ACT Practice Test 3

Mathematics Test

60 Minutes 60 Questions

Directions: Choose the correct solution to each question and fill in the corresponding bubble on your answer sheet.

Do not continue to spend time on questions if you get stuck. Solve as many questions as you can before returning to any if time permits.

You may use a calculator on this test for any question you choose. However, some questions may be better solved without a calculator.

Note: Unless otherwise stated, you can assume:

  1. Figures are NOT necessarily drawn to scale.
  2. Geometric figures are two dimensional.
  3. The term line indicates a straight line.
  4. The term average indicates arithmetic mean.

  1. Tanya used yards of fabric to make her dress, and she used yards of fabric to make her jacket. What was the total amount, in yards, that Tanya used for the complete outfit of dress and jacket?

  2. The product 5x 3y 5 · 6y 2 · 2xy is equivalent to which of the following?

    1. 13x 3y 7
    2. 13x 4y 8
    3. 60x 3y 7
    4. 60x 3y 10
    5. 60x 4y 8

  3. Brandon puts 6% of his $36,000 yearly salary into savings, in 12 equal monthly installments. Jacqui deposits $200 every month into savings. At the end of one full year, what is the difference, in dollars, between the amount of money that Jacqui saved and the amount of money that Brandon saved, not including any interest earned?

    1.    216
    2.    240
    3.    480
    4. 1,960
    5. 2,800

  4. A school is selling T-shirts as a fund-raiser. For the first 100 T-shirts that are sold, the school will earn $7 per shirt. For each additional shirt that is sold, the school will earn $10. How much will the school earn if 350 T-shirts are sold?

    1. $    245
    2. $    250
    3. $2,450
    4. $3,200
    5. $4,200

  5. For steel to be considered stainless steel, it must have a minimum of 10.5% chromium in the metal alloy. If a manufacturer has 262.5 pounds of chromium available, what is the maximum amount of stainless steel, in pounds, that the manufacturer can produce?

    1.        27.56
    2.      252
    3.      262.5
    4.   2,500
    5. 25,000

  6. A homeowner wants to put a wallpaper border around the top edge of all the walls of his kitchen. The ceiling is rectangular and measures 6.5 meters by 4 meters. What is the required length, in meters, of the border?

    1.   8
    2. 10.5
    3. 13
    4. 21
    5. 26

  7. Which of the following expressions is equivalent to w(x – (y + z)) ?

    1. wx – wy – wz
    2. wx – wy + wz
    3. wx – y + z
    4. wx – y – z
    5. wxy + wxz

  8. If 6n – 4 = 3n + 24, then n = ?

    1. 28

  9. To make the color he wants to paint his boat, Marco needs to mix 5 parts white paint with 3 parts blue paint. How many quarts of blue paint will he need to make 24 quarts of this color?

    1.   3
    2.   5
    3.   8
    4.   9
    5. 15

  10. What is the real-number value of if 5m2 = 45 ?

    1.   27
    2.   33
    3.   38.09
    4.   63
    5. 739.39

  11. The radius of a sphere is meters. What is the volume of the sphere, to the nearest cubic meter? Use the formula .

    1.   42
    2.   45
    3.   96
    4. 157
    5. 195

  12. There are 10 peanuts, 6 cashews, and 8 almonds in a bag of mixed nuts. If a nut is chosen at random from the bag, what is the probability that the nut is NOT a peanut?

    4. 10
    5. 14

  13. The number of people who shop at an electronics store during a given week is shown in the following matrix:

    The ratio of people from each age group who will purchase a product to the number of people in that age group who shop at the store is shown in the following matrix:

    Based on the matrices, how many people will make purchases?

    1.   15
    2.   41
    3.   53
    4. 133
    5. 205

    1. Use the following information to answer questions 14–15.

    2. The following table shows the current enrollment in all the English classes offered at King High School.

      Course Title Section Period Number of Students
      English I 1 5 29
      2 1 27
      3 2 22
      English II 1 3 26
      2 6 25
      3 4 24
      British Literature 1 6 23
      African-American Literature 1 2 26
      2 5 25

    3. What is the average number of students per section in English I ?

      1. 22
      2. 25
      3. 26
      4. 27
      5. 29

    4. The school has 2 computer labs with 30 computers each. There are 3 broken computers in one lab and 5 broken computers in the other lab. All broken computers are not available to be used by students. For which of the following class periods, if any, are there NOT enough computers available for each English student to use a computer without having to share?

      1. Period 2 only
      2. Period 5 only
      3. Period 6 only
      4. Periods 5 and 6 only
      5. None

  15. In the following table, every row, column, and diagonal must have equivalent sums. Which term or value belongs in the lower left cell for this to be true?

    m –4m 3m
    2m 0 –2m
    4m –m
    1. –4m
    2. –3m
    3. –3
    4.   0
    5.   m

  16. A standard coordinate plane is shown here, with the four quadrants labeled. Suppose point R, denoted by R(x,y), is graphed on this plane, such that x ≠ 0 and y ≠ 0.

    A standard coordinate plane with upper right quadrant labeled 1, upper left quadrant labeled 2, lower left quadrant labeled 3, and lower right quadrant labeled 4.

    If the product xy is a positive number, then point R is located in:

    1. Quadrant I only.
    2. Quadrant II only.
    3. Quadrant III only.
    4. Quadrant I or IV.
    5. Quadrant I or III.

  17. A party store has 54 packs of plates in stock. The packs are either sets of 8 or sets of 12. If the store has 496 total plates in stock, how many plates would a customer buy if he or she buys all of the packs of 12 that the store has in stock?

    1. 144
    2. 192
    3. 216
    4. 312
    5. 456

  18. At a university, there are 5 females for every 3 males. If there are 6,000 male students, how many students are female?

    1. 10,000
    2. 12,000
    3. 16,000
    4. 18,000
    5. 30,000

  19. In order to determine some information about local gas prices, Sally must find the average cost of gasoline per gallon in her neighborhood. She visits 4 gas stations near her house and finds that the price per gallon at each store is $2.15, $2.05, $2.15, and $1.97, respectively. Among these 4 stations, what is the average price of gasoline per gallon?

    1. $2.06
    2. $2.07
    3. $2.08
    4. $2.09
    5. $2.10

  20. Given f (x) = − 4x + 5 and , what is the value of ?

    1. –7
    2. –0.5
    3.   0 
    4.   3
    5.   6.5

  21. Which of the following equations is the slope-intercept form of 3x + 2y = 16 ?

  22. Which value is a solution to the equation x 2 + 75 = 20x ?

    1. –15
    2.   –5
    3.     0
    4.     3
    5.     5

  23. Given right triangle LMN, what is the value of cos N ?

    Right triangle L M N. Leg L M has length 5 m, leg M N has length 12 m, and hypotenuse L N has length 13 m.

  24. Chord AB passes through the center of circle O, shown here. If radius OC is perpendicular to chord AB and has a length of 7 centimeters, what is the length of chord BC, to the nearest tenth of a centimeter?

    A circle centered at O. Chord A B is horizontal and passes through the center of the circle. Radius O C is perpendicular to chord A B. Chord C B forms the hypotenuse of triangle O C B.
    1.   5.3
    2.   7.0
    3.   9.9
    4. 12.1
    5. 14.0

  25. If (a − b)2 = 36 and ab = 24, then a2 + b2 = ?

    1. –12
    2.   12
    3.   60
    4.   84
    5.   96

  26. An Olympic-sized pool is 50 meters long and 25 meters wide. It holds 14,375 cubic meters of water. If the pool is the same depth everywhere, about how many meters deep is the water in the pool?

    1. Less than 9
    2. Between 9 and 10
    3. Between 10 and 11
    4. Between 11 and 12
    5. More than 12

  27. Lucia is beginning a marathon-training program. During her first day of training, she wears a pedometer to get an idea of how far she can currently run. At the end of the run, the pedometer indicates that she took 24,288 steps. Lucia knows from experience that her average running stride (step) is 2.5 feet. Given that 1 mile = 5,280 feet, how many miles did she run on her first day of training? 

    1.   1.84
    2.   4.25
    3.   7.6
    4. 11.5
    5. 18.4

  28. The following bar graph shows the number of people at the spring prom, according to their grade level at the high school. According to the graph, what fraction of the people at the prom were sophomores?

    An untitled bar graph. The vertical axis is labeled from 0 to 250. The horizontal axis shows grade levels of Freshman, Sophomore, Junior, and Senior. The bar heights are Freshman equals 20, Sophomore equals 30, Junior equals 70, and Senior equals 200.

  29. The line that passes through the points (1,1) and (2,16) in the standard (x,y) coordinate plane is parallel to the line that passes through the points (–10,–5) and (a,25). What is the value of a ?

    1. –8
    2.   3
    3.   5
    4. 15
    5. 20

  30. What is the x-coordinate of the intersection point, in the (x,y) coordinate system, of the lines 2x + 3y = 8 and 5x + y = 7 ?

    1. –1
    2.   1
    3.   2
    5.   3

  31. For all pairs of real numbers a and b, where a = 2b – 8, b = ?

    1. a + 4
    2. 2a – 8
    3. 2a + 8

  32. What is the area, in square millimeters, of parallelogram RSUT, shown here? Assume that the dotted line meets side RT at a 90° angle.

    Parallelogram R S U T. Sides R T and S U are parallel and each have length 5 plus 9 m m. Sides R S and T U are parallel and each have length 13 m m. A segment of length 12 m m connects vertex S with side R T.
    1.   30
    2.   39
    3.   54
    4. 168
    5. 182

  33. What is the fifth term in the geometric sequence that begins −108, 36, −12, 4, … ?

    4.      0
    5.      1

  34. The following is a partial map of Centerville, showing 80 square miles: a total of 8 miles of Main Street and a total of 10 miles of Front Street. There is a fire station at the corner of Main and Front Streets, shown as point F. The town wants to build a new fire station exactly halfway between the hospital, at H, and the school, at S. What would be the driving directions to get from the current fire station to the new fire station, by way of Main and Elm streets? All streets and avenues shown intersect at right angles.

    A map showing 4 streets running parallel north and south and 3 streets running parallel east and west, intersecting the north, south streets. From left to right, the north to south streets are Oak Street, Elm Street, an unlabeled street, and Front Street. From top to bottom, the east to west streets are Main Street, Second Avenue, and Tenth Avenue. Main street and Front street intersect at a point labeled F. Tenth Avenue and the unlabeled street intersect at a point labeled S. The distance between Oak and Elm is 2 point 5 miles. The distance between Oak and point S is 5 miles, and the distance between point S and Front Street is 3 miles. The distance between Main and Second Avenue is 2 miles and the distance between second avenue and tenth avenue is 8 miles.
    1. 2.5 miles east, 4 miles north
    2. 2.5 miles west, 4 miles north
    3. 2.5 miles east, 6 miles south
    4. 5.5 miles west, 4 miles south
    5. 5.5 miles west, 6 miles south

  35. There are two consecutive even integers. The difference between four times the larger and twice the smaller is 36. If x represents the smaller integer, which of the following equations can be used to determine the smaller integer?

    1. 4x – 2x = 36
    2. 4(x + 2) – 2x = 36
    3. 4(x + 1) – 2x = 36  
    4. (x + 3) – 2x = 36
    5. 36 – 4x = 2x

  36. A 15-foot supporting wire is attached to a telephone pole 12 feet from the ground. The wire is then anchored to the ground. The telephone pole stands perpendicular to the ground. How far, in feet, is the anchor of the supporting wire from the base of the telephone pole?

    1.   3
    2.   6
    3.   9
    4. 12
    5. 15

  37. In the following figure, the sides of the square are tangent to the inner circle. If the area of the circle is 100π square units, what is the perimeter of the square?

    A circle inscribed in a square. The diameter of the circle is equal to the length of the side of the square.
    1.   20
    2.   40
    3.   80
    4. 100
    5. 100π

  38. Rectangles ABCD and EHGF, shown here, are similar. Using the given information, what is the length of side EH, to the nearest tenth of an inch?

    Two separate rectangles, A B C D and E H G F. In the first rectangle, side A D equals 3 inches and side C D equals 8 inches. In the second rectangle, side H G equals 2 inches.
    1. 0.8
    2. 1.3
    3. 5.3
    4. 7.0
    5. 8.0

  39. In parallelogram VWXY, shown here, points U, V, Y, and Z form a straight line. Given the angle measures as shown in the figure, what is the measure of angle ∠WYX ?

    Parallelogram V W X Y with bottom base V Y. A line is drawn along base V Y with point U to the left of V and point Z to the right of Y. Exterior angle U V W has a measure of 150 degrees. Diagonal W Y is drawn in, dividing the parallelogram into two congruent triangles. One interior angle, angle Y W X, of one of the triangles, has a measure of 50 degrees.
    1.   25°
    2.   30°
    3.   50°
    4. 100°
    5. 150°

  40. For all nonzero x and y, ?

    1. −12y3
    2. −12x2y2
    3. −12x2y3

  41. In the mayoral election, of the eligible voters at one site cast a vote. Three-fifths of the votes at this site were for candidate Martinez. If there are 3,500 eligible voters at this site, how many of them voted for Martinez?

    1.    417
    2. 1,575
    3. 2,100
    4. 2,625
    5. 4,725

  42. Given that a and b are positive integers and the greatest common factor of a4b2 and a3b is 54, what is a possible value for b ?

    1.   2
    2.   3
    3.   6
    4.   9
    5. 27

  43. If 40% of x is 70, then what is 180% of x ?

    1.   50.4
    2. 126
    3. 154
    4. 175
    5. 315

  44. Point M (2,3) and point N (6,5) are points on the coordinate plane. What is the length of segment MN ?

    4. 6 units
    5. 20 units

  45. If u and v are vectors such that  and , which of the following represents u + v ?

    1.  

  46. What is the equation of a circle in the coordinate plane with center (–2,3) and a radius of 9 units?

    1. (x – 2)2 + (y + 3)2 = 9
    2. (x + 2)2 + (y – 3)2 = 9
    3. (x – 2)2 + (y + 3)2 = 81
    4. (x + 2)2 + (y – 3)2 = 3
    5. (x + 2)2 + (y – 3)2 = 81

  47. In the complex number system, i 2 = –1. Given that is a complex number, what is the result of ?

  48. Which of the following represents as an integer?

    1.      12
    2.      24
    3.      64
    4.    256
    5. 2,048

  49. Fifty high school students were polled to see if they owned a cell phone and/or an MP3 player. A total of 35 of the students own a cell phone, and a total of 18 of the students own an MP3 player. How many students own both a cell phone and an MP3 player, assuming all 50 students own at least one of the two items?

    1.   0
    2.   3
    3. 17
    4. 32
    5. 53

  50. Scientists are modeling population trends and have noticed that when a certain population changes, the change is based on a linear function. When t = 21, the population is 3. When t = 35, the population is 5. Which of the following describes the population as a function of time t ?

  51. If 3 people all shake hands with each other, there are a total of 3 handshakes. If 4 people all shake hands with each other, there are a total of 6 handshakes. How many total handshakes will there be if 5 people all shake hands with each other?

    1.   7
    2.   9
    3. 10
    4. 11
    5. 12

  52. The following table shows the percentages of a county’s budget expenses by category. The remainder of the budget will be placed in the category Miscellaneous. If the same information is to be organized in a circle graph, what will be the degree measure of the central angle for the sector that represents Miscellaneous, rounded to the nearest degree?

    Budget Category Percentage of Budget
    Salaries 23
    Road Repair   5
    Employee Benefits 22
    Building Maintenance/Utilities 18
    1.   32
    2.   58
    3.   68
    4. 115
    5. 245

  53. If tan θ =  and  < θ < π, then sin θ = ?

    4.      
    5.      

  54. A portion of which of the following systems of inequalities can be approximated by the shaded region shown in the coordinate plane, assuming each grid line on the graph represents 1 unit?

    A shaded region on a coordinate plane bounded by a horizontal line and a diagonal line that intersect. The horizontal line crosses the y axis at 6. The shading is below this line. The diagonal line passes through the points zero comma negative 5 and 1 comma negative 2. The shading is to the left of this line.
    1. y ≤ 6 and y ≥ 3x – 5
    2. x ≤ 6 and y ≥ 3x – 5
    3. y ≤ 6 and y ≤ 3x – 5
    4. x ≤ 6 and y ≤ –3x – 5

  55. If f(x) = 3x + 5 and g(x) = 4x + 5, which of the following expressions represents f(g(x)) ?

    1. 7x + 5
    2. 7x + 10
    3. 12x + 15
    4. 12x + 20
    5. 12x + 25

  56. For what values of x does the graph of the quadratic function q(x) = 2x 2 + 7x − 4 cross the x-axis?

    1. x = −1,
    2. , x = −4
    3. , x = 4
    4. , x = −4
    5. x = 1,

  57. In a standard coordinate plane, what are the coordinates of Q′, the reflection of the point Q(r,s) over the y-axis?

    1. (–r,s)
    2. (–r,–s)
    3. (s,–r)
    4. (r,–s)
    5. (–s,r)

  58. If and , then what is the value of the trigonometric cofunction csc x ?

    1. −2
    5.   2

  59. Find cos(75°) given that cos(75°) = cos(30° + 45°). Use then following table and the formula: 

    cos(α + β) = cos(α)cos(β) – sin(α)sin(β) 

    θ sin θ cos θ tan θ
    30°
    45° 1
    60°