Proportionality — Study Guide

1

Ratios & Rates

Comparing quantities

🌍

Why this matters: Ratios and rates are everywhere — recipes, speeds, exchange rates, sports statistics. Understanding them lets you scale things up or down correctly.

What is a Ratio?

A ratio compares two quantities of the same kind using the same unit. It tells you "for every … there are …"

Ratio notation

a : b or a/b or "a to b"

A ratio has no units — both quantities must be in the same unit first.
A ratio can be simplified like a fraction: 6:9 = 2:3
Order matters: 2:3 is different from 3:2

What is a Rate?

A rate compares two quantities of different kinds (different units).

Examples of rates

60 km/h    (distance per time)
$3.50/kg    (price per mass)
120 beats/min    (count per time)

🔑

A unit rate has a denominator of 1 — it's the rate "per one unit." E.g. $2.40/kg means $2.40 per 1 kg.

Equivalent Ratios

Ratios are equivalent when they simplify to the same value — just like equivalent fractions.

2:3 = 4:6 = 6:9 = 10:15 (all equal 2/3)

To find an equivalent ratio, multiply or divide both parts by the same number.

Example 1 Simplify a ratio

›

Simplify the ratio 24:36.

1

Find GCF of 24 and 36. GCF = 12.

2

Divide both by 12: 24 ÷ 12 = 2, 36 ÷ 12 = 3.

Simplified ratio: 2 : 3

Example 2 Unit rate

›

A car travels 270 km in 3 hours. Find the unit rate (speed).

1

Write the rate: 270 km / 3 h

2

Divide: 270 ÷ 3 = 90

3

Add the units back.

Unit rate = 90 km/h

Checkpoint

A recipe calls for 3 cups of flour and 2 cups of sugar. If you want to make a double batch, how much of each ingredient do you need? What is the ratio of flour to sugar in simplest form?

Double batch: 6 cups flour, 4 cups sugar.
Ratio: 6 : 4 = 3 : 2 (same as original — doubling doesn't change the ratio).

2

Percentages & Percent Change

Out of 100

🛒

Why this matters: Discounts, taxes, tips, interest rates, test scores — percentages appear in real life constantly. Percent change tells you how much something grew or shrank.

Percent Basics

Percent means "out of 100." The symbol is %.

Converting between forms

Percent → decimal: divide by 100    (35% = 0.35)
Decimal → percent: multiply by 100    (0.72 = 72%)
Percent → fraction: put over 100, simplify    (40% = 40/100 = 2/5)

Finding a Percentage of a Number

Formula

Part = (Percent / 100) × Whole

e.g. 30% of 80 = (30/100) × 80 = 24

Percent Change

Formula

Percent Change = ((New − Old) / Old) × 100%

Positive → increase Negative → decrease

💡

Always divide by the original (old) value, not the new one. This is the most common mistake on tests!

Finding the Original Value

If you know the result after a percent change, you can work backwards.

After an increase of p%

Original = New Value ÷ (1 + p/100)

e.g. Price after 20% tax = $60. Original = 60 ÷ 1.20 = $50

Example 1 Percent change

›

A jacket costs $80. It goes on sale for $60. What is the percent decrease?

1

Identify: Old = 80, New = 60.

2

Apply formula: (60 − 80) / 80 × 100 = −20/80 × 100

3

= −0.25 × 100 = −25%

Percent decrease = 25%

Example 2 Finding the original

›

After a 15% increase, a phone costs $460. What was the original price?

1

After 15% increase: New = Original × 1.15

2

So: Original = 460 ÷ 1.15

3

= 400

Original price = $400

Checkpoint

A store sells a video game for $75, which includes a 25% markup over the wholesale price. What was the wholesale price?

Selling price = Wholesale × 1.25
Wholesale = 75 ÷ 1.25 = $60

3

Direct Proportion

As one grows, so does the other

📈

Why this matters: Direct proportion describes situations where two quantities always increase or decrease together at the same rate — like price and quantity, or distance and time at constant speed.

Definition

Two quantities are in direct proportion if their ratio is always constant.

Direct proportion

y = kx where k = constant of proportionality

k = y / x (always the same for any pair (x, y))

The graph is a straight line through the origin (0, 0)
If x doubles, y doubles. If x triples, y triples.
k is the slope of the line (rise/run)

How to Recognise Direct Proportion

Direct proportion — YES

Ratio y/x is the same for every pair
Graph is a line through origin
When x = 0, y = 0

Not direct proportion

Ratio y/x changes
Graph doesn't pass through origin
Has a fixed cost or starting value

Solving Direct Proportion Problems

Two methods — use whichever feels natural:

Method 1 — Unit rate

Find k = y/x, then multiply.

Method 2 — Cross multiply

x₁/y₁ = x₂/y₂
x₁ × y₂ = x₂ × y₁

Example 1 Finding a missing value

›

5 apples cost $3.25. How much do 12 apples cost?

1

Unit rate: $3.25 ÷ 5 = $0.65 per apple

2

12 apples: 12 × $0.65 = $7.80

12 apples cost $7.80

Example 2 Is it direct proportion?

›

Check if this table shows direct proportion:

x	y	y/x
2	6	3
5	15	3
8	24	3

1

Compute y/x for each row: 6/2 = 3, 15/5 = 3, 24/8 = 3.

2

Ratio is constant → yes, direct proportion. k = 3, so y = 3x.

Direct proportion: y = 3x

Checkpoint

A cyclist rides 36 km in 1.5 hours at constant speed. How far will she travel in 2.5 hours? What is k (the constant of proportionality)?

Speed (k): 36 ÷ 1.5 = 24 km/h
Distance in 2.5 h: 24 × 2.5 = 60 km

4

Inverse Proportion

As one grows, the other shrinks

⚖️

Why this matters: Speed and travel time, number of workers and time to finish a job, gears on a bike — these are all inverse proportions. As one quantity goes up, the other comes down.

Definition

Two quantities are in inverse proportion if their product is always constant.

Inverse proportion

x × y = k (constant product)

y = k / x (as x increases, y decreases)

The graph is a hyperbola (a curve that never touches the axes)
If x doubles, y is halved. If x triples, y becomes one third.

Direct vs Inverse — Side by Side

Direct proportion

y = kx
y/x = constant
Both increase together
Graph: straight line through origin

Inverse proportion

y = k/x
x × y = constant
One increases, other decreases
Graph: hyperbola (curve)

💡

Quick test: Multiply each x-y pair. If the products are all equal → inverse proportion. Divide y/x for each pair — if equal → direct proportion.

Example 1 Inverse proportion problem

›

4 workers can build a wall in 6 days. How long will it take 8 workers?

1

Recognise: more workers → fewer days = inverse proportion.

2

Find k: 4 × 6 = 24 (total worker-days).

3

With 8 workers: days = 24 ÷ 8 = 3.

8 workers take 3 days

Example 2 Identify from a table

›

Is this table direct or inverse proportion?

x	y	x × y	y/x
2	18	36	9
3	12	36	4
6	6	36	1

1

x × y is constant (36) → inverse proportion.

2

y/x is not constant → not direct proportion.

Inverse proportion: y = 36/x

Checkpoint

A car travels from Montreal to Quebec City at 80 km/h and takes 3 hours. If the driver travels at 120 km/h instead, how long will the trip take?

k = speed × time = 80 × 3 = 240 km (the distance is constant)
Time at 120 km/h = 240 ÷ 120 = 2 hours

5

Scale & Similar Figures

Maps, models, and geometry

🗺️

Why this matters: Architects draw blueprints, maps represent real distances, models reproduce real objects — all using scale factors. Similar figures have the same shape but different sizes.

Scale Factor

A scale factor is the ratio of a measurement in the drawing/model to the real measurement.

Scale factor

Scale = Drawing length / Real length

e.g. scale 1:50 means 1 cm on paper = 50 cm in reality

Converting

Real length = Drawing length ÷ Scale factor
Drawing length = Real length × Scale factor

Similar Figures

Two figures are similar (symbol: ~) if they have the same shape but different sizes.

Corresponding angles are equal
Corresponding sides are proportional (same ratio)
The ratio of corresponding sides is called the scale factor

Proportion of sides

AB/DE = BC/EF = AC/DF = k (for similar triangles ABC ~ DEF)

Area and Volume Scale

⚡

Important pattern: If the scale factor (linear) is k:
Area scales by k² Volume scales by k³

e.g. If lengths double (k = 2): area × 4, volume × 8

Example 1 Map scale

›

A map has a scale of 1:25 000. Two cities are 8 cm apart on the map. What is the real distance?

1

Scale 1:25 000 means 1 cm = 25 000 cm = 250 m.

2

Real distance = 8 × 25 000 = 200 000 cm

3

Convert: 200 000 cm = 2 000 m = 2 km

Real distance = 2 km

Example 2 Similar triangles

›

Triangle ABC ~ Triangle DEF. AB = 6 cm, DE = 9 cm, BC = 8 cm. Find EF.

1

Scale factor k = DE/AB = 9/6 = 1.5

2

EF = BC × k = 8 × 1.5 = 12

EF = 12 cm

Checkpoint

Two similar rectangles have a side length ratio of 3:5. If the area of the smaller rectangle is 27 cm², what is the area of the larger rectangle?

Linear scale factor: 3:5, so k = 5/3
Area scale factor: k² = (5/3)² = 25/9
Larger area: 27 × (25/9) = 675/9 = 75 cm²