Arithmetic Reasoning (AR)
Arithmetic Reasoning (AR) is one of the four AFQT subtests. It tests your ability to solve real-world math problems stated in words — the challenge is translating the sentence into the right calculation.
Practise Arithmetic Reasoning freeAlready know what's on it and still stuck? How to improve your Arithmetic Reasoning score.
What's on it
Ratios, proportions and rates (including distance–rate–time); percentages (discount, tax, tip, increase and decrease, commission); fractions and decimals in context; unit conversions; averages; simple interest; work/rate problems; mixtures; reading charts and tables; and some basic probability.
The translation habit
Read the whole problem first and decide exactly what's being asked. Write the numbers down, choose the operation, then compute. Check that the units of your answer (dollars, miles, hours) match the question.
Formulas to keep automatic
Distance = rate × time. Part = percent × whole, and percent change = (new − old) ÷ old. For combined work, rates add (or keep total worker-days constant). Simple interest I = P × r × t. Average = sum ÷ count.
Test-day strategy
Estimate first, then compute — it catches impossible answers. If a problem is awkward, work backwards from the answer choices. On the computer test AR is generously timed because the problems are multi-step, so read carefully rather than rushing.
Key takeaways
- One of the 4 AFQT subtests — it counts directly toward enlistment eligibility.
- It's arithmetic (roughly grade 6–8 math), but the wording makes it feel harder.
- Distance = rate × time and percent = part ÷ whole solve a huge share of questions.
Start a free Arithmetic Reasoning practice set — timed, with a worked explanation after every question.
Go to Arithmetic Reasoning practiceFree sample Arithmetic Reasoning questions
A few sample questions with worked answers. Our free Arithmetic Reasoning practice set has 200 questions in total — every one with a full explanation.
- A.10
- B.133
- C.12 ✓
- D.28
- A.1,900
- B.300
- C.2,100 ✓
- D.192
- A.3 hours
- B.5 hours
- C.2.5 hours ✓
- D.2 hours
- A.9.6 mph ✓
- B.9 mph
- C.10 mph
- D.20 mph
- A.$33.45
- B.$6.55 ✓
- C.$7.55
- D.$6.45
Arithmetic Reasoning cheat sheet
The formulas and facts worth memorising for this subtest. See all ASVAB cheat sheets →
- Distance = rate × time (rate = dist ÷ time, time = dist ÷ rate)
- Part = percent × whole; percent = part ÷ whole
- Percent change = (new − old) ÷ old
- Discount / markup: multiply by (1 − r) or (1 + r)
- Simple interest: I = P × r × t
- Average = sum ÷ count; unit rate = total ÷ number of units
- Split in ratio a:b:c → divide the total by (a + b + c) to get one 'part'
- Combined work: rates ADD (1/a + 1/b), or total worker-days = workers × days
- Average speed = total distance ÷ total time (NOT the average of the two speeds)
- 'half an hour' = 30 minutes — multiply, don't ÷ 2
- 'one in every N' → the rest = total − (total ÷ N)
- Compound depreciation: multiply by (1 − r) each year; don't just take r × years