AFOQT Arithmetic Reasoning practice and strategy

Arithmetic Reasoning format and timing

Source: AFOQT Information Pamphlet (Form T, updated 1 Aug 2015), Tables 1–2. Arithmetic Reasoning is subtest 2 of 12, in Part A. The test has been computer-only (the E-AFOQT) since 2023; whether that changed any timing isn't published, so these are the Form T figures our timed sets use. Our drills are written for scratch paper and no calculator; check your test site's rules.

What Arithmetic Reasoning tests

Arithmetic in context: reading a short story problem, setting it up and solving it by hand. In our sets the numbers stay workable on paper; the difficulty is the setup. Our bank covers rate, time and distance, work rates, percent change, discounts and interest, ratios and scale, weighted averages, unit conversion, mixtures and simple probability. No topic list is published, so that's our coverage, not a syllabus.

The technique: translate, estimate, solve, check

1. Read the question sentence first. What exactly is asked, and in what unit?
2. Write each number with its unit, and convert now so minutes and hours never meet in one equation.
3. Translate: “of” means multiply, “per” means divide, “is” means equals. Give the unknown one letter.
4. Estimate with round numbers. A rough answer often cuts the options to two.
5. Solve with the simplest arithmetic: cancel before you multiply, keep fractions as fractions.
6. Check the unit, check that the size is plausible and check that you answered the question asked, not a step on the way.

If the setup stalls after about 30 seconds, plug in the answer choices instead, starting with the middle value.

Three worked problems

Example 1. A drone flies 18 miles to a test range at 36 mph and returns the same 18 miles at 24 mph. What is its average speed for the whole trip?
A. 28.0 mph · B. 28.8 mph · C. 30.0 mph · D. 31.2 mph · E. 60.0 mph

Average speed is total distance ÷ total time, not the average of the two speeds. Out: 18 ÷ 36 = 0.5 h. Back: 18 ÷ 24 = 0.75 h. Total: 36 miles in 1.25 h = 28.8 mph (B). Check: 28.8 × 1.25 = 36. The trap is C, the plain average of 36 and 24. The drone spends longer at the slow speed, so the true average sits below 30.

Example 2. A flight of 12 cadets averages 80 on a navigation quiz. A second flight of 8 cadets averages 90. What is the average score of all 20 cadets?
A. 82 · B. 83 · C. 84 · D. 85 · E. 86

Weight each group by its size. Total points: 12 × 80 = 960 and 8 × 90 = 720, so 1,680 ÷ 20 = 84 (C). D averages the two averages as if the flights were the same size. Sense check: the bigger flight scored lower, so the answer must sit below the midpoint of 85.

Example 3. A store raises the price of a pair of boots by 20%, then puts them on sale at 20% off the new price. The sale price is $48. What was the original price?
A. $46.08 · B. $48.00 · C. $49.00 · D. $50.00 · E. $60.00

Each percent applies to a different base, so the two changes don't cancel. Original × 1.2 × 0.8 = original × 0.96 = $48, so the original is 48 ÷ 0.96 = $50 (D). Check: $50 → $60 → $48. B assumes the changes cancel, E undoes only the 20% cut, and A applies the net change in the wrong direction.

Common traps

• Averaging rates or averages. Average speed is total distance ÷ total time; combined averages are weighted by group size.
• Adding times in work problems. Add the rates (jobs per hour), then invert.
• Percent of the wrong base. Successive percents multiply; they don't cancel.
• Unit mix. Minutes against hours, cents against dollars. Wrong options are often built from these slips.
• Stopping one step early. You found the time, but the question asked for the distance.
• Closing speed. Opposite directions add the speeds; the same direction subtracts them.

Pacing plan: 25 in 29 minutes

Twenty-nine minutes for 25 problems is about 70 seconds each, but the spread is wide: a one-step problem might take 30 seconds, a multi-step one two minutes. Work in two passes if the testing software lets you go back within a section (whether the E-AFOQT does isn't published — read the on-screen instructions). First pass: solve everything you can set up within about 30 seconds; for the rest, enter your best estimate-based guess and move on. Checkpoint: about 13 answered by 15:00. Second pass, if allowed: revisit those guesses. Either way, keep the final minute to make sure nothing is blank.

Which composites it feeds

Arithmetic Reasoning counts toward Quantitative and Academic Aptitude. Quantitative carries a commissioning minimum of 10 for every line officer (DAFMAN 36-2664, Table A2.1) and is built from just two subtests, this one and Math Knowledge, so each problem carries real weight. It doesn't feed Pilot, CSO or ABM.

Composites are percentiles from 1 to 99 against a reference group. The norms aren't published, so any composite we show is an estimate. More in AFOQT scores explained.

Practice it here

Start with the five problems above. Then run a timed Arithmetic Reasoning set, 25 problems on the 29-minute clock, with scratch paper beside you. Each explanation names the technique and the slip behind every wrong option, so sort your misses into setup, unit and arithmetic errors. Then check your Quantitative estimate with the free diagnostic or a full mock.

How to solve AFOQT arithmetic word problems

1. Read the question line first and note the unit it asks for.
2. Write each number with its unit and convert to matching units.
3. Translate the words into one equation with one unknown.
4. Estimate with round numbers and cross out impossible options.
5. Solve, then check the unit, the size and the question asked.
6. Stuck after 30 seconds? Plug in the options, starting with the middle one.

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