📐 Concept diagram

00-07 — Ratios, Rates, and Proportions

Phase: 0 — Arithmetic & Number Foundations Subject: 00-07 Prerequisites: 00-01 — Whole Number Arithmetic, 00-02 — Fractions, 00-03 — Decimals, 00-04 — Percentages, 00-05 — Integers and Directed Numbers, 00-06 — Powers and Roots Next subject: 00-08 — Basic Number Theory

Learning Objectives

By the end of this subject, you will be able to:

1. Write, interpret, and simplify ratios in multiple notations (a:b, a/b, "a to b")
2. Apply the unitary method to solve proportion problems by finding the value of one unit first
3. Distinguish between direct and inverse proportion and solve problems involving both
4. Calculate and interpret unit rates (speed, density, price per item) from given data
5. Use scale factors to convert between real measurements and scaled representations (maps, models, diagrams)

Core Content

1. What Is a Ratio?

A ratio compares two or more quantities of the same kind. It tells you how much of one thing there is relative to another.

Notation: The ratio of a to b can be written as:

a : b ("a to b") a / b (as a fraction) "a to b"

⚠️ THIS IS CRITICAL — Ratios appear throughout all mathematics and real life: recipe scaling, map reading, financial analysis, chemical mixtures, probability, trigonometry (sine/cosine/tangent are ratios), and the slope of a line (Δy:Δx). Understanding ratios deeply is essential.

Examples: - In a class with 12 boys and 8 girls, the ratio of boys to girls is 12 : 8 - A recipe calls for flour and sugar in the ratio 3 : 1 - The aspect ratio of a screen is 16 : 9

Key idea: A ratio compares parts to parts (or parts to a whole). Unlike a fraction which often represents a part of a whole, a ratio can compare any two quantities.

2. Simplifying Ratios

Ratios can be simplified the same way fractions are simplified — divide both sides by their greatest common factor.

a : b simplifies to (a ÷ g) : (b ÷ g) where g = GCF(a, b)

Example 1: 12 : 8 GCF(12, 8) = 4 12 ÷ 4 = 3, 8 ÷ 4 = 2 Simplified: 3 : 2

Check: 12/8 = 3/2 ✓ (the ratios are equivalent)

Example 2: 45 : 30 GCF(45, 30) = 15 45 ÷ 15 = 3, 30 ÷ 15 = 2 Simplified: 3 : 2

Example 3 (three quantities): 24 : 36 : 60 GCF(24, 36, 60) = 12 24 ÷ 12 = 2, 36 ÷ 12 = 3, 60 ÷ 12 = 5 Simplified: 2 : 3 : 5

Rule: Always simplify ratios to their lowest terms using whole numbers. If a ratio contains fractions, multiply through by the LCM of the denominators.

Example 4 (with fractions): 1/2 : 3/4 LCM of denominators (2, 4) = 4 Multiply both sides by 4: (1/2 × 4) : (3/4 × 4) = 2 : 3

3. The Unitary Method

The unitary method is a problem-solving strategy: find the value of ONE unit first, then scale up to the required quantity.

⚠️ THIS IS CRITICAL — The unitary method is the single most reliable strategy for solving proportion problems. Learn it well.

Step 1: Find the value of 1 unit by dividing. Step 2: Multiply to find the value of the desired number of units.

Example: If 5 identical books cost £35, how much do 8 books cost?

Step 1 — Find the cost of 1 book: £35 ÷ 5 = £7 per book

Step 2 — Find the cost of 8 books: £7 × 8 = £56

Answer: £56

Why this works: We're assuming the relationship is proportional — each book costs the same. The unitary method explicitly uses this assumption.

Example with multiple rates: A car travels 240 km on 20 litres of fuel. How far can it travel on 35 litres?

Step 1 — Find distance per litre: 240 ÷ 20 = 12 km per litre

Step 2 — Distance for 35 litres: 12 × 35 = 420 km

Answer: 420 km

4. Direct Proportion

Two quantities are in direct proportion if:

• When one doubles, the other doubles
• When one is halved, the other is halved
• Their ratio stays constant

If y is directly proportional to x, then y = kx where k is the constant of proportionality.

Notation: y ∝ x (read "y is proportional to x")

Equivalent formulations: - y/x = k (constant) - y₁/x₁ = y₂/x₂

Example: 3 pens cost £4.50. How much do 7 pens cost?

Method 1 — Unitary method: 1 pen = £4.50 ÷ 3 = £1.50 7 pens = £1.50 × 7 = £10.50

Method 2 — Proportion equation: y/x = k → £4.50/3 = y/7 Cross-multiply: y = (£4.50 × 7) / 3 = £31.50 / 3 = £10.50

Method 3 — Scale factor: 7 pens is 7/3 times as many as 3 pens. Cost = £4.50 × (7/3) = £10.50

Answer: £10.50

How to recognise direct proportion: - "The more you buy, the more you pay" (at a fixed price per item) - "The faster you go, the more distance you cover" (in fixed time) - The graph of y vs x is a straight line through the origin

5. Inverse Proportion

Two quantities are in inverse proportion if:

• When one doubles, the other is halved
• When one is multiplied by n, the other is divided by n
• Their product stays constant

If y is inversely proportional to x, then y = k/x where k is the constant.

Notation: y ∝ 1/x

Equivalent formulation: x × y = k (constant product)

Example: 4 workers can build a wall in 6 hours. How long would 6 workers take?

Step 1 — Find the constant product: 4 workers × 6 hours = 24 worker-hours (this is k, the total work)

Step 2 — For 6 workers: 6 × time = 24 time = 24 ÷ 6 = 4 hours

Answer: 4 hours

Why does this make sense? More workers → less time. Twice as many workers → half the time (inverse relationship). The product (workers × time) is the total amount of work, which is fixed.

How to recognise inverse proportion: - "The more workers, the less time" (fixed total work) - "The faster you go, the less time a journey takes" (fixed distance) - "The more people sharing a prize, the less each gets" (fixed total) - The graph of y vs x is a hyperbola

6. Distinguishing Direct from Inverse Proportion

Test question: "If 5 machines produce 100 widgets per hour, how many widgets do 8 machines produce per hour?"

This is DIRECT proportion: more machines → more widgets. k = 100/5 = 20 widgets per machine per hour 8 machines produce 8 × 20 = 160 widgets per hour.

Test question: "If 5 machines take 8 hours to complete a job, how long would 8 machines take?"

This is INVERSE proportion: more machines → less time. k = 5 × 8 = 40 machine-hours 8 machines → 40/8 = 5 hours.

⚠️ Same numbers, different relationship. Always check whether doubling one quantity doubles or halves the other.

7. Unit Rates

A unit rate expresses how much of one quantity corresponds to one unit of another quantity. It's a ratio where the second term is 1.

Examples of unit rates: - Speed: 60 km per hour (60 km/h) — distance per 1 hour - Density: 7.8 g per cm³ (7.8 g/cm³) — mass per 1 unit volume - Price: £2.50 per litre — cost per 1 litre - Fuel efficiency: 15 km per litre — distance per 1 litre - Heart rate: 72 beats per minute — beats per 1 minute

Calculating unit rates:

If 300 km is travelled in 4 hours, speed = 300 ÷ 4 = 75 km/h.

If 5 litres of paint cover 60 m², coverage rate = 60 ÷ 5 = 12 m² per litre.

Using unit rates to compare values:

Which is the better buy: 500 g of coffee for £8 or 350 g for £5.95?

Coffee A: £8 ÷ 500 = £0.016 per gram = £1.60 per 100 g Coffee B: £5.95 ÷ 350 = £0.017 per gram = £1.70 per 100 g

Coffee A is the better value.

8. Scale Factors

A scale factor is the ratio between the size of a representation and the actual object.

Scale factor = measurement on model / measurement on actual object

Types of scale:

1. Ratio scale: 1 : 100 means 1 unit on the map = 100 units in real life
2. Scale statement: "1 cm represents 5 km"

⚠️ THIS IS CRITICAL — Scale factors are used in maps, architectural drawings, engineering plans, 3D printing, and computer graphics. They are an application of direct proportion.

Map reading example:

On a map with scale 1 : 50,000, two towns are 8 cm apart. What is the actual distance?

Actual distance = 8 cm × 50,000 = 400,000 cm Convert to km: 400,000 cm = 4,000 m = 4 km

Scale drawing example:

A building is 24 m long. On a plan with scale 1 : 200, how long is the building?

Plan length = 24 m ÷ 200 = 0.12 m = 12 cm

Area and volume scaling:

• If length is scaled by factor k, area scales by k²
• If length is scaled by factor k, volume scales by k³

Example: A 1 : 3 scale model of a cube. Model side: 2 cm → actual side: 6 cm Model area (one face): 2² = 4 cm² → actual area: 6² = 36 cm² = 4 × 9 = 4 × 3² ✓ Model volume: 2³ = 8 cm³ → actual volume: 6³ = 216 cm³ = 8 × 27 = 8 × 3³ ✓

9. Common Misconceptions

Misconception 1: "Ratio order doesn't matter." - Wrong! 3 : 2 ≠ 2 : 3. The ratio of boys to girls in a class with 15 boys and 10 girls is 15 : 10 = 3 : 2. The ratio of girls to boys is 10 : 15 = 2 : 3. Order matters.

Misconception 2: "All 'more → more' relationships are direct proportion." - Wrong! If you're 10 years old and 1.4 m tall, at age 20 you won't be 2.8 m tall. Age and height are NOT proportional — you don't keep growing at the same rate.

Misconception 3: "Inverse proportion means the quantities go in opposite directions." - Partially right but imprecise. The quantities must have a constant product, not just move in opposite directions. Time and temperature might move in opposite directions (cold morning → warm afternoon) but are NOT inversely proportional.

Misconception 4: "A scale of 1 : 100 means 1 cm² = 100 cm²." - Wrong! A scale factor of 1 : 100 is linear. 1 cm on the map = 100 cm in real life. 1 cm² on the map = 100² = 10,000 cm² in real life.

Key Terms

• Direct proportion
• Inverse proportion
• Ratios
• Scale factors
• The unitary method
• Unit rates

Worked Examples

Example 1: Simplifying and Using Ratios

Problem: A fruit salad contains apples, oranges, and grapes in the ratio 3 : 2 : 5. If there are 150 g of grapes, what is the total mass of the fruit salad?

Solution:

The ratio is 3 : 2 : 5 (apples : oranges : grapes).

Step 1 — Grapes correspond to 5 parts. Each part = 150 ÷ 5 = 30 g.

Step 2 — Total parts = 3 + 2 + 5 = 10

Step 3 — Total mass = 10 × 30 = 300 g.

Answer: 300 g

Check: Apples = 3 × 30 = 90 g, oranges = 2 × 30 = 60 g, grapes = 5 × 30 = 150 g 90 + 60 + 150 = 300 ✓

Example 2: Unitary Method with Multiple Steps

Problem: A machine produces 240 widgets in 3 hours. How many widgets would 4 such machines produce in 5 hours?

Solution:

Step 1 — Find production rate of 1 machine per hour: 240 widgets ÷ 3 hours = 80 widgets per hour per machine

Step 2 — Production of 4 machines per hour: 80 × 4 = 320 widgets per hour

Step 3 — Production in 5 hours: 320 × 5 = 1,600 widgets

Answer: 1,600 widgets

Alternative — using proportion directly: Rate ratio: (4 machines / 1 machine) × (5 hours / 3 hours) = (4 × 5) / 3 = 20/3 Widgets = 240 × 20/3 = 240 × 20 ÷ 3 = 4,800 ÷ 3 = 1,600 ✓

Example 3: Inverse Proportion

Problem: A tank can be filled by 3 pipes in 4 hours. How long would it take with 5 pipes (assuming all pipes have the same flow rate)?

Solution:

This is inverse proportion: more pipes → less time.

Step 1 — Find the constant (total pipe-hours): 3 pipes × 4 hours = 12 pipe-hours

Step 2 — Solve for time with 5 pipes: 5 × time = 12 time = 12 ÷ 5 = 2.4 hours = 2 hours 24 minutes

Answer: 2 hours 24 minutes

Intuition check: With more pipes, time should decrease. 2.4 < 4 ✓

Example 4: Scale Factor Application

Problem: On a map with scale 1 : 25,000, a rectangular park measures 3.2 cm by 5.0 cm. Find the actual area of the park in square kilometres.

Solution:

Step 1 — Actual dimensions: Length = 5.0 × 25,000 = 125,000 cm = 1,250 m = 1.25 km Width = 3.2 × 25,000 = 80,000 cm = 800 m = 0.8 km

Step 2 — Area: Area = 1.25 × 0.8 = 1.0 km²

Alternative using area scaling: Map area = 3.2 × 5.0 = 16 cm² Scale factor for area = 25,000² = 625,000,000 Actual area = 16 × 625,000,000 = 10,000,000,000 cm² = 10,000 m² × (1 km² / 1,000,000 m²)... Let's convert carefully: 10,000,000,000 cm² = 1,000,000 m² = 1 km² ✓

Answer: 1 km²

Practice Problems

(Answers are below. Try each problem before checking.)

Problem 1: Simplify the ratio 42 : 56 : 70.

Problem 2: If 8 identical pens cost £6.40, how much do 13 pens cost?

Problem 3: The ratio of red to blue to yellow beads in a necklace is 5 : 3 : 4. If there are 60 red beads, how many beads are in the necklace in total?

Problem 4: A car travels 350 km on 28 litres of fuel. How far can it travel on 40 litres of fuel?

Problem 5: 6 workers can paint a building in 10 days. How many days would 4 workers take (at the same rate)?

Problem 6: On a map with scale 1 : 500,000, two cities are 12 cm apart. What is the actual distance in kilometres?

Problem 7: Which is the better buy: 750 mL of juice for £2.40 or 1.25 L for £3.75?

Summary

1. Ratios compare quantities: a : b can be simplified by dividing by the GCF of all terms
2. The unitary method solves proportion problems by finding the value of one unit first, then scaling — it's the most reliable strategy
3. Direct proportion (y = kx): when one doubles, the other doubles; ratio y/x is constant. Inverse proportion (y = k/x): when one doubles, the other halves; product x × y is constant
4. Unit rates express how much per one unit (price per litre, speed in km/h) and are key for comparing values
5. Scale factors convert between model/representation and reality; ⚠️ area scales by k² and volume by k³, NOT by k

Pitfalls

• Reversing the ratio order. The ratio of boys to girls (3:2) is different from girls to boys (2:3). The order tells you which quantity comes first. Always check what the question asks for.
• Using linear scale factors for area and volume. A 1:100 scale map means 1 cm = 100 cm, but 1 cm² = 100² = 10,000 cm². Forgetting to square or cube the scale factor leads to dramatically wrong answers.
• Assuming all \"more X → more Y\" relationships are direct proportion. Height and shoe size both increase with age, but they are not proportional (doubling age doesn't double shoe size). Direct proportion requires a constant ratio y/x.
• Confusing direct and inverse proportion. Direct: y = kx (doubling x doubles y). Inverse: y = k/x (doubling x halves y). Check: \"more workers → less time\" = inverse; \"more items → more cost\" = direct.
• Adding ratios instead of using the unitary method for part-to-whole problems. If a ratio is 3:2:5 and you know one part, find 1 part first, then multiply by total parts. Adding the known quantity to the other ratios without finding unit value is unreliable.

Quiz

Answer each question, then read the explanation for your choice.

Q1: Simplify the ratio 48 : 60

A) 4 : 5 B) 8 : 10 C) 12 : 15 D) 5 : 4

Q2: If 7 identical books cost £31.50, how much do 5 books cost?

A) £22.50 B) £21.00 C) £25.20 D) £19.50

Q3: A pump fills a tank in 8 hours. How long would it take 4 identical pumps to fill the same tank?

A) 32 hours B) 2 hours C) 4 hours D) 12 hours

Q4: The ratio of flour to sugar in a recipe is 5 : 2. If 300 g of flour is used, how much sugar is needed?

A) 750 g B) 60 g C) 120 g D) 150 g

Q5: Which situation represents inverse proportion?

A) The cost of n apples at £0.30 each B) The distance travelled at constant speed over time C) The number of workers needed to complete a fixed job in fewer days D) The mass of n identical bricks

Q6: A car travels 180 km in 2.5 hours. What is its average speed in km/h?

A) 450 km/h B) 72 km/h C) 90 km/h D) 68 km/h

Q7: On a map with scale 1 : 200,000, a road measures 15 cm. What is the actual distance?

A) 30 km B) 3 km C) 300 km D) 0.3 km

Q8: A model car is built at a scale of 1 : 24. The model is 18 cm long. How long is the actual car?

A) 0.75 m B) 4.32 m C) 432 cm D) 1.33 m

Next Steps

Move on to 00-08 — Basic Number Theory to learn about divisibility rules, modular arithmetic, the Euclidean algorithm for GCD/LCM, and basic primality testing.

Q5: On a map with scale 1 : 50,000, a road measures 8 cm. What is the actual distance in km?

A) 0.4 km B) 4 km C) 40 km D) 400 km

Answer: B) 4 km

8 × 50,000 = 400,000 cm = 4,000 m = 4 km.