ASVAB math formulas: every formula for AR and MK, by topic

Every formula on this page has to be in your head before test day, because the ASVAB gives you no formula sheet and no calculator. Arithmetic Reasoning runs on percents, ratios, averages, rates, interest and basic probability. Mathematics Knowledge adds geometry and algebra. Each table below gives one topic's formulas, then a worked example.

Test Ninjas Research · Updated October 14, 2026

What the test gives you, and what it does not

The official ASVAB site says it in one line: "You cannot use a calculator when taking the ASVAB." Its FAQ gives the reason: the ASVAB is an employment test that measures whether you can apply principles to problems you might meet on the job, and that job may call for hand calculation. The site describes no formula sheet or reference page for any version of the test.

For the at-home PiCAT, the official tips tell you to have scratch paper and a pencil, and to finish without any outside help, including looking information up online.

No formula sheet, no calculator. Plan on the same conditions everywhere: you, a pencil, scratch paper and whatever you have memorized.

Both math subtests count toward your AFQT along with Word Knowledge and Paragraph Comprehension, so these formulas affect whether you can enlist, not only which jobs you qualify for. On the proctored computer test, Arithmetic Reasoning gives you 55 minutes for 15 scored questions, a little over 3.5 minutes each. Mathematics Knowledge gives you 31 minutes for 15, about 2 minutes each. That is time to set up a problem on paper, not to rebuild a formula from scratch.

Arithmetic Reasoning formulas

Arithmetic Reasoning is a word-problem test. The official samples include a tire rotating 552 times a minute, a $7,250 motorcycle losing 12% of its value in a year, and a faucet that runs 20 gallons in 5 seconds. Each is one formula from the tables below, dressed in a story.

A common setup error is a rate in miles per hour paired with a time in minutes, so learn the conversions with the formulas.

Everyday conversions

Unit	Equals
1 hour	60 minutes
1 foot	12 inches
1 yard	3 feet
1 mile	5,280 feet
1 pound	16 ounces
1 liter	1,000 milliliters

Reread the last sentence before you answer. Many wrong choices are the right number for the wrong quantity, such as the discount when the question asked for the sale price.

Percent and percent change

Percent formulas

Formula	Rule	When it appears
Find the part	part = percent (as a decimal) × whole	Discounts, tips, tax, "35% of 60"
Find the percent	percent = part ÷ whole × 100	"What percent of the 60 recruits passed"
Find the whole	whole = part ÷ percent (as a decimal)	"18 is 30% of what number"
Percent change	(new − original) ÷ original × 100	Raises, price changes, enrollment up or down
New amount	original × (1 + p/100) or original × (1 − p/100)	Markups, sale prices, depreciation
Original from new	new ÷ multiplier	"After a 20% raise she earns $18 an hour"
Successive changes	multiply the multipliers	"25% off, then another 10% off"

Worked example. A price rises from $40 to $50. The change is $10, and 10 ÷ 40 = 0.25, a 25% increase. Falling back to $40 is a 20% drop, because the same $10 is now divided by 50. Successive discounts multiply: 25% off and then 10% off an $80 jacket is 80 × 0.75 × 0.90 = $54, a total discount of 32.5%, not 35%. The percentages practice page has more drills.

Ratios and proportions

Ratio and proportion formulas

Formula	Rule	When it appears
Simplify a ratio	divide both terms by a common factor	36:12 = 3:1
Part to whole	the part's number ÷ the sum of the numbers	"What fraction of the group is..."
Split a total	total ÷ sum of the numbers = one share	"Divide $600 in the ratio 1:4:5"
Proportion	a/b = c/d, so ad = bc	Recipes, maps, "at this rate"
Unit rate	amount ÷ units	Price per ounce, gallons per second
Inverse variation	the product stays fixed	More workers, fewer days

Worked example. A unit of 64 splits 5:3 between soldiers and civilians. The parts add to 8, so one share is 64 ÷ 8 = 8: 40 soldiers and 24 civilians. If 3 gallons of paint cover 840 square feet, 1,400 square feet takes x gallons, where 3/840 = x/1,400. Cross-multiplying gives 840x = 4,200, so x = 5. If 30 workers need 12 days, the job is 360 worker-days, and 40 workers need 360 ÷ 40 = 9 days. The ratios practice page has more.

Averages and weighted averages

Average formulas

Formula	Rule	When it appears
Mean	sum ÷ count	Test scores, daily miles, monthly bills
Sum from mean	mean × count	"Five boxes average 14 pounds"
Score needed	target mean × new count − current sum	"What must she score on the next test"
Weighted average	sum of (value × weight) ÷ sum of the weights	Two classes combined, mixtures, grade points
Median	middle value when sorted, or the mean of the middle two	Data sets with one extreme value
Mode and range	most frequent value; greatest − least	Tables and lists of data

Worked example. Your first three scores are 84, 91 and 76, a sum of 251. Averaging 85 over four tests takes a sum of 85 × 4 = 340, so the fourth score must be 340 − 251 = 89. A class of 20 averaging 80 and a class of 30 averaging 90 combine to (1,600 + 2,700) ÷ 50 = 86. The tempting 85 ignores that the larger class carries more weight.

Distance, rate, time and work

Rate formulas

Formula	Rule	When it appears
Distance	d = r × t (also r = d ÷ t and t = d ÷ r)	Trips, convoys, runs
Average speed	total distance ÷ total time	A trip out and back at two speeds
Closing or separating	add the speeds	Two vehicles heading toward or away from each other
Chase, same direction	head start ÷ difference in speeds	One vehicle catching another
Work rate	1 job ÷ time to do it alone	"A pump fills the tank in 6 hours"
Working together	add the rates, then flip: T = AB ÷ (A + B) for two	Two crews, two pipes
Fill and drain	subtract the drain rate	An inlet and a leak
Minutes to hours	minutes ÷ 60	Any rate given per hour with time in minutes

Worked example. A driver covers 150 miles at 50 mph (3 hours) and returns at 75 mph (2 hours). The average speed is 300 miles ÷ 5 hours = 60 mph, not the 62.5 you get by averaging the two speeds. Pumps that fill a tank in 6 hours and in 3 hours have rates of 1/6 and 1/3. Together that is 1/6 + 2/6 = 1/2 of the tank per hour, so 2 hours, and the shortcut agrees: (6 × 3) ÷ (6 + 3) = 18 ÷ 9 = 2. The rate, time and work page has practice sets.

Simple interest

Interest formulas

Formula	Rule	When it appears
Simple interest	I = P × r × t, with t in years	Loans, savings accounts, bonds
Amount repaid	P + I	"How much will she owe in total"
Rate from interest	r = I ÷ (P × t)	"What annual rate did the account pay"
Months to years	months ÷ 12	Any loan stated in months
Compound yearly	multiply by (1 + r) each year	Questions that say "compounded"

Worked example. $3,000 borrowed at 4% simple interest for 18 months: t = 18 ÷ 12 = 1.5 years, so I = 3,000 × 0.04 × 1.5 = $180 and the total repaid is $3,180. Working backward, $96 of interest on $1,600 over 1.5 years means r = 96 ÷ 2,400 = 0.04, or 4%. $1,000 at 10% compounded yearly grows to $1,100 and then $1,210 after 2 years; simple interest gives $1,200.

Probability basics

Probability formulas

Formula	Rule	When it appears
One event	favorable outcomes ÷ total equally likely outcomes	Marbles, cards, a single die
Complement	P(not A) = 1 − P(A)	"What is the chance it is not green"
Independent events	multiply	Coin and die, draws with replacement
Without replacement	reduce the second fraction	Two draws from one bag
Either event, no overlap	add	"A 1 or a 6 on one roll"
At least one	1 − P(none)	Repeated tosses or tries
Counting principle	multiply the number of choices	Outfits, codes, menu combinations

Worked example. A bag holds 4 red and 6 blue marbles. Drawing two reds without replacement is 4/10 × 3/9 = 12/90 = 2/15. With replacement it is 4/10 × 4/10 = 4/25. The chance of at least one head in 3 coin tosses is 1 − 1/8 = 7/8. Two dice have 36 ordered outcomes, and 6 of them sum to 7, so P(7) = 6/36 = 1/6.

Mathematics Knowledge formulas

Mathematics Knowledge asks the math without the story. The official samples include the volume of a brick, a ratio to simplify, and an expression that collapses once you spot a difference of squares.

Perimeter and area

Perimeter and area formulas

Shape	Formula	When it appears
Rectangle	P = 2(l + w); A = lw	Rooms, fields, fencing, flooring
Square	P = 4s; A = s²	Tiles, square plots
Triangle	A = (1/2)bh, with h perpendicular to b	Roof gables, half of a rectangle
Parallelogram	A = bh, perpendicular height	Slanted figures
Trapezoid	A = (1/2)(b₁ + b₂)h	Lots and cross-sections with two parallel sides
Circle	C = 2πr = πd; A = πr²	Wheels, rugs, pipes, tracks
Square units	1 yd² = 9 ft²; 1 ft² = 144 in²	Carpet priced per square yard

Worked example. A 12 ft by 15 ft room has an area of 180 square feet, which is 180 ÷ 9 = 20 square yards, and a perimeter of 2(12 + 15) = 54 feet. A round rug 8 feet across has r = 4, so with π = 3.14 its area is 16π ≈ 50.24 square feet and its circumference is 8π ≈ 25.12 feet. A trapezoid with bases 8 and 14 and height 5 has area (8 + 14) ÷ 2 × 5 = 55. Halve a diameter before you use it as a radius: that one slip produces a ready-made wrong answer.

Volume

Volume formulas

Solid	Formula	When it appears
Box (rectangular solid)	V = lwh; cube V = s³	Bricks, crates, rooms, trenches
Cylinder	V = πr²h	Tanks, pipes, cans
Cone or pyramid	V = (1/3) × base area × h	Piles of sand, funnels
Sphere	V = (4/3)πr³	Balls, round tanks
Surface area of a box	2(lw + lh + wh)	Paint or wrap for a crate
Cubic units	1 yd³ = 27 ft³; 1 ft³ = 1,728 in³	Concrete ordered by the cubic yard
Scaling by k	lengths × k, area × k², volume × k³	"Each side is doubled"

Worked example. A cylindrical tank 4 feet across and 7 feet tall has r = 2, so V = π × 2² × 7 = 28π ≈ 87.92 cubic feet with π = 3.14. A cone with the same base and height holds a third of that, about 29.31 cubic feet. A sphere of radius 3 holds (4/3)π × 27 = 36π ≈ 113.04. A 3 by 4 by 5 box holds 60 cubic units and has a surface area of 2(12 + 15 + 20) = 94 square units.

Pythagorean theorem and common triples

Right triangle formulas

Formula	Rule	When it appears
Pythagorean theorem	a² + b² = c², c is the side opposite the right angle	Ladders, diagonals, shortest routes
Common triples	3-4-5, 5-12-13, 8-15-17, 7-24-25	Side lengths that come out whole
Multiples of triples	6-8-10, 9-12-15, 12-16-20, 10-24-26	Scaled versions of the same triangles
45-45-90 triangle	sides x, x, x√2	The diagonal of a square
30-60-90 triangle	sides x, x√3, 2x	Half of an equilateral triangle
Distance between points	√((x₂ − x₁)² + (y₂ − y₁)²)	Coordinate grid questions

Worked example. A 20-foot ladder stands with its foot 12 feet from a wall. The height it reaches is √(20² − 12²) = √(400 − 144) = √256 = 16 feet. Spotting 12-16-20 as 4 times 3-4-5 gets you there without squaring anything. The theorem works only for right triangles.

Angle facts

Angle formulas

Fact	Rule	When it appears
Complementary and supplementary	add to 90°; add to 180°	"Find the missing angle"
Vertical angles	equal	Two intersecting lines
Parallel lines and a transversal	corresponding and alternate interior angles are equal; same-side interior angles add to 180°	Diagrams with two parallel lines
Triangle	angles add to 180°; an exterior angle equals the two remote interior angles	Any triangle problem
Isosceles triangle	equal sides face equal angles	Two marked sides
Polygon interior sum	(n − 2) × 180°; each angle of a regular polygon is the sum ÷ n	Hexagons, octagons, stop signs
Exterior angles	add to 360° for any convex polygon	Regular polygon shortcuts

Worked example. A regular octagon has an interior sum of (8 − 2) × 180 = 1,080°, so each angle is 1,080 ÷ 8 = 135°. The exterior route agrees: 360 ÷ 8 = 45°, and 180 − 45 = 135°. If a triangle's angles are in the ratio 2:3:4, the 9 parts make 180°, so each part is 20° and the angles are 40°, 60° and 80°.

Exponent and radical rules

Rule	Formula	When it appears
Product	aᵐ × aⁿ = aᵐ⁺ⁿ	Simplifying x³ × x⁴
Quotient	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Fractions with powers
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(x³)⁴
Power of a product	(ab)ⁿ = aⁿbⁿ	(2x³)²
Zero and negative exponents	a⁰ = 1 and a⁻ⁿ = 1/aⁿ, for a not 0	5⁻² and 7⁰
Roots over products and quotients	√(ab) = √a × √b; √(a/b) = √a ÷ √b, for a and b not negative	Simplifying √72 or √(16/25)
No rule over sums	√a + √b is not √(a + b)	A common wrong choice
Scientific notation	a × 10ⁿ with a at least 1 and less than 10; add exponents to multiply	Very large or very small numbers

Worked example. (2x³)² × x⁴ = 4x⁶ × x⁴ = 4x¹⁰. √72 = √36 × √2 = 6√2. √36 + √64 is 6 + 8 = 14, not √100 = 10. In scientific notation, (3 × 10⁴)(5 × 10⁻⁷) = 15 × 10⁻³, which renormalizes to 1.5 × 10⁻².

Linear equations and slope

Line and equation formulas

Formula	Rule	When it appears
Solving	do the same thing to both sides; undo operations in reverse order	Every "solve for x" question
Inequalities	flip the sign when you multiply or divide by a negative	−3x ≥ 15
Slope	m = (y₂ − y₁) ÷ (x₂ − x₁)	Two points given
Slope-intercept form	y = mx + b	Reading slope and y-intercept
Point-slope form	y − y₁ = m(x − x₁)	A slope and one point given
Parallel and perpendicular	equal slopes; negative reciprocal slopes	"A line perpendicular to y = 2x + 1"
Intercepts	set x = 0 for the y-intercept; set y = 0 for the x-intercept	Where a line crosses an axis
Midpoint	((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)	The center of a segment

Worked example. The line through (2, 3) and (6, 11) has slope (11 − 3) ÷ (6 − 2) = 2. Substituting (2, 3) into y = 2x + b gives 3 = 4 + b, so b = −1 and the line is y = 2x − 1. Check the other point: 2 × 6 − 1 = 11. A perpendicular line would have slope −1/2. To solve 4(x + 2) − 3 = 2x + 11, distribute to get 4x + 5 = 2x + 11, so 2x = 6 and x = 3. Both sides then equal 17.

Factoring patterns

Pattern	Formula	When it appears
Greatest common factor	pull it out first	6x² + 9x = 3x(2x + 3)
Difference of squares	a² − b² = (a − b)(a + b)	x² − 49, or simplifying (x² − y²) ÷ (x − y)
Perfect square trinomial	a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)²	x² + 12x + 36
Trinomial x² + bx + c	two numbers with product c and sum b	x² + x − 12
Multiplying binomials	(x + p)(x + q) = x² + (p + q)x + pq	Expanding, or checking a factorization
Sum of squares	a² + b² does not factor over real numbers	A trap choice

Worked example. For x² + x − 12, you need two numbers with product −12 and sum 1: 4 and −3, so x² + x − 12 = (x + 4)(x − 3). The pattern 9x² − 25 is (3x)² − 5², so it factors as (3x − 5)(3x + 5). The same pattern speeds up mental arithmetic: 51 × 49 = 50² − 1² = 2,499. For the official sample (x² − y²) ÷ (x − y), factor the top and cancel to get x + y.

Quadratics

Quadratic formulas

Formula	Rule	When it appears
Standard form	ax² + bx + c = 0	Move every term to one side first
Zero-product rule	if pq = 0, then p = 0 or q = 0	After factoring
Square root method	x² = k with k positive gives x = ±√k	No x term, such as x² = 49
Quadratic formula	x = (−b ± √(b² − 4ac)) ÷ (2a)	Any quadratic, including ones that do not factor
Discriminant	b² − 4ac positive: 2 real roots; zero: 1 repeated root; negative: no real roots	"How many real solutions"
Roots of x² + bx + c	they add to −b and multiply to c	Building an equation from its roots

Worked example. x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0, so x = 2 or x = 3. The roots add to 5 and multiply to 6, matching −b and c. For 2x² + 3x − 2 = 0, the discriminant is 9 + 16 = 25, so x = (−3 ± 5) ÷ 4, which gives 1/2 or −2. It also factors, as (2x − 1)(x + 2). Check x = 1/2: 0.5 + 1.5 − 2 = 0.

Memorizing the formulas

Write the formulas from memory onto a blank page, topic by topic, and check them against these tables. The ones you miss go on tomorrow's page. Two weeks of that beats rereading a cheat sheet, because the test asks you to produce a formula, not recognize one.

Then practice on problems that do not name the formula. A percent change question never says "percent change." It says a price went from $40 to $50. Work timed sets on the math practice page, and take every miss back to the matching row above.

The same drill works for the technical subtests. Mechanical Comprehension and Electronics Information have formulas of their own, from levers and gear ratios to Ohm's law, collected on the mechanical and electrical formulas page.

Frequently Asked Questions

No. The official ASVAB site describes no formula sheet or reference page for any version of the test, and it states that you cannot use a calculator. Every formula you need for Arithmetic Reasoning and Mathematics Knowledge has to be memorized before test day.

No. The official ASVAB site says you cannot use a calculator, and its FAQ explains that the ASVAB is an employment test and hand calculation may be required on the job. The rule covers the proctored test and the at-home PiCAT, where the official tips also bar outside help such as looking things up online.

Percent and percent change, ratios and proportions, distance equals rate times time, work rates, averages and simple interest. Arithmetic Reasoning is a word-problem test, so the hard part is usually choosing which of these the story calls for. Once the setup is right, the arithmetic is short.

Exponent and radical rules, factoring patterns, solving linear equations, slope, quadratics, and the geometry set: perimeter, area, volume, the Pythagorean theorem and angle facts. The official site describes the subtest as knowledge of high school mathematics principles.

Learn it as a backup. Quadratics written for a no-calculator test often factor with small whole numbers, and factoring is faster when it works. The quadratic formula solves every quadratic, including the ones that do not factor neatly.

Follow the question. If it says to use 3.14 or 22/7, multiply it out. If the answer choices contain the symbol, leave pi as a symbol and keep the arithmetic exact.

On the proctored computer test, Arithmetic Reasoning gives you 55 minutes for 15 scored questions and Mathematics Knowledge gives you 31 minutes for 15. The paper version gives 36 minutes for 30 Arithmetic Reasoning questions and 24 minutes for 25 Mathematics Knowledge questions. Some examinees also get unscored tryout questions with a longer limit.

Sources

U.S. Department of Defense, official ASVAB website (officialasvab.com), Applicants: Frequently Asked Questions, page dated October 6, 2026, the answers on calculator use and on which four tests make up the AFQT.
U.S. Department of Defense, official ASVAB website, Applicants: What to Expect, the calculator rule, the subtest descriptions for Arithmetic Reasoning and Mathematics Knowledge, and the CAT-ASVAB and paper-and-pencil item counts and time limits, read October 2026.
U.S. Department of Defense, official ASVAB website, Sample Questions for Arithmetic Reasoning (AR) and Mathematics Knowledge (MK), read October 2026.
U.S. Department of Defense, official ASVAB website, Unproctored Administration of the ASVAB: PiCAT, the testing tips on scratch paper, outside assistance and calculators, read October 2026.
ASVAB course on this site, lessons AR-02 through AR-06 and MK-03 through MK-09, summary formula tables, checked October 2026 so the formulas and wording here match the lessons.