Proportionality — Cheat Sheet

RATIOS & RATES

Ratio

a:b — same type, no units

Rate

different units (km/h, $/kg)

Unit rate

rate per 1 unit

Ex: 240 km in 3 h → 80 km/h

SOLVING PROPORTIONS

Cross-mult

a/b = c/d → ad = bc

Ex: 3/4 = x/20
3×20 = 4×x → x = 15

SIMPLIFYING RATIOS

Divide both parts by GCF
12:18 ÷ 6 = 2:3
Order matters: 2:3 ≠ 3:2

PERCENTAGES

Find part

part = (%) / 100 × whole

Find %

% = (part / whole) × 100

Find whole

whole = part / (% ÷ 100)

PERCENT CHANGE

% change

((new − old) / old) × 100

% increase

new = old × (1 + rate)

% decrease

new = old × (1 − rate)

25% off $40: 40 × 0.75 = $30
$50 → $65: (15/50)×100 = 30% increase

WATCH OUT

Identify the ORIGINAL value first
20% off then 20% on ≠ same price

DIRECT PROPORTION

Formula

y = kx

Find k

k = y / x (constant ratio)

Graph: straight line through origin
If x doubles → y doubles

INVERSE PROPORTION

Formula

y = k/x (or xy = k)

Find k

k = x × y (constant product)

Graph: hyperbola curve
If x doubles → y halves

	Direct	Inverse
Formula	y = kx	y = k/x
Constant	y/x = k	xy = k
Graph	Line thru O	Hyperbola
x doubles	y doubles	y halves

SCALE & SIMILAR FIGURES

Scale factor

k = image / original

Map scale

1:n → 1 cm = n cm real

Similar figures: angles equal, sides proportional

Scale 1:50 000, 3.2 cm on map:
real = 3.2 × 50 000 = 160 000 cm = 1.6 km

Triangles 3,4,5 and 6,?,?
k = 6/3 = 2 → sides 8 and 10

COMMON MISTAKES

Direct vs inverse: does y go up or down?
Cross-mult: ad=bc, not ac=bd
Angles don't scale in similar figures
Convert map units before calculating