Below are answer explanations to the 4th full-length Math test released by ACT for the Digital ACT.

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4th Digital ACT Practice Test Math Answer Explanations

Question 1. The answer is B, 20.0.

  1. Each circle has a radius of 5 cm, so its diameter is 2(5) = 10 cm.
  2. The circles are tangent to each other and positioned side by side within the rectangle.
  3. The rectangle’s length is the sum of the two diameters: 10 + 10 = 20 cm.

Question 2. The answer is G, 23.8.

  1. The mean is the sum of all values divided by the number of values.
  2. Add the 10 amounts: 15 + 27 + 15 + 30 + 24 + 30 + 30 + 16 + 22 + 29 = 238.
  3. Divide by 10: 238 ÷ 10 = 23.8 gallons.

Question 3. The answer is C, 12.

  1. The party requires 5 liters of juice for every 12 guests.
  2. For 36 guests, calculate the total needed: (36 ÷ 12) × 5 = 15 liters.
  3. Subtract the 3 liters already available: 15 − 3 = 12 additional liters.

Question 4. The answer is J, $196.

  1. The profit function is P(t) = t³ + 3t² − 4.
  2. Substitute t = 5: P(5) = 5³ + 3(5²) − 4.
  3. Simplify: 125 + 75 − 4 = 196.
  4. The business’s yearly profit after five years is $196.

Question 5. The answer is A, (x − 4)(x + 7) = 0.

  1. An equation with solutions 4 and −7 must equal zero when x = 4 or x = −7.
  2. The corresponding factors are (x − 4) and (x + 7).
  3. Using the zero-product property, the equation is (x − 4)(x + 7) = 0.

Question 6. The answer is J, 30.

  1. The scale is 1 inch = 6⅔ feet.
  2. The picture measures 4½ inches.
  3. Multiply: 4½ × 6⅔ = (9/2)(20/3) = 30.
  4. The actual bus is approximately 30 feet long.

Question 7. The answer is C, the matrix with rows (5, 0) and (6, −1).

  1. To add matrices, add the entries in corresponding positions.
  2. Add the first row: 6 + (−1) = 5 and −2 + 2 = 0.
  3. Add the second row: 10 + (−4) = 6 and −5 + 4 = −1.
  4. The resulting matrix has first row (5, 0) and second row (6, −1).

Question 8. The answer is G, 4.

  1. A customer receives both gifts when their number is a multiple of both 24 and 60.
  2. Find the least common multiple: LCM(24, 60) = 120.
  3. Count the multiples of 120 up to 500: 120, 240, 360, and 480.
  4. Therefore, 4 customers receive both gifts.

Question 9. The answer is D, segment AB is parallel to segment HG.

  1. In a rectangular prism, corresponding edges running in the same direction are parallel.
  2. Segments AB and HG are opposite edges that run in the same direction.
  3. Therefore, AB is parallel to HG.

Question 10. The answer is J, a shift of 7 coordinate units up.

  1. The original function is g(x) = −2x².
  2. The transformed function is y = g(x) + 7 = −2x² + 7.
  3. Adding 7 outside the function increases every y-value by 7.
  4. Therefore, the graph shifts upward by 7 units.

Question 11. The answer is C, 876.

  1. The data follow a pattern close to y = 12x. For example, 36 ÷ 2.98 ≈ 12.08, and 324 ÷ 27 = 12.
  2. Use the prediction equation y = 12x.
  3. Substitute x = 73: y = 12(73).
  4. Calculate: y = 876.

Question 12. The answer is H, 8.

  1. The triangles are similar, with XZ corresponding to AC. Their lengths are 14 and 7, giving a scale factor of 14/7 = 2.
  2. Corresponding sides XY and AB must have the same ratio: x/(x − 4) = 2.
  3. Multiply: x = 2(x − 4).
  4. Solve: x = 2x − 8, so x = 8.

Question 13. The answer is B, $696.

  1. Bill originally saved $30 out of $200 each week, or 30/200 = 15% of his pay.
  2. His new weekly contribution is 15% of $240: 0.15(240) = $36.
  3. During the first 10 weeks, he saved 10(30) = $300.
  4. During Weeks 11–21, he saved 11(36) = $396.
  5. Add the contributions: $300 + $396 = $696.

Question 14. The answer is F, 35.

  1. Expand the left side: (x + k)² = x² + 2kx + k².
  2. Compare this with the given expression: x² + 70x + k².
  3. Set the coefficients of x equal: 2k = 70.
  4. Solve: k = 35.

Question 15. The answer is A, 7.

  1. The function is f(x) = 5/[2(x − 7)].
  2. A rational function is undefined when its denominator equals zero.
  3. Set the denominator equal to zero: 2(x − 7) = 0.
  4. Solve: x − 7 = 0, so x = 7.

Question 16. The answer is J, 4.

  1. Points five units from the origin lie on the circle x² + y² = 25.
  2. Points two units from the line y = 0 have y = 2 or y = −2.
  3. Substitute y² = 4 into the circle equation: x² + 4 = 25.
  4. Solve: x = ±√21.
  5. Each of the two possible x-values pairs with each of the two possible y-values, giving 4 points.

Question 17. The answer is B, 4/5.

  1. Substitute x = 2 into the expression: 2/[2 + 2/(x + 2)].
  2. Simplify the inner denominator: x + 2 = 4.
  3. Evaluate the complex fraction: 2/(2 + 2/4) = 2/(5/2).
  4. Multiply by the reciprocal: 2 × 2/5 = 4/5.

Question 18. The answer is J, (3x + 4)(3x − 1).

  1. Start with the trinomial: 9x² + 9x − 4.
  2. Split the middle term into 12x − 3x: 9x² + 12x − 3x − 4.
  3. Factor by grouping: 3x(3x + 4) − 1(3x + 4).
  4. Factor out the common binomial: (3x + 4)(3x − 1).

Question 19. The answer is D, 150.

  1. Find the total sample size: 15 + 23 + 25 + 9 + 3 = 75.
  2. Of these, 25 participants wear a large shirt, giving a proportion of 25/75 = 1/3.
  3. Apply this proportion to all 450 participants: 450 × 1/3 = 150.
  4. The estimated number of participants wearing large shirts is 150.

Question 20. The answer is G, 3x² + 1.

  1. Start with the expression (12x² + 4)/4.
  2. Divide each term in the numerator by 4: 12x²/4 + 4/4.
  3. Simplify: 3x² + 1.

Question 21. The answer is B, ∠RSV ≅ ∠UTV.

  1. The given information establishes ∠RVS ≅ ∠UVT and SV ≅ VT.
  2. To apply Angle-Side-Angle (ASA), the known side must lie between the two known angles.
  3. In triangle RSV, side SV lies between the angles at S and V.
  4. In triangle UTV, side VT lies between the angles at T and V.
  5. Therefore, the additional congruence needed is ∠RSV ≅ ∠UTV.

Question 22. The answer is H, $213.00.

  1. A 75% discount means the customer pays 25% of the original price.
  2. An additional 20% discount means the customer pays 80% of that reduced price.
  3. Multiply the remaining percentages: 0.25 × 0.80 = 0.20.
  4. The final price is 20% of the original: 0.20P = 42.60.
  5. Solve: P = 42.60/0.20 = $213.00.

Question 23. The answer is D, 48i.

  1. In a geometric sequence, consecutive terms have a constant ratio.
  2. Divide the second term by the first: −6/(3i) = 2i.
  3. Multiply the fourth term by this ratio: 24(2i) = 48i.
  4. Therefore, the fifth term is 48i.

Question 24. The answer is G, amplitude 6 and period 2π.

  1. The function is y = 6 cos x.
  2. The amplitude is the absolute value of the coefficient of cosine: |6| = 6.
  3. The period of y = a cos(bx) is 2π/|b|.
  4. Since b = 1, the period is 2π.
  5. Therefore, the amplitude is 6 and the period is 2π.

Question 25. The answer is B, 1 out of 60,000 flips results in a tail.

  1. A fair coin should produce tails approximately 50% of the time.
  2. Calculate the proportion of tails for Choice B: 1/60,000 ≈ 0.0000167.
  3. This is the smallest proportion of tails among the choices and is dramatically different from the expected 0.5.
  4. Therefore, Choice B provides the strongest evidence against the coin being fair among the listed results.

Question 26. The answer is J, 15.

  1. The show lasts 30 minutes.
  2. Commercials occupy 25% of the broadcast: 0.25(30) = 7.5 minutes.
  3. Convert to seconds: 7.5 × 60 = 450 seconds.
  4. Each commercial lasts 30 seconds, so 450 ÷ 30 = 15 commercials.

Question 27. The answer is B, 72.

  1. The smallest engine size in the table is 2.0 liters, corresponding to an actual weight of 1,086 kg.
  2. Substitute x = 2 into the prediction model: ŵ = 391x + 376.
  3. Calculate: ŵ = 391(2) + 376 = 1,158 kg.
  4. Find the difference: 1,158 − 1,086 = 72 kg.

Question 28. The answer is G, 13/19.

  1. Start with the equation (3x − y)/(x + y) = 5/8.
  2. Cross-multiply: 8(3x − y) = 5(x + y).
  3. Distribute: 24x − 8y = 5x + 5y.
  4. Combine like terms: 19x = 13y.
  5. Divide by 19y: x/y = 13/19.

Question 29. The answer is B, the sum of any rational number and any irrational number is irrational.

  1. A rational number can be expressed as a ratio of two integers, while an irrational number cannot.
  2. Suppose that adding a rational number to an irrational number produced a rational number.
  3. Subtracting the original rational number from that sum would then produce another rational number.
  4. However, this difference equals the original irrational number, creating a contradiction.
  5. Therefore, the sum must be irrational.

Question 30. The answer is F, 0.

  1. The equation is 1/√(x + 1) = 0.
  2. A fraction can equal zero only if its numerator is zero and its denominator is nonzero.
  3. Here, the numerator is always 1, so it can never equal zero.
  4. Therefore, the equation has no solutions.

Question 31. The answer is C, 12.

  1. Since one side is horizontal and another is vertical, the triangle is a right triangle.
  2. Find the horizontal distance between the given vertices: 3 − (−1) = 4 units.
  3. Find the vertical distance: 4 − (−2) = 6 units.
  4. Use the triangle area formula: A = (1/2)(4)(6) = 12 square units.

Question 32. The answer is G, −3/4 + i√55/4.

  1. Start with the equation 2x² + 3x + 8 = 0.
  2. Use the quadratic formula with a = 2, b = 3, and c = 8.
  3. Calculate the discriminant: b² − 4ac = 9 − 64 = −55.
  4. Substitute: x = [−3 ± √(−55)]/4.
  5. Rewrite √(−55) = i√55, giving x = −3/4 ± i√55/4.
  6. One of these solutions is −3/4 + i√55/4.

Question 33. The answer is D, √3 is between E and F.

  1. The number line from 0 to 3 is divided into 10 equal intervals.
  2. Each interval measures 3/10 = 0.3.
  3. Point E is at 1.5, and point F is at 1.8.
  4. Since √3 ≈ 1.732, it lies between E and F.

Question 34. The answer is H, 5/7.

  1. There are eight teams in the tournament, including your team.
  2. Your team can face any of the other seven teams in the first round.
  3. Two of those seven teams are Central teams, leaving five possible opponents that are not Central teams.
  4. The probability of avoiding a Central team is 5/7.

Question 35. The answer is C, y = 1.05^t.

  1. An exponential growth function has the form y = a(b^t), where b is a constant greater than 1.
  2. Choice C has a constant base of 1.05 and a variable exponent t.
  3. Since 1.05 > 1, the function increases by a constant percentage as t increases.
  4. Therefore, Choice C represents exponential growth.

Question 36. The answer is J, 314,432.

  1. The cube net consists of six squares. Together, these squares have 6(4) = 24 side lengths before accounting for shared edges.
  2. Five pairs of edges are shared within the net, so subtract 2(5) = 10 edges from the perimeter count.
  3. The perimeter is therefore 14 side lengths: 14s = 952.
  4. Solve for the cube’s side length: s = 952/14 = 68 cm.
  5. Calculate the volume: V = s³ = 68³ = 314,432 cubic centimeters.

Question 37. The answer is B, AC.

  1. Since △DEF ~ △GHI, corresponding sides have equal ratios: DE/GH = DF/GI.
  2. Multiply both sides by GI to obtain (DE)(GI)/GH = DF.
  3. Since △ABC ≅ △DEF, corresponding sides AC and DF are equal.
  4. Therefore, (DE)(GI)/GH = AC.

Question 38. The answer is H, (a, −b + 2).

  1. Start with the point P(a, b).
  2. Reflect across the x-axis by changing the sign of the y-coordinate: (a, −b).
  3. Translate two units upward by adding 2 to the y-coordinate.
  4. The resulting point is (a, −b + 2).

Question 39. The answer is A, x²/9 + y²/36 = 1.

  1. The first hyperbola, x²/9 − y²/4 = 1, has vertices at (−3, 0) and (3, 0).
  2. The second hyperbola, y²/36 − x²/25 = 1, has vertices at (0, −6) and (0, 6).
  3. An ellipse centered at the origin with horizontal radius 3 and vertical radius 6 passes through all four vertices.
  4. Use the ellipse equation x²/a² + y²/b² = 1.
  5. Substitute the squared radii: x²/9 + y²/36 = 1.

Question 40. The answer is F, 6² = 8² + 10² − 2(8)(10) cos x.

  1. In a triangle, the smallest angle lies opposite the shortest side.
  2. The shortest side is 6 cm, so the angle x must be opposite that side.
  3. Use the Law of Cosines: c² = a² + b² − 2ab cos x, where c is opposite angle x.
  4. Substitute c = 6, a = 8, and b = 10.
  5. The resulting equation is 6² = 8² + 10² − 2(8)(10) cos x.

Question 41. The answer is D, 22h + 0.08s − 33.

  1. Collier worked h hours, while Mandy worked h − 3 hours.
  2. Their combined hourly pay is 11h + 11(h − 3).
  3. Simplify: 11h + 11h − 33 = 22h − 33.
  4. Their combined commission is 8% of their combined sales, or 0.08s.
  5. Add hourly pay and commission: 22h + 0.08s − 33.

Question 42. The answer is H, the fourth-degree graph with two real x-intercepts.

  1. The polynomial has zeros of −2, 3, and i.
  2. Because the polynomial has real coefficients, the complex conjugate −i must also be a zero.
  3. Therefore, the polynomial has at least four zeros counting complex zeros, including exactly two real zeros at −2 and 3.
  4. The graph must cross the x-axis at one negative value and one positive value, with no other real intercepts.
  5. Choice H is consistent with a fourth-degree polynomial having those two real zeros and a conjugate pair of nonreal zeros.

Question 43. The answer is C, −11.

  1. The series is geometric, with first term a = −16.
  2. Find the common ratio: r = 8/(−16) = −1/2.
  3. Since |r| < 1, the infinite series converges, and its sum is S = a/(1 − r).
  4. Substitute: S = −16/[1 − (−1/2)] = −16/(3/2).
  5. Simplify: S = −32/3 ≈ −10.67, which is closest to −11.

Question 44. The answer is J, 124,416.

  1. Find the area in square feet: A = 24 × 36 = 864 ft².
  2. Since 1 foot = 12 inches, 1 square foot = 12² = 144 square inches.
  3. Convert the area: 864 × 144 = 124,416 square inches.

Question 45. The answer is D, 10⁴ − 9⁴.

  1. There are 10,000 integers from 0 through 9,999, which can be represented using four digits, including leading zeros.
  2. Each digit position has 10 possible values, giving 10⁴ total four-digit representations.
  3. To count numbers without any digit equal to 5, each position has only 9 choices, giving 9⁴ possibilities.
  4. Subtract these from the total to count numbers containing at least one 5: 10⁴ − 9⁴.
  5. This equals 10,000 − 6,561 = 3,439 integers.

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