Simplifying Complex Fractions Calculator

How to use this calculator

Type one token into each box — the top box is X, the bottom box is Y — to build the complex fraction $\frac{X}{Y}$. Each box accepts a whole number ("6"), a simple fraction ("5/6"), a mixed number ("2 1/3"), or a decimal ("0.5"). The calculator converts both tokens to simple fractions, treats the main fraction bar as division, multiplies by the reciprocal, reduces by the greatest common factor (GCF), and converts to a mixed number when the result is improper.

What makes this a "general" complex fraction

A complex fraction stacks one value on top of another, with the main bar between them meaning "divide":

$\frac{X}{Y}$

Many complex-fraction tools only accept a plain fraction in each slot. This calculator is more general: X and Y can each be a whole number, a fraction, a mixed number, or a decimal — any single token you'd naturally type for "the number on top" or "the number on the bottom."

Step 1: convert each token to a simple fraction

Before dividing, every token is rewritten as a simple (improper) fraction:

• A mixed number like $2\frac{1}{3}$ becomes $\frac{7}{3}$ (multiply the whole part by the denominator, add the numerator).
• A decimal like $0.5$ becomes $\frac{1}{2}$.
• A whole number like $6$ becomes $\frac{6}{1}$.
• A fraction like $\frac{5}{6}$ or $\frac{-6}{15}$ is reduced to lowest terms right away (the second one becomes $\frac{-2}{5}$).

The formula: multiply by the reciprocal

Once X and Y are both simple fractions $\frac{a}{b}$ and $\frac{c}{d}$, the main bar's "divide" reading gives:

$\frac{X}{Y} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Dividing by $\frac{c}{d}$ is the same as multiplying by its reciprocal $\frac{d}{c}$ — keep the top fraction, change ÷ to ×, and flip the bottom fraction upside down. Multiplying straight across (numerator × numerator, denominator × denominator) always gives a correct, if not yet reduced, result.

After multiplying: reduce, then convert to a mixed number

1. Find the GCF of the raw numerator and raw denominator, 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: $\dfrac{2\,1/3}{5/6}$

Convert: $2\frac{1}{3} = \frac{7}{3}$, and $\frac{5}{6}$ is already simple. Multiply by the reciprocal: $\frac{7}{3} \times \frac{6}{5} = \frac{42}{15}$. GCF(42, 15) = 3, so $\frac{14}{5}$. Since 14 ≥ 5, convert: $14 \div 5 = 2$ remainder $4$. The answer is $\frac{14}{5} = 2\frac{4}{5}$.

Example 2 — a decimal token: $\dfrac{0.5}{3/4}$

Convert: $0.5 = \frac{1}{2}$. Multiply by the reciprocal: $\frac{1}{2} \times \frac{4}{3} = \frac{4}{6}$. GCF(4, 6) = 2, so the answer is $\frac{2}{3}$.

Example 3 — a whole number on top: $\dfrac{6}{1/2}$

Convert: $6 = \frac{6}{1}$. Multiply by the reciprocal: $\frac{6}{1} \times \frac{2}{1} = \frac{12}{1}$. The answer is the whole number $12$.

Example 4 — negative tokens: $\dfrac{5\,1/3}{-6/15}$

Convert: $5\frac{1}{3} = \frac{16}{3}$, and $\frac{-6}{15}$ reduces to $\frac{-2}{5}$. Multiply by the reciprocal: $\frac{16}{3} \times \frac{5}{-2} = \frac{80}{-6} = -\frac{40}{3}$. The answer is $-\frac{40}{3} = -13\frac{1}{3}$.

Common results reference

Precision note

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