Add, subtract, multiply, and divide fractions with step-by-step solutions. Calculate LCD, simplify fractions, and convert to mixed numbers instantly.
Enter two fractions and click Calculate to see the result with detailed steps
A fractions calculator is an essential mathematical tool that performs arithmetic operations on fractions, providing accurate results with step-by-step solutions. Whether you need to add, subtract, multiply, or divide fractions, our calculator simplifies the process by automatically finding the least common denominator (LCD), performing the operation, and simplifying the result to its lowest terms.
Fractions represent parts of a whole and are fundamental in mathematics, cooking, construction, science, and everyday life. Understanding how to work with fractions is crucial for students, professionals, and anyone dealing with measurements, ratios, or proportions. Our calculator not only provides answers but also teaches the methodology through detailed step-by-step solutions.
A fraction represents a part of a whole or a ratio between two numbers. It consists of two components:
For example, in the fraction 3/4, the numerator is 3 (we have three parts) and the denominator is 4 (the whole is divided into four equal parts). This means we have three out of four parts, or three-quarters.
The numerator is less than the denominator (e.g., 3/4, 2/5, 7/10). The value is always less than 1.
The numerator is greater than or equal to the denominator (e.g., 7/4, 9/5). The value is 1 or greater.
A whole number combined with a proper fraction (e.g., 2 1/3, 5 3/4). Represents improper fractions in a more readable form.
Different fractions that represent the same value (e.g., 1/2 = 2/4 = 4/8). Created by multiplying or dividing both parts by the same number.
Adding fractions requires finding a common denominator. Here's the step-by-step process:
Determine the smallest number that both denominators divide into evenly.
Convert both fractions to equivalent fractions with the LCD as the denominator.
Keep the common denominator and add only the numerators.
Reduce the fraction to its lowest terms by dividing by the GCD.
Step 1: LCD of 4 and 3 = 12
Step 2: Convert: 1/4 = 3/12, 2/3 = 8/12
Step 3: Add: 3/12 + 8/12 = 11/12
Step 4: Result is already in simplest form: 11/12
Subtracting fractions follows the same process as addition, except you subtract the numerators instead of adding them:
Step 1: LCD of 6 and 4 = 12
Step 2: Convert: 5/6 = 10/12, 1/4 = 3/12
Step 3: Subtract: 10/12 − 3/12 = 7/12
Step 4: Result is already in simplest form: 7/12
Multiplying fractions is simpler than adding or subtracting because you don't need to find a common denominator:
Multiply the top numbers together.
Multiply the bottom numbers together.
Reduce to lowest terms.
Step 1: Multiply numerators: 2 × 3 = 6
Step 2: Multiply denominators: 3 × 4 = 12
Step 3: Result: 6/12
Step 4: Simplify: 6/12 = 1/2 (divide both by 6)
Dividing fractions uses the "multiply by the reciprocal" method:
Don't change the first fraction (the dividend).
Flip the second fraction (the divisor) by swapping numerator and denominator.
Follow the multiplication steps.
Reduce to lowest terms.
Step 1: Keep first fraction: 3/4
Step 2: Reciprocal of 2/5 is 5/2
Step 3: Multiply: 3/4 × 5/2 = 15/8
Step 4: Convert to mixed number: 1 7/8
The LCD is the smallest positive number that is a multiple of all denominators. It's essential for adding and subtracting fractions with different denominators.
The GCD (also called GCF - Greatest Common Factor) is the largest positive number that divides both the numerator and denominator evenly. It's used to simplify fractions to their lowest terms.
Simplifying (or reducing) a fraction means writing it in its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). A fraction is in simplest form when the GCD of the numerator and denominator is 1.
Step 1: Find GCD of 24 and 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
Step 2: Divide both numerator and denominator by GCD
24 ÷ 12 = 2
36 ÷ 12 = 3
Result: 24/36 = 2/3
Step 1: Divide numerator by denominator
Step 2: Quotient becomes the whole number
Step 3: Remainder becomes new numerator
Step 4: Keep same denominator
Example: 11/4
11 ÷ 4 = 2 remainder 3
Result: 2 3/4
Step 1: Multiply whole number by denominator
Step 2: Add the numerator to result
Step 3: New number is numerator
Step 4: Keep same denominator
Example: 3 2/5
(3 × 5) + 2 = 17
Result: 17/5
Recipes often use fractions for measurements. If you need to double a recipe calling for 2/3 cup of flour, you multiply: 2/3 × 2 = 4/3 = 1 1/3 cups.
Measuring materials requires fraction arithmetic. If you need three boards each 5 3/4 inches long, you calculate: 3 × 5 3/4 = 17 1/4 inches total.
Hours are divided into fractions. If a task takes 1/4 hour and another takes 1/3 hour, total time is: 1/4 + 1/3 = 3/12 + 4/12 = 7/12 hour (35 minutes).
Budgeting uses fractions. If you save 1/5 of your income for retirement and 1/10 for emergencies, you're saving: 1/5 + 1/10 = 2/10 + 1/10 = 3/10 of your income.
Medication dosages often involve fractions. A prescription might call for 1/2 tablet twice daily, meaning 1/2 × 2 = 1 tablet per day total.
Batting averages, shooting percentages, and win rates are fractions. A baseball player with 45 hits in 120 at-bats has an average of 45/120 = 3/8 = 0.375.
Wrong: 1/2 + 1/3 = 2/5 ❌
Correct: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 ✓
Incomplete: 2/3 × 3/4 = 6/12 (stopping here)
Complete: 2/3 × 3/4 = 6/12 = 1/2 ✓
Wrong: (1/2) ÷ (1/3) = (1÷1)/(2÷3) ❌
Correct: (1/2) ÷ (1/3) = (1/2) × (3/1) = 3/2 ✓
Always remember: numerator is on top, denominator is on bottom. The denominator tells you how many parts the whole is divided into.
Test your understanding with these practice problems. Use our calculator to check your answers!
LCD (Least Common Denominator) and LCM (Least Common Multiple) are the same concept applied to different contexts. LCD specifically refers to the LCM of the denominators when working with fractions. The calculation method is identical.
Find the LCD of both denominators, convert each fraction to an equivalent fraction with the LCD as the denominator, then add the numerators while keeping the denominator the same. Finally, simplify the result if possible.
Dividing by a fraction is the same as multiplying by its reciprocal. This is because division is the inverse operation of multiplication. For example, dividing by 2/3 is the same as multiplying by 3/2 (the reciprocal).
Use improper fractions when performing calculations (they're easier to work with mathematically). Convert to mixed numbers for final answers or when the result needs to be more intuitive to understand. For example, 1 1/2 cups is clearer than 3/2 cups in a recipe.
No, a denominator cannot be zero. Division by zero is undefined in mathematics. A fraction with a zero denominator has no mathematical meaning. However, a numerator can be zero (e.g., 0/5 = 0).
The negative sign can be placed in the numerator, denominator, or in front of the entire fraction (they're all equivalent: -3/4 = 3/-4 = -(3/4)). By convention, we usually keep the denominator positive and place the negative sign in the numerator or in front. When adding or subtracting, treat the negative sign like you would with regular numbers.
Find the GCD (Greatest Common Divisor) of the numerator and denominator, then divide both by that number. The Euclidean algorithm is the fastest method for finding the GCD, especially with large numbers. Our calculator does this automatically for you.
Divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75. Some fractions result in repeating decimals (e.g., 1/3 = 0.333...), while others terminate (e.g., 1/2 = 0.5).
Equivalent fractions are different fractions that represent the same value (e.g., 1/2 = 2/4 = 4/8). They're created by multiplying or dividing both numerator and denominator by the same number. They're crucial for adding and subtracting fractions with different denominators.
Our calculator is 100% accurate for all fraction operations. It uses precise mathematical algorithms to calculate GCD, LCD, perform operations, and simplify results. The step-by-step solutions show exactly how each calculation is performed, allowing you to verify the results and learn the process.
To compare 3/4 and 5/7, cross-multiply: 3×7=21 and 4×5=20. Since 21 > 20, we know 3/4 > 5/7.
Memorize common fractions: 1/2=0.5, 1/4=0.25, 1/3≈0.33, 3/4=0.75. This helps with quick mental math.
For 4/5 × 15/8, simplify first: cancel 4 and 8 (÷4), cancel 5 and 15 (÷5) = 1/1 × 3/2 = 3/2.
Any whole number can be written as a fraction with 1 as denominator: 5 = 5/1. This helps when mixing whole numbers with fractions.
Understanding fractions is a fundamental skill that extends far beyond the classroom. Whether you're scaling a recipe, measuring materials for a project, managing your finances, or helping your children with homework, fraction skills are invaluable. Our Fractions Calculator not only provides instant, accurate results but also teaches you the methodology through detailed step-by-step solutions.
By showing both the LCD method for addition/subtraction and the formula-based approach, our calculator helps you understand multiple solution strategies. The automatic simplification and mixed number conversion features ensure your answers are always in the most usable form.
Remember, practice makes perfect. Use our calculator to check your work, verify your methods, and build confidence in your fraction skills. Whether you're a student learning fractions for the first time, a professional needing quick calculations, or anyone in between, our tool is here to help you succeed.