🔢 Fraction Calculator
17 comprehensive fraction calculators with detailed step-by-step solutions
Add Fractions
📚 How to Add Fractions
Same Denominators: Add the numerators and keep the same denominator.
Different Denominators: Find the LCD (Least Common Denominator), convert both fractions to equivalent fractions with the LCD, then add the numerators.
Formula: a/b + c/d = (a×d + c×b)/(b×d)
Subtract Fractions
📚 How to Subtract Fractions
Same Denominators: Subtract the numerators and keep the same denominator.
Different Denominators: Find the LCD, convert both fractions, then subtract the numerators.
Formula: a/b - c/d = (a×d - c×b)/(b×d)
Multiply Fractions
📚 How to Multiply Fractions
Simple Rule: Multiply numerators together and denominators together.
Formula: a/b × c/d = (a×c)/(b×d)
Tip: You can cross-cancel before multiplying to simplify the calculation.
Divide Fractions
📚 How to Divide Fractions
Keep, Change, Flip: Keep the first fraction, change division to multiplication, flip the second fraction.
Formula: a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)
Remember: Dividing by a fraction is the same as multiplying by its reciprocal.
Simplify Fractions
📚 How to Simplify Fractions
Find GCD: Find the Greatest Common Divisor of numerator and denominator.
Divide Both: Divide both numerator and denominator by the GCD.
Result: The fraction is now in its simplest form (lowest terms).
Mixed Number to Improper Fraction
📚 How to Convert Mixed Numbers to Improper Fractions
Formula: a b/c = (a×c + b)/c
Steps: Multiply whole number by denominator, add numerator, keep same denominator.
Example: 2 3/4 = (2×4 + 3)/4 = 11/4
Improper Fraction to Mixed Number
📚 How to Convert Improper Fractions to Mixed Numbers
Divide: Divide numerator by denominator to get whole number and remainder.
Format: Quotient becomes whole number, remainder becomes new numerator.
Example: 11/4 = 2 3/4 (11÷4 = 2 remainder 3)
Fraction to Decimal
📚 How to Convert Fractions to Decimals
Division: Divide the numerator by the denominator.
Long Division: Use long division if needed for exact decimal places.
Types: Results can be terminating (0.25) or repeating (0.333...).
Decimal to Fraction
📚 How to Convert Decimals to Fractions
Count Decimal Places: Count digits after decimal point.
Create Fraction: Put decimal over power of 10 (10, 100, 1000, etc.).
Simplify: Reduce to lowest terms by finding GCD.
Example: 0.75 = 75/100 = 3/4
Fraction to Percent
📚 How to Convert Fractions to Percentages
Method 1: Divide numerator by denominator, then multiply by 100.
Method 2: Convert to equivalent fraction with denominator 100.
Formula: (a/b) × 100% = (a×100)/b %
Percent to Fraction
📚 How to Convert Percentages to Fractions
Remove % Sign: Write the number over 100.
Simplify: Reduce the fraction to lowest terms.
Example: 25% = 25/100 = 1/4
Compare Fractions
📚 How to Compare Fractions
Common Denominator Method: Convert to same denominator, compare numerators.
Cross Multiplication: For a/b vs c/d, compare a×d with b×c.
Decimal Method: Convert both to decimals and compare.
Equivalent Fractions
📚 How to Find Equivalent Fractions
Multiply or Divide: Multiply or divide both numerator and denominator by the same number.
Same Value: All equivalent fractions represent the same value.
Example: 1/2 = 2/4 = 3/6 = 4/8 = 5/10
LCM (Least Common Multiple)
📚 How to Find LCM
Definition: The smallest positive number that is divisible by both numbers.
Formula: LCM(a,b) = (a × b) ÷ GCD(a,b)
Use: Finding common denominators for adding/subtracting fractions.
GCF (Greatest Common Factor)
📚 How to Find GCF
Definition: The largest number that divides both numbers evenly.
Euclidean Algorithm: Keep dividing and using remainders until remainder is 0.
Use: Simplifying fractions to lowest terms.
Fraction Decomposition
📚 How Fraction Decomposition Works
Definition: Breaking down a fraction into simpler unit fractions (fractions with numerator 1).
Egyptian Fractions: Express any fraction as a sum of distinct unit fractions.
Example: 5/6 = 1/2 + 1/3
Greedy Algorithm: Uses the largest possible unit fraction at each step.
Partial Fraction Decomposition
📚 Partial Fraction Decomposition
Definition: Breaking down a rational function into simpler fractions for easier integration or analysis.
Linear Factors: For (ax+b)(cx+d), decompose as A/(ax+b) + B/(cx+d)
Example: 5x+1/((x+1)(x+2)) = A/(x+1) + B/(x+2)
Note: This calculator handles simple cases with linear factors.
📖 The Complete Guide to Fractions
What Are Fractions?
A fraction represents a part of a whole or a division of quantities. It consists of a numerator (top number) and a denominator (bottom number). The denominator tells us how many equal parts the whole is divided into, while the numerator tells us how many of those parts we're considering.
Fractions are fundamental to mathematics and appear everywhere in daily life, from cooking measurements to financial calculations.
Types of Fractions
- Proper Fractions: Numerator < Denominator (e.g., 3/5, 7/8)
- Improper Fractions: Numerator ≥ Denominator (e.g., 7/4, 9/3)
- Mixed Numbers: Whole number + fraction (e.g., 2 1/3, 5 2/7)
- Unit Fractions: Numerator = 1 (e.g., 1/8, 1/15)
- Complex Fractions: Fractions within fractions (e.g., (1/2)/(3/4))
- Equivalent Fractions: Different forms, same value (e.g., 1/2 = 2/4)
Why Learn Fractions?
Fractions are essential in everyday life and advanced mathematics. They help us:
- Cooking & Baking: Measure ingredients precisely and scale recipes
- Construction: Calculate measurements, materials, and proportions
- Finance: Understand interest rates, stock prices, and investments
- Time Management: Work with portions of hours and scheduling
- Science: Express ratios, concentrations, and probabilities
- Art & Design: Create proportional layouts and golden ratios
- Sports: Calculate statistics, batting averages, and performance metrics
Fraction Operations Explained
Addition & Subtraction
Requires common denominators. Find the LCD (Least Common Denominator), convert fractions, then add/subtract numerators. Always simplify the result.
Multiplication
Simply multiply numerators together and denominators together. No common denominator needed! Cross-cancellation can simplify calculations.
Division
Use "Keep, Change, Flip" - keep first fraction, change ÷ to ×, flip the second fraction (find its reciprocal).
Simplification
Find the GCD of numerator and denominator, then divide both by this number to get the lowest terms.
Converting Between Forms
Divide numerator by denominator
Multiply by 100, add % sign
Write over 100, then simplify
Multiply whole by denominator, add numerator
Pro Tips for Success
Common Fraction Equivalents
Real-World Applications
🍳 Cooking & Baking
Scaling recipes up or down, measuring ingredients precisely, adjusting serving sizes, converting between measurement systems.
🏗️ Construction
Measuring materials, calculating dimensions, working with blueprints, determining angles and proportions.
💰 Finance
Interest rates, stock prices, calculating portions of investments, loan payments, and financial returns.
⏰ Time Management
Understanding portions of hours, scheduling tasks, calculating elapsed time periods and productivity metrics.
🎨 Art & Design
Creating proportional layouts, golden ratio applications, scaling artwork, and design measurements.
⚕️ Medicine
Dosage calculations, concentration ratios, medical statistics, and treatment effectiveness rates.
Advanced Fraction Concepts
- Egyptian Fractions: Ancient method of expressing fractions as sums of unit fractions
- Continued Fractions: Representing numbers as nested fractions for approximations
- Partial Fractions: Decomposing complex rational expressions for calculus
- Farey Sequences: Ordered lists of reduced fractions between 0 and 1
- Mediant: Finding fractions between two given fractions
- Cross-Multiplication: Solving proportions and comparing fractions efficiently
Fraction Problem-Solving Strategies
Word Problems
Identify what the whole represents, determine what fraction is being described, and set up the equation carefully.
Multiple Steps
Break complex problems into smaller steps, solve each fraction operation separately, then combine results.
Estimation
Round fractions to nearby simple fractions (like 1/2, 1/4) to estimate answers before calculating exactly.
Pattern Recognition
Look for patterns in fraction sequences, equivalent forms, and recurring decimal representations.
Fractions in Science & Technology
- Chemistry: Molecular ratios, concentration calculations, stoichiometry
- Physics: Wave frequencies, probability calculations, measurement precision
- Computer Science: Algorithms, data compression, graphics rendering
- Engineering: Gear ratios, structural calculations, efficiency measurements
- Statistics: Probability distributions, sampling ratios, confidence intervals
- Music: Note frequencies, rhythm patterns, harmonic ratios
Common Fraction Mistakes to Avoid
📚 Complete Study Guide
Master the Basics
Understand what numerators and denominators represent, learn to identify different types of fractions, and practice reading fractions aloud.
Learn to Simplify
Practice finding GCD and reducing fractions to lowest terms. This skill is essential for all other fraction operations.
Practice Basic Operations
Start with same denominators, then move to different denominators. Master addition and subtraction before moving to multiplication and division.
Master Conversions
Practice converting between fractions, decimals, percentages, and mixed numbers. These skills connect fractions to other math concepts.
Apply to Real Problems
Work on word problems and real-world applications. This helps you understand when and how to use fractions in practical situations.
Explore Advanced Topics
Learn about complex fractions, partial fraction decomposition, and other advanced concepts for higher-level mathematics.