Fraction Calculator
Add, subtract, multiply or divide two fractions and get the result in lowest terms.
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Fraction 1
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About Fraction Calculator
This fraction calculator adds, subtracts, multiplies or divides two fractions and simplifies the result to lowest terms.
It is useful for homework, measurement math, recipe adjustments and other situations where working with fractions is clearer than converting everything to decimals.
Fraction Arithmetic Formulas
// Addition: a/b + c/d = (ad + bc) / bd // Subtraction: a/b - c/d = (ad - bc) / bd // Multiplication: a/b × c/d = ac / bd // Division: a/b ÷ c/d = a/b × d/c = ad / bc // Examples: // 1/3 + 1/4 = (1×4 + 1×3) / (3×4) = 7/12 // 2/5 - 1/3 = (2×3 - 1×5) / (5×3) = 1/15 // 3/4 × 2/5 = (3×2) / (4×5) = 6/20 = 3/10 // 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
GCD and LCM Relationship
| Concept | Formula | Example |
|---|---|---|
| GCD (Euclidean) | gcd(a,b) = gcd(b, a mod b) | gcd(48,18) = 6 |
| LCM from GCD | lcm(a,b) = |a×b| / gcd(a,b) | lcm(12,18) = 36 |
| Product Rule | a×b = gcd(a,b) × lcm(a,b) | 12×18 = 6×36 = 216 |
Fraction Simplification Examples
| Original | GCD | Simplified |
|---|---|---|
| 8/12 | 4 | 2/3 |
| 15/25 | 5 | 3/5 |
| 24/36 | 12 | 2/3 |
| 42/56 | 14 | 3/4 |
| 100/150 | 50 | 2/3 |
Frequently Asked Questions
- How do you add or subtract fractions with different denominators?
- To add/subtract fractions with different denominators, find a common denominator (usually the product or LCM), convert each fraction, then add/subtract numerators. Formula: a/b + c/d = (ad + bc)/bd. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. Result is simplified using GCD.
- How do you multiply and divide fractions?
- Multiply fractions: multiply numerators and denominators directly. a/b × c/d = ac/bd. Divide fractions: multiply by reciprocal. a/b ÷ c/d = a/b × d/c = ad/bc. Example: 2/3 × 3/4 = 6/12 = 1/2. Example: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9.
- What is the GCD method for simplifying fractions?
- After computing a result, divide both numerator and denominator by their Greatest Common Divisor. GCD is found using Euclidean algorithm: gcd(a,b) = gcd(b, a mod b). Example: 12/18 simplifies to 2/3 because gcd(12,18) = 6, so 12÷6=2 and 18÷6=3.
- What is the relationship between GCD and LCM?
- For any two positive integers a and b: a × b = gcd(a,b) × lcm(a,b). This means lcm(a,b) = |a×b| / gcd(a,b). Example: For 12 and 18, gcd=6, so lcm = (12×18)/6 = 36. This relationship is used to compute LCM efficiently.
- How do you handle negative fractions?
- A negative fraction can have the negative sign in numerator, denominator, or in front. Convention places it in front or in numerator. If denominator is negative, multiply both numerator and denominator by -1. Example: 3/(-4) = -3/4, (-3)/(-4) = 3/4.
- What are improper fractions and mixed numbers?
- Improper fractions have numerator ≥ denominator (e.g., 7/3). Mixed numbers combine whole number and fraction (e.g., 2 1/3). Conversion: 7/3 = 2 remainder 1, so 7/3 = 2 1/3. Reverse: 2 1/3 = (2×3+1)/3 = 7/3. This tool outputs improper fractions in simplified form.