Ratio, Percentages and Rates: Year 7 Study Guide
Ratios, percentages and rates compare quantities and are used in shopping, cooking and sport.
A ratio compares quantities of the same kind. Concrete mixed with sand and cement in the ratio 3 : 2 : 1 uses 3 parts sand, 2 parts cement and 1 part gravel, or whatever order is stated. Ratios are simplified by dividing by the HCF: 24 : 36 = 2 : 3. To share an amount, add the parts and find the value of one part.
A percentage is a fraction out of 100. Percentage change = change ÷ original × 100. A rate compares different units, such as km per hour or dollars per kilogram.
Sharing in a ratio
Share $450 between Lee and Kim in the ratio 4 : 5. Total parts = 9; one part = $50; Lee gets $200 and Kim gets $250. Check: 200 + 250 = 450. To scale a recipe up, multiply every part by the same amount.
| Step | Working | Result |
|---|---|---|
| Add the parts | 4 + 5 | 9 |
| One part | 450 ÷ 9 | $50 |
| Lee | 4 × 50 | $200 |
| Kim | 5 × 50 | $250 |
Percentage change and discounts
A bike drops from $480 to $408. The change is $72. Percentage change = 72 ÷ 480 × 100 = 15% decrease. A retailer adds 10% GST to a $70 item: $77. Be careful: a 20% increase followed by a 20% decrease does not return to the start (100 → 120 → 96).
New minus original.
Not the new value.
To get a percentage.
Interpret.
Rates
Speed = distance ÷ time. A cyclist travels 45 km in 1.5 hours: 30 km/h. To convert 72 km/h to m/s, multiply by 1000 and divide by 3600: 20 m/s. Best-buy problems compare unit prices: 500 g for $3.20 is 64c per 100 g; 750 g for $4.50 is 60c per 100 g, so the larger pack is cheaper.
Worked examples and common mistakes
Example: The ratio of boys to girls is 3 : 4 and there are 28 students: one part is 4, so 12 boys and 16 girls. Common mistakes: reversing the order of a ratio, working out percentage change from the new value and mixing units (minutes and hours). Always write units.
Check your understanding: model answers
1. 3 : 5. 2. 12 ÷ 50 = 24% increase. 3. 60 km/h.
✨ Key Takeaways
- Share in a ratio by finding one part.
- Percentage change uses the original amount.
- Rate = amount ÷ time.
- Compare unit prices.
💬 Questions for Discussion & Reflection
- Simplify 18 : 30.
- A price rises from $50 to $62. Find the percentage increase.
- Find the average speed: 150 km in 2.5 hours.