A kayak listed at $450 is on sale for 30% off. A 6% sales tax is added to the sale price. How much does the buyer pay?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.12; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
By the Fundamental Theorem of Arithmetic, 1,980 has exactly one prime factorization. Which is it?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.08; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
How many liters of pure water must be added to 12 liters of a 40% salt solution to dilute it to 30%?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.11; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
In the form a + bi, what is the quotient when 5 + 14i is divided by 3 − 2i?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination; study plan topic "Complex Numbers"
Show all 63 questions (59 more)
Number sense, continued
Two whole numbers have a least common multiple of 420 and a greatest common factor of 6. One of the numbers is 42. What is the other?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.09; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
Which number equals √((−6)²) + |2 − 9|?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.10; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
Which product of two irrational numbers is a rational number?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.04; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
Written as a fraction in lowest terms, what is the decimal 0.41666…, in which only the 6 repeats?American Board Mathematics Content Standards (preview), Domain 1 Number Sense, standard 1.1.02; Mathematics Exam Snapshot, content domain Number Sense, 12% of the examination
Algebra and functions 16
8x² − 10x − 3 factors over the integers as which product?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Factoring Polynomials"
A hardware store sells 3 hinges and 4 latches for $26.00, and 5 hinges and 2 latches for $20.00. What does one latch cost?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Solving Systems of Equations"
A sample holds 640 grams of a substance with a half-life of 5 days. How much of it is left after 20 days?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Exponential Functions"
The alternating series 20 − 10 + 5 − 5/2 + … goes on forever with the same ratio. What does it add up to?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Sequences and Series"
An outdoor amphitheater has 15 rows. The front row has 18 seats, and each row has 4 more seats than the row in front of it. How many seats are there in all?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Sequences and Series"
Given f(x) = 5 − 3x and g(x) = x² − 4x, what is f(g(−1))?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Compositions and Inverses of Functions"
In the expansion of (x + 3y)⁴, what is the coefficient of the x²y² term?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Binomial Expansion"
The inequality |2x − 5| ≤ 9 is true for exactly which values of x?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Absolute Values"
Line p has the equation 5x + 2y = 9. Which line is perpendicular to p and passes through (10, −3)?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Parallel and Perpendicular Lines"
The parabola y = −3x² + 24x − 41 has its highest point at which vertex?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Solving and Graphing Quadratic Equations"
Reduce (2x² − 8) ÷ (x² + 5x + 6) to lowest terms, for x ≠ −2 and x ≠ −3.American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Simplifying Rational Polynomials"
Simplify (2a⁻³b²)⁻² × 8a⁻⁴b for nonzero a and b.American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Laws of Integral Exponents"
What is the radius of the circle x² + y² − 8x + 18y − 24 = 0?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Conic Sections"
Which expression equals (125a⁹b⁻⁶)^(2/3) for positive a and b?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Laws of Fractional Exponents"
Which inequality gives every solution of 9 − 4(2x − 3) ≥ 5x − 18?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Solving Linear Equations and Inequalities"
Which real numbers are left out of the domain of r(x) = (x + 1) ÷ (x² − 5x − 14)?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Algebra and Functions, 26% of the examination; study plan topic "Relations and Functions"
Geometry and measurement 12
A conveyor belt moves at 90 centimeters per second. What is that speed in meters per minute?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A metal sphere of radius 6 cm is melted and recast, with no metal lost, as a cylinder of radius 4 cm. How tall is the cylinder?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A model of a statue is made of the same material at a scale of 1 : 5. The model weighs 2 kg. How much does the statue weigh?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A proof that the diagonals of parallelogram ABCD bisect each other at E shows triangles ABE and CDE congruent from two pairs of alternate interior angles and AB = CD. Which criterion does it use?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A regular polygon has 20 sides. By how many degrees does each interior angle exceed each exterior angle?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A rhombus has sides of 25 cm, and one of its diagonals is 14 cm long. How long is the other diagonal?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
A sprinkler at the center of a circular lawn of radius 15 m sweeps through an angle of 48°. How long is the curved edge of the sector it waters?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
An exterior angle at the base of an isosceles triangle measures 112°. What is the measure of the vertex angle?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
In triangle ABC, point D lies on side AB and point E on side AC, with DE parallel to BC. AD = 6, DB = 9 and DE = 8. How long is BC?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
Lines m and n are parallel. A transversal forms same-side interior angles of (7x − 12)° and (4x + 27)°. What is x?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
Point R(−8, 3) is turned 90° clockwise about the origin, and the image is then moved 6 units down. Where does it end?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
What is the length of the segment that joins (−2, −5) and (10, 30)?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Geometry and Measurement, 20% of the examination
Trigonometry 7
A triangle has sides of 7, 8 and 13. What is the measure of its largest angle?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
A wheel turns through 252°. What is that angle in radians?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
If tan θ = −8/15 and θ lies in the fourth quadrant, what is sin θ?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
On the interval [0, 2π), which values of x make √2 sin x + 1 = 0?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
What are the amplitude and the period of y = −4 sin(3x − π) + 1?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
What is the exact value of sec(3π/4)?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
Which expression equals cos⁴x − sin⁴x for every x?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Trigonometry, 11% of the examination
Probability, statistics and data analysis 8
A bakery box holds 9 plain bagels and 6 sesame bagels. Two bagels are taken at random, and the first is not put back. What is the probability of getting one of each kind?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
A carnival wheel has 16 equal sectors: 1 pays $24, 3 pay $8 each and the rest pay nothing. A spin costs $4. What is the expected net gain from one spin?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
A coach picks a captain, a co-captain and a manager, three different people, from a squad of 7. In how many ways can this be done?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
A least-squares line predicts a plant’s height in centimeters from its age in weeks: ŷ = 2.5x + 4. A 6-week-old plant is 17 cm tall. What is its residual?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
A survey of 80 shoppers finds that 50 used a coupon, 32 paid with cash and 20 did both. One of the shoppers who paid with cash is chosen at random. What is the probability that this shopper used a coupon?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
For the six values 3, 8, 8, 10, 14 and 29, how much larger is the mean than the median?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
Loaves from a bakery have weights that are normally distributed with mean 500 g and standard deviation 12 g. By the empirical rule, about what percent of loaves weigh between 476 g and 512 g?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
The quartiles of a set of quiz totals are Q1 = 41 and Q3 = 57. Under the 1.5 × IQR rule, what is the smallest whole-number total that counts as a high outlier?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Probability, Statistics, and Data Analysis, 12% of the examination
Linear algebra 4
Matrix A has rows [1, 2] and [3, −1]. Matrix B has rows [0, 4] and [5, −2]. What is the product AB?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Linear Algebra, 6% of the examination
The matrix with rows [2, 6] and [1, k] has no inverse for which value of k?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Linear Algebra, 6% of the examination
The vectors u = ⟨2, k, −3⟩ and v = ⟨4, 1, 6⟩ have a dot product of zero. What is k?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Linear Algebra, 6% of the examination
What is the determinant of the matrix with rows [4, −3] and [2, 5]?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Linear Algebra, 6% of the examination
Calculus 8
A particle’s position is s(t) = t³ − 9t² + 24t meters after t seconds. At which times is its velocity zero?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
A rancher has 84 meters of fence for a rectangular paddock beside a straight river, and the river side needs no fence. What is the largest area the paddock can have?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
At which x does f(x) = x³ − 6x² − 15x + 4 have a local minimum?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
The curve y = 6√x − 2 lies above the x-axis from x = 1 to x = 4. What is the area between the curve and the axis on that interval?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
If f(x) = x³ − 6x² + 4x, what is f′(3)?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
What is the derivative of (3x² + 1)⁴?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
What is the limit of (x² + 4x − 12) ÷ (x + 6) as x approaches −6?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
Which is an antiderivative of 6/x + e^(2x) for x > 0?American Board Mathematics Study Plan (preview), Mathematics Exam Snapshot, content domain Calculus, 13% of the examination
How this list was compiled: Our own questions, written to the Mathematics Exam Snapshot in the American Board’s Mathematics Study Plan preview, https://americanboard.org/wp-content/uploads/2025/06/Mathematics-Study-Plan-Preview.pdf, linked from https://americanboard.org/subjects/math/ and read 2026-10-12: 125 multiple-choice questions in 240 minutes, no calculator, in seven content domains (Number Sense 12%, Algebra and Functions 26%, Geometry and Measurement 20%, Trigonometry 11%, Probability, Statistics, and Data Analysis 12%, Linear Algebra 6%, Calculus 13%). This bank holds 8, 16, 12, 7, 8, 4 and 8. The Number Sense questions also follow the standards printed in the Mathematics Content Standards preview, https://americanboard.org/wp-content/uploads/2025/06/Mathematics-Standards-Preview.pdf (1.1.01 to 1.1.12), read 2026-10-12. Every answer was worked out a second time from the numbers in the question. Licence: Questions, choices and explanations are original work of this site.
How it works
This free ABCTE math practice test follows the American Board's mathematics exam: 63 questions of our own in its seven content domains, each worked out. Unofficial: not affiliated with or endorsed by the American Board for Certification of Teacher Excellence.
Read the full guide
What the exam covers
The board's Exam Snapshot lists seven domains: number sense 12%, algebra and functions 26%, geometry and measurement 20%, trigonometry 11%, probability, statistics and data analysis 12%, linear algebra 6% and calculus 13%. This bank holds 8, 16, 12, 7, 8, 4 and 8, the same shares.
Length and timing
The real exam is 125 multiple-choice questions in 240 minutes. Exam mode gives these 63 questions 121 minutes, the same pace.
What the questions follow
The board's free previews print its standards for number sense (1.1.01 to 1.1.12) and its topic lists for the first two domains. Questions there name the standard or topic; the others name only the domain.
Algebra and functions with geometry and measurement make up 46% of the exam.
The study plan says most topics are those of introductory college mathematics courses, with some more advanced topics included.
Questions and answers
What is a passing score on the ABCTE math exam?
The Exam Snapshot gives two marks on a scale from 0 to 500: Proficient at 272 and Distinguished at 340. The count of right answers is converted to that scale, so exam mode here shows a count and no verdict.
Can you use a calculator on the ABCTE math exam?
No. The board's study plan says use of a calculator is not permitted, and the Exam Snapshot lists no on-screen exhibits. Every question on this page can be worked by hand.
Where do you take the ABCTE math exam?
In person, at a Pearson VUE center, by the American Board's certification exams page. It says most subject exams are multiple choice and give results as soon as you finish.
What happens if you fail the ABCTE math exam?
You can take it again. The American Board's certification exams page, read October 12, 2026, says the first subject area exam is included in the program and a retake costs $195.