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MATH REVIEW

AFOQT Math Formula Sheet with Worked Examples

Review useful AFOQT math relationships, when to use them, and common mistakes. Work through six original examples, then practice Math Knowledge.

Use this as a review sheet

Start with the relationship you need, check its conditions, and solve the example before reading the steps. This is a selected set of review relationships, not an official or complete formula list. Print it with your browser’s Print command for studying at home; it is not a document to bring into the exam.

If you know a formula but struggle to choose it from a story, see Arithmetic Reasoning vs. Math Knowledge.

Fractions and percentages

a/b ÷ c/d = ad/bc

Symbols:
a/b is the first fraction; c/d is the nonzero divisor.
Use when:
Dividing one fraction by another. Multiply by the reciprocal of the divisor.
Conditions and units:
b, c, and d must be nonzero. Reduce the result when possible.
Avoid:
Flipping the first fraction instead of the divisor.

part = (percent ÷ 100) × whole

Symbols:
part is the requested amount; percent is the stated rate; whole is the starting total.
Use when:
Finding a percentage of a known quantity.
Conditions and units:
Convert the percent to a decimal or fraction before multiplying. Keep units consistent.
Avoid:
Multiplying by 18 instead of 0.18 for 18%.

percent change = (new − original) / original × 100%

Symbols:
original is the baseline; new is the changed value.
Use when:
Measuring increase or decrease relative to the starting value.
Conditions and units:
The original value must be nonzero. A negative result represents a decrease.
Avoid:
Dividing by the new value instead of the original value.

Ratios and proportions

a/b = c/d ⇔ ad = bc

Symbols:
a:b and c:d are two ratios written as fractions.
Use when:
Finding a missing value in an equivalent ratio or scale relationship.
Conditions and units:
b and d must be nonzero. Attach the matching units to corresponding positions.
Avoid:
Pairing unlike units or cross-multiplying values in the wrong order.

one share = total / (sum of ratio parts)

Symbols:
total is the whole amount; ratio parts are the numbers in the ratio.
Use when:
Splitting a total according to a stated ratio.
Conditions and units:
Use positive ratio parts with a nonzero sum. All parts must describe the same whole and use the same unit.
Avoid:
Dividing the total by only one side of the ratio.

Algebra

a(b + c) = ab + ac

Symbols:
a is the outside factor; b and c are the terms inside parentheses.
Use when:
Expanding an expression or factoring out a common factor.
Conditions and units:
The outside factor applies to every term inside the parentheses.
Avoid:
Multiplying only the first term inside the parentheses.

ax + b = c ⇒ x = (c − b)/a

Symbols:
x is the unknown; a is its nonzero coefficient; b and c are constants.
Use when:
Solving a one-variable linear equation.
Conditions and units:
a cannot be zero. Apply the same operation to both sides of the equation.
Avoid:
Changing a sign without applying the inverse operation to both sides.

m = (y₂ − y₁)/(x₂ − x₁)

Symbols:
m is slope; the two ordered pairs are (x₁, y₁) and (x₂, y₂).
Use when:
Finding the constant rate of change between two points.
Conditions and units:
x₂ − x₁ cannot be zero; a vertical line has undefined slope.
Avoid:
Reversing the subtraction order in only the numerator or denominator.

Exponents

aᵐ × aⁿ = aᵐ⁺ⁿ

Symbols:
a is the shared base; m and n are exponents.
Use when:
Multiplying powers that have the same base.
Conditions and units:
These review rules use integer exponents. The bases must match; a must be nonzero if an exponent is zero or negative.
Avoid:
Multiplying the exponents or applying the rule to unlike bases.

aᵐ / aⁿ = aᵐ⁻ⁿ

Symbols:
a is the shared nonzero base; m and n are exponents.
Use when:
Dividing powers that have the same base.
Conditions and units:
These review rules use integer exponents. a cannot be zero, and the bases must match.
Avoid:
Dividing the exponents instead of subtracting them.

(aᵐ)ⁿ = aᵐⁿ

Symbols:
a is the base; m is the inner exponent; n is the outer exponent.
Use when:
Raising an existing power to another power.
Conditions and units:
These review rules use integer exponents. Multiply the exponents; a must be nonzero if an exponent is zero or negative.
Avoid:
Using aᵐ⁺ⁿ, which belongs to multiplication of like bases.

Geometry

A = lw; P = 2l + 2w

Symbols:
A is area; P is perimeter; l is length; w is width.
Use when:
Finding the inside coverage or outside boundary of a rectangle.
Conditions and units:
Length and width must use the same linear unit. Area uses squared units.
Avoid:
Using the area formula when the problem asks for boundary length.

A = ½bh

Symbols:
A is area; b is the selected base; h is the perpendicular height.
Use when:
Finding the area of any triangle from a base and its perpendicular height.
Conditions and units:
Height must be perpendicular to the base. Area uses squared units.
Avoid:
Using a slanted side as the height or forgetting the factor of ½.

a² + b² = c²

Symbols:
a and b are the legs; c is the hypotenuse opposite the right angle.
Use when:
A triangle is known to be right and one side length is missing.
Conditions and units:
This relationship applies only to right triangles; all lengths use the same unit.
Avoid:
Treating an arbitrary side as c instead of identifying the hypotenuse.

Speed, distance, and time

d = rt; r = d/t; t = d/r

Symbols:
d is distance; r is average rate; t is elapsed time.
Use when:
A problem connects constant average speed, travel time, and distance.
Conditions and units:
For these travel formulas, use positive time and speed, and nonnegative distance. Rate and time units must match; convert minutes to hours when using miles per hour. Do not divide by zero time or zero speed.
Avoid:
Treating 1 hour 15 minutes as 1.15 hours instead of 1.25 hours.

Six worked examples

All answers are exact. Decimal money and distance answers below terminate; no rounding is required.

Example 1 · Fractions and percentages

What is 18% of 250?

  • A. 46
  • B. 44
  • C. 45
  • D. 4/5

Build the relationship: part = 18/100 × 250

  1. Reduce 18/100 to 9/50.
  2. Multiply 9/50 × 250 = 9 × 5.

Answer: 45

Common error: Using 1.8 instead of 0.18 makes the result 450, ten times too large.

Example 2 · Ratios and proportions

Solve 3/5 = x/40.

  • A. 8
  • B. 25
  • C. 23
  • D. 24

Build the relationship: 3 × 40 = 5x

  1. Cross-multiply to get 120 = 5x.
  2. Divide both sides by 5.

Answer: x = 24

Common error: Dividing 40 by 5 gives the scale factor 8; it still must be multiplied by 3 to get 24.

Example 3 · Algebra

Solve 4(x − 3) = 28.

  • A. 9
  • B. 10
  • C. 4
  • D. 11

Build the relationship: x − 3 = 28/4

  1. Divide both sides by 4 to get x − 3 = 7.
  2. Add 3 to both sides.

Answer: x = 10

Common error: Subtracting 3 from 7 instead of undoing −3 with addition produces 4.

Example 4 · Exponents

Simplify (2³ × 2⁴) / 2².

  • A. 8
  • B. 33
  • C. 31
  • D. 32

Build the relationship: 2³⁺⁴⁻² = 2⁵

  1. Add exponents when multiplying like bases: 3 + 4 = 7.
  2. Subtract the denominator exponent: 7 − 2 = 5, then evaluate 2⁵.

Answer: 32

Common error: Multiplying and dividing the exponents as (3 × 4) / 2 gives 6, then 2⁶ = 64. Like-base multiplication requires adding exponents, and division requires subtracting them.

Example 5 · Geometry

A triangle has a 10-inch base and a perpendicular height of 7 inches. What is its area?

  • A. 35 square inches
  • B. 17 square inches
  • C. 36 square inches
  • D. 34 square inches

Build the relationship: A = ½ × 10 × 7

  1. Confirm that 7 inches is perpendicular to the 10-inch base.
  2. Multiply and keep squared units: 70/2 = 35.

Answer: 35 square inches

Common error: Multiplying base by height but omitting ½ gives 70 square inches.

Example 6 · Speed, distance, and time

At a constant 55 miles per hour, how far do you travel in 1½ hours?

  • A. 163/2 miles
  • B. 165/2 miles
  • C. 110/3 miles
  • D. 167/2 miles

Build the relationship: d = 55 × 3/2

  1. Convert 1½ hours to the exact fraction 3/2 hours.
  2. Multiply: 55 × 3/2 = 165/2.

Answer: 82.5 miles

Common error: Dividing 55 by 1½ reverses the relationship and gives 36⅔ instead of distance.

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 Knowledge

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.

See our calculator-policy source checks and how to confirm your appointment rules

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.