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ASVAB Formula Sheet

You can't use a calculator or a formula sheet on the ASVAB, so these need to be in your head on test day. Print this page and study from it, or keep it open on your phone.

Arithmetic Reasoning

Percent of a number: part = percent × whole; Turn the percent into a decimal first: 15% of 80 = 0.15 × 80 = 12.
Percent change: (new − old) ÷ old × 100; Always divide by the starting value, not the new one.
Simple interest: I = P × r × t; P is the amount, r the yearly rate as a decimal, t the time in years. 18 months is 1.5.
Distance, rate and time: d = r × t; Rearranged: r = d ÷ t and t = d ÷ r. Keep the units matched (miles with hours).
Average speed: total distance ÷ total time; Do not average the two speeds. Add the distances, add the times, then divide.
Working together: 1/t = 1/a + 1/b; a and b are the times each takes alone. For two workers, t = (a × b) ÷ (a + b).
Average (mean): sum ÷ count; To find a missing score: needed total = target average × count, then subtract what you have.
Proportion: a/b = c/d → a × d = b × c; Cross-multiply to find the missing value. Keep the same units on the same level.
Probability: favourable outcomes ÷ all outcomes; For two independent events both happening, multiply their probabilities.
Unit price: total price ÷ quantity; Use it to compare deals: the lower unit price is the better buy.

Mathematics Knowledge

Rectangle: A = l × w P = 2l + 2w; A square is the same with l = w: A = s² and P = 4s.
Triangle: A = ½ × b × h; The height must be perpendicular to the base. The three angles add to 180°.
Circle: A = πr² C = 2πr = πd; r is the radius, d the diameter (d = 2r). Use π ≈ 3.14 or 22/7.
Parallelogram and trapezoid: A = b × h A = ½(b₁ + b₂) × h; For the trapezoid, b₁ and b₂ are the two parallel sides.
Volume: box: V = l × w × h cylinder: V = πr²h cube: V = s³; Volume is always area of the base × height for these shapes.
Pythagorean theorem: a² + b² = c²; Right triangles only; c is the longest side. Common sets: 3-4-5, 5-12-13, 8-15-17.
Special right triangles: 45-45-90 → 1 : 1 : √2 30-60-90 → 1 : √3 : 2; The ratios of the sides, shortest first.
Polygon angles: sum of interior angles = (n − 2) × 180°; n is the number of sides. A quadrilateral totals 360°, a hexagon 720°.
Slope: m = (y₂ − y₁) ÷ (x₂ − x₁); Rise over run. Parallel lines have equal slopes; perpendicular slopes multiply to −1.
Line: y = mx + b; m is the slope and b is where the line crosses the y-axis.
Distance and midpoint: d = √((x₂ − x₁)² + (y₂ − y₁)²) M = ((x₁ + x₂)/2, (y₁ + y₂)/2); The distance formula is the Pythagorean theorem on a grid.
Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a; Solves ax² + bx + c = 0. Try factoring first; it is usually faster.
Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ xᵃ ÷ xᵇ = xᵃ⁻ᵇ (xᵃ)ᵇ = xᵃᵇ; Also x⁰ = 1 and x⁻ⁿ = 1 ÷ xⁿ.
Factoring patterns: a² − b² = (a + b)(a − b) (a + b)² = a² + 2ab + b²; Spot the difference of two squares to factor in one step.

Electronics Information

Ohm's law: V = I × R; Volts = amps × ohms. Rearranged: I = V ÷ R and R = V ÷ I.
Power: P = V × I = I² × R = V² ÷ R; Watts. Pick the version that uses the two values you are given.
Resistors in series: R = R₁ + R₂ + R₃ …; One path. The current is the same through every part; the voltages add up.
Resistors in parallel: 1/R = 1/R₁ + 1/R₂ …; Two resistors: R = (R₁ × R₂) ÷ (R₁ + R₂). Equal resistors: one value ÷ how many. The total is always less than the smallest branch.
Energy used: energy = power × time; Kilowatt-hours = kilowatts × hours. A 2 kW heater for 3 hours uses 6 kWh.
Transformer: Vₛ ÷ Vₚ = Nₛ ÷ Nₚ; Voltage follows the turns ratio. More turns on the secondary steps the voltage up. AC only.

Mechanical Comprehension

Mechanical advantage: MA = load ÷ effort; How many times a machine multiplies your force. You pay for it in distance.
Lever: effort × effort arm = load × load arm; Arms are measured from the fulcrum. A longer effort arm means less effort.
Inclined plane: MA = length of ramp ÷ height; A longer, gentler ramp needs less force over more distance.
Wheel and axle: MA = wheel radius ÷ axle radius; A screwdriver handle and a doorknob work this way.
Pulleys: MA = number of rope segments holding the load; A single fixed pulley only changes direction (MA = 1).
Gears: speed₁ × teeth₁ = speed₂ × teeth₂; The smaller gear turns faster. Two meshed gears turn in opposite directions.
Pressure: P = F ÷ A; Force spread over area. In psi: pounds ÷ square inches.
Hydraulics: F₁ ÷ A₁ = F₂ ÷ A₂; Pressure is the same everywhere in the fluid, so a bigger piston gives a bigger force.
Work and power: W = F × d P = W ÷ t; No movement means no work. Power is how fast the work is done.
Torque: torque = force × distance from the pivot; A longer wrench gives more turning force for the same push.

General Science

Speed and acceleration: speed = d ÷ t acceleration = change in speed ÷ time; Near Earth, falling objects speed up by about 9.8 m/s every second.
Force and weight: F = m × a weight = m × g; Newton's second law. g ≈ 9.8 m/s². Mass stays the same anywhere; weight changes with gravity.
Momentum: p = m × v; Mass × velocity. Total momentum is the same before and after a collision.
Energy: kinetic = ½mv² potential = mgh; Doubling the speed gives four times the kinetic energy.
Density: density = mass ÷ volume; Anything less dense than water (1 g/cm³) floats in it.
Waves: speed = frequency × wavelength; Higher frequency means shorter wavelength at the same speed.
Temperature: °F = °C × 9/5 + 32 °C = (°F − 32) × 5/9; Water freezes at 0 °C (32 °F) and boils at 100 °C (212 °F).

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