Concept
Equivalent Fractions
Two fractions are equivalent when they name the exact same value, even though their numerator and denominator look nothing alike — 2/3, 4/6, and 12/18 are three different ways of writing one identical quantity. This calculator generates a full list of equivalents for any fraction you type, or checks whether two fractions you already have match, using the cross-multiplication test.
Fraction
Result
View step-by-step working
Method
Scaling a Fraction Without Changing Its Value
Worked Example — Scaling 2/3 Upward
Multiply the numerator and denominator by the same nonzero whole number, and the fraction's value never moves — only the numbers used to write it change.
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Start with the original fraction
2/3
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Multiply by 2/2
2×2 / 3×2 = 4/6
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Multiply by 3/3
2×3 / 3×3 = 6/9
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Multiply by 4/4
2×4 / 3×4 = 8/12
2/3 = 4/6 = 6/9 = 8/12
More Examples
Equivalent Fractions Across Different Denominators
Scale factor 2 and 3
1/2 = 2/4 = 3/6
Scale factor 2 and 3
3/5 = 6/10 = 9/15
Larger denominator
5/8 = 15/24
Scaled by a factor of 3 in one step.
Numerator of 1
1/4 = 3/12
A scale factor of 3 applied to both parts.
Two-digit result
7/10 = 14/20
Already scaled once
4/9 = 8/18 = 12/27
Verification
The Cross-Multiplication Test
Cross-multiplying is the fastest way to check whether two fractions — already written however they were given — represent the same value, with no need to simplify or find a common denominator first.
a/b = c/d if and only if a×d = c×b
Case 1 — A True Match
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Cross-multiply
3×8 = 24 · 6×4 = 24
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Compare the two products
24 = 24
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Conclusion
Equivalent
3/4 = 6/8
Case 2 — Not a Match
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Cross-multiply
3×7 = 21 · 5×4 = 20
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Compare the two products
21 ≠ 20
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Conclusion
Not equivalent
3/4 ≠ 5/7
Notice that neither fraction in either case needed to be simplified first. 6/8 is not written in lowest terms, yet the cross-multiplication test still confirms it matches 3/4 directly — the test works on any two fractions exactly as they're given.
Why It Matters
Where Equivalent Fractions Actually Get Used
Comparing two fractions with different denominators is genuinely hard to do at a glance — it's not obvious whether 5/8 or 7/11 is larger just by looking at the numbers. Rewriting both as equivalent fractions that share a common denominator turns the comparison into a simple contest between numerators, which is exactly why equivalence sits underneath every fraction comparison method taught in school.
Adding and subtracting fractions with different denominators depends on the same idea. 1/4 and 1/6 can't be combined directly, but rewriting each as an equivalent fraction over the shared denominator 12 — 3/12 and 2/12 — makes the arithmetic trivial. That rewriting step is precisely what the common denominator finder and the LCD calculator automate.
Simplifying works in the opposite direction along the same family of equivalent fractions. Every fraction belongs to an entire chain of equivalents stretching in both directions — scale up by multiplying, or reduce down by dividing out a shared factor — and the simplest form is just the one member of that chain where the numerator and denominator have no common factor left. The fraction simplifier finds that member automatically.
None of this changes what a numerator or denominator means. The numerator still counts parts and the denominator still says how many equal parts make one whole — scaling a fraction up or down just describes the identical quantity using a different-sized "part," which is why the underlying value never moves no matter which equivalent form is on the page.
Common Mistakes
Where Equivalent Fraction Reasoning Goes Wrong
Quick Reference
Equivalent Fractions for Common Values
Search or scroll this table for equivalents of frequently used fractions, scaled by factors of 2, 3, and 4 — useful for checking homework or double-checking a calculation by hand.
Questions
Frequently Asked Questions
Questions specific to equivalent fractions themselves, not already covered elsewhere on the site.