Add Fractions

Addition of two fractions with mixed numbers

Fraction Addition Calculator

What is calculated?

This calculator adds two fractions. Optionally, each fraction can include a whole number to create mixed numbers. The result is automatically simplified.

Enter fractions
+
Result
The result is shown as simplified fraction and decimal

Fraction Addition Info

Properties

Addition rules:

Common denominator Add numerators Simplify Mixed numbers

Note: For negative mixed numbers the sign applies to the whole number. Denominator must not be 0.

Examples
Simple: ²⁄₃ + ¹⁄₆ = ⁵⁄₆
Mixed: 1²⁄₃ + ¹⁄₆ = 1⁵⁄₆
Same denominator: ¹⁄₄ + ¹⁄₄ = ²⁄₄ = ¹⁄₂
Negative: -¹⁄₂ + ¹⁄₄ = -¹⁄₄

Formulas & Rules for Fraction Addition

Main formula
\[\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\] Find common denominator
Same denominator
\[\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}\] Add numerators
Mixed numbers
\[a\frac{b}{c} = \frac{ac + b}{c}\] Convert to improper fraction
Simplify
\[\frac{a \cdot k}{b \cdot k} = \frac{a}{b}\] Divide by gcd
Commutative
\[\frac{a}{b} + \frac{c}{d} = \frac{c}{d} + \frac{a}{b}\] Order doesn't matter
Associative
\[\left(\frac{a}{b} + \frac{c}{d}\right) + \frac{e}{f} = \frac{a}{b} + \left(\frac{c}{d} + \frac{e}{f}\right)\] Grouping is flexible
Identity element
\[\frac{a}{b} + 0 = \frac{a}{b} + \frac{0}{c} = \frac{a}{b}\] Adding zero changes nothing
Signs
\[-\frac{a}{b} + \frac{c}{d} = \frac{-ad + bc}{bd}\] Watch negative fractions

Step-by-step Example

Example: ²⁄₃ + ¹⁄₆
Step 1: Find common denominator
\[\text{lcm}(3, 6) = 6\]

The least common denominator of 3 and 6 is 6.

Step 2: Expand fractions
\[\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}\] \[\frac{1}{6} = \frac{1}{6}\] (already correct)
Step 3: Add numerators
\[\frac{4}{6} + \frac{1}{6} = \frac{4 + 1}{6} = \frac{5}{6}\]
Step 4: Check simplification
\[\text{gcd}(5, 6) = 1\] The fraction ⁵⁄₆ is already in simplest form.
More examples
With mixed numbers:
\[1\frac{2}{3} + \frac{1}{6} = \frac{5}{3} + \frac{1}{6} = \frac{10}{6} + \frac{1}{6} = \frac{11}{6} = 1\frac{5}{6}\]
Same denominator:
\[\frac{1}{4} + \frac{3}{4} = \frac{1+3}{4} = \frac{4}{4} = 1\]
With simplification:
\[\frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}\]
Tips
• Always use the least common denominator
• Simplify the result after addition
• Convert mixed numbers to improper fractions
• Watch signs for negative numbers
General steps for fraction addition
1. Find lcd
2. Expand fractions
3. Add numerators
4. Simplify result

These four steps always lead to the correct result.

Applications of fraction addition

Fraction addition is used in many practical areas:

Cooking & Kitchen
  • Summing recipe quantities
  • Ingredients with different units
  • Adjusting portion sizes
  • Using leftovers
Craft & Engineering
  • Calculating material amounts
  • Adding measurements and lengths
  • Tolerances and deviations
  • Accounting for waste
Math & Education
  • Foundation of fraction arithmetic
  • Understanding decimal fractions
  • Algebraic fraction expressions
  • Probability calculations
Everyday & Finance
  • Adding time portions
  • Share calculations
  • Discounts and percentages
  • Splitting costs

Mathematical context

Description

Fraction addition is a fundamental operation in arithmetic that extends addition of whole numbers to rational numbers. It preserves commutativity and associativity while requiring a common denominator. In higher mathematics, fraction addition underpins addition in the field of rational numbers.

Summary

Fraction addition combines arithmetic basics with practical applications. The procedure "find lcd - expand - add - simplify" ensures correct results and prepares for more advanced algebraic operations. Exact representation of parts is crucial in scientific, technical and everyday calculations.

More Fraction Calculators

Add Fractions  •  Compare Fractions  •  Decimal to Fraction  •  Divide Fractions  •  Fraction to Decimal  •  Fraction to Percent  •  Egyptian Fraction  •  Mixed number to improper Fraction  •  Multiply Fractions  •  Percent to Fraction  •  Simplify Fraction  •  Subtract Fractions  •