ACT Math study guide · Chapter 50
Mixed Challenge Practice
This bank extends the mixed practice at much greater volume. Each item either combines familiar mathematics from more than one Part of this book or presents a familiar skill in a less familiar representation. Content areas are labeled only in the solutions, never in the questions themselves, so the method has to be recognized from the problem instead of from a chapter heading. None of these numbers or contexts repeats anything used earlier in this book.
▸ Questions
1. A rectangle's length is 2 more than twice its width. If its area is 60, find its perimeter.
A) 30 B) 32 C) 38 D) 34
2. A shipping cost is modeled by f(t) = 2t + 10. Find the average of f(3), f(7), and f(11).
F) 20 G) 22 H) 18 J) 24
3. The equation x² + 4x + 13 = 0 has a complex solution with a negative imaginary part. Find the sum of its real and imaginary parts.
A) −5 B) 5 C) 3 D) −3
4. A right triangle has a hypotenuse of 26 and one leg of 10. Find the other leg.
F) 20 G) 16 H) 24 J) 28
5. In a survey of 250 people, 40% own a car. Of those car owners, 30% also own a bike. How many people own both?
A) 100 B) 30 C) 40 D) 70
6. A company's profit is modeled by P(x) = −3x² + 60x − 100. Find the maximum profit.
F) 150 G) 180 H) 220 J) 200
7. A line y = x − 2 intersects the circle x² + y² = 20 at two points. Find the sum of the x-coordinates.
A) −2 B) 4 C) −4 D) 2
8. Thirty students average 72 on a test; twenty other students average 85. Find the combined average of all fifty students.
F) 77.2 G) 78.5 H) 75 J) 80
9. Solve the system x + 2y = 8 and 3x − y = 3. Find x + y.
A) −1 B) 8 C) 5 D) 11
10. A right triangle has legs 9 and 12. Find sin(30°) multiplied by the hypotenuse.
F) 15 G) 7.5 H) 5 J) 10
11. A room measures 15 feet by 12 feet. Flooring costs $8.50 per square yard. Find the total flooring cost.
A) $102 B) $136 C) $170 D) $204
12. Solve x² − 5x + 6 = 0.
F) x = −3, −2 G) x = 3, −2 H) x = 3, 2 J) x = −3, 2
13. Two similar triangles have sides in ratio 4:10. The smaller triangle has a perimeter of 48. Find the larger triangle's perimeter.
A) 120 B) 96 C) 108 D) 132
14. A(t) = 500(1.08)ᵗ. Find the percent increase from t = 2 to t = 5, rounded to the nearest hundredth of a percent.
F) 15.00% G) 20.00% H) 25.97% J) 21.60%
15. A vector has components (6, 8). Find the area of a square whose side length equals the vector's magnitude.
A) 60 B) 80 C) 144 D) 100
16. A line of best fit is y = 1.5x + 20. Find x when y = 50.
F) 15 G) 25 H) 20 J) 30
17. A car's distance is modeled by f(t) = 15t, in miles per hour of driving time t. Find the time needed to travel 97.5 miles.
A) 5.5 B) 6.5 C) 7.5 D) 6.0
18. A circle of radius 10 contains a smaller concentric circle of radius 6. Find the probability that a randomly chosen point in the larger circle also lies in the smaller circle.
F) 0.16 G) 0.6 H) 0.06 J) 0.36
19. If (x + yi)(2 + 3i) = 16 + 11i for real numbers x and y, find x + y.
A) 3 B) 7 C) −1 D) 9
20. Let f(x) = 2x − 5. Find the distance from the point (f⁻¹(3), 0) to the origin.
F) 3 G) 4 H) 5 J) −4
21. A price is increased 20%, then decreased by a flat $15, giving a final price of $108. Find the original price.
A) $110.00 B) $90.00 C) $97.50 D) $102.50
22. A rectangular field measures 40 feet by 60 feet. Find the probability that a randomly chosen point in the field lies within 20 feet of one corner, rounded to the nearest thousandth.
F) 0.250 G) 0.131 H) 0.500 J) 0.065
23. Solve 3^(x+1) = 81.
A) 3 B) 2 C) 4 D) 5
24. A cylindrical tank has radius 3 feet and height 8 feet. Find its volume in gallons, rounded to the nearest hundredth. (1 cubic foot ≈ 7.48 gallons)
F) 1691.94 G) 1500.00 H) 1808.64 J) 1200.00
25. A garden's area is modeled by A(w) = w(20 − w), where w is its width. Find the maximum possible area.
A) 80 B) 100 C) 90 D) 110
26. An arithmetic sequence has a₁ = 5 and d = 4. Find the mean of its first five terms.
F) 13 G) 9 H) 11 J) 15
27. A scale drawing uses 1 inch to represent 15 feet. A wall is shown as 8.5 inches long. Find its actual length in yards.
A) 127.5 B) 42.5 C) 8.5 D) 15
28. Find the average of the two solutions to x² − 9x + 20 = 0.
F) 9 G) 4.5 H) 5 J) 4
29. The determinant of [[5, 2], [3, 4]] gives a scale factor applied to a triangle with area 6. Find the transformed area.
A) 24 B) 84 C) 36 D) 120
30. A shipping cost is c(w) = 5 if w ≤ 2, and c(w) = 5 + 1.5(w − 2) if w > 2, in dollars for weight w in pounds. Find the cost for a 7-pound package as a percent of a $20 budget.
F) 25% G) 40% H) 50% J) 62.5%
31. A cone has a volume of 100π cubic units and a height of 12. Find its radius.
A) 4 B) 6 C) 5 D) 10
32. Average test scores are modeled by f(x) = 90 − 2x, where x is weeks since the exam. Find the first whole-number week in which the average drops below 70, and the score at that week.
F) 70 G) 66 H) 68 J) 72
33. Two survey groups total 180 people. Group A is 20 fewer than 3 times Group B. Find the size of Group A.
A) 130 B) 50 C) 100 D) 140
34. Point P(2, 3) is reflected using the matrix [[1, 0], [0, −1]] to form P′. Find the distance from P to P′.
F) 0 G) 3 H) 6 J) 12
35. A sequence is defined by aₙ = 3n² − 2. Find a₄.
A) 46 B) 40 C) 34 D) 52
36. A raffle ticket costs $5. It pays $200 with probability 0.02. Find the expected net gain or loss per ticket.
F) $4 G) $1 H) −$4 J) −$1
37. In a right triangle, one angle measures 60° and the side adjacent to it is 5. Find the length of the side opposite that angle.
A) 5 B) 10 C) 5√3 D) 5√2
38. A line of best fit is y = 4x − 20. Find its x-intercept.
F) 5 G) 4 H) −5 J) 20
39. A population is modeled by P(t) = 200(2)^(t/5), with t in years. Find P(15).
A) 800 B) 1200 C) 2000 D) 1600
40. Find the sum of the two solutions to |x − 7| = 12.
F) 0 G) 14 H) −12 J) 19
41. A radioactive sample decays according to A(t) = 80(0.5)^(t/3), with t in years. Find A(9).
A) 40 B) 10 C) 20 D) 5
42. How many diagonals can be drawn in a regular hexagon?
F) 9 G) 6 H) 12 J) 15
43. Let u = (3, 1). If 2u + v = (10, 6), find v.
A) (−4, −4) B) (4, 4) C) (16, 4) D) (4, −4)
44. A triangle with area 12 is dilated by a scale factor of 3. Find the area of the dilated triangle.
F) 36 G) 108 H) 144 J) 84
45. In a sample of 40 items, 25 meet a quality standard. Find the sample proportion that meets the standard.
A) 0.4 B) 0.5 C) 0.625 D) 0.75
46. Solve √(3x − 2) = 4. Then find the value of 2x − 5.
F) 7 G) 11 H) 6 J) 16
47. A trapezoid has bases of (x + 2) and (x + 8) and a height of 4. If its area is 48, find x.
A) 7 B) 5.5 C) 6 D) 8
48. Two cell phone plans cost C₁(m) = 20 + 0.10m and C₂(m) = 35 + 0.05m for m minutes. Find the number of minutes at which the two plans cost the same.
F) 150 G) 300 H) 200 J) 350
49. Let f(x) = 2x + 7. If f(a) = 25, find a. Then let g(x) = a·x − 3 and find g(2).
A) 9 B) 18 C) 6 D) 15
50. Find the magnitude of the complex number z = 8 − 15i.
F) 7 G) 23 H) 15 J) 17
51. A garden is a 12-foot by 8-foot rectangle with a semicircle of diameter 8 feet attached to one side. Find the total area in square yards, rounded to the nearest hundredth.
A) 12.11 B) 121.13 C) 13.46 D) 15.00
52. Pipe A fills a tank in 6 hours; Pipe B fills the same tank in 3 hours. Working together, how long would it take to fill the tank?
F) 2 hours G) 1.5 hours H) 4.5 hours J) 3 hours
▸ Answers and Explanations
1: D. Algebra + Geometry. Let width=w, length=2w+2. w(2w+2)=60, so w²+w−30=0. Factoring: (w+6)(w−5)=0, so w=5. Length=2(5)+2=12. Perimeter=2(5+12)=34. Choice A adds only the width and length once instead of doubling their sum for the perimeter.
2: J. Functions + Statistics. f(3)=16, f(7)=24, f(11)=32. Average=(16+24+32)/3=24. Choice F reports f(3) alone instead of completing the average of all three values.
3: A. Number & Quantity + Functions. Discriminant=16−52=−36. Root=(−4−6i)/2=−2−3i. Sum of real and imaginary parts: −2+(−3)=−5. Choice G drops the sign on the imaginary part, computing −2+3 instead of −2+(−3).
4: H. Geometry. √(26²−10²)=√576=24. Choice F comes from an arithmetic slip while applying the Pythagorean theorem.
5: B. Statistics & Probability + Integrating Essential Skills. 250×0.40×0.30=30. Choice A reports the number of car owners alone (250×0.40=100) instead of completing the second percentage.
6: J. Functions + Integrating Essential Skills. Vertex at x=−60/(2·−3)=10. P(10)=−3(100)+600−100=200. Choice H comes from an arithmetic slip while evaluating P(10).
7: D. Geometry + Algebra. Substitute y=x−2: x²+(x−2)²=20, giving 2x²−4x−16=0, or x²−2x−8=0. Factoring: (x−4)(x+2)=0, so x=4 or x=−2. Sum=2. Choice A reverses the sign of one root while combining, computing −4+2 instead of 4+(−2).
8: F. Integrating Essential Skills. (30×72+20×85)/50=(2,160+1,700)/50=77.2. Choice G computes the simple average of 72 and 85 instead of the weighted average by group size.
9: C. Algebra. From the second equation, y=3x−3. Substituting: x+2(3x−3)=8, so 7x=14, x=2, y=3. x+y=5. Choice A comes from a sign error while solving the system.
10: G. Geometry + Number & Quantity. Hypotenuse=√(9²+12²)=15. sin(30°)×15=0.5×15=7.5. Choice F reports the hypotenuse alone instead of multiplying by sin(30°).
11: C. Geometry + Integrating Essential Skills. Area=15×12=180 square feet=20 square yards. 20×$8.50=$170. Choice A divides by 9 twice, an incorrect square-unit conversion.
12: H. Algebra. Factor: (x−2)(x−3)=0, giving x=2 or x=3. Choice G reverses the sign of both roots.
13: A. Congruence & Similarity. Scale factor=10/4=2.5. 48×2.5=120. Choice G inverts the scale factor, multiplying by 4/10 instead of 10/4.
14: H. Functions + Integrating Essential Skills. A(5)/A(2)−1=(500·1.08⁵)/(500·1.08²)−1≈0.2597, or 25.97%. Choice F comes from an arithmetic slip while computing the ratio of the two values.
15: D. Vectors & Matrices + Geometry. Magnitude=√(6²+8²)=10. Area=10²=100. Choice A treats the components as rectangle sides instead of finding the vector's magnitude first.
16: H. Bivariate Data. 50=1.5x+20, so 1.5x=30, x=20. Choice F misreads the slope as 2 instead of 1.5, computing (50−20)/2=15.
17: B. Functions + Integrating Essential Skills. 97.5/15=6.5. Choice A comes from an arithmetic slip while dividing 97.5 by 15.
18: J. Circles & Conics + Probability. Probability=(6/10)²=0.36, since area scales with the square of the radius ratio. Choice G applies the radius ratio directly (6/10=0.6) instead of squaring it for the area ratio.
19: A. Number & Quantity + Algebra. Expand: 2x+3xi+2yi+3yi²=(2x−3y)+(3x+2y)i=16+11i. Solving 2x−3y=16 and 3x+2y=11 gives x=5, y=−2. x+y=3. Choice G comes from a sign error while solving the system for x and y.
20: G. Inverse Functions + Coordinate Geometry. f⁻¹(x)=(x+5)/2, so f⁻¹(3)=4. The point (4,0) is 4 units from the origin. Choice F reports f(3) instead of f⁻¹(3), evaluating the original function instead of its inverse.
21: D. Integrating Essential Skills. (original×1.20)−15=108, so original×1.20=123, original=123/1.20=102.50. Choice F reports 123 directly instead of dividing by 1.20 to recover the original price.
22: G. Circles & Conics + Probability. One quarter-circle of radius 20 has area (1/4)π(20)²≈314.16. Field area=40×60=2,400. Probability=314.16/2,400≈0.131. Choice F reports the ratio of a quarter of the field's area to its full area (a rough visual estimate) instead of the actual quarter-circle calculation.
23: A. Exponential Equations. Rewrite 81 as 3⁴: 3^(x+1)=3⁴, so x+1=4, x=3. Choice G reports the exponent 4 minus 1 counted incorrectly, an off-by-one slip.
24: F. Area, Volume, and Composite Figures + Integrating Essential Skills. Volume=π(3)²(8)≈226.19 cubic feet. Converted: 226.19×7.48≈1,691.94 gallons. Choice G comes from an arithmetic slip while converting cubic feet to gallons.
25: B. Functions. The vertex of A(w)=−w²+20w occurs at w=10. A(10)=10(20−10)=100. Choice F comes from an arithmetic slip while evaluating A(10).
26: F. Sequences + Center and Spread. Terms: 5, 9, 13, 17, 21. Mean=(5+9+13+17+21)/5=13. Choice G misidentifies the second term (9) as the sequence's middle value instead of the third.
27: B. Congruence & Similarity + Integrating Essential Skills. 8.5×15=127.5 feet=42.5 yards. Choice A reports the actual length in feet instead of converting to the requested yards.
28: G. Quadratics. The sum of the roots of ax²+bx+c=0 is −b/a=9. The average is 9/2=4.5. Choice F reports the sum of the two roots instead of their average.
29: B. Vectors & Matrices + Congruence and Similarity. Determinant=5(4)−2(3)=14. Transformed area=14×6=84. Choice A comes from an arithmetic slip while computing the determinant.
30: J. Piecewise Functions + Integrating Essential Skills. c(7)=5+1.5(5)=12.50. As a percent of $20: 12.50/20×100=62.5%. Choice F reports the raw cost ($5) as a percent of the budget instead of the full piecewise cost.
31: C. Area, Volume, and Composite Figures. (1/3)πr²(12)=100π, so 4r²=100, r²=25, r=5. Choice A comes from an arithmetic slip while solving for r².
32: H. Functions + Integrating Essential Skills. f(10)=70 (not yet below); f(11)=68 (below 70). The first whole-number week is 11, with a score of 68. Choice F reports the score at week 10, the boundary value, instead of the first week the average actually drops below 70.
33: A. Nonroutine Multi-Step Chains. A=3B−20 and A+B=180, so 3B−20+B=180, 4B=200, B=50, A=3(50)−20=130. Choice G reports B, the smaller group, instead of the requested Group A.
34: H. Vectors & Matrices + Coordinate Geometry. The matrix reflects P(2,3) to P′(2,−3). Distance=√(0²+6²)=6. Choice F assumes the reflection leaves the point unchanged, giving a distance of zero instead of accounting for the flipped y-coordinate.
35: A. Sequences. a₄=3(4)²−2=3(16)−2=46. Choice G comes from an arithmetic slip while evaluating 3(4)².
36: J. Probability. Expected value=(200−5)(0.02)+(−5)(0.98)=3.90−4.90=−1.00. Choice F omits the ticket cost from the winning outcome, computing 200(0.02)−5(0.98) instead of (200−5)(0.02)+(−5)(0.98).
37: C. Right Triangles & Trigonometry. tan(60°)=opposite/adjacent, so opposite=5·tan(60°)=5√3. Choice J applies the 45-45-90 ratio (√2) instead of the 30-60-90 ratio (√3) for this angle.
38: F. Bivariate Data. Set y=0: 0=4x−20, so x=5. Choice G reports the coefficient of x instead of solving for the x-intercept.
39: D. Rational, Radical, Exponential Functions. P(15)=200(2)^(15/5)=200(2)³=1,600. Choice H comes from an arithmetic slip while evaluating the exponent 15/5.
40: G. Absolute Value. |x−7|=12 gives x=19 or x=−5. Sum=19+(−5)=14. Choice J reports the positive solution x=19 alone instead of summing both solutions.
41: B. Rational, Radical, Exponential Functions. A(9)=80(0.5)^(9/3)=80(0.5)³=10. Choice F comes from an arithmetic slip while evaluating the exponent 9/3.
42: F. Congruence & Similarity. Diagonals=n(n−3)/2=6(3)/2=9. Choice G reports n(n−3) without dividing by 2.
43: B. Vectors & Matrices. v=(10,6)−(6,2)=(4,4), since 2u=(6,2). Choice F drops the sign while isolating v, computing (10,6)+(6,2) instead of (10,6)−(6,2).
44: G. Congruence & Similarity. Area scales by the square of the linear factor: 3²=9. 12×9=108. Choice F applies the scale factor directly instead of squaring it for the area.
45: C. Data Collection. 25/40=0.625. Choice A reports an unrelated ratio instead of the requested proportion.
46: F. Rational, Radical, Exponential Functions. Squaring: 3x−2=16, so 3x=18, x=6. Then 2(6)−5=7. Choice H reports x itself (6) instead of completing the requested value of 2x−5.
47: A. Area, Volume, and Composite Figures. Trapezoid area=((x+2)+(x+8))/2×4=48. Simplifying: (2x+10)(2)=48, so 2x+10=24, 2x=14, x=7. Choice B comes from an arithmetic slip while solving the resulting linear equation.
48: G. Nonroutine Multi-Step Chains. 20+0.10m=35+0.05m, so 0.05m=15, m=300. Choice H averages the two per-minute rates (0.075) instead of subtracting them, giving 15/0.075=200.
49: D. Notation, Evaluation, and Composition. f(a)=25: 2a+7=25, a=9. g(x)=9x−3, so g(2)=9(2)−3=15. Choice F reports a itself instead of completing the calculation of g(2).
50: J. Number & Quantity. |8−15i|=√(8²+15²)=√289=17. Choice H reports one of the components directly instead of computing the full magnitude.
51: C. Area, Volume, and Composite Figures + Integrating Essential Skills. Rectangle=12×8=96. Semicircle (r=4): 0.5π(4)²≈25.13. Total≈121.13 square feet=13.46 square yards. Choice G reports the total in square feet instead of converting to the requested square yards.
52: F. Nonroutine Multi-Step Chains. Combined rate=1/6+1/3=1/2 tank per hour. Time=1÷(1/2)=2 hours. Choice G comes from an arithmetic slip while combining the two individual rates.