Compare Fractions

Compare two fractions or a fraction with a decimal number

Fraction Comparison Calculator

What is calculated?

This tool compares two fractions and determines which one is larger. You can also compare a fraction with a decimal number. Optionally, whole numbers can be provided for both fractions to form mixed numbers.

Enter fractions
First fraction
?

compare
Second fraction
Comparison result
Fractions are compared using cross-multiplication

Fraction Comparison Info

Properties

Comparison methods:

Cross-multiplication Common denominator Decimal conversion Mixed numbers

Note: For cross-multiplication compare a/b with c/d by a×d ? b×c.

Examples
Simple: ²⁄₃ > ¹⁄₂
Equal: ²⁄₄ = ¹⁄₂
Mixed: 1¹⁄₂ > ⁴⁄₃
Decimal: ³⁄₄ = 0.75


Formulas & Rules for Fraction Comparison

Cross-multiplication
\[\frac{a}{b} \text{ ? } \frac{c}{d}\] \[a \cdot d \text{ ? } b \cdot c\] Multiply across
Common denominator
\[\frac{a}{b} = \frac{a \cdot k}{b \cdot k}\] \[\frac{c}{d} = \frac{c \cdot j}{d \cdot j}\] Make same denominator
Decimal comparison
\[\frac{a}{b} = a \div b\] \[\frac{c}{d} = c \div d\] As decimal numbers
Comparison symbols
\[< \text{ smaller}\] \[= \text{ equal}\] \[> \text{ greater}\] Result symbols
Same denominators
\[\frac{a}{c} \text{ ? } \frac{b}{c}\] \[a \text{ ? } b\] Compare numerators only
Same numerators
\[\frac{a}{b} \text{ ? } \frac{a}{c}\] \[b \text{ ? } c \text{ (reversed)}\] Larger denominator = smaller fraction
Negative fractions
\[-\frac{a}{b} < 0 < \frac{c}{d}\] Negative fractions are smaller
Mixed numbers
\[n\frac{a}{b} = \frac{nb + a}{b}\] Convert to improper fraction first

Step-by-step Examples

Example 1: Compare ²⁄₃ and ³⁄₄
Method 1: Cross-multiplication
\[\frac{2}{3} \text{ ? } \frac{3}{4}\] \[2 \times 4 \text{ ? } 3 \times 3\] \[8 \text{ ? } 9\] \[8 < 9 \Rightarrow \frac{2}{3} < \frac{3}{4}\]
Method 2: Common denominator
\[\text{lcm}(3,4) = 12\] \[\frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12}\] \[\frac{8}{12} < \frac{9}{12}\]
Method 3: Decimal comparison
\[\frac{2}{3} = 0.6667\] \[\frac{3}{4} = 0.75\] \[0.6667 < 0.75\]
Example 2: Mixed numbers 1²⁄₃ and ⁴⁄₃
Step 1: Convert mixed number
\[1\frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{5}{3}\]
Step 2: Compare
\[\frac{5}{3} \text{ ? } \frac{4}{3}\] \[5 > 4 \Rightarrow \frac{5}{3} > \frac{4}{3}\] \[1\frac{2}{3} > \frac{4}{3}\]
Comparison tricks
Same denominator:
²⁄₅ vs ³⁄₅ → compare numerators only
Same numerator:
²⁄₃ vs ²⁄₅ → ²⁄₃ > ²⁄₅
Compare with 1:
⁵⁄₄ > 1 > ³⁄₄
Compare with ½:
³⁄₅ > ½ > ²⁄₅
Practical application
Pizza example:
²⁄₃ of a pizza vs ³⁄₄ of a pizza
→ ³⁄₄ is more!

Time example:
¹⁄₄ hour vs ¹⁄₃ hour
→ ¹⁄₃ hour is longer
Negative mixed numbers
\[-2\frac{2}{3}\] is interpreted as \[-(2\frac{2}{3})\]
The sign applies to the whole value.
Steps of fraction comparison
1. Convert mixed numbers
2. Choose comparison method
3. Calculate
4. Set comparison symbol

Cross-multiplication is usually the fastest method for comparing fractions.

Applications of fraction comparison

Fraction comparison is important in many everyday contexts:

Cooking & Household
  • Compare recipe quantities
  • Choose larger portions
  • Ingredient ratios
  • Estimate leftovers
Shopping & Prices
  • Price-performance comparison
  • Compare discounts
  • Evaluate larger packages
  • Assess offers
Time & Planning
  • Compare time portions
  • Evaluate work hours
  • Allocate breaks
  • Project planning
Education & Assessment
  • Compare grades
  • Performance shares
  • Success rates
  • Statistical analyses

Mathematical context

Description

Fraction comparison is a fundamental operation in mathematics establishing the order relation of rational numbers. It allows placing rational numbers on the number line and is essential for understanding magnitudes. Cross-multiplication uses fraction equivalence and demonstrates how algebraic manipulation leads to geometric understanding.

Summary

Fraction comparison connects abstract mathematical concepts with practical application. It underpins sorting algorithms, optimization problems, and statistical analysis. Understanding fraction comparison develops numerical intuition and logical reasoning. From everyday choices to scientific calculations, comparing fractions is an indispensable tool for quantitative thinking.




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  •