About this SAT Math practice test

Doing well on Digital SAT Math is partly about knowing when to let the built-in Desmos calculator do the heavy lifting instead of grinding through algebra by hand. This set follows College Board's four core math domains, with worked answers showing both the graphing shortcut and the standard algebraic method, so you can see which path is actually faster.

SAT Math Practice Test sample questions

These starter questions help you launch a math mock test quickly. Swap them with your own worksheet, notebook, or textbook questions any time.

1. 1. Solve for x: 3x − 7 = 2x + 4

2. 2. If f(x) = 2x² − 3x + 1, what is f(4)?

3. 3. A line passes through (2, 5) and (−1, −1). What is its slope?

4. 4. Solve the system: 2x + y = 11 and x − y = 1

5. 5. Which of the following is equivalent to (x + 3)(x − 3)?

   • A) x² + 6
   • B) x² − 9
   • C) x² − 6x − 9
   • D) x² + 6x − 9

6. 6. A car travels 150 miles in 2.5 hours. What is its average speed in miles per hour?

7. 7. The mean of a data set is 40. If each value is increased by 5, what is the new mean?

8. 8. In triangle ABC, angle A = 90°, BC = 13, and AB = 5. Find AC.

9. 9. What are the solutions to x² − 4x − 12 = 0?

10. 10. A store sells a jacket at a 25% discount. The sale price is $90. What was the original price?

11. 11. If 4x + 2y = 20 and x = 3, what is y?

12. 12. The scatter plot shows a linear relationship. The line of best fit passes through (0, 3) and (4, 11). What is the slope?

13. 13. sin(30°) × tan(45°) = ?

14. 14. A circle has area 36π cm². What is its circumference?

15. 15. If P(A) = 0.6 and P(B|A) = 0.5, what is P(A and B)?

16. 16. Solve the inequality for x: 5x + 8 ≤ 3x + 20. What is the largest integer value of x that satisfies the inequality?

17. 17. A ball is launched from a platform, and its height in feet after t seconds is modeled by h(t) = −16t² + 64t + 5. What is the maximum height the ball reaches, in feet?

18. 18. A movie theater charges $12 for an adult ticket and $8 for a child ticket. On Saturday, the theater sold 150 tickets in total for $1,540. How many adult tickets were sold?

19. 19. If f(x) = x² + 1 and g(x) = 2x − 3, what is the value of f(g(2))?

20. 20. A recipe that serves 8 people requires 3 cups of flour. Using the same ratio, how many cups of flour are needed to serve 20 people?

21. 21. A linear function g satisfies g(2) = 7 and g(5) = 16. What is the value of g(10)?

22. 22. Simplify the rational expression (x² − 9)/(x + 3) for all values of x ≠ −3.

23. 23. A cylindrical water tank has a radius of 3 feet and a height of 10 feet. What is the volume of the tank, in cubic feet, in terms of π?

    • A) 30π
    • B) 60π
    • C) 90π
    • D) 900π

24. 24. A chemist mixes a 10% acid solution with a 40% acid solution to create 30 liters of a 25% acid solution. How many liters of the 40% solution are needed?

25. 25. For which value of k does the equation x² + 6x + k = 0 have exactly one real solution?

    • A) k = 3
    • B) k = 6
    • C) k = 9
    • D) k = 12

26. 26. A line has a slope of −2 and passes through the point (3, 4). Which equation represents this line in slope-intercept form?

    • A) y = −2x + 10
    • B) y = −2x − 2
    • C) y = −2x + 4
    • D) y = 2x − 2

27. 27. What are all real solutions to the equation x³ − 4x = 0?

28. 28. A company's daily profit, in dollars, is modeled by P(x) = −5x² + 200x − 1200, where x is the number of units sold. How many units must be sold to maximize profit?

29. 29. Solve for x: (x + 2)/(x − 1) = 3

30. 30. Solve for x: |2x − 5| = 9

Answer key & quick explanations

Short answers for the sample questions above. Use this to self-check before generating a fresh AI-built mock test.

1. 1. Linear equation, 3x − 7 = 2x + 4

   x = 11

   Subtract 2x from both sides to get x − 7 = 4, then add 7 to both sides to find x = 11.

2. 2. Function evaluation, f(x) = 2x² − 3x + 1 at x = 4

   f(4) = 21

   Substituting x = 4 gives f(4) = 2(16) − 3(4) + 1 = 32 − 12 + 1 = 21.

3. 3. Slope through (2, 5) and (−1, −1)

   slope = 2

   Using the slope formula (y₂ − y₁)/(x₂ − x₁), we get (−1 − 5)/(−1 − 2) = −6/−3 = 2.

4. 4. System of equations, 2x + y = 11 and x − y = 1

   x = 4, y = 3

   Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting back into x − y = 1 gives y = 3.

5. 5. Expand (x + 3)(x − 3) [MCQ]

   B) x² − 9

   This is a difference of squares: (x + 3)(x − 3) = x² − 3x + 3x − 9 = x² − 9. The other options incorrectly keep or double the middle term.

6. 6. Average speed, 150 miles in 2.5 hours

   60 mph

   Average speed = distance ÷ time = 150 ÷ 2.5 = 60 miles per hour.

7. 7. Effect on mean when each value increases by 5

   45

   Adding a constant to every value in a data set increases the mean by that same constant, so the new mean is 40 + 5 = 45.

8. 8. Right triangle, hypotenuse BC = 13, leg AB = 5, find AC

   AC = 12

   Since angle A = 90°, BC is the hypotenuse. By the Pythagorean theorem, AC² = BC² − AB² = 169 − 25 = 144, so AC = 12.

9. 9. Quadratic equation, x² − 4x − 12 = 0

   x = 6 or x = −2

   Factoring gives (x − 6)(x + 2) = 0, since −6 × 2 = −12 and −6 + 2 = −4, so the solutions are x = 6 and x = −2.

10. 10. Original price before a 25% discount, sale price $90

    $120

    A 25% discount means the sale price is 75% of the original: 0.75P = 90, so P = 90 ÷ 0.75 = 120.

11. 11. Linear equation, 4x + 2y = 20 with x = 3

    y = 4

    Substituting x = 3 gives 12 + 2y = 20, so 2y = 8 and y = 4.

12. 12. Slope of line of best fit through (0, 3) and (4, 11)

    slope = 2

    Using the slope formula, (11 − 3)/(4 − 0) = 8/4 = 2.

13. 13. sin(30°) × tan(45°)

    0.5 (or 1/2)

    sin(30°) = 1/2 and tan(45°) = 1, so their product is (1/2)(1) = 1/2.

14. 14. Circumference of a circle with area 36π cm²

    12π cm

    Since area = πr² = 36π, r² = 36 so r = 6. Circumference = 2πr = 2π(6) = 12π cm.

15. 15. P(A and B) given P(A) = 0.6 and P(B|A) = 0.5

    0.3

    By the multiplication rule, P(A and B) = P(A) × P(B|A) = 0.6 × 0.5 = 0.3.

16. 16. Linear inequality, 5x + 8 ≤ 3x + 20 (largest integer x)

    x = 6

    Subtracting 3x from both sides gives 2x + 8 ≤ 20, then subtracting 8 gives 2x ≤ 12, so x ≤ 6. The largest integer satisfying this is 6.

17. 17. Maximum height of projectile, h(t) = −16t² + 64t + 5

    69 feet

    The maximum occurs at the vertex, t = −b/(2a) = −64/(2×−16) = 2 seconds. Substituting t = 2 gives h(2) = −16(4) + 64(2) + 5 = −64 + 128 + 5 = 69 feet.

18. 18. Ticket sales system, adult $12 / child $8, 150 tickets for $1,540

    85 adult tickets

    Let a = adult tickets and c = child tickets. From a + c = 150 and 12a + 8c = 1540, substitute c = 150 − a: 12a + 8(150 − a) = 1540 gives 4a + 1200 = 1540, so 4a = 340 and a = 85.

19. 19. Composite function, f(g(2)) where f(x) = x² + 1, g(x) = 2x − 3

    2

    First find g(2) = 2(2) − 3 = 1. Then f(1) = 1² + 1 = 2.

20. 20. Recipe ratio, 3 cups flour for 8 servings scaled to 20 servings

    7.5 cups

    Set up a proportion: 3/8 = x/20. Solving gives x = (3 × 20)/8 = 60/8 = 7.5 cups.

21. 21. Linear function, g(2) = 7 and g(5) = 16, find g(10)

    g(10) = 31

    The slope is (16 − 7)/(5 − 2) = 3, so g(x) = 3x + b. Using g(2) = 7: 6 + b = 7, so b = 1. Then g(10) = 3(10) + 1 = 31.

22. 22. Simplify rational expression (x² − 9)/(x + 3)

    x − 3

    Factor the numerator as a difference of squares: x² − 9 = (x − 3)(x + 3). Dividing by (x + 3) leaves x − 3, for all x ≠ −3.

23. 23. Volume of cylinder, radius 3 ft, height 10 ft [MCQ]

    C) 90π

    Volume = πr²h = π(3²)(10) = 90π cubic feet. Choice A forgets to square the radius, choice B uses the diameter instead of the radius, and choice D incorrectly squares the entire product.

24. 24. Mixture problem, 10% and 40% acid solutions to make 30 L of 25% solution

    15 liters

    Let x be liters of the 40% solution. Then 0.40x + 0.10(30 − x) = 0.25(30) gives 0.30x + 3 = 7.5, so 0.30x = 4.5 and x = 15 liters.

25. 25. Discriminant condition for one real solution, x² + 6x + k = 0 [MCQ]

    C) k = 9

    A quadratic has exactly one real solution when its discriminant equals zero: 6² − 4(1)(k) = 0, so 36 − 4k = 0 and k = 9. The other values all leave a nonzero discriminant, giving two real solutions.

26. 26. Slope-intercept equation, slope −2 through (3, 4) [MCQ]

    A) y = −2x + 10

    Using y = mx + b with m = −2 and the point (3, 4): 4 = −2(3) + b gives b = 10, so y = −2x + 10. Choice D has the correct point but the wrong slope, and B and C have incorrect y-intercepts.

27. 27. Cubic equation, x³ − 4x = 0

    x = −2, 0, 2

    Factor out x: x(x² − 4) = 0, then factor the difference of squares: x(x − 2)(x + 2) = 0. The solutions are x = 0, x = 2, and x = −2.

28. 28. Maximizing profit, P(x) = −5x² + 200x − 1200

    20 units

    Since this is a downward-opening parabola, profit is maximized at the vertex: x = −b/(2a) = −200/(2×−5) = 20 units.

29. 29. Rational equation, (x + 2)/(x − 1) = 3

    x = 2.5

    Multiply both sides by (x − 1): x + 2 = 3(x − 1), which gives x + 2 = 3x − 3, so 5 = 2x and x = 2.5. This value does not make the denominator zero, so it is valid.

30. 30. Absolute value equation, |2x − 5| = 9

    x = 7 or x = −2

    Split into two cases: 2x − 5 = 9 gives x = 7, and 2x − 5 = −9 gives x = −2. Both values satisfy the original equation.

Curriculum Mapping & Learning Guide

Use this breakdown to identify which skills each question tests and guide post-test review.

SAT Math units covered

1. Chapter 1: Algebra: Linear equations and inequalities
2. Chapter 2: Algebra: Systems of equations
3. Chapter 3: Algebra: Linear functions and graphs
4. Chapter 4: Advanced Math: Quadratic equations and parabolas
5. Chapter 5: Advanced Math: Polynomial and rational functions
6. Chapter 6: Advanced Math: Function notation and composition
7. Chapter 7: Problem-Solving: Ratios, rates, percentages
8. Chapter 8: Problem-Solving: Statistics (mean, median, spread)
9. Chapter 9: Problem-Solving: Probability and two-way tables
10. Chapter 10: Geometry: Area, volume, angle relationships
11. Chapter 11: Geometry: Pythagorean theorem and similarity
12. Chapter 12: Geometry: Trigonometry (sin, cos, tan)

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