Below are answer explanations to the 3rd full-length Math test released by ACT for the Digital ACT.

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3rd Digital ACT Practice Test Math Answer Explanations

Question 1. The answer is B, 2nd Avenue.

  1. The two locations are at Main Avenue and 4th Avenue, which are four blocks apart horizontally.
  2. The midpoint is halfway between these avenues, or two blocks from Main Avenue.
  3. Moving two blocks from Main Avenue places the midpoint on 2nd Avenue.

Question 2. The answer is J, 57.

  1. Find the total of the four numbers: 4 × 45 = 180.
  2. Add the three known numbers: 26 + 39 + 58 = 123.
  3. Subtract to find the missing number: 180 − 123 = 57.

Question 3. The answer is B, −287.

  1. Substitute a = 7, b = −4, and c = 5 into abc − ac² − ab.
  2. Evaluate: (7)(−4)(5) − (7)(25) − (7)(−4).
  3. Simplify: −140 − 175 + 28 = −287.

Question 4. The answer is H, (7, −4).

  1. Start with point B(5, 1).
  2. Translate two units right by adding 2 to the x-coordinate: 5 + 2 = 7.
  3. Translate five units down by subtracting 5 from the y-coordinate: 1 − 5 = −4.
  4. The new coordinates are (7, −4).

Question 5. The answer is B, 14.

  1. Find the original number of pumpkins: 8 × 7 = 56.
  2. Subtract the four harvested rows: 56 − 4(7) = 28.
  3. Half of the remaining pumpkins were damaged: 28 ÷ 2 = 14.

Question 6. The answer is H, 20.

  1. The interior angles of a triangle must sum to 180°.
  2. Write the equation: 6x + 2x + x = 180.
  3. Combine like terms: 9x = 180.
  4. Solve: x = 20.

Question 7. The answer is C, (−3, 0) and (7, 0).

  1. The x-intercepts occur where the graph crosses the x-axis, meaning y = 0.
  2. The graph crosses the x-axis at x = −3 and x = 7.
  3. Therefore, the x-intercepts are (−3, 0) and (7, 0).

Question 8. The answer is J, 3 + 12i.

  1. Simplify the first radical: √9 = 3.
  2. Rewrite the negative radical using i = √(−1): √(−144) = √144 × √(−1) = 12i.
  3. Add the terms: 3 + 12i.

Question 9. The answer is D, 792.

  1. The data follow a pattern close to y = 12x. For example, 36 ÷ 2.99 ≈ 12 and 348 ÷ 29 = 12.
  2. Since k must be an integer, use k = 12.
  3. Substitute x = 66 into y = 12x.
  4. Calculate: y = 12(66) = 792.

Question 10. The answer is G, −1.

  1. For |4x − 3| = 7, the expression inside the absolute value can equal 7 or −7.
  2. Solve both equations: 4x − 3 = 7 or 4x − 3 = −7.
  3. The solutions are x = 5/2 and x = −1.
  4. Of these, only −1 appears among the answer choices.

Question 11. The answer is A, 1 out of 170,000 flips is a tail.

  1. A fair coin should produce approximately 50% heads and 50% tails over many flips.
  2. Choice A produces a tail percentage of (1 ÷ 170,000) × 100 ≈ 0.00059%.
  3. This is the greatest departure from 50% among the options and occurs over the largest number of flips.
  4. Therefore, Choice A provides the strongest evidence that the coin is not fair.

Question 12. The answer is J, 8x²y − 4xy².

  1. Start with the expression: 3xy² − 4x²y + 12x²y − 7xy².
  2. Combine the x²y terms: −4x²y + 12x²y = 8x²y.
  3. Combine the xy² terms: 3xy² − 7xy² = −4xy².
  4. The simplified expression is 8x²y − 4xy².

Question 13. The answer is B, 44%.

  1. From the table, 101 customers like white cake, out of 229 total customers.
  2. Calculate the percentage: (101 ÷ 229) × 100 ≈ 44.1%.
  3. Round to the nearest whole percent: 44%.

Question 14. The answer is J, −83.

  1. The sequence is arithmetic with a common difference of −4.
  2. Use the arithmetic sequence formula: aₙ = a₁ + (n − 1)d.
  3. Substitute the first term, common difference, and term number: a₂₁ = −3 + (21 − 1)(−4).
  4. Simplify: −3 − 80 = −83.

Question 15. The answer is B, 19.

  1. The probability of a seventh grader responding “Yes” is 49/139, which is already in lowest terms. Therefore, the total number of seventh graders must be a multiple of 139, with a corresponding multiple of 49 answering “Yes.”
  2. Using the smallest possible total, there are 139 seventh graders, 49 of whom answered “Yes.”
  3. The table also shows that 71 seventh graders answered “I don’t know.”
  4. Subtract to find those who answered “No”: 139 − 49 − 71 = 19. Larger multiples would give values of y greater than any listed choice.

Question 16. The answer is F, 47.

  1. Expand the left side: (x + k)² = x² + 2kx + k².
  2. Compare with the given expression: x² + 94x + k².
  3. Set the coefficients of x equal: 2k = 94.
  4. Solve: k = 47.

Question 17. The answer is C, −5 + x/2.

  1. To divide functions, use (f/g)(x) = f(x)/g(x).
  2. Substitute the given functions: (−10 + x)/2.
  3. Divide each term by 2: −10/2 + x/2.
  4. Simplify: −5 + x/2.

Question 18. The answer is G, 0.5.

  1. Event A represents a windy day, and Event B represents Deon flying his kite.
  2. Since Deon only flies on windy days, whenever B occurs, A must also occur. Therefore, P(A and B) = P(B).
  3. He often flies his kite, so this probability is greater than 0. However, he does not fly on every windy day, so the probability is less than 1.
  4. Of the answer choices, 0.5 is the only possible value.

Question 19. The answer is A, 800.

  1. In the second sample, 70 of the 140 frogs were tagged, so the tagged proportion is 70/140 = 1/2.
  2. Assume this same proportion applies to the entire swamp population.
  3. Set up the proportion: 400/N = 1/2, where N is the total number of frogs.
  4. Solve: N = 800.

Question 20. The answer is J, x < 44.

  1. Start with the inequality: 7(x + 4) > 8(x − 2).
  2. Distribute: 7x + 28 > 8x − 16.
  3. Subtract 7x and add 16: 44 > x.
  4. Rewrite: x < 44.

Question 21. The answer is A, 11.

  1. Since PR bisects ∠QPS, it divides the 64° angle into two equal angles.
  2. Therefore, m∠RPS = 64° ÷ 2 = 32°.
  3. Set up the equation: 2x + 10 = 32.
  4. Solve: 2x = 22, so x = 11.

Question 22. The answer is F, 1/(2√2).

  1. Use the identity tan θ = sin θ / cos θ.
  2. Substitute the given values: tan θ = (1/3) ÷ (2√2/3).
  3. Multiply by the reciprocal: (1/3) × (3/(2√2)).
  4. Simplify: tan θ = 1/(2√2).

Question 23. The answer is B, (−1, −9).

  1. Rewrite the hyperbola in standard form by dividing by 100: (x + 1)²/25 − (y + 9)²/20 = 1.
  2. The center of a hyperbola in this form is (h, k), which is also the intersection of its asymptotes.
  3. Since x + 1 = x − (−1) and y + 9 = y − (−9), the center is (−1, −9).

Question 24. The answer is H, 10 + 2√21.

  1. Apply (a + b)² = a² + 2ab + b².
  2. Expand: (√3 + √7)² = 3 + 2√3√7 + 7.
  3. Simplify: 3 + 2√21 + 7 = 10 + 2√21.

Question 25. The answer is A, (−35/2, −6).

  1. From the graph, line j passes through (−10, 0) and (0, 8).
  2. Calculate its slope: m = (8 − 0)/(0 − (−10)) = 8/10 = 4/5.
  3. Write the equation: y = (4/5)x + 8.
  4. Line k is horizontal at y = −6, so set −6 = (4/5)x + 8.
  5. Solve: −14 = (4/5)x, giving x = −35/2. The intersection is (−35/2, −6).

Question 26. The answer is H, 5/12.

  1. Company A insures 28 patients with an FSA and 20 without one, for a total of 48 patients.
  2. Of these 48 patients, 20 do not have an FSA.
  3. The conditional probability is 20/48.
  4. Simplify: 20/48 = 5/12.

Question 27. The answer is B, 180°.

  1. Reflecting a point (x, y) across the y-axis changes it to (−x, y).
  2. A 180° rotation around the origin changes (−x, y) to (x, −y).
  3. This is exactly the result of reflecting the original point across the x-axis.
  4. Therefore, a clockwise rotation of 180° maps rectangle R″ onto rectangle R′.

Question 28. The answer is G, (−3, 8).

  1. A point of intersection must satisfy both f(x) = −x + 5 and g(x) = 4x² + 8x − 4.
  2. Substitute x = −3 into the first function: f(−3) = 3 + 5 = 8.
  3. Check the second function: g(−3) = 4(9) + 8(−3) − 4 = 36 − 24 − 4 = 8.
  4. Both functions equal 8 when x = −3, so (−3, 8) is a point of intersection.

Question 29. The answer is B, 6.

  1. Convert 5 gallons of water into cups: 5 × 4 × 4 = 80 cups.
  2. The ratio of concentrated solution to water is 3:40.
  3. Set up the proportion: x/80 = 3/40.
  4. Solve: x = 80(3/40) = 6 cups.

Question 30. The answer is H, 78.5.

  1. The two radii form a right angle, creating an isosceles right triangle with hypotenuse 5√2.
  2. In a 45°–45°–90° triangle, the hypotenuse is the leg length multiplied by √2. Therefore, the radius is 5.
  3. Use the circle area formula: A = πr² = π(5²) = 25π.
  4. Calculate: A ≈ 78.5 square inches.

Question 31. The answer is B, n = a/(bd) − m.

  1. Increasing m by n makes the equation true, so substitute m + n for m: a = b(m + n)d.
  2. Divide both sides by bd: a/(bd) = m + n.
  3. Subtract m from both sides: n = a/(bd) − m.

Question 32. The answer is F, △PRT.

  1. The diagonal of the square base is PR = 5√2.
  2. Because the pyramid is right, its 12 cm height lies in the plane of △PRT and is perpendicular to PR.
  3. Calculate the triangle’s area: A = (1/2)(5√2)(12) = 30√2 ≈ 42.4 cm².
  4. The other cross sections have smaller areas: △QRT ≈ 30.6 cm², △MNT = 30 cm², and square PQRS = 25 cm².
  5. Therefore, △PRT has the greatest area.

Question 33. The answer is D, 267.

  1. The large circle has radius 12 cm, so its area is π(12²) = 144π.
  2. The smaller unshaded circle has diameter 12 cm and radius 6 cm, so its area is π(6²) = 36π.
  3. The unshaded right triangle has legs of 12 cm each, giving an area of (1/2)(12)(12) = 72.
  4. Subtract the unshaded areas: 144π − 36π − 72 = 108π − 72 ≈ 267.3.
  5. The closest answer is 267 square centimeters.

Question 34. The answer is F, shifted in the negative x-direction 6 coordinate units.

  1. The original function is p(x) = −4x².
  2. The new function is r(x) = p(x + 6) = −4(x + 6)².
  3. Replacing x with x + 6 shifts the graph horizontally to the left by 6 units.
  4. Therefore, the graph shifts 6 units in the negative x-direction.

Question 35. The answer is C, 4.

  1. Match the upper-left entries of the matrices: 5a = 10, so a = 2.
  2. Match the upper-right entries: ab = 2, so b = 1.
  3. Match the lower-left entries: a = c, so c = 2.
  4. Match the lower-right entries: d = abc = (2)(1)(2) = 4.

Question 36. The answer is J, a − b = 1.

  1. The given inequality is |a − b − 1| > 0.
  2. An absolute value is greater than zero whenever the expression inside it is not equal to zero.
  3. Therefore, a − b − 1 ≠ 0, which means a − b ≠ 1.
  4. The statement a − b = 1 cannot be true.

Question 37. The answer is C, 31.9.

  1. Convert the marathon distance to kilometers: 26.2 × 1.6 = 41.92 km.
  2. Convert the 10,000-meter race to kilometers: 10,000 ÷ 1,000 = 10 km.
  3. Find the difference: 41.92 − 10 = 31.92 km.
  4. Round to the closest choice: 31.9 km.

Question 38. The answer is G, 15/17.

  1. Let θ = arcsin(8/17). Therefore, sin θ = 8/17.
  2. Construct a right triangle with opposite side 8 and hypotenuse 17.
  3. Use the Pythagorean theorem to find the adjacent side: √(17² − 8²) = √225 = 15.
  4. Therefore, cos θ = adjacent/hypotenuse = 15/17.

Question 39. The answer is C, 1,030.2.

  1. The height function is s(t) = −4.9t² + 44.1t + 931.
  2. Since the coefficient of t² is negative, the parabola opens downward, so its vertex gives the maximum height.
  3. Find the time at the vertex using t = −b/(2a): t = −44.1/[2(−4.9)] = 4.5 seconds.
  4. Substitute: s(4.5) = −4.9(4.5²) + 44.1(4.5) + 931 = 1,030.225.
  5. Round to the nearest tenth: 1,030.2 meters.

Question 40. The answer is J, 5.

  1. Work backward from h(f(g(a))) = 1.
  2. From the table, h(2) = 1, so f(g(a)) = 2.
  3. From the table, f(1) = 2, so g(a) = 1.
  4. Finally, g(5) = 1, so a = 5.

Question 41. The answer is A, 10ʸ/2.

  1. Start with y = log₁₀(2x).
  2. Rewrite in exponential form: 10ʸ = 2x.
  3. Divide both sides by 2: x = 10ʸ/2.

Question 42. The answer is J, x⁴ − 20x³ + 150x² − 500x + 625.

  1. Rewrite the expression as (x − 5)⁴ = [(x − 5)²]².
  2. Expand the binomial once: (x − 5)² = x² − 10x + 25.
  3. Square the resulting trinomial: (x² − 10x + 25)².
  4. Combine like terms to obtain x⁴ − 20x³ + 150x² − 500x + 625.

Question 43. The answer is A, 4√3.

  1. The side lengths of Triangle B are 1/2 the corresponding side lengths of Triangle A.
  2. Areas scale by the square of the side-length scale factor: (1/2)² = 1/4.
  3. Multiply the original area by 1/4: 16√3 × 1/4 = 4√3 square feet.

Question 44. The answer is H, 9.7.

  1. The expected value of a discrete random variable is the sum of each possible value multiplied by its probability.
  2. Set up the calculation: E(X) = 8(0.1) + 9(0.3) + 10(0.4) + 11(0.2).
  3. Multiply: E(X) = 0.8 + 2.7 + 4.0 + 2.2.
  4. Add: E(X) = 9.7.

Question 45. The answer is D, 1/3 meter per second per second.

  1. Average rate of change equals change in speed divided by change in time.
  2. The change in speed is 6 − 4 = 2 meters per second.
  3. Divide by the elapsed time: 2 ÷ 6 = 1/3.
  4. The average rate of change is 1/3 meter per second per second.

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