Fraction Calculator

Add, subtract, multiply, or divide fractions and get step-by-step solutions.
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Result & Steps

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How Fraction Operations Work

Understanding the mathematical processes behind fraction calculations helps you master these concepts and verify results manually.

🔢 Addition and Subtraction

Adding and subtracting fractions requires a common denominator. Here's the step-by-step process:

Step 1: Find the LCD

LCD (Least Common Denominator) is the smallest number that both denominators divide into evenly.

Example: For fractions 1/4 + 1/6

  • Multiples of 4: 4, 8, 12, 16, 20...
  • Multiples of 6: 6, 12, 18, 24...
  • LCD = 12
Step 2: Convert Fractions

Multiply numerator and denominator to get equivalent fractions:

  • 1/4 = (1×3)/(4×3) = 3/12
  • 1/6 = (1×2)/(6×2) = 2/12
  • Result: 3/12 + 2/12 = 5/12

✖️ Multiplication

Multiplying fractions is straightforward - multiply numerators together and denominators together:

Formula

a/b × c/d = (a×c)/(b×d)

Example

2/3 × 4/5 = (2×4)/(3×5) = 8/15

No common denominator needed!

➗ Division

Dividing fractions uses the "multiply by reciprocal" rule:

Formula

a/b ÷ c/d = a/b × d/c

Flip the second fraction and multiply

Example

2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12

Then simplify: 10/12 = 5/6

🔄 Simplification Process

Simplifying fractions involves finding the Greatest Common Divisor (GCD) and dividing both numerator and denominator by it:

Finding GCD

Example: Simplify 24/36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCD = 12 (largest common factor)

Simplification

Divide both parts by GCD:

24 ÷ 12 = 2

36 ÷ 12 = 3

Result: 24/36 = 2/3

💡 Quick Tips

  • 🎯 LCD shortcuts: If one denominator is a multiple of the other, the larger one is the LCD
  • ⚡ Prime denominators: If denominators are prime numbers, their LCD is their product
  • 🔍 GCD shortcuts: If numerator and denominator are both even, divide by 2 first
  • ✅ Check your work: The simplified fraction should equal the original when converted to decimal

🎓 Complete Example: 3/8 + 5/12

  1. Find LCD of 8 and 12:
    • 8 = 2³
    • 12 = 2² × 3
    • LCD = 2³ × 3 = 24
  2. Convert fractions:
    • 3/8 = (3×3)/(8×3) = 9/24
    • 5/12 = (5×2)/(12×2) = 10/24
  1. Add: 9/24 + 10/24 = 19/24
  2. Check if simplifiable:
    • GCD(19, 24) = 1
    • 19 is prime, doesn't divide 24
    • Already simplified!
Final Answer: 19/24

Step-by-Step Examples

Follow these detailed examples to master fraction operations with real calculations.

Example 1

Adding Fractions: 2/3 + 1/4

📝 Problem
2/3 + 1/4 = ?
Step 1: Find the LCD
  • Denominators: 3 and 4
  • Multiples of 3: 3, 6, 9, 12
  • Multiples of 4: 4, 8, 12
  • LCD = 12
Step 2: Convert to equivalent fractions
  • 2/3 = (2×4)/(3×4) = 8/12
  • 1/4 = (1×3)/(4×3) = 3/12
✅ Solution
Step 3: Add the fractions
8/12 + 3/12 = 11/12
Step 4: Check if simplifiable
  • GCD(11, 12) = ?
  • 11 is prime
  • 12 = 2² × 3
  • No common factors → GCD = 1
Final Answer: 11/12
Example 2

Subtracting Fractions: 5/6 - 1/4

📝 Problem
5/6 - 1/4 = ?
Step 1: Find the LCD
  • Denominators: 6 and 4
  • 6 = 2 × 3
  • 4 = 2²
  • LCD = 2² × 3 = 12
Step 2: Convert fractions
  • 5/6 = (5×2)/(6×2) = 10/12
  • 1/4 = (1×3)/(4×3) = 3/12
✅ Solution
Step 3: Subtract
10/12 - 3/12 = 7/12
Step 4: Simplify
  • Factors of 7: 1, 7
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • GCD(7, 12) = 1
  • Already in simplest form
Final Answer: 7/12
Example 3

Multiplying Fractions: 3/4 × 2/9

📝 Problem
3/4 × 2/9 = ?
Step 1: Multiply numerators
3 × 2 = 6
Step 2: Multiply denominators
4 × 9 = 36
Result before simplifying:
6/36
✅ Solution
Step 3: Find GCD to simplify
  • Factors of 6: 1, 2, 3, 6
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • GCD = 6
Step 4: Simplify
6 ÷ 6 = 1
36 ÷ 6 = 6
Final Answer: 1/6
Example 4

Dividing Fractions: 2/3 ÷ 4/5

📝 Problem
2/3 ÷ 4/5 = ?
Step 1: Convert to multiplication
2/3 ÷ 4/5 = 2/3 × ?
Flip the second fraction (reciprocal)
Step 2: Find reciprocal of 4/5
4/5 → 5/4
Step 3: Set up multiplication
2/3 × 5/4
✅ Solution
Step 4: Multiply
  • Numerator: 2 × 5 = 10
  • Denominator: 3 × 4 = 12
  • Result: 10/12
Step 5: Simplify
  • GCD(10, 12) = 2
  • 10 ÷ 2 = 5
  • 12 ÷ 2 = 6
Final Answer: 5/6

🎯 Practice Tips

  • Addition/Subtraction: Always find LCD first
  • Multiplication: Multiply straight across, then simplify
  • Division: Remember "Keep, Change, Flip"
  • Always: Check if your answer can be simplified
Try these examples with our calculator above to verify your manual calculations!

🌟 Real-World Use Cases

Discover how fractions are essential in everyday life through practical applications.

📚 Homework & Academic Applications

Math Homework Problems
Problem: "Sarah completed 2/3 of her math homework on Monday and 1/4 on Tuesday. What fraction of her homework has she completed?"
Solution: 2/3 + 1/4 = 8/12 + 3/12 = 11/12
Use Case: Mixed number operations, word problems, and real-life application of addition
Science & Statistics
Problem: "In a chemistry experiment, you need 3/8 of a solution but only have 1/2. How much extra do you have?"
Solution: 1/2 - 3/8 = 4/8 - 3/8 = 1/8 extra
Use Case: Laboratory calculations, measurement precision, and excess calculations
Geometry & Area
Problem: "A rectangular garden is 3/4 meters long and 2/3 meters wide. What's the area?"
Solution: 3/4 × 2/3 = 6/12 = 1/2 square meters
Use Case: Area calculations, geometric measurements, and spatial reasoning
Probability & Statistics
Problem: "If you study for 2/5 of your available time and sleep for 1/3, what fraction is left for other activities?"
Solution: 1 - (2/5 + 1/3) = 1 - (6/15 + 5/15) = 1 - 11/15 = 4/15
Use Case: Time management analysis, probability calculations, and resource allocation

🍳 Recipe & Cooking Applications

Recipe Scaling
Scenario: "A recipe calls for 3/4 cup of flour, but you want to make 1/2 of the recipe. How much flour do you need?"
Calculation: 3/4 × 1/2 = 3/8 cup of flour
Tip: Always multiply each ingredient by the scaling fraction to maintain proper ratios
Ingredient Combining
Scenario: "You have 1/3 cup of milk and add 1/4 cup more. How much milk do you have total?"
Calculation: 1/3 + 1/4 = 4/12 + 3/12 = 7/12 cup
Tip: Keep track of ingredient totals when adding multiple amounts
Portion Control
Scenario: "A cake recipe serves 8, but you want to serve 6 people. What fraction should you make?"
Calculation: 6 ÷ 8 = 6/8 = 3/4 of the original recipe
Tip: Divide desired servings by original servings to get the scaling factor
Nutritional Planning
Scenario: "Your daily protein goal is 1 cup. Breakfast provided 1/4 cup and lunch provided 1/3 cup. How much more do you need?"
Calculation: 1 - (1/4 + 1/3) = 1 - 7/12 = 5/12 cup remaining
Tip: Track nutritional intake using fractions for precise meal planning

🔧 Measurement & DIY Applications

Home Improvement
Scenario: "You need to cut a 7/8 inch board into pieces that are 1/4 inch thick. How many pieces can you get?"
Calculation: 7/8 ÷ 1/4 = 7/8 × 4/1 = 28/8 = 3.5 pieces (3 full pieces)
Tip: Always account for saw blade width and waste when calculating materials
Fabric & Sewing
Scenario: "A pattern requires 2/3 yard of fabric. You want to make 1.5 times the pattern. How much fabric do you need?"
Calculation: 2/3 × 3/2 = 6/6 = 1 yard total
Tip: Convert mixed numbers to improper fractions for easier calculation
Gardening & Landscaping
Scenario: "Your garden bed is 5/6 of an acre. You want to plant vegetables in 3/4 of it. How much area will you plant?"
Calculation: 5/6 × 3/4 = 15/24 = 5/8 of an acre
Tip: Plan garden layouts using fractions for efficient space utilization
Paint & Materials
Scenario: "You used 1/3 gallon of paint on the first coat and 1/4 gallon on the second. How much paint did you use total?"
Calculation: 1/3 + 1/4 = 4/12 + 3/12 = 7/12 gallon
Tip: Track material usage with fractions for accurate project budgeting

⚡ Quick Reference Guide

Academic Use
  • Word problems
  • Mixed operations
  • Geometry calculations
  • Time management
Cooking & Recipes
  • Scaling recipes
  • Combining ingredients
  • Portion control
  • Nutritional tracking
DIY & Measurements
  • Material calculations
  • Area planning
  • Project budgeting
  • Space optimization
💡 Pro Tip: Use our calculator above to solve any of these real-world fraction problems quickly and accurately!

❓ Frequently Asked Questions

Quick answers to common questions about fraction calculations and concepts.

To add fractions, make the denominators the same, add the numerators, and keep the denominator. Here's the process:

  1. Find the LCD (Least Common Denominator) of both fractions
  2. Convert both fractions to equivalent fractions with the LCD
  3. Add the numerators while keeping the common denominator
  4. Simplify the result if possible
Example: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2

Divide the numerator and denominator by their greatest common divisor (GCD) to simplify a fraction:

  1. Find the GCD of the numerator and denominator
  2. Divide both numbers by the GCD
  3. Check if further simplification is possible
Example: 12/18
• GCD(12, 18) = 6
• 12 ÷ 6 = 2, 18 ÷ 6 = 3
• Result: 2/3

Fractions are used in cooking, construction, science, and finance to represent parts of a whole:

🍳 Cooking & Recipes
  • Recipe measurements (1/2 cup, 3/4 teaspoon)
  • Scaling recipes up or down
  • Portion control and servings
🔧 Construction & DIY
  • Lumber measurements (2x4, 1/2 inch)
  • Area calculations
  • Material planning
💰 Finance
  • Interest rates (1/4% = 0.25%)
  • Stock prices ($50 1/2)
  • Budget allocations
🔬 Science & Medicine
  • Dosage calculations
  • Concentration ratios
  • Probability and statistics

A common denominator is a shared multiple of the denominators of two or more fractions. The least common denominator (LCD) is the smallest positive number that all denominators divide into evenly.

Example: For fractions 1/4 and 1/6
• Multiples of 4: 4, 8, 12, 16, 20...
• Multiples of 6: 6, 12, 18, 24...
• LCD = 12 because both 4 and 6 divide evenly into 12

Finding the LCD allows you to add or subtract fractions by converting them to equivalent fractions with the same denominator.

💡 Quick Tip: If one denominator is a multiple of the other, the larger number is the LCD!

A mixed number combines a whole number and a proper fraction (like 2 1/3), while an improper fraction has a numerator larger than or equal to its denominator (like 7/3). These represent the same value but in different forms.

Mixed Numbers
  • Easier to visualize (2 1/3 shows 2 whole units plus 1/3)
  • Better for everyday communication
  • Common in measurements
Example: 2 1/3 (two and one-third)
Improper Fractions
  • Easier to calculate with
  • Better for mathematical operations
  • Simpler for multiplication and division
Example: 7/3 (seven-thirds)
🔄 Converting: To convert 2 1/3 to improper fraction:
Multiply the whole number by the denominator, add the numerator: (2 × 3) + 1 = 7
Result: 7/3