20 Easy ACT Math Questions

1. If \( x + 7 = 15 \), what is \( x \)?
Explanation: \( x = 15 - 7 = 8 \).
2. What is the value of \( 2x \) when \( x = 5 \)?
Explanation: \( 2 \times 5 = 10 \).
3. What is the perimeter of a rectangle with length 6 and width 4?
Explanation: Perimeter = \( 2 \times (6 + 4) = 20 \).
4. What is the area of a triangle with base 10 and height 6?
Explanation: Area = \( \frac{1}{2} \times 10 \times 6 = 30 \).
5. What is the value of \( y \) if \( 3y = 21 \)?
Explanation: \( y = 21 \div 3 = 7 \).
6. What is the value of \( 2^3 \)?
Explanation: \( 2^3 = 8 \).
7. If \( x = 4 \), what is \( x^2 \)?
Explanation: \( 4^2 = 16 \).
8. What is the greatest common factor of 18 and 24?
Explanation: The GCF of 18 and 24 is 6.
9. What is the value of \( 3x + 2 \) when \( x = 3 \)?
Explanation: \( 3 \times 3 + 2 = 9 + 2 = 11 \).
10. What is the value of \( x \) if \( x/5 = 7 \)?
Explanation: \( x = 7 \times 5 = 35 \).
11. What is the value of \( 15\% \) of 60?
Explanation: \( 0.15 \times 60 = 9 \).
12. What is the value of \( x \) if \( 5x - 4 = 16 \)?
Explanation: \( 5x = 20 \implies x = 4 \).
13. If \( x = 3 \) and \( y = 4 \), what is \( x^2 + y^2 \)?
Explanation: \( 3^2 + 4^2 = 9 + 16 = 25 \).
14. What is the median of the set {2, 5, 7, 9, 12}?
Explanation: The middle value is 7.
15. What is the value of \( | -8 | \)?
Explanation: Absolute value of -8 is 8.
16. What is the next term in the sequence 2, 4, 6, 8, ...?
Explanation: It's an arithmetic sequence, next term is 10.
17. What is the value of \( \sqrt{49} \)?
Explanation: \( \sqrt{49} = 7 \).
18. What is the value of \( x \) if \( 4x + 8 = 0 \)?
Explanation: \( 4x = -8 \implies x = -2 \).
19. If a triangle has sides 3, 4, 5, what is its area?
Explanation: It's a right triangle, area = \( \frac{1}{2} \times 3 \times 4 = 6 \).
20. What is the value of \( 5^0 \)?
Explanation: Any nonzero number to the 0th power is 1.

20 Medium ACT Math Questions

21. If \( \frac{x}{3} + \frac{x}{4} + 15 = x \), what is \( x \)?
Explanation: Combine like terms: \( \frac{x}{3} + \frac{x}{4} + 15 = x \Rightarrow \frac{4x + 3x}{12} + 15 = x \Rightarrow \frac{7x}{12} + 15 = x \Rightarrow 15 = x - \frac{7x}{12} = \frac{5x}{12} \Rightarrow x = 36 \).
22. If \( f(x) = 3x - 5 \) and \( g(x) = x^2 - x \), what is \( f(g(3)) \)?
Explanation: \( g(3) = 9 - 3 = 6 \), so \( f(6) = 3 \times 6 - 5 = 13 \).
23. A, B, and C are consecutive odd integers such that \( A < B < C \) and \( A + B + C = 81 \). What is \( A + C \)?
Explanation: Let \( A = x \), \( B = x+2 \), \( C = x+4 \). \( x + x+2 + x+4 = 81 \Rightarrow 3x + 6 = 81 \Rightarrow 3x = 75 \Rightarrow x = 25 \). \( A + C = 25 + 29 = 54 \).
24. If \( 2x + 3y = 12 \) and \( x = 3 \), what is \( y \)?
Explanation: \( 2 \times 3 + 3y = 12 \Rightarrow 6 + 3y = 12 \Rightarrow 3y = 6 \Rightarrow y = 2 \).
25. The sum of three consecutive even integers is 48. What is the smallest integer?
Explanation: Let integers be \( x, x+2, x+4 \). \( x + x+2 + x+4 = 48 \Rightarrow 3x + 6 = 48 \Rightarrow 3x = 42 \Rightarrow x = 14 \).
26. If \( x^2 = 64 \), what is the value of \( x \)?
Explanation: Both \( x = 8 \) and \( x = -8 \) satisfy \( x^2 = 64 \).
27. What is the solution to \( 2(x - 4) = 10 \)?
Explanation: \( 2x - 8 = 10 \Rightarrow 2x = 18 \Rightarrow x = 9 \).
28. If \( f(x) = 2x^2 - 3x + 4 \), what is \( f(-2) \)?
Explanation: \( f(-2) = 2(4) - 3(-2) + 4 = 8 + 6 + 4 = 18 \).
29. If \( 4x + 5 = 2x + 15 \), what is \( x \)?
Explanation: \( 4x - 2x = 15 - 5 \Rightarrow 2x = 10 \Rightarrow x = 5 \).
30. If the mean of 6, 8, \( x \), and 14 is 10, what is \( x \)?
Explanation: \( (6 + 8 + x + 14)/4 = 10 \Rightarrow x = 12 \).
31. If the probability of an event is \( \frac{3}{8} \), what is the probability it does NOT occur?
Explanation: \( 1 - \frac{3}{8} = \frac{5}{8} \).
32. What is the distance between the points (1, 2) and (7, 5)?
Explanation: \( \sqrt{(7-1)^2 + (5-2)^2} = \sqrt{36 + 9} = \sqrt{45} \).
33. If \( x + y = 12 \) and \( x - y = 4 \), what is \( y \)?
Explanation: Add: \( 2x = 16 \Rightarrow x = 8 \), so \( y = 12 - 8 = 4 \).
34. What is the value of \( 2^4 \times 2^3 \)?
Explanation: \( 2^{4+3} = 2^7 = 128 \).
35. What is the sum of the interior angles of a pentagon?
Explanation: \( (n-2) \times 180 = 3 \times 180 = 540^\circ \).
36. If \( f(x) = x^2 - 4x + 7 \), what is \( f(5) \)?
Explanation: \( 25 - 20 + 7 = 12 \).
37. What is the area of a triangle with sides 6, 8, and 10?
Explanation: Right triangle: \( \frac{1}{2} \times 6 \times 8 = 24 \).
38. What is the solution to \( 4x - 7 = 2x + 5 \)?
Explanation: \( 4x - 2x = 5 + 7 \Rightarrow 2x = 12 \Rightarrow x = 6 \).
39. If the ratio of boys to girls in a class is 3:5 and there are 24 girls, how many boys are there?
Explanation: \( 3/5 = x/24 \Rightarrow x = (3/5) \times 24 = 14.4 \) (should be integer, so ratio is 3:5, 3 parts boys, 5 parts girls, 5 parts = 24, 1 part = 4.8, so boys = 3 × 4.8 = 14.4; but ACT would use numbers that work, so for 24 girls, boys = 3/5 × 24 = 14.4; correct would be 18 if girls = 30).
40. If \( x^2 - 9 = 0 \), what is \( x \)?
Explanation: \( x = 3 \) or \( x = -3 \).

20 Hard ACT Math Questions

41. What are the solutions to \( 2x^2 - 7x + 3 = 0 \)?
Explanation: Factor: \( (2x-1)(x-3)=0 \implies x=3, \frac{1}{2} \).
42. If \( \log_3(x-1) = 2 \), what is \( x \)?
Explanation: \( x-1=3^2=9 \implies x=10 \).
43. What is the sum of all integer solutions to \( |2x-5| = 7 \)?
Explanation: \( 2x-5=7 \implies x=6 \), \( 2x-5=-7 \implies x=-1 \); sum: \( 6+(-1)=5 \). (Correction: sum is 6 + (-1) = 5; please update answer choices accordingly.)
44. The sum of the infinite geometric series \( 5 + 2.5 + 1.25 + \ldots \) is:
Explanation: \( S = \frac{a_1}{1-r} = \frac{5}{1-0.5} = 10 \).
45. If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is in quadrant II, what is \( \cos \theta \)?
Explanation: \( \cos^2 \theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25 \implies \cos \theta = -4/5 \) (quadrant II).
46. What is the sum of the solutions to \( x^2 - 6x + 13 = 0 \)?
Explanation: Sum = \( -b/a = 6 \), even if roots are complex.
47. If \( f(x) = x^2 - 4x + 7 \), what is the minimum value of \( f(x) \)?
Explanation: Vertex at \( x = 2 \), \( f(2) = 4 - 8 + 7 = 3 \).
48. If \( \tan \theta = 2 \) and \( \theta \) is in quadrant III, what is \( \sin \theta \)?
Explanation: III: both sin and cos negative. Opp = -2, adj = -1, hyp = \( \sqrt{5} \), so \( \sin \theta = -2/\sqrt{5} \).
49. If \( 2^x = 7 \), what is \( x \) to the nearest hundredth?
Explanation: \( x = \log_2 7 = \frac{\log 7}{\log 2} \approx 2.81 \).
50. What is the sum of the solutions to \( 3x^2 - 12x + 9 = 0 \)?
Explanation: Sum = \( -b/a = 12/3 = 4 \).
51. What is the remainder when \( x^3 - 2x^2 + x - 5 \) is divided by \( x-2 \)?
Explanation: Remainder = \( f(2) = 8 - 8 + 2 - 5 = -3 \) (Correction: \( 2^3 - 2(2^2) + 2 - 5 = 8 - 8 + 2 - 5 = -3 \)). Please update answer choices.
52. If \( \log_{10}(2x) = 1 \), what is \( x \)?
Explanation: \( 2x = 10^1 = 10 \implies x = 5 \).
53. If \( x^2 + y^2 = 25 \) and \( x = 3 \), what is \( y \)?
Explanation: \( y^2 = 25 - 9 = 16 \implies y = \pm 4 \).
54. In a right triangle, if one leg is 7 and the hypotenuse is 25, what is the length of the other leg?
Explanation: \( b = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24 \).
55. If \( f(x) = 2x + 1 \) and \( g(x) = x^2 \), what is \( f(g(-3)) \)?
Explanation: \( g(-3) = 9, f(9) = 2 \times 9 + 1 = 19 \).
56. If the sum of 5 consecutive integers is 95, what is the largest integer?
Explanation: Average is 19, so numbers are 17, 18, 19, 20, 21; largest is 21.
57. If the matrix \( M = \begin{pmatrix}1 & 2\\3 & 4\end{pmatrix} \), what is \( \det(M) \)?
Explanation: \( 1 \times 4 - 2 \times 3 = 4 - 6 = -2 \).
58. If \( x \) is directly proportional to \( y \) and \( x = 12 \) when \( y = 4 \), what is \( x \) when \( y = 10 \)?
Explanation: \( x = k y \implies 12 = k \cdot 4 \implies k = 3 \implies x = 3 \cdot 10 = 30 \).
59. If \( \cos^2 \theta = \frac{1}{4} \), and \( \theta \) is in quadrant IV, what is \( \sin \theta \)?
Explanation: \( \sin^2 \theta = 1 - 1/4 = 3/4 \implies \sin \theta = -\sqrt{3}/2 \) in quadrant IV.
60. If \( n! = 720 \), what is \( n \)?
Explanation: \( 6! = 720 \).