Multiply fractions

Multiplication of two fractions with mixed numbers

Fraction Multiplication Calculator

What is calculated?

This calculator multiplies two fractions using the direct numerator×numerator, denominator×denominator method. Optionally include a whole number for each fraction. The result is simplified automatically.

Enter fractions
First factor
×
Second factor
Result
The result is shown as simplified fraction and decimal

Fraction Multiplication Info

Properties

Multiplication rules:

Numerator × Numerator Denominator × Denominator Simplify Commutative

Note: Multiplication is commutative: a × b = b × a. Simplifying before multiplication reduces work.

Examples
Simple: 2/3 × 1/6 = 2/18 = 1/9
Mixed: 2 2/3 × 1 1/2 = 4
With integer: 3/4 × 4 = 3
Simplify: 6/8 × 4/9 = 1/3

Formulas & Rules for Fraction Multiplication

Main formula
\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\] Direct multiplication
With integer
\[\frac{a}{b} \times n = \frac{a \times n}{b}\] Integer in numerator
Mixed numbers
\[a\frac{b}{c} = \frac{ac + b}{c}\] Convert to improper fraction
Simplify before multiply
\[\frac{a}{b} \times \frac{c}{d} = \frac{a}{d} \times \frac{c}{b}\] Cross-cancellation possible
Commutativity
\[\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}\] Order doesn't matter
Associativity
\[\left(\frac{a}{b} \times \frac{c}{d}\right) \times \frac{e}{f} = \frac{a}{b} \times \left(\frac{c}{d} \times \frac{e}{f}\right)\] Parentheses shiftable
Identity element
\[\frac{a}{b} \times 1 = \frac{a}{b} \times \frac{1}{1} = \frac{a}{b}\] Multiplication by 1
Signs
\[(-\frac{a}{b}) \times \frac{c}{d} = -\frac{ac}{bd}\] Observe sign rules

Step-by-step example

Example: 2 2/3 × 1 1/2
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 wholes to improper fractions.

Step 2: Set up multiplication
\[\frac{8}{3} \times \frac{3}{2}\]
Step 3: Simplify beforehand (optional)
\[\frac{8}{3} \times \frac{3}{2} = \frac{8 \times 3}{3 \times 2} = \frac{8}{2} = 4\] (The 3 cancels out)
Step 4: Verify result
\[\frac{24}{6} = 4\] The result is an integer: 4
More examples
Simple multiplication:
\[\frac{2}{3} \times \frac{1}{4} = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}\]
With integer:
\[\frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2}\]
Cross-cancellation:
\[\frac{6}{8} \times \frac{4}{9} = \frac{6 \times 4}{8 \times 9} = \frac{24}{72} = \frac{1}{3}\]
Practical tips
• Simplifying before multiplication saves steps
• Cross-cancellation is allowed
• Represent integers as fraction with denominator 1
• Always fully simplify the result
General steps for fraction multiplication
1. Convert mixed numbers
2. Simplify beforehand (optional)
3. Multiply
4. Simplify result

Fraction multiplication is the simplest fraction operation — numerator×numerator, denominator×denominator.

Applications of fraction multiplication

Fraction multiplication is used in many practical contexts:

Recipes & Cooking
  • Ingredient amounts for multiple portions
  • Calculate proportion of an ingredient
  • Concentration of spices
  • Adjust nutritional values proportionally
Craft & Construction
  • Calculate areas and volumes
  • Apply scales
  • Estimate material usage
  • Proportional scaling
Percentages & Shares
  • Compute a portion of a portion
  • Discounts and markups
  • Combine probabilities
  • Interest and returns
Science & Engineering
  • Physical calculations
  • Chemical reactions
  • Scaling factors
  • Measurement accuracy and tolerances

Mathematical context

Description

Fraction multiplication is the most direct and intuitive fraction operation. It extends integer multiplication seamlessly to rational numbers while preserving algebraic properties: commutativity, associativity, and distributivity. The simple rule "numerator×numerator, denominator×denominator" makes calculations with proportions manageable and underpins more advanced mathematical concepts.

Summary

Fraction multiplication combines mathematical elegance with practical applicability. As the only fraction operation that does not require a common denominator, it highlights the beauty of mathematical structure. From kitchen chemistry to quantum physics, it enables precise calculations with proportions, probabilities and scaling factors. Its simplicity belies its foundational role in algebra, analysis, and applied mathematics.

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  •