Free ACT Math Practice Test: 30 Sample Questions with Answers

Prepare for the ACT math section with 30 challenging practice questions. Detailed step-by-step solutions are available on our ACT solutions page. Answers are provided at the bottom.

1. What is the greatest common factor of 45, 135, and 270?
   A) 5
   B) 9
   C) 15
   D) 25
   E) 45
2. What is the value of \( 2x + \frac{1}{2} \) when \( 5x - 3 = -3x + 5 \)?
   A) \( -\frac{1}{2} \)
   B) \( -\frac{1}{4} \)
   C) 1
   D) \( 2\frac{1}{2} \)
   E) Value cannot be calculated
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 \( L \).
   A) \( 3L - 4 \)
   B) \( 4L - 4 \)
   C) \( 4L \)
   D) \( 3L - 2 \)
   E) \( 2L - 2 \)
4. The total surface area of all six sides of the rectangular box is 80 square inches. What is the volume of the box in cubic inches?
   
   A) 40
   B) 48
   C) 12
   D) 30
   E) 16
5. The points \( (-4, -4) \), \( (-1, -2) \), and \( (x, -8) \) are vertices of a right triangle with the right angle at \( (-1, -2) \). Find \( x \).
   A) 0
   B) 2
   C) 3
   D) -1
   E) -4
6. In the figure below, \( L_1 \) and \( L_2 \) are parallel lines. The correct relationship between angles \( y \) and \( x \) is:
   
   A) \( y = x \)
   B) \( x + y = 180^\circ \)
   C) \( x + y = 360^\circ \)
   D) \( y = x + 90^\circ \)
   E) \( y - x = 180^\circ \)
7. If \( (x + y)^2 = 144 \) and \( x^2 - y^2 = 24 \), and \( x, y > 0 \), what is \( x \)?
   A) 7
   B) 5
   C) 12
   D) 2
   E) 8
8. How many 3-digit numbers can be formed using digits 4, 5, 7, 9 with repetition allowed?
   A) \( 3^4 \)
   B) \( 3^3 \)
   C) \( 4^4 \)
   D) 12
   E) \( 4^3 \)
9. A boat travels 10 miles East, then 24 miles South to an island. How many miles is the direct distance from departure to island?
   A) 34
   B) 14
   C) 26
   D) \( 2\sqrt{119} \)
   E) 44
10. What is the slope of any line perpendicular to \( -5x + 3y = 9 \)?
    A) \( -\frac{3}{5} \)
    B) \( \frac{5}{3} \)
    C) \( \frac{3}{5} \)
    D) \( -\frac{5}{3} \)
    E) \( -\frac{1}{5} \)
11. Which is a factor of \( -2x^2 + 7x - 6 \)?
    A) \( -2x - 3 \)
    B) \( 2x + 2 \)
    C) \( x - 6 \)
    D) \( 2x - 2 \)
    E) \( -2x + 3 \)
12. Simplify: \( \sqrt{(-9)(-4)} + \sqrt{-4} \)
    A) \( 6 + 2i \)
    B) \( 6 - 2i \)
    C) 8
    D) 4
    E) \( 2 + 6i \)
13. Find the linear function \( f \) such that \( f(2) = 5 \) and \( f(3) = -5 \).
    A) \( f(x) = 2x + 1 \)
    B) \( f(x) = -10x - 5 \)
    C) \( f(x) = 3x - 1 \)
    D) \( f(x) = -10x + 25 \)
    E) \( f(x) = -10x - 25 \)
14. A circular garden has area \( 100\pi \) ft². What is its circumference?
    A) 20
    B) \( 10\pi \)
    C) \( 20\pi \)
    D) \( 400\pi \)
    E) \( 2500\pi \)
15. If \( \frac{5}{x} = 10 \) and \( \frac{2}{y} = 6 \), then \( \frac{x}{y} = \)?
    A) \( \frac{5}{3} \)
    B) \( \frac{3}{2} \)
    C) \( \frac{3}{5} \)
    D) \( \frac{2}{3} \)
    E) \( \frac{5}{2} \)
16. If \( \frac{2^{m-3}}{4^{2m}} = 8 \), then \( 2m - 1 = \)?
    A) 0
    B) 1
    C) 2
    D) -5
    E) -9
17. In the figure, \( ABC \) is a right triangle. Points \( B, C, D \) are collinear; \( D, E, F \) are collinear; and \( B, A, F \) are collinear. \( DC = DE \). Find angle \( AFE \) in degrees.
    
    A) \( 50^\circ \)
    B) \( 40^\circ \)
    C) \( 30^\circ \)
    D) \( 45^\circ \)
    E) \( 55^\circ \)
18. Which equation represents a line perpendicular to \( 3x - 6y = 9 \)?
    A) \( y = 2 \)
    B) \( 3x + 6y = 9 \)
    C) \( x - 2y = 3 \)
    D) \( 2x + 2y = 3 \)
    E) \( 2x + y = 7 \)
19. If \( a \) and \( b \) are real numbers, which expression is always positive?
    A) \( |a| \)
    B) \( |a + b| \)
    C) \( |a - b| + \frac{1}{2} \)
    D) \( a^2 + b^2 \)
    E) \( (a + b)^2 \)
20. The figure shows a right triangle and two semicircles on its legs. The hypotenuse is 8 cm. Find the shaded area in cm².
    
    A) 64
    B) \( 8\pi \)
    C) \( 64\pi \)
    D) \( 10\pi \)
    E) 16
21. The mean of \( a, b, c, d, e \) is 23. The mean including \( f \) is 22. What is \( f \)?
    A) 17
    B) 18
    C) 22
    D) 22.5
    E) 20
22. Which equation matches the graph?
    
    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. Given \( f(x) = x^2 + x \) and \( g(x) = \sqrt{x + 6} \), find \( g(f(2)) \).
    A) 3
    B) -3
    C) 7
    D) 6
    E) -6
24. The sum of three consecutive integers is 192. What is their product?
    A) 216,000
    B) 7,077,888
    C) 576
    D) 110,592
    E) 262,080
25. Find the x-coordinates of the intersection of \( y = x + 1 \) and \( x^2 + y^2 = 5 \).
    A) -2, 0
    B) 1, 2
    C) -2, 1
    D) -2, -1
    E) 1, 3
26. Simplify: \( \frac{1}{2} \sin(2x) (1 + \cot^2(x)) \)
    A) \( \tan(x) \)
    B) \( \sin(x) \)
    C) \( \cos(x) \)
    D) \( \cot(x) \)
    E) \( \sec(x) \)
27. Find the area of rectangle \( ABCD \).
    
    A) \( \frac{2500}{\sqrt{2}} \)
    B) 2500
    C) \( \frac{2500}{\sqrt{3}} \)
    D) 1250
    E) 5000
28. Solve for \( x \): \( \log_x(1024) = -5 \)
    A) \( \frac{1}{4} \)
    B) 4
    C) \( \frac{1}{2} \)
    D) \( \frac{1}{8} \)
    E) 2
29. Simplify: \( 6 \sqrt[3]{32} + 2 \sqrt[3]{108} \)
    A) 32
    B) \( 18 \sqrt[3]{2} \)
    C) \( 36 \sqrt[3]{2} \)
    D) \( 18 \sqrt[3]{4} \)
    E) \( 36 \sqrt[3]{4} \)
30. Evaluate: \( \frac{1}{(-5)^2} \)
    A) \( -\frac{1}{25} \)
    B) \( \frac{1}{25} \)
    C) 25
    D) -25
    E) \( \frac{1}{10} \)