The best way to improve your SAT math score is to work through practice problems and study the solutions, not just check the answer. This guide contains 20 SAT practice math problems organized by topic, each with a complete worked solution that shows you exactly how to get from the question to the correct answer.
These problems cover all four areas tested on the Digital SAT Math section: Algebra, Advanced Math, Problem Solving and Data Analysis, and Geometry and Trigonometry. Difficulty ranges from medium to hard so you can build skills and challenge yourself. Try each problem on your own first, then check the solution.
How to Use These Practice Problems
For the best results, follow this approach:
- Set a timer. Give yourself about 1.5 minutes per problem, which matches real SAT pacing.
- Write out your work. Do not solve problems in your head. Writing forces you to be precise and helps you catch errors.
- Try before you look. Attempt each problem fully before reading the solution. Struggling with a problem teaches you more than reading a solution passively.
- Study the method, not just the answer. The point is to learn the approach so you can apply it to similar problems on test day.
- Track your mistakes. Note which topics give you trouble so you know where to focus your study time.
Keep the SAT Math Formula Sheet nearby for reference. The SAT provides a reference sheet with basic geometry formulas during the test, but you should have all the common formulas memorized before test day.
Algebra (5 Problems)
Algebra questions make up roughly 35% of the SAT Math section. They test linear equations, systems of equations, inequalities, and interpreting linear functions.
Problem 1: Solving a Linear Equation
If 3(x + 4) = 2x + 17, what is the value of x?
Solution:
Distribute on the left side: 3x + 12 = 2x + 17
Subtract 2x from both sides: x + 12 = 17
Subtract 12 from both sides: x = 5
Answer: x = 5
Key takeaway: Distribute first, then isolate the variable. This is one of the most common SAT algebra patterns.
Problem 2: System of Equations
A store sells notebooks for each and pens for each. A student buys 8 items for a total of . How many notebooks did the student buy?
Solution:
Let n = notebooks and p = pens. Set up two equations:
n + p = 8 (total items)
4n + 2p = 22 (total cost)
From the first equation: p = 8 - n
Substitute into the second: 4n + 2(8 - n) = 22
4n + 16 - 2n = 22
2n = 6
n = 3
Answer: 3 notebooks
Key takeaway: Word problems that involve two unknowns almost always require a system of equations. Define your variables clearly, write two equations, then solve by substitution or elimination.
Problem 3: Linear Inequality
A delivery service charges a base fee plus .50 per mile. A customer has at most to spend. What is the maximum number of full miles the customer can travel?
Solution:
Set up the inequality: 5 + 1.50m ≤ 20
Subtract 5: 1.50m ≤ 15
Divide by 1.50: m ≤ 10
Answer: 10 miles
Key takeaway: Inequality problems are solved just like equations. The only difference is when you multiply or divide by a negative number (flip the sign). Here there is no sign flip needed.
Problem 4: Interpreting Slope
The equation y = 250 + 40x models the total cost y, in dollars, of renting a venue for x hours. What does the number 40 represent in this context?
A) The total cost for 40 hours
B) The initial fee for the venue
C) The hourly rental rate
D) The maximum number of hours
Answer: C
Solution: In y = mx + b form, the coefficient of x (the slope) represents the rate of change. Here, 40 is multiplied by x (hours), so it represents the cost per hour. The 250 is the y-intercept, which represents the initial fee (base cost at x = 0). The SAT frequently asks you to interpret slope and y-intercept in real-world contexts.
Problem 5: Absolute Value Equation
If |2x - 6| = 10, what is the sum of all possible values of x?
Solution:
An absolute value equation splits into two cases:
Case 1: 2x - 6 = 10, so 2x = 16, so x = 8
Case 2: 2x - 6 = -10, so 2x = -4, so x = -2
Sum: 8 + (-2) = 6
Answer: 6
Key takeaway: Always split absolute value equations into positive and negative cases. When the question asks for the sum of all solutions, this shortcut works: for |ax + b| = c, the two solutions are symmetric around x = -b/a, and their sum is always -2b/a. Here that is -2(-6)/2 = 6.
Advanced Math (5 Problems)
Advanced Math covers quadratics, polynomials, exponential functions, and nonlinear equations. These questions make up roughly 35% of the Math section and tend to be the most challenging.
Problem 6: Factoring a Quadratic
What are the solutions of x² - 5x - 14 = 0?
Solution:
Find two numbers that multiply to -14 and add to -5. Those numbers are -7 and 2.
Factor: (x - 7)(x + 2) = 0
Set each factor to zero: x = 7 or x = -2
Answer: x = 7 and x = -2
Key takeaway: For x² + bx + c = 0, find two numbers that multiply to c and add to b. If you cannot factor quickly, use the quadratic formula instead.
Problem 7: Quadratic Word Problem
A ball is thrown upward with a height modeled by h(t) = -16t² + 48t + 5, where h is height in feet and t is time in seconds. What is the maximum height reached by the ball?
Solution:
The maximum of a downward-opening parabola occurs at t = -b/(2a).
Here a = -16 and b = 48, so t = -48/(2 × -16) = -48/(-32) = 1.5 seconds.
Plug back in: h(1.5) = -16(1.5)² + 48(1.5) + 5 = -16(2.25) + 72 + 5 = -36 + 72 + 5 = 41
Answer: 41 feet
Key takeaway: The vertex formula t = -b/(2a) is essential for maximum/minimum problems. Memorize it. You will see this pattern on almost every SAT.
Problem 8: Exponential Growth
A population of bacteria doubles every 3 hours. If there are 500 bacteria initially, which expression gives the population after t hours?
A) 500(2)t
B) 500(2)t/3
C) 500(2)3t
D) 500(3)t/2
Answer: B
Solution: The general form for exponential growth is P = P0 × rt/d, where P0 is the initial amount, r is the growth factor, and d is the doubling period. Here: P0 = 500, r = 2 (doubles), and d = 3 hours. So P = 500(2)t/3. Check: at t = 3, P = 500(2)1 = 1000 (correct, it doubled). At t = 6, P = 500(2)2 = 2000 (doubled again).
Problem 9: Polynomial Division
If x² + 7x + 12 is divided by (x + 3), what is the result?
Solution:
Factor the numerator: x² + 7x + 12 = (x + 3)(x + 4)
Divide: (x + 3)(x + 4) / (x + 3) = x + 4
Answer: x + 4
Key takeaway: When dividing polynomials, always try factoring first. It is faster than long division and less error-prone. If the divisor is a factor of the numerator, the division is clean with no remainder.
Problem 10: Function Composition
If f(x) = 2x + 1 and g(x) = x² - 3, what is f(g(2))?
Solution:
Work from the inside out.
First find g(2): g(2) = (2)² - 3 = 4 - 3 = 1
Then find f(1): f(1) = 2(1) + 1 = 3
Answer: 3
Key takeaway: For composition, always evaluate the inner function first. A common mistake is plugging the entire g(x) expression into f(x) when you only need a specific value. When given a number, just compute step by step.
Problem Solving and Data Analysis (5 Problems)
This category covers ratios, percentages, probability, statistics, and data interpretation. It makes up roughly 15% of the Math section and often uses real-world scenarios.
Problem 11: Percent Change
A shirt originally costs . It goes on sale for 25% off, and then the sale price is taxed at 8%. What is the final price?
Solution:
Sale price: × 0.75 =
After tax: × 1.08 = .60
Answer: .60
Key takeaway: Apply discounts and taxes sequentially, not combined. A 25% discount followed by 8% tax is not the same as a 17% discount. The SAT loves testing whether you understand the order of operations with percentages.
Problem 12: Ratio Problem
In a class, the ratio of students who prefer math to those who prefer science is 5:3. If there are 40 students total, how many prefer science?
Solution:
Total parts: 5 + 3 = 8
Each part: 40 / 8 = 5 students
Science: 3 × 5 = 15
Answer: 15 students
Key takeaway: For ratio problems, add the parts to find the total ratio, then divide the actual total by the ratio total to find the value of one part.
Problem 13: Probability
A bag contains 4 red marbles, 6 blue marbles, and 5 green marbles. If one marble is drawn at random, what is the probability that it is NOT blue?
Solution:
Total marbles: 4 + 6 + 5 = 15
Not blue: 4 + 5 = 9
Probability: 9/15 = 3/5
Answer: 3/5 (or 0.6)
Key takeaway: P(not A) = 1 - P(A). You could also find P(blue) = 6/15 = 2/5, then subtract from 1: 1 - 2/5 = 3/5. Use whichever approach is faster.
Problem 14: Mean and Median
The test scores for 7 students are: 72, 85, 88, 90, 91, 93, 95. If the score of 72 is removed, by how much does the mean increase?
Solution:
Original mean: (72 + 85 + 88 + 90 + 91 + 93 + 95) / 7 = 614 / 7 = 87.71
New mean (without 72): (85 + 88 + 90 + 91 + 93 + 95) / 6 = 542 / 6 = 90.33
Increase: 90.33 - 87.71 = 2.62
Answer: approximately 2.62 points
Key takeaway: Removing an outlier (a value far from the mean) shifts the mean significantly. The SAT often asks how adding or removing a data point affects mean, median, or range. Mean is sensitive to outliers; median usually is not.
Problem 15: Interpreting a Scatterplot
A line of best fit for a scatterplot has the equation y = 1.2x + 15, where x is the number of hours studied and y is the test score. According to the model, what is the predicted score for a student who studies 10 hours?
Solution:
Substitute x = 10: y = 1.2(10) + 15 = 12 + 15 = 27
Answer: 27
Key takeaway: Line of best fit questions are just plug-and-chug once you identify what x and y represent. The SAT may also ask you to interpret the slope (1.2 = predicted score increase per additional hour studied) or the y-intercept (15 = predicted score with zero hours studied).
Geometry and Trigonometry (5 Problems)
Geometry and Trigonometry makes up roughly 15% of the SAT Math section. Questions cover area, volume, angles, triangles, circles, and basic trig. Review the formula sheet before working through these.
Problem 16: Area of a Triangle
A triangle has a base of 12 cm and a height of 9 cm. What is its area?
Solution:
Area = (1/2) × base × height = (1/2) × 12 × 9 = 54
Answer: 54 cm²
Key takeaway: This formula is on the SAT reference sheet, but you should have it memorized. The height must be perpendicular to the base, which is where students sometimes make errors on more complex triangle problems.
Problem 17: Circle Equation
A circle in the xy-plane has the equation (x - 3)² + (y + 2)² = 25. What is the radius of the circle?
Solution:
The standard form is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Here r² = 25, so r = 5. The center is (3, -2).
Answer: 5
Key takeaway: Know the standard circle equation cold. A common trap is giving r² (25) instead of r (5). The SAT also sometimes gives the equation in expanded form, requiring you to complete the square to find the center and radius.
Problem 18: Right Triangle Trigonometry
In a right triangle, one angle is 30 degrees and the hypotenuse is 10. What is the length of the side opposite the 30-degree angle?
Solution:
sin(30°) = opposite / hypotenuse
0.5 = opposite / 10
opposite = 5
Answer: 5
Key takeaway: Memorize the 30-60-90 triangle ratios (1 : √3 : 2) and the trig definitions (SOH CAH TOA). The side opposite 30 degrees is always half the hypotenuse. This is one of the most common trig setups on the SAT.
Problem 19: Volume of a Cylinder
A cylindrical water tank has a radius of 4 feet and a height of 10 feet. What is the volume of the tank in cubic feet? (Use π ≈ 3.14)
Solution:
Volume = πr²h = 3.14 × (4)² × 10 = 3.14 × 16 × 10 = 502.4
Answer: approximately 502.4 cubic feet (or 160π)
Key takeaway: The cylinder volume formula (πr²h) is provided on the SAT reference sheet. On the real test, you may leave your answer in terms of π or use 3.14 depending on what the answer choices look like. Always check whether the problem gives you radius or diameter.
Problem 20: Angle Relationships
Two parallel lines are cut by a transversal. One of the angles formed is 65 degrees. What is the measure of the supplementary angle on the same side of the transversal?
Solution:
Supplementary angles add up to 180 degrees.
180 - 65 = 115
Answer: 115 degrees
Key takeaway: When parallel lines are cut by a transversal, same-side interior angles are supplementary (add to 180). Alternate interior angles are equal. Know these relationships because the SAT tests them in multi-step geometry problems where you need to find several angles to reach the answer.
Strategies for Solving SAT Math Problems
Beyond knowing the math, these strategies help you work faster and more accurately on test day:
- Use Desmos. The Digital SAT has a built-in Desmos graphing calculator. Use it to graph equations, find intersections, and check your algebra. It is one of the most powerful tools available to you.
- Plug in answer choices. When answer choices are numbers and the problem asks for a specific value, try plugging them in. Start with the middle value to narrow down efficiently.
- Plug in your own numbers. For problems with variables in the answer choices, substitute a simple number (like 2 or 3) for the variable, calculate the result, and see which answer choice gives the same result.
- Read the question twice. Many wrong answers come from solving for the wrong thing. If the question asks for 2x, do not stop at x.
- Manage your time. You have about 1.5 minutes per question. If a problem is taking more than 2 minutes, mark it and move on. Come back with fresh eyes if time allows.
For a complete breakdown of every topic and concept tested, see the SAT Math Topics guide. For a deeper study plan, check out the SAT Math Study Guide.
What to Study Next
Now that you have worked through these 20 problems, your next steps depend on how you performed:
- Got most right (16+)? Move on to the Hardest SAT Math Questions for a challenge. Focus on speed and accuracy under timed conditions.
- Struggled with Algebra or Advanced Math (below 7/10)? Review the fundamentals with the SAT Math Formula Sheet and practice more problems in those categories using Larry Learns SAT Math quizzes.
- Struggled with Geometry/Trig or Data Analysis (below 3/5)? These topics have fewer questions on the SAT but are often the easiest points to pick up because the formulas are straightforward. A focused review session can yield quick gains.
Consistent practice is what drives improvement. Aim to work through 10-20 practice problems per study session, review every solution, and track which topics are improving over time.
Frequently Asked Questions About SAT Practice Math Problems
How many math questions are on the Digital SAT?
The Digital SAT Math section has 44 questions split across two modules. The first module has 22 questions (35 minutes), and the second module has 22 questions (35 minutes). The second module adjusts in difficulty based on your first-module performance.
What topics are tested most on SAT Math?
Algebra and Advanced Math together make up about 70% of the SAT Math section. Problem Solving and Data Analysis accounts for about 15%, and Geometry and Trigonometry makes up the remaining 15%. Focus your study time proportionally. See the SAT Math Topics breakdown for details.
Can I use a calculator on SAT Math?
Yes. The Digital SAT allows a calculator on all math questions. A built-in Desmos graphing calculator is provided, or you can bring your own approved calculator. See the SAT Math Calculator guide for tips on using Desmos effectively.
How do I improve my SAT math score quickly?
Focus on the topics where you are losing the most points. Take a practice test, identify your weakest areas, and drill those specific question types. Most students see the fastest gains from reviewing Algebra fundamentals and learning to use Desmos. A structured study plan makes the biggest difference. See the SAT Math Study Guide for a step-by-step approach.
Are these practice problems the same difficulty as the real SAT?
These problems range from medium to hard difficulty, which matches the majority of what you will see on the real SAT. The Digital SAT adapts its second module based on your performance, so if you do well on the first module, the second module will include harder questions. For the hardest tier of questions, see the Hardest SAT Math Questions guide.
Should I memorize formulas for SAT Math?
The SAT provides a reference sheet with basic geometry formulas (area of a circle, volume of a cylinder, Pythagorean theorem, etc.), but having formulas memorized saves time. You should absolutely memorize the quadratic formula, slope formula, and common trig ratios. See the SAT Math Formula Sheet for everything you need.
How many practice problems should I do per day?
Quality matters more than quantity. Aim for 10-20 problems per study session with full solution review. Working through 15 problems and studying every solution is more effective than rushing through 50 problems and only checking answers. Consistency over 4-6 weeks matters more than any single marathon session.
What is a good SAT Math score?
The average SAT Math score is around 520. A score of 600+ puts you above roughly 70% of test takers. For competitive colleges, aim for 700+. The SAT Math Scores guide covers averages, percentiles, and what different score ranges mean for college admissions.
