About Fractions
A fraction represents a part of a whole or a ratio between two numbers. It consists of two parts: the numerator (top number) which represents how many parts you have, and the denominator (bottom number) which represents how many equal parts the whole is divided into.
Types of Fractions
Proper Fractions
A proper fraction has a numerator smaller than its denominator (e.g., 3/4, 2/5). The value of a proper fraction is always less than 1.
Improper Fractions
An improper fraction has a numerator equal to or greater than its denominator (e.g., 5/3, 7/4). These can be converted to mixed numbers.
Mixed Numbers
A mixed number combines a whole number with a proper fraction (e.g., 2 1/3). It's another way to express an improper fraction. For example, 7/3 = 2 1/3.
Fraction Operations
Addition and Subtraction
To add or subtract fractions, they must have the same denominator (common denominator). If the denominators are different, find the least common multiple (LCM) of the denominators, convert both fractions, then add or subtract the numerators.
Formula: a/b ± c/d = (ad ± bc) / bd
Example: 1/2 + 1/3 = (1×3 + 1×2) / (2×3) = 5/6
Multiplication
Multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. You can simplify before or after multiplying.
Formula: a/b × c/d = (a×c) / (b×d)
Example: 2/3 × 3/4 = 6/12 = 1/2
Division
To divide fractions, multiply the first fraction by the reciprocal (flip) of the second fraction. This is often remembered as "multiply by the reciprocal" or "flip and multiply."
Formula: a/b ÷ c/d = a/b × d/c = (a×d) / (b×c)
Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Simplification
Simplifying (or reducing) a fraction means finding an equivalent fraction where the numerator and denominator are as small as possible. This is done by dividing both by their greatest common divisor (GCD).
Example: 12/16 = (12÷4) / (16÷4) = 3/4
Converting Between Forms
Mixed Number to Improper Fraction
Multiply the whole number by the denominator, add the numerator, and place the result over the denominator.
Example: 2 3/4 = (2×4 + 3) / 4 = 11/4
Improper Fraction to Mixed Number
Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator.
Example: 11/4 = 2 remainder 3 = 2 3/4
Real-World Applications
- Cooking: Recipe measurements (1/2 cup, 2/3 teaspoon)
- Construction: Measurements in inches (5/8", 3/4")
- Finance: Stock prices, interest rates, partial shares
- Time: Parts of an hour (1/4 hour = 15 minutes)
- Music: Note durations (quarter notes, eighth notes)
- Sewing: Fabric measurements and pattern adjustments
Tips for Working with Fractions
- Always simplify your final answer to lowest terms
- Check if you can simplify before performing operations to make calculations easier
- When adding or subtracting, finding the LCD (least common denominator) makes the process cleaner
- Remember that a whole number can be written as a fraction with denominator 1 (e.g., 5 = 5/1)
- A denominator of zero is undefined - fractions cannot have zero denominators
- Negative fractions can have the negative sign on the numerator, denominator, or in front
Common Fraction Equivalents
- 1/2 = 0.5 = 50%
- 1/3 ≈ 0.333... = 33.33%
- 1/4 = 0.25 = 25%
- 1/5 = 0.2 = 20%
- 1/8 = 0.125 = 12.5%
- 3/4 = 0.75 = 75%
- 2/3 ≈ 0.666... = 66.67%