What Is a Fraction?
A fraction represents a part of a whole. It is written as two numbers separated by a line: the numerator (top number) tells you how many parts you have, and the denominator (bottom number) tells you how many equal parts the whole is divided into.
For example, 3/4 means "three out of four equal parts." Fractions appear everywhere — in cooking (3/4 cup of flour), construction (7/8-inch drill bit), music (3/4 time), and school grades (scored 17/20 on a test). The fraction calculator above handles all four arithmetic operations instantly. The sections below explain the math behind each one.
Types of Fractions
| Type | Definition | Example | Value |
|---|---|---|---|
| Proper fraction | Numerator < denominator | 3/4 | Less than 1 |
| Improper fraction | Numerator ≥ denominator | 9/4 | 1 or greater |
| Mixed number | Whole number + proper fraction | 2 1/4 | 1 or greater |
| Unit fraction | Numerator is always 1 | 1/8 | Less than 1 |
| Equivalent fractions | Different notation, same value | 1/2 = 2/4 = 4/8 | 0.5 |
The Four Fraction Operations: Formulas
Each operation has its own formula. Addition and subtraction require a common denominator first. Multiplication and division do not.
Addition: a/b + c/d = (a×d + b×c) / (b×d)
Subtraction: a/b − c/d = (a×d − b×c) / (b×d)
Multiplication: a/b × c/d = (a×c) / (b×d)
Division: a/b ÷ c/d = (a×d) / (b×c)After every operation, simplify the result by dividing numerator and denominator by their greatest common divisor (GCD). Convert to decimal by dividing the numerator by the denominator.
Variable Definitions
| Variable | What It Means | Example |
|---|---|---|
| a | Numerator of the first fraction (top number) | 3 (in 3/4) |
| b | Denominator of the first fraction (bottom number) | 4 (in 3/4) |
| c | Numerator of the second fraction | 1 (in 1/4) |
| d | Denominator of the second fraction | 4 (in 1/4) |
| GCD | Greatest common divisor — largest number that divides evenly into both numerator and denominator | GCD(6, 9) = 3 |
| LCD | Least common denominator — smallest number both denominators divide into evenly | LCD(4, 6) = 12 |
Step-by-Step Worked Examples
Example 1 — Adding Fractions: Recipe Scaling
A recipe calls for 2/3 cup of flour for the batter and 1/4 cup for the coating. How much flour do you need in total?
Using the addition formula: a/b + c/d = (a×d + b×c) / (b×d)
| Step | Calculation | Result |
|---|---|---|
| Multiply the denominators for the common denominator | 3 × 4 | 12 |
| Convert 2/3 to twelfths | 2 × 4 = 8 → 8/12 | 8/12 |
| Convert 1/4 to twelfths | 1 × 3 = 3 → 3/12 | 3/12 |
| Add numerators | 8 + 3 | 11/12 |
| Check: GCD(11, 12) | 11 is prime; no common factor | Already simplified |
| Decimal equivalent | 11 ÷ 12 | 0.9167 |
You need 11/12 cup of flour (about 0.92 cups). This is slightly less than a full cup — useful to know when measuring.
Example 2 — Subtracting Fractions: Home Improvement
A wooden board measures 7/8 of an inch thick. You need to plane it down by 3/16 of an inch. What is the remaining thickness?
| Step | Calculation | Result |
|---|---|---|
| Apply subtraction formula: (a×d − b×c) / (b×d) | (7×16 − 8×3) / (8×16) | (112 − 24) / 128 |
| Subtract numerators | 112 − 24 | 88/128 |
| Find GCD(88, 128) | 88 = 8×11; 128 = 8×16 → GCD = 8 | 8 |
| Divide both by 8 | 88 ÷ 8 = 11; 128 ÷ 8 = 16 | 11/16 |
| Decimal equivalent | 11 ÷ 16 | 0.6875 inches |
After planing, the board will be 11/16 of an inch thick (0.6875 inches). In construction, this answer in sixteenths is exactly what you need — most measuring tapes mark sixteenths.
Example 3 — Multiplying Fractions: Scaling a Recipe Down
A full recipe uses 3/4 cup of butter, but you want to make only 2/3 of the recipe. How much butter do you need?
| Step | Calculation | Result |
|---|---|---|
| Multiply numerators | 3 × 2 | 6 |
| Multiply denominators | 4 × 3 | 12 |
| Unsimplified result | 6/12 | 6/12 |
| GCD(6, 12) | 6 divides both | 6 |
| Simplified | 6 ÷ 6 = 1; 12 ÷ 6 = 2 | 1/2 |
| Decimal | 1 ÷ 2 | 0.5 cups |
You need exactly 1/2 cup of butter. Notice how multiplication simplified beautifully here: the 3 in the numerator of 3/4 canceled with the 3 in the denominator of 2/3. This shortcut — called "cross-cancellation" — can be applied before multiplying to keep numbers smaller.
Example 4 — Dividing Fractions: Cutting Ribbon
A ribbon is 3/4 of a yard long. You want to cut pieces that are each 3/8 of a yard. How many pieces can you cut?
| Step | Calculation | Result |
|---|---|---|
| Division formula: (a×d) / (b×c) | (3×8) / (4×3) | 24/12 |
| GCD(24, 12) | 12 divides both | 12 |
| Simplified | 24 ÷ 12 = 2; 12 ÷ 12 = 1 | 2/1 = 2 |
| Decimal | 2 ÷ 1 | 2.0 |
You get exactly 2 pieces. Division by a fraction answers "how many groups of that fraction fit into the original amount." Here, two groups of 3/8 fit perfectly into 3/4.
How to Find a Common Denominator
Addition and subtraction both require fractions to share the same denominator before you can combine the numerators. There are two methods:
Method 1: Multiply the Denominators (Always Works)
The common denominator is always b × d. This is what the fraction calculator uses. It is fast but sometimes produces a large number that needs simplifying later.
Example: 1/6 + 1/4 → common denominator = 6 × 4 = 24
Convert: 4/24 + 6/24 = 10/24 → simplified = 5/12
Method 2: Least Common Multiple (Smaller Numbers)
The LCD (least common denominator) is the smallest number both denominators divide into evenly. For 6 and 4: multiples of 6 are 6, 12, 18… Multiples of 4 are 4, 8, 12… The LCM is 12.
Convert: 2/12 + 3/12 = 5/12 — the same answer, with smaller intermediate numbers.
| Denominator Pair | Product Method | LCD Method | Advantage of LCD |
|---|---|---|---|
| 4 and 6 | 24 | 12 | Smaller; simpler calculation |
| 5 and 7 | 35 | 35 | Same (no common factor) |
| 8 and 12 | 96 | 24 | Much smaller intermediate |
| 3 and 9 | 27 | 9 | 9 already contains 3 |
| 7 and 11 | 77 | 77 | Same (both prime) |
When denominators share no common factors (like 5 and 7, or 7 and 11), the product method and LCD method give the same result. When they do share factors, the LCD keeps numbers manageable — helpful for mental math or multi-step problems.
Simplifying Fractions: The GCD Method
A fraction is fully simplified (in "lowest terms") when the numerator and denominator share no common factor other than 1. The algorithm is:
- Find the GCD of the numerator and denominator using the Euclidean algorithm: repeatedly divide the larger number by the smaller and take the remainder until the remainder is 0. The last non-zero remainder is the GCD.
- Divide both the numerator and denominator by the GCD.
Example: Simplify 48/72
- 72 ÷ 48 = 1 remainder 24
- 48 ÷ 24 = 2 remainder 0 → GCD = 24
- 48 ÷ 24 = 2; 72 ÷ 24 = 3 → 2/3
Verify: no number other than 1 divides evenly into both 2 and 3. The fraction is fully simplified.
Converting Between Fractions, Decimals, and Percentages
| Fraction | Decimal | Percentage | Notes |
|---|---|---|---|
| 1/2 | 0.5 | 50% | Terminates |
| 1/3 | 0.3333… | 33.33% | Repeats |
| 1/4 | 0.25 | 25% | Terminates |
| 3/8 | 0.375 | 37.5% | Terminates |
| 2/3 | 0.6667 | 66.67% | Repeats (rounded to 4 dec.) |
| 5/6 | 0.8333… | 83.33% | Repeats |
| 7/8 | 0.875 | 87.5% | Terminates |
| 9/4 | 2.25 | 225% | Improper — value above 1 |
A fraction terminates in decimal form when the denominator (in simplified form) has only the prime factors 2 and/or 5. All other denominators produce repeating decimals. The fraction calculator rounds repeating decimals to 4 decimal places.
Mixed Numbers: Converting Before Calculating
The fraction calculator works with simple numerator/denominator pairs. To use it with mixed numbers, first convert to an improper fraction:
Improper Fraction = (Whole Number × Denominator + Numerator) / DenominatorExamples:
- 2 1/3 → (2 × 3 + 1) / 3 = 7/3
- 4 3/8 → (4 × 8 + 3) / 8 = 35/8
- 1 1/2 → (1 × 2 + 1) / 2 = 3/2
After calculating, convert the improper fraction result back to a mixed number by dividing numerator by denominator — the quotient is the whole number and the remainder becomes the new numerator.
Example: 17/5 → 17 ÷ 5 = 3 remainder 2 → 3 2/5
Common Fraction Mistakes (and How to Avoid Them)
- Adding denominators: Never add the bottom numbers. 1/2 + 1/3 ≠ 2/5. You must find a common denominator first. The correct answer is 5/6.
- Forgetting to simplify: An answer of 6/8 is technically correct but 3/4 is the expected form. Always divide by the GCD at the end.
- Dividing instead of multiplying by the reciprocal: To divide fractions, flip the second fraction and multiply — do not divide both numerators and denominators separately.
- Cross-multiplying incorrectly: Cross-multiplication is used to compare or solve proportion equations, not to add fractions. Keep the operations separate.
- Mishandling negative fractions: A negative sign can appear on the numerator or denominator — not both. Treat −a/b as (−a)/b when adding or subtracting.
How to Use the Fraction Calculator
The free fraction calculator performs all four operations and returns the simplified result, decimal, and percentage in one step:
- Enter the first fraction: Type the numerator (top number) in the "First Numerator" field and the denominator (bottom number) in "First Denominator." For 3/4, enter 3 and 4.
- Choose the operation: Select Add, Subtract, Multiply, or Divide from the dropdown.
- Enter the second fraction: Type the second numerator and denominator.
- Read the results: The calculator instantly shows the unsimplified result, simplified result (in numerator + denominator form), decimal equivalent, and percentage. No "Calculate" button needed — results update on every keystroke.
For mixed numbers, convert to improper fractions first using the formula above. For negative fractions, enter a negative numerator (e.g., −3 in the numerator field for −3/4). The calculator handles negative results automatically and ensures the denominator is always positive in the simplified output.