ASVAB Arithmetic Reasoning: Tips, Formulas and Practice
By Josh Preston, TopASVAB · Last reviewed · Free to read, no account needed
Arithmetic Reasoning (AR) is the ASVAB's word-problem subtest. Each question describes an everyday situation, such as a price, a wage, a trip or a recipe, and asks you to work out a number by hand: calculators are not allowed on the ASVAB. AR is one of the four subtests behind your AFQT score, so it counts toward enlistment eligibility as well as job qualification.
This guide covers what the subtest looks like, a four-step way to set up any word problem, formulas worth knowing, and five practice questions with worked answers.
What the AR subtest looks like
| Version | Questions | Time limit |
|---|---|---|
| Paper and pencil | 30 | 36 minutes (about 72 seconds a question) |
| Computer (CAT-ASVAB at a MEPS or MET site) | 15 scored, plus up to 15 unscored tryout questions | 55 minutes, or 113 when the subtest includes tryout questions |
On the CAT-ASVAB, each test taker gets tryout questions in 2 to 4 of the subtests, mixed in among the scored ones, so an AR section longer than 15 questions is normal.
The unproctored computer version, PiCAT, which you can take from home, has no time limits for individual subtests and no tryout questions. It is only for people who have never taken the ASVAB: a recruiter gives you an access code, you can take it once, and you must finish within 48 hours of starting. If your scores suggest you may be eligible, you then take a short proctored Verification Test at a MEPS or MET site to confirm them.
- No calculator. You work everything out by hand. The ASVAB program's reason is that doing math without a calculator can be part of the job, so the test measures it.
- Everyday arithmetic. The ASVAB program's four AR sample questions cover rates, ratios and percents. This guide also covers averages, simple interest, unit conversions, basic probability, and area and volume, which use the same everyday arithmetic.
- The computer version adapts. After a right answer the next question is usually harder, and after a wrong one it is usually easier, so seeing hard problems is not a bad sign.
- Leave nothing blank. On the paper test an unanswered question counts as wrong, so answer every question, guessing if you must. On the computer test, if time runs out before you finish, your score is set as if you had guessed at random on the questions you did not reach, so keep a steady pace.
A four-step method for word problems
- Find the question. Read the last sentence first. Note exactly what it asks for, and in what units: dollars, hours, feet.
- List what you know. Write each number with its label, such as "$64 = original price" and "25% = discount".
- Choose the relationship. Pick the formula or ratio that connects what you know to what you need, and write it down before you calculate.
- Estimate, calculate, check. Make a rough estimate, work the problem, then ask whether the answer makes sense. A sale price must be lower than the original price, and two people working together must finish faster than either one alone.
The estimate in step 4 often rules out one or two answer choices before any real arithmetic.
Arithmetic without a calculator
- Percents in your head. 10% moves the decimal point one place left ($64 becomes $6.40). 5% is half of that, 20% is double it, 15% is 10% plus 5%, and 1% moves the point two places.
- Common fractions as percents. 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/8 = 12.5%, and 1/3 is about 33.3%.
- Cancel before you multiply. 21 × 2 ÷ 7 is easier as 3 × 2.
- Work in one unit. Turn hours and minutes into minutes, or dollars into cents, before you calculate, then convert back.
- Let the choices guide you. If the choices are far apart, a rounded estimate may be enough. If they are close together, calculate exactly.
Formulas worth knowing
| To find | Use |
|---|---|
| A percent of a number | part = percent × whole (25% of 80 = 0.25 × 80 = 20) |
| Percent change | (new − old) ÷ old × 100 |
| A price after a discount | original price × (1 − discount rate) |
| Distance, speed or time | distance = speed × time |
| An average | sum of the values ÷ number of values |
| Average speed for a whole trip | total distance ÷ total time |
| Simple interest | principal × yearly rate × number of years |
| Two workers together | add their rates, in jobs per hour: 1 ÷ time together = 1 ÷ time A + 1 ÷ time B |
| Probability | favorable outcomes ÷ all possible outcomes, when each outcome is equally likely (a bag of 10 marbles with 3 red: 3/10 chance of drawing red) |
| One part of a ratio a : b | whole × a ÷ (a + b) |
| Area of a rectangle | length × width |
| Volume of a box | length × width × height |
Unit facts to know by heart:
- 12 inches = 1 foot, 3 feet = 1 yard, 5,280 feet = 1 mile
- 16 ounces = 1 pound, 2,000 pounds = 1 ton
- 2 pints = 1 quart, 4 quarts = 1 gallon
- 60 seconds = 1 minute, 60 minutes = 1 hour
Four traps to watch for
- A percent of what? A percent change is measured from the starting value. A price that rises from $50 to $60 has gone up 20%, but falling back from $60 to $50 is a drop of about 16.7%.
- Averaging speeds. Average speed is total distance ÷ total time, not the average of two speeds.
- Mixed units. 2 hours 30 minutes is 2.5 hours, not 2.3.
- Stopping one step early. Answer choices often include a number from the middle of the working, such as the discount instead of the sale price. Re-read the question before you choose.
Practice questions
Each question has one best answer. Work it out before you open the explanation.
Practice question 1
Practice question, not a real ASVAB itemA pair of work boots costs $64. During a sale the price is cut by 25%, and then 5% sales tax is added to the sale price. How much does the customer pay?
- A. $48.00
- B. $50.40
- C. $51.20
- D. $67.20
Show the answer and explanation
Answer: B. $50.40
The question asks for the total paid, after the discount and then the tax.
- Discount: 25% of $64 is $16, so the sale price is $64 − $16 = $48.
- Tax: 10% of $48 is $4.80, so 5% is half of that, $2.40.
- Total: $48 + $2.40 = $50.40.
Choice C nets the tax against the discount (a 20% cut from $64), but the tax is charged on the sale price. Choice A forgets the tax, and D adds tax to the full price.
Practice question 2
Practice question, not a real ASVAB itemA delivery driver covers 150 miles in 2 hours 30 minutes. At the same average speed, how long will a 240-mile trip take?
- A. 3 hours 12 minutes
- B. 3 hours 41 minutes
- C. 4 hours
- D. 4 hours 48 minutes
Show the answer and explanation
Answer: C. 4 hours
The question asks for a time, for 240 miles.
- Convert the time: 30 minutes is half an hour, so 2 hours 30 minutes is 2.5 hours.
- Speed: 150 ÷ 2.5 = 60 miles per hour. Check: 60 × 2.5 = 150.
- Time for 240 miles: 240 ÷ 60 = 4 hours.
A and B are time-unit slips: A ignores the 30 minutes (150 ÷ 2 = 75 miles per hour) and B reads 2 hours 30 minutes as 2.3 hours. D rounds the first drive's 2 hours 30 minutes up to 3 hours (150 ÷ 3 = 50 miles per hour).
Practice question 3
Practice question, not a real ASVAB itemA paint mix uses 2 parts blue to 5 parts white. How many quarts of blue paint are needed to make 21 quarts of the mix?
- A. 6
- B. 7
- C. 8.4
- D. 15
Show the answer and explanation
Answer: A. 6
A ratio of 2 : 5 means every 2 + 5 = 7 quarts of mix hold 2 quarts of blue.
- Batches: 21 ÷ 7 = 3.
- Blue: 3 × 2 = 6 quarts (and 3 × 5 = 15 quarts of white).
Choice B is the size of one batch (2 + 5 = 7 quarts), not the blue paint in it. D is the white paint. C treats 21 quarts as the white part alone (21 × 2 ÷ 5 = 8.4).
Practice question 4
Practice question, not a real ASVAB itemA repair shop charges $84 an hour for labor and bills in 15-minute blocks, rounding any part of a block up to a full block. A job takes 1 hour 40 minutes. What is the labor charge?
- A. $117.60
- B. $126.00
- C. $140.00
- D. $147.00
Show the answer and explanation
Answer: D. $147.00
The question asks for the billed charge, so round the time first.
- Blocks: 1 hour 40 minutes is 100 minutes. 100 ÷ 15 = 6 full blocks with 10 minutes left over, which counts as a seventh block.
- Price per block: $84 ÷ 4 = $21.
- Charge: 7 × $21 = $147.00.
Choice C charges the exact time (100 minutes is 5/3 of an hour, and 5/3 × $84 = $140). B bills only the 6 full blocks and drops the 10 minutes left over (6 × $21 = $126). A reads 1 hour 40 minutes as 1.4 hours (1.4 × $84 = $117.60).
Practice question 5
Practice question, not a real ASVAB itemA town is putting lamps along one side of a 96-meter path. There will be a lamp at both ends of the path and one every 8 meters between them. How many lamps are needed?
- A. 11
- B. 12
- C. 13
- D. 14
Show the answer and explanation
Answer: C. 13
96 ÷ 8 = 12 spaces between lamps. With a lamp at both ends there is one more lamp than there are spaces, so the path needs 12 + 1 = 13 lamps.
Check with a small case: a 16-meter path has lamps at 0, 8 and 16 meters, which is 2 spaces and 3 lamps.
Choice B counts the spaces instead of the lamps, and A counts only the lamps between the two ends. D takes the 12 spaces as 12 lamps in between and then adds the two end lamps, but only 11 lamps stand between the ends.
AR and your scores
AR counts directly in the AFQT composite (AR + MK + 2 × VE), so each standard-score point in AR adds one point to the composite: see how your AFQT score is calculated. AR also appears in several of the composite scores the services use for job qualification. For example, the Air Force's General (G) score combines AR with Verbal Expression (VE), the verbal score built from Word Knowledge and Paragraph Comprehension; the 2004 score-scale report gives its formula as AR + VE.
More practice
- The ASVAB program's own Arithmetic Reasoning sample questions.
- The Word Knowledge guide covers the verbal side of the AFQT.
Sources
- officialasvab.com, What to expect: question counts, tryout questions and time limits for the paper and proctored computer versions; who can take PiCAT, its untimed subtests and the Verification Test; and "You cannot use a calculator when taking the ASVAB."
- officialasvab.com, FAQs: why math is tested without a calculator.
- officialasvab.com, CAT-ASVAB: how the computer version chooses harder or easier questions.
- officialasvab.com, ASVAB Scoring: unanswered paper-test questions count as wrong, and the computer test scores unfinished questions as random guesses.
- officialasvab.com, Arithmetic Reasoning sample questions: the four sample questions (rates, a ratio and a percent).
- airforce.com, ASVAB: the subtests behind the Air Force's General (G) score.
- Segall, D. O. (2004). Development and Evaluation of the 1997 ASVAB Score Scale (Technical Report No. 2004-002). Defense Manpower Data Center. Section 2.4 (the AFQT composite) and section 2.5.1 (Air Force composites). PDF on officialasvab.com
- The practice questions and worked answers were written for this guide by TopASVAB.
Try it free
Try the free timed 5-question Arithmetic Reasoning practice. You'll see your result and a written explanation for every answer.
Free account: sign up with your email address and a password, then confirm your address from the email we send. No payment details needed.
More free guides
About this guide
Written by Josh Preston, who runs TopASVAB. Drafted with an AI assistant; every fact and practice question checked by Josh Preston. Last reviewed .
Practice questions, not real ASVAB items. No score is guaranteed.
TopASVAB is not affiliated with or endorsed by the U.S. Department of Defense or any military service.
TopASVAB will never message students or applicants first or ask for payment outside our website.
Spotted a mistake? Email support@topasvab.com.
From topasvab.com/guides/arithmetic-reasoning (ASVAB Arithmetic Reasoning: Tips, Formulas and Practice), last reviewed September 27, 2026. Practice material, not real ASVAB items. TopASVAB is not affiliated with or endorsed by the U.S. Department of Defense or any military service.