ACT Math Practice Test with Detailed Solutions (Exam 1) | The School of Mathematics

50:00

0 / 45

ACT Math · Full Practice Test

ACT Math Practice Test
with Detailed Solutions (Exam 1)

A full-length, 45-question ACT Math practice test that mirrors the real exam format and difficulty curve. Take it timed for a realistic score estimate, or work through it untimed and study every explanation. This is Exam 1 — for the original quiz version with score tracking, see Exam 1 on theschoolofmathematics.com.

45 Questions 50 Minutes Full Solutions

Algebra

Geometry

Stats & Prob.

Numbers & Trig

🔢

Questions 1–8

Numbers, Operations & Basic Algebra

Foundational tier

Order of Operations

What is the value of 8 + 2(5 − 3)²?

A 16

B 20

C 12

D 24

Solution

8 + 2(5-3)² = 8 + 2(2)² = 8 + 2(4) = 8 + 8 = 16

⚠️ PEMDAS: parentheses first, then exponents, then multiplication. Squaring the (2) before multiplying by 2 is essential.

Fractions

What is 2/3 + 3/4?

A 5/7

B 17/12

C 6/12

D 1

Solution

LCD = 12: 2/3 = 8/12, 3/4 = 9/12 8/12 + 9/12 = 17/12

💡 Never add fractions by adding numerators and denominators directly (5/7 is wrong) — always find a common denominator first.

Percentages

A shirt originally priced at $40 is discounted 25%. What is the sale price?

A $10

B $30

C $32

D $35

Solution

Multiplier for 25% off: 1 - 0.25 = 0.75 Sale price = $40 \times 0.75 = $30

💡 Use the multiplier method: multiply by (1 − discount rate) in one step instead of finding the discount amount separately.

Linear Equations

If 4x − 7 = 21, what is the value of x?

A 5

B 6

C 7

D 8

Solution

4x - 7 = 21 4x = 28 x = 7

💡 Always check your answer by substituting back: 4(7)−7 = 28−7 = 21 ⠅ correct.

Exponents

What is the value of (2³)(2²)?

A 16

B 32

C 64

D 10

Solution

2³ \times 2² = 2⁵⁺²⁵ = 2⁵³⁵ = 32

⚠️ When multiplying powers with the same base, ADD the exponents: a⁵ᵀ⁵ × a⁵ⁿ⁵ = a⁵ᵀ⁺ⁿ⁵. Don't multiply the exponents (that rule is for power of a power).

Ratios

The ratio of boys to girls in a class is 3:5. If there are 24 students total, how many are boys?

A 8

B 9

C 15

D 12

Solution

Total ratio parts = 3 + 5 = 8 Each part = 24 / 8 = 3 Boys = 3 \times 3 = 9

💡 Always add the ratio parts first to find the value of "one part," then multiply by the relevant number of parts.

Absolute Value

What is the solution set of |x − 4| = 9?

A x = 13 only

B x = 5 only

C x = 13 or x = −5

D x = −13 or x = 5

Solution

x - 4 = 9 or x - 4 = -9 x = 13 or x = -5

⚠️ Absolute value equations split into two cases: the expression equals +k or −k. Solve both.

Scientific Notation

What is 4.5 × 10³ written in standard form?

A 450

B 4,500

C 45,000

D 0.0045

Solution

4.5 \times 10³ = 4.5 \times 1000 = 4,500

📐

Questions 9–18

Algebra: Systems, Factoring & Quadratics

Building tier

Systems of Equations

If 2x + 3y = 14 and x − y = 2, what is the value of x?

A 3

B 4

C 5

D 6

Solution

From eq2: x = y + 2 Substitute into eq1: 2(y+2) + 3y = 14 2y + 4 + 3y = 14 5y = 10 \to y = 2 x = y + 2 = 4

⚠️ Substitution works well when one equation easily isolates a variable, as with x − y = 2 giving x = y + 2. Check: 2(4)+3(2)=8+6=14 ⠅

Factoring

What is the factored form of x² + 7x + 12?

A (x+3)(x+4)

B (x+2)(x+6)

C (x+1)(x+12)

D (x−3)(x−4)

Solution

Need two numbers that multiply to 12 and add to 7: 3 and 4 x² + 7x + 12 = (x+3)(x+4)

💡 Check: (x+3)(x+4) = x²+4x+3x+12 = x²+7x+12 ⠅ matches.

Quadratic Equations

What are the solutions to x² − 9 = 0?

A x = 3 only

B x = 9 only

C x = 3 or x = −3

D x = 9 or x = −9

Solution

x² = 9 x = \pm\sqrt9 = \pm3

⚠️ Don't forget the negative root! Taking a square root always produces two possible solutions.

Quadratic Formula

Using the quadratic formula, what are the solutions to x² + 2x − 8 = 0?

A x = 2 or x = −4

B x = −2 or x = 4

C x = 8 or x = −1

D x = 1 or x = −8

Solution

Factor: (x+4)(x-2) = 0 x = -4 or x = 2

💡 Factoring is faster than the quadratic formula whenever the trinomial factors cleanly — always try factoring first.

Function Notation

If f(x) = 2x² − 5, what is f(−3)?

A 13

B −13

C 23

D −23

Solution

f(-3) = 2(-3)² - 5 = 2(9) - 5 = 18 - 5 = 13

⚠️ (−3)² = +9, not −9. Square the negative input first before multiplying.

Slope

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

A 1

B 2

C 3

D 4

Solution

slope = (13-5)/(6-2) = 8/4 = 2

💡 Slope formula: (y₂−y₁)/(x₂−x₁). Keep point order consistent in numerator and denominator.

Inequalities

What is the solution to −2x + 5 ≤ 13?

A x ≤ −4

B x ≥ −4

C x ≤ 4

D x ≥ 4

Solution

-2x + 5 \leq 13 -2x \leq 8 x \geq -4 (flip sign when dividing by negative)

⚠️ Dividing both sides by a negative number flips the inequality direction. This is the most commonly tested inequality rule.

Rational Expressions

Simplify: (x² − 4) / (x + 2)

A x − 2

B x + 2

C x² − 2

D 2x − 4

Solution

x²-4 = (x+2)(x-2) (x+2)(x-2)/(x+2) = x-2, where x \neq -2

💡 Recognize the difference of squares pattern a²−b²=(a+b)(a−b) to factor quickly.

Arithmetic Sequences

An arithmetic sequence begins 5, 9, 13, 17, … What is the 10th term?

A 37

B 41

C 45

D 49

Solution

a₁ = 5, d = 4 aₙ = a₁ + (n-1)d a₁₀ = 5 + (9)(4) = 5 + 36 = 41

⚠️ For the 10th term, use (n−1) = 9 in the formula, not n = 10 directly — a common off-by-one error.

Direct Variation

y varies directly with x. If y = 12 when x = 4, what is y when x = 10?

A 24

B 28

C 30

D 36

Solution

y = kx 12 = k(4) \to k = 3 y = 3(10) = 30

📐

Questions 19–30

Geometry: Triangles, Circles & Coordinate Geometry

Mid tier

Triangle Angle Sum

A triangle has angles measuring 50° and 65°. What is the measure of the third angle?

A 55°

B 65°

C 75°

D 115°

Solution

Angles in a triangle sum to 180° Third angle = 180 - (50 + 65) = 180 - 115 = 65°

💡 Always subtract the sum of the two known angles from 180° — never from 360°, which is for quadrilaterals.

Pythagorean Theorem

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

A 13

B 15

C 18

D 21

Solution

c² = 9² + 12² = 81 + 144 = 225 c = \sqrt225 = 15

💡 This is a 3-4-5 triple scaled by 3 (9=3×3, 12=4×3, 15=5×3). Recognizing common triples saves calculation time.

Special Right Triangles

A 30-60-90 triangle has a shortest side of length 5. What is the length of the longest side (hypotenuse)?

A 5√2

B 5√3

C 10

D 15

Solution

30-60-90 side ratio: x : x\sqrt3 : 2x Shortest side x = 5 Hypotenuse = 2x = 10

⚠️ Memorize the ratio x : x√3 : 2x for 30-60-90 triangles. The hypotenuse is always double the shortest side.

Circle Area

A circle has a radius of 6 cm. What is its area in terms of π?

A 12π cm²

B 36π cm²

C 6π cm²

D 18π cm²

Solution

A = πr² = π(6)² = 36π cm²

💡 Don't confuse area (πr²) with circumference (2πr) — a common mix-up under time pressure.

Coordinate Geometry

What is the midpoint of the segment connecting (−2, 7) and (8, −3)?

A (3, 2)

B (3, −2)

C (6, 4)

D (5, 4)

Solution

Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) = ((-2+8)/2, (7+(-3))/2) = (6/2, 4/2) = (3, 2)

💡 Average the x-coordinates and average the y-coordinates separately.

Distance Formula

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

A 4

B 5

C 6

D 7

Solution

d = \sqrt[(4-1)² + (6-2)²] = \sqrt[3²+4²] = \sqrt[9+16] = \sqrt25 = 5

💡 This is a 3-4-5 right triangle in disguise — the distance formula is just the Pythagorean theorem applied to coordinates.

Volume

A rectangular box has dimensions 4 cm × 5 cm × 6 cm. What is its volume?

A 60 cm³

B 90 cm³

C 120 cm³

D 150 cm³

Solution

V = l \times w \times h = 4 \times 5 \times 6 = 120 cm³

💡 Volume of a rectangular box is simply the product of all three dimensions.

Similar Triangles

Two similar triangles have a scale factor of 3. If the smaller triangle has a side of length 5, what is the corresponding side on the larger triangle?

A 8

B 10

C 15

D 25

Solution

Corresponding side = 5 \times 3 = 15

💡 Scale factor multiplies all linear dimensions directly. Remember that AREA scales by the square of the factor (here, 9×), not the same factor.

Arc Length

A circle has radius 8. What is the length of an arc with a central angle of 45°?

A π units

B 2π units

C 4π units

D 8π units

Solution

Arc length = (θ/360) \times 2πr = (45/360) \times 2π(8) = (1/8) \times 16π = 2π

💡 Arc length is always a fraction of the full circumference, where the fraction is (central angle)/360°.

Polygon Angles

What is the sum of the interior angles of a hexagon (6 sides)?

A 540°

B 720°

C 900°

D 1080°

Solution

Sum of interior angles = (n-2) \times 180° = (6-2) \times 180° = 4 \times 180° = 720°

💡 Memorize this formula: (n−2)×180° works for any polygon with n sides.

Equation of a Circle

What is the equation of a circle with center (1, −3) and radius 4?

A (x−1)² + (y+3)² = 16

B (x+1)² + (y−3)² = 16

C (x−1)² + (y+3)² = 4

D (x−1)² − (y+3)² = 16

Solution

(x-h)² + (y-k)² = r² Center (1,-3): (x-1)² + (y-(-3))² = (x-1)²+(y+3)² Radius 4: r²=16 (x-1)² + (y+3)² = 16

⚠️ Watch the sign flip: center y-coordinate of −3 becomes (y+3) in the equation, and remember to square the radius.

Transformations

A point at (3, 4) is reflected across the x-axis. What are the new coordinates?

A (−3, 4)

B (3, −4)

C (−3, −4)

D (4, 3)

Solution

Reflection over the x-axis: (x, y) \to (x, -y) (3, 4) \to (3, -4)

📊

Questions 31–39

Statistics, Probability & Trigonometry

Advancing tier

Mean

The mean of 5 numbers is 12. If one of the numbers is removed and the mean of the remaining 4 numbers is 11, what was the removed number?

A 10

B 13

C 16

D 20

Solution

Original sum = 5 \times 12 = 60 Remaining sum = 4 \times 11 = 44 Removed number = 60 - 44 = 16

💡 Use Sum = Mean × Count to convert between means and totals — this is the key relationship for every averages problem.

Median

What is the median of the data set: 14, 8, 22, 11, 19, 8?

A 11

B 12.5

C 14

D 8

Solution

Sort: 8, 8, 11, 14, 19, 22 (6 values, even count) Median = average of middle two = (11+14)/2 = 12.5

⚠️ Always sort the data first. With an even number of values, average the two middle numbers — don't just pick one.

Basic Probability

A bag contains 5 red, 3 blue, and 2 green marbles. What is the probability of drawing a green marble?

A 1/5

B 1/10

C 2/5

D 1/2

Solution

Total marbles = 5+3+2 = 10 P(green) = 2/10 = 1/5

💡 P(event) = favorable outcomes / total outcomes. Always find the total first.

Compound Probability

A coin is flipped twice. What is the probability of getting heads both times?

A 1/2

B 1/4

C 1/3

D 3/4

Solution

P(H) = 1/2 each flip, independent events P(H and H) = 1/2 \times 1/2 = 1/4

💡 For independent events, multiply individual probabilities: P(A and B) = P(A) × P(B).

Weighted Average

A student scores 80 on a test worth 30% of the grade and 90 on a test worth 70% of the grade. What is the student's overall grade?

A 84

B 85

C 87

D 88

Solution

Weighted average = 80(0.30) + 90(0.70) = 24 + 63 = 87

⚠️ Don't just average 80 and 90 to get 85 — the weights are different, so each score must be multiplied by its own weight.

SOH-CAH-TOA

In a right triangle, the side opposite a 30° angle is 5, and the hypotenuse is 10. What is sin(30°)?

A 1/2

B √3/2

C 2

D 5

Solution

sin(θ) = opposite / hypotenuse sin(30°) = 5/10 = 1/2

💡 This matches the known exact value sin(30°) = 1/2 — a good way to verify your SOH-CAH-TOA setup.

Trig Identities

If sin(θ) = 3/5, what is cos(θ) given that θ is in the first quadrant?

A 2/5

B 3/5

C 4/5

D 1

Solution

sin²θ + cos²θ = 1 (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 \to cos²θ = 16/25 cosθ = 4/5 (positive since θ is in Quadrant I)

💡 This is a 3-4-5 right triangle: opposite=3, adjacent=4, hypotenuse=5.

Standard Deviation Concept

Data set A: {10, 10, 10, 10}. Data set B: {2, 8, 12, 18}. Both have the same mean. Which statement is true?

A Set A has a larger standard deviation

B Set B has a larger standard deviation

C Both have equal standard deviation

D Cannot be determined

Solution

Set A has zero spread (all values identical) \to standard deviation = 0 Set B has values spread far from the mean \to standard deviation > 0

💡 Standard deviation measures spread, not average. More spread out data means a larger standard deviation, regardless of the mean.

Combinations

A committee of 2 people is chosen from a group of 6. How many different committees are possible?

A 12

B 15

C 30

D 36

Solution

C(6,2) = 6!/(2!\times4!) = (6\times5)/(2\times1) = 30/2 = 15

🔥

Questions 40–45

Advanced Mixed-Topic Problems

Hardest tier — matches real ACT difficulty curve

Function Composition

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

A 2

B 5

C 26

D 1

Solution

g(2) = 2(2) - 3 = 1 f(g(2)) = f(1) = 1² + 1 = 2

⚠️ Work from the inside out: evaluate g(2) first, then plug that result into f.

Sector Area & Reasoning

A circular pizza has a radius of 10 inches and is cut into 8 equal slices. What is the area of one slice, in terms of π?

A 10π in²

B 12.5π in²

C 25π in²

D 100π in²

Solution

Total area = πr² = π(10)² = 100π One slice = 100π / 8 = 12.5π in²

💡 Find the total area first, then divide by the number of equal slices — no need to compute the central angle separately.

Systems & Word Problem

A theater sells adult tickets for $12 and child tickets for $7. If 50 tickets were sold for a total of $480, how many adult tickets were sold?

A 24

B 26

C 28

D 30

Solution

Let a = adult, c = child a + c = 50 \to c = 50-a 12a + 7(50-a) = 480 12a + 350 - 7a = 480 5a = 130 \to a = 26

💡 Check: 26 adults + 24 children = 50 tickets; 26(12)+24(7) = 312+168 = $480 ⠅

Logarithms

What is the value of x if log₂(x) = 5?

A 10

B 25

C 32

D 64

Solution

log₂(x) = 5 means 2⁵⁵ = x x = 2⁵ = 32

⚠️ Convert log form to exponential form: log_b(x) = y means b⁵ᴬ⁵ = x. This conversion is essential for every logarithm question.

Complex Numbers

What is the value of i⁵³⁵, where i = √−1?

A 1

B −1

C i

D −i

Solution

Powers of i cycle every 4: i¹=i, i²=-1, i³=-i, i⁴=1 13 \div 4 = 3 remainder 1 i¹³ = i¹ = i

💡 Divide the exponent by 4 and use the remainder to find the equivalent power in the cycle {i, −1, −i, 1}.

Multi-Step Geometry

A square has a diagonal of length 8√2. What is the area of the square?

A 32

B 64

C 128

D 16

Solution

For a square: diagonal = side \times \sqrt2 8\sqrt2 = side \times \sqrt2 \to side = 8 Area = side² = 8² = 64

💡 Recognize the diagonal formula side√2 for squares — it comes directly from the 45-45-90 triangle relationship.

Your Results

—

out of 45 questions

—

Correct

—

Incorrect

—

Time Used

Take the Official Quiz → More Practice →

Free · 2,000+ Questions · No Account Needed

Want More Full-Length Practice?

This is Exam 1 in our growing library of full ACT Math practice tests. The ACT Math QBank has 2,000+ additional questions organized by topic for targeted drilling.

Go to ACT Math QBank →

Free ACT Math Practice Test with Detailed Solutions

ACT Math Practice Test
with Detailed Solutions (Exam 1)

Want More Full-Length Practice?

Comments

Leave a comment