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ACT Math Diagnostic

Find Your Weak Spots

36 questions across 6 ACT Math skill areas. Your results will show exactly where to focus your practice time.

Pre-Algebra
Elem. Algebra
Inter. Algebra
Coord. Geometry
Plane Geometry
Trigonometry

40 minutes · No penalty for guessing

0 of 36 answered

1. What is the value of 3 + 4 × 2 − 6 ÷ 3?

  • A. 3
  • B. 7
  • C. 9
  • D. 11
  • E. 12
C. Order of operations (PEMDAS): multiply and divide first. 4 × 2 = 8 and 6 ÷ 3 = 2. Then 3 + 8 − 2 = 9.

2. A shirt originally priced at $40 is on sale for 25% off. What is the sale price?

  • F. $10
  • G. $15
  • H. $25
  • J. $30
  • K. $35
J. 25% of $40 is $10. Subtract the discount: $40 − $10 = $30.

3. The average of five numbers is 20. If four of the numbers are 15, 18, 22, and 25, what is the fifth number?

  • A. 18
  • B. 19
  • C. 20
  • D. 21
  • E. 22
C. The sum of all five numbers is 5 × 20 = 100. The four known numbers sum to 15 + 18 + 22 + 25 = 80. The fifth number is 100 − 80 = 20.

4. If the ratio of boys to girls in a class is 3:5, and there are 40 students total, how many girls are in the class?

  • F. 15
  • G. 20
  • H. 24
  • J. 25
  • K. 30
J. The total ratio parts are 3 + 5 = 8. Girls make up 5/8 of the class: (5/8) × 40 = 25.

5. What is the least common multiple of 12 and 18?

  • A. 6
  • B. 24
  • C. 36
  • D. 72
  • E. 216
C. 12 = 2² × 3 and 18 = 2 × 3². The LCM uses the highest power of each prime: 2² × 3² = 4 × 9 = 36.

6. If |x − 3| = 7, what is the sum of all possible values of x?

  • F. −4
  • G. 3
  • H. 6
  • J. 7
  • K. 10
H. |x − 3| = 7 gives two equations: x − 3 = 7 (so x = 10) and x − 3 = −7 (so x = −4). The sum is 10 + (−4) = 6.

7. If 3(x − 4) = 2x + 5, what is the value of x?

  • A. 9
  • B. 11
  • C. 13
  • D. 15
  • E. 17
E. Distribute: 3x − 12 = 2x + 5. Subtract 2x: x − 12 = 5. Add 12: x = 17.

8. Which of the following is equivalent to (2x³)²?

  • F. 2x6
  • G. 4x5
  • H. 4x6
  • J. 2x9
  • K. 4x9
H. Square both the coefficient and the variable: 2² = 4 and (x³)² = x6. So (2x³)² = 4x6.

9. If 2x + 7 > 15, which of the following represents all possible values of x?

  • A. x > 4
  • B. x > 8
  • C. x < 4
  • D. x > 11
  • E. x < 8
A. Subtract 7 from both sides: 2x > 8. Divide by 2: x > 4.

10. What are the solutions to x² − 5x + 6 = 0?

  • F. x = 1 and x = 6
  • G. x = 2 and x = 3
  • H. x = −2 and x = −3
  • J. x = −1 and x = 6
  • K. x = −2 and x = 3
G. Factor: (x − 2)(x − 3) = 0. So x = 2 or x = 3.

11. A number tripled and then decreased by 8 gives 25. What is the number?

  • A. 9
  • B. 10
  • C. 11
  • D. 12
  • E. 13
C. Set up the equation: 3x − 8 = 25. Add 8: 3x = 33. Divide by 3: x = 11.

12. If a = −2, what is the value of 3a³ − 2a + 1?

  • F. −27
  • G. −19
  • H. −15
  • J. 15
  • K. 27
G. Substitute: 3(−2)³ − 2(−2) + 1 = 3(−8) + 4 + 1 = −24 + 4 + 1 = −19.

13. What are the solutions to 2x² + 3x − 5 = 0?

  • A. x = 1 and x = −5/2
  • B. x = −1 and x = 5/2
  • C. x = 1 and x = 5/2
  • D. x = 2 and x = −5/2
  • E. x = −1 and x = −5/2
A. Using the quadratic formula with a = 2, b = 3, c = −5: discriminant = 9 + 40 = 49. x = (−3 ± 7)/4, giving x = 1 and x = −5/2.

14. If f(x) = 2x² − 3x + 1, what is f(−1)?

  • F. −4
  • G. 0
  • H. 2
  • J. 6
  • K. 8
J. f(−1) = 2(−1)² − 3(−1) + 1 = 2(1) + 3 + 1 = 6.

15. If log2(x) = 5, what is the value of x?

  • A. 10
  • B. 16
  • C. 25
  • D. 32
  • E. 64
D. log2(x) = 5 means 25 = x, so x = 32.

16. What is the solution to the system x + y = 7 and 2x − y = 5?

  • F. (2, 5)
  • G. (3, 4)
  • H. (4, 3)
  • J. (5, 2)
  • K. (6, 1)
H. Add the two equations: 3x = 12, so x = 4. Substitute back: 4 + y = 7, so y = 3.

17. What is the value of i² + i4, where i = √(−1)?

  • A. −2
  • B. −1
  • C. 0
  • D. 1
  • E. 2
C. i² = −1 and i4 = (i²)² = (−1)² = 1. So −1 + 1 = 0.

18. What is the sum of the first 5 terms of the arithmetic sequence 3, 7, 11, 15, …?

  • F. 45
  • G. 50
  • H. 55
  • J. 60
  • K. 65
H. The common difference is 4. The 5th term is 19. Sum = 3 + 7 + 11 + 15 + 19 = 55. (Or use S = n/2 × (first + last) = 5/2 × 22 = 55.)

19. What is the slope of the line passing through the points (2, 5) and (6, 13)?

  • A. 1/2
  • B. 2
  • C. 3
  • D. 4
  • E. 8
B. Slope = (13 − 5)/(6 − 2) = 8/4 = 2.

20. What is the distance between points (1, 2) and (4, 6)?

  • F. 3
  • G. 4
  • H. 5
  • J. 6
  • K. 7
H. Distance = √((4−1)² + (6−2)²) = √(9 + 16) = √25 = 5.

21. What is the midpoint of the segment connecting (−2, 8) and (6, 4)?

  • A. (2, 6)
  • B. (4, 12)
  • C. (2, 2)
  • D. (4, 6)
  • E. (−4, 2)
A. Midpoint = ((−2 + 6)/2, (8 + 4)/2) = (4/2, 12/2) = (2, 6).

22. The equation x² + y² = 49 describes a circle centered at the origin. What is the radius?

  • F. 5
  • G. 6
  • H. 7
  • J. 14
  • K. 49
H. The standard form is x² + y² = r². Since r² = 49, r = 7.

23. What is the y-intercept of the line 3x − 2y = 12?

  • A. −6
  • B. −4
  • C. 4
  • D. 6
  • E. 12
A. Set x = 0: −2y = 12, so y = −6.

24. A line has the equation y = 3x + 2. Which of the following lines is perpendicular to it?

  • F. y = 3x − 1
  • G. y = −3x + 4
  • H. y = (1/3)x + 5
  • J. y = −(1/3)x + 1
  • K. y = −(3/2)x + 3
J. The slope of the given line is 3. A perpendicular line has slope −1/3 (the negative reciprocal).

25. In a right triangle, the two legs measure 5 and 12. What is the length of the hypotenuse?

  • A. 10
  • B. 11
  • C. 13
  • D. 15
  • E. 17
C. By the Pythagorean theorem: c = √(5² + 12²) = √(25 + 144) = √169 = 13.

26. What is the area of a circle with a diameter of 10?

  • F. 10π
  • G. 20π
  • H. 25π
  • J. 50π
  • K. 100π
H. The radius is 10/2 = 5. Area = πr² = π(5)² = 25π.

27. The angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?

  • A. 40°
  • B. 60°
  • C. 72°
  • D. 80°
  • E. 90°
D. The angles sum to 180°: 2x + 3x + 4x = 180, so 9x = 180 and x = 20. The largest angle is 4(20) = 80°.

28. Two similar triangles have corresponding sides in a ratio of 3:5. If the area of the smaller triangle is 36 square units, what is the area of the larger triangle?

  • F. 60
  • G. 72
  • H. 90
  • J. 100
  • K. 120
J. For similar figures, the area ratio is the square of the side ratio: (5/3)² = 25/9. Larger area = 36 × 25/9 = 100.

29. What is the sum of the interior angles of a hexagon?

  • A. 360°
  • B. 540°
  • C. 720°
  • D. 900°
  • E. 1080°
C. Sum of interior angles = (n − 2) × 180° = (6 − 2) × 180° = 720°.

30. A rectangular box has dimensions 3, 4, and 5. What is its volume?

  • F. 12
  • G. 47
  • H. 60
  • J. 94
  • K. 120
H. Volume = length × width × height = 3 × 4 × 5 = 60.

31. In a right triangle, the side opposite angle θ is 3 and the hypotenuse is 5. What is sin θ?

  • A. 3/5
  • B. 4/5
  • C. 3/4
  • D. 5/3
  • E. 5/4
A. sin θ = opposite/hypotenuse = 3/5.

32. What is the value of cos(60°)?

  • F. 0
  • G. 1/4
  • H. 1/2
  • J. √2/2
  • K. √3/2
H. cos(60°) = 1/2. This is a standard unit circle value.

33. Which of the following is equivalent to sin²θ + cos²θ?

  • A. 0
  • B. 1
  • C. sin θ
  • D. tan θ
  • E. 2
B. The Pythagorean identity states that sin²θ + cos²θ = 1 for all values of θ.

34. Convert 3π/4 radians to degrees.

  • F. 120°
  • G. 135°
  • H. 150°
  • J. 210°
  • K. 270°
G. Multiply by 180/π: (3π/4) × (180/π) = 3 × 180/4 = 540/4 = 135°.

35. In a triangle, angle A = 30°, side a = 8, and angle B = 45°. Using the Law of Sines, what is the length of side b?

  • A. 4√2
  • B. 8
  • C. 8√2
  • D. 16
  • E. 16√2
C. a/sin A = b/sin B. So 8/sin 30° = b/sin 45°. Since sin 30° = 1/2 and sin 45° = √2/2: 16 = b/(√2/2), giving b = 16 × √2/2 = 8√2.

36. What is the period of the function y = sin(2x)?

  • F. π/2
  • G. π
  • H. 2π
  • J. 3π
  • K. 4π
G. The period of sin(bx) is 2π/b. Here b = 2, so the period is 2π/2 = π.

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