Ratios, Rates and Percent: Grade 6 Study Guide
Ratios compare quantities. They are the foundation for proportions, scale drawings and percentages.
A ratio compares two quantities. If a recipe uses 2 cups of flour for every 3 cups of milk, the ratio of flour to milk is 2 : 3 (or 2 to 3, or 2/3). Equivalent ratios are made by multiplying or dividing both parts by the same number: 2 : 3 = 4 : 6 = 6 : 9.
A rate is a ratio of two different units, such as 150 miles in 3 hours. A unit rate has 1 as its second term: 150 ÷ 3 = 50 miles per hour. A percent is a ratio out of 100: 35% means 35 per 100.
Tables and double number lines
A ratio table lists equivalent ratios. A double number line shows two quantities that grow together.
| Cups of flour | 2 | 4 | 6 | 10 |
|---|---|---|---|---|
| Cups of milk | 3 | 6 | 9 | 15 |
Unit rates
To find a unit rate, divide. 12 apples cost $9, so one apple costs $9 ÷ 12 = $0.75. Unit rates make fair comparison easy. Which is cheaper: 5 pencils for $2.00 or 8 for $3.04? Per pencil: $0.40 versus $0.38, so the second is cheaper.
$9 for 12 apples.
9 ÷ 12.
$0.75 per apple.
×20 apples = $15.
Percent
To find 30% of 80, convert to 30/100 × 80 = 24. Quick method: 10% of 80 is 8, so 30% is 3 × 8 = 24. To find the whole when a part is known: if 15 is 25% of a number, then 100% is 15 × 4 = 60.
| Percent | Fraction | Of 80 |
|---|---|---|
| 10% | 1/10 | 8 |
| 25% | 1/4 | 20 |
| 50% | 1/2 | 40 |
| 75% | 3/4 | 60 |
Worked examples and common mistakes
Example: A car travels 210 miles in 3.5 hours: 210 ÷ 3.5 = 60 miles per hour. Example: A shirt is $24 and 25% off: discount $6, price $18. Common mistakes: adding instead of multiplying to find equivalent ratios (2 : 3 is not 3 : 4), writing a ratio in the wrong order and forgetting units. Always say what each number measures.
Check your understanding: model answers
1. 2 : 3. 2. 48 ÷ 6 = 8 minutes per mile. 3. 100.
✨ Key Takeaways
- Equivalent ratios are made by multiplying or dividing both parts.
- A unit rate has a denominator of 1.
- Percent means per 100.
- Use tables and number lines to solve.
💬 Questions for Discussion & Reflection
- Write the ratio of 8 cats to 12 dogs in simplest form.
- A runner goes 6 miles in 48 minutes. Find the rate in minutes per mile.
- Find 40% of 250.