ACT Math Master Exam | 20 Essential Questions

Core Concepts & Formulas

Pre-Algebra & Number Theory

Topic 1

Memorize: Order of operations (PEMDAS), LCM/GCF, percent formula: Part = Percent × Whole, absolute value |x| = distance from 0.

Percent Change = ((New − Old) / Old) × 100%

Example

A shirt costs $40. After a 25% discount, what is the sale price?

✓ Sale price = 40 × (1 − 0.25) = 40 × 0.75 = $30

Algebra: Linear & Quadratic

Topic 2

Memorize: Slope = (y₂−y₁)/(x₂−x₁); Slope-intercept: y = mx + b; Quadratic formula: x = [−b ± √(b²−4ac)] / 2a; FOIL for (a+b)² = a²+2ab+b²

Quadratic Formula: x = [−b ± √(b² − 4ac)] / 2a

Example

Solve: x² − 5x + 6 = 0

✓ Factor: (x−2)(x−3) = 0 → x = 2 or x = 3

Functions & Graphs

Topic 3

Memorize: f(g(x)) = composite function; Domain: all valid x values; Range: all resulting y values; Vertex of parabola: x = −b/2a; Transformations: f(x)+k shifts up, f(x−h) shifts right.

Vertex: x = −b/(2a), y = f(−b/2a)

Example

If f(x) = 2x + 3 and g(x) = x², find f(g(2)).

✓ g(2) = 4, f(4) = 2(4)+3 = 11

Plane Geometry

Topic 4

Memorize: Triangle angles sum = 180°; Pythagorean theorem: a²+b²=c²; Special triangles: 30-60-90 (1:√3:2) and 45-45-90 (1:1:√2); Circle area = πr², circumference = 2πr.

Area: Triangle = ½bh | Trapezoid = ½(b₁+b₂)h | Circle = πr²

Example

A right triangle has legs 6 and 8. Find the hypotenuse.

✓ c = √(6²+8²) = √(36+64) = √100 = 10

Coordinate Geometry

Topic 5

Memorize: Distance = √[(x₂−x₁)²+(y₂−y₁)²]; Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2); Circle: (x−h)²+(y−k)²=r² has center (h,k) and radius r.

Distance = √[(Δx)² + (Δy)²] | Midpoint = (avg x, avg y)

Example

Find distance between (1,2) and (4,6).

✓ d = √[(4−1)²+(6−2)²] = √[9+16] = √25 = 5

Trigonometry

Topic 6

Memorize SOH-CAH-TOA: sin θ = opp/hyp; cos θ = adj/hyp; tan θ = opp/adj. Also: sin²θ + cos²θ = 1; Law of Sines: a/sinA = b/sinB = c/sinC.

SOH-CAH-TOA | sin²θ + cos²θ = 1

Example

In a right triangle, opposite = 3, hypotenuse = 5. Find sin θ.

✓ sin θ = 3/5 = 0.6

Statistics & Probability

Topic 7

Memorize: Mean = sum/count; Median = middle value; Mode = most frequent; Probability = favorable/total; P(A or B) = P(A)+P(B)−P(A∩B).

Mean = Σx/n | P(event) = favorable outcomes / total outcomes

Example

Data set: 3, 7, 7, 9, 14. Find mean and median.

✓ Mean = (3+7+7+9+14)/5 = 40/5 = 8; Median = 7

Advanced Algebra & Modeling

Topic 8

Memorize: Arithmetic sequence: aₙ = a₁+(n−1)d; Geometric sequence: aₙ = a₁·rⁿ⁻¹; Exponent rules: aᵐ·aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ.

Arithmetic: aₙ = a₁ + (n−1)d | Geometric: aₙ = a₁ · rⁿ⁻¹

Example

Find the 8th term of the sequence: 3, 6, 12, 24, ...

✓ Geometric, r=2: a₈ = 3·2⁷ = 3·128 = 384

Practice Questions

📐 Pre-Algebra & Number Theory

Pre-Algebra · Percentages

Easy

A store is offering a 30% discount on a jacket originally priced at $120. After the discount, a 10% sales tax is applied. What is the final price of the jacket?

1Discounted price = $120 × (1 − 0.30) = $120 × 0.70 = $84.00

2Apply 10% tax: $84.00 × 1.10 = $92.40

3Correct answer: B) $92.40

Final = Original × (1 − discount) × (1 + tax rate)

Pre-Algebra · Number Theory

Easy

What is the least common multiple (LCM) of 12, 18, and 24?

1Prime factorize: 12 = 2²×3; 18 = 2×3²; 24 = 2³×3

2LCM = highest power of each prime: 2³ × 3² = 8 × 9 = 72

3Verify: 72÷12=6 ✓, 72÷18=4 ✓, 72÷24=3 ✓

📊 Algebra

Algebra · Linear Equations

Easy

If 3x − 4 = 2x + 7, what is the value of x²?

13x − 4 = 2x + 7 → 3x − 2x = 7 + 4 → x = 11

2x² = 11² = 121

3Correct answer: B) 121

Algebra · Quadratics

Medium

The equation 2x² − 7x − 15 = 0 has two solutions. What is the sum of the two solutions?

1By Vieta's formulas, for ax²+bx+c=0, sum of roots = −b/a

2Here a=2, b=−7: Sum = −(−7)/2 = 7/2

3Verify by factoring: 2x²−7x−15 = (2x+3)(x−5) → roots: x = −3/2 and x = 5; Sum = −3/2 + 5 = 7/2 ✓

Sum of roots = −b/a | Product of roots = c/a

Algebra · Systems of Equations

Medium

If 3x + 2y = 16 and x − y = 2, what is the value of x + y?

1From eq.2: x = y + 2. Substitute into eq.1: 3(y+2)+2y = 16 → 3y+6+2y = 16 → 5y = 10 → y = 2

2x = y+2 = 2+2 = 4

3x+y = 4+2 = 6. Verify: 3(4)+2(2)=12+4=16 ✓; 4−2=2 ✓

4Correct answer: B) 6

Substitution: isolate one variable, substitute into other equation

📈 Functions & Graphs

Functions · Composition

Medium

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

1First compute f(3): f(3) = 3² − 3 = 9 − 3 = 6

2Then compute g(f(3)) = g(6): g(6) = 2(6)+1 = 12+1 = 13

3Correct answer: B) 13

g(f(x)): evaluate inner function first, then outer

Functions · Parabola Vertex

Medium

The function f(x) = −2x² + 8x − 3 has a maximum value at its vertex. What is the maximum value of f(x)?

1Vertex x-coordinate: x = −b/(2a) = −8/(2×−2) = −8/−4 = 2

2f(2) = −2(4)+8(2)−3 = −8+16−3 = 5

3Since a = −2 < 0, parabola opens down, so vertex is a maximum. Max value = 5

Vertex x = −b/(2a) | If a < 0, vertex is maximum

📐 Plane Geometry

Geometry · Triangles

Easy

In triangle ABC, angle A = 45°, angle B = 75°. A line segment from vertex C to side AB is drawn perpendicular to AB, creating two smaller triangles. What is the measure of angle C in the original triangle?

1Sum of angles in a triangle = 180°

2Angle C = 180° − 45° − 75° = 180° − 120° = 60°

3Correct answer: A) 60°

A + B + C = 180° for any triangle

Geometry · Circles

Medium

A circle has a circumference of 20π inches. A sector of this circle subtends a central angle of 72°. What is the area of this sector in square inches?

1Circumference = 2πr = 20π → r = 10

2Area of full circle = πr² = π(10)² = 100π

3Sector area = (θ/360°) × 100π = (72/360) × 100π = (1/5) × 100π = 20π

4Correct answer: A) 20π

Sector Area = (θ/360°) × πr²

Geometry · Special Triangles

Hard

An equilateral triangle has a perimeter of 24 cm. What is the area of the triangle in square centimeters?

1Side = 24/3 = 8 cm

2Height of equilateral triangle: h = (s√3)/2 = (8√3)/2 = 4√3

3Area = ½ × base × height = ½ × 8 × 4√3 = 16√3

4Correct answer: A) 16√3

Equilateral triangle area = (s²√3)/4 = (64√3)/4 = 16√3 ✓

🗺 Coordinate Geometry

Coordinate Geometry · Lines

Medium

Line ℓ passes through the points (−2, 5) and (4, −1). What is the equation of the line perpendicular to ℓ that passes through the point (1, 3)?

1Slope of ℓ: m = (−1−5)/(4−(−2)) = −6/6 = −1

2Perpendicular slope = negative reciprocal = +1

3Through (1,3) with slope 1: y−3 = 1(x−1) → y = x + 2

4Correct answer: A) y = x + 2

Perpendicular slopes: m₁ × m₂ = −1

Coordinate Geometry · Circles

Hard

What is the radius of the circle defined by x² + y² − 6x + 4y − 12 = 0?

1Complete the square: (x²−6x) + (y²+4y) = 12

2(x²−6x+9) + (y²+4y+4) = 12+9+4 → (x−3)² + (y+2)² = 25

3r² = 25 → r = 5. Center = (3, −2)

4Correct answer: B) 5

Complete the square to get (x−h)² + (y−k)² = r²

📡 Trigonometry

Trigonometry · SOH-CAH-TOA

Medium

In a right triangle, the side opposite to angle θ is 5 units and the side adjacent to angle θ is 12 units. What is the value of sin θ + cos θ?

1Hypotenuse = √(5²+12²) = √(25+144) = √169 = 13

2sin θ = opp/hyp = 5/13

3cos θ = adj/hyp = 12/13

4sin θ + cos θ = 5/13 + 12/13 = 17/13

sin = opp/hyp | cos = adj/hyp | tan = opp/adj

Trigonometry · Identities

Hard

If sin θ = 3/5 and θ is in the first quadrant, what is the value of tan θ?

1sin θ = 3/5, so opp = 3, hyp = 5

2adj = √(5²−3²) = √(25−9) = √16 = 4

3tan θ = opp/adj = 3/4. In Q1, tan is positive → 3/4

4Correct answer: B) 3/4

tan θ = sin θ / cos θ = (3/5)/(4/5) = 3/4

📊 Statistics & Probability

Statistics · Mean & Median

Medium

Six students scored: 72, 85, 90, 68, 95, and 80 on a test. If a 7th student joins and the mean score increases to 83, what did the 7th student score?

1Sum of 6 scores = 72+85+90+68+95+80 = 490

2New mean of 7 students = 83, so total sum = 83×7 = 581

37th student's score = 581 − 490 = 91

4Correct answer: C) 91

New score = (new mean × new count) − (old sum)

Probability · Compound Events

Hard

A bag contains 4 red, 3 blue, and 5 green marbles. Two marbles are drawn without replacement. What is the probability that both marbles are the same color?

1Total marbles = 4+3+5 = 12. Total ways to choose 2: C(12,2) = 66

2Both red: C(4,2) = 6 ways

3Both blue: C(3,2) = 3 ways

4Both green: C(5,2) = 10 ways

5Total same-color = 6+3+10 = 19. P = 19/66

P(same color) = [C(4,2)+C(3,2)+C(5,2)] / C(12,2) = 19/66

🔢 Advanced Algebra & Modeling

Advanced Algebra · Sequences

Medium

The 3rd term of a geometric sequence is 12 and the 6th term is 96. What is the 1st term of the sequence?

1a₃ = 12, a₆ = 96. a₆/a₃ = r³ = 96/12 = 8 → r = 2

2a₃ = a₁ × r² → 12 = a₁ × 4 → a₁ = 3

3Verify: 3, 6, 12, 24, 48, 96 ✓

4Correct answer: A) 3

aₙ = a₁ · rⁿ⁻¹ | r = ⁿ√(aₙ/a₁)

Advanced Algebra · Exponents

Hard

If 4^(x+1) = 8^(x−1), what is the value of x?

1Rewrite both sides as powers of 2: 4 = 2², 8 = 2³

22^(2(x+1)) = 2^(3(x−1)) → 2^(2x+2) = 2^(3x−3)

3Equate exponents: 2x+2 = 3x−3 → x = 5

4Check: 4^6 = 4096; 8^4 = 4096 ✓. Answer: B) 5

Convert to same base → equate exponents

Advanced Algebra · Logarithms

Hard

If log₂(x) + log₂(x − 2) = 3, what is the positive value of x?

1Combine logs: log₂(x(x−2)) = 3

2x(x−2) = 2³ = 8 → x²−2x−8 = 0

3Factor: (x−4)(x+2) = 0 → x = 4 or x = −2

4x = −2 is rejected (log of negative undefined). x = 4

5Verify: log₂(4)+log₂(2) = 2+1 = 3 ✓. Answer: B) 4

log(a) + log(b) = log(ab) | log₂(8) = 3

Mixed · Complex Word Problem

Hard

A rectangular swimming pool measures 20 meters long and 10 meters wide. A uniform tile border of width x meters is built around the pool. If the total area including the border is 2.5 times the area of the pool alone, what is the width x of the border?

1Pool area = 20 × 10 = 200 m²

2Total area = 2.5 × 200 = 500 m²

3Outer dimensions = (20+2x) × (10+2x) = 500

4Expand: 200 + 40x + 20x + 4x² = 500 → 4x²+60x−300 = 0 → x²+15x−75 = 0

5Quadratic formula: x = [−15 ± √(225+300)]/2 = [−15 ± √525]/2 ≈ [−15+22.91]/2 ≈ 3.96 ≈ 4 m

6Verify: (20+8)×(10+8) = 28×18 = 504 ≈ 500 (closest). Correct answer: D) 4 m

(L+2x)(W+2x) = 2.5·L·W | Solve with quadratic formula