Arithmetic Reasoning (AR)

Introduction

Arithmetic Reasoning tests whether you can take a real-world situation described in words and turn it into the right calculation. It is not a test of raw computation speed; it is a test of setup. Most wrong answers here come not from bad arithmetic but from solving the wrong problem — multiplying when you should divide, answering "how many are left" when the question asked "how many in all." Because AR is one of the four AFQT subtests that decide enlistment eligibility, the points you earn here are among the most valuable in the whole battery.

The winning habit is slow reading and deliberate setup: figure out what is being asked, pick out the numbers that matter, choose the operation that connects them, and track your units so the answer comes out in the form the question wants.

Key formulas & facts

Part A — Reading and setting up word problems

The first move is translation. Before you touch a number, restate in your own words what the question is asking you to find. Then locate the numbers that matter — and notice that some problems include a number you do not need, planted to see whether you understand the situation or just grab digits.

Match the wording to the operation. Words like total, combined, altogether, and sum signal addition; difference, how many more, and left signal subtraction; of (as in "25% of 240") usually signals multiplication; and per, each, and split evenly signal division. Finally, carry the units through the work. If the question asks for hours, your setup should be arranged so that miles cancel and hours remain. A setup that produces the right number in the wrong unit is usually a setup that is actually wrong.

Part B — Fractions, decimals, and percents

Fluency at moving between fractions, decimals, and percents lets you pick the easiest form for a calculation and sanity-check the result. To turn a percent into a decimal, divide by 100 (so 8% becomes 0.08). To find a percent of a number, multiply the whole by that decimal. To turn a fraction into a decimal, divide the numerator by the denominator (3/4 = 0.75). And the single most useful shortcut on this subtest: an increase or decrease by a percent is one multiplication. A 15% discount is a multiplication by 0.85; an 8% tax is a multiplication by 1.08. That one-step form is faster and less error-prone than computing the part and then adding or subtracting it.

Part C — Ratios, proportions, and rates

A ratio compares two quantities; reduce it like a fraction to see the relationship clearly. A proportion sets two ratios equal so you can scale up or down — place matching quantities in the same position on each side, then cross-multiply to solve for the unknown. A rate is a ratio with different units, and the unit rate (cost per pound, miles per hour) is the workhorse: once you have the per-one price, any quantity is a single multiplication away.

Part D — Averages, interest, and multi-step problems

Higher-value AR questions stack two or three steps. To find an average, add the values and divide by the count; to find a missing value when you know the average, first rebuild the total as average × count, then subtract the known values. Simple interest is principal × rate × time, computed only on the original amount. Work-rate problems multiply rate × number of workers × time to get total output. And on every multi-step problem, estimate to check: round the numbers, get a ballpark, and confirm your exact answer lands near it.

Worked examples

Example 1 — a rate problem. A train travels at a constant 60 miles per hour. How long to travel 180 miles?

Use time = distance ÷ rate. time = 180 miles ÷ 60 miles per hour = 3 hours. Notice the units: miles ÷ (miles per hour) leaves hours, exactly what the question asked for.

Example 2 — unit rate then scale. If 5 pounds of apples cost \$4.00, what do 12 pounds cost at the same rate?

First the unit rate: \$4.00 ÷ 5 lb = \$0.80 per pound. Then scale: \$0.80 × 12 lb = \$9.60. (A proportion, 4/5 = x/12, cross-multiplies to 5x = 48, x = 9.6 — the same answer.)

Example 3 — a missing value from an average. Four test scores average 85. Three of them are 80, 90, and 85. Find the fourth.

Rebuild the total: 85 × 4 = 340. The three known scores sum to 80 + 90 + 85 = 255. The fourth score is 340 − 255 = 85.

Example 4 — a percent increase (tax) in one step. A meal costs \$40 before an 8% tax. What is the total?

One multiplication: \$40 × 1.08 = \$43.20. (Checking the long way: 8% of 40 is \$3.20, and 40 + 3.20 = 43.20.)

Example 5 — a two-rate "opposite directions" problem. Two cars leave the same point in opposite directions at 40 mph and 55 mph. How far apart after 2 hours?

Opposite directions means the separation speeds add: 40 + 55 = 95 mph. In 2 hours: 95 × 2 = 190 miles apart.

Common traps

• Solving the wrong question — finding "how many girls" when the question asked "how many boys." Restate the question first.
• Grabbing an unneeded number. Some problems include a value you do not use. Select only what the question needs.
• Mixing units. Convert everything to the same unit (all feet, all dollars) before calculating.
• Adding a percent in two steps and slipping. Prefer the one-step multiplier (× 1.08 for +8%, × 0.85 for −15%).
• Forgetting to rebuild the total on average problems before finding a missing value.

Self-check

1. A worker assembles 8 units per hour. How many do 3 workers assemble in 5 hours? (8 × 3 × 5 = 120 units.)
2. What is 25% of 240? (0.25 × 240 = 60.)
3. A \$80 jacket is discounted 15%. Sale price? (80 × 0.85 = \$68.)
4. You deposit \$500 at 4% simple interest for 3 years. Interest earned? (500 × 0.04 × 3 = \$60.)
5. A map scale is 1 inch = 25 miles. Two towns are 4.5 inches apart. Actual distance? (4.5 × 25 = 112.5 miles.)