ACT Mathematics · Advanced Practice

Hard ACT Math
Exam 2025

20 Questions · All Topics · ACT-Style Difficulty

⏱ 60 min suggested
📊 36 max scaled score
🎯 5 topic areas
⚡ No calculator required
Elapsed Time
00:00
Core Concepts & Formulas
01
Pre-Algebra & Number Properties
FACTORS · RATIOS · PERCENTS · ABSOLUTE VALUE
Key Rules
  • |ab| = |a|·|b| and |a+b| ≤ |a|+|b| (triangle inequality)
  • % change = (new − old)/old × 100
  • If a/b = c/d, then ad = bc (cross-multiply)
  • For integers: (# of factors of n) = multiply (exponent+1) from prime factorization
|a − b| = distance between a and b on the number line
Quick Example
If |2x − 6| = 10, solve for all x.
Case 1: 2x − 6 = 10 → x = 8
Case 2: 2x − 6 = −10 → x = −2
Answer: x = 8 or x = −2
02
Algebra & Functions
LINEAR · QUADRATIC · SYSTEMS · INEQUALITIES
Must Memorize
Quadratic Formulax = (−b ± √(b²−4ac)) / 2a
Vertex Formy = a(x−h)² + k; vertex = (h, k)
Sum of rootsr₁ + r₂ = −b/a
Product of rootsr₁ · r₂ = c/a
Discriminantb²−4ac: >0 two real, =0 one real, <0 no real
Quick Example
For f(x) = x² − 5x + 6, the product of roots = 6/1 = 6, sum of roots = 5.
Roots: x = 2 and x = 3 ✓
03
Geometry & Measurement
TRIANGLES · CIRCLES · AREA · VOLUME · COORDINATE
Critical Formulas
Circle areaA = πr²
Arc lengthL = (θ/360°) × 2πr
Sector areaA = (θ/360°) × πr²
30-60-90 trianglesides: x, x√3, 2x
45-45-90 trianglesides: x, x, x√2
Distance formulad = √((x₂−x₁)² + (y₂−y₁)²)
MidpointM = ((x₁+x₂)/2, (y₁+y₂)/2)
Slopem = (y₂−y₁)/(x₂−x₁)
Quick Example
A 30-60-90 triangle has hypotenuse = 10. Find the short leg.
Hypotenuse = 2x → x = 5. Short leg = 5, long leg = 5√3.
Short leg = 5
04
Trigonometry
SOH-CAH-TOA · UNIT CIRCLE · IDENTITIES · LAW
Must Memorize
sin²θ + cos²θ= 1
tan θ= sin θ / cos θ
Law of Sinesa/sin A = b/sin B = c/sin C
Law of Cosinesc² = a² + b² − 2ab·cos C
sin 30° = cos 60°= 1/2
sin 45° = cos 45°= √2/2
sin 60° = cos 30°= √3/2
Quick Example
If sin θ = 3/5 and θ is in Quadrant I, find cos θ.
cos²θ = 1 − (3/5)² = 1 − 9/25 = 16/25 → cos θ = 4/5
cos θ = 4/5
05
Statistics & Probability
MEAN · MEDIAN · STANDARD DEV · COMBINATIONS
Key Concepts
  • Mean = sum of values / number of values
  • Median = middle value when sorted (average of two middle values if even count)
  • P(A and B) = P(A) × P(B) if A, B independent
  • P(A or B) = P(A) + P(B) − P(A and B)
  • Combinations: C(n,r) = n! / [r!(n−r)!]
  • Permutations: P(n,r) = n! / (n−r)!
Quick Example
How many ways to choose 3 students from 7? C(7,3) = 7!/(3!×4!) = (7×6×5)/(3×2×1) = 35
35 ways
Practice Questions
ACT-Style · 5 Answer Choices · Select & Submit
0/0
Correct
1
PRE-ALGEBRA
If |3x + 9| = 21, what are all possible values of x?
2
ALGEBRA
For the quadratic equation 2x² − 7x + k = 0, the sum of its roots is 7/2 and the product of its roots is 3. What is the value of k?
3
COORDINATE GEOMETRY
The equation of a circle is x² + y² − 6x + 4y − 12 = 0. What is the radius of this circle?
4
TRIGONOMETRY
In right triangle ABC, angle C = 90°, sin A = 5/13. What is the value of tan A?
5
STATISTICS
A data set has 5 values: 8, 12, x, 18, 22. If the mean of the set is 15, what is the median?
6
ALGEBRA — FUNCTIONS
If f(x) = 3x² − 2x + 1 and g(x) = x − 2, what is f(g(3))?
7
GEOMETRY — TRIANGLES
In triangle PQR, angle P = 45°, angle Q = 30°, and side QR = 6. What is the length of side PR? (Use Law of Sines)
8
PRE-ALGEBRA — EXPONENTS
Which of the following is equivalent to (23 × 42) / (82)?
9
ALGEBRA — SYSTEMS
If the system 3x + 2y = 16 and x − y = 2 has a solution (x, y), what is the value of x + y?
10
GEOMETRY — CIRCLES
A central angle of 120° intercepts an arc of a circle with radius 9. What is the area of the sector formed by this angle?
11
ALGEBRA — INEQUALITIES
What is the solution set for |2x − 3| < 7?
12
PROBABILITY
A bag contains 4 red, 3 blue, and 5 green marbles. Two marbles are drawn without replacement. What is the probability that both are red?
13
ALGEBRA — QUADRATIC
The vertex of the parabola y = −2x² + 8x − 5 is at point (h, k). What is the value of k?
14
TRIGONOMETRY — IDENTITIES
Which expression is equivalent to (sin²θ + cos²θ) / cos²θ?
15
GEOMETRY — 3D
A cone has a radius of 6 cm and height of 8 cm. What is the slant height of the cone?
16
ALGEBRA — RATIONAL EXPRESSIONS
Simplify: (x² − 9) / (x² − x − 6)
17
COORDINATE GEOMETRY — LINES
Lines ℓ and m are perpendicular. Line ℓ passes through (2, 5) and (6, 3). Which of the following could be the equation of line m?
18
COMBINATIONS & COUNTING
From a group of 8 students, a committee of 3 is to be selected. How many different committees are possible?
19
TRIGONOMETRY — GRAPHS
The function y = 3 sin(2x) has amplitude A and period T. What are the values of A and T?
20
ALGEBRA — COMPLEX SYSTEMS
If x² + y² = 25 and xy = 12, what is the value of (x + y)²?
Final Score
0
/ 20
Answer Key & Explanations
All 20 questions · Step-by-step solutions