Comprehensive fraction calculator for addition, subtraction, multiplication, and division of fractions. Includes mixed numbers, decimal conversion, simplification, and step-by-step solutions.
Fractions are fundamental mathematical concepts representing parts of a whole. They consist of a numerator (top number) indicating how many parts you have, and a denominator (bottom number) showing how many equal parts make up the whole.
Our comprehensive fraction calculator handles all basic operations, mixed numbers, decimal conversions, and provides step-by-step solutions to help you understand the process.
Understanding different types of fractions is essential for mastering fraction arithmetic and real-world applications.
Each fraction operation follows specific rules and procedures that ensure accurate results.
Addition and Subtraction: Fractions must have the same denominator before adding or subtracting. Find the least common denominator (LCD), convert both fractions, then add or subtract the numerators while keeping the denominator the same.
Multiplication: Multiply numerators together and denominators together. The operation is straightforward: (a/b) × (c/d) = (ac)/(bd). Always simplify the result to its lowest terms.
Division: Multiply the first fraction by the reciprocal (inverse) of the second fraction. To divide (a/b) ÷ (c/d), calculate (a/b) × (d/c) = (ad)/(bc).
When adding or subtracting fractions with different denominators, you must first find a common denominator:
Least Common Multiple (LCM) Method: Find the smallest number that both denominators divide into evenly. This produces the simplest equivalent fractions.
Cross-Multiplication Method: Multiply the numerator and denominator of each fraction by the denominator of the other fraction. This always works but may not give the simplest form.
Simplifying fractions reduces them to their lowest terms by dividing both numerator and denominator by their greatest common factor (GCF):
Finding the GCF: List the factors of both numbers and identify the largest number that divides both evenly. For example, with 12/16, the GCF is 4, so 12/16 = 3/4.
Prime Factorization Method: Break down both numbers into prime factors and multiply the common factors to find the GCF.
Converting between fractions, decimals, and mixed numbers is essential for different mathematical contexts:
Fraction to Decimal: Divide the numerator by the denominator using long division or a calculator. For example, 3/4 = 3 ÷ 4 = 0.75.
Decimal to Fraction: Write the decimal as a fraction with a power of 10 as the denominator, then simplify. For example, 0.6 = 6/10 = 3/5.
Mixed Number Conversions: To convert to improper: multiply the whole number by the denominator and add the numerator. To convert from improper: divide the numerator by the denominator.
Cooking and Recipes: Adjusting ingredient quantities, scaling recipes up or down, and measuring partial portions.
Construction and Carpentry: Measuring materials, calculating dimensions, and working with architectural plans that use fractional measurements.
Finance: Calculating partial payments, interest rates, and proportional distributions in budgeting and investment scenarios.
Time Management: Breaking down tasks into fractional time periods and calculating partial completion percentages.
Adding Denominators: Never add denominators when adding fractions. Only add numerators after finding a common denominator.
Cross-Multiplication Errors: When finding common denominators, ensure you multiply both the numerator and denominator of each fraction.
Improper Simplification: Always check if your final answer can be simplified further by finding the GCF.
Division Confusion: Remember to multiply by the reciprocal when dividing fractions, not the original second fraction.
Complex Fractions: Fractions that contain fractions in the numerator, denominator, or both. Simplify by treating the main fraction line as a division operation.
Unit Fractions: Fractions with numerator 1 (e.g., 1/2, 1/3, 1/7). These are building blocks for understanding all other fractions.
Equivalent Fractions: Different fractions that represent the same value (e.g., 1/2, 2/4, 3/6). Understanding equivalence is crucial for fraction operations.