FREE SAT MATH PRACTICE

Free SAT Math Practice Questions with Full Solutions

Work through SAT Math questions that test structure, reasoning, timing decisions, and common Digital SAT traps. Each question includes a full solution, a common mistake, and a method note.

Work through focused questions and review each solution. The diagnostic identifies weak domains; a full practice test checks accuracy and timing across mixed topics.

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Quick SAT Math recognition tips

SAT Math often rewards recognizing the structure of a question before calculating. These tips will help you use the free questions below more effectively.

Look for structure first. Factored quadratics, exponential forms, and similar solids often have a shortcut hidden in the setup.
Do not over-expand. Many SAT questions are faster if you use roots, shifts, signs, or scale factors instead of expanding everything.
Use Desmos selectively. Desmos is powerful, but not every question needs it. Sometimes the fastest route is algebraic recognition.
Watch the wording. Phrases like “one possible value,” “no solution,” and “similar solids” change the solving route.

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Algebra

Practice linear equations, systems, and the meaning of slopes and intercepts. After solving, check whether you translated the situation into the right equation.

SAT Math no solution linear equation find k

Linear Equations Hard Free Response Find the condition for a linear equation to have no solution
5(kx − n) = −6514x − 8518

In the given equation, k and n are constants and n > 1. The equation has no solution. What is the value of k?

Correct answer: −1314

Full solution Expand the left side: 5(kx − n) = 5kx − 5n. For a linear equation to have no solution, the coefficients of x must be equal but the constants must be different. Therefore, 5k = −6514. Divide both sides by 5: k = (−6514) ÷ 5 = −6570 = −1314. Since n > 1, the constant −5n is less than −5, which is not equal to −8518, so the constants are different.
Common mistake A common mistake is trying to solve for x. But the question says the equation has no solution, so the goal is to match x-coefficients and keep constants unequal.
Method note No solution in a linear equation means same slope or coefficient, different constant.
SAT note SAT note: This is a structure question. Do not over-solve; compare the two sides.
Desmos note Desmos is not needed.

Systems of linear equations practice question

Algebra Medium Multiple Choice Systems of linear equations
A school sells adult tickets for $12 and student tickets for $7. A total of 80 tickets brings in $760. How many adult tickets were sold?
A. 24
B. 32
C. 40
D. 56

Correct answer: Choice C

40

Full solution Let a be the number of adult tickets. Then 80 − a student tickets were sold. Solve 12a + 7(80 − a) = 760: 5a = 200, so a = 40.
Common mistake Dividing $760 by one ticket price ignores that the two ticket types have different prices.
Method note Name the requested quantity first, then express the other quantity as the total minus that number.

Linear equations with no solution practice question

Algebra Medium Multiple Choice Linear equations with no solution
For what value of k does the equation 3(2x − 5) = kx + 9 have no solution?
A. −6
B. 3
C. 6
D. 9

Correct answer: Choice C

6

Full solution Expand to 6x − 15 = kx + 9. When k = 6, the x terms cancel and the equation becomes −15 = 9, which is false. Therefore no x satisfies it.
Common mistake A zero coefficient after subtraction does not always mean infinitely many solutions; compare the constant terms.
Method note No solution occurs when the variable coefficients match but the constants do not.

Linear functions and slope practice question

Algebra Medium Multiple Choice Linear functions and slope
A line passes through (2, 7) and (8, 19). What is its y-intercept?
A. −3
B. 3
C. 7
D. 11

Correct answer: Choice B

3

Full solution The slope is (19 − 7)/(8 − 2) = 2. Write y = 2x + b and substitute (2, 7): 7 = 4 + b, so b = 3.
Common mistake The y-coordinate 7 is not the y-intercept because the given point has x = 2.
Method note Find the slope from two points, then substitute either point into slope-intercept form.

Advanced Math

Work with equivalent expressions, nonlinear functions, and their structure. Compare a direct algebraic route with a graph only when the graph helps answer the exact question.

SAT Math minimum value of shifted quadratic function

Quadratics and Functions Medium Multiple Choice Find the minimum value of a shifted quadratic function
f(x) = 3x² + 48x + 193

The function g is defined by g(x) = f(x + 7). What is the minimum value of g(x)?
A. −15
B. −8
C. 1
D. 8

Correct answer: Choice C

1

Full solution The transformation g(x) = f(x + 7) shifts the graph horizontally, but it does not change the minimum value. Find the minimum of f(x) = 3x² + 48x + 193. The vertex occurs at x = −b/(2a) = −48/(2·3) = −8. Then f(−8) = 3(64) + 48(−8) + 193 = 192 − 384 + 193 = 1. Therefore, the minimum value of g(x) is also 1.
Common mistake A common mistake is trying to substitute x + 7 into the whole expression and expand. That works, but it is slower and increases the chance of arithmetic errors.
Method note Horizontal shifts change where the minimum occurs, not the minimum value itself.
SAT note SAT note: Recognize function shifts quickly. The test often rewards avoiding unnecessary expansion.
Desmos note Can be solved using Desmos, but it can also be solved by finding the vertex and noticing that g(x) is only a horizontal transformation of f(x), so the minimum value stays the same. Desmos is more helpful here if your algebra foundation is weak or if remem

SAT Math free response factored quadratic sign pattern

Quadratics and Functions Hard Free Response Use signs of a factored quadratic to locate integer roots
f(x) = (x − a)(x − b)

The function f is defined by the given equation, where a and b are integer constants. If f(17) > 0, f(20) < 0, and f(23) > 0, what is one possible value of a + b?

Correct answer: 40

Full solution The expression (x − a)(x − b) changes sign between its two roots and is positive outside the roots. Since f(17) > 0, f(20) < 0, and f(23) > 0, one root must be between 17 and 20 and the other must be between 20 and 23. Because a and b are integers, possible roots are one from {18,19} and one from {21,22}. Possible sums are 18+21=39, 18+22=40, 19+21=40, or 19+22=41. A clean possible answer is 40.
Common mistake A common mistake is assuming the roots are exactly 17, 20, and 23. The signs only tell where the roots must lie.
Method note For factored quadratics with positive leading coefficient, the function is positive outside the roots and negative between them.
SAT note SAT note: Free-response questions may have multiple valid answers. Enter any accepted value.
Desmos note Desmos can help visualize the sign pattern, but interval reasoning is faster.

SAT Math cubic zeros find coefficient a

Polynomial Functions Medium Multiple Choice Use zeros of a monic cubic to identify the coefficient of x²
g(x) = x³ + ax² + bx + c

The function g is defined by the given equation, where a, b, and c are integer constants. The zeros of the function are −2, −7, and 6. What is the value of a?
A. −84
B. −3
C. 3
D. 84

Correct answer: Choice C

3

Full solution For a monic cubic with zeros r1, r2, and r3, the coefficient of x2 is the negative of the sum of the zeros. The zeros are −2, −7, and 6. Their sum is −2 + (−7) + 6 = −3. Therefore, a = −(−3) = 3.
Common mistake A common mistake is multiplying the roots and choosing 84 or −84. The product relates to the constant term, not the coefficient of x2.
Method note For x3 + ax2 + bx + c, use a = −(sum of zeros).
SAT note SAT note: This is much faster using root-sum structure than expanding the full polynomial.
Desmos note Desmos is not needed.

SAT Math exponential function displays 1 over c coefficient of x

Exponential Functions Hard Multiple Choice Connect exponential growth factor to equivalent exponent forms
For the function f, for each increase in the value of x by c, where c is a positive constant, the value of f(x) increases by a factor of 27. Which of the following equivalent forms of the function f displays 1c as a coefficient of x?
A. f(x) = 35(3)(12)x
B. f(x) = 35(33)(16)x
C. f(x) = 35(9)(14)x
D. f(x) = 35(27(13)x)12

Correct answer: Choice B

f(x) = 35(33)(16)x

Full solution If increasing x by c multiplies f(x) by 27, then a natural form is f(x) = 35(27)(1c)x. Since 27 = 33, this can be written as 35(33)(1c)x. Choice B is 35(33)(16)x, so it displays 1c as the coefficient of x, with c = 6.
Common mistake A common mistake is choosing an equivalent-looking expression without checking whether the displayed coefficient of x is actually 1c.
Method note Look for the form where the base represents the full growth factor and the exponent contains (1c)x.
SAT note SAT note: This is an exponent-structure question. The wording is more important than computation.
Desmos note Can be solved by Desmos, but it usually takes more time. The better route is to understand the structure of exponential growth questions because this pattern is common on the SAT.

SAT Math exponential function passes through two points find b

Exponential Functions Hard Multiple Choice Use substitution in an exponential function with unknown constants
The function f is defined by f(x) = ax − b, where a and b are constants. In the xy-plane, the graph of y = f(x) passes through the points (c, 5) and (2c, 137), where c is a constant. Which of the following could be the value of b?
A. 7
B. 16
C. 132
D. 142

Correct answer: Choice A

7

Full solution Let ac = t. From the point (c,5), we get t − b = 5, so t = b + 5. From the point (2c,137), we get a2c − b = 137. Since a2c = (ac)2 = t2, we have t2 − b = 137. Substitute b = t − 5: t2 − (t − 5) = 137, so t2 − t − 132 = 0. Factor: (t − 12)(t + 11) = 0. Since t = ac is positive, t = 12. Then b = t − 5 = 7.
Common mistake A common mistake is treating a2c as 2ac. It should be (ac)2.
Method note Use substitution when the exponents are related. Letting ac = t turns the problem into a quadratic.
SAT note SAT note: This is a strong example of structure over brute force.
Desmos note Desmos is not useful unless values are already known. Algebraic substitution is the clean route.

Equivalent expressions practice question

Advanced Math Medium Multiple Choice Equivalent expressions
For x ≠ 3, which expression is equivalent to (x² − 9)/(x − 3)?
A. x − 3
B. x + 3
C. x² + 3
D. x + 6

Correct answer: Choice B

x + 3

Full solution Factor the numerator as (x − 3)(x + 3). Cancel x − 3 only for x ≠ 3, leaving x + 3.
Common mistake Canceling the 3 terms separately across addition or subtraction is not a valid algebraic step.
Method note Factor before canceling, and keep the original domain restriction visible.

Problem Solving and Data Analysis

Interpret rates, percentages, probability, and data in context. State what each number measures before calculating or drawing a conclusion from a sample.

One-variable data and spread practice question

Problem Solving and Data Analysis Medium Multiple Choice One-variable data and spread
Data set A is 8, 10, 12. Data set B is 2, 10, 18. Both sets have mean 10. Which statement about their standard deviations is true?
A. A has the larger standard deviation.
B. B has the larger standard deviation.
C. They have the same standard deviation.
D. The standard deviations cannot be compared without a formula.

Correct answer: Choice B

B has the larger standard deviation.

Full solution Both sets center at 10. The values in B are 8 units from 10 at the extremes, while the values in A are only 2 units from 10. B is more spread out, so its standard deviation is larger.
Common mistake Equal means do not imply equal spread. Compare distances from the common mean.
Method note This comparison needs an understanding of spread rather than a full calculation.

Two-variable data and regression practice question

Problem Solving and Data Analysis Medium Multiple Choice Two-variable data and regression
A line of best fit for a data set is y = 4.5x + 12, where x is the number of weeks after a campaign starts and y is the predicted number of sign-ups. What does 4.5 represent?
A. 12 predicted sign-ups each week
B. An increase of about 4.5 predicted sign-ups for each additional week
C. 4.5 sign-ups before the campaign starts
D. A total of 4.5 sign-ups after 12 weeks

Correct answer: Choice B

An increase of about 4.5 predicted sign-ups for each additional week

Full solution The coefficient of x is the slope. For a one-week increase in x, the model predicts an increase of 4.5 in y, measured in sign-ups.
Common mistake The intercept 12 describes the prediction at week 0; it is not the weekly change.
Method note Attach the units of y per unit of x to a regression slope.

Percentages and compounding practice question

Problem Solving and Data Analysis Medium Multiple Choice Percentages and compounding
A quantity increases by 20% and then decreases by 20%. The final value is what percent of the original value?
A. 80%
B. 96%
C. 100%
D. 104%

Correct answer: Choice B

96%

Full solution Start with an original value of 100. After a 20% increase it is 120; a 20% decrease of 120 is 24, leaving 96. Equivalently, 1.20 × 0.80 = 0.96.
Common mistake The equal percentage changes do not cancel because the second percentage uses a different base.
Method note Multiply change factors when percentages occur in sequence.

Conditional probability practice question

Problem Solving and Data Analysis Medium Multiple Choice Conditional probability
A club has 30 members: 18 play chess, and 6 of those 18 also play tennis. If a chess player is selected at random, what is the probability that the player also plays tennis?
A. 15
B. 13
C. 25
D. 35

Correct answer: Choice B

13

Full solution The condition restricts the sample space to the 18 chess players. Of those, 6 also play tennis, so the probability is 618 = 13.
Common mistake Using 30 as the denominator ignores the condition that the selected member plays chess.
Method note After the word “if,” identify the restricted group before calculating.

Ratios and unit rates practice question

Problem Solving and Data Analysis Medium Multiple Choice Ratios and unit rates
A machine produces 45 parts in 6 minutes at a constant rate. At that rate, how many parts does it produce in 14 minutes?
A. 90
B. 100
C. 105
D. 120

Correct answer: Choice C

105

Full solution The unit rate is 456 = 7.5 parts per minute. Multiply by 14 minutes: 7.5 × 14 = 105 parts.
Common mistake Multiplying 45 by 14 without dividing by 6 treats a six-minute output as a one-minute output.
Method note Carry the units through the rate calculation.

Statistical inference from samples practice question

Problem Solving and Data Analysis Medium Multiple Choice Statistical inference from samples
A random sample of 200 voters finds that 60% support a proposal. Which conclusion is justified by the sample alone?
A. Exactly 60% of all voters support the proposal.
B. The sample provides an estimate of support among all voters.
C. Every group of 200 voters would give the same result.
D. The proposal is certain to pass.

Correct answer: Choice B

The sample provides an estimate of support among all voters.

Full solution A random sample can estimate a population proportion, but sampling variation means the population value need not equal the sample result exactly.
Common mistake A sample percentage is an estimate, not an exact census or a guarantee about an outcome.
Method note Distinguish the observed sample statistic from the unknown population proportion.

Geometry and Trigonometry

Sketch and label the given relationships before using a formula. Check which lengths or angles the question actually asks for.

SAT Math right triangle area question with radicals

Geometry and Measurement Medium Multiple Choice Identify the legs of a right triangle and calculate area
A right triangle has sides of length 3√2, 5√2, and √68 units. What is the area of the triangle, in square units?
A. 8√2 + √68
B. 30√68
C. 30
D. 15

Correct answer: Choice D

15

Full solution Because the triangle is right, the longest side is the hypotenuse. Compare the side lengths: √68 is the largest. Check the legs: (3√2)2 + (5√2)2 = 18 + 50 = 68, so the legs are 3√2 and 5√2. The area is 12 × 3√2 × 5√2 = 12 × 15 × 2 = 15.
Common mistake A common mistake is multiplying the hypotenuse by one of the legs. For a right triangle, area uses the two perpendicular legs, not the hypotenuse.
Method note First identify the hypotenuse by finding the longest side. Then use Area = 12 × leg × leg.
SAT note SAT note: This is a route-choice question. A fast check is to square the sides and confirm which two sides form the legs.
Desmos note Desmos is not needed. Mental math and the Pythagorean relationship are faster.

SAT Math unit circle cos p over sin p free response

Geometry and Trigonometry Hard Free Response Use unit circle coordinates to evaluate a trigonometric ratio
In the xy-plane, a unit circle with center at the origin O contains point A with coordinates (1, 0) and point B with coordinates (5√34, −3√34). If the measure of angle AOB is p radians, what is the value of cos psin p?

Correct answer: 53

Full solution Point A = (1,0) lies on the positive x-axis. Point B has coordinates (5√34, −3√34). The angle p is the angle between the ray to A and the ray to B. The cosine of this angle is based on the horizontal component, so cos p = 5/√34. The sine of the angle measure is the positive magnitude of the vertical change from the positive x-axis, so sin p = 3/√34. Therefore, cos psin p = (5√34)/(3√34) = 5/3.
Common mistake A common mistake is using the negative y-coordinate directly and getting −5/3. The question asks for the measure of angle AOB, so the angle measure is positive.
Method note Use the unit circle coordinates as cosine and sine magnitudes for the angle from the positive x-axis.
SAT note SAT note: This is mainly a coordinate-trigonometry recognition question, not a long trigonometry problem.
Desmos note Desmos is not needed.

SAT Math quadrilateral diagonal length radical free response

Advanced Geometry Hard Free Response Use intersecting diagonals and distance relationships in a kite-like quadrilateral
LMNK
Note: Figure not drawn to scale.

In quadrilateral KLMN shown, KL = 3, LM = 3, KN = 27, and MN = 27. Diagonals KM and LN intersect at point G, where GK = 1 and GM = 1. If the length of diagonal LN is √p + √w, where p and w are integers, what is the value of p + w?

Correct answer: 736

Full solution Since GK = 1 and GM = 1, point G is the midpoint of diagonal KM, so KM = 2. Also, KL = LM = 3, so point L is equidistant from K and M. Similarly, KN = MN = 27, so point N is also equidistant from K and M. Therefore, both L and N lie on the perpendicular bisector of KM, which passes through G. In right triangle KGL, KL = 3 and GK = 1, so GL = √(3² − 1²) = √8. In right triangle KGN, KN = 27 and GK = 1, so GN = √(27² − 1²) = √728. Thus LN = √8 + √728, so p + w = 8 + 728 = 736.
Common mistake A common mistake is assuming the diagram is drawn to scale or trying to estimate the diagonal visually. The note says the figure is not drawn to scale.
Method note Use equal side lengths to identify a perpendicular bisector relationship.
SAT note SAT note: This is a high-value geometry reasoning question. Draw the diagonals and use right triangles.
Desmos note Desmos is not needed.

SAT Math similar cylinders volume and surface area scale factor

Geometry and Measurement Hard Free Response Use similarity scale factors for volume and surface area
The table shows the volume of two similar solids, right circular cylinder A and right circular cylinder B.

Volume (cubic units)
Right circular cylinder A32π
Right circular cylinder B864π

The radius of right circular cylinder A is 2 units. The surface area of right circular cylinder A is kπ square units, and the surface area of right circular cylinder B is nπ square units, where k and n are constants. What is the value of n − k? The surface area of a right circular cylinder with radius r and height h is 2πr2 + 2πrh.

Correct answer: 320

Full solution Cylinder A has volume 32π and radius 2. Since V = πr2h, we get 32π = π(22)h, so h = 8. The surface area of A is 2π(22) + 2π(2)(8) = 8π + 32π = 40π, so k = 40. The volume ratio from A to B is 864π32π = 27. For similar solids, volume scale factor is the cube of the linear scale factor, so the linear scale factor is 3. Surface area scales by the square of the linear scale factor, so the surface area scale factor is 9. Therefore, cylinder B has surface area 9 × 40π = 360π, so n = 360. Thus n − k = 360 − 40 = 320.
Common mistake A common mistake is multiplying surface area by 27. Volume scales cubically, but surface area scales by the square of the linear scale factor.
Method note For similar solids: linear scale factor affects length, its square affects area, and its cube affects volume.
SAT note SAT note: This is a similarity scaling question hidden inside cylinder geometry.
Desmos note Desmos is not needed. In fact, it can make this question slower. Use similarity ratios: volume scales by the cube of the linear scale factor, and surface area scales by the square of the linear scale factor.

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