ASVAB Math

ASVAB Math Practice Test: 12 AR and MK Questions Solved

Twelve original ASVAB Arithmetic Reasoning and Mathematics Knowledge questions, solved without a calculator, plus how they shape your AFQT.

Professor Chacha October 10, 2026 6 min read 4 views
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The ASVAB has two math subtests, Arithmetic Reasoning (AR) and Mathematics Knowledge (MK), and you can't use a calculator on either. They matter more than their share of the test suggests: AR and MK are two of the four subtests that make up your AFQT score, the score that decides whether you can enlist. Below is how the math sections work, then 12 original practice questions, six of each type, solved step by step.

The two math subtests

According to the official ASVAB site, most applicants take the computer version (CAT-ASVAB), which adapts to your answers. The paper version is longer:

Subtest What it tests Computer (CAT-ASVAB) Paper
Arithmetic Reasoning (AR) Word problems: rates, percents, averages, money, time and distance 15 questions, 55 minutes 30 questions, 36 minutes
Mathematics Knowledge (MK) Concepts: algebra, exponents, number properties, geometry 15 questions, 31 minutes 25 questions, 24 minutes

The computer test can take a little longer if it includes unscored tryout questions. Either way, the pace on the computer version is generous: more than 3 minutes per AR question. On paper, it is closer to one minute each, so practice speed if you are taking the paper test.

Why the math counts for so much: the AFQT

Your AFQT score comes from four subtests: Arithmetic Reasoning, Mathematics Knowledge, Word Knowledge, and Paragraph Comprehension. It is reported as a percentile from 1 to 99, which compares you with a reference group of 18- to 23-year-olds. Each branch of the military sets its own minimum AFQT and its own job requirements, so ask your recruiter for the current numbers.

In practice, half of the subtests behind your AFQT are math. Raising your math is one of the most direct ways to raise your AFQT.

Arithmetic Reasoning practice

AR is all word problems. The arithmetic itself is simple; the skill is turning the words into the right operation.

1.A recruit runs 3.5 miles a day for 6 days. How many miles does she run in total?

  1. A21
  2. B18.5
  3. C9.5
  4. D24
Show solution \[3.5 \times 6 = 21\]

Without a calculator: \(3 \times 6 = 18\) and \(0.5 \times 6 = 3\), so \(18 + 3 = 21\).

Common trap: Adding 3.5 and 6 instead of multiplying, which gives 9.5. "Each day for 6 days" means the same distance six times.

Answer: A (21)

2.A platoon has 48 members. Three-eighths of them are assigned to a training exercise. How many members is that?

  1. A30
  2. B16
  3. C18
  4. D24
Show solution

Find one-eighth, then multiply by 3.

\[\begin{gathered}48 \div 8 = 6 \\ 6 \times 3 = 18\end{gathered}\]

Common trap: Answering 30, the number not assigned (\(\tfrac{5}{8}\) of 48). Check which part the question asks for.

Answer: C (18)

3.A truck uses 12 gallons of fuel to travel 300 miles. At the same rate, how many gallons does it need for 450 miles?

  1. A37.5
  2. B18
  3. C15
  4. D20
Show solution

The truck gets \(300 \div 12 = 25\) miles per gallon.

\[450 \div 25 = 18 \text{ gallons}\]

Common trap: Dividing 450 by 12 and getting 37.5. That mixes up miles and gallons. Find the rate first, then use it.

Answer: B (18)

4.Sam earns $18.50 an hour for the first 40 hours of the week and time-and-a-half for overtime. Last week Sam worked 46 hours. What did Sam earn?

  1. A$851.00
  2. B$906.50
  3. C$1,276.50
  4. D$166.50
Show solution

Regular pay plus overtime pay at 1.5 times the hourly rate.

\[\begin{gathered}40 \times 18.50 = 740 \\ 6 \times 27.75 = 166.50 \\ 740 + 166.50 = 906.50\end{gathered}\]

Common trap: Paying all 46 hours at the regular rate ($851) or all of them at time-and-a-half ($1,276.50). Only the 6 hours over 40 get the higher rate.

Answer: B ($906.50)

5.Your scores on three practice tests are 72, 85, and 90. What score do you need on the fourth test for an average of 84?

  1. A84
  2. B80
  3. C92
  4. D89
Show solution

An average of 84 over four tests means a total of \(4 \times 84 = 336\).

\[\begin{gathered}336 - (72 + 85 + 90) \\ = 336 - 247 = 89\end{gathered}\]

Common trap: Answering 84. Your first three scores average a little less than 84, so the fourth must be higher to make up the gap.

Answer: D (89)

6.A convoy drives at 45 miles per hour for 2.5 hours, then at 60 miles per hour for 1.5 hours. How far does it travel in total?

  1. A202.5 miles
  2. B210 miles
  3. C105 miles
  4. D420 miles
Show solution

Find each distance with \(d = rt\), then add.

\[\begin{gathered}45 \times 2.5 = 112.5 \\ 60 \times 1.5 = 90 \\ 112.5 + 90 = 202.5\end{gathered}\]

Common trap: Averaging the speeds, \(\tfrac{45 + 60}{2} = 52.5\), and multiplying by 4 hours to get 210. The convoy spent different amounts of time at each speed, so the simple average does not apply.

Answer: A (202.5 miles)

Mathematics Knowledge practice

MK tests the rules: solving equations, multiplying expressions, exponents, prime numbers, and basic geometry.

7.Solve for \(x\): \(4x - 7 = 2x + 9\)

  1. A1
  2. B4
  3. C8
  4. D16
Show solution \[\begin{gathered}4x - 2x = 9 + 7 \\ \Rightarrow\; 2x = 16 \\ \Rightarrow\; x = 8\end{gathered}\]

Common trap: Stopping at \(2x = 16\) and answering 16. Finish by dividing both sides by 2.

Answer: C (8)

8.Multiply: \((x + 5)(x - 3)\)

  1. A\(x^2 - 15\)
  2. B\(x^2 + 2x - 15\)
  3. C\(x^2 + 8x - 15\)
  4. D\(x^2 - 2x - 15\)
Show solution

Multiply every term by every term (FOIL), then combine the middle terms.

\[x^2 - 3x + 5x - 15 = x^2 + 2x - 15\]

Common trap: Multiplying only the first and last terms, which gives \(x^2 - 15\). The outer and inner products make the middle term.

Answer: B (\(x^2 + 2x - 15\))

9.A triangle has a base of 12 inches and a height of 9 inches. What is its area?

  1. A\(108 \text{ in}^2\)
  2. B\(21 \text{ in}^2\)
  3. C\(42 \text{ in}^2\)
  4. D\(54 \text{ in}^2\)
Show solution \[A = \tfrac{1}{2}bh = \tfrac{1}{2}(12)(9) = 54\]

Common trap: Forgetting the \(\tfrac{1}{2}\) and answering 108, which is the area of a rectangle with those sides.

Answer: D (\(54 \text{ in}^2\))

10.Which of these numbers is prime?

  1. A51
  2. B57
  3. C59
  4. D91
Show solution

Test each for small factors:

\(51 = 3 \times 17\), \(57 = 3 \times 19\), and \(91 = 7 \times 13\). 59 has no factors other than 1 and itself, so it is prime.

Common trap: Picking 51, 57, or 91 because they "look" prime. A quick test: if the digits add up to a multiple of 3, the number is divisible by 3 (\(5 + 1 = 6\), \(5 + 7 = 12\)).

Answer: C (59)

11.Evaluate: \(\sqrt{144} + 3^2\)

  1. A21
  2. B18
  3. C15
  4. D81
Show solution \[\begin{gathered}\sqrt{144} = 12 \\ 3^2 = 9 \\ 12 + 9 = 21\end{gathered}\]

Common trap: Reading \(3^2\) as \(3 \times 2 = 6\), which gives 18. An exponent means repeated multiplication: \(3 \times 3\).

Answer: A (21)

12.Two angles are supplementary. One of them measures \(65^\circ\). What is the measure of the other?

  1. A\(25^\circ\)
  2. B\(295^\circ\)
  3. C\(115^\circ\)
  4. D\(35^\circ\)
Show solution

Supplementary angles add up to \(180^\circ\).

\[180^\circ - 65^\circ = 115^\circ\]

Common trap: Using \(90^\circ\), which gives \(25^\circ\). That is the complement. Memory aid: complementary comes before supplementary in the alphabet, and 90 comes before 180.

Answer: C (\(115^\circ\))

Answer key

  1. 1 A
  2. 2 C
  3. 3 B
  4. 4 B
  5. 5 D
  6. 6 A
  7. 7 C
  8. 8 B
  9. 9 D
  10. 10 C
  11. 11 A
  12. 12 C

How to prepare without a calculator

  • Drill the basics by hand: fractions, decimals, and percents. Our lesson on percent word problems covers the types AR uses most.
  • Know the rules MK checks: exponent rules, factoring and multiplying, and the common area and volume formulas, which MK expects you to know.
  • Translate word problems into equations. Many AR questions are two-step problems like the ones in our lesson on systems of equations word problems.
  • Use the official resources. The official ASVAB site has sample questions and information about the test; your recruiter can also give you access to practice materials.

Teacher's note: Without a calculator, the fastest method is round, compute, adjust. For \(18.50 \times 40\), think \(18 \times 40 = 720\), then add \(0.5 \times 40 = 20\) to get 740. Breaking a number into easy pieces is faster and safer than long multiplication, and it is a skill that improves noticeably with a week of daily practice.

Tags: AFQT Arithmetic reasoning ASVAB Practice test
Professor Chacha
Professor Chacha Math teacher and educational psychologist

Math teacher and educational psychologist with more than 20 years of classroom experience. He writes every practice question on this site from scratch and solves it step by step, the way he explains it to his own students.

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