About

Comparing fractions with unlike denominators is a non-trivial arithmetic operation that accounts for a significant share of errors in standardized math assessments. The naive approach of converting to decimals introduces floating-point rounding artifacts. This tool uses cross-multiplication: given fractions ab and cd, it compares a × d against c × b, yielding exact integer comparisons with zero rounding error. It also computes the least common denominator via the Euclidean GCD algorithm and returns both fractions in simplified form.

Limitation: this tool operates on proper and improper fractions with integer components. It does not handle continued fractions or irrational numerators. Denominators of zero are algebraically undefined and are rejected. For mixed numbers, convert to improper fractions first: multiply the whole number by the denominator, add the numerator.

Formulas

The comparison of two fractions ab and cd (where b, d > 0) is performed via cross-multiplication:

ab ? cd ⇔ a × d ? c × b

When a denominator is negative, the tool normalizes by multiplying both numerator and denominator by −1 so that the denominator is always positive before cross-multiplying.

The Greatest Common Divisor is computed using the Euclidean algorithm:

GCD(a, b) =

{

a if b = 0GCD(b, a mod b) otherwise

The Least Common Multiple for finding a common denominator:

LCM(b, d) = |b × d|GCD(b, d)

Where a = numerator of the first fraction, b = denominator of the first fraction, c = numerator of the second fraction, d = denominator of the second fraction.

Reference Data

Fraction Pair	Cross Products	Result	Common Denominator Form	Decimal Approx.
12 vs 13	3 vs 2	12 > 13	36 vs 26	0.5 vs 0.333
34 vs 56	18 vs 20	34 < 56	912 vs 1012	0.75 vs 0.833
25 vs 410	20 vs 20	25 = 410	410 vs 410	0.4 vs 0.4
78 vs 1112	84 vs 88	78 < 1112	2124 vs 2224	0.875 vs 0.917
59 vs 37	35 vs 27	59 > 37	3563 vs 2763	0.556 vs 0.429
−34 vs −23	−9 vs −8	−34 < −23	−912 vs −812	−0.75 vs −0.667
17 vs 213	13 vs 14	17 < 213	1391 vs 1491	0.143 vs 0.154
911 vs 79	81 vs 77	911 > 79	8199 vs 7799	0.818 vs 0.778
1317 vs 1519	247 vs 255	1317 < 1519	247323 vs 255323	0.765 vs 0.789
615 vs 820	120 vs 120	615 = 820	25 vs 25	0.4 vs 0.4
05 vs 11000	0 vs 5	05 < 11000	01000 vs 11000	0 vs 0.001
−12 vs 12	−2 vs 2	−12 < 12	−12 vs 12	−0.5 vs 0.5

Frequently Asked Questions

Cross-multiplication transforms the comparison into an integer problem. Instead of dividing (which produces repeating decimals like 0.333... for 13), the tool computes a × d and c × b, both of which are exact integers. The comparison between integers is always exact in JavaScript for values up to 2⁵³ − 1.

A negative denominator changes the sign of the fraction. The tool normalizes fractions before comparison by ensuring the denominator is always positive. If b < 0, both a and b are multiplied by −1. This is critical because cross-multiplication assumes positive denominators. Without normalization, 1−3 would compare incorrectly.

Yes. Improper fractions (where the numerator exceeds the denominator, e.g. 73) are handled identically since cross-multiplication works on any integers. A fraction with a zero numerator equals 0 regardless of denominator. The tool correctly identifies 05 = 099.

The LCD converts both fractions to equivalent forms with identical denominators, making visual comparison intuitive. For 34 vs 56, the LCD is 12, yielding 912 vs 1012. This is calculated via LCM(b, d) = |b × d| ÷ GCD(b, d).

JavaScript integers are exact up to 9,007,199,254,740,991 (2⁵³ − 1). Since the tool cross-multiplies, the effective safe limit for individual numerators and denominators is approximately 94,906,265 (the square root of 2⁵³). Beyond this, the cross products may lose integer precision. The tool accepts inputs in the range −999,999,999 to 999,999,999.

The visual bars represent the absolute decimal value of each fraction as a percentage of the larger absolute value. The larger fraction always fills 100% of the bar width. The smaller fraction's bar width is computed as (|smaller| ÷ |larger|) × 100%. When fractions are equal, both bars fill to 100%. Negative fractions show bars extending leftward.