Divide Fractions

Division of two fractions with mixed numbers

Fraction Division Calculator

What is calculated?

This calculator divides two fractions using the reciprocal method. Optionally, a whole number can be provided for each fraction. The result is simplified automatically.

Enter fractions
Dividend (first fraction)
÷
Divisor (second fraction)
Result
The result is shown as simplified fraction and decimal

Fraction Division Info

Properties

Division rules:

Form reciprocal Then multiply Simplify Division by 0 forbidden

Note: Division is not commutative: a ÷ b ≠ b ÷ a. The divisor must never be 0.

Examples
Simple: ²⁄₃ ÷ ¹⁄₂ = ⁴⁄₃
Mixed: 2²⁄₃ ÷ 1¹⁄₂ = 1⁷⁄₉
Integer divisor: ¾ ÷ 2 = ³⁄₈
Reciprocal: ½ ÷ ⅓ = ³⁄₂

Formulas & Rules for Fraction Division

Main formula
\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\] Reciprocal method
Reciprocal
\[\text{reciprocal of } \frac{c}{d} \text{ is } \frac{d}{c}\] Swap numerator and denominator
Division by integer
\[\frac{a}{b} \div n = \frac{a}{b} \times \frac{1}{n} = \frac{a}{bn}\] Multiply n into denominator
Mixed numbers
\[a\frac{b}{c} = \frac{ac + b}{c}\] Convert to improper fraction
Not commutative
\[\frac{a}{b} \div \frac{c}{d} \neq \frac{c}{d} \div \frac{a}{b}\] Order matters!
Identity element
\[\frac{a}{b} \div 1 = \frac{a}{b} \div \frac{1}{1} = \frac{a}{b}\] Dividing by 1 changes nothing
Inverse element
\[\frac{a}{b} \div \frac{a}{b} = 1\] Self-division yields 1
Division by zero
\[\frac{a}{b} \div 0 = \text{undefined}\] Division by 0 forbidden!

Step-by-step Example

Example: 2²⁄₃ ÷ 1¹⁄₂
Step 1: Convert mixed numbers
\[2\frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{8}{3}\] \[1\frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{3}{2}\]

Convert whole parts to improper fractions.

Step 2: Set up division
\[\frac{8}{3} \div \frac{3}{2}\]
Step 3: Form reciprocal and multiply
\[\frac{8}{3} \div \frac{3}{2} = \frac{8}{3} \times \frac{2}{3} = \frac{8 \times 2}{3 \times 3} = \frac{16}{9}\]
Step 4: Convert to mixed number
\[\frac{16}{9} = 1\frac{7}{9}\] (since 16 ÷ 9 = 1 remainder 7)
More examples
Simple division:
\[\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}\]
Division by integer:
\[\frac{3}{4} \div 2 = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8}\]
With simplification:
\[\frac{6}{8} \div \frac{3}{4} = \frac{6}{8} \times \frac{4}{3} = \frac{24}{24} = 1\]
Important rules
• Division by 0 is impossible
• Reciprocal = swap numerator and denominator
• Division = multiply by reciprocal
• Always simplify the result
General steps for fraction division
1. Convert mixed numbers
2. Form reciprocal of divisor
3. Multiply
4. Simplify result

The reciprocal method turns any fraction division into a multiplication.

Applications of fraction division

Fraction division is used in many practical situations:

Sharing & Portions
  • Cutting cake into equal pieces
  • Number of portions from a total
  • Fabric amounts for cutting
  • Time allocation for tasks
Speed & Efficiency
  • Calculate work rate
  • Determine production rate
  • Consumption per unit
  • Time per operation
Recipes & Mixtures
  • Ingredients for fewer portions
  • Solution concentrations
  • Adjust mixing ratios
  • Calculate dilutions
Finance & Prices
  • Calculate price per unit
  • Determine shares of costs
  • Convert interest rates
  • Currency conversions

Mathematical context

Description

Fraction division extends division to rational numbers using the elegant reciprocal method. By converting any division into multiplication by the reciprocal, a complex operation is reduced to familiar rules. This method showcases the power of mathematical transformations and is fundamental for algebraic operations with rational functions.

Summary

Fraction division demonstrates how mathematical operations can be simplified through clever transformations. The reciprocal method turns any division into a multiplication, making complex computations manageable. As a foundation for algebra, analysis and applied mathematics, fraction division connects abstract concepts with practical problem solving.

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  •