Fraction Calculator
Calculate and simplify fractions
Result
Pro Tip
The calculator automatically simplifies fractions to their lowest terms by finding the greatest common divisor!
Privacy & Security
Your calculations are completely private. All fraction operations are performed locally in your browser - no data is transmitted, stored, or tracked. Your work remains confidential and secure.
What is a Fraction Calculator?
A fraction calculator is a specialized mathematical tool that performs arithmetic operations on fractions - numbers representing parts of a whole, expressed as one integer (numerator) over another integer (denominator). Unlike decimal calculations where operations follow straightforward rules, fraction arithmetic requires finding common denominators for addition and subtraction, cross-multiplication for division, and simplification to express results in lowest terms. A fraction calculator automates these complex steps, ensuring accurate results while showing the work involved, making it invaluable for students learning fraction operations, professionals needing precise fractional measurements, cooks converting recipe quantities, craftspeople working with standard measurements, and anyone working with parts and ratios. Fractions appear throughout mathematics, science, cooking, carpentry, music, and everyday life - understanding how to manipulate them is a fundamental numeracy skill. The calculator handles proper fractions (numerator smaller than denominator, like 3/4), improper fractions (numerator larger than denominator, like 7/4), and mixed numbers (whole number plus fraction, like 1¾). It performs all four basic operations: addition combines fractions to find the total (2/5 + 1/3 = 11/15), subtraction finds the difference (3/4 - 1/2 = 1/4), multiplication finds fractional parts of fractions (2/3 × 3/4 = 1/2), and division determines how many times one fraction fits into another (3/4 ÷ 1/2 = 3/2 or 1½). Each operation follows specific mathematical rules that the calculator applies automatically. Addition and subtraction require converting fractions to common denominators before combining numerators - for example, 1/3 + 1/4 requires converting to 4/12 + 3/12 = 7/12. Multiplication is simpler - multiply numerators together and denominators together: 2/3 × 3/5 = 6/15, which simplifies to 2/5. Division involves flipping the second fraction (finding its reciprocal) and multiplying: 2/3 ÷ 4/5 becomes 2/3 × 5/4 = 10/12 = 5/6. After each operation, the calculator simplifies results by dividing numerator and denominator by their greatest common divisor (GCD), ensuring the fraction is in lowest terms. Understanding fractions builds foundational mathematical thinking about ratios, proportions, percentages, probability, and algebra.
Key Features
All Four Operations
Add, subtract, multiply, and divide fractions with step-by-step solution display
Automatic Simplification
Results are automatically reduced to lowest terms using greatest common divisor calculation
Mixed Number Support
Input and display results as mixed numbers (like 2¾) or improper fractions (like 11/4)
Common Denominator Finding
Automatically finds least common denominator for addition and subtraction operations
Multiple Fraction Operations
Chain multiple fraction operations together in a single expression
Step-by-Step Solutions
Shows detailed work for each operation, perfect for learning and homework verification
Decimal Conversion
Converts fraction results to decimal equivalents for practical applications
Validation & Error Detection
Identifies invalid fractions like zero denominators and provides helpful error messages
How to Use the Fraction Calculator
Enter First Fraction
Type the numerator and denominator of your first fraction. For mixed numbers, enter the whole number, numerator, and denominator separately. Example: 2 3/4 or 11/4.
Select Operation
Choose whether you want to add (+), subtract (-), multiply (×), or divide (÷) the fractions. The calculator handles each operation according to proper mathematical rules.
Enter Second Fraction
Input the second fraction in the same format as the first. You can use proper fractions, improper fractions, or mixed numbers in any combination.
Calculate Result
Click the equals button or press Enter to compute the result. The calculator performs the operation following standard fraction arithmetic rules.
View Simplified Answer
See the result in simplified form (lowest terms) along with any equivalent representations. Results may display as proper fractions, improper fractions, or mixed numbers.
Review Solution Steps
Examine the step-by-step solution showing how the calculator found common denominators, performed operations, and simplified the result. Perfect for learning or homework checking.
Fraction Calculation Tips
- Always Simplify Final Answers: Reduce fractions to lowest terms after every operation. Teachers expect it, and simplified fractions are easier to understand and use.
- Convert Mixed Numbers for Operations: When multiplying or dividing mixed numbers, convert to improper fractions first. It makes calculations much simpler and less error-prone.
- Find LCD Before Adding/Subtracting: Always find the least common denominator before adding or subtracting fractions with different denominators. Don't add numerators and denominators separately!
- Remember 'Keep-Change-Flip' for Division: When dividing fractions, keep the first fraction, change division to multiplication, and flip the second fraction. This method never fails.
- Use Cross-Cancellation When Multiplying: Before multiplying fractions, look for common factors in any numerator with any denominator and cancel them. This prevents large numbers requiring simplification later.
- Check Answers with Decimals: Convert your fraction answer to decimal and see if it makes sense. If 1/2 + 1/4 gives you 5/8, converting to 0.625 confirms it's reasonable (0.5 + 0.25).
Frequently Asked Questions
How do you add and subtract fractions?
Adding and subtracting fractions requires a common denominator - both fractions must have the same bottom number before you can combine them. If the denominators are already the same, simply add or subtract the numerators and keep the denominator unchanged. For example, 3/7 + 2/7 = 5/7 (add numerators: 3+2=5, keep denominator: 7). Similarly, 5/8 - 1/8 = 4/8 = 1/2 (subtract numerators: 5-1=4, keep denominator: 8, then simplify). When denominators differ, you must first convert both fractions to equivalent fractions with a common denominator. The least common denominator (LCD) is the smallest number divisible by both original denominators. For example, adding 1/3 + 1/4: the LCD of 3 and 4 is 12. Convert 1/3 to 4/12 (multiply numerator and denominator by 4) and 1/4 to 3/12 (multiply numerator and denominator by 3). Now add: 4/12 + 3/12 = 7/12. For subtraction with different denominators, follow the same process: 3/4 - 1/6 requires LCD of 12. Convert: 3/4 = 9/12 and 1/6 = 2/12. Subtract: 9/12 - 2/12 = 7/12. After adding or subtracting, always simplify the result if possible. For instance, 2/6 + 1/6 = 3/6, which simplifies to 1/2 by dividing numerator and denominator by 3. For mixed numbers, either convert to improper fractions first, or add/subtract the whole numbers and fractions separately. For example, 2⅓ + 1¼: convert to improper fractions (7/3 + 5/4), find LCD of 12, convert (28/12 + 15/12 = 43/12), convert back to mixed number (3 7/12). Understanding common denominators is crucial because fractions represent parts of a whole - you can only combine parts measured in the same-sized units. Adding 1/3 (one-third) to 1/4 (one-fourth) requires expressing both in common units (twelfths), just as you can't add 2 meters to 3 feet without first converting to common units.
How do you multiply and divide fractions?
Multiplying fractions is more straightforward than addition or subtraction - you simply multiply numerators together and denominators together, then simplify. For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2 (simplified by dividing numerator and denominator by 6). No common denominator is needed for multiplication. For another example: 3/5 × 2/7 = 6/35 (already in simplest form since 6 and 35 share no common factors). When multiplying with whole numbers, treat the whole number as a fraction over 1: 5 × 2/3 = 5/1 × 2/3 = 10/3 = 3⅓. For mixed numbers, convert to improper fractions first: 2½ × 1⅓ becomes 5/2 × 4/3 = 20/6 = 10/3 = 3⅓. A useful shortcut when multiplying is cross-cancellation (simplifying before multiplying): for 4/9 × 3/8, notice that 4 and 8 share a factor of 4, and 3 and 9 share a factor of 3. Simplify: 4÷4=1, 8÷4=2, 3÷3=1, 9÷3=3, giving 1/3 × 1/2 = 1/6. This prevents working with large numbers that need simplification later. Dividing fractions uses the 'keep-change-flip' or 'multiply by the reciprocal' method: keep the first fraction, change division to multiplication, flip the second fraction (swap its numerator and denominator). For example, 3/4 ÷ 1/2 becomes 3/4 × 2/1 = 6/4 = 3/2 = 1½. The logic is that dividing by 1/2 is the same as asking 'how many halves fit into 3/4?' - the answer is 1½ halves. Another example: 2/5 ÷ 3/7 becomes 2/5 × 7/3 = 14/15. For whole number division: 6 ÷ 2/3 becomes 6/1 × 3/2 = 18/2 = 9 (which makes sense: if you divide something into thirds, you get more pieces; 6 divided into two-thirds-sized pieces yields 9 pieces). For mixed numbers in division, convert to improper fractions first: 2¼ ÷ 1½ becomes 9/4 ÷ 3/2, then 9/4 × 2/3 = 18/12 = 3/2 = 1½. Understanding fraction multiplication and division is essential for solving ratio problems, scaling recipes, calculating portions, and working with rates and proportions.
How do you simplify fractions to lowest terms?
Simplifying (or reducing) a fraction means expressing it in lowest terms where the numerator and denominator have no common factors other than 1, making the fraction as simple as possible while representing the same value. The process involves finding the greatest common divisor (GCD) of the numerator and denominator, then dividing both by that GCD. For example, to simplify 12/18: find factors of 12 (1,2,3,4,6,12) and 18 (1,2,3,6,9,18); the greatest common factor is 6. Divide both numerator and denominator by 6: 12÷6=2, 18÷6=3, giving 2/3. The fraction 12/18 and 2/3 are equivalent (represent the same value), but 2/3 is in lowest terms. For another example, simplify 45/60: GCD of 45 and 60 is 15. Divide both by 15: 45÷15=3, 60÷15=4, giving 3/4. A quick simplification method is to repeatedly divide by small common factors you notice: 45/60 is even, so divide both by 2 giving 22.5/30 (wait, must use common factors that divide evenly), or notice both end in 5, so both divisible by 5: 9/12, then both divisible by 3: 3/4. Another approach is the Euclidean algorithm for finding GCD, though this is typically automated in calculators. You can verify simplification is complete by checking whether the numerator and denominator share any common factors - if they're coprime (only common factor is 1), the fraction is fully simplified. Some fractions are already in lowest terms: 3/7, 5/8, 7/13 - their numerators and denominators share no common factors. Improper fractions can also be simplified: 12/8 simplifies to 3/2 (GCD of 4), which can be expressed as mixed number 1½. When adding, subtracting, multiplying, or dividing fractions, always simplify the final answer. Simplified fractions are easier to understand, compare, and use in further calculations. In practical contexts, simplified fractions often have intuitive meaning: 2/3 is immediately recognizable as 'two-thirds,' while 14/21 (equivalent but unsimplified) is less intuitive. Teachers expect homework answers in simplified form, and standardized tests typically require it. Learning to simplify fractions builds number sense and understanding of factors, divisibility, and equivalent representations.
What is the difference between proper fractions, improper fractions, and mixed numbers?
These three forms represent fractional quantities differently, each with advantages in different contexts. A proper fraction has a numerator (top number) smaller than its denominator (bottom number), resulting in a value less than 1. Examples include 1/2, 3/4, 7/8, 5/9 - all representing parts of a whole. Proper fractions are intuitive for representing portions: 3/4 of a pizza, 5/8 of an inch, 2/3 of a cup. An improper fraction has a numerator equal to or greater than its denominator, resulting in a value equal to or greater than 1. Examples include 5/4, 7/3, 11/5, 8/8. While called 'improper,' these fractions are mathematically valid and often easier for calculations. The value 5/4 represents five quarters, or 1¼. Improper fractions are preferred in algebra and many calculations because they maintain single-fraction form without the complexity of mixed numbers. A mixed number (or mixed fraction) combines a whole number with a proper fraction, expressing values greater than 1 in a form that shows the whole and fractional parts separately. Examples include 1¼, 2⅔, 5⅞. Mixed numbers are intuitive for measurements and practical quantities: a recipe calls for 2¾ cups of flour, or a board measures 3½ feet long. Converting between forms: to convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place over the original denominator. For example, 2¾ = (2×4+3)/4 = 11/4. To convert an improper fraction to a mixed number, divide the numerator by the denominator - the quotient becomes the whole number, the remainder becomes the new numerator, and the denominator stays the same. For example, 17/5: 17÷5 = 3 remainder 2, so 17/5 = 3⅖. When to use each form: proper fractions for values less than 1; improper fractions for calculations (easier to multiply, divide, add, subtract) and algebra; mixed numbers for practical applications, measurements, and contexts where whole units are meaningful. Many fraction calculators work internally with improper fractions but display results as mixed numbers for readability. Understanding all three forms and how to convert between them is essential for fraction fluency and practical mathematics.
How do you find the least common denominator?
The least common denominator (LCD), also called the least common multiple (LCM) of the denominators, is the smallest positive number divisible by all denominators involved in a fraction operation. Finding the LCD is essential for adding or subtracting fractions with different denominators. Several methods exist for finding LCD. The listing multiples method lists multiples of each denominator until you find the smallest common multiple. For denominators 4 and 6: multiples of 4 are 4,8,12,16,20... and multiples of 6 are 6,12,18,24...; the first common multiple is 12, so LCD = 12. This method works well for small numbers but becomes tedious with large denominators. The prime factorization method finds the LCD by breaking each denominator into prime factors, then multiplying each prime factor the maximum number of times it appears in any denominator. For example, to find LCD of 12 and 18: 12 = 2²×3 and 18 = 2×3²; take the highest power of each prime: 2² and 3², giving LCD = 4×9 = 36. For denominators 8, 12, and 15: 8=2³, 12=2²×3, 15=3×5; maximum powers are 2³, 3¹, 5¹, so LCD = 8×3×5 = 120. The greatest common divisor method uses the formula LCD(a,b) = (a×b)/GCD(a,b) where GCD is the greatest common divisor. For denominators 6 and 9: GCD(6,9) = 3, so LCD = (6×9)/3 = 54/3 = 18. This method is efficient for two denominators and is easily extended to more denominators by finding LCD of the first two, then finding LCD of that result and the third denominator, and so on. For three or more denominators, you can find LCD pairwise: LCD(4,6,9) = LCD(LCD(4,6),9) = LCD(12,9) = 36. In practice, once you find the LCD, you convert each fraction to an equivalent fraction with the LCD as denominator. For adding 1/4 + 1/6 with LCD=12: convert 1/4 to 3/12 (multiply numerator and denominator by 3) and 1/6 to 2/12 (multiply numerator and denominator by 2), then add: 3/12 + 2/12 = 5/12. Understanding LCD is crucial not just for fraction arithmetic but also for comparing fractions, solving equations with fractions, and working with rational expressions in algebra. While calculators automate LCD finding, understanding the concept builds number sense and mathematical reasoning.
How are fractions used in everyday life?
Fractions appear throughout daily life in contexts ranging from cooking and construction to time management and financial decisions, making fraction literacy a practical essential skill. In cooking and baking, recipes use fractions extensively: 2½ cups flour, ¾ teaspoon salt, ⅓ cup sugar. When scaling recipes (doubling a recipe or halving it), you multiply or divide fractions: doubling a recipe requiring ¾ cup milk means calculating ¾ × 2 = 3/2 = 1½ cups. In construction and carpentry, measurements use fractions of inches: cutting a board 3⅝ inches wide, or drilling a hole 5/16 inches in diameter. Carpenters add fractions when calculating total lengths: 2¾ inches + 1⅝ inches + 3½ inches for total board length. In time management, fractions represent parts of hours: ¼ hour = 15 minutes, ½ hour = 30 minutes, ¾ hour = 45 minutes. Calculating time durations often involves fractions. In finance, fractions appear in interest rates (3¾% APR), stock prices (historically in eighths and sixteenths, now decimalized but conceptually fractional), and portions of currency (½ dollar, ¼ dollar). In shopping, discounts involve fractions: ⅓ off means paying ⅔ of original price; figuring sale prices requires fraction calculations. In fuel and distance calculations, ⅞ tank of gas or 2⅓ gallons used involve fractions. In medicine, drug dosages use fractions: ½ tablet, or 1½ teaspoons of liquid medicine. In probability and statistics, fractions express chances and portions: ⅙ chance of rolling a particular number on a die, or understanding that 3/10 of survey respondents equals 30%. In music, note durations are fractional: whole notes, half notes (½), quarter notes (¼), eighth notes (⅛), sixteenth notes (1/16) represent fractions of a measure's duration. In map reading and scale, fractions represent ratios: a map scale of 1:24000 means 1 inch represents 24000 inches in reality, involving fractional calculations for distance. In sports statistics, batting averages, completion percentages, and shooting percentages are often expressed as fractions or their decimal equivalents. In gardening and landscaping, spacing recommendations use fractions: plant bulbs 6½ inches apart, or space rows 2¾ feet apart. Understanding fractions enables accurate recipe execution, precise construction, correct medication dosing, smart financial decisions, and countless other practical activities that make up everyday life.
What are equivalent fractions and why are they important?
Equivalent fractions are different fractions that represent the same value or quantity, even though their numerators and denominators differ. For example, 1/2, 2/4, 3/6, 4/8, 5/10, and 50/100 are all equivalent - they all represent half of a whole. You create equivalent fractions by multiplying (or dividing) both the numerator and denominator by the same non-zero number, which maintains the fraction's value. Multiplying 1/2 by 2/2 (which equals 1, so doesn't change the value) gives 2/4; multiplying by 3/3 gives 3/6, and so on. Similarly, dividing both numerator and denominator of 6/8 by 2 gives 3/4; dividing by 3 would give 2/2.67 (not a whole number, so not valid for creating equivalent fractions with integer numerators and denominators). Equivalent fractions are essential for several reasons. First, they enable adding and subtracting fractions with different denominators - you convert fractions to equivalent forms with common denominators before combining. For example, 1/3 + 1/4 requires converting to equivalent fractions 4/12 + 3/12 = 7/12. Second, equivalent fractions allow comparing fractions: which is larger, 3/5 or 5/8? Convert to common denominator 40: 24/40 vs 25/40, showing 5/8 is slightly larger. Third, simplifying fractions to lowest terms uses equivalent fractions in reverse - dividing numerator and denominator by common factors to find the simplest equivalent form. Fourth, equivalent fractions demonstrate a fundamental mathematical principle: different representations can express the same value, crucial for algebraic thinking. The concept of equivalent fractions extends to proportions and ratios used in scaling, mixing, and rate problems: a recipe ratio of 2 cups flour to 3 cups liquid (2/3) is equivalent to 4 cups flour to 6 cups liquid (4/6) when doubling the recipe. Understanding equivalence helps recognize that 50/100 = ½ = 0.5 = 50% - all equivalent representations in different formats. Visual models help understand equivalence: a circle divided into 2 parts with 1 shaded (1/2) shows the same shaded area as a circle divided into 4 parts with 2 shaded (2/4). Equivalent fractions are foundational to fraction arithmetic, proportion reasoning, unit conversion, and algebraic manipulation - mastering them is essential for mathematical fluency.
How do you convert fractions to decimals and percentages?
Converting between fractions, decimals, and percentages allows flexible representation of quantities in whichever format is most useful for a given context. To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. Some fractions produce terminating decimals (finite number of digits): 1/2 = 0.5, 3/8 = 0.375, 7/20 = 0.35. Others produce repeating decimals (infinite repeating pattern): 1/3 = 0.333... (written as 0.3̄ with bar over repeating digit), 2/9 = 0.222... = 0.2̄, 1/7 = 0.142857142857... = 0.1̄42857̄. For mixed numbers, convert the whole number and fraction separately: 2¾ = 2 + 0.75 = 2.75. To convert a fraction to a percentage, first convert to decimal, then multiply by 100 (or move decimal point two places right). For example, 3/4 = 0.75 = 75%. Or use the direct method: divide numerator by denominator and multiply by 100: (3÷4)×100 = 75%. For 7/8: (7÷8)×100 = 87.5%. Some common fraction-percentage equivalents are worth memorizing: 1/2 = 50%, 1/4 = 25%, 3/4 = 75%, 1/5 = 20%, 1/10 = 10%, 1/3 ≈ 33.33%, 2/3 ≈ 66.67%, 1/8 = 12.5%. Converting from decimal to fraction involves writing the decimal digits as numerator over the appropriate power of 10, then simplifying. For example, 0.6 = 6/10 = 3/5 (simplified); 0.75 = 75/100 = 3/4 (simplified); 0.125 = 125/1000 = 1/8 (simplified). For repeating decimals, the process is more complex: 0.333... = 1/3, 0.666... = 2/3. Converting from percentage to fraction means writing the percentage as numerator over 100, then simplifying: 40% = 40/100 = 2/5; 85% = 85/100 = 17/20; 12.5% = 12.5/100 = 125/1000 = 1/8. These conversions are essential across many applications: percentages are preferred for proportions, discounts, interest rates, and statistics (75% of people...); decimals are used in measurements, money ($2.75), and calculations; fractions are used in cooking (¾ cup), construction (⅝ inch), and contexts requiring exact values without rounding. Understanding all three formats and converting fluently between them provides mathematical flexibility and enables appropriate format selection for each context.
Why Use Our Fraction Calculator?
Fraction arithmetic involves complex rules for finding common denominators, cross-multiplication, and simplification that are easy to get wrong. Our fraction calculator handles all these steps automatically while showing the work, making it perfect for students checking homework, cooks converting recipes, craftspeople calculating measurements, or anyone needing accurate fraction operations. With automatic simplification, mixed number support, step-by-step solutions, and instant results, you'll solve fraction problems confidently and learn the underlying mathematics. No registration required - just enter your fractions and get simplified, accurate results immediately.