Fraction Calculator
Add, subtract, multiply, and divide fractions. Simplify fractions and convert to decimals.
Simplified Fraction
3/4
Step-by-Step Solution
- 1Find common denominator: 2 × 4 = 8
- 2Convert first fraction: 1/2 = 4/8
- 3Convert second fraction: 1/4 = 2/8
- 4Add: 4/8 + 2/8 = 6/8
- ✓Simplify: 6/8 = 3/4
Common Fraction Equivalents
About This Calculator
The Fraction Calculator performs addition, subtraction, multiplication, and division of fractions with detailed step-by-step solutions that show exactly how to solve each problem. Whether you're adding fractions with different denominators, multiplying mixed numbers, simplifying complex fractions, or converting between improper fractions and mixed numbers, this calculator provides instant results with educational explanations. Fractions are fundamental to mathematics and appear everywhere in daily life—from cooking recipes that call for 3/4 cup of flour, to measuring lumber in eighths of an inch, to understanding financial concepts like half-price sales. Many students and adults struggle with fraction operations because they require multiple steps and conceptual understanding. Our calculator not only gives you the correct answer but teaches you the process: finding common denominators, cross-multiplying, reducing to lowest terms, and converting between formats. Enter any two fractions—including mixed numbers and negative fractions—to see the answer in fully simplified form with complete working shown. Master fractions with this comprehensive calculator that makes complex operations simple and understandable.
How to Use the Fraction Calculator
- 1Enter the first fraction by typing the numerator (top) and denominator (bottom).
- 2For mixed numbers like 2 1/3, enter the whole number, numerator, and denominator separately.
- 3Select the operation: addition (+), subtraction (-), multiplication (×), or division (÷).
- 4Enter the second fraction in the same format.
- 5Click Calculate to see the result in simplified form.
- 6Review the step-by-step solution to understand how the answer was found.
- 7Use the simplify button to reduce any fraction to its lowest terms.
- 8Toggle between improper fractions and mixed numbers with the convert button.
Fraction Operations Explained
Addition and Subtraction: Requires a common denominator (same bottom number).
Steps:
- Find the Least Common Denominator (LCD)
- Convert each fraction to equivalent fraction with LCD
- Add or subtract the numerators
- Keep the denominator the same
- Simplify if possible
Example: 1/4 + 2/3
- LCD of 4 and 3 = 12
- 1/4 = 3/12 (multiply top and bottom by 3)
- 2/3 = 8/12 (multiply top and bottom by 4)
- 3/12 + 8/12 = 11/12 ✓
Multiplication: Multiply straight across—no common denominator needed!
Steps:
- Multiply the numerators together
- Multiply the denominators together
- Simplify the result
Example: 2/3 × 3/4
- (2 × 3) / (3 × 4) = 6/12 = 1/2 ✓
Division: Flip the second fraction and multiply.
Steps:
- Keep the first fraction
- Change division to multiplication
- Flip the second fraction (reciprocal)
- Multiply and simplify
Example: 2/3 ÷ 4/5
- 2/3 × 5/4 = 10/12 = 5/6 ✓
Finding Common Denominators
Least Common Denominator (LCD): The smallest number that both denominators divide into evenly.
Method 1: List Multiples For denominators 4 and 6:
- Multiples of 4: 4, 8, 12, 16, 20, 24...
- Multiples of 6: 6, 12, 18, 24...
- LCD = 12
Method 2: Prime Factorization
- 4 = 2²
- 6 = 2 × 3
- LCD = 2² × 3 = 12
Method 3: Largest Denominator Sometimes the larger denominator is a multiple of the smaller:
- 4 and 8 → LCD = 8 (8 is already a multiple of 4)
- 3 and 12 → LCD = 12
Common LCD Reference Table:
| Denominators | LCD | Converting |
|---|---|---|
| 2, 3 | 6 | ×3, ×2 |
| 2, 4 | 4 | ×2, ×1 |
| 2, 5 | 10 | ×5, ×2 |
| 3, 4 | 12 | ×4, ×3 |
| 3, 5 | 15 | ×5, ×3 |
| 4, 5 | 20 | ×5, ×4 |
| 4, 6 | 12 | ×3, ×2 |
| 5, 6 | 30 | ×6, ×5 |
| 6, 8 | 24 | ×4, ×3 |
| 8, 12 | 24 | ×3, ×2 |
Why Common Denominators Matter: Think of fractions as pizza slices. You can only combine slices if they're the same size. 1/4 of a pizza and 1/3 of a pizza are different-sized slices—we need to "re-cut" both pizzas into equal-sized pieces (twelfths) before combining.
Simplifying Fractions
What is Simplifying? Reducing a fraction to its lowest terms by dividing both numerator and denominator by their Greatest Common Factor (GCF).
Example: Simplify 12/18
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- GCF = 6
- 12/18 = (12÷6)/(18÷6) = 2/3 ✓
Quick Simplification Tests:
| If Both Are... | Divide By... | Example |
|---|---|---|
| Even numbers | 2 | 8/12 → 4/6 → 2/3 |
| End in 0 or 5 | 5 | 15/25 → 3/5 |
| Digit sums ÷3 | 3 | 9/12 → 3/4 |
| End in 0 | 10 | 30/50 → 3/5 |
Step-by-Step Method:
- Check if both numbers are even → divide by 2
- Check if both are divisible by 3 → divide by 3
- Check if both end in 0 or 5 → divide by 5
- Repeat until no common factors remain
Example: Simplify 24/36
- 24/36 → 12/18 (÷2)
- 12/18 → 6/9 (÷2)
- 6/9 → 2/3 (÷3) ✓
Common Simplified Fractions:
| Fraction | Simplified |
|---|---|
| 2/4 | 1/2 |
| 3/6 | 1/2 |
| 4/8 | 1/2 |
| 2/6 | 1/3 |
| 3/9 | 1/3 |
| 4/12 | 1/3 |
| 3/4 | 3/4 (already simplified) |
| 6/8 | 3/4 |
Mixed Numbers and Improper Fractions
Definitions:
- Mixed Number: A whole number plus a fraction (2 1/3)
- Improper Fraction: Numerator ≥ denominator (7/3)
- Proper Fraction: Numerator < denominator (2/3)
Converting Mixed Numbers to Improper Fractions: Formula: (Whole × Denominator + Numerator) / Denominator
Example: Convert 2 1/3 to improper
- (2 × 3 + 1) / 3 = 7/3 ✓
| Mixed Number | Improper Fraction |
|---|---|
| 1 1/2 | 3/2 |
| 1 3/4 | 7/4 |
| 2 1/3 | 7/3 |
| 2 1/2 | 5/2 |
| 3 1/4 | 13/4 |
| 3 2/3 | 11/3 |
Converting Improper Fractions to Mixed Numbers: Divide numerator by denominator. Quotient = whole, Remainder = new numerator.
Example: Convert 11/4 to mixed
- 11 ÷ 4 = 2 remainder 3
- Answer: 2 3/4 ✓
Operations with Mixed Numbers: Convert to improper fractions first, then perform the operation.
Example: 2 1/2 + 1 3/4
- Convert: 5/2 + 7/4
- LCD = 4: 10/4 + 7/4
- Add: 17/4
- Convert back: 4 1/4 ✓
Cross-Cancellation (Shortcut for Multiplication)
What is Cross-Cancellation? Simplifying BEFORE multiplying by canceling common factors diagonally.
Example Without Cross-Cancel: 3/8 × 4/9 = 12/72 = 1/6 (needs simplification)
Example With Cross-Cancel: 3/8 × 4/9
- 3 and 9 share factor 3: (1/8 × 4/3)
- 4 and 8 share factor 4: (1/2 × 1/3)
- Result: 1/6 ✓ (already simplified!)
Visual Method:
3 4 1 1
- × - → - × - = 1/6
8 9 2 3
↗ ↖
When to Cross-Cancel:
- Only with multiplication (NOT addition/subtraction)
- Can also use with division after flipping the second fraction
- Look for numbers that share common factors
Benefits:
- Keeps numbers smaller
- Avoids large products
- Result is already simplified
Practice:
| Original | Cross-Cancel | Answer |
|---|---|---|
| 5/12 × 8/15 | (1/3 × 2/3) | 2/9 |
| 6/7 × 14/18 | (1/1 × 2/3) | 2/3 |
| 9/16 × 8/27 | (1/2 × 1/3) | 1/6 |
Fractions, Decimals, and Percents
Conversion Reference Table:
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333... | 33.33% |
| 2/3 | 0.666... | 66.67% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 2/5 | 0.4 | 40% |
| 3/5 | 0.6 | 60% |
| 4/5 | 0.8 | 80% |
| 1/8 | 0.125 | 12.5% |
| 3/8 | 0.375 | 37.5% |
| 5/8 | 0.625 | 62.5% |
| 7/8 | 0.875 | 87.5% |
| 1/10 | 0.1 | 10% |
| 1/100 | 0.01 | 1% |
Conversion Methods:
Fraction to Decimal: Divide numerator by denominator.
- 3/4 = 3 ÷ 4 = 0.75
Decimal to Fraction: Put over appropriate power of 10, simplify.
- 0.75 = 75/100 = 3/4
Fraction to Percent: Convert to decimal, multiply by 100.
- 3/4 = 0.75 = 75%
Repeating Decimals:
| Fraction | Decimal | Pattern |
|---|---|---|
| 1/3 | 0.333... | 3 repeats |
| 1/6 | 0.1666... | 6 repeats |
| 1/7 | 0.142857... | 142857 repeats |
| 1/9 | 0.111... | 1 repeats |
| 1/11 | 0.0909... | 09 repeats |
Real-World Fraction Applications
Cooking and Recipes:
Doubling a recipe that calls for 2/3 cup:
- 2/3 × 2 = 4/3 = 1 1/3 cups
Halving 3/4 cup:
- 3/4 × 1/2 = 3/8 cup
Measuring Cups Available:
| Size | Equivalent |
|---|---|
| 1 cup | 1 |
| 3/4 cup | 3/4 |
| 2/3 cup | 2/3 |
| 1/2 cup | 1/2 |
| 1/3 cup | 1/3 |
| 1/4 cup | 1/4 |
| 1/8 cup | 1/8 |
Construction and Measurement: Inches are divided into fractions:
- 1/16", 1/8", 3/16", 1/4", 5/16", 3/8", 7/16", 1/2", etc.
Adding lumber lengths: 5 3/8" + 7 5/8" = 5 + 7 + 3/8 + 5/8 = 12 + 8/8 = 13"
Music:
| Note | Fraction of Whole Note |
|---|---|
| Whole | 1 |
| Half | 1/2 |
| Quarter | 1/4 |
| Eighth | 1/8 |
| Sixteenth | 1/16 |
Time:
- 1/4 hour = 15 minutes
- 1/3 hour = 20 minutes
- 1/2 hour = 30 minutes
- 3/4 hour = 45 minutes
Comparing and Ordering Fractions
Method 1: Common Denominator Convert both to same denominator, compare numerators.
Compare 3/4 and 2/3:
- 3/4 = 9/12
- 2/3 = 8/12
- 9/12 > 8/12, so 3/4 > 2/3 ✓
Method 2: Cross Multiplication Multiply diagonally and compare products.
Compare 3/4 and 2/3:
- 3 × 3 = 9 (goes with first fraction)
- 4 × 2 = 8 (goes with second fraction)
- 9 > 8, so 3/4 > 2/3 ✓
Method 3: Convert to Decimals
- 3/4 = 0.75
- 2/3 = 0.667
- 0.75 > 0.667, so 3/4 > 2/3 ✓
Benchmark Fractions: Use 1/2 as a reference point:
- Greater than 1/2: 3/4, 2/3, 5/8, 4/7
- Less than 1/2: 1/4, 1/3, 3/8, 2/5
- Equal to 1/2: 2/4, 3/6, 4/8, 5/10
Ordering Fractions: Arrange 1/2, 2/3, 3/4, 1/4 from least to greatest:
- Convert: 6/12, 8/12, 9/12, 3/12
- Order: 3/12 < 6/12 < 8/12 < 9/12
- Answer: 1/4 < 1/2 < 2/3 < 3/4 ✓
Pro Tips
- 💡Always simplify your final answer to lowest terms—teachers and tests expect this.
- 💡Convert mixed numbers to improper fractions before calculating, then convert back at the end.
- 💡Use cross-cancellation when multiplying to keep numbers smaller and avoid simplifying later.
- 💡Check your answer by converting to decimals: 1/4 + 1/4 = 2/4 = 0.5 ✓
- 💡When dividing, remember "Keep, Change, Flip"—keep first, change ÷ to ×, flip second.
- 💡For common denominators, use the larger denominator if it is a multiple of the smaller one.
- 💡The numerator tells "how many parts," the denominator tells "what size parts."
- 💡Negative fractions: put the negative sign in front for clarity, not in the denominator.
- 💡To compare fractions quickly, cross-multiply and compare the products.
- 💡In cooking, 1/3 cup + 1/3 cup + 1/3 cup = 1 cup—use this to check your work.
- 💡Equivalent fractions: multiply (or divide) top AND bottom by the same number.
- 💡When stuck, convert to decimals to check if your fraction answer makes sense.
Frequently Asked Questions
Addition combines "like things"—you can only add fractions with the same-sized pieces. 1/4 and 1/3 are different-sized slices, so we convert to same-sized pieces (twelfths) first. Multiplication is different—it finds "a fraction OF another fraction," which works by multiplying straight across. Think of 1/2 × 1/3 as "half of one-third," which is 1/6.
