ASVAB AFQT Comprehensive Condensed Study Guide

AFQT Map & Preparation Framework

  • Scope and Purpose:

    • Adapts core content from ASVAB AFQT For Dummies by Rod Powers (2010).

    • Focuses exclusively on the four AFQT-qualifying subtests: Arithmetic Reasoning (AR), Word Knowledge (WK), Paragraph Comprehension (PC), and Mathematics Knowledge (MK).

    • Strips out administrative policy details, military occupational background, resource lists, and full-length practice exams to form a concentrated, evergreen review.

    • Maintained under 100100 pages for digital study tool compatibility.

  • Color-Coding Study System:

    • BLUE - CONCEPT: Defines core ideas, principles, and foundational terminology.

    • RED - MUST MEMORIZE: Highlights essential formulas, operational rules, and non-negotiable mathematical/verbal facts.

    • GREEN - TEST MOVE: Denotes high-speed test-taking strategies, shortcut techniques, and checking steps.

    • ORANGE - TRAP: Flags common distractor patterns, frequent mistakes, and cognitive pitfalls.

    • PURPLE - WORD PART: Pinpoints vocabulary building blocks, including Latin/Greek roots, prefixes, and suffixes.

  • AFQT Subtest Core Breakdown:

    • Arithmetic Reasoning (AR): Evaluates mathematical word problems. The primary objective is to convert verbal descriptions into mathematical equations, execute solutions, and verify units and logical plausibility.

    • Word Knowledge (WK): Evaluates vocabulary comprehension. The primary objective is to identify direct synonyms using contextual clues and structural word analysis.

    • Paragraph Comprehension (PC): Evaluates reading comprehension. The primary objective is to locate main ideas, extract specific supporting details, analyze implied meanings, and determine contextual word meanings.

    • Mathematics Knowledge (MK): Evaluates high school mathematics principles. The primary objective is to solve direct computational problems using algebraic rules, geometric formulas, and arithmetic properties.

  • Strategic Study Prioritization:

    • Avoid equal time distribution across all four subtests simply because they are weighted equally. Focus heavily on documented weak areas.

    • Rereading familiar topics produces low study yields. Maximum score improvements result from targeting error patterns, slow problem types, weak formulas, and unfamiliar vocabulary.

    • The 5-Step Score-Building Loop:

    1. Diagnose: Complete short mixed problem sets and categorize every error by specific topic rather than subtest.

    2. Repair: Thoroughly review the underlying rule, principle, or formula associated with each missed item.

    3. Practice: Complete 33 to 55 new practice problems of that exact type without referencing study guides.

    4. Re-test: Re-evaluate the target skill after a deliberate delay to verify retention.

    5. Track Patterns: Continually monitor recurring errors to eliminate systemic score leaks.

General Study & Test Strategies

  • SQR3 Method for Technical Review:

    • Survey: Fast-scan headings, formulas, worked example steps, charts, and bold terms prior to reading.

    • Question: Formulate specific questions regarding expected test applications and question formats.

    • Read: Read sections actively with the targeted goal of answering self-generated questions.

    • Recall: Close study materials and state the rule, formula, or solution steps purely from memory.

    • Review: Periodically re-assess notes and missed practice items to ensure long-term retention.

  • High-Yield Note-Taking Principles:

    • Structural Chunking: Format study notes into distinct, digestible blocks containing one rule, one formula, one worked example, and one distractor pattern.

    • Associative Learning: Connect unfamiliar concepts to existing mental models to improve recall speed.

    • Conciseness: Write succinct definitions; avoid lengthy card paragraphs.

    • Error Analysis in Math: Record the exact operational step where an error occurred rather than just writing the correct final answer.

    • Vocabulary Card Structure: Maintain one target word per card, paired with a short, highly memorable definition.

  • Test-Day Execution Guidelines:

    • Avoid last-minute cramming; reserve the final evening prior to the exam for brief formula review and adequate sleep.

    • Arrive well-nourished but not overly full, minimizing environmental stress.

    • Thoroughly read every set of section directions, regardless of prior familiarity.

    • Actively scan for critical negative qualifiers that invert prompt meanings, including NOTNOT, EXCEPTEXCEPT, LEASTLEAST, and FALSEFALSE.

    • Examine all four answer choices fully when prompts ask for the best or most correct option.

    • Guessing and Pacing Strategy: Employ process of elimination immediately. Make an educated guess and move forward rather than expending excessive time on a single item. Leaving unanswered questions is strictly sub-optimal.

    • Answer Pattern Fallacy: Do not alter correct choices based on answer letter repetition. Repeating letters occur naturally, and each question operates independently.

Word Knowledge (WK) Mastery

  • Fundamental Testing Mechanism:

    • WK operates as a direct synonym evaluation presented either as isolated words or embedded in context sentences.

    • The task requires selecting the option closest in meaning to the target word as used in that specific context.

  • The 5-Step WK Solution Method:

    1. Read the target word within its context sentence and formulate a broad replacement concept.

    2. Identify the part of speech (noun, verb, adjective, or adverb) to eliminate structurally incompatible choices.

    3. Dissect the target word into structural components: Root + Prefix + Suffix.

    4. Eliminate antonyms (which are routinely inserted as distractors) and completely unrelated terms.

    5. Choose the closest synonym option, avoiding options that merely sound natural in a similar sentence.

  • The Homonym Trap:

    • Distractors frequently exploit familiar spellings or secondary phonetic meanings. Always defer to the precise meaning demanded by the contextual sentence.

  • Essential Vocabulary Roots:

    • anthrop: human (e.g., anthropology)

    • bibli: book (e.g., bibliography)

    • brev: short (e.g., abbreviate)

    • cede / ceed: go, yield (e.g., recede)

    • chrom: color (e.g., monochrome)

    • circum: around (e.g., circumnavigate)

    • cogn: know (e.g., cognitive)

    • corp: body (e.g., corporate)

    • dict: speak / say (e.g., dictionary)

    • domin: rule (e.g., dominate)

    • flu / flux: flow (e.g., influx)

    • form: shape (e.g., formation)

    • fract / frag: break (e.g., fragment)

    • graph: write / record (e.g., biography)

    • junct: join (e.g., junction)

    • liber: free (e.g., liberate)

    • lum: light (e.g., illuminate)

    • oper: work (e.g., cooperate)

    • port: carry (e.g., portable)

    • scrib / script: write (e.g., describe, script)

    • sens / sent: feel / think (e.g., sensation)

    • tract: pull (e.g., traction)

    • voc / vok: call / voice (e.g., vocal)

  • High-Frequency Prefixes and Suffixes:

    • anti-: against (e.g., antibody)

    • bi-: two (e.g., bilateral)

    • de-: away / down (e.g., decrease)

    • extra-: outside / beyond (e.g., extracurricular)

    • hyper-: over / excessive (e.g., hyperactive)

    • mal-: bad / wrong (e.g., malfunction)

    • multi-: many (e.g., multiple)

    • omni-: all (e.g., omnipresent)

    • re-: back / again (e.g., return, redo)

    • semi-: half / partly (e.g., semicircle)

    • super-: above / more (e.g., superior)

    • tele-: far (e.g., telephone)

    • trans-: across (e.g., transport)

    • un-: not (e.g., uncertain)

    • -able / -ible: capable of (e.g., readable)

    • -ful: full of (e.g., helpful)

    • -less: without (e.g., fearless)

    • -ment: action / result (e.g., movement)

    • -ness: state / quality (e.g., kindness)

    • -tion / -ion: act / process / result (e.g., creation)

    • -ous: having (e.g., dangerous)

    • -ist: person / characteristic (e.g., artist)

    • -ity: quality / state (e.g., clarity)

    • -ic / -ical: related to (e.g., historic, logical)

  • Vocabulary Retention System:

    • Read regular prose to build passive contextual familiarity.

    • When encountering unfamiliar terms, record a 4-part profile: Brief Definition + Direct Synonym + Direct Antonym + Original Practice Sentence.

    • Flashcard Protocol: Actively test recall, isolate incorrect cards into a secondary pile, and repeat until mastery is achieved.

    • Pronounce words aloud to engage auditory memory encoding.

    • Focus strictly on simplified definitions that enable instant test-day recognition.

  • Word Knowledge Quick Practice:

    1. Abundant most nearly means:

    • (A) scarce

    • (B) plentiful

    • (C) hidden

    • (D) narrow

    1. Reluctant most nearly means:

    • (A) unwilling

    • (B) cheerful

    • (C) exact

    • (D) noisy

    1. Concise most nearly means:

    • (A) brief

    • (B) doubtful

    • (C) expensive

    • (D) careless

    1. Obsolete most nearly means:

    • (A) current

    • (B) outdated

    • (C) dangerous

    • (D) complex

    1. The medic remained calm during the emergency. Calm most nearly means:

    • (A) composed

    • (B) careless

    • (C) rapid

    • (D) strict

    1. Diminish most nearly means:

    • (A) reduce

    • (B) repeat

    • (C) repair

    • (D) divide

    1. Vital most nearly means:

    • (A) optional

    • (B) essential

    • (C) temporary

    • (D) obvious

    1. Hostile most nearly means:

    • (A) friendly

    • (B) unfriendly

    • (C) uncertain

    • (D) silent

    1. The panel reached a unanimous decision. Unanimous most nearly means:

    • (A) delayed

    • (B) secret

    • (C) agreed by all

    • (D) controversial

    1. Rigid most nearly means:

      • (A) flexible

      • (B) stiff

      • (C) empty

      • (D) ordinary

  • WK Practice Answer Key:

    • 1: B - plentiful

    • 2: A - unwilling

    • 3: A - brief

    • 4: B - outdated

    • 5: A - composed

    • 6: A - reduce

    • 7: B - essential

    • 8: B - unfriendly

    • 9: C - agreed by all

    • 10: B - stiff

Paragraph Comprehension (PC) Mastery

  • Core Comprehension Capacities:

    • Main Idea Identification: Isolating the overarching thesis of a passage.

    • Detail Recall: Locating explicit factual data points.

    • Logical Inference: Drawing defensible conclusions without adding external premises.

  • The Four Question Archetypes:

    • Direct Detail ("Treasure Hunt"): Prompts ask for facts directly stated. Method: Scan the passage for matching keywords, proper names, or numerical values.

    • Main Point: Prompts ask for the primary message. Method: Analyze the opening sentence, concluding sentence, and core recurring concepts.

    • Word in Context ("Dictionary"): Prompts ask for exact meaning of a word in passage context. Method: Substitute each choice into the sentence to verify coherence.

    • Inference ("Deep Thinking"): Prompts ask for unstated logical extensions. Method: Select options strictly supported by textual facts; reject choices requiring outside knowledge.

  • Distinguishing Main Idea from Supporting Details:

    • Main Idea: The broad, central point unifying the entire narrative structure.

    • Supporting Details: Specific evidence, statistics, examples, or explanations introduced to justify the main idea.

    • Evaluation Test: If an answer choice describes only a single sentence or example, it is a supporting detail. The true main idea must cover the text as a whole.

  • Reading Speed Techniques:

    • Skimming: Used to quickly grasp overall structure. Read the first sentence, key transitional phrases, and the final sentence.

    • Scanning: Used to locate specific facts. Move eyes rapidly across lines targeting specific names, numbers, dates, or technical terms.

    • Read the prompt question before reading the passage to determine whether skimming, scanning, or detailed reading is required.

    • Treat each prompt independently when multiple questions reference a single reading passage.

  • Common PC Traps & Tactical Adjustments:

    • Outside Knowledge Trap: Never select an option based on real-world knowledge if it is not explicitly anchored in the text. External facts are incorrect distractors in PC.

    • "Most Right" Strategy: Fully evaluate all options. Distractors are often partially correct statements. The correct selection is directly supported by textual evidence.

  • Paragraph Comprehension Quick Practice:

  • Passage 1:   Sleep does more than rest the body. During sleep, the brain organizes new information and strengthens memories formed during the day. People who consistently sleep too little may therefore have more difficulty remembering material they studied, even when they spent the same number of hours reading it.

    • 1. What is the main idea?

    • (A) Reading is better at night

    • (B) Sleep helps learning and memory

    • (C) The body needs less rest than the brain

    • (D) Studying longer always improves memory

    • 2. Which statement is supported?

    • (A) Memory is strengthened during sleep

    • (B) Sleep replaces studying

    • (C) Everyone needs the same sleep schedule

    • (D) Reading causes fatigue

    • 3. The word consistently most nearly means:

    • (A) occasionally

    • (B) regularly

    • (C) suddenly

    • (D) secretly

  • Passage 1 Answer Key:

    • 1: B - Sleep helps learning and memory

    • 2: A - Memory is strengthened during sleep

    • 3: B - regularly

  • Passage 2:   Preventive maintenance is usually cheaper than waiting for equipment to fail. A small inspection may reveal loose parts, low fluid levels, or unusual wear before those problems cause a breakdown. Regular checks do not prevent every failure, but they reduce avoidable downtime and often extend the useful life of equipment.

    • 1. What is the main point?

    • (A) Inspections cost too much

    • (B) Equipment always fails

    • (C) Preventive maintenance can reduce breakdowns and cost

    • (D) Fluid levels are the only maintenance issue

    • 2. Which is a reasonable inference?

    • (A) Small problems can become larger problems if ignored

    • (B) Maintenance prevents every failure

    • (C) Old equipment never benefits from inspections

    • (D) Downtime is free

    • 3. According to the passage, regular checks often:

    • (A) guarantee perfect equipment

    • (B) extend useful life

    • (C) eliminate the need for repairs

    • (D) increase downtime

  • Passage 2 Answer Key:

    • 1: C - Preventive maintenance can reduce breakdowns and cost

    • 2: A - Small problems can become larger problems if ignored

    • 3: B - extend useful life

Mathematics Knowledge (MK) Core Concepts

  • Fundamental Terminology:

    • sum: Result of addition.

    • difference: Result of subtraction.

    • product: Result of multiplication.

    • quotient: Result of division.

    • factor: An integer that divides another integer evenly.

    • multiple: Product of a given integer multiplied by any integer.

    • prime number: A positive integer strictly greater than 11 possessing exactly two positive factors: 11 and itself.

    • variable: A letter representing an unknown numeric value.

    • coefficient: A numerical factor multiplying a variable.

    • expression: A mathematical combination of numbers, variables, and operators lacking an equals sign.

    • equation: A statement asserting the equality of two mathematical expressions.

    • inequality: A mathematical sentence comparing expressions using <<, >>, ≠\neq, ≠\ne, <=\text{<=}, or >=\text{>=}.

  • Mandatory Order of Operations:

    1. Parentheses and Brackets (evaluate innermost grouping symbols first).

    2. Exponents and Radical Roots.

    3. Multiplication and Division (evaluated in strict left-to-right sequence).

    4. Addition and Subtraction (evaluated in strict left-to-right sequence).

  • Mental Math & Distributive Property:

    • Break complex multiplication into linear algebraic components:     7×38=7×(40−2)=(7×40)−(7×2)=280−14=2667 \times 38 = 7 \times (40 - 2) = (7 \times 40) - (7 \times 2) = 280 - 14 = 266

    • Algebraic Form: a(b+c)=ab+aca(b + c) = ab + ac

  • Factors, Primes, GCF, and LCM:

    • Prime Factorization: Reducing a composite number into pure prime multipliers (e.g., 60=2×2×3×560 = 2 \times 2 \times 3 \times 5).

    • Greatest Common Factor (GCF): The largest factor common to two or more integers; utilized for reducing fractional expressions.

    • Least Common Multiple (LCM): The smallest positive integer divisible by two or more numbers; utilized for establishing common fractional denominators.

    • Divisibility Tests:

    • Divisible by 22: Number is even (ends in 0,2,4,6,80, 2, 4, 6, 8).

    • Divisible by 33: Sum of digits equals a multiple of 33.

    • Divisible by 55: Final digit is 00 or 55.

    • Divisible by 1010: Final digit is 00

  • Complete Fraction Rules:

    • Simplification: Divide both numerator and denominator by their GCF (e.g., 1824=18/ 624/ 6=34\frac{18}{24} = \frac{18 \text{/ } 6}{24 \text{/ } 6} = \frac{3}{4}).

    • Multiplication: Multiply numerators directly, multiply denominators directly, then simplify (e.g., 23×58=1024=512\frac{2}{3} \times \frac{5}{8} = \frac{10}{24} = \frac{5}{12}).

    • Division: Multiply the dividend by the reciprocal of the divisor (e.g., 23/ 45=23×54=1012=56\frac{2}{3} \text{/ } \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}).

    • Addition/Subtraction (Identical Denominators): Combine numerators over the existing denominator (e.g., 38+28=58\frac{3}{8} + \frac{2}{8} = \frac{5}{8}).

    • Addition/Subtraction (Different Denominators): Convert fractions to equivalent forms using the LCM as the common denominator before combining numerators (e.g., 13+14=412+312=712\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}).

    • Comparison Strategy: Perform cross-multiplication. To compare 35\frac{3}{5} and 58\frac{5}{8}, calculate 3×8=243 \times 8 = 24 and 5×5=255 \times 5 = 25. Since 25>2425 > 24, 58>35\frac{5}{8} > \frac{3}{5}.

    • Fraction to Decimal Conversion: Divide numerator by denominator (e.g., 38=3/ 8=0.375\frac{3}{8} = 3 \text{/ } 8 = 0.375).

    • Common Fraction Trap: Never add denominators when adding or subtracting fractions.

  • Ratios, Proportions, Decimals, and Percents:

    • Proportions: Equations asserting equality between ratios. Solve via cross-multiplication: 35=x20→5x=60→x=12\frac{3}{5} = \frac{x}{20} \rightarrow 5x = 60 \rightarrow x = 12.

    • Ratio Total Shortcut: If a ratio is 3:53:5 and total quantity is 6464:     3+5=8 total parts3 + 5 = 8 \text{ total parts}     648=8 per part\frac{64}{8} = 8 \text{ per part}     Quantities=3×8=24 and 5×8=40\text{Quantities} = 3 \times 8 = 24 \text{ and } 5 \times 8 = 40

    • Decimal Operations:

    • Addition/Subtraction: Align decimal points vertically before calculating.

    • Multiplication: Compute as whole numbers, then place decimal point equal to total decimal digits in factors.

    • Division: Shift divisor decimal point to create an integer; shift dividend decimal point the identical number of positions.

    • Percent Operations:

    • Decimal to Percent: Multiply by 100100 (shift decimal 22 places right).

    • Percent to Decimal: Divide by 100100 (shift decimal 22 places left).

    • Part Formula: part=percent (as decimal)×whole\text{part} = \text{percent (as decimal)} \times \text{whole}

    • Percent Formula: percent=partwhole×100%\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\text{\%}

    • Percent Change Formula: Percent Change=changeoriginal value×100%\text{Percent Change} = \frac{\text{change}}{\text{original value}} \times 100\text{\%}

    • Computational Examples: 35%35\text{\%} of 240=0.35×240=84240 = 0.35 \times 240 = 84. To find what percent 1818 is of 7272: 1872=0.25=25%\frac{18}{72} = 0.25 = 25\text{\%}.

  • Signed Numbers, Exponents, and Roots:

    • Operations with Signed Numbers:

    • Adding like signs: Add absolute values and retain sign.

    • Adding opposing signs: Subtract absolute values and retain sign of larger magnitude.

    • Subtraction: Add the additive inverse (a−b=a+(−b)a - b = a + (-b)).

    • Multiplication/Division: Same signs yield positive; differing signs yield negative.

    • Laws of Exponents:

    • Multiplication (same base): am×an=am+na^m \times a^n = a^{m+n}

    • Division (same base): aman=am−n\frac{a^m}{a^n} = a^{m-n}

    • Power of a Power: (am)n=am×n(a^m)^n = a^{m \times n}

    • Perfect Squares: 1,4,9,16,25,36,49,64,81,100,121,1441, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

    • Scientific Notation: Formatted as a×10ba \times 10^b where 1<=a<101 \text{<=} a < 10 (e.g., 4,500,000=4.5×1064,500,000 = 4.5 \times 10^6).

  • Algebraic Principles & Inequalities:

    • Combine like terms exclusively (3x+5x=8x3x + 5x = 8x; 3x+53x + 5 cannot be combined).

    • Expand terms via distribution before combining (4(x+3)=4x+124(x + 3) = 4x + 12).

    • Isolate variables by applying inverse operations systematically to both sides.

    • Inequality Inversion Rule: Reverse the inequality symbol whenever multiplying or dividing both sides by a negative scalar (e.g., −3x>12→x<−4-3x > 12 \rightarrow x < -4).

    • Quadratic Equations:

    • Standard Form: ax2+bx+c=0ax^2 + bx + c = 0

    • Factoring Example: x2+5x+6=0→(x+2)(x+3)=0→x=−2 or −3x^2 + 5x + 6 = 0 \rightarrow (x + 2)(x + 3) = 0 \rightarrow x = -2 \text{ or } -3

    • Quadratic Formula: x = \frac{-b \text{+-} \text{\sqrt{b^2 - 4ac}}}{2a}

  • Geometry Formula Sheet:

    • Straight Line Angle: 180o180^\text{o}

    • Full Circle Angle Sum: 360o360^\text{o}

    • Triangle Interior Angle Sum: 180o180^\text{o}

    • Rectangle/Square Interior Angles: 90o90^\text{o} each

    • Pythagorean Theorem: a2+b2=c2a^2 + b^2 = c^2

    • Square Rules:

    • Perimeter: P=4sP = 4s

    • Area: A=s2A = s^2

    • Diagonal: d = s\text{\sqrt{2}}

    • Rectangle Rules:

    • Perimeter: P=2l+2wP = 2l + 2w

    • Area: A=lwA = lw

    • Diagonal: d = \text{\sqrt{l^2 + w^2}}

    • Triangle Rules:

    • Perimeter: P=s1+s2+s3P = s_1 + s_2 + s_3

    • Area: A=12bhA = \frac{1}{2}bh

    • Circle Rules:

    • Radius/Diameter: r=d2r = \frac{d}{2}, d=2rd = 2r

    • Circumference: C = 2\text{\pi}r = \text{\pi}d

    • Area: A = \text{\pi}r^2

    • Approximation Rule: Use \text{\pi} \text{~=} 3.14 unless explicitly instructed to maintain \text{\pi}.

    • Three-Dimensional Solid Rules:

    • Cube Volume: V=s3V = s^3

    • Cube Surface Area: SA=6s2SA = 6s^2

    • Rectangular Box Volume: V=lwhV = lwh

    • Rectangular Box Surface Area: SA=2lw+2wh+2lhSA = 2lw + 2wh + 2lh

    • Cylinder Volume: V = \text{\pi}r^2h

  • MK Test Moves:

    • Maintain clear scratchwork to avoid sign errors, decimal shifts, or transcription mistakes.

    • Plug answer choices directly into algebraic expressions to verify solutions when formal algebra is slow.

    • Eliminate logically impossible options (such as negative area metrics) prior to computation.

  • MK Quick Practice:

    1. 18−3(4)+22=?18 - 3(4) + 2^2 = ?

    • (A) 66

    • (B) 1010

    • (C) 1414

    • (D) 2222

    1. Greatest common factor of 1818 and 2424?

    • (A) 33

    • (B) 66

    • (C) 99

    • (D) 1212

    1. 34+58=?\frac{3}{4} + \frac{5}{8} = ?

    • (A) 1181\frac{1}{8}

    • (B) 1381\frac{3}{8}

    • (C) 1581\frac{5}{8}

    • (D) 1116\frac{11}{16}

    1. 23/ 45=?\frac{2}{3} \text{/ } \frac{4}{5} = ?

    • (A) 56\frac{5}{6}

    • (B) 815\frac{8}{15}

    • (C) 310\frac{3}{10}

    • (D) 107\frac{10}{7}

    1. 0.3750.375 as a fraction in lowest terms is:

    • (A) 35\frac{3}{5}

    • (B) 38\frac{3}{8}

    • (C) 58\frac{5}{8}

    • (D) 720\frac{7}{20}

    1. 35%35\text{\%} of 240=?240 = ?

    • (A) 7272

    • (B) 8080

    • (C) 8484

    • (D) 9696

    1. 1818 is what percent of 7272?

    • (A) 18%18\text{\%}

    • (B) 20%20\text{\%}

    • (C) 25%25\text{\%}

    • (D) 40%40\text{\%}

    1. −6+14−9=?-6 + 14 - 9 = ?

    • (A) −1-1

    • (B) 11

    • (C) 1111

    • (D) 1717

    1. Simplify 34×323^4 \times 3^2

    • (A) 363^6

    • (B) 969^6

    • (C) 383^8

    • (D) 666^6

    1. Solve 5x−7=285x - 7 = 28

      • (A) 55

      • (B) 77

      • (C) 99

      • (D) 1111

    2. Solve −3x>12-3x > 12

      • (A) x>−4x > -4

      • (B) x<−4x < -4

      • (C) x>4x > 4

      • (D) x<4x < 4

    3. A rectangle is 99 by 1414. Area?

      • (A) 2323

      • (B) 4646

      • (C) 126126

      • (D) 252252

    4. Circle radius 55. Approximate circumference?

      • (A) 15.715.7

      • (B) 2525

      • (C) 31.431.4

      • (D) 78.578.5

    5. Right triangle legs 66 and 88. Hypotenuse?

      • (A) 99

      • (B) 1010

      • (C) 1212

      • (D) 1414

    6. Rectangular box 4×3×74 \times 3 \times 7. Volume?

      • (A) 1414

      • (B) 2828

      • (C) 5656

      • (D) 8484

  • MK Practice Answer Key:

    • 1: B - 1010

    • 2: B - 66

    • 3: B - 1381\frac{3}{8}

    • 4: A - 56\frac{5}{6}

    • 5: B - 38\frac{3}{8}

    • 6: C - 8484

    • 7: C - 25%25\text{\%}

    • 8: A - −1-1

    • 9: A - 363^6

    • 10: B - 77

    • 11: B - x<−4x < -4

    • 12: C - 126126

    • 13: C - 31.431.4

    • 14: B - 1010

    • 15: D - 8484

Arithmetic Reasoning (AR) Word Problems

  • Nature of AR Questions:

    • AR evaluates mathematical translation skills rather than advanced math concepts. The core task is extracting mathematical relationships from narrative sentences.

  • The 4-Step AR Method:

    1. Identify the Target Question: Explicitly state what value must be solved; assign a variable directly to this unknown.

    2. Extract Essential Facts: List given numerical values, operational relationships, and unit labels. Cross out irrelevant narrative fluff.

    3. Translate Words to Equations: Convert verbal statements into algebraic expressions, ratios, tables, or geometric figures.

    4. Execute & Validate: Compute the solution, attach appropriate units, and confirm logical plausibility.

  • Scratchwork Techniques:

    • Draw visual diagrams immediately when dealing with geometric shapes, travel paths, directional movements, or spatial configurations.

    • Strike through distractor information. Prompts often contain unnecessary numbers designed to mislead candidates.

  • Keyword Translation Reference:

    • Addition Clues: increased by, more than, combined, together, total, sum, added to. (Note: "xx more than yy" translates to y+xy + x).

    • Subtraction Clues: decreased by, minus, less, difference, less than, fewer than. (Note: "77 less than yy" translates to y−7y - 7; term order reverses).

    • Multiplication Clues: of, times, multiplied by, product, factor of. (Note: "15%15\text{\%} of xx" translates to 0.15×x0.15 \times x).

    • Division Clues: per, average, ratio, quotient. (Note: Average=sum of termsnumber of terms\text{Average} = \frac{\text{sum of terms}}{\text{number of terms}}).

    • Operational Caution: Keywords are indicators, not absolute rules. Always parse the full context of a sentence.

  • Mandatory Unit Standards:

    • Standard Conversions:

    • 12 inches=1 foot12 \text{ inches} = 1 \text{ foot}

    • 3 feet=1 yard3 \text{ feet} = 1 \text{ yard}

    • 60 seconds=1 minute60 \text{ seconds} = 1 \text{ minute}

    • 60 minutes=1 hour60 \text{ minutes} = 1 \text{ hour}

    • 100 cents=$1100 \text{ cents} = \$1

    • Keep units uniform across all intermediate steps. Convert mixed units before executing operations.

  • Solution Templates for Common Problem Types:

    • Age Problems: Define unknown variables precisely.

    • Example: A father is three times his son's age, and their ages total 5252.

    • Let son's age =s= s; Father's age =3s= 3s

    • s+3s=52→4s=52→s=13s + 3s = 52 \rightarrow 4s = 52 \rightarrow s = 13

    • Number Problems:

    • Consecutive Integers: n,n+1,n+2n, n+1, n+2

    • Consecutive Even Integers: n,n+2,n+4n, n+2, n+4

    • Consecutive Odd Integers: n,n+2,n+4n, n+2, n+4

    • Coin & Value Problems:

    • Multiply total item count by item value. Maintain unified units.

    • Example: If qq quarters and dd dimes total $4.20\$4.20, set up 0.25q+0.10d=4.200.25q + 0.10d = 4.20

    • Percent Applications:

    • Discount: Sale Price=Original−(Rate×Original)\text{Sale Price} = \text{Original} - (\text{Rate} \times \text{Original})

    • Tax/Tip/Increase: New Value=Original+(Rate×Original)\text{New Value} = \text{Original} + (\text{Rate} \times \text{Original})

    • Decrease: New Value=Original−(Rate×Original)\text{New Value} = \text{Original} - (\text{Rate} \times \text{Original})

    • Distance & Rate Problems (d=rtd = rt):

    • Formulas: d=r×td = r \times t, r=dtr = \frac{d}{t}, t=drt = \frac{d}{r}

    • Opposite Direction Trajectory: Add speeds to find combined rate of separation (Rate=r1+r2\text{Rate} = r_1 + r_2).

    • Same Direction Trajectory: Subtract speeds to calculate rate of separation or catch-up (Rate=r1−r2\text{Rate} = r_1 - r_2).

    • Perpendicular Travel: Path distances form right triangle legs; solve hypotenuse using a2+b2=c2a^2 + b^2 = c^2.

    • Worked Example: Two vehicles leave a point in opposite directions at 45 mph45\text{ mph} and 55 mph55\text{ mph}. Distance after 2 hours=(45+55)×2=100×2=200 miles2\text{ hours} = (45 + 55) \times 2 = 100 \times 2 = 200\text{ miles}.

    • Combined Work Problems:

    • Calculate individual hourly work rates as fractional fractions of a complete job.

    • Individual Rate =1time to complete job= \frac{1}{\text{time to complete job}}

    • Combined Rate Formula: Ratetotal=Rate1+Rate2\text{Rate}_{\text{total}} = \text{Rate}_1 + \text{Rate}_2

    • Combined Time Formula: Timetotal=1Ratetotal\text{Time}_{\text{total}} = \frac{1}{\text{Rate}_{\text{total}}}

    • Worked Example: Worker A finishes in 6 hours6\text{ hours} (rate=16\text{rate} = \frac{1}{6}), Worker B finishes in 3 hours3\text{ hours} (rate=13\text{rate} = \frac{1}{3}). Combined rate =16+13=12= \frac{1}{6} + \frac{1}{3} = \frac{1}{2}. Combined time =2 hours= 2\text{ hours}.

    • Simple Interest:

    • Formula: I=P×r×tI = P \times r \times t

    • Variables: I=InterestI = \text{Interest}, P=PrincipalP = \text{Principal}, r=Annual Interest Rate (as decimal)r = \text{Annual Interest Rate (as decimal)}, t=Time in yearst = \text{Time in years}.

    • Worked Example: $1,500\$1,500 at 4%4\text{\%} for 3 years3\text{ years} =1500×0.04×3=$180= 1500 \times 0.04 \times 3 = \$180

    • Weighted Mixture Problems:

    • Structure: Amount1×Concentration1+Amount2×Concentration2=Amounttotal×Concentrationfinal\text{Amount}_1 \times \text{Concentration}_1 + \text{Amount}_2 \times \text{Concentration}_2 = \text{Amount}_{\text{total}} \times \text{Concentration}_{\text{final}}

    • Worked Example: Combine 10 lb10\text{ lb} of $2/lb\$2/\text{lb} product with x lbx\text{ lb} of $5/lb\$5/\text{lb} product to form $3/lb\$3/\text{lb} mixture:       (10×2)+(x×5)=(10+x)×3(10 \times 2) + (x \times 5) = (10 + x) \times 3       20+5x=30+3x20 + 5x = 30 + 3x       2x=10→x=5 lb2x = 10 \rightarrow x = 5\text{ lb}

  • AR Test Moves:

    • Focus strictly on isolating required facts and ignoring distractor narrative.

    • Estimate computed values prior to arithmetic execution to quickly flag decimal errors.

    • Substitute multiple choice values directly into problem parameters as a backup strategy.

  • AR Quick Practice:

    1. An $80\$80 jacket is discounted 25%25\text{\%}. Sale price?

    • (A) $20\$20

    • (B) $55\$55

    • (C) $60\$60

    • (D) $75\$75

    1. An employee earns $18\$18 per hour for 7.57.5 hours. Pay before deductions?

    • (A) $120\$120

    • (B) $126\$126

    • (C) $135\$135

    • (D) $144\$144

    1. A car travels 55 mph55\text{ mph} for 3.2 hours3.2\text{ hours}. Distance?

    • (A) 165 mi165\text{ mi}

    • (B) 176 mi176\text{ mi}

    • (C) 181 mi181\text{ mi}

    • (D) 220 mi220\text{ mi}

    1. Two vehicles leave the same point in opposite directions at 45 mph45\text{ mph} and 55 mph55\text{ mph}. After 2 hours2\text{ hours}, how far apart?

    • (A) 100 mi100\text{ mi}

    • (B) 180 mi180\text{ mi}

    • (C) 200 mi200\text{ mi}

    • (D) 220 mi220\text{ mi}

    1. Two vehicles move in the same direction at 65 mph65\text{ mph} and 50 mph50\text{ mph}. After 3 hours3\text{ hours}, how much farther has the faster vehicle traveled?

    • (A) 15 mi15\text{ mi}

    • (B) 30 mi30\text{ mi}

    • (C) 45 mi45\text{ mi}

    • (D) 115 mi115\text{ mi}

    1. One worker can complete a job in 6 hours6\text{ hours} and another in 3 hours3\text{ hours}. Together?

    • (A) 1.5 h1.5\text{ h}

    • (B) 2 h2\text{ h}

    • (C) 3 h3\text{ h}

    • (D) 4.5 h4.5\text{ h}

    1. A father is three times his son's age. Their ages total 5252. Son's age?

    • (A) 1111

    • (B) 1313

    • (C) 1717

    • (D) 3939

    1. Someone has twice as many quarters as dimes and $4.20\$4.20 total. How many quarters?

    • (A) 77

    • (B) 1010

    • (C) 1414

    • (D) 2121

    1. 10 lb10\text{ lb} at $2/lb\$2/\text{lb} is mixed with x lbx\text{ lb} at $5/lb\$5/\text{lb} to make a $3/lb\$3/\text{lb} mixture. x=?x = ?

    • (A) 22

    • (B) 55

    • (C) 88

    • (D) 1010

    1. Simple interest on $1,500\$1,500 at 4%4\text{\%} for 3 years3\text{ years}?

      • (A) $60\$60

      • (B) $120\$120

      • (C) $180\$180

      • (D) $240\$240

    2. 5 feet5\text{ feet} plus 30 inches30\text{ inches} equals:

      • (A) 5.3 ft5.3\text{ ft}

      • (B) 6.5 ft6.5\text{ ft}

      • (C) 7 ft7\text{ ft}

      • (D) 7.5 ft7.5\text{ ft}

    3. An hourly wage of $20\$20 gets a 15%15\text{\%} raise. New wage?

      • (A) $21.50\$21.50

      • (B) $22\$22

      • (C) $23\$23

      • (D) $25\$25

    4. A rectangle has perimeter 5050 and length 1515. Width?

      • (A) 55

      • (B) 1010

      • (C) 1515

      • (D) 2020

    5. Three consecutive integers sum to 5151. Middle integer?

      • (A) 1616

      • (B) 1717

      • (C) 1818

      • (D) 1919

    6. A group is divided in a 3:53:5 ratio and totals 6464. Smaller group?

      • (A) 1616

      • (B) 2424

      • (C) 3232

      • (D) 4040

  • AR Practice Answer Key:

    • 1: C - $60\$60

    • 2: C - $135\$135

    • 3: B - 176 mi176\text{ mi}

    • 4: C - 200 mi200\text{ mi}

    • 5: C - 45 mi45\text{ mi}

    • 6: B - 2 h2\text{ h}

    • 7: B - 1313

    • 8: C - 14 quarters14\text{ quarters}

    • 9: B - 5 lb5\text{ lb}

    • 10: C - $180\$180

    • 11: D - 7.5 ft7.5\text{ ft}

    • 12: C - $23\$23

    • 13: B - 1010

    • 14: B - 1717

    • 15: B - 2424

Comprehensive Review Sheets & Practice Checkpoints

  • Master Formula Summary:

    • Order of Operations: Parentheses →\rightarrow Exponents/Roots →\rightarrow Multiply/Divide →\rightarrow Add/Subtract

    • Percent of Value: part=rate×whole\text{part} = \text{rate} \times \text{whole}

    • Percent Rate: rate=partwhole\text{rate} = \frac{\text{part}}{\text{whole}}

    • Percent Change: Percent Change=changeoriginal×100%\text{Percent Change} = \frac{\text{change}}{\text{original}} \times 100\text{\%}

    • Proportions: ab=cd→ad=bc\frac{a}{b} = \frac{c}{d} \rightarrow ad = bc

    • Average: Average=sum of valuesnumber of values\text{Average} = \frac{\text{sum of values}}{\text{number of values}}

    • Travel Distance: d=rtd = rt

    • Simple Interest: I=PrtI = P r t

    • Work Rates: rate=1time\text{rate} = \frac{1}{\text{time}}; Combined Rate=rate1+rate2\text{Combined Rate} = \text{rate}_1 + \text{rate}_2

    • Square: P=4sP = 4s, A=s2A = s^2, d = s\text{\sqrt{2}}

    • Rectangle: P=2l+2wP = 2l + 2w, A=lwA = lw, d = \text{\sqrt{l^2 + w^2}}

    • Triangle: P=s1+s2+s3P = s_1 + s_2 + s_3, A=12bhA = \frac{1}{2}bh, angles sum to 180o\text{angles sum to } 180^\text{o}

    • Right Triangle: a2+b2=c2a^2 + b^2 = c^2

    • Circle: d=2rd = 2r, C = 2\text{\pi}r = \text{\pi}d, A = \text{\pi}r^2

    • Cube: V=s3V = s^3, SA=6s2SA = 6s^2

    • Rectangular Box: V=lwhV = lwh, SA=2lw+2wh+2lhSA = 2lw + 2wh + 2lh

    • Cylinder: V = \text{\pi}r^2h

    • Inequalities: Reverse inequality direction when multiplying or dividing by a negative number.

    • Quadratic Formula: x = \frac{-b \text{+-} \text{\sqrt{b^2 - 4ac}}}{2a}

  • Diagnostic Error Checklist:

    • Missed WK Item: Did I check contextual usage? Did I select an antonym distractor? Could root analysis have resolved it?

    • Missed PC Item: Did I rely on outside facts? Did I pick a minor detail instead of the main idea?

    • Missed Fraction Item: Did I fail to find a common denominator? Did I forget to flip the divisor fraction?

    • Missed Decimal/Percent Item: Was decimal placement inaccurate? Did I convert percentages to decimals prior to multiplication?

    • Missed Algebra Item: Did I fail to distribute signs correctly? Did I perform operations unequally across the equals sign?

    • Missed Inequality Item: Did I fail to invert the inequality direction after operating with a negative value?

    • Missed Geometry Item: Did I confuse perimeter, area, and volume formulas? Are units squared or cubed correctly?

    • Missed AR Word Problem Item: Did I solve for the wrong variable? Did I forget unit conversions?

    • Missed Travel/Rate Item: Were time units inconsistent with rate units? Did I fail to combine speeds appropriately?

    • General Error: Did I miss critical qualifiers like NOTNOT, EXCEPTEXCEPT, or LEASTLEAST?

  • 7-Day Intensive Review Protocol:

    • Day 1: Diagnostic Mixed Set. Identify weak areas and build a custom formula/error reference sheet.

    • Day 2: Arithmetic Reasoning Focus. Practice problem translation, unit conversions, percent changes, and rate setups. Complete 1515 target word problems.

    • Day 3: Mathematics Knowledge Focus. Master fractions, decimals, algebra steps, and operational order. Complete 1515 target math problems.

    • Day 4: Word Knowledge Focus. Study Latin/Greek roots, prefixes, suffixes, and active flashcard sets.

    • Day 5: Paragraph Comprehension Focus. Practice timed passage sets targeting main ideas, details, and context meanings.

    • Day 6: Error Correction Day. Re-solve every missed problem from the week without referencing solutions.

    • Day 7: Final Review Day. Review formula summary sheet, word roots, common traps, and ensure adequate sleep. Do not study new material.

  • 20-Question Mixed Checkpoint Practice:

    1. WK: Mitigate most nearly means:

    • (A) worsen

    • (B) reduce

    • (C) copy

    • (D) reveal

    1. WK: Durable most nearly means:

    • (A) long-lasting

    • (B) colorful

    • (C) temporary

    • (D) narrow

    1. WK: The commander gave an explicit order. Explicit most nearly means:

    • (A) unclear

    • (B) detailed and clear

    • (C) optional

    • (D) secret

    1. WK: Adverse most nearly means:

    • (A) favorable

    • (B) harmful

    • (C) equal

    • (D) ordinary

    1. PC: A town replaced old streetlights with efficient LED fixtures. Electricity use fell, maintenance calls became less frequent, and the brighter lights improved visibility at intersections. What is the main idea?

    • (A) LEDs are expensive

    • (B) The town benefited from replacing old streetlights

    • (C) Intersections are dangerous

    • (D) Electricity use always rises

    1. PC: From the same passage, which statement is supported?

    • (A) Maintenance calls increased

    • (B) Electricity use fell

    • (C) The town removed all intersections

    • (D) LEDs use more power

    1. PC: "The training schedule was compressed into three days." Compressed most nearly means:

    • (A) shortened/condensed

    • (B) forgotten

    • (C) repeated

    • (D) delayed

    1. PC: A manual states that batteries last longer when stored in a cool, dry location. Which inference is best supported?

    • (A) Heat and moisture can reduce battery life

    • (B) Batteries never fail

    • (C) Freezing always improves batteries

    • (D) Storage location does not matter

    1. MK: 6+2(5−1)2=?6 + 2(5 - 1)^2 = ?

    • (A) 2222

    • (B) 3232

    • (C) 3838

    • (D) 7070

    1. MK: 56−14=?\frac{5}{6} - \frac{1}{4} = ?

      • (A) 12\frac{1}{2}

      • (B) 712\frac{7}{12}

      • (C) 23\frac{2}{3}

      • (D) 38\frac{3}{8}

    2. MK: Solve 4x+9=334x + 9 = 33

      • (A) 44

      • (B) 55

      • (C) 66

      • (D) 88

    3. MK: A circle has diameter 1212. Radius?

      • (A) 33

      • (B) 66

      • (C) 1212

      • (D) 2424

    4. MK: A right triangle has hypotenuse 1313 and one leg 55. Other leg?

      • (A) 88

      • (B) 1010

      • (C) 1212

      • (D) 1818

    5. MK: 0.420.42 as a percent is:

      • (A) 4.2%4.2\text{\%}

      • (B) 42%42\text{\%}

      • (C) 420%420\text{\%}

      • (D) 0.42%0.42\text{\%}

    6. AR: A 120120-page manual is read at 1515 pages per day. Days needed?

      • (A) 66

      • (B) 88

      • (C) 99

      • (D) 1010

    7. AR: A tank contains 1818 gallons and gains 33 gallons per minute. How much after 77 minutes?

      • (A) 2121

      • (B) 3636

      • (C) 3939

      • (D) 126126

    8. AR: A runner travels 66 miles in 4848 minutes. At the same pace, how many miles in 22 hours?

      • (A) 1010

      • (B) 1212

      • (C) 1515

      • (D) 1818

    9. AR: A $50\$50 item has 8%8\text{\%} sales tax. Total?

      • (A) $54\$54

      • (B) $58\$58

      • (C) $62\$62

      • (D) $46\$46

    10. AR: A box is 6 ft6\text{ ft} long, 4 ft4\text{ ft} wide, and 3 ft3\text{ ft} high. Volume?

      • (A) 13 ft313\text{ ft}^3

      • (B) 24 ft324\text{ ft}^3

      • (C) 72 ft372\text{ ft}^3

      • (D) 144 ft3144\text{ ft}^3

    11. AR: Two machines finish a job in 4 hours4\text{ hours} and 12 hours12\text{ hours} separately. Together, how long?

      • (A) 2 h2\text{ h}

      • (B) 3 h3\text{ h}

      • (C) 4 h4\text{ h}

      • (D) 8 h8\text{ h}

  • Mixed Checkpoint Answer Key:

    • 1: B - reduce

    • 2: A - long-lasting

    • 3: B - detailed and clear

    • 4: B - harmful

    • 5: B - The town benefited from replacing old streetlights

    • 6: B - Electricity use fell

    • 7: A - shortened/condensed

    • 8: A - Heat and moisture can reduce battery life

    • 9: C - 3838 (Calculation: 6+2(4)2=6+2(16)=6+32=386 + 2(4)^2 = 6 + 2(16) = 6 + 32 = 38)

    • 10: B - 712\frac{7}{12} (Calculation: 1012−312=712\frac{10}{12} - \frac{3}{12} = \frac{7}{12})

    • 11: C - 66 (Calculation: 4x=24→x=64x = 24 \rightarrow x = 6)

    • 12: B - 66 (Calculation: r=122=6r = \frac{12}{2} = 6)

    • 13: C - 1212 (Calculation: 52+b2=132→25+b2=169→b2=144→b=125^2 + b^2 = 13^2 \rightarrow 25 + b^2 = 169 \rightarrow b^2 = 144 \rightarrow b = 12)

    • 14: B - 42%42\text{\%}

    • 15: B - 8 days8\text{ days} (Calculation: 12015=8\frac{120}{15} = 8)

    • 16: C - 39 gallons39\text{ gallons} (Calculation: 18+(3×7)=18+21=3918 + (3 \times 7) = 18 + 21 = 39)

    • 17: C - 15 miles15\text{ miles} (Pace: 48 min6 mi=8 min/mi\frac{48\text{ min}}{6\text{ mi}} = 8\text{ min/mi}. In 2 hrs2\text{ hrs} (120 mins120\text{ mins}): 1208=15 miles\frac{120}{8} = 15\text{ miles})

    • 18: A - $54\$54 (Calculation: 50+(0.08×50)=50+4=5450 + (0.08 \times 50) = 50 + 4 = 54)

    • 19: C - 72 ft372\text{ ft}^3 (Calculation: V=6×4×3=72V = 6 \times 4 \times 3 = 72)

    • 20: B - 3 h3\text{ h} (Calculation: Combined Rate=14+112=412=13→Time=3 hours\text{Combined Rate} = \frac{1}{4} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3} \rightarrow \text{Time} = 3\text{ hours})

  • Score Diagnostic Matrix:

    • 17–2017\text{--}20 Correct: Exceptional readiness. Retain speed and polish minor error edge cases.

    • 13–1613\text{--}16 Correct: Solid foundation. Repair targeted topic weaknesses identified in missed questions.

    • 9–129\text{--}12 Correct: Moderate leaks. Re-study foundational MK core rules, word problem translations, and reading tactics.

    • 0–80\text{--}8 Correct: Substantial reconstruction required. Pause mixed practice and study subtests systematically one topic family at a time.