Below are answer explanations to the full-length Math test released by ACT for the ACT 2025. 

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ACT 2025 Practice Test Math Answer Explanations

Question 1. The answer is D, 90.

1. Plus the values given into the formula for mean:

  • \frac{100 + 60 + 80 + 30 + x}{5} = 72

2. Multiply both sides by 5:

  • 270 + x = 360

3. Subtract:

  • x = 90

Question 2. The answer is D, 19.

1. Add all the farms that are not within any of the four parts of circle C:

12 + 2 + 5 = 19

Question 3. The answer is B, “He should increase the area of the red section by decreasing the area of the blue section.”

1. Each section should be landed on 25\% \times 500 = 0.25 \times 500 = 125 times.

2. Looking at the table, sections yellow and green are close enough to 125 to not fix.

3. The results of the spins for sections red and blue are too low and too high, respectively.

4. To fix the imbalance, the area of the blue section should be decreased so the area of the red section is increased.

Question 4. The answer is A, “18.2 degrees“.

1. Add the given angles together to 180 degrees, using angle A = x:

x + 143.6° + x = 180°

2. Subtract:

2x = 36.4°

3. Divide by 2:

x = angle A = 18.2°

Question 5. The answer is C, “(x - 6)(x + 5)“.

1. The factored form of x² - x - 30 needs factors of -30 that add to -1.

2. Out of all potential factors of -30, only -6 and 5 satisfy the requirement.

3. Use the satisfactory factors to find:

x² - x - 30 = (x - 6)(x + 5)

4. FOIL the factored form to prove the equation above:

(x - 6)(x + 5) = x² + 5x - 6x - 30 = x² - x - 30

Question 6. The answer is D, Matrix: [-20 10; 0 -25].

1. Multiply each element in the matrix by 5:

Matrix multiplication and result

Question 7. The answer is B, “15”.

1. There must be a common factor of 30 and 75 as the number of members in the work group.

2. Of all the answer choices, only 15 is a common factor of 30 and 75:

15 × 2 = 30 and 15 × 5 = 75

Question 8. The answer is C, 6√105.

1. Plug the given skid length into the speed formula:

Speed = √(35 × 108)

2. Multiply:

35 × 108 = 3780 ⇒ √3780

3. Rationalize the radical:

√3780 = √(36 × 105) = 6√105

Question 9. The answer is D, (6y + 1)/5.

1. Add 1 to both sides:

6y + 1 = 5x

2. Divide both sides by 5:

x = (6y + 1)/5

Question 10. The answer is C, “44”.

1. Convert miles to feet:

30 miles/hour = 30 × 5,280 = 158,400 feet/hour

2. Convert hours to seconds:

1 hour = 60 × 60 = 3,600 seconds

3. Divide to find feet per second:

158,400 feet / 3,600 seconds = 44 feet/second

Question 11. The answer is B, “100.9”.

1. Substitute t = 3 into the equation:

h(3) = -4.9(3)^2 + 30(3) + 55 = -4.9(9) + 90 + 55 = -44.1 + 90 + 55 = 100.9

Question 12. The answer is D, 10 over 30.

1. List the prime numbers from 1 to 30:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

There are 10 prime numbers in total.

2. Calculate the probability:

Probability = 10/30

Question 13. The answer is A, 12 over 13.

1. Given:

sin α = opposite/hypotenuse = 5/13

and

tan α = opposite/adjacent = 5/12

Identify the side lengths of the triangle:

  • Opposite = 5
  • Adjacent = 12
  • Hypotenuse = 13

2. Use the side lengths to find:

cos α = adjacent/hypotenuse = 12/13

Question 14. The answer is D, “There is no such value for y.”

1. Solve the first equation for y:

2x - y = 7 ⇒ y = 2x - 7

2. Substitute into the second equation:

-4x + 2(2x - 7) = 2; -4x + 4x - 14 = 2; 0x - 14 = 2 ⇒ -14 = 2

This contradiction means the system has no solution.

Question 15. The answer is A, y^3 + 21y^2 + 147y + 343.

1. Use the binomial expansion:

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

2. Plug in a = y, b = 7 and simplify:

(y + 7)^3 = y^3 + 3y^2(7) + 3y(49) + 343 = y^3 + 21y^2 + 147y + 343

Question 16. The answer is B, “36”.

1. Let the integers be:

5x, 3x, 2x

2. The sum of the integers is given to be 180:

5x + 3x + 2x = 180 ⇒ 10x = 180 ⇒ x = 18

3. Plug x back in to find the least integer:

2x = 2(18) = 36

Question 17. The answer is A, “-5x^2 - 4xy“.

1. Distribute the minus sign:

x^2 - y^2 - 6x^2 - 4xy + y^2

2. Combine like terms:

x^2 - 6x^2 = -5x^2; -y^2 + y^2 = 0; -4xy = -4xy

3. The expression simplifies to:

-5x^2 - 4xy

Question 18. The answer is D, 3 + 4i.

1. Rationalize the radicals using the identity given:

√9 = 3, √-16 = i√16 = 4i

2. Add the results:

3 + 4i = 3 + 4i

Question 19. The answer is B, “343”.

1. The formula for the nth term of an arithmetic sequence is:

t_n = a + (n - 1)d

where:

  • a = 7 (first term)
  • d = 21 - 7 = 14 (common difference)

2. Plug these values into the formula:

t₍₂₅₎ = 7 + (25 - 1)14 = 7 + 24·14 = 7 + 336 = 343

Question 20. The answer is C, “15.0”.

1. Use similar triangles to set up a proportion:

(height of fence)/(shadow of fence) = (height of flagpole)/(shadow of flagpole)

4.0/2.4 = h/9.0 ⇒ 4.0×9.0 = 2.4×h ⇒ 36 = 2.4h ⇒ h = 36/2.4 = 15

Question 21. The answer is C, “17”.

1. Let x be the horizontal leg of triangle DEC and use the Pythagorean Theorem:

CE² = DE² + x²; 25² = 7² + x² ⇒ 625 = 49 + x² ⇒ x² = 576 ⇒ x = 24

2. Let y = BC, the full height of the rectangle.

3. Apply Pythagorean Theorem again given triangle BEC has vertical leg y - 7 and horizontal leg 24:

BE² = (y - 7)² + 24²; 26² = (y - 7)² + 576 ⇒ 676 = (y - 7)² + 576; 100 = (y - 7)² ⇒ y - 7 = 10 ⇒ y = 17

Question 22. The answer is B, “6 cups”.

1. Convert 5 gallons of water to cups:

1 gallon = 4 quarts, 1 quart = 4 cups ⇒ 5 × 4 × 4 = 80 cups of water

2. Use the ratio 3/40 = x/80 and solve for x:

40x = 3 × 80 = 240 ⇒ x = 240 / 40 = 6

Question 23. The answer is C,  2 × 10^7.

1. Evaluate:

f(5) = 7e^15 + 1

2. Estimate e^15 using:

e^15 ≈ 3,261,848

3. Plug in the estimation and multiply:

f(5) ≈ 7 × 3,261,848 + 1 ≈ 22,833,937

4. The closest answer of the possible choices is:

2 × 10^7 = 20,000,000 (closest)

Question 24. The answer is B,  2π/9.

1. To convert degrees to radians, multiply by π/180:

40° × (π / 180) = 40π / 180 = 2π / 9

Question 25. The answer is A, “shifted downward 4 coordinate units”.

1. Subtracting 4 from the function output shifts the graph down 4 units:

y = f(x) - 4 ⇒ vertical shift downward by 4 units

2. No horizontal shift or stretching/compression is involved.

Question 26. The answer is B,  √[2(d - c)/a].

1. Subtract c from both sides:

(1/2)ab² = d - c

2. Multiply both sides by 2:

ab² = 2(d - c)

3. Divide both sides by a:

b² = [2(d - c)]/a

4. Take the square root of both sides:

b = √[2(d - c)/a]

Question 27. The answer is B, 31 over 479.

1. From the table:

  • Total number of trucks: 479
  • Number of black trucks: 31

2. Find the probability:

Probability = Black Trucks / Total Trucks = 31/479

Question 28. The answer is A, “54”.

1. Because it is inscribed in a circle, each side of the regular hexagon is equal in length to the radius of the circle.

2. Find the side lengths:

Diameter = 18 ⇒ Radius = 18/2 = 9

3. Use the side lengths to find the perimeter:

Perimeter = 6 × side length = 6 × 9 = 54

Question 29. The answer is C, “$144,160″.

1. Let the annual raise be r.

2. Find the value of the annual raise:

34000 + 3r = 38080 ⇒ 3r = 4080 ⇒ r = 1360

3. Calculate earnings for each year:

Earnings over 4 years

4. Sum to find the total earnings over 4 years:

Total = 34000 + 35360 + 36720 + 38080 = 144160

Question 30. The answer is D, “4”.

1. Points that are 5 units from the origin lie on the circle:

x² + y² = 25

2. Points that are 2 units from the line y = 0 must lie on the lines:

y = 2 or y = -2

3. Plug y = 2 into the circle equation:

x² + 2² = 25 ⇒ x² = 21 ⇒ x = ±√21

4. Plug y = -2 into the circle equation:

x² + (-2)² = 25 ⇒ x² = 21 ⇒ x = ±√21

5. These equations give us four points total:

  • (√21, 2)
  • (-√21, 2)
  • (√21, -2)
  • (-√21, -2)

Question 31. The answer is C, “BC < AC < AB“.

1. Given:

∠A < ∠B < ∠C

2. The sides opposite angles in triangles are directly proportional, meaning the shortest side of a triangle is opposite the smallest angle:

Opposite ∠A → side BC, Opposite ∠B → side AC, Opposite ∠C → side AB

3. From least to greatest:

BC < AC < AB

Question 32. The answer is D, “128”.

1. Let x be the total number of pies sold.

2. Create an equation of total pies sold and solve for x:

(1/4)x + (1/2)x + 24 + 8 = x ⇒ (3/4)x + 32 = x ⇒ x - (3/4)x = 32 ⇒ (1/4)x = 32 ⇒ x = 128

Question 33. The answer is C, “43”.

1. The sum of interior angles in a quadrilateral is 360°.

2. Add the given interior angle values and solve for x:

2(3x + 5) + 2(x + 3) = 360 ⇒ 6x + 10 + 2x + 6 = 360 ⇒ 8x + 16 = 360 ⇒ 8x = 344 ⇒ x = 43

Question 34. The answer is B, “57”.

“The answer is B. 57”

1. Use the given definition of the sequence and n = 5 to find the fifth term:

a₅ = a₄ + (5 - 1)² = 16 + 16 = 32

2. With the fifth term, plug in n = 6 to find the answer:

a₆ = a₅ + (6 - 1)² = 32 + 25 = 57

Question 35. The answer is C, “48”.

1. Find the decimal forms of the given boundaries:

-65/6 ≈ -10.83 and 75/2 = 37.5

2. The smallest integer greater than -10.8 is -10, and the largest integer less than 37.5 is 37.

3. Subtract to find the number of integers between:

37 - (-10) + 1 = 48

Question 36. The answer is D, “P(A and B).

1. Mutually exclusive means that the events cannot happen at the same time.

2. So:

P(A and B) = 0

Question 37. The answer is C, “–2 and 3”.

1. Complete polynomial long division given (x - 3) is a factor:

p(x) = (x - 3)(x² + 4x + 4) = (x - 3)(x + 2)²

2. Using the completely factored form, the zeros are:

x = 3, x = -2 (with multiplicity 2)

Question 38. The answer is D, “4.5”.

1. Given 18 days, the median is the average of the 9th and 10th values.

2. Find the cumulative frequencies: 1

3. The 9th value is 4, and the 10th value is 5.

4. Find the average:

Median = (4 + 5) / 2 = 4.5

Question 39. The answer is C, “2.4”.

1. Calculate the weighted average using the weights of the given values:

Mean = ((1×10) + (2×40) + (3×50)) / 100 = (10 + 80 + 150) / 100 = 240 / 100 = 2.4

Question 40. The answer is D, “a nonnegative real number.”

1. The equation is:

√x = y, where
y ∈ ℝ.

2. The principal square root √x is only defined as a real number when x ≥ 0. If x < 0, then √x is imaginary, which contradicts the condition that y is real.

3. Therefore, x must be a nonnegative real number, as even real numbers, rational numbers, and integers can all be negative.

Question 41. The answer is D,  1/8.

1. For a fraction to grow, the numerator must increase and/or the denominator must decrease.

2. To make the largest possible value, maximize m and minimize
n and
p:

  • 1 ≤ m ≤ 4, choose m = 4
  • 4 ≤ n ≤ 6, choose n = 4
  • 8 ≤ p ≤ 10, choose p = 8

3. Plug in new values and simplify:

(4/4)(1/8) = 1 × 1/8 = 1/8

Question 42. The answer is A, “0, 0, 10, 10”.

1. Compute the standard deviation for every answer choice, starting with A: 0, 0, 10, 10:

Mean = (0+0+10+10)/4 = 5; Deviations = (-5)^2, (-5)^2, (5)^2, (5)^2 = 25, 25, 25, 25; Variance = (sum)/4 = 25; SD = √25 = 5

2. B: 0, 1, 9, 10:

Mean = (0+1+9+10)/4 = 5; Deviations = 25, 16, 16, 25; Variance = 82/4 = 20.5; SD ≈ √20.5 ≈ 4.53

3. C: 2, 3, 5, 7:

Mean = 17/4 = 4.25 (smaller spread); SD < 5

4. D: 5, 5, 5, 5:

All values equal ⇒ SD = 0

5. Option A has the greatest spread and therefore the largest standard deviation.

Question 43. The answer is C,  162π.

1. Volume of a cylinder is given by:

V = πr²h

2. Find the radius r:

Diameter = 18 ⇒ r = 18/2 = 9

3. Due to the displacement of water, the height is h = 2.

4. Plug in h and r and solve:

V = π(9)²(2) = π·81·2 = 162π

Question 44. The answer is D, “5”.

1. Work backwards using the given y value to find a.

2. h(x) = 1 when x = 2, where x = f(g(a)), thus:

2 = f(g(a))

3. f(x) = 2 when x = 1, where x = g(a), thus:

1 = g(a)

4. g(x) = 1 when x = 5.

5. Thus, h(f(g(a))) = 1 when a = 5.

Question 45. The answer is A, “$1.25”.

1. Let the probability of landing facedown be x, and the probability of landing faceup is 3x:

x + 3x = 1 ⇒ 4x = 1 ⇒ x = 1/4

2. Find the expected value by multiplying by the values of each flip:

E = 1.00 × (3/4) + 2.00 × (1/4) = 3/4 + 2/4 = 5/4 = 1.25

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