  E  3  C  5  A.  71  B. C. D.  3 2 11  E.  -2  D  2...
38. If 2   - x    F. G. H. J. K.  x   4, what is the value of   x     2   -4 -2 0 2 4  39. Which of the following lines is...
42. If x    x   2, what is the value of  3    5      F. G. H. J. K. 43. If  8 10 12 14 None of the above 15k   1 and x   4...
46. Which of the following is equivalent to cos x sin x     sin x cos x F.  cos x   sin x sin x cos x  G.  1 sin x cos x  ...
50. If f x    x2     5 and g x    5x, what is the value of f g 3       g f 3    F. G. H. J. K.  400 240 200 40 0  51. What...
54. Which of the following is not a solution for  5     2x    3  F. G. H. J. K.  -2 -1 0 2 5  55. Which of the following f...
2  58. 8 3     2    1     F.  1 16  1 2  G. H. J. K.  2 4 16  59. If ABCD is a square and CDE is an equilateral triangle, ...
PRACTICE TEST C 60 minutes     60 questions  Directions  Answer each question. Choose the correct answer from the 5 choice...
4.  The area of the above figure is  F. G.  205   180   25    H.  180    J. K.  90   25   50   180   1  5. 1   1   A. B. C...
7.  The ordered pair  -4, 0  lies in which quadrant  A. IV B. III C. II D. I E. none of the above  8. If f x    5 and g x ...
11. If 3x   9y   18, what is the value of x   A. B. C. D. E.  18     9y 3y   6 6y   3 -3y   6 -6y   3  12. If g2     g    ...
15. A merchant purchased a line of handbags for  30 each. She wants to offer a 10  discount off the listed selling price a...
18. The figure below shows two parallel lines, l1 and l2, cut by transversal t. What is the measure of angle x  t x y  F. ...
22. In a certain high school, 30  of the students are on the honor roll, and 40  of the students are boys. If 15  of the s...
25. If 3 x   2    5 x     8 , what is the value of x 2  A. B. C. D. E.  23 25 40 46 50  26. Let  y  be defined as -y2     ...
29. If x     0, x2   y, and 2y   x, what is the value of x   A.  1  B.  1 2  C.  1 4  D.  1 8  E.  1 16  30. If Ax   By   ...
32. If twice a number increased by 5 equals 3 , what is the number  5  F.  33.  5 11  -  G.  11 5  H.  5 11  J.  - 11  K. ...
35. How much should Kelly score on her third test to get an average of 91, if her previous scores are x   88 and 90     x ...
38. In the triangle below, what is the value of x  60    6  F. G. H. J. K.  x  3  1 2 3 2 3  39. In the figure below, what...
41. If the radius of a circle is increased by 10 , by what percent is the area increased  A. B. C. D. E.  10  20  21  32  ...
45. In which of the following equations does the product of the roots equal 14  x2     2x   14   0 2x2   x   7   0 x2   14...
48. In the circle, DE is a diameter and EF   OE. What is the measure of arc DF   F  D  F. G. H. J. K.  O  E  100   120   1...
52. What is the simplest form of 48 4  1   6  12      27     F. G. H.  -2 3   2 2 4 2 - 3  J. K.  2 3 0  53. In the triang...
57. What is the center of the circle 2x2   2y2     4x   16y   128  A.  1, -2  B.  1, 4  C.  1, -4  D.  -1, -2  E.  -1, -4 ...
59. In the figure below, what is the length of AB   A 2  2 2 2 2  A. B. C. D. E.  10 12 14 15 20  4  B  60. In the figure ...
PRE-ALGEBRA SKILL BUILDER ONE Operations with Whole Numbers If a numerical expression does not contain parentheses, first ...
Solution Change to improper fractions and multiply by the reciprocal of the divisor  11 7 11 6           3 6 3 7 2 11 6  1...
Divide both sides by 5  5x 10   5 5 x 2 Example If 3a   1   2a   6, then a     Solution Combine like terms  5a   1   6 Sub...
Orientation Exercises 1.  A. B. C. D. E. 2. 2  A. B. C. D. E.  12 13 17 20 38  B. C. D. E.  A. B. C. D. E.  A. B. C. D. E....
Practice Exercise 1 1.  2    3    2       A. 7 B. 8 C. 10 D. 12 E. 18  6. What is the decimal equivalent of 1    2. 16   2...
Practice Test 1 1. 12   24    6    3     A 2 B. 9 C. 18 D. 24 E. 48  6. What is the decimal equivalent of 7   A. B. C. D. ...
SKILL BUILDER TWO Averages  Probability  Example John got a 75 on his first math test, 85 on his second, 80 on his third, ...
Total price    1.25 .99   9.99  12.23  Change 6  to .06.Multiply the decimal by the total price to get the sales tax   12....
2 3 5     or 1 mile, the cost of the first 5 5 5 mile    1.80   1.50    3.30. The trip is five miles  therefore, the four ...
Orientation Exercises 1. The average height of 5 basketball players at South High School is 6 feet 2 inches. If four of th...
Practice Exercise 2 1. The video store rented 42 videotapes on Monday, 35 on Tuesday, 51 on Wednesday, and 32 on Thursday....
9. Mr. King left  1 of his estate to his wife and 2  1 of the remainder to his granddaughter. 3 What part of his estate is...
Practice Test 2 1. What is the average of all the multiples of 5 from 5 to 50, inclusive  A. 5 B. 11 C. 21 D. 26.5 E. 27.5...
9. Betsy s softball team won 47 games, lost 15 games, and tied none. What fractional part of the games played did the team...
SKILL BUILDER THREE Exponents An exponent tells you how many times to write the base in a multiplication problem. For exam...
Products of Even and Odd Numbers The product of two even numbers is always an even number. 4    8   32 The product of two ...
Factorials For a positive integer n, the product of all the positive integers less than or equal to n is called a factoria...
Orientation Exercises 1. 43     32   80     A. 11 B. 14 C. 47 D. 53 E. 56 2. What is the sum of all the ODD integers from ...
Practice Exercise 3 1. What is the value of  .02 2   A. .4 B. .04 C. .004 D. .0004 E. .22 2. 4 3     A. B. C. D. E.  3 2  ...
Practice Test 3 1. 8 1     5 0   3 2     A. 9 B. 12 C. 13 D. 14 E. 16  1 7. 8   3 4  A. 1  2  2. The numerical value of 11...
ELEMENTARY ALGEBRA SKILL BUILDER FOUR Multiplication of Signed Numbers  Evaluation of Algebraic Expressions  Rules for Mul...
e. The sum of a number and one-fourth of the number sum    indicates addition,    of    indicates multiplication f.  Examp...
Orientation Exercises 1.  What is the product of  -7  and   8   A. -56 D. 1 B. C.  2.  3.  -1 -1  1 7  E.  56      18 y 3 ...
Practice Exercise 4 1.  2.  3.  What is the product of            3        and  -15       5     A.  -25  D.  15 3  B. C.  ...
Practice Test 4 1.  2.  3.  Find  -4             1         5             1        .     4     A.  -2  D.  1 9  B.  -1  E. ...
12.    A number n equals 7 more than half the number    can be written as  A.  n 7  1n  B.  n   7n    C.  n   7n    D.  n ...
SKILL BUILDER FIVE Evaluating a Formula  Equations Containing Fractions  A formula is an instruction written in the symbol...
Set each factor equal to zero 3x     2   0 3x   2 x  2  x 3  0 x   -3  3  x   -3 is the smaller of the two solutions.  Mis...
Orientation Exercises 1.  x  y z is used to find 3 the average  A  of three numbers x, y, and z. What is the average of 86...
Practice Exercise 5 1.  Using the formula P   2 l   w , find P if l   40 and w   20. A. 60 B. 80 C. 100 D. 120 E. 140  6. ...
11. Which of the following is not equal to the other three  A. 25  of 80 B. 1 of 100 C. D. E.  5  40    0.5 2 100 20  12. ...
Practice Test 5 1.  If C   A. B. C. D. E.  2.  3.  7.  If s   3 and t   2, then find w if  8.  st  3   s   t   2   16.50  ...
11. A bookstore owner wishes to generate  5,000 in profit each month. Each hardcover  h  generates  5.25 in profit, and ea...
SKILL BUILDER SIX Fundamental Operations with Monomials and Polynomials Example  4x   5y     2x     11y      Solution Comb...
Orientation Exercises 1.  2.  3.  4.  5.   x     4  x   4    A. x2   16 B. x2     16 C. x2     8x     16 D. x2     8 E. x2...
Practice Exercise 6 1.  2.  3.  4.  A rectangle has a length of x   7 and a width of 2x     3. If its perimeter is 32, wha...
Practice Test 6 1.  2.  3.  Maria has  x   14  pencils, Brian has  x   10  pencils, and Scott has  x   6  pencils. All the...
INTERMEDIATE ALGEBRA SKILL BUILDER SEVEN Linear Inequalities in One Variable Solving an algebraic inequality is similar to...
Operations with Integer Exponents  Fractional Exponents  Rule 1. The exponent of the product of two powers of the same bas...
The slope and y intercepts are m   0  horizontal line  b   -5  Example Write a linear equation that is parallel to y   3x ...
Orientation Exercises 1.  Solve for x  2x     3   11 A. x   4 D. x   -7 B. x   7 E. x   -4 C. x   -7  2.  Solve for x  7  ...
13.  6    108  4    10-3    A. B. C. D. E.  2.4    105 2.4    106 2.4    107 2.4    104 2.4    1011  15. Which graph below...
Practice Exercise 7 1.  2.  Solve for x  2   3 5     x    8 A. x   3 B. x   3 C. x   -9 D. x   -3 E. x   -3 Which of the f...
10. Find the y-intercept of the line with the equation 2x   y   5. A. B. C.  -5 -2 -1  D. E.  0 5  2  11. The slope of a l...
Practice Test 7 1.  Solve for a   3      a     2    3   a A. a   1 B. a   -1 C. a   1 D. a   1 E. a   -1  2.  Which of the...
12. Which of the following lines does NOT 2 have a slope equal to   3 A. 2x     3y   0 B.  3y   12   2x  C.  2x   3y   9  ...
SKILL BUILDER EIGHT Systems of Two Linear Equations with Two Variables If there are two unknown quantities in a problem, t...
Example 2      2           x     9        . Find the product             3x     9           4 x     1        Solution 1  2...
      5 3 3     5   5 9     5 15   5     3     5 15    15     5 15 The division of two radicals of the same order uses a p...
Example Express 5     36     2     36 as one term. Solution Extract the i    5i 36     2i 36   5i     6     2i     6   30i...
Observe that the value of b2     4ac, the radicand 2     b    b     4ac indicates the 2a sort of roots to expect.  in the ...
Orientation Exercises 1.  2.  Which of the following systems of equations does NOT have a solution  A. 2x   4y   26 D. 2x ...
Practice Exercise 8 1.  Solve the following system  a     12b   5 2a     10b   66 A. B. C. D. E.  2.  3.  Simplify  E. 4. ...
Practice Test 8 1.  The sum of the reciprocals of two numbers is 8 and the difference of their  6.  15  A. B. C.  reciproc...
COORDINATE GEOMETRY SKILL BUILDER NINE Graphing on the Number Line   any number greater than -1 and less than or equal to ...
Example   1, 4 ,  4, 8 ,  5, 8 ,  8, 9   is an example of a function. Domain    1, 4, 5, 8  Range    4, 8, 9  No vertical ...
Example If given the equation of a line, transform it into the form y   mx   b where m represents the slope. 3x   2y   6 2...
Orientation Exercises 1.  The diagram represents the graph of what set of numbers      -3 -2 A. B. C. D. E.  2.  3.    -1 ...
Practice Exercise 9 1.  The figure represents the graph of what set of numbers   A. B. C. D. E. 2.  6.  B. C.   natural nu...
Practice Test 9 1.  The figure represents the graph of what set of numbers   A. B. C. D. E.  2.  3.   natural numbers   wh...
7.  What is the length of line segment AB    A. B. C. D. E. 8.  4 2 4 3 2 6 8 2  9.  Point P  0, 3  is the center of a cir...
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since...
Example Graph y   1. Solution If y is the only variable in the equation, it is a horizontal line one unit above the x-axis...
Identify and graph each of the following. Example y   x2 Parabola  x y  0 0    1   2 1 4  y   x2   2 Parabola  x y  0 2   ...
Let the negative variable equal zero and x2  1 4  Example  x     2 2    y     1 2   36 Circle center at  2, 1  radius   36...
Example Solve the pair of equations x y 4 -x   y   2  x  y     4 y        x   4 1 m     1 b 4 The graph is the line and al...
Orientation Exercises 1.  Which ordered pair represents a point that lies on the given graph   5.  Identify the graph of t...
9.  Which graph below is the graph of xy   2   10. The inequality graphed below is   A. A. B. C. D. E.  B.  C.  D.  E.  11...
Practice Exercise 10 1.  An equation of the y-axis is  A. B. C. D. E.  2.  x 0 y 0 x y 0 x   y 0 The y-axis has no equatio...
8.  The shaded region in the figure represents the solution set of   9.  If xy   12  hyperbola  and x2   y2   25  circle ,...
Practice Test 10 1.  Which of the following equations is linear  A. x2   y2   16 B. y   x2   16 C. x   y   16 D. xy   16 E...
7.  Which of the following equations could represent the graph   10. The graph below is the quadratic function f x    x2  ...
PLANE GEOMETRY SKILL BUILDER ELEVEN Lines, Segments, and Rays  AD and DA are the same segment.  The following examples sho...
Classifying Angles Angles are classified according to the number of degrees contained in the angle. Type of Angle acute an...
Example If BA     AC, find the number of degrees in angle x.  A  C  Solution Since BA     AC,    BAC   90  . Therefore, x ...
The arrows in the diagram indicate that the lines are parallel. The symbol    means    is parallel to     AB    CD .      ...
Example l    m and m    a   100  . Find the number of degrees in angles b, c, d, e, f, g, and h.  Example AB    ED,    B  ...
Orientation Exercises 1.  Which rays form the sides of    ABC                                                      A. AB ,...
Practice Exercise 11 1.  Three points, R, S, and T are collinear. Point S lies between R and T. If RS   and RS   48, find ...
9.  The height of the triangle below is 10 units. What is its area   10. The measure of the smaller angle in figure below ...
Practice Test 11 1.  In the figure, U, V, W, and X are collinear. UX is 50 units long, UW is 22 units long, and VX is 29 u...
9.  The perimeter of the triangle below is   A. B. C. D. E.  10. A letter carrier must go from point A to point B through ...
SKILL BUILDER TWELVE Properties of Triangles  Classification by Angles  Classification by Sides  Triangles can be classifi...
Example Classify each triangle pictured by the angles shown.  Solution   A   AB     AC and vertex m   A   48  . Thus  48  ...
An altitude of a triangle is a segment from one vertex perpendicular to the opposite side.  Altitude BD  An exterior angle...
A trapezoid is a quadrilateral with exactly one pair of parallel sides.  An isosceles trapezoid is a trapezoid whose nonpa...
Other theorems used to show triangles similar are SAS similarity theorem SSS similarity theorem where the angles are congr...
Solution x 2   42 x2   16 x2 x    62   36   20   20  x   4 5  x  2 5 NOTE  There are sets of numbers that satisfy the Pyth...
Solution PQ is the side opposite the 30   angle and its measure is one-half the hypotenuse PR. Thus, PR   8. QR, the side ...
Orientation Exercises 1.  In the figure,     P measures 30  . PS is a line segment and PQ   QT   TR   RS. Find the number ...
Practice Exercise 12 1.  In the figure, the three triangles are equilateral and share a common vertex. Find the value of x...
7.  In the figure,    RST is a right triangle. Hypotenuse RS is 8 units long, and side RT is 4 units long. How many units ...
Practice Test 12 1.  In the figure below,     ORS has one vertex at the center of a circle and two vertices on the circle....
7.  In the figure, what is the length, in inches, of a side of square ABCD   9.  If each side of an equilateral triangle h...
SKILL BUILDER THIRTEEN Basic Properties of a Circle  Radius, Diameter, and Circumference  Concentric circles are circles t...
Solution C     d 22    d 66   7 1     1             3  Multiplying both 22     7         7            22        66   7    ...
Example O is the center of the circle with a radius of 4   . Find the length of     AB intercepted by a central angle of 4...
Solution The sum of the measures of the minor and major arcs is 360  .  Solution 5 of 360. x is 12  x    3x   20    360 x ...
Solution A regular hexagon is a polygon whose six sides and six angles are congruent. Its central angle measures 60    360...
Example O is the center. Chords BC and AC form an inscribed angle of 30  . Find the central angle whose measure is x.  Are...
Trapezoid  1 h b1   b2  where h   altitude and b1 2 and b2 are the lengths of the parallel bases 1       6 15   7  2 1    ...
Orientation Exercises 1.  Following are the distances, in feet, of five points from the center of a circle. Point R   2.75...
7.  In the figure below,    PQR is an inscribed    . angle. Find the measure of PR  A. B. C. D. E. 8.  30   15   60   90  ...
Practice Exercise 13 1.  In the diagram, which line segment is a diameter   4.  O is the center. AB    CD . Find y.  A. A....
7.  The figure below is a circle inscribed within a square. The area of the shaded region is   A. B. C. D. E. 8.  The figu...
Practice Test 13 1.  In the figure,    RST is inscribed in circle O. If RS   6 units and ST   8 units, then the length of ...
6.  The diameter of the small circle is 10. The diameter of the larger circle is 14. The area of the shaded region is   A....
TRIGONOMETRY SKILL BUILDER FOURTEEN Right Triangle Trigonometry The following six ratios are called trigonometric function...
Solution sin A    1. Construct the angle 2. Find the third side opp 3. Tan equals adj  3     0.866 2 1 cos 60       0.500 ...
Trigonometric Identities There are eight fundamental relationships. These relationships among the trigonometric functions ...
Solution Expand the formula  cos 30     45      cos 30   cos 45       sin 30   sin 45   Substitute values  3 2 1 2        ...
Example Draw the graph of y   2 sin Solution  1 x. 2 1 2 2 2      1 4     amplitude   period  2       Example  3 Draw the ...
Orientation Exercises 1.  The lengths of the sides of the triangle shown are 5, 12, and 13 units and A is the measure of o...
9.  If tan   is in Quadrant III, find tan   in terms of cos   . 1     cos 2    cos     A. B.       1     cos 2     C.  164...
Practice Exercise 14 1.  Evaluate 2 sin 30       tan 45     sec 60   A. B. C.  2.  3.  2 3  D.  B.  3  E.  C.  3 3  Simpli...
Practice Test 14 1.  If the sine of an angle in quadrant I is what is the cosine of the angle  3 D. A. 3 2 5 B. C.  2.  3....