Mixed Numbers Calculator

How to use this calculator

Enter a whole number, a fraction (numerator/denominator), or a full mixed number (both boxes filled) on each side, then choose an operation. The calculator converts each side to an improper fraction, applies the matching formula, reduces the result by its greatest common factor (GCF), and converts back to a mixed number when the answer is improper.

The core idea: everything becomes a fraction

A mixed number is a whole number paired with a proper fraction, like $1\,\frac{1}{2}$ (one and a half) or $3\,\frac{5}{8}$. Before any arithmetic happens, this calculator converts every mixed number (and every whole number) into an improper fraction using one formula:

$w\,\frac{n}{d} = \frac{(w \times d) + n}{d}$

In words: multiply the whole number by the denominator, add the numerator, and place the result over the same denominator. A plain whole number $w$ is treated as $\frac{w}{1}$; a plain fraction is used as-is.

Example: $2\,\frac{1}{4} = \frac{(2\times4)+1}{4} = \frac{9}{4}$.

The sign convention for negative mixed numbers

This trips people up because it looks like ordinary addition of a negative whole number and a positive fraction, but a mixed number is not a sum of two independently-signed parts — it's one signed quantity written in two pieces. This calculator always applies the sign to the whole mixed value before converting.

The four formulas (applied to the improper fractions)

Once both sides are improper fractions $\frac{a}{b}$ and $\frac{c}{d}$, the calculator reuses the same cross-multiplication formulas as a plain fractions calculator — no common denominator needs to be found first.

Addition

$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Subtraction

$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$

Multiplication

$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Division — keep the first fraction, change ÷ to ×, and flip the second:

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

After every operation: reduce, then convert back to a mixed number

1. Find the GCF of the raw numerator and denominator and divide both by it.
2. Convert back to a mixed number if the reduced numerator's absolute value is at least the denominator: divide with remainder, and the quotient becomes the whole part.

Worked examples

Example 1 — addition: $1\frac{1}{2} + 2\frac{1}{4}$

Convert: $1\frac{1}{2} = \frac{3}{2}$ and $2\frac{1}{4} = \frac{9}{4}$. Apply the formula: $\frac{(3\times4)+(2\times9)}{2\times4} = \frac{30}{8}$. Reduce by GCF(30, 8) = 2: $\frac{15}{4}$. Convert back: $15 \div 4 = 3$ remainder $3$. The answer is $\frac{15}{4} = 3\frac{3}{4}$.

Example 2 — a negative mixed number: $-1\frac{1}{2} + 2$

Convert: $-1\frac{1}{2} = -\frac{3}{2}$ (the sign applies to the whole value) and $2 = \frac{2}{1}$. Apply the formula: $\frac{(-3\times1)+(2\times2)}{2\times1} = \frac{1}{2}$. The answer is $\frac{1}{2}$ — already reduced, no mixed-number form needed.

Example 3 — division: $3 \div 1\frac{1}{2}$

Convert: $3 = \frac{3}{1}$ and $1\frac{1}{2} = \frac{3}{2}$. Keep, change, flip: $\frac{3}{1} \times \frac{2}{3} = \frac{6}{3}$. Reduce: the answer is $\frac{6}{3} = 2$, a whole number.

Example 4 — multiplication: $2\frac{1}{2} \times 1\frac{1}{3}$

Convert: $2\frac{1}{2} = \frac{5}{2}$ and $1\frac{1}{3} = \frac{4}{3}$. Multiply straight across: $\frac{5\times4}{2\times3} = \frac{20}{6}$. Reduce by GCF(20, 6) = 2: $\frac{10}{3}$. Convert back: $10 \div 3 = 3$ remainder $1$. The answer is $\frac{10}{3} = 3\frac{1}{3}$.

Common results reference

Precision note

Every computation uses arbitrary-precision integers (BigInt), not floating-point numbers — so the improper-fraction conversion, the cross-multiplied result, the GCF, and the final reduced (or mixed) answer are always exact, no matter how large the whole numbers, numerators, or denominators are.