CalConverters

Complex Fraction Calculator

Simplify a complex fraction (a/b)/(c/d) into a single fraction, step by step, using "keep, change, flip" and automatic mixed-number conversion.

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How to use this calculator

Enter four whole numbers — a and b for the top fraction, c and d for the bottom fraction — to build the complex fraction a/bc/d\frac{a/b}{c/d}. The calculator reads the main fraction bar as division, applies "keep, change, flip," multiplies straight across, reduces by the greatest common factor (GCF), and converts to a mixed number when the result is improper.

Tip: Every input box accepts negative whole numbers too — type the minus sign directly, like -3 or -4, in any of the four boxes.

The core idea: the fraction bar means "divide"

A complex fraction stacks one fraction on top of another:

  ab    cd  \frac{\;\frac{a}{b}\;}{\;\frac{c}{d}\;}

The big horizontal bar separating the two fractions works exactly like any other fraction bar — it means division. So this complex fraction is really asking:

ab÷cd\frac{a}{b} \div \frac{c}{d}

The formula: keep, change, flip

Dividing by a fraction is defined as multiplying by its reciprocal (the fraction flipped upside down). The memory trick is Keep, Change, Flip:

  1. Keep the top fraction as it is.
  2. Change the ÷ sign to ×.
  3. Flip the bottom fraction — swap its numerator and denominator.

a/bc/d=ab÷cd=ab×dc=adbc\frac{a/b}{c/d} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}

No common denominator is needed anywhere in this process — multiplying straight across (numerator × numerator, denominator × denominator) always gives a correct, if not yet fully reduced, result.

Tip: Because dividing by cd\frac{c}{d} is the same as multiplying by dc\frac{d}{c}, the bottom fraction's numerator (c) can never be zero — flipping a zero numerator would create a zero denominator.

After multiplying: reduce, then convert to a mixed number

  1. Find the GCF of the raw numerator (ad) and raw denominator (bc), then divide both by it — this is the fully reduced fraction.
  2. Convert to a mixed number if the reduced numerator's absolute value is at least the denominator (and the denominator isn't 1): divide with remainder, and the quotient becomes the whole part.

Worked examples

Example 1 — the default case: 1/23/4\dfrac{1/2}{3/4}

Keep, change, flip: 12×43=46\frac{1}{2} \times \frac{4}{3} = \frac{4}{6}. GCF(4, 6) = 2, so 4÷26÷2=23\frac{4\div2}{6\div2} = \frac{2}{3}. The answer is 23\frac{2}{3} — already a proper fraction.

Example 2 — a mixed-number result: 2/34/9\dfrac{2/3}{4/9}

Keep, change, flip: 23×94=1812\frac{2}{3} \times \frac{9}{4} = \frac{18}{12}. GCF(18, 12) = 6, so 32\frac{3}{2}. Since 3 ≥ 2, convert: 3÷2=13 \div 2 = 1 remainder 11. The answer is 32=112\frac{3}{2} = 1\frac{1}{2}.

Example 3 — a whole-number result: 5/11/2\dfrac{5/1}{1/2}

Keep, change, flip: 51×21=101\frac{5}{1} \times \frac{2}{1} = \frac{10}{1}. GCF(10, 1) = 1. The answer is the whole number 1010.

Example 4 — a negative complex fraction: −1/23/4\dfrac{-1/2}{3/4}

Keep, change, flip: −12×43=−46\frac{-1}{2} \times \frac{4}{3} = \frac{-4}{6}. GCF(4, 6) = 2, so −4÷26÷2=−23\frac{-4\div2}{6\div2} = \frac{-2}{3}. The answer is −23-\frac{2}{3}.

Common results reference

Complex fractionKeep, change, flipAnswer
(1/2)/(3/4)1/2 × 4/32/3
(2/3)/(4/9)2/3 × 9/41 1/2
(5/1)/(1/2)5/1 × 2/110
(−1/2)/(3/4)−1/2 × 4/3−2/3

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the cross product, the GCF, and the reduced fraction are always exact, no matter how large the four numbers are.

Frequently asked questions

What is a complex fraction?
A complex fraction is a fraction whose numerator, denominator, or both are themselves fractions — like a/bc/d\frac{a/b}{c/d}. The main fraction bar in the middle means "divide," so a complex fraction is really a division problem in disguise.
How do you simplify a complex fraction?
Read the main bar as division, then apply "keep, change, flip": keep the top fraction, change ÷ to ×, and flip (take the reciprocal of) the bottom fraction. a/bc/d=ab÷cd=ab×dc=adbc\frac{a/b}{c/d} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}.
Why do you flip the bottom fraction?
Dividing by a number is the same as multiplying by its reciprocal. Since cd\frac{c}{d} is the divisor, dividing by it is equivalent to multiplying by dc\frac{d}{c} — its reciprocal (flipped) form.
Can any of the four numbers be zero?
The numerator of the top fraction (a) can be zero — that just makes the whole complex fraction equal 0. But the two denominators (b, d) and the bottom fraction's numerator (c) can never be zero: a zero denominator makes a fraction undefined, and a zero numerator on the bottom fraction would put a zero in the denominator once it's flipped.
Why did I get a mixed number in the answer?
Whenever the simplified result is improper (its numerator, ignoring sign, is at least as large as the denominator) and the denominator isn't 1, the calculator also shows the equivalent mixed number. 32\frac{3}{2} becomes 1121\frac{1}{2} because 3÷2=13 \div 2 = 1 remainder 11.
Does this work with negative numbers?
Yes. Enter a minus sign on any of the four boxes. A single negative sign anywhere in the complex fraction makes the final answer negative; two negative signs cancel out, exactly like signed whole numbers.
What's the difference between a complex fraction and a regular fraction?
A regular fraction has whole numbers as its numerator and denominator, like 34\frac{3}{4}. A complex fraction has a fraction (or fractions) in its numerator, denominator, or both, like 1/23/4\frac{1/2}{3/4} — it needs an extra simplification step before it looks like a regular fraction.

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