ACT Math Practice Questions
with Detailed Solutions - sample 2

Math questions, with detailed solutions, are presented. The questions are those in ACT sample 2.

  1. What is the greatest common factor (GCF) of 45, 135 and 270?

    Solution

    Use prime factorization to rewrite 45, 135 and 210 as follows

    45 = 1 * 32 * 5

    135 = 1 * 33 * 5

    270 = 1 * 2 * 33 * 5

    The GCF = 5 * 32 = 45

  2. What is the value of 2x + 1/2 when 5x - 3 = -3x + 5

    Solution

    We first solve the equation 5x - 3 = -3x + 5

    5x + 3x = 5 + 3

    8x = 8

    x = 1

    We now calculate 2x + 1/2

    2x + 1/2 = 2(1) + 1/2 = 2 1/2

  3. The width W of a rectangle is 2 inches less than half its length L. Express the perimeter P of the rectangle in terms of the length L.

    Solution

    We first express the width W in terms of the length L

    W = L/2 - 2

    We now express P in terms of L as follows

    P = 2(W + L) = 2(L/2 - 2 + L) = L - 4 + 2L = 3L -4

  4. The total surface area of all six sides of the rectangular box below is equal to 80 square inches. What is the volume of the rectangular box in inches cubed?

    act problem 52.



    Solution

    Let us first find the total area of all six sides:

    Area of left and right sides: 4x + 4x = 8x

    Area of front and back sides: 4x + 4x = 8x

    Area of top and bottom sides: 16 + 16 = 32

    Total area = 16x + 32 = 80

    Solve for x

    16x = 80 - 32

    16x = 48

    x = 3 in

    Volume of the box is

    4 * 4 * 3 = 48 inches cubed


  5. The points A(-4 , -4), B(-1 , -2) and C(x , -8) are the vertices of a right triangle with the right angle at (-1 , -2). Find the value of x.

    Solution

    Since the right angle is at point B, BA and BC are the legs of the right triangle and AC is the hypotenuse. Find squares of distances AB, BC and AC.

    AB2 = (-1 + 4)2 + (-2 + 4)2 = 13

    BC2 = (x + 1)2 + (-8 + 2)2 = (x + 1)2 + 36

    AC2 = (x + 4)2 + (-8 + 4)2 = (x + 4)2 + 16

    Apply Pythagora's theorem.

    AB2 + BC2 = AC2

    13 + (x + 1)2 + 36 = (x + 4)2 + 16

    Expand and simplify.

    13 + x2 + 2x + 1 + 36 = x2 + 8x + 16 + 16

    -6x = -18

    x = 3

  6. In the figure below, L1 and L2 are parallel lines. The correct relationship between angles y and x is

    act problem, sample 2, problem 6 .



    Solution

    The angle supplementary to angle x on the right side of L3 is equal to y since they are corresponding angles. Hence

    x + y = 180o

  7. If (x + y)2 = 144 and x2 - y2 = 24, then what is x if x and y are both positive?

    Solution

    Take the square root of both sides in (x + y)2 = 144 to obtain

    x + y = 12

    Factor the left side of x2 - y2 = 24 and write

    (x + y)(x - y) = 24

    Use x + y = 12 obtained above to write

    12(x - y) = 24

    Simplify

    x - y = 2

    Solve for x the system of equation: x + y = 12 and x - y = 2. Add the right and left sides of the two equations top obtain

    2x = 14

    x = 7

  8. How many 3 digit numbers can we make using the digits 4, 5, 7 and 9 and where repetition is allowed?

    Solution

    We have four digits. The numbers we need to make have 3 digits. There are 4 choices for the first digit, 4 choices for the second digit and 4 choices for the third digit since repetition is allowed. Hence the total number of 3 digit numbers that can be made is equal to

    4 * 4 * 4 = 43

  9. A boat travels 10 miles East and then 24 miles South to an island. How many miles are there from the point of departure of the boat to the island?

    Solution

    In traveling East and then South, the boat is moving along the legs of 10 and 24 miles of a right triangle. The distance d then would be the hypotenuse which can be found using Pythagora's theorem.

    d2 = 102 + 242 = 676

    d = 26 miles.

  10. What is the slope of any line perpendicular to the line -5x + 3y = 9?

    Solution

    Write the equation of the given line and find the slope.

    -5x + 3y = 9

    3y = 5x + 9

    y = (5/3)x + 3

    slope = 5/3

    slope m of perpendicular is such that (5/3) * m = -1. solve for m.

    m = -3/5

  11. Which of the following is a factor of the polynomial -2x2 + 7x - 6?

    A) -2x - 3
    B) 2x + 2
    C) x - 6
    D) 2x - 2
    E) -2x + 3

    Solution

    Factor the given expression.

    -2x2 + 7x - 6 = (-2x + 3)(x - 2)

    -2x + 3 is a factor of -2x2 + 7x - 6.
  12. √(-9)(-4) + √(-4) = ?

    Solution

    Simplify both terms as follows.

    √(-9)(-4) = √36 = 6

    √(-4) = √((-1)4) = √(-1) √4 = i *2 = 2i

    Hence.

    √(-9)(-4) + √(-4) = 6 + 2 i , where i = √(-1)

  13. Find the linear function f such that f(2) = 5 and f(3) = -5.

    Solution

    A linear function f has the form.

    f(x) = m x + b

    Use f(2) = 5 and f(3) = -5 to write

    f(2) = 2 m + b = 5 and f(3) = 3 m + b = -5

    Solve the system of equations: 2 m + b = 5 and 3 m + b = -5. Subtract right sides an left sides to eliminate b as follows

    (3 m + b) - (2 m + b) = - 5 - 5

    m = -10

    b = 5 - 2 m = 5 - 2 (-10) = 5 + 20 = 25

    f(x) = - 10 x + 25

  14. A circular garden has an area of 100π feet squared. What is the circumference of the garden in feet?

    Solution

    The area of the circular garden of radius R is equal to

    π R2 = 100π

    Solve for radius R

    R = 10 feet

    The circumference of the circular garden is equal to

    2 π R = 20 π

  15. If 5/x = 10 and 2/y = 6 then x/y = ?

    Solution

    If 5/x = 10, then

    x/5 = 1/10

    We now multiply the left sides and right sides of x/5 = 1/10 and 2/y = 6 to obtain

    (x/5)(2/y) = (1/10)(6)

    (x/y)(2/5) = 6/10

    x/y = (6/10)(5/2) = 3/2

  16. If 2m-3 / 42m = 8, then 2m - 1 = ?

    Solution

    Write 42m and 8 in exponential form with exponent 2

    42m = (22)2m = 24m

    8 = 23

    Use the above in the given equation

    2m - 3 / 24m = 23

    2m - 3 - 4m = 23

    Which gives

    m - 3 - 4 m = 3

    -3m = 6

    m = - 2

    2m - 1 = 2(- 2) - 1 = - 5

  17. In the figure below, ABC is a right triangle. Points B, C and D are collinear; points D, E and F are also collinear and so are points B, A and F. The length of segments DC and DE are equal. What is the size, in degrees, of angle AFE?

    act problem, sample 2, problem 17.



    Solution

    Since ABC is a right triangle, then

    angle BCA = 90 - 50 = 40

    Since DE and DC have equal lengths, triangle DCE is isosceles and therefore

    angle DEC = 40

    Angles AEF and DEC are vertical and therefore equal. Hence

    angle AEF = 40

    Since angle BAE is equal to 90 then angle FAE is also equal to 90 and therefore

    angle AFE = 90 - 40 = 50

  18. Which of the following is an equation of a line perpendicular to the line with equation 3x - 6y = 9?

    A) y = 2
    B) 3x + 6y = 9
    C) x - 2y = 3
    D) 2x + 2y = 3
    E) 2x + y = 7

    Solution

    Find slope of line 3x - 6y = 9.

    3x - 6y = 9

    - 6y = - 3x + 9

    y = (1/2) x - 3/2

    slope = 1/2

    Find slope of the given lines.

    A) y = 2 , slope = 0
    B) 3x + 6y = 9 , 6y = - 3x + 9 , y = (-1/2)x + 3/2 , slope = -1/2
    C) x - 2y = 3 , -2y = - x + 2 , y = (1/2) x - 1 , slope = 1/2
    D) 2x + 2y = 3 , 2y = - 2x + 3 , y = - x + 3/2 , slope = -1
    E) 2x + y = 7 , y = -2x + 7 , slope = -2

    If we multiply the slope of the given line which is 1/2 by the slope in E) which -2 the answer is -1 and therefore the line in E) is perpendicular to the given line.

  19. If a and b are any real numbers, then which of the following expressions is always positive?
    A) |a|
    B) |a + b|
    C) |a - b| + 1/2
    D) a2 + b2
    E) (a + b)2
  20. The geometric figure below consists of a right triangle and 2 semicircles. The diameters of the semicircles are the sides of the triangle. What is the area of the shaded region in square centimeters if the length of the hypotenuse of the triangle is 8 centimeters?

    act problem, sample 2, problem 20.


    A) 64
    B) 8π
    C) 64π
    D) 10π
    E) 16
  21. The mean of the numbers a, b, c, d and e is 23. The mean of the numbers a, b, c, d, e and f is 22. What is the value of f?
    A) 23
    B) 18
    C) 22
    D) 22.5
    E) 20
  22. Which of the following equations corresponds to the graph shown below?

    act problem, sample 2, problem 22 .


    A) y = (x - 3)2 - 1
    B) y = -(x - 3)2 + 1
    C) y = (x - 3)2 + 1
    D) y = x - 3
    E) y = -(x - 3)2 - 1
  23. Functions f and g are defined by f(x) = x2 + x and g(x) = √(x + 6). What is the value of g(f(2))?
    A) 3
    B) -3
    C) 7
    D) 6
    E) -6
  24. The sum of three consecutive integers is equal to 192. What is the product of these numbers?
    A) 216000
    B) 7077888
    C) 576
    D) 110592
    E) 262080
  25. What are the x-coordinates of the points intersection of the line with equation y = x + 1 and the circle with equation x2 + y2 = 5
    A) -2 , 0
    B) 1 , 2
    C) -2 , 1
    D) -2 , -1
    E) 1 , 3
  26. (1/2)sin(2x)(1 + cot2(x)) =
    A) tan(x)
    B) sin(x)
    C) cos(x)
    D) cot(x)
    E) sec(x)
  27. Find the area of the rectangle ABCD shown in the figure below.

    act problem, sample 2, problem 27.


    A) 2500 / √2
    B) 2500
    C) 2500 / √3
    D) 1250
    E) 5000
  28. Solve for x: logx(1024) = -5
    A) 1/4
    B) 4
    C) 1/2
    D) 1/8
    E) 2
  29. Simplify: 6 3√32 + 2 3√108
    A) 32
    B) 18 3√2
    C) 36 3√2
    D) 18 3√4
    E) 36 3√4
  30. Evaluate: 1 / (-5)2
    A) - 1 / 25
    B) 1 / 25
    C) 25
    D) -25
    E) 1 / 10