Every formula in the course

part = percent × base

Any percent problem: two are given, find the third

Example 15% of 80 = 0.15 × 80 = 12

percent change = change ÷ original

Always divide by where you started

Example 80 → 100: 20 ÷ 80 = 25%

I = P × r × t

Simple interest: principal × yearly rate (as a decimal) × time in years (months ÷ 12)

Example $500 at 4% for 3 years: 500 × 0.04 × 3 = $60

total = P + I

What you have or owe at the end

Example 500 + 60 = $560

a/b = c/d → a × d = b × c

Direct proportion: same units in the same places, then cross-multiply

Example 150 miles on 5 gallons, x miles on 8: 5x = 150 × 8, x = 240

first × first = second × second

Inverse proportion: when one goes up, the other goes down

Example 6 hours at 50 mph = x hours at 60 mph: 6 × 50 = 60x, x = 5

d = r × t

Distance = rate × time

Example 45 mph for 2 hours = 90 miles

average speed = total distance ÷ total time

Never the average of the two speeds

Example 100 miles in 4 hours = 25 mph

1/a + 1/b = 1/T

Work together: add the rates (jobs per hour), then flip. Works for any number of workers

Example 3 h and 4 h: 1/3 + 1/4 = 7/12, so T = 12/7 ≈ 1.7 hours

T = (a × b) ÷ (a + b)

Shortcut for exactly two workers (a and b are their hours alone)

Example 6 h and 3 h: 18 ÷ 9 = 2 hours

pure part = amount × percent

Mixtures: track the pure part of each liquid

Example 20 gallons at 10% has 2 gallons pure

a × p + x × q = (a + x) × r

How much to add: the pure parts going in = the pure part of the mix

Example 9 L at 15% + x L at 45% = 25%: x = 4.5 L

mean = sum ÷ count

The average

Example (4 + 6 + 11) ÷ 3 = 7

sum = mean × count

Work backward from an average

Example 5 numbers averaging 8 add to 40

median = the middle value, after sorting

With an even count, the mean of the two middle values

Example 2, 5, 9, 12 → (5 + 9) ÷ 2 = 7

term n = first + (n − 1) × step

A term of an arithmetic sequence

Example 3, 7, 11 …: term 10 = 3 + 9 × 4 = 39

probability = favorable ÷ total

A number from 0 to 1

Example 3 red of 12 marbles: 3/12 = 1/4

P(not A) = 1 − P(A)

The chance it does NOT happen

Example 3 of 8 spaces are red: P(not red) = 1 − 3/8 = 5/8

P(A and B) = P(A) × P(B)

Two independent events

Example Two heads: 1/2 × 1/2 = 1/4

a/d + b/d = (a + b)/d

Add only with the same denominator

Example 2/9 + 4/9 = 6/9 = 2/3

a/b × c/d = (a × c)/(b × d)

Multiply straight across

Example 2/3 × 3/4 = 6/12 = 1/2

a/b ÷ c/d = a/b × d/c

Divide: keep, change, flip

Example 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2

whole a/b → (whole × b + a)/b

Mixed number to improper fraction, before you multiply or divide

Example 2 1/2 = (2 × 2 + 1)/2 = 5/2

LCM = each prime, the most times it shows up in one number

The common denominator for adding fractions

Example LCM of 4 and 6 = 2 × 2 × 3 = 12

a − (−b) = a + b

Subtracting a negative adds

Example 5 − (−3) = 8

quotient × divisor + remainder = dividend

Check any division by hand

Example 17 ÷ 5 = 3 r 2: 3 × 5 + 2 = 17

aᵐ × aⁿ = aᵐ⁺ⁿ

Multiply powers of the same base: add the exponents

Example 2³ × 2² = 2⁵ = 32

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Divide: subtract the exponents

Example x⁷ ÷ x³ = x⁴

(aᵐ)ⁿ = aᵐⁿ

Power of a power: multiply the exponents

Example (x²)³ = x⁶

a⁰ = 1 · a⁻ⁿ = 1/aⁿ

Zero and negative exponents (a negative exponent does not make the number negative)

Example 5⁻² = 1/25

(a + b)(c + d) = ac + ad + bc + bd

FOIL: first, outer, inner, last

Example (x + 2)(x + 3) = x² + 5x + 6

a² − b² = (a + b)(a − b)

Difference of squares

Example x² − 49 = (x + 7)(x − 7)

slope = (y₂ − y₁) ÷ (x₂ − x₁)

Rise over run between two points

Example (1, 3) and (5, 11): 8 ÷ 4 = 2

midpoint = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

The point halfway between two points

Example (2, 4) and (8, 10) → (5, 7)

y = mx + b

m is the slope; b is where the line crosses the y-axis

Example y = 2x + 1: slope 2, crosses at (0, 1)

distance = √((x₂ − x₁)² + (y₂ − y₁)²)

The Pythagorean theorem drawn on the grid

Example (1, 2) and (7, 10): √(36 + 64) = 10

rectangle: A = l × w, P = 2l + 2w

Area and perimeter

Example 8 by 5: A = 40, P = 26

triangle: A = ½ × base × height

The height is perpendicular to the base

Example base 10, height 6: A = 30

a² + b² = c²

Pythagorean theorem: c is the side across from the right angle

Example 6, 8 → c = 10

triangle angles add to 180°

In any triangle

Example 50° and 60° → the third is 70°

sum of angles = (n − 2) × 180°

Inside any polygon with n sides

Example hexagon: 4 × 180° = 720°

C = 2πr, A = πr²

Circumference and area of a circle (π ≈ 3.14)

Example r = 5: C ≈ 31.4, A ≈ 78.5

box: V = l × w × h

Volume of a rectangular box

Example 4 × 3 × 2 = 24

cylinder: V = πr²h

Circle area × height

Example r = 2, h = 5: V = 20π ≈ 62.8

cone: V = ⅓πr²h

One third of the matching cylinder

Example r = 3, h = 4: V = 12π

sphere: V = ⁴⁄₃πr³

Volume of a ball

Example r = 3: V = 36π ≈ 113

sector = (angle ÷ 360) × πr²

A slice of a circle

Example 90° slice of r = 4: ¼ × 16π = 4π

12 inches = 1 foot · 3 feet = 1 yard · 5,280 feet = 1 mile

Length

Example 4 feet = 48 inches

16 ounces = 1 pound · 2,000 pounds = 1 ton

Weight

Example 3 pounds = 48 ounces

2 cups = 1 pint · 2 pints = 1 quart · 4 quarts = 1 gallon

Liquid volume

Example 2 gallons = 8 quarts

60 seconds = 1 minute · 60 minutes = 1 hour · 24 hours = 1 day

Time

Example 2.5 hours = 150 minutes

to a smaller unit, multiply; to a bigger unit, divide

Converting

Example feet → inches: × 12

AFQT = AR + MK + 2 × VE

VE is the verbal score from Word Knowledge and Paragraph Comprehension

Example Verbal counts twice