In the equation 3(x + 4) = 3x + k, k is a constant. For what value of k does the equation have infinitely many solutions?
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About this test
This is a free 15-question sampler in the style of the digital SAT Math section. It covers all four content areas, algebra, advanced math, problem-solving and data analysis, and geometry and trigonometry, with multiple choice, select all that apply and fill in the blank. Keep a calculator open like you would on test day, answer all 15 questions, then read the explanation for every one.
All 15 questions at a glance
Prefer to read before you play? Every question in the sampler is listed below. Answers and explanations stay collapsed until you open them.
1.In the equation 3(x + 4) = 3x + k, k is a constant. For what value of k does the equation have infinitely many solutions?
- A. 4
- B. 12
- C. 3
- D. 16
Answer: B. 12
Distribute on the left to get 3x + 12 = 3x + k. The 3x terms match on both sides, so the two sides are identical, and the equation is true for every x, exactly when k = 12. Any other k leaves a false statement such as 12 = 4, which means no solutions at all.
2.A streaming service charges a one-time setup fee plus a fixed price each month. The total cost, in dollars, after m months is C = 14m + 35. What does the number 35 represent in this context?
- A. The price charged each month
- B. The total cost after 35 months
- C. The cost of the first month alone
- D. The one-time setup fee
Answer: D. The one-time setup fee
In a linear model the constant is the value when the variable is zero, so 35 is what you pay at m = 0, before any month has passed. That is the setup fee. The coefficient 14 is the monthly price, and the first month alone would cost 14 plus the fee, not 35.
3.A school store sells pencils for $0.50 each and notebooks for $3 each. A student buys 12 items in total and pays $18.50. How many notebooks did the student buy?
- A. 7
- B. 12
- C. 5
- D. 3
Answer: C. 5
Let n be the notebooks and p the pencils. Then p + n = 12 and 0.50p + 3n = 18.50. Substitute p = 12 - n to get 6 - 0.50n + 3n = 18.50, so 2.5n = 12.5 and n = 5. The student bought 7 pencils, and 7 is the trap answer, since it is the number of pencils rather than notebooks.
4.A pump moves water at a constant rate of 3.6 liters per minute. How many liters of water does it move in 2.5 hours?
- A. 9
- B. 90
- C. 540
- D. 900
Answer: C. 540
The rate is per minute, so convert the time first: 2.5 hours is 150 minutes. Then 3.6 liters per minute times 150 minutes is 540 liters. Multiplying 3.6 by 2.5 gives 9, which mixes the units and answers a different question.
5.Which expression is equivalent to (x + 4)^2 - (x - 4)^2?
- A. 16x
- B. 16
- C. 8x
- D. 2x^2 + 32
Answer: A. 16x
Expand both squares: (x^2 + 8x + 16) - (x^2 - 8x + 16). The x^2 terms and the 16s cancel, and 8x - (-8x) leaves 16x. If you are unsure, test a value: with x = 2 the original is 36 - 4 = 32, and 16x gives 32.
6.Complete the sentence.
The line 4x + 3y = 24 crosses the x-axis at (_____, 0) and crosses the y-axis at (0, _____).
Options for blank 1: 3, 6, 8, 24
Options for blank 2: 4, 6, 8, 24
Answer: 6, 8
To find the x-intercept, set y = 0: 4x = 24, so x = 6. To find the y-intercept, set x = 0: 3y = 24, so y = 8. A common slip is to swap them, which gives 8 for the x-intercept, so say out loud which variable you are setting to zero.
7.Which of the following ordered pairs (x, y) are solutions to the system y > 2x - 1 and y ≤ -x + 5?
Select all that apply.
- A. (2, 3)
- B. (1, 4)
- C. (3, 1)
- D. (0, 0)
- E. (-1, 4)
Answer: B. (1, 4), D. (0, 0), E. (-1, 4)
A pair is a solution only if it satisfies both inequalities, so test each one in each. (1, 4) works because 4 > 1 and 4 ≤ 4, and the second inequality allows equality. (0, 0) and (-1, 4) pass both too. (2, 3) fails because the first inequality is strict and 3 > 3 is false, and (3, 1) fails it badly since 1 > 5 is false.
8.A data set has 7 values and a mean of 12. A new value of 20 is added to the set. What is the mean of the 8 values?
- A. 13
- B. 12.5
- C. 14
- D. 16
Answer: A. 13
Work with sums, not averages. Seven values with a mean of 12 add up to 84, and adding 20 makes 104. Divide by the new count: 104 / 8 = 13. Averaging 12 and 20 to get 16 ignores that the old values outnumber the new one.
9.The product of two consecutive positive integers is 156. What is the larger of the two integers?
- A. 12
- B. 13
- C. 14
- D. 78
Answer: B. 13
Call the smaller integer n, so n(n + 1) = 156 and n^2 + n - 156 = 0. This factors as (n - 12)(n + 13) = 0, and the positive solution is n = 12. The larger integer is 13, and 12 times 13 is 156. The answer 12 is the smaller integer, which the question did not ask for.
10.A collectible is worth 250 dollars today, and its value increases by 8 percent each year. Which function gives its value V, in dollars, t years from now?
- A. V = 250(0.08)^t
- B. V = 250(1.8)^t
- C. V = 250 + 0.08t
- D. V = 250(1.08)^t
Answer: D. V = 250(1.08)^t
A percent increase repeated every period is exponential, and the growth factor is 1 plus the rate: 1 + 0.08 = 1.08. At t = 0 the value is 250, and each year multiplies it by 1.08. The third option adds the same small amount each year, which would be linear growth, and 1.8 would mean 80 percent growth.
11.Complete the sentence.
In the xy-plane, the line y = x + 2 and the parabola y = x^2 intersect at two points. The x-coordinates of those points are _____ and _____.
Options for blank 1: -2, -1, 1, 2
Options for blank 2: -1, 1, 2, 4
Answer: -1, 2
Where the graphs meet, the y-values are equal, so x^2 = x + 2, which rearranges to x^2 - x - 2 = 0 and factors as (x - 2)(x + 1) = 0. The solutions are x = -1 and x = 2. On test day you can also graph both equations in Desmos and read off the intersections.
12.For the function f(x) = x^2 - 6x + 5, which of the following statements are true?
Select all that apply.
- A. The graph crosses the x-axis at x = 1 and x = 5.
- B. The vertex of the graph is at (-3, -4).
- C. The minimum value of f(x) is -4.
- D. The maximum value of f(x) is 5.
- E. The graph crosses the y-axis at (0, 5).
Answer: A. The graph crosses the x-axis at x = 1 and x = 5., C. The minimum value of f(x) is -4., E. The graph crosses the y-axis at (0, 5).
The function factors as (x - 1)(x - 5), so it crosses the x-axis at 1 and 5. The vertex sits halfway between them at x = 3, where f(3) = 9 - 18 + 5 = -4, so the vertex is (3, -4) and -4 is the minimum, since the parabola opens upward. The y-intercept is f(0) = 5. A parabola that opens upward has no maximum, and the vertex has x = 3, not -3.
13.In right triangle PQR, the right angle is at Q, angle P measures 30 degrees and PR = 18. What is the length of QR?
- A. 9√3
- B. 9
- C. 12
- D. 18
Answer: B. 9
PR is the hypotenuse and QR is the side opposite the 30 degree angle, so sin 30 degrees = QR / 18. Since sin 30 degrees is 1/2, QR = 9. This is the 30-60-90 pattern: the side opposite 30 degrees is half the hypotenuse, and the side opposite 60 degrees is that short side times the square root of 3.
14.Complete the sentence.
In the xy-plane, the circle (x - 3)^2 + (y + 2)^2 = 25 has its center at (3, _____) and a radius of _____.
Options for blank 1: -3, -2, 2, 3
Options for blank 2: 5, 10, 25, 50
Answer: -2, 5
A circle with center (h, k) and radius r has the equation (x - h)^2 + (y - k)^2 = r^2. The term (y + 2) is (y - (-2)), so k = -2, and the sign flips from what you see. The right side is r^2 = 25, so the radius is 5, not 25.
15.A line of best fit for a scatterplot of daily high temperature x, in degrees Fahrenheit, and cups of hot cocoa sold y at a stand is y = -2.4x + 230. Which of the following statements are supported by the model?
Select all that apply.
- A. For each 1 degree increase in temperature, the model predicts about 2.4 fewer cups sold.
- B. Higher temperatures cause fewer people to want hot cocoa.
- C. The model predicts about 134 cups sold on a 40 degree day.
- D. The model gives a reasonable prediction for a 100 degree day.
- E. Exactly 134 cups were sold on every 40 degree day.
- F. The model predicts 0 cups sold at a temperature of about 96 degrees.
Answer: A. For each 1 degree increase in temperature, the model predicts about 2.4 fewer cups sold., C. The model predicts about 134 cups sold on a 40 degree day., F. The model predicts 0 cups sold at a temperature of about 96 degrees.
The slope of -2.4 means 2.4 fewer cups per extra degree, and the line gives -2.4(40) + 230 = 134 at 40 degrees. Setting y = 0 gives x = 230 / 2.4, about 96 degrees. A line of best fit shows an association, not a cause, and it predicts an average, so it cannot say what happened every day. At 100 degrees it predicts -10 cups, which is impossible, so the model should not be pushed that far.
What's on the digital SAT Math section
The digital SAT Math section has 44 questions and 70 minutes, split into two modules of 35 minutes each. That is about 95 seconds per question. The test is section-adaptive: how you do on the first module decides whether the second module is harder or easier, and inside each module the questions run from easiest to hardest.
About three in four questions are four-choice multiple choice. The rest are student-produced response questions, where you type the answer instead of picking it, and some of those have more than one correct answer even though you enter only one. Roughly 30 percent of the questions are set in a real-world context, so expect word problems about prices, rates and data. A calculator is allowed on the whole section: either the Desmos calculator built into the Bluebook testing app or a handheld one that meets the College Board's calculator policy.
The questions fall into four content areas. Algebra has 13 to 15 questions: linear equations and inequalities, linear functions and systems of two linear equations. Advanced Math has 13 to 15: equivalent expressions, nonlinear equations and functions, and systems that include a curve. Problem-Solving and Data Analysis has 5 to 7: ratios, rates, percentages, statistics, probability and judging claims from studies. Geometry and Trigonometry has 5 to 7: area and volume, lines, angles and triangles, right-triangle trigonometry and circles.
How to use this sampler, and how to get the PDF
This sampler follows the real mix in miniature: 5 algebra questions, 5 advanced math, 3 on problem-solving and data analysis, and 2 on geometry and trigonometry. The real test has only four-choice and typed-answer questions, so two formats here are stand-ins. Fill in the blank replaces typing, with a dropdown for each blank, and select all that apply is an extra that trains you to check every statement instead of hunting for one right answer.
Set a timer for 24 minutes, which is 15 questions at 95 seconds each, keep a calculator open and work in one sitting. Then read the explanation for every question, including the ones you got right. Review by content area, not by single question: if you missed the circle equation and the right-triangle question, spend your next session on geometry and trigonometry. If you were right but slow, note which questions took more than two minutes.
You can also take this test on paper. Use the download icon at the top of the test or the Download test as PDF button on the score screen: the file has all 15 questions with their answer choices, and the answer key follows on its own page. The download needs a free StudyPDF account, and playing online is free. Be careful with SAT math PDFs from other places. Many come from the older paper test, which had a separate no-calculator section and no adaptive modules, so the format and the pacing do not match the digital test.
How StudyPDF builds full practice tests from your prep book
Fifteen questions can show you which areas are shaky. They cannot get you ready on their own, and a generic question bank has no idea which topics your class covered or which formulas you keep forgetting.
StudyPDF works from your own material. Upload your prep book, class notes or your teacher's slides, then use the New material form to build a quiz or a practice exam from those exact pages. Every question cites the page it came from, you can focus a set on one area such as systems of equations or circles, and you can track which concepts you keep missing.
You do not need a file to start. Name a topic instead, for example SAT quadratics or percent word problems, and StudyPDF sets up practice on it. Starting is free.
Written by the StudyPDF team. Last updated August 19, 2026.
Good to know
Is this SAT math practice test free?
Yes. All 15 questions, the score screen and every explanation are free to play in your browser. A free StudyPDF account is only needed to download the PDF or to generate more practice from your own study materials.
How many questions and how much time are on the SAT Math section?
The digital SAT Math section has 44 questions and 70 minutes, in two modules of 35 minutes each. The second module is harder or easier depending on how you did on the first, and within a module the questions go from easiest to hardest.
Can I use a calculator on the SAT Math section?
Yes, on the whole section. You can use the Desmos calculator built into the Bluebook testing app or a handheld calculator that meets the College Board's calculator policy, so check that policy before test day.
Are these real SAT questions?
No. The StudyPDF team wrote every question and explanation to follow the content areas of the digital SAT Math section. They are practice questions, not real exam questions, and StudyPDF is not affiliated with the College Board.