Equivalent Fractions Calculator

Background.

This equivalent fractions calculator does something Quanta's general fraction calculator has no concept of at all: given a single fraction, it generates an entire family of equal-valued fractions — the ×2, ×3, ×4, and ×5 multiples — and it can solve for the exact equivalent fraction at any denominator you name. Enter 3/4 with a target denominator of 20 and it returns 6/8, 9/12, 12/16, and 15/20 as the generated set, plus targetNumerator = 15 confirming that 3/4 and 15/20 are the same value written two different ways.

Quanta's fraction calculator (math/fraction) performs binary arithmetic — add, subtract, multiply, divide — on two fractions you supply; it has no single-fraction 'show me equivalents' mode, no scale-factor output, and no way to solve 'what is 3/4 over a denominator of 20?' That is an entirely different question from arithmetic, and this calculator answers it directly.

What makes two fractions equivalent? Multiplying (or dividing) both the numerator and denominator of a fraction by the same non-zero number never changes its value, because you are really just multiplying by k/k, which always equals 1. 3/4 and 6/8 are equivalent because 6/8 = (3×2)/(4×2) = 3/4 × 2/2 = 3/4 × 1 = 3/4. This is the same principle behind the least common denominator used to add fractions, behind reducing a fraction to lowest terms (which is exactly this process run in reverse — dividing instead of multiplying), and behind converting a fraction to a percentage or a different unit denominator.

This calculator does the reduction first — finding the fraction's lowest-terms form via the Euclidean algorithm — because every equivalent fraction, including the one you started with, is a multiple of that unique reduced form. From there, generating the ×2 through ×5 set is direct multiplication, and solving for a specific target denominator is one division (to find the scale factor) followed by one multiplication (to find the matching numerator). If the target denominator you name is not an exact multiple of the reduced denominator, no equivalent fraction with a whole-number numerator exists at that denominator, and the calculator says so plainly rather than returning a fractional numerator that defeats the purpose of a whole-number fraction.

This calculator is built for the exact moment a fraction needs to be rewritten to match something else: converting a recipe measurement to match a measuring cup marked only in twelfths, finding a common denominator by hand before adding two fractions, checking a homework answer that asks for 'three equivalent fractions,' or lining up a ratio with a specific scale on a technical drawing. Whether your starting fraction is already reduced or not, and whether your target denominator is small or large, this page and the calculator above generate the full equivalent family with primary-source citations to OpenStax and Euclid.

What is equivalent fractions calculator?

Two fractions are equivalent when they represent the same rational number, even though they are written with different numerators and denominators — for example, 1/2, 2/4, 3/6, and 50/100 are all equivalent, because each equals exactly one-half. Equivalent fractions are generated from any starting fraction n/d by multiplying both the numerator and the denominator by the same non-zero whole number k, since (n×k)/(d×k) = n/d × k/k = n/d × 1 = n/d — multiplying by k/k never changes the value because k/k always equals 1 for non-zero k. Every fraction has infinitely many equivalent forms, one for every positive integer k, but exactly one of them is in lowest terms — the form where the numerator and denominator share no common factor greater than 1, found by dividing both by their greatest common divisor. Conversely, given a target denominator, an equivalent fraction with that denominator exists with a whole-number numerator if and only if the target denominator is an integer multiple of the original fraction's lowest-terms denominator; the required numerator is then the reduced numerator multiplied by that same integer multiple (the scale factor).

How to use this calculator.

1. 01Enter your starting fraction's numerator and denominator. The denominator must be non-zero.
2. 02Enter a target denominator — the specific denominator you want an equivalent fraction for. It must be a whole-number multiple of your fraction's lowest-terms denominator.
3. 03Read the reduced form (simplifiedNumerator, simplifiedDenominator) to see your fraction in lowest terms, and the gcdUsed to see what it was divided by to get there.
4. 04Read the scale factor and the primary result, targetNumerator, to see the numerator of the equivalent fraction at your chosen target denominator.
5. 05Look at the generated set — the ×2 through ×5 numerator/denominator pairs — for a ready-made list of equivalent fractions without needing to specify a target denominator at all.
6. 06If the calculator reports an error about the target denominator, it means no whole-number equivalent exists there — try a target denominator that is a multiple of the reduced denominator shown in the error message.

The formula.

The calculator first reduces the entered fraction to lowest terms, dividing both the numerator and denominator by g = gcd(|numerator|, |denominator|), found via the Euclidean algorithm (Euclid, Elements, Book VII, Proposition 2): repeatedly replace the larger of two numbers with the remainder of dividing it by the smaller, until the remainder is zero — the last non-zero remainder is the GCD. Reducing first matters because every equivalent fraction, including any unreduced form you might have entered, is itself a whole-number multiple of the unique lowest-terms fraction; starting from the reduced form guarantees every generated equivalent is a genuine, correctly-scaled member of the family. Generating the fixed ×2 through ×5 set is then direct: multiply both the reduced numerator and reduced denominator by 2, then by 3, then by 4, then by 5, each time producing a valid equivalent fraction because (n×k)/(d×k) = (n/d)×(k/k) = (n/d)×1 = n/d for any non-zero integer k — multiplying by k/k never changes a fraction's value. Solving for a specific target denominator reverses that logic: if the target denominator T is an integer multiple of the reduced denominator d, the scale factor is k = T/d, an integer, and the matching numerator is simply the reduced numerator multiplied by that same k. If T is not an integer multiple of d, no whole-number numerator can produce an exact equivalent fraction with denominator T, because the defining property of equivalence — multiplying both terms by the identical integer — would be violated; the calculator therefore rejects a non-multiple target denominator explicitly, with the reduced denominator stated in the error so you know what multiple to try instead.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax. Prealgebra 2e, §4.1 'Visualize Fractions'. Rice University, peer-reviewed open-access textbook — the equivalent-fractions concept and the 'multiply by 1' justification this calculator implements.
2. [2]Euclid. Elements, Book VII, Proposition 2 (c. 300 BCE). The Euclidean algorithm for the greatest common divisor, used to reduce the starting fraction before generating equivalents.
3. [3]NIST Digital Library of Mathematical Functions, §27.1 ('Functions of Number Theory') — divisibility and the greatest common divisor underlying the reduction and multiple-of-denominator checks.