ACT Math Hard Mode

20 official-style questions covering all ACT Math topics. Modeled on the hardest questions from real ACT exams.

20
Questions
35
Minutes
36
Max Score
Pre-AlgebraQ 1–2
Elementary AlgebraQ 3–4
Intermediate AlgebraQ 5–7
Coordinate GeometryQ 8–10
Plane GeometryQ 11–13
TrigonometryQ 14–15
FunctionsQ 16–17
Statistics / ProbabilityQ 18–19
Number TheoryQ 20
Matrices & SequencesBonus

Key Concepts & Formulas

Pre-Algebra Ratios, Percents, Absolute Value

% change = (new − old) / old × 100
|a| = b → a = b or a = −b
LCM / GCF: use prime factorization
Ex: If |2x − 3| = 7, then 2x−3=7 → x=5, or 2x−3=−7 → x=−2

Algebra Quadratics & Systems

Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a
Discriminant: b²−4ac > 0 (2 real), = 0 (1 real), < 0 (no real)
Sum of roots = −b/a    Product of roots = c/a
Ex: x²−5x+6=0 → roots 2 and 3; sum=5=5/1✓ product=6✓

Intermediate Algebra Functions, Inequalities, Radicals

Radical rule: √(a·b) = √a·√b
Exponent rule: a^m · a^n = a^(m+n), (a^m)^n = a^(mn)
Rational inequality: flip sign when multiplying by negative
Ex: √50 = √(25·2) = 5√2

Coordinate Geometry Lines, Circles, Distance

Slope = (y₂−y₁)/(x₂−x₁)
Distance = √((x₂−x₁)² + (y₂−y₁)²)
Circle: (x−h)² + (y−k)² = r²
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Ex: Circle (x−2)²+(y+3)²=25 → center (2,−3), radius 5

Plane Geometry Triangles, Circles, Area

Triangle area = ½·b·h
Similar triangles: corresponding sides in proportion
Arc length = (θ/360)·2πr    Sector area = (θ/360)·πr²
30-60-90: sides 1 : √3 : 2    45-45-90: sides 1 : 1 : √2
Ex: 30-60-90 triangle, short leg=3 → long leg=3√3, hyp=6

Trigonometry SOH CAH TOA, Identities

sin θ = opp/hyp   cos θ = adj/hyp   tan θ = opp/adj
sin²θ + cos²θ = 1
Law of Sines: a/sinA = b/sinB = c/sinC
Law of Cosines: c² = a² + b² − 2ab·cosC
Ex: tan 45° = 1; sin 30° = 0.5; cos 60° = 0.5

Statistics Mean, Median, Std Dev

Mean = sum / count
Weighted mean = Σ(weight × value) / Σ(weights)
Probability = favorable outcomes / total outcomes
P(A and B) = P(A)·P(B) [if independent]
Ex: P(rolling 2 sixes in a row) = (1/6)×(1/6) = 1/36
QUESTION 1 OF 20
Pre-Algebra
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Answer Review & Solutions