40 Mathematics Knowledge (MK) Practice Questions & Answers
Every Mathematics Knowledge (MK) practice question from the ASVAB Practice Test, with the correct answer and a short explanation.
Start practice test →1. Evaluate: −3² + (−4)(−5) − 12 ÷ 4
- A.32
- B.11
- C.8✓ Answer
- D.26
Under the order of operations (PEMDAS) the exponent applies only to the 3 because there are no parentheses, so −3² = −9, not +9. Then multiplication and division are done left to right: (−4)(−5) = 20 and 12 ÷ 4 = 3, giving −9 + 20 − 3 = 8. Choosing 26 means the negative sign was squared along with the 3.
Ref QB-JTF7S8Source: ASVAB CEP Overview p.19, MK content area "number theory"; order of operations (PEMDAS) and sign rules for multiplication of two negativesReport a problem with this question
2. Which of the following numbers is prime?
- A.91
- B.87
- C.51
- D.61✓ Answer
A prime number has exactly two distinct positive factors, 1 and itself. 51 = 3 × 17, 87 = 3 × 29 and 91 = 7 × 13 are composite (the digit sums of 51 and 87 are divisible by 3). 61 has no factor of 2, 3, 5 or 7, and since 8² = 64 > 61 it is enough to test primes up to √61, so 61 is prime.
Ref QB-ZAXPZ6Source: ASVAB CEP Overview p.19, MK content area "number theory"; definition of prime vs. composite and the divisibility-by-3 digit-sum ruleReport a problem with this question
3. Simplify: √(48/3)
- A.4√3
- B.8
- C.16
- D.4✓ Answer
The order of operations requires the division inside the radical to be done first: 48 ÷ 3 = 16, and √16 = 4. The distractor 4√3 comes from simplifying √48 while ignoring the division, and 16 is the value one step before taking the root — no calculator is needed if the division is done first.
Ref QB-HT4VK7Source: ASVAB CEP Overview p.19, MK content area "numeration (radicals)"; DoD/OPA no-calculator policy note on order of operations inside a radicalReport a problem with this question
4. Simplify: (2x³)⁴
- A.8x¹²
- B.16x⁷
- C.2x¹²
- D.16x¹²✓ Answer
The power-of-a-product rule raises every factor inside the parentheses to the outside power: 2⁴ = 16 and (x³)⁴ = x^(3·4) = x¹², so the result is 16x¹². Adding the exponents (x⁷) or leaving the coefficient un-raised (2x¹²) are the standard errors.
Ref QB-YAKCU5Source: ASVAB CEP Overview p.19, MK content area "numeration (exponents)"; laws of exponents (ab)ⁿ = aⁿbⁿ and (xᵃ)ᵇ = x^(ab)Report a problem with this question
5. (4 × 10⁵)(2 × 10⁻³) =
- A.8 × 10⁸
- B.8 × 10²✓ Answer
- C.6 × 10²
- D.8 × 10⁻¹⁵
When multiplying numbers in scientific notation, multiply the coefficients and add the exponents of 10: 4 × 2 = 8 and 5 + (−3) = 2, giving 8 × 10². Adding the coefficients produces 6, and multiplying the exponents produces the impossible 10⁻¹⁵.
Ref QB-GTRHE6Source: ASVAB CEP Overview p.19, MK content area "numeration (scientific notation)"; product rule for exponents xᵃ·xᵇ = x^(a+b)Report a problem with this question
6. If 30% of x is 42, then x =
- A.12.6
- B.70
- C.140✓ Answer
- D.126
The percent relationship is part/whole = percent/100, so 0.30x = 42 and x = 42 ÷ 0.30 = 140. The distractor 12.6 is 30% of 42, which reverses the relationship and treats 42 as the whole instead of the part.
Ref QB-XA82P4Source: ASVAB CEP Overview p.19, MK content area "numeration (percents)"; percent proportion part/whole = percent/100Report a problem with this question
7. 2/3 ÷ 4/9 =
- A.2/3
- B.8/27
- C.3/2✓ Answer
- D.27/8
Dividing by a fraction is the same as multiplying by its reciprocal: 2/3 × 9/4 = 18/12, which reduces to 3/2. The distractor 8/27 comes from multiplying straight across without inverting the second fraction.
Ref QB-FTD795Source: ASVAB CEP Overview p.19, MK content area "numeration (fractions)"; division of fractions by multiplying by the reciprocalReport a problem with this question
8. Solve for x: 5x − 7 = 3x + 9
- A.x = 16
- B.x = 8✓ Answer
- C.x = −8
- D.x = 1
Isolate the variable with inverse operations: subtracting 3x from both sides gives 2x − 7 = 9, adding 7 gives 2x = 16, and dividing by 2 gives x = 8. The distractor 16 is the value one step before the final division — a classic trap when rushing.
Ref QB-6BQCVCSource: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; solving one-variable linear equations by inverse operationsReport a problem with this question
9. Solve the inequality: −4x + 3 > 19
- A.x > 4
- B.x < −4✓ Answer
- C.x < 4
- D.x > −4
Subtracting 3 from both sides gives −4x > 16. Dividing both sides by a negative number reverses the direction of the inequality, so x < −4. Forgetting to flip the sign is one of the most common MK errors and produces the distractor x > −4.
Ref QB-PTVHFDSource: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; inequality rule — reverse the sign when multiplying or dividing by a negativeReport a problem with this question
10. For 3a − 5b ≠ 0, simplify: (9a² − 25b²)/(3a − 5b)
- A.9a² + 25b²
- B.3a² + 5b²
- C.3a + 5b✓ Answer
- D.3a − 5b
The numerator is a difference of squares: 9a² − 25b² = (3a + 5b)(3a − 5b). Because 3a − 5b ≠ 0, that common factor may be cancelled, leaving 3a + 5b. The stated restriction is exactly what makes the cancellation legal — division by zero would otherwise be undefined.
Ref QB-EDSKFUSource: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; difference-of-squares factoring a² − b² = (a + b)(a − b)Report a problem with this question
11. (2x − 3)(x + 5) =
- A.2x² + 7x + 15
- B.2x² + 7x − 15✓ Answer
- C.2x² + 13x − 15
- D.2x² − 7x − 15
Using FOIL: 2x·x = 2x², 2x·5 = 10x, −3·x = −3x, and −3·5 = −15. Combining like terms gives 10x − 3x = 7x, so the product is 2x² + 7x − 15. The constant is negative because a negative times a positive is negative.
Ref QB-WVKEVTSource: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; distributive property applied as FOIL for binomial productsReport a problem with this question
12. What are the solutions of x² − 7x + 12 = 0?
- A.1 and 12
- B.3 and 4✓ Answer
- C.2 and 6
- D.−3 and −4
Factor by finding two numbers whose product is +12 and whose sum is −7: those are −3 and −4, so x² − 7x + 12 = (x − 3)(x − 4). The zero-product property then gives x = 3 or x = 4. The factors carry minus signs, but the roots themselves are positive.
Ref QB-BM6AM2Source: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; factoring trinomials and the zero-product propertyReport a problem with this question
13. If 2x + y = 11 and x − y = 1, then x =
- A.3
- B.7
- C.5
- D.4✓ Answer
Adding the two equations eliminates y because +y and −y cancel: 3x = 12, so x = 4 (substituting back gives y = 3). The distractor 3 is the value of y — misreading which variable is asked for is a frequent MK mistake.
Ref QB-T4Z422Source: ASVAB CEP Overview p.19, MK content area "algebraic operations and equations"; solving a system of two linear equations by eliminationReport a problem with this question
14. What is the slope of the line through (−2, 5) and (4, −7)?
- A.−2✓ Answer
- B.2
- C.1/2
- D.−1/2
Slope is rise over run: m = (y₂ − y₁)/(x₂ − x₁) = (−7 − 5)/(4 − (−2)) = −12/6 = −2. The slope is negative because y decreases as x increases; the distractor −1/2 comes from inverting rise and run.
Ref QB-CD3X4SSource: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; coordinate-geometry slope formula m = (y₂ − y₁)/(x₂ − x₁)Report a problem with this question
15. A square has a perimeter of 36 cm. What is its area?
- A.324 cm²
- B.18 cm²
- C.36 cm²
- D.81 cm²✓ Answer
A square's perimeter is P = 4s, so s = 36 ÷ 4 = 9 cm, and its area is A = s² = 9² = 81 cm². Area uses squared units; confusing perimeter with area (or squaring half the perimeter, which gives 324) is a common source of error.
Ref QB-UVVRHRSource: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; square perimeter P = 4s and area A = s²Report a problem with this question
16. A circle has a circumference of 12π. What is its area?
- A.12π
- B.6π
- C.36π✓ Answer
- D.144π
From C = 2πr, 12π = 2πr gives r = 6. Then A = πr² = π(6²) = 36π. The distractor 144π comes from using 12 as the radius instead of first solving for r, and 6π mistakes the radius itself for the area.
Ref QB-DDE9ATSource: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; circle formulas C = 2πr and A = πr²Report a problem with this question
17. What is the volume of a cylinder with radius 3 and height 10?
- A.30π
- B.90π✓ Answer
- C.180π
- D.900π
A cylinder's volume is the base area times the height: V = πr²h = π(3²)(10) = 90π. The distractor 30π omits squaring the radius, and 180π uses 2rh (the lateral-surface pattern) instead of r².
Ref QB-VV84QSSource: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; cylinder volume V = πr²hReport a problem with this question
18. In a right triangle the hypotenuse is 13 and one leg is 5. How long is the other leg?
- A.12✓ Answer
- B.18
- C.8
- D.√194
The Pythagorean theorem states a² + b² = c², where c is the hypotenuse, so b² = 13² − 5² = 169 − 25 = 144 and b = 12 — the familiar 5-12-13 triple. The distractor √194 comes from adding the squares instead of subtracting, and 8 from subtracting the sides directly.
Ref QB-EM8A56Source: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; Pythagorean theorem a² + b² = c² and the 5-12-13 tripleReport a problem with this question
19. The interior angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?
- A.60°
- B.80°✓ Answer
- C.90°
- D.40°
The interior angles of any triangle sum to 180°, so 2x + 3x + 4x = 180, giving 9x = 180 and x = 20°. The largest angle is 4x = 80°. Note the triangle is acute, so 90° cannot be correct.
Ref QB-W435J5Source: ASVAB CEP Overview p.19, MK content area "geometry and measurement"; triangle interior-angle sum = 180°Report a problem with this question
20. A bag contains 4 red marbles and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?
- A.2/15✓ Answer
- B.3/25
- C.4/25
- D.2/5
Probability is favorable outcomes over total outcomes, and because the draws are without replacement they are dependent: the first draw is 4/10 and the second is 3/9, since both the red count and the total drop by one. Multiplying gives 12/90 = 2/15. The distractor 4/25 assumes replacement, (4/10)².
Ref QB-K5R3NPSource: ASVAB CEP Overview p.19, MK content area "probability"; multiplication rule for dependent events without replacementReport a problem with this question
21. Solve for x: 4(x - 3) = 2x + 10
- A.x = -1
- B.x = 22
- C.x = 11✓ Answer
- D.x = 13/2
Distribute the 4 to BOTH terms inside the parentheses: 4x - 12 = 2x + 10. Subtracting 2x and adding 12 to both sides gives 2x = 22, so x = 11. Choosing 13/2 comes from distributing only to the x (4x - 3), and -1 comes from a sign error when moving the -12.
Ref QB-4NW88QSource: Distributive property and properties of equality (high-school algebra, ASVAB MK domain per officialasvab.com)Report a problem with this question
22. Solve the inequality: -5x + 2 ≤ 17
- A.x ≥ -3✓ Answer
- B.x ≤ -3
- C.x ≥ 3
- D.x ≤ 3
Subtract 2 from both sides to get -5x ≤ 15, then divide by -5. Dividing (or multiplying) an inequality by a negative number reverses the inequality sign, so x ≥ -3. Answer x ≤ -3 is the classic mistake of forgetting to flip the sign.
Ref QB-M54ETQSource: Multiplication/division property of inequalities: the sign reverses when multiplying or dividing by a negativeReport a problem with this question
23. If a = -2 and b = 3, what is the value of 2a² - ab?
- A.2
- B.-2
- C.22
- D.14✓ Answer
Substitute carefully: a² = (-2)² = 4, so 2a² = 8; and -ab = -(-2)(3) = +6. The total is 8 + 6 = 14. Getting 2 comes from dropping the sign on -ab (8 - 6); -2 comes from treating a² as -4; 22 comes from squaring 2a instead of a.
Ref QB-5N9KDRSource: Evaluating algebraic expressions by substitution; rules for signed-number multiplicationReport a problem with this question
24. If 3x + 2y = 12 and y = x + 1, what is the value of y?
- A.1
- B.3✓ Answer
- C.5
- D.2
Substitute y = x + 1 into the first equation: 3x + 2(x + 1) = 12, so 5x + 2 = 12 and x = 2. Then y = x + 1 = 3. Answering 2 confuses x for y — the question asks for y, a deliberate trap on substitution problems.
Ref QB-H52EAMSource: Solving systems of linear equations by substitution (high-school algebra)Report a problem with this question
25. Simplify: 3x² + 5x - 2 - (x² - 4x + 6)
- A.2x² + x + 4
- B.2x² + 9x - 8✓ Answer
- C.2x² + 9x + 4
- D.2x² + x - 8
The minus sign distributes to EVERY term in the parentheses: 3x² - x² = 2x², 5x - (-4x) = 9x, and -2 - 6 = -8, giving 2x² + 9x - 8. The distractors come from applying the minus only to the first term or forgetting it on the middle or last term.
Ref QB-2N7KVMSource: Distributing a negative over parentheses and combining like terms (polynomial subtraction)Report a problem with this question
26. (3x + 4)(2x - 5) =
- A.6x² + 7x - 20
- B.6x² - 7x - 20✓ Answer
- C.6x² - 23x - 20
- D.6x² - 20
Use FOIL: First 6x², Outer -15x, Inner +8x, Last -20. The middle terms combine to -15x + 8x = -7x, giving 6x² - 7x - 20. Choosing 6x² - 20 multiplies only First and Last; -23x comes from adding -15 and -8 instead of -15 and +8.
Ref QB-J5DRFNSource: Binomial multiplication (FOIL): (a+b)(c+d) = ac + ad + bc + bdReport a problem with this question
27. What are the solutions of 2x² + 5x - 3 = 0?
- A.x = 1/2 or x = -3✓ Answer
- B.x = 1 or x = -6
- C.x = -1/2 or x = 3
- D.x = 1/2 or x = 3
By the quadratic formula, x = (-5 ± √(25 + 24))/(2·2) = (-5 ± 7)/4, giving x = 2/4 = 1/2 or x = -12/4 = -3. Equivalently, the trinomial factors as (2x - 1)(x + 3) = 0. Dividing by 2 instead of 2a = 4 gives the wrong pair 1 and -6; flipping both signs gives -1/2 and 3.
Ref QB-3NJW2PSource: Quadratic formula: x = (-b ± √(b² - 4ac))/(2a)Report a problem with this question
28. Factor completely: 4y² - 49
- A.(4y - 7)(y + 7)
- B.(2y - 7)(2y - 7)
- C.It cannot be factored
- D.(2y - 7)(2y + 7)✓ Answer
This is a difference of squares: 4y² = (2y)² and 49 = 7², so a² - b² = (a - b)(a + b) gives (2y - 7)(2y + 7). (2y - 7)² would expand to 4y² - 28y + 49, and (4y - 7)(y + 7) produces an unwanted middle term. A binomial with no middle term is often mistakenly called unfactorable — the difference-of-squares pattern is exactly the disguised case the test likes.
Ref QB-F5BRXHSource: Difference of squares pattern: a² - b² = (a - b)(a + b)Report a problem with this question
29. Simplify: (x⁵ · x³) ÷ x⁴
- A.x⁴✓ Answer
- B.x¹²
- C.x¹¹
- D.x²
Multiplying same-base powers ADDS exponents (5 + 3 = 8) and dividing SUBTRACTS them (8 - 4 = 4), so the result is x⁴. x² comes from dividing the exponents (8 ÷ 4), x¹¹ from multiplying 5 × 3 first, and x¹² from adding all three exponents.
Ref QB-YMGWHJSource: Laws of exponents: xᵃ·xᵇ = xᵃ⁺ᵇ and xᵃ/xᵇ = xᵃ⁻ᵇReport a problem with this question
30. If log₃ x = 4, what is x?
- A.64
- B.81✓ Answer
- C.12
- D.7
A logarithm is an exponent: log₃ x = 4 means 3⁴ = x, so x = 81. Choosing 64 swaps base and exponent (4³), 12 multiplies 3 × 4, and 7 adds them — all misreadings of the log-to-exponential conversion.
Ref QB-MGGY5DSource: Definition of logarithm: log_b x = y ⇔ bʸ = xReport a problem with this question
31. Simplify: √72
- A.4√3
- B.6√2✓ Answer
- C.12√6
- D.36√2
Pull out the largest perfect-square factor: 72 = 36 × 2, so √72 = √36 · √2 = 6√2. Writing 36√2 forgets to take the square root of 36; 12√6 pulls the factor 12 out without square-rooting it; 4√3 equals √48, a wrong factorization of 72.
Ref QB-4ZATJCSource: Simplifying radicals: √(a·b) = √a·√b with the largest perfect-square factor (radical items are an official sample style, officialasvab.com)Report a problem with this question
32. Solve: |x - 3| = 7
- A.x = 4 or x = -10
- B.x = 10 only
- C.x = 7 or x = -7
- D.x = 10 or x = -4✓ Answer
Absolute value measures distance from zero, so x - 3 can equal 7 OR -7. That gives x = 10 or x = -4. Taking only the positive case misses a solution; x = ±7 solves |x| = 7, ignoring the -3 inside the bars.
Ref QB-KG5NYBSource: Absolute value equations: |A| = k (k > 0) ⇒ A = k or A = -kReport a problem with this question
33. Evaluate: 24 - 6 × 2 + 8 ÷ 4
- A.11
- B.14✓ Answer
- C.38
- D.10
By order of operations, do multiplication and division first: 6 × 2 = 12 and 8 ÷ 4 = 2. Then work left to right: 24 - 12 + 2 = 14. Getting 11 comes from computing strictly left to right; 10 comes from adding 12 + 2 before subtracting; 38 from subtracting 24 - 6 first.
Ref QB-3ZXFDBSource: Order of operations (PEMDAS): multiplication/division before addition/subtraction, then left to rightReport a problem with this question
34. Written as a fraction in lowest terms, 0.375 equals:
- A.3/8✓ Answer
- B.1/3
- C.3/5
- D.5/8
0.375 = 375/1000; dividing numerator and denominator by their GCF of 125 gives 3/8. 5/8 is 0.625 (the complement of 0.375), 3/5 is 0.6 (a place-value slip), and 1/3 is the repeating decimal 0.333..., not 0.375.
Ref QB-JGSATASource: Decimal-to-fraction conversion: write over a power of 10 and reduce by the GCFReport a problem with this question
35. Solve the proportion: 4/7 = 20/y
- A.y = 28
- B.y = 23
- C.y = 35✓ Answer
- D.y = 24
Cross-multiply: 4y = 7 × 20 = 140, so y = 35. Equivalently, 4 was multiplied by 5 to get 20, so 7 must also be multiplied by 5. Choosing 23 adds 16 to 7 (additive instead of multiplicative reasoning); 28 is 7 × 4; 24 is 20 + 4.
Ref QB-2ZK58ASource: Ratio and proportion via cross-multiplication (an official MK sample-question style, officialasvab.com)Report a problem with this question
36. A triangle has a base of 12 and a height of 7. What is its area?
- A.19
- B.42✓ Answer
- C.38
- D.84
The area of a triangle is ½ × base × height = ½ × 12 × 7 = 42. Answering 84 uses the rectangle formula (base × height) and forgets the ½ — the most common triangle-area error; 19 just adds the dimensions and 38 doubles that sum like a perimeter.
Ref QB-HGEYM9Source: Triangle area formula A = ½bh (formulas are not provided on the ASVAB — must be memorized)Report a problem with this question
37. Two angles are supplementary. If one measures 47°, what is the other?
- A.313°
- B.133°✓ Answer
- C.47°
- D.43°
Supplementary angles sum to 180°, so the other angle is 180° - 47° = 133°. Answering 43° uses the complementary rule (sum to 90°), 313° subtracts from a full 360° circle, and 47° confuses supplementary angles with vertical angles, which are equal.
Ref QB-ZY8T29Source: Angle rules: supplementary angles sum to 180°; complementary angles sum to 90°Report a problem with this question
38. A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?
- A.21
- B.6√3
- C.3√7
- D.15✓ Answer
By the Pythagorean theorem, c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15 (a 3-4-5 triangle scaled by 3). Answering 21 simply adds the legs; 3√7 = √63 subtracts the squares as if finding a leg; 6√3 = √108 multiplies 9 × 12 under the radical.
Ref QB-GG3NF8Source: Pythagorean theorem: a² + b² = c² for right trianglesReport a problem with this question
39. What is the volume of a rectangular box measuring 8 by 5 by 3?
- A.158
- B.40
- C.16
- D.120✓ Answer
The volume of a rectangular prism is length × width × height = 8 × 5 × 3 = 120 cubic units. 158 is the surface area 2(40 + 24 + 15); 40 is only the base area 8 × 5; 16 merely adds the three dimensions.
Ref QB-YYVFV7Source: Volume of a rectangular prism V = lwh (volume is an official MK sample-question style, officialasvab.com)Report a problem with this question
40. What is the y-intercept of the line 3x + 2y = 8?
- A.-3/2
- B.8
- C.4✓ Answer
- D.8/3
Solve for y to get slope-intercept form: 2y = -3x + 8, so y = -(3/2)x + 4, and the y-intercept (the value of y when x = 0) is 4. Answering 8 reads the constant without dividing by 2; -3/2 is the slope; 8/3 is the x-intercept.
Ref QB-DZQRKFSource: Slope-intercept form y = mx + b: b is the y-intercept, found by setting x = 0Report a problem with this question
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