Core Concepts Explained
- Improper Fraction: A fraction where the numerator is equal to or larger than the denominator (e.g., 7⁄4).
- Mixed Number: A whole number and a fraction combined (e.g., 1 3⁄4).
- Like Denominators: Fractions that have the same bottom number (denominator).
- Unlike Denominators: Fractions with different denominators.
- Reciprocal: Flip the numerator and denominator (e.g., reciprocal of 2⁄3 is 3⁄2).
- Estimating: Rounding fractions to 0, 1/2, or 1 to estimate sums or differences.
- Ratio: Comparison of two numbers using division, written as 3:2 or 3/2.
- Rate: A type of ratio comparing quantities with different units (e.g., miles/hour).
- Proportion: An equation showing two ratios are equivalent.
- Cross Multiplying: In a proportion a/b = c/d, cross multiply: ad = bc.
- Ratio Table: A table of equivalent ratios, useful for scaling up/down.
Key Fraction Operations
- To change an improper fraction to a mixed number: Divide numerator by denominator, the quotient is the whole number, remainder is the new numerator.
- To change a mixed number to an improper fraction: Multiply the whole number by denominator, add numerator, put over original denominator.
- Adding/Subtracting like denominators: Add/subtract numerators, keep denominator.
- Adding/Subtracting unlike denominators: Find LCD, convert both, then add/subtract.
- Multiplying fractions: Multiply numerators, multiply denominators, reduce if possible.
- Dividing fractions: Multiply by reciprocal of divisor.
Worked Examples
Changing Improper Fractions to Mixed Numbers
Example: 11⁄4 = 2 3⁄4
11 ÷ 4 = 2 remainder 3, so answer is 2 3⁄4
Example: 11⁄4 = 2 3⁄4
11 ÷ 4 = 2 remainder 3, so answer is 2 3⁄4
Changing Mixed Numbers to Improper Fractions
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 (simplified)
Example: 5⁄9 - 2⁄9 = 3⁄9 = 1⁄3 (simplified)
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
Multiplying Fractions
Example: 3⁄5 × 2⁄7 = 6⁄35
Example: 3⁄5 × 2⁄7 = 6⁄35
Dividing Fractions
Example: 4⁄5 ÷ 2⁄3 = 4⁄5 × 3⁄2 = 12⁄10 = 6⁄5
Example: 4⁄5 ÷ 2⁄3 = 4⁄5 × 3⁄2 = 12⁄10 = 6⁄5
Ratio Table
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
Proportion with Cross-Multiplying
Example: 3⁄4 = x⁄8
3×8 = 4×x ⇒ x = 6
Example: 3⁄4 = x⁄8
3×8 = 4×x ⇒ x = 6
Pre-Test: Check Your Understanding!
Questions & Answers
What is the difference between a fraction and a ratio?
A fraction compares a part to a whole, while a ratio compares two quantities (parts to parts or part to whole).
How do I find a common denominator?
Find the least common multiple (LCM) of the denominators and rewrite each fraction with that denominator.
When dividing by a fraction, why do we multiply by its reciprocal?
Dividing by a number is the same as multiplying by its reciprocal, so this converts division into multiplication.
What does it mean to reduce a fraction?
To write the fraction in lowest terms by dividing numerator and denominator by their greatest common factor (GCF).
How do I estimate a sum or difference of mixed numbers?
Round each mixed number to the nearest whole number or half, then add or subtract.