Fraction Calculator - Free Math Tool to Add, Subtract, Multiply and Divide Fractions with Automatic Simplification
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Complete Guide to Fraction Operations
š What is a Fraction?
A fraction represents a part of a whole, expressed as one number (numerator) divided by another (denominator):
- Proper fraction: Numerator < Denominator (1/2, 3/4)
- Improper fraction: Numerator ā„ Denominator (5/4, 7/3)
- Mixed number: Whole + Fraction (1 1/2, 2 1/3)
- Equivalent fractions: Same value (1/2 = 2/4 = 3/6)
š¢ Fraction Terminology
Fraction Operation Rules
ā Addition
Steps: 1) Find common denominator, 2) Convert both fractions, 3) Add numerators, 4) Keep denominator, 5) Simplify
ā Subtraction
Steps: 1) Find common denominator, 2) Convert both fractions, 3) Subtract numerators, 4) Keep denominator, 5) Simplify
āļø Multiplication
Steps: 1) Multiply numerators, 2) Multiply denominators, 3) Simplify. Tip: Cross-cancel common factors first to simplify calculation.
ā Division
Steps: 1) Keep first fraction, 2) Change Ć· to Ć, 3) Flip second fraction (reciprocal), 4) Multiply, 5) Simplify
Common Fractions Reference
| Fraction | Decimal | Percent | Common Use |
|---|---|---|---|
| 1/2 | 0.5 | 50% | Half, recipes, measurements |
| 1/3 | 0.333... | 33.3% | One-third, splitting 3 ways |
| 1/4 | 0.25 | 25% | Quarter, 15 minutes, coins |
| 1/5 | 0.2 | 20% | Fifth, discounts, tips |
| 1/8 | 0.125 | 12.5% | Eighth, measurements, music |
| 3/4 | 0.75 | 75% | Three-quarters, 45 minutes |
| 2/3 | 0.666... | 66.7% | Two-thirds, majority |
ā Fraction Tips & Tricks
- āCross-multiply to compare: For a/b vs c/d, compare aĆd vs bĆc
- āSimplify before multiplying: Cancel common factors first to avoid large numbers
- āUse benchmarks: Compare to 0, 1/2, or 1 to estimate reasonableness
- āRemember "Keep, Change, Flip": For division, keep first, change to Ć, flip second
- āConvert mixed to improper: Multiply whole by denominator, add numerator
- āFind LCD efficiently: For small numbers, list multiples; for large, use prime factorization
ā ļø Common Fraction Mistakes
- āAdding denominators: Never add denominators directly - find common denominator first
- āForgetting to simplify: Always reduce to lowest terms for final answer
- āDivision confusion: Don't flip the first fraction, only flip the second (divisor)
- āMixed number operations: Convert to improper fractions first before calculating
- āZero denominator: Never use 0 as denominator - division by zero is undefined
- āSign errors: Track negative signs carefully - one negative makes result negative
š Real-World Fraction Applications
š³ Cooking & Recipes
Scaling recipes up or down, converting measurements (1/2 cup + 1/4 cup = 3/4 cup), adjusting ingredient ratios for different serving sizes.
šØ Construction & Carpentry
Measuring lumber (2Ć4 boards), calculating material needs, reading tape measures (7 1/8 inches), determining spacing between studs or joists.
ā° Time Management
Understanding time (1/4 hour = 15 minutes), calculating work hours (7 1/2 hours), dividing time blocks, planning schedules and deadlines.
š° Finance & Shopping
Calculating discounts (1/3 off sale), splitting bills among friends, understanding stock shares, comparing product sizes and unit prices.
šµ Music Theory
Note durations (whole, half, quarter, eighth notes), time signatures (3/4, 4/4), understanding rhythm patterns and beat divisions in compositions.
š Data & Statistics
Expressing probabilities (1/6 chance), understanding proportions in surveys, representing population segments, calculating risk ratios.
Frequently Asked Questions
How do you add fractions with different denominators?
To add fractions with different denominators: 1) Find a common denominator (usually the least common multiple of both denominators), 2) Convert each fraction to an equivalent fraction with the common denominator, 3) Add the numerators while keeping the denominator the same, 4) Simplify if possible. Example: 1/2 + 1/3 ā Find common denominator (6) ā 3/6 + 2/6 = 5/6. Our calculator does all steps automatically.
How do you multiply fractions?
To multiply fractions: 1) Multiply the numerators together, 2) Multiply the denominators together, 3) Simplify the result. Formula: a/b Ć c/d = (a Ć c)/(b Ć d). Example: 2/3 Ć 3/4 = (2 Ć 3)/(3 Ć 4) = 6/12 = 1/2 (simplified). You can simplify before multiplying by cross-canceling common factors to make calculation easier.
How do you divide fractions?
To divide fractions, multiply by the reciprocal (flip) of the second fraction. Formula: a/b Ć· c/d = a/b Ć d/c = (a Ć d)/(b Ć c). Example: 1/2 Ć· 1/4 = 1/2 Ć 4/1 = 4/2 = 2. Remember the saying 'Keep, Change, Flip' - Keep the first fraction, Change division to multiplication, Flip the second fraction. Then multiply as normal.
How do you simplify fractions?
To simplify a fraction, divide both the numerator and denominator by their Greatest Common Divisor (GCD). Example: 12/18 ā GCD(12, 18) = 6 ā 12Ć·6 / 18Ć·6 = 2/3. To find the GCD, list factors of both numbers or use the Euclidean algorithm. A fraction is fully simplified when the GCD of numerator and denominator is 1 (they share no common factors except 1). Our calculator automatically simplifies all results.
What is a mixed number?
A mixed number combines a whole number and a proper fraction, written like 2 1/2 (two and one-half). It represents an improper fraction (where numerator > denominator) in a more readable form. To convert improper fraction to mixed number: divide numerator by denominator ā whole number is the quotient, remainder becomes the new numerator, denominator stays the same. Example: 7/3 = 2 1/3 (7Ć·3 = 2 remainder 1). To convert mixed to improper: multiply whole number by denominator, add numerator ā 2 1/3 = (2Ć3 + 1)/3 = 7/3.
How do you convert fractions to decimals?
To convert a fraction to decimal, divide the numerator by the denominator. Example: 3/4 = 3 Ć· 4 = 0.75. Some fractions convert to terminating decimals (end after finite digits) like 1/2 = 0.5. Others become repeating decimals like 1/3 = 0.333... (indicated as 0.3Ģ). Common conversions to memorize: 1/2 = 0.5, 1/4 = 0.25, 1/3 ā 0.333, 1/5 = 0.2, 1/8 = 0.125. Our calculator shows the decimal equivalent automatically.
What is the least common denominator (LCD)?
The Least Common Denominator (LCD) is the smallest number that is a multiple of all denominators in a set of fractions. It's used when adding or subtracting fractions with different denominators. The LCD is the Least Common Multiple (LCM) of the denominators. Example: For 1/4 and 1/6, multiples of 4 are (4, 8, 12, 16...) and multiples of 6 are (6, 12, 18...). The LCD is 12. While any common multiple works, using the LCD keeps numbers smaller and simplifies calculations.
Can fractions be negative?
Yes, fractions can be negative. The negative sign can be placed in three positions: -1/2, 1/-2, or -(1/2) - all represent the same value. By convention, we typically write the negative sign in front of the entire fraction or in the numerator, not the denominator. Example: -3/4 (standard) rather than 3/-4. When multiplying or dividing fractions, follow sign rules: negative Ć positive = negative, negative Ć negative = positive. Example: -1/2 Ć 2/3 = -2/6 = -1/3, but -1/2 Ć -2/3 = 2/6 = 1/3.
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