📋 Table of Contents
1. What Is a Fraction?
A fraction represents a part of a whole or a ratio between two quantities. It consists of two parts: the numerator (top number, indicating how many parts we have) and the denominator (bottom number, indicating the total number of equal parts the whole is divided into).
For example, ¾ means: the whole is divided into 4 equal parts, and we have 3 of them. The fraction bar (vinculum) represents division: ¾ = 3 ÷ 4 = 0.75. Fractions are fundamentally about sharing and measuring — two skills humans needed long before algebra or calculus were invented.
Memory trick: Denominator = Down (bottom). Numerator = North (top). Or: the denominator tells you the type of piece (fourths, eighths), the numerator tells you how many.
2. Types of Fractions
A proper fraction has a numerator smaller than the denominator (value < 1): ½, ¾, 5/8. An improper fraction has a numerator equal to or larger than the denominator (value ≥ 1): 7/4, 5/3, 8/8. A mixed number combines a whole number and a proper fraction: 1¾, 2⅓, 5⅝. Improper fractions and mixed numbers represent the same value — you can convert between them.
To convert an improper fraction to a mixed number: divide the numerator by the denominator. The quotient is the whole number; the remainder over the denominator is the fractional part. Example: 11/4 → 11 ÷ 4 = 2 remainder 3 → 2¾.
To convert a mixed number to an improper fraction: multiply the whole number by the denominator, then add the numerator, keeping the same denominator. Example: 3⅖ → (3 × 5) + 2 = 17 → 17/5.
3. Equivalent Fractions & Simplification
Equivalent fractions have the same value but different numerators and denominators: ½ = 2/4 = 3/6 = 4/8. To create equivalent fractions, multiply (or divide) both numerator and denominator by the same nonzero number. This is the fundamental rule of fractions — it does not change the value, only the form.
Simplifying (reducing) a fraction means dividing both numerator and denominator by their greatest common factor (GCF). A fraction in lowest terms (simplest form) has a GCF of 1. Example: 12/18 → GCF(12,18) = 6 → divide both by 6 → 2/3.
The Least Common Denominator (LCD) — needed for adding and subtracting unlike fractions — is the Least Common Multiple (LCM) of the denominators. Finding the LCD: for 1/4 and 1/6, LCM(4,6) = 12, so LCD = 12.
4. Adding and Subtracting Fractions
To add or subtract fractions with the same denominator: add/subtract the numerators, keep the denominator, then simplify. Example: 3/8 + 2/8 = 5/8; 7/10 − 3/10 = 4/10 = 2/5.
To add or subtract fractions with different denominators: (1) Find the LCD. (2) Convert each fraction: multiply numerator and denominator by the factor needed. (3) Add/subtract numerators. (4) Simplify.
Example: 1/3 + 1/4 → LCD = 12 → 4/12 + 3/12 = 7/12. ✓
To add/subtract mixed numbers: handle the whole numbers and fractions separately, then combine. If the fraction part of a subtraction would go negative, "borrow" 1 from the whole number column (converting it to an improper fraction component).
5. Multiplying and Dividing Fractions
To multiply fractions: multiply numerators together and denominators together, then simplify. No common denominator needed! Example: 2/3 × 3/4 = 6/12 = 1/2. Tip: cross-cancel first — cancel any numerator factor with any denominator factor to keep numbers small.
To divide fractions: multiply by the reciprocal of the divisor (the second fraction). The reciprocal flips the fraction: the reciprocal of 4/5 is 5/4. Memory trick: KCF — Keep, Change, Flip. Keep the first fraction; Change ÷ to ×; Flip the second fraction.
Example: 2/3 ÷ 4/5 → 2/3 × 5/4 → 10/12 → 5/6. ✓
To multiply or divide mixed numbers: always convert to improper fractions first, then apply the operation, then convert back if desired.
Why does "multiply by the reciprocal" work? Division is the inverse of multiplication. Dividing by 4/5 is the same as asking "what do I multiply by to get 1?" The answer is 5/4 (since 4/5 × 5/4 = 1). So dividing by 4/5 = multiplying by 5/4.
6. Fractions, Decimals, and Percents
Fraction to decimal: divide numerator by denominator. 3/4 = 3 ÷ 4 = 0.75. Some fractions terminate (3/4 = 0.75); others repeat (1/3 = 0.333…). Fractions whose denominators have only factors of 2 and 5 always terminate.
Decimal to fraction: use the place value as the denominator. 0.75 → 75/100 → 3/4. For repeating decimals, use algebra: let x = 0.333…, then 10x = 3.333…, subtract: 9x = 3, so x = 3/9 = 1/3.
Percent means "per hundred." Fraction ↔ percent: multiply by 100. 3/4 = 0.75 = 75%. Percent ↔ fraction: divide by 100. 35% = 35/100 = 7/20. The three forms — fraction, decimal, percent — are interchangeable and each has contexts where it is most useful.
7. Fractions in the Real World
Fractions are everywhere. In cooking, recipes use ½ cup, ¾ teaspoon. In construction, lumber is sold in 1/4-inch increments. In music, time signatures (4/4, 3/4, 6/8) are fraction-like ratios describing beats per measure. In finance, interest rates and stock prices use fractions and decimals. In medicine, drug dosages are calculated from fractions of body weight.
Understanding fractions underlies proportional reasoning — the ability to scale quantities up and down. Doubling a recipe, calculating a 30% discount, figuring out how many tiles fit in a room, or comparing unit prices at the grocery store all require fluent fraction thinking. Proportion is also the gateway to algebra, trigonometry, and calculus.
8. Key Terms Glossary
Numerator
The top number in a fraction, indicating how many parts we have.
Denominator
The bottom number in a fraction, indicating the total number of equal parts in the whole.
Proper fraction
Numerator < denominator; value < 1. Examples: ½, ¾, 5/8.
Improper fraction
Numerator ≥ denominator; value ≥ 1. Examples: 7/4, 5/3.
Mixed number
A whole number plus a proper fraction: 2¾, 1⅓.
Equivalent fraction
Fractions with the same value in different forms: ½ = 2/4 = 4/8.
Simplest form
A fraction reduced so GCF(numerator, denominator) = 1. E.g., 6/8 → 3/4.
Reciprocal
The multiplicative inverse: flip numerator and denominator. Reciprocal of 3/4 is 4/3. Used in fraction division.
LCD
Least Common Denominator: the smallest denominator both fractions can be converted to for adding/subtracting.
Percent
"Per hundred." Fraction × 100 = percent. 3/4 = 75%. Used in discounts, taxes, and statistics.
9. Frequently Asked Questions
How do you add fractions with different denominators?
Find the LCD, convert each fraction, then add numerators. Example: 1/3 + 1/4 → LCD=12 → 4/12 + 3/12 = 7/12.
How do you multiply fractions?
Multiply numerators together and denominators together, then simplify. Cross-cancel first to keep numbers small. Example: 2/3 × 3/4 = 6/12 = 1/2.
How do you divide fractions?
Keep-Change-Flip (KCF): keep first fraction, change ÷ to ×, flip second fraction. Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
How do you convert a fraction to a decimal?
Divide numerator by denominator: 3/4 = 3 ÷ 4 = 0.75. Denominators with only factors of 2 and 5 terminate; others repeat (1/3 = 0.333...).
What is the difference between proper, improper, and mixed number fractions?
Proper (< 1): numerator < denominator (3/4). Improper (≥ 1): numerator ≥ denominator (7/4). Mixed number: whole number + proper fraction (1¾). Improper and mixed numbers represent the same value — convert between them as needed.