Part A: Numeracy
3
Ratios & Proportions
Master ratio sharing, unit rates, map scale conversions, and proportional reasoning
Learning Objectives
- Simplify ratios to lowest terms and maintain term order
- Divide classroom resource budgets and student cohorts using multi-part ratios
- Calculate unit rates, travel speeds, and material consumption rates
- Convert map distances and architectural plans using representative fraction scales
- Solve direct and inverse proportional reasoning problems in school scenarios
Chapter Progress: 0%
1. Ratio Basics & Simplifying
What is a Ratio?
A ratio compares quantities of the same kind. It expresses how much of one item exists relative to another item. Ratios are written with a colon (a : b) or as a fraction (a / b). Ratios have no units because both quantities share the same unit of measurement.
Example 1
Simplifying Student Ratios in Classroom Cohorts
❓ Question: In a primary school art class, there are 15 boys and 25 girls. Express the ratio of boys to girls in simplest form.
📝 Step-by-Step Method:
1
Write down the initial ratio in requested sequence.
2
Find the highest common factor (HCF) of 15 and 25.
3
Divide both terms by their HCF (5).
🎯 Final Answer: 3 : 5 (For every 3 boys, there are 5 girls)
💡 Why It Is Correct: Dividing both terms by 5 simplifies the ratio 15:25 to 3:5 while keeping relative proportions identical.
2. Sharing Quantities in Multi-Part Ratios
The Unitary Method for Ratios
Value of 1 Part = Total Quantity ÷ Total Ratio Parts
Total Parts = Sum of all ratio terms (a + b + c)
Individual Share = Ratio Term × Value of 1 Part
Example
To share $150 in ratio 2:3: Total parts = 5. Value of 1 part = $150 ÷ 5 = $30. Shares = $60 and $90.
Example 2
Sharing Classroom Resource Budget by Ratio
❓ Question: A budget of $150 for art supplies is shared between Year 5 and Year 6 in the ratio 2 : 3. How much money does each year level receive?
📝 Step-by-Step Method:
1
Calculate total parts by adding the terms in the ratio.
2
Determine the monetary value of 1 part.
3
Multiply each ratio term by the value of one part.
🎯 Final Answer: Year 5 receives $60 and Year 6 receives $90
💡 Why It Is Correct: With 5 total parts, each part is worth $30. 2 parts equal $60 and 3 parts equal $90. Check: $60 + $90 = $150.
3. Rates, Speeds & Unit Rates
Ratios vs Rates Comparison
| Feature | Ratio | Rate | Teaching Example |
|---|---|---|---|
| Units | Same units (cancel out) | Different units (retained) | 1 : 25 teacher to students vs $3.50/L fuel |
| Notation | Written with colon (a:b) | Written with slash (km/h) | 3:5 ratio vs 80 km/h average speed |
| Purpose | Compares parts of a whole | Compares two distinct quantities | Budget splits vs hourly teacher pay rates |
Example 3
Calculating Average Excursion Bus Speed
❓ Question: A school excursion bus travels 240 kilometres to a science museum in 3 hours. What was its average speed in km/h?
📝 Step-by-Step Method:
1
State speed formula: Speed = Distance ÷ Time.
2
Substitute distance (240 km) and time (3 hours).
3
Perform the division.
🎯 Final Answer: 80 km/h
💡 Why It Is Correct: Dividing total distance (240 km) by total travel time (3 h) yields an average travel rate of 80 km/h.
4. Map Scales & Scale Drawings
Example 4
Calculating Real-World Distance from Regional Map Scale
❓ Question: On a regional school district map with a scale of 1 : 50,000, two partner schools are 4 cm apart. What is the actual distance between the schools in kilometres?
📝 Step-by-Step Method:
1
Multiply map distance by scale factor to find real distance in cm.
2
Convert centimetres to metres (divide by 100).
3
Convert metres to kilometres (divide by 1,000).
🎯 Final Answer: 2 kilometres
💡 Why It Is Correct: 4 cm × 50,000 = 200,000 cm = 2,000 m = 2 km.
Chapter 3 Practice Quiz
Chapter Summary: Chapter 3: Ratios & Proportions Summary
Key Concepts
Important Formulas
Top Tips
Chapter 3 Flashcards
1 / 4Ratios
How do you find the value of 1 part in a ratio problem?
Divide the total amount by the sum of all ratio parts.