Word problems that test whether you can turn a sentence into an equation.
Arithmetic Reasoning measures your ability to solve practical word problems using the four operations, fractions, percentages, ratios and measurement. It is the reasoning half of the math section: the arithmetic itself is never hard, but the problem has to be decoded before any number crunching starts.
Every AR item is a short scenario - a motor pool, a field kitchen, a bus schedule, a pay check - followed by one question and four answers. There is no formula to recall and no calculator. What is being tested is whether you can identify the relationship being described, choose the right operation, and carry it through two or three steps without losing track of what the question asked for.
Because AR is one of the four AFQT subtests, it is also the single largest time block in a practice session: you get more seconds per item here than anywhere else on the test. Slow, careful reading is a legitimate strategy in a way it is not in Word Knowledge.
Section 2 of 10 in our timed mock template. This is one of the four subtests in the AFQT, so it affects enlistment eligibility directly.
Deeper AFQT page
There is a separate page on how Arithmetic Reasoning drives the AFQT, with a week-by-week training routine: Arithmetic Reasoning and the AFQT.
The shapes below are what our bank is written against, and what the section is built around. None of them are reproduced from the official test.
Specific, not generic: what the items in this topic test, and the one or two moves that solve most of them.
Reduce the ratio to one part first. If trucks to vans is 3 to 2 and there are 18 trucks, one part is 6, so the 2 parts of vans are 12. Most wrong answers on these items come from multiplying by the ratio upside down, so before you commit, ask yourself which quantity is supposed to be bigger.
15 original questions in this topic set.
Decide immediately whether you need the change or the result. A 15 percent discount asks for 85 percent of the price, not 15 percent of it. For percent-of problems, translate "of" into multiplication and "is" into equals: "40 is what percent of 250" becomes 40 = p x 250.
15 original questions in this topic set.
Keep distance, rate and time in the same units and on the same line of your sketch. The classic trap is average speed over two legs: it is total distance divided by total time, never the simple average of the two speeds, because you spend more time at the slower speed.
15 original questions in this topic set.
Convert each worker to a rate per hour, add the rates, then invert. One person finishes in 4 hours (1/4 job per hour) and another in 6 (1/6), so together they do 5/12 per hour and finish in 12/5 hours. Answering 5 - the sum of the rates - is the most common wrong choice.
15 original questions in this topic set.
Simple interest is I = prt and it is always computed on the original principal in these items. Read whether the question wants the interest earned or the total balance at the end; the two are almost always both offered as options.
15 original questions in this topic set.
When something is spent, lost or used in stages, work with what remains, not what was taken. If you spend half and then half of what is left, you have a quarter remaining. Line up decimal points before adding, and convert to fractions only when the denominators are friendly.
15 original questions in this topic set.
Total equals count times mean. That one identity solves nearly every average problem: to find the missing score needed for a target mean, multiply the target by the number of items and subtract what you already have.
15 original questions in this topic set.
Multi-step items are long, not difficult. Write the given numbers with their units, note the final question in the margin, and answer the question that was actually asked. Distractors are usually correct intermediate results, which is why re-reading the last sentence is worth the five seconds.
15 original questions in this topic set.
Sampled from the Arithmetic Reasoning bank below, spread across its topics so you can see the range. The correct option is marked, the method is written out, and the tempting wrong options are explained - the same format you get in the app the moment you answer.
A base motor pool has trucks and vans in a ratio of 3 to 2. If there are 18 trucks, how many vans are there?
The ratio 3:2 means the 18 trucks represent 3 equal parts, so one part is 18 ÷ 3 = 6 vehicles. Vans make up 2 parts, so there are 2 × 6 = 12 vans.
A jacket costs $80. It is discounted by 25%. What is the sale price?
A 25% discount means the customer pays 75% of the original price: 80 × 0.75 = $60. Equivalently, the discount is 80 × 0.25 = $20, and 80 − 20 = $60.
A convoy travels at 60 miles per hour for 3 hours. How far does it travel?
Distance equals rate times time: 60 × 3 = 180 miles.
Four workers build a wall in 9 days. Working at the same rate, how many days would 6 workers need to build the same wall?
The job requires 4 × 9 = 36 worker-days. With 6 workers, the time is 36 ÷ 6 = 6 days.
On the clock
AR gives you the most seconds per item of any subtest, roughly 2 minutes and 25 seconds each on our practice template. Budget about 60 seconds to set the problem up and the rest to compute and check; if an item has not resolved by 3 minutes, flag it, pick the closest plausible answer and move on.
Free Arithmetic Reasoning practice on this site, graded instantly with explanations. Timed sections, wrong-answer review and the full library are in the app.