Fraction Calculator & Simplifier
Perform operations on proper, improper, and mixed fractions with automated decimal & percent conversion.
Complete Guide to Fraction Arithmetic: Proper, Improper & Mixed Numbers
Fractions represent parts of a whole where the numerator expresses the number of equal parts taken and the denominator defines the total number of parts. Performing arithmetic operations across differing denominators requires converting to common multiples (LCD) and reducing fractions to their lowest terms using Greatest Common Divisors (GCD).
Addition & Subtraction (LCD Rule)
When denominators differ, find the Least Common Denominator (LCD) or use cross-multiplication:
Formula: (a/b) ± (c/d) = (a·d ± b·c) / (b·d)
Example: 3/4 + 2/5 = (15 + 8) / 20 = 23/20
Simplification: Divide by GCD(23, 20) = 1 → 23/20
Multiplication & Division (KFC Rule)
Multiplication multiplies straight across. Division uses the Keep-Flip-Change (KFC) reciprocal rule:
Multiplication: (a/b) × (c/d) = (a·c) / (b·d)
Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a·d) / (b·c)
Mixed Number: Whole = ⌊num/den⌋, Remainder = num % den
Common Fraction, Decimal & Percentage Equivalents
| Fraction | Decimal Equivalent | Percentage | Classification |
|---|---|---|---|
| 1/2 | 0.500 | 50.0% | Proper Fraction |
| 1/4 | 0.250 | 25.0% | Proper Fraction |
| 3/4 | 0.750 | 75.0% | Proper Fraction |
| 1/3 | 0.333... | 33.33% | Repeating Decimal |
| 5/4 | 1.250 | 125.0% | Improper Fraction (1 1/4) |
Frequently Asked Questions (FAQ)
What is the difference between a proper and an improper fraction?
A proper fraction has a numerator smaller than its denominator (e.g., 3/4, value < 1). An improper fraction has a numerator greater than or equal to its denominator (e.g., 7/4, value ≥ 1) and can be converted into a mixed number (1 3/4).
How do you simplify a fraction to its lowest terms?
Find the Greatest Common Divisor (GCD) of both the numerator and the denominator, then divide both numbers by that GCD. For example, in 12/16, the GCD is 4, simplifying the fraction to 3/4.
Can a fraction have a denominator of zero (0)?
No, division by zero is mathematically undefined. A denominator represents the number of parts a whole is split into, which cannot be zero.