Subtract fractions

Subtraction of two fractions with mixed numbers

Fraction Subtraction Calculator

What is calculated?

This calculator subtracts two fractions. Optionally include a whole number part for each fraction to form mixed numbers. The result is simplified automatically.

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

Fraction Subtraction Info

Properties

Subtraction rules:

Common denominator Subtract numerators Simplify Observe signs

Note: For negative mixed numbers the sign applies to the whole value. Subtraction is not commutative: a - b ≠ b - a.

Examples
Simple: 2/3 − 1/6 = 1/2
Mixed: 1 2/3 − 1/6 = 1 1/2
Same denominator: 3/4 − 1/4 = 2/4 = 1/2
Negative: 1/2 − 3/4 = −1/4


Formulas & Rules for Fraction Subtraction

Main formula
\[\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}\] Form common denominator
Same denominator
\[\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}\] Subtract 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
Not commutative
\[\frac{a}{b} - \frac{c}{d} \neq \frac{c}{d} - \frac{a}{b}\] Order matters!
Identity element
\[\frac{a}{b} - 0 = \frac{a}{b} - \frac{0}{c} = \frac{a}{b}\] Subtracting zero changes nothing
Inverse element
\[\frac{a}{b} - \frac{a}{b} = 0\] Subtracting itself yields zero
Signs
\[\frac{a}{b} - \left(-\frac{c}{d}\right) = \frac{a}{b} + \frac{c}{d}\] Minus times minus gives plus

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: Subtract numerators
\[\frac{4}{6} - \frac{1}{6} = \frac{4 - 1}{6} = \frac{3}{6}\]
Step 4: Simplify
\[\frac{3}{6} = \frac{3 \div 3}{6 \div 3} = \frac{1}{2}\] \[\text{gcd}(3, 6) = 3\]
More examples
With mixed numbers:
\[2\frac{1}{4} - \frac{3}{4} = \frac{9}{4} - \frac{3}{4} = \frac{6}{4} = 1\frac{1}{2}\]
Negative result:
\[\frac{1}{4} - \frac{3}{4} = \frac{1-3}{4} = -\frac{2}{4} = -\frac{1}{2}\]
Different denominators:
\[\frac{3}{4} - \frac{1}{3} = \frac{9}{12} - \frac{4}{12} = \frac{5}{12}\]
Important notes
• Subtraction is not commutative
• Observe sign for negative results
• Always use least common denominator
• Simplify the result
General steps for fraction subtraction
1. Find common denominator
2. Expand fractions
3. Subtract numerators
4. Simplify result

These four steps always lead to the correct result.

Applications of fraction subtraction

Fraction subtraction is used in many practical contexts:

Time & Planning
  • Compute remaining time
  • Work time minus breaks
  • Reduce project duration
  • Compare time shares
Quantities & Materials
  • Remaining quantities after consumption
  • Material savings
  • Losses and shrinkage
  • Weight differences
Finance & Business
  • Cost reductions
  • Discounts and allowances
  • Losses and profits
  • Market share losses
Science & Engineering
  • Measurement deviations
  • Tolerance calculations
  • Loss analyses
  • Efficiency comparisons

Mathematical context

Description

Fraction subtraction extends integer subtraction to rational numbers and is a fundamental fraction operation. Unlike addition, subtraction is not commutative, which requires attention to operand order. Subtraction of rational numbers follows the same structural rules as addition but with the sign of the subtrahend reversed.

Summary

Fraction subtraction combines arithmetic precision with algebraic properties and teaches handling non-commutative operations. It is essential for computing differences, losses and deviations in scientific and practical contexts. As a foundation for more complex algebraic operations, it shows how structured procedures lead to correct results even with signed values.




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  •