CalConverters

Equivalent Fractions Calculator

Find fractions equal in value to any fraction by multiplying the numerator and denominator by the same number, with 20 equivalents and every step shown.

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How to use this calculator

Enter a numerator and denominator. The calculator first reduces the fraction to lowest terms (if it isn't already), then multiplies that reduced fraction's numerator and denominator by 2, 3, 4, … all the way to 21 — generating 20 fractions that all equal the same value as your input.

Negative fractions and a zero numerator both work correctly — the step-by-step breakdown and the scrollable table below the answer show exactly how each case is handled.

Tip: If your fraction isn't already in lowest terms (like 4/6), the calculator reduces it first and treats that reduced form (2/3) as the starting point — the reduced fraction is itself one of the equivalents, just written with smaller numbers.

The core formula

Multiply both the numerator and denominator by the same nonzero integer k:

ab=a×kb×k,k=1,2,3,…\frac{a}{b} = \frac{a \times k}{b \times k}, \quad k = 1, 2, 3, \dots

This works because k ÷ k = 1, and multiplying any value by 1 leaves it unchanged — only the way it's written changes, never the value itself.

Worked example — generating equivalents of 3/4

34×22=68,34×33=912,34×44=1216\frac{3}{4} \times \frac{2}{2} = \frac{6}{8}, \quad \frac{3}{4} \times \frac{3}{3} = \frac{9}{12}, \quad \frac{3}{4} \times \frac{4}{4} = \frac{12}{16}

⇒34=68=912=1216=1520=1824=…\Rightarrow \frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16} = \frac{15}{20} = \frac{18}{24} = \dots

Reducing any of these back down — for example 1824÷66=34\frac{18}{24} \div \frac{6}{6} = \frac{3}{4} — returns the original fraction, confirming they're all equal in value.

Mixed numbers and whole numbers

  • A mixed number like 1 1/2 converts to an improper fraction first (3/2), then multiplies by k/k the same way.
  • A whole number like 5 is written as 5/1 first, then multiplied: 10/2, 15/3, 20/4, ….

When the numerator is zero or the fraction is negative

Tip: 0/5 reduces to 0/1, and every equivalent generated from it — 0/2, 0/3, 0/4, … — is still exactly 0. A negative fraction like −2/3 keeps its sign in the numerator throughout: −4/6, −6/9, −8/12, ….

Both cases follow the same rule as any other fraction — only the specific numbers change, never the underlying logic of multiplying numerator and denominator by the same k.

Why this matters: common denominators and comparing fractions

Equivalent fractions are the tool behind adding, subtracting, and comparing fractions with different denominators. To compare 12\frac{1}{2} and 25\frac{2}{5}, rewrite both with a common denominator using equivalents: 12=510\frac{1}{2} = \frac{5}{10} and 25=410\frac{2}{5} = \frac{4}{10} — now it's clear that 510>410\frac{5}{10} > \frac{4}{10}, so 12>25\frac{1}{2} > \frac{2}{5}.

This is also the reverse operation of simplifying a fraction: simplifying divides numerator and denominator by their GCF to reach the smallest form; generating equivalents multiplies by k to reach larger, equal forms.

Common equivalent fractions reference

FractionEquivalent (×2)Equivalent (×3)Equivalent (×4)
1/22/43/64/8
1/32/63/94/12
1/52/103/154/20
2/34/66/98/12
3/46/89/1212/16

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the reduced fraction and all 20 generated equivalents are always exact, no matter how large the numerator or denominator is.

Frequently asked questions

What is an equivalent fraction?
An equivalent fraction is a fraction that looks different but represents the same value. 12\frac{1}{2}, 24\frac{2}{4}, and 36\frac{3}{6} are all equivalent — they all mean "half."
How do you find an equivalent fraction?
Multiply the numerator and denominator by the same nonzero whole number. For 12\frac{1}{2}, multiplying by 2 gives 24\frac{2}{4}, multiplying by 3 gives 36\frac{3}{6}, and so on.
Why does multiplying numerator and denominator by the same number not change the value?
Because kk=1\frac{k}{k} = 1 for any nonzero k, and multiplying by 1 never changes a value. ab×kk=a×kb×k\frac{a}{b} \times \frac{k}{k} = \frac{a \times k}{b \times k} is just ab\frac{a}{b} written with bigger numbers.
What are the equivalent fractions of 4/6?
4/6 is not in lowest terms — its GCF is 2, so it reduces to 23\frac{2}{3}. From there, the equivalents are 4/6, 6/9, 8/12, 10/15, and so on, generated by multiplying 23\frac{2}{3} by 2, 3, 4, ….
Can a fraction with a zero numerator have equivalent fractions?
Yes. 0/5 reduces to 0/1, and every equivalent — 0/2, 0/3, 0/4, … — still equals 0. Only the denominator changes; the value never does.
Do equivalent fractions work with negative fractions?
Yes — the sign stays with the numerator. −2/3 is already in lowest terms, and its equivalents are −4/6, −6/9, −8/12, and so on, all equal to the same negative value.
What is the difference between simplifying a fraction and finding equivalent fractions?
They're opposite operations. Simplifying divides the numerator and denominator by their GCF to get smaller numbers; finding equivalents multiplies by k to get bigger numbers that still equal the same value.
What are equivalent fractions used for?
They let you rewrite fractions with a common denominator so you can add, subtract, or compare them — for example, turning 1/2 and 1/3 into 3/6 and 2/6 before adding.

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