Fractions Calculator
Result:
Our comprehensive fractions calculator performs all four basic operations on fractions and mixed numbers with detailed step-by-step solutions. Whether you're adding, subtracting, multiplying, or dividing fractions, get accurate results with complete explanations.
Complete Fractions Calculator
This all-in-one fractions calculator handles every type of fraction operation including proper fractions, improper fractions, mixed numbers, and whole numbers. Perfect for students, teachers, and professionals.
Understanding Fraction Operations
Fraction operations form the foundation of mathematical literacy and are essential for success in algebra, geometry, and higher mathematics. Each operation—addition, subtraction, multiplication, and division—follows specific rules and procedures that, once mastered, make working with fractions intuitive and straightforward.
What are Fractions?
Definition: Numbers representing parts of a whole
Parts: Numerator (top) and Denominator (bottom)
Example: In 3/4, numerator is 3, denominator is 4
Meaning: 3 parts out of 4 equal parts
Types of Fractions
Proper: Numerator < Denominator (3/4)
Improper: Numerator ≥ Denominator (7/4)
Mixed: Whole number + fraction (1 3/4)
Unit: Numerator = 1 (1/4)
Addition of Fractions
Adding fractions requires finding a common denominator to combine the parts:
Same Denominators
Example: 2/7 + 3/7
Rule: Add numerators, keep the same denominator
Calculation: (2 + 3)/7 = 5/7
Visual: 🔵🔵 + 🔵🔵🔵 = 🔵🔵🔵🔵🔵 (all sevenths)
Check: 2/7 + 3/7 = 5/7 ✓
Different Denominators
Example: 1/4 + 1/6
Step 1: Find LCD of 4 and 6
Multiples of 4: 4, 8, 12, 16...
Multiples of 6: 6, 12, 18...
LCD = 12
Step 2: Convert fractions
1/4 = (1×3)/(4×3) = 3/12
1/6 = (1×2)/(6×2) = 2/12
Step 3: Add
3/12 + 2/12 = 5/12
Answer: 5/12
Mixed Numbers Addition
Example: 2 1/3 + 1 2/5
Method 1: Add parts separately
Whole parts: 2 + 1 = 3
Fraction parts: 1/3 + 2/5
LCD of 3 and 5 = 15
1/3 = 5/15, 2/5 = 6/15
5/15 + 6/15 = 11/15
Result: 3 + 11/15 = 3 11/15
Method 2: Convert to improper fractions
2 1/3 = 7/3, 1 2/5 = 7/5
7/3 + 7/5 = 35/15 + 21/15 = 56/15 = 3 11/15
Subtraction of Fractions
Subtracting fractions follows similar rules to addition but requires attention to borrowing:
Same Denominators
Example: 5/8 - 2/8
Rule: Subtract numerators, keep the same denominator
Calculation: (5 - 2)/8 = 3/8
Check: 3/8 + 2/8 = 5/8 ✓
Different Denominators
Example: 3/4 - 1/3
Step 1: Find LCD of 4 and 3 = 12
Step 2: Convert fractions
3/4 = 9/12, 1/3 = 4/12
Step 3: Subtract
9/12 - 4/12 = 5/12
Borrowing in Mixed Numbers
Example: 3 1/4 - 1 3/4
Problem: 1/4 < 3/4, so we need to borrow
Step 1: Borrow 1 from whole number
3 1/4 = 2 + 1 + 1/4 = 2 + 4/4 + 1/4 = 2 5/4
Step 2: Subtract
2 5/4 - 1 3/4 = (2-1) + (5/4 - 3/4) = 1 + 2/4 = 1 1/2
Multiplication of Fractions
Multiplying fractions is straightforward - multiply across:
Basic Multiplication
Example: 2/3 × 4/5
Rule: Multiply numerators together, multiply denominators together
Calculation: (2×4)/(3×5) = 8/15
Check: 2/3 ≈ 0.667, 4/5 = 0.8, 8/15 ≈ 0.533 ≈ 0.667×0.8 ✓
Multiplication with Simplification
Example: 6/8 × 4/9
Method 1: Multiply then simplify
(6×4)/(8×9) = 24/72
GCD(24, 72) = 24
24/72 = 1/3
Method 2: Cancel before multiplying
6/8 × 4/9 = (6÷2)/(8÷2) × (4÷4)/(9÷1) = 3/4 × 1/9
Then: (3×1)/(4×9) = 3/36 = 1/12
Note: Method 2 has an error - let's redo correctly:
6/8 × 4/9: Cancel 6 and 9 (÷3), 8 and 4 (÷4)
Result: 2/2 × 1/3 = 1/3 ✓
Mixed Numbers Multiplication
Example: 2 1/4 × 1 1/3
Step 1: Convert to improper fractions
2 1/4 = (2×4+1)/4 = 9/4
1 1/3 = (1×3+1)/3 = 4/3
Step 2: Multiply
9/4 × 4/3 = (9×4)/(4×3) = 36/12 = 3
Answer: 3 (whole number)
Division of Fractions
Dividing fractions uses the "multiply by reciprocal" rule:
Basic Division
Example: 3/4 ÷ 2/5
Rule: Multiply by the reciprocal of the second fraction
Reciprocal of 2/5: 5/2
Calculation: 3/4 × 5/2 = (3×5)/(4×2) = 15/8
Convert to mixed: 15/8 = 1 7/8
Check: 1 7/8 × 2/5 should equal 3/4
(15/8) × (2/5) = 30/40 = 3/4 ✓
Division with Whole Numbers
Example: 2/3 ÷ 4
Step 1: Write whole number as fraction
4 = 4/1
Step 2: Find reciprocal of 4/1
Reciprocal: 1/4
Step 3: Multiply
2/3 × 1/4 = 2/12 = 1/6
Comprehensive Examples by Operation
Addition Examples
1/2 + 1/3 = 3/6 + 2/6 = 5/6
3/8 + 1/4 = 3/8 + 2/8 = 5/8
2 1/5 + 1 3/10 = 2 2/10 + 1 3/10 = 3 5/10 = 3 1/2
Subtraction Examples
5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2
7/8 - 1/2 = 7/8 - 4/8 = 3/8
3 1/4 - 1 5/8 = 3 2/8 - 1 5/8 = 2 10/8 - 1 5/8 = 1 5/8
Multiplication Examples
1/3 × 2/5 = 2/15
3/4 × 8/9 = 24/36 = 2/3
2 1/2 × 1 1/3 = 5/2 × 4/3 = 20/6 = 3 1/3
Division Examples
2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 2 2/3
5/8 ÷ 5/6 = 5/8 × 6/5 = 30/40 = 3/4
3 ÷ 2/5 = 3/1 × 5/2 = 15/2 = 7 1/2
Order of Operations with Fractions
Follow PEMDAS/BODMAS when multiple operations are present:
Example: 1/2 + 1/3 × 2/5
Step 1: Multiplication first (PEMDAS)
1/3 × 2/5 = 2/15
Step 2: Then addition
1/2 + 2/15
LCD of 2 and 15 = 30
15/30 + 4/30 = 19/30
Answer: 19/30
Real-World Applications
Cooking and Recipes
Fraction operations are essential in culinary mathematics:
Example: A recipe calls for 2 1/3 cups flour. You want to make 1 1/2 times the recipe.
Calculation: 2 1/3 × 1 1/2 = 7/3 × 3/2 = 21/6 = 3 1/2 cups flour
Application: Scale ingredients proportionally for different serving sizes
Calculation: 2 1/3 × 1 1/2 = 7/3 × 3/2 = 21/6 = 3 1/2 cups flour
Application: Scale ingredients proportionally for different serving sizes
Construction and Carpentry
Precise measurements require fraction calculations:
Example: A board is 8 3/4 inches long. You need to cut it into 3 equal pieces.
Calculation: 8 3/4 ÷ 3 = 35/4 ÷ 3/1 = 35/4 × 1/3 = 35/12 = 2 11/12 inches per piece
Application: Ensure accurate cuts and material usage
Calculation: 8 3/4 ÷ 3 = 35/4 ÷ 3/1 = 35/4 × 1/3 = 35/12 = 2 11/12 inches per piece
Application: Ensure accurate cuts and material usage
Time and Scheduling
Time calculations often involve fractions of hours:
Example: A project takes 2 1/4 hours. You've completed 3/4 of it. How much time remains?
Calculation: Remaining = 2 1/4 - (2 1/4 × 3/4) = 9/4 - (9/4 × 3/4) = 9/4 - 27/16 = 36/16 - 27/16 = 9/16 hours
Conversion: 9/16 hours = 9/16 × 60 = 33.75 minutes ≈ 34 minutes
Calculation: Remaining = 2 1/4 - (2 1/4 × 3/4) = 9/4 - (9/4 × 3/4) = 9/4 - 27/16 = 36/16 - 27/16 = 9/16 hours
Conversion: 9/16 hours = 9/16 × 60 = 33.75 minutes ≈ 34 minutes
Common Mistakes and Solutions
❌ Common Mistakes
- Adding denominators when adding fractions
- Not finding common denominators
- Forgetting to simplify final answers
- Incorrect order of operations
- Not converting mixed numbers for multiplication/division
- Dividing instead of multiplying by reciprocal
✅ Best Practices
- Always find LCD for addition/subtraction
- Convert mixed numbers for multiplication/division
- Simplify fractions at each step when possible
- Check answers by converting to decimals
- Use cross-multiplication to verify equivalence
- Remember PEMDAS for complex expressions
Practice Problems
Test your fraction operation skills:
Solution:
LCD of 5 and 3 = 15
2/5 = 6/15, 1/3 = 5/15
6/15 + 5/15 = 11/15
Answer: 11/15
LCD of 5 and 3 = 15
2/5 = 6/15, 1/3 = 5/15
6/15 + 5/15 = 11/15
Answer: 11/15
Solution:
Convert to eighths: 3 1/4 = 3 2/8
Need to borrow: 3 2/8 = 2 10/8
2 10/8 - 1 5/8 = 1 5/8
Answer: 1 5/8
Convert to eighths: 3 1/4 = 3 2/8
Need to borrow: 3 2/8 = 2 10/8
2 10/8 - 1 5/8 = 1 5/8
Answer: 1 5/8
Solution:
Multiply across: (2×3)/(3×8) = 6/24
Simplify: 6/24 = 1/4
Answer: 1/4
Multiply across: (2×3)/(3×8) = 6/24
Simplify: 6/24 = 1/4
Answer: 1/4
Solution:
Multiply by reciprocal: 4/5 × 3/2
Calculate: (4×3)/(5×2) = 12/10
Simplify: 12/10 = 6/5 = 1 1/5
Answer: 1 1/5
Multiply by reciprocal: 4/5 × 3/2
Calculate: (4×3)/(5×2) = 12/10
Simplify: 12/10 = 6/5 = 1 1/5
Answer: 1 1/5
Tips for Success
Find Common Denominators
Master LCD techniques for addition and subtraction operations.
Convert Mixed Numbers
Always convert to improper fractions for multiplication and division.
Simplify Regularly
Reduce fractions to lowest terms throughout your calculations.
Key Takeaways
- Addition/Subtraction: Find common denominators, then add/subtract numerators
- Multiplication: Multiply numerators together, multiply denominators together
- Division: Multiply by the reciprocal of the second fraction
- Mixed Numbers: Convert to improper fractions for multiplication and division
- Always simplify your final answers to lowest terms
- Check your work using decimal conversions or inverse operations
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