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MATH REVIEW

AFOQT Arithmetic Reasoning vs. Math Knowledge

Arithmetic Reasoning emphasizes solving word problems; Math Knowledge emphasizes mathematical principles. Compare four original pairs and choose what to practice.

How the skills overlap

Arithmetic Reasoning asks you to identify quantities and relationships in a written scenario, then calculate the requested result. Math Knowledge focuses on arithmetic, algebra, geometry, and related mathematical principles. It can also include algebra-based word problems, so “Math Knowledge never has words” is not a useful rule.

The paired examples below deliberately share a concept. They help you separate a calculation error from a setup error. They illustrate study skills and are not official questions or a guarantee of the format of any exam item.

Pair 1: Percent calculation

Math Knowledge focus

Find 35% of 80.

  • A. 12
  • B. 29
  • C. 27
  • D. 28

Build the relationship: 35/100 × 80

  1. Reduce 35/100 to 7/20.
  2. Multiply 7/20 × 80 = 28.

Answer: 28

Common error: Using the percent number itself ignores the whole of 80.

Arithmetic Reasoning focus

A jacket costs $80 and is discounted 35%. What is the sale price before tax?

  • A. $28.00
  • B. $53.00
  • C. $51.00
  • D. $52.00

Build the relationship: $80 − (35/100 × $80)

  1. The discount is $28.
  2. Subtract the discount from the original price: $80 − $28.

Answer: $52.00

Common error: $28 is the discount amount, not the requested sale price.

What each item requires
Math Knowledge asks for the percent operation directly. Arithmetic Reasoning makes you decide whether the question wants the discount or the remaining price.
Why the word problem can go wrong
A learner can calculate 35% correctly and still answer $28 because the wording asks for a second step.
What to practice next
Practice labeling the requested quantity before calculating: discount, original price, or final price.

Pair 2: Ratio simplification

Math Knowledge focus

Simplify the ratio 18:24.

  • A. -1:4
  • B. 3:4
  • C. 1:2
  • D. 7:4

Build the relationship: 18 ÷ 6 : 24 ÷ 6

  1. The greatest common factor is 6.
  2. Divide both terms by 6 to get 3:4.

Answer: 3:4

Common error: Dividing 18 by 18 but 24 by 12 changes the relationship and produces 1:2. Both terms must be divided by the same nonzero factor.

Arithmetic Reasoning focus

A 28-person team has pilots and engineers in a 3:4 ratio. How many pilots are on the team?

  • A. 7 pilots
  • B. 13 pilots
  • C. 11 pilots
  • D. 12 pilots

Build the relationship: 3 + 4 = 7 parts; 28/7 = 4 people per part; 3 × 4 pilots

  1. Add ratio parts to get 7 total shares.
  2. Each share is 4 people, so the pilot share is 3 × 4.

Answer: 12 pilots

Common error: 16 is the engineer share. The question asks for the 3-part pilot share.

What each item requires
The direct item checks whether you can reduce a ratio. The word problem checks whether you can turn ratio parts into shares of a total.
Why the word problem can go wrong
Knowing that 18:24 becomes 3:4 does not automatically tell you to add the parts before dividing a total.
What to practice next
Practice drawing ratio boxes and writing the total number of shares before using the given total.

Pair 3: Linear equation

Math Knowledge focus

Solve 5x + 7 = 32.

  • A. 6
  • B. 4
  • C. 5
  • D. 25

Build the relationship: 5x = 32 − 7

  1. Subtract 7 from both sides to get 5x = 25.
  2. Divide both sides by 5.

Answer: x = 5

Common error: Adding 7 instead of subtracting it gives 39/5.

Arithmetic Reasoning focus

A ride costs a $7 base fee plus $5 per mile. The total is $32. How many miles was the ride?

  • A. 6 miles
  • B. 4 miles
  • C. 5 miles
  • D. 25 miles

Build the relationship: 7 + 5m = 32

  1. Translate the base fee and per-mile charge into 7 + 5m.
  2. Subtract 7 and divide the remaining $25 by $5 per mile.

Answer: 5 miles

Common error: Dividing the full $32 by $5 ignores the fixed $7 base fee.

What each item requires
Both questions reduce to the same equation. Math Knowledge supplies it; Arithmetic Reasoning requires you to build it from the pricing rule.
Why the word problem can go wrong
A solver may handle 5x + 7 = 32 easily but divide $32 by $5 before accounting for the fixed fee.
What to practice next
Underline fixed amounts and per-unit rates, then write the equation before doing arithmetic.

Pair 4: Formula substitution

Math Knowledge focus

Using d = rt, find d when r = 48 and t = 5/4.

  • A. 192/5
  • B. 61
  • C. 59
  • D. 60

Build the relationship: d = 48 × 5/4

  1. Substitute the given values.
  2. Reduce 48/4 to 12, then multiply by 5.

Answer: d = 60

Common error: Dividing by 5/4 instead of multiplying gives 192/5.

Arithmetic Reasoning focus

An aircraft travels at an average 48 miles per hour for 1 hour 15 minutes. How far does it travel?

  • A. 55.2 miles
  • B. 61 miles
  • C. 59 miles
  • D. 60 miles

Build the relationship: 1 hour 15 minutes = 5/4 hours; d = 48 × 5/4

  1. Convert 15 minutes to 15/60 = 1/4 hour.
  2. Multiply 48 miles per hour by 5/4 hours.

Answer: 60 miles

Common error: Reading 1 hour 15 minutes as 1.15 hours gives 55.2 miles; minutes are sixtieths of an hour, not hundredths.

What each item requires
The direct item tests substitution. The word problem adds the decision to convert minutes into a fraction of an hour.
Why the word problem can go wrong
Formula recall can be perfect while the input units are wrong, producing a precise but incorrect result.
What to practice next
Write units beside every number and convert them before substituting into a rate formula.

Choose your next practice by the error

  • If the equation is supplied and you still miss it, review the concept and calculation. The formula sheet with worked examples gives conditions as well as formulas.
  • If the calculation works after someone writes the equation for you, practice extracting the requested quantity, fixed amounts, rates, and units from the words.
  • If both are accurate but slow, use the existing no-calculator drill and method demonstrations after checking that each shortcut applies.

Put the review into practice

Use worked solutions to label each miss: concept, setup, calculation, or units. Review that skill, then try another set. Your practice accuracy describes this set; it does not predict an AFOQT score, percentile, or passing probability.

Practice Math KnowledgePractice Arithmetic Reasoning

Arithmetic Reasoning opens the existing 12-question timed sample drill.

Sources and verification

Source notes updated: . This records a wording update, not a new full content review.

Reviewed October 6, 2026. The topic scope follows Pearson VUE’s descriptions of arithmetic, algebra, geometry, exponents, ratios, percentages, and time-distance. The Air Force pamphlet supports the distinction between arithmetic problem solving and mathematical principles; its older testing schedule and paper-based procedures are not used as current rules. All examples here are original study material.

Policy sources checked: 2026-10-11. The Pearson VUE AFOQT page and the Air Force practice pamphlet reviewed do not state a calculator rule for current AFOQT appointments. Calculator policy remains unverified; confirm it with your test control officer or testing provider. Scratch materials are addressed separately below.

The Air Force practice pamphlet describes supplied scratch paper alongside pencil-and-answer-sheet procedures (page 4). That statement does not verify the materials supplied at a current computer-based appointment. Current permission to bring formula references has not been verified. This page is for preparation and does not authorize bringing notes into the exam.

AFOQT Lab is an independent study aid and is not affiliated with the U.S. Air Force or Pearson VUE.