ASVAB LEARNING PLATFORM

1. WORD PROBLEM STRATEGY

Arithmetic Reasoning tests your ability to use math to solve practical, real-world problems. Every question is a word problem. Use this systematic approach:

Operation Keywords

2. FRACTIONS, DECIMALS & PERCENTS

Fraction Operations

• Add/Subtract fractions: Find the Least Common Denominator (LCD), convert, then add/subtract numerators. Keep the denominator. Reduce if possible.
• Multiply fractions: Multiply numerators × numerators, denominators × denominators. Cross-cancel before multiplying if possible. 3/4 × 8/9 = (3×8)/(4×9) = 24/36 = 2/3
• Divide fractions: Keep-Change-Flip (KCF). Keep the first fraction, change ÷ to ×, flip the second fraction. 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
• Mixed numbers: Convert to improper fractions first. 2⅓ = 7/3

Percent Problems — Three Types

Converting Between Forms

3. RATIOS, PROPORTIONS & RATES

Ratios

• A ratio compares two quantities. Can be written as 3:4, 3 to 4, or 3/4.
• Always simplify ratios. 12:16 = 3:4
• Part-to-part vs. part-to-whole: In a class of 30 students, 18 are female. Female:Male = 18:12 = 3:2 (part:part). Female:Total = 18:30 = 3:5 (part:whole).

Proportions

• A proportion states two ratios are equal: a/b = c/d
• Solve by cross-multiplying: ad = bc
• Example: "If 3 items cost $7.50, how much do 8 items cost?" → 3/7.50 = 8/x → 3x = 60 → x = $20

Rates

• Unit rate: Rate per ONE unit. 240 miles / 4 hours = 60 miles per hour.
• Distance = Rate × Time (D = RT): The most important formula for AR. Always align units.
• Work rate problems: If person A completes a job in 3 hours and person B in 6 hours, working together: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2 of the job per hour → 2 hours together.

4. INTEGERS, EXPONENTS & ROOTS

Order of Operations — PEMDAS

Exponents

• x^n means multiply x by itself n times. 2^5 = 2×2×2×2×2 = 32
• x^0 = 1 for any non-zero x. x^1 = x.
• Multiplying same base: add exponents. x^3 × x^4 = x^7
• Dividing same base: subtract exponents. x^5 ÷ x^2 = x^3
• Power of a power: multiply exponents. (x^2)^3 = x^6
• Negative exponent: reciprocal. 2^−3 = 1/2^3 = 1/8

Square Roots

• √x = the number that multiplied by itself gives x. √49 = 7 because 7×7 = 49
• Perfect squares to memorize: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
• √(ab) = √a × √b. Simplify: √72 = √(36×2) = 6√2

5. GEOMETRY FOR AR WORD PROBLEMS

Key Formulas

6. AVERAGES, PROBABILITY & STATISTICS

Averages (Mean)

• Mean: Sum of values ÷ number of values.
• Reverse-average problem: "A soldier's average score on 4 tests is 82. What must she score on the 5th test to average 85?" → Total needed = 85×5 = 425. Total so far = 82×4 = 328. Need 425−328 = 97.
• Weighted average: When groups have different sizes. Average = (sum of all values) / (total count).

Simple Probability

1. NUMBER THEORY & PROPERTIES

Types of Numbers

• Natural numbers: Counting numbers — 1, 2, 3, 4... (no zero, no negatives)
• Whole numbers: 0, 1, 2, 3, 4... (includes zero)
• Integers: ...−3, −2, −1, 0, 1, 2, 3... (includes negatives)
• Rational numbers: Any number expressible as a fraction a/b (b≠0). Includes decimals that terminate or repeat.
• Irrational numbers: Cannot be expressed as a fraction. Decimals that never terminate or repeat. Examples: π, √2, √3.
• Prime numbers: Divisible only by 1 and themselves. First 10: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Note: 1 is NOT prime.
• Composite numbers: Have more than two factors. Examples: 4, 6, 8, 9, 10...

Divisibility Rules

GCF and LCM

• GCF (Greatest Common Factor): The largest number that divides evenly into all given numbers. Factor each number and find the largest common factor. GCF(12, 18) = 6.
• LCM (Least Common Multiple): The smallest positive number that is a multiple of all given numbers. LCM(4, 6) = 12.
• Shortcut: GCF × LCM = Product of the two numbers. GCF(12,18)=6, LCM=36. 6×36=216=12×18 ✓

2. ALGEBRA FUNDAMENTALS

Solving Equations

• Isolate the variable using inverse operations. Balance the equation — do the same thing to both sides.
• One-step: 5x = 35 → x = 7. Two-step: 3x + 4 = 19 → 3x = 15 → x = 5.
• With fractions: (2/3)x = 8 → x = 8 × (3/2) = 12
• Literal equations: Solve for a variable in terms of others. D = RT → R = D/T

Inequalities

• Solve like equations. CRITICAL: When multiplying or dividing BOTH SIDES by a negative number, REVERSE the inequality sign.
• −2x < 10 → divide by −2 → x > −5 (sign reversed!)
• Graphing: open circle (< or >) = endpoint NOT included; closed circle (≤ or ≥) = endpoint included.

Polynomials & Factoring

• Like terms: Same variable and same exponent. 3x² and 7x² are like terms; 3x² and 3x are NOT.
• Distributive property: a(b + c) = ab + ac. Example: 3(2x − 5) = 6x − 15
• FOIL (First, Outer, Inner, Last): (x + 3)(x − 2) = x² − 2x + 3x − 6 = x² + x − 6
• Factor out GCF: 12x³ − 8x² = 4x²(3x − 2)
• Factor trinomials: x² + 5x + 6 → find two numbers that multiply to 6, add to 5 → (x+2)(x+3)
• Difference of squares: a² − b² = (a+b)(a−b). Example: x² − 25 = (x+5)(x−5)

Quadratic Equations

3. LINEAR EQUATIONS & COORDINATE GEOMETRY

Graphing Lines

Systems of Equations

• Substitution: Solve one equation for one variable; plug into the other equation.
• Elimination (Addition): Add or subtract equations to cancel one variable. Multiply one/both equations first if needed.
• Graphically: The solution is the intersection point of the two lines. Parallel lines = no solution; same line = infinite solutions.

4. GEOMETRY

Angles & Triangles

• Angles in a triangle sum to 180°. Exterior angle = sum of the two remote interior angles.
• Scalene: All sides different. Isosceles: Two equal sides (base angles equal). Equilateral: All sides and angles equal (60° each).
• Right triangle: One 90° angle. Pythagorean theorem applies. Special right triangles: 30-60-90 (sides: 1 : √3 : 2) and 45-45-90 (sides: 1 : 1 : √2).
• Similar triangles: Same angles, proportional sides. If triangles are similar with ratio 3:2, their areas are in ratio 9:4.

Circles

Quadrilaterals & Polygons

• Parallelogram: Opposite sides parallel and equal. A = base × height.
• Rectangle: All angles 90°. A = lw. Diagonals equal in length.
• Square: All sides equal, all angles 90°. A = s². Diagonals equal and perpendicular bisectors.
• Rhombus: All sides equal. Diagonals are perpendicular bisectors of each other. A = ½d₁d₂.
• Trapezoid: One pair of parallel sides. A = ½(b₁ + b₂)h.
• Sum of interior angles of any polygon with n sides = (n−2) × 180°. Triangle: 180°. Quadrilateral: 360°. Pentagon: 540°. Hexagon: 720°.

5. FUNCTIONS, SEQUENCES & PROBABILITY

Functions

• A function assigns exactly ONE output to each input. f(x) notation: "f of x".
• Evaluate: f(x) = 2x² − 3x + 1. Find f(4): f(4) = 2(16) − 12 + 1 = 21.
• Domain: all valid input values. Range: all resulting output values.

Sequences

• Arithmetic sequence: Each term differs from the previous by a constant difference (d). 2, 5, 8, 11... (d=3). nth term: a_n = a₁ + (n−1)d
• Geometric sequence: Each term is multiplied by a constant ratio (r). 2, 6, 18, 54... (r=3). nth term: a_n = a₁ × r^(n−1)

Permutations & Combinations

1. WORD KNOWLEDGE FORMAT

WK questions come in two formats: (1) standalone synonym questions and (2) underlined words in sentences. Both test whether you know what words mean.

• Synonym question: "CANDID most nearly means: A) hostile B) honest C) careful D) ambiguous" → Answer: B
• Context sentence: "Her meticulous work impressed her supervisor. Meticulous most nearly means: A) rapid B) careless C) detailed D) creative" → Answer: C

2. ESSENTIAL PREFIXES

3. ESSENTIAL ROOTS

4. KEY SUFFIXES

1. QUESTION TYPES & STRATEGIES

PC tests how well you understand written material. Passages are 50–200 words on various topics. There are five main question types, each requiring a slightly different approach.

2. READING TECHNIQUES FOR SPEED

Active Reading Strategy

1. Skim the question first — know what you're looking for before reading the passage.
2. Read the first and last sentence of each paragraph — they usually contain the main idea.
3. Underline key words as you read (on paper test). On CAT, note them mentally.
4. Eliminate clearly wrong answers — remove answers that contradict the passage or are too extreme.
5. Return to the passage for detail questions — don't answer from memory.

Main Idea vs. Topic vs. Theme

• Topic: The subject in one word or phrase. ("climate change")
• Main idea: What the passage says ABOUT the topic. ("Climate change is accelerating faster than predicted.")
• Theme: Broader universal message. (Often in fiction)
• Wrong main idea answers are often: too narrow (one specific detail), too broad (more than the passage covers), or contradicted by the passage.

3. INFERENCE & IMPLICIT MEANING

Inference questions are the most challenging PC question type. The answer is never stated directly — you must read between the lines.

• Valid inferences are supported by evidence in the passage — not just possible, but indicated by the text.
• Avoid answers that are too extreme (always, never, everyone, impossible) unless the passage supports it.
• Avoid answers that bring in outside knowledge not mentioned in the passage.
• Look for cause-and-effect relationships, contrasts, and patterns within the passage.

4. VOCABULARY IN CONTEXT

Context Clue Types

• Definition clue: The word is explained immediately after. "She was parsimonious, or extremely frugal with money."
• Synonym clue: A similar word appears nearby. "The child was querulous, always whining and complaining."
• Antonym/Contrast clue: An opposite is signaled by "but," "however," "unlike." "Unlike her gregarious brother, she was reserved."
• Example clue: Examples help define the word. "Many mammals — lions, tigers, and bears — are carnivorous."

High-Frequency ASVAB Vocabulary

1. LIFE SCIENCE

Cell Biology

• Cell Theory: all organisms are made of cells; cells are the basic unit of life; all cells come from existing cells.
• Prokaryotes (no nucleus): bacteria, archaea. Eukaryotes (have nucleus): plants, animals, fungi, protists.
• Mitochondria = powerhouse (makes ATP). Nucleus = DNA storage. Chloroplast = photosynthesis. Ribosome = protein synthesis. Cell membrane = controls entry/exit.
• Photosynthesis: CO₂ + H₂O + light → glucose + O₂ (chloroplasts)
• Cellular respiration: glucose + O₂ → CO₂ + H₂O + ATP (mitochondria)

Genetics

• DNA → RNA → Protein (Central Dogma). DNA is in the nucleus as chromosomes.
• Dominant (A) expressed if 1+ copies present. Recessive (a) expressed only if homozygous (aa).
• Mitosis: 2 identical cells (growth). Meiosis: 4 haploid gametes (reproduction).
• Natural selection: favorable traits increase over generations → evolution.

Human Body Systems

2. CHEMISTRY

Atomic Structure & Periodic Table

• Atom: protons (+, nucleus), neutrons (neutral, nucleus), electrons (−, orbital shells).
• Atomic number = protons. Mass number = protons + neutrons.
• Periodic table organized by atomic number. Groups (columns) have similar properties due to same number of valence electrons.
• Metals (left): shiny, conductive, malleable. Nonmetals (right): poor conductors. Metalloids (staircase): properties of both.

Chemical Bonds & Reactions

• Ionic bond: Transfer of electrons (metal + nonmetal). NaCl, MgO.
• Covalent bond: Sharing of electrons (nonmetal + nonmetal). H₂O, CO₂, O₂.
• Law of Conservation of Mass: Matter can't be created or destroyed in a reaction. Equations must balance.
• Exothermic: Releases heat (combustion, oxidation). Endothermic: Absorbs heat (photosynthesis, melting).

States of Matter & pH

3. PHYSICS

Newton's Laws & Forces

• 1st Law (Inertia): An object at rest stays at rest; in motion stays in motion — unless a net force acts on it.
• 2nd Law: F = ma (Force = mass × acceleration). More mass = more force needed for same acceleration.
• 3rd Law: For every action, equal and opposite reaction. Rocket thrust, gun recoil, swimming.
• Gravity: F = Gm₁m₂/r² (Universal Law of Gravitation). All objects attract each other.
• Friction: Opposes motion between surfaces. Static friction > kinetic friction.

Energy & Waves

4. EARTH & SPACE SCIENCE

• Earth's layers: Crust → Mantle (convection currents) → Outer Core (liquid iron) → Inner Core (solid)
• Plate tectonics: Convergent (collision → mountains/trenches), Divergent (separation → ridges), Transform (sliding → earthquakes)
• Rock types: Igneous (cooled magma), Sedimentary (layered deposits), Metamorphic (heat + pressure)
• Water cycle: Evaporation → Condensation → Precipitation → Runoff/Infiltration
• Atmosphere: Troposphere (weather) → Stratosphere (ozone) → Mesosphere → Thermosphere → Exosphere
• Solar system: Mercury, Venus, Earth, Mars | Jupiter, Saturn, Uranus, Neptune. Inner = rocky, outer = gas giants.
• Stars: Energy from nuclear fusion (hydrogen → helium). Sun is a medium-sized yellow dwarf star.
• Moon phases: New → Waxing Crescent → First Quarter → Waxing Gibbous → Full → Waning Gibbous → Third Quarter → Waning Crescent

1. ELECTRICAL FUNDAMENTALS

Basic Electrical Quantities

Ohm's Law — The Most Important EI Formula

2. SERIES VS. PARALLEL CIRCUITS

Series Circuits

• All components on a single path. Same current flows through every component.
• Total resistance: R_total = R₁ + R₂ + R₃ (adds up)
• Voltage: Distributes across components (V_total = V₁ + V₂ + V₃)
• Current: Same everywhere (I is constant)
• If one component breaks: Entire circuit stops working (like old Christmas lights)

Parallel Circuits

• Multiple paths for current. Voltage is the same across each branch.
• Total resistance: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ (total is LESS than any single resistor)
• Voltage: Same across all branches
• Current: Divides between branches (adds up to total)
• If one component breaks: Other branches continue working (household wiring)

3. ELECTRONIC COMPONENTS

4. AC vs. DC & ELECTRICAL SAFETY

Direct Current (DC) vs. Alternating Current (AC)

• DC: Flows in one direction only. Batteries, solar cells, fuel cells. Used in electronics, vehicles, digital devices.
• AC: Reverses direction periodically (60 Hz in U.S. = 60 cycles/second). Power plants and wall outlets provide AC. Used for homes, industry, motors. More efficient to transmit over long distances.
• Transformers only work with AC (they need a changing magnetic field to operate).

Conductors, Insulators & Semiconductors

• Conductors: Allow electrons to flow freely. Examples: copper, gold, silver, aluminum, iron. Copper is most commonly used in wiring.
• Insulators: Resist electron flow. Examples: rubber, glass, plastic, wood, dry air. Used to protect against shock.
• Semiconductors: Conduct under certain conditions. Examples: silicon, germanium. Basis of transistors, chips, diodes.

Electrical Safety Concepts

• Grounding: Connects circuit to earth, preventing dangerous voltage buildup.
• Fuse/Circuit breaker: Protects circuits from overload. Fuses melt (must replace); breakers trip (can reset).
• GFCI (Ground Fault Circuit Interrupter): Detects leakage current and shuts off power instantly. Required near water.
• Kilowatt-hour (kWh): Unit of electrical energy. What your utility bill measures. 1 kWh = 1,000 watts for 1 hour.

1. THE INTERNAL COMBUSTION ENGINE

4-Stroke Cycle (Otto Cycle)

1. Intake stroke: Piston moves DOWN, intake valve opens, air-fuel mixture drawn in.
2. Compression stroke: Piston moves UP, both valves closed, mixture compressed.
3. Power (combustion) stroke: Spark plug fires, fuel ignites, expanding gases push piston DOWN.
4. Exhaust stroke: Piston moves UP, exhaust valve opens, burned gases expelled.

Engine Components

2. ENGINE SUPPORT SYSTEMS

Cooling System

• Purpose: prevent overheating by removing excess heat from the engine.
• Components: radiator (dissipates heat to air), coolant/antifreeze (carries heat), water pump (circulates coolant), thermostat (regulates temperature), radiator cap (pressurizes system).
• Overheating symptoms: temperature gauge rises, steam from hood, coolant loss.

Lubrication System

• Purpose: reduce friction between moving metal parts, carry away heat and contaminants.
• Oil pump circulates oil from the oil pan through galleries to all engine parts.
• Oil filter removes contaminants. Regular oil changes prevent engine wear and sludge buildup.
• Low oil pressure is a serious warning — stop engine immediately.

Fuel System

• Fuel pump (electric in most modern cars): draws fuel from tank, pressurizes it.
• Fuel filter: removes dirt and contaminants from fuel before injection.
• Fuel injectors: precisely controlled by ECU (engine control unit) to deliver correct fuel amount.
• Carburetor (older vehicles): mixes air and fuel mechanically — replaced by fuel injection in modern vehicles.

Exhaust System

• Exhaust manifold → catalytic converter (reduces harmful emissions: HC, CO, NOx) → muffler (reduces sound) → tailpipe.
• O₂ sensors monitor exhaust content and report to ECU to optimize fuel mixture.

3. DRIVETRAIN & BRAKES

Transmission & Drivetrain

• Clutch (manual): Disconnects engine from transmission for gear changes.
• Transmission: Changes gear ratios to optimize engine power vs. speed. Automatic: uses fluid coupling and planetary gears. Manual: uses clutch + gear lever.
• Driveshaft: Transfers torque from transmission to differential.
• Differential: Allows wheels to spin at different speeds during turns.
• Drive types: FWD (front-wheel drive), RWD (rear-wheel drive), AWD/4WD (all four wheels driven).

Brake Systems

• Disc brakes: Caliper squeezes brake pads against a rotor disc. More efficient, better heat dissipation. Used on front (and often all four) wheels.
• Drum brakes: Brake shoes press outward against a drum. Less efficient but simpler. Often used on rear wheels of economy cars.
• Hydraulic brakes: Brake pedal force multiplied by brake fluid through master cylinder and brake lines to each wheel.
• ABS (Anti-lock Braking System): Prevents wheels from locking up during hard braking by rapidly pulsing brake pressure, maintaining steering control.

4. HAND TOOLS & SHOP KNOWLEDGE

Measuring Tools

• Micrometer: Measures very small dimensions to 0.001" accuracy. Used for shaft diameters, wall thickness.
• Vernier caliper: Measures internal and external dimensions, depth. More versatile than micrometer.
• Feeler gauge: Measures gaps (spark plug gap, valve clearance) using thin metal blades of known thickness.
• Torque wrench: Tightens fasteners to a specific torque value to prevent over- or under-tightening.

Hand Tools

Shop Safety Rules

• Always wear safety glasses when grinding, cutting, or working with chemicals.
• Use the RIGHT tool for the job — wrong tool causes damage and injury.
• Disconnect the battery before working on electrical systems.
• Never work under a vehicle supported only by a jack — always use jack stands.
• Keep flammable materials away from sparks and heat sources.
• Torque fasteners to spec — too tight can strip or crack; too loose can cause failure.

1. LEVERS

A lever is a rigid bar that rotates around a fixed point called the fulcrum. Levers provide mechanical advantage — allowing a small effort to move a larger load.

Three Classes of Levers

2. PULLEYS & WHEEL-AND-AXLE

Pulleys

• Fixed pulley: Changes direction of force only. MA = 1. (Flagpole)
• Movable pulley: The load is attached to the pulley. MA = 2. (One rope section supports the load)
• Block and tackle: Combination of fixed and movable pulleys. MA = number of rope sections supporting the movable block.

Wheel and Axle

• A large wheel attached to a smaller axle. Turning the wheel produces more torque at the axle.
• MA = Radius of wheel ÷ Radius of axle
• Examples: steering wheel, screwdriver, doorknob, winch, bicycle pedals/gears.

3. INCLINED PLANE, WEDGE & SCREW

Inclined Plane (Ramp)

• A flat surface at an angle. Reduces the force needed to raise a load, but increases the distance over which force is applied.
• MA = Length of slope ÷ Height = 1/sin(θ)
• A longer, shallower ramp = greater MA = less effort needed.

Wedge

• Two inclined planes back to back. Converts downward force into horizontal splitting force.
• A sharper (thinner) wedge has greater MA. Examples: axe blade, chisel, knife, nail.

Screw

• An inclined plane wrapped around a cylinder. The tighter (smaller) the thread pitch, the greater the MA.
• MA = Circumference of handle ÷ Distance between threads (pitch)
• Examples: bolt, jar lid, auger, vise, jack screw.

4. GEARS, SPRINGS & PRESSURE

Gears

• Meshing gears always rotate in opposite directions (unless an idler gear is used).
• Larger gear rotates slower but with more torque. Smaller gear rotates faster with less torque.
• Gear Ratio = Teeth on Driven gear ÷ Teeth on Driver gear
• If driver has 10 teeth and driven has 30 teeth: ratio = 3:1 → driven turns 3× slower but with 3× the torque.

Springs

• Hooke's Law: F = kx (Force = spring constant × displacement). Springs stretch proportionally to force applied.
• Springs in series: same force, but stretch adds. Springs in parallel: forces add, same stretch.
• Compression spring: resists compression. Extension spring: resists extension. Torsion spring: resists twisting.

Pressure & Fluid Mechanics

1. UNDERSTANDING THE SUBTEST

Assembling Objects (AO) tests your spatial reasoning — your ability to mentally visualize and manipulate objects. It is used primarily by the Navy for certain ratings.

Two Question Types

• Connector problems: You see a shape and a dot on it. You must find which assembled picture shows the shapes connected at those exact points.
• Assembly problems: You see several pieces. You must identify which final shape they assemble into.

2. SPATIAL REASONING STRATEGIES

Connector Problem Strategies

• Note the shape of each piece first — eliminate answers with wrong shapes.
• Note the position of the dot on each piece. Is it on an edge, corner, or interior?
• Check that the connection point in the answer matches the dot positions exactly.
• Both pieces must maintain their original shape — they shouldn't stretch or deform.

Assembly Problem Strategies

• Count the number of pieces. Eliminate answers that look like they need more or fewer pieces.
• Identify the most distinctive piece (unusual shape) and look for it in each answer choice.
• Check that the total area of pieces roughly matches the area of the assembled figure.
• Pieces can be rotated but NOT flipped (mirror-imaged). Eliminate flipped shapes.

3. MENTAL ROTATION & VISUALIZATION

Building Spatial Skills

• Mental rotation: Visualize an object from a different angle. Practice by looking at 3D objects and imagining them from above, below, and each side.
• Pattern folding: Visualize a 2D flat pattern folding into a 3D shape. Common on AO: which 3D box can be formed from this unfolded flat net?
• Part-to-whole visualization: Given the parts, what does the assembled whole look like? Check size proportions.

Common Traps to Avoid

• Mirror images look correct but are wrong if pieces can't be flipped.
• Connection point is at the wrong position on the shape (e.g., middle of edge vs. end of edge).
• Assembled shape has correct pieces but wrong relative orientation.
• One piece is scaled differently than in the problem.