Practice ACT Math Test: 15 Timed Questions With Worked Solutions

• Questions: 15 (mixed topics, easy to hard)
• Time: 17 minutes
• Calculator: Allowed (graphing or scientific)
• Guessing: No penalty for wrong answers, so answer every question
• Format: 4 answer choices per question (A, B, C, D)

Question 1

A movie ticket costs $12.50. If you buy tickets for yourself and 3 friends, what is the total cost?

A) $37.50    B) $43.75    C) $50.00    D) $56.25

Question 2

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

A) 1/2    B) 2    C) 4    D) 8

Question 3

If 5(x − 3) = 2x + 6, what is the value of x?

A) 3    B) 5    C) 7    D) 9

Question 4

A rectangular garden is 18 feet long and 12 feet wide. What is the length of the diagonal, in feet?

A) 6√13    B) 6√5    C) 30    D) 15

Question 5

A store marks up the wholesale price of a jacket by 60%. If the retail price is $120, what was the wholesale price?

A) $48    B) $72    C) $75    D) $80

Question 6

The expression (3x²y)(4xy³) is equivalent to:

A) 7x³y&sup4;    B) 12x²y³    C) 12x³y&sup4;    D) 12x³y³

Question 7

In a circle with radius 8, what is the area of a sector with a central angle of 90°?

A) 8π    B) 16π    C) 32π    D) 64π

Question 8

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

A) 7    B) 13    C) 15    D) 26

Question 9

A bag contains 4 red marbles, 6 blue marbles, and 5 green marbles. If you draw one marble at random, what is the probability that it is NOT blue?

A) 2/5    B) 3/5    C) 2/3    D) 3/4

Question 10

What is the solution set for the equation x² − 9 = 0?

A) {3}    B) {−3}    C) {−3, 3}    D) {−9, 9}

Question 11

A triangle has vertices at (0, 0), (6, 0), and (3, 4). What is the area of the triangle?

A) 10    B) 12    C) 18    D) 24

Question 12

For what value of k does the system of equations below have no solution?

2x + 3y = 7

6x + ky = 10

A) 3    B) 6    C) 9    D) 21

Question 13

In a right triangle, one leg is 7 and the hypotenuse is 25. What is the length of the other leg?

A) 18    B) 24    C) 26    D) √576

Question 14

If sin(θ) = 4/5 and θ is in the first quadrant, what is tan(θ)?

A) 3/4    B) 4/3    C) 3/5    D) 5/4

Question 15

The function f(x) = −2(x − 3)² + 8 has a maximum value. What is it, and at what x-value does it occur?

A) Maximum of 8 at x = 3    B) Maximum of 8 at x = −3    C) Maximum of 3 at x = 8    D) Maximum of −8 at x = 3

Solution 1 — Answer: C) $50.00

Topic: Pre-Algebra (arithmetic)

You are buying tickets for yourself AND 3 friends, so that is 4 tickets total (a common trap is choosing 3).

4 × $12.50 = $50.00

Solution 2 — Answer: B) 2

Topic: Coordinate Geometry (slope)

Slope = (y&sub2; − y&sub1;) / (x&sub2; − x&sub1;) = (13 − 5) / (6 − 2) = 8/4 = 2

Solution 3 — Answer: C) 7

Topic: Elementary Algebra (linear equations)

Distribute: 5x − 15 = 2x + 6

Subtract 2x from both sides: 3x − 15 = 6

Add 15 to both sides: 3x = 21

Divide by 3: x = 7

Check: 5(7 − 3) = 5(4) = 20. And 2(7) + 6 = 20. Correct.

Solution 4 — Answer: A) 6√13

Topic: Plane Geometry (Pythagorean theorem)

The diagonal of a rectangle forms a right triangle with the sides.

d² = 18² + 12² = 324 + 144 = 468

d = √468 = √(36 × 13) = 6√13

Simplify the radical by factoring out perfect squares: 468 = 4 × 117 = 4 × 9 × 13 = 36 × 13.

Solution 5 — Answer: C) $75

Topic: Pre-Algebra (percentages)

A 60% markup means the retail price is 160% of wholesale.

Wholesale × 1.60 = $120

Wholesale = $120 / 1.60 = $75

Common trap: taking 60% of 120 ($72), which gives you the markup amount, not the wholesale price.

Solution 6 — Answer: C) 12x³y&sup4;

Topic: Elementary Algebra (exponent rules)

Multiply coefficients: 3 × 4 = 12

Multiply x terms: x² × x = x³ (add exponents: 2 + 1 = 3)

Multiply y terms: y × y³ = y&sup4; (add exponents: 1 + 3 = 4)

Result: 12x³y&sup4;

Solution 7 — Answer: B) 16π

Topic: Plane Geometry (circles/sectors)

Full circle area = πr² = π(8)² = 64π

A 90° sector is 90/360 = 1/4 of the full circle.

Sector area = 64π / 4 = 16π

Solution 8 — Answer: B) 13

Topic: Intermediate Algebra (function composition)

Work from the inside out.

First find g(2): g(2) = 2² + 1 = 4 + 1 = 5

Then find f(5): f(5) = 3(5) − 2 = 15 − 2 = 13

Key concept: f(g(2)) means "plug 2 into g first, then plug that result into f."

Solution 9 — Answer: B) 3/5

Topic: Pre-Algebra (probability)

Total marbles = 4 + 6 + 5 = 15

NOT blue = 4 red + 5 green = 9

P(not blue) = 9/15 = 3/5

Alternatively: P(not blue) = 1 − P(blue) = 1 − 6/15 = 1 − 2/5 = 3/5.

Solution 10 — Answer: C) {−3, 3}

Topic: Intermediate Algebra (difference of squares)

x² − 9 = 0 is a difference of squares: x² − 3² = 0

Factor: (x + 3)(x − 3) = 0

Set each factor to zero: x = −3 or x = 3

A common trap is forgetting the negative solution. Always check for both when solving quadratics.

Solution 11 — Answer: B) 12

Topic: Coordinate Geometry (area)

For a triangle with one side along the x-axis, use: Area = (1/2) × base × height.

The base runs from (0, 0) to (6, 0), so base = 6.

The third vertex (3, 4) has a y-coordinate of 4, so height = 4.

Area = (1/2) × 6 × 4 = 12

Solution 12 — Answer: C) 9

Topic: Intermediate Algebra (systems of equations)

A system has no solution when the lines are parallel (same slope, different y-intercept).

Multiply the first equation by 3: 6x + 9y = 21

Compare with: 6x + ky = 10

For parallel lines, the coefficients of x and y must be proportional but the constants must not be. So k = 9 makes the left sides identical (6x + 9y), but 21 ≠ 10, meaning no solution exists.

Solution 13 — Answer: B) 24

Topic: Plane Geometry (Pythagorean theorem)

a² + b² = c², where c = 25 (hypotenuse) and a = 7.

7² + b² = 25²

49 + b² = 625

b² = 576

b = 24

This is the 7-24-25 Pythagorean triple. Note that answer D (√576) equals 24, so both B and D give the same number, but B is the simplified form the ACT expects.

Solution 14 — Answer: B) 4/3

Topic: Trigonometry (trig ratios)

sin(θ) = opposite/hypotenuse = 4/5, so opposite = 4 and hypotenuse = 5.

Find adjacent using Pythagorean theorem: adj = √(5² − 4²) = √(25 − 16) = √9 = 3

tan(θ) = opposite/adjacent = 4/3

This uses the 3-4-5 right triangle. Recognizing common Pythagorean triples saves time on the ACT.

Solution 15 — Answer: A) Maximum of 8 at x = 3

Topic: Intermediate Algebra (vertex form)

f(x) = −2(x − 3)² + 8 is in vertex form: f(x) = a(x − h)² + k, where the vertex is at (h, k).

Here, h = 3 and k = 8, so the vertex is at (3, 8).

Since a = −2 (negative), the parabola opens downward, making the vertex a maximum.

The maximum value is 8, occurring at x = 3.