![\"Four](\"https://res.cloudinary.com/hlt-media/image/upload/v1779274278/hlt-mmm2/generated/mmm2-hlt-mastery-inline-educational-illustration-mpdxyrjp.webp\")
Before you memorize a single formula, understand the battlefield. These two subtests look similar but fight differently.
\n| Arithmetic Reasoning (AR) | Mathematics Knowledge (MK) | |
|---|---|---|
| Questions | 30 | 25 |
| Time Limit | 36 minutes | 24 minutes |
| Time Per Question | ~72 seconds | ~58 seconds |
| Format | Word problems | Straight math (no story) |
| Tests Your | Reasoning + setup + calculation | Pure formula recall + execution |
| Key Topics | Percentages, ratios, d=rt, work rate, interest | Algebra, geometry, exponents, angles |
| AFQT Weight | Yes — 25% of AFQT | Yes — 25% of AFQT |
| Biggest Trap | Misreading the question | Mixing up similar formulas |
Why this matters: AR + MK = 50% of your AFQT. Your AFQT percentile determines branch eligibility — Army needs 31+, Air Force needs 36+, Space Force needs 70+. Higher scores = more MOS/AFSC/rate options. These two sections are where you move the needle fastest.\n
AR is all word problems. The math itself isn't hard — the challenge is translating English into equations. Here's every formula you'll need, organized by problem type.
\nThese show up on almost every AR section. Drill them until they're automatic.
Part = Percent × Whole → or rearranged: Percent = (Part ÷ Whole) × 100((New − Old) ÷ Old) × 100Original Price × (1 − Discount%)Original Price × (1 + Rate%)Quick example: A $80 jacket is 25% off. Price = $80 × (1 − 0.25) = $80 × 0.75 = $60.
\nIf you see \"for every\" or \"per\" in a word problem, you're in ratio territory.
a/b = c/da × d = b × cQuick example: If 3 widgets cost $12, how much do 7 cost? → 3/12 = 7/x → 3x = 84 → x = $28.
\nThe king of AR word problems. If something is moving, this is your formula.
Distance = Rate × Time (d = rt)r = d ÷ tt = d ÷ rQuick example: You drive 180 miles at 60 mph. Time = 180 ÷ 60 = 3 hours.
\n\"Person A can do a job in X hours, Person B in Y hours. How long together?\" This formula handles it.
1/t₁ + 1/t₂ = 1/TQuick example: Alex paints a room in 4 hours, Jamie in 6 hours. Together: 1/4 + 1/6 = 3/12 + 2/12 = 5/12. T = 12/5 = 2.4 hours.
\nI = P × r × t (Principal × rate × time)A = P + I or A = P(1 + rt)Profit = Revenue − CostMean = Sum of all values ÷ Number of valuesAR Trap Alert: Word problems love to hide extra information or switch units mid-problem. Always check: Are the units consistent? Did they give you hours but ask for minutes? Read the question twice before you calculate once.\n
Don't stare at a word problem wondering where to start. Follow this decision tree every single time — it works whether the problem is about money, distance, or paint cans.
\nUse this copyable decision path instead of a blurry image. Start with the question, choose the problem family, then check units before you answer.
| Step | What to look for | Move |
|---|---|---|
| 1. Read | The final sentence and the units requested | Do not calculate until you know what the question asks. |
| 2. Classify | Percent, ratio, distance/rate/time, work rate, or money over time | Pick the matching formula family. |
| 3. Set up | Numbers that belong in the formula vs. extra distractors | Write the equation before doing arithmetic. |
| 4. Solve | No-calculator arithmetic | Keep numbers simple; cancel or reduce when possible. |
| 5. Check | Units and real-world size | Convert minutes/hours, inches/feet, or percent/decimal before choosing. |
The AR Attack Sequence:
MK strips away the word problem wrapper and tests you on raw math. You either know the formula or you don't. No partial credit for vibes.
\ny = mx + b (m = slope, b = y-intercept)m = (y₂ − y₁) ÷ (x₂ − x₁)x = (−b ± √(b² − 4ac)) ÷ 2a(a+b)(c+d) = ac + ad + bc + bda² − b² = (a+b)(a−b)These are free points if you memorize the patterns.
xᵃ × xᵇ = x^(a+b) — multiplying same base? Add exponents.(xᵃ)ᵇ = x^(a×b) — power of a power? Multiply exponents.xᵃ ÷ xᵇ = x^(a−b) — dividing same base? Subtract exponents.x⁰ = 1 — anything to the zero power equals 1 (except 0⁰).x⁻ⁿ = 1/xⁿ — negative exponent? Flip it to a fraction.A = length × widthA = ½ × base × heightA = πr²A = ½(b₁ + b₂) × hA = base × heightP = 2l + 2wP = a + b + c (sum of all sides)C = 2πr or C = πdV = l × w × hV = πr²hV = (4/3)πr³V = (1/3)πr²ha² + b² = c² (right triangles only — c is the hypotenuse)180°360°180°90°equalHigh-Yield Shortcut: Memorize the Pythagorean triples (3-4-5, 5-12-13, 8-15-17). When you see a right triangle on the MK section, check if the given sides match a triple or a multiple of one. Instant answer — no calculation needed.\n
Geometry questions give you a shape and ask you to find something. This flowchart gets you to the right formula in seconds.
\nPick the formula by matching the shape to what the problem asks you to find.
| If the shape is… | And they ask for… | Use |
|---|---|---|
| Circle | Area | A = πr² |
| Circle | Circumference | C = 2πr |
| Triangle | Area | A = ½bh |
| Right triangle | Missing side | a² + b² = c² |
| Rectangle | Area or perimeter | A = lw or P = 2l + 2w |
| Box | Volume | V = lwh |
| Cylinder | Volume | V = πr²h |
The Geometry Decision Path:
A = πr²C = 2πrA = ½bha² + b² = c²A = lwV = lwhV = πr²hFormulas are useless if you can't recall them under pressure. These memory tricks lock them in.
\n| P | Parentheses first |
| E | Exponents next |
| M | Multiplication (left to right with Division) |
| D | Division (left to right with Multiplication) |
| A | Addition (left to right with Subtraction) |
| S | Subtraction (left to right with Addition) |
Remember it as: Please Excuse My Dear Aunt Sally. Note: Multiplication and Division are equal priority (left to right), and so are Addition and Subtraction. Don't multiply before dividing just because M comes before D.\n
| SOH | Sine = Opposite ÷ Hypotenuse |
| CAH | Cosine = Adjacent ÷ Hypotenuse |
| TOA | Tangent = Opposite ÷ Adjacent |
Your time budget:\n
• AR: 36 minutes ÷ 30 questions = 72 seconds each
• MK: 24 minutes ÷ 25 questions = 58 seconds each
The rule: If you've spent 90 seconds on a single question, pick your best guess and move. Getting stuck on one question means leaving easier points on the table. On the CAT version, you can't go back — so guessing and moving is always better than burning time.
A car travels 180 miles in 3 hours. The answer choices are in miles per hour. Before solving, name the category: this is a rate-time setup, so use distance = rate × time and solve for rate.
Remember: The ASVAB is NOT pass/fail. Every point you score opens more doors — more branches, better MOS options, better signing bonuses. The math sections are where disciplined prep pays off the fastest. You've got the formulas. Now put in the reps.","body_text":"The ASVAB does not give you a formula sheet or a calculator. This AR + MK cheat sheet collects the formulas, shortcuts, and decision steps that move your AFQT score fastest — drill the formulas and practice mental math until it is automatic.\n\nAR vs. MK: Know Your Enemy\n\nBefore you memorize a single formula, understand the battlefield. These two subtests look similar but fight differently.\n\nArithmetic Reasoning vs. Mathematics Knowledge — side by side\nDimension | Arithmetic Reasoning (AR) | Mathematics Knowledge (MK)\nQuestions: 30 | 25\nTime Limit: 36 minutes | 24 minutes\nTime Per Question: ~72 seconds | ~58 seconds\nFormat: Word problems | Straight math (no story)\nTests Your: Reasoning + setup + calculation | Pure formula recall + execution\nKey Topics: Percentages, ratios, d=rt, work rate, interest | Algebra, geometry, exponents, angles\nAFQT Weight: Yes — 25% of AFQT | Yes — 25% of AFQT\nBiggest Trap: Misreading the question | Mixing up similar formulas\n\nSeconds per question: your real time budget\nFrom your AR/MK time limits — AR allows ~72 seconds per question; MK only ~58. Source: HLT Mastery.\n\n50% — of your AFQT comes from AR + MK combined\n\nWhy this matters\nAR + MK = 50% of your AFQT. Your AFQT percentile determines branch eligibility — Army needs 31+, Air Force needs 36+, Space Force needs 70+. Higher scores = more MOS/AFSC/rate options. These two sections are where you move the needle fastest.\n\nArithmetic Reasoning: The Formula Arsenal\n\nAR is all word problems. The math itself isn't hard — the challenge is translating English into equations. Here's every formula you'll need, organized by problem type.\n\nPercentages & Percent Change\n\nThese show up on almost every AR section. Drill them until they're automatic.\n\n• Finding a percent: Part = Percent × Whole → or rearranged: Percent = (Part ÷ Whole) × 100\n• Percent change: ((New − Old) ÷ Old) × 100\n• Discount price: Original Price × (1 − Discount%)\n• Tax/markup price: Original Price × (1 + Rate%)\n\nQuick example: A $80 jacket is 25% off. Price = $80 × (1 − 0.25) = $80 × 0.75 = $60.\n\nRatios & Proportions\n\nIf you see \"for every\" or \"per\" in a word problem, you're in ratio territory.\n\n• Proportion setup: a/b = c/d\n• Cross multiply to solve: a × d = b × c\n\nQuick example: If 3 widgets cost $12, how much do 7 cost? → 3/12 = 7/x → 3x = 84 → x = $28.\n\nDistance, Rate & Time\n\nThe king of AR word problems. If something is moving, this is your formula.\n\n• Core formula: Distance = Rate × Time (d = rt)\n• Rearranged for rate: r = d ÷ t\n• Rearranged for time: t = d ÷ r\n\nQuick example: You drive 180 miles at 60 mph. Time = 180 ÷ 60 = 3 hours.\n\nWork Rate Problems\n\n\"Person A can do a job in X hours, Person B in Y hours. How long together?\" This formula handles it.\n\n• Combined work rate: 1/t₁ + 1/t₂ = 1/T\n\nQuick example: Alex paints a room in 4 hours, Jamie in 6 hours. Together: 1/4 + 1/6 = 3/12 + 2/12 = 5/12. T = 12/5 = 2.4 hours.\n\nSimple Interest & Money\n\n• Simple interest: I = P × r × t (Principal × rate × time)\n• Total amount: A = P + I or A = P(1 + rt)\n• Profit: Profit = Revenue − Cost\n• Average (Mean): Mean = Sum of all values ÷ Number of values\n\nAR Trap Alert\nWord problems love to hide extra information or switch units mid-problem. Always check: Are the units consistent? Did they give you hours but ask for minutes? Read the question twice before you calculate once.\n\nFlowchart: How to Attack Any AR Word Problem\n\nDon't stare at a word problem wondering where to start. Follow this decision path every single time — it works whether the problem is about money, distance, or paint cans. Start with the question, choose the problem family, then check units before you answer.\n\nArithmetic Reasoning decision path\nStep | What to look for | Move\n1. Read: The final sentence and the units requested | Do not calculate until you know what the question asks.\n2. Classify: Percent, ratio, distance/rate/time, work rate, or money over time | Pick the matching formula family.\n3. Set up: Numbers that belong in the formula vs. extra distractors | Write the equation before doing arithmetic.\n4. Solve: No-calculator arithmetic | Keep numbers simple; cancel or reduce when possible.\n5. Check: Units and real-world size | Convert minutes/hours, inches/feet, or percent/decimal before choosing.\n\n1. Read the full problem — Don't start calculating halfway through.\n2. Identify what they're asking for — Circle or underline the actual question.\n3. Spot the problem type — Is something moving? → d=rt. Percent involved? → Part/Whole. Two workers? → Work rate. Money over time? → I=Prt.\n4. Set up the equation — Translate words into math symbols.\n5. Solve — Do the arithmetic carefully — no calculator means no room for sloppy mistakes.\n6. Check units — If they asked for hours and you got minutes, convert before you answer.\n7. Sanity check — Does the answer make real-world sense?\n\nMathematics Knowledge: The Formula Vault\n\nMK strips away the word problem wrapper and tests you on raw math. You either know the formula or you don't. No partial credit for vibes.\n\nAlgebra Essentials\n\n• Slope-intercept form: y = mx + b (m = slope, b = y-intercept)\n• Slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁)\n• Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a\n• FOIL method: (a+b)(c+d) = ac + ad + bc + bd\n• Difference of squares: a² − b² = (a+b)(a−b)\n\nExponent Rules\n\nThese are free points if you memorize the patterns.\n\n• xᵃ × xᵇ = x^(a+b) — multiplying same base? Add exponents.\n• (xᵃ)ᵇ = x^(a×b) — power of a power? Multiply exponents.\n• xᵃ ÷ xᵇ = x^(a−b) — dividing same base? Subtract exponents.\n• x⁰ = 1 — anything to the zero power equals 1 (except 0⁰).\n• x⁻ⁿ = 1/xⁿ — negative exponent? Flip it to a fraction.\n\nGeometry: Areas\n\n• Rectangle: A = length × width\n• Triangle: A = ½ × base × height\n• Circle: A = πr²\n• Trapezoid: A = ½(b₁ + b₂) × h\n• Parallelogram: A = base × height\n\nGeometry: Perimeters & Circumference\n\n• Rectangle perimeter: P = 2l + 2w\n• Triangle perimeter: P = a + b + c (sum of all sides)\n• Circle circumference: C = 2πr or C = πd\n\nGeometry: Volumes\n\n• Rectangular solid (box): V = l × w × h\n• Cylinder: V = πr²h\n• Sphere: V = (4/3)πr³\n• Cone: V = (1/3)πr²h\n\nGeometry: Angles & Triangles\n\n• Pythagorean theorem: a² + b² = c² (right triangles only — c is the hypotenuse)\n• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 (and their multiples: 6-8-10, 9-12-15)\n• Angles in a triangle: always sum to 180°\n• Angles in a quadrilateral: always sum to 360°\n• Supplementary angles: sum to 180°\n• Complementary angles: sum to 90°\n• Vertical angles: always equal\n\nHigh-Yield Shortcut\nMemorize the Pythagorean triples (3-4-5, 5-12-13, 8-15-17). When you see a right triangle on the MK section, check if the given sides match a triple or a multiple of one. Instant answer — no calculation needed.\n\nFlowchart: Which Geometry Formula Do I Need?\n\nGeometry questions give you a shape and ask you to find something. This flowchart gets you to the right formula in seconds — match the shape to what the problem asks you to find.\n\nGeometry formula selector\nIf the shape is… | And they ask for… | Use\nCircle: Area | A = πr²\nCircle: Circumference | C = 2πr\nTriangle: Area | A = ½bh\nRight triangle: Missing side | a² + b² = c²\nRectangle: Area or perimeter | A = lw or P = 2l + 2w\nBox: Volume | V = lwh\nCylinder: Volume | V = πr²h\n\n1. What shape is it? — Circle, triangle, rectangle, cylinder, or other?\n2. What are they asking for? — Area, perimeter/circumference, volume, or angle?\n3. Match shape + ask to formula — Circle + area → A = πr²\n Circle + circumference → C = 2πr\n Triangle + area → A = ½bh\n Triangle + missing side (right triangle) → a² + b² = c²\n Rectangle + area → A = lw\n Box + volume → V = lwh\n Cylinder + volume → V = πr²h\n4. Plug in known values and solve — Substitute the numbers you have into the chosen formula and work it out.\n\nMnemonics That Stick\n\nFormulas are useless if you can't recall them under pressure. These memory tricks lock them in.\n\nMaster the simple fundamentals and they compose into bigger problem-solving — the same few formulas, assembled. — Simple geometric shapes — circle, triangle, square, rectangle — assembling left to right into one larger composite structure\n\nPEMDAS — Order of Operations\nRemember it as Please Excuse My Dear Aunt Sally :\n P — Parentheses first\n E — Exponents next\n M — Multiplication (left to right with Division)\n D — Division (left to right with Multiplication)\n A — Addition (left to right with Subtraction)\n S — Subtraction (left to right with Addition)\n\n Multiplication and Division are equal priority (left to right), and so are Addition and Subtraction. Don't multiply before dividing just because M comes before D.\n\nSOH-CAH-TOA — Trigonometry Ratios (if tested)\nSOH — Sine = Opposite ÷ Hypotenuse\n CAH — Cosine = Adjacent ÷ Hypotenuse\n TOA — Tangent = Opposite ÷ Adjacent\n\nMore Memory Hacks\n\n• d = rt → \"Dirt\" — Distance = Rate × Time. Think of it as a dirt road you're driving down.\n• Pythagorean Theorem → \"a² + b² = c² ... A Bug plus a Bee = a Cat squared\" — silly, but it sticks.\n• Area of a circle → \"Apple pies are round\" = A = πr² (A pie are squared).\n• Circumference → \"Cherry pies delight\" = C = πd.\n\nBeat the Clock: Time Management\n\nYour time budget\nAR: 36 minutes ÷ 30 questions = 72 seconds each\n MK: 24 minutes ÷ 25 questions = 58 seconds each\n\n The rule: If you've spent 90 seconds on a single question, pick your best guess and move. Getting stuck on one question means leaving easier points on the table. On the CAT version, you can't go back — so guessing and moving is always better than burning time.\n\nBefore you calculate, identify the trap: units, percentages, ratios, or rate and time. Most ASVAB math misses come from choosing the wrong setup, not from hard arithmetic. For example, a car travels 180 miles in 3 hours and the answer choices are in miles per hour — that is a rate-time setup, so use distance = rate × time and solve for rate.\n\n5 Rapid-Fire Test Day Tips\n\n• Always answer every question. The ASVAB has NO penalty for guessing. A blank answer is a guaranteed zero — a guess has at least a 25% chance. Never leave a question unanswered.\n• Eliminate two wrong answers first. If you can knock out two choices, your guess odds jump from 25% to 50%. Look for answers that are obviously too large, too small, or have the wrong sign.\n• Plug in answer choices. On MK algebra problems, sometimes it's faster to plug each answer choice into the equation and see which one works rather than solving from scratch.\n• Estimate before you calculate. Round numbers to make quick mental math. If the answer choices are $47, $52, $98, and $112, and your estimate lands around $50, you've already narrowed it to two choices.\n• Watch your units. The #1 AR careless mistake: calculating correctly but in the wrong unit. If the question asks for feet and you solved in inches, you'll pick the wrong answer with total confidence.\n\nThe ASVAB is NOT pass/fail. Every point you score opens more doors — more branches, better MOS options, better signing bonuses. — HLT Mastery\n\nThe math sections are where disciplined prep pays off the fastest. You've got the formulas. Now put in the reps.","og":{"title":"ASVAB Math Cheat Sheet 2026: AR + MK Formulas, Flowcharts & Shortcuts","description":"A focused ASVAB math cheat sheet for Arithmetic Reasoning and Mathematics Knowledge: formulas, flowcharts, mnemonics, and no-calculator shortcuts for AFQT gains.","image":"https://res.cloudinary.com/hlt-media/image/upload/f_auto,q_auto,dpr_auto,c_fill,g_auto,ar_40:21,w_1200/v1779271701/hlt-mmm2/generated/mmm2-create-articlehero-for-asvab-learners-mpdwfjc0.webp"}}