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
ACER Authentic Problem: Student-to-Teacher Ratio (Sept 2023 Q10)
❓ Question: A Government primary school employed 40 teachers in 1980. The average number of students per teacher for the school was equal to the relevant average value shown in the table (20.2). What was the total number of students enrolled at the school in 1980?
📄 Source Passage / Reference Data:
STUDENTS PER TEACHER
In 1980, the average student-to-teacher ratio in Government primary schools was 20.2 students per teacher.
📝 Step-by-Step Method:
1
Identify the given ratio terms: 20.2 students per 1 teacher.
2
Multiply the ratio rate by the number of teachers (40).
3
Compute the product.
🔍 Evidence Extracted: "40 teachers × 20.2 students/teacher = 808 students"
🎯 Final Answer: 808 students
💡 Why It Is Correct: With an average ratio of 20.2 students to each teacher, 40 teachers account for 40 × 20.2 = 808 students enrolled.
⚡ LANTITE Exam Strategy: Official ACER Tip: Rate multiplication is direct: Enrolment = Teachers × (Students per Teacher).
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
ACER Authentic Problem: Calculating Staffing from Cohort Size (Sept 2023 Q11)
❓ Question: A Catholic secondary school had a total enrolment of 1098 students in 2021. The average number of students per teacher for the school was equal to the table value (12.2). What was the total number of teachers at the school in 2021?
📄 Source Passage / Reference Data:
STUDENTS PER TEACHER
In 2021, Catholic secondary schools operated at an average ratio of 12.2 students per teacher.
📝 Step-by-Step Method:
1
State the inverse relationship: Number of Teachers = Total Students ÷ Students per Teacher.
2
Clear the decimal by multiplying both numerator and denominator by 10.
3
Perform the division.
🔍 Evidence Extracted: "1098 students ÷ 12.2 = 90 teachers"
🎯 Final Answer: 90 teachers
💡 Why It Is Correct: Dividing total students (1098) by the ratio (12.2) gives exactly 90 teachers.
⚡ LANTITE Exam Strategy: Official ACER Solution: When dividing by a decimal, scale both terms by 10 to make integer mental division much simpler.
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
ACER Authentic Problem: Unit Energy Rate Conversion (July 2017 Q5)
❓ Question: Based on the information that an average adult diet is 8400 kJ (or 2000 kcal), how many kilojoules (kJ) are equal to 1 kilocalorie (kcal)?
📄 Source Passage / Reference Data:
NUTRITION INFORMATION
* Percentage daily intakes are based on an average adult diet of 8400 kJ (or 2000 kcal).
📝 Step-by-Step Method:
1
Set up the rate equation between kJ and kcal.
2
Divide both sides by 2000 to find the rate per 1 kcal.
3
Simplify the division.
🔍 Evidence Extracted: "8400 ÷ 2000 = 4.2 kJ per kcal"
🎯 Final Answer: 4.2 kJ
💡 Why It Is Correct: Dividing 8400 kJ by 2000 kcal yields an exact conversion factor of 4.2 kilojoules per kilocalorie.
⚡ LANTITE Exam Strategy: LANTITE Unit Rate Strategy: "Per 1 unit" always implies dividing total quantity by the reference count.
4. Map Scales & Scale Drawings
Example 4
ACER Authentic Problem: Comparing Upfront vs Periodic Rates (Sample Q2)
❓ Question: For a 12-month ‘Gym and Swim’ membership, how much more does it cost to pay by monthly debit ($66/month) rather than upfront ($773)?
📄 Source Passage / Reference Data:
GYM COSTS
| Membership | Gym and Swim ($) |
|---|---|
| 12 Months (upfront) | 773 |
| 12 Months (monthly debit) | 66 |
📝 Step-by-Step Method:
1
Calculate annual cost of paying monthly across 12 months.
2
Compute the product (10 × 66 = 660, 2 × 66 = 132 → 660 + 132).
3
Subtract the upfront lump sum cost.
🔍 Evidence Extracted: "$792 monthly total - $773 upfront = $19 difference"
🎯 Final Answer: $19
💡 Why It Is Correct: Paying monthly costs 12 × $66 = $792 per year. Upfront payment costs $773. The monthly option costs $19 more.
⚡ LANTITE Exam Strategy: Financial Maths: Multiply the periodic installment rate by the number of periods before finding the variance against upfront cost.
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.