Pre-Algebra & Number Theory
CoreKey Formulas to Memorize
• LCM(a,b) = a·b / GCD(a,b)
• Percent Change = (New − Old)/Old × 100
• Simple Interest: I = Prt
• Compound Interest: A = P(1 + r/n)^(nt)
• Percent Change = (New − Old)/Old × 100
• Simple Interest: I = Prt
• Compound Interest: A = P(1 + r/n)^(nt)
Critical Facts
Order of operations: PEMDAS — Parentheses, Exponents, Multiplication/Division (L→R), Addition/Subtraction (L→R)
Absolute value: |x| ≥ 0 always. |−5| = 5
A prime number has exactly 2 factors: 1 and itself. 1 is NOT prime.
Quick Example
If a price increases from $80 to $100, what is the percent increase?
→ (100 − 80)/80 × 100 = 25%
Algebra & Functions
High YieldMust-Know Formulas
• Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a
• Vertex Form: f(x) = a(x−h)² + k → vertex = (h, k)
• Discriminant: D = b²−4ac
D > 0: 2 real roots | D = 0: 1 root | D < 0: no real roots
• Slope: m = (y₂−y₁)/(x₂−x₁)
• Slope-Intercept: y = mx + b
• Point-Slope: y − y₁ = m(x − x₁)
• Vertex Form: f(x) = a(x−h)² + k → vertex = (h, k)
• Discriminant: D = b²−4ac
D > 0: 2 real roots | D = 0: 1 root | D < 0: no real roots
• Slope: m = (y₂−y₁)/(x₂−x₁)
• Slope-Intercept: y = mx + b
• Point-Slope: y − y₁ = m(x − x₁)
Function Rules
f(g(x)) means apply g first, then f — composition order matters!
For inverse functions: swap x and y, then solve for y
Even function: f(−x) = f(x). Odd function: f(−x) = −f(x)
Quick Example
Find the vertex of f(x) = 2x² − 8x + 3
→ h = −(−8)/(2·2) = 2, k = 2(4)−16+3 = −5 → Vertex: (2, −5)
Geometry & Coordinate Geometry
25% of ACTEssential Formulas
TRIANGLES: Area = ½bh | Heron's: A = √(s(s-a)(s-b)(s-c))
CIRCLE: Area = πr² | Circumference = 2πr
Arc Length = (θ/360°)·2πr | Sector Area = (θ/360°)·πr²
3D: Sphere Vol = (4/3)πr³ | Cylinder Vol = πr²h
Distance = √((x₂-x₁)²+(y₂-y₁)²)
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Circle eq: (x−h)²+(y−k)² = r²
CIRCLE: Area = πr² | Circumference = 2πr
Arc Length = (θ/360°)·2πr | Sector Area = (θ/360°)·πr²
3D: Sphere Vol = (4/3)πr³ | Cylinder Vol = πr²h
Distance = √((x₂-x₁)²+(y₂-y₁)²)
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Circle eq: (x−h)²+(y−k)² = r²
Key Angle Facts
Triangle angles sum to 180°. Exterior angle = sum of two non-adjacent interior angles
Special triangles: 30-60-90 (sides 1:√3:2) | 45-45-90 (sides 1:1:√2)
Parallel lines cut by transversal: alternate interior angles are equal
Trigonometry
12% of ACTSOH-CAH-TOA + Identities
sin θ = opp/hyp | cos θ = adj/hyp | tan θ = opp/adj
sin²θ + cos²θ = 1 (Pythagorean Identity)
tan θ = sin θ / cos θ
Law of Sines: a/sin A = b/sin B = c/sin C
Law of Cosines: c² = a² + b² − 2ab·cos C
Values: sin 30°=½ | sin 45°=√2/2 | sin 60°=√3/2
sin²θ + cos²θ = 1 (Pythagorean Identity)
tan θ = sin θ / cos θ
Law of Sines: a/sin A = b/sin B = c/sin C
Law of Cosines: c² = a² + b² − 2ab·cos C
Values: sin 30°=½ | sin 45°=√2/2 | sin 60°=√3/2
Key Facts
Radian: π radians = 180°. So 1 radian ≈ 57.3°
ASTC rule: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4) are positive
Statistics, Probability & Data
CoreFormulas
Mean = Σx / n
Median = middle value when sorted
Mode = most frequent value
P(A) = favorable outcomes / total outcomes
P(A and B) = P(A)·P(B) [if 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)!
Median = middle value when sorted
Mode = most frequent value
P(A) = favorable outcomes / total outcomes
P(A and B) = P(A)·P(B) [if 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)!
Key Facts
Range = Max − Min. Standard deviation measures spread from mean.
In a normal distribution: 68% within 1σ, 95% within 2σ, 99.7% within 3σ
35 minutes · 20 questions