Fraction Calculator
The comprehensive online calculator for fraction arithmetic. Solve addition, subtraction, multiplication, division, mixed numbers, and simplest-form reductions with step-by-step common denominator derivations.
Interactive Fraction Calculator
Perform exact fraction arithmetic with step-by-step common denominators and simplest-form reduction.
Step-by-Step Solution Breakdown
How to Calculate and Simplify Fractions
A fraction represents part of a whole expressed as a/b (numerator divided by denominator). To add or subtract fractions, convert them to a common denominator (LCD) and combine numerators. To multiply, multiply numerators and denominators straight across. To divide, multiply the first fraction by the reciprocal of the second (Keep, Change, Flip).
Anatomy & Classifications of Fractions
A fraction represents a numerical division of a whole into equal parts. Written in the standard algebraic format a / b, every fraction consists of two core components:
- Numerator (a): The top number, representing how many equal parts are selected or counted.
- Denominator (b): The bottom number, representing the total number of equal parts that constitute one whole unit. The denominator can never be zero (b ≠ 0).
The value is strictly less than 1 (e.g., 3/4, 5/8, 7/12).
The value is greater than or equal to 1 (e.g., 9/4, 15/8, 7/7).
Combines an integer and a fraction (e.g., 2 1/4 = 9/4).
The Four Arithmetic Operations on Fractions
Unlike ordinary integer operations, fraction arithmetic relies on proportional scaling and modular balancing across numerators and denominators:
Addition & Subtraction
Fractions can only be added or subtracted when they share an identical common denominator. If denominators differ, scale both fractions to their Least Common Denominator (LCD):
Multiplication
Fraction multiplication does not require common denominators. Multiply numerators directly and denominators directly:
Division (Keep, Change, Flip)
Dividing by a fraction is equivalent to multiplying by its reciprocal. Keep the dividend, change division to multiplication, and flip the divisor:
Finding the Least Common Denominator (LCD) & Simplification
The Least Common Denominator (LCD) is the smallest positive integer that is divisible by all denominators in a calculation:
The 3-Step LCD Algorithm
- Find Prime Factorizations: Break down each denominator into its prime factor components (e.g., 12 = 2² × 3, and 18 = 2 × 3²).
- Take Highest Powers: Multiply each unique prime factor raised to its highest occurring power: LCM = 2² × 3² = 4 × 9 = 36.
- Scale Both Fractions: Multiply the numerator and denominator of each fraction by the factor needed to reach the LCD.
Simplifying via the Greatest Common Divisor (GCD)
A fraction is in simplest form when GCD(a, b) = 1. Dividing both parts by their GCD ensures minimal integer representation:
Mixed Numbers & Decimal Equivalents
| Fraction | Mixed Number | Exact Decimal | Percentage |
|---|---|---|---|
| 1/2 | — | 0.5 | 50% |
| 3/4 | — | 0.75 | 75% |
| 5/4 | 1 1/4 | 1.25 | 125% |
| 7/3 | 2 1/3 | 2.3333... | 233.33% |
| 9/8 | 1 1/8 | 1.125 | 112.5% |
Real-World Applications & Industry Use Cases
Scaling a recipe requiring 3/4 cup of flour by 2 1/2 batches: 3/4 × 5/2 = 15/8 = 1 7/8 cups.
Subtracting a saw kerf of 1/8" from a 3 3/16" board: 51/16 − 2/16 = 49/16 = 3 1/16".
Time signatures like 3/4 or 6/8 dictate note durations: combining whole (1), half (1/2), and quarter (1/4) notes.
Dividing profits between partners owning 3/8 and 1/4 (2/8) yields a combined equity of 5/8 (62.5%).
Step-by-Step Worked Examples
Calculate: 5/6 + 3/8
1. Find LCD(6, 8) = 24.
2. Convert: (5×4)/(6×4) = 20/24, and (3×3)/(8×3) = 9/24.
3. Add numerators: 20/24 + 9/24 = 29/24.
4. Result: 1 5/24 (Decimal: 1.2083).
Calculate: 2 1/3 × 1 3/4
1. Convert to improper: 2 1/3 = 7/3, and 1 3/4 = 7/4.
2. Multiply: (7 × 7) / (3 × 4) = 49/12.
3. Result: 4 1/12 (Decimal: 4.0833).
Common Calculation Pitfalls
Writing 1/2 + 1/3 = 2/5 is incorrect. Always find the LCD first (3/6 + 2/6 = 5/6).
Only the divisor (the second fraction) should be inverted in division (Keep, Change, Flip).
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
Mathematics & curriculum specialists. Audited against standard algebraic and arithmetic principles.