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ASVAB math practice test with worked solutions

This page has 20 original practice questions: 10 Arithmetic Reasoning word problems and 10 Mathematics Knowledge items, each with a full hand solution. Work them without a calculator, because the official ASVAB does not allow one, then use the review steps below to decide what to study next.

Last updated · Published by Certloom Learning · Facts checked against the official pages listed under Sources on November 14, 2026. Practice items are original.

What math is on the ASVAB?

Two subtests carry the math: Arithmetic Reasoning (AR) and Mathematics Knowledge (MK). The official subtest table describes AR as the ability to solve arithmetic word problems and MK as knowledge of high school mathematics principles (Official ASVAB: ASVAB Subtests). Both count toward the AFQT, together with Word Knowledge and Paragraph Comprehension (Official ASVAB: Understanding ASVAB Scores).

In practice, AR is a reading-and-setup test. The arithmetic is usually short once you know what the question asks for. Common types include rates and distance, percents and discounts, ratios and recipes, averages, work problems, simple interest and unit conversions. MK has no story around it. It asks you to solve an equation, simplify an exponent or root, use a geometry formula, factor a quadratic or handle signs correctly.

The test below follows that split: questions 1 to 10 are AR word problems and questions 11 to 20 are MK items. The mix of topics follows the kinds of skills the official descriptions name. It is not a copy of any official form, and the real CAT-ASVAB adapts its questions to you, so no fixed set of 20 items can reproduce it exactly.

How should you take this practice test?

Take it once, on paper, with no calculator and a timer running in the background. That setup is closest to what you will do on test day.

  1. Get two or three sheets of scratch paper and a pencil. The official CAT-ASVAB page mentions asking for "more scratch paper or another pencil" during the test, so practice writing your work down (Official ASVAB: The CAT-ASVAB (recruiter page)).
  2. Start a stopwatch, but do not race it on the first attempt. Note the time when you finish question 10 and again at question 20.
  3. For each item, write the question's target in a few words ("gallons for 140 miles") before you touch the numbers.
  4. Pick an answer for every item before you open any explanation. On the computer test you cannot go back, so practice committing.
  5. Mark any item where you guessed with a small "G". A lucky guess still counts as a gap to study.
  6. Only after all 20 are done, open the explanations and score yourself.

If you want a second, cleaner attempt later, cover your old answers and redo only the items you missed or marked G, a few days after the first try.

How much time do you get per math question?

On the proctored CAT-ASVAB, the time limits are generous: 55 minutes for 15 scored AR questions and 31 minutes for 15 scored MK questions when no tryout questions are given (Official ASVAB: The CAT-ASVAB). The paper version is tighter per question. The table divides each official time limit by its question count.

Subtest and versionQuestionsTime limitAverage time per question
AR, CAT-ASVAB (no tryout items)15 scored55 minutesabout 3 minutes 40 seconds
MK, CAT-ASVAB (no tryout items)15 scored31 minutesabout 2 minutes 4 seconds
AR, paper and pencil3036 minutes72 seconds
MK, paper and pencil2524 minutesabout 58 seconds

Sources: CAT table on Official ASVAB: The CAT-ASVAB; paper table on Official ASVAB: What to Expect When You Take the ASVAB. The CAT page also says nearly all examinees finish each subtest in time.

For this page, a useful target is the paper pace, since it is the tighter of the two. Ten AR items at 72 seconds each is 12 minutes. Ten MK items at about 58 seconds each is about 9 minutes 36 seconds, so call it 10 minutes. If your first untimed run took far longer, that is normal. Speed comes after the method is reliable.

What skills do the 20 questions cover?

Each item checks one skill, so a miss points to one lesson. Use this map when you review.

QuestionsSkillWhere to study it
1, 5Rates and ratiosRatios and rates
2, 8Discounts, tax and percent changePercentages, interest and percent change
3, 6Work and distance problemsDistance, rate and time
4AveragesAverages and probability
7Simple interestInterest and percent change
9Area and unit conversionGeometry review
10Age problems written as equationsWord problems
11, 15, 16, 20Equations, factoring, inequalities, substitutionAlgebra
12, 18, 19Exponents, scientific notation, rootsExponents and roots
13, 14, 17Circles, right triangles, anglesGeometry review

If you missed several AR items for different reasons, start with word problem types. If you missed formulas, keep the math formulas page open while you redo them.

How do you set up an AR word problem quickly?

Write the target, find the rate or relationship, then compute. Most AR mistakes happen in the setup, not the arithmetic.

Example: A printer makes 18 pages in 45 seconds. How many pages does it make in 3 minutes?

  1. Target: pages in 3 minutes.
  2. Put the time in one unit: 3 minutes is 180 seconds.
  3. 180 seconds is 4 times 45 seconds (45 x 4 = 180).
  4. Pages scale the same way: 18 x 4 = 72.
  5. Size check: 3 minutes is longer than 45 seconds, so the answer must be more than 18. It is.

Answer: 72 pages

The step that saves the most points is the unit check. Many wrong choices on AR are numbers you get by mixing minutes with seconds, feet with yards, or a percent with a dollar amount. Question 9 below is built on exactly that mistake.

How do you check an MK answer without a calculator?

Plug your answer back in, or estimate. MK items usually have one exact answer, so a fast check is often possible.

Example: Solve 5x + 9 = 2x + 30.

  1. Subtract 2x from both sides: 3x + 9 = 30.
  2. Subtract 9 from both sides: 3x = 21.
  3. Divide by 3: x = 7.
  4. Check: left side 5(7) + 9 = 44; right side 2(7) + 30 = 44. Both sides match.

Answer: x = 7

For roots and exponents, check by working backward. If you think the square root of 72 is 6 times the square root of 2, square it: 36 x 2 = 72. For geometry, ask whether the size is sensible: a hypotenuse must be longer than either leg, and a supplementary angle pair must add to 180 degrees.

Why do the wrong answer choices look so convincing?

Because good distractors are the numbers you get from common mistakes. On this page, every wrong choice was built that way, and real multiple-choice tests work on the same principle.

Look at question 2. If you take 15% off and forget the tax, you get $34.00, which is one of the choices. If you combine the discount and the tax into a single 9% cut, you get $36.40, which is also there. Question 8 offers about 13%, which is what you get when you divide the price change by the new price instead of the original. Question 16 offers x > -3, the answer you get when you forget to flip the inequality sign.

So finding your number among the choices does not prove it is right. Two habits help:

  • Before you look at the choices, estimate the answer's size and sign. A sale price must be below the original price; a combined work time must be shorter than the faster worker's time.
  • After you choose, run one quick check: plug the answer back in, reverse the operation or compare it with your estimate.

When you review, read the explanation for every distractor you picked. Knowing which mistake produced it tells you exactly what to fix.

How do you review your answers after the test?

Score the test, then sort every miss and every guess by cause before you redo anything. The cause decides what you study.

  1. Read wrong: you solved a different question, such as finding the discount instead of the sale price. Fix it by writing the target before computing.
  2. Setup wrong: you understood the question but built the wrong relationship, such as adding work times instead of adding work rates. Study that problem type and redo two fresh examples.
  3. Rule gap: you did not know a fact, such as how to multiply powers with the same base. Learn the rule, then write one example in your own words.
  4. Slip: the method was right and an arithmetic step was wrong. Slow down on that step type and add a check.

Keep a short error log with the question number, the cause and the fix. The mistake review lesson shows a fuller version. Two or three days later, redo only the logged items from a blank page. If you get them right with clear work, move on.

Count your guesses separately from your misses. Five right answers with two lucky guesses means you know three, and your study plan should reflect that.

What should you study next after this test?

Study the skill behind your most common mistake first, then take the test again in a week. A raw count on 20 practice items cannot tell you your AFQT, which is a percentile computed from four subtests' standard scores (Official ASVAB: Understanding ASVAB Scores). What it can show is where your method breaks.

When math feels steady, take the verbal practice test too, since Word Knowledge and Paragraph Comprehension count toward the AFQT as well.

Practice questions

Work each one on paper first, then open the explanation. These are original AptiStep items, not real ASVAB questions.

Question 1

Question 1 (AR). A car uses 3 gallons of gas to travel 84 miles. At the same rate, how many gallons does it need to travel 140 miles?

  1. 4.5 gallons
  2. 5 gallons
  3. 5.5 gallons
  4. 6 gallons
Show answer and explanation

Answer: B. 5 gallons

Step 1: find the unit rate. 84 miles divided by 3 gallons is 28 miles per gallon. Step 2: divide the new distance by that rate. 140 divided by 28 is 5. Check: 5 gallons x 28 miles per gallon = 140 miles. The answer is 5 gallons.

Question 2

Question 2 (AR). A jacket costs $40. It is marked 15% off, and then a 6% sales tax is added to the sale price. What is the final cost?

  1. $34.00
  2. $36.40
  3. $36.04
  4. $42.40
Show answer and explanation

Answer: C. $36.04

Step 1: 15% of $40 is $6, so the sale price is $34. Step 2: 6% of $34 is $2.04 (1% is $0.34, times 6). Step 3: $34 + $2.04 = $36.04. $34.00 forgets the tax, $36.40 subtracts 9% from $40 in one step, and $42.40 adds tax to the full price.

Question 3

Question 3 (AR). Ana can paint a fence in 6 hours. Ben can paint the same fence in 3 hours. How long will it take them working together?

  1. 2 hours
  2. 3 hours
  3. 4.5 hours
  4. 9 hours
Show answer and explanation

Answer: A. 2 hours

Add rates, not times. Ana paints 1/6 of the fence per hour and Ben paints 1/3, which is 2/6. Together they paint 3/6, or 1/2, of the fence per hour, so the whole fence takes 2 hours. Check: the team must be faster than Ben alone, so the answer must be less than 3 hours.

Question 4

Question 4 (AR). Your first four quiz scores are 82, 90, 76 and 88. What score do you need on the fifth quiz to average exactly 85?

  1. 84
  2. 85
  3. 91
  4. 89
Show answer and explanation

Answer: D. 89

Step 1: an average of 85 over 5 quizzes needs a total of 85 x 5 = 425 points. Step 2: the first four add to 82 + 90 + 76 + 88 = 336. Step 3: 425 - 336 = 89. Check: (336 + 89) / 5 = 425 / 5 = 85.

Question 5

Question 5 (AR). A recipe uses 3 cups of flour for every 2 cups of sugar. How many cups of sugar go with 12 cups of flour?

  1. 6 cups
  2. 8 cups
  3. 10 cups
  4. 18 cups
Show answer and explanation

Answer: B. 8 cups

Step 1: 12 cups of flour is 4 times the 3 cups in the recipe (12 / 3 = 4). Step 2: multiply the sugar by the same factor: 2 x 4 = 8 cups. 18 cups comes from flipping the ratio (12 x 3 / 2), which gives more sugar than flour and does not match the recipe.

Question 6

Question 6 (AR). Two trains leave the same station at the same time and travel in opposite directions, one at 45 miles per hour and the other at 55 miles per hour. After how many hours are they 350 miles apart?

  1. 3.5 hours
  2. 3 hours
  3. 4 hours
  4. 7 hours
Show answer and explanation

Answer: A. 3.5 hours

Moving in opposite directions, the distance between them grows by 45 + 55 = 100 miles every hour. Time = distance / rate = 350 / 100 = 3.5 hours. Check: in 3.5 hours one train goes 157.5 miles and the other 192.5 miles, and 157.5 + 192.5 = 350.

Question 7

Question 7 (AR). You put $2,400 in an account that pays 5% simple interest per year. How much interest do you earn in 3 years?

  1. $120
  2. $378.30
  3. $2,760
  4. $360
Show answer and explanation

Answer: D. $360

Simple interest is principal x rate x time. Step 1: 5% of $2,400 is $120 per year. Step 2: $120 x 3 years = $360. $120 is only one year, $2,760 is the new balance rather than the interest, and $378.30 is what compound interest would give.

Question 8

Question 8 (AR). The price of a tool rose from $80 to $92. By what percent did the price increase?

  1. 12%
  2. 15%
  3. about 13%
  4. 115%
Show answer and explanation

Answer: B. 15%

Percent change = change / original. Step 1: the change is 92 - 80 = $12. Step 2: divide by the original price, $80: 12 / 80 = 0.15, which is 15%. About 13% comes from dividing by the new price (12 / 92), and 12% confuses dollars with percent.

Question 9

Question 9 (AR). A room is 12 feet by 15 feet. Carpet costs $3 per square yard. How much does it cost to carpet the room?

  1. $60
  2. $180
  3. $540
  4. $20
Show answer and explanation

Answer: A. $60

Step 1: area = 12 x 15 = 180 square feet. Step 2: one square yard is 3 feet x 3 feet = 9 square feet, so 180 / 9 = 20 square yards. Step 3: 20 x $3 = $60. $540 forgets to convert to square yards, and $180 divides by 3 instead of 9.

Question 10

Question 10 (AR). A mother is 3 times as old as her son. In 12 years she will be twice as old as he is. How old is the son now?

  1. 6
  2. 24
  3. 12
  4. 36
Show answer and explanation

Answer: C. 12

Let s be the son's age now, so the mother is 3s. In 12 years: 3s + 12 = 2(s + 12). Expand: 3s + 12 = 2s + 24. Subtract 2s and 12: s = 12. Check: now 12 and 36; in 12 years 24 and 48, and 48 is twice 24. 36 is the mother's age, not the son's.

Question 11

Question 11 (MK). Solve for x: 3x - 7 = 2x + 5.

  1. x = 12
  2. x = -2
  3. x = 2
  4. x = -12
Show answer and explanation

Answer: A. x = 12

Step 1: subtract 2x from both sides: x - 7 = 5. Step 2: add 7 to both sides: x = 12. Check: 3(12) - 7 = 29 and 2(12) + 5 = 29. The value -2 comes from subtracting 7 instead of adding it.

Question 12

Question 12 (MK). Which expression equals (2^3)(2^4)?

  1. 2^12
  2. 2^7
  3. 4^7
  4. 4^12
Show answer and explanation

Answer: B. 2^7

When you multiply powers with the same base, keep the base and add the exponents: 2^3 x 2^4 = 2^(3+4) = 2^7, which is 128. Multiplying the exponents gives 2^12, and multiplying the bases gives 4^7; both are common slips. Check: 8 x 16 = 128.

Question 13

Question 13 (MK). What is the area of a circle with a radius of 5 inches?

  1. 10π square inches
  2. 5π square inches
  3. 25π square inches
  4. 100π square inches
Show answer and explanation

Answer: C. 25π square inches

The area of a circle is π times the radius squared. Step 1: square the radius: 5 x 5 = 25. Step 2: the area is 25π square inches. 10π is the circumference formula (2πr), and 100π uses the diameter, 10, in place of the radius.

Question 14

Question 14 (MK). A right triangle has legs of 6 and 8. How long is the hypotenuse?

  1. 14
  2. 12
  3. 48
  4. 10
Show answer and explanation

Answer: D. 10

Use the Pythagorean theorem: a^2 + b^2 = c^2. Step 1: 6^2 + 8^2 = 36 + 64 = 100. Step 2: c is the square root of 100, which is 10. Check: the hypotenuse must be longer than 8 and shorter than 6 + 8 = 14, and 10 fits. 14 just adds the legs.

Question 15

Question 15 (MK). Which expression is equal to x^2 - 5x + 6?

  1. (x - 2)(x - 3)
  2. (x + 2)(x + 3)
  3. (x - 1)(x - 6)
  4. (x + 1)(x - 6)
Show answer and explanation

Answer: A. (x - 2)(x - 3)

Find two numbers that multiply to +6 and add to -5. Those are -2 and -3. So x^2 - 5x + 6 = (x - 2)(x - 3). Check by expanding: x^2 - 3x - 2x + 6 = x^2 - 5x + 6. (x + 1)(x - 6) expands to x^2 - 5x - 6, which has the wrong sign on 6.

Question 16

Question 16 (MK). Solve the inequality -2x + 4 > 10.

  1. x > -3
  2. x < 3
  3. x < -3
  4. x > -7
Show answer and explanation

Answer: C. x < -3

Step 1: subtract 4 from both sides: -2x > 6. Step 2: divide both sides by -2, and flip the inequality sign because you divided by a negative: x < -3. Check with x = -4: -2(-4) + 4 = 12, and 12 > 10 is true. Forgetting to flip gives x > -3.

Question 17

Question 17 (MK). Two angles are supplementary. One angle is 4 times the other. What is the larger angle?

  1. 72 degrees
  2. 144 degrees
  3. 36 degrees
  4. 135 degrees
Show answer and explanation

Answer: B. 144 degrees

Supplementary angles add to 180 degrees. Let the smaller angle be a, so a + 4a = 180, 5a = 180 and a = 36. The larger angle is 4 x 36 = 144 degrees. Check: 36 + 144 = 180. 72 degrees comes from using 90 (complementary) instead of 180.

Question 18

Question 18 (MK). What is (3 x 10^4)(2 x 10^-6)?

  1. 6 x 10^-24
  2. 5 x 10^-2
  3. 6 x 10^10
  4. 6 x 10^-2
Show answer and explanation

Answer: D. 6 x 10^-2

Multiply the front numbers and add the exponents. Step 1: 3 x 2 = 6. Step 2: 10^4 x 10^-6 = 10^(4 + (-6)) = 10^-2. The product is 6 x 10^-2, which is 0.06. Multiplying the exponents gives 10^-24, and subtracting them, 4 - (-6), gives 10^10.

Question 19

Question 19 (MK). Which is equal to the square root of 72?

  1. 6 times the square root of 2
  2. 8 times the square root of 3
  3. 4 times the square root of 2
  4. 2 times the square root of 6
Show answer and explanation

Answer: A. 6 times the square root of 2

Find the largest perfect square factor of 72. 72 = 36 x 2, and the square root of 36 is 6, so the square root of 72 is 6 times the square root of 2. Check by squaring: 6^2 x 2 = 36 x 2 = 72. The other choices square to 192, 32 and 24.

Question 20

Question 20 (MK). If a = -3 and b = 2, what is the value of a^2 - 4ab?

  1. -15
  2. 33
  3. 15
  4. -33
Show answer and explanation

Answer: B. 33

Substitute carefully with parentheses. Step 1: a^2 = (-3)^2 = 9, which is positive. Step 2: 4ab = 4 x (-3) x 2 = -24. Step 3: 9 - (-24) = 9 + 24 = 33. Writing a^2 as -9 gives 15, and treating -4ab as -24 gives -15.

Frequently asked questions

Are these real ASVAB questions?

No. They are original practice items written for AptiStep. The official site has its own sample items on the sample questions page, which are worth doing for format.

Can I use a calculator on the ASVAB math sections?

No. The official What to Expect page says you cannot use a calculator, and the FAQ explains that hand calculation may be needed on the job (official FAQ).

What score on this test means I will pass the ASVAB?

None. A score on 20 practice items does not convert to an AFQT percentile, and each branch sets its own rules. Use your misses to choose lessons, and see minimum scores by branch for published minimums.

Where can I practice more math?

Sources

Checked November 14, 2026. AptiStep is an independent study site; confirm anything that affects your enlistment with a recruiter.

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