Math Practice Bank

40 original digital-SAT math questions — all four domains, with verified worked solutions

11 min read

How to use: work Section 1 (questions only) timed, Desmos allowed; then check Section 2 for the answer + full worked solution (with the fast Desmos method where it helps). 30 multiple-choice + 10 grid-in, labelled by domain and difficulty. Every answer was verified twice. Grid-ins: no mixed numbers — enter an improper fraction or decimal (negatives allowed).

#SECTION 1 — QUESTIONS

#Domain: Algebra

1. (Algebra — Easy) Solve for x: 3(x − 4) = 2x + 5. (A) −7 (B) 7 (C) 17 (D) 9

2. (Algebra — Easy) (grid-in) If (2/3)x − 5 = 7, what is the value of x?

3. (Algebra — Easy) A submarine's depth d, in meters below the surface, after t minutes is d = 15 + 6t. Which is the best interpretation of the 6? (A) The depth when t = 0 (B) The submarine descends 6 meters each minute (C) The total depth at the end (D) The minutes to reach 15 meters

4. (Algebra — Medium) Given 4x + 2y = 18 and 3x − 2y = 10, what is x + y? (A) 1 (B) 3 (C) 4 (D) 5

5. (Algebra — Medium) (grid-in) 2 muffins and 3 cookies cost $13; 4 muffins and 1 cookie cost $16. What is the cost, in dollars, of one muffin?

6. (Algebra — Medium) Which describes all solutions to −3x + 7 ≥ 1? (A) x ≤ 2 (B) x ≥ 2 (C) x ≤ −2 (D) x ≥ −2

7. (Algebra — Medium) A line passes through (2, 5) and (6, 13). Which is its equation? (A) y = (1/2)x + 4 (B) y = 2x + 5 (C) y = 2x + 1 (D) y = 2x − 1

8. (Algebra — Medium) Line ℓ is y = (3/4)x − 2. Line m is perpendicular to ℓ. What is the slope of m? (A) 3/4 (B) −4/3 (C) 4/3 (D) −3/4

9. (Algebra — Hard) The system kx + 6y = 10 and 4x + 3y = 7 has no solution. What is k? (A) −8 (B) 2 (C) 12 (D) 8

10. (Algebra — Hard) In 6x + a = 2(3x + 5), a is a constant. If the equation has infinitely many solutions, what is a? (A) 10 (B) 5 (C) 0 (D) −10

11. (Algebra — Medium) A tank holds 500 L and drains 20 L per minute. Which equation gives the volume V after t minutes? (A) V = 20t − 500 (B) V = 500 − 20t (C) V = 500 + 20t (D) V = 20 − 500t

12. (Algebra — Hard) (grid-in) A theater sold 200 tickets for $2,160 total. Adult tickets are $15, student tickets $8. How many adult tickets were sold?

#Domain: Advanced Math

13. (Advanced — Easy) If f(x) = 2x² − 3x + 1, what is f(−2)? (A) −1 (B) 3 (C) 11 (D) 15

14. (Advanced — Easy) (grid-in) If 2ˣ = 32, what is x?

15. (Advanced — Easy) What are the solutions to x² − 5x + 6 = 0? (A) 2 and 3 (B) −2 and −3 (C) 1 and 6 (D) −1 and −6

16. (Advanced — Medium) f(x) = x² − 6x + 11. What is the minimum value of f(x)? (A) 3 (B) 2 (C) 6 (D) 11

17. (Advanced — Medium) (grid-in) A ball's height (ft) after t seconds is h(t) = −16t² + 48t. After how many seconds does it return to the ground?

18. (Advanced — Medium) A colony of 500 cells doubles every 3 hours. Which gives the number P after t hours? (A) P = 500·2^(t/3) (B) P = 500·2^(3t) (C) P = 500·3^(t/2) (D) P = 500 + 2t

19. (Advanced — Medium) For x ≠ 3, which is equivalent to (x² − 9)/(x − 3)? (A) x − 3 (B) x + 3 (C) x + 9 (D) x² − 3

20. (Advanced — Medium) One solution to x² − 10x + 21 = 0 is x = 3. What is the other? (A) −3 (B) 4 (C) 7 (D) −7

21. (Advanced — Hard) In x² + 6x + c = 0, c is a constant. If the equation has exactly one real solution, what is c? (A) 3 (B) 6 (C) 36 (D) 9

22. (Advanced — Hard) If 3^(2x − 1) = 27, what is x? (A) 1 (B) 5 (C) 2 (D) 3

23. (Advanced — Hard) (grid-in) The graphs of y = x² − 4 and y = 2x − 1 intersect at two points. What is the least of the two x-coordinates?

24. (Advanced — Hard) Which is equivalent to 4x² + 12x + 9? (A) (2x + 3)² (B) (2x − 3)² (C) (4x + 3)² (D) (2x + 9)²

#Domain: Problem-Solving & Data Analysis

25. (PSDA — Easy) An $80 jacket is discounted 25%. What is the sale price? (A) $55 (B) $60 (C) $20 (D) $65

26. (PSDA — Easy) A car travels 150 miles in 2.5 hours at constant speed. Average speed (mph)? (A) 50 (B) 62.5 (C) 60 (D) 375

27. (PSDA — Medium) A stock price rose from $40 to $50. What was the percent increase? (A) 25% (B) 20% (C) 10% (D) 50%

28. (PSDA — Medium) A recipe uses 3 cups flour per 2 cups sugar. With 12 cups flour, how many cups of sugar keep the ratio? (A) 6 (B) 10 (C) 18 (D) 8

29. (PSDA — Medium) For the set 4, 7, 7, 10, 12, the value 40 is added. Which best describes the effect on mean and median? (A) The mean increases more than the median (B) The median increases more than the mean (C) Both stay the same (D) The mean stays the same, the median increases

30. (PSDA — Medium) A table of 100 students: Studied → Passed 45, Failed 5 (50); Did not study → Passed 20, Failed 30 (50); Totals Passed 65, Failed 35. If a student is chosen at random from those who passed, what is the probability the student studied? (A) 45/100 (B) 45/50 (C) 45/65 (D) 65/100

31. (PSDA — Medium) (grid-in) A bag has 5 red, 3 blue, 2 green marbles. One is drawn at random. What is the probability it is NOT blue?

32. (PSDA — Hard) A poll of 400 voters finds 52% support, margin of error 3 points. Best conclusion? (A) Exactly 52% of all voters support it (B) The true percent is plausibly between 49% and 55% (C) Between 49% and 55% of the 400 sampled support it (D) The measure will definitely pass

33. (PSDA — Hard) (grid-in) A chemist has 10 L of a 20% acid solution. How many liters of pure water must be added to make an 8% acid mixture?

#Domain: Geometry & Trigonometry

34. (Geo/Trig — Easy) Two complementary angles; one is 35°. The other? (A) 55° (B) 145° (C) 65° (D) 45°

35. (Geo/Trig — Easy) Two interior angles of a triangle are 40° and 75°. The third? (A) 75° (B) 115° (C) 55° (D) 65°

36. (Geo/Trig — Medium) A right circular cylinder has radius 3, height 10. Volume? (A) 30π (B) 90π (C) 900π (D) 60π

37. (Geo/Trig — Medium) (grid-in) Triangle ABC ~ triangle DEF, AB ↔ DE. If AB = 6, DE = 9, BC = 8, what is EF?

38. (Geo/Trig — Medium) In a right triangle, an acute angle θ has opposite side 5 and hypotenuse 13. What is cos θ? (A) 5/13 (B) 13/12 (C) 12/13 (D) 5/12

39. (Geo/Trig — Hard) A circle is (x − 2)² + (y + 3)² = 25. What is its circumference? (A) 10π (B) 25π (C) 5π (D) 50π

40. (Geo/Trig — Hard) (grid-in) a and a + 15 are the degree measures of two acute angles, and sin(2a)° = cos((a + 15)°). What is a?

#SECTION 2 — ANSWER KEY & FULL SOLUTIONS

#Algebra

1. C) 17. 3x − 12 = 2x + 5 → x = 17. (D=9 forgets to distribute; A=−7 sign error.) 2. 18. (2/3)x = 12 → x = 18. 3. B). The coefficient of t is the rate of change per minute; 15 (constant) is the depth at t=0 (that's A). 4. D) 5. Add the equations → 7x = 28 → x = 4; then y = 1; x + y = 5. (Desmos: intersection (4,1).) (C=4 is x alone.) 5. 3.5. m + system: 2m + 3c = 13, 4m + c = 16 → c = 16 − 4m → 2m + 48 − 12m = 13 → −10m = −35 → m = 3.5 (c = 2). Enter 3.5 or 7/2. 6. A) x ≤ 2. −3x ≥ −6 → divide by −3 and FLIP → x ≤ 2. (B forgets to flip.) 7. C) y = 2x + 1. slope = (13−5)/(6−2) = 2; y − 5 = 2(x − 2) → y = 2x + 1. Check x=6 → 13. ✓ 8. B) −4/3. Perpendicular = negative reciprocal of 3/4. 9. D) 8. No solution → parallel: k/4 = 6/3 = 2 → k = 8; constants 10/7 ≠ 2, so distinct lines (truly no solution). (B=2 is the ratio, not k.) 10. A) 10. RHS = 6x + 10; infinitely many solutions needs identical sides → a = 10. 11. B) V = 500 − 20t. Start 500, lose 20 each minute. (C fills instead of drains.) 12. 80. a + s = 200, 15a + 8s = 2160 → 15a + 8(200 − a) = 2160 → 7a = 560 → a = 80. Check: 80·15 + 120·8 = 2160. ✓

#Advanced Math

13. D) 15. 2(4) − 3(−2) + 1 = 8 + 6 + 1 = 15. (B=3 mishandles −3(−2).) 14. 5. 32 = 2⁵ → x = 5. 15. A) 2 and 3. (x − 2)(x − 3) = 0. (Desmos: x-intercepts 2, 3.) 16. B) 2. vertex x = −b/2a = 3; f(3) = 9 − 18 + 11 = 2; opens up → minimum. (A=3 is the x-coordinate; D=11 is f(0).) (Desmos: min point (3,2).) 17. 3. −16t² + 48t = 0 → −16t(t − 3) = 0 → t = 0 or 3; returns to ground at t = 3. 18. A) P = 500·2^(t/3). Doubles once per 3-hour period; # periods = t/3. Check t=3 → 1000. ✓ 19. B) x + 3. (x−3)(x+3)/(x−3) = x + 3. 20. C) 7. (x − 3)(x − 7) = 0 → other root 7. (Sum of roots = 10 → 3 + 7.) 21. D) 9. One real solution → discriminant 0 → 36 − 4c = 0 → c = 9. (Desmos: slider c until tangent to x-axis.) 22. C) 2. 27 = 3³ → 2x − 1 = 3 → x = 2. 23. −1. x² − 4 = 2x − 1 → x² − 2x − 3 = 0 → (x − 3)(x + 1) → x = −1, 3; least is −1. Enter -1. 24. A) (2x + 3)². = 4x² + 12x + 9 (perfect-square trinomial: √(4x²)=2x, √9=3, 2·2x·3 = 12x). ✓

#Problem-Solving & Data Analysis

25. B) $60. Pay 75%: 80 × 0.75 = 60. 26. C) 60. 150 ÷ 2.5 = 60. (D multiplies instead of divides.) 27. A) 25%. (50 − 40)/40 = 0.25. (B uses the wrong base 10/50.) 28. D) 8. 3/2 = 12/s → 3s = 24 → s = 8. (Flour scaled ×4, from 3 to 12, so sugar ×4 = 8.) (C inverts the ratio.) 29. A). Before: mean 8, median 7. After adding 40: mean = 80/6 ≈ 13.33, median = (7+10)/2 = 8.5. The outlier lifts the mean (+5.33) far more than the median (+1.5). 30. C) 45/65. "Given passed" → denominator = 65 passers; 45 of them studied → 45/65. (B conditions on "studied"; A uses the whole 100.) 31. 7/10 (or 0.7). Not blue = (5 + 2)/10 = 7/10. 32. B). 52% ± 3 → the true population percent plausibly lies in 49–55%. (A over-pins the population; C misapplies it to the sample; D over-claims.) 33. 15. Acid stays 0.20 × 10 = 2 L. Require 2/(10 + x) = 0.08 → 2 = 0.8 + 0.08x → x = 15. Check 2/25 = 0.08. ✓

#Geometry & Trigonometry

34. A) 55°. Complementary → 90 − 35 = 55. (B treats them as supplementary.) 35. D) 65°. 180 − 40 − 75 = 65. (B forgets to subtract from 180.) 36. B) 90π. πr²h = π·9·10 = 90π. 37. 12. 6/9 = 8/EF → 6·EF = 72 → EF = 12 (scale factor 1.5). 38. C) 12/13. adjacent = √(13² − 5²) = 12; cos θ = 12/13. (A is sin θ.) 39. A) 10π. r² = 25 → r = 5; circumference = 2πr = 10π. (B is the area.) 40. 25. sin(2a) = cos(a + 15) → complementary angles → 2a + (a + 15) = 90 → 3a = 75 → a = 25. Check: 50° + 40° = 90°. ✓ (Desmos in degree mode: curves meet at a = 25.)

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