Key Concepts
- Improper Fractions & Mixed Numbers: An improper fraction has a numerator equal to or larger than the denominator. A mixed number combines a whole number and a fraction.
- Adding & Subtracting Fractions: Add/subtract numerators if denominators are the same. Find a common denominator if not.
- Mixed Numbers: Add or subtract whole parts and fractional parts, using common denominators if needed.
- Estimating: Round fractions or mixed numbers to estimate sums/differences.
- Multiplying Fractions: Multiply numerators, multiply denominators, reduce if needed. Multiply mixed numbers by converting to improper fractions first.
- Reciprocals: Flip numerator and denominator. Used in dividing fractions.
- Dividing Fractions: Multiply by the reciprocal of the divisor.
- Ratios: Compare two numbers, written as a:b, a/b, or "a to b".
- Proportions: Two equal ratios. Use cross-multiplying to solve.
- Ratio Tables & Rates: Show equivalent ratios or compare two quantities with different units (rate).
- Problem Solving: Use the above methods to solve real-world problems involving fractions, ratios, and rates.
Concepts to Focus On
- Finding common denominators
- Reducing fractions
- Recognizing and working with ratios and proportions
- Converting between improper fractions and mixed numbers
- Applying operations to mixed numbers and fractions
Worked Examples
Improper Fraction to Mixed Number
Example: 11⁄4 = 2 3⁄4
11 ÷ 4 = 2 remainder 3; so, 2 3⁄4
Example: 11⁄4 = 2 3⁄4
11 ÷ 4 = 2 remainder 3; so, 2 3⁄4
Mixed Number to Improper Fraction
Example: 2 2⁄5 = (2×5+2)/5 = 12⁄5
Example: 2 2⁄5 = (2×5+2)/5 = 12⁄5
Adding Fractions with Like Denominators
Example: 3⁄8 + 4⁄8 = 7⁄8
Example: 3⁄8 + 4⁄8 = 7⁄8
Subtracting Fractions with Like Denominators
Example: 5⁄9 - 2⁄9 = 3⁄9 = 1⁄3
Example: 5⁄9 - 2⁄9 = 3⁄9 = 1⁄3
Adding Fractions with Unlike Denominators
Example: 1⁄4 + 1⁄6 = 3⁄12 + 2⁄12 = 5⁄12
Example: 1⁄4 + 1⁄6 = 3⁄12 + 2⁄12 = 5⁄12
Adding Mixed Numbers with Unlike Denominators
Example: 2 1⁄3 + 1 1⁄6 = 3 1⁄2
Example: 2 1⁄3 + 1 1⁄6 = 3 1⁄2
Estimating Sums of Mixed Numbers
Example: 2 3⁄4 + 3 1⁄4 ≈ 3 + 3 = 6
Example: 2 3⁄4 + 3 1⁄4 ≈ 3 + 3 = 6
Multiplying Fractions and Whole Numbers
Example: 3 × 2⁄5 = 6⁄5 = 1 1⁄5
Example: 3 × 2⁄5 = 6⁄5 = 1 1⁄5
Multiplying Fractions: Reciprocals
Example: Reciprocal of 3⁄4 is 4⁄3
Example: Reciprocal of 3⁄4 is 4⁄3
Dividing Fractions by Whole Numbers
Example: 3⁄5 ÷ 2 = 3⁄5 × 1⁄2 = 3⁄10
Example: 3⁄5 ÷ 2 = 3⁄5 × 1⁄2 = 3⁄10
Ratios and Proportions
Example: If 2 apples cost $6, how much for 5 apples?
2:6 = 5:x → x = (5×6)/2 = $15
Example: If 2 apples cost $6, how much for 5 apples?
2:6 = 5:x → x = (5×6)/2 = $15
Ratio Table
Example:
Example:
| Apples | Cost |
|---|---|
| 2 | 6 |
| 4 | 12 |
| 6 | 18 |