Dividing Fractions Calculator

Background.

This dividing fractions calculator applies — and shows — the 'keep, change, flip' method every pre-algebra classroom teaches: keep the first fraction exactly as it is, change the division sign to multiplication, and flip the second fraction upside down (swap its numerator and denominator) before multiplying. Enter 2/3 ÷ 4/5 and the calculator does not just print '5/6' — it reports reciprocalNumerator = 5 and reciprocalDenominator = 4 (the literal flip of 4/5 into 5/4), the pre-simplified product numerator 10 and denominator 12, the gcd used to reduce (2), and finally the simplified answer 5/6.

Quanta's general-purpose fraction calculator (math/fraction) also divides fractions, but it computes the equivalent expression (n1×d2)/(d1×n2) in one internal step and returns only the final simplified quotient — the reciprocal of the second fraction, the actual 'flip' that gives the method its name, is never a declared, readable value anywhere in its output. This calculator exists specifically to make that flip visible: the second fraction's numerator and denominator, swapped, shown as their own numbers, followed by the multiplication and simplification steps.

Why does flipping and multiplying work for division? Dividing by a number is defined as multiplying by that number's reciprocal — its multiplicative inverse, the number that, when multiplied by the original, gives 1. For a fraction c/d (with c ≠ 0), the reciprocal is d/c, because (c/d) × (d/c) = (c×d)/(d×c) = 1. So dividing by c/d is mathematically identical to multiplying by d/c, and 'keep, change, flip' is simply that identity applied mechanically: keep the dividend, change ÷ to ×, flip the divisor to its reciprocal. This is exactly the same rule as 'invert and multiply,' the phrase used interchangeably in most algebra textbooks.

The method has three steps, and this calculator reports each one. First, flip the second fraction: its denominator becomes the new numerator, and its numerator becomes the new denominator. Second, multiply the first fraction by that flipped fraction, straight across — numerator times numerator, denominator times denominator. Third, normalise the sign so the denominator is positive, and simplify the result by dividing both terms by their greatest common divisor, found via the Euclidean algorithm.

This calculator also handles the details that complicate manual division: a negative dividend or divisor (sign is normalised after the flip-and-multiply step, so the denominator you see in the final answer is always positive), a zero numerator on the fraction being divided (the answer is always 0, correctly reduced to 0/1), and — critically — a zero numerator on the divisor, which this calculator explicitly rejects with a clear error, because dividing by 0/anything is dividing by zero itself, which is undefined. Whether you're checking homework on keep-change-flip, splitting a measured quantity into fractional portions, or want to see the reciprocal step made explicit for your own numbers, this page and the calculator above cover the single operation of fraction division completely.

What is dividing fractions calculator?

Dividing one fraction by another, n1/d1 ÷ n2/d2, means finding the value x such that x × (n2/d2) = n1/d1. Because every non-zero rational number c/d has a multiplicative inverse — its reciprocal d/c, satisfying (c/d) × (d/c) = 1 — division by a fraction is defined as multiplication by that fraction's reciprocal: n1/d1 ÷ n2/d2 = n1/d1 × d2/n2 = (n1 × d2)/(d1 × n2). This is the algebraic justification for the 'keep, change, flip' or 'invert and multiply' rule taught in every arithmetic course: keep the dividend unchanged, change the operation from division to multiplication, and flip the divisor into its reciprocal by swapping its numerator and denominator. The reciprocal is undefined when the original numerator is zero, because 0/d has no multiplicative inverse (there is no fraction that, multiplied by 0/d, gives 1) — which is exactly why dividing by a fraction whose numerator is zero is disallowed, mirroring ordinary division by zero. Once the reciprocal is taken and the multiplication is carried out, the result is reduced to lowest terms exactly as with any other fraction: divide both the numerator and denominator by their greatest common divisor, found via the Euclidean algorithm, and normalise the sign so the denominator is always reported as positive. When the resulting quotient's magnitude is 1 or greater, it can equivalently be expressed as a mixed number.

How to use this calculator.

1. 01Enter the first fraction (the number being divided) as a numerator and denominator in the first pair of fields.
2. 02Enter the second fraction (the number you are dividing by) as a numerator and denominator in the second pair of fields. Its numerator must be non-zero — dividing by zero is undefined.
3. 03Read the reciprocal outputs to see the literal 'flip' of the second fraction: its denominator becomes the reciprocal numerator, and its numerator becomes the reciprocal denominator.
4. 04Check the pre-simplified product numerator and denominator to see the result of multiplying the first fraction by that reciprocal, before any reduction.
5. 05Read the gcd used to simplify, and the primary result — the simplified quotient's numerator and denominator.
6. 06Check the decimal value for a quick sense of magnitude, and the mixed-number outputs (isMixed, wholePart, fractionalNumerator) whenever the quotient's magnitude is 1 or greater.
7. 07If you enter a divisor with a numerator of zero, the calculator stops and explains why — that divisor equals zero, and division by zero is undefined for any dividend.

The formula.

The calculator implements keep-change-flip in three explicit, individually-reported steps, following OpenStax Prealgebra 2e §4.2. Step one — flip: the reciprocal of n2/d2 is d2/n2, valid whenever n2 ≠ 0, because (n2/d2) × (d2/n2) = (n2×d2)/(d2×n2) = 1 by definition of a multiplicative inverse. If n2 = 0, no reciprocal exists (there is no fraction that, multiplied by 0/d2, yields 1), so the calculator rejects that input explicitly rather than propagating an undefined result. Step two — keep, change: multiply the first fraction by the reciprocal found in step one, straight across: numerator times numerator over denominator times denominator, giving (n1 × d2) over (d1 × n2). This is the algebraic content of 'dividing by X equals multiplying by 1/X,' applied with X = n2/d2. Step three — simplify: normalise the sign of the raw product so its denominator is positive (if d1 × n2 came out negative, both the numerator and denominator are multiplied by −1, which does not change the fraction's value), then divide both terms by their greatest common divisor, 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. The resulting fraction is, by definition of the GCD, in lowest terms. If its magnitude is at least 1, the mixed-number form follows immediately: the whole part is the floor of the decimal value, and the remaining fractional numerator is the reduced numerator minus the whole part times the reduced denominator.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax. Prealgebra 2e, §4.2 'Multiply and Divide Fractions'. Rice University, peer-reviewed open-access textbook — the reciprocal/keep-change-flip method this calculator implements.
2. [2]Euclid. Elements, Book VII, Proposition 2 (c. 300 BCE). The Euclidean algorithm for the greatest common divisor, used for the final simplification step.
3. [3]NIST Digital Library of Mathematical Functions, §27.1 ('Functions of Number Theory') — the greatest common divisor and its multiplicative properties.