Key Takeaways
- Percent means "out of 100"
- Part = Percent × Whole (for finding a percent of a number)
- Ratios compare two quantities; proportions show equal ratios
- Cross-multiplication solves proportions
- Mental math shortcuts help with common percentages
Percentages, Ratios, and Proportions
Understanding percentages, ratios, and proportions is essential for many real-world math problems on the ParaPro.
Understanding Percentages
Percent means "per hundred" or "out of 100."
| Form | Example |
|---|---|
| Percent | 25% |
| Decimal | 0.25 |
| Fraction | ¼ |
Converting Between Percent, Decimal, and Fraction
Percent → Decimal: Divide by 100 (move decimal left 2 places)
- 45% = 0.45
Decimal → Percent: Multiply by 100 (move decimal right 2 places)
- 0.08 = 8%
Percent → Fraction: Write over 100, simplify
- 75% = 75/100 = ¾
Fraction → Percent: Convert to decimal, then percent
- ⅖ = 0.4 = 40%
Finding a Percentage of a Number
Formula: Part = Percent × Whole
Example: What is 20% of 150?
- 0.20 × 150 = 30
Finding What Percent One Number Is of Another
Formula: Percent = Part ÷ Whole × 100
Example: 30 is what percent of 150?
- 30 ÷ 150 = 0.2 × 100 = 20%
Understanding Ratios
A ratio compares two quantities. Written as:
- 3 to 4
- 3:4
- ¾
Example: If there are 12 boys and 8 girls, the ratio of boys to girls is 12:8 or simplified 3:2.
Simplifying Ratios
Divide both numbers by their greatest common factor (GCF):
- 12:8 → GCF is 4 → 3:2
Understanding Proportions
A proportion states that two ratios are equal.
Solving Proportions (Cross-Multiplication)
Example: Solve for x: 3/4 = x/20
Cross-multiply: 3 × 20 = 4 × x
- 60 = 4x
- x = 15
Real-World Applications
| Application | Type | Example |
|---|---|---|
| Sale prices | Percent | 25% off a $40 item |
| Recipes | Ratio | Double a recipe |
| Maps | Proportion | 1 inch = 10 miles |
| Test scores | Percent | 18 out of 20 correct |
Common Percentage Calculations
| Percent | Decimal | Mental Math Trick |
|---|---|---|
| 10% | 0.10 | Move decimal left 1 place |
| 50% | 0.50 | Divide by 2 |
| 25% | 0.25 | Divide by 4 |
| 1% | 0.01 | Move decimal left 2 places |
| 20% | 0.20 | Divide by 5 |
Classroom Application
Help students with percentages, ratios, and proportions by:
- Relating to real situations (shopping, cooking, sports stats)
- Using visual models (pie charts, bar models)
- Teaching cross-multiplication step-by-step
- Practicing mental math shortcuts for common percentages
- Creating ratio tables for visual comparison
What is 15% of 80?
Solve the proportion: 5/8 = x/24