Free ASVAB Mathematics Knowledge Practice Test
The ASVAB Mathematics Knowledge Practice Test helps you review the math rules and skills used on the real ASVAB. This part of the test focuses on high school math rather than long word problems.
You may need to solve an equation, simplify an expression, work with exponents, compare numbers, or use a geometry formula.
ASVAB Mathematics Knowledge Practice Test
Algebra and Expressions
Simplify: (8x^2 + 3x – 5) – (3x^2 – 2x + 4)
Distribute the minus sign through the second polynomial, then combine like terms: 8x^2 – 3x^2 = 5x^2, 3x + 2x = 5x, and -5 – 4 = -9.
Equations and Inequalities
Solve for x: 5(x – 2) + 3 = 3x + 11
Expand to get 5x – 7 = 3x + 11. Subtract 3x and add 7: 2x = 18, so x = 9.
Exponents Roots and Scientific Notation
Simplify: (x^4)(x^7) / x^3
Add exponents when multiplying and subtract when dividing: 4 + 7 – 3 = 8.
Fractions Ratios and Percents
What is 5/6 – 1/4?
Use a common denominator of 12: 5/6 = 10/12 and 1/4 = 3/12. The difference is 7/12.
Number Properties and Operations
Evaluate: 30 – 4(3 + 2) + 18 / 3
Do parentheses first: 3 + 2 = 5. Then multiplication and division: 4 x 5 = 20 and 18 / 3 = 6. Finally, 30 – 20 + 6 = 16.
Geometry and Measurement
A rectangle has length 12 units and area 84 square units. What is its perimeter?
The width is 84 / 12 = 7. Perimeter = 2(12 + 7) = 38 units.
Coordinate Geometry and Functions
What is the slope of the line through (2, 3) and (6, 11)?
Slope = change in y divided by change in x = (11 – 3) / (6 – 2) = 8/4 = 2.
Data Probability and Applied Math
What is the median of 4, 7, 9, 12, 15, and 18?
There are six values, so average the two middle numbers: (9 + 12)/2 = 10.5.
Algebra and Expressions
Factor completely: 6a(a – 3) + 5(a – 3)
Both terms contain the common factor (a – 3). Factoring it out leaves 6a + 5.
Equations and Inequalities
Solve for x: (3/4)x + 5 = 14
Subtract 5 to get (3/4)x = 9. Multiply both sides by 4/3, giving x = 12.
Exponents Roots and Scientific Notation
Simplify: (2a^3)^2
Square both the coefficient and the power: 2^2 = 4 and (a^3)^2 = a^6.
Fractions Ratios and Percents
What is (3/5) divided by (9/10)?
Multiply by the reciprocal: (3/5)(10/9) = 30/45 = 2/3.
Number Properties and Operations
What is the least common multiple of 8, 12, and 18?
The prime factors require 2^3 and 3^2. Their product is 8 x 9 = 72.
Geometry and Measurement
What is the area of a triangle with base 14 units and height 9 units?
Triangle area is one-half times base times height: (1/2)(14)(9) = 63.
Coordinate Geometry and Functions
If f(x) = 3x – 4, what is f(5)?
Substitute 5 for x: f(5) = 3(5) – 4 = 15 – 4 = 11.
Data Probability and Applied Math
What is the mean of 8, 11, 13, 16, and 17?
Add the values to get 65, then divide by 5. The mean is 13.
Algebra and Expressions
Simplify: 4(2x – 3) – 3(x + 1)
Distribute first: 8x – 12 – 3x – 3. Combining like terms gives 5x – 15.
Equations and Inequalities
Solve for y: 2(3y + 1) = 4y + 18
Expand: 6y + 2 = 4y + 18. Subtract 4y and 2 to get 2y = 16, so y = 8.
Exponents Roots and Scientific Notation
What is the value of 5^-2?
A negative exponent means take the reciprocal: 5^-2 = 1 / 5^2 = 1/25.
Fractions Ratios and Percents
Which ratio is equivalent to 42:56?
Divide both terms by their greatest common factor, 14. That gives 3:4.
Number Properties and Operations
How many positive integers less than 80 are divisible by both 4 and 6?
A number divisible by both 4 and 6 must be divisible by their LCM, 12. The values below 80 are 12, 24, 36, 48, 60, and 72.
Geometry and Measurement
Using pi = 3.14, what is the circumference of a circle with radius 5 units?
Circumference = 2pi r = 2(3.14)(5) = 31.4 units.
Coordinate Geometry and Functions
Which equation has slope -2 and y-intercept 5?
Slope-intercept form is y = mx + b. Here m = -2 and b = 5.
Data Probability and Applied Math
A bag contains 5 red, 3 blue, and 2 green counters. What is the probability of selecting a blue counter at random?
There are 10 counters total and 3 are blue, so the probability is 3/10.
Algebra and Expressions
Evaluate 2m^2 – 3mn + n^2 when m = 3 and n = -2.
Substitute the values: 2(3^2) – 3(3)(-2) + (-2)^2 = 18 + 18 + 4 = 40.
Equations and Inequalities
If 7p – 5 = 3p + 19, what is the value of 2p?
First solve for p: 4p = 24, so p = 6. Then 2p = 12.
Exponents Roots and Scientific Notation
Evaluate: (-4)^3
Multiply -4 three times. Two negative factors make a positive 16, and 16 times -4 equals -64.
Fractions Ratios and Percents
What is 35% of 240?
Convert 35% to 0.35 and multiply: 0.35 x 240 = 84.
Number Properties and Operations
Which of the following is a prime number?
61 has no positive factors other than 1 and 61. The other choices are divisible by 3.
Geometry and Measurement
What is the volume of a rectangular prism that is 8 units long, 5 units wide, and 3 units high?
Volume of a rectangular prism is length x width x height: 8 x 5 x 3 = 120.
Coordinate Geometry and Functions
Which equation represents a vertical line through x = -3?
Every point on a vertical line has the same x-coordinate. A vertical line through -3 is written x = -3.
Data Probability and Applied Math
What is the simple interest on $500 at 4% per year for 2 years?
Simple interest = principal x rate x time = 500 x 0.04 x 2 = 40.
Algebra and Expressions
If x is not equal to 5, simplify (x^2 – 25) / (x – 5).
Factor the numerator as a difference of squares: x^2 – 25 = (x – 5)(x + 5). Cancel the common factor x – 5.
Equations and Inequalities
Which inequality is the solution to 4x – 7 < 13?
Add 7 to both sides to get 4x < 20. Divide by 4, giving x < 5.
Exponents Roots and Scientific Notation
Evaluate: sqrt(9^2 + 12^2)
Compute the powers first: 81 + 144 = 225. The square root of 225 is 15.
Fractions Ratios and Percents
63 is 70% of what number?
Write 63 = 0.70x. Dividing 63 by 0.70 gives x = 90.
Number Properties and Operations
Which number is irrational?
sqrt(2) cannot be written as a ratio of two integers. The other choices are rational numbers.
Geometry and Measurement
Two angles of a triangle measure 47 degrees and 68 degrees. What is the third angle?
The angles of a triangle total 180 degrees. 180 – 47 – 68 = 65 degrees.
Coordinate Geometry and Functions
For the line y = (1/2)x + 2, what is x when y = 8?
Substitute y = 8: 8 = (1/2)x + 2. Subtract 2 to get 6 = x/2, so x = 12.
Data Probability and Applied Math
Convert 2.7 kilometers to meters.
One kilometer equals 1,000 meters. Multiply 2.7 by 1,000 to get 2,700 meters.
Algebra and Expressions
Which expression is equivalent to 3(2y + 5) – 4y?
Distribute the 3 to get 6y + 15, then subtract 4y. The result is 2y + 15.
Equations and Inequalities
If x + y = 11 and x – y = 3, what is x?
Add the equations: 2x = 14. Therefore x = 7.
Exponents Roots and Scientific Notation
Write 7.2 x 10^-3 as a decimal.
A power of 10^-3 moves the decimal point three places to the left: 7.2 becomes 0.0072.
Fractions Ratios and Percents
Which value is greatest?
Convert each value to a decimal: 5/8 = 0.625, 0.61 = 0.610, 62% = 0.620, and 3/5 = 0.600. The greatest is 0.625.
Number Properties and Operations
Simplify: 7! / 5!
Cancel 5! from numerator and denominator: 7! / 5! = 7 x 6 = 42.
Geometry and Measurement
Two parallel lines are cut by a transversal. If one interior angle is 112 degrees, what is the same-side interior angle?
Same-side interior angles formed by parallel lines are supplementary. 180 – 112 = 68 degrees.
Coordinate Geometry and Functions
What is the midpoint of the segment joining (-2, 5) and (6, 1)?
Average the x-coordinates and y-coordinates: (-2 + 6)/2 = 2 and (5 + 1)/2 = 3.
Data Probability and Applied Math
A value increases from 120 to 150. What is the percent increase?
The increase is 30. Divide by the original value: 30/120 = 0.25, or 25%.
MATH SKILL
What Mathematics Knowledge Tests
Mathematics Knowledge, or MK, measures your understanding of high school math principles.
The questions focus on math rules, number skills, algebra, geometry, and related concepts.
Algebra
Solve equations, simplify expressions, and work with variables.
Geometry
Use area, perimeter, volume, angles, and shape rules.
Number Skills
Work with fractions, decimals, percentages, ratios, and signs.
Exponents
Use powers, roots, and common exponent rules correctly.
Order of Operations
Follow the correct order when several operations appear together.
Number Properties
Recognize factors, multiples, primes, absolute value, and number types.
Basic Data
Find the mean, median, and simple values from number sets.
Powers of 10
Work with scientific notation and place-value changes.
CURRENT TEST FORMAT
Mathematics Knowledge Test Format
The CAT-ASVAB and paper ASVAB use different question counts and time limits.
| Test Version | Questions | Time Limit | Approx. Time |
|---|---|---|---|
| CAT-ASVAB | Questions 15 scored | Time Limit 31 minutes | Approx. Time About 2 min 4 sec |
| Paper ASVAB | Questions 25 | Time Limit 24 minutes | Approx. Time About 58 sec |
The CAT-ASVAB may include unscored tryout questions. If tryout questions are added, extra time is provided for that subtest.
KNOW THE DIFFERENCE
Mathematics Knowledge Is Not the Same as
Arithmetic Reasoning
Both sections use math, but the type of thinking is different.
Mathematics Knowledge
Focuses on math rules and procedures. You may solve an equation, simplify an expression, work with exponents, or use a geometry formula.
Arithmetic Reasoning
Focuses on word problems. You first decide what the problem is asking, then choose the math needed to solve it.
CIRE TOPICS
Build Strong Algebra and Geometry Skills
These areas require you to know the rule before you calculate.
Algebra
Practice solving for a variable, combining like terms, working with negative signs, using parentheses, and simplifying expressions one step at a time.
Geometry
Know the difference between area, perimeter, and volume. Read what the question asks before choosing a formula.
Exponents and Roots
Review positive and negative exponents, powers of 10, common square roots, and the rules for multiplying or dividing powers.
Fractions and Ratios
Be ready to compare fractions, decimals, percentages, and ratios by changing them into a form that is easier to compare.
QUICK RULES
Know the Rule Before You Start the Math
A short problem can still go wrong if you use the wrong rule.
PEMDAS
Work through parentheses, exponents, multiplication or division, then addition or subtraction.
aᵐ × aⁿ
When multiplying powers with the same base, add the exponents.
√81 = 9
Know common perfect squares so basic root questions do not slow you down.
TEST RULES
You Cannot Use a Calculator on the ASVAB
Practice basic calculations by hand so the test does not feel different from your study sessions.
- Multiplication and division
- Fraction work
- Decimal movement
- Common perfect squares
- Simple percentage calculations
BEFORE YOUR CALCULATOR
Check What the Question Actually Wants
Many math mistakes happen because the student solves the wrong thing correctly.
If the question asks for perimeter, do not stop after finding area.
If it asks for x, make sure the value you choose is actually x and not another number from your work.
If it asks for the greatest or least value, compare every valid option before selecting an answer.
CHECK YOUR WORK
Use Estimation to Catch Bad Answers
A rough estimate can show whether your final answer is reasonable.
Estimate first
If you expect an answer near 50 and your exact calculation gives 500, stop and check the steps.
Use the answer choices
Choices can help you judge the size of the answer, but they should be used to check your work, not replace it.
AFQT AND TIMING
Why Mathematics Knowledge Deserves
Focused Practice
Mathematics Knowledge contributes to the AFQT, and the CAT version gives a little over two minutes per
scored question on average.
AFQT importance
The AFQT uses Arithmetic Reasoning, Mathematics Knowledge, Paragraph Comprehension, and Word Knowledge.
Practice the current pace
For CAT-style practice, try 15 Mathematics Knowledge questions in 31 minutes. Easy questions should leave more time for harder ones.
REVIEW MISTAKE
Find the Step Where Your Math Went
Wrong
Do not review only the correct answer. Identify the type of error so you know what to study next.
FAQs