Reference
Fraction and Mixed Number Rules
Every formula elsewhere on this site is built on top of a small set of rules — boundaries that decide what a fraction is even allowed to be, and what has to be true before an operation can be carried out at all. This page collects them in one place, alongside the terminology they depend on. Try any two values below to see the rules apply in real time.
Type the whole number, numerator, and denominator into their own fields. For a negative value such as −2 3/4, use the sign toggle or type −2 in the whole-number field.
Result
View step-by-step working
Terminology
The Parts of a Mixed Number
Every rule on this page refers back to these four terms, so it's worth pinning them down first.
Numerator
The top number in a fraction. It counts how many parts are being taken — in 3/4, the numerator 3 says "three of these parts."
Denominator
The bottom number in a fraction. It names how many equal parts one whole is divided into — in 3/4, the denominator 4 says "each whole is cut into four."
Whole number
The integer part of a mixed number, sitting beside the fraction rather than inside it — the 2 in 2 3/4, understood as 2 + 3/4.
Proper / improper
A fraction is proper when the numerator is smaller than the denominator (3/4), and improper when it's equal to or larger (7/4) — naming one whole unit or more.
Core Rules
The Rules Every Operation Has to Obey
Worked Example
The Equivalence Rule in Action
Worked Example — Is 3/4 Equivalent to 9/12?
The equivalence rule gives a direct test: does the same nonzero number multiply the numerator and the denominator?
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Start from the simpler fraction
3/4
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Find the multiplier on the denominator
4 × 3 = 12
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Apply the same multiplier to the numerator
3 × 3 = 9
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Compare to the second fraction
9/12 matches exactly
3/4 = 9/12 — confirmed equivalent, since both numbers were scaled by the same factor of 3
Why It Matters
Rules vs. Formulas
A formula is a recipe — a sequence of steps built to produce a correct answer, like a/b + c/d = (ad + bc)/bd for addition. A rule is something stricter: a boundary that decides whether an answer is even eligible to be correct in the first place. Formulas are built to respect rules; rules don't depend on any particular formula existing at all.
This is why the rules on this page apply everywhere on the site, even to pages that never mention "rules" by name. The addition formula only works because it quietly obeys the common-denominator rule. The simplifying calculator only stops where it stops because lowest terms is defined by a GCF of 1. Learn the rule once here, and every formula elsewhere on the site becomes something you can verify, not just something you trust.
Rules are also what make an answer wrong even when the arithmetic inside it was performed correctly. Adding 1/2 + 1/3 as 2/5 involves real addition — 1 + 1 = 2 and 2 + 3 = 5 — but it breaks the common-denominator rule, so the result is incorrect regardless of how carefully the addition itself was carried out. The rule, not the arithmetic, is what flags the error.
Because rules define the boundaries and formulas do the work inside them, it's the rules that transfer to unfamiliar problems. A new formula might need to be looked up, but the equivalence rule, the zero-denominator restriction, and the reciprocal rule apply identically whether the fraction in front of you is 1/2 or 117/238.
Common Mistakes
Where Fraction Rules Get Broken
Questions
Frequently Asked Questions
Questions about fraction terminology and the general rules themselves, not the operation-specific mechanics covered on other pages.